josh
a395fbe7de
preamble 已用 \DeclareMathOperator*{\argmax}{arg\,max} 定义算子，应整体使用
而非 \arg\max（后者下标只落在 max 下、且非整体算子）。全文 12 处统一为
\argmax/\argmin。
972 lines
26 KiB
TeX
972 lines
26 KiB
TeX
\section{常用 DoA 估计方法}
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\label{sec:doa_methods}
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本附录汇总 MVDR、\gls{music}、\gls{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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\label{subsec:mvdr}
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Consider a narrowband far-field source model impinging on an $M$-element array. The received signal at snapshot $t$ is given by
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%
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\begin{equation}
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\mathbf{y}_t
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=
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\sum_{k=1}^{K}
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\mathbf{a}(\theta_k) s_k(t)
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+
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\mathbf{n}_t
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=
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\mathbf{A}(\boldsymbol{\theta}) \mathbf{s}_t + \mathbf{n}_t,
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\qquad t = 1,\dots,T,
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\label{eq:mvdr_signal_model}
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\end{equation}
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%
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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 $\theta$.
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For a uniform linear array (ULA), the steering vector can be expressed as
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%
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\begin{equation}
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\mathbf{a}(\theta)
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=
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\left[
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1,\,
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e^{-j 2\pi \frac{d}{\lambda_{\mathrm{c}}}\sin\theta},\,
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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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\right]^{\Tran},
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\label{eq:mvdr_steering_vector}
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\end{equation}
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%
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where $d$ is the inter-element spacing and $\lambda_{\mathrm{c}}$ is the carrier wavelength.
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The covariance matrix of the received data is defined as
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%
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\begin{equation}
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\mathbf{R}_y
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=
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\mathbb{E}\left\{
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\mathbf{y}_t \mathbf{y}_t^{\Herm}
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\right\}.
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\label{eq:mvdr_covariance}
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\end{equation}
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%
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In practice, $\mathbf{R}_y$ is replaced by the sample covariance matrix
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%
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\begin{equation}
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\widehat{\mathbf{R}}_y
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=
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\frac{1}{T}
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\sum_{t=1}^{T}
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\mathbf{y}_t \mathbf{y}_t^{\Herm}.
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\label{eq:mvdr_sample_covariance}
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\end{equation}
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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
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%
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\begin{equation}
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z_t(\theta)
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=
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\mathbf{w}^{\Herm}(\theta)\mathbf{y}_t.
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\label{eq:mvdr_output}
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\end{equation}
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%
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The MVDR beamformer minimizes the output power while maintaining a distortionless response in the look direction. Accordingly, the weight vector is obtained by solving
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\begin{equation}
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\min_{\mathbf{w}(\theta)}
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\quad
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\mathbf{w}^{\Herm}(\theta)\mathbf{R}_y\mathbf{w}(\theta)
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\qquad
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\text{s.t.}
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\quad
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\mathbf{w}^{\Herm}(\theta)\mathbf{a}(\theta)=1.
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\label{eq:mvdr_optimization}
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\end{equation}
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%
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Using the method of Lagrange multipliers, the optimal solution is given by
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\begin{equation}
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\mathbf{w}_{\mathrm{MVDR}}(\theta)
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=
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\frac{
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\mathbf{R}_y^{-1}\mathbf{a}(\theta)
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}{
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\mathbf{a}^{\Herm}(\theta)\mathbf{R}_y^{-1}\mathbf{a}(\theta)
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}.
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\label{eq:mvdr_weight}
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\end{equation}
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Substituting \eqref{eq:mvdr_weight} into the beamformer output power yields
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%
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\begin{equation}
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P_{\mathrm{MVDR}}(\theta)
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=
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\frac{1}{
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\mathbf{a}^{\Herm}(\theta)\mathbf{R}_y^{-1}\mathbf{a}(\theta)
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}.
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\label{eq:mvdr_spectrum}
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\end{equation}
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%
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In practical implementation, the sample covariance matrix is used, leading to
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\begin{equation}
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\widehat{P}_{\mathrm{MVDR}}(\theta)
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=
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\frac{1}{
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\mathbf{a}^{\Herm}(\theta)\widehat{\mathbf{R}}_y^{-1}\mathbf{a}(\theta)
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}.
