From f665c5e2598bb77233563ce25c3dbb5a0178a9a7 Mon Sep 17 00:00:00 2001 From: Josh Gao Date: Sun, 14 Jun 2026 16:35:12 +0800 Subject: [PATCH] =?UTF-8?q?meta:=20=E9=87=8D=E6=9E=84=E8=B5=B7=E7=82=B9?= =?UTF-8?q?=E5=9F=BA=E7=BA=BF?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 新增 .gitignore 项目规则、CLAUDE.md（含 git 规范）、 agent_instruction.md 重构指导。作为 agent_instruction.md 阶段 1 之前的回滚锚点。 --- .gitignore | 344 + CLAUDE.md | 101 + agent_instruction.md | 507 + doa_doppler_est.tex | 178 + ...pplerEciDualSatSingleSourceUpa2048Snap.pdf | Bin 0 -> 131915 bytes ...doaEcefDualSatSingleSourceUpa20000Snap.pdf | Bin 0 -> 96621 bytes ...oaEcefSingleSatSingleSourceUla2000Snap.pdf | Bin 0 -> 109585 bytes ...oaEcefSingleSatSingleSourceUpa2000Snap.pdf | Bin 0 -> 86513 bytes fig/dualSatSingleSourceScene.bmp | Bin 0 -> 777246 bytes fig/multiSatSingleSourceScene.bmp | Bin 0 -> 852742 bytes parts/00-roadmap.tex | 116 + parts/01-assumptions.tex | 556 + parts/02-modeling.tex | 472 + parts/98-future work.tex | 82 + parts/99-literature.tex | 37 + parts/A1-background.tex | 267 + parts/A2-doaEstMethod.tex | 831 + parts/A3-detDoaCrb.tex | 412 + parts/A4-stoDoaCrb.tex | 307 + parts/A5-detDoaDopplerCrb.tex | 238 + parts/A6-crbStatPilotDoaDopper.tex | 543 + parts/A7-crbDynPilotDoaDopperCont.tex | 922 + parts/A8-crbDynPilotDoaDopperInd.tex | 395 + parts/detDoaDopplerEst.tex | 158 + parts/detDoaEst.tex | 108 + parts/multiArrayDetDoaDopplerDelayEst.tex | 199 + parts/multiArrayDynDetDoaDopplerEst.tex | 850 + parts/multiArrayStatDetDoaDopplerEst.tex | 286 + parts/multiArrayStoDoaEst.tex | 619 + parts/stoDoaDopplerEst.tex | 5 + parts/stoDoaEst.tex | 0 ref.bib | 26370 ++++++++++++++++ 32 files changed, 34903 insertions(+) create mode 100644 .gitignore create mode 100644 CLAUDE.md create mode 100644 agent_instruction.md create mode 100644 doa_doppler_est.tex create mode 100644 fig/doaDopplerEciDualSatSingleSourceUpa2048Snap.pdf create mode 100644 fig/doaEcefDualSatSingleSourceUpa20000Snap.pdf create mode 100644 fig/doaEcefSingleSatSingleSourceUla2000Snap.pdf create mode 100644 fig/doaEcefSingleSatSingleSourceUpa2000Snap.pdf create mode 100644 fig/dualSatSingleSourceScene.bmp create mode 100644 fig/multiSatSingleSourceScene.bmp create mode 100644 parts/00-roadmap.tex create mode 100644 parts/01-assumptions.tex create mode 100644 parts/02-modeling.tex create mode 100644 parts/98-future work.tex create mode 100644 parts/99-literature.tex create mode 100644 parts/A1-background.tex create mode 100644 parts/A2-doaEstMethod.tex create mode 100644 parts/A3-detDoaCrb.tex create mode 100644 parts/A4-stoDoaCrb.tex create mode 100644 parts/A5-detDoaDopplerCrb.tex create mode 100644 parts/A6-crbStatPilotDoaDopper.tex create mode 100644 parts/A7-crbDynPilotDoaDopperCont.tex create mode 100644 parts/A8-crbDynPilotDoaDopperInd.tex create mode 100644 parts/detDoaDopplerEst.tex create mode 100644 parts/detDoaEst.tex create mode 100644 parts/multiArrayDetDoaDopplerDelayEst.tex create mode 100644 parts/multiArrayDynDetDoaDopplerEst.tex create mode 100644 parts/multiArrayStatDetDoaDopplerEst.tex create mode 100644 parts/multiArrayStoDoaEst.tex create mode 100644 parts/stoDoaDopplerEst.tex create mode 100644 parts/stoDoaEst.tex create mode 100644 ref.bib diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..7d5085d --- /dev/null +++ b/.gitignore @@ -0,0 +1,344 @@ +## Core latex/pdflatex auxiliary files: +*.aux +*.lof +*.log +*.lot +*.fls +*.out +*.toc +*.fmt +*.fot +*.cb +*.cb2 +.*.lb + +## Intermediate documents: +*.dvi +*.xdv +*-converted-to.* +# these rules might exclude image files for figures etc. +# *.ps +# *.eps +# *.pdf + +## Generated if empty string is given at "Please type another file name for output:" +.pdf + +## Bibliography auxiliary files (bibtex/biblatex/biber): +*.bbl +*.bbl-SAVE-ERROR +*.bcf +*.bcf-SAVE-ERROR +*.blg +*-blx.aux +*-blx.bib +*.run.xml + +## Build tool auxiliary files: +*.fdb_latexmk +*.synctex +*.synctex(busy) +*.synctex.gz +*.synctex.gz(busy) +*.pdfsync +*.rubbercache +rubber.cache + +## Build tool directories for auxiliary files +# latexrun +latex.out/ + +## Auxiliary and intermediate files from other packages: +# algorithms +*.alg +*.loa + +# achemso +acs-*.bib + +# amsthm +*.thm + +# beamer +*.nav +*.pre +*.snm +*.vrb + +# changes +*.soc + +# comment +*.cut + +# cprotect +*.cpt + +# elsarticle (documentclass of Elsevier journals) +*.spl + +# endnotes +*.ent + +# fixme +*.lox + +# feynmf/feynmp +*.mf +*.mp +*.t[1-9] +*.t[1-9][0-9] +*.tfm + +#(r)(e)ledmac/(r)(e)ledpar +*.end +*.?end +*.[1-9] +*.[1-9][0-9] +*.[1-9][0-9][0-9] +*.[1-9]R +*.[1-9][0-9]R +*.[1-9][0-9][0-9]R +*.eledsec[1-9] +*.eledsec[1-9]R +*.eledsec[1-9][0-9] +*.eledsec[1-9][0-9]R +*.eledsec[1-9][0-9][0-9] +*.eledsec[1-9][0-9][0-9]R + +# glossaries +*.acn +*.acr +*.glg* +*.glo* +*.gls* +*.glsdefs +*.lzo +*.lzs +*.slg +*.slo +*.sls +*.ignoreme* + +# uncomment this for glossaries-extra (will ignore makeindex's style files!) +*.ist + +# gnuplot +*.gnuplot +*.table + +# gnuplottex +*-gnuplottex-* + +# gregoriotex +*.gaux +*.glog +*.gtex + +# htlatex +*.4ct +*.4tc +*.idv +*.lg +*.trc +*.xref + +# hypdoc +*.hd + +# hyperref +*.brf + +# knitr +*-concordance.tex +# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files +# *.tikz +*-tikzDictionary + +# listings +*.lol + +# luatexja-ruby +*.ltjruby + +# makeidx +*.idx +*.ilg +*.ind + +# minitoc +*.maf +*.mlf +*.mlt +*.mtc[0-9]* +*.slf[0-9]* +*.slt[0-9]* +*.stc[0-9]* + +# minted +_minted* +*.pyg + +# morewrites +*.mw + +# newpax +*.newpax + +# nomencl +*.nlg +*.nlo +*.nls + +# pax +*.pax + +# pdfpcnotes +*.pdfpc + +# sagetex +*.sagetex.sage +*.sagetex.py +*.sagetex.scmd + +# scrwfile +*.wrt + +# svg +svg-inkscape/ + +# sympy +*.sout +*.sympy +sympy-plots-for-*.tex/ + +# pdfcomment +*.upa +*.upb + +# pythontex +*.pytxcode +pythontex-files-*/ + +# tcolorbox +*.listing + +# thmtools +*.loe + +# TikZ & PGF +*.dpth +*.md5 +*.auxlock + +# titletoc +*.ptc + +# todonotes +*.tdo + +# vhistory +*.hst +*.ver + +# easy-todo +*.lod + +# xcolor +*.xcp + +# xmpincl +*.xmpi + +# xindy +*.xdy + +# xypic precompiled matrices and outlines +*.xyc +*.xyd + +# endfloat +*.ttt +*.fff + +# Latexian +TSWLatexianTemp* + +## Editors: +# WinEdt +*.bak +*.sav + +# Texpad +.texpadtmp + +# LyX +*.lyx~ + +# Kile +*.backup + +# gummi +.*.swp + +# KBibTeX +*~[0-9]* + +# TeXnicCenter +*.tps + +# auto folder when using emacs and auctex +./auto/* +*.el + +# expex forward references with \gathertags +*-tags.tex + +# standalone packages +*.sta + +# Makeindex log files +*.lpz + +# xwatermark package +*.xwm + +# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib +# option is specified. Footnotes are the stored in a file with suffix Notes.bib. +# Uncomment the next line to have this generated file ignored. +#*Notes.bib + +## System generated files +# -- macOS +.DS_Store +.AppleDouble +.LSOverride +._* + +# -- Windows +Thumbs.db +ehthumbs.db +Desktop.ini + +# -- Linux +*~ + +# IDEs and editors' files +*.idea +*.vscode +*.sublime-workspace +*.sublime-project + +## ----- Project-specific ---------------------------------------------------- +# 根目录的编译输出 PDF 与历史快照（fig/*.pdf 不受影响） +/doa_doppler_est.pdf +/doa_doppler_est_*.pdf + +# 打包归档 +/*.zip + +# TeXstudio 偶发产生的字面量目录 +/%OUTDIR%/ + +# 重构期间的临时归档目录（用 git mv 后真正删除前的暂存） +/archive/ \ No newline at end of file diff --git a/CLAUDE.md b/CLAUDE.md new file mode 100644 index 0000000..2ccbe84 --- /dev/null +++ b/CLAUDE.md @@ -0,0 +1,101 @@ +# 项目说明 + +LaTeX 学习笔记：相控阵 DoA + Doppler 估计与 CRB 推导。作者 Josh Gao，单人项目。 + +## 结构 + +- 主文件：`doa_doppler_est.tex`（preamble 与 `\input` 列表） +- 章节源：`parts/*.tex` +- 文献：`ref.bib` +- 图片：`fig/` +- 重构指导：`agent_instruction.md`（一次性方案，**非常驻规则**） + +## 当前进行中的重构 + +完整方案在 `agent_instruction.md`，分 6 个阶段。开始任一阶段前必须 `@agent_instruction.md` 读取该阶段细节，**不要凭记忆执行**。一次会话只做一个阶段，做完即停等用户编译验证。 + +## 通用规则 + +- **不要编译 LaTeX**：xelatex/pdflatex/latexmk 均不调用，用户在 TeXstudio 手动编译。agent 只改文件。 +- **不要读编译产物**：`*.aux`、`*.log`、`*.bbl`、`*.bcf`、`*.synctex.gz`、`*.run.xml`、`*.toc`、`*.ist`、`*.glo`、`*.acn`、根目录历史 PDF（`doa_doppler_est_*.pdf`）、`*.zip`。这些都在 `.gitignore` 里。 +- **不要重排 `ref.bib`**：除非明确要求，保持现有条目顺序与格式。 +- **不要动 preamble 的宏包定义**：主文件 `doa_doppler_est.tex` 的 `\input` 列表可以按重构需要调整，宏包加载段不动。 +- **删除/重命名文件用 `git mv`**：保留 history，diff 更小。彻底删除前先 `git mv` 到 `archive/` 子目录（已 gitignore）暂存。 +- **推导卡壳留 `% TODO:` 注释**，不要编造 placeholder 文字。 +- **语言**：中文为主；公式、符号、文件路径、命令用英文。 +- **不要总结改动**：用户看 diff，不需要 trailing summary。 + +--- + +# Git commit 规范 + +## 格式 + +``` +(): <中文 subject，命令式，≤50 字> + +<可选 body，每行 ≤72 字> +``` + +## type（仅 4 类） + +| type | 用途 | +|------|------| +| `struct` | 文件移动/合并/拆分、`\input` 顺序调整 | +| `content` | 实质性内容：新增/改写推导、过渡段、文献、图表 | +| `fix` | 修错：typo、公式错、断引用 | +| `meta` | 元文件：CLAUDE.md、agent_instruction.md、.gitignore、.gitattributes | + +## scope（细分维度） + +- 章节定位：`sec1`–`sec7`、`app-A`…`app-G`、`preamble` +- 跨章节专题：`symbol`、`bib`、`fig` + +## body 强制规则 + +- `type=struct` **必须**有 body，说明拆/合的依据 +- `scope=symbol` **必须**有 body，说明符号冲突或废弃原因 +- 其他情况自由，但建议解释"为什么"而非"什么" + +## subject 规则 + +- 中英混合：type 用英文，scope/subject 用中文 +- 命令式动词开头："新增 / 删除 / 合并 / 重命名 / 修正" +- 不写"添加了 / 已完成"等完成时 +- 不加句号 + +## 其他硬规则 + +- **每阶段至少一个 commit**，不跨阶段堆改动 +- **结构改动与内容改动分开 commit**——"重命名 A6 → E" (`struct`) 和 "E.2 新增多用户推导" (`content`) 必须是两个 commit +- **不写 `Co-Authored-By` 尾注**——个人学习笔记 +- **不加阶段标记**（如 `Stage: 1/6`）——subject 自带阶段语义即可 + +## 示例 + +``` +struct(app): 合并 A3+A4+A5 为附录 D + +三者共享 FIM 推导框架（确定/随机信号 + DoA/DoA-Doppler）， +分立反而割裂对比。合并后子节顺序 D.1 确定 DoA、D.2 确定 +DoA-Doppler、D.3 随机 DoA（对照支线）。 +``` + +``` +struct(symbol): Q_ℓ 全局替换为 Ω_ℓ + +Q 保留给波束成形人工噪声协方差矩阵（后续第 7 节）。 +影响 parts/ 下全部 .tex 文件及附录 A 符号表。 +``` + +``` +content(sec5): 新增时间戳误差信号模型与 EFIM 框架 +``` + +``` +fix(sec3): 修正参考星 Doppler 偏差号 +``` + +``` +meta: 新增 agent_instruction.md 重构指导文档 +``` diff --git a/agent_instruction.md b/agent_instruction.md new file mode 100644 index 0000000..b923da3 --- /dev/null +++ b/agent_instruction.md @@ -0,0 +1,507 @@ +# DoA-Doppler 学习笔记重构指导 + +本文档用于指导 agent 对 `doa_doppler_est` 仓库进行整体重构。所有改造目标、原则与具体操作均在本文档内自包含，agent 在执行前应完整阅读本文档，并以本文档为唯一交付依据。 + +--- + +## 一、任务背景与改造目标 + +### 1.1 文档当前的定位与问题 + +`doa_doppler_est` 仓库目前承担了两个互相冲突的角色： + +- **学习笔记**：按"我是怎么学会这件事的"逻辑组织，应从简到繁、有叙事性。 +- **推导参考库**：按"这个结论在哪里"逻辑组织，应可跳读、有索引性。 + +当前文档把两种角色混在一起，导致： + +1. `00-roadmap.tex` 作为正文第一节出现，是参考库的索引逻辑，读者还没看到任何实质内容就要消化大量元信息，显得突兀。 +2. `01-assumptions.tex` 把全部符号表前置到第二节，读者一进门就是六张表格，阅读体验不连贯。 +3. 正文章节之间缺少过渡段落，读起来像独立技术报告的拼贴。 +4. 正文模型默认单用户假设（$K=1$），与新论文的多用户扩展不一致。 +5. 附录按"一节正文对应一节附录"机械拆分，部分附录（A3/A4/A5、A7/A8）内容紧密相关，分开反而割裂。 + +### 1.2 改造的整体目标 + +本次改造以**学习笔记为主要角色，参考库功能降级为服务手段**。具体目标： + +1. 将 roadmap 改为简短的不编号前言，移除元信息密集型章节。 +2. 将完整符号表移入附录，正文在首次出现时按需定义。 +3. 正文按"每节只推进一个维度"的原则重新组织递进路径。 +4. 引入两个新章节（多用户扩展、时间戳误差模型），对应新论文方向。 +5. 合并紧密相关的附录，避免过度拆分。 +6. 符号体系预留波束成形命名空间，但不展开。 + +### 1.3 研究背景的进一步说明 + +- 第一篇论文（已完成）：多帧连续相位 DoA-Doppler 估计，对应当前文档中的多帧 CP 模型。 +- 第二篇论文（进行中）：多用户、多星、阵列天线、部分相干（timestamp error）场景下的参数估计，目标是引入轻量 AI 方法做 nuisance 补偿。 +- 用户假设单天线、卫星侧多天线。 +- 第二篇论文采用单帧连续信号模型，**不**沿用第一篇的多帧结构。 +- 后续可能扩展到波束成形与鲁棒安全设计，本次仅预留命名空间。 + +--- + +## 二、整体改造原则 + +执行本次改造时，所有局部决定必须服从以下原则： + +### 2.1 学习笔记优先原则 + +正文按"我是怎么一步步把问题变复杂的"逻辑组织。每节开头有 3–5 句过渡段，明确说明上一节的限制如何被解除、本节新增什么结构。 + +### 2.2 符号跟随问题引入原则 + +正文不再前置完整符号表。每个符号在它**第一次出现**的那一节首次定义，定义紧贴使用场景。完整符号索引移入附录 A 供查阅，不进入正文阅读流。 + +### 2.3 一节一维度原则 + +正文每节只沿一个维度推进问题，其他维度保持最简形态。维度包括：阵列规模、参数空间、信号模型、观测时间结构、接收节点数、同步条件、处理目标。**避免一节里同时引入多个新维度**。 + +### 2.4 正文止于集中似然原则 + +正文只写到"集中似然的最终形式"，不展开矩阵代入与逐步化简： + +- 信号模型：完整写出 +- 负对数似然定义：完整写出 +- 对干扰参数的集中化（消去复增益、幅度、噪声方差）：给出结论，标注"推导见附录 X" +- EFIM 的 Schur complement 表达式：给出结论 +- 最终估计器表达式：给出结论 + +详细推导全部移入附录。这样正文每节推导密度一致，读者能流畅跟进。 + +### 2.5 附录合并原则 + +同一参数空间下不同模型的 CRB 推导可以合并在同一附录里。不要求"一节正文对应一节附录"。合并依据是**是否共享推导框架**，共享则合并，不共享则独立。 + +### 2.6 随机信号模型不进入主线 + +第二篇论文及后续主线均依赖确定信号假设（已知训练序列/导频）。原因：随机信号模型把信源波形视为零均值复高斯过程，似然只依赖于信号子空间，无法分离 Doppler 与空间相位。随机信号 DoA 估计保留在正文第 1 节末尾的一个简短对比小节中，详细 CRB 推导保留在附录 D.3 供对照参考，**不进入第 2 节及之后的主线**。 + +--- + +## 三、新的正文章节结构 + +### 3.1 总体结构 + +``` +前言（不编号，约半页） + - 笔记目标与范围 + - 内容递进逻辑 + - 如何使用本文档（指向附录 A 的符号索引） + +第1节 多快拍确定信号 DoA 估计 +第2节 加入 Doppler——单阵列确定信号 DoA-Doppler 估计 +第3节 多星协作——单用户全局参数化 +第4节 多用户扩展 +第5节 时间戳误差与部分相干观测【新论文核心】 +第6节 多帧动态模型与连续相位结构【第一篇论文】 +第7节 波束成形（预留，本次不展开） + +附录 A 完整符号索引 +附录 B 数学基础 +附录 C 经典 DoA 方法（MVDR/MUSIC/ESPRIT） +附录 D 单阵列 CRB 汇总 +附录 E 多星静态模型 CRB +附录 F 时间戳误差 EFIM 分析【新增】 +附录 G 多帧动态模型 CRB +``` + +### 3.2 各节内容指引 + +下面给出每节的过渡段落主旨、首次引入的符号、关键结论与对应附录。**过渡段落只给主旨方向，由 agent 根据现有推导内容润色成稿。** + +#### 第 1 节：多快拍确定信号 DoA 估计 + +**起点说明**：这是整个笔记的实质起点。不再写单快拍场景，因为 MLE 在单快拍下退化平凡，缺乏实质内容。直接从多快拍出发。 + +**过渡段主旨**：开门见山引入阵列信号处理的基本问题——多个信源从远场入射到 $M$ 元阵列，已知 $T$ 个快拍内的训练序列波形，目标是估计 DoA。 + +**首次引入符号**：$M$、$K$、$T$、$\mathbf{y}_t$、$\mathbf{Y}$、$\mathbf{a}(\boldsymbol\theta_k)$、$\mathbf{A}(\boldsymbol\Theta)$、$\mathbf{S}$、$\zeta_k$、$\sigma_n^2$、$\mathbf{n}_t$、$\mathbf{N}$、$\mathcal P_\mathbf{A}^\perp$。 + +**信号模型**：$\mathbf{Y} = \mathbf{A}(\boldsymbol\Theta)\mathbf{S} + \mathbf{N}$。 + +**关键结论**：DML 集中似然对复增益和噪声方差集中化后，估计准则化为投影残差最小化。引出投影矩阵 $\mathcal P_\mathbf{A}^\perp$ 作为后续所有章节复用的核心工具。 + +**末尾对比小节**：用一段话说明随机信号模型的基本思路和它在引入 Doppler 后的根本限制（相位模糊），指出本文主线选择确定信号模型的原因。详细 SML 推导见附录 D.3。 + +**对应附录**：D.1（确定信号 DoA CRB），D.3（随机信号 DoA CRB，作为对比）。 + +#### 第 2 节：加入 Doppler——单阵列确定信号 DoA-Doppler 估计 + +**过渡段主旨**：第 1 节假设信源静止，时间相位可以被吸收进 $\mathbf{S}$。当信源带有 Doppler 时，时间相位变成 $e^{j\nu_k t}$，无法再被吸收。本节引入 Doppler 参数，说明这是唯一新增的维度。 + +**首次引入符号**：$\nu_k$、$\mathbf{q}_k$（时间模板）、$\mathbf{b}_{\mathrm{st},k}$（空时导向向量）、$\mathbf{B}_\mathrm{st}$。 + +**信号模型**：$\mathbf{y}(t) = \sum_k \zeta_k \mathbf{a}(\boldsymbol\theta_k) s_k(t) e^{j\nu_k t} + \mathbf{n}(t)$。 + +**关键结论**：对 $(\boldsymbol\theta_k, \nu_k)$ 的联合集中似然搜索；空时导向矩阵 $\mathbf{B}_\mathrm{st}$ 的 Kronecker 结构；DoA-Doppler 耦合对单独估计任一参数造成的信息损失。 + +**对应附录**：D.2（确定信号 DoA-Doppler CRB）。 + +#### 第 3 节：多星协作——单用户全局参数化 + +**过渡段主旨**：第 2 节的根本限制是单阵列孔径决定 DoA 分辨率上限，单链路 Doppler 只提供一个标量约束。当多颗卫星从不同轨道位置观测同一用户时，它们的几何多样性提供额外信息维度。本节建立多星统一参数化。 + +**首次引入符号**：$L$、$\ell$、$\ell_\mathrm{ref}$、$\boldsymbol\xi = [\alpha,\beta]^\Tran$、$\boldsymbol\Omega_\ell$（旋转矩阵，**不再用 $\mathbf{Q}_\ell$**）、$\mathbf{u}(\boldsymbol\xi)$、$\mathbf{u}_\ell^\mathrm{loc}$、$\mathbf{a}_\ell(\boldsymbol\xi)$、$\nu_\mathrm{ref}$、$\Delta\nu_\ell(\boldsymbol\xi)$、$\boldsymbol\eta_\mathrm{stat}$。 + +**核心论点**：直接把 $L$ 颗星的 Doppler 全部作为独立参数会让主参数维度膨胀到 $2+L$，但这 $L$ 个 Doppler 受卫星轨道几何约束，并非独立。参考星参数化利用这个约束把主参数降回 3 维。这个动机要在过渡段说清楚。 + +**信号模型**：$\mathbf{y}_\ell(t) = \zeta_\ell \mathbf{a}_\ell(\boldsymbol\xi) s(t) e^{j\nu_\ell(\boldsymbol\xi,\nu_\mathrm{ref})t} + \mathbf{n}_\ell(t)$，单用户。 + +**关键结论**：多星联合 MLE 通过对各星残差能量加权求和，信息聚合机制由参考星参数化诱导。 + +**对应附录**：E.1（多星静态单用户 CRB）。 + +#### 第 4 节：多用户扩展 + +**过渡段主旨**：第 3 节假设单用户。多用户场景在数学上是把 $K$ 从 1 推广到一般值，但它带来一个第 3 节不存在的新问题：用户分离的可辨识性。当两个用户的 DoA 或 Doppler 过于接近时，集中似然曲面会出现伪极值。这个问题是后续引入 AI 方法的重要动机之一，本节用几何直觉说清楚，不需要严格证明。 + +**首次引入符号**：$\boldsymbol\xi_k$、$s_k(t)$、$\zeta_{\ell,k}$、$\nu_{\mathrm{ref},k}$、$\boldsymbol\eta_\mathrm{mu}$。 + +**信号模型**：$\mathbf{y}_\ell(t) = \sum_{k=1}^K \zeta_{\ell,k} \mathbf{a}_\ell(\boldsymbol\xi_k) s_k(t) e^{j\nu_{\ell,k}(\boldsymbol\xi_k,\nu_{\mathrm{ref},k})t} + \mathbf{n}_\ell(t)$。 + +**主参数**：$\boldsymbol\eta_\mathrm{mu} = [\boldsymbol\xi_1^\Tran, \nu_{\mathrm{ref},1}, \ldots, \boldsymbol\xi_K^\Tran, \nu_{\mathrm{ref},K}]^\Tran \in \mathbb{R}^{3K}$。 + +**关键结论**：多用户集中似然的矩阵形式；用户分离的可辨识性条件（几何陈述，不展开证明）。 + +**对应附录**：E.2（多星静态多用户 CRB，作为 E.1 的扩展）。 + +#### 第 5 节：时间戳误差与部分相干观测【新论文核心】 + +**过渡段主旨**：前面四节都假设各星之间完全相干。实际系统中，星间时钟不可能完美同步，存在未知时间戳误差 $\delta_\ell$。本节把同步条件这个维度从"完全相干"推进到"部分相干"。 + +**首次引入符号**：$\delta_\ell$（采样点数为单位）、$\delta\tau_\ell$（秒为单位）、$\boldsymbol\delta = [\delta_1,\ldots,\delta_{L-1}]^\Tran$（参考星 $\delta_{\ell_\mathrm{ref}} = 0$）。 + +**扰动信号模型**： +$$\mathbf{y}_\ell(t) = \sum_k \zeta_{\ell,k} \mathbf{a}_\ell(\boldsymbol\xi_k) s_k(t - \delta_\ell) e^{j\nu_{\ell,k}(t - \delta_\ell)} + \mathbf{n}_\ell(t)$$ + +**关键观察**：$\delta_\ell$ 与 $\nu_{\ell,k}$ 耦合产生一个未知相位偏移 $-\nu_{\ell,k}\delta_\ell$。这是部分相干观测的核心结构。 + +**EFIM 分析**：将 $\{\delta_\ell\}_{\ell \neq \ell_\mathrm{ref}}$ 作为干扰参数通过 Schur complement 消去，得到对 $\boldsymbol\eta_\mathrm{mu}$ 的有效信息。解析表达式推不完整的地方，明确说明"以下用数值 EFIM 计算"，给出计算框架。 + +**适用性分析**：$|\delta_\ell|$ 的量级对模型的影响——小误差可一阶近似，大误差需要完整处理。这部分对应论文中"方法适用边界"的讨论。 + +**对应附录**：F（时间戳误差 EFIM 分析，新增）。 + +#### 第 6 节：多帧动态模型与连续相位结构【第一篇论文】 + +**过渡段主旨**：前五节都在单帧（静态）框架内推进。第一篇论文的方向是另一个独立的维度扩展：多帧观测带来跨帧相位连续性这一新信息。本节在第 3 节（单用户多星）基础上引入多帧结构，**不**叠加第 4、5 节的多用户和时间戳误差扩展（避免一次引入太多变量）。 + +**首次引入符号**：$P$、$p$、$\tau_p$、$t_{p,t}$、$\dot\nu_\mathrm{ref}$、$\Delta\dot\nu_\ell(\boldsymbol\xi)$、$\chi_\ell$、$\vartheta_{\ell,p}$、$\rho_{\ell,p}^\mathrm{cp}$、$\rho_{\ell,p}^\mathrm{ip}$。 + +**CP 模型与 IP 基线**：连续相位模型保留 $\chi_\ell$ 作为同星公共相位；独立相位基线用 $\vartheta_{\ell,p}$ 作为块相位。两者的 EFIM 对比量化跨帧相位连续性携带的信息量。 + +**Doppler rate 处理**：$\dot\nu_\mathrm{ref}$ 作为干扰参数，已知/未知两种情形对应 CP-K/CP-U。 + +**对应附录**：G（多帧动态 CRB，由原 A7+A8 合并）。 + +#### 第 7 节：波束成形（预留） + +本次改造**不展开**第 7 节内容，但在主文件 `doa_doppler_est.tex` 中以注释形式预留位置，并在前言和符号索引中提及波束成形的接口符号（$\mathbf{w}_{\ell,k}$、$\mathbf{h}_{\ell,k}$、$\mathbf{Q}$ 等）。 + +--- + +## 四、符号体系统一方案 + +### 4.1 必须执行的符号修改 + +**唯一硬冲突**：将旋转矩阵 $\mathbf{Q}_\ell$ 改为 $\boldsymbol\Omega_\ell$。原因：$\mathbf{Q}$ 在波束成形领域是人工噪声协方差的强约定符号，必须释放。 + +具体操作： +1. 在 `01-assumptions.tex`（移入附录 A 之后）的 DoA 与几何符号表中，将 $\mathbf{Q}_\ell$ 改为 $\boldsymbol\Omega_\ell$。 +2. 在 `02-modeling.tex` 中全局替换所有 $\mathbf{Q}_\ell$ 为 $\boldsymbol\Omega_\ell$。 +3. 在所有 `parts/*.tex` 和 `parts/A*.tex` 中全局替换。 +4. 在符号使用规则表中新增一条："旋转矩阵使用 $\boldsymbol\Omega_\ell$，$\mathbf{Q}$ 保留给波束成形人工噪声协方差矩阵。" + +### 4.2 现有符号表的补充行 + +下列符号应在附录 A 对应表格中补充。这些是估计侧与波束成形侧的桥接符号，提前定义可避免后续章节重复定义。 + +**在"索引、维度与数据堆叠"表末尾补充：** + +| 符号 | 含义 | +|------|------| +| $N_e$ / $e$ | $N_e$ 为窃听者总数；$e$ 为窃听者索引，$e=1,\ldots,N_e$。单窃听者场景中可省略下标。 | + +**在"DoA、阵列响应与几何符号"表末尾补充：** + +| 符号 | 含义 | +|------|------| +| $\mathbf{h}_{\ell,k} \in \mathbb{C}^{M_\ell}$ | 第 $k$ 个用户（单天线）到第 $\ell$ 颗卫星的上行等效信道向量，远场窄带假设下 $\mathbf{h}_{\ell,k} = \zeta_{\ell,k}\mathbf{a}_\ell(\boldsymbol\xi_k)$。参数估计侧使用 $\mathbf{a}_\ell$ 和 $\zeta_{\ell,k}$；波束成形侧合并为 $\mathbf{h}_{\ell,k}$。 | +| $\mathbf{H}_\ell \in \mathbb{C}^{M_\ell \times K}$ | 第 $\ell$ 颗卫星的上行信道矩阵，第 $k$ 列为 $\mathbf{h}_{\ell,k}$。 | + +**新增"时间戳误差"小节或并入"Doppler 与相位参数"表：** + +| 符号 | 含义 | +|------|------| +| $\delta_\ell$ | 第 $\ell$ 颗卫星的时间戳误差，单位采样点数。参考星 $\delta_{\ell_\mathrm{ref}} = 0$，其余作为干扰参数。 | +| $\delta\tau_\ell$ | 对应物理时间误差（秒），$\delta_\ell = \delta\tau_\ell / T_\mathrm{s}$。 | + +**在"CRB、FIM 与有效信息符号"表末尾补充：** + +| 符号 | 含义 | +|------|------| +| $\boldsymbol\epsilon_{\xi,k} = \hat{\boldsymbol\xi}_k - \boldsymbol\xi_k$ | 第 $k$ 个用户的 DoA 估计误差向量。 | +| $\mathbf{C}_{\xi,k}$ | 对应的 CRB 协方差矩阵，EFIM 对 $\boldsymbol\xi_k$ 块的逆，作为波束成形鲁棒设计的理论上界。 | + +### 4.3 新增波束成形预留表 + +在附录 A 的所有现有表之后新增一张表"波束成形与安全通信符号"。本次只填充，不在正文使用： + +| 符号 | 含义 | +|------|------| +| $\mathbf{w}_{\ell,k} \in \mathbb{C}^{M_\ell}$ | 第 $\ell$ 颗卫星对第 $k$ 个用户的波束成形向量。上行/下行语境由所在节说明确定，同时出现时加上标 $\mathrm{ul}$ 或 $\mathrm{dl}$。 | +| $\mathbf{W}_\ell \in \mathbb{C}^{M_\ell \times K}$ | 第 $\ell$ 颗卫星的波束成形矩阵，第 $k$ 列为 $\mathbf{w}_{\ell,k}$。 | +| $\mathbf{Q} \in \mathbb{C}^{M_\ell \times M_\ell}$ | 人工噪声协方差矩阵，$\mathbf{Q} \succeq \mathbf{0}$。仅在安全波束成形节中使用。 | +| $P_t$ | 发射总功率约束。 | +| $P_\ell$ | 第 $\ell$ 颗卫星的功率约束。 | +| $\mathrm{SINR}_k$ | 第 $k$ 个合法用户的接收信噪干扰比。 | +| $\mathcal{K} = \{1,\ldots,K\}$ | 合法用户集合（花体字母，与用户总数 $K$ 区分）。 | +| $\mathcal{E} = \{1,\ldots,N_e\}$ | 窃听者集合。 | +| $\mathbf{h}_{\ell,e} \in \mathbb{C}^{M_\ell}$ | 第 $e$ 个窃听者（单天线）到第 $\ell$ 颗卫星的信道向量。 | +| $C_s^{(k)}$ | 对第 $k$ 个合法用户的保密速率。 | +| $\Delta\mathbf{h}_{\ell,k}$ | 信道估计误差向量，$\|\Delta\mathbf{h}_{\ell,k}\|_2 \leq \epsilon_k$ 用于鲁棒设计。 | +| $\epsilon_k$ | 第 $k$ 个用户的信道不确定性半径。 | + +### 4.4 全文符号一致性检查 + +修改完成后需要做一次全文符号一致性检查，重点检查： + +1. 所有 $\mathbf{Q}_\ell$ 是否已全部替换为 $\boldsymbol\Omega_\ell$。 +2. 单用户场景的 $\zeta_\ell$、$\boldsymbol\xi$、$\nu_\mathrm{ref}$ 与多用户场景的 $\zeta_{\ell,k}$、$\boldsymbol\xi_k$、$\nu_{\mathrm{ref},k}$ 在各自上下文中没有交叉污染。 +3. 单帧记号 $\mathbf{y}_\ell(t)$ 与多帧记号 $\mathbf{y}_{\ell,p}^\mathrm{cp}(t)$ 在第 6 节后没有混用。 +4. 投影算子统一使用 $\mathcal P$，Fisher 信息矩阵统一使用 $\mathcal I$，Jacobian 统一使用 $\mathbf{J}$。 + +--- + +## 五、附录合并与重构方案 + +### 5.1 新附录结构 + +``` +附录 A 完整符号索引 + 来源：完整迁移 01-assumptions.tex 的所有符号表 + 调整：补充 §4.2 和 §4.3 的新增符号 + 新增：在最末加入"波束成形与安全通信符号"表 + +附录 B 数学基础 + 来源：保留原 A1-background.tex，不动 + +附录 C 经典 DoA 方法（MVDR/MUSIC/ESPRIT） + 来源：保留原 A2-doaEstMethod.tex，不动 + +附录 D 单阵列 CRB 汇总 + 来源：合并原 A3-detDoaCrb.tex + A4-stoDoaCrb.tex + A5-detDoaDopplerCrb.tex + 子节结构： + D.1 确定信号 DoA CRB ← 原 A3 + D.2 确定信号 DoA-Doppler CRB ← 原 A5 + D.3 随机信号 DoA CRB ← 原 A4（支线对照） + 合并理由：三者共享 FIM 推导框架，合并便于对比 + +附录 E 多星静态模型 CRB + 来源：原 A6-crbStatPilotDoaDopper.tex + 调整：扩展为单用户和多用户两个子节 + 子节结构： + E.1 单用户多星 CRB ← 原 A6 主体 + E.2 多用户多星 CRB ← 新增推导，对应正文第 4 节 + +附录 F 时间戳误差 EFIM 分析（新增） + 新增内容，对应正文第 5 节 + 子节结构： + F.1 FIM 分块结构 + F.2 Schur complement 消去 δ_ℓ + F.3 数值 EFIM 计算框架 + F.4 特殊情况的解析推导（K=1, L=2 等可推导的情况） + +附录 G 多帧动态模型 CRB + 来源：合并原 A7-crbDynPilotDoaDopperCont.tex + A8-crbDynPilotDoaDopperInd.tex + 子节结构： + G.1 连续相位 CP 模型 EFIM ← 原 A7 + G.2 独立相位 IP 基线 EFIM ← 原 A8 + G.3 CP/IP 信息差异的 Schur 分析 + 合并理由：CP 与 IP 模型本来就是对比关系，合并有助于理解信息差异 +``` + +### 5.2 附录合并的具体操作 + +**附录 D 合并步骤**： +1. 创建新文件 `parts/D-singleArrayCrb.tex`。 +2. 写入合并后的章节标题与导言段（说明本附录涵盖三种单阵列 CRB 情形）。 +3. 依次插入原 A3、A5、A4 的内容（注意顺序：先 DoA、再 DoA-Doppler、最后随机信号对照）。 +4. 调整原 A3/A5/A4 内的 `\section`/`\subsection` 层级。 +5. 删除原 A3/A4/A5 文件。 + +**附录 E 扩展步骤**： +1. 将原 A6 重命名为 `parts/E-multiSatStaticCrb.tex`。 +2. 将原 A6 内容作为 E.1。 +3. 新增 E.2 多用户扩展推导（agent 根据正文第 4 节的信号模型推导）。 + +**附录 F 新建步骤**： +1. 创建 `parts/F-timestampErrCrb.tex`。 +2. 根据正文第 5 节的信号模型推导 FIM 分块结构。 +3. F.4 的解析推导可以从 $K=1, L=2$ 的最简情形入手，作为 agent 推导的起点。 + +**附录 G 合并步骤**： +1. 创建 `parts/G-multiFrameCrb.tex`。 +2. 写入合并后的章节标题与导言段（说明 CP 与 IP 是对比关系）。 +3. 依次插入原 A7、A8 内容。 +4. 新增 G.3 对比分析子节。 +5. 删除原 A7/A8 文件。 + +--- + +## 六、文件操作清单 + +### 6.1 需要新建的文件 + +| 文件路径 | 用途 | +|----------|------| +| `parts/preface.tex` | 不编号前言，替代原 `00-roadmap.tex` 的入口角色 | +| `parts/sec1_singleArrayDoa.tex` | 正文第 1 节 | +| `parts/sec2_singleArrayDoaDoppler.tex` | 正文第 2 节（重组原 detDoaDopplerEst.tex） | +| `parts/sec3_multiSatSingleUser.tex` | 正文第 3 节（重组原 multiArrayStatDetDoaDopplerEst.tex） | +| `parts/sec4_multiSatMultiUser.tex` | 正文第 4 节（新增） | +| `parts/sec5_timestampError.tex` | 正文第 5 节（新增，新论文核心） | +| `parts/sec6_multiFrameCp.tex` | 正文第 6 节（重组原 multiArrayDynDetDoaDopplerEst.tex） | +| `parts/D-singleArrayCrb.tex` | 附录 D（由 A3+A4+A5 合并） | +| `parts/F-timestampErrCrb.tex` | 附录 F（新增） | +| `parts/G-multiFrameCrb.tex` | 附录 G（由 A7+A8 合并） | + +### 6.2 需要重命名/移动的文件 + +| 原路径 | 新路径 | 操作类型 | +|--------|--------|----------| +| `parts/01-assumptions.tex` | `parts/A-symbolIndex.tex` | 重命名（迁移到附录 A） | +| `parts/A1-background.tex` | `parts/B-mathBackground.tex` | 重命名 | +| `parts/A2-doaEstMethod.tex` | `parts/C-classicalDoa.tex` | 重命名 | +| `parts/A6-crbStatPilotDoaDopper.tex` | `parts/E-multiSatStaticCrb.tex` | 重命名并扩展 | + +### 6.3 需要删除的文件 + +合并后删除以下文件： + +- `parts/00-roadmap.tex`（由 `preface.tex` 替代，内容部分移入维护说明） +- `parts/02-modeling.tex`（内容拆分到正文各节首次引入处） +- `parts/detDoaEst.tex`（合并入 `sec1_singleArrayDoa.tex`） +- `parts/stoDoaEst.tex`（核心论述迁移到 sec1 末尾对比小节，详细推导保留在附录 D.3） +- `parts/detDoaDopplerEst.tex`（合并入 `sec2_singleArrayDoaDoppler.tex`） +- `parts/stoDoaDopplerEst.tex`（已注释，可直接删除） +- `parts/multiArrayStoDoaEst.tex`（不进入主线，删除） +- `parts/multiArrayStatDetDoaDopplerEst.tex`（合并入 `sec3_multiSatSingleUser.tex`） +- `parts/multiArrayDynDetDoaDopplerEst.tex`（合并入 `sec6_multiFrameCp.tex`） +- `parts/multiArrayDetDoaDopplerDelayEst.tex`（与新论文方向不符，移入仓库根目录的 `archive/` 子目录保留备查） +- `parts/A3-detDoaCrb.tex`、`parts/A4-stoDoaCrb.tex`、`parts/A5-detDoaDopplerCrb.tex`（合并入附录 D 后删除） +- `parts/A7-crbDynPilotDoaDopperCont.tex`、`parts/A8-crbDynPilotDoaDopperInd.tex`（合并入附录 G 后删除） +- `parts/98-future work.tex`（内容并入前言末尾的"后续方向"简短段落） +- `parts/99-literature.tex`（已注释，可直接删除） + +### 6.4 需要修改的文件 + +**`doa_doppler_est.tex`（主文件）**： + +修改 `\input` 顺序为： + +```latex +\input{parts/preface} + +\input{parts/sec1_singleArrayDoa} +\input{parts/sec2_singleArrayDoaDoppler} +\input{parts/sec3_multiSatSingleUser} +\input{parts/sec4_multiSatMultiUser} +\input{parts/sec5_timestampError} +\input{parts/sec6_multiFrameCp} +% \input{parts/sec7_beamforming} % 预留，暂不展开 + +\appendix +\renewcommand{\thesection}{附录~\Alph{section}} +...（保持原附录编号设置） + +\input{parts/A-symbolIndex} +\input{parts/B-mathBackground} +\input{parts/C-classicalDoa} +\input{parts/D-singleArrayCrb} +\input{parts/E-multiSatStaticCrb} +\input{parts/F-timestampErrCrb} +\input{parts/G-multiFrameCrb} +``` + +同时更新 abstract 内容，使其反映新的章节结构。 + +--- + +## 七、实施顺序 + +为降低风险，按以下阶段执行。每完成一个阶段，先做编译测试，确认 LaTeX 不报错、交叉引用不断裂，再进入下一阶段。 + +### 阶段 1：附录重整（最低风险） + +1. 创建 `parts/A-symbolIndex.tex`，从 `01-assumptions.tex` 迁移内容并补充 §4.2、§4.3 的新增符号。 +2. 重命名 `A1` → `B-mathBackground.tex`，`A2` → `C-classicalDoa.tex`。 +3. 合并 A3+A4+A5 为 `parts/D-singleArrayCrb.tex`。 +4. 重命名 A6 为 `parts/E-multiSatStaticCrb.tex`（暂不扩展多用户部分）。 +5. 合并 A7+A8 为 `parts/G-multiFrameCrb.tex`。 +6. 修改主文件的附录部分 `\input`，编译测试。 + +**验收标准**：编译通过；所有从正文指向附录的交叉引用仍能解析；附录结构与本文档 §5.1 一致。 + +### 阶段 2：执行 $\mathbf{Q}_\ell \to \boldsymbol\Omega_\ell$ 全局替换 + +1. 全局搜索 `\mathbf{Q}_\ell`、`\mathbf{Q}_{\ell}`、`\mathbf Q_\ell` 等所有写法。 +2. 替换为 `\boldsymbol\Omega_\ell`。 +3. 在附录 A 的符号表中更新对应条目。 +4. 在符号规则表中新增释义。 + +**验收标准**：全文不再出现旋转矩阵的 $\mathbf{Q}$ 写法；编译通过。 + +### 阶段 3：新建前言并删除 roadmap + +1. 创建 `parts/preface.tex`，写入约半页的简短前言（笔记目标、内容递进逻辑、如何使用本文档）。 +2. 将原 `00-roadmap.tex` 中"放置规则"和"文件定位"的部分移入维护说明（可放在仓库根目录的 `MAINTENANCE.md`）。 +3. 删除 `00-roadmap.tex`。 +4. 修改主文件 `\input` 顺序。 + +**验收标准**：编译通过；前言长度不超过一页；正文第一节直接开始 DoA 估计的实质内容。 + +### 阶段 4：正文章节重组（最高风险，需谨慎） + +按第 1、2、3、6 节顺序重组现有内容（这四节内容已经存在，只是需要重新组织、加过渡段、精简推导）： + +1. 创建 `sec1_singleArrayDoa.tex`：以 `detDoaEst.tex` 为基础，加过渡段，把详细 CRB 推导引用到附录 D.1，末尾加随机信号对比小节。 +2. 创建 `sec2_singleArrayDoaDoppler.tex`：以 `detDoaDopplerEst.tex` 为基础，加过渡段。 +3. 创建 `sec3_multiSatSingleUser.tex`：以 `multiArrayStatDetDoaDopplerEst.tex` 为基础，强化参考星参数化的动机说明。 +4. 创建 `sec6_multiFrameCp.tex`：以 `multiArrayDynDetDoaDopplerEst.tex` 为基础，加过渡段，把详细 CRB 推导引用到附录 G。 +5. 删除被合并的原文件。 +6. 修改主文件 `\input` 顺序。 + +**验收标准**：每节开头有过渡段；正文不再展开矩阵代入；详细推导都在附录中；编译通过；所有交叉引用解析。 + +### 阶段 5：新增多用户与时间戳误差章节 + +1. 创建 `sec4_multiSatMultiUser.tex`，按 §3.2 第 4 节的指引撰写。 +2. 创建 `sec5_timestampError.tex`，按 §3.2 第 5 节的指引撰写。 +3. 扩展附录 E 加入 E.2 多用户 CRB。 +4. 新建附录 F（时间戳误差 EFIM）。 +5. 修改主文件 `\input` 顺序。 + +**验收标准**：编译通过；新增章节与前面章节符号一致；附录 F 至少包含 F.1 分块结构与 F.2 Schur complement，F.3 数值框架，F.4 可暂留 placeholder 等待具体推导。 + +### 阶段 6：全文检查与抛光 + +1. 全文符号一致性检查（按 §4.4 清单）。 +2. 全文交叉引用检查。 +3. 各节过渡段流畅性检查。 +4. 编译产生最终 PDF，由用户复核。 + +--- + +## 八、给 agent 的执行须知 + +1. **保守优先**：本次改造工作量大，每阶段都要先编译测试。出现编译错误立即回滚到上一阶段稳定版本，不要带错误继续推进。 +2. **保留备份**：执行任何文件删除前，先 git commit 当前状态，不要直接 `rm` 无备份。建议给被删除的文件先复制到仓库根目录的 `archive/` 子目录。 +3. **过渡段写作风格**：每节开头的过渡段应做到"上一节的什么限制 → 本节解除哪个限制 → 引入什么新结构"，避免空泛的承上启下。 +4. **正文不要展开矩阵代入**：正文一旦出现"将 ... 代入 ... 得到 ..."这样的字眼，立刻检查是否应该把这一步推到附录。 +5. **新章节的符号必须在附录 A 注册**：第 4、5 节新增的符号（$\boldsymbol\eta_\mathrm{mu}$、$\delta_\ell$ 等）必须同步更新附录 A 的符号表。 +6. **遇到推导卡壳**：如果某个具体推导（如附录 F.4 的解析特殊情况）卡住，先留 placeholder 注释，把章节结构和其他可完成的部分先做完，留待后续讨论。 +7. **语言风格**：中文为主，公式、符号、文件路径用英文；保持与原有文档的写作风格一致。 +8. **不创造新的元概念**：不要新增"放置规则"、"阅读路线图"等元信息章节。这些信息全部归入前言或仓库根目录的 `MAINTENANCE.md`。 diff --git a/doa_doppler_est.tex b/doa_doppler_est.tex new file mode 100644 index 0000000..b08ca32 --- /dev/null +++ b/doa_doppler_est.tex @@ -0,0 +1,178 @@ +\documentclass[11pt,a4paper]{article} + +% packages +\usepackage{xeCJK} +\usepackage{geometry} % 用于设置上下左右页边距 +\geometry{ + left=1.9cm, + right=1.9cm, + top=2.54cm, + bottom=2.54cm +} + +\usepackage{indentfirst} +\setlength{\parindent}{2em} + +\usepackage{amsmath,amssymb,amsfonts,bm} +\usepackage{array} +\usepackage{longtable} +\numberwithin{equation}{section} +\DeclareMathOperator*{\argmax}{arg\,max} +\DeclareMathOperator*{\argmin}{arg\,min} +\DeclareMathOperator*{\diag}{diag} +\DeclareMathOperator*{\maximize}{maximize} +\DeclareMathOperator*{\minimize}{minimize} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\vect}{vec} +\newcommand{\Tran}{\mathsf{T}} +\newcommand{\Herm}{\mathsf{H}} +\newcommand{\diff}{\mathop{}\!\mathrm{d}} + +\usepackage{arydshln} + +\usepackage[ + hidelinks, + bookmarksopen = true, + bookmarksnumbered, + hypertexnames = false, + colorlinks = true, + allcolors = blue + ]{hyperref} + +\usepackage{bookmark} + +\usepackage[ + backend=biber, + style=ieee, + maxcitenames=1, + mincitenames=1, + url=false, + doi=false, + isbn=false]{biblatex} +% \addbibresource{IEEEabrv.bib} +\addbibresource{ref.bib} + +\usepackage[inline]{enumitem} +\setlist[itemize]{leftmargin=*,itemsep=0pt, topsep=0pt, parsep=0pt, partopsep=0pt} +\setlist[enumerate]{leftmargin=*,itemsep=0pt, topsep=0pt, parsep=0pt, partopsep=0pt} + +\usepackage[most]{tcolorbox} + +\newenvironment{mdquote}{ +\begin{tcolorbox}[ + colback=gray!12, + colframe=gray!40, + boxrule=0pt, + left=6pt, + right=6pt, + top=6pt, + bottom=6pt, + arc=2pt, + breakable, + before skip=4pt, + after skip=4pt + ] + \setlength{\parindent}{2em} + }{ +\end{tcolorbox} +} + +\usepackage{caption} +\captionsetup[figure]{font=small} + +\usepackage{titlesec} +\titlespacing*{\section}{0pt}{*0}{*0} % 左边缩进 = 0pt，上间距 = 0，下间距 = 0 +\titlespacing*{\subsection}{0pt}{*0}{*0} % 左边缩进 = 0pt，上间距 = 0，下间距 = 0 +\titlespacing*{\subsubsection}{0pt}{*0}{*0} % 左边缩进 = 0pt，上间距 = 0，下间距 = 0 + +\usepackage[acronym]{glossaries} +\makeglossaries% +\glsdisablehyper% +\setacronymstyle{short-long} +\newacronym{cml}{CML}{Conditional Maximum Likelihood，条件最大似然} +\newacronym{crb}{CRB}{Cramér–Rao Bound，克拉美罗界} +\newacronym{crlb}{CRLB}{Cramér–Rao Lower Bound，克拉美罗下界} +\newacronym{cdml}{CDML}{Concentrated Deterministic Maximum Likelihood，集中确定最大似然} +\newacronym{cpi}{CPI}{Coherent Processing Interval，相干处理间隔} +\newacronym{csml}{CSML}{Concentrated Stochastic Maximum Likelihood，集中随机最大似然} +\newacronym{dml}{DML}{Deterministic Maximum Likelihood，确定最大似然} +\newacronym{doa}{DoA}{Direction-of-Arrival，波达方向} +\newacronym{ecef}{ECEF}{Earth-centered, Earth-fixed，地心地固} +\newacronym{eci}{ECI}{Earth-centered Inertial，地心惯性} +\newacronym{esprit}{ESPRIT}{Estimation of Signal Parameters via Rotational Invariant Techniques，旋转不变信号参数估计法} +\newacronym{fdoa}{FDOA}{Frequency Difference of Arrival，到达频率差} +\newacronym{fim}{FIM}{Fisher Information Matrix，费舍尔信息矩阵} +\newacronym{leo}{LEO}{Low Earth Orbir，低地球轨道} +\newacronym{mle}{MLE}{Maximum Likelihood Estimation，最大似然估计} +\newacronym{mse}{MSE}{Mean Square Error，均方误差} +\newacronym{music}{MUSIC}{MUtiple SIgnal Classification，多重信号分类} +\newacronym{pdf}{PDF}{Probability Density Function，概率密度函数} +\newacronym{rmse}{RMSE}{Root-Mean Square Error，均方根误差} +\newacronym{sml}{SML}{Stochastic Maximum Likelihood，随机最大似然} +\newacronym{tdoa}{TDOA}{Time Difference of Arrival，到达时间差} +\newacronym{ula}{ULA}{Uniform Linear Array，均匀线性阵列} +\newacronym{uml}{UML}{Unconditional Maximum Likelihood，无条件最大似然} +\newacronym{upa}{UPA}{Uniform Planar Array，均匀平面阵列} + +\title{DoA 与 Doppler 估计} +\author{Josh Gao} +\date{\today} + +\linespread{1.5} +\allowdisplaybreaks[4] +\setlength{\abovedisplayskip}{0pt} +\setlength{\belowdisplayskip}{0pt} + +\begin{document} + +\maketitle + +\begin{abstract} + 使用通信信号估计来波方向与 Doppler 参数，是通信感知一体化和低轨卫星协作感知中的基础问题。 + 本文以一份可持续扩展的推导笔记为目标，先在单阵列确定信号与随机信号模型下整理经典 \gls{doa} 估计、DoA--Doppler 联合估计及其 \gls{crb}，再将模型推进到单用户多星协作场景。 + 多星主线采用统一的全局方向参数 $\boldsymbol\xi=[\alpha,\beta]^\Tran$ 与参考星链路归一化 Doppler 参数 $\nu_{\mathrm{ref}}$，先建立静态 DoA--Doppler 基线模型，再引入多帧索引、Doppler rate $\dot\nu_{\mathrm{ref}}$ 和跨帧连续相位结构，进一步比较 relaxed 连续相位模型与独立相位基线模型的信息差异。 + 在此基础上，本文还给出 DoA--Doppler--delay 扩展和相应附录推导，使单阵列基础、多星主线、对比模型与 \gls{crb} 分析在同一符号体系下衔接。 +\end{abstract} + +\input{parts/00-roadmap.tex} +\input{parts/01-assumptions.tex} +\input{parts/02-modeling.tex} +\input{parts/detDoaEst.tex} +\input{parts/stoDoaEst.tex} +\input{parts/detDoaDopplerEst.tex} +% \input{parts/stoDoaDopplerEst.tex} +\input{parts/multiArrayStoDoaEst.tex} +\input{parts/multiArrayStatDetDoaDopplerEst.tex} +\input{parts/multiArrayDynDetDoaDopplerEst.tex} +\input{parts/multiArrayDetDoaDopplerDelayEst.tex} +% \input{parts/98-future work.tex} +% \input{parts/99-literature.tex} + +\appendix +\renewcommand{\thesection}{附录~\Alph{section}} +\renewcommand{\thesubsection}{\Alph{section}.\arabic{subsection}} +\renewcommand{\thesubsubsection}{\Alph{section}.\arabic{subsection}.\arabic{subsubsection}} +\renewcommand{\theequation}{\Alph{section}.\arabic{equation}} + +\newpage +\input{parts/A1-background.tex} +\newpage +\input{parts/A2-doaEstMethod.tex} +\newpage +\input{parts/A3-detDoaCrb.tex} +\newpage +\input{parts/A4-stoDoaCrb.tex} +\newpage +\input{parts/A5-detDoaDopplerCrb.tex} +\newpage +\input{parts/A6-crbStatPilotDoaDopper.tex} +\newpage +\input{parts/A7-crbDynPilotDoaDopperCont.tex} +\newpage +\input{parts/A8-crbDynPilotDoaDopperInd.tex} + +% \newpage +% \renewcommand*{\bibfont}{\footnotesize} +% \printbibliography\ + +\end{document} diff --git a/fig/doaDopplerEciDualSatSingleSourceUpa2048Snap.pdf b/fig/doaDopplerEciDualSatSingleSourceUpa2048Snap.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0b6f9f66abdd8ecdfc36a2b7fe6b6eb997efe23c GIT binary patch literal 131915 zcmeEv2|!cF_P77nD(b)CF-tBBHVr5d~2o z3Ic+VSWyD91Z9a3Ak_!~!eR&^kdWk?n+55seXo75wf24ORPNl#y>n;g&Tr0{-#IgP z=OfY71_o1A_j~vqcU4}!W~!>mNmnO7#7X7VLeYd> z6O1S9pI`z0Zke!tf-;C7o!|1Owa=98sMKA_@_BR9h|F)|LTBW_kmy3#Mc}FX9p&nnBXGLA9~UzP@+92 z5ib7Du970{1rJshX9s1n5oDq*E3^l~+tkSq)WTFxLrp_lO;bZtT}@L*OJkkd=2dEH ztJX|4GMeh@?E=+VV|?9%Pq-?pnmGA69Y?r>$2qyX`YLOJ3-*n>VbjlU*c#+#Zsi9a 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zem{MVB0v$K2v7tl0u+Ith`?u?_Ggzr@r!hHbT(|*(5qLkpE74-*4dabKV>m=5{dvt J;Fm$*e*uX~(eVHP literal 0 HcmV?d00001 diff --git a/parts/00-roadmap.tex b/parts/00-roadmap.tex new file mode 100644 index 0000000..31a1ca7 --- /dev/null +++ b/parts/00-roadmap.tex @@ -0,0 +1,116 @@ +\section{阅读路线图与文件定位} +\label{sec:roadmap} + +本笔记的核心目标，是在同一套符号体系下，把单阵列 \gls{doa}、单阵列 DoA--Doppler、多星静态 DoA--Doppler、多星动态连续相位 DoA--Doppler，以及相应 \gls{crb} 推导组织成一条可追踪的递进链条。 +因此，本文并不把所有文件都压缩成单一论文主线，而是区分主线模型、对比模型、扩展模型和附录推导四类内容。 +阅读时应首先判断当前文件的定位：它是在定义全文共用记号，还是在主线模型上增加新结构，或者作为附录推导与对比材料。 +除特别说明外，当前多星 DoA--Doppler 主线以第~\ref{sec:assumptionsAndNotation} 节和第~\ref{subsec:multiArrayCommonNotation} 小节为符号来源。 + +\subsection{全文的层次递进} + +本文按如下层次展开： +\begin{enumerate} + \item \textbf{统一基础层：}第~\ref{sec:assumptionsAndNotation} 节统一基本假设、索引、几何参数、Doppler 参数、参考星参数化、连续相位/独立相位记号，以及 \gls{fim}/\gls{crb} 记号；第~\ref{subsec:single_arr_doa_est}--\ref{subsec:multiArrayCommonNotation} 小节给出单阵列和多星几何建模入口。 + \item \textbf{单阵列基线层：}确定信号 \gls{doa}、随机信号 \gls{doa} 和单阵列 DoA--Doppler 估计用于建立似然函数、集中似然和投影矩阵的基本模板。 + \item \textbf{多星主线层：}在统一全局 DoA $\boldsymbol\xi=[\alpha,\beta]^\Tran$ 与参考星 Doppler $\nu_{\mathrm{ref}}$ 的基础上，先建立静态多星 DoA--Doppler 模型，再引入多帧索引、Doppler rate $\dot\nu_{\mathrm{ref}}$ 和跨帧连续相位结构。 + \item \textbf{对比与扩展层：}relaxed 连续相位模型用于降低集中似然处理复杂度，独立相位模型用于量化丢弃跨帧绝对相位连续性后的信息损失，DoA--Doppler--delay 模型则是在静态主线上的时延扩展。 + \item \textbf{附录推导层：}附录集中放置基础概率公式、经典算法整理和各类 \gls{crb} 推导，使正文专注于模型结构和估计问题本身。 +\end{enumerate} + +这一路径体现的逻辑是：先有统一符号，再有单阵列基线；先有静态多星模型，再有动态多帧模型；先有连续相位主模型，再有 relaxed 处理和独立相位对比。 + +\subsection{推荐阅读顺序} + +若希望快速把握当前多星 DoA--Doppler 主线，建议按如下顺序阅读： +\begin{enumerate} + \item \texttt{parts/01-assumptions.tex}：全文最高优先级的符号表与假设表； + \item \texttt{parts/02-modeling.tex}：单阵列模型、多阵列模型和多星 DoA--Doppler 统一几何入口； + \item \texttt{parts/multiArrayStatDetDoaDopplerEst.tex}：静态多星确定信号 DoA--Doppler 基线模型； + \item \texttt{parts/multiArrayDynDetDoaDopplerEst.tex}：动态多帧连续相位主模型、relaxed concentrated MLE 与独立相位对比； + \item \texttt{parts/A6-crbStatPilotDoaDopper.tex}：静态多星模型对应的 \gls{crb}； + \item \texttt{parts/A7-crbDynPilotDoaDopperCont.tex}：连续相位与 relaxed 连续相位模型对应的 \gls{crb}； + \item \texttt{parts/A8-crbDynPilotDoaDopperInd.tex}：独立相位基线模型对应的 \gls{crb}。 +\end{enumerate} + +若只关心单阵列 \gls{doa} 与经典方法，可阅读： +\begin{enumerate} + \item \texttt{parts/01-assumptions.tex}； + \item \texttt{parts/02-modeling.tex} 中的单阵列部分； + \item \texttt{parts/detDoaEst.tex} 和 \texttt{parts/stoDoaEst.tex}； + \item \texttt{parts/A1-background.tex}--\texttt{parts/A4-stoDoaCrb.tex}。 +\end{enumerate} + +\subsection{各文件定位} + +\subsubsection{统一基础与单阵列基线} +\begin{itemize} + \item \texttt{parts/01-assumptions.tex}：定义全文符号、参数分组、模型标签和阅读规则。若其他文件的局部表述与本节冲突，以本节为准。 + \item \texttt{parts/02-modeling.tex}：给出单阵列、多阵列和多星主线共用的建模入口，是后续多星正文和附录默认依赖的母定义。 + \item \texttt{parts/detDoaEst.tex}：单阵列确定信号 \gls{doa} 估计，用于建立条件似然、投影矩阵和集中似然模板。 + \item \texttt{parts/stoDoaEst.tex}：单阵列随机信号 \gls{doa} 估计，用于和确定信号模型形成统计假设对照。 + \item \texttt{parts/detDoaDopplerEst.tex}：单阵列确定信号 DoA--Doppler 估计，用于说明 Doppler 需要导频/训练结构才能从未知信号中分离出来。 + \item \texttt{parts/stoDoaDopplerEst.tex}：随机信号 DoA--Doppler 的预留支线，当前不作为多星主线的一部分。 +\end{itemize} + +\subsubsection{多阵列/多星正文} +\begin{itemize} + \item \texttt{parts/multiArrayStoDoaEst.tex}：多阵列随机 \gls{doa} 模型，主要作为几何映射和随机模型支线，不是当前 DoA--Doppler 主线的必经文件。 + \item \texttt{parts/multiArrayStatDetDoaDopplerEst.tex}：\textbf{多星静态 DoA--Doppler 主线基线}。主参数为 $\boldsymbol\eta_{\mathrm{stat}}=[\boldsymbol\xi^\Tran,\nu_{\mathrm{ref}}]^\Tran$，各星导向矢量和 Doppler 均由统一几何映射诱导。 + \item \texttt{parts/multiArrayDynDetDoaDopplerEst.tex}：\textbf{多星动态多帧 DoA--Doppler 主线文件}。主估计目标仍为 $\boldsymbol\eta_{\mathrm{dyn}}=\boldsymbol\eta_{\mathrm{stat}}$，而 $\dot\nu_{\mathrm{ref}}$ 与连续相位公共项作为干扰参数进入相位模板。 + \item \texttt{parts/multiArrayDetDoaDopplerDelayEst.tex}：DoA--Doppler--delay 扩展模型。它在 $\boldsymbol\eta_{\mathrm{stat}}$ 的基础上加入 $\tau_{\mathrm{ref}}$，属于扩展主题，而不是动态连续相位主线的前置步骤。 +\end{itemize} + +\subsubsection{附录推导} +\begin{itemize} + \item \texttt{parts/A1-background.tex}：概率分布和 \gls{crb} 基础公式； + \item \texttt{parts/A2-doaEstMethod.tex}：MVDR、MUSIC、ESPRIT 及其空时扩展方法整理； + \item \texttt{parts/A3-detDoaCrb.tex}：单阵列确定信号 \gls{doa} 的 \gls{crb}； + \item \texttt{parts/A4-stoDoaCrb.tex}：单阵列随机信号 \gls{doa} 的 \gls{crb}； + \item \texttt{parts/A5-detDoaDopplerCrb.tex}：单阵列确定信号 DoA--Doppler 的 \gls{crb}； + \item \texttt{parts/A6-crbStatPilotDoaDopper.tex}：静态多星 DoA--Doppler 的 \gls{crb}； + \item \texttt{parts/A7-crbDynPilotDoaDopperCont.tex}：动态连续相位与 relaxed 连续相位模型的 \gls{crb}； + \item \texttt{parts/A8-crbDynPilotDoaDopperInd.tex}：动态独立相位基线模型的 \gls{crb}。 +\end{itemize} + +\subsection{当前主线与支线边界} + +为避免阅读时失焦，可将当前内容分为两组： +\begin{itemize} + \item \textbf{当前主线：}单用户多星协作场景下，基于参考星参数化的静态 DoA--Doppler 模型、动态连续相位多帧模型、relaxed concentrated MLE，以及与独立相位模型的性能对比。 + \item \textbf{支线与扩展：}随机信号 \gls{doa} 模型、单阵列经典算法、delay 扩展和附录推导。 +\end{itemize} + +支线内容并非无关内容，而是为主线提供背景、对照或扩展入口。阅读或修改时，应避免把支线中的局部记号反向带回当前多星主线。 + +\subsection{优先遵守的符号约定} + +阅读和新增内容应优先遵守以下约定： +\begin{enumerate} + \item 多星主线的全局 DoA 统一写为 $\boldsymbol\xi=[\alpha,\beta]^\Tran$。 + \item 参考星索引统一写为 $\ell_{\mathrm{ref}}$，除特别说明外默认 $\ell_{\mathrm{ref}}=0$。 + \item 参考星链路归一化 Doppler 与 Doppler rate 统一写为 $\nu_{\mathrm{ref}}$ 与 $\dot\nu_{\mathrm{ref}}$；物理 Hz 量只在几何解释中写作 $\nu^{\mathrm{Hz}}$ 与 $\dot\nu^{\mathrm{Hz}}$。 + \item 各星 Doppler 与 Doppler rate 由 + \[ + \nu_\ell=f_{\nu,\ell}(\boldsymbol\xi,\nu_{\mathrm{ref}}), + \qquad + \dot\nu_\ell=f_{\dot\nu,\ell}(\boldsymbol\xi,\dot\nu_{\mathrm{ref}}) + \] + 给出，而不是把每颗卫星的链路参数全部作为独立主参数。 + \item 多星阵列导向矢量统一写为 $\mathbf a_\ell(\boldsymbol\xi)$，它只表示阵列响应，不表示动力学加速度。 + \item 多帧离散时间统一写为 $t_{p,t}=\tau_p+t$，其中 $p$ 为帧索引，$t$ 为帧内采样索引。 + \item 连续相位模型中的同星公共相位写为 $\chi_\ell$，独立相位模型中的块相位写为 $\vartheta_{\ell,p}$。 + \item Fisher 信息矩阵写为 $\mathcal I$，有效 Fisher 信息写为 $\mathcal I_{\mathrm e}$；$\mathbf J$ 只用于 Jacobian 或导数矩阵。 +\end{enumerate} + +\subsection{新增或修改内容时的放置原则} + +新增内容前，建议先判断其属于哪一类： +\begin{enumerate} + \item 若是在统一符号、坐标、时间索引或参数映射层面补定义，应放入第~\ref{sec:assumptionsAndNotation} 节或第~\ref{subsec:multiArrayCommonNotation} 小节。 + \item 若是在静态多星 DoA--Doppler 基线中增加新观测或新干扰参数，应先说明它如何改变 $\boldsymbol\eta_{\mathrm{stat}}$ 或干扰参数集合。 + \item 若是在动态多帧模型中增加新结构，应说明它是改变连续相位模板、改变 relaxed 处理，还是新增一个对比基线。 + \item 若只是补充求导、矩阵分块或 Schur 补细节，应优先放入附录，避免正文重复推导。 + \item 若是备用推导或对比材料，应显式标注其与当前主线定义的关系。 +\end{enumerate} + +按照上述原则，本文可以保持“大而全”的推导库形态，同时让读者清楚知道：哪些内容定义主线，哪些内容解释主线，哪些内容用于对比或备查。 diff --git a/parts/01-assumptions.tex b/parts/01-assumptions.tex new file mode 100644 index 0000000..fa4b45f --- /dev/null +++ b/parts/01-assumptions.tex @@ -0,0 +1,556 @@ + +\section{基本假设与符号约定} +\label{sec:assumptionsAndNotation} + +本节汇总本文正文和附录中反复使用的基本假设、符号、上下标和模型标记。 +各节若需要引入局部临时符号，应说明其与本节符号的对应关系，避免同一物理量在不同估计方法中发生语义漂移。 + +\subsection{基本物理与统计假设} + +除特别说明外，后续单阵列、多阵列、静态 DoA--Doppler、多帧动态 DoA--Doppler 以及相应 \gls{crb} 推导均默认采用如下基本假设。 + +\begin{enumerate} + \item \textbf{信号源假设} + \begin{enumerate} + \item 信号源满足远场平面波假设，因此阵列响应可用方向相关导向矢量描述。 + \item 信号源满足窄带阵列假设，即信号扫过阵列孔径时包络近似不变，阵元间差异主要体现为相位差。 + \item 不同信号源在空间上可分，且信号源数量 $K$ 已知。 + \item 在确定信号模型，即 \gls{dml} 或 \gls{cml} 模型中，信源波形视为未知但确定的非随机量。 + \item 在随机信号模型，即 \gls{sml} 或 \gls{uml} 模型中，信源波形视为随机过程，其二阶统计量满足相应模型要求。 + \item 在 \gls{music} 类子空间方法中，默认信源协方差矩阵非奇异；若进一步要求信源互不相关，应在具体方法处单独说明。 + \end{enumerate} + + \item \textbf{噪声假设} + \begin{enumerate} + \item 噪声为加性噪声，并与信源相互独立。 + \item 单阵列基础模型中，噪声通常取为空间白、时间白、零均值圆对称复高斯噪声。 + \item 噪声方差统一记为 $\sigma_n^2$；多阵列或多卫星模型写为 $\sigma_{n,\ell}^2$。 + \item 多阵列或多卫星模型中，允许不同阵列或不同卫星具有不同噪声方差 $\sigma_{n,\ell}^2$，但同一阵列或同一卫星在一个短观测块或短观测窗口内保持不变。 + \item 典型噪声向量写为 $\mathbf n_t\sim\mathcal{CN}(\mathbf 0,\sigma_n^2\mathbf I_M)$、$\mathbf n_{\ell,t}\sim\mathcal{CN}(\mathbf 0,\sigma_{n,\ell}^2\mathbf I_{M_\ell})$ 或 $\mathbf n_{\ell,p}(t)\sim\mathcal{CN}(\mathbf 0,\sigma_{n,\ell}^2\mathbf I_{M_\ell})$。 + \end{enumerate} + + \item \textbf{阵列假设} + \begin{enumerate} + \item 阵元数应大于待分辨信号源数，以保证基本空间分辨能力。 + \item 基于样本协方差矩阵的随机信号估计方法通常要求快拍数足够大；若需要样本协方差矩阵满秩，常见条件为快拍数不小于阵元数或信源数相关维度。 + \item 阵列几何在一个观测块或短多帧窗口内视为已知；多星模型中卫星位置、速度和姿态在参考时刻由星历或几何模型给定。 + \end{enumerate} +\end{enumerate} + +\subsection{数学记号约定} + +全文默认采用表~\ref{tab:mathNotation} 中的数学记号约定。 +该表使用 \texttt{longtable} 环境，可以在符号项较多时自动跨页。 +带有文字含义的上下标统一使用直立体，例如 $T_{\mathrm{s}}$、$\lambda_{\mathrm{c}}$、$\ell_{\mathrm{ref}}$ 和 $\|\cdot\|_{\mathrm{F}}$。 +普通变量下标保持斜体，例如第 $k$ 个信源或第 $p$ 帧。 + +\begingroup +\small +\setlength{\LTpre}{4pt} +\setlength{\LTpost}{4pt} +\renewcommand{\arraystretch}{1.15} +\begin{longtable}{>{\raggedright\arraybackslash}p{0.30\textwidth}>{\raggedright\arraybackslash}p{0.60\textwidth}} + \caption{数学记号约定} + \label{tab:mathNotation}\\ + \hline + \textbf{符号} & \textbf{含义}\\ + \hline + \endfirsthead + \multicolumn{2}{l}{\small\tablename~\thetable{} 数学记号约定（续）}\\ + \hline + \textbf{符号} & \textbf{含义}\\ + \hline + \endhead + \hline + \multicolumn{2}{r}{\small 续下页}\\ + \endfoot + \hline + \endlastfoot + $a$ & 标量，除特别说明外用斜体小写或希腊字母表示。\\ + $\mathbf a\in\mathbb C^{m}$ & 向量，通常为列向量；英文字母向量用小写粗体，希腊字母向量用粗体希腊字母，例如 $\boldsymbol\mu$。\\ + $\mathbf A\in\mathbb C^{m\times n}$ & 矩阵，用大写粗体字母表示；协方差、旋转矩阵和选择矩阵也遵循该规则。\\ + $\mathcal A$ & 集合、函数族、算子或上下文相关对象；不要用它表示普通矩阵。\\ + $(\cdot)^\Tran$ & 转置。\\ + $(\cdot)^\Herm$ & 共轭转置。\\ + $(\cdot)^*$ & 复共轭。\\ + $\Re\{\cdot\}$ 和 $\Im\{\cdot\}$ & 取实部和取虚部。\\ + $\|\cdot\|_2$ & 向量二范数。\\ + $\|\cdot\|_{\mathrm{F}}$ & 矩阵 Frobenius 范数；下标 $\mathrm{F}$ 使用直立体。\\ + $\vect(\cdot)$ & 按列向量化算子。 本文推荐使用自定义 operator $\vect$；它在渲染上等价于 $\vect$，但便于全文统一。\\ + $\diag(\cdot)$ & 由输入构造对角矩阵。\\ + $\tr(\cdot)$ & 矩阵迹。 \\ + $\otimes$ & Kronecker 积。\\ + $\mathbf I_M$ & $M\times M$ 单位矩阵。\\ + $\mathbf 0$ & 与上下文维度匹配的零向量或零矩阵。\\ + $\mathbb R$ 和 $\mathbb C$ & 实数域与复数域。\\ + $\mathcal{CN}(\boldsymbol\mu,\mathbf R)$ & 均值向量为 $\boldsymbol\mu$、协方差矩阵为 $\mathbf R$ 的圆对称复高斯分布。\\ + $\argmax$ 和 $\argmin$ & 极大化与极小化解集；若上下文默认唯一解，则直接作为估计量使用。\\ +\end{longtable} +\endgroup + +\subsection{索引、维度与数据堆叠} + +全文中最常用的索引、维度和数据矩阵汇总于表~\ref{tab:indexDimensionNotation}。 +可以合并说明的总数与索引放在同一栏中，以减少符号表重复。 + +\begingroup +\small +\setlength{\LTpre}{4pt} +\setlength{\LTpost}{4pt} +\renewcommand{\arraystretch}{1.15} +\begin{longtable}{>{\raggedright\arraybackslash}p{0.30\textwidth}>{\raggedright\arraybackslash}p{0.60\textwidth}} + \caption{索引、维度与数据堆叠符号约定} + \label{tab:indexDimensionNotation}\\ + \hline + \textbf{符号} & \textbf{含义}\\ + \hline + \endfirsthead + \multicolumn{2}{l}{\small\tablename~\thetable{} 索引、维度与数据堆叠符号约定（续）}\\ + \hline + \textbf{符号} & \textbf{含义}\\ + \hline + \endhead + \hline + \multicolumn{2}{r}{\small 续下页}\\ + \endfoot + \hline + \endlastfoot + $K$ / $k$ & $K$ 为信号源或用户总数；$k$ 为信号源或用户索引，通常取 $k=1,\ldots,K$。\\ + $M$ / $m$ & $M$ 为单阵列阵元总数；$m$ 为阵元索引。 若未显式使用 $m$，则阵元索引由向量或矩阵维度隐含。\\ + $M_x,M_y$ & \gls{upa} 在 $x$ 轴和 $y$ 轴上的阵元数，常有 $M=M_xM_y$。\\ + $L$ / $\ell$ & $L$ 为多阵列或多卫星系统中的接收阵列数或协作卫星数；$\ell$ 为阵列、卫星或链路索引，通常取 $\ell=0,1,\ldots,L-1$。\\ + $M_\ell$ & 第 $\ell$ 个阵列或第 $\ell$ 颗卫星的阵元数。\\ + $T$ / $t$ & $T$ 为快拍数、单个观测块内采样点数或单帧保留样本数；$t$ 为快拍索引、块内采样索引或帧内采样索引，通常取 $t=0,1,\ldots,T-1$。\\ + $P$ / $p$ & $P$ 为多帧动态模型中的联合帧数；$p$ 为帧索引，通常取 $p=0,1,\ldots,P-1$。\\ + $\tau_p$ & 第 $p$ 帧起始点相对于参考时刻的离散采样偏移，默认 $\tau_0=0$。\\ + $t_{p,t}$ & 第 $p$ 帧第 $t$ 个采样点对应的全局离散时间索引，定义为 $t_{p,t}\triangleq\tau_p+t$。\\ + $T_{\mathrm{s}}$ & 采样周期；文字下标 $\mathrm{s}$ 使用直立体。\\ + $\mathbf y_t\in\mathbb C^{M}$ & 单阵列在第 $t$ 个快拍的接收向量。\\ + $\mathbf y_{\ell,t}\in\mathbb C^{M_\ell}$ & 第 $\ell$ 个阵列在第 $t$ 个快拍的接收向量。\\ + $\mathbf y_\ell(t)\in\mathbb C^{M_\ell}$ & 静态多星模型中第 $\ell$ 颗卫星在采样点 $t$ 的接收向量。\\ + $\mathbf y_{\ell,p}^{\mathrm{cp}}(t)$ 和 $\mathbf y_{\ell,p}^{\mathrm{ip}}(t)$ & 动态多帧模型中第 $\ell$ 颗卫星、第 $p$ 帧、第 $t$ 个采样点的连续相位和独立相位接收向量，维度通常为 $M_\ell\times1$。\\ + $\mathbf Y\in\mathbb C^{M\times T}$ & 单阵列所有快拍按列堆叠得到的观测矩阵。\\ + $\mathbf Y_\ell$ & 第 $\ell$ 个阵列或第 $\ell$ 颗卫星的观测矩阵，维度由具体模型确定。\\ + $\mathbf Y_{\ell,p}^{\mathrm{cp}}$ 和 $\mathbf Y_{\ell,p}^{\mathrm{ip}}$ & 第 $\ell$ 颗卫星第 $p$ 帧在连续相位或独立相位模型下的矩阵观测，典型维度为 $M_\ell\times T$。\\ + $\tilde{\mathbf y}_\ell$ 和 $\tilde{\mathbf y}_{\ell,p}$ & 对相应观测矩阵按列向量化后的观测向量。\\ + $\mathbf N$、$\mathbf N_\ell$、$\mathbf N_{\ell,p}$ & 与相应观测矩阵维度匹配的噪声矩阵。\\ + $\tilde{\mathbf n}_\ell$ 和 $\tilde{\mathbf n}_{\ell,p}$ & 对相应噪声矩阵按列向量化后的噪声向量，例如 $\tilde{\mathbf n}_\ell\triangleq\vect(\mathbf N_\ell)$、$\tilde{\mathbf n}_{\ell,p}\triangleq\vect(\mathbf N_{\ell,p})$。\\ +\end{longtable} +\endgroup + +\subsection{DoA、阵列响应与几何符号} + +单阵列基础模型和多星主线模型均使用方向参数，但二者的参数层级不同。 +本文采用表~\ref{tab:doaGeometryNotation} 中的约定，并统一使用 $\boldsymbol\xi=[\alpha,\beta]^\Tran$ 表示参考时刻全局 DoA。 + +\begingroup +\small +\setlength{\LTpre}{4pt} +\setlength{\LTpost}{4pt} +\renewcommand{\arraystretch}{1.15} +\begin{longtable}{>{\raggedright\arraybackslash}p{0.30\textwidth}>{\raggedright\arraybackslash}p{0.60\textwidth}} + \caption{DoA、阵列响应与几何符号约定} + \label{tab:doaGeometryNotation}\\ + \hline + \textbf{符号} & \textbf{含义}\\ + \hline + \endfirsthead + \multicolumn{2}{l}{\small\tablename~\thetable{} DoA、阵列响应与几何符号约定（续）}\\ + \hline + \textbf{符号} & \textbf{含义}\\ + \hline + \endhead + \hline + \multicolumn{2}{r}{\small 续下页}\\ + \endfoot + \hline + \endlastfoot + $\bm\theta_k$ & 第 $k$ 个信源的一般方向参数。 对于 \gls{ula}，它可退化为一维角度 $\theta_k$；对于 \gls{upa} 或三维场景，它可表示二维角度向量。\\ + $\bm\Theta$ & 所有信源方向参数的集合，定义为 $\bm\Theta\triangleq[\bm\theta_1,\ldots,\bm\theta_K]$。\\ + $\boldsymbol\xi=[\alpha,\beta]^\Tran$ & 单用户多星主线中的参考时刻全局 DoA，其中 $\alpha$ 为全局方位角，$\beta$ 为全局俯仰角。\\ + $\boldsymbol\xi_k=[\alpha_k,\beta_k]^\Tran$ & 第 $k$ 个用户或信源的全局 DoA；单用户主线中通常省略下标 $k$。\\ + $\mathbf u(\boldsymbol\xi)\in\mathbb R^{3}$ & 全局坐标系下由 $\boldsymbol\xi$ 确定的单位方向向量。\\ + $\mathbf s_\ell\in\mathbb R^{3}$ 和 $\mathbf v_\ell\in\mathbb R^{3}$ & 第 $\ell$ 颗卫星在参考时刻的全局位置向量和速度向量。\\ + $\mathbf e_{1,\ell}$、$\mathbf e_{2,\ell}$、$\mathbf e_{3,\ell}$ & 第 $\ell$ 颗卫星的本地阵列坐标基。\\ + $\boldsymbol{\Omega}_\ell\in\mathbb R^{3\times3}$ & 第 $\ell$ 颗卫星从本地坐标系到全局坐标系的旋转矩阵。因此 $\boldsymbol{\Omega}_\ell^\Tran\mathbf u(\boldsymbol\xi)$ 给出全局方向在该卫星本地坐标系下的表达。（注：原符号 $\mathbf{Q}_\ell$ 已改为 $\boldsymbol{\Omega}_\ell$，$\mathbf{Q}$ 保留给波束成形章节的人工噪声协方差矩阵。）\\ + $\mathbf u_\ell^{\mathrm{loc}}\in\mathbb R^{3}$ & 第 $\ell$ 颗卫星本地坐标系下的单位来向向量。\\ + $\alpha_\ell^{\mathrm{loc}}$ 和 $\beta_\ell^{\mathrm{loc}}$ & 第 $\ell$ 颗卫星本地坐标系下的方位角与俯仰角。\\ + $\mathbf a(\bm\theta_k)\in\mathbb C^{M}$ & 单阵列中第 $k$ 个信源方向对应的导向矢量。\\ + $\mathbf a_\ell(\boldsymbol\xi)\in\mathbb C^{M_\ell}$ & 多星主线中第 $\ell$ 颗卫星的本地阵列导向矢量。 该符号只表示阵列响应，不表示动力学加速度。\\ + $\mathbf A(\bm\Theta)\in\mathbb C^{M\times K}$ & 单阵列导向矩阵，通常由 $[\mathbf a(\bm\theta_1),\ldots,\mathbf a(\bm\theta_K)]$ 构成。\\ + $\mathbf A_\ell(\bm\Theta)$ & 第 $\ell$ 个阵列或卫星对应的局部导向矩阵。\\ + $d$、$d_x$、$d_y$ & 阵元间距；$d$ 用于 \gls{ula}，$d_x,d_y$ 用于 \gls{upa}。\\ + $\lambda_{\mathrm{c}}$ & 载波波长，专门用于阵列相位项。\\ + $u_{x,k}$ 和 $u_{y,k}$ & \gls{upa} 平面内的方向余弦分量。\\ + $\mathbf B_{\mathrm{st}}(\bm\Theta,\boldsymbol\nu)$ & 空时导向矩阵或空时 steering/dictionary matrix，由各信源的空时导向列向量组成；该符号避免与 CRB 协方差矩阵 $\mathbf C$ 冲突。\\ + $\mathbf b_{\mathrm{st},k}(\bm\theta_k,\nu_k)$ & 第 $k$ 个信源的空时导向列向量，通常由空间导向矢量与 Doppler 时间导向矢量 Kronecker 组合得到。此处下标 $\mathrm{st}$ 表示 space--time。\\ +\end{longtable} +\endgroup + +\subsection{信号、Doppler 与时延符号} + +本文统一使用 $\nu$ 表示 Doppler 参数；理论推导和似然函数中的 $\nu$ 默认表示离散归一化角 Doppler，单位为 rad/sample。 +若必须给出物理 Hz 单位，应显式写为 $\nu^{\mathrm{Hz}}$，避免与归一化模型参数混淆。 +相关符号汇总于表~\ref{tab:signalDopplerDelayNotation}。 + +\begingroup +\small +\setlength{\LTpre}{4pt} +\setlength{\LTpost}{4pt} +\renewcommand{\arraystretch}{1.15} +\begin{longtable}{>{\raggedright\arraybackslash}p{0.30\textwidth}>{\raggedright\arraybackslash}p{0.60\textwidth}} + \caption{信号、Doppler 与时延符号约定} + \label{tab:signalDopplerDelayNotation}\\ + \hline + \textbf{符号} & \textbf{含义}\\ + \hline + \endfirsthead + \multicolumn{2}{l}{\small\tablename~\thetable{} 信号、Doppler 与时延符号约定（续）}\\ + \hline + \textbf{符号} & \textbf{含义}\\ + \hline + \endhead + \hline + \multicolumn{2}{r}{\small 续下页}\\ + \endfoot + \hline + \endlastfoot + $x_k(t)$ & 第 $k$ 个信号源在第 $t$ 个快拍或采样点的复包络。\\ + $\mathbf x_t\in\mathbb C^K$ & 所有信源在第 $t$ 个快拍的信源向量。\\ + $\mathbf X$ & 所有快拍的信源矩阵。\\ + $s(t)$ & 单个观测块内的已知训练序列。\\ + $s_k(t)$ & 单阵列 DoA--Doppler 模型中第 $k$ 个源的已知训练序列；若采用常包络单音模型，则 $s_k(t)\equiv 1$。\\ + $s_p(t)$ & 多帧模型中第 $p$ 帧使用的已知训练序列。\\ + $\nu_\ell$ & 第 $\ell$ 条链路的归一化角 Doppler，单位为 rad/sample。\\ + $\dot\nu_\ell$ & 第 $\ell$ 条链路的归一化 Doppler rate，单位为 rad/sample$^2$。\\ + $\nu_\ell^{\mathrm{Hz}}$ & 第 $\ell$ 条链路的物理 Doppler，单位为 Hz；仅在需要物理单位解释时使用。\\ + $\dot\nu_\ell^{\mathrm{Hz}}$ & 第 $\ell$ 条链路的物理 Doppler rate，单位为 Hz/s；仅在需要物理单位解释时使用。\\ + $\boldsymbol\nu=[\nu_1,\ldots,\nu_K]^\Tran$ & 单阵列 DoA--Doppler 模型中所有信源归一化角 Doppler 的集合。