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4f9f1b8847
D/E/F/G/H/I/J 每篇开头置一张 mdquote 卡片,统一标出: 对应正文哪节模型(\appref/\ref 一跳、不重写信号模型)、主参数 与干扰参数(定义仍指回正文)、本附录给出的界/推导对象、本附录 局部关键或易混符号(就地补注、末尾指回附录 A 完整索引)。 卡片抬头单独一行,其余字段进 bullet;局部覆盖 itemize 为 leftmargin=1.5em/itemindent=0pt 的悬挂缩进,长字段换行续行对齐 正文下、不跑到 bullet 左。 多为把既有开头标准化进框(H/J 折叠原开头首句、I 卡片下保留 复杂度对比、E 折入 psi 与 FIM/投影记号声明),不新建符号表、不 重写模型。A(本身是索引)、B(通用数学背景)、C(经典方法叙述) 不加。
998 lines
47 KiB
TeX
998 lines
47 KiB
TeX
\section{单阵列 CRB 汇总}
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\label{sec:singleArrayCrb}
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\begin{mdquote}
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\noindent\textbf{本附录设定}：单阵列模型下三种 \gls{crb} 的汇总——确定信号 \gls{doa}、确定信号 \gls{doa}--Doppler、随机信号 \gls{doa}，共享 \gls{fim} 构造 + Schur 补框架，集中便于交叉对照。
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\begin{itemize}[leftmargin=1.5em,itemindent=0pt]
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\item \emph{主参数} \gls{doa} 角（含 Doppler 模型再加 $\nu$）；\emph{干扰参数} 确定信号模型为复增益与噪声方差、随机信号模型为信号与噪声协方差。
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\item \emph{本附录给出} 三种模型的 \gls{fim} 逐元构造与 Schur 补消去干扰参数的闭式 \gls{crb}（通用复 Gauss \gls{crb} 与 Slepian--Bangs 公式见 \cite{kayFundamentalsStatisticalSignal1993,stoicaMusicMaximumLikelihood1989,stoicaPerformanceStudyConditional1990}）；前两者是多星主线的直接基础，随机信号 \gls{doa} 作对照支线保留。
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\item \emph{本附录局部符号} $\mathcal I$：\gls{fim}；$\mathbf I$：单位阵；$\mathcal P$：投影矩阵。完整索引见\appref{sec:assumptionsAndNotation}。
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\end{itemize}
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\end{mdquote}
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\subsection{确定信号 DoA CRB}
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\label{sec:det_doa_crb}
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本附录对应第~\ref{sec:det_doa_est} 节的单阵列确定信号 \gls{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 \gls{fim}并通过 Schur 补消去确定信号快拍等干扰参数。
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\subsubsection{一阶偏导求取}
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\paragraph{对噪声方差的一阶偏导}
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\begin{equation}
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\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.
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\end{equation}
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\paragraph{对信源的一阶偏导}
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\begin{mdquote}
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复向量的偏导数（Wirtinger 微分）：令复数 $z=x+jy$，则其 Wirtinger 微分定义为
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%
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\begin{equation}
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\label{eq:wirtinger_derivative}
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\begin{cases}
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\frac{\partial}{\partial z} = \frac{1}{2}\left(\frac{\partial}{\partial x} - j\frac{\partial}{\partial y} \right)
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\\
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\frac{\partial}{\partial z^*} = \frac{1}{2}\left(\frac{\partial}{\partial x} + j\frac{\partial}{\partial y} \right)
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\end{cases}
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\quad\Rightarrow\quad
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\begin{cases}
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\frac{\partial}{\partial x} = \frac{\partial}{\partial z} + \frac{\partial}{\partial z^*}
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\\
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\frac{\partial}{\partial y} = j\left(\frac{\partial}{\partial z} - \frac{\partial}{\partial z^*} \right)
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\end{cases},
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\end{equation}
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%
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且有
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\begin{subequations}
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\begin{equation}
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\frac{\partial z^*}{\partial z}
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= \frac{1}{2}\left(\frac{\partial}{\partial x} - j\frac{\partial}{\partial y} \right) (x - jy)
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= \frac{1}{2}\left(\frac{\partial x}{\partial x} - j\frac{\partial (-jy)}{\partial y} \right) = 0,
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\end{equation}
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%
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\begin{equation}
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\frac{\partial z}{\partial z^*}
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= \frac{1}{2}\left(\frac{\partial}{\partial x} + j\frac{\partial}{\partial y} \right) (x + jy)
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= \frac{1}{2}\left(\frac{\partial x}{\partial x} + j\frac{\partial (jy)}{\partial y} \right) = 0,
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\end{equation}
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\end{subequations}
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%
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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。
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\end{mdquote}
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首先按 Wirtinger 微分求取信源 $\mathbf{x}_t$ 的偏导
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%
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\begin{align*}
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\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
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= - \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
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\\
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&= \frac{1}{\sigma} \left( \mathbf{y}_t^\Herm \mathbf{A} - \mathbf{x}_t^\Herm \mathbf{A}^\Herm \mathbf{A} \right)
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= \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}.