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\label{eq:mvdr_sample_spectrum}
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\end{equation}
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%
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The \gls{doa} estimates are then obtained by locating the $K$ dominant peaks of the MVDR spatial spectrum, namely,
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%
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\begin{equation}
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\hat{\theta}_k
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=
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\argmax_{\theta}
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\widehat{P}_{\mathrm{MVDR}}(\theta),
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\qquad k=1,\dots,K.
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\label{eq:mvdr_estimation}
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\end{equation}
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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.
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\subsection{MUSIC Algorithm\texorpdfstring{~\cite{schmidtMultipleEmitterLocation1986}}{}}
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\label{subsec:music}
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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
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%
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\begin{equation}
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\mathbf{y}_t
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=
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\sum_{k=1}^{K}
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\mathbf{a}(\theta_k) s_k(t)
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+
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\mathbf{n}_t
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=
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\mathbf{A}(\boldsymbol{\theta}) \mathbf{s}_t + \mathbf{n}_t,
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\qquad t = 1,\dots,T,
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\label{eq:music_signal_model}
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\end{equation}
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%
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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
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%
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\begin{equation}
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\mathbf{A}(\boldsymbol{\theta})
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=
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\left[
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\mathbf{a}(\theta_1),\dots,\mathbf{a}(\theta_K)
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\right]
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\in \mathbb{C}^{M \times K}
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\label{eq:music_array_manifold}
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\end{equation}
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%
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is the array manifold matrix.
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For a uniform linear array (ULA), the steering vector corresponding to the angle $\theta_k$ is given by
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%
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\begin{equation}
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\mathbf{a}(\theta_k)
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=
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\left[
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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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\dots,\,
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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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\label{eq:music_steering_vector}
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\end{equation}
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%
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where $d$ denotes the inter-element spacing and $\lambda_{\mathrm{c}}$ is the carrier wavelength.
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Assuming that the source signals are mutually uncorrelated and that the noise is spatially white with variance $\sigma$, the array covariance matrix is
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\begin{equation}
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\mathbf{R}_y
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=
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\mathbb{E}\{\mathbf{y}_t \mathbf{y}_t^{\Herm}\}
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=
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\mathbf{A}(\boldsymbol{\theta}) \mathbf{R}_s \mathbf{A}^{\Herm}(\boldsymbol{\theta})
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+
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\sigma \mathbf{I}_M,
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\label{eq:music_covariance}
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\end{equation}
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%
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where
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%
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\begin{equation}
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\mathbf{R}_s
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=
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\mathbb{E}\{\mathbf{s}_t \mathbf{s}_t^{\Herm}\}
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\label{eq:music_source_covariance}
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\end{equation}
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%
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is the source covariance matrix.
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In practice, $\mathbf{R}_y$ is approximated by the sample covariance matrix
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%
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\begin{equation}
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\widehat{\mathbf{R}}_y
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=
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\frac{1}{T}
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\sum_{t=1}^{T}
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\mathbf{y}_t \mathbf{y}_t^{\Herm}.
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\label{eq:music_sample_covariance}
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\end{equation}
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%
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By performing eigenvalue decomposition, one obtains
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%
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\begin{equation}
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\widehat{\mathbf{R}}_y
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=
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\mathbf{U}_s \boldsymbol{\Lambda}_s \mathbf{U}_s^{\Herm}
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+
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\mathbf{U}_n \boldsymbol{\Lambda}_n \mathbf{U}_n^{\Herm},
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\label{eq:music_evd}
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\end{equation}
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%
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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.
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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,
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%
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\begin{equation}
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\mathbf{U}_n^{\Herm} \mathbf{a}(\theta_k)
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=
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\mathbf{0},
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\qquad k=1,\dots,K.
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\label{eq:music_orthogonality}
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\end{equation}
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%
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Based on this orthogonality property, the \gls{music} pseudospectrum is constructed as
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%
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\begin{equation}
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P_{\mathrm{MUSIC}}(\theta)
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=
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\frac{1}
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{\mathbf{a}^{\Herm}(\theta)\,
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\mathbf{U}_n \mathbf{U}_n^{\Herm}\,
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\mathbf{a}(\theta)}.