\\ + $\mathbf D_t(\boldsymbol\nu)$ & 单阵列 DoA--Doppler 模型中的 Doppler 相位对角矩阵。\\ + $\tau_\ell$ & 第 $\ell$ 条链路上的离散时延。 若它由参考星时延诱导得到，应在参考星参数化中说明其映射。\\ +\end{longtable} +\endgroup + +物理 Doppler 与归一化角 Doppler 的关系统一写为 +\begin{equation} + \nu = 2\pi\nu^{\mathrm{Hz}} T_{\mathrm{s}}, + \qquad + \dot\nu = 2\pi\dot\nu^{\mathrm{Hz}} T_{\mathrm{s}}^2. +\end{equation} +因此，正文估计参数默认使用 $(\nu,\dot\nu)$，而 $(\nu^{\mathrm{Hz}},\dot\nu^{\mathrm{Hz}})$ 仅用于物理单位解释。 + +\subsection{参考星参数化与参数分组} +\label{subsec:assumptionRefSatParam} + +多星 DoA--Doppler 主线采用参考星链路参数化，而不是星群质心或星群平均速度参数化。 +该约定用于保证单星基线、多星静态模型、多星多帧动态模型以及 CRB 推导均在同一参数域下比较。 +相关符号汇总于表~\ref{tab:refSatParamNotation}。 + +\begingroup +\small +\setlength{\LTpre}{4pt} +\setlength{\LTpost}{4pt} +\renewcommand{\arraystretch}{1.15} +\begin{longtable}{>{\raggedright\arraybackslash}p{0.30\textwidth}>{\raggedright\arraybackslash}p{0.60\textwidth}} + \caption{参考星参数化与参数分组符号约定} + \label{tab:refSatParamNotation}\\ + \hline + \textbf{符号} & \textbf{含义}\\ + \hline + \endfirsthead + \multicolumn{2}{l}{\small\tablename~\thetable{} 参考星参数化与参数分组符号约定（续）}\\ + \hline + \textbf{符号} & \textbf{含义}\\ + \hline + \endhead + \hline + \multicolumn{2}{r}{\small 续下页}\\ + \endfoot + \hline + \endlastfoot + $\ell_{\mathrm{ref}}$ & 参考星索引，除特别说明外默认 $\ell_{\mathrm{ref}}=0$。\\ + $\nu_{\mathrm{ref}}$ & 参考星链路上的归一化角 Doppler。\\ + $\dot\nu_{\mathrm{ref}}$ & 参考星链路上的归一化 Doppler rate。\\ + + $\tau_{\mathrm{ref}}$ & 参考星链路上的离散时延，用于 DoA--Doppler--delay 扩展模型。\\ + $f_{\nu,\ell}(\cdot)$ & 由全局 DoA 与参考星归一化 Doppler 诱导得到第 $\ell$ 条链路归一化 Doppler 的几何映射。\\ + $f_{\dot\nu,\ell}(\cdot)$ & 由全局 DoA 与参考星归一化 Doppler rate 诱导得到第 $\ell$ 条链路归一化 Doppler rate 的几何映射。\\ + $f_{\tau,\ell}(\cdot)$ & 由全局 DoA 与参考星链路时延诱导得到第 $\ell$ 条链路时延的几何映射。 该符号属于参考星参数化，而不是一般信号符号。\\ + $\Delta\nu_\ell(\boldsymbol\xi)$ & 第 $\ell$ 条链路相对于参考星链路的归一化 Doppler 偏差。\\ + $\Delta\dot\nu_\ell(\boldsymbol\xi)$ & 第 $\ell$ 条链路相对于参考星链路的归一化 Doppler-rate 偏差。\\ + $\boldsymbol\eta$ & 将一个模型中的所有待估计主参数堆叠得到的通用参数向量。 本文用 $\boldsymbol\eta$ 表示估计参数集合，避免与一般 DoA 参数 $\bm\theta_k$ 混淆。\\ + $\boldsymbol\eta_{\mathrm{stat}}$ & 静态多星 DoA--Doppler 主参数，统一写为 $\boldsymbol\eta_{\mathrm{stat}}=[\boldsymbol\xi^\Tran,\nu_{\mathrm{ref}}]^\Tran$。\\ + $\boldsymbol\eta_{\tau}$ & DoA--Doppler--delay 扩展模型主参数，可写为 $\boldsymbol\eta_{\tau}=[\boldsymbol\xi^\Tran,\tau_{\mathrm{ref}},\nu_{\mathrm{ref}}]^\Tran$。\\ + $\boldsymbol\eta_{\mathrm{dyn}}$ & 动态多帧模型中作为主估计目标的参数，默认与 $\boldsymbol\eta_{\mathrm{stat}}$ 相同；$\dot\nu_{\mathrm{ref}}$ 放入干扰参数侧。\\ + $\boldsymbol\kappa$ & 完整实参数向量，专用于把主参数 $\boldsymbol\eta$ 与干扰参数 $\boldsymbol\psi$ 一起堆叠的场合，例如 $\boldsymbol\kappa=[\boldsymbol\eta^\Tran,\boldsymbol\psi^\Tran]^\Tran$。全文不再用 $\boldsymbol\eta$ 表示完整参数向量。\\ +\end{longtable} +\endgroup + +对应映射统一写为 +\begin{equation} + \nu_\ell + = + f_{\nu,\ell}(\boldsymbol\xi,\nu_{\mathrm{ref}}), + \qquad + \dot\nu_\ell + = + f_{\dot\nu,\ell}(\boldsymbol\xi,\dot\nu_{\mathrm{ref}}). +\end{equation} +等价地，在仅需强调相对偏差时，可写为 +\begin{equation} + \nu_\ell + = + \nu_{\mathrm{ref}}+ + \Delta\nu_\ell(\boldsymbol\xi), + \qquad + \dot\nu_\ell + = + \dot\nu_{\mathrm{ref}}+ + \Delta\dot\nu_\ell(\boldsymbol\xi). +\end{equation} +动态多帧模型中，$\dot\nu_{\mathrm{ref}}$ 默认作为干扰参数，而不是主估计目标。 + +\subsection{连续相位、独立相位与增益参数} + +多帧动态模型中存在三类容易混淆的相位和增益符号，需按表~\ref{tab:phaseGainNotation} 统一区分。 +由于 $\alpha$ 已用于方位角，链路复系数不使用 $\alpha_\ell$ 表示；同时，$h$ 仅在明确讨论信道响应时使用。未知复标量增益统一使用 $\zeta$。当需要显式刻画幅度与相位结构时，将复增益分解为 $\zeta=\rho e^{j\vartheta}$，其中 $\rho\in\mathbb R_{\ge0}$ 为非负实幅度，$\vartheta$ 为相位。本文不再使用 $b$ 表示复幅度，以避免与时间模板或空时导向列混淆。 + +\begingroup +\small +\setlength{\LTpre}{4pt} +\setlength{\LTpost}{4pt} +\renewcommand{\arraystretch}{1.15} +\begin{longtable}{>{\raggedright\arraybackslash}p{0.30\textwidth}>{\raggedright\arraybackslash}p{0.60\textwidth}} + \caption{连续相位、独立相位与增益符号约定} + \label{tab:phaseGainNotation}\\ + \hline + \textbf{符号} & \textbf{含义}\\ + \hline + \endfirsthead + \multicolumn{2}{l}{\small\tablename~\thetable{} 连续相位、独立相位与增益符号约定（续）}\\ + \hline + \textbf{符号} & \textbf{含义}\\ + \hline + \endhead + \hline + \multicolumn{2}{r}{\small 续下页}\\ + \endfoot + \hline + \endlastfoot + $\phi_{\ell,p}^{\mathrm{cp}}(t)$ & 连续相位模型中第 $\ell$ 颗卫星、第 $p$ 帧、第 $t$ 个采样点的归一化相位模板，采用全局时间索引 $t_{p,t}$。该符号专用于相位模板，避免与干扰参数向量 $\boldsymbol\psi$ 混淆。\\ + $\phi_{\ell,p}^{\mathrm{ip}}(t)$ & 独立相位模型中第 $\ell$ 颗卫星、第 $p$ 帧、第 $t$ 个采样点的帧内局部相位模板，采用帧内索引 $t$。\\ + $\bar\nu_{\ell,p}$ & 独立相位模型中第 $\ell$ 颗卫星第 $p$ 帧的等效局部归一化 Doppler，定义为 $\bar\nu_{\ell,p}\triangleq\nu_\ell+\dot\nu_\ell\tau_p$。 此处保留横线，以强调它不是独立的帧级待估参数。\\ + $\chi_\ell$ & 连续相位结构化模型中第 $\ell$ 颗卫星在参考时刻的公共相位。 该相位在同一颗卫星的不同帧之间共享。\\ + $\vartheta_{\ell,p}$ & 独立相位模型中第 $\ell$ 颗卫星第 $p$ 帧的独立未知相位。 它是确定性未知参数，而不是 Bayesian 随机相位先验。\\ + $\rho_{\ell,p}^{\mathrm{cp}}$ 和 $\rho_{\ell,p}^{\mathrm{ip}}$ & CP 与 IP 模型中第 $(\ell,p)$ 个复增益极坐标分解下的非负实幅度；它们分别属于不同模型，不表示额外的自由复参数。\\ + $\rho$ & 通用极坐标分解下的非负实幅度符号；在具体 CP/IP 模型中分别写为 $\rho_{\ell,p}^{\mathrm{cp}}$ 与 $\rho_{\ell,p}^{\mathrm{ip}}$，不再单独使用无上标的 $\rho_{\ell,p}$ 表示模型参数。\\ + $\zeta_\ell$ & 静态单块模型或 delay 扩展模型中第 $\ell$ 条链路的未知复干扰系数；新写内容不使用 $h_\ell$，以避免与信道响应混淆。\\ + $\zeta_{\ell,p}^{\mathrm{cp}}$ & structured continuous-phase 模型中第 $(\ell,p)$ 个观测块的结构化复增益简写，定义为 $\zeta_{\ell,p}^{\mathrm{cp}}\triangleq \rho_{\ell,p}^{\mathrm{cp}}e^{j\chi_\ell}$；它不作为自由复参数估计，其相位部分在同一颗卫星的不同帧之间共享。\\ + $\zeta_{\ell,p}^{\mathrm{cp,rel}}$ & relaxed continuous-phase 模型中第 $(\ell,p)$ 个观测块的自由复增益。 relaxed 版本应显式保留上标 $\mathrm{cp,rel}$，以区别于结构化 CP 和 IP 模型。\\ + $\zeta_{\ell,p}^{\mathrm{ip}}$ & 独立相位模型中第 $(\ell,p)$ 个观测块的自由复增益。CRB 推导中可写为极坐标分解 $\zeta_{\ell,p}^{\mathrm{ip}}\triangleq \rho_{\ell,p}^{\mathrm{ip}}e^{j\vartheta_{\ell,p}}$。\\ +\end{longtable} +\endgroup + +连续相位与独立相位模板分别统一写为 +\begin{equation} + \phi_{\ell,p}^{\mathrm{cp}}(t) + = + \nu_\ell t_{p,t}+ + \frac{1}{2}\dot\nu_\ell t_{p,t}^2, + \qquad + t_{p,t}=\tau_p+t, +\end{equation} +以及 +\begin{equation} + \phi_{\ell,p}^{\mathrm{ip}}(t) + = + \bar\nu_{\ell,p}t+ + \frac{1}{2}\dot\nu_\ell t^2, + \qquad + \bar\nu_{\ell,p}=\nu_\ell+\dot\nu_\ell\tau_p. +\end{equation} + +\subsection{模板向量、匹配统计量与残差符号} + +后续 MLE、concentrated likelihood 和 CRB 推导中反复使用的模板、统计量和残差符号汇总于表~\ref{tab:templateResidualNotation}。 + +\begingroup +\small +\setlength{\LTpre}{4pt} +\setlength{\LTpost}{4pt} +\renewcommand{\arraystretch}{1.15} +\begin{longtable}{>{\raggedright\arraybackslash}p{0.30\textwidth}>{\raggedright\arraybackslash}p{0.60\textwidth}} + \caption{模板向量、匹配统计量与残差符号约定} + \label{tab:templateResidualNotation}\\ + \hline + \textbf{符号} & \textbf{含义}\\ + \hline + \endfirsthead + \multicolumn{2}{l}{\small\tablename~\thetable{} 模板向量、匹配统计量与残差符号约定（续）}\\ + \hline + \textbf{符号} & \textbf{含义}\\ + \hline + \endhead + \hline + \multicolumn{2}{r}{\small 续下页}\\ + \endfoot + \hline + \endlastfoot + $\mathbf q_\ell$ & 单块静态或 delay 扩展模型中第 $\ell$ 条链路的时间模板向量，维度通常为 $T\times1$。\\ + $\mathbf q_{\ell,p}^{\mathrm{cp}}$ & 连续相位多帧模型中第 $\ell$ 颗卫星第 $p$ 帧的时间模板向量。\\ + $\mathbf q_{\ell,p}^{\mathrm{ip}}$ & 独立相位多帧模型中第 $\ell$ 颗卫星第 $p$ 帧的时间模板向量。\\ + $\mathbf g_\ell$ & 单块模型中由时间模板与阵列导向矢量 Kronecker 组合得到的向量化模板。\\ + $\mathbf g_{\ell,p}^{\mathrm{cp}}$ & 连续相位模型中由 $\mathbf q_{\ell,p}^{\mathrm{cp}}\otimes\mathbf a_\ell$ 构造的向量化模板。\\ + $\mathbf g_{\ell,p}^{\mathrm{ip}}$ & 独立相位模型中由 $\mathbf q_{\ell,p}^{\mathrm{ip}}\otimes\mathbf a_\ell$ 构造的向量化模板。\\ + $z_{\ell,p}^{\mathrm{cp}}$ 和 $z_{\ell,p}^{\mathrm{ip}}$ & 连续相位和独立相位模型中的匹配统计量。\\ + $d_{\ell,p}^{\mathrm{cp}}$ 和 $d_{\ell,p}^{\mathrm{ip}}$ & 连续相位和独立相位模型中由导向矢量范数与时间模板范数相乘得到的归一化因子。\\ + $\mathcal L(\cdot)$ & 负对数似然函数或省略常数后的优化目标；具体参数由所在模型决定。\\ + $\mathcal L_{\mathrm{cp}}$ & 连续相位结构化模型的负对数似然函数。\\ + $\mathcal L_{\mathrm{cp}}^{\mathrm{pro}}$ & 对噪声方差等参数轮廓化后的连续相位负对数似然。\\ + $\mathcal E_\ell$ & 静态单块多星模型中，消去复增益 $\zeta_\ell$ 后的第 $\ell$ 颗卫星残差能量。\\ + $\mathcal E_{\tau,\ell}$ & DoA--Doppler--delay 扩展模型中，消去复增益 $\zeta_\ell$ 后的第 $\ell$ 颗卫星残差能量。\\ + $\mathcal R_\ell$ & 第 $\ell$ 颗卫星的聚合残差。\\ + $\mathcal R_\ell^{\mathrm{cp}}$ & 连续相位模型中第 $\ell$ 颗卫星的聚合残差。\\ + $\mathcal R_\ell^{\mathrm{cp,str}}$ & 连续相位结构化模型在消去幅度后的每星聚合残差。\\ + $\mathcal R_\ell^{\mathrm{cp,rel}}$ & relaxed 连续相位模型的每星聚合残差。\\ + $\mathcal R_\ell^{\mathrm{ip}}$ & 独立相位模型的每星聚合残差。\\ + $\mathcal E_{\ell,p}^{\mathrm{cp,str}}$、$\mathcal E_{\ell,p}^{\mathrm{cp,rel}}$ 和 $\mathcal E_{\ell,p}^{\mathrm{ip}}$ & 第 $(\ell,p)$ 个观测块在相应集中模型下的块残差。\\ +\end{longtable} +\endgroup + +\subsection{CRB、FIM 与有效信息符号} + +附录中的确定信号、随机信号、静态 DoA--Doppler 与动态 DoA--Doppler CRB 推导使用表~\ref{tab:crbFimNotation} 中的统一统计符号。 +本文约定 Fisher 信息矩阵使用 $\mathcal I$，Jacobian 矩阵保留 $\mathbf J$。 + +\begingroup +\small +\setlength{\LTpre}{4pt} +\setlength{\LTpost}{4pt} +\renewcommand{\arraystretch}{1.15} +\begin{longtable}{>{\raggedright\arraybackslash}p{0.30\textwidth}>{\raggedright\arraybackslash}p{0.60\textwidth}} + \caption{CRB、FIM 与有效信息符号约定} + \label{tab:crbFimNotation}\\ + \hline + \textbf{符号} & \textbf{含义}\\ + \hline + \endfirsthead + \multicolumn{2}{l}{\small\tablename~\thetable{} CRB、FIM 与有效信息符号约定（续）}\\ + \hline + \textbf{符号} & \textbf{含义}\\ + \hline + \endhead + \hline + \multicolumn{2}{r}{\small 续下页}\\ + \endfoot + \hline + \endlastfoot + $\mathcal I$ & Fisher 信息矩阵。\\ + $\mathcal I_{\eta\eta}$、$\mathcal I_{\eta\psi}$、$\mathcal I_{\psi\psi}$ & 按主参数与干扰参数分块后的 Fisher 信息矩阵子块。块标签直接作为下标书写。\\ + $\mathcal I_{\mathrm{e}}$ & 消去干扰参数后的有效 Fisher 信息矩阵，即 EFIM。\\ + $\mathbf C$ & CRB 协方差矩阵，通常为 Fisher 信息矩阵或有效 Fisher 信息矩阵的逆。\\ + $\mathbf J$ & Jacobian 矩阵或导数矩阵，不再用于表示 Fisher 信息矩阵。\\ + $\boldsymbol\psi$ & 通用干扰参数向量。 在不同上下文中，它可包含信源复包络、复干扰系数、相位、Doppler rate 或噪声方差。\\ + $\boldsymbol\psi_\mu$ & 只影响均值向量的干扰参数子集。CRB 附录中若噪声方差与均值参数 Fisher 块解耦，则 Schur 补通常只显式保留 $\boldsymbol\psi_\mu$。\\ + $\boldsymbol\psi_{\mathrm{cp}}$ & 连续相位模型的干扰参数集合，通常包含 $\dot\nu_{\mathrm{ref}}$、$\{\chi_\ell\}$、$\{\rho_{\ell,p}^{\mathrm{cp}}\}$ 与 $\{\sigma_{n,\ell}^2\}$。\\ + $\boldsymbol\psi_{\mathrm{ip}}$ & 独立相位模型的干扰参数集合，通常包含 $\dot\nu_{\mathrm{ref}}$、自由复增益 $\{\zeta_{\ell,p}^{\mathrm{ip}}\}$ 或其等价极坐标参数 $\{\vartheta_{\ell,p},\rho_{\ell,p}^{\mathrm{ip}}\}$，以及 $\{\sigma_{n,\ell}^2\}$。\\ + $\mathcal P_{\mathbf B}$ 和 $\mathcal P_{\mathbf B}^{\perp}$ & 到矩阵 $\mathbf B$ 列空间及其正交补的投影算子，定义为 $\mathcal P_{\mathbf B}=\mathbf B(\mathbf B^\Herm\mathbf B)^{-1}\mathbf B^\Herm$ 和 $\mathcal P_{\mathbf B}^{\perp}=\mathbf I-\mathcal P_{\mathbf B}$。当 $\mathbf B=\mathbf A$ 时得到空间导向矩阵投影；当 $\mathbf B=\mathbf B_{\mathrm{st}}$ 时得到空时导向矩阵投影。\\ + $\mathbf R$ 或 $\mathbf R_y$ & 随机信号模型中的观测协方差矩阵。\\ + $\mathbf R_x$ & 随机信号模型中的信源协方差矩阵。\\ +\end{longtable} +\endgroup + +若将完整参数分为主参数 $\boldsymbol\eta$ 与干扰参数 $\boldsymbol\psi$，则有效 Fisher 信息通常可写为 Schur complement 形式 +\begin{equation} + \mathcal I_{\mathrm{e}} + = + \mathcal I_{\eta\eta} + - + \mathcal I_{\eta\psi} + \mathcal I_{\psi\psi}^{-1} + \mathcal I_{\psi\eta}. +\end{equation} +在动态连续相位模型中，若专门分析未知 $\dot\nu_{\mathrm{ref}}$ 带来的主参数信息损失，则 $\dot\nu_{\mathrm{ref}}$ 应放在干扰参数侧，而不是重新并入主参数向量。 + +\subsection{上下标、模型标记与阅读规则} + +为避免同一主符号因上下标不同而产生误读，全文采用表~\ref{tab:modelMarkerNotation} 中的阅读规则。 + +\begingroup +\small +\setlength{\LTpre}{4pt} +\setlength{\LTpost}{4pt} +\renewcommand{\arraystretch}{1.15} +\begin{longtable}{>{\raggedright\arraybackslash}p{0.30\textwidth}>{\raggedright\arraybackslash}p{0.60\textwidth}} + \caption{上下标与模型标记约定} + \label{tab:modelMarkerNotation}\\ + \hline + \textbf{标记} & \textbf{含义}\\ + \hline + \endfirsthead + \multicolumn{2}{l}{\small\tablename~\thetable{} 上下标与模型标记约定（续）}\\ + \hline + \textbf{标记} & \textbf{含义}\\ + \hline + \endhead + \hline + \multicolumn{2}{r}{\small 续下页}\\ + \endfoot + \hline + \endlastfoot + 下标 $k$ & 信号源或用户索引。\\ + 下标 $\ell$ & 阵列、卫星或链路索引。\\ + 下标 $p$ & 帧索引。\\ + 下标 $t$ & 快拍或采样索引；在 $t_{p,t}$ 中第二个 $t$ 表示帧内采样索引。\\ + 下标 $\mathrm{ref}$ & 参考星链路相关参数，而不是参考时刻。\\ + 上标 $\mathrm{loc}$ & 卫星本地坐标系中的量。\\ + 上标 $\mathrm{cp}$ & continuous-phase，即连续相位模型。\\ + 上标 $\mathrm{ip}$ & independent-phase，即独立相位基线模型。\\ + 上标 $\mathrm{str}$ & structured，即保留结构化公共相位或幅度约束后的模型或残差。\\ + 上标 $\mathrm{rel}$ & relaxed，即放松块复干扰系数 结构后的模型或残差。\\ + 上标 $\mathrm{pro}$ & profiled，即已经对部分干扰参数作轮廓化处理的似然或目标函数。\\ + 帽号 $\hat{\cdot}$ & 估计量。\\ + 横线 $\bar{\cdot}$ & 局部化或等效化后的量，例如 $\bar\nu_{\ell,p}$ 表示第 $p$ 帧内的局部归一化 Doppler。\\ + 波浪号 $\tilde{\cdot}$ & 向量化或变换后的量，例如 $\tilde{\mathbf y}_{\ell,p}=\vect(\mathbf Y_{\ell,p})$。\\ +\end{longtable} +\endgroup + +全文写作遵守表~\ref{tab:notationWritingRules} 中的符号使用规则。 + +\begingroup +\small +\setlength{\LTpre}{4pt} +\setlength{\LTpost}{4pt} +\renewcommand{\arraystretch}{1.15} +\begin{longtable}{>{\raggedright\arraybackslash}p{0.10\textwidth}>{\raggedright\arraybackslash}p{0.80\textwidth}} + \caption{符号使用规则} + \label{tab:notationWritingRules}\\ + \hline + \textbf{序号} & \textbf{规则}\\ + \hline + \endfirsthead + \multicolumn{2}{l}{\small\tablename~\thetable{} 符号使用规则（续）}\\ + \hline + \textbf{序号} & \textbf{规则}\\ + \hline + \endhead + \hline + \multicolumn{2}{r}{\small 续下页}\\ + \endfoot + \hline + \endlastfoot + 1 & 带有文字含义的上下标使用 $\mathrm{\cdot}$，例如 $T_{\mathrm{s}}$、$\lambda_{\mathrm{c}}$、$\ell_{\mathrm{ref}}$、$\mathcal I_{\mathrm{e}}$、$\|\cdot\|_{\mathrm{F}}$。 普通索引如 $k,\ell,p,t$ 保持斜体。\\ + 2 & 普通标量变量使用斜体；向量使用小写粗体字母或粗体希腊字母；矩阵使用大写粗体字母。\\ + 3 & 不用 $\lambda$ 表示载波波长；载波波长一律写为 $\lambda_{\mathrm{c}}$。 全局方位角使用 $\alpha$。\\ + 4 & 不用 $\mathbf a_\ell$ 表示卫星动力学加速度；动力学加速度若需要出现，应写为 $\dot{\mathbf v}_\ell$ 或其他明确符号。\\ + 5 & Doppler 与 Doppler rate 统一使用 $\nu$ 与 $\dot\nu$；正文估计参数默认是归一化角频率量。若必须写物理 Hz 单位，使用 $\nu^{\mathrm{Hz}}$ 与 $\dot\nu^{\mathrm{Hz}}$。全文使用 $\nu$ 体系。\\ + 6 & 参考星与参考时刻是两个不同概念；$\mathrm{ref}$ 下标只表示参考星链路，参考时刻应通过 $t=0$、$\tau_0=0$ 或文字说明表达。\\ + 7 & 连续相位模型中的公共相位 $\chi_\ell$ 与独立相位模型中的块相位 $\vartheta_{\ell,p}$ 不能混用。\\ + 8 & Fisher 信息矩阵使用 $\mathcal I$，其分块下标直接书写，例如 $\mathcal I_{\eta\psi}$；$\mathbf J$ 保留给 Jacobian 或导数矩阵。\\ + 9 & 投影算子使用 $\mathcal P$，例如 $\mathcal P_{\mathbf A}^{\perp}$。\\ + 10 & relaxed continuous-phase 相关目标、残差和复增益应保留上标 $\mathrm{cp,rel}$，避免与 structured CP 或 IP 混淆。\\ + 11 & CRB 协方差矩阵保留 $\mathbf C$；空时导向矩阵不再使用 $\mathbf C$，统一写为 $\mathbf B_{\mathrm{st}}$。\\ + 12 & 未知复标量增益统一使用 $\zeta$；$\rho$ 只表示极坐标分解下的非负实幅度，不表示额外的自由复幅度参数。CP 与 IP 的幅度分别写为 $\rho_{\ell,p}^{\mathrm{cp}}$ 和 $\rho_{\ell,p}^{\mathrm{ip}}$。\\ + 13 & 旋转矩阵统一写为 $\boldsymbol{\Omega}_\ell$，不再使用 $\mathbf{Q}_\ell$；$\mathbf{Q}$ 保留给波束成形章节中的人工噪声协方差矩阵，避免符号冲突。\\ +\end{longtable} +\endgroup diff --git a/parts/02-modeling.tex b/parts/02-modeling.tex new file mode 100644 index 0000000..a3dde4f --- /dev/null +++ b/parts/02-modeling.tex @@ -0,0 +1,472 @@ +\section{问题建模} + +本节作为全文模型层的统一入口。除单阵列基础模型沿用一般方向参数 $\bm\theta_k$ 外，后续单用户多星主线均采用第~\ref{sec:assumptionsAndNotation} 节中的新符号体系：全局 DoA 写为 $\boldsymbol\xi=[\alpha,\beta]^\Tran$，参考星链路归一化 Doppler 写为 $\nu_{\mathrm{ref}}$，Doppler rate 写为 $\dot\nu_{\mathrm{ref}}$。 + +\subsection{单阵列 DoA 估计问题} +\label{subsec:single_arr_doa_est} + +假设有 $K$ 个信源，来向参数记为 +\begin{equation} + \bm{\Theta} + \triangleq + \left[\bm{\theta}_1,\ldots,\bm{\theta}_K\right], +\end{equation} +快拍数（或采样点数）为 $T$，采样间隔为 $T_{\mathrm{s}}$。则 $M$ 元阵列在第 $t$ 个快拍的接收向量可表示为 +\begin{align} + \label{eq:single_arr_doa_model} + \mathbf y_t + = + \mathbf A(\bm{\Theta})\mathbf x_t + \mathbf n_t, + \qquad t=0,1,\ldots,T-1, +\end{align} +其中： +\begin{itemize} + \item 观测向量：$\mathbf y_t \in \mathbb C^{M\times 1}$； + \item 导向矩阵： + \begin{equation*} + \mathbf A(\bm{\Theta}) + \triangleq + \left[\mathbf a(\bm{\theta}_1),\ldots,\mathbf a(\bm{\theta}_K)\right] + \in \mathbb C^{M\times K}, + \end{equation*} + 其中 $\mathbf a(\bm{\theta}_k)$ 为第 $k$ 个信源对应的导向矢量； + \item 信源向量：$\mathbf x_t=[x_1(t),\ldots,x_K(t)]^\Tran\in\mathbb C^{K\times 1}$； + \item 噪声向量：$\mathbf n_t\sim\mathcal{CN}(\mathbf 0,\sigma_n^2\mathbf I_M)$。 +\end{itemize} + +将所有快拍按列堆叠，可得矩阵形式 +\begin{equation} + \mathbf Y = \mathbf A(\bm{\Theta})\mathbf X + \mathbf N, +\end{equation} +其中 $\mathbf Y=[\mathbf y_0,\ldots,\mathbf y_{T-1}]\in\mathbb C^{M\times T}$， +$\mathbf X=[\mathbf x_0,\ldots,\mathbf x_{T-1}]\in\mathbb C^{K\times T}$， +$\mathbf N=[\mathbf n_0,\ldots,\mathbf n_{T-1}]\in\mathbb C^{M\times T}$。 + +为统一表示，将所有未知方向参数收集为 +\begin{equation*} + \boldsymbol\eta + \triangleq + \vect(\bm\Theta) + = + \left[\bm\theta_1^\Tran,\ldots,\bm\theta_K^\Tran\right]^\Tran. +\end{equation*} +于是单阵列 \gls{doa} 估计问题可表述为 +\begin{equation} + \widehat{\boldsymbol\eta} + = + \argmin_{\boldsymbol\eta}\;\mathcal J(\boldsymbol\eta), +\end{equation} +其中 $\mathcal J(\cdot)$ 为依赖于具体方法的代价函数，例如 \gls{mle}、\gls{music} 或 \gls{esprit} 对应的准则函数。 + + +\subsection{多阵列（多卫星） DoA 估计问题} +\label{subsec:multi_arr_doa_est} + +假设存在 $L$ 个相互独立的接收阵列，第 $\ell$ 个阵列具有 $M_\ell$ 个阵元。所有信源的真实来向仍由全局方向参数 +\begin{equation} + \bm\Theta + = + \left[\bm\theta_1,\ldots,\bm\theta_K\right] +\end{equation} +描述，但由于各阵列所在位置和姿态不同，同一全局方向在各阵列上的局部来向一般不同。记第 $\ell$ 个阵列的局部来向映射为 +\begin{equation*} + \bm\theta_{\ell,k}=f_\ell(\bm\theta_k), +\end{equation*} +其中 $f_\ell(\cdot)$ 由几何关系唯一确定。则第 $\ell$ 个阵列在第 $t$ 个快拍的接收向量可写为 +\begin{align} + \label{eq:multi_arr_doa_model} + \mathbf y_{\ell,t} + = + \mathbf A_\ell(\bm\Theta)\mathbf x_t + \mathbf n_{\ell,t}, + \qquad + \ell=0,1,\ldots,L-1,\; + t=0,1,\ldots,T-1, +\end{align} +其中 +\begin{itemize} + \item $\mathbf y_{\ell,t}\in\mathbb C^{M_\ell\times 1}$； + \item + $ + \mathbf A_\ell(\bm\Theta) + \triangleq + \left[ + \mathbf a_\ell(\bm\theta_{\ell,1}),\ldots,\mathbf a_\ell(\bm\theta_{\ell,K}) + \right] + \in \mathbb C^{M_\ell\times K}; + $ + \item $\mathbf x_t\in\mathbb C^{K\times 1}$ 为所有信源在第 $t$ 个快拍的复包络； + \item $\mathbf n_{\ell,t}\sim\mathcal{CN}(\mathbf 0,\sigma_{n,\ell}^2\mathbf I_{M_\ell})$，不同阵列之间相互独立。 +\end{itemize} + +将每个阵列的观测堆叠，可得 +\begin{equation} + \mathbf Y_\ell + = + \mathbf A_\ell(\bm\Theta)\mathbf X + \mathbf N_\ell, + \qquad \ell=0,1,\ldots,L-1, +\end{equation} +其中 $\mathbf Y_\ell\in\mathbb C^{M_\ell\times T}$，$\mathbf X\in\mathbb C^{K\times T}$，$\mathbf N_\ell\in\mathbb C^{M_\ell\times T}$。 + +与单阵列情形相同，我们关心的未知参数仍为全局方向 +\begin{equation*} + \boldsymbol\eta + \triangleq + \vect(\bm\Theta) + = + \left[\bm\theta_1^\Tran,\ldots,\bm\theta_K^\Tran\right]^\Tran. +\end{equation*} +在噪声独立、各阵列观测条件独立的假设下，多阵列 \gls{doa} 估计问题可写为 +\begin{equation} + \widehat{\boldsymbol\eta} + = + \argmin_{\boldsymbol\eta}\;\mathcal J_{\mathrm{joint}}(\boldsymbol\eta) + = + \argmin_{\boldsymbol\eta}\;\sum_{\ell=0}^{L-1}\mathcal J_\ell(\boldsymbol\eta), +\end{equation} +其中 $\mathcal J_\ell(\cdot)$ 为第 $\ell$ 个阵列对应的代价函数。换言之，多阵列 \gls{doa} 估计仍以同一组全局方向参数为中心，只是通过局部几何映射把该组参数一致地约束到所有阵列的局部坐标系中。 + + +\subsection{单阵列 DoA--Doppler 估计问题} +\label{subsec:single_arr_doa_doppler_est} + +在 \ref{subsec:single_arr_doa_est} 的基础上进一步引入 Doppler。本文默认 $\nu_k$ 表示第 $k$ 个信源的\emph{归一化角 Doppler}，单位为 rad/sample；若需要物理 Hz 单位，应显式写作 $\nu_k^{\mathrm{Hz}}$。若在观测时间窗口内第 $k$ 个信源的归一化 Doppler 可近似看作常数，记 +\begin{equation} + \boldsymbol\nu + \triangleq + [\nu_1,\ldots,\nu_K]^\Tran, +\end{equation} +则单阵列接收模型可写为 +\begin{align} + \label{eq:single_arr_doa_doppler_model} + \mathbf y_t + = + \mathbf A(\bm\Theta)\mathbf D_t(\boldsymbol\nu)\mathbf x_t + \mathbf n_t, + \qquad t=0,1,\ldots,T-1, +\end{align} +其中 +\begin{equation} + \mathbf D_t(\boldsymbol\nu) + \triangleq + \diag\!\big( + e^{j\nu_1 t}, + \ldots, + e^{j\nu_K t} + \big) + \in\mathbb C^{K\times K}. +\end{equation} + +将所有快拍堆叠，可得 +\begin{equation} + \mathbf Y + = + \mathbf A(\bm\Theta) + \left[ + \mathbf D_0\mathbf x_0,\ldots,\mathbf D_{T-1}\mathbf x_{T-1} + \right] + + \mathbf N. +\end{equation} + +此时待估参数除了方向外，还包括归一化 Doppler。将所有未知量收集为扩展参数向量 +\begin{equation*} + \boldsymbol\eta + \triangleq + \begin{bmatrix} + \vect(\bm\Theta)\\ + \boldsymbol\nu + \end{bmatrix} + = + \left[ + \bm\theta_1^\Tran,\ldots,\bm\theta_K^\Tran, + \nu_1,\ldots,\nu_K + \right]^\Tran. +\end{equation*} +于是单阵列 \gls{doa}--Doppler 联合估计问题可表述为 +\begin{equation} + \widehat{\boldsymbol\eta} + = + \argmin_{\boldsymbol\eta}\;\mathcal J(\boldsymbol\eta), +\end{equation} +其中 $\mathcal J(\cdot)$ 的具体形式取决于所采用的估计算法，例如 \gls{mle}、二维 \gls{music} 或空间--时间 \gls{esprit} 等。 + + +\subsection{协作卫星 DoA--Doppler 统一符号约定} +\label{subsec:multiArrayCommonNotation} + +从 \S\ref{sec:multiArrayStatDetDoaDopplerEst} 开始，本文聚焦于\emph{单用户、多卫星}的协作 DoA--Doppler 估计问题。为避免静态模型、动态模型以及各附录中重复定义并引入符号漂移，本节统一后续各节共用的几何、时间与参数记号。除特别说明外，后文均沿用本节定义。 + +\subsubsection{全局/局部 DoA 映射} +\label{subsec:globalLocalDoaMap} + +后文统一采用全局方向参数 +\begin{equation} + \boldsymbol\xi + \triangleq + \begin{bmatrix} + \alpha\\ + \beta + \end{bmatrix} +\end{equation} +描述参考时刻的用户方向，并定义对应的单位方向向量为 +\begin{equation} + \mathbf u(\boldsymbol\xi) + = + \begin{bmatrix} + \cos\beta\cos\alpha\\ + \cos\beta\sin\alpha\\ + \sin\beta + \end{bmatrix}, + \qquad + \|\mathbf u(\boldsymbol\xi)\|_2=1. +\end{equation} +其中 $\alpha$ 表示全局方位角，$\beta$ 表示全局俯仰角。若需要回到多用户背景，则可写为 $\boldsymbol\xi_k=[\alpha_k,\beta_k]^\Tran$；后续单用户主线中不再显式保留该下标。 + +设第 $\ell$ 颗卫星在参考时刻的位置与速度分别为 $\mathbf s_\ell$ 和 $\mathbf v_\ell$。定义其本地阵列坐标基为 +\begin{equation} + \mathbf e_{3,\ell}=-\frac{\mathbf s_\ell}{\|\mathbf s_\ell\|_2}, + \qquad + \mathbf e_{2,\ell}=\frac{\mathbf s_\ell\times\mathbf v_\ell}{\|\mathbf s_\ell\times\mathbf v_\ell\|_2}, + \qquad + \mathbf e_{1,\ell}=\mathbf e_{2,\ell}\times\mathbf e_{3,\ell}, +\end{equation} +并记从本地坐标系到全局坐标系的旋转矩阵为 +\begin{equation} + \boldsymbol{\Omega}_\ell + \triangleq + \begin{bmatrix} + \mathbf e_{1,\ell},\mathbf e_{2,\ell},\mathbf e_{3,\ell} + \end{bmatrix} + \in\mathbb R^{3\times 3}. +\end{equation} +于是，第 $\ell$ 颗卫星本地坐标系下的来向向量为 +\begin{equation} + \mathbf u_\ell^{\mathrm{loc}} + = + \boldsymbol{\Omega}_\ell^\Tran\mathbf u(\boldsymbol\xi) + = + \begin{bmatrix} + u^{\mathrm{loc}}_{\ell,1},u^{\mathrm{loc}}_{\ell,2},u^{\mathrm{loc}}_{\ell,3} + \end{bmatrix}^\Tran. +\end{equation} +对应的局部方位角与俯仰角统一记为 +\begin{equation} + \alpha_\ell^{\mathrm{loc}} + = + \operatorname{atan2}\!\big(u^{\mathrm{loc}}_{\ell,2},u^{\mathrm{loc}}_{\ell,1}\big), + \qquad + \beta_\ell^{\mathrm{loc}} + = + \arcsin\!\big(u^{\mathrm{loc}}_{\ell,3}\big). +\end{equation} +因此，第 $\ell$ 颗卫星的阵列导向矢量统一定义为 +\begin{equation} + \mathbf a_\ell(\boldsymbol\xi) + \triangleq + \mathbf a_\ell\!\big(\alpha_\ell^{\mathrm{loc}},\beta_\ell^{\mathrm{loc}}\big) + \in\mathbb C^{M_\ell\times 1}. +\end{equation} +后续静态、多帧连续相位以及独立相位模型都统一使用该记号。特别地，$\mathbf a_\ell(\boldsymbol\xi)$ 始终只表示\emph{阵列导向矢量}，不再用于表示轨道动力学中的加速度量。 + +\subsubsection{归一化 Doppler、Doppler rate 与时间索引} +\label{subsec:dopplerNormalization} + +后文在物理解释层使用实际 Doppler 与实际 Doppler rate，分别记为 +\begin{equation} + \nu^{\mathrm{Hz}}, + \qquad + \dot\nu^{\mathrm{Hz}}, +\end{equation} +单位分别为 Hz 与 Hz/s；在离散信号模型、似然函数和 \gls{crb} 推导中，统一使用归一化角 Doppler 与归一化 Doppler rate，分别记为 +\begin{equation} + \nu, + \qquad + \dot\nu, +\end{equation} +单位分别为 rad/sample 与 rad/sample$^2$。它们满足关系 +\begin{equation} + \nu = 2\pi\nu^{\mathrm{Hz}} T_{\mathrm{s}}, + \qquad + \dot\nu = 2\pi\dot\nu^{\mathrm{Hz}} T_{\mathrm{s}}^2. +\end{equation} +因此，$(\nu^{\mathrm{Hz}},\dot\nu^{\mathrm{Hz}})$ 与 $(\nu,\dot\nu)$ 只是单位不同，而不是两组相互独立的参数。后续模型若未显式带有上标 $\mathrm{Hz}$，均默认使用归一化量。 + +在单观测块静态模型中，块内离散采样索引统一记为 +\begin{equation} + t=0,1,\ldots,T-1. +\end{equation} +在多帧动态模型中，帧索引记为 +\begin{equation} + p=0,1,\ldots,P-1, +\end{equation} +第 $p$ 帧起始点相对于参考时刻的离散采样偏移记为 $\tau_p$，并约定 $\tau_0=0$。相应地，第 $p$ 帧第 $t$ 个采样对应的全局离散时间索引统一写为 +\begin{equation} + t_{p,t} + \triangleq + \tau_p+t, + \qquad + p=0,\ldots,P-1,\;t=0,\ldots,T-1. +\end{equation} +后文中，括号内的 $t$ 始终表示\emph{帧内局部采样索引}，而 $t_{p,t}$ 表示其对应的\emph{全局离散时间位置}。 + +\subsubsection{参考卫星参数化} +\label{subsec:refSatParameterization} + +为统一多星协作模型与单星基线模型，后续各节固定选择一颗卫星作为\emph{参考卫星}，其索引记为 +\begin{equation} + \ell_{\mathrm{ref}}, +\end{equation} +除特别说明外，默认取 $\ell_{\mathrm{ref}}=0$。后文中涉及 ``ref'' 的 Doppler 参数，均指\emph{参考卫星链路}上的参数，而不再使用星群质心或星群平均速度定义参考 Doppler。 + +据此，后续静态 DoA--Doppler 主参数统一写为 +\begin{equation} + \boldsymbol\eta_{\mathrm{stat}} + = + \begin{bmatrix} + \boldsymbol\xi\\ + \nu_{\mathrm{ref}} + \end{bmatrix}, +\end{equation} +其中 $\nu_{\mathrm{ref}}$ 为参考时刻、参考卫星链路上的归一化 Doppler。在动态多帧模型中，主估计目标默认仍写为 +\begin{equation} + \boldsymbol\eta_{\mathrm{dyn}} + = + \boldsymbol\eta_{\mathrm{stat}} + = + \begin{bmatrix} + \boldsymbol\xi\\ + \nu_{\mathrm{ref}} + \end{bmatrix}, +\end{equation} +而参考卫星链路上的归一化 Doppler rate $\dot\nu_{\mathrm{ref}}$ 默认放入干扰参数侧。若进一步加入参考星链路时延，则 DoA--Doppler--delay 扩展模型的主参数可写为 +\begin{equation} + \boldsymbol\eta_{\tau} + = + \begin{bmatrix} + \boldsymbol\xi\\ + \tau_{\mathrm{ref}}\\ + \nu_{\mathrm{ref}} + \end{bmatrix}. +\end{equation} + +给定全局方向 $\boldsymbol\xi$ 以及参考卫星链路参数 $(\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})$ 后，其余卫星的局部链路参数统一通过几何映射表示为 +\begin{equation} + \nu_\ell + = + f_{\nu,\ell}(\boldsymbol\xi,\nu_{\mathrm{ref}}), + \qquad + \dot\nu_\ell + = + f_{\dot\nu,\ell}(\boldsymbol\xi,\dot\nu_{\mathrm{ref}}), + \qquad \ell=0,1,\ldots,L-1. +\end{equation} +等价地，也可写成 +\begin{equation} + \nu_\ell + = + \nu_{\mathrm{ref}}+\Delta\nu_\ell(\boldsymbol\xi), + \qquad + \dot\nu_\ell + = + \dot\nu_{\mathrm{ref}}+\Delta\dot\nu_\ell(\boldsymbol\xi), +\end{equation} +其中 $\Delta\nu_\ell(\boldsymbol\xi)$ 与 $\Delta\dot\nu_\ell(\boldsymbol\xi)$ 均由已知几何、速度与姿态关系确定。 + +因此，从 \S\ref{sec:multiArrayStatDetDoaDopplerEst} 到各附录，后文统一遵循以下原则： +\begin{enumerate} + \item 全局方向统一用 $\boldsymbol\xi=[\alpha,\beta]^\Tran$； + \item 局部来向角统一写成 $\alpha_\ell^{\mathrm{loc}}$ 与 $\beta_\ell^{\mathrm{loc}}$； + \item 局部导向矢量统一写成 $\mathbf a_\ell(\boldsymbol\xi)$； + \item 物理量层用 $(\nu^{\mathrm{Hz}},\dot\nu^{\mathrm{Hz}})$，离散模型层用 $(\nu,\dot\nu)$； + \item Doppler 参考量统一定义在参考卫星链路上，而不是星群平均参考点上； + \item 动态模型中 $\dot\nu_{\mathrm{ref}}$ 默认是干扰参数，主估计目标仍为 $\boldsymbol\eta_{\mathrm{dyn}}=[\boldsymbol\xi^\Tran,\nu_{\mathrm{ref}}]^\Tran$； + \item 多帧模型的时间索引统一用 $(p,t,\tau_p,t_{p,t})$ 表示。 +\end{enumerate} + + +\subsection{阵列模型} + +\subsubsection{ULA} +\label{subsubsec:single_arr_ula_model} + +对于 \gls{ula}，假设阵元间距为 $d$，工作波长为 $\lambda_{\mathrm{c}}$，第 $k$ 个信源的到达角为标量 $\theta_k$。由于 \gls{ula} 只有一个角度维度的分辨能力，这里采用一维角度参数化，其导向矢量可写为 +\begin{equation} + \mathbf a_{\mathrm{ULA}}(\theta_k) + = + \begin{bmatrix} + 1, + e^{-j\frac{2\pi}{\lambda_{\mathrm{c}}}d\sin\theta_k}, + \ldots, + e^{-j\frac{2\pi}{\lambda_{\mathrm{c}}}(M-1)d\sin\theta_k} + \end{bmatrix}^\Tran + \in\mathbb C^{M\times 1}. +\end{equation} +对应的导向矩阵为 +\begin{equation} + \mathbf A_{\mathrm{ULA}}(\bm\Theta) + \triangleq + \left[ + \mathbf a_{\mathrm{ULA}}(\theta_1),\ldots,\mathbf a_{\mathrm{ULA}}(\theta_K) + \right] + \in\mathbb C^{M\times K}. +\end{equation} + +\subsubsection{UPA} +\label{subsubsec:single_arr_upa_model} + +对于 \gls{upa}，假设阵列位于 $xOy$ 平面内，沿 $x$ 轴和 $y$ 轴分别等间距排布，水平方向阵元数为 $M_x$，垂直方向阵元数为 $M_y$，阵元间距分别为 $d_x$ 与 $d_y$，总阵元数为 $M=M_xM_y$。后文统一采用与全局方向一致的角度记号 +\begin{equation} + \bm\theta_k + \triangleq + \begin{bmatrix} + \alpha_k\\ + \beta_k + \end{bmatrix}, +\end{equation} +其中 $\alpha_k$ 表示方位角，$\beta_k$ 表示俯仰角。定义平面内两个方向余弦为 +\begin{equation} + u_{x,k}\triangleq\cos\beta_k\cos\alpha_k, + \qquad + u_{y,k}\triangleq\cos\beta_k\sin\alpha_k. +\end{equation} +则在 $x$ 和 $y$ 方向上的一维导向矢量分别为 +\begin{subequations} + \begin{align} + \mathbf a_x(\bm\theta_k) + &= + \begin{bmatrix} + 1, + e^{-j\frac{2\pi}{\lambda_{\mathrm{c}}}d_xu_{x,k}}, + \ldots, + e^{-j\frac{2\pi}{\lambda_{\mathrm{c}}}(M_x-1)d_xu_{x,k}} + \end{bmatrix}^\Tran + \in\mathbb C^{M_x\times 1},\\ + \mathbf a_y(\bm\theta_k) + &= + \begin{bmatrix} + 1, + e^{-j\frac{2\pi}{\lambda_{\mathrm{c}}}d_yu_{y,k}}, + \ldots, + e^{-j\frac{2\pi}{\lambda_{\mathrm{c}}}(M_y-1)d_yu_{y,k}} + \end{bmatrix}^\Tran + \in\mathbb C^{M_y\times 1}. + \end{align} +\end{subequations} +因此，\gls{upa} 的导向矢量可写为 +\begin{equation} + \mathbf a_{\mathrm{UPA}}(\bm\theta_k) + = + \mathbf a_y(\bm\theta_k)\otimes\mathbf a_x(\bm\theta_k) + \in\mathbb C^{M\times 1}, +\end{equation} +对应的导向矩阵为 +\begin{equation} + \mathbf A_{\mathrm{UPA}}(\bm\Theta) + \triangleq + \left[ + \mathbf a_{\mathrm{UPA}}(\bm\theta_1),\ldots,\mathbf a_{\mathrm{UPA}}(\bm\theta_K) + \right] + \in\mathbb C^{M\times K}. +\end{equation} + +上述 ULA/UPA 表达只给出阵列导向矢量的标准形式；在多卫星协作场景下，后续各节统一先通过 \ref{subsec:globalLocalDoaMap} 中的全局--局部坐标映射得到 $(\alpha_\ell^{\mathrm{loc}},\beta_\ell^{\mathrm{loc}})$，再代入相应阵列类型的导向矢量表达式。 + +至此，全文模型所需的单阵列基线、多阵列几何映射和多星参考星参数化已经集中定义完毕。后续单阵列章节只使用本节前半部分的 $\bm\Theta$ 与 $\boldsymbol\nu$ 记号；多星 DoA--Doppler 主线则默认沿用 $\boldsymbol\xi$、$\nu_{\mathrm{ref}}$、$\dot\nu_{\mathrm{ref}}$ 以及 $\mathbf a_\ell(\boldsymbol\xi)$，不再在每个正文或附录中重新定义几何模型。 diff --git a/parts/98-future work.tex b/parts/98-future work.tex new file mode 100644 index 0000000..d35f602 --- /dev/null +++ b/parts/98-future work.tex @@ -0,0 +1,82 @@ +\section{后续研究方向} + +在当前多星协作 DoA-Doppler 联合估计工作的基础上，后续可以从以下几个方向进一步扩展。 + +\subsection{扩展到定位场景：面向 DoA-Doppler-时延联合估计} + +在当前多星协作 DoA-Doppler 联合估计工作的基础上，后续可进一步引入传播时延参数，研究多星场景下的 DoA-Doppler-时延联合估计问题，建立多星联合观测模型，对三类参数进行统一估计，并利用它们与用户位置、速度之间的几何关系，进一步实现定位或测速。 + +主要目标包括： +\begin{itemize} + \item 基于 MLE 的高精度联合估计； + \item 基于 空时二维/三维谱搜索的子空间方法（如空时 MUSIC）； + \item 面向高维参数空间的低复杂度迭代算法或交替优化算法； + \item 相应的 CRB 推导，用于刻画 DoA-Doppler-时延联合估计及定位的性能下界。 +\end{itemize} + +但是引入时延后，待估参数维度明显上升，计算复杂度会显著增加，会给算法设计带来更大挑战。 + +\subsection{考虑星间传输受限：分布式处理与低比特量化} + +当前模型通常默认多星接收数据能够无损、实时地汇聚到中心节点进行全相干处理，这一假设在实际卫星系统中偏理想化。由于星间链路带宽、时延和计算资源受限，后续可研究在受限传输条件下的联合估计问题。 + +可以考虑分布式处理，也即各卫星本地完成部分信号处理，仅上传局部统计量、局部似然信息或低维特征，在中心节点进行融合；或者低比特量化，也即引入 1-bit ADC、混合精度 ADC 或压缩采样机制，以减少需要传输的原始观测数据量。 + +主要目标包括： +\begin{itemize} + \item 传输开销与估计性能之间的权衡关系； + \item 分布式/量化条件下 MLE、MUSIC 或迭代融合算法的设计； + \item 相应的量化 CRB、分布式 CRB 或性能损失分析。 +\end{itemize} + +\subsection{考虑模型失配与鲁棒联合估计} + +当前分析通常建立在理想信号模型之上，但实际系统中往往存在模型失配。后续可以研究在模型不准确条件下的鲁棒 DoA-Doppler 联合估计。 + +可考虑的失配因素包括： +\begin{itemize} + \item 阵列流形失配，如阵元位置偏差、幅相误差、互耦等； + \item 卫星平台相关误差，如姿态误差、轨道参数误差、基线误差； + \item 时频同步误差，如星间时钟偏差、频偏； + \item 信道扰动，如相位抖动或短时不稳定效应。 +\end{itemize} + +这一方向通常可表述为带不确定集约束的优化问题，往往是非凸的，因此需要设计近似求解或迭代优化算法，并通过仿真验证鲁棒性提升。鲁棒算法适合形成篇幅较短但问题明确的论文，适合发 letter。 + +\subsection{面向超大规模阵列的随机矩阵理论分析} + +当卫星端阵列规模、观测快拍数或协作卫星数进一步增大时，传统有限维分析可能难以准确刻画系统行为。此时可以引入随机矩阵理论，研究大系统条件下的渐近性能。 + +该方向可以分析的性能主要包括： + +\begin{itemize} + \item \textbf{参数估计性能界}：例如大规模阵列下 DoA-Doppler 联合估计的渐近 CRB 或 Fisher 信息矩阵的确定性等价； + \item \textbf{子空间方法性能}：例如 MUSIC 谱峰分辨能力、噪声子空间泄漏、信号子空间与噪声子空间分离条件； + \item \textbf{协方差估计误差影响}：研究有限快拍条件下样本协方差偏差对参数估计的影响； + \item \textbf{可分辨性与相变现象}：分析两个相近信号在大系统条件下何时可分辨、何时出现分辨失效； + \item \textbf{量化或压缩条件下的性能极限}：例如 1-bit 量化、大规模阵列与有限快拍共同作用下的性能退化规律。 +\end{itemize} + +随机矩阵理论的优点是不仅可以解释仿真现象，还可以给出简洁的渐近表达式，用于揭示阵列规模、快拍数、信噪比和协作卫星数之间的基本规律；但对问题建模要求较高，需要首先明确所考虑的大系统极限形式。 + +\subsection{其他可进一步考虑的扩展方向} + +除上述方向外，还可考虑以下扩展： + +\begin{itemize} + \item \textbf{多用户场景}：由单用户扩展到多用户 DoA-Doppler 联合估计，并研究参数配对与用户分离问题； + \item \textbf{动态跟踪场景}：由静态估计扩展到时序跟踪，结合状态空间模型研究 DoA-Doppler 连续跟踪； + \item \textbf{宽带或近场模型}：针对超大孔径阵列或宽带信号，研究窄带远场假设失效时的联合估计问题； + \item \textbf{异步协作场景}：考虑多星间并非严格相干同步，而存在未知时延、相位差和频率偏差时的联合估计。 +\end{itemize} + +\subsection{与物理层安全的结合} + +在上述方向基础上，需要进一步引入物理层安全（Physical Layer Security, PLS）的概念，使参数估计与安全传输形成结合。可能的切入方式包括： + +\begin{itemize} + \item \textbf{基于参数估计的安全波束赋形}：利用多星估计得到的用户方向或位置参数，设计面向合法用户的协作波束，并抑制潜在窃听方向； + \item \textbf{鲁棒安全设计}：在 DoA、Doppler 或位置存在估计误差时，联合优化波束赋形与人工噪声，提高系统的保密性能； + \item \textbf{基于物理特征的认证}：利用多星观测到的 DoA-Doppler-Delay 组合作为物理层特征，实现合法用户识别与伪装检测； + \item \textbf{受限传输下的隐私保护融合}：在分布式估计或压缩传输场景中，同时考虑“估计性能--传输开销--信息泄露”之间的权衡。 +\end{itemize} diff --git a/parts/99-literature.tex b/parts/99-literature.tex new file mode 100644 index 0000000..0f0684a --- /dev/null +++ b/parts/99-literature.tex @@ -0,0 +1,37 @@ +\section{初步文献调研} + +\subsection{多星全相干合并建模分析} + +在理想条件下，本文采用的多星协同 \gls{doa} 估计并不是在观测数据层面直接将多颗卫星阵列拼接成一个“超级阵列”，而是基于单星随机信号模型，通过共享一组全局角度参数，在似然函数中实现相干合并。多星最大似然估计的对数似然可写为各单星代价函数的加和，其形式与单星随机信号 \gls{mle} 基本一致，只是将本地角度替换为 $\mathcal{P}_\ell(\boldsymbol{\Theta}_\mathrm{ref})$。在实现上，可以先对每颗卫星独立构造协方差矩阵与本地阵列流形，再在全局角度域中将这些单星代价相干叠加以完成联合估计。 + +从统计结构来看，该方法相当于在单星 \gls{mle} 之外增加“全局到本地角度映射”与“多星求和”两个步骤，其核心估计机制仍与传统随机信号 \gls{mle} 保持一致。因此，在完全同步与完美标定的假设下，多星全相干融合的主要工作更多体现在几何关系的建模与联合性能分析，而非估计算法框架的改变。 + +这一现象与部分校准阵列和分布式子阵列的既有研究结果相吻合~\cite{pesaventoDirectionFindingPartly2002, seeDirectionofArrivalEstimationPartly2004, zhangDirectionFindingPartly2023, suDirectionFindingPartly2025}。当子阵间的相对相位、阵元位置及姿态信息均已校准时，多子阵协同 \gls{doa} 估计通常可以等价为单一大阵列的估计问题。本文的全相干建模与此类方法在本质上是一致的，因此整体表达形式也较为直接。 + +从性能角度来看，对于两颗结构相同且噪声功率一致的卫星阵列，其联合 Fisher 信息近似为单星的两倍，从而使单参数的 \gls{crb} 约降低一半，等效于获得约 $3$\,dB 的理论性能提升。此外，当多颗卫星处于不同轨道位置时，等效阵列孔径进一步增大，可在某些几何结构下获得额外的精度提升。因此，全相干模型在后续工作中可作为多星协同性能的理论上界。 + +\subsection{分布式阵列相关文献的调研结果} + +虽然全相干模型提供了一个便于分析的理论基线，但在实际星座系统中完全满足相关假设并不容易。星间时钟偏移、载频漂移、姿态扰动和链路延迟等因素都会破坏严格相干性，使得直接采用全相干建模具有局限性。因此，有必要参考分布式阵列的相关研究，探讨在不完美条件下如何有效利用多星观测信息。 + +现阶段，与卫星通信直接结合的 \gls{doa} 估计研究仍然有限，更多的成果来自阵列信号处理和分布式子阵网络的研究体系。以下内容给出若干具有代表性的方向。 + +\subsubsection{基于部分校准阵列和弱同步模型的多星 DoA} + +多星系统可视为时变的分布式阵列网络。现有关于部分标定阵列和分布式子阵列的研究~\cite{seeDirectionofArrivalEstimationPartly2004, zhangDOAEstimationExploiting2024, chowdhuryDirectionofArrivalEstimationExploiting2021, wangEnhancedDOAEstimation2024} 表明，当子阵间存在未知相位漂移或阵元位置误差时，通过对子阵内部与子阵间的约束结构进行建模，仍可在一定程度上恢复虚拟全孔径的角度估计能力。 + +基于这一思路，在卫星场景中可以构建“星内完全标定、星间弱同步”的模型，并进一步研究在星间信息交换受限的条件下如何实现稳定且性能接近虚拟全孔径的联合估计。 + +\subsubsection{基于角域稀疏性的 DoA 估计} + +在密集 \gls{leo} 星座下，卫星可覆盖范围极大，但实际地面来向往往只有有限数量，因此角域呈现明显稀疏性。压缩感知与稀疏表示~\cite{chowdhuryDirectionofArrivalEstimationExploiting2021, liHighResolutionDOA2025} 提供了一类适合这类结构的 \gls{doa} 估计方法，尤其适用于快拍数有限、信噪比较低或仅有少量卫星参与协作的情况。 + +此外，多星在轨道运行中具有不断变化的相对几何结构，这些时变特征可进一步作为额外约束，引入到“多星多时刻”场景的时–空联合稀疏估计框架中。 + +\subsubsection{DoA、TDOA 与 FDOA 的联合参数估计} + +现有卫星定位研究~\cite{haoInterferenceGeolocationSatellite2021, elgamoudiSurveyRecentTechniques2021} 主要依赖 \gls{tdoa} 和 \gls{fdoa} 测量，而基于 \gls{doa} 的方法通常仅在单星场景有所涉及。多星场景使三类观测量可以在同一几何框架下结合，从而获得更稳健的联合估计能力。 + +构建联合似然函数并推导联合 \gls{crb} 能够揭示不同观测量之间的互补关系。例如，\gls{doa} 估计可以降低对高精度时钟同步的依赖，而 \gls{tdoa}/\gls{fdoa} 提供了对目标位置与运动状态更敏感的测量；几何结构随时间变化也能为多参量融合带来额外增益。 + +在星间链路受限时，还可考虑分布式求解策略，即各卫星仅交换有限的中间信息，通过迭代的方式逼近集中式最优解。典型方法包括基于“共识（consensus）”思想的分布式最大似然方法~\cite{chenDistributedAlgorithmsArray2021}：各卫星先利用本地观测进行更新，再通过少量信息交换使整个系统在迭代中逐步趋于一致。这类方法通信开销可控，适用于未来大规模密集星座的协作定位任务。 diff --git a/parts/A1-background.tex b/parts/A1-background.tex new file mode 100644 index 0000000..57be527 --- /dev/null +++ b/parts/A1-background.tex @@ -0,0 +1,267 @@ +\section{背景知识} +\label{sec:background} + +本附录只整理后续似然函数与 \gls{crb} 推导反复使用的概率分布、Fisher 信息和 CRB 基础公式。这里的广义参数 $\theta$ 或 $\bm\theta$ 不代表多星主线中的全局 DoA；多星主线的方向参数仍以 $\boldsymbol\xi=[\alpha,\beta]^\Tran$ 为准。 + +\subsection{高斯变量的 PDF} + +\subsubsection{一维实高斯随机变量} + +一维实高斯随机变量 $X \sim \mathcal{N} \left( \mu, \sigma \right)$~\footnote{本文使用 $\sigma$ 表示高斯随机变量的方差。} 的 \gls{pdf} 为 +% +\begin{align} + \label{eq:1d_real_gauss_pdf} + f_X \left( x \mid \mu, \sigma \right) + = \frac{1}{ \sqrt{2\pi\sigma} } + \exp \left[ -\frac{ \left( x - \mu \right)^2 }{ 2\sigma } \right]. +\end{align} + + +\subsubsection{一维圆对称复高斯随机变量} + +\begin{mdquote} + 圆对称复高斯（Circularly Symmetric Complex Gaussian）随机变量 + $Y \sim \mathcal{CN} \left( \mu, \sigma \right)$ 的实部和虚部是独立的实高斯随机变量。 + 即令 $Y = Y^{(r)} + j Y^{(i)}$，有 + $Y^{(r)} \sim \mathcal{N} \left( \mu^{(r)}, \dfrac{\sigma}{2} \right)$， + $Y^{(i)} \sim \mathcal{N} \left( \mu^{(i)}, \dfrac{\sigma}{2} \right)$， + 其中 $\mu = \mu^{(r)} + j \mu^{(i)}$。 +\end{mdquote} + +一维圆对称复高斯随机变量 $Y \sim \mathcal{CN} \left( \mu, \sigma \right)$ 的 \gls{pdf} 为 +% +\begin{align} + \label{eq:1d_cplx_gauss_pdf} + f_Y \left( y \mid \mu, \sigma \right) + &= f_{Y^{(r)}} \left( y^{(r)} \,\middle|\, \mu^{(r)}, \frac{ \sigma }{ 2 } \right) + \times + f_{Y^{(i)}} \left( y^{(i)} \,\middle|\, \mu^{(i)}, \frac{ \sigma }{ 2 } \right) \nonumber + \\ + &= \frac{1}{ \pi \sigma } + \exp \left( + -\frac{ + \left( y^{(r)} - \mu^{(r)} \right)^2 + + + \left( y^{(i)} - \mu^{(i)} \right)^2 + }{ + \sigma + } + \right) + = \frac{1}{ \pi \sigma } + \exp \left( + -\frac{ \left| y - \mu \right|^2 }{ \sigma } + \right). +\end{align} +% +其中 $\left|\, \cdot \,\right|$ 表示取复数的模。 + + +\subsubsection{\texorpdfstring{$M$}{M} 维圆对称复高斯随机变量} + +$M$ 维圆对称复高斯随机变量 +$\mathbf{Z} \sim \mathcal{CN} \left( \bm{\mu}, \mathbf{R} \right)$ 的 \gls{pdf} 为 +% +\begin{align} + \label{eq:general_complex_gaussian_pdf} + f_{\mathbf{Z}} \left( \mathbf{z} \mid \bm{\mu}, \mathbf{R} \right) + = \frac{1}{ \pi^M \det \left( \mathbf{R} \right) } + \exp \left[ + -\left( \mathbf{z} - \bm{\mu} \right)^\Herm + \mathbf{R}^{-1} + \left( \mathbf{z} - \bm{\mu} \right) + \right], +\end{align} +% +其中 $\bm{\mu} \in \mathbb{C}^{M \times 1}$ 为均值向量，$\mathbf{R} \in \mathbb{C}^{M \times M}$ 为协方差矩阵，$\det \left( \mathbf{R} \right)$ 为归一化项，使 \gls{pdf} 的积分为 1。 + +在\emph{各维独立且方差 $\sigma$ 相同}的情形下，协方差矩阵退化为 $\mathbf{R} = \sigma \mathbf{I}_M$，即 $\mathbf{Z} \sim \mathcal{CN} \left( \bm{\mu}, \sigma \mathbf{I}_M \right)$，则 \gls{pdf} 简化为 +% +\begin{align} + \label{eq:independent_complex_gaussian_pdf} + f_{\mathbf{Z}} \left( \mathbf{z} \mid \bm{\mu}, \sigma \right) + = \prod_{i=1}^M + f_{Z_i} \left( z_i \mid \mu_i, \sigma \right) + = \frac{1}{ \left( \pi \sigma \right)^M } + \exp \left[ + -\frac{ \left\| \mathbf{z} - \bm{\mu} \right\|^2 }{ \sigma } + \right]. +\end{align} + + +\subsection{CRB 的定义\texorpdfstring{~\cite{kayFundamentalsStatisticalSignal1993}}{}} + +\begin{mdquote} + 本节的 $\theta$ 和 $\bm{\theta}$ 是广义的参数和参数向量，并非 DoA 估计问题中的角度。 +\end{mdquote} + +\Gls{crb}，亦称 \gls{crlb}，在参数估计理论中给出了\emph{无偏估计量方差的下界}。 + +\subsubsection{单参量的 CRB} + +假设随机变量 $X$ 的 \gls{pdf} 为 $f_X \left( x; \theta \right)$，其中 $\theta \in \mathbb{R}$ 是待估计参量，且满足正则条件 +% +\begin{align} + \mathbb{E}_X \left[ + \frac{ + \partial \ln f_X \left( x; \theta \right) + }{ + \partial \theta + } + \right] + = 0, \quad \forall\, \theta , +\end{align} +% +则任何\emph{无偏估计} $\hat\theta$ 的方差必然满足 +% +\begin{align} + \mathrm{var} ( \hat\theta ) + \geqslant + \frac{1}{-\mathbb{E}_X \left[ + \dfrac{ + \partial^2 \ln f_X \left( x; \theta \right) + }{ + \partial \theta^2 + } + \right]} + = + \frac{1}{ \mathcal{I} \left( \theta \right) } = \mathrm{CRB}(\theta), +\end{align} +% +其中 $\mathcal{I} \left( \theta \right) + \triangleq + -\mathbb{E}_X \left[ + \dfrac{ + \partial^2 \ln f_X \left( x; \theta \right) + }{ + \partial \theta^2 + } + \right] +$ 称为 Fisher 信息，$\dfrac{1}{\mathcal{I} \left( \theta \right)}$ 称为 \gls{crb}。 + +\subsubsection{多参量的 CRB} + +假设随机变量 $X$ 的 \gls{pdf} 为 $f_X \left( x; \bm{\theta} \right)$，其中 $\bm{\theta} = \left[ \theta_1, \ldots, \theta_K \right]^\Tran \in \mathbb{R}^{K \times 1}$ 是待估计参量，且满足正则条件 +% +\begin{align} + \mathbb{E}_X \left[ + \frac{ + \partial \ln f_X \left( x; \bm{\theta} \right) + }{ + \partial \bm{\theta} + } + \right] + = \mathbf{0}, + \quad \forall\, \bm{\theta} , +\end{align} +% +则任何\emph{无偏估计} $\hat{\theta}_i$ 的方差必然满足 +% +\begin{align} + \mathrm{var} ( \hat{\theta}_i ) + \geqslant \left[ \mathcal{I}^{-1} \left( \bm{\theta} \right) \right]_{ii} + = \left[ \mathrm{CRB}(\bm{\theta}) \right]_{ii} . +\end{align} +% +其中 $\mathcal{I} \left( \bm{\theta} \right)$ 是 $K \times K$ 的 \gls{fim}，其元素定义为 +% +\begin{align} + \left[ \mathcal{I} \left( \bm{\theta} \right) \right]_{ij} + \triangleq + -\mathbb{E}_X \left[ + \frac{ + \partial^2 \ln f_X \left( x; \bm{\theta} \right) + }{ + \partial \theta_i \, \partial \theta_j + } + \right], + \quad i = 1, \ldots, K;\, j = 1, \ldots, K . +\end{align} + +此外，可以推导得到 +% +\begin{align} + \mathbb{E}_X \left[ + \frac{ + \partial^2 \ln f_X \left( x; \bm{\theta} \right) + }{ + \partial \theta_i \, \partial \theta_j + } + \right] + = -\mathbb{E}_X \left[ + \frac{ + \partial \ln f_X \left( x; \bm{\theta} \right) + }{ + \partial \theta_i + } + \frac{ + \partial \ln f_X \left( x; \bm{\theta} \right) + }{ + \partial \theta_j + } + \right] . +\end{align} +% +因此，\gls{fim} 的元素还可以写作 +% +\begin{align} + \label{eq:fisher_element} + \left[ \mathcal{I} \left( \bm{\theta} \right) \right]_{ij} + = \mathbb{E}_X \left[ + \frac{ + \partial \ln f_X \left( x; \bm{\theta} \right) + }{ + \partial \theta_i + } + \frac{ + \partial \ln f_X \left( x; \bm{\theta} \right) + }{ + \partial \theta_j + } + \right], + \quad i = 1, \ldots, K;\, j = 1, \ldots, K . +\end{align} +% + +令$ +\bm{\Psi} +\triangleq +\left[ +\dfrac{ +\partial \ln f_X \left( x; \bm{\theta} \right) +}{ +\partial \theta_1 +}, +\ldots, +\dfrac{ +\partial \ln f_X \left( x; \bm{\theta} \right) +}{ +\partial \theta_K +} +\right]^\Tran , +$ 则 \gls{fim}可以表示为 +% +\begin{equation} + \label{eq:compact_fisher_matrix} + \mathcal{I} \left( \bm{\theta} \right) + = \mathbb{E}_X \left[ + \bm{\Psi} \bm{\Psi}^\Tran + \right] . +\end{equation} + +\subsubsection{一般高斯随机过程的 CRB\texorpdfstring{~\cite{stoicaPerformanceStudyConditional1990}}{}} +\label{subsubsec:general_gauss_crb} + +假设 $M$ 维圆对称复高斯随机变量 $\mathbf{Z} \sim \mathcal{CN} (\bm{\mu} (\bm{\theta}), \mathbf{R}(\bm{\theta}))$，其中均值矢量 $\bm{\mu} (\bm{\theta}) \in \mathbb{C}^{M \times 1}$ 和协方差矩阵 $\mathbf{R}(\bm{\theta})\in \mathbb{C}^{M \times M}$ 由实参数向量 $\bm{\theta}$ 唯一确定，也即复高斯随机变量 $\mathbf{Z}$ 的分布由实参数向量 $\bm{\theta}$ 唯一确定，则 $\bm{\theta}$ 的任何无偏估计的协方差矩阵的 \gls{crb} 可以表示为 +% +\begin{equation} + \label{eq:general_gauss_crb} + [(\mathrm{CRB})^{-1}]_{ij} = \tr \left[ + \mathbf{R}^{-1} (\bm{\theta}) \frac{\partial \mathbf{R} (\bm{\theta})}{\partial \theta_i} \mathbf{R}^{-1} (\bm{\theta}) \frac{\partial \mathbf{R} (\bm{\theta})}{\partial \theta_j} + \right] + 2 \Re \left\{ + \frac{\partial \bm{\mu}^\Herm (\bm{\theta})}{\partial \theta_i} \mathbf{R}^{-1} (\bm{\theta}) \frac{\partial \bm{\mu} (\bm{\theta})}{\partial \theta_j} + \right\}, +\end{equation} +% +其中 $\theta_i$ 表示 $\bm{\theta}$ 的第 $i$ 个分量。 + +以上公式构成后续各类 CRB 推导的公共工具箱。 \ref{sec:det_doa_crb}--\ref{sec:det_doa_doppler_crb} 将先把这些公式用于单阵列基线， \ref{sec:A6_multiArrayStatDetDoaDopplerCrb}--\ref{sec:A8_dynIndependentPhaseCrb} 再用于多星主线模型。 diff --git a/parts/A2-doaEstMethod.tex b/parts/A2-doaEstMethod.tex new file mode 100644 index 0000000..580f049 --- /dev/null +++ b/parts/A2-doaEstMethod.tex @@ -0,0 +1,831 @@ +\section{常用 DoA 估计方法} +\label{sec:doa_methods} + +本附录汇总 MVDR、MUSIC、ESPRIT 及其空时扩展形式，主要用于补充算法背景。这里的算法记号服务于经典单阵列或空时阵列模型，不改变正文多星主线中 $\boldsymbol\xi$、$\nu_{\mathrm{ref}}$ 和 $\dot\nu_{\mathrm{ref}}$ 的参数分组。 + +\subsection{MVDR Algorithm} +\label{subsec:mvdr} + +Consider a narrowband far-field source model impinging on an $M$-element array. The received signal at snapshot $t$ is given by +\begin{equation} +\mathbf{y}_t += +\sum_{k=1}^{K} +\mathbf{a}(\theta_k) s_k(t) ++ +\mathbf{n}_t += +\mathbf{A}(\boldsymbol{\theta}) \mathbf{s}_t + \mathbf{n}_t, +\qquad t = 1,\dots,T, +\label{eq:mvdr_signal_model} +\end{equation} +where $\mathbf{y}_t \in \mathbb{C}^{M \times 1}$ is the received array snapshot, $\mathbf{s}_t \in \mathbb{C}^{K \times 1}$ is the source signal vector, $\mathbf{n}_t \in \mathbb{C}^{M \times 1}$ is the additive noise vector, and $\mathbf{a}(\theta) \in \mathbb{C}^{M \times 1}$ denotes the array steering vector corresponding to angle $\theta$. + +For a uniform linear array (ULA), the steering vector can be expressed as +\begin{equation} +\mathbf{a}(\theta) += +\left[ +1,\, +e^{j 2\pi \frac{d}{\lambda_{\mathrm{c}}}\sin\theta},\, +\dots,\, +e^{j 2\pi \frac{d}{\lambda_{\mathrm{c}}}(M-1)\sin\theta} +\right]^{\Tran}, +\label{eq:mvdr_steering_vector} +\end{equation} +where $d$ is the inter-element spacing and $\lambda_{\mathrm{c}}$ is the carrier wavelength. + +The covariance matrix of the received data is defined as +\begin{equation} +\mathbf{R}_y += +\mathbb{E}\left\{ +\mathbf{y}_t \mathbf{y}_t^{\Herm} +\right\}. +\label{eq:mvdr_covariance} +\end{equation} +In practice, $\mathbf{R}_y$ is replaced by the sample covariance matrix +\begin{equation} +\widehat{\mathbf{R}}_y += +\frac{1}{T} +\sum_{t=1}^{T} +\mathbf{y}_t \mathbf{y}_t^{\Herm}. +\label{eq:mvdr_sample_covariance} +\end{equation} + +For a given look direction $\theta$, the output of a linear beamformer with weight vector $\mathbf{w}(\theta) \in \mathbb{C}^{M \times 1}$ is +\begin{equation} +z_t(\theta) += +\mathbf{w}^{\Herm}(\theta)\mathbf{y}_t. +\label{eq:mvdr_output} +\end{equation} +The MVDR beamformer minimizes the output power while maintaining a distortionless response in the look direction. Accordingly, the weight vector is obtained by solving +\begin{equation} +\min_{\mathbf{w}(\theta)} +\quad +\mathbf{w}^{\Herm}(\theta)\mathbf{R}_y\mathbf{w}(\theta) +\qquad +\text{s.t.} +\quad +\mathbf{w}^{\Herm}(\theta)\mathbf{a}(\theta)=1. +\label{eq:mvdr_optimization} +\end{equation} +Using the method of Lagrange multipliers, the optimal solution is given by +\begin{equation} +\mathbf{w}_{\mathrm{MVDR}}(\theta) += +\frac{ +\mathbf{R}_y^{-1}\mathbf{a}(\theta) +}{ +\mathbf{a}^{\Herm}(\theta)\mathbf{R}_y^{-1}\mathbf{a}(\theta) +}. +\label{eq:mvdr_weight} +\end{equation} + +Substituting \eqref{eq:mvdr_weight} into the beamformer output power yields +\begin{equation} +P_{\mathrm{MVDR}}(\theta) += +\frac{1}{ +\mathbf{a}^{\Herm}(\theta)\mathbf{R}_y^{-1}\mathbf{a}(\theta) +}. +\label{eq:mvdr_spectrum} +\end{equation} +In practical implementation, the sample covariance matrix is used, leading to +\begin{equation} +\widehat{P}_{\mathrm{MVDR}}(\theta) += +\frac{1}{ +\mathbf{a}^{\Herm}(\theta)\widehat{\mathbf{R}}_y^{-1}\mathbf{a}(\theta) +}. +\label{eq:mvdr_sample_spectrum} +\end{equation} +The DoA estimates are then obtained by locating the $K$ dominant peaks of the MVDR spatial spectrum, namely, +\begin{equation} +\hat{\theta}_k += +\arg\max_{\theta} +\widehat{P}_{\mathrm{MVDR}}(\theta), +\qquad k=1,\dots,K. +\label{eq:mvdr_estimation} +\end{equation} + +The MVDR algorithm adaptively suppresses interference and noise by minimizing the array output power subject to the distortionless constraint, and thus achieves high-resolution spatial spectrum estimation with improved interference rejection capability. + + +\subsection{MUSIC Algorithm} +\label{subsec:music} + +Consider the same narrowband far-field array signal model as in \eqref{eq:mvdr_signal_model}. The received signal at snapshot $t$ can be expressed as +\begin{equation} +\mathbf{y}_t += +\sum_{k=1}^{K} +\mathbf{a}(\theta_k) s_k(t) ++ +\mathbf{n}_t += +\mathbf{A}(\boldsymbol{\theta}) \mathbf{s}_t + \mathbf{n}_t, +\qquad t = 1,\dots,T, +\label{eq:music_signal_model} +\end{equation} +where $\mathbf{y}_t \in \mathbb{C}^{M \times 1}$ is the received array snapshot, $\mathbf{s}_t = [s_1(t),\dots,s_K(t)]^{\Tran} \in \mathbb{C}^{K \times 1}$ is the source signal vector, $\mathbf{n}_t \in \mathbb{C}^{M \times 1}$ is the additive noise vector, and +\begin{equation} +\mathbf{A}(\boldsymbol{\theta}) += +\left[ +\mathbf{a}(\theta_1),\dots,\mathbf{a}(\theta_K) +\right] +\in \mathbb{C}^{M \times K} +\label{eq:music_array_manifold} +\end{equation} +is the array manifold matrix. + +For a uniform linear array (ULA), the steering vector corresponding to the angle $\theta_k$ is given by +\begin{equation} +\mathbf{a}(\theta_k) += +\left[ +1,\, +e^{j 2\pi \frac{d}{\lambda_{\mathrm{c}}}\sin\theta_k},\, +\dots,\, +e^{j 2\pi \frac{d}{\lambda_{\mathrm{c}}}(M-1)\sin\theta_k} +\right]^{\Tran}, +\label{eq:music_steering_vector} +\end{equation} +where $d$ denotes the inter-element spacing and $\lambda_{\mathrm{c}}$ is the carrier wavelength. + +Assuming that the source signals are mutually uncorrelated and that the noise is spatially white with variance $\sigma_n^2$, the array covariance matrix is +\begin{equation} +\mathbf{R}_y += +\mathbb{E}\{\mathbf{y}_t \mathbf{y}_t^{\Herm}\} += +\mathbf{A}(\boldsymbol{\theta}) \mathbf{R}_s \mathbf{A}^{\Herm}(\boldsymbol{\theta}) ++ +\sigma_n^2 \mathbf{I}_M, +\label{eq:music_covariance} +\end{equation} +where +\begin{equation} +\mathbf{R}_s += +\mathbb{E}\{\mathbf{s}_t \mathbf{s}_t^{\Herm}\} +\label{eq:music_source_covariance} +\end{equation} +is the source covariance matrix. + +In practice, $\mathbf{R}_y$ is approximated by the sample covariance matrix +\begin{equation} +\widehat{\mathbf{R}}_y += +\frac{1}{T} +\sum_{t=1}^{T} +\mathbf{y}_t \mathbf{y}_t^{\Herm}. +\label{eq:music_sample_covariance} +\end{equation} +By performing eigenvalue decomposition, one obtains +\begin{equation} +\widehat{\mathbf{R}}_y += +\mathbf{U}_s \boldsymbol{\Lambda}_s \mathbf{U}_s^{\Herm} ++ +\mathbf{U}_n \boldsymbol{\Lambda}_n \mathbf{U}_n^{\Herm}, +\label{eq:music_evd} +\end{equation} +where $\mathbf{U}_s \in \mathbb{C}^{M \times K}$ and $\mathbf{U}_n \in \mathbb{C}^{M \times (M-K)}$ denote the signal subspace and the noise subspace, respectively. + +Under ideal conditions, the steering vectors associated with the true source directions lie in the signal subspace and are therefore orthogonal to the noise subspace, namely, +\begin{equation} +\mathbf{U}_n^{\Herm} \mathbf{a}(\theta_k) += +\mathbf{0}, +\qquad k=1,\dots,K. +\label{eq:music_orthogonality} +\end{equation} +Based on this orthogonality property, the MUSIC pseudospectrum is constructed as +\begin{equation} +P_{\mathrm{MUSIC}}(\theta) += +\frac{1} +{\mathbf{a}^{\Herm}(\theta)\, +\mathbf{U}_n \mathbf{U}_n^{\Herm}\, +\mathbf{a}(\theta)}. +\label{eq:music_spectrum} +\end{equation} +The DoA estimates are then obtained by locating the $K$ dominant peaks of the pseudospectrum, i.e., +\begin{equation} +\hat{\theta}_k += +\arg\max_{\theta} +P_{\mathrm{MUSIC}}(\theta), +\qquad k=1,\dots,K. +\label{eq:music_estimation} +\end{equation} + +The MUSIC algorithm achieves high-resolution DoA estimation by exploiting the orthogonality between the array steering vectors and the noise subspace, and has been widely used in subspace-based array signal processing. + + +\subsection{ESPRIT Algorithm} +\label{subsec:esprit} + +Consider the same narrowband far-field array signal model as in \eqref{eq:mvdr_signal_model}, i.e., +\begin{equation} +\mathbf{y}_t += +\sum_{k=1}^{K} +\mathbf{a}(\theta_k) s_k(t) ++ +\mathbf{n}_t += +\mathbf{A}(\boldsymbol{\theta}) \mathbf{s}_t + \mathbf{n}_t, +\qquad t = 1,\dots,T, +\label{eq:esprit_signal_model} +\end{equation} +where $\mathbf{A}(\boldsymbol{\theta}) = [\mathbf{a}(\theta_1), \dots, \mathbf{a}(\theta_K)] \in \mathbb{C}^{M \times K}$ is the array manifold matrix. For a uniform linear array (ULA), the steering vector is given by +\begin{equation} +\mathbf{a}(\theta_k) += +\left[ +1,\, +e^{j \mu_k},\, +\dots,\, +e^{j (M-1)\mu_k} +\right]^{\Tran}, +\label{eq:esprit_steering_vector} +\end{equation} +where +\begin{equation} +\mu_k += +2\pi \frac{d}{\lambda_{\mathrm{c}}} \sin\theta_k, +\label{eq:esprit_spatial_frequency} +\end{equation} +with $d$ denoting the inter-element spacing and $\lambda_{\mathrm{c}}$ the carrier wavelength. + +The ESPRIT algorithm exploits the translational invariance structure of the array. To this end, two maximally overlapped subarrays are constructed from the original ULA. Define the selection matrices +\begin{equation} +\mathbf{J}_1 += +\left[ +\mathbf{I}_{M-1} \ \mathbf{0} +\right] +\in \mathbb{C}^{(M-1)\times M}, +\qquad +\mathbf{J}_2 += +\left[ +\mathbf{0} \ \mathbf{I}_{M-1} +\right] +\in \mathbb{C}^{(M-1)\times M}. +\label{eq:esprit_selection_matrices} +\end{equation} +Then the corresponding subarray manifold matrices are +\begin{equation} +\mathbf{A}_1 = \mathbf{J}_1 \mathbf{A}, +\qquad +\mathbf{A}_2 = \mathbf{J}_2 \mathbf{A}. +\label{eq:esprit_subarray_manifolds} +\end{equation} +Due to the shift invariance of the ULA, $\mathbf{A}_1$ and $\mathbf{A}_2$ satisfy +\begin{equation} +\mathbf{A}_2 += +\mathbf{A}_1 \bm{\Phi}, +\label{eq:esprit_shift_invariance} +\end{equation} +where +\begin{equation} +\bm{\Phi} += +\operatorname{diag} +\left( +e^{j\mu_1},\dots,e^{j\mu_K} +\right) +\label{eq:esprit_phi} +\end{equation} +is the rotational invariance matrix. + +As in subspace-based methods, the sample covariance matrix is first formed as +\begin{equation} +\widehat{\mathbf{R}}_y += +\frac{1}{T} +\sum_{t=1}^{T} +\mathbf{y}_t \mathbf{y}_t^{\Herm}. +\label{eq:esprit_sample_covariance} +\end{equation} +By eigenvalue decomposition, +\begin{equation} +\widehat{\mathbf{R}}_y += +\mathbf{U}_s \boldsymbol{\Lambda}_s \mathbf{U}_s^{\Herm} ++ +\mathbf{U}_n \boldsymbol{\Lambda}_n \mathbf{U}_n^{\Herm}, +\label{eq:esprit_evd} +\end{equation} +where $\mathbf{U}_s \in \mathbb{C}^{M \times K}$ denotes the estimated signal subspace. Since the signal subspace spans the same column space as $\mathbf{A}$, there exists a nonsingular matrix $\mathbf{T} \in \mathbb{C}^{K \times K}$ such that +\begin{equation} +\mathbf{U}_s = \mathbf{A} \mathbf{T}. +\label{eq:esprit_subspace_relation} +\end{equation} +Applying the selection matrices to $\mathbf{U}_s$ yields +\begin{equation} +\mathbf{U}_{s,1} += +\mathbf{J}_1 \mathbf{U}_s, +\qquad +\mathbf{U}_{s,2} += +\mathbf{J}_2 \mathbf{U}_s. +\label{eq:esprit_subspace_partitions} +\end{equation} +Substituting \eqref{eq:esprit_subspace_relation} into \eqref{eq:esprit_subspace_partitions} and using \eqref{eq:esprit_shift_invariance}, one obtains +\begin{equation} +\mathbf{U}_{s,2} += +\mathbf{U}_{s,1}\, +\bm{\Psi}, +\label{eq:esprit_invariance_subspace} +\end{equation} +where +\begin{equation} +\bm{\Psi} += +\mathbf{T}^{-1} \bm{\Phi} \mathbf{T}. +\label{eq:esprit_psi} +\end{equation} +Therefore, $\bm{\Psi}$ is similar to $\bm{\Phi}$ and has the same eigenvalues $\{e^{j\mu_k}\}_{k=1}^{K}$. + +In practice, $\bm{\Psi}$ can be estimated by solving the least-squares problem +\begin{equation} +\widehat{\bm{\Psi}} += +\arg\min_{\bm{\Psi}} +\left\| +\mathbf{U}_{s,2} - \mathbf{U}_{s,1}\bm{\Psi} +\right\|_{\mathrm{F}}^2, +\label{eq:esprit_ls_problem} +\end{equation} +whose closed-form solution is +\begin{equation} +\widehat{\bm{\Psi}} += +\mathbf{U}_{s,1}^{\dagger}\mathbf{U}_{s,2}, +\label{eq:esprit_ls_solution} +\end{equation} +where $(\cdot)^{\dagger}$ denotes the Moore--Penrose pseudoinverse. + +Let $\hat{\kappa}_k$ denote the $k$th eigenvalue of $\widehat{\bm{\Psi}}$. Then the spatial frequency is estimated as +\begin{equation} +\hat{\mu}_k = \angle(\hat{\kappa}_k), +\qquad k = 1,\dots,K, +\label{eq:esprit_mu_estimate} +\end{equation} +where $\angle(\cdot)$ denotes the phase angle of a complex number. Finally, the DoA estimate is obtained from +\begin{equation} +\hat{\theta}_k += +\arcsin\!