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\end{align*}
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%
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对其进行维度修正（取转置）\footnote{%
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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
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%
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\begin{equation}
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\label{eq:first_order_derivative_source}
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\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^*.
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\end{equation}
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%
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然后类似求取信源的共轭转置 $\mathbf{x}_t^\Herm $ 的偏导，得
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%
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\begin{equation}
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\label{eq:first_order_derivative_source_hermit}
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\frac{\partial \ln \mathcal{L}}{\partial \mathbf{x}_t^\Herm } = \frac{1}{\sigma} \mathbf{A}^\Herm \mathbf{n}_t.
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\end{equation}
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%
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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
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%
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\begin{equation}
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\frac{\partial \ln \mathcal{L}}{\partial \mathbf{x}_t^{(r)}}
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= \frac{1}{\sigma} \left( \mathbf{A}^\Tran\mathbf{n}_t^* + \mathbf{A}^\Herm \mathbf{n}_t \right)
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= \frac{2}{\sigma} \Re\{\mathbf{A}^\Herm \mathbf{n}_t\},
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\end{equation}%
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%
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\begin{equation}
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\frac{\partial \ln \mathcal{L}}{\partial \mathbf{x}_t^{(i)}}
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= \frac{1}{\sigma} j\left( \mathbf{A}^\Tran\mathbf{n}_t^* - \mathbf{A}^\Herm \mathbf{n}_t \right)
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= \frac{2}{\sigma} \Im\{\mathbf{A}^\Herm \mathbf{n}_t\}.
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\end{equation}
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\paragraph{对角度的一阶偏导}
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% align 不走 $$，紧贴标题会多出一行空白；保留 align 以让编号贴末行，\tightalign 抵消之。
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\tightalign
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\begin{align}
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\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
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\\
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&= \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
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\\
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&= \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].
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\end{align}
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%
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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 \gls{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 \gls{efim}，但最终矩阵维度和 Hadamard 表达需要按角度分量重新展开。}
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%
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\begin{equation}
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\frac{\partial\ln \mathcal{L}}{\partial \bm{\theta}}
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= \begin{bmatrix}
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\frac{\partial}{\partial \theta_1}
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\\
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\vdots
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\\
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\frac{\partial}{\partial \theta_K}
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\end{bmatrix} \ln \mathcal{L}
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= \frac{2}{\sigma} \sum_{t=1}^T \Re\left[ \mathbf{X}^\Herm _t \mathbf{D}^\Herm \mathbf{n}_t \right].
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\end{equation}
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\subsubsection{二阶偏导及其均值}
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\paragraph{噪声方差的二阶偏导及均值}
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\begin{mdquote}
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\vspace{-1em}
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%
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\begin{equation*}
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\mathbb{E} \left[ \mathbf{n}^\Herm_t \mathbf{n}_t \right] = M \sigma.
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\end{equation*}
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%
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\begin{equation*}
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\mathbb{E} \left[ \mathbf{n}^\Herm_t \mathbf{n}_t \mathbf{n}^\Herm_s \mathbf{n}_s \right] = \begin{cases}
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M^2 \sigma^2, & \text{for}\ t\ne s \\
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M(M+1)\sigma^2, & \text{for}\ t = s
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\end{cases}.
