fix(equation): 与正文连续的公式补 % 隔行占位
按新约定，给紧贴正文（无空行）的 \begin{equation}/\end{equation} 两侧各
插入一行 % 占位，全文共 675 处，覆盖 14 个文件。自成段落（前后留空行）的
公式不加。% 占位不影响渲染，diff 纯为新增 % 行，编译两遍输出不变。
This commit is contained in:
@@ -8,6 +8,7 @@
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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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@@ -20,9 +21,11 @@ Consider a narrowband far-field source model impinging on an $M$-element array.
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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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@@ -34,9 +37,11 @@ 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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@@ -45,7 +50,9 @@ The covariance matrix of the received data is defined as
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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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@@ -56,13 +63,16 @@ In practice, $\mathbf{R}_y$ is replaced by the sample covariance matrix
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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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%
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\begin{equation}
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\min_{\mathbf{w}(\theta)}
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\quad
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@@ -73,7 +83,9 @@ The MVDR beamformer minimizes the output power while maintaining a distortionles
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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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%
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\begin{equation}
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\mathbf{w}_{\mathrm{MVDR}}(\theta)
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=
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@@ -86,6 +98,7 @@ Using the method of Lagrange multipliers, the optimal solution is given by
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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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@@ -94,7 +107,9 @@ P_{\mathrm{MVDR}}(\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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%
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\begin{equation}
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\widehat{P}_{\mathrm{MVDR}}(\theta)
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=
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@@ -103,7 +118,9 @@ In practical implementation, the sample covariance matrix is used, leading to
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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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@@ -120,6 +137,7 @@ The MVDR algorithm adaptively suppresses interference and noise by minimizing th
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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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@@ -132,7 +150,9 @@ Consider the same narrowband far-field array signal model as in \eqref{eq:mvdr_s
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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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@@ -142,9 +162,11 @@ where $\mathbf{y}_t \in \mathbb{C}^{M \times 1}$ is the received array snapshot,
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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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@@ -156,9 +178,11 @@ 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 $\sigma_n^2$, the array covariance matrix is
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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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@@ -169,16 +193,20 @@ Assuming that the source signals are mutually uncorrelated and that the noise is
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\sigma_n^2 \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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@@ -187,7 +215,9 @@ In practice, $\mathbf{R}_y$ is approximated by the sample covariance matrix
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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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@@ -196,9 +226,11 @@ By performing eigenvalue decomposition, one obtains
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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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@@ -206,7 +238,9 @@ Under ideal conditions, the steering vectors associated with the true source dir
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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 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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@@ -216,7 +250,9 @@ P_{\mathrm{MUSIC}}(\theta)
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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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@@ -233,6 +269,7 @@ The MUSIC algorithm achieves high-resolution \gls{doa} estimation by exploiting
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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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@@ -245,7 +282,9 @@ Consider the same narrowband far-field array signal model as in \eqref{eq:mvdr_s
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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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@@ -257,16 +296,20 @@ 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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%
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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 ESPRIT algorithm exploits the translational invariance structure of the array. To this end, two maximally overlapped subarrays are constructed from the original ULA. Define the selection matrices
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%
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\begin{equation}
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\mathbf{J}_1
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=
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@@ -283,21 +326,27 @@ The ESPRIT algorithm exploits the translational invariance structure of the arra
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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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%
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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 ULA, $\mathbf{A}_1$ and $\mathbf{A}_2$ satisfy
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%
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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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@@ -307,9 +356,11 @@ 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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@@ -318,7 +369,9 @@ As in subspace-based methods, the sample covariance matrix is first formed as
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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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@@ -327,12 +380,16 @@ By eigenvalue decomposition,
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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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@@ -343,7 +400,9 @@ Applying the selection matrices to $\mathbf{U}_s$ yields
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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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@@ -351,16 +410,20 @@ Substituting \eqref{eq:esprit_subspace_relation} into \eqref{eq:esprit_subspace_
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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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@@ -370,22 +433,28 @@ In practice, $\bm{\Psi}$ can be estimated by solving the least-squares problem
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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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|
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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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%
|
||||
\begin{equation}
|
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\hat{\theta}_k
|
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=
|
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@@ -403,6 +472,7 @@ Compared with MUSIC, the ESPRIT algorithm avoids explicit spectral search and di
|
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\label{subsec:st_music}
|
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|
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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}
|
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=
|
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@@ -415,7 +485,9 @@ Consider a narrowband far-field source model observed by an $M$-element array ov
|
||||
\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)
|
||||
=
|
||||
@@ -427,7 +499,9 @@ e^{-j 2\pi \frac{d}{\lambda_{\mathrm{c}}}(M-1)\sin\theta_k}
|
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\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)
|
||||
=
|
||||
@@ -439,9 +513,11 @@ 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
|
||||
@@ -454,7 +530,9 @@ By vectorizing \eqref{eq:st_music_block_model}, one obtains
|
||||
\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
|
||||
@@ -462,9 +540,11 @@ where $\mathbf{n}_{\ell} \triangleq \vect(\mathbf{N}_{\ell})$ and
|
||||
\in \mathbb{C}^{MT \times 1}
|
||||
\label{eq:st_steering_vector}
|
||||
\end{equation}
|
||||
%
|
||||
is the spatio-temporal steering vector, with $\otimes$ denoting the Kronecker product. Hence, the joint angle-Doppler estimation problem can be recast as a standard subspace estimation problem in the augmented spatio-temporal domain.