\left( +\frac{\lambda_{\mathrm{c}}}{2\pi d}\hat{\mu}_k +\right), +\qquad k = 1,\dots,K. +\label{eq:esprit_theta_estimate} +\end{equation} + +Compared with MUSIC, the ESPRIT algorithm avoids explicit spectral search and directly retrieves the DoA parameters from the rotational invariance structure of the signal subspace, thereby offering lower computational complexity while maintaining high estimation accuracy. + + +\subsection{Spatio-Temporal MUSIC Algorithm} +\label{subsec:st_music} + +Consider a narrowband far-field source model observed by an $M$-element array over $T$ temporal samples within each data block. Let $\mathbf{Y}_{\ell} \in \mathbb{C}^{M \times T}$ denote the $\ell$th received data block, where $\ell = 1, \dots, L$. For $K$ uncorrelated sources, the received signal can be written as +\begin{equation} +\mathbf{Y}_{\ell} += +\sum_{k=1}^{K} +\zeta_{k,\ell}\, +\mathbf{a}(\theta_k)\, +\mathbf q_k^\Tran(\nu_k) ++ +\mathbf{N}_{\ell}, +\qquad \ell = 1,\dots,L, +\label{eq:st_music_block_model} +\end{equation} +where $\zeta_{k,\ell} \in \mathbb{C}$ is the complex gain of the $k$th source in the $\ell$th block, $\mathbf{a}(\theta_k) \in \mathbb{C}^{M \times 1}$ is the spatial steering vector associated with the direction of arrival (DoA) $\theta_k$, $\mathbf q_k(\nu_k) \in \mathbb{C}^{T \times 1}$ is the temporal steering vector associated with the normalized angular Doppler $\nu_k$, and $\mathbf{N}_{\ell}$ is the additive noise matrix. For a uniform linear array (ULA), the spatial steering vector is given by +\begin{equation} +\mathbf{a}(\theta_k) += +\left[ +1,\, +e^{j 2\pi \frac{d}{\lambda_{\mathrm{c}}}\sin\theta_k},\, +\dots,\, +e^{j 2\pi \frac{d}{\lambda_{\mathrm{c}}}(M-1)\sin\theta_k} +\right]^\Tran, +\label{eq:spatial_steering} +\end{equation} +where $d$ is the inter-element spacing and $\lambda_{\mathrm{c}}$ is the carrier wavelength. The temporal steering vector is defined as +\begin{equation} +\mathbf q_k(\nu_k) += +\left[ +1,\, +e^{j\nu_k},\, +\dots,\, +e^{j(T-1)\nu_k} +\right]^\Tran, +\label{eq:temporal_steering} +\end{equation} +where $\nu_k$ denotes the normalized angular Doppler in rad/sample. + +By vectorizing \eqref{eq:st_music_block_model}, one obtains +\begin{equation} +\mathbf{y}_{\ell} +\triangleq +\vect(\mathbf{Y}_{\ell}) += +\sum_{k=1}^{K} +\zeta_{k,\ell}\, +\mathbf b_{\mathrm{st},k}(\theta_k,\nu_k) ++ +\mathbf{n}_{\ell}, +\label{eq:vec_model} +\end{equation} +where $\mathbf{n}_{\ell} \triangleq \vect(\mathbf{N}_{\ell})$ and +\begin{equation} +\mathbf b_{\mathrm{st},k}(\theta_k,\nu_k) +\triangleq +\mathbf q_k(\nu_k) \otimes \mathbf{a}(\theta_k) +\in \mathbb{C}^{MT \times 1} +\label{eq:st_steering_vector} +\end{equation} +is the spatio-temporal steering vector, with $\otimes$ denoting the Kronecker product. Hence, the joint angle-Doppler estimation problem can be recast as a standard subspace estimation problem in the augmented spatio-temporal domain. + +Assuming that the source signals are mutually uncorrelated and the noise is spatially and temporally white with variance $\sigma_n^2$, the covariance matrix of $\mathbf{y}_{\ell}$ is given by +\begin{equation} +\mathbf{R}_y += +\mathbb{E}\{\mathbf{y}_{\ell}\mathbf{y}_{\ell}^\Herm\} += +\mathbf{B}_{\mathrm{st}}\mathbf{R}_{\zeta}\mathbf{B}_{\mathrm{st}}^\Herm ++ +\sigma_n^2 \mathbf{I}_{MT}, +\label{eq:covariance_st} +\end{equation} +where +\begin{equation} +\mathbf{B}_{\mathrm{st}} += +\left[ +\mathbf b_{\mathrm{st},1}(\theta_1,\nu_1), +\dots, +\mathbf b_{\mathrm{st},K}(\theta_K,\nu_K) +\right] +\in \mathbb{C}^{MT \times K} +\label{eq:Bst_matrix} +\end{equation} +and $\mathbf{R}_{\zeta} = \mathbb{E}\{\boldsymbol{\zeta}_{\ell}\boldsymbol{\zeta}_{\ell}^\Herm\}$ is the source covariance matrix, with $\boldsymbol{\zeta}_{\ell} = [\zeta_{1,\ell},\dots,\zeta_{K,\ell}]^\Tran$. + +In practice, $\mathbf{R}_y$ is estimated from $L$ data blocks as +\begin{equation} +\widehat{\mathbf{R}}_y += +\frac{1}{L} +\sum_{\ell=1}^{L} +\mathbf{y}_{\ell}\mathbf{y}_{\ell}^\Herm. +\label{eq:sample_covariance} +\end{equation} +Performing eigenvalue decomposition yields +\begin{equation} +\widehat{\mathbf{R}}_y += +\mathbf{U}_s \boldsymbol{\Lambda}_s \mathbf{U}_s^\Herm ++ +\mathbf{U}_n \boldsymbol{\Lambda}_n \mathbf{U}_n^\Herm, +\label{eq:eig_decomp} +\end{equation} +where $\mathbf{U}_s \in \mathbb{C}^{MT \times K}$ and $\mathbf{U}_n \in \mathbb{C}^{MT \times (MT-K)}$ denote the signal and noise subspaces, respectively. Under ideal conditions, each true spatio-temporal steering vector lies in the signal subspace and is therefore orthogonal to the noise subspace, i.e., +\begin{equation} +\mathbf{U}_n^\Herm\mathbf b_{\mathrm{st},k}(\theta_k,\nu_k) = \mathbf{0}, +\qquad k=1,\dots,K. +\label{eq:orthogonality_condition} +\end{equation} +Based on this orthogonality property, the spatio-temporal MUSIC pseudospectrum is constructed as +\begin{equation} +P_{\mathrm{ST\text{-}MUSIC}}(\theta,\nu) += +\frac{1} +{\mathbf b_{\mathrm{st}}^\Herm(\theta,\nu)\, +\mathbf{U}_n\mathbf{U}_n^\Herm\, +\mathbf b_{\mathrm{st}}(\theta,\nu)}. +\label{eq:st_music_spectrum} +\end{equation} +The joint estimates of DoA and Doppler are then obtained by searching for the $K$ dominant peaks of \eqref{eq:st_music_spectrum}, namely, +\begin{equation} +(\hat{\theta}_k,\hat{\nu}_k) += +\arg\max_{\theta,\nu} +P_{\mathrm{ST\text{-}MUSIC}}(\theta,\nu), +\qquad k=1,\dots,K. +\label{eq:st_music_estimation} +\end{equation} + +Compared with conventional MUSIC, the spatio-temporal MUSIC algorithm exploits both the spatial array manifold and the temporal Doppler structure, thereby enabling joint angle-Doppler estimation in a unified subspace framework. + + +\subsection{Spatio-Temporal ESPRIT Algorithm} +\label{subsec:st_esprit} + +Consider the same spatio-temporal signal model as in \eqref{eq:st_music_block_model}. For the $\ell$th data block, the received signal matrix is given by +\begin{equation} +\mathbf{Y}_{\ell} += +\sum_{k=1}^{K} +\zeta_{k,\ell}\, +\mathbf{a}(\theta_k)\, +\mathbf q_k^{\Tran}(\nu_k) ++ +\mathbf{N}_{\ell}, +\qquad \ell = 1,\dots,L, +\label{eq:st_esprit_block_model} +\end{equation} +where $\mathbf{a}(\theta_k)\in\mathbb{C}^{M\times 1}$ and $\mathbf q_k(\nu_k)\in\mathbb{C}^{T\times 1}$ denote the spatial and temporal steering vectors, respectively. For a uniform linear array (ULA), the spatial steering vector is +\begin{equation} +\mathbf{a}(\theta_k) += +\left[ +1,\, +e^{j\mu_k},\, +\dots,\, +e^{j(M-1)\mu_k} +\right]^{\Tran}, +\label{eq:st_esprit_spatial_steering} +\end{equation} +where +\begin{equation} +\mu_k += +2\pi \frac{d}{\lambda_{\mathrm{c}}}\sin\theta_k, +\label{eq:st_esprit_spatial_freq} +\end{equation} +and the temporal steering vector is +\begin{equation} +\mathbf q_k(\nu_k) += +\left[ +1,\, +e^{j\nu_k},\, +\dots,\, +e^{j(T-1)\nu_k} +\right]^{\Tran}, +\label{eq:st_esprit_temporal_steering} +\end{equation} +where $\nu_k$ denotes the normalized angular Doppler in rad/sample. + +By vectorizing \eqref{eq:st_esprit_block_model}, one obtains +\begin{equation} +\mathbf{y}_{\ell} +\triangleq +\vect(\mathbf{Y}_{\ell}) += +\sum_{k=1}^{K} +\zeta_{k,\ell}\, +\mathbf b_{\mathrm{st},k}(\theta_k,\nu_k) ++ +\mathbf{n}_{\ell}, +\label{eq:st_esprit_vec_model} +\end{equation} +where +\begin{equation} +\mathbf b_{\mathrm{st},k}(\theta_k,\nu_k) += +\mathbf q_k(\nu_k) \otimes \mathbf{a}(\theta_k) +\in \mathbb{C}^{MT\times 1}, +\label{eq:st_esprit_st_steering} +\end{equation} +and $\mathbf{n}_{\ell}=\vect(\mathbf{N}_{\ell})$. Define the spatio-temporal manifold matrix as +\begin{equation} +\mathbf{B}_{\mathrm{st}} += +\left[ +\mathbf b_{\mathrm{st},1}(\theta_1,\nu_1), +\dots, +\mathbf b_{\mathrm{st},K}(\theta_K,\nu_K) +\right] +\in\mathbb{C}^{MT\times K}. +\label{eq:st_esprit_Bst} +\end{equation} + +The sample covariance matrix is estimated as +\begin{equation} +\widehat{\mathbf{R}}_y += +\frac{1}{L} +\sum_{\ell=1}^{L} +\mathbf{y}_{\ell}\mathbf{y}_{\ell}^{\Herm}. +\label{eq:st_esprit_sample_covariance} +\end{equation} +Performing eigenvalue decomposition yields +\begin{equation} +\widehat{\mathbf{R}}_y += +\mathbf{U}_s \boldsymbol{\Lambda}_s \mathbf{U}_s^{\Herm} ++ +\mathbf{U}_n \boldsymbol{\Lambda}_n \mathbf{U}_n^{\Herm}, +\label{eq:st_esprit_evd} +\end{equation} +where $\mathbf{U}_s\in\mathbb{C}^{MT\times K}$ denotes the estimated signal subspace. Since $\mathbf{U}_s$ and $\mathbf{B}_{\mathrm{st}}$ span the same column space, there exists a nonsingular matrix $\mathbf{T}\in\mathbb{C}^{K\times K}$ such that +\begin{equation} +\mathbf{U}_s = \mathbf{B}_{\mathrm{st}}\mathbf{T}. +\label{eq:st_esprit_subspace_relation} +\end{equation} + +The spatio-temporal ESPRIT algorithm exploits the shift invariance structure in both the spatial and temporal dimensions. First, define the spatial selection matrices +\begin{equation} +\mathbf{J}_{s,1} += +\left[ +\mathbf{I}_{M-1}\ \mathbf{0} +\right] +\in\mathbb{C}^{(M-1)\times M}, +\qquad +\mathbf{J}_{s,2} += +\left[ +\mathbf{0}\ \mathbf{I}_{M-1} +\right] +\in\mathbb{C}^{(M-1)\times M}, +\label{eq:st_esprit_spatial_selection} +\end{equation} +and the temporal selection matrices +\begin{equation} +\mathbf{J}_{t,1} += +\left[ +\mathbf{I}_{T-1}\ \mathbf{0} +\right] +\in\mathbb{C}^{(T-1)\times T}, +\qquad +\mathbf{J}_{t,2} += +\left[ +\mathbf{0}\ \mathbf{I}_{T-1} +\right] +\in\mathbb{C}^{(T-1)\times T}. +\label{eq:st_esprit_temporal_selection} +\end{equation} + +Correspondingly, the augmented selection matrices acting on the vectorized spatio-temporal observation are constructed as +\begin{equation} +\widetilde{\mathbf{J}}_{s,1} += +\mathbf{I}_T \otimes \mathbf{J}_{s,1}, +\qquad +\widetilde{\mathbf{J}}_{s,2} += +\mathbf{I}_T \otimes \mathbf{J}_{s,2}, +\label{eq:st_esprit_augmented_spatial_selection} +\end{equation} +and +\begin{equation} +\widetilde{\mathbf{J}}_{t,1} += +\mathbf{J}_{t,1} \otimes \mathbf{I}_M, +\qquad +\widetilde{\mathbf{J}}_{t,2} += +\mathbf{J}_{t,2} \otimes \mathbf{I}_M. +\label{eq:st_esprit_augmented_temporal_selection} +\end{equation} + +Using the Kronecker product structure in \eqref{eq:st_esprit_st_steering}, the spatio-temporal manifold matrix satisfies the spatial shift invariance relation +\begin{equation} +\widetilde{\mathbf{J}}_{s,2}\mathbf{B}_{\mathrm{st}} += +\widetilde{\mathbf{J}}_{s,1}\mathbf{B}_{\mathrm{st}}\bm{\Phi}_s, +\label{eq:st_esprit_spatial_invariance} +\end{equation} +where +\begin{equation} +\bm{\Phi}_s += +\operatorname{diag} +\left( +e^{j\mu_1},\dots,e^{j\mu_K} +\right), +\label{eq:st_esprit_Phi_s} +\end{equation} +and the temporal shift invariance relation +\begin{equation} +\widetilde{\mathbf{J}}_{t,2}\mathbf{B}_{\mathrm{st}} += +\widetilde{\mathbf{J}}_{t,1}\mathbf{B}_{\mathrm{st}}\bm{\Phi}_t, +\label{eq:st_esprit_temporal_invariance} +\end{equation} +where +\begin{equation} +\bm{\Phi}_t += +\operatorname{diag} +\left( +e^{j\nu_1},\dots,e^{j\nu_K} +\right). +\label{eq:st_esprit_Phi_t} +\end{equation} + +Applying the same selection operations to the signal subspace $\mathbf{U}_s$ gives +\begin{equation} +\mathbf{U}_{s,1}^{(s)} += +\widetilde{\mathbf{J}}_{s,1}\mathbf{U}_s, +\qquad +\mathbf{U}_{s,2}^{(s)} += +\widetilde{\mathbf{J}}_{s,2}\mathbf{U}_s, +\label{eq:st_esprit_spatial_subspace_partitions} +\end{equation} +and +\begin{equation} +\mathbf{U}_{s,1}^{(t)} += +\widetilde{\mathbf{J}}_{t,1}\mathbf{U}_s, +\qquad +\mathbf{U}_{s,2}^{(t)} += +\widetilde{\mathbf{J}}_{t,2}\mathbf{U}_s. +\label{eq:st_esprit_temporal_subspace_partitions} +\end{equation} +Substituting \eqref{eq:st_esprit_subspace_relation} into \eqref{eq:st_esprit_spatial_subspace_partitions} and \eqref{eq:st_esprit_temporal_subspace_partitions}, and using \eqref{eq:st_esprit_spatial_invariance}--\eqref{eq:st_esprit_temporal_invariance}, one obtains +\begin{equation} +\mathbf{U}_{s,2}^{(s)} += +\mathbf{U}_{s,1}^{(s)} \bm{\Psi}_s, +\label{eq:st_esprit_Psi_s_relation} +\end{equation} +with +\begin{equation} +\bm{\Psi}_s += +\mathbf{T}^{-1}\bm{\Phi}_s\mathbf{T}, +\label{eq:st_esprit_Psi_s} +\end{equation} +and +\begin{equation} +\mathbf{U}_{s,2}^{(t)} += +\mathbf{U}_{s,1}^{(t)} \bm{\Psi}_t, +\label{eq:st_esprit_Psi_t_relation} +\end{equation} +with +\begin{equation} +\bm{\Psi}_t += +\mathbf{T}^{-1}\bm{\Phi}_t\mathbf{T}. +\label{eq:st_esprit_Psi_t} +\end{equation} + +The matrices $\bm{\Psi}_s$ and $\bm{\Psi}_t$ can be estimated in the least-squares sense as +\begin{equation} +\widehat{\bm{\Psi}}_s += +\left(\mathbf{U}_{s,1}^{(s)}\right)^{\dagger} +\mathbf{U}_{s,2}^{(s)}, +\qquad +\widehat{\bm{\Psi}}_t += +\left(\mathbf{U}_{s,1}^{(t)}\right)^{\dagger} +\mathbf{U}_{s,2}^{(t)}, +\label{eq:st_esprit_LS_solution} +\end{equation} +where $(\cdot)^{\dagger}$ denotes the Moore--Penrose pseudoinverse. Let $\hat{\kappa}_{s,k}$ and $\hat{\kappa}_{t,k}$ denote the paired eigenvalues of $\widehat{\bm{\Psi}}_s$ and $\widehat{\bm{\Psi}}_t$, respectively. Then the spatial and temporal frequencies are estimated as +\begin{equation} +\hat{\mu}_k += +\angle(\hat{\kappa}_{s,k}), +\qquad +\hat{\nu}_k += +\angle(\hat{\kappa}_{t,k}), +\qquad k=1,\dots,K, +\label{eq:st_esprit_mu_nu_estimation} +\end{equation} +where $\angle(\cdot)$ denotes the phase angle of a complex number. Finally, the DoA estimate and, if needed, the corresponding physical Doppler conversion are given by +\begin{equation} +\hat{\theta}_k += +\arcsin\!\left( +\frac{\lambda_{\mathrm{c}}}{2\pi d}\hat{\mu}_k +\right), +\qquad +\widehat{\nu}^{\mathrm{Hz}}_k += +\frac{\hat{\nu}_k}{2\pi T_{\mathrm{s}}}, +\qquad k=1,\dots,K. +\label{eq:st_esprit_theta_fd_estimation} +\end{equation} + +Compared with spatio-temporal MUSIC, the spatio-temporal ESPRIT algorithm avoids two-dimensional spectral search and directly retrieves the DoA and Doppler parameters from the rotational invariance structure of the spatio-temporal signal subspace, thereby achieving lower computational complexity. + +这些经典方法提供了谱搜索和子空间估计的算法背景，但本文多星主线的重点仍是似然建模、参考星参数化和 Fisher 信息分析。因此，附录中的方法可作为实现参考，而不是当前符号体系的定义来源。 diff --git a/parts/A3-detDoaCrb.tex b/parts/A3-detDoaCrb.tex new file mode 100644 index 0000000..2b3a9b6 --- /dev/null +++ b/parts/A3-detDoaCrb.tex @@ -0,0 +1,412 @@ +\section{确定信号 DoA 估计的 CRB 推导} +\label{sec:det_doa_crb} + +本附录对应第~\ref{sec:det_doa_est} 节的单阵列确定信号 DoA 模型。根据 \gls{fim} 的定义 $\eqref{eq:fisher_element}$，需要先求出对数似然函数 $\eqref{eq:det_ml_log_likelihood}$ 对各类未知量的一阶偏导，再构造 Fisher 信息矩阵并通过 Schur 补消去确定信号快拍等干扰参数。 + +\subsection{一阶偏导求取} + +\subsubsection{对噪声方差的一阶偏导} + +\begin{equation} +\frac{\partial \ln \mathcal{L}}{\partial \sigma} = -\frac{MT}{\sigma}+\frac{1}{\sigma^2} \sum_{t=1}^T \mathbf{n}^\Herm _t\mathbf{n}_t. +\end{equation} + +\subsubsection{对信源的一阶偏导} + +\begin{mdquote} +复向量的偏导数（Wirtinger 微分）：令复数 $z=x+jy$，则其 Wirtinger 微分定义为 +% +\begin{equation} + \label{eq:wirtinger_derivative} + \begin{cases} + \dfrac{\partial}{\partial z} = \dfrac{1}{2}\left(\dfrac{\partial}{\partial x} - j\dfrac{\partial}{\partial y} \right) + \\ + \dfrac{\partial}{\partial z^*} = \dfrac{1}{2}\left(\dfrac{\partial}{\partial x} + j\dfrac{\partial}{\partial y} \right) + \end{cases} + \quad\Rightarrow\quad + \begin{cases} + \dfrac{\partial}{\partial x} = \dfrac{\partial}{\partial z} + \dfrac{\partial}{\partial z^*} + \\ + \dfrac{\partial}{\partial y} = j\left(\dfrac{\partial}{\partial z} - \dfrac{\partial}{\partial z^*} \right) + \end{cases}, +\end{equation} +% +且有 +\begin{subequations} + \begin{equation} + \frac{\partial z^*}{\partial z} + = \dfrac{1}{2}\left(\dfrac{\partial}{\partial x} - j\dfrac{\partial}{\partial y} \right) (x - jy) + = \frac{1}{2}\left(\frac{\partial x}{\partial x} - j\frac{\partial (-jy)}{\partial y} \right) = 0, + \end{equation} + % + \begin{equation} + \frac{\partial z}{\partial z^*} + = \dfrac{1}{2}\left(\dfrac{\partial}{\partial x} + j\dfrac{\partial}{\partial y} \right) (x + jy) + = \frac{1}{2}\left(\frac{\partial x}{\partial x} + j\frac{\partial (jy)}{\partial y} \right) = 0, + \end{equation} +\end{subequations} +% +也即在 Wirtinger 微分定义下，复变量 $z$ 及其复共轭 $z^*$ 可以看做看作两个不相关的变量，对其中一个求导时，将另一项看作常量即可。 +\end{mdquote} + +首先按 Wirtinger 微分求取信源 $\mathbf{x}_t$ 的偏导 +% +\begin{align*} +\frac{\partial \ln \mathcal{L}}{\partial \mathbf{x}_t} &= - \frac{\partial }{\partial \mathbf{x}_t} \left[\frac{1}{\sigma} \left( \mathbf{y}_t-\mathbf{A}\mathbf{x}_t \right)^\Herm \left( \mathbf{y}_t-\mathbf{A}\mathbf{x}_t \right) \right] \nonumber += - \frac{\partial }{\partial \mathbf{x}_t} \left[\frac{1}{\sigma} \left( \mathbf{y}_t^\Herm - \mathbf{x}_t^\Herm \mathbf{A}^\Herm \right)\left( \mathbf{y}_t-\mathbf{A}\mathbf{x}_t \right) \right] \nonumber +\\ +&= \frac{1}{\sigma} \left( \mathbf{y}_t^\Herm \mathbf{A} - \mathbf{x}_t^\Herm \mathbf{A}^\Herm \mathbf{A} \right) += \frac{1}{\sigma} \left( \mathbf{y}_t-\mathbf{A}\mathbf{x}_t \right)^\Herm \mathbf{A} = \frac{1}{\sigma} \mathbf{n}_t^\Herm \mathbf{A}. +\end{align*} +% +对其进行维度修正（取转置）\footnote{% +由于按 Jacobian 定义对列向量 $\mathbf{x}_t$ 求导得到的是 $1\times K$ 的行向量，而后续推导采用列向量形式的梯度表示，因此对结果取转置以统一维度。该转置仅源于导数布局的不同约定，不影响最终结果。}，得到列向量 +% +\begin{equation} +\label{eq:first_order_derivative_source} +\frac{\partial \ln \mathcal{L}}{\partial \mathbf{x}_t} = \frac{1}{\sigma} \left(\mathbf{n}_t^\Herm \mathbf{A} \right)^\Tran = \frac{1}{\sigma} \mathbf{A}^\Tran\mathbf{n}_t^*. +\end{equation} +% +然后类似求取信源的共轭转置 $\mathbf{x}_t^\Herm $ 的偏导，得 +% +\begin{equation} +\label{eq:first_order_derivative_source_hermit} +\frac{\partial \ln \mathcal{L}}{\partial \mathbf{x}_t^\Herm } = \frac{1}{\sigma} \mathbf{A}^\Herm \mathbf{n}_t. +\end{equation} +% +然后将式 $\eqref{eq:first_order_derivative_source}$ 和 $\eqref{eq:first_order_derivative_source_hermit}$ 代入 Writinger 微分 $\eqref{eq:wirtinger_derivative}$，可得信源实部 $\mathbf{x}_t^{(r)}$ 和虚部 $\mathbf{x}_t^{(i)}$ 的一阶偏导 +% +\begin{equation} +\frac{\partial \ln \mathcal{L}}{\partial \mathbf{x}_t^{(r)}} += \frac{1}{\sigma} \left( \mathbf{A}^\Tran\mathbf{n}_t^* + \mathbf{A}^\Herm \mathbf{n}_t \right) += \frac{2}{\sigma} \Re\{\mathbf{A}^\Herm \mathbf{n}_t\}, +\end{equation}% +% +\begin{equation} +\frac{\partial \ln \mathcal{L}}{\partial \mathbf{x}_t^{(i)}} += \frac{1}{\sigma} j\left( \mathbf{A}^\Tran\mathbf{n}_t^* - \mathbf{A}^\Herm \mathbf{n}_t \right) += \frac{2}{\sigma} \Im\{\mathbf{A}^\Herm \mathbf{n}_t\}. +\end{equation} + +\subsubsection{对角度的一阶偏导} + +\begin{align} +\frac{\partial\ln \mathcal{L}}{\partial \theta_k} &= -\frac{1}{\sigma} \sum_{t=1}^T \left[ - \mathbf{x}_t^\Herm \frac{\mathrm{d}\mathbf{A}^\Herm }{\mathrm{d}\theta_k } (\mathbf{y}_t - \mathbf{A} \mathbf{x}_t) - (\mathbf{y}_t - \mathbf{A} \mathbf{x}_t)^\Herm \frac{\mathrm{d}\mathbf{A}}{\mathrm{d}\theta_k } \mathbf{x}_t \right] \nonumber +\\ +&= \frac{1}{\sigma} \sum_{t=1}^T \left[ \mathbf{x}_t^\Herm \frac{\mathrm{d}\mathbf{A}^\Herm }{\mathrm{d}\theta_k } \mathbf{n}_t + \mathbf{n}_t^\Herm \frac{\mathrm{d}\mathbf{A}}{\mathrm{d}\theta_k } \mathbf{x}_t \right] \nonumber +\\ +&= \frac{2}{\sigma} \sum_{t=1}^T \Re\left[ \mathbf{x}^\Herm _t \frac{\mathrm{d}\mathbf{A}^\Herm }{\mathrm{d}\theta_k } \mathbf{n}_t \right]. +\end{align} +% +令 $\mathbf{d}(\theta_k) \triangleq \dfrac{\diff}{\diff\theta_k} \mathbf{a} (\theta_k) \in \mathbb{C}^{M\times 1}$，$\mathbf{d}^\Herm(\theta_k) \triangleq \dfrac{\diff}{\diff\theta_k} \mathbf{a}^\Herm (\theta_k)$，$\mathbf{D} = [\mathbf{d}(\theta_1), \ldots, \mathbf{d}(\theta_K)] \in \mathbb{C}^{M\times K}$，$\mathbf{X}_t \triangleq \diag ( x_1(t), \allowbreak \ldots, \allowbreak x_K (t) ) \in \mathbb{C}^{K\times K}$，则~\footnote{对于多维 DoA 参数，需要将 $\mathbf{D}$ 扩展为关于所有角度分量的 Jacobian。此时仍可通过相同的 Schur 补思想得到等效 Fisher 信息矩阵，但最终矩阵维度和 Hadamard 表达需要按角度分量重新展开。} +% +\begin{equation} +\frac{\partial\ln \mathcal{L}}{\partial \bm{\theta}} += \begin{bmatrix} + \dfrac{\partial}{\partial \theta_1} + \\ + \vdots + \\ + \dfrac{\partial}{\partial \theta_K} +\end{bmatrix} \ln \mathcal{L} += \frac{2}{\sigma} \sum_{t=1}^T \Re\left[ \mathbf{X}^\Herm _t \mathbf{D}^\Herm \mathbf{n}_t \right]. +\end{equation} + + +\subsection{二阶偏导及其均值} + +\subsubsection{噪声方差的二阶偏导及均值} + +\begin{mdquote} +\vspace{-1em} +\begin{equation*} + \mathbb{E} \left[ \mathbf{n}^\Herm_t \mathbf{n}_t \right] = M \sigma_n^2 = M \sigma. +\end{equation*} +% +\begin{equation*} + \mathbb{E} \left[ \mathbf{n}^\Herm_t \mathbf{n}_t \mathbf{n}^\Herm_s \mathbf{n}_s \right] = \begin{cases} + M^2 \sigma^2, & \text{for}\ t\ne s \\ + M(M+1)\sigma^2, & \text{for}\ t = s + \end{cases}. +\end{equation*} +\end{mdquote} +% +\begin{align} +\mathbb{E}\left\{\left[\frac{\partial \ln \mathcal{L}}{\partial \sigma}\right]^2\right\} +=& \mathbb{E}\left[\left(-\frac{MT}{\sigma}+\frac{1}{\sigma^2} \sum_{t=1}^T \mathbf{n}^\Herm _t\mathbf{n}_t\right)^2 \right] +\nonumber \\ +=& \frac{M^2T^2}{\sigma^2} - \frac{2MT}{\sigma^3} \sum_{t=1}^T \mathbb{E}\left[\mathbf{n}^\Herm _t\mathbf{n}_t\right] + \frac{1}{\sigma^4} \sum_{t=1}^T \sum_{s=1}^T \mathbb{E}\left[ \mathbf{n}^\Herm _t\mathbf{n}_t\mathbf{n}^\Herm _s\mathbf{n}_s \right] +\nonumber \\ +=& \frac{M^2T^2}{\sigma^2} - \frac{2M^2T^2}{\sigma^2} + \frac{1}{\sigma^4}[(T^2-T)M^2\sigma^2 + TM(M+1)\sigma^2] +\nonumber \\ +=& \frac{MT}{\sigma^2}. +\end{align} + +\subsubsection{噪声方差和其他参量的二阶混合偏导及均值} + +由于不同时刻（$t\ne s$）的噪声不相关，相同时刻（$t=s$）高斯变量的三阶矩为零，因此有 +% +\begin{equation} +\mathbb{E}\left[ \mathbf{n}_t^\Herm \mathbf{n}_t \mathbf{n}_s^\Tran \right] = \mathbf{0}, +\end{equation} +% +因此，噪声方差和其他参量的混合二阶偏导均为零。 + +\subsubsection{信源实虚部的二阶偏导、二阶混合偏导及均值} + +\begin{mdquote} +\begin{enumerate} + \item 关于噪声的结论：噪声是时间平稳的圆对称复高斯随机过程，也即 + % + \begin{equation*} + \mathbb{E}\left[\mathbf{n}_t\mathbf{n}_s^\Herm \right] + = \sigma \mathbf I_M \delta_{ts}, + \end{equation*} + % + \begin{equation*} + \mathbb{E}[\mathbf{n}_t\mathbf{n}_s^\Tran] = 0,\quad \forall t,s. + \end{equation*} + + \item 关于实虚部乘法运算的结论（将复数按实虚部展开后可得） + \begin{align*} + &\Re\{\mathbf{x}\} \Re\{\mathbf{y}^\Tran\} = \frac{1}{2}[\Re\{\mathbf{x} \mathbf{y}^\Tran\} + \Re\{\mathbf{x} \mathbf{y}^\Herm \}], + \\ + &\Im\{\mathbf{x}\} \Im\{\mathbf{y}^\Tran\} = -\frac{1}{2}[\Re\{\mathbf{x} \mathbf{y}^\Tran\} - \Re\{\mathbf{x} \mathbf{y}^\Herm \}], + \\ + &\Re\{\mathbf{x}\} \Im\{\mathbf{y}^\Tran\} = \frac{1}{2}[\Im\{\mathbf{x} \mathbf{y}^\Tran\} - \Im\{\mathbf{x} \mathbf{y}^\Herm \}]. + \end{align*} +\end{enumerate} +\end{mdquote} +% +\begin{subequations} + \begin{align} + %信源实部的二阶偏导 + \mathbb{E}\left\{\left[\frac{\partial \ln \mathcal{L}}{\partial \mathbf{x}_t^{(r)}}\right]\left[\frac{\partial \ln \mathcal{L}}{\partial \mathbf{x}_s^{(r)}}\right]^\Tran\right\} + &= \frac{1}{\sigma^2} \mathbb{E} \left[ \left( \mathbf{A}^\Tran \mathbf{n}_t^* + \mathbf{A}^\Herm \mathbf{n}_t \right)\left( \mathbf{A}^\Tran\mathbf{n}_s^* + \mathbf{A}^\Herm \mathbf{n}_s \right)^\Tran \right] \nonumber + \\ + =& \frac{1}{\sigma^2} \mathbb{E}\left[\left( \mathbf{A}^\Tran \mathbf{n}_t^* \mathbf{n}_s^\Herm \mathbf{A} + \mathbf{A}^\Tran \mathbf{n}_t^* \mathbf{n}_s^\Tran \mathbf{A}^\ast + \mathbf{A}^\Herm \mathbf{n}_t \mathbf{n}_s^\Herm \mathbf{A} + \mathbf{A}^\Herm \mathbf{n}_t \mathbf{n}_s^\Tran \mathbf{A}^\ast \right)\right] \nonumber + \\ + =& \frac{1}{\sigma^2} \mathbb{E} \left[ \left( \mathbf{A}^\Tran \mathbf{n}_t^* \mathbf{n}_s^\Tran \mathbf{A}^\ast + \mathbf{A}^\Herm \mathbf{n}_t \mathbf{n}_s^\Herm \mathbf{A} \right) \right] + = \frac{2}{\sigma^2}\Re\left\{\mathbf{A}^\Herm \mathbb{E}\left[ \mathbf{n}_t \mathbf{n}_s^\Herm \right] \mathbf{A}\right\} \nonumber + \\ + =& \frac{2}{\sigma} \Re\left[\mathbf{A}^\Herm \mathbf{A}\right]\delta(t-s). + \end{align} + %信源实虚部的混合二阶偏导 + \begin{equation} + \mathbb{E}\left\{\left[\frac{\partial \ln \mathcal{L}}{\partial \mathbf{x}_t^{(r)}}\right]\left[\frac{\partial \ln \mathcal{L}}{\partial \mathbf{x}_s^{(i)}}\right]^\Tran\right\} + = \frac{4}{\sigma^2} \left(-\frac{1}{2}\right)\Im\left\{\mathbb{E}\left[\mathbf{A}^\Herm \mathbf{n}_t \mathbf{n}_s^\Herm \mathbf{A} \right]\right\} + = -\frac{2}{\sigma} \Im\left[\mathbf{A}^\Herm \mathbf{A}\right]\delta(t-s). + \end{equation} + %信源虚部的二阶偏导 + \begin{equation} + \mathbb{E}\left\{\left[\frac{\partial \ln \mathcal{L}}{\partial \mathbf{x}_t^{(i)}}\right]\left[\frac{\partial \ln \mathcal{L}}{\partial \mathbf{x}_s^{(i)}}\right]^\Tran\right\} + = \frac{4}{\sigma^2} \frac{1}{2}\Re\left\{\mathbb{E}\left[\mathbf{A}^\Herm \mathbf{n}_t \mathbf{n}_s^\Herm \mathbf{A} \right]\right\} + = \frac{2}{\sigma} \Re\left[\mathbf{A}^\Herm \mathbf{A}\right]\delta(t-s). + \end{equation} +\end{subequations} + + +\subsubsection{信源实虚部与角度的混合二阶偏导及均值} + +\begin{subequations} + \begin{equation} + \mathbb{E}\left\{\left[\frac{\partial \ln \mathcal{L}}{\partial \mathbf{x}_t^{(r)}}\right]\left[\frac{\partial\ln \mathcal{L}}{\partial \bm{\theta}}\right]^\Tran\right\} + = \frac{4}{\sigma^2} \sum_{t=1}^T\frac{1}{2}\Re\{\mathbf{A}^\Herm \mathbb{E}\left[ \mathbf{n}_t\mathbf{n}_t^\Herm \right] \mathbf{D} \mathbf{X}_t\} + = \frac{2}{\sigma}\Re\left[ \mathbf{A}^\Herm \mathbf{D} \mathbf{X}_t \right]. + \end{equation} + % + \begin{equation} + \mathbb{E}\left\{\left[\frac{\partial \ln \mathcal{L}}{\partial \mathbf{x}_t^{(i)}}\right]\left[\frac{\partial\ln \mathcal{L}}{\partial \bm{\theta}}\right]^\Tran\right\} + = \frac{4}{\sigma^2} \sum_{t=1}^T\left(-\frac{1}{2}\right)\Im\{\mathbf{A}^\Herm \mathbb{E}\left[ \mathbf{n}_t\mathbf{n}_s^\Herm \right] \mathbf{D} \mathbf{X}_t\} + = \frac{2}{\sigma}\Im\left[ \mathbf{A}^\Herm \mathbf{D} \mathbf{X}_t \right]. + \end{equation} +\end{subequations} + + +\subsubsection{角度的二阶偏导及均值} + +\begin{equation} +\mathbb{E}\left\{\left[\frac{\partial\ln \mathcal{L}}{\partial \bm{\theta}}\right]\left[\frac{\partial\ln \mathcal{L}}{\partial \bm{\theta}}\right]^\Tran\right\} += \frac{4}{\sigma^2} \sum_{t=1}^T \sum_{s=1}^T \frac{1}{2} \Re\{\mathbf{X}_t^\Herm \mathbf{D}^\Herm \mathbb{E}\left[ \mathbf{n}_t\mathbf{n}_s^\Herm \right] \mathbf{D} \mathbf{X}_s\} += \frac{2}{\sigma} \sum_{t=1}^T \Re\left[ \mathbf{X}_t^\Herm \mathbf{D}^\Herm \mathbf{D} \mathbf{X}_t \right]. +\end{equation} + +\subsection{Fisher 信息矩阵/CRB 协方差矩阵构造} + +\begin{mdquote} + \begin{enumerate} + \item 复矩阵求逆法则：令 $\mathbf{H}$ 是非奇异矩阵，且有 $\mathbf{G}\triangleq\mathbf{H}^{-1}$，则 + \begin{equation} + \left[ \begin{matrix} + \Re\{\mathbf{H}\} & -\Im\{\mathbf{H}\} \\ + \Im\{\mathbf{H}\} & \Re\{\mathbf{H}\} \\ + \end{matrix}\right] ^{-1} = \left[ \begin{matrix} + \Re\{\mathbf{G}\} & -\Im\{\mathbf{G}\} \\ + \Im\{\mathbf{G}\} & \Re\{\mathbf{G}\} \\ + \end{matrix}\right]. + \end{equation} + + \item Schur 补在 \gls{fim} 中的应用：当 \gls{fim} 被分块为 + % + \begin{equation} + \mathcal{I} = \left[ \begin{matrix} + \mathcal{I}_{\theta\theta} & \mathcal{I}_{\theta\psi} \\ + \mathcal{I}_{\psi\theta} & \mathcal{I}_{\psi\psi} + \end{matrix} \right], + \end{equation} + % + 其中 $\bm{\theta}$ 为目标参数，$\bm{\psi}$ 为扰动（nuisance）参数，可以通过 Schur 补获得 CRB + % + \begin{equation} + \label{eq:schur_complement} + \mathrm{CRB}^{-1}(\bm{\theta}) + = \mathcal{I}_{\mathrm e} + = \mathcal{I}_{\theta\theta} - \mathcal{I}_{\theta\psi} \mathcal{I}_{\psi\psi}^{-1} \mathcal{I}_{\psi\theta}. + \end{equation} + \end{enumerate} +\end{mdquote} + +\Gls{fim} 可以写为 +% +\begin{align} + &\mathcal{I} \left( \sigma, \mathbf{x}_{1}^{{(r)}}, \mathbf{x}_{1}^{{(i)}}, \ldots, \mathbf{x}_{T}^{{(r)}}, \mathbf{x}_{T}^{{(i)}},\bm{\theta } \right) \nonumber + \\ + =& \left[ \begin{array}{c:c:c} + \text{CRB}^{-1}(\sigma) & 0 & 0 \\ \hdashline + 0 & \begin{array}{c:c:c} + \begin{matrix} + \Re\{\mathbf{H}\}\quad & -\Im\{\mathbf{H}\}\quad \\ + \Im\{\mathbf{H}\}\quad & \Re\{\mathbf{H}\}\quad \\ + \end{matrix} & & 0 \\ + \hdashline + & \ddots & \\ + \hdashline + 0 & & \begin{matrix} + \Re\{\mathbf{H}\}\quad & -\Im\{\mathbf{H}\}\quad\,\, \\ + \Im\{\mathbf{H}\}\quad & \Re\{\mathbf{H}\}\quad \\ + \end{matrix} \\ + \end{array} & \begin{matrix} + \Re\{\bm{\Delta}_1\} \\ \Im\{\bm{\Delta}_1\} \\ \hdashline \vdots \\ \hdashline \Re\{\bm{\Delta}_T\} \\ \Im\{\bm{\Delta}_T\} + \end{matrix} \\ + \hdashline + 0 & \begin{array}{cc:c:cc} + \Re\{\bm{\Delta}_1\}^\Tran & + \Im\{\bm{\Delta}_1\}^\Tran & + \cdots & + \Re\{\bm{\Delta}_T\}^\Tran & + \Im\{\bm{\Delta}_T\}^\Tran + \end{array} &\bm{\Gamma} + \end{array} + \right], +\end{align} +% +其中 $\text{CRB}(\sigma) = \dfrac{\sigma^2}{MT}$，$\mathbf{H} = \dfrac{2}{\sigma}\mathbf{A}^\Herm \mathbf{A}$，$\bm{\Delta}_t = \dfrac{2}{\sigma} \mathbf{A}^\Herm \mathbf{D} \mathbf{X}_t$，$\bm{\Gamma} = \dfrac{2}{\sigma}\displaystyle\sum_{t=1}^T \Re\left[ \mathbf{X}_t^\Herm \mathbf{D}^\Herm \mathbf{D} \mathbf{X}_t \right]$。注意到，由于噪声方差和其他各参量的联合二阶偏导数为零，因此噪声方差这个参量不会通过 Schur 补影响 $\bm{\theta}$，也即可以只考虑除噪声方差之外的参量构成的 \gls{fim}。 + +我们的目标是通过 Schur 补消去确定信号快拍 $\{\mathbf x_t\}_{t=1}^T$，得到关于 $\bm\theta$ 的等效 Fisher 信息矩阵；随后对该等效 Fisher 信息矩阵求逆，得到 $\bm\theta$ 的 CRB 协方差矩阵。首先定义几个辅助变量，令 +% +\begin{equation*} + \mathbf{F}_{\psi\psi}=\left[ \begin{array}{c:c:c} + \begin{matrix} + \Re\{\mathbf{H}\} & -\Im\{\mathbf{H}\} \\ + \Im\{\mathbf{H}\} & \Re\{\mathbf{H}\} \\ + \end{matrix} + & & 0 \\ + \hdashline + & \ddots & \\ + \hdashline + 0 & & \begin{matrix} + \Re\{\mathbf{H}\} & -\Im\{\mathbf{H}\} \\ + \Im\{\mathbf{H}\} & \Re\{\mathbf{H}\} \\ + \end{matrix} \\ + \end{array} \right],\quad + % + \mathbf{F}_{\psi\theta} = \left[ \begin{array}{c} + \Re\{\bm{\Delta}_1\} \\ + \Im\{\bm{\Delta}_1\} \\ + \hdashline + \vdots \\ + \hdashline + \Re\{\bm{\Delta}_T\} \\ + \Im\{\bm{\Delta}_T\} + \end{array}\right], +\end{equation*} +% +\begin{equation*} + \mathbf{F}_{\theta\psi} = \left[ + \begin{array}{c:c:c} + \begin{matrix} + \Re\{\bm{\Delta}_1\}^\Tran & \Im\{\bm{\Delta}_1\}^\Tran + \end{matrix} + & + \cdots + & + \begin{matrix} + \Re\{\bm{\Delta}_T\}^\Tran & \Im\{\bm{\Delta}_T\}^\Tran + \end{matrix} + \end{array} \right]. +\end{equation*} +% +因此，我们有 +% +\begin{align} + \mathrm{CRB}^{-1}(\bm{\theta}) = \bm{\Gamma}-\mathbf{F}_{\theta\psi}\mathbf{F}_{\psi\psi}^{-1}\mathbf{F}_{\psi\theta} +\end{align} +% +观察矩阵结构，通过分块矩阵乘法，令 $\mathbf{G} = \mathbf{H}^{-1}$，可以有~\footnote{将右侧按照复数乘法展开即可得到。} +% +\begin{equation*} + \left[ \begin{matrix} + \Re\{\mathbf{G}\} & -\Im\{\mathbf{G}\} \\ + \Im\{\mathbf{G}\} & \Re\{\mathbf{G}\} \\ + \end{matrix} \right] \left[ \begin{matrix} + \Re\{\bm{\Delta}_t\} \\ + \Im\{\bm{\Delta}_t\} \\ + \end{matrix} \right] = \left[ \begin{matrix} + \Re\{\mathbf{G}\}\Re\{\bm{\Delta}_t\} - \Im\{\mathbf{G}\}\Im\{\bm{\Delta}_t\} \\ + \Im\{\mathbf{G}\}\Re\{\bm{\Delta}_t\} + \Re\{\mathbf{G}\}\Im\{\bm{\Delta}_t\} \\ + \end{matrix} \right] = \left[ \begin{matrix} + \Re\{\mathbf{G}\bm{\Delta}_t\} \\ + \Im\{\mathbf{G}\bm{\Delta}_t\} \\ + \end{matrix} \right], +\end{equation*} +% +以及~\footnote{将右侧展开，$\Re\{\bm{\Delta}_t^\Herm \mathbf{G}\bm{\Delta}_t\} = (\Re\{\bm{\Delta}_t\} - j \Im\{\bm{\Delta}_t\})^\Tran (\Re\{\mathbf{G}\bm{\Delta}_t\} + j \Im\{\mathbf{G}\bm{\Delta}_t\}) $，即可得到。} +% +\begin{equation*} +\left[ \begin{matrix} + \Re\{\bm{\Delta}_t\}^\Tran & \Im\{\bm{\Delta}_t\}^\Tran \\ + \end{matrix} \right] \left[ \begin{matrix} + \Re\{\mathbf{G}\bm{\Delta}_t\} \\ + \Im\{\mathbf{G}\bm{\Delta}_t\} \\ + \end{matrix} \right] = \Re\{\bm{\Delta}_t^\Herm \mathbf{G}\bm{\Delta}_t\}, +\end{equation*} +% +因此，角度的 \gls{crb} 可以表示为 +% +\begin{align} + \mathrm{CRB}^{-1}(\bm{\theta}) &= \bm{\Gamma} - \sum_{t=1}^T \Re\{\bm{\Delta}_t^\Herm \mathbf{G}\bm{\Delta}_t\} \nonumber + = \dfrac{2}{\sigma} \sum_{t=1}^T \Re\left\{ \mathbf{X}_t^\Herm \mathbf{D}^\Herm \mathbf{D} \mathbf{X}_t - \mathbf{X}_t^\Herm \mathbf{D}^\Herm \mathbf{A}(\mathbf{A}^\Herm \mathbf{A})^{-1}\mathbf{A}^\Herm \mathbf{D} \mathbf{X}_t \right\} \nonumber + \\ + &= \dfrac{2}{\sigma} \sum_{t=1}^T \Re\left\{ \mathbf{X}_t^\Herm \mathbf{D}^\Herm \left[ \mathbf{I} - \mathbf{A}(\mathbf{A}^\Herm \mathbf{A})^{-1}\mathbf{A}^\Herm \right] \mathbf{D} \mathbf{X}_t \right\}. +\end{align} + +参考 \cite{stoicaMusicMaximumLikelihood1989} 进一步定义阵列导向矩阵的列空间的投影矩阵 $\mathcal{P}_{\mathbf{A}} = \mathbf{A}(\mathbf{A}^\Herm \mathbf{A})^{-1}\mathbf{A}^\Herm$，及其正交补空间的投影矩阵 $\mathcal{P}_{\mathbf{A}}^\perp = \mathbf{I} -\mathcal{P}_{\mathbf{A}}$，投影矩阵下的等效信息矩阵 $\mathcal{H} = \mathbf{D}^\Herm \mathcal{P}_{\mathbf{A}}^\perp \mathbf{D}$，确定信号的经验协方差 $\mathbf R_{x,\mathrm{det}} = \dfrac{1}{T} \displaystyle\sum_{t=1}^T \mathbf{x}_t \mathbf{x}_t^\Herm$，则关于角度的等效 Fisher 信息矩阵可以表示为 +% +\begin{align} + \mathcal I_{\mathrm e}(\bm\theta) + = \mathrm{CRB}^{-1}(\bm{\theta}) + = \frac{2T}{\sigma} \Re\left\{ \mathcal{H} \odot \mathbf R_{x,\mathrm{det}}^\Tran \right\}. +\end{align} +% +因此，条件（conditional）\gls{crb} 或确定（deterministic）\gls{crb} 为 +% +\begin{align} + \mathcal{B}_c + := \mathrm{CRB}(\bm{\theta}) + = \frac{\sigma}{2T} + \left[\Re\left\{ \mathcal{H} \odot \mathbf R_{x,\mathrm{det}}^\Tran \right\}\right]^{-1}, +\end{align} +% +其中 $\odot$ 表示 Hadamard 积。 + +该结果是单阵列确定信号 DoA 的条件 CRB 基线。后续单阵列 DoA--Doppler 和多星 DoA--Doppler 的 CRB 推导会沿用“先构造完整 Fisher 信息，再对干扰参数做 Schur 补”的处理方式。 diff --git a/parts/A4-stoDoaCrb.tex b/parts/A4-stoDoaCrb.tex new file mode 100644 index 0000000..5a33a54 --- /dev/null +++ b/parts/A4-stoDoaCrb.tex @@ -0,0 +1,307 @@ +\section{随机信号 DoA 估计的 CRB 推导} +\label{sec:sto_doa_crb} + +本附录对应第~\ref{sec:sto_doa_est} 节的单阵列随机信号 DoA 模型。由 \ref{subsubsec:general_gauss_crb} 可知，零均值圆对称复高斯随机过程的 \gls{crb} 可以表示为 +% +\begin{equation} + [\mathcal I(\boldsymbol\kappa)]_{ij} + = T\Re\left\{\tr \left[ + \mathbf{R}^{-1} (\boldsymbol\kappa) \frac{\partial \mathbf{R} (\boldsymbol\kappa)}{\partial \kappa_i} \mathbf{R}^{-1} (\boldsymbol\kappa) \frac{\partial \mathbf{R} (\boldsymbol\kappa)}{\partial \kappa_j} + \right]\right\}. +\end{equation} +% +因此可以通过类似\ref{sec:det_doa_crb} 的方式，按照第 \ref{sec:sto_doa_est} 节的建模方式，求出接收信号协方差矩阵 $\mathbf{R} = \mathbf{A} \mathbf{R}_x \mathbf{A}^\Herm + \sigma \mathbf{I}_M$ 对各个参量的一阶导数，构建 \gls{fim}。 +随机信号 \gls{doa} 估计问题的参数向量可以写为 +% +\begin{equation*} + \boldsymbol\kappa = \begin{bmatrix} + \boldsymbol\theta \\ \boldsymbol\psi + \end{bmatrix}, + \quad + \boldsymbol\psi = \begin{bmatrix} + \operatorname{vech}_{\mathbb R}(\mathbf{R}_x) \\ \sigma + \end{bmatrix}, +\end{equation*} +% +其中 $\boldsymbol\theta$ 为待估 \gls{doa} 参数向量。严格地说，$\mathbf R_x=\mathbf R_x^\Herm$ 应采用非冗余实参数化；为避免引入额外记号，下文仅用 $\boldsymbol\psi$ 表示信源协方差与噪声方差组成的干扰参数集合，$\sigma \triangleq \sigma_n^2$ 为噪声方差。 +由 $\eqref{eq:schur_complement}$ 的 Schur 补可得 \gls{doa} 的等效 \gls{fim} 为 +% +\begin{equation*} + \mathcal{I}_{\mathrm e} + = \mathcal{I}_{\theta\theta} - \mathcal{I}_{\theta\psi} \mathcal{I}_{\psi\psi}^{-1} \mathcal{I}_{\psi\theta}. +\end{equation*} +% +但是在随机信号模型中，直接计算 Fisher 块需要显式处理 $\mathbf{R}^{-1}$ 与各导数的耦合项，代数推导代价很高，因此参考 \cite{stoicaPerformanceStudyConditional1990}，通过协方差扰动子空间的结构，将干扰参数 $\boldsymbol\psi$ 所能引起的协方差扰动张成一个线性子空间，并将 $\mathbf{R}_i \triangleq \dfrac{\partial \mathbf{R} (\boldsymbol\kappa)}{\partial \kappa_i}$ 中落在该子空间内的分量投影消去，从而得到与 Schur 补等价的投影消元形式。 + +\subsection{干扰参数扰动子空间} + +在随机信号模型下，接收信号协方差矩阵为 +% +\begin{equation*} + \mathbf{R} = \mathbf{A} \mathbf{R}_x \mathbf{A}^\Herm + \sigma \mathbf{I}_M. +\end{equation*} +% +目标参数为 \gls{doa} 参数 $\bm{\theta}$，但协方差矩阵不仅依赖 $\bm{\theta}$，还依赖信源协方差 $\mathbf{R}_x$ 和噪声功率 $\sigma$。 +因此，当接收信号协方差发生微小变化时，这种变化可能来自 \gls{doa} 的变化、信源功率或源相关性的变化、噪声功率的变化。 +\emph{为了区分这些不同来源的变化，有必要分析对于不同参数的微小扰动，会使协方差矩阵沿哪些方向发生变化。} + +设参数发生一个小扰动，即 +% +\begin{equation*} + \mathbf{R}_x \to \mathbf{R}_x + \Delta \mathbf{R}_x, + \quad + \sigma \to \sigma + \Delta \sigma, +\end{equation*} +% +则协方差的变化为 +% +\begin{equation*} + \Delta \mathbf{R} = \mathbf{A} (\Delta\mathbf{R}_x) \mathbf{A}^\Herm + (\Delta\sigma) \mathbf{I}_M. +\end{equation*} +% +可以看出，协方差的扰动具有固定的结构，包括第一类扰动 $\mathbf{A}(\ \cdot \ )\mathbf{A}^\Herm$ 和第二类扰动 $\mathbf{I}_M$。 +也即干扰参数 $(\mathbf{R}_x, \sigma)$ 只能使协方差沿某些\emph{特定方向}变化，而不可能产生任意形式的矩阵变化。 + +将所有由干扰参数产生的协方差扰动集合定义为 +% +\begin{equation} + \mathcal{S}_{\boldsymbol\psi} + = \left\{ \mathbf{A} \mathbf{X} \mathbf{A}^\Herm + \mu \mathbf{I}_M \mid \mathbf{X} = \mathbf{X}^\Herm, \mu \in \mathbb{R} \right\} + \subseteq \mathrm{span}_{\mathbb R} \left\{ \mathbf{a}_i \mathbf{a}_j^\Herm,\; j\le i,\; \mathbf I_M \right\}, +\end{equation} +% +这个集合包含了所有可能由信源协方差和噪声功率变化引起的接收信号协方差变化。由于 $\Delta\mathbf R_x$ 必须保持 Hermitian，上式中的张成应理解为 Hermitian 矩阵集合上的实线性张成，本文称该集合为\emph{干扰参数扰动子空间}。 +它描述了协方差矩阵由于干扰参数变化所能移动的所有方向。 + +Schur 补 $\mathcal{I}_{\mathrm e} += \mathcal{I}_{\theta\theta} - \mathcal{I}_{\theta\psi} \mathcal{I}_{\psi\psi}^{-1} \mathcal{I}_{\psi\theta}$ 的本质是从 $\mathcal{I}_{\theta\theta}$ 中剔除可以被干扰参数“解释”的变化部分，从而获得关于 \gls{doa} 的独立信息。 +因此 Schur 补在几何上等价于将 $\mathbf{R}_{\theta_i}$ 在扰动子空间 $\mathcal{S}_{\boldsymbol\psi}$ 上的分量消除，只保留与该子空间正交的部分。 + +\subsection{Fisher 块构建} + +考虑等式 +% +\begin{equation*} + \tr (\mathbf{X}^\Herm \mathbf{Y}) = \vect (\mathbf{X})^\Herm \vect (\mathbf{Y}), +\end{equation*} +% +\begin{equation*} + \tr (\mathbf{R}^{-1} \mathbf{X} \mathbf{R}^{-1} \mathbf{Y}) + = \vect (\mathbf{X})^\Herm (\mathbf{R}^{-\Tran} \otimes \mathbf{R}^{-1}) \vect (\mathbf{Y}) + = \vect (\mathbf{X})^\Herm \mathbf{W} \vect (\mathbf{Y}), +\end{equation*} +% +其中 $\otimes$ 表示 Kronecker 积，$\mathbf{W} \triangleq \mathbf{R}^{-\Tran} \otimes \mathbf{R}^{-1}$。由于这里所有参数均为实参数，且各协方差导数为 Hermitian，迹项理论上为实数；下文显式保留 $\Re\{\cdot\}$ 以避免复数数值舍入误差。则考虑 $T$ 个快拍时，Fisher 信息 +% +\begin{align} + [\mathcal{I}]_{ij} + &= T\Re\left\{\tr (\mathbf{R}^{-1} \mathbf{R}_i \mathbf{R}^{-1} \mathbf{R}_j)\right\} + \nonumber \\ + &= T\Re\left\{\vect (\mathbf{R}_i)^\Herm (\mathbf{R}^{-\Tran} \otimes \mathbf{R}^{-1}) \vect (\mathbf{R}_j)\right\} + = T\Re\left\{\vect (\mathbf{R}_i)^\Herm \mathbf{W} \vect (\mathbf{R}_j)\right\}. +\end{align} + +定义 $\mathbf{G}_{\bm{\theta}} \triangleq [\vect (\mathbf{R}_{\theta_1}), \ldots, \vect (\mathbf{R}_{\theta_K})]$，$\mathbf{G}_{\boldsymbol\psi} \triangleq [\vect (\mathbf{R}_{\psi_1}), \ldots, \vect (\mathbf{R}_{\psi_q})]$，则 \gls{fim} 的各项可以写为 +% +\begin{subequations} +\begin{equation} + [\mathcal{I}_{\theta\theta}]_{ij} + = T\Re\left\{\tr (\mathbf{R}^{-1} \mathbf{R}_{\theta_i} \mathbf{R}^{-1} \mathbf{R}_{\theta_j})\right\} + = T\Re\left\{\vect (\mathbf{R}_{\theta_i})^\Herm \mathbf{W} \vect (\mathbf{R}_{\theta_j})\right\}, +\end{equation} +% +\begin{equation} + [\mathcal{I}_{\theta\psi}]_{ij} + = T\Re\left\{\tr (\mathbf{R}^{-1} \mathbf{R}_{\theta_i} \mathbf{R}^{-1} \mathbf{R}_{\psi_j})\right\} + = T\Re\left\{\vect (\mathbf{R}_{\theta_i})^\Herm \mathbf{W} \vect (\mathbf{R}_{\psi_j})\right\}, +\end{equation} +% +\begin{equation} + [\mathcal{I}_{\psi\psi}]_{ij} + = T\Re\left\{\tr (\mathbf{R}^{-1} \mathbf{R}_{\psi_i} \mathbf{R}^{-1} \mathbf{R}_{\psi_j})\right\} + = T\Re\left\{\vect (\mathbf{R}_{\psi_i})^\Herm \mathbf{W} \vect (\mathbf{R}_{\psi_j})\right\}, +\end{equation} +\end{subequations} +% +写成矩阵形式，有 +% +\begin{subequations} +\begin{equation} + \mathcal{I}_{\theta\theta} = T\Re\left\{\mathbf{G}_{\bm{\theta}}^\Herm \mathbf{W} \mathbf{G}_{\bm{\theta}}\right\}, +\end{equation} +% +\begin{equation} + \mathcal{I}_{\theta\psi} = T\Re\left\{\mathbf{G}_{\bm{\theta}}^\Herm \mathbf{W} \mathbf{G}_{\boldsymbol\psi}\right\}, +\end{equation} +% +\begin{equation} + \mathcal{I}_{\psi\psi} = T\Re\left\{\mathbf{G}_{\boldsymbol\psi}^\Herm \mathbf{W} \mathbf{G}_{\boldsymbol\psi}\right\}, +\end{equation} +\end{subequations} +% +代入 Schur 补，有 +% +\begin{equation} + \mathcal{I}_{\mathrm e} + = \mathcal{I}_{\theta\theta} - \mathcal{I}_{\theta\psi} \mathcal{I}_{\psi\psi}^{-1} \mathcal{I}_{\psi\theta} + = T \mathbf{G}_{\bm{\theta}}^\Herm \mathbf{W} \left( \mathbf{I} - \mathcal{P}^{(\mathbf{W})}_{\boldsymbol\psi} \right) \mathbf{G}_{\bm{\theta}}, +\end{equation} +% +其中 $\mathcal{P}^{(\mathbf{W})}_{\boldsymbol\psi} \triangleq \mathbf{G}_{\boldsymbol\psi} (\mathbf{G}_{\boldsymbol\psi}^\Herm \mathbf{W} \mathbf{G}_{\boldsymbol\psi})^{-1} \mathbf{G}_{\boldsymbol\psi}^\Herm \mathbf{W}$ 是干扰空间的 $\mathbf{W}$ 加权投影矩阵。 + +\subsection{有效 Fisher 信息求取} + +下面先给出每个信源对应一个标量角参数 $\theta_k$ 的标准 stochastic CRB 形式。若某个信源使用二维或更高维方向参数，只需将下文的单索引 $k$ 扩展为参数索引 $(k,r)$，并把 $\mathbf d_k$ 替换为 $\partial\mathbf a(\bm\theta_k)/\partial[\bm\theta_k]_r$；Fisher 信息矩阵的元素级表达保持相同。 + +接收信号协方差矩阵对角度的一阶偏导为 +% +\begin{equation*} + \mathbf{R}_{\theta_k} + \triangleq \frac{\partial\mathbf{R}}{\partial\theta_k} + = \frac{\diff \mathbf{A}}{\diff \theta_k} \mathbf{R}_x \mathbf{A}^\Herm + \mathbf{A} \mathbf{R}_x \left( \frac{\diff \mathbf{A}}{\diff \theta_k} \right)^\Herm. +\end{equation*} +% +令 $\mathbf{d}_k := \mathbf{d}(\theta_k) \triangleq \dfrac{\diff}{\diff\theta_k} \mathbf{a} (\theta_k) \in \mathbb{C}^{M\times 1}$，$\mathbf{d}_k^\Herm \triangleq \dfrac{\diff}{\diff\theta_k} \mathbf{a}^\Herm (\theta_k)$，$\mathbf{R}_x = [\mathbf r_{x,1}, \ldots, \mathbf r_{x,K}]$，则 +% +\begin{equation} + \label{eq:par_recv_cov_angle} + \mathbf{R}_{\theta_k} + = \mathbf{d}_k \mathbf r_{x,k}^\Herm \mathbf{A}^\Herm + \mathbf{A} \mathbf r_{x,k} \mathbf{d}_k^\Herm. +\end{equation} + +令阵列流形矩阵的列向量（也即导向矢量）张成的信号子空间为 $\mathrm{span}(\mathbf{A})$，则其投影矩阵和其正交补空间的投影矩阵分别为 +% +\begin{equation*} + \mathcal{P}_\mathbf{A} = \mathbf{A} \left( \mathbf{A}^\Herm \mathbf{A} \right)^{-1} \mathbf{A}^\Herm, + \quad + \mathcal{P}_\mathbf{A}^\perp = \mathbf{I} - \mathbf{A} \left( \mathbf{A}^\Herm \mathbf{A} \right)^{-1} \mathbf{A}^\Herm, +\end{equation*} +% +则 $\mathbf{d}_k$ 可以正交投影分解为 +% +\begin{equation} + \label{eq:steering_vec_diff_orth_decompose} + \mathbf{d}_k + = \mathcal{P}_\mathbf{A} \mathbf{d}_k + \mathcal{P}_\mathbf{A}^\perp \mathbf{d}_k + = \mathbf{A} \beta_k + \mathbf{d}_k^\perp, +\end{equation} +% +其中 $\beta_k \triangleq \left( \mathbf{A}^\Herm \mathbf{A} \right)^{-1} \mathbf{A}^\Herm \mathbf{d}_k$，$\mathbf{d}_k^\perp = \mathcal{P}_\mathbf{A}^\perp \mathbf{d}_k$。 + +将 $\eqref{eq:steering_vec_diff_orth_decompose}$ 代入 $\eqref{eq:par_recv_cov_angle}$，有 +% +\begin{equation} + \mathbf{R}_{\theta_k} + = \underset{\in \mathrm{span}\{ \mathbf{A}(\ \cdot \ )\mathbf{A}^\Herm \} \subset \mathcal{S}_{\boldsymbol\psi}}{\underbrace{\mathbf{A} \left( \beta_k \mathbf r_{x,k}^\Herm + \mathbf r_{x,k} \beta_k^\Herm \right) \mathbf{A}^\Herm}} + + + \underset{\triangleq \mathbf{R}_{\theta_k}^{\mathrm{eff}}}{\underbrace{\mathbf{d}_k^\perp \mathbf r_{x,k}^\Herm \mathbf{A}^\Herm + \mathbf{A} \mathbf r_{x,k} \left( \mathbf{d}_k^\perp \right)^\Herm}}, +\end{equation} +% +第一项位于干扰参数扰动子空间 $\mathcal{S}_{\boldsymbol\psi}$ 中，会被加权投影意义下的 $\mathcal P_{\boldsymbol\psi}^{(\mathbf W),\perp}$ 消去，也即 +% +\begin{equation} + \mathcal P_{\boldsymbol\psi}^{(\mathbf W),\perp} \vect (\mathbf{R}_{\theta_k}) + = \vect \left( \mathbf{R}_{\theta_k}^{\mathrm{eff}} \right), +\end{equation} +% +因此，有效 Fisher 信息可以写作 +% +\begin{equation} + \label{eq:sto_doa_eff_fisher_info} + [\mathcal{I}_{\mathrm e}]_{ij} + = T\Re\left\{\left[\vect \left( \mathbf{R}_{\theta_i}^{\mathrm{eff}} \right) \right]^\Herm \mathbf{W} \vect \left( \mathbf{R}_{\theta_j}^{\mathrm{eff}} \right)\right\} + = T\Re\left\{\tr \left( \mathbf{R}^{-1} \mathbf{R}_{\theta_i}^{\mathrm{eff}} \mathbf{R}^{-1} \mathbf{R}_{\theta_j}^{\mathrm{eff}} \right)\right\}. +\end{equation} +% +令 $\mathbf{R}_k^{(1)} = \mathbf{d}_k^\perp \mathbf r_{x,k}^\Herm \mathbf{A}^\Herm$，$\mathbf{R}_k^{(2)} = \mathbf{A} \mathbf r_{x,k} \left( \mathbf{d}_k^\perp \right)^\Herm$，则 +% +\begin{equation} + \mathbf{R}_{\theta_k}^{\mathrm{eff}} = \mathbf{R}_k^{(1)} + \mathbf{R}_k^{(2)}. +\end{equation} +% +按此形式对 $\eqref{eq:sto_doa_eff_fisher_info}$ 进行展开，有 +% +\begin{align} + [\mathcal{I}_{\mathrm e}]_{ij} + =& T \tr \left( \mathbf{R}^{-1} \mathbf{R}_{\theta_i}^{\mathrm{eff}} \mathbf{R}^{-1} \mathbf{R}_{\theta_j}^{\mathrm{eff}} \right) + = T \tr \left( \mathbf{R}^{-1} \left( \mathbf{R}_i^{(1)} + \mathbf{R}_i^{(2)} \right) \mathbf{R}^{-1} \left( \mathbf{R}_j^{(1)} + \mathbf{R}_j^{(2)} \right) \right) + \nonumber \\ + =& \underset{\triangleq T_{11}}{\underbrace{T \tr \left( \mathbf{R}^{-1} \mathbf{R}_i^{(1)} \mathbf{R}^{-1} \mathbf{R}_j^{(1)} \right)}} + + \underset{\triangleq T_{12}}{\underbrace{T \tr \left( \mathbf{R}^{-1} \mathbf{R}_i^{(1)} \mathbf{R}^{-1} \mathbf{R}_j^{(2)} \right)}} + + \underset{\triangleq T_{21}}{\underbrace{T \tr \left( \mathbf{R}^{-1} \mathbf{R}_i^{(2)} \mathbf{R}^{-1} \mathbf{R}_j^{(1)} \right)}} + \nonumber \\ + &+ \underset{\triangleq T_{22}}{\underbrace{T \tr \left( \mathbf{R}^{-1} \mathbf{R}_i^{(2)} \mathbf{R}^{-1} \mathbf{R}_j^{(2)} \right)}} +\end{align} + +由 Woodbury 公式 $(\mathbf{A} + \mathbf{UBV})^{-1} = \mathbf{A}^{-1} - \mathbf{A}^{-1} \mathbf{U} \left( \mathbf{B}^{-1} + \mathbf{VA}^{-1} \mathbf{U} \right)^{-1} \mathbf{VA}^{-1}$，接收信号协方差矩阵 $\mathbf{R} = \mathbf{A} \mathbf{R}_x \mathbf{A}^\Herm + \sigma \mathbf{I}_M$ 的逆可以写为 +% +\begin{equation} + \mathbf{R}^{-1} = \frac{1}{\sigma} \mathbf{I}_M - \frac{1}{\sigma^2} \mathbf{A} \left( \mathbf{R}_x^{-1} + \frac{1}{\sigma} \mathbf{A}^\Herm \mathbf{A} \right)^{-1} \mathbf{A}^\Herm. +\end{equation} +% +由于 $\mathbf{A}^\Herm \mathcal{P}_\mathbf{A}^\perp = 0$，$\mathcal{P}_\mathbf{A}^\perp \mathbf{A} = 0$，因此 +% +\begin{equation} + \mathbf{A}^\Herm \mathbf{R}^{-1} \mathcal{P}_\mathbf{A}^\perp = 0, + \quad + \mathcal{P}_\mathbf{A}^\perp \mathbf{R}^{-1} \mathbf{A} = 0. +\end{equation} +% +接下来计算 $[\mathcal{I}_{\mathrm e}]_{ij}$ 的各分量。