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\label{eq:music_spectrum}
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\end{equation}
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%
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The \gls{doa} estimates are then obtained by locating the $K$ dominant peaks of the pseudospectrum, i.e.,
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%
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\begin{equation}
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\hat{\theta}_k
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=
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\argmax_{\theta}
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P_{\mathrm{MUSIC}}(\theta),
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\qquad k=1,\dots,K.
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\label{eq:music_estimation}
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\end{equation}
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The \gls{music} algorithm achieves high-resolution \gls{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.
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\subsection{ESPRIT Algorithm\texorpdfstring{~\cite{royESPRITEstimationSignalParameters1989}}{}}
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\label{subsec:esprit}
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Consider the same narrowband far-field array signal model as in \eqref{eq:mvdr_signal_model}, i.e.,
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%
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\begin{equation}
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\mathbf{y}_t
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=
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\sum_{k=1}^{K}
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\mathbf{a}(\theta_k) s_k(t)
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+
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\mathbf{n}_t
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=
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\mathbf{A}(\boldsymbol{\theta}) \mathbf{s}_t + \mathbf{n}_t,
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\qquad t = 1,\dots,T,
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\label{eq:esprit_signal_model}
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\end{equation}
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%
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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
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%
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\begin{equation}
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\mathbf{a}(\theta_k)
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=
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\left[
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1,\,
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e^{-j \mu_k},\,
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\dots,\,
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e^{-j (M-1)\mu_k}
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\right]^{\Tran},
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\label{eq:esprit_steering_vector}
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\end{equation}
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%
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where
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\begin{equation}
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\mu_k
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=
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2\pi \frac{d}{\lambda_{\mathrm{c}}} \sin\theta_k,
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\label{eq:esprit_spatial_frequency}
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\end{equation}
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%
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with $d$ denoting the inter-element spacing and $\lambda_{\mathrm{c}}$ the carrier wavelength.
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The \gls{esprit} algorithm exploits the translational invariance structure of the array. To this end, two maximally overlapped subarrays are constructed from the original \gls{ula}. Define the selection matrices
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\begin{equation}
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\mathbf{J}_1
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=
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\left[
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\mathbf{I}_{M-1} \ \mathbf{0}
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\right]
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\in \mathbb{C}^{(M-1)\times M},
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\qquad
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\mathbf{J}_2
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=
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\left[
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\mathbf{0} \ \mathbf{I}_{M-1}
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\right]
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\in \mathbb{C}^{(M-1)\times M}.
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\label{eq:esprit_selection_matrices}
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\end{equation}
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%
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Then the corresponding subarray manifold matrices are
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\begin{equation}
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\mathbf{A}_1 = \mathbf{J}_1 \mathbf{A},
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\qquad
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\mathbf{A}_2 = \mathbf{J}_2 \mathbf{A}.
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\label{eq:esprit_subarray_manifolds}
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\end{equation}
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%
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Due to the shift invariance of the \gls{ula}, $\mathbf{A}_1$ and $\mathbf{A}_2$ satisfy
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\begin{equation}
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\mathbf{A}_2
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=
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\mathbf{A}_1 \bm{\Phi},
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\label{eq:esprit_shift_invariance}
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\end{equation}
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%
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where
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%
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\begin{equation}
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\bm{\Phi}
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=
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\operatorname{diag}
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\left(
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e^{-j\mu_1},\dots,e^{-j\mu_K}
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\right)
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\label{eq:esprit_phi}
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\end{equation}
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%
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is the rotational invariance matrix.
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As in subspace-based methods, the sample covariance matrix is first formed as
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%
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\begin{equation}
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\widehat{\mathbf{R}}_y
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=
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\frac{1}{T}
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\sum_{t=1}^{T}
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\mathbf{y}_t \mathbf{y}_t^{\Herm}.
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\label{eq:esprit_sample_covariance}
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\end{equation}
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%
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By eigenvalue decomposition,
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%
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\begin{equation}
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\widehat{\mathbf{R}}_y
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=
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\mathbf{U}_s \boldsymbol{\Lambda}_s \mathbf{U}_s^{\Herm}
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+
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\mathbf{U}_n \boldsymbol{\Lambda}_n \mathbf{U}_n^{\Herm},
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\label{eq:esprit_evd}
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\end{equation}
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%
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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
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%
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\begin{equation}
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\mathbf{U}_s = \mathbf{A} \mathbf{T}.