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\end{equation*}
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\end{mdquote}
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%
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\begin{align}
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\mathbb{E}\left\{\left[\frac{\partial \ln \mathcal{L}}{\partial \sigma}\right]^2\right\}
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=& \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]
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\nonumber \\
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=& \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]
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\nonumber \\
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=& \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]
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\nonumber \\
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=& \frac{MT}{\sigma^2}.
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\end{align}
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\paragraph{噪声方差和其他参量的二阶混合偏导及均值}
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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
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%
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\begin{equation}
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\mathbb{E}\left[ \mathbf{n}_t^\Herm \mathbf{n}_t \mathbf{n}_s^\Tran \right] = \mathbf{0},
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\end{equation}
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%
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因此，噪声方差和其他参量的混合二阶偏导均为零。
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\paragraph{信源实虚部的二阶偏导、二阶混合偏导及均值}
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\begin{mdquote}
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\begin{enumerate}
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\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
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%
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\begin{equation*}
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\mathbb{E}\left[\mathbf{n}_t\mathbf{n}_s^\Herm \right]
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= \sigma \mathbf I_M \delta_{ts},
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\end{equation*}
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%
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\begin{equation*}
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\mathbb{E}[\mathbf{n}_t\mathbf{n}_s^\Tran] = 0,\quad \forall t,s.
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\end{equation*}
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\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）
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\begin{align*}
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&\Re\{\mathbf{x}\} \Re\{\mathbf{y}^\Tran\} = \frac{1}{2}[\Re\{\mathbf{x} \mathbf{y}^\Tran\} + \Re\{\mathbf{x} \mathbf{y}^\Herm \}],
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\\
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&\Im\{\mathbf{x}\} \Im\{\mathbf{y}^\Tran\} = -\frac{1}{2}[\Re\{\mathbf{x} \mathbf{y}^\Tran\} - \Re\{\mathbf{x} \mathbf{y}^\Herm \}],
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\\
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&\Re\{\mathbf{x}\} \Im\{\mathbf{y}^\Tran\} = \frac{1}{2}[\Im\{\mathbf{x} \mathbf{y}^\Tran\} - \Im\{\mathbf{x} \mathbf{y}^\Herm \}].
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\end{align*}
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\end{enumerate}
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\end{mdquote}
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%
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\begin{subequations}
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\begin{align}
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%信源实部的二阶偏导
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\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\}
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&= \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
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\\
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=& \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
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\\
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=& \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]
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= \frac{2}{\sigma^2}\Re\left\{\mathbf{A}^\Herm \mathbb{E}\left[ \mathbf{n}_t \mathbf{n}_s^\Herm \right] \mathbf{A}\right\} \nonumber
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\\
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=& \frac{2}{\sigma} \Re\left[\mathbf{A}^\Herm \mathbf{A}\right]\delta(t-s).
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\end{align}
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%信源实虚部的混合二阶偏导
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\begin{equation}
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\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\}
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= \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\}
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= -\frac{2}{\sigma} \Im\left[\mathbf{A}^\Herm \mathbf{A}\right]\delta(t-s).
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\end{equation}
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%信源虚部的二阶偏导
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\begin{equation}
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\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\}
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= \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\}
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= \frac{2}{\sigma} \Re\left[\mathbf{A}^\Herm \mathbf{A}\right]\delta(t-s).
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\end{equation}
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\end{subequations}
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\paragraph{信源实虚部与角度的混合二阶偏导及均值}
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\begin{subequations}
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\begin{equation}
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\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\}
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= \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\}
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= \frac{2}{\sigma}\Re\left[ \mathbf{A}^\Herm \mathbf{D} \mathbf{X}_t \right].
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\end{equation}
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%
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\begin{equation}
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\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\}
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= \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\}
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= \frac{2}{\sigma}\Im\left[ \mathbf{A}^\Herm \mathbf{D} \mathbf{X}_t \right].