令 $\mathbf{U} = \mathbf{R}_x \mathbf{A}^\Herm \mathbf{R}^{-1} \mathbf{A}\mathbf{R}_x$，则 +% +\begin{subequations} +\begin{align} + T_{11} + &= T \tr \left( \mathbf{R}^{-1} \mathbf{R}_i^{(1)} \mathbf{R}^{-1} \mathbf{R}_j^{(1)} \right) + = T \tr \left( \mathbf{R}^{-1} \left( \mathbf{d}_i^\perp \mathbf r_{x,i}^\Herm \mathbf{A}^\Herm \right) \mathbf{R}^{-1} \left( \mathbf{d}_j^\perp \mathbf r_{x,j}^\Herm \mathbf{A}^\Herm \right) \right) + \nonumber \\ + &= T \tr \left( \mathbf r_{x,j}^\Herm \mathbf{A}^\Herm \mathbf{R}^{-1} \mathbf{d}_i^\perp \mathbf r_{x,i}^\Herm \mathbf{A}^\Herm \mathbf{R}^{-1} \mathbf{d}_j^\perp \right) + = T \tr \left( \mathbf r_{x,j}^\Herm \mathbf{A}^\Herm \mathbf{R}^{-1} \mathcal{P}_\mathbf{A}^\perp \mathbf{d}_i \mathbf r_{x,i}^\Herm \mathbf{A}^\Herm \mathbf{R}^{-1} \mathcal{P}_\mathbf{A}^\perp \mathbf{d}_j \right) = 0, + \\ + T_{12} + &= T \tr \left( \mathbf{R}^{-1} \mathbf{R}_i^{(1)} \mathbf{R}^{-1} \mathbf{R}_j^{(2)} \right) + = T \tr \left( \mathbf{R}^{-1} \left( \mathbf{d}_i^\perp \mathbf r_{x,i}^\Herm \mathbf{A}^\Herm \right) \mathbf{R}^{-1} \left( \mathbf{A} \mathbf r_{x,j} \left( \mathbf{d}_j^\perp \right)^\Herm \right) \right) + \nonumber \\ + &= T \left( \left( \mathbf{d}_j^\perp \right)^\Herm \mathbf{R}^{-1} \mathbf{d}_i^\perp \mathbf r_{x,i}^\Herm \mathbf{A}^\Herm \mathbf{R}^{-1} \mathbf{A} \mathbf r_{x,j} \right) + = T \left( \mathbf{d}_j^\perp \right)^\Herm \mathbf{R}^{-1} \mathbf{d}_i^\perp [\mathbf{U}]_{ij}, + \\ + T_{21} + &= T \tr \left( \mathbf{R}^{-1} \mathbf{R}_i^{(2)} \mathbf{R}^{-1} \mathbf{R}_j^{(1)} \right) + = T \left( \mathbf{d}_i^\perp \right)^\Herm \mathbf{R}^{-1} \mathbf{d}_j^\perp [\mathbf{U}]_{ji}, + \\ + T_{22} + &= T \tr \left( \mathbf{R}^{-1} \mathbf{R}_i^{(2)} \mathbf{R}^{-1} \mathbf{R}_j^{(2)} \right) + = 0, +\end{align} +\end{subequations} +% +因此，有效 Fisher 信息可以进一步写作 +% +\begin{align} + [\mathcal{I}_{\mathrm e}]_{ij} + &= T_{11} + T_{12} + T_{21} + T_{22} + = 2 T \Re \left\{ \left( \mathbf{d}_i^\perp \right)^\Herm \mathbf{R}^{-1} \mathbf{d}_j^\perp [\mathbf{U}]_{ji} \right\} + \nonumber \\ + &= 2 T \Re \left\{ \left( \mathcal{P}_\mathbf{A}^\perp \mathbf{d}_i \right)^\Herm \mathbf{R}^{-1} \mathcal{P}_\mathbf{A}^\perp \mathbf{d}_j [\mathbf{U}]_{ji} \right\} + = \frac{2 T}{\sigma} \Re \left\{ \mathbf{d}_i^\Herm \mathcal{P}_\mathbf{A}^\perp \mathbf{d}_j [\mathbf{U}]_{ji} \right\}. +\end{align} +% +令 $\mathcal{H} = \mathbf{D}^\Herm \mathcal{P}_\mathbf{A}^\perp \mathbf{D}$，$\mathbf{D} = [\mathbf{d}_1, \ldots, \mathbf{d}_K]$，则 +% +\begin{equation} + [\mathcal{I}_{\mathrm e}]_{ij} = \frac{2T}{\sigma} \Re \left\{ [\mathcal{H}]_{ij} [\mathbf{U}]_{ji} \right\}, +\end{equation} +% +用矩阵形式表示为 +\begin{equation} + \mathcal{I}_{\mathrm e} = \frac{2T}{\sigma} \Re \left\{ \mathcal{H} \odot \mathbf{U}^\Tran \right\}. +\end{equation} +% +其中 $\odot$ 表示 Hadamard 积。因此，无条件（unconditional）\gls{crb} 或随机（stochastic）\gls{crb} 可以表示为 +% +\begin{align} + \mathcal{B}_u := \mathcal{I}_{\mathrm e}^{-1} = \frac{\sigma}{2T} \left(\Re \left\{ \mathcal{H} \odot \mathbf{U}^\Tran \right\}\right)^{-1}. +\end{align} +% + +该随机信号 CRB 与\ref{sec:det_doa_crb} 的确定信号 CRB 共同给出单阵列 DoA 的两类性能基线。多星主线后续采用确定信号/导频化模型，因此其 CRB 推导更接近\ref{sec:det_doa_crb} 和\ref{sec:det_doa_doppler_crb} 的处理方式。 diff --git a/parts/A5-detDoaDopplerCrb.tex b/parts/A5-detDoaDopplerCrb.tex new file mode 100644 index 0000000..dafa86b --- /dev/null +++ b/parts/A5-detDoaDopplerCrb.tex @@ -0,0 +1,238 @@ +\section{单阵列确定信号 DoA--Doppler 估计的 CRB 推导} +\label{sec:det_doa_doppler_crb} + +本附录对应第~\ref{sec:det_doa_doppler_est} 节的单阵列确定信号 DoA--Doppler 模型，是从纯空间 CRB 过渡到空时 CRB 的中间层。由式~\eqref{eq:vec_det_doa_doppler_recv_sig}，向量化观测满足 +\begin{equation} + \tilde{\mathbf y} + = + \mathbf B_{\mathrm{st}}(\bm\Theta,\boldsymbol\nu)\boldsymbol\zeta + + + \tilde{\mathbf n}, + \qquad + \tilde{\mathbf n}\sim\mathcal{CN}(\mathbf 0,\sigma\mathbf I_{MT}), +\end{equation} +其中 $\sigma\triangleq\sigma_n^2$。因此 +\begin{equation} + \tilde{\mathbf y}\sim\mathcal{CN}\big(\boldsymbol\mu(\boldsymbol\kappa),\sigma\mathbf I_{MT}\big), + \qquad + \boldsymbol\mu(\boldsymbol\kappa) + = + \mathbf B_{\mathrm{st}}\boldsymbol\zeta. +\end{equation} +本附录使用完整参数向量 +\begin{equation} + \boldsymbol\kappa + = + \begin{bmatrix} + \sigma, + (\boldsymbol\zeta^{(r)})^\Tran, + (\boldsymbol\zeta^{(i)})^\Tran, + \boldsymbol\nu^\Tran, + \bm\theta^\Tran + \end{bmatrix}^\Tran, +\end{equation} +其中 $\boldsymbol\zeta^{(r)}=\Re\{\boldsymbol\zeta\}$、$\boldsymbol\zeta^{(i)}=\Im\{\boldsymbol\zeta\}$。这里的 $\bm\theta$ 表示单阵列一维 DoA 情形下的角度向量；若使用二维 DoA 参数，可将相应导数矩阵按同样方式扩展。 + + +\subsection{FIM 元素的通式} + +由于协方差 $\sigma\mathbf I_{MT}$ 只依赖噪声方差，而其他参数只通过均值向量进入，因此均值参数之间的 Fisher 信息为 +\begin{equation} + [\mathcal I(\boldsymbol\kappa)]_{ij} + = + \frac{2}{\sigma} + \Re\left\{ + \left(\frac{\partial\boldsymbol\mu}{\partial\kappa_i}\right)^\Herm + \left(\frac{\partial\boldsymbol\mu}{\partial\kappa_j}\right) + \right\}, +\end{equation} +噪声方差自身的信息为 +\begin{equation} + [\mathcal I(\boldsymbol\kappa)]_{\sigma\sigma} + = + \frac{MT}{\sigma^2}, +\end{equation} +且噪声方差与均值参数的 Fisher 交叉项为零。因此后续只需构造均值参数部分的 FIM。 + +\subsection{偏导数求取} + +定义时间索引向量和对角矩阵 +\begin{equation} + \mathbf t + \triangleq + [0,1,\ldots,T-1]^\Tran, + \qquad + \mathbf T_d + \triangleq + \diag(0,1,\ldots,T-1). +\end{equation} +由 $[\mathbf q_k(\nu_k)]_t=s_k(t)e^{j\nu_k t}$ 可得 +\begin{equation} + \dot{\mathbf q}_k + \triangleq + \frac{\partial\mathbf q_k(\nu_k)}{\partial\nu_k} + = + j\mathbf T_d\mathbf q_k(\nu_k). +\end{equation} +再定义空间导向矢量和空时导向列向量的一阶导数为 +\begin{equation} + \dot{\mathbf a}_k + \triangleq + \frac{\partial\mathbf a(\theta_k)}{\partial\theta_k}, +\end{equation} +\begin{equation} + \dot{\mathbf b}_{\mathrm{st},\theta_k} + \triangleq + \frac{\partial\mathbf b_{\mathrm{st},k}}{\partial\theta_k} + = + \mathbf q_k(\nu_k)\otimes\dot{\mathbf a}_k, + \qquad + \dot{\mathbf b}_{\mathrm{st},\nu_k} + \triangleq + \frac{\partial\mathbf b_{\mathrm{st},k}}{\partial\nu_k} + = + \dot{\mathbf q}_k\otimes\mathbf a(\theta_k). +\end{equation} +进一步写成矩阵形式 +\begin{equation} + \mathbf B_{\mathrm{st},\theta} + \triangleq + \left[ + \dot{\mathbf b}_{\mathrm{st},\theta_1},\ldots, + \dot{\mathbf b}_{\mathrm{st},\theta_K} + \right] + \in\mathbb C^{MT\times K}, +\end{equation} +\begin{equation} + \mathbf B_{\mathrm{st},\nu} + \triangleq + \left[ + \dot{\mathbf b}_{\mathrm{st},\nu_1},\ldots, + \dot{\mathbf b}_{\mathrm{st},\nu_K} + \right] + \in\mathbb C^{MT\times K}. +\end{equation} +由此，对复增益、Doppler 和角度的一阶偏导为 +\begin{equation} + \mathbf J_{\boldsymbol\zeta^{(r)}} + = + \mathbf B_{\mathrm{st}}, + \qquad + \mathbf J_{\boldsymbol\zeta^{(i)}} + = + j\mathbf B_{\mathrm{st}}, +\end{equation} +\begin{equation} + \mathbf J_{\boldsymbol\nu} + = + \mathbf B_{\mathrm{st},\nu}\mathbf D_{\zeta}, + \qquad + \mathbf J_{\bm\theta} + = + \mathbf B_{\mathrm{st},\theta}\mathbf D_{\zeta}, +\end{equation} +其中 $\mathbf D_\zeta=\diag(\zeta_1,\ldots,\zeta_K)$。 + +\subsection{消去复增益后的 DoA--Doppler EFIM} + +令 +\begin{equation} + \boldsymbol\eta_{\theta\nu} + \triangleq + \left[\bm\theta^\Tran,\boldsymbol\nu^\Tran\right]^\Tran, + \qquad + \mathbf J_{\boldsymbol\eta} + \triangleq + \left[ + \mathbf J_{\bm\theta}, + \mathbf J_{\boldsymbol\nu} + \right]. +\end{equation} +完整均值参数 Jacobian 可写为 +\begin{equation} + \mathbf J_{\mathrm{mean}} + = + \left[ + \mathbf B_{\mathrm{st}}, + j\mathbf B_{\mathrm{st}}, + \mathbf J_{\boldsymbol\eta} + \right], + \qquad + \mathcal I_{\mathrm{mean}} + = + \frac{2}{\sigma}\Re\{\mathbf J_{\mathrm{mean}}^\Herm\mathbf J_{\mathrm{mean}}\}. +\end{equation} +由于 $\mathbf B_{\mathrm{st}}$ 与 $j\mathbf B_{\mathrm{st}}$ 对应未知复增益的实部和虚部，消去复增益后，$\boldsymbol\eta_{\theta\nu}$ 的 EFIM 为 +\begin{equation} + \label{eq:single_det_doa_doppler_efim_theta_nu} + \mathcal I_{\mathrm e}(\boldsymbol\eta_{\theta\nu}) + = + \frac{2}{\sigma}\Re\left\{ + \begin{bmatrix} + \mathbf D_\zeta^\Herm\mathbf B_{\mathrm{st},\theta}^\Herm \\ + \mathbf D_\zeta^\Herm\mathbf B_{\mathrm{st},\nu}^\Herm + \end{bmatrix} + \mathcal P_{\mathbf B_{\mathrm{st}}}^\perp + \begin{bmatrix} + \mathbf B_{\mathrm{st},\theta}\mathbf D_\zeta & + \mathbf B_{\mathrm{st},\nu}\mathbf D_\zeta + \end{bmatrix} + \right\}, +\end{equation} +其中 +\begin{equation} + \mathcal P_{\mathbf B_{\mathrm{st}}}^\perp + = + \mathbf I_{MT} + - + \mathbf B_{\mathrm{st}} + (\mathbf B_{\mathrm{st}}^\Herm\mathbf B_{\mathrm{st}})^{-1} + \mathbf B_{\mathrm{st}}^\Herm. +\end{equation} +将式~\eqref{eq:single_det_doa_doppler_efim_theta_nu} 分块写为 +\begin{equation} + \mathcal I_{\mathrm e}(\boldsymbol\eta_{\theta\nu}) + = + \begin{bmatrix} + \mathcal I_{\theta\theta} & \mathcal I_{\theta\nu} \\ + \mathcal I_{\nu\theta} & \mathcal I_{\nu\nu} + \end{bmatrix}. +\end{equation} + +\subsection{仅关心 DoA 或 Doppler 时的 CRB} + +若同时估计 DoA 和 Doppler，则 +\begin{equation} + \operatorname{CRB}(\boldsymbol\eta_{\theta\nu}) + = + \mathcal I_{\mathrm e}^{-1}(\boldsymbol\eta_{\theta\nu}). +\end{equation} +若只关心 DoA，并把 Doppler 视为实值干扰参数，则应在式~\eqref{eq:single_det_doa_doppler_efim_theta_nu} 的实值 EFIM 上对 $\boldsymbol\nu$ 做 Schur 补： +\begin{equation} + \operatorname{CRB}(\bm\theta) + = + \left( + \mathcal I_{\theta\theta} + - + \mathcal I_{\theta\nu}\mathcal I_{\nu\nu}^{-1}\mathcal I_{\nu\theta} + \right)^{-1}. +\end{equation} +类似地，若只关心 Doppler，并把 DoA 视为实值干扰参数，则 +\begin{equation} + \operatorname{CRB}(\boldsymbol\nu) + = + \left( + \mathcal I_{\nu\nu} + - + \mathcal I_{\nu\theta}\mathcal I_{\theta\theta}^{-1}\mathcal I_{\theta\nu} + \right)^{-1}. +\end{equation} +需要注意，$\boldsymbol\nu$ 是实值干扰参数，因此不能把 +$\mathbf B_{\mathrm{st},\nu}\mathbf D_\zeta$ 当作具有任意复系数的未知子空间并直接写成复投影 +$\mathcal P_{[\mathbf B_{\mathrm{st}},\mathbf B_{\mathrm{st},\nu}\mathbf D_\zeta]}^\perp$。 +正确做法是先消去复增益，再在实值参数 EFIM 中对 Doppler 做 Schur 补。 +若 Doppler 已知，则 DoA 的 CRB 为 $\mathcal I_{\theta\theta}^{-1}$；若 Doppler 未知，则上式中的 Schur 补体现了 DoA--Doppler 耦合带来的信息损失。 + +当 $\boldsymbol\nu$ 已知且固定为 $\mathbf0$ 时，本模型退化为带已知时间导频权重的确定信号 DoA 模型；若 $\boldsymbol\nu$ 未知，即使在真实值 $\boldsymbol\nu=\mathbf0$ 处，Doppler 仍会通过导数 $\dot{\mathbf q}_k=j\mathbf T_d\mathbf q_k$ 作为干扰参数影响 DoA 的 CRB。 + +该单阵列空时 CRB 说明 DoA 与 Doppler 的耦合会通过 Schur 补造成信息损失。多星静态和动态 CRB 在结构上延续这一思想，但将独立的 $\nu_k$ 替换为参考星参数化下的 $\nu_{\mathrm{ref}}$，并进一步处理多星复增益、连续相位和 Doppler rate 等干扰参数。 diff --git a/parts/A6-crbStatPilotDoaDopper.tex b/parts/A6-crbStatPilotDoaDopper.tex new file mode 100644 index 0000000..2a8e192 --- /dev/null +++ b/parts/A6-crbStatPilotDoaDopper.tex @@ -0,0 +1,543 @@ +\section{多阵列确定信号 DoA--Doppler 估计的 CRB 推导} +\label{sec:A6_multiArrayStatDetDoaDopplerCrb} + +本附录沿用第~\ref{sec:multiArrayStatDetDoaDopplerEst} 节，尤其是第~\ref{subsec:staticBlockModel} 小节中的静态多阵列确定信号 DoA--Doppler 观测模型与参数定义，只补充 \gls{crb} 推导所需的 Jacobian、\gls{fim} 分块与 Schur 补步骤。主参数仍记为 +\begin{equation} + \boldsymbol\eta_{\mathrm{stat}} + = + \begin{bmatrix} + \alpha \\ + \beta \\ + \nu_{\mathrm{ref}} + \end{bmatrix} + \in \mathbb R^{3}. +\end{equation} +与\ref{sec:det_doa_doppler_crb} 的单阵列情形相比，这里需要先联合堆叠 $L$ 颗协作卫星的观测，再将各星复干扰系数视为干扰参数并通过 Schur 补得到 $\boldsymbol\eta_{\mathrm{stat}}$ 的等效 \gls{fim}。本文统一采用 $\mathcal I$ 表示 Fisher 信息矩阵，采用 $\mathbf I$ 表示单位阵，采用 $\mathcal P$ 表示投影矩阵。 +正文中的静态干扰参数 $\boldsymbol\psi_{\mathrm{stat}}$ 同时包含复增益和噪声方差。本附录在构造主参数 $\boldsymbol\eta_{\mathrm{stat}}$ 的 EFIM 时，只显式保留影响均值向量的复增益干扰参数；噪声方差只进入协方差，并与均值参数 Fisher 块解耦，因此不改变下面关于复增益的 Schur 补形式。换言之，下文得到的是在给定真实噪声方差或将其作为解耦协方差参数时，关于 $\boldsymbol\eta_{\mathrm{stat}}$ 的等效 Fisher 信息。 + + +\subsection{联合观测模型与 Fisher 信息矩阵通式} + +沿用第~\ref{subsec:staticBlockModel} 小节中的向量化观测模型，第 $\ell$ 颗卫星的均值向量可写为 $\zeta_\ell \mathbf g_\ell(\boldsymbol\eta_{\mathrm{stat}})$。为便于后续统一构造 \gls{fim}，本小节先将所有卫星观测按块联合堆叠。 + +对于第 $\ell$ 颗卫星，向量化后的观测模型为 +\begin{equation} + \tilde{\mathbf y}_\ell + = + \zeta_\ell \mathbf g_\ell(\boldsymbol\eta_{\mathrm{stat}}) + + + \tilde{\mathbf n}_\ell, + \qquad + \tilde{\mathbf n}_\ell \sim \mathcal{CN}(\mathbf 0,\sigma_{n,\ell}^2 \mathbf I_{M_\ell T}), + \quad \ell=0,1,\ldots,L-1, +\end{equation} +其中 +\begin{equation} + \mathbf g_\ell(\boldsymbol\eta_{\mathrm{stat}}) + \triangleq + \mathbf q_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}}) \otimes \mathbf a_\ell(\boldsymbol\xi) + \in \mathbb C^{M_\ell T \times 1}, +\end{equation} +$\zeta_\ell\in\mathbb C$ 为第 $\ell$ 条链路的未知复干扰系数，$\sigma_{n,\ell}^2$ 为第 $\ell$ 颗卫星对应的噪声方差。 + +将所有卫星的观测联合堆叠，定义 +\begin{equation} + \tilde{\mathbf y} + \triangleq + \begin{bmatrix} + \tilde{\mathbf y}_0 \\ + \vdots \\ + \tilde{\mathbf y}_{L-1} + \end{bmatrix}, + \qquad + \boldsymbol\mu(\boldsymbol\kappa_{\mathrm{stat}}) + \triangleq + \begin{bmatrix} + \zeta_0 \mathbf g_0(\boldsymbol\eta_{\mathrm{stat}}) \\ + \vdots \\ + \zeta_{L-1} \mathbf g_{L-1}(\boldsymbol\eta_{\mathrm{stat}}) + \end{bmatrix}, + \qquad + \tilde{\mathbf n} + \triangleq + \begin{bmatrix} + \tilde{\mathbf n}_0 \\ + \vdots \\ + \tilde{\mathbf n}_{L-1} + \end{bmatrix}, +\end{equation} +则有 +\begin{equation} + \tilde{\mathbf y} = \boldsymbol\mu(\boldsymbol\kappa_{\mathrm{stat}}) + \tilde{\mathbf n}, + \qquad + \tilde{\mathbf n} \sim \mathcal{CN}(\mathbf 0,\mathbf R_n), +\end{equation} +其中 +\begin{equation} + \mathbf R_n + = + \operatorname{blkdiag} + \big( + \sigma_{n,0}^2 \mathbf I_{M_0 T}, + \ldots, + \sigma_{n,L-1}^2 \mathbf I_{M_{L-1} T} + \big). +\end{equation} + +将各星复干扰系数按实部和虚部分解为 +\begin{equation} + \zeta_\ell = \zeta_\ell^{(r)} + j\zeta_\ell^{(i)}, +\end{equation} +并定义实参数化后的均值干扰参数向量 +\begin{equation} + \boldsymbol\psi_{\zeta,\mathrm{stat}} + \triangleq + \begin{bmatrix} + \zeta_0^{(r)} & \zeta_0^{(i)} & \cdots & \zeta_{L-1}^{(r)} & \zeta_{L-1}^{(i)} + \end{bmatrix}^{\Tran} + \in \mathbb R^{2L}. +\end{equation} +于是总实参数向量可写为 +\begin{equation} + \boldsymbol\kappa_{\mathrm{stat}} + \triangleq + \begin{bmatrix} + \boldsymbol\eta_{\mathrm{stat}}^{\Tran} & \boldsymbol\psi_{\zeta,\mathrm{stat}}^{\Tran} + \end{bmatrix}^{\Tran} + \in \mathbb R^{2L+3}. +\end{equation} + +由于当前均值参数仅通过均值项 $\boldsymbol\mu(\boldsymbol\kappa_{\mathrm{stat}})$ 进入，而协方差矩阵 $\mathbf R_n$ 与 $\boldsymbol\kappa_{\mathrm{stat}}$ 无关，因此对任意两个实参数 $\kappa_i,\kappa_j$，Fisher 信息矩阵元素满足 +\begin{equation} + [\mathcal I(\boldsymbol\kappa_{\mathrm{stat}})]_{ij} + = + 2\Re\!\left\{ + \frac{\partial \boldsymbol\mu^{\Herm}(\boldsymbol\kappa_{\mathrm{stat}})}{\partial \kappa_i} + \mathbf R_n^{-1} + \frac{\partial \boldsymbol\mu(\boldsymbol\kappa_{\mathrm{stat}})}{\partial \kappa_j} + \right\}. + \label{eq:A6_fim_general} +\end{equation} + +\subsection{\texorpdfstring{关于 $\boldsymbol\eta_{\mathrm{stat}}$ 的导数}{关于 eta_stat 的导数}} + +由第~\ref{sec:multiArrayStatDetDoaDopplerEst} 节，单位方向向量为 +\begin{equation} + \mathbf u(\boldsymbol\xi) + = + \begin{bmatrix} + \cos\beta\cos\alpha \\ + \cos\beta\sin\alpha \\ + \sin\beta + \end{bmatrix}, +\end{equation} +其关于全局角度参数的一阶导数为 +\begin{equation} + \mathbf u_{\alpha} + \triangleq + \frac{\partial \mathbf u}{\partial \alpha} + = + \begin{bmatrix} + -\cos\beta\sin\alpha \\ + \cos\beta\cos\alpha \\ + 0 + \end{bmatrix}, + \qquad + \mathbf u_{\beta} + \triangleq + \frac{\partial \mathbf u}{\partial \beta} + = + \begin{bmatrix} + -\sin\beta\cos\alpha \\ + -\sin\beta\sin\alpha \\ + \cos\beta + \end{bmatrix}. +\end{equation} + +根据第~\ref{subsec:refSatParameterization} 小节的参考星参数化，静态模型中第 $\ell$ 颗卫星对应的归一化 Doppler 统一写成 +\begin{equation} + \nu_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}}) + = + f_{\nu,\ell}(\boldsymbol\xi,\nu_{\mathrm{ref}}) + = + \nu_{\mathrm{ref}}+\Delta\nu_\ell(\boldsymbol\xi), +\end{equation} +其中 $\Delta\nu_\ell(\boldsymbol\xi)$ 由已知几何、速度与姿态关系确定，且对参考星 $\ell_{\mathrm{ref}}$ 有 +\begin{equation} + f_{\nu,\ell_{\mathrm{ref}}}(\boldsymbol\xi,\nu_{\mathrm{ref}}) + = + \nu_{\mathrm{ref}}. +\end{equation} +因此其一阶导数统一写为 +\begin{equation} + \frac{\partial \nu_\ell}{\partial \alpha} + = + \frac{\partial \Delta\nu_\ell(\boldsymbol\xi)}{\partial \alpha}, + \qquad + \frac{\partial \nu_\ell}{\partial \beta} + = + \frac{\partial \Delta\nu_\ell(\boldsymbol\xi)}{\partial \beta}, + \qquad + \frac{\partial \nu_\ell}{\partial \nu_{\mathrm{ref}}}=1. + \label{eq:A6_domega} +\end{equation} +上式利用了正文中采用的 +$\nu_\ell=\nu_{\mathrm{ref}}+\Delta\nu_\ell(\boldsymbol\xi)$ +这一参考星链路参数化形式。特别地，对参考星有 +$\Delta\nu_{\ell_{\mathrm{ref}}}(\boldsymbol\xi)=0$，因而 +$\partial\nu_{\ell_{\mathrm{ref}}}/\partial\alpha=0$、 +$\partial\nu_{\ell_{\mathrm{ref}}}/\partial\beta=0$。若改用别的等价参数化，只需将上述三个 Doppler 链式导数替换为对应结果即可。 + +再定义第 $\ell$ 颗卫星本地坐标系下的来向向量 +\begin{equation} + \mathbf u_\ell^{\mathrm{loc}} + = + \boldsymbol{\Omega}_\ell^{\Tran}\mathbf u + = + \begin{bmatrix} + u_{\ell,1}^{\mathrm{loc}} & u_{\ell,2}^{\mathrm{loc}} & u_{\ell,3}^{\mathrm{loc}} + \end{bmatrix}^{\Tran}. +\end{equation} +相应的局部角度为 +\begin{equation} + \alpha_\ell^{\mathrm{loc}} + = + \operatorname{atan2}\!\big(u_{\ell,2}^{\mathrm{loc}},u_{\ell,1}^{\mathrm{loc}}\big), + \qquad + \beta_\ell^{\mathrm{loc}} + = + \arcsin\!\big(u_{\ell,3}^{\mathrm{loc}}\big). +\end{equation} + +对任意 $x\in\{\alpha,\beta\}$，有 +\begin{equation} + \frac{\partial \mathbf u_\ell^{\mathrm{loc}}}{\partial x} + = + \boldsymbol{\Omega}_\ell^{\Tran}\mathbf u_x. +\end{equation} +进一步可得 +\begin{equation} + \frac{\partial \alpha_\ell^{\mathrm{loc}}}{\partial x} + = + \frac{ + u_{\ell,1}^{\mathrm{loc}}\,\dfrac{\partial u_{\ell,2}^{\mathrm{loc}}}{\partial x} + - + u_{\ell,2}^{\mathrm{loc}}\,\dfrac{\partial u_{\ell,1}^{\mathrm{loc}}}{\partial x} + }{ + \big(u_{\ell,1}^{\mathrm{loc}}\big)^2 + \big(u_{\ell,2}^{\mathrm{loc}}\big)^2 + }, + \qquad + \frac{\partial \beta_\ell^{\mathrm{loc}}}{\partial x} + = + \frac{\dfrac{\partial u_{\ell,3}^{\mathrm{loc}}}{\partial x}}{\sqrt{1-\big(u_{\ell,3}^{\mathrm{loc}}\big)^2}}. + \label{eq:A6_local_angle_chain} +\end{equation} + +记局部阵列导向矢量关于局部方位角和局部俯仰角的一阶导数分别为 +\begin{equation} + \dot{\mathbf a}_{\ell,\alpha} + \triangleq + \frac{\partial \mathbf a_\ell}{\partial \alpha_\ell^{\mathrm{loc}}}, + \qquad + \dot{\mathbf a}_{\ell,\beta} + \triangleq + \frac{\partial \mathbf a_\ell}{\partial \beta_\ell^{\mathrm{loc}}}. +\end{equation} +则对任意 $x\in\{\alpha,\beta\}$，根据链式法则有 +\begin{equation} + \mathbf a_{\ell,x} + \triangleq + \frac{\partial \mathbf a_\ell(\boldsymbol\xi)}{\partial x} + = + \dot{\mathbf a}_{\ell,\alpha}\frac{\partial \alpha_\ell^{\mathrm{loc}}}{\partial x} + + + \dot{\mathbf a}_{\ell,\beta}\frac{\partial \beta_\ell^{\mathrm{loc}}}{\partial x}. + \label{eq:A6_da_chain} +\end{equation} + +由第~\ref{sec:multiArrayStatDetDoaDopplerEst} 节的离散时间模板定义， +\begin{equation} + [\mathbf q_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}})]_t + = + s(t)e^{j\nu_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}})t}, + \qquad t=0,1,\ldots,T-1. +\end{equation} +因此其关于 $\nu_\ell$ 的导数为 +\begin{equation} + \dot{\mathbf q}_\ell + \triangleq + \frac{\partial \mathbf q_\ell}{\partial \nu_\ell} + = + j\,\operatorname{diag}(0,1,\ldots,T-1)\,\mathbf q_\ell. + \label{eq:A6_qdot} +\end{equation} +从而对任意 $x\in\{\alpha,\beta\}$，有 +\begin{equation} + \mathbf q_{\ell,x} + \triangleq + \frac{\partial \mathbf q_\ell}{\partial x} + = + \dot{\mathbf q}_\ell\frac{\partial \nu_\ell}{\partial x}, + \qquad + \mathbf q_{\ell,\nu} + \triangleq + \frac{\partial \mathbf q_\ell}{\partial \nu_{\mathrm{ref}}} + = + \dot{\mathbf q}_\ell\frac{\partial \nu_\ell}{\partial \nu_{\mathrm{ref}}} + = + \dot{\mathbf q}_\ell. + \label{eq:A6_dq} +\end{equation} +其中最后一步同样利用了 +$\partial \nu_\ell/\partial \nu_{\mathrm{ref}}=1$。 + +对空时导向向量 +$\mathbf g_\ell=\mathbf q_\ell\otimes \mathbf a_\ell$ +应用 Kronecker 积求导法则，可得 +\begin{equation} + \mathbf d_{\ell,\alpha} + \triangleq + \frac{\partial \mathbf g_\ell}{\partial \alpha} + = + \mathbf q_\ell \otimes \mathbf a_{\ell,\alpha} + + + \mathbf q_{\ell,\alpha} \otimes \mathbf a_\ell, +\end{equation} +\begin{equation} + \mathbf d_{\ell,\beta} + \triangleq + \frac{\partial \mathbf g_\ell}{\partial \beta} + = + \mathbf q_\ell \otimes \mathbf a_{\ell,\beta} + + + \mathbf q_{\ell,\beta} \otimes \mathbf a_\ell, +\end{equation} +\begin{equation} + \mathbf d_{\ell,\nu} + \triangleq + \frac{\partial \mathbf g_\ell}{\partial \nu_{\mathrm{ref}}} + = + \mathbf q_{\ell,\nu} \otimes \mathbf a_\ell + = + \dot{\mathbf q}_\ell \otimes \mathbf a_\ell. +\end{equation} +进一步定义 +\begin{equation} + \mathbf D_\ell + \triangleq + \begin{bmatrix} + \mathbf d_{\ell,\alpha} & \mathbf d_{\ell,\beta} & \mathbf d_{\ell,\nu} + \end{bmatrix} + \in \mathbb C^{M_\ell T \times 3}. + \label{eq:A6_Dl} +\end{equation} + +\subsection{Fisher 信息矩阵的分块构造} + +对第 $\ell$ 颗卫星，其均值向量为 +\begin{equation} + \boldsymbol\mu_\ell(\boldsymbol\kappa_{\mathrm{stat}}) + = + \zeta_\ell\mathbf g_\ell(\boldsymbol\eta_{\mathrm{stat}}). +\end{equation} +因此，其关于主参数向量 $\boldsymbol\eta_{\mathrm{stat}}$ 的 Jacobian 为 +\begin{equation} + \mathbf J_{\eta,\ell} + \triangleq + \frac{\partial \boldsymbol\mu_\ell}{\partial \boldsymbol\eta_{\mathrm{stat}}} + = + \zeta_\ell \mathbf D_\ell + \in \mathbb C^{M_\ell T \times 3}, + \label{eq:A6_Jeta} +\end{equation} +而其关于本星复干扰系数实部与虚部的 Jacobian 为 +\begin{equation} + \mathbf J_{\psi,\ell} + \triangleq + \frac{\partial \boldsymbol\mu_\ell}{\partial + \begin{bmatrix} + \zeta_\ell^{(r)} & \zeta_\ell^{(i)} + \end{bmatrix}^{\Tran}} + = + \begin{bmatrix} + \mathbf g_\ell & j\mathbf g_\ell + \end{bmatrix} + \in \mathbb C^{M_\ell T \times 2}. + \label{eq:A6_Jpsi} +\end{equation} + +由于不同卫星噪声相互独立，总 Fisher 信息矩阵可按卫星求和。记总 Fisher 信息矩阵关于 $(\boldsymbol\eta_{\mathrm{stat}},\boldsymbol\psi_{\zeta,\mathrm{stat}})$ 的分块形式为 +\begin{equation} + \mathcal I(\boldsymbol\kappa_{\mathrm{stat}}) + = + \begin{bmatrix} + \mathcal I_{\eta\eta} & \mathcal I_{\eta\psi} \\ + \mathcal I_{\psi\eta} & \mathcal I_{\psi\psi} + \end{bmatrix}, +\end{equation} +则根据 \eqref{eq:A6_fim_general} 有 +\begin{equation} + \mathcal I_{\eta\eta} + = + \sum_{\ell=0}^{L-1} + \frac{2}{\sigma_{n,\ell}^2} + \Re\!\left\{ + \mathbf J_{\eta,\ell}^{\Herm}\mathbf J_{\eta,\ell} + \right\}, + \label{eq:A6_Itt} +\end{equation} +\begin{equation} + \mathcal I_{\eta\psi} + = + \begin{bmatrix} + \dfrac{2}{\sigma_{n,0}^2}\Re\!\left\{\mathbf J_{\eta,0}^{\Herm}\mathbf J_{\psi,0}\right\} + & \cdots & + \dfrac{2}{\sigma_{n,L-1}^2}\Re\!\left\{\mathbf J_{\eta,L-1}^{\Herm}\mathbf J_{\psi,L-1}\right\} + \end{bmatrix}, + \label{eq:A6_Itb} +\end{equation} +\begin{equation} + \mathcal I_{\psi\psi} + = + \operatorname{blkdiag} + \left( + \frac{2}{\sigma_{n,0}^2}\Re\!\left\{\mathbf J_{\psi,0}^{\Herm}\mathbf J_{\psi,0}\right\}, + \ldots, + \frac{2}{\sigma_{n,L-1}^2}\Re\!\left\{\mathbf J_{\psi,L-1}^{\Herm}\mathbf J_{\psi,L-1}\right\} + \right). + \label{eq:A6_Ibb} +\end{equation} + +\subsection{消去复干扰系数后的等效 Fisher 信息矩阵} + +由于 $\boldsymbol\psi_{\zeta,\mathrm{stat}}$ 是均值干扰参数，主参数向量 $\boldsymbol\eta_{\mathrm{stat}}$ 的等效 Fisher 信息矩阵为 +\begin{equation} + \mathcal I_{\mathrm e}(\boldsymbol\eta_{\mathrm{stat}}) + = + \mathcal I_{\eta\eta} + - + \mathcal I_{\eta\psi} + \mathcal I_{\psi\psi}^{-1} + \mathcal I_{\psi\eta}. + \label{eq:A6_schur} +\end{equation} + +对每颗卫星，复增益实部和虚部对应的 Jacobian 为 +$\begin{bmatrix}\mathbf g_\ell & j\mathbf g_\ell\end{bmatrix}$。虽然这两个列向量对应的是实参数化下的二维实线性干扰子空间，但其等价于由 $\mathbf g_\ell$ 张成的复一维子空间；相应的 Schur 补可以写成对该复一维子空间的正交投影形式。定义投影矩阵及其正交补为 +\begin{equation} + \mathcal P_{\mathbf g_\ell} + \triangleq + \mathbf g_\ell(\mathbf g_\ell^{\Herm}\mathbf g_\ell)^{-1}\mathbf g_\ell^{\Herm}, + \qquad + \mathcal P_{\mathbf g_\ell}^{\perp} + \triangleq + \mathbf I_{M_\ell T} - \mathcal P_{\mathbf g_\ell}. + \label{eq:A6_proj} +\end{equation} +将 \eqref{eq:A6_Jeta}--\eqref{eq:A6_proj} 代入 \eqref{eq:A6_schur} 并整理，可得 +\begin{equation} + \mathcal I_{\mathrm e}(\boldsymbol\eta_{\mathrm{stat}}) + = + 2\sum_{\ell=0}^{L-1} + \frac{1}{\sigma_{n,\ell}^2} + \Re\!\left\{ + \mathbf J_{\eta,\ell}^{\Herm} + \mathcal P_{\mathbf g_\ell}^{\perp} + \mathbf J_{\eta,\ell} + \right\}. + \label{eq:A6_Ieff_proj} +\end{equation} +再利用 $\mathbf J_{\eta,\ell}=\zeta_\ell\mathbf D_\ell$，得到紧凑形式 +\begin{equation} + \mathcal I_{\mathrm e}(\boldsymbol\eta_{\mathrm{stat}}) + = + 2\sum_{\ell=0}^{L-1} + \frac{|\zeta_\ell|^2}{\sigma_{n,\ell}^2} + \Re\!\left\{ + \mathbf D_\ell^{\Herm} + \mathcal P_{\mathbf g_\ell}^{\perp} + \mathbf D_\ell + \right\}. + \label{eq:A6_Ieff_final} +\end{equation} +式~\eqref{eq:A6_Ieff_final} 即为静态多阵列确定信号 DoA--Doppler 联合估计的等效 \gls{fim}。 + +\subsection{CRB 表达式} + +因此，联合参数向量 $\boldsymbol\eta_{\mathrm{stat}}=[\alpha,\beta,\nu_{\mathrm{ref}}]^{\Tran}$ 的 \gls{crb} 为 +\begin{equation} + \operatorname{CRB}(\boldsymbol\eta_{\mathrm{stat}}) + = + \mathcal I_{\mathrm e}^{-1}(\boldsymbol\eta_{\mathrm{stat}}). + \label{eq:A6_crb_joint} +\end{equation} + +若仅关心角度参数 +\begin{equation} + \boldsymbol\xi + \triangleq + \begin{bmatrix} + \alpha \\ + \beta + \end{bmatrix}, +\end{equation} +则将 \eqref{eq:A6_Ieff_final} 按 $(\boldsymbol\xi,\nu_{\mathrm{ref}})$ 分块为 +\begin{equation} + \mathcal I_{\mathrm e}(\boldsymbol\eta_{\mathrm{stat}}) + = + \begin{bmatrix} + \mathcal I_{\mathrm{ang},\mathrm{ang}} & \mathcal I_{\mathrm{ang},\nu} \\ + \mathcal I_{\nu,\mathrm{ang}} & \mathcal I_{\nu\nu} + \end{bmatrix}, +\end{equation} +其中 $\mathcal I_{\mathrm{ang},\mathrm{ang}}\in\mathbb R^{2\times 2}$，$\mathcal I_{\mathrm{ang},\nu}\in\mathbb R^{2\times 1}$，则 DoA 参数的等效 \gls{crb} 为 +\begin{equation} + \operatorname{CRB}(\boldsymbol\xi) + = + \left( + \mathcal I_{\mathrm{ang},\mathrm{ang}} + - + \mathcal I_{\mathrm{ang},\nu}\,\mathcal I_{\nu\nu}^{-1}\,\mathcal I_{\nu,\mathrm{ang}} + \right)^{-1}. + \label{eq:A6_crb_ang} +\end{equation} +类似地，仅参考归一化 Doppler 参数 $\nu_{\mathrm{ref}}$ 的 \gls{crb} 为 +\begin{equation} + \operatorname{CRB}(\nu_{\mathrm{ref}}) + = + \left( + \mathcal I_{\nu\nu} + - + \mathcal I_{\nu,\mathrm{ang}}\,\mathcal I_{\mathrm{ang},\mathrm{ang}}^{-1}\,\mathcal I_{\mathrm{ang},\nu} + \right)^{-1}. + \label{eq:A6_crb_omega} +\end{equation} + +\subsection{备注} + +\begin{enumerate} + \item 若所有卫星噪声方差相同，即 $\sigma_{n,\ell}^2=\sigma^2$，则 \eqref{eq:A6_Ieff_final} 退化为 + \begin{equation} + \mathcal I_{\mathrm e}(\boldsymbol\eta_{\mathrm{stat}}) + = + \frac{2}{\sigma^2} + \sum_{\ell=0}^{L-1} + |\zeta_\ell|^2 + \Re\!\left\{ + \mathbf D_\ell^{\Herm}\mathcal P_{\mathbf g_\ell}^{\perp}\mathbf D_\ell + \right\}. + \end{equation} + + \item 若进一步将噪声方差 $\{\sigma_{n,\ell}^2\}_{\ell=0}^{L-1}$ 也并入待估参数，则协方差项仅对这些方差参数自身有贡献，且与均值参数块解耦。具体而言，若把第 $\ell$ 条链路的方差参数取为 $\sigma_{n,\ell}^2$，则有 + \begin{equation} + [\mathcal I(\boldsymbol\kappa_{\mathrm{stat}})]_{\sigma_{n,\ell}^2,\sigma_{n,\ell}^2} + = + \frac{M_\ell T}{\sigma_{n,\ell}^4}, + \qquad + [\mathcal I(\boldsymbol\kappa_{\mathrm{stat}})]_{\sigma_{n,\ell}^2,i}=0, + \quad i\neq \sigma_{n,\ell}^2, + \end{equation} + 因此不会改变 \eqref{eq:A6_Ieff_final} 给出的 $\boldsymbol\eta_{\mathrm{stat}}$ 的等效 \gls{fim}。 +\end{enumerate} + +因此，附录 A6 与正文静态多星模型一一对应：正文负责给出观测模型、参数分组和集中似然，本文附录负责给出 Jacobian、Fisher 信息分块与 Schur 补形式。动态模型的连续相位和独立相位 CRB 分别在 \ref{sec:A7_dynDoaDopplerCrb} 与 \ref{sec:A8_dynIndependentPhaseCrb} 中继续展开。 diff --git a/parts/A7-crbDynPilotDoaDopperCont.tex b/parts/A7-crbDynPilotDoaDopperCont.tex new file mode 100644 index 0000000..700f750 --- /dev/null +++ b/parts/A7-crbDynPilotDoaDopperCont.tex @@ -0,0 +1,922 @@ +\section{动态 DoA--Doppler 估计在连续相位与 relaxed 连续相位模型下的 CRB 推导} +\label{sec:A7_dynDoaDopplerCrb} + +本附录沿用第~\ref{sec:dynamicDoaDopplerEst} 节中的动态多帧记号，只讨论两类与正文主线严格对应的模型： +\begin{enumerate} + \item 第~\ref{subsec:dynamicCpModel} 小节中的连续相位模型（structured continuous-phase, CP）； + \item 第~\ref{subsec:dynamicRelaxedModel} 小节中的 relaxed 连续相位模型，即保留连续相位模板，但将每个观测块的复干扰系数 $\zeta_{\ell,p}^{\mathrm{cp,rel}}$ 视为自由参数。 +\end{enumerate} +主参数统一定义为 +\begin{equation} + \boldsymbol\eta_{\mathrm{dyn}} + = + \begin{bmatrix} + \alpha \\ + \beta \\ + \nu_{\mathrm{ref}} + \end{bmatrix}, + \qquad + \boldsymbol\eta_{\mathrm{dyn}}\in\mathbb R^3, +\end{equation} +并将 $\dot\nu_{\mathrm{ref}}$ 作为干扰参数处理。为避免重复建模，这里只保留 \gls{crb} 推导所需的最小定义、导数记号与 \gls{fim}/EFIM 结构。 + +\subsection{沿用正文模型的最小定义与导数记号} + +沿用第~\ref{subsec:dynamicCpModel} 小节中的连续相位模板与块级向量化方式。对任意卫星 $\ell=0,\ldots,L-1$ 与帧 $p=0,\ldots,P-1$，定义连续相位模板 +\begin{equation} + \phi^{\mathrm{cp}}_{\ell,p}(t) + = + \nu_\ell t_{p,t} + \frac{1}{2}\dot\nu_\ell t_{p,t}^2, + \qquad + t_{p,t}=\tau_p+t, + \qquad + t=0,\ldots,T-1, +\end{equation} +其中 +\begin{equation} + \nu_\ell = f_{\nu,\ell}(\boldsymbol\xi,\nu_{\mathrm{ref}}), + \qquad + \dot\nu_\ell = f_{\dot\nu,\ell}(\boldsymbol\xi,\dot\nu_{\mathrm{ref}}). +\end{equation} +按时间堆叠后定义 +\begin{equation} + \mathbf q^{\mathrm{cp}}_{\ell,p} + \triangleq + \begin{bmatrix} + s_p(0)e^{j\phi^{\mathrm{cp}}_{\ell,p}(0)} \\ + s_p(1)e^{j\phi^{\mathrm{cp}}_{\ell,p}(1)} \\ + \vdots \\ + s_p(T-1)e^{j\phi^{\mathrm{cp}}_{\ell,p}(T-1)} + \end{bmatrix} + \in\mathbb C^{T\times 1}, +\end{equation} +以及向量化后的空时模板 +\begin{equation} + \mathbf g^{\mathrm{cp}}_{\ell,p} + \triangleq + \mathbf q^{\mathrm{cp}}_{\ell,p}\otimes \mathbf a_\ell(\boldsymbol\xi) + \in\mathbb C^{M_\ell T\times 1}. +\end{equation} + +对角度参数 $x\in\{\alpha,\beta\}$，记导向矢量导数为 +\begin{equation} + \mathbf a_{\ell,x} + \triangleq + \frac{\partial \mathbf a_\ell(\boldsymbol\xi)}{\partial x}. +\end{equation} +由链式法则可得时间模板导数 +\begin{equation} + \frac{\partial \mathbf q^{\mathrm{cp}}_{\ell,p}}{\partial x} + = + j\left[ + \frac{\partial \nu_\ell}{\partial x} + \operatorname{diag}(t_{p,0},\ldots,t_{p,T-1}) + + + \frac{1}{2} + \frac{\partial \dot\nu_\ell}{\partial x} + \operatorname{diag}(t_{p,0}^2,\ldots,t_{p,T-1}^2) + \right] + \mathbf q^{\mathrm{cp}}_{\ell,p}, + \label{eq:A7_qx_cp} +\end{equation} +并且 +\begin{equation} + \frac{\partial \mathbf q^{\mathrm{cp}}_{\ell,p}}{\partial \nu_{\mathrm{ref}}} + = + j\operatorname{diag}(t_{p,0},\ldots,t_{p,T-1})\mathbf q^{\mathrm{cp}}_{\ell,p}, + \label{eq:A7_qomega_cp} +\end{equation} +\begin{equation} + \frac{\partial \mathbf q^{\mathrm{cp}}_{\ell,p}}{\partial \dot\nu_{\mathrm{ref}}} + = + \frac{j}{2}\operatorname{diag}(t_{p,0}^2,\ldots,t_{p,T-1}^2)\mathbf q^{\mathrm{cp}}_{\ell,p}, + \label{eq:A7_qgamma_cp} +\end{equation} +其中已利用正文中的几何参数化：$\partial \nu_\ell/\partial \nu_{\mathrm{ref}}=1$，$\partial \dot\nu_\ell/\partial \dot\nu_{\mathrm{ref}}=1$。这里的显式导数采用正文中的加性参考星参数化，$\nu_\ell=\nu_{\mathrm{ref}}+\Delta\nu_\ell(\boldsymbol\xi)$ 且 $\dot\nu_\ell=\dot\nu_{\mathrm{ref}}+\Delta\dot\nu_\ell(\boldsymbol\xi)$；若改用一般映射，只需将上述两个导数替换为对应链式法则项。 + +于是空时模板对主参数与 $\dot\nu_{\mathrm{ref}}$ 的导数分别为 +\begin{equation} + \mathbf d^{\mathrm{cp}}_{\ell,p,\alpha} + \triangleq + \frac{\partial \mathbf g^{\mathrm{cp}}_{\ell,p}}{\partial \alpha} + = + \mathbf q^{\mathrm{cp}}_{\ell,p}\otimes \mathbf a_{\ell,\alpha} + + + \frac{\partial \mathbf q^{\mathrm{cp}}_{\ell,p}}{\partial \alpha}\otimes \mathbf a_\ell, +\end{equation} +\begin{equation} + \mathbf d^{\mathrm{cp}}_{\ell,p,\beta} + \triangleq + \frac{\partial \mathbf g^{\mathrm{cp}}_{\ell,p}}{\partial \beta} + = + \mathbf q^{\mathrm{cp}}_{\ell,p}\otimes \mathbf a_{\ell,\beta} + + + \frac{\partial \mathbf q^{\mathrm{cp}}_{\ell,p}}{\partial \beta}\otimes \mathbf a_\ell, +\end{equation} +\begin{equation} + \mathbf d^{\mathrm{cp}}_{\ell,p,\nu} + \triangleq + \frac{\partial \mathbf g^{\mathrm{cp}}_{\ell,p}}{\partial \nu_{\mathrm{ref}}} + = + \frac{\partial \mathbf q^{\mathrm{cp}}_{\ell,p}}{\partial \nu_{\mathrm{ref}}}\otimes \mathbf a_\ell, +\end{equation} +\begin{equation} + \mathbf d^{\mathrm{cp}}_{\ell,p,\dot\nu} + \triangleq + \frac{\partial \mathbf g^{\mathrm{cp}}_{\ell,p}}{\partial \dot\nu_{\mathrm{ref}}} + = + \frac{\partial \mathbf q^{\mathrm{cp}}_{\ell,p}}{\partial \dot\nu_{\mathrm{ref}}}\otimes \mathbf a_\ell. +\end{equation} +记 +\begin{equation} + \mathbf D^{\mathrm{cp}}_{\eta,\ell,p} + \triangleq + \begin{bmatrix} + \mathbf d^{\mathrm{cp}}_{\ell,p,\alpha} & + \mathbf d^{\mathrm{cp}}_{\ell,p,\beta} & + \mathbf d^{\mathrm{cp}}_{\ell,p,\nu} + \end{bmatrix} + \in\mathbb C^{M_\ell T\times 3}, +\end{equation} +以及 +\begin{equation} + \mathbf d^{\mathrm{cp}}_{\dot\nu,\ell,p} + \triangleq + \mathbf d^{\mathrm{cp}}_{\ell,p,\dot\nu} + \in\mathbb C^{M_\ell T\times 1}. +\end{equation} + +将所有 $(\ell,p)$ 的向量化观测按任意固定顺序堆叠为 $\tilde{\mathbf y}$。为避免块顺序引起歧义，记 +\begin{equation} + \mathcal B + \triangleq + \{(\ell,p):\ell=0,\ldots,L-1, + \ p=0,\ldots,P-1\}, +\end{equation} +则对应噪声协方差为 +\begin{equation} + \mathbf R_n + = + \operatorname{blkdiag} + \Big( + \{\sigma_{n,\ell}^2\mathbf I_{M_\ell T}\}_{(\ell,p)\in\mathcal B} + \Big), + \label{eq:A7_Rn_cp_stack} +\end{equation} +其中每个 $(\ell,p)$ 观测块对应一个 $M_\ell T\times M_\ell T$ 的对角噪声协方差块；同一卫星的不同帧共享同一 $\sigma_{n,\ell}^2$。 +由于这里所有待估参数仅通过均值向量进入，而 $\mathbf R_n$ 与这些参数无关，因此对任意实参数 $\kappa_i,\kappa_j$，Fisher 信息矩阵元素统一写为 +\begin{equation} + [\mathcal I(\boldsymbol\kappa)]_{ij} + = + 2\Re\left\{ + \frac{\partial \boldsymbol\mu^{\Herm}(\boldsymbol\kappa)}{\partial \kappa_i} + \mathbf R_n^{-1} + \frac{\partial \boldsymbol\mu(\boldsymbol\kappa)}{\partial \kappa_j} + \right\}. + \label{eq:A7_FIM_general} +\end{equation} + +\subsection{连续相位模型下的 CRB} + +连续相位模型写为 +\begin{equation} + \tilde{\mathbf y}^{\mathrm{cp}}_{\ell,p} + = + \zeta_{\ell,p}^{\mathrm{cp}} + \mathbf g^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}}) + + \tilde{\mathbf n}_{\ell,p}, + \qquad + \tilde{\mathbf n}_{\ell,p}\sim\mathcal{CN}(\mathbf 0,\sigma_{n,\ell}^2\mathbf I_{M_\ell T}). +\end{equation} +记 +\begin{equation} + \zeta_{\ell,p}^{\mathrm{cp}}\triangleq \rho_{\ell,p}^{\mathrm{cp}}e^{j\chi_\ell},\qquad \rho_{\ell,p}^{\mathrm{cp}}\in\mathbb R_{\ge0}. +\end{equation} +在常规局部 \gls{crb} 推导中默认真实工作点满足 $\rho_{\ell,p}^{\mathrm{cp}}>0$，从而幅度--相位参数化在该点局部可微且相位参数可辨识；若某些块的幅度为零，则对应相位不可辨识，应在数值实现中删除该相位列或采用广义逆形式的 EFIM。 +则第 $(\ell,p)$ 个观测块的均值为 +\begin{equation} + \boldsymbol\mu^{\mathrm{cp}}_{\ell,p} + = + \zeta_{\ell,p}^{\mathrm{cp}}\mathbf g^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}}). +\end{equation} + +定义连续相位模型的干扰参数向量为 +\begin{equation} + \boldsymbol\psi_{\mathrm{cp},\mu} + = + \Big[ + \dot\nu_{\mathrm{ref}}, + \{\chi_\ell\}_{\ell=0}^{L-1}, + \{\rho_{\ell,p}^{\mathrm{cp}}:\ell=0,\ldots,L-1,\ p=0,\ldots,P-1\} + \Big]^{\Tran}, +\end{equation} +并令总参数向量为 +\begin{equation} + \boldsymbol\kappa_{\mathrm{cp}} + = + \begin{bmatrix} + \boldsymbol\eta_{\mathrm{dyn}}^{\Tran} & \boldsymbol\psi_{\mathrm{cp},\mu}^{\Tran} + \end{bmatrix}^{\Tran}. +\end{equation} + +对单个观测块，有 +\begin{equation} + \frac{\partial \boldsymbol\mu^{\mathrm{cp}}_{\ell,p}}{\partial \boldsymbol\eta_{\mathrm{dyn}}} + = + \zeta_{\ell,p}^{\mathrm{cp}}\mathbf D^{\mathrm{cp}}_{\eta,\ell,p}, + \label{eq:A7_Jeta_cp_block} +\end{equation} +\begin{equation} + \frac{\partial \boldsymbol\mu^{\mathrm{cp}}_{\ell,p}}{\partial \dot\nu_{\mathrm{ref}}} + = + \zeta_{\ell,p}^{\mathrm{cp}}\mathbf d^{\mathrm{cp}}_{\dot\nu,\ell,p}, + \label{eq:A7_Jgamma_cp_block} +\end{equation} +以及对任意 $m\in\{0,\ldots,L-1\}$ 与任意 $(m,n)$， +\begin{equation} + \frac{\partial \boldsymbol\mu^{\mathrm{cp}}_{\ell,p}}{\partial \chi_m} + = + \begin{cases} + j \zeta_{\ell,p}^{\mathrm{cp}}\mathbf g^{\mathrm{cp}}_{\ell,p}, & m=\ell,\\[0.4em] + \mathbf 0, & m\neq \ell, + \end{cases} + \label{eq:A7_Jphi_cp_block} +\end{equation} +\begin{equation} + \frac{\partial \boldsymbol\mu^{\mathrm{cp}}_{\ell,p}}{\partial \rho_{m,n}^{\mathrm{cp}}} + = + \begin{cases} + e^{j\chi_\ell}\mathbf g^{\mathrm{cp}}_{\ell,p}, & (m,n)=(\ell,p),\\[0.4em] + \mathbf 0, & (m,n)\neq(\ell,p). + \end{cases} + \label{eq:A7_Jb_cp_block} +\end{equation} + +将所有观测块联合堆叠后，记相应 Jacobian 为 +\begin{equation} + \mathbf J^{\mathrm{cp}}_{\eta} + \triangleq + \frac{\partial \boldsymbol\mu^{\mathrm{cp}}}{\partial \boldsymbol\eta_{\mathrm{dyn}}}, + \qquad + \mathbf J^{\mathrm{cp}}_{\psi} + \triangleq + \frac{\partial \boldsymbol\mu^{\mathrm{cp}}}{\partial \boldsymbol\psi_{\mathrm{cp},\mu}}. +\end{equation} +于是根据 \eqref{eq:A7_FIM_general}，连续相位模型的 FIM 分块为 +\begin{equation} + \mathcal I_{\eta\eta}^{\mathrm{cp}} + = + 2\Re\left\{ + (\mathbf J^{\mathrm{cp}}_{\eta})^{\Herm} + \mathbf R_n^{-1} + \mathbf J^{\mathrm{cp}}_{\eta} + \right\}, +\end{equation} +\begin{equation} + \mathcal I_{\eta\psi}^{\mathrm{cp}} + = + 2\Re\left\{ + (\mathbf J^{\mathrm{cp}}_{\eta})^{\Herm} + \mathbf R_n^{-1} + \mathbf J^{\mathrm{cp}}_{\psi} + \right\}, +\end{equation} +\begin{equation} + \mathcal I_{\psi\psi}^{\mathrm{cp}} + = + 2\Re\left\{ + (\mathbf J^{\mathrm{cp}}_{\psi})^{\Herm} + \mathbf R_n^{-1} + \mathbf J^{\mathrm{cp}}_{\psi} + \right\}. +\end{equation} +因此，主参数 $\boldsymbol\eta_{\mathrm{dyn}}$ 在连续相位模型下的等效 Fisher 信息矩阵为 +\begin{equation} + \mathcal I_{\mathrm e}^{\mathrm{cp}}(\boldsymbol\eta_{\mathrm{dyn}}) + = + \mathcal I_{\eta\eta}^{\mathrm{cp}} + - + \mathcal I_{\eta\psi}^{\mathrm{cp}} + \big(\mathcal I_{\psi\psi}^{\mathrm{cp}}\big)^{-1} + \mathcal I_{\psi\eta}^{\mathrm{cp}}, + \label{eq:A7_Ieff_cp} +\end{equation} +其中上式默认 $\mathcal I_{\psi\psi}^{\mathrm{cp}}$ 在考虑的局部工作点非奇异。若由于低信噪比、零幅度块或几何退化导致干扰参数块秩亏，则数值仿真中应将 $\big(\mathcal I_{\psi\psi}^{\mathrm{cp}}\big)^{-1}$ 替换为 Moore--Penrose 广义逆 $\big(\mathcal I_{\psi\psi}^{\mathrm{cp}}\big)^{\dagger}$，并将所得结果解释为广义 EFIM。 +从而其 CRB 为 +\begin{equation} + \operatorname{CRB}_{\mathrm{cp}}(\boldsymbol\eta_{\mathrm{dyn}}) + = + \big(\mathcal I_{\mathrm e}^{\mathrm{cp}}(\boldsymbol\eta_{\mathrm{dyn}})\big)^{-1}. + \label{eq:A7_CRB_cp} +\end{equation} + +需要强调的是，与 relaxed 情形不同，连续相位模型中的 $\{\chi_\ell\}$ 在同一卫星的多帧之间共享，而 $\{\rho_{\ell,p}^{\mathrm{cp}}\}$ 为实参数，因此其 Schur 补一般不能进一步化成逐块独立的简单投影求和式。式~\eqref{eq:A7_Ieff_cp} 是与正文模型严格一致且最稳妥的写法。 + +\subsection{Known-rate/unknown-rate EFIM 关系与单星 DoA 正交性} +\label{subsec:A7_knownUnknownRateOrthogonality} + +为解释 known-rate 与 unknown-rate 下 DoA bound 可能重合的现象，本小节把 Doppler-rate 干扰参数从其他幅度/相位干扰参数中单独分离出来。 +记 +\begin{equation} + \boldsymbol\psi_{\mathrm{cp},0} + = + \Big[ + \{\chi_\ell\}_{\ell=0}^{L-1}, + \{\rho_{\ell,p}^{\mathrm{cp}}:\ell=0,\ldots,L-1,\ p=0,\ldots,P-1\} + \Big]^\Tran +\end{equation} +为不含 $\dot\nu_{\mathrm{ref}}$ 的幅度/相位干扰参数。 +若 $\dot\nu_{\mathrm{ref}}$ 已知，则先消去 $\boldsymbol\psi_{\mathrm{cp},0}$ 后，主参数 $\boldsymbol\eta_{\mathrm{dyn}}$ 的 known-rate EFIM 可写为 +\begin{equation} + \mathcal I_{\mathrm e}^{\mathrm{cp,K}} + = + \mathcal I_{\eta\eta}^{(0)} + - + \mathcal I_{\eta\psi_0}^{(0)} + \big(\mathcal I_{\psi_0\psi_0}^{(0)}\big)^\dagger + \mathcal I_{\psi_0\eta}^{(0)}, + \label{eq:A7_CP_K_EFIM} +\end{equation} +其中上标 $(0)$ 表示只按参数 $(\boldsymbol\eta_{\mathrm{dyn}},\boldsymbol\psi_{\mathrm{cp},0})$ 构造的 FIM 分块。 +若进一步把 $\dot\nu_{\mathrm{ref}}$ 作为干扰参数，则可在消去 $\boldsymbol\psi_{\mathrm{cp},0}$ 后，将扩展 EFIM 按 $(\boldsymbol\eta_{\mathrm{dyn}},\dot\nu_{\mathrm{ref}})$ 分块为 +\begin{equation} + \mathcal I_{\mathrm e}^{\mathrm{cp,ext}} + = + \begin{bmatrix} + \mathcal I_{\eta\eta}^{\mathrm K} & \mathbf i_{\eta\dot\nu} \\ + \mathbf i_{\dot\nu\eta} & i_{\dot\nu\dot\nu} + \end{bmatrix}. + \label{eq:A7_CP_ext_EFIM_partition} +\end{equation} +于是 unknown-rate EFIM 为 +\begin{equation} + \mathcal I_{\mathrm e}^{\mathrm{cp,U}} + = + \mathcal I_{\eta\eta}^{\mathrm K} + - + \mathbf i_{\eta\dot\nu} + i_{\dot\nu\dot\nu}^{\dagger} + \mathbf i_{\dot\nu\eta}. + \label{eq:A7_CP_U_EFIM_from_K} +\end{equation} +式~\eqref{eq:A7_CP_U_EFIM_from_K} 中的第二项不是一个必然很大的性能退化项，而是对 $\boldsymbol\eta_{\mathrm{dyn}}$ 的 residualized score 与 $\dot\nu_{\mathrm{ref}}$ 的 residualized nuisance score 之间耦合程度的 accounting。 +若该残差耦合很弱，则 known-rate 与 unknown-rate bound 可以非常接近。 + +进一步将 known-rate EFIM 和 rate-coupling 向量按 $\boldsymbol\eta_{\mathrm{dyn}}=[\boldsymbol\xi^\Tran,\nu_{\mathrm{ref}}]^\Tran$ 分块： +\begin{equation} + \mathcal I_{\eta\eta}^{\mathrm K} + = + \begin{bmatrix} + \mathbf A_{\xi\xi}^{\mathrm K} & \mathbf a_{\xi\nu}^{\mathrm K} \\ + \mathbf a_{\nu\xi}^{\mathrm K} & a_{\nu\nu}^{\mathrm K} + \end{bmatrix}, + \qquad + \mathbf i_{\eta\dot\nu} + = + \begin{bmatrix} + \mathbf i_{\xi\dot\nu} \\ + i_{\nu\dot\nu} + \end{bmatrix}. + \label{eq:A7_CP_K_partition_xi_nu_rate} +\end{equation} +如果 +\begin{equation} + \mathbf i_{\xi\dot\nu}=\mathbf 0, + \qquad + \mathbf a_{\xi\nu}^{\mathrm K}=\mathbf 0, + \label{eq:A7_SS_xi_rate_orthogonality_condition} +\end{equation} +则消去 $\dot\nu_{\mathrm{ref}}$ 只会改变 $\nu_{\mathrm{ref}}$ 对应的标量块，而不会改变 DoA 的 CRB 子块。 +换言之，DoA 的 known-rate 与 unknown-rate CRB 完全一致。 + +在 single-satellite multi-frame case 中，若唯一卫星即 reference satellite，则 +\begin{equation} + f_{\nu,\ell_{\mathrm{ref}}}(\boldsymbol\xi,\nu_{\mathrm{ref}}) + = + \nu_{\mathrm{ref}}, + \qquad + f_{\dot\nu,\ell_{\mathrm{ref}}}(\boldsymbol\xi,\dot\nu_{\mathrm{ref}}) + = + \dot\nu_{\mathrm{ref}}. + \label{eq:A7_ref_link_mapping_no_xi} +\end{equation} +因此 $\partial \nu_{\ell_{\mathrm{ref}}}/\partial x=0$ 且 $\partial \dot\nu_{\ell_{\mathrm{ref}}}/\partial x=0$ 对 $x\in\{\alpha,\beta\}$ 成立。 +由式~\eqref{eq:A7_qx_cp} 可知，reference-link 的 DoA 导数不含 Doppler-mapping-induced temporal phase derivative，而只含空间阵列导向导数。 +在幅度和公共相位干扰方向被投影消去后，该 DoA score 的残差部分是与 $\mathbf a_{\ell_{\mathrm{ref}}}(\boldsymbol\xi)$ 正交的空间切向量；相反，$\nu_{\mathrm{ref}}$ 和 $\dot\nu_{\mathrm{ref}}$ 的残差 score 仍具有沿 reference steering vector 的时间相位结构。 +在标准窄带阵列流形下，二者的 Fisher 内积为零，因此式~\eqref{eq:A7_SS_xi_rate_orthogonality_condition} 成立。 +这解释了 single-satellite multi-frame known-rate 与 unknown-rate DoA CRB 的严格重合。 + +在 multi-satellite case 中，非参考星通常满足 +\begin{equation} + \nu_\ell + = + \nu_{\mathrm{ref}}+\Delta\nu_\ell(\boldsymbol\xi), + \qquad + \dot\nu_\ell + = + \dot\nu_{\mathrm{ref}}+\Delta\dot\nu_\ell(\boldsymbol\xi), + \qquad + \ell\neq\ell_{\mathrm{ref}}. +\end{equation} +此时 DoA 导数包含由 $\partial\Delta\nu_\ell/\partial x$ 与 $\partial\Delta\dot\nu_\ell/\partial x$ 诱导的时间相位分量。 +因此，$\mathbf i_{\xi\dot\nu}$ 和 $\mathbf a_{\xi\nu}^{\mathrm K}$ 一般不再严格为零，known-rate 与 unknown-rate DoA CRB 的完全相等性也不应被作为一般结论。 +若数值上 multi-satellite DoA bound 仍与 known-rate benchmark 基本重合，更稳妥的解释是：在所选 representative geometry 下，DoA residual score 与 residualized Doppler-rate nuisance direction 的耦合较弱。 +这一定量现象适合作为 EFIM accounting 的 sanity check，而不应升级为主要理论贡献。 + +\subsection{relaxed 连续相位模型下的 CRB} + +沿用第~\ref{subsec:dynamicRelaxedModel} 小节中的 relaxed 连续相位处理，每个观测块的复干扰系数 $\zeta_{\ell,p}^{\mathrm{cp,rel}}$ 视为自由参数，即 +\begin{equation} + \tilde{\mathbf y}^{\mathrm{cp}}_{\ell,p} + = + \zeta_{\ell,p}^{\mathrm{cp,rel}} + \mathbf g^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}}) + + \tilde{\mathbf n}_{\ell,p}, + \qquad + \zeta_{\ell,p}^{\mathrm{cp,rel}}=\zeta^{(r)}_{\ell,p}+j\zeta^{(i)}_{\ell,p}. +\end{equation} +这里模板仍然是 $\mathbf g^{\mathrm{cp}}_{\ell,p}$，因此该模型只是放松了块复系数结构，并未放弃跨帧连续相位模板。 + +令扩展参数向量为 +\begin{equation} + \boldsymbol\eta_{\mathrm{ext}} + \triangleq + \begin{bmatrix} + \boldsymbol\eta_{\mathrm{dyn}} \\ + \dot\nu_{\mathrm{ref}} + \end{bmatrix} + = + \begin{bmatrix} + \alpha \\ + \beta \\ + \nu_{\mathrm{ref}} \\ + \dot\nu_{\mathrm{ref}} + \end{bmatrix}, +\end{equation} +并记 +\begin{equation} + \mathbf D^{\mathrm{rel}}_{\mathrm{ext},\ell,p} + \triangleq + \begin{bmatrix} + \mathbf d^{\mathrm{cp}}_{\ell,p,\alpha} & + \mathbf d^{\mathrm{cp}}_{\ell,p,\beta} & + \mathbf d^{\mathrm{cp}}_{\ell,p,\nu} & + \mathbf d^{\mathrm{cp}}_{\dot\nu,\ell,p} + \end{bmatrix} + \in\mathbb C^{M_\ell T\times 4}. +\end{equation} +则单个观测块均值 +\begin{equation} + \boldsymbol\mu^{\mathrm{rel}}_{\ell,p} + = + \zeta_{\ell,p}^{\mathrm{cp,rel}}\mathbf g^{\mathrm{cp}}_{\ell,p}(\boldsymbol\eta_{\mathrm{ext}}) +\end{equation} +对 $\boldsymbol\eta_{\mathrm{ext}}$ 的 Jacobian 为 +\begin{equation} + \mathbf J^{\mathrm{rel}}_{\mathrm{ext},\ell,p} + \triangleq + \frac{\partial \boldsymbol\mu^{\mathrm{rel}}_{\ell,p}}{\partial \boldsymbol\eta_{\mathrm{ext}}} + = + \zeta_{\ell,p}^{\mathrm{cp,rel}}\mathbf D^{\mathrm{rel}}_{\mathrm{ext},\ell,p}, +\end{equation} +对块复干扰系数的实部与虚部的 Jacobian 为 +\begin{equation} + \mathbf J^{\mathrm{rel}}_{\zeta,\ell,p} + \triangleq + \frac{\partial \boldsymbol\mu^{\mathrm{rel}}_{\ell,p}}{\partial + \begin{bmatrix} + \zeta^{(r)}_{\ell,p} & \zeta^{(i)}_{\ell,p} + \end{bmatrix}^{\Tran}} + = + \begin{bmatrix} + \mathbf g^{\mathrm{cp}}_{\ell,p} & j\mathbf g^{\mathrm{cp}}_{\ell,p} + \end{bmatrix}. +\end{equation} + +由此构造对应的 FIM 分块，并对所有块复系数执行 Schur 补，可得扩展参数 $\boldsymbol\eta_{\mathrm{ext}}$ 的等效 Fisher 信息矩阵 +\begin{equation} + \mathcal I_{\mathrm e}^{\mathrm{rel}}(\boldsymbol\eta_{\mathrm{ext}}) + = + 2\sum_{\ell=0}^{L-1}\sum_{p=0}^{P-1} + \frac{|\zeta_{\ell,p}^{\mathrm{cp,rel}}|^2}{\sigma_{n,\ell}^2} + \Re\left\{ + \big(\mathbf D^{\mathrm{rel}}_{\mathrm{ext},\ell,p}\big)^{\Herm} + \mathcal P^{\perp}_{\mathbf g^{\mathrm{cp}}_{\ell,p}} + \mathbf D^{\mathrm{rel}}_{\mathrm{ext},\ell,p} + \right\}, + \label{eq:A7_Ieff_rel_ext} +\end{equation} +其中 +\begin{equation} + \mathcal P_{\mathbf g^{\mathrm{cp}}_{\ell,p}} + \triangleq + \mathbf g^{\mathrm{cp}}_{\ell,p} + \big((\mathbf g^{\mathrm{cp}}_{\ell,p})^{\Herm}\mathbf g^{\mathrm{cp}}_{\ell,p}\big)^{-1} + (\mathbf g^{\mathrm{cp}}_{\ell,p})^{\Herm}, +\end{equation} +\begin{equation} + \mathcal P^{\perp}_{\mathbf g^{\mathrm{cp}}_{\ell,p}} + \triangleq + \mathbf I_{M_\ell T}-\mathcal P_{\mathbf g^{\mathrm{cp}}_{\ell,p}}. +\end{equation} + +若关心的是联合参数 $\boldsymbol\eta_{\mathrm{ext}}=[\boldsymbol\eta_{\mathrm{dyn}}^{\Tran},\dot\nu_{\mathrm{ref}}]^{\Tran}$ 的 CRB，则直接有 +\begin{equation} + \operatorname{CRB}_{\mathrm{rel}}(\boldsymbol\eta_{\mathrm{ext}}) + = + \big(\mathcal I_{\mathrm e}^{\mathrm{rel}}(\boldsymbol\eta_{\mathrm{ext}})\big)^{-1}. + \label{eq:A7_CRB_rel_ext} +\end{equation} + +若进一步仍只把 $\boldsymbol\eta_{\mathrm{dyn}}$ 视为感兴趣参数，而将 $\dot\nu_{\mathrm{ref}}$ 作为干扰参数，则将 \eqref{eq:A7_Ieff_rel_ext} 按 +\begin{equation} + \mathcal I_{\mathrm e}^{\mathrm{rel}}(\boldsymbol\eta_{\mathrm{ext}}) + = + \begin{bmatrix} + \mathcal I_{\eta\eta}^{\mathrm{rel}} & \mathcal I_{\eta\dot\nu}^{\mathrm{rel}} \\ + \mathcal I_{\dot\nu\eta}^{\mathrm{rel}} & \mathcal I_{\dot\nu\dot\nu}^{\mathrm{rel}} + \end{bmatrix} +\end{equation} +分块，则对应的主参数等效 Fisher 信息矩阵为 +\begin{equation} + \mathcal I_{\mathrm e,\eta}^{\mathrm{rel}}(\boldsymbol\eta_{\mathrm{dyn}}) + = + \mathcal I_{\eta\eta}^{\mathrm{rel}} + - + \mathcal I_{\eta\dot\nu}^{\mathrm{rel}} + \big(\mathcal I_{\dot\nu\dot\nu}^{\mathrm{rel}}\big)^{-1} + \mathcal I_{\dot\nu\eta}^{\mathrm{rel}}, + \label{eq:A7_Ieff_rel_eta} +\end{equation} +于是 +\begin{equation} + \operatorname{CRB}_{\mathrm{rel}}(\boldsymbol\eta_{\mathrm{dyn}}) + = + \big(\mathcal I_{\mathrm e,\eta}^{\mathrm{rel}}(\boldsymbol\eta_{\mathrm{dyn}})\big)^{-1}. + \label{eq:A7_CRB_rel_eta} +\end{equation} + + + + + +\subsection{EFIM 病态与 DoA--Doppler 耦合的投影机制} +\label{subsec:A7_efimIllConditioning} + +式~\eqref{eq:A7_Ieff_cp} 与式~\eqref{eq:A7_Ieff_rel_eta} 都是对完整 Fisher 信息矩阵中的 +干扰参数执行 Schur 补后得到的 EFIM。本小节给出一个与具体 CP/IP 模型无关的局部线性解释， +用于说明多星场景中角度 CRB 显著改善但 $\nu_{\mathrm{ref}}$ 边际 CRB 改善有限、同时 EFIM 条件数 +变大的原因。 + +将所有观测块堆叠后的均值 Jacobian 写为 +\begin{equation} + \mathbf J + = + \begin{bmatrix} + \mathbf J_{\eta} & \mathbf J_{\psi} + \end{bmatrix}, + \qquad + \mathbf J_{\eta} + \triangleq + \frac{\partial\boldsymbol\mu}{\partial\boldsymbol\eta_{\mathrm{dyn}}}, + \qquad + \mathbf J_{\psi} + \triangleq + \frac{\partial\boldsymbol\mu}{\partial\boldsymbol\psi_\mu}. +\end{equation} +这里 $\boldsymbol\psi_\mu$ 表示只影响均值向量的干扰参数子集；噪声方差参数若并入完整参数， +由于与均值参数 Fisher 块解耦，不影响下面的投影解释。令 +\begin{equation} + \bar{\mathbf J}_{\eta} + \triangleq + \mathbf R_n^{-1/2}\mathbf J_{\eta}, + \qquad + \bar{\mathbf J}_{\psi} + \triangleq + \mathbf R_n^{-1/2}\mathbf J_{\psi} +\end{equation} +为白化 Jacobian。则 Schur 补可写成 +\begin{equation} + \mathcal I_{\mathrm{e}} + = + 2\Re\left\{ + \bar{\mathbf J}_{\eta}^{\Herm} + \mathcal P_{\bar{\mathbf J}_{\psi}}^{\perp} + \bar{\mathbf J}_{\eta} + \right\}, + \qquad + \mathcal P_{\bar{\mathbf J}_{\psi}}^{\perp} + = + \mathbf I- + \bar{\mathbf J}_{\psi} + \left(\bar{\mathbf J}_{\psi}^{\Herm}\bar{\mathbf J}_{\psi}\right)^{\dagger} + \bar{\mathbf J}_{\psi}^{\Herm}. + \label{eq:A7_efim_projection_interpret} +\end{equation} +因此，对任意实扰动 $\delta\boldsymbol\eta$，有 +\begin{equation} + \delta\boldsymbol\eta^\Tran + \mathcal I_{\mathrm{e}} + \delta\boldsymbol\eta + = + 2\left\| + \mathcal P_{\bar{\mathbf J}_{\psi}}^{\perp} + \bar{\mathbf J}_{\eta}\delta\boldsymbol\eta + \right\|_2^2. + \label{eq:A7_efim_quadratic_projection} +\end{equation} +由此得到如下判据： +\begin{equation} + \mathcal I_{\mathrm{e}}\ \text{奇异} + \quad\Longleftrightarrow\quad + \exists\ \delta\boldsymbol\eta\neq\mathbf 0, + \quad + \mathcal P_{\bar{\mathbf J}_{\psi}}^{\perp} + \bar{\mathbf J}_{\eta}\delta\boldsymbol\eta + = + \mathbf 0, + \label{eq:A7_exact_absorbed_direction} +\end{equation} +也即 +\begin{equation} + \bar{\mathbf J}_{\eta}\delta\boldsymbol\eta + \in + \operatorname{range}(\bar{\mathbf J}_{\psi}). + \label{eq:A7_range_absorbed_direction} +\end{equation} +近奇异情形对应于 +\begin{equation} + \bar{\mathbf J}_{\eta}\delta\boldsymbol\eta + \approx + \bar{\mathbf J}_{\psi}\delta\boldsymbol\psi, + \label{eq:A7_near_nuisance_absorption} +\end{equation} +即主参数扰动产生的一阶均值变化可以由干扰参数扰动近似表示。 + +为了把这种近似表示写成可度量的量，定义最小投影角 +\begin{equation} + \sin^2\theta_{\min} + \triangleq + \min_{\delta\boldsymbol\eta\neq\mathbf 0} + \frac{ + \left\| + \mathcal P_{\bar{\mathbf J}_{\psi}}^{\perp} + \bar{\mathbf J}_{\eta}\delta\boldsymbol\eta + \right\|_2^2 + }{ + \left\| + \bar{\mathbf J}_{\eta}\delta\boldsymbol\eta + \right\|_2^2 + }. + \label{eq:A7_min_projection_angle} +\end{equation} +式~\eqref{eq:A7_efim_quadratic_projection} 表明，EFIM 在某个方向上的有效信息正比于该方向 +score 在干扰子空间正交补上的残余能量。因此，当 $\theta_{\min}\approx0$ 时， +$\mathcal I_{\mathrm{e}}$ 会出现小特征值；若同时多星几何使其他方向的信息快速增加， +则 $\kappa(\mathcal I_{\mathrm{e}})$ 会显著变大。 + +接下来把上述投影解释专门用于 DoA--Doppler 耦合。将 EFIM 按 +$\boldsymbol\eta_{\mathrm{dyn}}=[\boldsymbol\xi^\Tran,\nu_{\mathrm{ref}}]^\Tran$ 分块为 +\begin{equation} + \mathcal I_{\mathrm{e}} + = + \begin{bmatrix} + \mathcal I_{\xi\xi} & \mathcal I_{\xi\nu} \\ + \mathcal I_{\nu\xi} & \mathcal I_{\nu\nu} + \end{bmatrix}. + \label{eq:A7_efim_xi_nu_partition} +\end{equation} +若 $\boldsymbol\xi$ 已知，则参考 Doppler 的 scalar bound 为 +\begin{equation} + C_{\nu}^{\mathrm{sc}} + = + \mathcal I_{\nu\nu}^{-1}. + \label{eq:A7_nu_scalar_bound} +\end{equation} +若 $\boldsymbol\xi$ 未知，则 $\nu_{\mathrm{ref}}$ 的边际 Fisher 信息为 +\begin{equation} + \mathcal I_{\nu|\xi} + = + \mathcal I_{\nu\nu} + - + \mathcal I_{\nu\xi} + \mathcal I_{\xi\xi}^{\dagger} + \mathcal I_{\xi\nu}, + \label{eq:A7_nu_given_xi_schur} +\end{equation} +对应边际 CRB 为 +\begin{equation} + C_{\nu}^{\mathrm{mar}} + = + \mathcal I_{\nu|\xi}^{-1}. + \label{eq:A7_nu_marginal_bound} +\end{equation} +为遵守第~\ref{sec:assumptionsAndNotation} 节中 $\rho$ 只表示幅度的约定，这里不用 $\rho$ 表示相关系数， +而定义局部 DoA--Doppler 耦合比例 +\begin{equation} + \gamma_{\xi,\nu}^{2} + \triangleq + \frac{ + \mathcal I_{\nu\xi} + \mathcal I_{\xi\xi}^{\dagger} + \mathcal I_{\xi\nu} + }{ + \mathcal I_{\nu\nu} + }. + \label{eq:A7_xi_nu_coupling_fraction} +\end{equation} +在 $\mathcal I_{\mathrm{e}}$ 半正定且 $\mathcal I_{\nu\nu}>0$ 的正则情形下， +$0\leq\gamma_{\xi,\nu}^{2}\leq1$，并且 +\begin{equation} + \mathcal I_{\nu|\xi} + = + \mathcal I_{\nu\nu}(1-\gamma_{\xi,\nu}^{2}), + \qquad + C_{\nu}^{\mathrm{mar}} + = + \frac{1}{\mathcal I_{\nu\nu}(1-\gamma_{\xi,\nu}^{2})}. + \label{eq:A7_nu_info_after_coupling} +\end{equation} +于是边际化 DoA 所造成的 Doppler 标准差放大量为 +\begin{equation} + \frac{\sqrt{C_{\nu}^{\mathrm{mar}}}} + {\sqrt{C_{\nu}^{\mathrm{sc}}}} + = + \frac{1}{\sqrt{1-\gamma_{\xi,\nu}^{2}}}. + \label{eq:A7_nu_std_inflation} +\end{equation} +这说明即使多星使 $\mathcal I_{\nu\nu}$ 增大，只要 +$\mathcal I_{\nu\xi}\mathcal I_{\xi\xi}^{\dagger}\mathcal I_{\xi\nu}$ 同时较大， +Schur 补后的 $\mathcal I_{\nu|\xi}$ 仍然可能增长缓慢。 + +若用上标 $(L)$ 表示 $L$ 颗协作卫星下的量，则参考 Doppler 的多星边际增益可写为 +\begin{equation} + G_{\nu}(L) + = + \frac{\sqrt{C_{\nu}^{\mathrm{mar},(1)}}} + {\sqrt{C_{\nu}^{\mathrm{mar},(L)}}} + = + \sqrt{ + \frac{ + \mathcal I_{\nu\nu}^{(L)} + \left(1-\gamma_{\xi,\nu}^{2,(L)}\right) + }{ + \mathcal I_{\nu\nu}^{(1)} + \left(1-\gamma_{\xi,\nu}^{2,(1)}\right) + } + }. + \label{eq:A7_multisat_nu_gain} +\end{equation} +因此，$\nu_{\mathrm{ref}}$ 的多星增益由两个因子共同决定：一个是 scalar 信息 +$\mathcal I_{\nu\nu}^{(L)}$ 的能量叠加，另一个是耦合惩罚 +$1-\gamma_{\xi,\nu}^{2,(L)}$。这解释了为什么两星或少数多星情形中，角度 CRB 可能因 +$\mathcal I_{\xi\xi}$ 增强而明显降低，而 $\nu_{\mathrm{ref}}$ 的边际 CRB 只出现温和改善。 + +对本文的动态连续相位模型，造成 $\gamma_{\xi,\nu}^{2}$ 较大的主要来源有三类。第一，幅度与相位 +干扰参数张成了与 $\mathbf g_{\ell,p}^{\mathrm{cp}}$ 和 +$j\mathbf g_{\ell,p}^{\mathrm{cp}}$ 相关的局部子空间，主参数导数中落在该子空间的分量会被 +式~\eqref{eq:A7_efim_projection_interpret} 投影掉。第二，短观测窗口内 +$\nu_{\mathrm{ref}}$ 和 $\dot\nu_{\mathrm{ref}}$ 都表现为低阶时间相位项， +$\dot\nu_{\mathrm{ref}}$ 被作为干扰参数消去后会削弱公共 Doppler 截距方向。第三， +$\boldsymbol\xi$ 不仅通过 $\mathbf a_\ell(\boldsymbol\xi)$ 改变阵列流形，也通过 +$\Delta\nu_\ell(\boldsymbol\xi)$ 和 $\Delta\dot\nu_\ell(\boldsymbol\xi)$ 改变各星相对 Doppler， +这些几何差分项可以与公共 $\nu_{\mathrm{ref}}$ 扰动形成近线性组合。 + +在数值实现中，建议同时报告 +$\sqrt{C_{\nu}^{\mathrm{mar}}}$、$\sqrt{C_{\nu}^{\mathrm{sc}}}$、 +$\gamma_{\xi,\nu}^{2}$、$\theta_{\min}$ 以及尺度归一化后的条件数。若 +$\sqrt{C_{\nu}^{\mathrm{sc}}}$ 随星数增加而改善，但 +$\sqrt{C_{\nu}^{\mathrm{mar}}}$ 改善有限，同时 $\gamma_{\xi,\nu}^{2}$ 较大， +则应解释为 DoA--Doppler Schur 补损失，而不是 Doppler 模板本身没有信息。 + + + + +\subsection{周期帧窗口与 Doppler tooth ambiguity} +\label{subsec:A7_frameTimeCombTooth} + +本附录前面得到的 EFIM 是局部二阶量；它默认估计点位于同一个 Doppler tooth 的局部邻域内。 +为了描述全局 profile likelihood 的离散多峰结构，需要单独分析帧起始时刻集合所诱导的谱窗口。 +按照表~\ref{tab:indexDimensionNotation} 的约定，$\tau_p$ 表示第 $p$ 帧起始点相对于参考时刻的离散采样偏移， +归一化角 Doppler 偏移记为 $\Delta\nu$。若只保留由帧起始时刻造成的公共相位项，则跨帧相干项可写为 +\begin{equation} + W_{\mathcal P}(\Delta\nu) + \triangleq + \sum_{p\in\mathcal P}w_p\exp\{j\Delta\nu\tau_p\}, + \qquad w_p\geq0. + \label{eq:A7_frame_window_general} +\end{equation} +这里 $w_p$ 可以看作在局部线性化和幅度消元后，第 $p$ 帧对 Doppler 匹配统计量的等效权重。 +对于周期帧，$\tau_p=p\Delta\tau_{\mathrm{f}}$。若 $w_p$ 在短窗口内近似常数，则 +\begin{equation} + W_{\mathcal P}(\Delta\nu) + \propto + \sum_{p=0}^{P-1}\exp\{j\Delta\nu p\Delta\tau_{\mathrm{f}}\} + = + \exp\left\{j\frac{(P-1)\Delta\nu\Delta\tau_{\mathrm{f}}}{2}\right\} + \frac{ + \sin(P\Delta\nu\Delta\tau_{\mathrm{f}}/2) + }{ + \sin(\Delta\nu\Delta\tau_{\mathrm{f}}/2) + }. + \label{eq:A7_frame_window_dirichlet} +\end{equation} +因此所有满足 +\begin{equation} + \Delta\nu + = + k\Delta\nu_{\mathrm{tooth}}, + \qquad + \Delta\nu_{\mathrm{tooth}} + \triangleq + \frac{2\pi}{\Delta\tau_{\mathrm{f}}}, + \qquad k\in\mathbb Z, + \label{eq:A7_tooth_spacing_discrete} +\end{equation} +的 Doppler 偏移在帧起始点上具有相同相位。换算到物理 Doppler，利用 +$\nu=2\pi\nu^{\mathrm{Hz}}T_{\mathrm s}$，可得 +\begin{equation} + \Delta\nu_{\mathrm{tooth}}^{\mathrm{Hz}} + = + \frac{1}{\Delta\tau_{\mathrm{f}}T_{\mathrm s}}. + \label{eq:A7_tooth_spacing_hz} +\end{equation} +这解释了为什么周期多帧目标函数会出现以 $1/(\Delta\tau_{\mathrm{f}}T_{\mathrm s})$ 为间隔的 +Doppler comb branches。 + +有限长度的 pilot block 破坏了这种完全等价性。令每帧保留 $T$ 个采样点，块内相关因子为 +\begin{equation} + R_T(\Delta\nu) + \triangleq + \frac{1}{T}\sum_{t=0}^{T-1}\exp\{j\Delta\nu t\} + = + \frac{\exp\{j(T-1)\Delta\nu/2\}}{T} + \frac{\sin(T\Delta\nu/2)}{\sin(\Delta\nu/2)}. + \label{eq:A7_block_dirichlet_corr} +\end{equation} +于是相邻 tooth 的可分离性可由 +\begin{equation} + D_{\mathrm{blk}} + \triangleq + 1- + \left|R_T(\Delta\nu_{\mathrm{tooth}})\right|^2 + \label{eq:A7_block_tooth_separation} +\end{equation} +度量。若 $T/\Delta\tau_{\mathrm{f}}\ll1$，则 +\begin{equation} + D_{\mathrm{blk}} + \approx + 1-\operatorname{sinc}^{2}\left(\frac{T}{\Delta\tau_{\mathrm{f}}}\right) + \approx + \frac{\pi^2}{3} + \left(\frac{T}{\Delta\tau_{\mathrm{f}}}\right)^2, + \qquad + \operatorname{sinc}(x)=\frac{\sin(\pi x)}{\pi x}. + \label{eq:A7_block_small_ratio_approx} +\end{equation} +因此，增加块长主要增强的是相邻 Doppler tooth 的块内区分度，而不是必然依赖 +$\dot\nu_{\mathrm{ref}}$ 在单块内部产生可观测的二次相位曲率。 + +该分析也解释了周期窗口与非周期 subset 的不同作用。若采用非周期子集 +$\mathcal T_{\mathrm{sub}}=\{\tau_{p_1},\ldots,\tau_{p_Q}\}$，则 +\begin{equation} + W_{\mathrm{sub}}(\Delta\nu) + = + \sum_{q=1}^{Q}w_q\exp\{j\Delta\nu\tau_{p_q}\}. + \label{eq:A7_subset_frame_window} +\end{equation} +当 $\mathcal T_{\mathrm{sub}}$ 不再满足严格等间隔结构时，式~\eqref{eq:A7_tooth_spacing_discrete} +给出的候选偏移不再必然同时对齐所有帧相位，因而可压低周期 comb 的旁瓣。由此可将实际求解分解为两类问题： +非周期 subset 更适合执行 tooth acquisition，完整周期窗口则更适合在 tooth 已选对时执行同 tooth 内相干 refine。 +这也说明，wrong-tooth error 是全局 profile likelihood 的 branch 选择问题；而 +式~\eqref{eq:A7_efim_projection_interpret}--\eqref{eq:A7_nu_info_after_coupling} 描述的是给定 branch 内的局部可辨识性。 + +\subsection{数值实现中的一致性检查} + +为了直接指导后续仿真，建议在实现 A7 的 CRB 时按如下顺序检查：首先按照式~\eqref{eq:A7_qx_cp}--\eqref{eq:A7_qgamma_cp} 构造每个 $(\ell,p)$ 块的解析 Jacobian，并用中心差分验证 +\begin{equation} + \frac{\partial\boldsymbol\mu^{\mathrm{cp}}_{\ell,p}}{\partial x} + \approx + \frac{\boldsymbol\mu^{\mathrm{cp}}_{\ell,p}(x+\epsilon)-\boldsymbol\mu^{\mathrm{cp}}_{\ell,p}(x-\epsilon)}{2\epsilon}, + \qquad + x\in\{\alpha,\beta,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}},\chi_\ell,\rho_{\ell,p}^{\mathrm{cp}}\}. +\end{equation} +其次，先由完整实参数 Jacobian 构造 +\begin{equation} + \mathcal I_{\mathrm{full}}^{\mathrm{cp}} + = + 2\Re\{(\mathbf J_{\mathrm{full}}^{\mathrm{cp}})^{\Herm}\mathbf R_n^{-1}\mathbf J_{\mathrm{full}}^{\mathrm{cp}}\}, +\end{equation} +再按 $[\boldsymbol\eta_{\mathrm{dyn}}^{\Tran},\boldsymbol\psi_{\mathrm{cp},\mu}^{\Tran}]^{\Tran}$ 分块执行 Schur 补得到式~\eqref{eq:A7_Ieff_cp}。最后，对 relaxed 连续相位模型，应同时用完整 FIM 消元和投影式~\eqref{eq:A7_Ieff_rel_ext} 计算一次，两者应在数值精度内一致；该检查可以有效排除 $\mathbf g^{\mathrm{cp}}_{\ell,p}$、$\mathbf D^{\mathrm{rel}}_{\mathrm{ext},\ell,p}$、噪声方差系数和 Kronecker 堆叠顺序的实现错误。 + +\subsection{关于噪声方差参数的说明} + +上面的推导默认 $\{\sigma_{n,\ell}^2\}$ 已知，或者已通过预估获得。若将 $\{\sigma_{n,\ell}^2\}$ 也并入待估参数，由于本附录中的确定信号模型只通过均值向量依赖于 $\boldsymbol\eta_{\mathrm{dyn}}$、$\dot\nu_{\mathrm{ref}}$ 及各类增益/相位参数，而协方差只依赖于 $\{\sigma_{n,\ell}^2\}$，因此噪声方差参数与均值参数在 Fisher 信息矩阵中是正交的。对第 $\ell$ 颗卫星，有 +\begin{equation} + \big[\mathcal I(\boldsymbol\kappa)\big]_{\sigma_{n,\ell}^2,\sigma_{n,\ell}^2} + = + \frac{M_\ell P T}{\sigma_{n,\ell}^4}. +\end{equation} +因此，无论在连续相位模型还是 relaxed 连续相位模型下，将 $\sigma_{n,\ell}^2$ 视为未知量都只会在 FIM 中附加与其他参数解耦的对角块，不改变式~\eqref{eq:A7_Ieff_cp}、\eqref{eq:A7_Ieff_rel_ext} 和 \eqref{eq:A7_Ieff_rel_eta} 的形式。 + +因此，附录 A7 给出的不是新的动态模型，而是正文连续相位模型和 relaxed 连续相位处理的 Fisher 信息表达。它与 \ref{sec:A8_dynIndependentPhaseCrb} 的比较应围绕“是否保留跨帧绝对相位连续性”展开，而不是围绕不同的主参数定义展开。 diff --git a/parts/A8-crbDynPilotDoaDopperInd.tex b/parts/A8-crbDynPilotDoaDopperInd.tex new file mode 100644 index 0000000..bac9622 --- /dev/null +++ b/parts/A8-crbDynPilotDoaDopperInd.tex @@ -0,0 +1,395 @@ +\section{动态 DoA--Doppler 估计在独立相位模型下的 CRB 推导} +\label{sec:A8_dynIndependentPhaseCrb} + +本附录仅对应第~\ref{subsec:independentPhaseModel} 小节中的独立相位基线模型。这里的``独立相位''是指：每个观测块 $(\ell,p)$ 具有各自的\emph{独立未知初始相位} $\vartheta_{\ell,p}$，这些相位不再受同一颗卫星公共相位的结构约束；但它们仍是\emph{确定性未知参数}，而不是按某个先验分布生成的随机变量。因此，本附录与正文及 \ref{sec:A7_dynDoaDopplerCrb} 一样，采用\emph{确定性参数 CRB}，而不再引入 hybrid CRB。 + +主参数统一定义为 +\begin{equation} + \boldsymbol\eta_{\mathrm{dyn}} + = + \begin{bmatrix} + \alpha \\ + \beta \\ + \nu_{\mathrm{ref}} + \end{bmatrix}, + \qquad + \boldsymbol\eta_{\mathrm{dyn}}\in\mathbb R^3, +\end{equation} +并继续将 $\dot\nu_{\mathrm{ref}}$ 作为干扰参数处理。为避免与正文和 \ref{sec:A7_dynDoaDopplerCrb} 重复，这里只保留独立相位模型对应的最小定义、导数记号与 EFIM 结构。 + +\subsection{沿用正文独立相位模型的最小定义与导数记号} + +沿用第~\ref{subsec:independentPhaseModel} 小节中的块级独立相位建模。对任意卫星 $\ell=0,\ldots,L-1$ 与帧 $p=0,\ldots,P-1$，定义局部归一化 Doppler 为 +\begin{equation} + \bar\nu_{\ell,p} + \triangleq + \nu_\ell + \dot\nu_\ell\tau_p, + \label{eq:A8_IP_baromega} +\end{equation} +其中 +\begin{equation} + \nu_\ell = f_{\nu,\ell}(\boldsymbol\xi,\nu_{\mathrm{ref}}), + \qquad + \dot\nu_\ell = f_{\dot\nu,\ell}(\boldsymbol\xi,\dot\nu_{\mathrm{ref}}). + \label{eq:A8_IP_omegagamma_map} +\end{equation} +于是独立相位模板写为 +\begin{equation} + \phi^{\mathrm{ip}}_{\ell,p}(t) + = + \bar\nu_{\ell,p} t + \frac{1}{2}\dot\nu_\ell t^2, + \qquad + t=0,\ldots,T-1. + \label{eq:A8_IP_phase_template} +\end{equation} +对应的时间模板与空时模板分别定义为 +\begin{equation} + \mathbf q^{\mathrm{ip}}_{\ell,p} + \triangleq + \begin{bmatrix} + s_p(0)e^{j\phi^{\mathrm{ip}}_{\ell,p}(0)} \\ + s_p(1)e^{j\phi^{\mathrm{ip}}_{\ell,p}(1)} \\ + \vdots \\ + s_p(T-1)e^{j\phi^{\mathrm{ip}}_{\ell,p}(T-1)} + \end{bmatrix} + \in\mathbb C^{T\times 1}, + \label{eq:A8_IP_qdef} +\end{equation} +\begin{equation} + \mathbf g^{\mathrm{ip}}_{\ell,p} + \triangleq + \mathbf q^{\mathrm{ip}}_{\ell,p}\otimes \mathbf a_\ell(\boldsymbol\xi) + \in\mathbb C^{M_\ell T\times 1}. + \label{eq:A8_IP_gdef} +\end{equation} +于是向量化观测模型为 +\begin{equation} + \tilde{\mathbf y}^{\mathrm{ip}}_{\ell,p} + = + \zeta_{\ell,p}^{\mathrm{ip}} + \mathbf g^{\mathrm{ip}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}}) + + \tilde{\mathbf n}_{\ell,p}, + \label{eq:A8_IP_model} +\end{equation} +其中 +\begin{equation} + \tilde{\mathbf n}_{\ell,p}\sim\mathcal{CN}(\mathbf 0,\sigma_{n,\ell}^2\mathbf I_{M_\ell T}), + \qquad + \ell=0,\ldots,L-1,\qquad + p=0,\ldots,P-1. +\end{equation} +正文 MLE 中可直接把 $\zeta_{\ell,p}^{\mathrm{ip}}$ 作为自由复增益消元；本附录为了构造实参数 Fisher 信息矩阵，将其写作极坐标形式 +\begin{equation} + \zeta_{\ell,p}^{\mathrm{ip}} + = + \rho_{\ell,p}^{\mathrm{ip}}e^{j\vartheta_{\ell,p}}, + \qquad + \rho_{\ell,p}^{\mathrm{ip}}\in\mathbb R_{\ge0}. +\end{equation} +这里的 $\vartheta_{\ell,p}$ 是第 $(\ell,p)$ 个观测块的独立未知初始相位；与连续相位模型相比，跨帧绝对相位连续性已经被这些块级相位参数吸收。 + +对角度参数 $x\in\{\alpha,\beta\}$，记 +\begin{equation} + \mathbf a_{\ell,x} + \triangleq + \frac{\partial \mathbf a_\ell(\boldsymbol\xi)}{\partial x}. +\end{equation} +由链式法则可得 +\begin{equation} + \frac{\partial \mathbf q^{\mathrm{ip}}_{\ell,p}}{\partial x} + = + j\left[ + \left( + \frac{\partial \nu_\ell}{\partial x} + + + \tau_p\frac{\partial \dot\nu_\ell}{\partial x} + \right) + \operatorname{diag}(0,1,\ldots,T-1) + + + \frac{1}{2} + \frac{\partial \dot\nu_\ell}{\partial x} + \operatorname{diag}(0^2,1^2,\ldots,(T-1)^2) + \right] + \mathbf q^{\mathrm{ip}}_{\ell,p}. + \label{eq:A8_IP_qx} +\end{equation} +又由于 +\begin{equation} + \frac{\partial \nu_\ell}{\partial \nu_{\mathrm{ref}}}=1, + \qquad + \frac{\partial \dot\nu_\ell}{\partial \dot\nu_{\mathrm{ref}}}=1, +\end{equation} +这里同样采用正文中的加性参考星参数化；若改用一般映射，则 $\partial\nu_\ell/\partial\nu_{\mathrm{ref}}$ 与 $\partial\dot\nu_\ell/\partial\dot\nu_{\mathrm{ref}}$ 应替换为对应的链式法则结果。 +因此 +\begin{equation} + \frac{\partial \mathbf q^{\mathrm{ip}}_{\ell,p}}{\partial \nu_{\mathrm{ref}}} + = + j\operatorname{diag}(0,1,\ldots,T-1)\mathbf q^{\mathrm{ip}}_{\ell,p}, + \label{eq:A8_IP_qomega} +\end{equation} +\begin{equation} + \frac{\partial \mathbf q^{\mathrm{ip}}_{\ell,p}}{\partial \dot\nu_{\mathrm{ref}}} + = + j\left[ + \tau_p\operatorname{diag}(0,1,\ldots,T-1) + + + \frac{1}{2}\operatorname{diag}(0^2,1^2,\ldots,(T-1)^2) + \right] + \mathbf q^{\mathrm{ip}}_{\ell,p}. + \label{eq:A8_IP_qgamma} +\end{equation} + +于是模板导数可统一写为 +\begin{equation} + \mathbf d^{\mathrm{ip}}_{\ell,p,\alpha} + \triangleq + \frac{\partial \mathbf g^{\mathrm{ip}}_{\ell,p}}{\partial \alpha} + = + \mathbf q^{\mathrm{ip}}_{\ell,p}\otimes \mathbf a_{\ell,\alpha} + + + \frac{\partial \mathbf q^{\mathrm{ip}}_{\ell,p}}{\partial \alpha}\otimes \mathbf a_\ell, +\end{equation} +\begin{equation} + \mathbf d^{\mathrm{ip}}_{\ell,p,\beta} + \triangleq + \frac{\partial \mathbf g^{\mathrm{ip}}_{\ell,p}}{\partial \beta} + = + \mathbf q^{\mathrm{ip}}_{\ell,p}\otimes \mathbf a_{\ell,\beta} + + + \frac{\partial \mathbf q^{\mathrm{ip}}_{\ell,p}}{\partial \beta}\otimes \mathbf a_\ell, +\end{equation} +\begin{equation} + \mathbf d^{\mathrm{ip}}_{\ell,p,\nu} + \triangleq + \frac{\partial \mathbf g^{\mathrm{ip}}_{\ell,p}}{\partial \nu_{\mathrm{ref}}} + = + \frac{\partial \mathbf q^{\mathrm{ip}}_{\ell,p}}{\partial \nu_{\mathrm{ref}}}\otimes \mathbf a_\ell, +\end{equation} +\begin{equation} + \mathbf d^{\mathrm{ip}}_{\ell,p,\dot\nu} + \triangleq + \frac{\partial \mathbf g^{\mathrm{ip}}_{\ell,p}}{\partial \dot\nu_{\mathrm{ref}}} + = + \frac{\partial \mathbf q^{\mathrm{ip}}_{\ell,p}}{\partial \dot\nu_{\mathrm{ref}}}\otimes \mathbf a_\ell. +\end{equation} +为后续紧凑书写，定义 +\begin{equation} + \mathbf D^{\mathrm{ip}}_{\eta\dot\nu,\ell,p} + \triangleq + \begin{bmatrix} + \mathbf d^{\mathrm{ip}}_{\ell,p,\alpha} & + \mathbf d^{\mathrm{ip}}_{\ell,p,\beta} & + \mathbf d^{\mathrm{ip}}_{\ell,p,\nu} & + \mathbf d^{\mathrm{ip}}_{\ell,p,\dot\nu} + \end{bmatrix} + \in\mathbb C^{M_\ell T\times 4}. + \label{eq:A8_IP_Detadotnu} +\end{equation} + +\subsection{完整参数的 Fisher 信息矩阵} + +定义独立相位模型的干扰参数向量为 +\begin{equation} + \boldsymbol\psi_{\mathrm{ip},\mu} + = + \Big[ + \dot\nu_{\mathrm{ref}}, + \{\vartheta_{\ell,p}:\ell=0,\ldots,L-1,\ p=0,\ldots,P-1\}, + \{\rho_{\ell,p}^{\mathrm{ip}}:\ell=0,\ldots,L-1,\ p=0,\ldots,P-1\} + \Big]^{\Tran}, +\end{equation} +并令总参数向量为 +\begin{equation} + \boldsymbol\kappa_{\mathrm{ip}} + = + \begin{bmatrix} + \boldsymbol\eta_{\mathrm{dyn}}^{\Tran} & \boldsymbol\psi_{\mathrm{ip},\mu}^{\Tran} + \end{bmatrix}^{\Tran}. +\end{equation} +为简化书写，这里先假定 $\{\sigma_{n,\ell}^2\}$ 已知；关于未知噪声方差的情形将在附录末尾补充说明。 + +对任意实参数 $\kappa_i$ 与 $\kappa_j$，Fisher 信息矩阵元素统一写为 +\begin{equation} + [\mathcal I(\boldsymbol\kappa_{\mathrm{ip}})]_{ij} + = + 2\Re\left\{ + \frac{\partial \boldsymbol\mu^{\Herm}(\boldsymbol\kappa_{\mathrm{ip}})}{\partial \kappa_i} + \mathbf R_n^{-1} + \frac{\partial \boldsymbol\mu(\boldsymbol\kappa_{\mathrm{ip}})}{\partial \kappa_j} + \right\}, + \label{eq:A8_IP_FIM_general} +\end{equation} +其中 $\boldsymbol\mu(\boldsymbol\kappa_{\mathrm{ip}})$ 为联合均值向量，$\mathbf R_n$ 为联合噪声协方差矩阵。 + +对单个观测块 $(\ell,p)$，其均值为 +\begin{equation} + \boldsymbol\mu^{\mathrm{ip}}_{\ell,p} + = + \rho_{\ell,p}^{\mathrm{ip}}e^{j\vartheta_{\ell,p}}\mathbf g^{\mathrm{ip}}_{\ell,p}. +\end{equation} +于是有 +\begin{equation} + \frac{\partial \boldsymbol\mu^{\mathrm{ip}}_{\ell,p}}{\partial + \begin{bmatrix} + \alpha & \beta & \nu_{\mathrm{ref}} & \dot\nu_{\mathrm{ref}} + \end{bmatrix}^{\Tran}} + = + \rho_{\ell,p}^{\mathrm{ip}}e^{j\vartheta_{\ell,p}} + \mathbf D^{\mathrm{ip}}_{\eta\dot\nu,\ell,p}, + \label{eq:A8_IP_Jetadotnu_block} +\end{equation} +\begin{equation} + \frac{\partial \boldsymbol\mu^{\mathrm{ip}}_{\ell,p}}{\partial \vartheta_{\ell,p}} + = + j\rho_{\ell,p}^{\mathrm{ip}}e^{j\vartheta_{\ell,p}}\mathbf g^{\mathrm{ip}}_{\ell,p}, + \label{eq:A8_IP_Jphase_block} +\end{equation} +\begin{equation} + \frac{\partial \boldsymbol\mu^{\mathrm{ip}}_{\ell,p}}{\partial \rho_{\ell,p}^{\mathrm{ip}}} + = + e^{j\vartheta_{\ell,p}}\mathbf g^{\mathrm{ip}}_{\ell,p}. + \label{eq:A8_IP_Jamp_block} +\end{equation} + +注意到式~\eqref{eq:A8_IP_Jphase_block} 与式~\eqref{eq:A8_IP_Jamp_block} 在实参数意义下所张成的干扰子空间，等价于复向量 $\mathbf g^{\mathrm{ip}}_{\ell,p}$ 所张成的一维复子空间。因此，对每个观测块的 $(\vartheta_{\ell,p},\rho_{\ell,p}^{\mathrm{ip}})$ 执行 Schur 补后，可直接得到关于联合参数 +\begin{equation} + \begin{bmatrix} + \boldsymbol\eta_{\mathrm{dyn}}^{\Tran} & \dot\nu_{\mathrm{ref}} + \end{bmatrix}^{\Tran} +\end{equation} +的块级等效 Fisher 信息矩阵： +\begin{equation} + \mathcal I^{\mathrm{ip}}_{\mathrm{ext},\ell,p} + = + 2\frac{(\rho_{\ell,p}^{\mathrm{ip}})^2}{\sigma_{n,\ell}^2} + \Re\left\{ + \big(\mathbf D^{\mathrm{ip}}_{\eta\dot\nu,\ell,p}\big)^{\Herm} + \mathcal P^{\perp}_{\mathbf g^{\mathrm{ip}}_{\ell,p}} + \mathbf D^{\mathrm{ip}}_{\eta\dot\nu,\ell,p} + \right\}, + \label{eq:A8_IP_FIM_block_eff} +\end{equation} +其中 +\begin{equation} + \mathcal P_{\mathbf g^{\mathrm{ip}}_{\ell,p}} + \triangleq + \mathbf g^{\mathrm{ip}}_{\ell,p} + \big((\mathbf g^{\mathrm{ip}}_{\ell,p})^{\Herm}\mathbf g^{\mathrm{ip}}_{\ell,p}\big)^{-1} + (\mathbf g^{\mathrm{ip}}_{\ell,p})^{\Herm}, +\end{equation} +\begin{equation} + \mathcal P^{\perp}_{\mathbf g^{\mathrm{ip}}_{\ell,p}} + \triangleq + \mathbf I_{M_\ell T}-\mathcal P_{\mathbf g^{\mathrm{ip}}_{\ell,p}}. +\end{equation} + +对全部卫星与全部帧求和，可得 +\begin{equation} + \mathcal I^{\mathrm{ip}}_{\mathrm{ext}} + = + \sum_{\ell=0}^{L-1}\sum_{p=0}^{P-1} + \mathcal I^{\mathrm{ip}}_{\mathrm{ext},\ell,p} + = + 2\sum_{\ell=0}^{L-1}\sum_{p=0}^{P-1} + \frac{(\rho_{\ell,p}^{\mathrm{ip}})^2}{\sigma_{n,\ell}^2} + \Re\left\{ + \big(\mathbf D^{\mathrm{ip}}_{\eta\dot\nu,\ell,p}\big)^{\Herm} + \mathcal P^{\perp}_{\mathbf g^{\mathrm{ip}}_{\ell,p}} + \mathbf D^{\mathrm{ip}}_{\eta\dot\nu,\ell,p} + \right\}. + \label{eq:A8_IP_FIM_etadotnu} +\end{equation} + +\subsection{\texorpdfstring{主参数 $\boldsymbol\eta_{\mathrm{dyn}}$ 的等效 Fisher 信息与 CRB}{主参数 eta_dyn 的等效 Fisher 信息与 CRB}} + +若把 $\dot\nu_{\mathrm{ref}}$ 也视为感兴趣参数，则 +\begin{equation} + \operatorname{CRB}_{\mathrm{ip}}(\boldsymbol\eta_{\mathrm{dyn}},\dot\nu_{\mathrm{ref}}) + = + \big(\mathcal I^{\mathrm{ip}}_{\mathrm{ext}}\big)^{-1}. + \label{eq:A8_IP_CRB_etadotnu} +\end{equation} + +若仍只把 $\boldsymbol\eta_{\mathrm{dyn}}$ 作为主参数，而把 $\dot\nu_{\mathrm{ref}}$ 作为干扰参数，则将 +\eqref{eq:A8_IP_FIM_etadotnu} 按如下方式分块： +\begin{equation} + \mathcal I^{\mathrm{ip}}_{\mathrm{ext}} + = + \begin{bmatrix} + \mathcal I^{\mathrm{ip}}_{\eta\eta} & \mathcal I^{\mathrm{ip}}_{\eta\dot\nu} \\ + \mathcal I^{\mathrm{ip}}_{\dot\nu\eta} & \mathcal I^{\mathrm{ip}}_{\dot\nu\dot\nu} + \end{bmatrix}, + \label{eq:A8_IP_FIM_partition} +\end{equation} +于是主参数 $\boldsymbol\eta_{\mathrm{dyn}}$ 的等效 Fisher 信息矩阵为 +\begin{equation} + \mathcal I^{\mathrm{ip}}_{\mathrm e}(\boldsymbol\eta_{\mathrm{dyn}}) + = + \mathcal I^{\mathrm{ip}}_{\eta\eta} + - + \mathcal I^{\mathrm{ip}}_{\eta\dot\nu} + \big(\mathcal I^{\mathrm{ip}}_{\dot\nu\dot\nu}\big)^{-1} + \mathcal I^{\mathrm{ip}}_{\dot\nu\eta}, + \label{eq:A8_IP_EFIM_eta} +\end{equation} +从而 +\begin{equation} + \operatorname{CRB}_{\mathrm{ip}}(\boldsymbol\eta_{\mathrm{dyn}}) + = + \big(\mathcal I^{\mathrm{ip}}_{\mathrm e}(\boldsymbol\eta_{\mathrm{dyn}})\big)^{-1}. + \label{eq:A8_IP_CRB_eta} +\end{equation} + +式~\eqref{eq:A8_IP_EFIM_eta} 给出的正是独立相位模型下，未知 $\dot\nu_{\mathrm{ref}}$ 对主参数 $(\alpha,\beta,\nu_{\mathrm{ref}})$ 的信息损失形式。与正文研究主线一致，这里的关键不在于再引入更高维动态状态，而在于比较：在相同短窗条件下，连续相位模型和独立相位模型分别能为主参数保留多少有效信息。 + +\subsection{与 A7 中 relaxed 连续相位结果的关系} + +式~\eqref{eq:A8_IP_FIM_etadotnu} 给出的扩展 Fisher 信息矩阵的投影型结构与 \ref{sec:A7_dynDoaDopplerCrb} 中 relaxed 连续相位模型的结果形式一致，但两者对应的\emph{时间模板不同}： +\begin{itemize} + \item \ref{sec:A7_dynDoaDopplerCrb} 使用的是连续相位模板 $\mathbf g^{\mathrm{cp}}_{\ell,p}$，其中通过 $t_{p,t}=\tau_p+t$ 保留了跨帧绝对相位连续性； + \item 本附录使用的是独立相位模板 $\mathbf g^{\mathrm{ip}}_{\ell,p}$，其中帧间常相位项已被 $\vartheta_{\ell,p}$ 吸收，只保留帧内局部相位演化。 +\end{itemize} +因此，两类模型的差异不在 Schur 补的代数形式本身，而在模板里是否保留跨帧连续相位结构。这也正是正文中``连续相位 vs.~独立相位''对比的统计本质。 + +从 EFIM 病态的角度看，独立相位模型与 relaxed 连续相位模型共享同一类投影解释。若定义白化 Jacobian +\begin{equation} + \bar{\mathbf J}^{\mathrm{ip}}_{\eta} + \triangleq + \mathbf R_n^{-1/2} + \frac{\partial\boldsymbol\mu^{\mathrm{ip}}}{\partial\boldsymbol\eta_{\mathrm{dyn}}}, + \qquad + \bar{\mathbf J}^{\mathrm{ip}}_{\psi} + \triangleq + \mathbf R_n^{-1/2} + \frac{\partial\boldsymbol\mu^{\mathrm{ip}}}{\partial\boldsymbol\psi_{\mathrm{ip},\mu}}, +\end{equation} +则式~\eqref{eq:A8_IP_EFIM_eta} 等价于 +\begin{equation} + \mathcal I^{\mathrm{ip}}_{\mathrm{e}} + = + 2\Re\left\{ + (\bar{\mathbf J}^{\mathrm{ip}}_{\eta})^{\Herm} + \mathcal P^{\perp}_{\bar{\mathbf J}^{\mathrm{ip}}_{\psi}} + \bar{\mathbf J}^{\mathrm{ip}}_{\eta} + \right\}. + \label{eq:A8_IP_EFIM_projection} +\end{equation} +因此，独立相位模型下的弱可辨识方向仍由 +$\mathcal P^{\perp}_{\bar{\mathbf J}^{\mathrm{ip}}_{\psi}}\bar{\mathbf J}^{\mathrm{ip}}_{\eta}$ +的秩和最小奇异方向决定。与 A7 中的 relaxed 连续相位模型相比，二者的主要区别不在 +Schur 补代数形式，而在 $\mathbf g^{\mathrm{ip}}_{\ell,p}$ 使用局部时间索引并允许每个 +$(\ell,p)$ 块具有独立相位 $\vartheta_{\ell,p}$。这些自由相位会吸收跨帧绝对相位所携带的公共 +Doppler 截距信息，因此通常不会缓解 DoA--Doppler--rate 耦合，而是构成一个更保守的相干性基线。 + + +\subsection{关于噪声方差参数的说明} + +若进一步将 $\{\sigma_{n,\ell}^2\}$ 也并入待估参数，由于它们只进入协方差矩阵而不进入均值函数，因此与前述均值参数在 Fisher 信息矩阵中仍然解耦。对第 $\ell$ 颗卫星，有 +\begin{equation} + \big[\mathcal I(\cdot)\big]_{\sigma_{n,\ell}^2,\sigma_{n,\ell}^2} + = + \frac{M_\ell P T}{\sigma_{n,\ell}^4}. +\end{equation} +因此，将 $\sigma_{n,\ell}^2$ 视为未知量不会改变式~\eqref{eq:A8_IP_FIM_etadotnu} 与式~\eqref{eq:A8_IP_EFIM_eta} 的形式，只是在完整 Fisher 信息矩阵中附加一个与其他参数解耦的对角块。 + +至此，动态多帧主线的两类性能界已经形成闭环： \ref{sec:A7_dynDoaDopplerCrb} 对应连续相位/relaxed 连续相位模型，本附录对应独立相位基线模型。二者共享同一主参数 $\boldsymbol\eta_{\mathrm{dyn}}$，因而可以直接比较相位连续性对 DoA--Doppler 信息保留能力的影响。 diff --git a/parts/detDoaDopplerEst.tex b/parts/detDoaDopplerEst.tex new file mode 100644 index 0000000..83feac3 --- /dev/null +++ b/parts/detDoaDopplerEst.tex @@ -0,0 +1,158 @@ +\section{确定信号 DoA--Doppler 估计} +\label{sec:det_doa_doppler_est} + +本节在单阵列确定信号 DoA 模板上加入归一化角 Doppler $\nu_k$，是从纯空间阵列模型过渡到空时导向模型的桥梁。若仍把发送信号 $x_k(t)$ 在每一时刻都视为自由未知量，则 Doppler 项 $e^{j\nu_k t}$ 会被吸收进未知的 $x_k(t)$ 中，从而使 $\nu_k$ 不可辨识并导致 \gls{crb} 奇异。因此，本节沿用第~\ref{subsec:single_arr_doa_doppler_est} 小节的归一化 Doppler 约定，并显式引入已知导频序列 $s_k(t)$。当 $s_k(t)\equiv1$ 时，下面模型退化为常包络导频或单音模型。 + +\subsection{似然函数求取} + +定义第 $k$ 个信源的导频加权时间模板为 +\begin{equation} + \mathbf q_k(\nu_k) + \triangleq + \begin{bmatrix} + s_k(0)e^{j\nu_k\cdot0} \\ + s_k(1)e^{j\nu_k\cdot1} \\ + \vdots \\ + s_k(T-1)e^{j\nu_k(T-1)} + \end{bmatrix} + \in\mathbb C^{T\times1}. +\end{equation} +令 $\mathbf a_k\triangleq\mathbf a(\bm\theta_k)$，并将第 $k$ 个信源的未知复增益记为 $\zeta_k$，则所有快拍可写为 +\begin{equation} + \mathbf Y + = + \sum_{k=1}^{K}\zeta_k\mathbf a(\bm\theta_k)\mathbf q_k^\Tran(\nu_k) + + + \mathbf N, +\end{equation} +其中 $\mathbf Y=[\mathbf y_0,\ldots,\mathbf y_{T-1}]\in\mathbb C^{M\times T}$，$\mathbf N=[\mathbf n_0,\ldots,\mathbf n_{T-1}]\in\mathbb C^{M\times T}$。 + +进一步定义第 $k$ 个信源的空时导向列向量和整体空时字典为 +\begin{equation} + \mathbf b_{\mathrm{st},k}(\bm\theta_k,\nu_k) + \triangleq + \mathbf q_k(\nu_k)\otimes\mathbf a(\bm\theta_k) + \in\mathbb C^{MT\times1}, +\end{equation} +\begin{equation} + \mathbf B_{\mathrm{st}}(\bm\Theta,\boldsymbol\nu) + \triangleq + \left[ + \mathbf b_{\mathrm{st},1},\ldots,\mathbf b_{\mathrm{st},K} + \right] + \in\mathbb C^{MT\times K}. +\end{equation} +定义信源复增益向量 $\boldsymbol\zeta=[\zeta_1,\ldots,\zeta_K]^\Tran$，则接收信号可以进一步表示为 +\begin{equation} + \label{eq:vec_det_doa_doppler_recv_sig} + \tilde{\mathbf y} + \triangleq + \vect(\mathbf Y) + = + \mathbf B_{\mathrm{st}}(\bm\Theta,\boldsymbol\nu)\boldsymbol\zeta + + + \tilde{\mathbf n}, + \qquad + \tilde{\mathbf n}\triangleq\vect(\mathbf N). +\end{equation} + +由于噪声在时间和空间上均不相关，且 $\tilde{\mathbf n}\sim\mathcal{CN}(\mathbf 0,\sigma\mathbf I_{MT})$，其中 $\sigma\triangleq\sigma_n^2$，有 +\begin{equation} + \tilde{\mathbf y} + \sim + \mathcal{CN}\big(\mathbf B_{\mathrm{st}}\boldsymbol\zeta,\sigma\mathbf I_{MT}\big). +\end{equation} +因此，观测似然函数为 +\begin{equation} + L(\sigma,\boldsymbol\zeta,\boldsymbol\nu,\bm\Theta) + = + (\pi\sigma)^{-MT} + \exp\left[ + -\frac{1}{\sigma} + \left\|\tilde{\mathbf y}-\mathbf B_{\mathrm{st}}\boldsymbol\zeta\right\|_2^2 + \right], +\end{equation} +对应对数似然函数为 +\begin{equation} + \label{eq:singlelog_likelihood} + \ln\mathcal L(\sigma,\boldsymbol\zeta,\boldsymbol\nu,\bm\Theta) + = + \mathrm{Const} + -MT\ln\sigma + -\frac{1}{\sigma} + \left\|\tilde{\mathbf y}-\mathbf B_{\mathrm{st}}\boldsymbol\zeta\right\|_2^2. +\end{equation} + +\subsection{确定/条件最大似然} + +\subsubsection{原始问题建模} + +\Gls{dml}/\gls{cml} 优化问题可以表述为 +\begin{alignat}{2} + \label{eq:det_ml_doa_doppler_opt} + &\minimize_{\boldsymbol\zeta,\boldsymbol\nu,\bm\Theta,\sigma} \quad && + MT\ln\sigma + + + \frac{1}{\sigma}\left\|\tilde{\mathbf y}-\mathbf B_{\mathrm{st}}\boldsymbol\zeta\right\|_2^2 + \nonumber\\ + &\text{subject to} && \bm\Theta\in\varTheta. +\end{alignat} + +\subsubsection{集中似然表达} + +我们关心的是信号来向 $\bm\Theta$ 和 Doppler $\boldsymbol\nu$。假设 $\mathbf B_{\mathrm{st}}(\bm\Theta,\boldsymbol\nu)$ 列满秩，则对固定的 $(\bm\Theta,\boldsymbol\nu)$，复增益和噪声方差的极大似然估计为 +\begin{subequations} +\begin{equation} + \widehat{\boldsymbol\zeta}(\bm\Theta,\boldsymbol\nu) + = + \left(\mathbf B_{\mathrm{st}}^\Herm\mathbf B_{\mathrm{st}}\right)^{-1} + \mathbf B_{\mathrm{st}}^\Herm\tilde{\mathbf y}, +\end{equation} +\begin{equation} + \widehat\sigma(\bm\Theta,\boldsymbol\nu) + = + \frac{1}{MT}\tilde{\mathbf y}^\Herm + \left(\mathbf I_{MT}-\mathcal P_{\mathbf B_{\mathrm{st}}}\right) + \tilde{\mathbf y}, +\end{equation} +\end{subequations} +其中 +\begin{equation} + \mathcal P_{\mathbf B_{\mathrm{st}}} + \triangleq + \mathbf B_{\mathrm{st}} + \left(\mathbf B_{\mathrm{st}}^\Herm\mathbf B_{\mathrm{st}}\right)^{-1} + \mathbf B_{\mathrm{st}}^\Herm. +\end{equation} +代入式~\eqref{eq:det_ml_doa_doppler_opt} 后，可以得到 $\bm\Theta$ 和 $\boldsymbol\nu$ 的集中确定最大似然估计 +\begin{equation} + \label{eq:con_det_ml_doa_doppler_opt} + \left(\widehat{\bm\Theta}_{\mathrm{CDML}},\widehat{\boldsymbol\nu}_{\mathrm{CDML}}\right) + = + \argmin_{\bm\Theta,\boldsymbol\nu} + \tilde{\mathbf y}^\Herm + \left(\mathbf I_{MT}-\mathcal P_{\mathbf B_{\mathrm{st}}}\right) + \tilde{\mathbf y}. +\end{equation} +也即 $(\bm\Theta,\boldsymbol\nu)$ 的估计可以通过最小化 +\begin{equation} + \mathcal J(\bm\Theta,\boldsymbol\nu) + \triangleq + \tilde{\mathbf y}^\Herm + \left(\mathbf I_{MT}-\mathcal P_{\mathbf B_{\mathrm{st}}(\bm\Theta,\boldsymbol\nu)}\right) + \tilde{\mathbf y} +\end{equation} +得到。 + +若不关心 Doppler，可将其视为干扰参数并先做 profile 处理： +\begin{equation} + \widehat{\bm\Theta}_{\mathrm{CDML}} + = + \argmin_{\bm\Theta} + \min_{\boldsymbol\nu} + \tilde{\mathbf y}^\Herm + \left(\mathbf I_{MT}-\mathcal P_{\mathbf B_{\mathrm{st}}}\right) + \tilde{\mathbf y}. +\end{equation} + +由此可见，Doppler 不是简单附加在 DoA 估计之后的独立标量，而是通过时间模板与空间导向矢量共同形成空时子空间。多星静态 DoA--Doppler 模型将在这一思想上进一步加入卫星几何约束：各链路 Doppler 不再独立搜索，而由同一个 $\boldsymbol\xi$ 和参考星参数 $\nu_{\mathrm{ref}}$ 统一诱导。 diff --git a/parts/detDoaEst.tex b/parts/detDoaEst.tex new file mode 100644 index 0000000..8f9d9ed --- /dev/null +++ b/parts/detDoaEst.tex @@ -0,0 +1,108 @@ +\section{确定信号 DoA 估计\texorpdfstring{~\cite{stoicaMusicMaximumLikelihood1989}}{}} +\label{sec:det_doa_est} + +本节是单阵列基线层的第一个似然推导：在第~\ref{subsec:single_arr_doa_est} 小节的观测模型基础上，将源信号在每个快拍处的取值视为确定性未知量，从而得到条件/确定最大似然问题。该推导的重点不在多星几何，而在建立投影矩阵、集中似然和干扰参数消元的基本模板；对应的 \gls{crb} 推导放在 \ref{sec:det_doa_crb} 中。 + +\subsection{似然函数求取} + +本节沿用式~\eqref{eq:single_arr_doa_model} 的单阵列 DoA 观测模型，并采用全文统一的快拍索引 $t=0,1,\ldots,T-1$。为简化本节公式，记 +\begin{equation} + \sigma \triangleq \sigma_n^2. +\end{equation} +由于噪声 $\mathbf n_t\sim\mathcal{CN}(\mathbf 0,\sigma\mathbf I_M)$，且 $\mathbf x_t$ 为确定性未知信号快拍，有 +\begin{equation} + \mathbf y_t\sim\mathcal{CN}\big(\mathbf A(\bm\Theta)\mathbf x_t,\sigma\mathbf I_M\big). +\end{equation} +因此，根据式~\eqref{eq:independent_complex_gaussian_pdf}，所有快拍的似然函数为 +\begin{equation} + L(\sigma,\{\mathbf x_t\}_{t=0}^{T-1},\bm\Theta\mid \{\mathbf y_t\}_{t=0}^{T-1}) + = + (\pi\sigma)^{-MT} + \exp\left[ + -\frac{1}{\sigma} + \sum_{t=0}^{T-1}\left\|\mathbf y_t-\mathbf A(\bm\Theta)\mathbf x_t\right\|_2^2 + \right]. +\end{equation} +对应的对数似然函数为 +\begin{equation} + \label{eq:det_ml_log_likelihood} + \ln \mathcal L + = + \mathrm{Const} + -MT\ln\sigma + -\frac{1}{\sigma}\sum_{t=0}^{T-1}\left\|\mathbf y_t-\mathbf A(\bm\Theta)\mathbf x_t\right\|_2^2. +\end{equation} +其中待估计参数包括噪声方差 $\sigma$，确定信号快拍 $\{\mathbf x_t\}_{t=0}^{T-1}$ 的实部和虚部，以及多源来向集合 $\bm\Theta$。 + +\subsection{确定/条件最大似然估计} + +\subsubsection{原始问题建模} + +\Gls{dml}/\gls{cml} 优化问题可以表述为 +\begin{alignat}{2} + \label{eq:det_ml_doa_opt} + &\minimize_{\bm\Theta,\{\mathbf x_t\},\sigma} \quad && + MT\ln\sigma + + + \frac{1}{\sigma}\sum_{t=0}^{T-1}\left\|\mathbf y_t-\mathbf A(\bm\Theta)\mathbf x_t\right\|_2^2 + \nonumber\\ + &\text{subject to} && \bm\Theta\in\varTheta, +\end{alignat} +其中 $\varTheta$ 为 $\bm\Theta$ 的可行域。 + +\subsubsection{集中似然表达} + +我们关心的是信号的来向 $\bm\Theta$，因此将对数似然函数中的其他参量，包括噪声方差 $\sigma$ 与信号快拍 $\{\mathbf x_t\}_{t=0}^{T-1}$，均用 $\bm\Theta$ 和观测 $\{\mathbf y_t\}_{t=0}^{T-1}$ 表示。 + +对固定的 $\bm\Theta$，令 $\partial\ln\mathcal L/\partial\mathbf x_t=0$，可得 +\begin{equation} + \widehat{\mathbf x}_t(\bm\Theta) + = + \left[\mathbf A^\Herm\mathbf A\right]^{-1}\mathbf A^\Herm\mathbf y_t. +\end{equation} +令 $\partial\ln\mathcal L/\partial\sigma=0$，并代入 $\widehat{\mathbf x}_t$，可得 +\begin{align} + \widehat\sigma(\bm\Theta) + &= + \frac{1}{MT}\sum_{t=0}^{T-1} + \left\|\mathbf y_t-\mathbf A\widehat{\mathbf x}_t\right\|_2^2 \nonumber\\ + &= + \frac{1}{MT}\sum_{t=0}^{T-1} + \mathbf y_t^\Herm + \left(\mathbf I_M-\mathcal P_{\mathbf A}\right) + \mathbf y_t, +\end{align} +其中 +\begin{equation} + \mathcal P_{\mathbf A} + \triangleq + \mathbf A(\mathbf A^\Herm\mathbf A)^{-1}\mathbf A^\Herm +\end{equation} +为导向矩阵列空间的投影算子。进一步定义样本协方差矩阵 +\begin{equation} + \widehat{\mathbf R} + \triangleq + \frac{1}{T}\sum_{t=0}^{T-1}\mathbf y_t\mathbf y_t^\Herm, +\end{equation} +则 +\begin{equation} + \widehat\sigma(\bm\Theta) + = + \frac{1}{M}\tr\left[\left(\mathbf I_M-\mathcal P_{\mathbf A}\right)\widehat{\mathbf R}\right]. +\end{equation} +代入式~\eqref{eq:det_ml_log_likelihood} 并忽略与 $\bm\Theta$ 无关的项后，$\bm\Theta$ 的 \gls{cdml} 可写为 +\begin{equation} + \widehat{\bm\Theta}_{\mathrm{CDML}} + = + \argmin_{\bm\Theta} + \tr\left[\left(\mathbf I_M-\mathcal P_{\mathbf A(\bm\Theta)}\right)\widehat{\mathbf R}\right]. +\end{equation} +也即 $\bm\Theta$ 的 \gls{cdml} 可以通过最小化 +\begin{equation} + \mathcal J(\bm\Theta) + \triangleq + \tr\left[\left(\mathbf I_M-\mathcal P_{\mathbf A(\bm\Theta)}\right)\widehat{\mathbf R}\right] +\end{equation} +得到。 + +本节得到的形式说明：当信号快拍被视为确定性干扰参数时，DoA 信息通过阵列导向子空间的投影残差体现。下一节的随机信号模型将改变信源统计假设，但仍沿用同一个单阵列几何入口。 diff --git a/parts/multiArrayDetDoaDopplerDelayEst.tex b/parts/multiArrayDetDoaDopplerDelayEst.tex new file mode 100644 index 0000000..6a8c710 --- /dev/null +++ b/parts/multiArrayDetDoaDopplerDelayEst.tex @@ -0,0 +1,199 @@ +\section{基于参考星链路参数化的多阵列确定信号 DoA--Doppler--时延估计扩展} +\label{sec:multiArrayDetDoaDopplerDelayEst} + +本节在静态多星 DoA--Doppler 主线的基础上进一步加入参考星链路时延 $\tau_{\mathrm{ref}}$。因此，本节不是重新定义一套几何模型，而是在第~\ref{subsec:multiArrayCommonNotation} 小节的全局方向 $\boldsymbol\xi$、局部导向矢量 $\mathbf a_\ell(\boldsymbol\xi)$ 和参考星 Doppler 参数化基础上，额外引入各星链路时延映射。 + +\subsection{信号模型} + +与前述静态 DoA--Doppler 模型一致，固定选择参考卫星 $\ell_{\mathrm{ref}}$。在加入时延后，DoA--Doppler--delay 扩展模型的主参数统一写为 +\begin{equation} + \boldsymbol\eta_{\tau} + = + \begin{bmatrix} + \boldsymbol\xi\\ + \tau_{\mathrm{ref}}\\ + \nu_{\mathrm{ref}} + \end{bmatrix}, +\end{equation} +其中 $\tau_{\mathrm{ref}}$ 与 $\nu_{\mathrm{ref}}$ 分别为参考卫星链路上的时延和归一化 Doppler。 + +第 $\ell$ 颗卫星的链路时延与归一化 Doppler 分别由参考星参数化写为 +\begin{equation} + \tau_\ell + = + f_{\tau,\ell}(\boldsymbol\xi,\tau_{\mathrm{ref}}), + \qquad + \nu_\ell + = + f_{\nu,\ell}(\boldsymbol\xi,\nu_{\mathrm{ref}}), + \qquad + \ell=0,1,\ldots,L-1, +\end{equation} +并满足参考星约束 +\begin{equation} + f_{\tau,\ell_{\mathrm{ref}}}(\boldsymbol\xi,\tau_{\mathrm{ref}}) + = + \tau_{\mathrm{ref}}, + \qquad + f_{\nu,\ell_{\mathrm{ref}}}(\boldsymbol\xi,\nu_{\mathrm{ref}}) + = + \nu_{\mathrm{ref}}. +\end{equation} +对 $\ell\neq\ell_{\mathrm{ref}}$，上述映射由已知卫星位置、速度、姿态以及参考星链路参数诱导得到。 + +考虑已知训练序列 $s(t)$。加入链路时延后，第 $\ell$ 颗卫星在第 $t$ 个采样点的接收向量写为 +\begin{equation} + \mathbf y_\ell(t) + = + \zeta_\ell + \mathbf a_\ell(\boldsymbol\xi) + s\!\bigl(t-\tau_\ell(\boldsymbol\xi,\tau_{\mathrm{ref}})\bigr) + e^{j\nu_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}})t} + + + \mathbf n_\ell(t), + \qquad t=0,1,\ldots,T-1, +\end{equation} +当 $\tau_\ell$ 非整数时，$s(t-\tau_\ell)$ 表示连续基带训练波形经过分数时延后在第 $t$ 个采样时刻的取值，等价地可由插值或分数时延滤波实现；若仅考虑整数采样时延，则应额外约束 $\tau_\ell\in\mathbb Z$。 +其中为书写紧凑记 +\begin{equation} + \tau_\ell(\boldsymbol\xi,\tau_{\mathrm{ref}}) + \triangleq + f_{\tau,\ell}(\boldsymbol\xi,\tau_{\mathrm{ref}}), + \qquad + \nu_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}}) + \triangleq + f_{\nu,\ell}(\boldsymbol\xi,\nu_{\mathrm{ref}}). +\end{equation} +这里 $\zeta_\ell\in\mathbb C$ 为未知复干扰系数，噪声满足 +\begin{equation} + \mathbf n_\ell(t) + \sim + \mathcal{CN}(\mathbf 0,\sigma_{n,\ell}^2\mathbf I_{M_\ell}). +\end{equation} + +定义包含时延和 Doppler 的时间模板 +\begin{equation} + \mathbf q_\ell(\boldsymbol\xi,\tau_{\mathrm{ref}},\nu_{\mathrm{ref}}) + \triangleq + \begin{bmatrix} + s\!\bigl(0-\tau_\ell(\boldsymbol\xi,\tau_{\mathrm{ref}})\bigr)e^{j\nu_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}})\cdot0}\\ + s\!\bigl(1-\tau_\ell(\boldsymbol\xi,\tau_{\mathrm{ref}})\bigr)e^{j\nu_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}})\cdot1}\\ + \vdots\\ + s\!\bigl(T-1-\tau_\ell(\boldsymbol\xi,\tau_{\mathrm{ref}})\bigr)e^{j\nu_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}})(T-1)} + \end{bmatrix}. +\end{equation} +相应地，矩阵观测模型为 +\begin{equation} + \mathbf Y_\ell + = + \zeta_\ell + \mathbf a_\ell(\boldsymbol\xi) + \mathbf q_\ell^\Tran(\boldsymbol\xi,\tau_{\mathrm{ref}},\nu_{\mathrm{ref}}) + + + \mathbf N_\ell. +\end{equation} +向量化后，定义 +\begin{equation} + \tilde{\mathbf y}_\ell + \triangleq + \vect(\mathbf Y_\ell), + \qquad + \tilde{\mathbf n}_\ell + \triangleq + \vect(\mathbf N_\ell), + \qquad + \mathbf g_\ell(\boldsymbol\xi,\tau_{\mathrm{ref}},\nu_{\mathrm{ref}}) + \triangleq + \mathbf q_\ell(\boldsymbol\xi,\tau_{\mathrm{ref}},\nu_{\mathrm{ref}}) + \otimes + \mathbf a_\ell(\boldsymbol\xi), +\end{equation} +可得 +\begin{equation} + \tilde{\mathbf y}_\ell + = + \zeta_\ell\mathbf g_\ell(\boldsymbol\xi,\tau_{\mathrm{ref}},\nu_{\mathrm{ref}}) + + + \tilde{\mathbf n}_\ell. +\end{equation} +本节中的干扰参数可写为 +\begin{equation} + \boldsymbol\psi_{\tau} + = + \Big[ + \{\zeta_\ell\}_{\ell=0}^{L-1}, + \{\sigma_{n,\ell}^2\}_{\ell=0}^{L-1} + \Big]^\Tran. +\end{equation} + +\subsection{联合似然与集中式估计} + +在星间噪声独立、且第 $\ell$ 颗卫星噪声方差为 $\sigma_{n,\ell}^2$ 的条件下，省略与参数无关的常数项后，联合负对数似然为 +\begin{equation} + \mathcal L_{\tau}(\boldsymbol\eta_{\tau},\boldsymbol\psi_{\tau}) + = + \sum_{\ell=0}^{L-1} + \left[ + M_\ell T\log\sigma_{n,\ell}^2 + + + \frac{1}{\sigma_{n,\ell}^2} + \left\| + \tilde{\mathbf y}_\ell + - + \zeta_\ell\mathbf g_\ell(\boldsymbol\xi,\tau_{\mathrm{ref}},\nu_{\mathrm{ref}}) + \right\|_2^2 + \right]. +\end{equation} + +对于固定的 $(\boldsymbol\xi,\tau_{\mathrm{ref}},\nu_{\mathrm{ref}})$ 和 $\{\sigma_{n,\ell}^2\}$，$\zeta_\ell$ 的最小二乘解为 +\begin{equation} + \hat\zeta_\ell(\boldsymbol\xi,\tau_{\mathrm{ref}},\nu_{\mathrm{ref}}) + = + \frac{ + \mathbf g_\ell^\Herm(\boldsymbol\xi,\tau_{\mathrm{ref}},\nu_{\mathrm{ref}})\tilde{\mathbf y}_\ell + }{ + \|\mathbf g_\ell(\boldsymbol\xi,\tau_{\mathrm{ref}},\nu_{\mathrm{ref}})\|_2^2 + }. +\end{equation} +代回后，第 $\ell$ 颗卫星的集中残差为 +\begin{equation} + \mathcal E_{\tau,\ell}(\boldsymbol\xi,\tau_{\mathrm{ref}},\nu_{\mathrm{ref}}) + \triangleq + \|\tilde{\mathbf y}_\ell\|_2^2 + - + \frac{ + \left| + \mathbf g_\ell^\Herm(\boldsymbol\xi,\tau_{\mathrm{ref}},\nu_{\mathrm{ref}})\tilde{\mathbf y}_\ell + \right|^2 + }{ + \|\mathbf g_\ell(\boldsymbol\xi,\tau_{\mathrm{ref}},\nu_{\mathrm{ref}})\|_2^2 + }. +\end{equation} +若进一步对噪声方差进行轮廓化，则在星间异方差假设下，DoA--Doppler--delay 扩展模型的集中估计问题为 +\begin{equation} + \widehat{\boldsymbol\eta}_{\tau} + = + \arg\min_{\boldsymbol\xi,\tau_{\mathrm{ref}},\nu_{\mathrm{ref}}} + \sum_{\ell=0}^{L-1} + M_\ell T + \log \mathcal E_{\tau,\ell}(\boldsymbol\xi,\tau_{\mathrm{ref}},\nu_{\mathrm{ref}}). +\end{equation} +若各星噪声方差已知，则也可采用加权集中似然形式 +\begin{equation} + \widehat{\boldsymbol\eta}_{\tau} + = + \arg\max_{\boldsymbol\xi,\tau_{\mathrm{ref}},\nu_{\mathrm{ref}}} + \sum_{\ell=0}^{L-1} + \frac{1}{\sigma_{n,\ell}^2} + \frac{ + \left| + \mathbf g_\ell^\Herm(\boldsymbol\xi,\tau_{\mathrm{ref}},\nu_{\mathrm{ref}})\tilde{\mathbf y}_\ell + \right|^2 + }{ + \|\mathbf g_\ell(\boldsymbol\xi,\tau_{\mathrm{ref}},\nu_{\mathrm{ref}})\|_2^2 + }. +\end{equation} + +该扩展模型与静态 DoA--Doppler 模型的区别在于：$\boldsymbol\eta_{\tau}$ 在 $\boldsymbol\eta_{\mathrm{stat}}$ 的基础上显式加入 $\tau_{\mathrm{ref}}$，并通过 $f_{\tau,\ell}(\boldsymbol\xi,\tau_{\mathrm{ref}})$ 将各星时延耦合在同一个参考星参数化框架中。因此，delay 扩展不会改变多星 DoA--Doppler 主线的符号体系，只是在同一层级上增加一类可观测链路参数。 + +因此，阅读顺序上应先理解静态 DoA--Doppler 和动态连续相位主线，再把本节视为在同一参考星参数化框架下增加时延观测的一种横向扩展。若后续继续推导该模型的 CRB，也应沿用 $\boldsymbol\eta_{\tau}$ 与干扰参数分组，而不要重新引入一套独立的几何符号。 diff --git a/parts/multiArrayDynDetDoaDopplerEst.tex b/parts/multiArrayDynDetDoaDopplerEst.tex new file mode 100644 index 0000000..870bc81 --- /dev/null +++ b/parts/multiArrayDynDetDoaDopplerEst.tex @@ -0,0 +1,850 @@ +\section{多星动态多帧连续相位 DoA--Doppler 估计问题模型} +\label{sec:dynamicDoaDopplerEst} + +\subsection{离散时间统一符号与参考星参数化} + +本节在第~\ref{sec:multiArrayStatDetDoaDopplerEst} 节静态多星模型的基础上引入多帧索引、Doppler rate 和跨帧连续相位结构。主估计目标仍为 +\begin{equation} + \boldsymbol\eta_{\mathrm{dyn}} + = + \boldsymbol\eta_{\mathrm{stat}} + = + \begin{bmatrix} + \boldsymbol\xi\\ + \nu_{\mathrm{ref}} + \end{bmatrix}, +\end{equation} +而参考卫星链路上的归一化 Doppler rate $\dot\nu_{\mathrm{ref}}$ 作为干扰参数处理。因此，动态模型并不改变主参数域，而是通过多帧相位结构增加可利用信息。 + +设共有 $L$ 颗协作卫星，卫星索引为 $\ell=0,1,\ldots,L-1$。观测窗口包含 $P$ 帧数据，帧索引为 +\begin{equation} + p=0,1,\ldots,P-1. +\end{equation} +第 $p$ 帧内的离散采样点索引为 +\begin{equation} + t=0,1,\ldots,T-1, +\end{equation} +其中 $T$ 为单帧采样点数。设第 $p$ 帧起始点相对于参考时刻的离散采样偏移为 $\tau_p$，并约定 $\tau_0=0$。为统一表示跨帧连续相位，定义第 $p$ 帧第 $t$ 个采样对应的全局离散时间索引为 +\begin{equation} + t_{p,t}\triangleq \tau_p+t, + \qquad + p=0,\ldots,P-1,\;t=0,\ldots,T-1. +\end{equation} +括号中的 $t$ 始终表示帧内局部采样索引，而 $t_{p,t}$ 表示对应的全局离散时间位置。 + +沿用第~\ref{subsec:globalLocalDoaMap} 小节的全局/局部 DoA 映射，第 $\ell$ 颗卫星的阵列导向矢量统一写为 $\mathbf a_\ell(\boldsymbol\xi)$。由于本文关注短观测窗口，后续将该导向矢量视为在整个窗口内保持不变。 + +参考卫星链路上的归一化 Doppler 与归一化 Doppler rate 分别定义为 +\begin{equation} + \nu_{\mathrm{ref}}\triangleq \nu_{\ell_{\mathrm{ref}}}, + \qquad + \dot\nu_{\mathrm{ref}}\triangleq \dot\nu_{\ell_{\mathrm{ref}}}. +\end{equation} +基于多星几何关系，本文不把每颗卫星的 $(\nu_\ell,\dot\nu_\ell)$ 都作为独立待估参数，而是统一采用 +\begin{equation} + \nu_\ell + = + f_{\nu,\ell}(\boldsymbol\xi,\nu_{\mathrm{ref}}), + \qquad + \dot\nu_\ell + = + f_{\dot\nu,\ell}(\boldsymbol\xi,\dot\nu_{\mathrm{ref}}), + \qquad + \ell=0,\ldots,L-1, +\end{equation} +其中 +\begin{equation} + f_{\nu,\ell_{\mathrm{ref}}}(\boldsymbol\xi,\nu_{\mathrm{ref}})=\nu_{\mathrm{ref}}, + \qquad + f_{\dot\nu,\ell_{\mathrm{ref}}}(\boldsymbol\xi,\dot\nu_{\mathrm{ref}})=\dot\nu_{\mathrm{ref}}. +\end{equation} +等价地，也可写为 +\begin{equation} + \nu_\ell + = + \nu_{\mathrm{ref}}+\Delta\nu_\ell(\boldsymbol\xi), + \qquad + \dot\nu_\ell + = + \dot\nu_{\mathrm{ref}}+\Delta\dot\nu_\ell(\boldsymbol\xi), +\end{equation} +其中 $\Delta\nu_\ell(\boldsymbol\xi)$ 和 $\Delta\dot\nu_\ell(\boldsymbol\xi)$ 由已知几何状态计算得到。 + +\subsection{连续相位多帧离散信号模型} +\label{subsec:dynamicCpModel} + +在短观测窗口下，本文采用如下动态近似： +\begin{enumerate} + \item DoA 在整个观测窗口内近似保持不变，因此第 $\ell$ 颗卫星的导向矢量记为 $\mathbf a_\ell(\boldsymbol\xi)$； + \item 第 $\ell$ 条链路在观测窗口内的归一化相位采用二次相位模型； + \item 主参数为 $\boldsymbol\eta_{\mathrm{dyn}}=[\boldsymbol\xi^\Tran,\nu_{\mathrm{ref}}]^\Tran$，而 $\dot\nu_{\mathrm{ref}}$ 作为干扰参数处理； + \item 各星链路参数 $(\nu_\ell,\dot\nu_\ell)$ 通过几何关系由 $(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})$ 诱导得到。 +\end{enumerate} + +于是，第 $\ell$ 颗卫星在第 $p$ 帧第 $t$ 个采样点的归一化连续相位可写为 +\begin{equation} + \phi^{\mathrm{cp}}_{\ell,p}(t) + = + \nu_\ell t_{p,t} + + + \frac{1}{2}\dot\nu_\ell t_{p,t}^2, + \qquad + t_{p,t}=\tau_p+t. +\end{equation} +若第 $p$ 帧所用已知训练序列记为 $s_p(t)$，并定义结构化连续相位复增益 +\begin{equation} + \zeta_{\ell,p}^{\mathrm{cp}} + \triangleq + \rho_{\ell,p}^{\mathrm{cp}}e^{j\chi_\ell}, + \qquad + \rho_{\ell,p}^{\mathrm{cp}}\in\mathbb R_{\ge0}, +\end{equation} +则连续相位模型下的接收向量为 +\begin{equation} + \mathbf y^{\mathrm{cp}}_{\ell,p}(t) + = + \zeta_{\ell,p}^{\mathrm{cp}} + \mathbf a_\ell(\boldsymbol\xi) + s_p(t) + \exp\left\{ + j\left[ + \nu_\ell t_{p,t} + + + \frac{1}{2}\dot\nu_\ell t_{p,t}^2 + \right] + \right\} + + + \mathbf n_{\ell,p}(t), +\end{equation} +其中 $\zeta_{\ell,p}^{\mathrm{cp}}$ 只是结构化复增益的简写，并不作为自由复参数独立估计；其自由度由非负实幅度 $\rho_{\ell,p}^{\mathrm{cp}}$ 和同星共享相位 $\chi_\ell$ 给出。$\chi_\ell$ 为第 $\ell$ 颗卫星在参考时刻的公共相位，因此同一颗卫星在不同帧之间共享该相位。噪声满足 $\mathbf n_{\ell,p}(t)\sim\mathcal{CN}(\mathbf 0,\sigma_{n,\ell}^2\mathbf I_{M_\ell})$。不同卫星的噪声方差允许不同，但对同一颗卫星，在短观测窗口内各帧各采样点共享同一个噪声方差 $\sigma_{n,\ell}^2$。 + +将单帧数据按时间堆叠，定义 +\begin{equation} + \mathbf q^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}}) + \triangleq + \begin{bmatrix} + s_p(0)e^{j\phi^{\mathrm{cp}}_{\ell,p}(0)} \\ + s_p(1)e^{j\phi^{\mathrm{cp}}_{\ell,p}(1)} \\ + \vdots \\ + s_p(T-1)e^{j\phi^{\mathrm{cp}}_{\ell,p}(T-1)} + \end{bmatrix} + \in\mathbb C^{T\times 1}. +\end{equation} +则第 $\ell$ 颗卫星在第 $p$ 帧的矩阵观测可写为 +\begin{equation} + \mathbf Y^{\mathrm{cp}}_{\ell,p} + = + \zeta_{\ell,p}^{\mathrm{cp}} + \mathbf a_\ell(\boldsymbol\xi) + \big(\mathbf q^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})\big)^\Tran + + + \mathbf N_{\ell,p}, +\end{equation} +其中 $\mathbf Y^{\mathrm{cp}}_{\ell,p}\in\mathbb C^{M_\ell\times T}$，$\mathbf N_{\ell,p}\in\mathbb C^{M_\ell\times T}$ 的各列独立满足 $\mathcal{CN}(\mathbf 0,\sigma_{n,\ell}^2\mathbf I_{M_\ell})$。进一步向量化，定义 +\begin{equation} + \tilde{\mathbf y}^{\mathrm{cp}}_{\ell,p} + \triangleq + \vect(\mathbf Y^{\mathrm{cp}}_{\ell,p}), + \qquad + \tilde{\mathbf n}_{\ell,p} + \triangleq + \vect(\mathbf N_{\ell,p}), +\end{equation} +则有 +\begin{equation} + \tilde{\mathbf y}^{\mathrm{cp}}_{\ell,p} + = + \zeta_{\ell,p}^{\mathrm{cp}} + \mathbf g^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}}) + + + \tilde{\mathbf n}_{\ell,p}, +\end{equation} +其中 +\begin{equation} + \mathbf g^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}}) + \triangleq + \mathbf q^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}}) + \otimes + \mathbf a_\ell(\boldsymbol\xi). +\end{equation} +该模型的关键在于：相位项通过 $t_{p,t}=\tau_p+t$ 显式保留了跨帧绝对时间信息，因此多帧处理带来的增益不仅来自能量叠加，也来自跨帧连续相位轨迹的一致性。 + +\subsection{独立相位基线模型} +\label{subsec:independentPhaseModel} + +为量化跨帧绝对相位连续性带来的收益，本小节引入独立相位基线模型。该模型保留每帧内部的相干结构，但允许不同帧具有独立未知相位，因此不再利用连续相位模型中的同星跨帧公共相位 $\chi_\ell$。 + +将连续相位模型中的相位项展开，可得 +\begin{equation} + \nu_\ell t_{p,t}+\frac{1}{2}\dot\nu_\ell t_{p,t}^2 + = + \underbrace{\nu_\ell\tau_p+\frac{1}{2}\dot\nu_\ell\tau_p^2}_{\text{帧间常相位项}} + + + \underbrace{(\nu_\ell+\dot\nu_\ell\tau_p)t+\frac{1}{2}\dot\nu_\ell t^2}_{\text{帧内局部相位演化}}. +\end{equation} +若允许每一帧具有独立未知相位，则上述帧间常相位项可被每帧相位吸收。定义第 $\ell$ 颗卫星在第 $p$ 帧的局部归一化 Doppler 为 +\begin{equation} + \bar\nu_{\ell,p} + \triangleq + \nu_\ell+\dot\nu_\ell\tau_p. +\end{equation} +令 $\zeta_{\ell,p}^{\mathrm{ip}}\in\mathbb C$ 表示第 $(\ell,p)$ 个观测块的自由复增益，其相位吸收了帧间常相位项。于是独立相位模型可写为 +\begin{equation} + \mathbf y^{\mathrm{ip}}_{\ell,p}(t) + = + \zeta_{\ell,p}^{\mathrm{ip}} + \mathbf a_\ell(\boldsymbol\xi) + s_p(t) + \exp\left\{ + j\left[ + \bar\nu_{\ell,p}t+\frac{1}{2}\dot\nu_\ell t^2 + \right] + \right\} + + + \mathbf n_{\ell,p}(t). +\end{equation} +其中 $\zeta_{\ell,p}^{\mathrm{ip}}$ 是每个观测块的自由复干扰系数。若需要显式写出其实参数形式，可令 $\zeta_{\ell,p}^{\mathrm{ip}}=\rho_{\ell,p}^{\mathrm{ip}}e^{j\vartheta_{\ell,p}}$，其中 $\rho_{\ell,p}^{\mathrm{ip}}\in\mathbb R_{\ge0}$，$\vartheta_{\ell,p}$ 为块独立相位。对应该模型的时间模板记为 +\begin{equation} + \mathbf q^{\mathrm{ip}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}}) + \triangleq + \begin{bmatrix} + s_p(0)e^{j\phi^{\mathrm{ip}}_{\ell,p}(0)} \\ + s_p(1)e^{j\phi^{\mathrm{ip}}_{\ell,p}(1)} \\ + \vdots \\ + s_p(T-1)e^{j\phi^{\mathrm{ip}}_{\ell,p}(T-1)} + \end{bmatrix}, +\end{equation} +其中 +\begin{equation} + \phi^{\mathrm{ip}}_{\ell,p}(t) + \triangleq + \bar\nu_{\ell,p}t+\frac{1}{2}\dot\nu_\ell t^2. +\end{equation} +并定义独立相位模型的空时模板 +\begin{equation} + \mathbf g^{\mathrm{ip}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}}) + \triangleq + \mathbf q^{\mathrm{ip}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}}) + \otimes + \mathbf a_\ell(\boldsymbol\xi). +\end{equation} +相应的向量化模型为 +\begin{equation} + \tilde{\mathbf y}^{\mathrm{ip}}_{\ell,p} + = + \zeta_{\ell,p}^{\mathrm{ip}} + \mathbf g^{\mathrm{ip}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}}) + + + \tilde{\mathbf n}_{\ell,p}. +\end{equation} + +\subsection{连续相位模型下的结构化最大似然估计} + +本小节直接沿用第~\ref{subsec:dynamicCpModel} 小节的连续相位观测模型。其中 $\zeta_{\ell,p}^{\mathrm{cp}}\triangleq\rho_{\ell,p}^{\mathrm{cp}}e^{j\chi_\ell}$ 只是结构化复增益的简写；在 structured CP 模型中，$\zeta_{\ell,p}^{\mathrm{cp}}$ 不作为自由复参数估计，其自由度由非负实幅度 $\rho_{\ell,p}^{\mathrm{cp}}$ 与同星共享相位 $\chi_\ell$ 给出。 + +根据第~\ref{subsec:refSatParameterization} 小节的参数分工，定义连续相位模型的干扰参数向量为 +\begin{equation} + \boldsymbol\psi_{\mathrm{cp}} + = + \Big[ + \dot\nu_{\mathrm{ref}}, + \{\chi_\ell\}_{\ell=0}^{L-1}, + \{\rho_{\ell,p}^{\mathrm{cp}}:\ell=0,\ldots,L-1,\ p=0,\ldots,P-1\}, + \{\sigma_{n,\ell}^2\}_{\ell=0}^{L-1} + \Big]^\Tran. +\end{equation} +于是结构化连续相位模型对应的最大似然估计问题为 +\begin{equation} + (\widehat{\boldsymbol\eta}_{\mathrm{dyn}},\widehat{\boldsymbol\psi}_{\mathrm{cp}}) + = + \arg\max_{\boldsymbol\eta_{\mathrm{dyn}},\boldsymbol\psi_{\mathrm{cp}}} + p\left(\{\mathbf Y^{\mathrm{cp}}_{\ell,p}\}_{\ell,p};\boldsymbol\eta_{\mathrm{dyn}},\boldsymbol\psi_{\mathrm{cp}}\right). +\end{equation} + +在 ``不同卫星噪声方差不同，但同一卫星短窗内方差不变'' 的假设下，去掉与参数无关的常数项后，联合负对数似然函数可写为 +\begin{equation} + \mathcal L_{\mathrm{cp}}(\boldsymbol\eta_{\mathrm{dyn}},\boldsymbol\psi_{\mathrm{cp}}) + = + \sum_{\ell=0}^{L-1}M_\ell PT\log\sigma_{n,\ell}^2 + + + \sum_{\ell=0}^{L-1}\frac{1}{\sigma_{n,\ell}^2} + \sum_{p=0}^{P-1} + \left\| + \mathbf Y^{\mathrm{cp}}_{\ell,p} + - + \rho_{\ell,p}^{\mathrm{cp}}e^{j\chi_\ell} + \mathbf a_\ell(\boldsymbol\xi) + \big(\mathbf q^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})\big)^\Tran + \right\|_{\mathrm{F}}^2. +\end{equation} + +\subsubsection{对卫星噪声方差的轮廓化处理} + +对固定的其余参数，第 $\ell$ 颗卫星噪声方差 $\sigma_{n,\ell}^2$ 的极大似然估计为 +\begin{equation} + \hat\sigma_{n,\ell}^2 + = + \frac{1}{M_\ell PT} + \sum_{p=0}^{P-1} + \left\| + \mathbf Y^{\mathrm{cp}}_{\ell,p} + - + \rho_{\ell,p}^{\mathrm{cp}}e^{j\chi_\ell} + \mathbf a_\ell(\boldsymbol\xi) + \big(\mathbf q^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})\big)^\Tran + \right\|_{\mathrm{F}}^2. +\end{equation} +将其代回后，可得到对 $\{\sigma_{n,\ell}^2\}$ 轮廓化后的负对数似然： +\begin{equation} + \mathcal L_{\mathrm{cp}}^{\mathrm{pro}}(\boldsymbol\eta_{\mathrm{dyn}},\dot\nu_{\mathrm{ref}},\{\chi_\ell\},\{\rho_{\ell,p}^{\mathrm{cp}}\}) + = + \sum_{\ell=0}^{L-1}M_\ell PT + \log \mathcal R^{\mathrm{cp}}_\ell(\boldsymbol\eta_{\mathrm{dyn}},\dot\nu_{\mathrm{ref}},\chi_\ell,\{\rho_{\ell,p}^{\mathrm{cp}}\}_{p=0}^{P-1}), +\end{equation} +其中 +\begin{equation} + \mathcal R^{\mathrm{cp}}_\ell(\boldsymbol\eta_{\mathrm{dyn}},\dot\nu_{\mathrm{ref}},\chi_\ell,\{\rho_{\ell,p}^{\mathrm{cp}}\}_{p=0}^{P-1}) + \triangleq + \sum_{p=0}^{P-1} + \left\| + \mathbf Y^{\mathrm{cp}}_{\ell,p} + - + \rho_{\ell,p}^{\mathrm{cp}}e^{j\chi_\ell} + \mathbf a_\ell(\boldsymbol\xi) + \big(\mathbf q^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})\big)^\Tran + \right\|_{\mathrm{F}}^2. +\end{equation} +因此，在异方差条件下，连续相位结构化 MLE 可等价写为 +\begin{equation} + (\widehat{\boldsymbol\eta}_{\mathrm{dyn}},\widehat{\dot\nu}_{\mathrm{ref}},\{\hat\chi_\ell\},\{\hat\rho_{\ell,p}^{\mathrm{cp}}\}) + = + \arg\min_{\boldsymbol\eta_{\mathrm{dyn}},\dot\nu_{\mathrm{ref}},\{\chi_\ell\},\{\rho_{\ell,p}^{\mathrm{cp}}\}} + \sum_{\ell=0}^{L-1}M_\ell PT\log \mathcal R^{\mathrm{cp}}_\ell. +\end{equation} +若在实际实现中将 $\sigma_{n,\ell}^2$ 视为已知或预估量，则上式也可退化为按 $1/\sigma_{n,\ell}^2$ 加权的最小二乘形式。 + +\subsubsection{对幅度参数的闭式消元} + +为进一步集中似然，定义匹配统计量 +\begin{equation} + z^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}}) + \triangleq + \mathbf a_\ell^\Herm(\boldsymbol\xi) + \mathbf Y^{\mathrm{cp}}_{\ell,p} + \big(\mathbf q^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})\big)^*, +\end{equation} +以及归一化因子 +\begin{equation} + d^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}}) + \triangleq + \|\mathbf a_\ell(\boldsymbol\xi)\|_2^2 + \big\|\mathbf q^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})\big\|_2^2. +\end{equation} +则对固定的 $(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}},\chi_\ell)$，第 $(\ell,p)$ 个非负幅度模值的最优解为 +\begin{equation} + \hat\rho_{\ell,p}^{\mathrm{cp}} + = + \left[ + \frac{\Re\left\{e^{-j\chi_\ell}z^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})\right\}} + {d^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})} + \right]_{+}, +\end{equation} +其中 $[x]_{+}\triangleq \max\{x,0\}$。代回残差后，有 +\begin{equation} + \mathcal E^{\mathrm{cp,str}}_{\ell,p} + \triangleq + \|\mathbf Y^{\mathrm{cp}}_{\ell,p}\|_{\mathrm{F}}^2 + - + \frac{ + \left[ + \Re\left\{e^{-j\chi_\ell}z^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})\right\} + \right]_{+}^{2} + }{d^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})}. +\end{equation} +定义第 $\ell$ 颗卫星聚合残差为 +\begin{equation} + \mathcal R^{\mathrm{cp,str}}_\ell + \triangleq + \sum_{p=0}^{P-1}\mathcal E^{\mathrm{cp,str}}_{\ell,p}. +\end{equation} +于是，经过 $\{\sigma_{n,\ell}^2\}$ 和 $\{\rho_{\ell,p}^{\mathrm{cp}}\}$ 集中后的结构化 MLE 可写为 +\begin{equation} + \widehat{\boldsymbol\eta}_{\mathrm{dyn}} + = + \arg\min_{\boldsymbol\xi,\nu_{\mathrm{ref}}} + \min_{\dot\nu_{\mathrm{ref}},\{\chi_\ell\}} + \sum_{\ell=0}^{L-1}M_\ell PT\log \mathcal R^{\mathrm{cp,str}}_\ell. +\end{equation} +该表达式体现了本文动态主线：$\boldsymbol\eta_{\mathrm{dyn}}=[\boldsymbol\xi^\Tran,\nu_{\mathrm{ref}}]^\Tran$ 为主参数，$\dot\nu_{\mathrm{ref}}$ 和 $\{\chi_\ell\}$ 为干扰参数，而 $\{\rho_{\ell,p}^{\mathrm{cp}}\}$ 与 $\{\sigma_{n,\ell}^2\}$ 已被闭式或轮廓化消除。 + +\subsection{连续相位模型下的 relaxed concentrated MLE} +\label{subsec:dynamicRelaxedModel} + +若仅关心主参数搜索的可实现性，也可进一步引入自由复干扰系数 $\zeta_{\ell,p}^{\mathrm{cp,rel}}$，从而暂时放松 ``同一卫星跨帧共享公共相位'' 这一结构约束。