fix(app-C): 经典方法空间导向统一为 -j 相位约定
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。
This commit is contained in:
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-15
@@ -2,6 +2,7 @@
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\label{sec:doa_methods}
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\label{sec:doa_methods}
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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。
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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。
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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。
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\subsection{MVDR Algorithm\texorpdfstring{~\cite{caponHighResolutionFrequencyWavenumberSpectrum1969}}{}}
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\subsection{MVDR Algorithm\texorpdfstring{~\cite{caponHighResolutionFrequencyWavenumberSpectrum1969}}{}}
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\label{subsec:mvdr}
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\label{subsec:mvdr}
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@@ -27,9 +28,9 @@ For a uniform linear array (ULA), the steering vector can be expressed as
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=
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=
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\left[
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\left[
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1,\,
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1,\,
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e^{j 2\pi \frac{d}{\lambda_{\mathrm{c}}}\sin\theta},\,
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e^{-j 2\pi \frac{d}{\lambda_{\mathrm{c}}}\sin\theta},\,
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\dots,\,
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\dots,\,
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e^{j 2\pi \frac{d}{\lambda_{\mathrm{c}}}(M-1)\sin\theta}
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e^{-j 2\pi \frac{d}{\lambda_{\mathrm{c}}}(M-1)\sin\theta}
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\right]^{\Tran},
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\right]^{\Tran},
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\label{eq:mvdr_steering_vector}
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\label{eq:mvdr_steering_vector}
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\end{equation}
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\end{equation}
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@@ -149,9 +150,9 @@ For a uniform linear array (ULA), the steering vector corresponding to the angle
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=
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=
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\left[
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\left[
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1,\,
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1,\,
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e^{j 2\pi \frac{d}{\lambda_{\mathrm{c}}}\sin\theta_k},\,
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e^{-j 2\pi \frac{d}{\lambda_{\mathrm{c}}}\sin\theta_k},\,
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\dots,\,
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\dots,\,
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e^{j 2\pi \frac{d}{\lambda_{\mathrm{c}}}(M-1)\sin\theta_k}
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e^{-j 2\pi \frac{d}{\lambda_{\mathrm{c}}}(M-1)\sin\theta_k}
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\right]^{\Tran},
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\right]^{\Tran},
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\label{eq:music_steering_vector}
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\label{eq:music_steering_vector}
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\end{equation}
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\end{equation}
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@@ -250,9 +251,9 @@ where $\mathbf{A}(\boldsymbol{\theta}) = [\mathbf{a}(\theta_1), \dots, \mathbf{a
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=
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=
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\left[
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\left[
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1,\,
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1,\,
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e^{j \mu_k},\,
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e^{-j \mu_k},\,
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\dots,\,
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\dots,\,
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e^{j (M-1)\mu_k}
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e^{-j (M-1)\mu_k}
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\right]^{\Tran},
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\right]^{\Tran},
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\label{eq:esprit_steering_vector}
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\label{eq:esprit_steering_vector}
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\end{equation}
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\end{equation}
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@@ -302,7 +303,7 @@ where
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=
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=
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\operatorname{diag}
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\operatorname{diag}
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\left(
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\left(
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e^{j\mu_1},\dots,e^{j\mu_K}
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e^{-j\mu_1},\dots,e^{-j\mu_K}
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\right)
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\right)
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\label{eq:esprit_phi}
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\label{eq:esprit_phi}
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\end{equation}
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\end{equation}
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@@ -357,7 +358,7 @@ where
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\mathbf{T}^{-1} \bm{\Phi} \mathbf{T}.
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\mathbf{T}^{-1} \bm{\Phi} \mathbf{T}.
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\label{eq:esprit_psi}
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\label{eq:esprit_psi}
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\end{equation}
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\end{equation}
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Therefore, $\bm{\Psi}$ is similar to $\bm{\Phi}$ and has the same eigenvalues $\{e^{j\mu_k}\}_{k=1}^{K}$.
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Therefore, $\bm{\Psi}$ is similar to $\bm{\Phi}$ and has the same eigenvalues $\{e^{-j\mu_k}\}_{k=1}^{K}$.
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In practice, $\bm{\Psi}$ can be estimated by solving the least-squares problem
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In practice, $\bm{\Psi}$ can be estimated by solving the least-squares problem
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\begin{equation}
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\begin{equation}
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@@ -380,7 +381,7 @@ where $(\cdot)^{\dagger}$ denotes the Moore--Penrose pseudoinverse.
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Let $\hat{\kappa}_k$ denote the $k$th eigenvalue of $\widehat{\bm{\Psi}}$. Then the spatial frequency is estimated as
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Let $\hat{\kappa}_k$ denote the $k$th eigenvalue of $\widehat{\bm{\Psi}}$. Then the spatial frequency is estimated as
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\begin{equation}
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\begin{equation}
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\hat{\mu}_k = \angle(\hat{\kappa}_k),
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\hat{\mu}_k = -\angle(\hat{\kappa}_k),
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\qquad k = 1,\dots,K,
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\qquad k = 1,\dots,K,
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\label{eq:esprit_mu_estimate}
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\label{eq:esprit_mu_estimate}
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\end{equation}
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\end{equation}
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@@ -420,9 +421,9 @@ where $\zeta_{k,\ell} \in \mathbb{C}$ is the complex gain of the $k$th source in
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=
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=
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\left[
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\left[
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1,\,
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1,\,
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e^{j 2\pi \frac{d}{\lambda_{\mathrm{c}}}\sin\theta_k},\,
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e^{-j 2\pi \frac{d}{\lambda_{\mathrm{c}}}\sin\theta_k},\,
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\dots,\,
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\dots,\,
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e^{j 2\pi \frac{d}{\lambda_{\mathrm{c}}}(M-1)\sin\theta_k}
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e^{-j 2\pi \frac{d}{\lambda_{\mathrm{c}}}(M-1)\sin\theta_k}
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\right]^\Tran,
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\right]^\Tran,
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\label{eq:spatial_steering}
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\label{eq:spatial_steering}
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\end{equation}
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\end{equation}
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@@ -557,9 +558,9 @@ where $\mathbf{a}(\theta_k)\in\mathbb{C}^{M\times 1}$ and $\mathbf q_k(\nu_k)\in
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=
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=
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\left[
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\left[
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1,\,
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1,\,
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e^{j\mu_k},\,
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e^{-j\mu_k},\,
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\dots,\,
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\dots,\,
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e^{j(M-1)\mu_k}
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e^{-j(M-1)\mu_k}
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\right]^{\Tran},
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\right]^{\Tran},
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\label{eq:st_esprit_spatial_steering}
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\label{eq:st_esprit_spatial_steering}
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\end{equation}
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\end{equation}
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@@ -713,7 +714,7 @@ where
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=
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=
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\operatorname{diag}
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\operatorname{diag}
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\left(
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\left(
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e^{j\mu_1},\dots,e^{j\mu_K}
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e^{-j\mu_1},\dots,e^{-j\mu_K}
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\right),
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\right),
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\label{eq:st_esprit_Phi_s}
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\label{eq:st_esprit_Phi_s}
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\end{equation}
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\end{equation}
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@@ -803,7 +804,7 @@ where $(\cdot)^{\dagger}$ denotes the Moore--Penrose pseudoinverse. Let $\hat{\k
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\begin{equation}
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\begin{equation}
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\hat{\mu}_k
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\hat{\mu}_k
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=
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=
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\angle(\hat{\kappa}_{s,k}),
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-\angle(\hat{\kappa}_{s,k}),
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\qquad
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\qquad
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\hat{\nu}_k
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\hat{\nu}_k
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=
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=
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