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\label{eq:esprit_subspace_relation}
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\end{equation}
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%
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Applying the selection matrices to $\mathbf{U}_s$ yields
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%
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\begin{equation}
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\mathbf{U}_{s,1}
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=
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\mathbf{J}_1 \mathbf{U}_s,
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\qquad
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\mathbf{U}_{s,2}
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=
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\mathbf{J}_2 \mathbf{U}_s.
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\label{eq:esprit_subspace_partitions}
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\end{equation}
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%
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Substituting \eqref{eq:esprit_subspace_relation} into \eqref{eq:esprit_subspace_partitions} and using \eqref{eq:esprit_shift_invariance}, one obtains
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%
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\begin{equation}
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\mathbf{U}_{s,2}
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=
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\mathbf{U}_{s,1}\,
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\bm{\Psi},
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\label{eq:esprit_invariance_subspace}
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\end{equation}
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%
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where
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%
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\begin{equation}
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\bm{\Psi}
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=
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\mathbf{T}^{-1} \bm{\Phi} \mathbf{T}.
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\label{eq:esprit_psi}
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\end{equation}
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%
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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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%
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\begin{equation}
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\widehat{\bm{\Psi}}
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=
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\argmin_{\bm{\Psi}}
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\left\|
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\mathbf{U}_{s,2} - \mathbf{U}_{s,1}\bm{\Psi}
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\right\|_{\mathrm{F}}^2,
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\label{eq:esprit_ls_problem}
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\end{equation}
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%
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whose closed-form solution is
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%
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\begin{equation}
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\widehat{\bm{\Psi}}
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=
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\mathbf{U}_{s,1}^{\dagger}\mathbf{U}_{s,2},
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\label{eq:esprit_ls_solution}
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\end{equation}
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%
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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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%
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\begin{equation}
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\hat{\mu}_k = -\angle(\hat{\kappa}_k),
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\qquad k = 1,\dots,K,
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\label{eq:esprit_mu_estimate}
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\end{equation}
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%
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where $\angle(\cdot)$ denotes the phase angle of a complex number. Finally, the \gls{doa} estimate is obtained from
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%
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\begin{equation}
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\hat{\theta}_k
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=
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\arcsin\left(
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\frac{\lambda_{\mathrm{c}}}{2\pi d}\hat{\mu}_k
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\right),
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\qquad k = 1,\dots,K.
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\label{eq:esprit_theta_estimate}
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\end{equation}
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Compared with \gls{music}, the \gls{esprit} algorithm avoids explicit spectral search and directly retrieves the \gls{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\texorpdfstring{~\cite{waxSpatioTemporalSpectralAnalysis1984}}{}}
|
|
\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 (\gls{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 $\sigma$, 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 \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 \gls{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 \gls{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)
|
|
=
|
|
\argmax_{\theta,\nu}
|
|
P_{\mathrm{ST\text{-}MUSIC}}(\theta,\nu),
|
|
\qquad k=1,\dots,K.
|
|
\label{eq:st_music_estimation}
|
|
\end{equation}
|
|
|
|
Compared with conventional \gls{music}, the spatio-temporal \gls{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\texorpdfstring{~\cite{royESPRITEstimationSignalParameters1989,waxSpatioTemporalSpectralAnalysis1984}}{}}
|
|
\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 \gls{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}
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\end{equation}
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%
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where $\angle(\cdot)$ denotes the phase angle of a complex number. Finally, the \gls{doa} estimate and, if needed, the corresponding physical Doppler conversion are given by
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%
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\begin{equation}
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\hat{\theta}_k
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=
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\arcsin\left(
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\frac{\lambda_{\mathrm{c}}}{2\pi d}\hat{\mu}_k
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\right),
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\qquad
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\widehat{\nu}^{\mathrm{Hz}}_k
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=
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\frac{\hat{\nu}_k}{2\pi T_{\mathrm{s}}},
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\qquad k=1,\dots,K.
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\label{eq:st_esprit_theta_fd_estimation}
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\end{equation}
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Compared with spatio-temporal \gls{music}, the spatio-temporal \gls{esprit} algorithm avoids two-dimensional spectral search and directly retrieves the \gls{doa} and Doppler parameters from the rotational invariance structure of the spatio-temporal signal subspace, thereby achieving lower computational complexity.
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|
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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。
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