模型可写为 +\begin{equation} + \mathbf Y^{\mathrm{cp}}_{\ell,p} + = + \zeta_{\ell,p}^{\mathrm{cp,rel}} + \mathbf a_\ell(\boldsymbol\xi) + \big(\mathbf q^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})\big)^\Tran + + + \mathbf N_{\ell,p}. +\end{equation} +对固定的 $(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})$，其最小二乘解为 +\begin{equation} + \hat\zeta_{\ell,p}^{\mathrm{cp,rel}} + = + \frac{z^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})} + {d^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})}. +\end{equation} +代回后得到块残差 +\begin{equation} + \mathcal E^{\mathrm{cp,rel}}_{\ell,p} + \triangleq + \|\mathbf Y^{\mathrm{cp}}_{\ell,p}\|_{\mathrm{F}}^2 + - + \frac{ + \left|z^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})\right|^2 + }{d^{\mathrm{cp}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})}. +\end{equation} +定义对应的每星聚合残差 +\begin{equation} + \mathcal R^{\mathrm{cp,rel}}_\ell + \triangleq + \sum_{p=0}^{P-1}\mathcal E^{\mathrm{cp,rel}}_{\ell,p}. +\end{equation} +于是，在异方差假设下的 relaxed concentrated MLE 可写为 +\begin{equation} + \widehat{\boldsymbol\eta}_{\mathrm{dyn,rel}} + = + \arg\min_{\boldsymbol\xi,\nu_{\mathrm{ref}}} + \min_{\dot\nu_{\mathrm{ref}}} + \sum_{\ell=0}^{L-1}M_\ell PT\log \mathcal R^{\mathrm{cp,rel}}_\ell. +\end{equation} +需要强调的是，relaxed concentrated MLE 只是放松了块复系数结构，并没有改变连续相位模板本身，因此它\emph{不是}独立相位模型。 + +\subsection{独立相位模型下的 concentrated MLE} + +本小节沿用第~\ref{subsec:independentPhaseModel} 小节的独立相位基线模型。这里的 +$\zeta_{\ell,p}^{\mathrm{ip}}\in\mathbb C$ 是每个观测块的自由复系数， +它吸收了帧间常相位项，因此和 CP 模型中的共享公共相位 $\chi_\ell$ 不同。正文 MLE 直接对自由复增益进行闭式消元；只有在 CRB 附录构造实参数 Fisher 信息矩阵时，才将其写作 +$\zeta_{\ell,p}^{\mathrm{ip}}=\rho_{\ell,p}^{\mathrm{ip}}e^{j\vartheta_{\ell,p}}$。 + +对于独立相位模型，可把 $\zeta_{\ell,p}^{\mathrm{ip}}$ 视为每个观测块的自由复干扰系数，则对应模型可写为 +\begin{equation} + \mathbf Y^{\mathrm{ip}}_{\ell,p} + = + \zeta_{\ell,p}^{\mathrm{ip}} + \mathbf a_\ell(\boldsymbol\xi) + \big(\mathbf q^{\mathrm{ip}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})\big)^\Tran + + + \mathbf N_{\ell,p}. +\end{equation} +相应地，定义 +\begin{equation} + z^{\mathrm{ip}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}}) + \triangleq + \mathbf a_\ell^\Herm(\boldsymbol\xi) + \mathbf Y^{\mathrm{ip}}_{\ell,p} + \big(\mathbf q^{\mathrm{ip}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})\big)^*, +\end{equation} +\begin{equation} + d^{\mathrm{ip}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}}) + \triangleq + \|\mathbf a_\ell(\boldsymbol\xi)\|_2^2 + \big\|\mathbf q^{\mathrm{ip}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})\big\|_2^2. +\end{equation} +其自由复干扰系数的闭式解为 +\begin{equation} + \hat\zeta_{\ell,p}^{\mathrm{ip}} + = + \frac{z^{\mathrm{ip}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})} + {d^{\mathrm{ip}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})}. +\end{equation} +代回后得到块残差 +\begin{equation} + \mathcal E^{\mathrm{ip}}_{\ell,p} + \triangleq + \|\mathbf Y^{\mathrm{ip}}_{\ell,p}\|_{\mathrm{F}}^2 + - + \frac{\left|z^{\mathrm{ip}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})\right|^2} + {d^{\mathrm{ip}}_{\ell,p}(\boldsymbol\xi,\nu_{\mathrm{ref}},\dot\nu_{\mathrm{ref}})}. +\end{equation} +定义每星聚合残差 +\begin{equation} + \mathcal R^{\mathrm{ip}}_\ell + \triangleq + \sum_{p=0}^{P-1}\mathcal E^{\mathrm{ip}}_{\ell,p}. +\end{equation} +则独立相位模型下的集中似然估计问题为 +\begin{equation} + \widehat{\boldsymbol\eta}_{\mathrm{dyn,ip}} + = + \arg\min_{\boldsymbol\xi,\nu_{\mathrm{ref}}} + \min_{\dot\nu_{\mathrm{ref}}} + \sum_{\ell=0}^{L-1}M_\ell PT\log \mathcal R^{\mathrm{ip}}_\ell. +\end{equation} +与连续相位模型相比，该准则不再利用跨帧绝对相位关系，因此更适合作为对比基线，而不宜作为本文主模型。 + +\subsection{模型与估计问题的几点说明} + +为避免后续推导中的歧义，需要对本文建模方式进一步说明如下。 +\begin{enumerate} + \item 本文从信号模型到 MLE 全程采用归一化离散域表达。实际物理量 $(\nu^{\mathrm{Hz}},\dot\nu^{\mathrm{Hz}})$ 只在几何解释层面出现，而真正参与似然与估计的是归一化量 $(\nu,\dot\nu)$； + \item 主参数是 $\boldsymbol\eta_{\mathrm{dyn}}=[\boldsymbol\xi^\Tran,\nu_{\mathrm{ref}}]^\Tran$，而 $\dot\nu_{\mathrm{ref}}$ 作为干扰参数处理。这与 ``先完成 DoA--参考 Doppler 联合估计，再分析 unknown rate 影响'' 的主线一致； + \item 多星噪声建模采用 ``星间异方差、星内短窗同方差'' 假设，即每颗卫星有自己的噪声方差 $\sigma_{n,\ell}^2$。这比 ``全星同一噪声方差'' 更符合实际链路差异，又不会显著破坏后续 MLE 与 CRB 推导的可处理性； + \item 连续相位模型、relaxed concentrated MLE 和独立相位模型是三个不同层面的对象：前者是本文的主信号模型；中者只是对块复系数结构的放松，仍保留连续相位模板；后者则真正放弃了跨帧绝对相位连续性，因此只能作为对比基线。 +\end{enumerate} + +与后续 known-rate/unknown-rate \gls{crb} 对比相关，还需要注意一个结构性现象。 +在 single-satellite multi-frame 情形中，若该卫星即 reference satellite，则参考链路满足 +\begin{equation} + f_{\nu,\ell_{\mathrm{ref}}}(\boldsymbol\xi,\nu_{\mathrm{ref}}) + = + \nu_{\mathrm{ref}}, + \qquad + f_{\dot\nu,\ell_{\mathrm{ref}}}(\boldsymbol\xi,\dot\nu_{\mathrm{ref}}) + = + \dot\nu_{\mathrm{ref}}. +\end{equation} +因此，reference-link 的 Doppler 和 Doppler rate 不通过几何映射对 $\boldsymbol\xi$ 产生额外相位导数。 +在幅度和公共相位干扰方向被消去后，DoA 导数主要保留为空间阵列流形的残差切向量，而 $\nu_{\mathrm{ref}}$ 与 $\dot\nu_{\mathrm{ref}}$ 的导数是沿 reference steering vector 的时间相位切向量。 +在标准窄带阵列模型下，这两个残差方向在 Fisher 内积意义下正交，因此 single-satellite multi-frame 的 DoA bound 在 known-rate 与 unknown-rate 设置下可以完全一致。 +该结论应被理解为 reference-satellite parameterization 的 sanity check，而不是新的主要创新点。 +在 multi-satellite 情形中，非参考星的 $\Delta\nu_\ell(\boldsymbol\xi)$ 和 $\Delta\dot\nu_\ell(\boldsymbol\xi)$ 会把时间相位导数引入 DoA score，因此严格正交性一般不再成立；若数值上 DoA bound 仍几乎重合，则说明该 representative geometry 下 DoA 方向与 residualized Doppler-rate nuisance 之间的耦合较弱。 + +因此，本文后续算法与性能分析都围绕如下统一问题展开：在短观测窗口下，基于多星多帧连续相位观测，利用结构化或 relaxed concentrated MLE 估计主参数 $\boldsymbol\eta_{\mathrm{dyn}}=[\boldsymbol\xi^\Tran,\nu_{\mathrm{ref}}]^\Tran$，并将 $\dot\nu_{\mathrm{ref}}$ 作为干扰参数进行 profile 处理和 calibrated information accounting；同时，以独立相位模型作为对比基线，定量说明保留跨帧连续相位的必要性。对应的 Fisher 信息和 CRB 推导分别放在 \ref{sec:A7_dynDoaDopplerCrb} 与 \ref{sec:A8_dynIndependentPhaseCrb} 中。 + + +\subsection{多星增益与 Doppler 信息可辨识性的投影解释} +\label{subsec:multiSatGainFdIdentifiability} + +从 Fisher 信息的角度看，多星协作带来的增益并不是对 +$\boldsymbol\xi$ 与 $\nu_{\mathrm{ref}}$ 的等比例放大。更精确地说，新增观测只增加主参数 score +在干扰参数 score 正交补上的投影长度。令 +\begin{equation} + \boldsymbol\eta_{\mathrm{dyn}} + = + \begin{bmatrix} + \boldsymbol\xi^\Tran & \nu_{\mathrm{ref}} + \end{bmatrix}^\Tran, + \qquad + \boldsymbol\psi_\mu + = + \text{只影响均值向量的干扰参数}, +\end{equation} +其中 $\boldsymbol\psi_\mu$ 可包含 $\dot\nu_{\mathrm{ref}}$、相位和幅度等局部参数。 +设白化后的均值 Jacobian 为 +\begin{equation} + \bar{\mathbf J}_{\eta} + \triangleq + \mathbf R_n^{-1/2} + \frac{\partial\boldsymbol\mu}{\partial\boldsymbol\eta_{\mathrm{dyn}}}, + \qquad + \bar{\mathbf J}_{\psi} + \triangleq + \mathbf R_n^{-1/2} + \frac{\partial\boldsymbol\mu}{\partial\boldsymbol\psi_\mu}. +\end{equation} +则消去干扰参数后的 EFIM 可写成投影 Gram 矩阵 +\begin{equation} + \mathcal I_{\mathrm{e}} + = + 2\Re\left\{ + \bar{\mathbf J}_{\eta}^{\Herm} + \mathcal P^{\perp}_{\bar{\mathbf J}_{\psi}} + \bar{\mathbf J}_{\eta} + \right\}, + \qquad + \mathcal P^{\perp}_{\bar{\mathbf J}_{\psi}} + = + \mathbf I- + \bar{\mathbf J}_{\psi} + \left(\bar{\mathbf J}_{\psi}^{\Herm}\bar{\mathbf J}_{\psi}\right)^\dagger + \bar{\mathbf J}_{\psi}^{\Herm}. + \label{eq:main_dyn_efim_projection} +\end{equation} +因此，EFIM 奇异的判据不是 $\bar{\mathbf J}_{\eta}$ 本身是否满秩，而是投影后的主参数 +Jacobian 是否满秩： +\begin{equation} + \mathcal I_{\mathrm{e}}\ \text{奇异} + \quad\Longleftrightarrow\quad + \operatorname{rank}\left( + \mathcal P^{\perp}_{\bar{\mathbf J}_{\psi}}\bar{\mathbf J}_{\eta} + \right) + < + \dim(\boldsymbol\eta_{\mathrm{dyn}}). + \label{eq:main_dyn_efim_rank_condition} +\end{equation} +等价地，存在非零方向 $\delta\boldsymbol\eta\neq\mathbf 0$，使得 +\begin{equation} + \bar{\mathbf J}_{\eta}\delta\boldsymbol\eta + \in + \operatorname{range}(\bar{\mathbf J}_{\psi}), + \label{eq:main_dyn_exact_nuisance_range} +\end{equation} +或在近奇异情形下满足 +\begin{equation} + \bar{\mathbf J}_{\eta}\delta\boldsymbol\eta + \approx + \bar{\mathbf J}_{\psi}\delta\boldsymbol\psi . + \label{eq:main_dyn_near_nuisance_range} +\end{equation} +这表明，某个主参数扰动的一阶均值变化若能被 $\dot\nu_{\mathrm{ref}}$、相位或幅度等干扰参数 +近似吸收，则该方向在边际化后只保留很少有效信息。 + +为了刻画 ``近吸收'' 的程度，可定义局部最小投影角 +\begin{equation} + \sin^2\theta_{\min} + \triangleq + \min_{\delta\boldsymbol\eta\neq\mathbf 0} + \frac{ + \left\| + \mathcal P^{\perp}_{\bar{\mathbf J}_{\psi}} + \bar{\mathbf J}_{\eta}\delta\boldsymbol\eta + \right\|_2^2 + }{ + \left\| + \bar{\mathbf J}_{\eta}\delta\boldsymbol\eta + \right\|_2^2 + }. + \label{eq:main_dyn_min_projection_angle} +\end{equation} +当 $\theta_{\min}\approx 0$ 时，说明主参数 score 子空间中存在接近干扰子空间的方向， +此时 $\lambda_{\min}(\mathcal I_{\mathrm{e}})$ 会很小。多星几何可以快速增大与 +$\boldsymbol\xi$ 有关的主特征值，但若式~\eqref{eq:main_dyn_min_projection_angle} 中的最小投影角 +仍然很小，则 EFIM 条件数 +\begin{equation} + \kappa(\mathcal I_{\mathrm{e}}) + = + \frac{\lambda_{\max}(\mathcal I_{\mathrm{e}})} + {\lambda_{\min}(\mathcal I_{\mathrm{e}})} +\end{equation} +仍可能随着多星几何增强而变大。 + +进一步，将 EFIM 按 +$\boldsymbol\eta_{\mathrm{dyn}}=[\boldsymbol\xi^\Tran,\nu_{\mathrm{ref}}]^\Tran$ 分块： +\begin{equation} + \mathcal I_{\mathrm{e}} + = + \begin{bmatrix} + \mathcal I_{\xi\xi} & \mathcal I_{\xi\nu} \\ + \mathcal I_{\nu\xi} & \mathcal I_{\nu\nu} + \end{bmatrix}, + \qquad + \mathcal I_{\xi\xi}\in\mathbb R^{2\times2},\quad + \mathcal I_{\nu\nu}\in\mathbb R . + \label{eq:main_dyn_efim_xi_nu_partition} +\end{equation} +若 $\boldsymbol\xi$ 已知，则 $\nu_{\mathrm{ref}}$ 的 scalar 信息为 $\mathcal I_{\nu\nu}$； +若 $\boldsymbol\xi$ 未知，则 $\nu_{\mathrm{ref}}$ 的边际信息为 Schur 补 +\begin{equation} + \mathcal I_{\nu|\xi} + = + \mathcal I_{\nu\nu} + - + \mathcal I_{\nu\xi} + \mathcal I_{\xi\xi}^{\dagger} + \mathcal I_{\xi\nu}. + \label{eq:main_dyn_nu_schur_after_xi} +\end{equation} +为避免与幅度符号 $\rho_{\ell,p}^{\mathrm{cp}}$ 混淆，本文不用 $\rho$ 表示相关系数；在本小节中局部定义 +DoA--Doppler 耦合比例 +\begin{equation} + \gamma_{\xi,\nu}^{2} + \triangleq + \frac{ + \mathcal I_{\nu\xi} + \mathcal I_{\xi\xi}^{\dagger} + \mathcal I_{\xi\nu} + }{ + \mathcal I_{\nu\nu} + }, + \qquad + 0\leq \gamma_{\xi,\nu}^{2}\leq 1, + \label{eq:main_dyn_xi_nu_coupling_fraction} +\end{equation} +则 +\begin{equation} + \mathcal I_{\nu|\xi} + = + \mathcal I_{\nu\nu}(1-\gamma_{\xi,\nu}^{2}), + \qquad + \operatorname{CRB}_{\nu} + = + \frac{1}{\mathcal I_{\nu\nu}(1-\gamma_{\xi,\nu}^{2})}. + \label{eq:main_dyn_nu_crb_coupling_fraction} +\end{equation} +式~\eqref{eq:main_dyn_nu_crb_coupling_fraction} 给出了更直接的解释：多星确实可能增加 +$\mathcal I_{\nu\nu}$，但若同时使 $\gamma_{\xi,\nu}^{2}$ 较大，则 Schur 补后的 +$\mathcal I_{\nu|\xi}$ 仍然增长有限。相对于 DoA 已知的 scalar bound，DoA 未知时的 Doppler +标准差放大量为 +\begin{equation} + \frac{\sqrt{\operatorname{CRB}_{\nu}}} + {1/\sqrt{\mathcal I_{\nu\nu}}} + = + \frac{1}{\sqrt{1-\gamma_{\xi,\nu}^{2}}}. + \label{eq:main_dyn_nu_std_inflation} +\end{equation} +因此，pair 场景中即使 scalar Doppler 信息有增加，只要 $\gamma_{\xi,\nu}^{2}$ 较大， +边际 Doppler CRB 仍会被显著放大。 + +若用上标 $(L)$ 表示 $L$ 颗协作卫星下的 EFIM 分块，则相对于单星的参考 Doppler 标准差增益可写为 +\begin{equation} + G_{\nu}(L) + \triangleq + \frac{\sqrt{\operatorname{CRB}_{\nu}^{(1)}}} + {\sqrt{\operatorname{CRB}_{\nu}^{(L)}}} + = + \sqrt{ + \frac{ + \mathcal I_{\nu\nu}^{(L)} + \left(1-\gamma_{\xi,\nu}^{2,(L)}\right) + }{ + \mathcal I_{\nu\nu}^{(1)} + \left(1-\gamma_{\xi,\nu}^{2,(1)}\right) + } + }. + \label{eq:main_dyn_multisat_nu_gain} +\end{equation} +只有当 DoA--Doppler 耦合比例不随 $L$ 增大而恶化时，$G_{\nu}(L)$ 才会接近能量叠加所对应的 +$\sqrt L$ 量级；若新增卫星主要增强 $\mathcal I_{\xi\xi}$ 的几何方向，而 +$\gamma_{\xi,\nu}^{2,(L)}$ 仍然较大，则 $\nu_{\mathrm{ref}}$ 的边际增益会明显小于 +角度 CRB 的改善。由此可见，实际 \gls{leo} 多星场景中的多星收益应分开理解： +$\boldsymbol\xi$ 的收益主要来自空间视线和阵列流形几何的多样性；$\nu_{\mathrm{ref}}$ 的收益主要来自 +能量叠加和跨帧相干结构，并受到式~\eqref{eq:main_dyn_nu_schur_after_xi} 中 DoA--Doppler +Schur 补损失的限制。 + +\subsection{周期多帧观测下的 Doppler comb 与 tooth ambiguity} +\label{subsec:fdCombToothAmbiguity} + +上一小节的 EFIM 分析描述的是已落入正确局部 basin 后的 in-tooth 曲率；它不能解释 +profile likelihood 在 Doppler 方向上的全局离散多峰结构。该多峰结构来自帧起始时刻集合 +本身。令参与相干匹配的帧集合为 $\mathcal P$，并记第 $p$ 帧起始点的离散采样偏移为 +$\tau_p$。若参考 Doppler 存在归一化角频率偏移 $\Delta\nu$，则跨帧公共相位因子为 +\begin{equation} + \exp\{j\Delta\nu\tau_p\}. +\end{equation} +因此，沿 $\nu_{\mathrm{ref}}$ 方向的集中目标中自然包含 frame-time spectral window +\begin{equation} + W_{\mathcal P}(\Delta\nu) + \triangleq + \sum_{p\in\mathcal P} w_p + \exp\{j\Delta\nu\tau_p\}, + \label{eq:main_fd_comb_window} +\end{equation} +其中 $w_p\geq0$ 表示第 $p$ 帧在局部线性化下的等效权重。若使用等间隔周期帧， +\begin{equation} + \tau_p=p\Delta\tau_{\mathrm{f}}, + \qquad p=0,\ldots,P-1, +\end{equation} +并且 $w_p$ 近似一致，则式~\eqref{eq:main_fd_comb_window} 退化为离散 Dirichlet 型窗口 +\begin{equation} + W_{\mathcal P}(\Delta\nu) + \approx + \exp\left\{j\frac{(P-1)\Delta\nu\Delta\tau_{\mathrm{f}}}{2}\right\} + \frac{ + \sin\left(P\Delta\nu\Delta\tau_{\mathrm{f}}/2\right) + }{ + \sin\left(\Delta\nu\Delta\tau_{\mathrm{f}}/2\right) + }. + \label{eq:main_fd_comb_dirichlet} +\end{equation} +于是当 +\begin{equation} + \Delta\nu + = + k\Delta\nu_{\mathrm{tooth}}, + \qquad + \Delta\nu_{\mathrm{tooth}} + \triangleq + \frac{2\pi}{\Delta\tau_{\mathrm{f}}}, + \qquad k\in\mathbb Z, + \label{eq:main_fd_comb_tooth_spacing_norm} +\end{equation} +有 +\begin{equation} + \exp\{j\Delta\nu\tau_p\} + = + \exp\{j2\pi kp\} + =1, + \qquad \forall p. +\end{equation} +因此，周期多帧观测会在参考 Doppler 方向上产生间隔为 +$\Delta\nu_{\mathrm{tooth}}$ 的 comb-like objective branches。若换算成物理 Hz 单位，并记相邻帧 +物理间隔为 $T_{\mathrm{f}}^{\mathrm{sec}}=\Delta\tau_{\mathrm{f}}T_{\mathrm s}$，则 +\begin{equation} + \Delta\nu_{\mathrm{tooth}}^{\mathrm{Hz}} + = + \frac{1}{T_{\mathrm{f}}^{\mathrm{sec}}}. + \label{eq:main_fd_comb_tooth_spacing_hz} +\end{equation} +该 comb 是全局 profile likelihood 的周期窗口效应，与局部 EFIM 曲率并不矛盾：EFIM 只刻画 +某一个 tooth 内的二阶曲率，而不保证全局唯一 tooth。 + +相邻 tooth 是否容易区分，还取决于单帧 pilot block 的有限长度。对于长度为 $T$ 个采样点的 +块，归一化 Doppler 偏移 $\Delta\nu$ 在块内产生的归一化相关为 +\begin{equation} + R_T(\Delta\nu) + \triangleq + \frac{1}{T}\sum_{t=0}^{T-1}\exp\{j\Delta\nu t\} + = + \frac{\exp\{j(T-1)\Delta\nu/2\}}{T} + \frac{\sin(T\Delta\nu/2)}{\sin(\Delta\nu/2)}. + \label{eq:main_fd_block_corr} +\end{equation} +因此，相邻 tooth 的块内相干损失可用 +\begin{equation} + D_{\mathrm{blk}} + \triangleq + 1- + \left|R_T(\Delta\nu_{\mathrm{tooth}})\right|^2 + \label{eq:main_fd_block_tooth_loss} +\end{equation} +刻画。当 $T\ll\Delta\tau_{\mathrm{f}}$ 时，由 +$\Delta\nu_{\mathrm{tooth}}=2\pi/\Delta\tau_{\mathrm{f}}$ 得到近似 +\begin{equation} + D_{\mathrm{blk}} + \approx + 1- + \operatorname{sinc}^{2}\left(\frac{T}{\Delta\tau_{\mathrm{f}}}\right) + \approx + \frac{\pi^2}{3}\left(\frac{T}{\Delta\tau_{\mathrm{f}}}\right)^2, + \label{eq:main_fd_block_loss_small_ratio} +\end{equation} +其中 $\operatorname{sinc}(x)=\sin(\pi x)/(\pi x)$。因此，增大 pilot block length 可以增强相邻 +Doppler tooth 的分离度；但这并不意味着单个块内的 Doppler-rate 二次相位已经提供了强曲率。 +在短同步块 \,\gls{leo} 场景中，$\dot\nu_{\mathrm{ref}}$ 对可辨识性的主要影响仍来自跨帧相位演化 +以及被作为干扰参数消去后的 Schur 补损失。 + +非周期帧子集可以用同一个窗口解释。若选取子集 +\begin{equation} + \mathcal T_{\mathrm{sub}} + = + \{\tau_{p_1},\ldots,\tau_{p_Q}\}, +\end{equation} +则对应窗口为 +\begin{equation} + W_{\mathrm{sub}}(\Delta\nu) + = + \sum_{q=1}^{Q}w_q + \exp\{j\Delta\nu\tau_{p_q}\}. + \label{eq:main_fd_subset_window} +\end{equation} +当 $\mathcal T_{\mathrm{sub}}$ 破坏严格等间隔结构时， +$\Delta\nu=k\Delta\nu_{\mathrm{tooth}}$ 不再必然使所有帧相位同时对齐，周期 comb sidelobes 会被压低。 +因此，非周期 subset 更适合用于 tooth acquisition；而完整周期数据在 tooth 选对之后，由于保留了更多 +相干能量，更适合同 tooth 内 refine。若定义第 $k$ 个 Doppler tooth 内的 profile optimum 为 +$\widehat{\boldsymbol\eta}_k$，则 tooth 可分离性可写为 +\begin{equation} + D_{\mathrm{tooth}} + \triangleq + \min_{k\in\mathbb Z\setminus\{0\}} + \mathcal L^{\mathrm{pro}}_{\mathrm{cp}}(\widehat{\boldsymbol\eta}_{k}) + - + \mathcal L^{\mathrm{pro}}_{\mathrm{cp}}(\widehat{\boldsymbol\eta}_{0}). + \label{eq:main_fd_tooth_discriminability} +\end{equation} +这里 $k=0$ 表示正确 tooth。若 $D_{\mathrm{tooth}}$ 较小或为负，则局部 CRB 即使很小，MLE 仍可能 +落在错误 tooth；若 $D_{\mathrm{tooth}}$ 足够大，则后续的 EFIM/CRB 才能作为同 tooth 内估计误差的局部描述。 + diff --git a/parts/multiArrayStatDetDoaDopplerEst.tex b/parts/multiArrayStatDetDoaDopplerEst.tex new file mode 100644 index 0000000..f3a9db7 --- /dev/null +++ b/parts/multiArrayStatDetDoaDopplerEst.tex @@ -0,0 +1,286 @@ +\section{静态多阵列确定信号 DoA--Doppler 估计问题模型} +\label{sec:multiArrayStatDetDoaDopplerEst} + +\subsection{离散时间统一符号与参考星参数化} + +本节在第~\ref{subsec:multiArrayCommonNotation} 小节统一几何与参考卫星参数化的基础上，考虑单观测块内的多星确定信号 DoA--Doppler 联合估计问题。与单阵列模型相比，本节的新增结构是：各卫星局部导向矢量均由同一个全局方向 $\boldsymbol\xi=[\alpha,\beta]^\Tran$ 诱导，各链路归一化 Doppler 均由参考卫星链路参数 $\nu_{\mathrm{ref}}$ 诱导。 + +设共有 $L$ 颗协作卫星，卫星索引为 +\begin{equation} + \ell=0,1,\ldots,L-1. +\end{equation} +静态模型仅考虑单个观测块（单帧、单个 \gls{cpi} 或单个处理块）内的估计问题。块内离散采样点索引为 +\begin{equation} + t=0,1,\ldots,T-1, +\end{equation} +其中 $T$ 为该观测块内的采样点数。本文不再额外引入连续时间变量；若需要与实际物理量建立联系，仅通过采样周期 $T_{\mathrm{s}}$ 进行一次性转换。 + +固定选择一颗卫星作为参考卫星，其索引记为 $\ell_{\mathrm{ref}}$，除特别说明外默认取 $\ell_{\mathrm{ref}}=0$。参考卫星链路上的归一化 Doppler 为 +\begin{equation} + \nu_{\mathrm{ref}}\triangleq \nu_{\ell_{\mathrm{ref}}}. +\end{equation} +根据第~\ref{subsec:refSatParameterization} 小节的约定，第 $\ell$ 条链路的归一化 Doppler 由 +\begin{equation} + \nu_\ell + = + f_{\nu,\ell}(\boldsymbol\xi,\nu_{\mathrm{ref}}), + \qquad + \ell=0,1,\ldots,L-1, +\end{equation} +给出，并满足 +\begin{equation} + f_{\nu,\ell_{\mathrm{ref}}}(\boldsymbol\xi,\nu_{\mathrm{ref}}) + = + \nu_{\mathrm{ref}}. +\end{equation} +等价地，也可写成 +\begin{equation} + \nu_\ell + = + \nu_{\mathrm{ref}}+\Delta\nu_\ell(\boldsymbol\xi), +\end{equation} +其中 $\Delta\nu_\ell(\boldsymbol\xi)$ 由已知卫星位置、速度、姿态和参考星选择确定。因此，静态多星 DoA--Doppler 主参数统一为 +\begin{equation} + \boldsymbol\eta_{\mathrm{stat}} + = + \begin{bmatrix} + \boldsymbol\xi\\ + \nu_{\mathrm{ref}} + \end{bmatrix}, +\end{equation} +而每颗卫星的 $\nu_\ell$ 只作为由 $(\boldsymbol\xi,\nu_{\mathrm{ref}})$ 诱导的局部链路量出现。 + +\subsection{静态离散信号模型} +\label{subsec:staticBlockModel} + +考虑单用户上行已知训练序列 $s(t)$。在单个观测块内，假设： +\begin{enumerate} + \item DoA 在整个观测块内保持不变； + \item 每条链路的 Doppler 在该观测块内近似为常数； + \item 每颗卫星对应一个未知复干扰系数 $\zeta_\ell$； + \item 不同卫星的噪声方差允许不同，但同一颗卫星在该观测块内噪声方差保持不变。 +\end{enumerate} +于是，第 $\ell$ 颗卫星在第 $t$ 个采样点的阵列接收向量可写为 +\begin{equation} + \mathbf y_\ell(t) + = + \zeta_\ell + \mathbf a_\ell(\boldsymbol\xi) + s(t) + e^{j\nu_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}})t} + + + \mathbf n_\ell(t), + \qquad t=0,1,\ldots,T-1, +\end{equation} +其中为书写紧凑记 +\begin{equation} + \nu_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}}) + \triangleq + f_{\nu,\ell}(\boldsymbol\xi,\nu_{\mathrm{ref}}). +\end{equation} +这里 $\zeta_\ell\in\mathbb C$ 为第 $\ell$ 条链路的未知复干扰系数，噪声满足 +\begin{equation} + \mathbf n_\ell(t) + \sim + \mathcal{CN}(\mathbf 0,\sigma_{n,\ell}^2\mathbf I_{M_\ell}), + \qquad \ell=0,1,\ldots,L-1. +\end{equation} +这里采用\emph{星间异方差}假设，即不同卫星的噪声方差 $\sigma_{n,\ell}^2$ 可以不同。 + +将第 $\ell$ 颗卫星在整个观测块内的数据按时间堆叠，定义时间模板 +\begin{equation} + \mathbf q_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}}) + \triangleq + \begin{bmatrix} + s(0)e^{j\nu_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}})\cdot 0} \\ + s(1)e^{j\nu_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}})\cdot 1} \\ + \vdots \\ + s(T-1)e^{j\nu_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}})(T-1)} + \end{bmatrix} + \in\mathbb C^{T\times 1}. +\end{equation} +相应地，第 $\ell$ 颗卫星的矩阵观测可写为 +\begin{equation} + \mathbf Y_\ell + = + \zeta_\ell + \mathbf a_\ell(\boldsymbol\xi) + \mathbf q_\ell^\Tran(\boldsymbol\xi,\nu_{\mathrm{ref}}) + + + \mathbf N_\ell, +\end{equation} +其中 $\mathbf Y_\ell\in\mathbb C^{M_\ell\times T}$，$\mathbf N_\ell\in\mathbb C^{M_\ell\times T}$ 的各列独立满足 $\mathcal{CN}(\mathbf 0,\sigma_{n,\ell}^2\mathbf I_{M_\ell})$。 + +为便于后续似然函数构建和 \gls{crb} 推导，进一步向量化定义 +\begin{equation} + \tilde{\mathbf y}_\ell + \triangleq + \vect(\mathbf Y_\ell), + \qquad + \tilde{\mathbf n}_\ell + \triangleq + \vect(\mathbf N_\ell), +\end{equation} +并定义空时导向向量 +\begin{equation} + \mathbf g_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}}) + \triangleq + \mathbf q_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}}) + \otimes + \mathbf a_\ell(\boldsymbol\xi) + \in\mathbb C^{M_\ell T\times 1}. +\end{equation} +则向量化观测模型可统一写为 +\begin{equation} + \tilde{\mathbf y}_\ell + = + \zeta_\ell\mathbf g_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}}) + + + \tilde{\mathbf n}_\ell. +\end{equation} +与第~\ref{subsec:refSatParameterization} 小节一致，本文把 +\begin{equation} + \boldsymbol\psi_{\mathrm{stat}} + = + \Big[ + \{\zeta_\ell\}_{\ell=0}^{L-1}, + \{\sigma_{n,\ell}^2\}_{\ell=0}^{L-1} + \Big]^\Tran +\end{equation} +作为静态模型的干扰参数。若在后续 \gls{crb} 推导中需要显式使用实参数化，则可进一步将 $\zeta_\ell$ 分解为其实部和虚部。 + +\subsection{联合似然函数与结构化最大似然估计} + +本小节不再重新定义新的观测模型，而是直接沿用上一小节的向量化形式 $\tilde{\mathbf y}_\ell=\zeta_\ell\mathbf g_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}})+\tilde{\mathbf n}_\ell$。因此，下文中的 $\mathbf g_\ell$、$\zeta_\ell$、$\sigma_{n,\ell}^2$ 与模型小节完全一致；待估主参数为 $\boldsymbol\eta_{\mathrm{stat}}$，干扰参数为 $\boldsymbol\psi_{\mathrm{stat}}$。 + +由于不同卫星噪声相互独立，且每颗卫星内部噪声服从圆对称复高斯分布，因此多星联合似然函数可由各星观测块的概率密度相乘得到。去掉与参数无关的常数项后，对应的联合负对数似然函数为 +\begin{equation} + \mathcal L_{\mathrm{stat}}(\boldsymbol\eta_{\mathrm{stat}},\boldsymbol\psi_{\mathrm{stat}}) + = + \sum_{\ell=0}^{L-1} + \left[ + M_\ell T\log\sigma_{n,\ell}^2 + + + \frac{1}{\sigma_{n,\ell}^2} + \left\| + \tilde{\mathbf y}_\ell + - + \zeta_\ell\mathbf g_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}}) + \right\|_2^2 + \right]. +\end{equation} +因此，结构化最大似然估计问题可写为 +\begin{equation} + (\widehat{\boldsymbol\eta}_{\mathrm{stat}},\widehat{\boldsymbol\psi}_{\mathrm{stat}}) + = + \arg\min_{\boldsymbol\eta_{\mathrm{stat}},\boldsymbol\psi_{\mathrm{stat}}} + \mathcal L_{\mathrm{stat}}(\boldsymbol\eta_{\mathrm{stat}},\boldsymbol\psi_{\mathrm{stat}}). +\end{equation} + +\subsubsection{对复干扰系数参数的闭式消元} + +对于固定的 $(\boldsymbol\xi,\nu_{\mathrm{ref}})$ 和 $\{\sigma_{n,\ell}^2\}$，第 $\ell$ 条链路复干扰系数 $\zeta_\ell$ 的最小二乘解为 +\begin{equation} + \hat\zeta_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}}) + = + \frac{\mathbf g_\ell^\Herm(\boldsymbol\xi,\nu_{\mathrm{ref}})\tilde{\mathbf y}_\ell} + {\|\mathbf g_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}})\|_2^2}. +\end{equation} +将其代回后，对应的每星残差可写为 +\begin{equation} + \mathcal E_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}}) + \triangleq + \|\tilde{\mathbf y}_\ell\|_2^2 + - + \frac{ + \left| + \mathbf g_\ell^\Herm(\boldsymbol\xi,\nu_{\mathrm{ref}})\tilde{\mathbf y}_\ell + \right|^2 + }{\|\mathbf g_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}})\|_2^2}. +\end{equation} +于是对 $\{\zeta_\ell\}$ 集中后的负对数似然可写为 +\begin{equation} + \mathcal L_{\mathrm{stat,c}}(\boldsymbol\xi,\nu_{\mathrm{ref}},\{\sigma_{n,\ell}^2\}) + = + \sum_{\ell=0}^{L-1} + \left[ + M_\ell T\log\sigma_{n,\ell}^2 + + + \frac{1}{\sigma_{n,\ell}^2} + \mathcal E_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}}) + \right]. +\end{equation} + +\subsubsection{对噪声方差的轮廓化处理} + +若进一步把每颗卫星的噪声方差视为未知参数，则对固定的 $(\boldsymbol\xi,\nu_{\mathrm{ref}})$，其极大似然估计为 +\begin{equation} + \hat\sigma_{n,\ell}^2(\boldsymbol\xi,\nu_{\mathrm{ref}}) + = + \frac{1}{M_\ell T} + \mathcal E_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}}), + \qquad + \ell=0,1,\ldots,L-1. +\end{equation} +代回后，可得到完全轮廓化的目标函数 +\begin{equation} + \mathcal L_{\mathrm{stat}}^{\mathrm{pro}}(\boldsymbol\xi,\nu_{\mathrm{ref}}) + = + \sum_{\ell=0}^{L-1} + M_\ell T + \log \mathcal E_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}}), +\end{equation} +其中同样忽略了与参数无关的常数项。因此，在星间异方差假设下，静态多阵列 DoA--Doppler 问题的轮廓最大似然估计可写为 +\begin{equation} + \widehat{\boldsymbol\eta}_{\mathrm{stat}} + = + \arg\min_{\boldsymbol\xi,\nu_{\mathrm{ref}}} + \sum_{\ell=0}^{L-1} + M_\ell T + \log \mathcal E_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}}). +\end{equation} +若实际中各星噪声方差 $\sigma_{n,\ell}^2$ 已知或可由预处理阶段获得，则也可采用加权集中似然形式 +\begin{equation} + \widehat{\boldsymbol\eta}_{\mathrm{stat}} + = + \arg\max_{\boldsymbol\xi,\nu_{\mathrm{ref}}} + \sum_{\ell=0}^{L-1} + \frac{1}{\sigma_{n,\ell}^2} + \frac{ + \left| + \mathbf g_\ell^\Herm(\boldsymbol\xi,\nu_{\mathrm{ref}})\tilde{\mathbf y}_\ell + \right|^2 + }{\|\mathbf g_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}})\|_2^2}. +\end{equation} +该表达式表明：在多星协作条件下，联合估计的本质是在统一的全局方向参数 $\boldsymbol\xi$ 和参考归一化 Doppler $\nu_{\mathrm{ref}}$ 下，对各卫星空时匹配输出进行加权融合；其中权重既受几何映射影响，也受不同卫星噪声水平影响。 + +\subsection{仿真结果说明} + +图 \ref{fig:dualSatSingleSourceScene} 使用 STK 给出了双星单用户场景，用户位于两颗卫星的交叠覆盖区中，卫星视角为 $\pm 55^\circ$。 + +图 \ref{fig:doaDopplerEciDualSatSingleSourceUpa2048Snap} 给出了在存在 Doppler 的情况下，发送导频信号并采用上述 \gls{mle} 方法时，仅进行 DoA 估计与联合进行 DoA--Doppler 估计的二维地表坐标估计结果。结果表明，在经度和纬度两个维度上，DoA--Doppler 联合估计均优于仅 DoA 估计，但与 \gls{crb} 之间仍存在一定差距，这一差距可能与地面坐标到卫星局部角度的非线性映射有关。对于 Doppler 参数估计，其结果基本达到 \gls{crb}。 + +图 \ref{fig:multiSatSingleSourceScene} 使用 STK，按照用户仰角 $15^\circ$ 和卫星视角 $\pm 55^\circ$ 给出了多星单用户场景。在此基础上，可进一步针对不同卫星组合计算 \gls{crb}，从而实现基于 \gls{crb} 的选星。 + +\begin{figure}[!h] + \centering + \includegraphics[width=0.9\textwidth]{fig/dualSatSingleSourceScene.bmp} + \caption{静态多星 DoA--Doppler 主线基线：双星单源几何场景示意} + \label{fig:dualSatSingleSourceScene} +\end{figure} + +\begin{figure}[!h] + \centering + \includegraphics[width=0.95\textwidth]{fig/doaDopplerEciDualSatSingleSourceUpa2048Snap.pdf} + \caption{静态多星 DoA--Doppler 主线基线：ECI 坐标系下双星 UPA、单信源场景中，单独估计 DoA 与联合估计 DoA--Doppler 的 MSE 对比} + \label{fig:doaDopplerEciDualSatSingleSourceUpa2048Snap} +\end{figure} + +\begin{figure}[!h] + \centering + \includegraphics[width=0.85\textwidth]{fig/multiSatSingleSourceScene.bmp} + \caption{静态多星 DoA--Doppler 主线基线：多星单源几何场景示意} + \label{fig:multiSatSingleSourceScene} +\end{figure} + +本节完成了多星主线中的静态基线：同一个全局方向 $\boldsymbol\xi$ 同时决定各星局部导向矢量，而参考星 Doppler $\nu_{\mathrm{ref}}$ 通过几何映射决定各星链路 Doppler。下一节将在保持主参数 $\boldsymbol\eta_{\mathrm{dyn}}=\boldsymbol\eta_{\mathrm{stat}}$ 不变的前提下，引入多帧索引、Doppler rate 和跨帧连续相位结构。 diff --git a/parts/multiArrayStoDoaEst.tex b/parts/multiArrayStoDoaEst.tex new file mode 100644 index 0000000..12e7b16 --- /dev/null +++ b/parts/multiArrayStoDoaEst.tex @@ -0,0 +1,619 @@ +\section{多阵列随机信号 DoA 估计} +\label{sec:multiArrayStoDoaEst} + +本节是多阵列随机 DoA 支线，用于说明多颗卫星或多个阵列如何通过几何映射共享同一组全局未知参数。它与后续多星 DoA--Doppler 主线在“多阵列几何约束”上相通，但不引入参考星 Doppler 参数 $\nu_{\mathrm{ref}}$，因此阅读主线时可将本节视为多阵列 DoA 的背景和对照。 + +\subsection{参考阵列角度参数化的多阵列随机 DoA 模型} +\label{subsec:projection_operator} + +在多阵列（多卫星）协同 \gls{doa} 估计中，不同阵列由于位置与指向各不相同，即便面对同一组地面目标，其局部观测角度也不一致。因此，需要建立从某一参考阵列的角度参数到任意阵列角度参数的映射，确保所有阵列在 \gls{mle} 中共享一套全局未知参数。本节给出这一映射的几何构造，并定义角度映射算子。 + +\subsubsection{参考阵列与角度映射} + +为便于参数表示，将第 $r$ 个阵列（例如 $r=0$）作为参考阵列，并将其局部来向集合记为 +% +\begin{align} + \bm{\Theta}_\mathrm{ref} + \triangleq + \left[\bm{\theta}_{\mathrm{ref},0},\ldots,\bm{\theta}_{\mathrm{ref},K-1}\right], +\end{align} +% +对于其余阵列 $\ell \neq r$，通过几何关系（卫星轨道、姿态以及坐标变换）构造从参考阵列角度到第 $\ell$ 个阵列局部角度的已知角度映射算子 $\mathcal T_\ell(\,\cdot\,)$，使得 +% +\begin{align} + \bm{\Theta}_\ell + = \mathcal T_\ell(\bm{\Theta}_\mathrm{ref}),\qquad \ell=0,\ldots,L-1, +\end{align} +% +则第 $\ell$ 个阵列的导向矩阵可以写成 +% +\begin{align} + \mathbf{A}_\ell(\bm{\Theta}) + = + \mathbf{A}_\ell\left(\bm{\Theta}_\ell\right) + = + \mathbf{A}_\ell\left(\mathcal T_\ell(\bm{\Theta}_\mathrm{ref})\right), +\end{align} +% +从而所有阵列的协方差矩阵 $\{\mathbf{R}_\ell\}$ 最终都只依赖于同一组参考角度参数 $\bm{\Theta}_\mathrm{ref}$。 + +\subsubsection{卫星--地面几何关系} + +设地面目标、第 $\ell$ 个卫星阵列、参考阵列（例如卫星 $0$）的 \gls{ecef} 坐标分别为 $\mathbf{p}_g, \mathbf{p}_\ell, \mathbf{p}_{\mathrm{ref}} \in \mathbb{R}^3$，从卫星 $\ell$ 指向地面目标的全局方向向量定义为 +\begin{align} + \mathbf{u}_{\ell}^{\mathrm{global}} + = + \frac{\mathbf{p}_g - \mathbf{p}_\ell}{ + \|\mathbf{p}_g - \mathbf{p}_\ell\|} + \in \mathbb{R}^3 . +\end{align} + +几何上，$\mathbf{u}_{\ell}^{\mathrm{global}}$ 表示地面目标在 \gls{ecef} 坐标系下的方向，而 \gls{doa} 估计所依赖的角度参数必须定义在卫星阵列的局部坐标系内。因此需要构造卫星局部坐标系并将全局方向向量旋转至其坐标系下。 + +\subsubsection{局部坐标系的建立} + +对于卫星 $\ell$，首先定义其局部坐标系在 \gls{ecef} 坐标系下的坐标轴全局方向向量 $\{\mathbf{x}_\ell, \mathbf{y}_\ell, \mathbf{z}_\ell\}$。不失一般性，假设阵列法向指向地心，也即取局部 $\mathbf{z}$ 轴为 +% +\begin{align*} + \mathbf{z}_\ell = -\frac{\mathbf{p}_\ell}{\|\mathbf{p}_\ell\|}, +\end{align*} +% +然后选取一个与 $\mathbf{z}_\ell$ 不平行的向量 +% +\begin{align*} + \mathbf{v} = \begin{cases} + [1,0,0]^\Tran, & |z_{\ell,x}| < 0.9,\\ + [0,1,0]^\Tran, & \text{otherwise}, + \end{cases} +\end{align*} +% +其中 $z_{\ell,x}$ 表示 $\mathbf{z}_\ell$ 在 \gls{ecef} 坐标线 $x$ 方向的分量，以此构造局部 $x$ 轴 +% +\begin{align*} + \mathbf{x}_\ell = \frac{\mathbf{v} \times \mathbf{z}_\ell}{ \|\mathbf{v} \times \mathbf{z}_\ell\| }, +\end{align*} +% +最后，局部 $y$ 轴由右手法则给出： +% + \begin{align*} + \mathbf{y}_\ell = \mathbf{z}_\ell \times \mathbf{x}_\ell . +\end{align*} +% +由此，从\emph{局部坐标系}到\emph{全局坐标系}的旋转矩阵可写为 +% +\begin{align} + \boldsymbol{\Omega}_\ell + = \begin{bmatrix} + \mathbf{x}_\ell, \mathbf{y}_\ell, \mathbf{z}_\ell + \end{bmatrix} + \in \mathbb{R}^{3\times 3}, +\end{align} +% +其中列向量为局部坐标轴在全局坐标系下的方向。 + +利用该旋转矩阵可将 \gls{ecef} 坐标系下的全局方向向量旋转到阵列的局部坐标系中 +% +\begin{align} + \mathbf{u}_{\ell}^{\mathrm{local}} + = \boldsymbol{\Omega}_\ell^\Tran \mathbf{u}_{\ell}^{\mathrm{global}} + = \begin{bmatrix} + u_{\ell,x}, u_{\ell,y}, u_{\ell,z} + \end{bmatrix}^\Tran. +\end{align} +% +由此得到以阵列为参考系的局部方位角与俯仰角分别为 +\begin{align} + \alpha_\ell^{\mathrm{loc}} = \operatorname{atan2}(u_{\ell,y}, u_{\ell,x}),\qquad + \beta_\ell^{\mathrm{loc}} = \arcsin(u_{\ell,z}). +\end{align} + +\subsubsection{角度映射算子} + +我们希望在似然函数中使用统一的一组未知方向参数来描述所有阵列的视线方向。为此，选定参考阵列（例如卫星 $0$），将其局部方向参数 +% +\begin{align*} + \bm{\theta}_{\mathrm{ref}} = (\alpha_{\mathrm{ref}}^{\mathrm{loc}}, \beta_{\mathrm{ref}}^{\mathrm{loc}}) +\end{align*} +% +作为全局未知参数。对应的局部单位方向向量为 +% +\begin{align} + \mathbf{u}_{\mathrm{ref}}^{\mathrm{local}}(\bm{\theta}_{\mathrm{ref}}) + = + \begin{bmatrix} + \cos\beta_{\mathrm{ref}}^{\mathrm{loc}} \cos\alpha_{\mathrm{ref}}^{\mathrm{loc}} \\[1mm] + \cos\beta_{\mathrm{ref}}^{\mathrm{loc}} \sin\alpha_{\mathrm{ref}}^{\mathrm{loc}} \\[1mm] + \sin\beta_{\mathrm{ref}}^{\mathrm{loc}} + \end{bmatrix} . +\end{align} +% +利用局部到全局的旋转矩阵 \(\boldsymbol{\Omega}_{\mathrm{ref}}\)，可得到参考阵列视线在 \gls{ecef} 坐标系下的方向 +% +\begin{align} + \mathbf{u}_{\mathrm{ref}}^{\mathrm{global}}(\bm{\theta}_{\mathrm{ref}}) + = \boldsymbol{\Omega}_{\mathrm{ref}} + \mathbf{u}_{\mathrm{ref}}^{\mathrm{local}}(\bm{\theta}_{\mathrm{ref}}). +\end{align} +% +接下来即可根据参考阵列的 \gls{ecef} 坐标 $\mathbf{p}_{\mathrm{ref}}$ 与视线方向 $\mathbf{u}_{\mathrm{ref}}^{\mathrm{global}}(\bm{\theta}_{\mathrm{ref}})$ 恢复地面目标的空间位置。 + +\begin{enumerate} + \item 假定目标位于地球表面，并将地球近似为半径为 $\mathcal{R}_\mathrm{E}$ 的球体，则地面点 $\mathbf{p}_g$ 的坐标可表示为 + % + \begin{align} + \mathbf{p}_g + = \mathbf{p}_{\mathrm{ref}} + + r_{\mathrm{ref}}\, + \mathbf{u}_{\mathrm{ref}}^{\mathrm{global}}(\bm{\theta}_{\mathrm{ref}}), + \end{align} + % + 其中距离参数 $r_{\mathrm{ref}} > 0 $由射线与球面 $\|\mathbf{p}_g\| = \mathcal{R}_\mathrm{E}$ 的交点条件确定。由于 $\mathbf{u}_{\mathrm{ref}}^{\mathrm{global}}$ 为单位向量，有 + % + \begin{align*} + \left\|\mathbf{p}_{\mathrm{ref}} + r_{\mathrm{ref}}\mathbf{u}_{\mathrm{ref}}^{\mathrm{global}}\right\|^2 + = \mathcal{R}_\mathrm{E}^2 + \quad\Longrightarrow\quad + r_{\mathrm{ref}}^2 + + 2\,\mathbf{p}_{\mathrm{ref}}^\Tran \mathbf{u}_{\mathrm{ref}}^{\mathrm{global}} r_{\mathrm{ref}} + + \left\|\mathbf{p}_{\mathrm{ref}}\right\|^2 - \mathcal{R}_\mathrm{E}^2 + = 0, + \end{align*} + % + 从而 + % + \begin{align} + r_{\mathrm{ref}} + = + - \mathbf{p}_{\mathrm{ref}}^\Tran + \mathbf{u}_{\mathrm{ref}}^{\mathrm{global}}(\bm{\theta}_{\mathrm{ref}}) + - \sqrt{ + \left(\mathbf{p}_{\mathrm{ref}}^\Tran + \mathbf{u}_{\mathrm{ref}}^{\mathrm{global}}(\bm{\theta}_{\mathrm{ref}})\right)^2 + - \left(\|\mathbf{p}_{\mathrm{ref}}\|^2 - \mathcal{R}_\mathrm{E}^2\right) + }, + \end{align} + % + 其中取使地面点位于参考阵列视线前方的正根。 + + \item 假定目标位于地球椭球表面，其 \gls{ecef} 坐标满足 + % + \begin{align} + \mathbf{p}_g^\Tran \mathbf{B}\,\mathbf{p}_g = 1, + \end{align} + % + 其中 $\mathbf{B}$ 为描述地球椭球的对称正定矩阵。 例如，对于赤道半轴为 $a$、极半轴为 $b$ 的轴对称椭球，有 $\mathbf{B} = \operatorname{diag}\bigl(1/a^2,\,1/a^2,\,1/b^2\bigr)$，则地面点可写为 + % + \begin{align} + \mathbf{p}_g + = \mathbf{p}_{\mathrm{ref}} + + r_{\mathrm{ref}}\, + \mathbf{u}_{\mathrm{ref}}^{\mathrm{global}}(\bm{\theta}_{\mathrm{ref}}), + \end{align} + % + 其中距离参数 $r_{\mathrm{ref}} > 0$ 由射线与椭球的交点条件 $\mathbf{p}_g^\Tran \mathbf{B}\,\mathbf{p}_g = 1$ 确定。将上述射线方程代入可得关于 $r_{\mathrm{ref}}$ 的二次方程 + % + \begin{align} + A r_{\mathrm{ref}}^2 + B r_{\mathrm{ref}} + C = 0, + \end{align} + % + 其中 $A = (\mathbf{u}_{\mathrm{ref}}^{\mathrm{global}})^\Tran \mathbf{B} \mathbf{u}_{\mathrm{ref}}^{\mathrm{global}}$，$B = 2\mathbf{p}_{\mathrm{ref}}^\Tran \mathbf{B} \mathbf{u}_{\mathrm{ref}}^{\mathrm{global}}$，$C = \mathbf{p}_{\mathrm{ref}}^\Tran \mathbf{B} \mathbf{p}_{\mathrm{ref}} - 1$，从而 + % + \begin{align} + r_{\mathrm{ref}} + = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}, + \end{align} + 其中取沿视线方向、对应地面点位于参考阵列视线前方的正根。 + +\end{enumerate} + +在得到 $\mathbf{p}_g$ 后，第 $\ell$ 个阵列（卫星）指向该地面点的全局单位方向向量为 +% +\begin{align} + \mathbf{u}_\ell^{\mathrm{global}}(\bm{\theta}_{\mathrm{ref}}) + = + \frac{ + \mathbf{p}_g - \mathbf{p}_\ell + }{ + \left\|\mathbf{p}_g - \mathbf{p}_\ell\right\| + }. +\end{align} +% +利用第 $\ell$ 个阵列的旋转矩阵 $\boldsymbol{\Omega}_\ell$ 可得到其局部坐标系中的方向 +% +\begin{align} + \mathbf{u}_\ell^{\mathrm{local}}(\bm{\theta}_{\mathrm{ref}}) + = \boldsymbol{\Omega}_\ell^\Tran + \mathbf{u}_\ell^{\mathrm{global}}(\bm{\theta}_{\mathrm{ref}}) +\end{align} +% +由此，第 $\ell$ 个阵列在自身局部坐标系下的方位角与俯仰角可写为 +% +\begin{align} + \alpha_\ell^{\mathrm{loc}}(\bm{\theta}_{\mathrm{ref}}) + = \operatorname{atan2} \left( + \left[\mathbf{u}_\ell^{\mathrm{local}}(\bm{\theta}_{\mathrm{ref}})\right]_y, + \left[\mathbf{u}_\ell^{\mathrm{local}}(\bm{\theta}_{\mathrm{ref}})\right]_x + \right),\quad + \beta_\ell^{\mathrm{loc}}(\bm{\theta}_{\mathrm{ref}}) + = \arcsin\left( + \left[\mathbf{u}_\ell^{\mathrm{local}}(\bm{\theta}_{\mathrm{ref}})\right]_z + \right). +\end{align} +% +于是可以定义从参考阵列角度到第 $\ell$ 个阵列角度的\emph{非线性}角度映射算子 +% +\begin{align} + \bm{\theta}_\ell + = \mathcal T_\ell(\bm{\theta}_{\mathrm{ref}}) + \triangleq + \Bigl( + \alpha_\ell^{\mathrm{loc}}(\bm{\theta}_{\mathrm{ref}}),\, + \beta_\ell^{\mathrm{loc}}(\bm{\theta}_{\mathrm{ref}}) + \Bigr). +\end{align} +% +可以看出，$\mathcal T_\ell(\,\cdot\,)$ 由参考阵列与第 $\ell$ 个阵列的 \gls{ecef} 坐标 $\mathbf{p}_{\mathrm{ref}}, \mathbf{p}_\ell$、对应的姿态矩阵 $\boldsymbol{\Omega}_{\mathrm{ref}}, \boldsymbol{\Omega}_\ell$，以及地球模型参数 $\mathcal{R}_\mathrm{E}$ 唯一确定。因此，在似然函数中可以仅以参考阵列的方向参数$\bm{\theta}_{\mathrm{ref}}$ 作为未知量，并通过几何映射 $\mathcal T_\ell$ 将其映射到各个阵列的局部角度，实现多阵列的统一参数化建模。 + +\subsection{参考阵列角度参数化下的多阵列似然函数求取} + +假设信源服从圆对称复高斯分布 $\mathbf{x}_t \sim \mathcal{CN} \left( 0,\mathbf R_x \right)$，为简化讨论，假设各信源相互不相关，则 $\mathbf R_x$ 为对角矩阵，且其协方差矩阵非奇异 $\det \left( \mathbf R_x \right) \ne 0$，时域不相关 +% +\begin{equation} + \begin{cases} + \mathbb{E} \left[\mathbf{x}_t \mathbf{x}_s^\Herm \right] = \mathbf R_x \delta(t-s) + \\ + \mathbb{E} \left[ \mathbf{x}_t \mathbf{x}_s^\Tran \right] = 0 + \end{cases}, \quad\forall t,s. +\end{equation} +% +由于噪声 $\mathbf{n}_{\ell,t} \sim \mathcal{CN}(0,\sigma_{n,\ell}^2 \mathbf{I}_{M_\ell})$，且与 $\mathbf{x}_t$ 不相关。则在给定 $(\bm{\Theta},\mathbf R_x,\{\sigma_{n,\ell}^2\})$ 的条件下，$\{\mathbf{y}_{\ell,t}\}$ 为相互独立的圆对称复高斯向量，也即 $\mathbf{y}_{\ell,t} \sim \mathcal{CN}(\mathbf{0},\mathbf{R}_\ell)$，因而联合似然函数为 +% +\begin{align} + L\left(\{\mathbf{R}_\ell\},\bm{\Theta}\middle|\{\mathbf{y}_{\ell,t}\}\right) + &= \prod_{\ell=0}^{L-1} \prod_{t=0}^{T-1} + \frac{1}{\pi^{M_\ell}\det(\mathbf{R}_\ell)} + \exp\!\left(-\mathbf{y}_{\ell,t}^\Herm \mathbf{R}_\ell^{-1}\mathbf{y}_{\ell,t}\right) + \nonumber\\ + &= \prod_{\ell=0}^{L-1} + \pi^{-M_\ell T} \left[\det(\mathbf{R}_\ell)\right]^{-T} + \exp\!\left( + - T\,\tr\left(\mathbf{R}_\ell^{-1}\hat{\mathbf{R}}_\ell\right) + \right). +\end{align} +% +其中第 $\ell$ 个阵列的样本协方差矩阵为 $\hat{\mathbf{R}}_\ell \triangleq \displaystyle\frac{1}{T} \sum_{t=0}^{T-1} \mathbf{y}_{\ell,t}\mathbf{y}_{\ell,t}^\Herm$，对应的对数似然函数为 +% +\begin{equation} + \ln \mathcal{L}\left(\{\mathbf{R}_\ell\},\bm{\Theta}\right) + = \mathrm{Const} + - T \sum_{\ell=0}^{L-1} \log\det(\mathbf{R}_\ell) + - T \sum_{\ell=0}^{L-1} \tr\left(\mathbf{R}_\ell^{-1}\hat{\mathbf{R}}_\ell\right). +\end{equation} + +\subsubsection{原始问题建模} + +引入参考局部来向集合 $\bm{\Theta}_\mathrm{ref}$ 将多阵列对数似然函数写为 +% +\begin{equation} + \ln \mathcal{L}\left(\{\mathbf{R}_\ell\},\bm{\Theta}_\mathrm{ref}\right) + = \mathrm{Const} + - T \sum_{\ell=0}^{L-1} \log\det(\mathbf{R}_\ell) + - T \sum_{\ell=0}^{L-1} \tr\left(\mathbf{R}_\ell^{-1}\hat{\mathbf{R}}_\ell\right), +\end{equation} +% +其中 $\mathbf{R}_\ell + = + \mathbf{A}_\ell\left(\mathcal T_\ell(\bm{\Theta}_\mathrm{ref})\right) + \,\mathbf R_x\, + \mathbf{A}_\ell^\Herm\left(\mathcal T_\ell(\bm{\Theta}_\mathrm{ref})\right) + + \sigma_{n,\ell}^2 \mathbf{I}_{M_\ell}. +$ 则多阵列的 \gls{sml}/\gls{uml} 优化问题可以表述为 +% +\begin{alignat}{2} + \label{eq:multi_arr_sto_ml_doa_opt} + &\underset{\bm{\Theta}_\mathrm{ref},\,\mathbf R_x,\,\{\sigma_{n,\ell}^2\}}{\mathrm{minimize}} \quad + && \sum_{\ell=0}^{L-1} + \left[ + \log\det(\mathbf{R}_\ell) + + \tr\left(\mathbf{R}_\ell^{-1}\hat{\mathbf{R}}_\ell\right) + \right] \nonumber\\ + &\text{subject to} \quad + && \mathbf{R}_\ell + = + \mathbf{A}_\ell\left(\mathcal T_\ell(\bm{\Theta}_\mathrm{ref})\right) + \,\mathbf R_x\, + \mathbf{A}_\ell^\Herm\left(\mathcal T_\ell(\bm{\Theta}_\mathrm{ref})\right) + + \sigma_{n,\ell}^2 \mathbf{I}_{M_\ell},\ \ell=0,\ldots,L-1, \nonumber\\ + &&& \bm{\Theta}_\mathrm{ref} \in \varTheta_\mathrm{ref},\quad + \mathbf R_x \succeq \mathbf{0},\quad + \sigma_{n,\ell}^2 > 0. +\end{alignat} +% +可以看到，相比 $\eqref{eq:sto_ml_doa_opt}$ 的单阵列情形，多阵列的随机最大似然只是将代价函数在阵列索引 $\ell$ 上累加，同时通过映射算子 $\mathcal T_\ell(\,\cdot\,)$ 将所有阵列的导向矩阵统一为同一组参考角度参数 $\bm{\Theta}_\mathrm{ref}$，从而保证最终的参数估计结果只有一组角度。 + +\subsubsection{集中似然表达} + +严格而言，多阵列随机模型中各阵列共享同一信号协方差矩阵 $\mathbf R_x$。为了得到与单阵列情形形式一致、便于实现的集中似然形式，本文采用一种常见的近似处理：对每个阵列分别应用单阵列的集中化步骤，得到局部的 $\widehat{\mathbf R}_{x,\ell}(\bm{\Theta}_\mathrm{ref})$ 与 $\widehat{\sigma}_{n,\ell}^2(\bm{\Theta}_\mathrm{ref})$，再在 $\ell$ 维度上进行联合搜索。 + + +与单阵列情形类似，我们只关心来向参数 $\bm{\Theta}_\mathrm{ref}$，因此希望将对数似然函数中的其他参量（包括信号协方差 $\mathbf R_x$ 以及噪声方差 $\{\sigma_{n,\ell}^2\}$）用 $\bm{\Theta}_\mathrm{ref}$ 和观测 $\{\mathbf{y}_{\ell,t}\}$ 来表征。对 \eqref{eq:multi_arr_sto_ml_doa_opt} 中 $\mathbf R_x$ 和 $\{\sigma_{n,\ell}^2\}$ 分别求偏导并令其为零，可以得到与单阵列情形类似的极大似然估计分别为 +% +\begin{equation} + \widehat{\mathbf R}_{x,\ell}(\bm{\Theta}_\mathrm{ref}) = (\mathbf{A}_\ell^\Herm \mathbf{A}_\ell)^{-1} \mathbf{A}_\ell^\Herm \hat{\mathbf{R}}_\ell \mathbf{A}_\ell (\mathbf{A}_\ell^\Herm \mathbf{A}_\ell)^{-1} - \widehat{\sigma}_{n,\ell}^2(\bm{\Theta}_\mathrm{ref}) (\mathbf{A}_\ell^\Herm \mathbf{A}_\ell)^{-1}, +\end{equation} +% +\begin{equation} + \widehat{\sigma}_{n,\ell}^2(\bm{\Theta}_\mathrm{ref}) = \frac{1}{M_\ell-K} \operatorname{tr}\left[ \left(\mathbf{I}_{M_\ell} - \mathbf{A}_\ell(\mathbf{A}_\ell^\Herm \mathbf{A}_\ell)^{-1} \mathbf{A}_\ell^\Herm \right) \hat{\mathbf{R}}_\ell \right], +\end{equation} +% +将这些估计回代到多阵列对数似然函数中，$\displaystyle\sum_{\ell}\tr(\mathbf{R}_\ell^{-1}\hat{\mathbf{R}}_\ell)$ 在最优点处变为常数项，可被消去，从而得到关于 $\bm{\Theta}_\mathrm{ref}$ 的集中随机最大似然估计 +% +\begin{equation} + \bm{\Theta}_{\mathrm{CSML}} + = \argmin_{\bm{\Theta}_\mathrm{ref}} + \sum_{\ell=0}^{L-1} + \log\det\!\left[ + \mathbf{A}_\ell\left(\mathcal T_\ell(\bm{\Theta}_\mathrm{ref})\right) + \widehat{\mathbf R}_{x,\ell}(\bm{\Theta}_\mathrm{ref}) + \mathbf{A}_\ell^\Herm\left(\mathcal T_\ell(\bm{\Theta}_\mathrm{ref})\right) + + \widehat{\sigma}_{n,\ell}^2(\bm{\Theta}_\mathrm{ref})\mathbf{I}_{M_\ell} + \right]. +\end{equation} +% +换言之，多阵列的 \gls{csml} 估计可以通过最小化联合代价函数 +% +\begin{equation} + \mathcal{J}_\mathrm{joint}(\bm{\Theta}_\mathrm{ref}) + \triangleq + \sum_{\ell=0}^{L-1} + \log\det\!\left[ + \mathbf{A}_\ell\left(\mathcal T_\ell(\bm{\Theta}_\mathrm{ref})\right) + \widehat{\mathbf R}_{x,\ell}(\bm{\Theta}_\mathrm{ref}) + \mathbf{A}_\ell^\Herm\left(\mathcal T_\ell(\bm{\Theta}_\mathrm{ref})\right) + + \widehat{\sigma}_{n,\ell}^2(\bm{\Theta}_\mathrm{ref})\mathbf{I}_{M_\ell} + \right] +\end{equation} +% +而得到，其中 $\bm{\Theta}_\mathrm{ref}$ 是唯一的待估角度参数向量，其通过已知的角度映射算子 $\{\mathcal T_\ell(\,\cdot\,)\}$ 统一约束了所有阵列的局部来向，从而自然实现了多星观测在同一角度域下的联合最大似然融合。 + +\subsection{仿真结果} + +下面基于多阵列角度映射进行仿真。算法的流程是： +% +\begin{enumerate} + \item 选择其中一个阵列为参考阵列； + \item 从参考阵列的视角选择初始角度 $\bm{\theta}_\mathrm{ref}^{(0)}$，计算得到代价函数； + \item 从参考阵列的 \gls{ecef} 坐标 $\mathbf{p}_\mathrm{ref}$ 出发，沿着 $\bm{\theta}_\mathrm{ref}^{(0)}$ 方向做射线，求得射线和地表的交点（也即假设用户位于地表）； + \item 由交点和第 $\ell$ 个阵列的 \gls{ecef} 坐标 $\mathbf{p}_g$ 和 $\mathbf{p}_\ell$，求得第 $\ell$ 个阵列视角下的信号来向 $\bm{\theta}_\ell^{(0)}$，计算得到代价函数； + \item 将所有阵列的代价函数相加，得到总的代价函数； + \item 遍历可行角度集，找到使代价函数最小的角度，即为目标角度。 +\end{enumerate} + +图 \ref{fig:doaEcefDualSatSingleSourceUpa20000Snap} 给出了基于若干算法得到的估计结果，分别是： +% +\begin{enumerate} + \item 单阵列 \gls{music}，仅进行角度估计，作为估计性能的基线； + \item 多阵列平均 \gls{music}，两个阵列独立进行 \gls{doa} 估计，第二个阵列估计得到的角度结合阵列坐标得到和地表的交点，再变换到第一个阵列的视角下，两个角度取平均作为估计结果； + \item 单阵列 \gls{mle}：按照上述算法流程，但是仅包含一颗卫星，同样会进行角度 $\to$ 地表坐标 $\to$ 角度的变换； + \item 多阵列平均 \gls{mle}：两颗卫星分别按照算法流程进行处理，第二个阵列的估计结果类似多阵列 \gls{music} 统一到第一个阵列的视角下； + \item 多阵列联合 \gls{mle}：按照上述算法流程进行处理，包含两颗卫星。 +\end{enumerate} + +\begin{figure}[!h] + \centering + \includegraphics[width=0.8\textwidth]{fig/doaEcefDualSatSingleSourceUpa20000Snap.pdf} + \caption{多阵列随机 DoA 支线：ECEF 坐标系下单/双星 UPA、单信源场景中 MUSIC 与 MLE 的 DoA MSE（20000 快拍）} + \label{fig:doaEcefDualSatSingleSourceUpa20000Snap} +\end{figure} + +从估计结果来看，\gls{mle} 的估计结果不理想，初步怀疑求射线和地表交点这一步会引入较大的非线性误差，下一步的仿真考虑遍历地面坐标，只进行坐标 $\to$ 角度的映射，看能否改善估计误差。 + +\subsection{地面坐标参数化的多阵列随机 DoA 模型} + +假设信源服从圆对称复高斯分布 $\mathbf{x}_t \sim \mathcal{CN} \left( 0,\mathbf R_x \right)$，为简化讨论，假设各信源相互不相关，则 $\mathbf R_x$ 为对角矩阵，且协方差矩阵非奇异 $\det \left( \mathbf R_x \right) \ne 0$，时域不相关 +% +\begin{equation} + \begin{cases} + \mathbb{E} \left[\mathbf{x}_t \mathbf{x}_s^\Herm \right] = \mathbf R_x \,\delta(t-s),\\[1mm] + \mathbb{E} \left[ \mathbf{x}_t \mathbf{x}_s^\Tran \right] = 0 + \end{cases}, \quad\forall t,s. +\end{equation} +% +噪声满足 $\mathbf{n}_{\ell,t} \sim \mathcal{CN}(0,\sigma_{n,\ell}^2 \mathbf{I}_{M_\ell})$，且与 $\mathbf{x}_t$ 不相关。 +在给定 $(\bm{\Xi},\mathbf R_x,\{\sigma_{n,\ell}^2\})$ 的条件下，$\{\mathbf{y}_{\ell,t}\}$ 为相互独立的圆对称复高斯向量，也即 $\mathbf{y}_{\ell,t} \sim \mathcal{CN}(\mathbf{0},\mathbf{R}_\ell)$，联合似然函数为 +% +\begin{align} + L\left(\{\mathbf{R}_\ell\},\bm{\Xi}\middle|\{\mathbf{y}_{\ell,t}\}\right) + &= \prod_{\ell=0}^{L-1} \prod_{t=0}^{T-1} + \frac{1}{\pi^{M_\ell}\det(\mathbf{R}_\ell)} + \exp\!\left(-\mathbf{y}_{\ell,t}^\Herm \mathbf{R}_\ell^{-1}\mathbf{y}_{\ell,t}\right) + \nonumber\\ + &= \prod_{\ell=0}^{L-1} + \pi^{-M_\ell T} \left[\det(\mathbf{R}_\ell)\right]^{-T} + \exp\!\left( + - T\,\tr\left(\mathbf{R}_\ell^{-1}\hat{\mathbf{R}}_\ell\right) + \right), +\end{align} +% +其中第 $\ell$ 个阵列的样本协方差矩阵为 +% +\begin{equation} + \hat{\mathbf{R}}_\ell + \triangleq \frac{1}{T} \sum_{t=0}^{T-1} \mathbf{y}_{\ell,t}\mathbf{y}_{\ell,t}^\Herm, +\end{equation} +% +对应的对数似然函数为 +% +\begin{equation} + \ln \mathcal{L}\left(\{\mathbf{R}_\ell\},\bm{\Xi}\right) + = \mathrm{Const} + - T \sum_{\ell=0}^{L-1} \log\det(\mathbf{R}_\ell) + - T \sum_{\ell=0}^{L-1} \tr\left(\mathbf{R}_\ell^{-1}\hat{\mathbf{R}}_\ell\right). +\end{equation} + +\subsubsection{原始问题建模} + +为建立多阵列协同估计中的统一参数化，本文以地面目标的空间位置作为全局未知量。 +设第 $k$ 个地面目标在 \gls{ecef} 坐标系下的位置为 +$\mathbf{p}_{g,k} \in \mathbb{R}^3$，将所有目标位置堆叠为 +% +\begin{align} + \bm{P}_g + \triangleq + \bigl[\mathbf{p}_{g,0},\ldots,\mathbf{p}_{g,K-1}\bigr] + \in \mathbb{R}^{3\times K}, +\end{align} +% +并将其矢量化后记为 +% +\begin{align} + \bm{\Xi} \triangleq \vect(\bm{P}_g) \in \mathbb{R}^{3K}, +\end{align} +% +作为多阵列随机信号模型中的统一未知参数。 + +给定卫星阵列的 \gls{ecef} 坐标 $\{\mathbf{p}_\ell\}$ 与姿态矩阵 $\{\boldsymbol{\Omega}_\ell\}$，地面目标 $\mathbf{p}_{g,k}$ 相对于第 $\ell$ 个阵列的视线方向由 +% +\begin{align} + \mathbf{u}_{\ell,k}^{\mathrm{global}}(\bm{P}_g) + &= \frac{\mathbf{p}_{g,k} - \mathbf{p}_\ell} + {\|\mathbf{p}_{g,k} - \mathbf{p}_\ell\|},\\ + \mathbf{u}_{\ell,k}^{\mathrm{local}}(\bm{P}_g) + &= \boldsymbol{\Omega}_\ell^\Tran \mathbf{u}_{\ell,k}^{\mathrm{global}}(\bm{P}_g), +\end{align} +% +从而得到第 $\ell$ 个阵列视角下第 $k$ 个目标的局部方位角与俯仰角 +% +\begin{subequations} +\begin{align} + \alpha_{\ell,k}^{\mathrm{loc}}(\bm{P}_g) + &= \operatorname{atan2}\!\left( + [\mathbf{u}_{\ell,k}^{\mathrm{local}}(\bm{P}_g)]_y,\, + [\mathbf{u}_{\ell,k}^{\mathrm{local}}(\bm{P}_g)]_x + \right),\\ + \beta_{\ell,k}^{\mathrm{loc}}(\bm{P}_g) + &= \arcsin\!\left( + [\mathbf{u}_{\ell,k}^{\mathrm{local}}(\bm{P}_g)]_z + \right). +\end{align} +\end{subequations} +% +记第 $\ell$ 个阵列的局部来向集合为 +% +\begin{align} + \bm{\Theta}_\ell(\bm{P}_g) + \triangleq + \bigl[ + \bm{\theta}_{\ell,0}(\bm{P}_g),\ldots, + \bm{\theta}_{\ell,K-1}(\bm{P}_g) + \bigr],\quad + \bm{\theta}_{\ell,k}(\bm{P}_g) + \triangleq + \bigl(\alpha_{\ell,k}^{\mathrm{loc}}(\bm{P}_g),\beta_{\ell,k}^{\mathrm{loc}}(\bm{P}_g)\bigr), +\end{align} +% +则第 $\ell$ 个阵列的导向矩阵可写为 +% +\begin{align} + \mathbf{A}_\ell(\bm{P}_g) + \triangleq + \mathbf{A}_\ell\bigl(\bm{\Theta}_\ell(\bm{P}_g)\bigr). +\end{align} +% +在此记号下，第 $\ell$ 个阵列的信号协方差矩阵为 +% +\begin{align} + \mathbf{R}_\ell(\bm{P}_g,\mathbf R_x,\sigma_{n,\ell}^2) + = + \mathbf{A}_\ell(\bm{P}_g)\, + \mathbf R_x\, + \mathbf{A}_\ell^\Herm(\bm{P}_g) + + \sigma_{n,\ell}^2 \mathbf{I}_{M_\ell}. +\end{align} +% +因此，多阵列的随机最大似然/无约束最大似然优化问题可表述为 +% +\begin{alignat}{2} + \label{eq:multi_arr_sto_ml_geo_opt} + &\underset{\bm{P}_g,\,\mathbf R_x,\,\{\sigma_{n,\ell}^2\}}{\mathrm{minimize}} \quad + && \sum_{\ell=0}^{L-1} + \left[ + \log\det\bigl(\mathbf{R}_\ell(\bm{P}_g,\mathbf R_x,\sigma_{n,\ell}^2)\bigr) + + \tr\Bigl(\mathbf{R}_\ell^{-1}(\bm{P}_g,\mathbf R_x,\sigma_{n,\ell}^2)\,\hat{\mathbf{R}}_\ell\Bigr) + \right] \nonumber\\ + &\text{subject to} \quad + && \bm{P}_g \in \mathcal{G},\quad + \mathbf R_x \succeq \mathbf{0},\quad + \sigma_{n,\ell}^2 > 0,\ \ell=0,\ldots,L-1, +\end{alignat} +% +其中 $\mathcal{G}$ 表示地面目标位置的可行集合（例如 WGS-84 椭球表面或给定覆盖区域）。 +可以看到，相比 \eqref{eq:sto_ml_doa_opt} 的单阵列情形，多阵列的随机最大似然只是将代价函数在阵列索引 $\ell$ 上累加，同时通过几何关系 +$\bm{P}_g \mapsto \bm{\Theta}_\ell(\bm{P}_g)$ +将所有阵列的导向矩阵统一为同一组地面目标坐标参数，从而保证最终的参数估计结果在几何上对应唯一的一组地面位置。 + +\subsubsection{集中似然表达} + +严格而言，多阵列随机模型中各阵列共享同一信号协方差矩阵 $\mathbf R_x$。 +为了得到与单阵列情形形式一致、便于实现的集中似然形式，本文采用一种常见的近似处理：对每个阵列分别应用单阵列的集中化步骤，得到局部的 +$\widehat{\mathbf R}_{x,\ell}(\bm{P}_g)$ 与 +$\widehat{\sigma}_{n,\ell}^2(\bm{P}_g)$，再在 $\ell$ 维度上进行联合搜索。 + +与单阵列情形类似，我们只关心地面目标坐标参数 $\bm{P}_g$，因此希望将对数似然函数中的其他参量（包括信号协方差 $\mathbf R_x$ 以及噪声方差 $\{\sigma_{n,\ell}^2\}$）用 $\bm{P}_g$ 和观测 $\{\mathbf{y}_{\ell,t}\}$ 来表征。 +在上述近似框架下，对每个阵列的局部似然函数分别对 $\mathbf R_x$ 和 $\sigma_{n,\ell}^2$ 求偏导并令其为零，可得到单阵列形式的闭式极大似然估计 +% +\begin{equation} + \widehat{\mathbf R}_{x,\ell}(\bm{P}_g) + = \bigl(\mathbf{A}_\ell^\Herm(\bm{P}_g)\,\mathbf{A}_\ell(\bm{P}_g)\bigr)^{-1} + \mathbf{A}_\ell^\Herm(\bm{P}_g)\,\hat{\mathbf{R}}_\ell\, + \mathbf{A}_\ell(\bm{P}_g)\, + \bigl(\mathbf{A}_\ell^\Herm(\bm{P}_g)\,\mathbf{A}_\ell(\bm{P}_g)\bigr)^{-1} + - \widehat{\sigma}_{n,\ell}^2(\bm{P}_g)\, + \bigl(\mathbf{A}_\ell^\Herm(\bm{P}_g)\,\mathbf{A}_\ell(\bm{P}_g)\bigr)^{-1}, +\end{equation} +\begin{equation} + \widehat{\sigma}_{n,\ell}^2(\bm{P}_g) + = \frac{1}{M_\ell-K} \operatorname{tr}\!\left[ + \Bigl(\mathbf{I}_{M_\ell} + - \mathbf{A}_\ell(\bm{P}_g) + \bigl(\mathbf{A}_\ell^\Herm(\bm{P}_g)\,\mathbf{A}_\ell(\bm{P}_g)\bigr)^{-1} + \mathbf{A}_\ell^\Herm(\bm{P}_g) + \Bigr)\, + \hat{\mathbf{R}}_\ell + \right]. +\end{equation} +% +将这些局部极大似然估计回代到多阵列对数似然函数中， +$\sum_{\ell}\tr(\mathbf{R}_\ell^{-1}\hat{\mathbf{R}}_\ell)$ +在最优点处变为与 $\bm{P}_g$ 无关的常数项，可被消去，从而得到关于地面目标坐标参数的集中随机最大似然估计 +% +\begin{equation} + \bm{P}_{g,\mathrm{CSML}} + = \argmin_{\bm{P}_g} + \sum_{\ell=0}^{L-1} + \log\det\!\left[ + \mathbf{A}_\ell(\bm{P}_g)\, + \widehat{\mathbf R}_{x,\ell}(\bm{P}_g)\, + \mathbf{A}_\ell^\Herm(\bm{P}_g) + + \widehat{\sigma}_{n,\ell}^2(\bm{P}_g)\,\mathbf{I}_{M_\ell} + \right]. +\end{equation} +% +换言之，多阵列的 \gls{csml} 估计可以通过最小化联合代价函数 +% +\begin{equation} + \mathcal{J}_\mathrm{joint}(\bm{P}_g) + \triangleq + \sum_{\ell=0}^{L-1} + \log\det\!\left[ + \mathbf{A}_\ell(\bm{P}_g)\, + \widehat{\mathbf R}_{x,\ell}(\bm{P}_g)\, + \mathbf{A}_\ell^\Herm(\bm{P}_g) + + \widehat{\sigma}_{n,\ell}^2(\bm{P}_g)\,\mathbf{I}_{M_\ell} + \right] +\end{equation} +% +而得到，其中 $\bm{P}_g$ 是唯一的待估地面坐标参数，其通过几何映射 +$\bm{P}_g \mapsto \{\bm{\Theta}_\ell(\bm{P}_g)\}$ 统一约束了所有阵列的局部来向，从而自然实现了基于地面位置参数的多星观测联合最大似然融合。 + +本节的关键作用是展示“多个局部阵列观测由同一个全局几何参数约束”的思想。后续多星静态 DoA--Doppler 主线会保留这一共享全局参数的结构，但把待估目标改写为 $\boldsymbol\eta_{\mathrm{stat}}=[\boldsymbol\xi^\Tran,\nu_{\mathrm{ref}}]^\Tran$，并显式利用各星链路 Doppler 的参考星参数化。 diff --git a/parts/stoDoaDopplerEst.tex b/parts/stoDoaDopplerEst.tex new file mode 100644 index 0000000..eddc530 --- /dev/null +++ b/parts/stoDoaDopplerEst.tex @@ -0,0 +1,5 @@ +\section{随机信号 DoA-Doppler 估计} + +\subsection{CRB 推导} + +\subsection{随机/无条件最大似然} \ No newline at end of file diff --git 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Lett.}, + volume = {21}, + number = {11}, + pages = {2273--2277}, + issn = {1548-5757}, + doi = {10.1109/LAWP.2022.3185208}, + urldate = {2024-07-28}, + langid = {american} +} + +@article{linDynamicBeamPattern2022, + title = {Dynamic {{Beam Pattern}} and {{Bandwidth Allocation Based}} on {{Multi-Agent Deep Reinforcement Learning}} for {{Beam Hopping Satellite Systems}}}, + author = {Lin, Zhiyuan and Ni, Zuyao and Kuang, Linling and Jiang, Chunxiao and Huang, Zhen}, + year = 2022, + month = apr, + journal = {IEEE Trans. Veh. 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Commun.}, + volume = {71}, + number = {1}, + pages = {282--295}, + issn = {1558-0857}, + doi = {10.1109/TCOMM.2022.3226190}, + urldate = {2024-07-27}, + langid = {american} +} + +@article{linNGSOSatellitesBeam2023, + title = {{{NGSO Satellites Beam Hopping Strategy Based}} on {{Load Balancing}} and {{Interference Avoidance}} for {{Coexistence With GSO Systems}}}, + author = {Lin, Zhiyuan and Ni, Zuyao and Kuang, Linling and Jiang, Chunxiao and Huang, Zhen}, + year = 2023, + month = jan, + journal = {IEEE Commun. Lett.}, + volume = {27}, + number = {1}, + pages = {278--282}, + issn = {1558-2558}, + doi = {10.1109/LCOMM.2022.3213912}, + urldate = {2024-08-04}, + langid = {american} +} + +@article{linRobustMultiObjectiveBeamforming2019, + title = {Robust {{Multi-Objective Beamforming}} for {{Integrated Satellite}} and {{High Altitude Platform Network With Imperfect Channel State Information}}}, + author = {Lin, Zhi and Lin, Min and Huang, Yongming and de Cola, Tomaso and Zhu, Wei-Ping}, + year = 2019, + month = dec, + journal = {IEEE Trans. Signal Process.