From ad9e9364de0ae6b98c24625a568ed4f01fe842b0 Mon Sep 17 00:00:00 2001
From: Josh Gao <josh_gao@foxmail.com>
Date: Mon, 15 Jun 2026 19:53:56 +0800
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附录 C 原用 e^{+j} 空间导向，与正文 e^{-j} 约定相反，
同名 a(θ) 在正文/附录含义不一致。统一改为 e^{-j}：
MVDR/MUSIC/ST-MUSIC 谱对该符号取实、峰位不变；
ESPRIT/ST-ESPRIT 同步翻 Φ 与本征值，并把空间频率反解
改为 μ̂=-∠(κ̂)，保证 θ̂=arcsin(λμ̂/2πd) 结果不变。
时间导向 q_k 仍保留无导频纯 Fourier 形式，已加注说明
其与正文含导频 q_k 的差异、属附录局部记号。
---
 parts/C-classicalDoa.tex | 31 ++++++++++++++++---------------
 1 file changed, 16 insertions(+), 15 deletions(-)

diff --git a/parts/C-classicalDoa.tex b/parts/C-classicalDoa.tex
index bb0eaa5..7f16609 100644
--- a/parts/C-classicalDoa.tex
+++ b/parts/C-classicalDoa.tex
@@ -2,6 +2,7 @@
 \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；综合性的子空间方法回顾可参见~\cite{krimTwoDecadesArray1996,vantreesOptimumArrayProcessing2002}。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。
+本附录的空间导向矢量已与正文统一采用 $e^{-j(\cdot)}$ 相位约定；但空时方法中的时间导向矢量 $\mathbf q_k(\nu_k)$ 取无导频加权的纯 Fourier 形式 $[1,e^{j\nu_k},\ldots,e^{j(T-1)\nu_k}]^\Tran$（经典子空间法不假设已知导频），与正文第~\ref{sec:det_doa_doppler_est} 节含已知导频 $s_k(t)$ 的同名 $\mathbf q_k$ 含义不同，仅在本附录局部使用。
 
 \subsection{MVDR Algorithm\texorpdfstring{~\cite{caponHighResolutionFrequencyWavenumberSpectrum1969}}{}}
 \label{subsec:mvdr}
@@ -27,9 +28,9 @@ For a uniform linear array (ULA), the steering vector can be expressed as
 =
 \left[
 1,\,
-e^{j 2\pi \frac{d}{\lambda_{\mathrm{c}}}\sin\theta},\,
+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}
+e^{-j 2\pi \frac{d}{\lambda_{\mathrm{c}}}(M-1)\sin\theta}
 \right]^{\Tran},
 \label{eq:mvdr_steering_vector}
 \end{equation}
@@ -149,9 +150,9 @@ For a uniform linear array (ULA), the steering vector corresponding to the angle
 =
 \left[
 1,\,
-e^{j 2\pi \frac{d}{\lambda_{\mathrm{c}}}\sin\theta_k},\,
+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}
+e^{-j 2\pi \frac{d}{\lambda_{\mathrm{c}}}(M-1)\sin\theta_k}
 \right]^{\Tran},
 \label{eq:music_steering_vector}
 \end{equation}
@@ -250,9 +251,9 @@ where $\mathbf{A}(\boldsymbol{\theta}) = [\mathbf{a}(\theta_1), \dots, \mathbf{a
 =
 \left[
 1,\,
-e^{j \mu_k},\,
+e^{-j \mu_k},\,
 \dots,\,
-e^{j (M-1)\mu_k}
+e^{-j (M-1)\mu_k}
 \right]^{\Tran},
 \label{eq:esprit_steering_vector}
 \end{equation}
@@ -302,7 +303,7 @@ where
 =
 \operatorname{diag}
 \left(
-e^{j\mu_1},\dots,e^{j\mu_K}
+e^{-j\mu_1},\dots,e^{-j\mu_K}
 \right)
 \label{eq:esprit_phi}
 \end{equation}
@@ -357,7 +358,7 @@ where
 \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 $\{e^{j\mu_k}\}_{k=1}^{K}$.
+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}
@@ -380,7 +381,7 @@ 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),
+\hat{\mu}_k = -\angle(\hat{\kappa}_k),
 \qquad k = 1,\dots,K,
 \label{eq:esprit_mu_estimate}
 \end{equation}
@@ -420,9 +421,9 @@ where $\zeta_{k,\ell} \in \mathbb{C}$ is the complex gain of the $k$th source in
 =
 \left[
 1,\,
-e^{j 2\pi \frac{d}{\lambda_{\mathrm{c}}}\sin\theta_k},\,
+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}
+e^{-j 2\pi \frac{d}{\lambda_{\mathrm{c}}}(M-1)\sin\theta_k}
 \right]^\Tran,
 \label{eq:spatial_steering}
 \end{equation}
@@ -557,9 +558,9 @@ where $\mathbf{a}(\theta_k)\in\mathbb{C}^{M\times 1}$ and $\mathbf q_k(\nu_k)\in
 =
 \left[
 1,\,
-e^{j\mu_k},\,
+e^{-j\mu_k},\,
 \dots,\,
-e^{j(M-1)\mu_k}
+e^{-j(M-1)\mu_k}
 \right]^{\Tran},
 \label{eq:st_esprit_spatial_steering}
 \end{equation}
@@ -713,7 +714,7 @@ where
 =
 \operatorname{diag}
 \left(
-e^{j\mu_1},\dots,e^{j\mu_K}
+e^{-j\mu_1},\dots,e^{-j\mu_K}
 \right),
 \label{eq:st_esprit_Phi_s}
 \end{equation}
@@ -803,7 +804,7 @@ where $(\cdot)^{\dagger}$ denotes the Moore--Penrose pseudoinverse. Let $\hat{\k
 \begin{equation}
 \hat{\mu}_k
 =
-\angle(\hat{\kappa}_{s,k}),
+-\angle(\hat{\kappa}_{s,k}),
 \qquad
 \hat{\nu}_k
 =