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\end{equation}
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\end{subequations}
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\paragraph{角度的二阶偏导及均值}
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\begin{equation}
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\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\}
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= \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\}
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= \frac{2}{\sigma} \sum_{t=1}^T \Re\left[ \mathbf{X}_t^\Herm \mathbf{D}^\Herm \mathbf{D} \mathbf{X}_t \right].
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\end{equation}
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\subsubsection{Fisher 信息矩阵/CRB 协方差矩阵构造}
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\begin{mdquote}
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\begin{enumerate}
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\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
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\begin{equation}
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\left[ \begin{matrix}
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\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 \gls{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} &\mathcal{I}_{\theta\theta}
|
||
\end{array}
|
||
\right],
|
||
\label{eq:single_det_fim_full}
|
||
\end{align}
|
||
%
|
||
其中 $\text{CRB}(\sigma) = \dfrac{\sigma^2}{MT}$，$\mathbf{H} = \dfrac{2}{\sigma}\mathbf{A}^\Herm \mathbf{A}$，$\bm{\Delta}_t = \dfrac{2}{\sigma} \mathbf{A}^\Herm \mathbf{D} \mathbf{X}_t$，$\mathcal{I}_{\theta\theta} = \dfrac{2}{\sigma}\displaystyle\sum_{t=1}^T \Re\left[ \mathbf{X}_t^\Herm \mathbf{D}^\Herm \mathbf{D} \mathbf{X}_t \right]$。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}。
|
||
|
||
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 \gls{efim}；随后对该 \gls{efim} 求逆，得到 $\bm\theta$ 的 \gls{crb} 协方差矩阵。将 \eqref{eq:single_det_fim_full} 按 $(\bm\theta,\{\mathbf x_t\})$ 重新分块，对应的 \gls{fim} 子块为
|
||
%
|
||
\begin{equation*}
|
||
\mathcal{I}_{\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
|
||
%
|
||
\mathcal{I}_{\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*}
|
||
\mathcal{I}_{\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*}
|
||
%
|
||
因此有
|
||
%
|
||
\begin{align}
|
||
\mathrm{CRB}^{-1}(\bm{\theta}) = \mathcal{I}_{\theta\theta}-\mathcal{I}_{\theta\psi}\mathcal{I}_{\psi\psi}^{-1}\mathcal{I}_{\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}) &= \mathcal{I}_{\theta\theta} - \sum_{t=1}^T \Re\{\bm{\Delta}_t^\Herm \mathbf{G}\bm{\Delta}_t\} \nonumber
|
||
= \frac{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
|
||
\\
|
||
&= \frac{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 \gls{efim} 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 \gls{doa} 的条件 \gls{crb} 基线。后续单阵列 \gls{doa}--Doppler 和多星 \gls{doa}--Doppler 的 \gls{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。
|
||
|
||
\subsection{确定信号 DoA--Doppler CRB}
|
||
\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 \gls{doa}--Doppler 模型，是从纯空间 \gls{crb} 过渡到空时 \gls{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 $\sigma$ 为噪声方差。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 \gls{doa} 情形下的角度向量；若使用二维 \gls{doa} 参数，可将相应导数矩阵按同样方式扩展。
|
||
|
||
|
||
\subsubsection{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 \gls{fim}。
|
||
|
||
\subsubsection{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}
|
||
\frac{\partial\mathbf q_k(\nu_k)}{\partial\nu_k}
|
||
=
|
||
j\mathbf T_d\mathbf q_k(\nu_k).