}, + volume = {67}, + number = {24}, + pages = {6384--6396}, + issn = {1941-0476}, + doi = {10.1109/TSP.2019.2952045}, + urldate = {2024-08-04}, + langid = {american} +} + +@inproceedings{linRobustSecrecyEnergy2019, + title = {Robust {{Secrecy Energy Efficient Beamforming}} in {{Satellite Communication Systems}}}, + booktitle = {{{ICC}} 2019 - 2019 {{IEEE International Conference}} on {{Communications}} ({{ICC}})}, + author = {Lin, Zhi and Yin, Chunyan and Ouyang, Jian and Wu, Xiaohuan and Panagopoulos, Athanasios D.}, + year = 2019, + month = may, + pages = {1--5}, + issn = {1938-1883}, + doi = {10.1109/ICC.2019.8761056}, + urldate = {2025-03-13} +} + +@article{linRobustSecureBeamforming2018, + title = {Robust {{Secure Beamforming}} for {{5G Cellular Networks Coexisting With Satellite Networks}}}, + author = {Lin, Zhi and Lin, Min and Wang, Jun-Bo and Huang, Yongming and Zhu, Wei-Ping}, + year = 2018, + month = apr, + journal = {IEEE J. Sel. Areas Commun.}, + volume = {36}, + number = {4}, + pages = {932--945}, + issn = {1558-0008}, + doi = {10.1109/JSAC.2018.2824760}, + urldate = {2025-01-16}, + langid = {american} +} + +@article{linRobustSecureBeamforming2019, + title = {Robust {{Secure Beamforming}} for {{Multibeam Satellite Communication Systems}}}, + author = {Lin, Zhi and Lin, Min and Ouyang, Jian and Zhu, Wei-Ping and Panagopoulos, Athanasios D. and Alouini, Mohamed-Slim}, + year = 2019, + month = jun, + journal = {IEEE Trans. Veh. Technol.}, + volume = {68}, + number = {6}, + pages = {6202--6206}, + issn = {1939-9359}, + doi = {10.1109/TVT.2019.2913793}, + urldate = {2024-12-03}, + langid = {american} +} + +@article{linRobustSecureBeamforming2021, + title = {Robust {{Secure Beamforming}} for {{Wireless Powered Cognitive Satellite-Terrestrial Networks}}}, + author = {Lin, Zhi and Lin, Min and Zhu, Wei-Ping and Wang, Jun-Bo and Cheng, Julian}, + year = 2021, + month = jun, + journal = {IEEE Trans. on Cogn. Commun. Netw.}, + volume = {7}, + number = {2}, + pages = {567--580}, + issn = {2332-7731}, + doi = {10.1109/TCCN.2020.3016096}, + urldate = {2024-06-11}, + langid = {american} +} + +@article{linSatelliteTerrestrialCoordinatedMultiSatellite2024, + title = {Satellite-{{Terrestrial Coordinated Multi-Satellite Beam Hopping Scheduling Based}} on {{Multi-Agent Deep Reinforcement Learning}}}, + author = {Lin, Zhiyuan and Ni, Zuyao and Kuang, Linling and Jiang, Chunxiao and Huang, Zhen}, + year = 2024, + month = aug, + journal = {IEEE Trans. Wireless Commun.}, + volume = {23}, + number = {8}, + pages = {10091--10103}, + issn = {1558-2248}, + doi = {10.1109/TWC.2024.3368689}, + urldate = {2025-03-31} +} + +@article{linSecrecyEnergyEfficientHybrid2021, + title = {Secrecy-{{Energy Efficient Hybrid Beamforming}} for {{Satellite-Terrestrial Integrated Networks}}}, + author = {Lin, Zhi and Lin, Min and Champagne, Benoit and Zhu, Wei-Ping and {Al-Dhahir}, Naofal}, + year = 2021, + month = sep, + journal = {IEEE Trans. Commun.}, + volume = {69}, + number = {9}, + pages = {6345--6360}, + issn = {1558-0857}, + doi = {10.1109/TCOMM.2021.3088898}, + urldate = {2024-08-04}, + langid = {american} +} + +@article{linSecureBeamformingAntiJamming2025, + title = {Secure {{Beamforming}} and {{Anti-Jamming Coalition Formation}} for {{Air-Terrestrial Integrated Ad-Hoc Networks}}}, + author = {Lin, Xin and Liu, Aijun and Han, Chen and Sun, Yifu and Lin, Zhi and An, Kang and Gao, Xiqi and Wang, Jiangzhou}, + year = 2025, + month = aug, + journal = {IEEE Transactions on Wireless Communications}, + volume = {24}, + number = {8}, + pages = {6722--6736}, + issn = {1558-2248}, + doi = {10.1109/TWC.2025.3555600}, + urldate = {2025-09-02} +} + +@article{linSecureBeamformingCognitive2021, + title = {Secure {{Beamforming}} for {{Cognitive Satellite Terrestrial Networks With Unknown Eavesdroppers}}}, + author = {Lin, Zhi and Lin, Min and Champagne, Benoit and Zhu, Wei-Ping and {Al-Dhahir}, Naofal}, + year = 2021, + month = jun, + journal = {IEEE Syst. J.}, + volume = {15}, + number = {2}, + pages = {2186--2189}, + issn = {1937-9234}, + doi = {10.1109/JSYST.2020.2983309}, + urldate = {2024-08-01}, + langid = {american} +} + +@article{linSLNRBasedSecureEnergy2023, + title = {{{SLNR-Based Secure Energy Efficient Beamforming}} in {{Multibeam Satellite Systems}}}, + author = {Lin, Zhi and An, Kang and Niu, Hehao and Hu, Yihua and Chatzinotas, Symeon and Zheng, Gan and Wang, Jiangzhou}, + year = 2023, + month = apr, + journal = {IEEE Trans. Aerosp. Electron. 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Lett.}, + volume = {9}, + number = {10}, + pages = {1730--1733}, + issn = {2162-2345}, + doi = {10.1109/LWC.2020.3002846}, + urldate = {2024-08-04}, + langid = {american} +} + +@article{liuFirstOrderPerturbationAnalysis2008, + title = {First-{{Order Perturbation Analysis}} of {{Singular Vectors}} in {{Singular Value Decomposition}}}, + author = {Liu, Jun and Liu, Xiangqian and Ma, Xiaoli}, + year = 2008, + month = jul, + journal = {IEEE Trans. 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Technol.}, + volume = {73}, + number = {4}, + pages = {5162--5178}, + issn = {1939-9359}, + doi = {10.1109/TVT.2023.3329168}, + urldate = {2024-12-19} +} + +@inproceedings{liuLargeScaleSmallSatellite2020, + title = {Large-{{Scale Small Satellite Network Simulator}}: {{Design}} and {{Evaluation}}}, + shorttitle = {Large-{{Scale Small Satellite Network Simulator}}}, + booktitle = {2020 3rd {{Int}}. {{Conf}}. {{Hot Inf}}. - {{Centric Netw}}.}, + author = {Liu, Mengjie and Gui, Yongqiang and Li, Jian and Lu, Hancheng}, + year = 2020, + month = dec, + pages = {194--199}, + doi = {10.1109/HotICN50779.2020.9350838}, + urldate = {2025-02-13} +} + +@article{liuMachineLearningTimeofArrival2023, + title = {Machine {{Learning}} for {{Time-of-Arrival Estimation With 5G Signals}} in {{Indoor Positioning}}}, + author = {Liu, Zhaoliang and Chen, Liang and Zhou, Xin and Jiao, Zhenhang and Guo, Guangyi and Chen, Ruizhi}, + year = 2023, + month = jun, + journal = {IEEE Internet Things J.}, + volume = {10}, + number = {11}, + pages = {9782--9795}, + issn = {2327-4662}, + doi = {10.1109/JIOT.2023.3234123}, + urldate = {2024-07-11} +} + +@article{liuMultiAgentDeepReinforcement2024, + title = {A {{Multi-Agent Deep Reinforcement Learning-Based Handover Scheme}} for {{Mega-Constellation Under Dynamic Propagation Conditions}}}, + author = {Liu, Haotian and Wang, Yichen and Li, Peixuan and Cheng, Julian}, + year = 2024, + month = oct, + journal = {IEEE Transactions on Wireless Communications}, + volume = {23}, + number = {10}, + pages = {13579--13596}, + issn = {1558-2248}, + doi = {10.1109/TWC.2024.3407358}, + urldate = {2025-08-28} +} + +@article{liuOptimalBeamformingHybrid2020, + title = {Optimal {{Beamforming}} for {{Hybrid Satellite Terrestrial Networks With Nonlinear PA}} and {{Imperfect CSIT}}}, + author = {Liu, Chengxiao and Feng, Wei and Chen, Yunfei and Wang, Cheng-Xiang and Ge, Ning}, + year = 2020, + month = mar, + journal = {IEEE Wireless Commun. Lett.}, + volume = {9}, + number = {3}, + pages = {276--280}, + issn = {2162-2345}, + doi = {10.1109/LWC.2019.2952124}, + urldate = {2024-08-04}, + langid = {american} +} + +@article{liuPerformanceAnalysisNGSO2024, + title = {Performance {{Analysis}} of {{NGSO Satellite Communication Systems With Flexible Beams}}}, + author = {Liu, Bingkun and Kuang, Linling and Lu, Jianhua}, + year = 2024, + month = jul, + journal = {IEEE Internet of Things Journal}, + volume = {11}, + number = {14}, + pages = {24726--24738}, + issn = {2327-4662}, + doi = {10.1109/JIOT.2024.3379586}, + urldate = {2025-09-02} +} + +@article{liuReconfigurableIntelligentSurfaces2021, + title = {Reconfigurable {{Intelligent Surfaces}}: {{Principles}} and {{Opportunities}}}, + shorttitle = {Reconfigurable {{Intelligent Surfaces}}}, + author = {Liu, Yuanwei and Liu, Xiao and Mu, Xidong and Hou, Tianwei and Xu, Jiaqi and Di Renzo, Marco and {Al-Dhahir}, Naofal}, + year = 2021, + journal = {IEEE Commun. Surveys Tuts.}, + volume = {23}, + number = {3}, + pages = {1546--1577}, + issn = {1553-877X}, + doi = {10.1109/COMST.2021.3077737}, + urldate = {2023-12-29}, + langid = {english} +} + +@article{liuRemarksSpatialSmoothing2015, + title = {Remarks on the {{Spatial Smoothing Step}} in {{Coarray MUSIC}}}, + author = {Liu, Chun-Lin and Vaidyanathan, P. P.}, + year = 2015, + month = sep, + journal = {IEEE Signal Process. Lett.}, + volume = {22}, + number = {9}, + pages = {1438--1442}, + issn = {1558-2361}, + doi = {10.1109/LSP.2015.2409153}, + urldate = {2023-12-09}, + langid = {english} +} + +@inproceedings{liuResearchHandoverStrategy2021, + title = {Research on {{Handover Strategy}} of {{LEO Satellite Network}}}, + booktitle = {2021 2nd {{Int}}. {{Conf}}. {{Big Data Informatization Educ}}.}, + author = {Liu, Zhiguo and Zha, Xueli and Ren, Xiaobo and Yao, Qiaoyu}, + year = 2021, + month = apr, + pages = {188--194}, + doi = {10.1109/ICBDIE52740.2021.00050}, + urldate = {2025-01-13} +} + +@article{liuRISEmpoweredSatelliteAerialTerrestrialNetworks2024, + title = {{{RIS-Empowered Satellite-Aerial-Terrestrial Networks With PD-NOMA}}}, + author = {Liu, Rui and Guo, Kefeng and Li, Xingwang and Dev, Kapal and Ali Khowaja, Sunder and Tsiftsis, Theodoros A. and Song, Houbing}, + year = 2024, + journal = {IEEE Communications Surveys \& Tutorials}, + volume = {26}, + number = {4}, + pages = {2258--2289}, + issn = {1553-877X}, + doi = {10.1109/COMST.2024.3393612}, + urldate = {2025-09-03} +} + +@article{liuRobustAdaptiveWideband2019, + title = {Robust {{Adaptive Wideband Beamforming Based}} on {{Time Frequency Distribution}}}, + author = {Liu, Yaqi and Liu, Chengcheng and Hu, Dexiu and Zhao, Yongjun}, + year = 2019, + month = aug, + journal = {IEEE Trans. Signal Process.}, + volume = {67}, + number = {16}, + pages = {4370--4382}, + issn = {1941-0476}, + doi = {10.1109/TSP.2019.2929924}, + urldate = {2025-03-03} +} + +@inproceedings{liuRobustBeamformingDownlink2018, + title = {Robust {{Beamforming}} for {{Downlink 3D-MIMO Systems}} with L11-Norm {{Bounded CSI Uncertainty}}}, + booktitle = {2018 10th {{International Conference}} on {{Wireless Communications}} and {{Signal Processing}} ({{WCSP}})}, + author = {Liu, Kai and Feng, Hui and Yang, Tao and Hu, Bo}, + year = 2018, + month = oct, + pages = {1--5}, + issn = {2472-7628}, + doi = {10.1109/WCSP.2018.8555867}, + urldate = {2024-07-09} +} + +@article{liuRobustEnergyEfficientHybrid2022, + title = {Robust {{Energy-Efficient Hybrid Beamforming Design}} for {{Massive MIMO LEO Satellite Communication Systems}}}, + author = {Liu, Yang and Li, Changqing and Li, Jiong and Feng, Lu}, + year = 2022, + journal = {IEEE Access}, + volume = {10}, + pages = {63085--63099}, + issn = {2169-3536}, + doi = {10.1109/ACCESS.2022.3181178}, + urldate = {2025-01-16}, + langid = {american} +} + +@article{liuRobustMIMOBeamforming2017, + title = {Robust {{MIMO Beamforming}} for {{Cellular}} and {{Radar Coexistence}}}, + author = {Liu, Fan and Masouros, Christos and Li, Ang and Ratnarajah, Tharmalingam}, + year = 2017, + month = jun, + journal = {IEEE Wireless Commun. 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Mag.}, + volume = {40}, + number = {5}, + pages = {106--121}, + issn = {1558-0792}, + doi = {10.1109/MSP.2023.3272881}, + urldate = {2023-10-17}, + lccn = {1} +} + +@article{liuSpaceAirGroundIntegratedNetwork2018, + title = {Space-{{Air-Ground Integrated Network}}: {{A Survey}}}, + shorttitle = {Space-{{Air-Ground Integrated Network}}}, + author = {Liu, Jiajia and Shi, Yongpeng and Fadlullah, Zubair Md. and Kato, Nei}, + year = 2018, + journal = {IEEE Commun. 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Signal Process.}, + volume = {64}, + number = {16}, + pages = {4203--4217}, + issn = {1941-0476}, + doi = {10.1109/TSP.2016.2558167}, + urldate = {2023-10-21}, + langid = {american}, + lccn = {1} +} + +@article{liuSurveyFundamentalLimits2022, + title = {A {{Survey}} on {{Fundamental Limits}} of {{Integrated Sensing}} and {{Communication}}}, + author = {Liu, An and Huang, Zhe and Li, Min and Wan, Yubo and Li, Wenrui and Han, Tony Xiao and Liu, Chenchen and Du, Rui and Tan, Danny Kai Pin and Lu, Jianmin and Shen, Yuan and Colone, Fabiola and Chetty, Kevin}, + year = 2022, + journal = {IEEE Commun. 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Mag.}, + volume = {40}, + number = {4}, + pages = {80--91}, + issn = {1558-0792}, + doi = {10.1109/MSP.2023.3258060}, + urldate = {2023-10-17}, + langid = {english}, + lccn = {1} +} + +@article{liuUltraStarLightweightSimulator2023, + title = {{{UltraStar}}: {{A Lightweight Simulator}} of {{Ultra-Dense LEO Satellite Constellation Networking}} for {{6G}}}, + shorttitle = {{{UltraStar}}}, + author = {Liu, Xiaoyu and Ma, Ting and Tang, Zhixuan and Qin, Xiaohan and Zhou, Haibo and Shen, Xuemin Sherman}, + year = 2023, + month = mar, + journal = {IEEE/CAA J. 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Technol.}, + volume = {70}, + number = {9}, + pages = {9103--9117}, + issn = {1939-9359}, + doi = {10.1109/TVT.2021.3100562}, + urldate = {2024-07-24}, + langid = {american} +} + +@article{wangJointOptimizationBeamHopping2022, + title = {Joint {{Optimization}} of {{Beam-Hopping Design}} and {{NOMA-Assisted Transmission}} for {{Flexible Satellite Systems}}}, + author = {Wang, Anyue and Lei, Lei and Lagunas, Eva and {P{\'e}rez-Neira}, Ana I. and Chatzinotas, Symeon and Ottersten, Bj{\"o}rn}, + year = 2022, + month = oct, + journal = {IEEE Trans. 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Wireless Commun.}, + pages = {1--1}, + issn = {1558-2248}, + doi = {10.1109/TWC.2024.3508741}, + urldate = {2024-12-23} +} + +@article{wangResourceEfficiencyOptimization2021, + title = {Resource {{Efficiency Optimization}} for {{Robust Beamforming}} in {{Multi-Beam Satellite Communications}}}, + author = {Wang, Wenjin and Gao, Linna and Ding, Rui and Lei, Jizhao and You, Li and Chan, Chien Aun and Gao, Xiqi}, + year = 2021, + month = jul, + journal = {IEEE Trans. Veh. Technol.}, + volume = {70}, + number = {7}, + pages = {6958--6968}, + issn = {1939-9359}, + doi = {10.1109/TVT.2021.3087744}, + urldate = {2024-08-04}, + langid = {american} +} + +@article{wangRetrodirectiveFrequencyDiverse2019, + title = {Retrodirective {{Frequency Diverse Array Focusing}} for {{Wireless Information}} and {{Power Transfer}}}, + author = {Wang, Wen-Qin}, + year = 2019, + month = jan, + journal = {IEEE J. Sel. 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Technol.}, + volume = {71}, + number = {1}, + pages = {1010--1014}, + issn = {1939-9359}, + doi = {10.1109/TVT.2021.3124928}, + urldate = {2025-01-16}, + langid = {american} +} + +@article{wangSeamlessHandoverLEO2023, + title = {Seamless {{Handover}} in {{LEO Based Non-Terrestrial Networks}}: {{Service Continuity}} and {{Optimization}}}, + shorttitle = {Seamless {{Handover}} in {{LEO Based Non-Terrestrial Networks}}}, + author = {Wang, Feng and Jiang, Dingde and Wang, Zhihao and Chen, Jianguang and Quek, Tony Q. 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S. and Ding, Yanwu and Shen, Dan and Pham, Khanh and Blasch, Erik and Chen, Genshe}, + year = 2024, + month = may, + pages = {1--5}, + issn = {2375-5318}, + doi = {10.1109/RadarConf2458775.2024.10548034}, + urldate = {2025-12-04} +} + +@inproceedings{zhangDoaEstimationSensor1993, + title = {Doa {{Estimation}} with {{Sensor Gain}}, {{Phase}} and {{Position Perturbations}}}, + booktitle = {Proc. {{IEEE}} 1993 {{Nat}}. {{Aerosp}}. {{Electron}}. {{Conf}}.}, + author = {Zhang, Ming and Zhu, Zhao-Da}, + year = 1993, + month = may, + pages = {67-69 vol.1}, + doi = {10.1109/NAECON.1993.290894}, + urldate = {2024-08-02}, + langid = {american} +} + +@article{zhangDualBandDualPolarizationHorn2022, + title = {Dual-{{Band Dual-Polarization Horn Antenna Array Based}} on {{Orthomode Transducers With High Isolation}} for {{Satellite Communication}}}, + author = {Zhang, Jie and Li, Xiuping and Qi, Zihang and Huang, Yuhan and Zhu, Hua}, + year = 2022, + month = oct, + journal = {IEEE Trans. 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Areas Commun.}, + volume = {43}, + number = {1}, + pages = {107--121}, + issn = {1558-0008}, + doi = {10.1109/JSAC.2024.3460059}, + urldate = {2024-12-20} +} + +@book{ZhangJuZhenFenXiYuYingYong2013, + title = {{矩阵分析与应用}}, + author = {张, 贤达}, + year = 2013, + edition = {第2版}, + publisher = {清华大学出版社}, + address = {北京}, + isbn = {978-7-302-33859-8}, + langid = {chinese} +} + +@article{zhangKaBandPlanarSharedAperture2023, + title = {K-/{{Ka-Band Planar Shared-Aperture Beam-Scanning Array Antenna}} for {{Simultaneous Transmitting}} and {{Receiving Low Earth Orbit Satellite Communication Terminal}}}, + author = {Zhang, Jin Fan and Cheng, Yu Jian}, + year = 2023, + month = aug, + journal = {IEEE Trans. Antennas Propag.}, + volume = {71}, + number = {8}, + pages = {6617--6627}, + issn = {1558-2221}, + doi = {10.1109/TAP.2023.3284163}, + urldate = {2024-08-04} +} + +@phdthesis{ZhangKaiShuZhiNengTianXianKuaiSuWenJianFangXiangTuZongHeYuKangGanRaoJiShuYanJiu2023, + type = {{硕士}}, + title = {{智能天线快速稳健方向图综合与抗干扰技术研究}}, + author = {{张开墅}}, + year = 2023, + doi = {10.27241/d.cnki.gnjgu.2021.000994}, + urldate = {2023-12-10}, + langid = {chinese}, + school = {南京理工大学} +} + +@article{zhangLEOMegaConstellationTerrestrialCommunications2025, + title = {{{LEO Mega-Constellation-Terrestrial Communications Suffering Poisson Arc Hardcore Distributed Space Interference}}}, + author = {Zhang, Haoxing and Miao, Xiaqing and Ni, Zihan and Wang, Shuai and Pan, Gaofeng and Cavdar, Cicek and An, Jianping}, + year = 2025, + journal = {IEEE Transactions on Wireless Communications}, + pages = {1--1}, + issn = {1558-2248}, + doi = {10.1109/TWC.2024.3523950}, + urldate = {2025-04-09} +} + +@article{zhangMassiveHybridAntenna2015, + title = {Massive Hybrid Antenna Array for Millimeter-Wave Cellular Communications}, + author = {Zhang, Jian A. and Huang, Xiaojing and Dyadyuk, Val and Guo, Y. Jay}, + year = 2015, + month = feb, + journal = {IEEE Wireless Commun.}, + volume = {22}, + number = {1}, + pages = {79--87}, + issn = {1558-0687}, + doi = {10.1109/MWC.2015.7054722}, + urldate = {2024-12-17} +} + +@article{zhangMultiSatelliteCooperativeNetworks2024, + title = {Multi-{{Satellite Cooperative Networks}}: {{Joint Hybrid Beamforming}} and {{User Scheduling Design}}}, + shorttitle = {Multi-{{Satellite Cooperative Networks}}}, + author = {Zhang, Xuan and Sun, Shu and Tao, Meixia and Huang, Qin and Tang, Xiaohu}, + year = 2024, + month = jul, + journal = {IEEE Trans. Wireless Commun.}, + volume = {23}, + number = {7}, + pages = {7938--7952}, + issn = {1558-2248}, + doi = {10.1109/TWC.2023.3346463}, + urldate = {2024-07-27}, + langid = {american} +} + +@article{zhangOptimalHybridBeamforming2019, + title = {Optimal {{Hybrid Beamforming Design}} for {{Millimeter-Wave Massive Multi-User MIMO Relay Systems}}}, + author = {Zhang, Yang and Du, Jianhe and Chen, Yuanzhi and Han, Meng and Li, Xingwang}, + year = 2019, + journal = {IEEE Access}, + volume = {7}, + pages = {157212--157225}, + issn = {2169-3536}, + doi = {10.1109/ACCESS.2019.2949786}, + urldate = {2024-07-09} +} + +@article{zhangOutageAnalysisCooperative2022, + title = {Outage {{Analysis}} of {{Cooperative Satellite-Aerial-Terrestrial Networks With Spatially Random Terminals}}}, + author = {Zhang, Haoxing and Pan, Gaofeng and Ke, Sheng and Wang, Shuai and An, Jianping}, + year = 2022, + month = jul, + journal = {IEEE Trans. Commun.}, + volume = {70}, + number = {7}, + pages = {4972--4987}, + issn = {1558-0857}, + doi = {10.1109/TCOMM.2022.3179768}, + urldate = {2025-04-09} +} + +@article{zhangPassiveMaritimeSurveillance2020, + title = {Passive {{Maritime Surveillance Based}} on {{Low Earth Orbit Satellite Constellations}}}, + author = {Zhang, Qianyun and Li, Shijie and Wu, Biyi and Wang, Jingchao}, + year = 2020, + month = dec, + journal = {IEEE Wireless Communications}, + volume = {27}, + number = {6}, + pages = {61--67}, + issn = {1558-0687}, + doi = {10.1109/MWC.001.2000135}, + urldate = {2025-03-12}, + langid = {american} +} + +@article{zhangPotentialGameBasedRadio2021, + title = {Potential {{Game-Based Radio Resource Allocation}} in {{Uplink Multibeam Satellite IoT Networks}}}, + author = {Zhang, Xiaokai and Zhang, Bangning and Guo, Daoxing and An, Kang and Qi, Shuai and Wu, Gang}, + year = 2021, + month = dec, + journal = {IEEE Trans. Aerosp. Electron. Syst.}, + volume = {57}, + number = {6}, + pages = {4269--4279}, + issn = {1557-9603}, + doi = {10.1109/TAES.2021.3100335}, + urldate = {2024-12-23} +} + +@article{zhangProspectiveMultipleAntenna2020, + title = {Prospective {{Multiple Antenna Technologies}} for {{Beyond 5G}}}, + author = {Zhang, Jiayi and Bj{\"o}rnson, Emil and Matthaiou, Michail and Ng, Derrick Wing Kwan and Yang, Hong and Love, David J.}, + year = 2020, + month = aug, + journal = {IEEE J. Sel. Areas Commun.}, + volume = {38}, + number = {8}, + pages = {1637--1660}, + issn = {1558-0008}, + doi = {10.1109/JSAC.2020.3000826}, + urldate = {2024-07-09}, + langid = {american} +} + +@article{zhangResourceAllocationTerrestrialSatelliteBased2022, + title = {Resource {{Allocation}} in {{Terrestrial-Satellite-Based Next Generation Multiple Access Networks With Interference Cooperation}}}, + author = {Zhang, Yaomin and Zhang, Haijun and Zhou, Huan and Long, Keping and Karagiannidis, George K.}, + year = 2022, + month = apr, + journal = {IEEE J. Sel. Areas Commun.}, + volume = {40}, + number = {4}, + pages = {1210--1221}, + issn = {1558-0008}, + doi = {10.1109/JSAC.2022.3145810}, + urldate = {2024-12-23} +} + +@article{zhangRobustBeamformingEnhancing2021, + title = {Robust {{Beamforming}} for {{Enhancing Security}} in {{Multibeam Satellite Systems}}}, + author = {Zhang, Jian and Lin, Min and Ouyang, Jian and Zhu, Wei-Ping and De Cola, Tomaso}, + year = 2021, + month = jul, + journal = {IEEE Commun. Lett.}, + volume = {25}, + number = {7}, + pages = {2161--2165}, + issn = {1558-2558}, + doi = {10.1109/LCOMM.2021.3073026}, + urldate = {2024-06-11}, + langid = {american} +} + +@article{zhangRobustBeamformingMultibeam2019, + title = {Robust {{Beamforming}} for {{Multibeam Satellite Communication}} in the {{Face}} of {{Phase Perturbations}}}, + author = {Zhang, Xin and Wang, Jingjing and Jiang, Chunxiao and Yan, Chaoxing and Ren, Yong and Hanzo, Lajos}, + year = 2019, + month = mar, + journal = {IEEE Trans. Veh. Technol.}, + volume = {68}, + number = {3}, + pages = {3043--3047}, + issn = {1939-9359}, + doi = {10.1109/TVT.2019.2896245}, + urldate = {2024-08-04}, + langid = {american} +} + +@article{ZhangRuiWeiXingDaoHangGaoJingDuKangGanRaoSuanFaYanJiu2023, + title = {{卫星导航高精度抗干扰算法研究}}, + author = {{张锐} and {徐兵} and {谢斌斌}}, + year = 2023, + journal = {现代导航}, + volume = {14}, + number = {5}, + pages = {339--344}, + issn = {1674-7976}, + urldate = {2023-12-10}, + langid = {chinese} +} + +@article{zhangSecureUplinkTransmissions2023, + title = {On {{Secure Uplink Transmissions}} in {{Satellite-Aerial Systems}}}, + author = {Zhang, Haoxing and Yue, Pingyue and Wang, Shuai and Pan, Gaofeng and An, Jianping}, + year = 2023, + month = aug, + journal = {IEEE Transactions on Aerospace and Electronic Systems}, + volume = {59}, + number = {4}, + pages = {4666--4673}, + issn = {1557-9603}, + doi = {10.1109/TAES.2022.3230380}, + urldate = {2025-04-09}, + langid = {american} +} + +@article{zhangSecureWirelessCommunications2019, + title = {Secure {{Wireless Communications Based}} on {{Compressive Sensing}}: {{A Survey}}}, + shorttitle = {Secure {{Wireless Communications Based}} on {{Compressive Sensing}}}, + author = {Zhang, Yushu and Xiang, Yong and Zhang, Leo Yu and Rong, Yue and Guo, Song}, + year = 2019, + journal = {IEEE Commun. Surveys Tuts.}, + volume = {21}, + number = {2}, + pages = {1093--1111}, + issn = {1553-877X}, + doi = {10.1109/COMST.2018.2878943}, + urldate = {2023-10-21}, + langid = {english} +} + +@inproceedings{zhangSparsityBasedDOAEstimation2013, + title = {Sparsity-{{Based DOA Estimation Using Co-Prime Arrays}}}, + booktitle = {2013 {{IEEE International Conference}} on {{Acoustics}}, {{Speech}} and {{Signal Processing}}}, + author = {Zhang, Yimin D. and Amin, Moeness G. and Himed, Braham}, + year = 2013, + month = may, + pages = {3967--3971}, + issn = {2379-190X}, + doi = {10.1109/ICASSP.2013.6638403}, + urldate = {2023-12-09}, + langid = {english} +} + +@inproceedings{zhangTDOAAssistedDirectPosition2021, + title = {A {{TDOA-Assisted Direct Position Determination}} for {{Efficient Geolocalization Using LEO Satellites}}}, + booktitle = {2021 {{IEEE}} 20th {{International Conference}} on {{Trust}}, {{Security}} and {{Privacy}} in {{Computing}} and {{Communications}} ({{TrustCom}})}, + author = {Zhang, Qianyun and Li, Shijie and Shi, Jiting and Guan, Zhenyu}, + year = 2021, + month = oct, + pages = {1323--1327}, + issn = {2324-9013}, + doi = {10.1109/TrustCom53373.2021.00183}, + urldate = {2025-03-12}, + langid = {american} +} + +@article{zhangTensorBasedChannelEstimation2025, + title = {Tensor-{{Based Channel Estimation}} for {{Extremely Large-Scale MIMO-OFDM With Dynamic Metasurface Antennas}}}, + author = {Zhang, Ruoyu and Chen, Guangyi and Cheng, Lei and Guan, Xinrong and Wu, Qingqing and Wu, Wen and Zhang, Rui}, + year = 2025, + month = jul, + journal = {IEEE Transactions on Wireless Communications}, + volume = {24}, + number = {7}, + pages = {6052--6068}, + issn = {1558-2248}, + doi = {10.1109/TWC.2025.3551144}, + urldate = {2025-09-08} +} + +@article{zhangUnifiedNOMAFramework2023, + title = {A {{Unified NOMA Framework}} in {{Beam-Hopping Satellite Communication Systems}}}, + author = {Zhang, Xuyang and Yue, Xinwei and Li, Tian and Han, Zhihao and Wang, Yafei and Ding, Yong and Liu, Rongke}, + year = 2023, + month = oct, + journal = {IEEE Trans. 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Syst.}, + volume = {59}, + number = {5}, + pages = {5390--5404}, + issn = {1557-9603}, + doi = {10.1109/TAES.2023.3260059}, + urldate = {2024-12-23} +} + +@article{zhangUserActivityDetection2020, + title = {User {{Activity Detection}} and {{Channel Estimation}} for {{Grant-Free Random Access}} in {{LEO Satellite-Enabled Internet}} of {{Things}}}, + author = {Zhang, Zhaoji and Li, Ying and Huang, Chongwen and Guo, Qinghua and Liu, Lei and Yuen, Chau and Guan, Yong Liang}, + year = 2020, + month = sep, + journal = {IEEE Internet Things J.}, + volume = {7}, + number = {9}, + pages = {8811--8825}, + issn = {2327-4662}, + doi = {10.1109/JIOT.2020.2997336}, + urldate = {2024-12-23}, + langid = {american} +} + +@article{zhangVariablePhaseShiftBasedRFBasebandCodesign2005, + title = {Variable-{{Phase-Shift-Based RF-Baseband Codesign}} for {{MIMO Antenna Selection}}}, + author = {Zhang, Xinying and Molisch, A.F. and Kung, Sun-Yuan}, + year = 2005, + month = nov, + journal = {IEEE Trans. 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Veh. Technol.}, + volume = {70}, + number = {3}, + pages = {2322--2336}, + issn = {1939-9359}, + doi = {10.1109/TVT.2021.3057547}, + urldate = {2024-07-17}, + langid = {american} +} + +@phdthesis{ZhaoAnQiJiYuJiQiXueXiHeSuiJiJuZhenLiLunDeDaZhenLieDOAGuJiFangFaYanJiu2020, + type = {{硕士学位论文}}, + title = {{基于机器学习和随机矩阵理论的大阵列DOA估计方法研究}}, + author = {{赵安琪}}, + year = 2020, + doi = {10.27162/d.cnki.gjlin.2020.005804}, + urldate = {2024-03-11}, + collaborator = {{姜宏}}, + langid = {chinese}, + school = {吉林大学} +} + +@inproceedings{zhaoBeamHoppingLEO2022, + title = {Beam {{Hopping}} for {{LEO Satellite}}:{{Challenges}} and {{Opportunities}}}, + shorttitle = {Beam {{Hopping}} for {{LEO Satellite}}}, + booktitle = {2022 {{Int}}. {{Conf}}. {{Culture}} - {{Oriented Sci}}. {{Technol}}.}, + author = {Zhao, Qi and Hu, Yiqing and Pang, Zhenyu and Ren, Derong}, + year = 2022, + month = aug, + pages = {319--324}, + doi = {10.1109/CoST57098.2022.00072}, + urldate = {2024-12-23}, + langid = {american} +} + +@article{zhaoBeamTrackingUAV2018, + title = {Beam {{Tracking}} for {{UAV Mounted SatCom}} On-the-{{Move With Massive Antenna Array}}}, + author = {Zhao, Jianwei and Gao, Feifei and Wu, Qihui and Jin, Shi and Wu, Yi and Jia, Weimin}, + year = 2018, + month = feb, + journal = {IEEE J. Sel. Areas Commun.}, + volume = {36}, + number = {2}, + pages = {363--375}, + issn = {1558-0008}, + doi = {10.1109/JSAC.2018.2804239}, + urldate = {2024-07-08}, + langid = {american} +} + +@article{zhaoDemandAwareBeamHopping2025, + title = {Demand-{{Aware Beam Hopping}} and {{Power Allocation}} for {{Load Balancing}} in {{Digital Twin Empowered LEO Satellite Networks}}}, + author = {Zhao, Ruili and Cai, Jun and Luo, Jiangtao and Gao, Junpeng and Ran, Yongyi}, + year = 2025, + month = jun, + journal = {IEEE Transactions on Wireless Communications}, + volume = {24}, + number = {6}, + pages = {5084--5098}, + issn = {1558-2248}, + doi = {10.1109/TWC.2025.3545745}, + urldate = {2025-09-01} +} + +@article{zhaoDynamicResourceAllocation2024, + title = {Dynamic {{Resource Allocation}} for {{Multi-Satellite Cooperation Networks}}: {{A Decentralized Scheme Under Statistical CSI}}}, + shorttitle = {Dynamic {{Resource Allocation}} for {{Multi-Satellite Cooperation Networks}}}, + author = {Zhao, Meihui and Yu, Hanxiao and Pan, Jianxiong and Jin, Yifeng and Lv, Guocheng and Jin, Ye}, + year = 2024, + journal = {IEEE Access}, + volume = {12}, + pages = {15419--15437}, + issn = {2169-3536}, + doi = {10.1109/ACCESS.2024.3356058}, + urldate = {2025-03-03} +} + +@article{zhaoGenerativeAISecure2025, + title = {Generative {{AI}} for {{Secure Physical Layer Communications}}: {{A Survey}}}, + shorttitle = {Generative {{AI}} for {{Secure Physical Layer Communications}}}, + author = {Zhao, Changyuan and Du, Hongyang and Niyato, Dusit and Kang, Jiawen and Xiong, Zehui and Kim, Dong In and Shen, Xuemin and Letaief, Khaled B.}, + year = 2025, + month = feb, + journal = {IEEE Trans. Cogn. Commun. Netw.}, + volume = {11}, + number = {1}, + pages = {3--26}, + issn = {2332-7731}, + doi = {10.1109/TCCN.2024.3438379}, + urldate = {2025-03-08}, + langid = {american} +} + +@article{zhaoIntegratedOTFSNOMAFramework2025, + title = {An {{Integrated OTFS-NOMA Framework}} for {{Multi-Beam LEO Systems}}: {{Reliability}} and {{Capacity Analysis}}}, + shorttitle = {An {{Integrated OTFS-NOMA Framework}} for {{Multi-Beam LEO Systems}}}, + author = {Zhao, Xiaohui and Lei, Lei and Wei, Zhiqiang and Fang, Hai and Wang, Wenjie and Chatzinotas, Symeon}, + year = 2025, + journal = {IEEE Transactions on Wireless Communications}, + pages = {1--1}, + issn = {1558-2248}, + doi = {10.1109/TWC.2025.3594792}, + urldate = {2025-08-25} +} + +@article{zhaoMultiSatelliteCooperativeLoadBalancing2024, + title = {Multi-{{Satellite Cooperative Load-Balancing Scheme Based}} on {{Dynamic Beam Coverage}} for {{LEO Beam Hopping Systems}}}, + author = {Zhao, Xiaoyan and Wang, Cheng and Cai, Songsong and Chen, Ruoqi and Wen, Jiarui and Xu, Lexi and Wang, Weidong}, + year = 2024, + month = oct, + journal = {IEEE Wireless Communications Letters}, + volume = {13}, + number = {10}, + pages = {2892--2896}, + issn = {2162-2345}, + doi = {10.1109/LWC.2024.3452115}, + urldate = {2025-09-01} +} + +@article{zhaoOutageConstrainedRobust2021, + title = {Outage {{Constrained Robust Secure Beamforming}} in {{Cognitive Satellite-Aerial Networks}}}, + author = {Zhao, Bai and Lin, Min and Cheng, Ming and Zhu, Wei-Ping and {Al-Dhahir}, Naofal}, + year = 2021, + month = aug, + journal = {IEEE Commun. Lett.}, + volume = {25}, + number = {8}, + pages = {2708--2712}, + issn = {1558-2558}, + doi = {10.1109/LCOMM.2021.3077686}, + urldate = {2025-01-16}, + langid = {american} +} + +@article{zhaoRobustFrameBasedBeamforming2024, + title = {Robust {{Frame-Based Beamforming Scheme}} to {{Enhance Secure Multigroup Multicast Transmission}} for {{Multibeam Satellite Systems}}}, + author = {Zhao, Bai and Lin, Min and Li, Fei and Cheng, Ming and {Al-Dhahir}, Naofal}, + year = 2024, + month = mar, + journal = {IEEE Trans. Veh. Technol.}, + volume = {73}, + number = {3}, + pages = {4407--4411}, + issn = {1939-9359}, + doi = {10.1109/TVT.2023.3324437}, + urldate = {2024-12-19}, + langid = {american} +} + +@article{zhaoTimeVaryingChannel2018, + title = {Time {{Varying Channel Tracking With Spatial}} and {{Temporal BEM}} for {{Massive MIMO Systems}}}, + author = {Zhao, Jianwei and Xie, Hongxiang and Gao, Feifei and Jia, Weimin and Jin, Shi and Lin, Hai}, + year = 2018, + month = aug, + journal = {IEEE Trans. Wireless Commun.}, + volume = {17}, + number = {8}, + pages = {5653--5666}, + issn = {1558-2248}, + doi = {10.1109/TWC.2018.2848259}, + urldate = {2024-12-20} +} + +@phdthesis{ZhengBaiFuDuoGuanCeMoXingWeiBuYouHuaQianXiShuXinHaoChongGouFangFaYanJiu2023, + type = {{博士学位论文}}, + title = {{多观测模型尾部优化欠稀疏信号重构方法研究}}, + author = {{郑佰福}}, + year = 2023, + doi = {10.27389/d.cnki.gxadu.2022.000176}, + urldate = {2024-03-11}, + collaborator = {{李世东} and {廖桂生}}, + langid = {chinese}, + school = {西安电子科技大学} +} + +@article{zhengCooperativeMultiSatelliteMultiRIS2024, + title = {Cooperative {{Multi-Satellite}} and {{Multi-RIS Beamforming}}: {{Enhancing LEO SatCom}} and {{Mitigating LEO-GEO Intersystem Interference}}}, + shorttitle = {Cooperative {{Multi-Satellite}} and {{Multi-RIS Beamforming}}}, + author = {Zheng, Ziyuan and Jing, Wenpeng and Lu, Zhaoming and Wu, Qingqing and Zhang, Haijun and Gesbert, David}, + year = 2024, + journal = {IEEE J. Sel. Areas Commun.}, + pages = {1--1}, + issn = {1558-0008}, + doi = {10.1109/JSAC.2024.3460068}, + urldate = {2024-12-19}, + langid = {american} +} + +@article{zhengCovarianceMatrixReconstruction2018, + title = {Covariance {{Matrix Reconstruction}} with {{Interference Steering Vector}} and {{Power Estimation}} for {{Robust Adaptive Beamforming}}}, + author = {Zheng, Zhi and Zheng, Yan and Wang, Wen-Qin and Zhang, Hongbo}, + year = 2018, + month = sep, + journal = {IEEE Trans. Veh. Technol.}, + volume = {67}, + number = {9}, + pages = {8495--8503}, + issn = {1939-9359}, + doi = {10.1109/TVT.2018.2849646}, + urldate = {2024-01-03}, + langid = {english} +} + +@article{zhengDiversityMultiplexingFundamental2003, + title = {Diversity and {{Multiplexing}}: {{A Fundamental Tradeoff}} in {{Multiple-Antenna Channels}}}, + shorttitle = {Diversity and {{Multiplexing}}}, + author = {Zheng, Lizhong and Tse, D.N.C.}, + year = 2003, + month = may, + journal = {IEEE Trans. Inf. 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P.}, + year = 2021, + journal = {IEEE Transactions on Signal Processing}, + volume = {69}, + pages = {4607--4622}, + issn = {1941-0476}, + doi = {10.1109/TSP.2021.3101015}, + urldate = {2025-12-04}, + langid = {american} +} + +@inproceedings{chenEnergyEfficientJointHandover2024, + title = {Energy-{{Efficient Joint Handover}} and {{Beam Switching Scheme}} for {{Multi-LEO Networks}}}, + booktitle = {2024 {{IEEE}} 99th {{Vehicular Technology Conference}} ({{VTC2024-Spring}})}, + author = {Chen, Sz-Han and Shen, Li-Hsiang and Feng, Kai-Ten and Yang, Lie-Liang and Wu, Jen-Ming}, + year = 2024, + month = jun, + pages = {1--7}, + issn = {2577-2465}, + doi = {10.1109/VTC2024-Spring62846.2024.10683088}, + urldate = {2025-08-26} +} + +@article{chenFastEfficientPhaseOnly2024, + title = {Fast and {{Efficient Phase-Only Beam Nulling}} for {{NGSO Satellite Communication Systems}}}, + author = {Chen, Zhen and Kuang, Linling and Liu, Bingkun and Ni, Zuyao}, + year = 2024, + month = dec, + journal = {IEEE Wireless Communications Letters}, + volume = {13}, + number = {12}, + pages = {3261--3265}, + issn = {2162-2345}, + doi = {10.1109/LWC.2024.3386852}, + urldate = {2025-09-02} +} + +@article{chengSVDAidedMultiBeamDirectional2020, + title = {{{SVD-Aided Multi-Beam Directional Modulation Scheme Based}} on {{Frequency Diverse Array}}}, + author = {Cheng, Qian and Fusco, Vincent and Zhu, Jiang and Wang, Shilian and Gu, Chao}, + year = 2020, + month = mar, + journal = {IEEE Wireless Commun. 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IEEE}, + volume = {78}, + number = {8}, + pages = {1327--1343}, + issn = {1558-2256}, + doi = {10.1109/5.58320}, + urldate = {2024-01-06}, + langid = {english} +} + +@article{comptonPowerInversionAdaptiveArray1979, + title = {The {{Power-Inversion Adaptive Array}}: {{Concept}} and {{Performance}}}, + shorttitle = {The {{Power-Inversion Adaptive Array}}}, + author = {Compton, R.T.}, + year = 1979, + month = nov, + journal = {IEEE Trans. Aerosp. Electron. 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All rights reserved.}, + langid = {english} +} + +@article{liBroadbandCircularlyPolarized2022, + title = {Broadband {{Circularly Polarized Magnetoelectric Dipole Antenna}} and {{Array}} for {{K-Band}} and {{Ka-Band Satellite Communications}}}, + author = {Li, Jiawang and Hu, Yun and Xiang, Lei and Kong, Weiyang and Hong, Wei}, + year = 2022, + month = jul, + journal = {IEEE Trans. Antennas Propag.}, + volume = {70}, + number = {7}, + pages = {5907--5912}, + issn = {1558-2221}, + doi = {10.1109/TAP.2022.3140507}, + urldate = {2024-08-04} +} + +@article{liCapacityAnalysisTerrestrial2021, + title = {Capacity {{Analysis}} of {{Terrestrial Antenna Array}} in {{Distributed Satellite MIMO Communication System}}}, + author = {Li, Cong and Zhu, Hongpeng and Cai, Jun and Hu, Jing and Li, Guang and Li, Guangxia}, + year = 2021, + month = may, + journal = {IEEE Trans. Veh. Technol.}, + volume = {70}, + number = {5}, + pages = {4435--4450}, + issn = {1939-9359}, + doi = {10.1109/TVT.2021.3068389}, + urldate = {2024-07-08}, + langid = {american} +} + +@article{liCapacityComparisonBeam2025, + title = {Capacity {{Comparison}} of {{Beam Coverage Modes}} in {{ZF Satellite MIMO Downlink With LOS Channels}}}, + author = {Li, Cong and Zhang, Jian and Zhu, Hongpeng and Bian, Dongming and Hu, Jing and Cheng, Jian and Li, Guangxia}, + year = 2025, + month = jun, + journal = {IEEE Transactions on Aerospace and Electronic Systems}, + volume = {61}, + number = {3}, + pages = {7236--7250}, + issn = {1557-9603}, + doi = {10.1109/TAES.2025.3537068}, + urldate = {2025-08-24} +} + +@article{liChannelEstimationLEO2023, + title = {Channel {{Estimation}} for {{LEO Satellite Massive MIMO OFDM Communications}}}, + author = {Li, Ke-Xin and Gao, Xiqi and Xia, Xiang-Gen}, + year = 2023, + month = nov, + journal = {IEEE Trans. Wireless Commun.}, + volume = {22}, + number = {11}, + pages = {7537--7550}, + issn = {1558-2248}, + doi = {10.1109/TWC.2023.3252895}, + urldate = {2024-12-20} +} + +@article{liChannelEstimationPerformance2017, + title = {Channel {{Estimation}} and {{Performance Analysis}} of {{One-Bit Massive MIMO Systems}}}, + author = {Li, Yongzhi and Tao, Cheng and {Seco-Granados}, Gonzalo and Mezghani, Amine and Swindlehurst, A. Lee and Liu, Liu}, + year = 2017, + month = aug, + journal = {IEEE Trans. Signal Process.}, + volume = {65}, + number = {15}, + pages = {4075--4089}, + issn = {1941-0476}, + doi = {10.1109/TSP.2017.2706179}, + urldate = {2023-12-25}, + langid = {english} +} + +@article{liComparisonBeamformingTechniques2003, + title = {Comparison of {{Beamforming Techniques}} for {{W-CDMA Communication Systems}}}, + author = {Li, Hsueh-Jyh and Liu, Ta-Yung}, + year = 2003, + month = jul, + journal = {IEEE Trans. Veh. Technol.}, + volume = {52}, + number = {4}, + pages = {752--760}, + issn = {1939-9359}, + doi = {10.1109/TVT.2002.804864}, + urldate = {2024-05-08}, + langid = {american} +} + +@article{liDoublyConstrainedRobust2004, + title = {Doubly {{Constrained Robust Capon Beamformer}}}, + author = {Li, Jian and Stoica, P. and Wang, Zhisong}, + year = 2004, + month = sep, + journal = {IEEE Trans. Signal Process.}, + volume = {52}, + number = {9}, + pages = {2407--2423}, + issn = {1941-0476}, + doi = {10.1109/TSP.2004.831998}, + urldate = {2023-11-26} +} + +@article{liDownlinkTransmitDesign2022, + title = {Downlink {{Transmit Design}} for {{Massive MIMO LEO Satellite Communications}}}, + author = {Li, Ke-Xin and You, Li and Wang, Jiaheng and Gao, Xiqi and Tsinos, Christos G. and Chatzinotas, Symeon and Ottersten, Bj{\"o}rn}, + year = 2022, + month = feb, + journal = {IEEE Trans. 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Wireless Commun.}, + pages = {1--1}, + issn = {1558-2248}, + doi = {10.1109/TWC.2025.3527962}, + urldate = {2025-04-03} +} + +@article{liImprovedOnOrbitRelative2022, + title = {An {{Improved On-Orbit Relative Radiometric Calibration Method}} for {{Agile High-Resolution Optical Remote-Sensing Satellites With Sensor Geometric Distortion}}}, + author = {Li, Litao and Zhang, Guo and Jiang, Yonghua and Shen, Xin}, + year = 2022, + journal = {IEEE Trans. Geosci. Remote Sens.}, + volume = {60}, + pages = {1--15}, + issn = {1558-0644}, + doi = {10.1109/TGRS.2021.3078815}, + urldate = {2025-02-13} +} + +@misc{lillicrapContinuousControlDeep2019, + title = {Continuous Control with Deep Reinforcement Learning}, + author = {Lillicrap, Timothy P. and Hunt, Jonathan J. and Pritzel, Alexander and Heess, Nicolas and Erez, Tom and Tassa, Yuval and Silver, David and Wierstra, Daan}, + year = 2019, + month = jul, + number = {arXiv:1509.02971}, + eprint = {1509.02971}, + primaryclass = {cs}, + publisher = {arXiv}, + doi = {10.48550/arXiv.1509.02971}, + urldate = {2025-04-07}, + archiveprefix = {arXiv} +} + +@article{liMIMOTransmissionLEO2022, + title = {{{LoS MIMO Transmission}} for {{LEO Satellite Communication Systems}}}, + author = {Li, Lingxuan and Chen, Tingting and Wang, Wenjin and Song, Xiaohang and You, Li and Gao, Xiqi}, + year = 2022, + month = oct, + journal = {China Commun.}, + volume = {19}, + number = {10}, + pages = {180--193}, + issn = {1673-5447}, + doi = {10.23919/JCC.2022.00.026}, + urldate = {2024-07-14}, + langid = {american} +} + +@article{liMultipolarizedPlanarPhased2022, + title = {A {{Multipolarized Planar Phased Array}} for {{LEO SATCOM Applications}}}, + author = {Li, Ke-Xin and Guo, Zi-Jun and Hao, Zhang-Cheng}, + year = 2022, + month = nov, + journal = {IEEE Antennas Wireless Propag. Lett.}, + volume = {21}, + number = {11}, + pages = {2273--2277}, + issn = {1548-5757}, + doi = {10.1109/LAWP.2022.3185208}, + urldate = {2024-07-28}, + langid = {american} +} + +@article{linDynamicBeamPattern2022, + title = {Dynamic {{Beam Pattern}} and {{Bandwidth Allocation Based}} on {{Multi-Agent Deep Reinforcement Learning}} for {{Beam Hopping Satellite Systems}}}, + author = {Lin, Zhiyuan and Ni, Zuyao and Kuang, Linling and Jiang, Chunxiao and Huang, Zhen}, + year = 2022, + month = apr, + journal = {IEEE Trans. Veh. Technol.}, + volume = {71}, + number = {4}, + pages = {3917--3930}, + issn = {1939-9359}, + doi = {10.1109/TVT.2022.3145848}, + urldate = {2024-08-04}, + langid = {american} +} + +@article{linExploitingOnOrbitCharacteristics2024, + title = {Exploiting {{On-Orbit Characteristics}} for {{Joint Parameter}} and {{Channel Tracking}} in {{LEO Satellite Communications}}}, + author = {Lin, Chenlan and Chen, Xiaoming and Zhang, Zhaoyang}, + year = 2024, + month = dec, + journal = {IEEE Trans. Wireless Commun.}, + volume = {23}, + number = {12}, + pages = {19789--19803}, + issn = {1558-2248}, + doi = {10.1109/TWC.2024.3486709}, + urldate = {2024-12-20}, + langid = {american} +} + +@article{linFSFMUSICJoint2006, + title = {{{FSF MUSIC}} for {{Joint DOA}} and {{Frequency Estimation}} and {{Its Performance Analysis}}}, + author = {Lin, Jen-Der and Fang, Wen-Hsien and Wang, Yung-Yi and Chen, Jiunn-Tsair}, + year = 2006, + month = dec, + journal = {IEEE Transactions on Signal Processing}, + volume = {54}, + number = {12}, + pages = {4529--4542}, + issn = {1941-0476}, + doi = {10.1109/TSP.2006.882112}, + urldate = {2025-11-26}, + langid = {american} +} + +@article{linJointBeamformingSecure2018, + title = {Joint {{Beamforming}} for {{Secure Communication}} in {{Cognitive Satellite Terrestrial Networks}}}, + author = {Lin, Min and Lin, Zhi and Zhu, Wei-Ping and Wang, Jun-Bo}, + year = 2018, + month = may, + journal = {IEEE J. Sel. Areas Commun.}, + volume = {36}, + number = {5}, + pages = {1017--1029}, + issn = {1558-0008}, + doi = {10.1109/JSAC.2018.2832819}, + urldate = {2024-08-01}, + langid = {american} +} + +@article{linLEORelayAidedGEO2023, + title = {{{LEO Relay-Aided GEO Satellite-Terrestrial Transmissions}}}, + author = {Lin, Xinghan and Zhang, Haoxing and Pan, Gaofeng and Wang, Shuai and An, Jianping}, + year = 2023, + month = dec, + journal = {IEEE Transactions on Vehicular Technology}, + volume = {72}, + number = {12}, + pages = {16899--16904}, + issn = {1939-9359}, + doi = {10.1109/TVT.2023.3299964}, + urldate = {2025-04-09} +} + +@article{linLowComplexityReturnLinkBeamHopping2025, + title = {A {{Low-Complexity Return-Link Beam-Hopping Scheduling}} for {{NGSO Mega-Constellations With Dynamic Topology}} and {{Uneven Traffic}}}, + author = {Lin, Zhiyuan and Kuang, Linling and Liu, Bingkun and Jiang, Chunxiao and Jin, Jin}, + year = 2025, + month = jun, + journal = {IEEE Internet of Things Journal}, + volume = {12}, + number = {11}, + pages = {16695--16710}, + issn = {2327-4662}, + doi = {10.1109/JIOT.2025.3532928}, + urldate = {2025-08-28} +} + +@article{linMultiSatelliteBeamHopping2023, + title = {Multi-{{Satellite Beam Hopping Based}} on {{Load Balancing}} and {{Interference Avoidance}} for {{NGSO Satellite Communication Systems}}}, + author = {Lin, Zhiyuan and Ni, Zuyao and Kuang, Linling and Jiang, Chunxiao and Huang, Zhen}, + year = 2023, + month = jan, + journal = {IEEE Trans. Commun.}, + volume = {71}, + number = {1}, + pages = {282--295}, + issn = {1558-0857}, + doi = {10.1109/TCOMM.2022.3226190}, + urldate = {2024-07-27}, + langid = {american} +} + +@article{linNGSOSatellitesBeam2023, + title = {{{NGSO Satellites Beam Hopping Strategy Based}} on {{Load Balancing}} and {{Interference Avoidance}} for {{Coexistence With GSO Systems}}}, + author = {Lin, Zhiyuan and Ni, Zuyao and Kuang, Linling and Jiang, Chunxiao and Huang, Zhen}, + year = 2023, + month = jan, + journal = {IEEE Commun. Lett.}, + volume = {27}, + number = {1}, + pages = {278--282}, + issn = {1558-2558}, + doi = {10.1109/LCOMM.2022.3213912}, + urldate = {2024-08-04}, + langid = {american} +} + +@article{linRobustMultiObjectiveBeamforming2019, + title = {Robust {{Multi-Objective Beamforming}} for {{Integrated Satellite}} and {{High Altitude Platform Network With Imperfect Channel State Information}}}, + author = {Lin, Zhi and Lin, Min and Huang, Yongming and de Cola, Tomaso and Zhu, Wei-Ping}, + year = 2019, + month = dec, + journal = {IEEE Trans. Signal Process.}, + volume = {67}, + number = {24}, + pages = {6384--6396}, + issn = {1941-0476}, + doi = {10.1109/TSP.2019.2952045}, + urldate = {2024-08-04}, + langid = {american} +} + +@inproceedings{linRobustSecrecyEnergy2019, + title = {Robust {{Secrecy Energy Efficient Beamforming}} in {{Satellite Communication Systems}}}, + booktitle = {{{ICC}} 2019 - 2019 {{IEEE International Conference}} on {{Communications}} ({{ICC}})}, + author = {Lin, Zhi and Yin, Chunyan and Ouyang, Jian and Wu, Xiaohuan and Panagopoulos, Athanasios D.}, + year = 2019, + month = may, + pages = {1--5}, + issn = {1938-1883}, + doi = {10.1109/ICC.2019.8761056}, + urldate = {2025-03-13} +} + +@article{linRobustSecureBeamforming2018, + title = {Robust {{Secure Beamforming}} for {{5G Cellular Networks Coexisting With Satellite Networks}}}, + author = {Lin, Zhi and Lin, Min and Wang, Jun-Bo and Huang, Yongming and Zhu, Wei-Ping}, + year = 2018, + month = apr, + journal = {IEEE J. Sel. Areas Commun.}, + volume = {36}, + number = {4}, + pages = {932--945}, + issn = {1558-0008}, + doi = {10.1109/JSAC.2018.2824760}, + urldate = {2025-01-16}, + langid = {american} +} + +@article{linRobustSecureBeamforming2019, + title = {Robust {{Secure Beamforming}} for {{Multibeam Satellite Communication Systems}}}, + author = {Lin, Zhi and Lin, Min and Ouyang, Jian and Zhu, Wei-Ping and Panagopoulos, Athanasios D. and Alouini, Mohamed-Slim}, + year = 2019, + month = jun, + journal = {IEEE Trans. Veh. Technol.}, + volume = {68}, + number = {6}, + pages = {6202--6206}, + issn = {1939-9359}, + doi = {10.1109/TVT.2019.2913793}, + urldate = {2024-12-03}, + langid = {american} +} + +@article{linRobustSecureBeamforming2021, + title = {Robust {{Secure Beamforming}} for {{Wireless Powered Cognitive Satellite-Terrestrial Networks}}}, + author = {Lin, Zhi and Lin, Min and Zhu, Wei-Ping and Wang, Jun-Bo and Cheng, Julian}, + year = 2021, + month = jun, + journal = {IEEE Trans. on Cogn. Commun. Netw.}, + volume = {7}, + number = {2}, + pages = {567--580}, + issn = {2332-7731}, + doi = {10.1109/TCCN.2020.3016096}, + urldate = {2024-06-11}, + langid = {american} +} + +@article{linSatelliteTerrestrialCoordinatedMultiSatellite2024, + title = {Satellite-{{Terrestrial Coordinated Multi-Satellite Beam Hopping Scheduling Based}} on {{Multi-Agent Deep Reinforcement Learning}}}, + author = {Lin, Zhiyuan and Ni, Zuyao and Kuang, Linling and Jiang, Chunxiao and Huang, Zhen}, + year = 2024, + month = aug, + journal = {IEEE Trans. Wireless Commun.