|
||||
|
||||
Assuming that the source signals are mutually uncorrelated and the noise is spatially and temporally white with variance $\sigma_n^2$, the covariance matrix of $\mathbf{y}_{\ell}$ is given by
|
||||
%
|
||||
\begin{equation}
|
||||
\mathbf{R}_y
|
||||
=
|
||||
@@ -475,7 +555,9 @@ Assuming that the source signals are mutually uncorrelated and the noise is spat
|
||||
\sigma_n^2 \mathbf{I}_{MT},
|
||||
\label{eq:covariance_st}
|
||||
\end{equation}
|
||||
%
|
||||
where
|
||||
%
|
||||
\begin{equation}
|
||||
\mathbf{B}_{\mathrm{st}}
|
||||
=
|
||||
@@ -487,9 +569,11 @@ where
|
||||
\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
|
||||
=
|
||||
@@ -498,7 +582,9 @@ In practice, $\mathbf{R}_y$ is estimated from $L$ data blocks as
|
||||
\mathbf{y}_{\ell}\mathbf{y}_{\ell}^\Herm.
|
||||
\label{eq:sample_covariance}
|
||||
\end{equation}
|
||||
%
|
||||
Performing eigenvalue decomposition yields
|
||||
%
|
||||
\begin{equation}
|
||||
\widehat{\mathbf{R}}_y
|
||||
=
|
||||
@@ -507,13 +593,17 @@ Performing eigenvalue decomposition yields
|
||||
\mathbf{U}_n \boldsymbol{\Lambda}_n \mathbf{U}_n^\Herm,
|
||||
\label{eq:eig_decomp}
|
||||
\end{equation}
|
||||
%
|
||||
where $\mathbf{U}_s \in \mathbb{C}^{MT \times K}$ and $\mathbf{U}_n \in \mathbb{C}^{MT \times (MT-K)}$ denote the signal and noise subspaces, respectively. Under ideal conditions, each true spatio-temporal steering vector lies in the signal subspace and is therefore orthogonal to the noise subspace, i.e.,
|
||||
%
|
||||
\begin{equation}
|
||||
\mathbf{U}_n^\Herm\mathbf b_{\mathrm{st},k}(\theta_k,\nu_k) = \mathbf{0},
|
||||
\qquad k=1,\dots,K.