|
||
\end{equation}
|
||
%
|
||
空时导向列向量对各源自身角度 $\theta_k$ 与 Doppler $\nu_k$ 的一阶导数为
|
||
%
|
||
\begin{equation}
|
||
\frac{\partial\mathbf b_{\mathrm{st},k}}{\partial\theta_k}
|
||
=
|
||
\mathbf q_k(\nu_k)\otimes\frac{\partial\mathbf a(\theta_k)}{\partial\theta_k},
|
||
\qquad
|
||
\frac{\partial\mathbf b_{\mathrm{st},k}}{\partial\nu_k}
|
||
=
|
||
\frac{\partial\mathbf q_k(\nu_k)}{\partial\nu_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[
|
||
\frac{\partial\mathbf b_{\mathrm{st},1}}{\partial\theta_1},\ldots,
|
||
\frac{\partial\mathbf b_{\mathrm{st},K}}{\partial\theta_K}
|
||
\right]
|
||
\in\mathbb C^{MT\times K},
|
||
\end{equation}
|
||
\begin{equation}
|
||
\mathbf B_{\mathrm{st},\nu}
|
||
\triangleq
|
||
\left[
|
||
\frac{\partial\mathbf b_{\mathrm{st},1}}{\partial\nu_1},\ldots,
|
||
\frac{\partial\mathbf b_{\mathrm{st},K}}{\partial\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 $\mathbf D_\zeta=\diag(\zeta_1,\ldots,\zeta_K)$。
|
||
|
||
\subsubsection{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 \gls{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}
|
||
|
||
\subsubsection{仅关心 DoA 或 Doppler 时的 CRB}
|
||
|
||
若同时估计 \gls{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 \gls{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 \gls{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}.
|
||
\label{eq:single_det_doa_crb_schur}
|
||
\end{equation}
|
||
%
|
||
类似地，若只关心 Doppler，并把 \gls{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 \gls{efim} 中对 Doppler 做 Schur 补。
|
||
若 Doppler 已知，则 \gls{doa} 的 \gls{crb} 为 $\mathcal I_{\theta\theta}^{-1}$；若 Doppler 未知，则 \eqref{eq:single_det_doa_crb_schur} 中的 Schur 补体现了 \gls{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 \gls{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 $\dfrac{\partial\mathbf q_k}{\partial\nu_k}=j\mathbf T_d\mathbf q_k$ 作为干扰参数影响 \gls{doa} 的 \gls{crb}。
|
||
|
||
该单阵列空时 \gls{crb} 说明 \gls{doa} 与 Doppler 的耦合会通过 Schur 补造成信息损失。多星静态和动态 \gls{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。
|
||
|
||
\subsection{随机信号 DoA CRB（对照支线）}
|
||
\label{sec:sto_doa_crb}
|
||
本附录对应第~\ref{sec:det_doa_est} 节末尾第~\ref{subsec:sml_contrast} 小节的随机信号 \gls{doa} 对照支线。由\appref{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\appref{sec:det_doa_crb} 的方式，按照第~\ref{subsec:sml_contrast} 小节中给出的随机信号建模方式，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，$\sigma$ 为噪声方差。
|
||
由 $\eqref{eq:schur_complement}$ 的 Schur 补可得 \gls{doa} 的 \gls{efim} 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。
|
||
|
||
\subsubsection{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\},
|
||
\label{eq:single_sml_perturbation_set}
|
||
\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，\eqref{eq:single_sml_perturbation_set} 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。
|
||
|
||
\subsubsection{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。
|
||
|
||
\subsubsection{\texorpdfstring{\gls{efim} 求取}{EFIM 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 \gls{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 $\dfrac{\partial\mathbf a(\bm\theta_k)}{\partial[\bm\theta_k]_r}$；\gls{fim}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} \mathbf c_k + \mathbf{d}_k^\perp,
|
||
\end{equation}
|
||
%
|
||
其中 $\mathbf c_k \triangleq \left( \mathbf{A}^\Herm \mathbf{A} \right)^{-1} \mathbf{A}^\Herm \mathbf{d}_k$，$\mathbf{d}_k^\perp = \mathcal{P}_\mathbf{A}^\perp \mathbf{d}_k$。
|
||
|
||
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( \mathbf c_k \mathbf r_{x,k}^\Herm + \mathbf r_{x,k} \mathbf c_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，\gls{efim} 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 \\
|
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&+ \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}
|
||
|
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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，\gls{efim} 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 \gls{crb} 与\appref{sec:det_doa_crb} 的确定信号 \gls{crb} 共同给出单阵列 \gls{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 \gls{crb} 推导更接近\appref{sec:det_doa_crb} 和\appref{sec:det_doa_doppler_crb} 的处理方式。 |