}, + volume = {23}, + number = {8}, + pages = {10091--10103}, + issn = {1558-2248}, + doi = {10.1109/TWC.2024.3368689}, + urldate = {2025-03-31} +} + +@article{linSecrecyEnergyEfficientHybrid2021, + title = {Secrecy-{{Energy Efficient Hybrid Beamforming}} for {{Satellite-Terrestrial Integrated Networks}}}, + author = {Lin, Zhi and Lin, Min and Champagne, Benoit and Zhu, Wei-Ping and {Al-Dhahir}, Naofal}, + year = 2021, + month = sep, + journal = {IEEE Trans. Commun.}, + volume = {69}, + number = {9}, + pages = {6345--6360}, + issn = {1558-0857}, + doi = {10.1109/TCOMM.2021.3088898}, + urldate = {2024-08-04}, + langid = {american} +} + +@article{linSecureBeamformingAntiJamming2025, + title = {Secure {{Beamforming}} and {{Anti-Jamming Coalition Formation}} for {{Air-Terrestrial Integrated Ad-Hoc Networks}}}, + author = {Lin, Xin and Liu, Aijun and Han, Chen and Sun, Yifu and Lin, Zhi and An, Kang and Gao, Xiqi and Wang, Jiangzhou}, + year = 2025, + month = aug, + journal = {IEEE Transactions on Wireless Communications}, + volume = {24}, + number = {8}, + pages = {6722--6736}, + issn = {1558-2248}, + doi = {10.1109/TWC.2025.3555600}, + urldate = {2025-09-02} +} + +@article{linSecureBeamformingCognitive2021, + title = {Secure {{Beamforming}} for {{Cognitive Satellite Terrestrial Networks With Unknown Eavesdroppers}}}, + author = {Lin, Zhi and Lin, Min and Champagne, Benoit and Zhu, Wei-Ping and {Al-Dhahir}, Naofal}, + year = 2021, + month = jun, + journal = {IEEE Syst. J.}, + volume = {15}, + number = {2}, + pages = {2186--2189}, + issn = {1937-9234}, + doi = {10.1109/JSYST.2020.2983309}, + urldate = {2024-08-01}, + langid = {american} +} + +@article{linSLNRBasedSecureEnergy2023, + title = {{{SLNR-Based Secure Energy Efficient Beamforming}} in {{Multibeam Satellite Systems}}}, + author = {Lin, Zhi and An, Kang and Niu, Hehao and Hu, Yihua and Chatzinotas, Symeon and Zheng, Gan and Wang, Jiangzhou}, + year = 2023, + month = apr, + journal = {IEEE Trans. Aerosp. Electron. Syst.}, + volume = {59}, + number = {2}, + pages = {2085--2088}, + issn = {1557-9603}, + doi = {10.1109/TAES.2022.3190238}, + urldate = {2024-04-15}, + langid = {american} +} + +@article{linStackedIntelligentMetasurface2024, + title = {Stacked {{Intelligent Metasurface Enabled LEO Satellite Communications Relying}} on {{Statistical CSI}}}, + author = {Lin, Shining and An, Jiancheng and Gan, Lu and Debbah, M{\'e}rouane and Yuen, Chau}, + year = 2024, + month = may, + journal = {IEEE Wireless Commun. Lett.}, + volume = {13}, + number = {5}, + pages = {1295--1299}, + issn = {2162-2345}, + doi = {10.1109/LWC.2024.3368238}, + urldate = {2025-04-03} +} + +@article{liPatternDesignPower2023, + title = {Pattern {{Design}} and {{Power Management}} for {{Cognitive LEO Beaming Hopping Satellite-Terrestrial Networks}}}, + author = {Li, Tong and Yao, Rugui and Fan, Ye and Zuo, Xiaoya and Miridakis, Nikolaos I. and Tsiftsis, Theodoros A.}, + year = 2023, + month = dec, + journal = {IEEE Trans. on Cogn. Commun. Netw.}, + volume = {9}, + number = {6}, + pages = {1531--1545}, + issn = {2332-7731}, + doi = {10.1109/TCCN.2023.3299576}, + urldate = {2024-12-23} +} + +@article{liPhasedArrayMaximum2010, + title = {The {{Phased Array Is}} the {{Maximum SNR Active Array}} [{{Lecture Notes}}]}, + author = {Li, Jian and Stoica, Petre}, + year = 2010, + month = mar, + journal = {IEEE Signal Process. Mag.}, + volume = {27}, + number = {2}, + pages = {143--144}, + issn = {1558-0792}, + doi = {10.1109/MSP.2009.935418}, + urldate = {2023-10-19}, + lccn = {1} +} + +@article{liPhysicalLayerSecuritySpace2020, + title = {Physical-{{Layer Security}} in {{Space Information Networks}}: {{A Survey}}}, + shorttitle = {Physical-{{Layer Security}} in {{Space Information Networks}}}, + author = {Li, Bin and Fei, Zesong and Zhou, Caiqiu and Zhang, Yan}, + year = 2020, + month = jan, + journal = {IEEE Internet Things J.}, + volume = {7}, + number = {1}, + pages = {33--52}, + issn = {2327-4662}, + doi = {10.1109/JIOT.2019.2943900}, + urldate = {2024-12-30} +} + +@article{liRecentAdvancesIntelligent2023, + title = {Recent {{Advances}} in {{Intelligent Processing}} of {{Satellite Video}}: {{Challenges}}, {{Methods}}, and {{Applications}}}, + shorttitle = {Recent {{Advances}} in {{Intelligent Processing}} of {{Satellite Video}}}, + author = {Li, Shengyang and Sun, Xian and Gu, Yanfeng and Lv, Yixuan and Zhao, Manqi and Zhou, Zhuang and Guo, Weilong and Sun, Yuhan and Wang, Han and Yang, Jian}, + year = 2023, + journal = {IEEE J. 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Signal Process.}, + volume = {51}, + number = {7}, + pages = {1702--1715}, + issn = {1941-0476}, + doi = {10.1109/TSP.2003.812831}, + urldate = {2023-11-26} +} + +@article{liRobustDOAEstimation2022, + title = {Robust {{DOA Estimation}} in {{Satellite Systems}} in {{Presence}} of {{Coherent Signals Subject}} to {{Low SNR}}}, + author = {Li, Yunfeng and Huang, Yonghui and Ren, Jian and Liu, Ying and Pedersen, Gert Fr{\o}lund and Shen, Ming}, + year = 2022, + journal = {IEEE Access}, + volume = {10}, + pages = {109983--109993}, + issn = {2169-3536}, + doi = {10.1109/ACCESS.2022.3213712}, + urldate = {2025-03-03} +} + +@article{liStackedIntelligentMetasurfaces2024, + title = {Stacked {{Intelligent Metasurfaces}} for {{Holographic MIMO-Aided Cell-Free Networks}}}, + author = {Li, Qingchao and {El-Hajjar}, Mohammed and Xu, Chao and An, Jiancheng and Yuen, Chau and Hanzo, Lajos}, + year = 2024, + month = nov, + journal = {IEEE Trans. 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Lett.}, + volume = {24}, + number = {1}, + pages = {7--11}, + issn = {1558-2361}, + doi = {10.1109/LSP.2016.2632750}, + urldate = {2024-04-26}, + langid = {american} +} + +@article{liuDownlinkMIMOLTEadvanced2012, + title = {Downlink {{MIMO}} in {{LTE-advanced}}: {{SU-MIMO}} vs. {{MU-MIMO}}}, + shorttitle = {Downlink {{MIMO}} in {{LTE-advanced}}}, + author = {Liu, Lingjia and Chen, Runhua and Geirhofer, Stefan and Sayana, Krishna and Shi, Zhihua and Zhou, Yongxing}, + year = 2012, + month = feb, + journal = {IEEE Commun. Mag.}, + volume = {50}, + number = {2}, + pages = {140--147}, + issn = {1558-1896}, + doi = {10.1109/MCOM.2012.6146493}, + urldate = {2024-07-12} +} + +@article{liuDynamicBeamShut2020, + title = {A {{Dynamic Beam Shut Off Algorithm}} for {{LEO Multibeam Satellite Constellation Network}}}, + author = {Liu, Shuaijun and Lin, Jinkun and Xu, Lexi and Gao, Xianming and Liu, Lixiang and Jiang, Linyan}, + year = 2020, + month = oct, + journal = {IEEE Wireless Commun. Lett.}, + volume = {9}, + number = {10}, + pages = {1730--1733}, + issn = {2162-2345}, + doi = {10.1109/LWC.2020.3002846}, + urldate = {2024-08-04}, + langid = {american} +} + +@article{liuFirstOrderPerturbationAnalysis2008, + title = {First-{{Order Perturbation Analysis}} of {{Singular Vectors}} in {{Singular Value Decomposition}}}, + author = {Liu, Jun and Liu, Xiangqian and Ma, Xiaoli}, + year = 2008, + month = jul, + journal = {IEEE Trans. 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Technol.}, + volume = {73}, + number = {4}, + pages = {5162--5178}, + issn = {1939-9359}, + doi = {10.1109/TVT.2023.3329168}, + urldate = {2024-12-19} +} + +@inproceedings{liuLargeScaleSmallSatellite2020, + title = {Large-{{Scale Small Satellite Network Simulator}}: {{Design}} and {{Evaluation}}}, + shorttitle = {Large-{{Scale Small Satellite Network Simulator}}}, + booktitle = {2020 3rd {{Int}}. {{Conf}}. {{Hot Inf}}. - {{Centric Netw}}.}, + author = {Liu, Mengjie and Gui, Yongqiang and Li, Jian and Lu, Hancheng}, + year = 2020, + month = dec, + pages = {194--199}, + doi = {10.1109/HotICN50779.2020.9350838}, + urldate = {2025-02-13} +} + +@article{liuMachineLearningTimeofArrival2023, + title = {Machine {{Learning}} for {{Time-of-Arrival Estimation With 5G Signals}} in {{Indoor Positioning}}}, + author = {Liu, Zhaoliang and Chen, Liang and Zhou, Xin and Jiao, Zhenhang and Guo, Guangyi and Chen, Ruizhi}, + year = 2023, + month = jun, + journal = {IEEE Internet Things J.}, + volume = {10}, + number = {11}, + pages = {9782--9795}, + issn = {2327-4662}, + doi = {10.1109/JIOT.2023.3234123}, + urldate = {2024-07-11} +} + +@article{liuMultiAgentDeepReinforcement2024, + title = {A {{Multi-Agent Deep Reinforcement Learning-Based Handover Scheme}} for {{Mega-Constellation Under Dynamic Propagation Conditions}}}, + author = {Liu, Haotian and Wang, Yichen and Li, Peixuan and Cheng, Julian}, + year = 2024, + month = oct, + journal = {IEEE Transactions on Wireless Communications}, + volume = {23}, + number = {10}, + pages = {13579--13596}, + issn = {1558-2248}, + doi = {10.1109/TWC.2024.3407358}, + urldate = {2025-08-28} +} + +@article{liuOptimalBeamformingHybrid2020, + title = {Optimal {{Beamforming}} for {{Hybrid Satellite Terrestrial Networks With Nonlinear PA}} and {{Imperfect CSIT}}}, + author = {Liu, Chengxiao and Feng, Wei and Chen, Yunfei and Wang, Cheng-Xiang and Ge, Ning}, + year = 2020, + month = mar, + journal = {IEEE Wireless Commun. Lett.}, + volume = {9}, + number = {3}, + pages = {276--280}, + issn = {2162-2345}, + doi = {10.1109/LWC.2019.2952124}, + urldate = {2024-08-04}, + langid = {american} +} + +@article{liuPerformanceAnalysisNGSO2024, + title = {Performance {{Analysis}} of {{NGSO Satellite Communication Systems With Flexible Beams}}}, + author = {Liu, Bingkun and Kuang, Linling and Lu, Jianhua}, + year = 2024, + month = jul, + journal = {IEEE Internet of Things Journal}, + volume = {11}, + number = {14}, + pages = {24726--24738}, + issn = {2327-4662}, + doi = {10.1109/JIOT.2024.3379586}, + urldate = {2025-09-02} +} + +@article{liuReconfigurableIntelligentSurfaces2021, + title = {Reconfigurable {{Intelligent Surfaces}}: {{Principles}} and {{Opportunities}}}, + shorttitle = {Reconfigurable {{Intelligent Surfaces}}}, + author = {Liu, Yuanwei and Liu, Xiao and Mu, Xidong and Hou, Tianwei and Xu, Jiaqi and Di Renzo, Marco and {Al-Dhahir}, Naofal}, + year = 2021, + journal = {IEEE Commun. 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Lett.}, + volume = {22}, + number = {9}, + pages = {1438--1442}, + issn = {1558-2361}, + doi = {10.1109/LSP.2015.2409153}, + urldate = {2023-12-09}, + langid = {english} +} + +@inproceedings{liuResearchHandoverStrategy2021, + title = {Research on {{Handover Strategy}} of {{LEO Satellite Network}}}, + booktitle = {2021 2nd {{Int}}. {{Conf}}. {{Big Data Informatization Educ}}.}, + author = {Liu, Zhiguo and Zha, Xueli and Ren, Xiaobo and Yao, Qiaoyu}, + year = 2021, + month = apr, + pages = {188--194}, + doi = {10.1109/ICBDIE52740.2021.00050}, + urldate = {2025-01-13} +} + +@article{liuRISEmpoweredSatelliteAerialTerrestrialNetworks2024, + title = {{{RIS-Empowered Satellite-Aerial-Terrestrial Networks With PD-NOMA}}}, + author = {Liu, Rui and Guo, Kefeng and Li, Xingwang and Dev, Kapal and Ali Khowaja, Sunder and Tsiftsis, Theodoros A. and Song, Houbing}, + year = 2024, + journal = {IEEE Communications Surveys \& Tutorials}, + volume = {26}, + number = {4}, + pages = {2258--2289}, + issn = {1553-877X}, + doi = {10.1109/COMST.2024.3393612}, + urldate = {2025-09-03} +} + +@article{liuRobustAdaptiveWideband2019, + title = {Robust {{Adaptive Wideband Beamforming Based}} on {{Time Frequency Distribution}}}, + author = {Liu, Yaqi and Liu, Chengcheng and Hu, Dexiu and Zhao, Yongjun}, + year = 2019, + month = aug, + journal = {IEEE Trans. Signal Process.}, + volume = {67}, + number = {16}, + pages = {4370--4382}, + issn = {1941-0476}, + doi = {10.1109/TSP.2019.2929924}, + urldate = {2025-03-03} +} + +@inproceedings{liuRobustBeamformingDownlink2018, + title = {Robust {{Beamforming}} for {{Downlink 3D-MIMO Systems}} with L11-Norm {{Bounded CSI Uncertainty}}}, + booktitle = {2018 10th {{International Conference}} on {{Wireless Communications}} and {{Signal Processing}} ({{WCSP}})}, + author = {Liu, Kai and Feng, Hui and Yang, Tao and Hu, Bo}, + year = 2018, + month = oct, + pages = {1--5}, + issn = {2472-7628}, + doi = {10.1109/WCSP.2018.8555867}, + urldate = {2024-07-09} +} + +@article{liuRobustEnergyEfficientHybrid2022, + title = {Robust {{Energy-Efficient Hybrid Beamforming Design}} for {{Massive MIMO LEO Satellite Communication Systems}}}, + author = {Liu, Yang and Li, Changqing and Li, Jiong and Feng, Lu}, + year = 2022, + journal = {IEEE Access}, + volume = {10}, + pages = {63085--63099}, + issn = {2169-3536}, + doi = {10.1109/ACCESS.2022.3181178}, + urldate = {2025-01-16}, + langid = {american} +} + +@article{liuRobustMIMOBeamforming2017, + title = {Robust {{MIMO Beamforming}} for {{Cellular}} and {{Radar Coexistence}}}, + author = {Liu, Fan and Masouros, Christos and Li, Ang and Ratnarajah, Tharmalingam}, + year = 2017, + month = jun, + journal = {IEEE Wireless Commun. 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Mag.}, + volume = {40}, + number = {5}, + pages = {106--121}, + issn = {1558-0792}, + doi = {10.1109/MSP.2023.3272881}, + urldate = {2023-10-17}, + lccn = {1} +} + +@article{liuSpaceAirGroundIntegratedNetwork2018, + title = {Space-{{Air-Ground Integrated Network}}: {{A Survey}}}, + shorttitle = {Space-{{Air-Ground Integrated Network}}}, + author = {Liu, Jiajia and Shi, Yongpeng and Fadlullah, Zubair Md. and Kato, Nei}, + year = 2018, + journal = {IEEE Commun. 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Signal Process.}, + volume = {64}, + number = {16}, + pages = {4203--4217}, + issn = {1941-0476}, + doi = {10.1109/TSP.2016.2558167}, + urldate = {2023-10-21}, + langid = {american}, + lccn = {1} +} + +@article{liuSurveyFundamentalLimits2022, + title = {A {{Survey}} on {{Fundamental Limits}} of {{Integrated Sensing}} and {{Communication}}}, + author = {Liu, An and Huang, Zhe and Li, Min and Wan, Yubo and Li, Wenrui and Han, Tony Xiao and Liu, Chenchen and Du, Rui and Tan, Danny Kai Pin and Lu, Jianmin and Shen, Yuan and Colone, Fabiola and Chetty, Kevin}, + year = 2022, + journal = {IEEE Commun. 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Mag.}, + volume = {40}, + number = {4}, + pages = {80--91}, + issn = {1558-0792}, + doi = {10.1109/MSP.2023.3258060}, + urldate = {2023-10-17}, + langid = {english}, + lccn = {1} +} + +@article{liuUltraStarLightweightSimulator2023, + title = {{{UltraStar}}: {{A Lightweight Simulator}} of {{Ultra-Dense LEO Satellite Constellation Networking}} for {{6G}}}, + shorttitle = {{{UltraStar}}}, + author = {Liu, Xiaoyu and Ma, Ting and Tang, Zhixuan and Qin, Xiaohan and Zhou, Haibo and Shen, Xuemin Sherman}, + year = 2023, + month = mar, + journal = {IEEE/CAA J. 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Commun.}, + volume = {69}, + number = {6}, + pages = {3984--4000}, + issn = {1558-0857}, + doi = {10.1109/TCOMM.2021.3062376}, + urldate = {2024-06-24} +} + +@article{wangJointBeamformingIllumination2024, + title = {Joint {{Beamforming}} and {{Illumination Pattern Design}} for {{Beam-Hopping LEO Satellite Communications}}}, + author = {Wang, Jing and Qi, Chenhao and Yu, Shui and Mao, Shiwen}, + year = 2024, + month = dec, + journal = {IEEE Trans. Wireless Commun.}, + volume = {23}, + number = {12}, + pages = {18940--18950}, + issn = {1558-2248}, + doi = {10.1109/TWC.2024.3463002}, + urldate = {2024-12-19} +} + +@article{wangJointBeamformingIntegrated2021, + title = {Joint {{Beamforming}} for {{Integrated Mmwave Satellite-Terrestrial Self-Backhauled Networks}}}, + author = {Wang, Qingming and Zhang, Hua and Wang, Jun-Bo and Yang, Fan and Li, Geoffrey Ye}, + year = 2021, + month = sep, + journal = {IEEE Trans. Veh. Technol.}, + volume = {70}, + number = {9}, + pages = {9103--9117}, + issn = {1939-9359}, + doi = {10.1109/TVT.2021.3100562}, + urldate = {2024-07-24}, + langid = {american} +} + +@article{wangJointOptimizationBeamHopping2022, + title = {Joint {{Optimization}} of {{Beam-Hopping Design}} and {{NOMA-Assisted Transmission}} for {{Flexible Satellite Systems}}}, + author = {Wang, Anyue and Lei, Lei and Lagunas, Eva and {P{\'e}rez-Neira}, Ana I. and Chatzinotas, Symeon and Ottersten, Bj{\"o}rn}, + year = 2022, + month = oct, + journal = {IEEE Trans. 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Wireless Commun.}, + volume = {18}, + number = {3}, + pages = {1637--1650}, + issn = {1558-2248}, + doi = {10.1109/TWC.2019.2894823}, + urldate = {2025-01-16}, + langid = {american} +} + +@article{wangMultiObjectiveRobustBeamforming2023, + title = {Multi-{{Objective Robust Beamforming}} for {{Integrated Satellite}} and {{Aerial Networks Supporting Heterogeneous Services}}}, + author = {Wang, Zining and Lin, Min and Ouyang, Jian and Wang, Jun-Bo and Zhu, Wei-Ping and {Al-Dhahir}, Naofal}, + year = 2023, + month = oct, + journal = {IEEE Trans. Wireless Commun.}, + volume = {22}, + number = {10}, + pages = {6870--6882}, + issn = {1558-2248}, + doi = {10.1109/TWC.2023.3246460}, + urldate = {2024-12-19}, + langid = {american} +} + +@article{wangNewOnOrbitGeometric2018, + title = {A {{New On-Orbit Geometric Self-Calibration Approach}} for the {{High-Resolution Geostationary Optical Satellite GaoFen4}}}, + author = {Wang, Mi and Cheng, Yufeng and Tian, Yuan and He, Luxiao and Wang, Yanli}, + year = 2018, + month = may, + journal = {IEEE J. Sel. Topics Appl. Earth Observ. Remote Sens.}, + volume = {11}, + number = {5}, + pages = {1670--1683}, + issn = {2151-1535}, + doi = {10.1109/JSTARS.2018.2814205}, + urldate = {2025-02-13}, + langid = {american} +} + +@article{wangOverviewEnhancedMassive2019, + title = {An {{Overview}} of {{Enhanced Massive MIMO With Array Signal Processing Techniques}}}, + author = {Wang, Mingjin and Gao, Feifei and Jin, Shi and Lin, Hai}, + year = 2019, + month = sep, + journal = {IEEE J. Sel. 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Sci.}, + volume = {64}, + number = {6}, + pages = {161301}, + issn = {1869-1919}, + doi = {10.1007/s11432-019-2928-9}, + urldate = {2024-07-11}, + langid = {english} +} + +@article{wangPerformanceAnalysisCoarrayBased2018, + title = {Performance {{Analysis}} of {{Coarray-Based MUSIC}} in the {{Presence}} of {{Sensor Location Errors}}}, + author = {Wang, Mianzhi and Zhang, Zhen and Nehorai, Arye}, + year = 2018, + month = jun, + journal = {IEEE Transactions on Signal Processing}, + volume = {66}, + number = {12}, + pages = {3074--3085}, + issn = {1941-0476}, + doi = {10.1109/TSP.2018.2824283}, + urldate = {2025-04-14} +} + +@article{wangResourceAllocationLoad2024, + title = {Resource {{Allocation}} and {{Load Balancing}} for {{Beam Hopping Scheduling}} in {{Satellite-Terrestrial Communications}}: {{A Cooperative Satellite Approach}}}, + shorttitle = {Resource {{Allocation}} and {{Load Balancing}} for {{Beam Hopping Scheduling}} in {{Satellite-Terrestrial Communications}}}, + author = {Wang, Guanhua and Yang, Fang and Song, Jian and Han, Zhu}, + year = 2024, + journal = {IEEE Trans. Wireless Commun.}, + pages = {1--1}, + issn = {1558-2248}, + doi = {10.1109/TWC.2024.3508741}, + urldate = {2024-12-23} +} + +@article{wangResourceEfficiencyOptimization2021, + title = {Resource {{Efficiency Optimization}} for {{Robust Beamforming}} in {{Multi-Beam Satellite Communications}}}, + author = {Wang, Wenjin and Gao, Linna and Ding, Rui and Lei, Jizhao and You, Li and Chan, Chien Aun and Gao, Xiqi}, + year = 2021, + month = jul, + journal = {IEEE Trans. Veh. Technol.}, + volume = {70}, + number = {7}, + pages = {6958--6968}, + issn = {1939-9359}, + doi = {10.1109/TVT.2021.3087744}, + urldate = {2024-08-04}, + langid = {american} +} + +@article{wangRetrodirectiveFrequencyDiverse2019, + title = {Retrodirective {{Frequency Diverse Array Focusing}} for {{Wireless Information}} and {{Power Transfer}}}, + author = {Wang, Wen-Qin}, + year = 2019, + month = jan, + journal = {IEEE J. Sel. 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Technol.}, + volume = {71}, + number = {1}, + pages = {1010--1014}, + issn = {1939-9359}, + doi = {10.1109/TVT.2021.3124928}, + urldate = {2025-01-16}, + langid = {american} +} + +@article{wangSeamlessHandoverLEO2023, + title = {Seamless {{Handover}} in {{LEO Based Non-Terrestrial Networks}}: {{Service Continuity}} and {{Optimization}}}, + shorttitle = {Seamless {{Handover}} in {{LEO Based Non-Terrestrial Networks}}}, + author = {Wang, Feng and Jiang, Dingde and Wang, Zhihao and Chen, Jianguang and Quek, Tony Q. S.}, + year = 2023, + month = feb, + journal = {IEEE Transactions on Communications}, + volume = {71}, + number = {2}, + pages = {1008--1023}, + issn = {1558-0857}, + doi = {10.1109/TCOMM.2022.3229014}, + urldate = {2025-09-23} +} + +@article{wangSpacePhasedArray2020, + title = {Space {{Phased Array Antenna Developments}}: {{A Perspective}} on {{Structural Design}}}, + shorttitle = {Space {{Phased Array Antenna Developments}}}, + author = {Wang, Congsi and Wang, Yan and Lian, Peiyuan and Xue, Song and Xu, Qian and Shi, Yu and Jia, Yu and Du, Biao and Liu, Jing and Tang, Baofu}, + year = 2020, + month = jul, + journal = {IEEE Aerosp. Electron. Syst. 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Vincent}, + year = 2024, + journal = {IEEE Commun. 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S. and Ding, Yanwu and Shen, Dan and Pham, Khanh and Blasch, Erik and Chen, Genshe}, + year = 2024, + month = may, + pages = {1--5}, + issn = {2375-5318}, + doi = {10.1109/RadarConf2458775.2024.10548034}, + urldate = {2025-12-04} +} + +@inproceedings{zhangDoaEstimationSensor1993, + title = {Doa {{Estimation}} with {{Sensor Gain}}, {{Phase}} and {{Position Perturbations}}}, + booktitle = {Proc. {{IEEE}} 1993 {{Nat}}. {{Aerosp}}. {{Electron}}. {{Conf}}.}, + author = {Zhang, Ming and Zhu, Zhao-Da}, + year = 1993, + month = may, + pages = {67-69 vol.1}, + doi = {10.1109/NAECON.1993.290894}, + urldate = {2024-08-02}, + langid = {american} +} + +@article{zhangDualBandDualPolarizationHorn2022, + title = {Dual-{{Band Dual-Polarization Horn Antenna Array Based}} on {{Orthomode Transducers With High Isolation}} for {{Satellite Communication}}}, + author = {Zhang, Jie and Li, Xiuping and Qi, Zihang and Huang, Yuhan and Zhu, Hua}, + year = 2022, + month = oct, + journal = {IEEE Trans. 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Areas Commun.}, + volume = {43}, + number = {1}, + pages = {107--121}, + issn = {1558-0008}, + doi = {10.1109/JSAC.2024.3460059}, + urldate = {2024-12-20} +} + +@book{ZhangJuZhenFenXiYuYingYong2013, + title = {{矩阵分析与应用}}, + author = {张, 贤达}, + year = 2013, + edition = {第2版}, + publisher = {清华大学出版社}, + address = {北京}, + isbn = {978-7-302-33859-8}, + langid = {chinese} +} + +@article{zhangKaBandPlanarSharedAperture2023, + title = {K-/{{Ka-Band Planar Shared-Aperture Beam-Scanning Array Antenna}} for {{Simultaneous Transmitting}} and {{Receiving Low Earth Orbit Satellite Communication Terminal}}}, + author = {Zhang, Jin Fan and Cheng, Yu Jian}, + year = 2023, + month = aug, + journal = {IEEE Trans. Antennas Propag.}, + volume = {71}, + number = {8}, + pages = {6617--6627}, + issn = {1558-2221}, + doi = {10.1109/TAP.2023.3284163}, + urldate = {2024-08-04} +} + +@phdthesis{ZhangKaiShuZhiNengTianXianKuaiSuWenJianFangXiangTuZongHeYuKangGanRaoJiShuYanJiu2023, + type = {{硕士}}, + title = {{智能天线快速稳健方向图综合与抗干扰技术研究}}, + author = {{张开墅}}, + year = 2023, + doi = {10.27241/d.cnki.gnjgu.2021.000994}, + urldate = {2023-12-10}, + langid = {chinese}, + school = {南京理工大学} +} + +@article{zhangLEOMegaConstellationTerrestrialCommunications2025, + title = {{{LEO Mega-Constellation-Terrestrial Communications Suffering Poisson Arc Hardcore Distributed Space Interference}}}, + author = {Zhang, Haoxing and Miao, Xiaqing and Ni, Zihan and Wang, Shuai and Pan, Gaofeng and Cavdar, Cicek and An, Jianping}, + year = 2025, + journal = {IEEE Transactions on Wireless Communications}, + pages = {1--1}, + issn = {1558-2248}, + doi = {10.1109/TWC.2024.3523950}, + urldate = {2025-04-09} +} + +@article{zhangMassiveHybridAntenna2015, + title = {Massive Hybrid Antenna Array for Millimeter-Wave Cellular Communications}, + author = {Zhang, Jian A. and Huang, Xiaojing and Dyadyuk, Val and Guo, Y. Jay}, + year = 2015, + month = feb, + journal = {IEEE Wireless Commun.}, + volume = {22}, + number = {1}, + pages = {79--87}, + issn = {1558-0687}, + doi = {10.1109/MWC.2015.7054722}, + urldate = {2024-12-17} +} + +@article{zhangMultiSatelliteCooperativeNetworks2024, + title = {Multi-{{Satellite Cooperative Networks}}: {{Joint Hybrid Beamforming}} and {{User Scheduling Design}}}, + shorttitle = {Multi-{{Satellite Cooperative Networks}}}, + author = {Zhang, Xuan and Sun, Shu and Tao, Meixia and Huang, Qin and Tang, Xiaohu}, + year = 2024, + month = jul, + journal = {IEEE Trans. Wireless Commun.}, + volume = {23}, + number = {7}, + pages = {7938--7952}, + issn = {1558-2248}, + doi = {10.1109/TWC.2023.3346463}, + urldate = {2024-07-27}, + langid = {american} +} + +@article{zhangOptimalHybridBeamforming2019, + title = {Optimal {{Hybrid Beamforming Design}} for {{Millimeter-Wave Massive Multi-User MIMO Relay Systems}}}, + author = {Zhang, Yang and Du, Jianhe and Chen, Yuanzhi and Han, Meng and Li, Xingwang}, + year = 2019, + journal = {IEEE Access}, + volume = {7}, + pages = {157212--157225}, + issn = {2169-3536}, + doi = {10.1109/ACCESS.2019.2949786}, + urldate = {2024-07-09} +} + +@article{zhangOutageAnalysisCooperative2022, + title = {Outage {{Analysis}} of {{Cooperative Satellite-Aerial-Terrestrial Networks With Spatially Random Terminals}}}, + author = {Zhang, Haoxing and Pan, Gaofeng and Ke, Sheng and Wang, Shuai and An, Jianping}, + year = 2022, + month = jul, + journal = {IEEE Trans. Commun.}, + volume = {70}, + number = {7}, + pages = {4972--4987}, + issn = {1558-0857}, + doi = {10.1109/TCOMM.2022.3179768}, + urldate = {2025-04-09} +} + +@article{zhangPassiveMaritimeSurveillance2020, + title = {Passive {{Maritime Surveillance Based}} on {{Low Earth Orbit Satellite Constellations}}}, + author = {Zhang, Qianyun and Li, Shijie and Wu, Biyi and Wang, Jingchao}, + year = 2020, + month = dec, + journal = {IEEE Wireless Communications}, + volume = {27}, + number = {6}, + pages = {61--67}, + issn = {1558-0687}, + doi = {10.1109/MWC.001.2000135}, + urldate = {2025-03-12}, + langid = {american} +} + +@article{zhangPotentialGameBasedRadio2021, + title = {Potential {{Game-Based Radio Resource Allocation}} in {{Uplink Multibeam Satellite IoT Networks}}}, + author = {Zhang, Xiaokai and Zhang, Bangning and Guo, Daoxing and An, Kang and Qi, Shuai and Wu, Gang}, + year = 2021, + month = dec, + journal = {IEEE Trans. Aerosp. Electron. Syst.}, + volume = {57}, + number = {6}, + pages = {4269--4279}, + issn = {1557-9603}, + doi = {10.1109/TAES.2021.3100335}, + urldate = {2024-12-23} +} + +@article{zhangProspectiveMultipleAntenna2020, + title = {Prospective {{Multiple Antenna Technologies}} for {{Beyond 5G}}}, + author = {Zhang, Jiayi and Bj{\"o}rnson, Emil and Matthaiou, Michail and Ng, Derrick Wing Kwan and Yang, Hong and Love, David J.}, + year = 2020, + month = aug, + journal = {IEEE J. Sel. Areas Commun.}, + volume = {38}, + number = {8}, + pages = {1637--1660}, + issn = {1558-0008}, + doi = {10.1109/JSAC.2020.3000826}, + urldate = {2024-07-09}, + langid = {american} +} + +@article{zhangResourceAllocationTerrestrialSatelliteBased2022, + title = {Resource {{Allocation}} in {{Terrestrial-Satellite-Based Next Generation Multiple Access Networks With Interference Cooperation}}}, + author = {Zhang, Yaomin and Zhang, Haijun and Zhou, Huan and Long, Keping and Karagiannidis, George K.}, + year = 2022, + month = apr, + journal = {IEEE J. Sel. Areas Commun.}, + volume = {40}, + number = {4}, + pages = {1210--1221}, + issn = {1558-0008}, + doi = {10.1109/JSAC.2022.3145810}, + urldate = {2024-12-23} +} + +@article{zhangRobustBeamformingEnhancing2021, + title = {Robust {{Beamforming}} for {{Enhancing Security}} in {{Multibeam Satellite Systems}}}, + author = {Zhang, Jian and Lin, Min and Ouyang, Jian and Zhu, Wei-Ping and De Cola, Tomaso}, + year = 2021, + month = jul, + journal = {IEEE Commun. Lett.}, + volume = {25}, + number = {7}, + pages = {2161--2165}, + issn = {1558-2558}, + doi = {10.1109/LCOMM.2021.3073026}, + urldate = {2024-06-11}, + langid = {american} +} + +@article{zhangRobustBeamformingMultibeam2019, + title = {Robust {{Beamforming}} for {{Multibeam Satellite Communication}} in the {{Face}} of {{Phase Perturbations}}}, + author = {Zhang, Xin and Wang, Jingjing and Jiang, Chunxiao and Yan, Chaoxing and Ren, Yong and Hanzo, Lajos}, + year = 2019, + month = mar, + journal = {IEEE Trans. Veh. Technol.}, + volume = {68}, + number = {3}, + pages = {3043--3047}, + issn = {1939-9359}, + doi = {10.1109/TVT.2019.2896245}, + urldate = {2024-08-04}, + langid = {american} +} + +@article{ZhangRuiWeiXingDaoHangGaoJingDuKangGanRaoSuanFaYanJiu2023, + title = {{卫星导航高精度抗干扰算法研究}}, + author = {{张锐} and {徐兵} and {谢斌斌}}, + year = 2023, + journal = {现代导航}, + volume = {14}, + number = {5}, + pages = {339--344}, + issn = {1674-7976}, + urldate = {2023-12-10}, + langid = {chinese} +} + +@article{zhangSecureUplinkTransmissions2023, + title = {On {{Secure Uplink Transmissions}} in {{Satellite-Aerial Systems}}}, + author = {Zhang, Haoxing and Yue, Pingyue and Wang, Shuai and Pan, Gaofeng and An, Jianping}, + year = 2023, + month = aug, + journal = {IEEE Transactions on Aerospace and Electronic Systems}, + volume = {59}, + number = {4}, + pages = {4666--4673}, + issn = {1557-9603}, + doi = {10.1109/TAES.2022.3230380}, + urldate = {2025-04-09}, + langid = {american} +} + +@article{zhangSecureWirelessCommunications2019, + title = {Secure {{Wireless Communications Based}} on {{Compressive Sensing}}: {{A Survey}}}, + shorttitle = {Secure {{Wireless Communications Based}} on {{Compressive Sensing}}}, + author = {Zhang, Yushu and Xiang, Yong and Zhang, Leo Yu and Rong, Yue and Guo, Song}, + year = 2019, + journal = {IEEE Commun. Surveys Tuts.}, + volume = {21}, + number = {2}, + pages = {1093--1111}, + issn = {1553-877X}, + doi = {10.1109/COMST.2018.2878943}, + urldate = {2023-10-21}, + langid = {english} +} + +@inproceedings{zhangSparsityBasedDOAEstimation2013, + title = {Sparsity-{{Based DOA Estimation Using Co-Prime Arrays}}}, + booktitle = {2013 {{IEEE International Conference}} on {{Acoustics}}, {{Speech}} and {{Signal Processing}}}, + author = {Zhang, Yimin D. and Amin, Moeness G. and Himed, Braham}, + year = 2013, + month = may, + pages = {3967--3971}, + issn = {2379-190X}, + doi = {10.1109/ICASSP.2013.6638403}, + urldate = {2023-12-09}, + langid = {english} +} + +@inproceedings{zhangTDOAAssistedDirectPosition2021, + title = {A {{TDOA-Assisted Direct Position Determination}} for {{Efficient Geolocalization Using LEO Satellites}}}, + booktitle = {2021 {{IEEE}} 20th {{International Conference}} on {{Trust}}, {{Security}} and {{Privacy}} in {{Computing}} and {{Communications}} ({{TrustCom}})}, + author = {Zhang, Qianyun and Li, Shijie and Shi, Jiting and Guan, Zhenyu}, + year = 2021, + month = oct, + pages = {1323--1327}, + issn = {2324-9013}, + doi = {10.1109/TrustCom53373.2021.00183}, + urldate = {2025-03-12}, + langid = {american} +} + +@article{zhangTensorBasedChannelEstimation2025, + title = {Tensor-{{Based Channel Estimation}} for {{Extremely Large-Scale MIMO-OFDM With Dynamic Metasurface Antennas}}}, + author = {Zhang, Ruoyu and Chen, Guangyi and Cheng, Lei and Guan, Xinrong and Wu, Qingqing and Wu, Wen and Zhang, Rui}, + year = 2025, + month = jul, + journal = {IEEE Transactions on Wireless Communications}, + volume = {24}, + number = {7}, + pages = {6052--6068}, + issn = {1558-2248}, + doi = {10.1109/TWC.2025.3551144}, + urldate = {2025-09-08} +} + +@article{zhangUnifiedNOMAFramework2023, + title = {A {{Unified NOMA Framework}} in {{Beam-Hopping Satellite Communication Systems}}}, + author = {Zhang, Xuyang and Yue, Xinwei and Li, Tian and Han, Zhihao and Wang, Yafei and Ding, Yong and Liu, Rongke}, + year = 2023, + month = oct, + journal = {IEEE Trans. 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Syst.}, + volume = {59}, + number = {5}, + pages = {5390--5404}, + issn = {1557-9603}, + doi = {10.1109/TAES.2023.3260059}, + urldate = {2024-12-23} +} + +@article{zhangUserActivityDetection2020, + title = {User {{Activity Detection}} and {{Channel Estimation}} for {{Grant-Free Random Access}} in {{LEO Satellite-Enabled Internet}} of {{Things}}}, + author = {Zhang, Zhaoji and Li, Ying and Huang, Chongwen and Guo, Qinghua and Liu, Lei and Yuen, Chau and Guan, Yong Liang}, + year = 2020, + month = sep, + journal = {IEEE Internet Things J.}, + volume = {7}, + number = {9}, + pages = {8811--8825}, + issn = {2327-4662}, + doi = {10.1109/JIOT.2020.2997336}, + urldate = {2024-12-23}, + langid = {american} +} + +@article{zhangVariablePhaseShiftBasedRFBasebandCodesign2005, + title = {Variable-{{Phase-Shift-Based RF-Baseband Codesign}} for {{MIMO Antenna Selection}}}, + author = {Zhang, Xinying and Molisch, A.F. and Kung, Sun-Yuan}, + year = 2005, + month = nov, + journal = {IEEE Trans. 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Veh. Technol.}, + volume = {70}, + number = {3}, + pages = {2322--2336}, + issn = {1939-9359}, + doi = {10.1109/TVT.2021.3057547}, + urldate = {2024-07-17}, + langid = {american} +} + +@phdthesis{ZhaoAnQiJiYuJiQiXueXiHeSuiJiJuZhenLiLunDeDaZhenLieDOAGuJiFangFaYanJiu2020, + type = {{硕士学位论文}}, + title = {{基于机器学习和随机矩阵理论的大阵列DOA估计方法研究}}, + author = {{赵安琪}}, + year = 2020, + doi = {10.27162/d.cnki.gjlin.2020.005804}, + urldate = {2024-03-11}, + collaborator = {{姜宏}}, + langid = {chinese}, + school = {吉林大学} +} + +@inproceedings{zhaoBeamHoppingLEO2022, + title = {Beam {{Hopping}} for {{LEO Satellite}}:{{Challenges}} and {{Opportunities}}}, + shorttitle = {Beam {{Hopping}} for {{LEO Satellite}}}, + booktitle = {2022 {{Int}}. {{Conf}}. {{Culture}} - {{Oriented Sci}}. {{Technol}}.}, + author = {Zhao, Qi and Hu, Yiqing and Pang, Zhenyu and Ren, Derong}, + year = 2022, + month = aug, + pages = {319--324}, + doi = {10.1109/CoST57098.2022.00072}, + urldate = {2024-12-23}, + langid = {american} +} + +@article{zhaoBeamTrackingUAV2018, + title = {Beam {{Tracking}} for {{UAV Mounted SatCom}} On-the-{{Move With Massive Antenna Array}}}, + author = {Zhao, Jianwei and Gao, Feifei and Wu, Qihui and Jin, Shi and Wu, Yi and Jia, Weimin}, + year = 2018, + month = feb, + journal = {IEEE J. Sel. Areas Commun.}, + volume = {36}, + number = {2}, + pages = {363--375}, + issn = {1558-0008}, + doi = {10.1109/JSAC.2018.2804239}, + urldate = {2024-07-08}, + langid = {american} +} + +@article{zhaoDemandAwareBeamHopping2025, + title = {Demand-{{Aware Beam Hopping}} and {{Power Allocation}} for {{Load Balancing}} in {{Digital Twin Empowered LEO Satellite Networks}}}, + author = {Zhao, Ruili and Cai, Jun and Luo, Jiangtao and Gao, Junpeng and Ran, Yongyi}, + year = 2025, + month = jun, + journal = {IEEE Transactions on Wireless Communications}, + volume = {24}, + number = {6}, + pages = {5084--5098}, + issn = {1558-2248}, + doi = {10.1109/TWC.2025.3545745}, + urldate = {2025-09-01} +} + +@article{zhaoDynamicResourceAllocation2024, + title = {Dynamic {{Resource Allocation}} for {{Multi-Satellite Cooperation Networks}}: {{A Decentralized Scheme Under Statistical CSI}}}, + shorttitle = {Dynamic {{Resource Allocation}} for {{Multi-Satellite Cooperation Networks}}}, + author = {Zhao, Meihui and Yu, Hanxiao and Pan, Jianxiong and Jin, Yifeng and Lv, Guocheng and Jin, Ye}, + year = 2024, + journal = {IEEE Access}, + volume = {12}, + pages = {15419--15437}, + issn = {2169-3536}, + doi = {10.1109/ACCESS.2024.3356058}, + urldate = {2025-03-03} +} + +@article{zhaoGenerativeAISecure2025, + title = {Generative {{AI}} for {{Secure Physical Layer Communications}}: {{A Survey}}}, + shorttitle = {Generative {{AI}} for {{Secure Physical Layer Communications}}}, + author = {Zhao, Changyuan and Du, Hongyang and Niyato, Dusit and Kang, Jiawen and Xiong, Zehui and Kim, Dong In and Shen, Xuemin and Letaief, Khaled B.}, + year = 2025, + month = feb, + journal = {IEEE Trans. Cogn. Commun. Netw.}, + volume = {11}, + number = {1}, + pages = {3--26}, + issn = {2332-7731}, + doi = {10.1109/TCCN.2024.3438379}, + urldate = {2025-03-08}, + langid = {american} +} + +@article{zhaoIntegratedOTFSNOMAFramework2025, + title = {An {{Integrated OTFS-NOMA Framework}} for {{Multi-Beam LEO Systems}}: {{Reliability}} and {{Capacity Analysis}}}, + shorttitle = {An {{Integrated OTFS-NOMA Framework}} for {{Multi-Beam LEO Systems}}}, + author = {Zhao, Xiaohui and Lei, Lei and Wei, Zhiqiang and Fang, Hai and Wang, Wenjie and Chatzinotas, Symeon}, + year = 2025, + journal = {IEEE Transactions on Wireless Communications}, + pages = {1--1}, + issn = {1558-2248}, + doi = {10.1109/TWC.2025.3594792}, + urldate = {2025-08-25} +} + +@article{zhaoMultiSatelliteCooperativeLoadBalancing2024, + title = {Multi-{{Satellite Cooperative Load-Balancing Scheme Based}} on {{Dynamic Beam Coverage}} for {{LEO Beam Hopping Systems}}}, + author = {Zhao, Xiaoyan and Wang, Cheng and Cai, Songsong and Chen, Ruoqi and Wen, Jiarui and Xu, Lexi and Wang, Weidong}, + year = 2024, + month = oct, + journal = {IEEE Wireless Communications Letters}, + volume = {13}, + number = {10}, + pages = {2892--2896}, + issn = {2162-2345}, + doi = {10.1109/LWC.2024.3452115}, + urldate = {2025-09-01} +} + +@article{zhaoOutageConstrainedRobust2021, + title = {Outage {{Constrained Robust Secure Beamforming}} in {{Cognitive Satellite-Aerial Networks}}}, + author = {Zhao, Bai and Lin, Min and Cheng, Ming and Zhu, Wei-Ping and {Al-Dhahir}, Naofal}, + year = 2021, + month = aug, + journal = {IEEE Commun. Lett.}, + volume = {25}, + number = {8}, + pages = {2708--2712}, + issn = {1558-2558}, + doi = {10.1109/LCOMM.2021.3077686}, + urldate = {2025-01-16}, + langid = {american} +} + +@article{zhaoRobustFrameBasedBeamforming2024, + title = {Robust {{Frame-Based Beamforming Scheme}} to {{Enhance Secure Multigroup Multicast Transmission}} for {{Multibeam Satellite Systems}}}, + author = {Zhao, Bai and Lin, Min and Li, Fei and Cheng, Ming and {Al-Dhahir}, Naofal}, + year = 2024, + month = mar, + journal = {IEEE Trans. Veh. Technol.}, + volume = {73}, + number = {3}, + pages = {4407--4411}, + issn = {1939-9359}, + doi = {10.1109/TVT.2023.3324437}, + urldate = {2024-12-19}, + langid = {american} +} + +@article{zhaoTimeVaryingChannel2018, + title = {Time {{Varying Channel Tracking With Spatial}} and {{Temporal BEM}} for {{Massive MIMO Systems}}}, + author = {Zhao, Jianwei and Xie, Hongxiang and Gao, Feifei and Jia, Weimin and Jin, Shi and Lin, Hai}, + year = 2018, + month = aug, + journal = {IEEE Trans. Wireless Commun.}, + volume = {17}, + number = {8}, + pages = {5653--5666}, + issn = {1558-2248}, + doi = {10.1109/TWC.2018.2848259}, + urldate = {2024-12-20} +} + +@phdthesis{ZhengBaiFuDuoGuanCeMoXingWeiBuYouHuaQianXiShuXinHaoChongGouFangFaYanJiu2023, + type = {{博士学位论文}}, + title = {{多观测模型尾部优化欠稀疏信号重构方法研究}}, + author = {{郑佰福}}, + year = 2023, + doi = {10.27389/d.cnki.gxadu.2022.000176}, + urldate = {2024-03-11}, + collaborator = {{李世东} and {廖桂生}}, + langid = {chinese}, + school = {西安电子科技大学} +} + +@article{zhengCooperativeMultiSatelliteMultiRIS2024, + title = {Cooperative {{Multi-Satellite}} and {{Multi-RIS Beamforming}}: {{Enhancing LEO SatCom}} and {{Mitigating LEO-GEO Intersystem Interference}}}, + shorttitle = {Cooperative {{Multi-Satellite}} and {{Multi-RIS Beamforming}}}, + author = {Zheng, Ziyuan and Jing, Wenpeng and Lu, Zhaoming and Wu, Qingqing and Zhang, Haijun and Gesbert, David}, + year = 2024, + journal = {IEEE J. Sel. Areas Commun.}, + pages = {1--1}, + issn = {1558-0008}, + doi = {10.1109/JSAC.2024.3460068}, + urldate = {2024-12-19}, + langid = {american} +} + +@article{zhengCovarianceMatrixReconstruction2018, + title = {Covariance {{Matrix Reconstruction}} with {{Interference Steering Vector}} and {{Power Estimation}} for {{Robust Adaptive Beamforming}}}, + author = {Zheng, Zhi and Zheng, Yan and Wang, Wen-Qin and Zhang, Hongbo}, + year = 2018, + month = sep, + journal = {IEEE Trans. Veh. Technol.}, + volume = {67}, + number = {9}, + pages = {8495--8503}, + issn = {1939-9359}, + doi = {10.1109/TVT.2018.2849646}, + urldate = {2024-01-03}, + langid = {english} +} + +@article{zhengDiversityMultiplexingFundamental2003, + title = {Diversity and {{Multiplexing}}: {{A Fundamental Tradeoff}} in {{Multiple-Antenna Channels}}}, + shorttitle = {Diversity and {{Multiplexing}}}, + author = {Zheng, Lizhong and Tse, D.N.C.}, + year = 2003, + month = may, + journal = {IEEE Trans. Inf. Theory}, + volume = {49}, + number = {5}, + pages = {1073--1096}, + issn = {1557-9654}, + doi = {10.1109/TIT.2003.810646}, + urldate = {2024-07-09}, + langid = {american} +} + +@article{zhengDualDrivenLeaning2024, + title = {Dual {{Driven Leaning}} for {{Joint Activity Detection}} and {{Channel Estimation}} in {{Multibeam LEO Satellite Communications}}}, + author = {Zheng, Shuntian and Wu, Sheng and Jia, Haoge and Zhao, Jingjing and Shi, Yuanming and Jiang, Chunxiao}, + year = 2024, + month = oct, + journal = {IEEE Journal of Selected Topics in Signal Processing}, + volume = {18}, + number = {7}, + pages = {1194--1209}, + issn = {1941-0484}, + doi = {10.1109/JSTSP.2024.3461308}, + urldate = {2025-08-25} +} + +@article{zhengIntelligentReflectingSurfaceAided2022, + title = {Intelligent {{Reflecting Surface-Aided LEO Satellite Communication}}: {{Cooperative Passive Beamforming}} and {{Distributed Channel Estimation}}}, + shorttitle = {Intelligent {{Reflecting Surface-Aided LEO Satellite Communication}}}, + author = {Zheng, Beixiong and Lin, Shaoe and Zhang, Rui}, + year = 2022, + month = oct, + journal = {IEEE J. Sel. Areas Commun.}, + volume = {40}, + number = {10}, + pages = {3057--3070}, + issn = {1558-0008}, + doi = {10.1109/JSAC.2022.3196119}, + urldate = {2024-06-11}, + langid = {english} +} + +@article{zhengMassiveMIMOChannel2014, + title = {Massive {{MIMO Channel Models}}: {{A Survey}}}, + shorttitle = {Massive {{MIMO Channel Models}}}, + author = {Zheng, Kan and Ou, Suling and Yin, Xuefeng}, + year = 2014, + journal = {Int. J. Antennas Propag.}, + volume = {2014}, + number = {1}, + pages = {848071}, + issn = {1687-5877}, + doi = {10.1155/2014/848071}, + urldate = {2024-07-12}, + copyright = {Copyright \copyright{} 2014 Kan Zheng et al.}, + langid = {english} +} + +@article{zhengMultiAntennaCovertCommunications2019, + title = {Multi-{{Antenna Covert Communications}} in {{Random Wireless Networks}}}, + author = {Zheng, Tong-Xing and Wang, Hui-Ming and Ng, Derrick Wing Kwan and Yuan, Jinhong}, + year = 2019, + month = mar, + journal = {IEEE Trans. Wireless Commun.}, + volume = {18}, + number = {3}, + pages = {1974--1987}, + issn = {1558-2248}, + doi = {10.1109/TWC.2019.2900915}, + urldate = {2024-06-24}, + langid = {american} +} + +@inproceedings{zhengMultigatewayCooperationMultibeam2012, + title = {Multi-Gateway Cooperation in Multibeam Satellite Systems}, + booktitle = {2012 {{IEEE}} 23rd {{International Symposium}} on {{Personal}}, {{Indoor}} and {{Mobile Radio Communications}} - ({{PIMRC}})}, + author = {Zheng, Gan and Chatzinotas, Symeon and Ottersten, Bj{\"o}rn}, + year = 2012, + month = sep, + pages = {1360--1364}, + issn = {2166-9589}, + doi = {10.1109/PIMRC.2012.6362558}, + urldate = {2025-02-27} +} + +@article{zhengPhysicalLayerSecurity2012, + title = {Physical {{Layer Security}} in {{Multibeam Satellite Systems}}}, + author = {Zheng, Gan and Arapoglou, Pantelis-Daniel and Ottersten, Bjorn}, + year = 2012, + month = feb, + journal = {IEEE Trans. Wireless Commun.}, + volume = {11}, + number = {2}, + pages = {852--863}, + issn = {1558-2248}, + doi = {10.1109/TWC.2011.120911.111460}, + urldate = {2024-12-12} +} + +@inproceedings{zhengRISAidedLEOSatCom2024, + title = {{{RIS-Aided LEO SatCom}} with {{LEO-GEO Inter-System Interference Mitigation}}: {{Joint Multi-Satellite Multi-RIS Beamforming}}}, + shorttitle = {{{RIS-Aided LEO SatCom}} with {{LEO-GEO Inter-System Interference Mitigation}}}, + booktitle = {2024 {{IEEE Wireless Communications}} and {{Networking Conference}} ({{WCNC}})}, + author = {Zheng, Ziyuan and Jing, Wenpeng and Lu, Zhaoming and Wen, Xiangming and Li, Wei}, + year = 2024, + month = apr, + pages = {1--6}, + issn = {1558-2612}, + doi = {10.1109/WCNC57260.2024.10570610}, + urldate = {2025-08-24} +} + +@article{zhengSparseArrayBeamforming2021, + title = {Sparse {{Array Beamforming Design}} for {{Coherently Distributed Sources}}}, + author = {Zheng, Zhi and Fu, Yueping and Wang, Wen-Qin}, + year = 2021, + month = may, + journal = {IEEE Trans. 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Surveys Tuts.}, + volume = {17}, + number = {3}, + pages = {1738--1760}, + issn = {1553-877X}, + doi = {10.1109/COMST.2015.2425294}, + urldate = {2024-07-12}, + langid = {american} +} + +@article{zhengTrafficAwareResourceManagement2024, + title = {Traffic-{{Aware Resource Management}} of {{Beam Hopping}} in {{Satellite-Enabled Internet}} of {{Things}}}, + author = {Zheng, Shuang and Zhang, Xing and Zhang, Jaixin and Wang, Peng and Wang, Wenbo}, + year = 2024, + month = nov, + journal = {IEEE Internet of Things Journal}, + volume = {11}, + number = {21}, + pages = {34504--34518}, + issn = {2327-4662}, + doi = {10.1109/JIOT.2024.3432901}, + urldate = {2025-08-26} +} + +@article{zhongLEOSatelliteMultibeam2021, + title = {{{LEO Satellite Multibeam Coverage Area Division}} and {{Beamforming Method}}}, + author = {Zhong, Kai Yuan and Cheng, Yu Jian and Yang, Hai Ning and Zheng, Bin}, + year = 2021, + month = nov, + journal = {IEEE Antennas Wireless Propag. Lett.}, + volume = {20}, + number = {11}, + pages = {2115--2119}, + issn = {1548-5757}, + doi = {10.1109/LAWP.2021.3099104}, + urldate = {2024-07-28}, + langid = {american} +} + +@article{zhouAccurateDOAEstimation2020, + title = {Accurate {{DOA Estimation With Adjacent Angle Power Difference}} for {{Indoor Localization}}}, + author = {Zhou, Xin and Chen, Liang and Yan, Jun and Chen, Ruizhi}, + year = 2020, + journal = {IEEE Access}, + volume = {8}, + pages = {44702--44713}, + issn = {2169-3536}, + doi = {10.1109/ACCESS.2020.2977371}, + urldate = {2024-07-11} +} + +@article{zhouActiveTerminalIdentification2023, + title = {Active {{Terminal Identification}}, {{Channel Estimation}}, and {{Signal Detection}} for {{Grant-Free NOMA-OTFS}} in {{LEO Satellite Internet-of-Things}}}, + author = {Zhou, Xingyu and Ying, Keke and Gao, Zhen and Wu, Yongpeng and Xiao, Zhenyu and Chatzinotas, Symeon and Yuan, Jinhong and Ottersten, Bj{\"o}rn}, + year = 2023, + month = apr, + journal = {IEEE Trans. Wireless Commun.}, + volume = {22}, + number = {4}, + pages = {2847--2866}, + issn = {1558-2248}, + doi = {10.1109/TWC.2022.3214862}, + urldate = {2024-12-23} +} + +@article{zhouDirectionofArrivalEstimationCoprime2018, + title = {Direction-of-{{Arrival Estimation}} for {{Coprime Array}} via {{Virtual Array Interpolation}}}, + author = {Zhou, Chengwei and Gu, Yujie and Fan, Xing and Shi, Zhiguo and Mao, Guoqiang and Zhang, Yimin D.}, + year = 2018, + month = nov, + journal = {IEEE Transactions on Signal Processing}, + volume = {66}, + number = {22}, + pages = {5956--5971}, + issn = {1941-0476}, + doi = {10.1109/TSP.2018.2872012}, + urldate = {2025-12-04}, + langid = {american} +} + +@article{zhouDualPolarizedTightlyCoupled2019, + title = {Dual-{{Polarized Tightly Coupled Dipole Array}} for {{UHF}}--{{X-Band Satellite Applications}}}, + author = {Zhou, Wenyang and Chen, Yikai and Yang, Shiwen}, + year = 2019, + month = mar, + journal = {IEEE Antennas Wireless Propag. Lett.}, + volume = {18}, + number = {3}, + pages = {467--471}, + issn = {1548-5757}, + doi = {10.1109/LAWP.2019.2894348}, + urldate = {2024-08-04}, + langid = {american} +} + +@article{zhouHandoverAnalysisUltraDense2023, + title = {Handover {{Analysis}} in {{Ultra-Dense LEO Satellite Networks With Beamforming Methods}}}, + author = {Zhou, He and Li, Jianguo and Yang, Kai and Zhou, Haibo and An, Jianping and Han, Zhu}, + year = 2023, + month = mar, + journal = {IEEE Trans. Veh. Technol.}, + volume = {72}, + number = {3}, + pages = {3676--3690}, + issn = {1939-9359}, + doi = {10.1109/TVT.2022.3221447}, + urldate = {2025-01-13}, + langid = {american} +} + +@article{zhouMachineLearningBasedResource2021, + title = {Machine {{Learning-Based Resource Allocation}} in {{Satellite Networks Supporting Internet}} of {{Remote Things}}}, + author = {Zhou, Di and Sheng, Min and Wang, Yixin and Li, Jiandong and Han, Zhu}, + year = 2021, + month = oct, + journal = {IEEE Trans. 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Acoust., Speech, Signal Process.}, + volume = {36}, + number = {10}, + pages = {1553--1560}, + issn = {0096-3518}, + doi = {10.1109/29.7543}, + urldate = {2023-12-02} +} + +@article{zoltowskiRealTimeFrequency2D1994, + title = {Real-{{Time Frequency}} and 2-{{D Angle Estimation}} with {{Sub-Nyquist Spatio-Temporal Sampling}}}, + author = {Zoltowski, M.D. and Mathews, C.P.}, + year = 1994, + month = oct, + journal = {IEEE Trans. Signal Process.}, + volume = {42}, + number = {10}, + pages = {2781--2794}, + issn = {1941-0476}, + doi = {10.1109/78.324743}, + urldate = {2023-10-17}, + langid = {american}, + lccn = {1} +}

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