|
||||
\label{eq:orthogonality_condition}
|
||||
\end{equation}
|
||||
%
|
||||
Based on this orthogonality property, the spatio-temporal MUSIC pseudospectrum is constructed as
|
||||
%
|
||||
\begin{equation}
|
||||
P_{\mathrm{ST\text{-}MUSIC}}(\theta,\nu)
|
||||
=
|
||||
@@ -523,7 +613,9 @@ P_{\mathrm{ST\text{-}MUSIC}}(\theta,\nu)
|
||||
\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)
|
||||
=
|
||||
@@ -540,6 +632,7 @@ Compared with conventional MUSIC, the spatio-temporal MUSIC algorithm exploits b
|
||||
\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}
|
||||
=
|
||||
@@ -552,7 +645,9 @@ Consider the same spatio-temporal signal model as in \eqref{eq:st_music_block_mo
|
||||
\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)
|
||||
=
|
||||
@@ -564,14 +659,18 @@ 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)
|
||||
=
|
||||
@@ -583,9 +682,11 @@ 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
|
||||
@@ -598,7 +699,9 @@ By vectorizing \eqref{eq:st_esprit_block_model}, one obtains
|
||||
\mathbf{n}_{\ell},
|
||||
\label{eq:st_esprit_vec_model}
|
||||
\end{equation}
|
||||
%
|
||||
where
|
||||
%
|
||||
\begin{equation}
|
||||
\mathbf b_{\mathrm{st},k}(\theta_k,\nu_k)
|
||||
=
|
||||
@@ -606,7 +709,9 @@ where
|
||||
\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}}
|
||||
=
|
||||
@@ -620,6 +725,7 @@ and $\mathbf{n}_{\ell}=\vect(\mathbf{N}_{\ell})$. Define the spatio-temporal man
|
||||
\end{equation}
|
||||
|
||||
The sample covariance matrix is estimated as
|
||||
%
|
||||
\begin{equation}
|
||||
\widehat{\mathbf{R}}_y
|
||||
=
|
||||
@@ -628,7 +734,9 @@ The sample covariance matrix is estimated as
|
||||
\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
|
||||
=
|
||||
@@ -637,13 +745,16 @@ Performing eigenvalue decomposition yields
|
||||
\mathbf{U}_n \boldsymbol{\Lambda}_n \mathbf{U}_n^{\Herm},
|
||||
\label{eq:st_esprit_evd}
|
||||
\end{equation}
|
||||
%
|
||||
where $\mathbf{U}_s\in\mathbb{C}^{MT\times K}$ denotes the estimated signal subspace. Since $\mathbf{U}_s$ and $\mathbf{B}_{\mathrm{st}}$ span the same column space, there exists a nonsingular matrix $\mathbf{T}\in\mathbb{C}^{K\times K}$ such that
|
||||
%
|
||||
\begin{equation}
|
||||
\mathbf{U}_s = \mathbf{B}_{\mathrm{st}}\mathbf{T}.
|
||||
\label{eq:st_esprit_subspace_relation}
|
||||
\end{equation}
|
||||
|
||||
The spatio-temporal ESPRIT algorithm exploits the shift invariance structure in both the spatial and temporal dimensions. First, define the spatial selection matrices
|
||||
%
|
||||
\begin{equation}
|
||||
\mathbf{J}_{s,1}
|
||||
=
|
||||
@@ -660,7 +771,9 @@ The spatio-temporal ESPRIT algorithm exploits the shift invariance structure in
|
||||
\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}
|
||||
=
|
||||
@@ -679,6 +792,7 @@ and the temporal selection matrices
|
||||
\end{equation}
|
||||
|
||||
Correspondingly, the augmented selection matrices acting on the vectorized spatio-temporal observation are constructed as
|
||||
%
|
||||
\begin{equation}
|
||||
\widetilde{\mathbf{J}}_{s,1}
|
||||
=
|
||||
@@ -689,7 +803,9 @@ Correspondingly, the augmented selection matrices acting on the vectorized spati
|
||||
\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}
|
||||
=
|
||||
@@ -702,13 +818,16 @@ and
|
||||
\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
|
||||
=
|
||||
@@ -718,14 +837,18 @@ 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
|
||||
=
|
||||
@@ -737,6 +860,7 @@ e^{j\nu_1},\dots,e^{j\nu_K}
|
||||
\end{equation}
|
||||
|
||||
Applying the same selection operations to the signal subspace $\mathbf{U}_s$ gives
|
||||
%
|
||||
\begin{equation}
|
||||
\mathbf{U}_{s,1}^{(s)}
|
||||
=
|
||||
@@ -747,7 +871,9 @@ Applying the same selection operations to the signal subspace $\mathbf{U}_s$ giv
|
||||
\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)}
|
||||
=
|
||||
@@ -758,28 +884,36 @@ and
|
||||
\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
|
||||
=
|
||||
@@ -788,6 +922,7 @@ with
|
||||
\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
|
||||
=
|
||||
@@ -800,7 +935,9 @@ The matrices $\bm{\Psi}_s$ and $\bm{\Psi}_t$ can be estimated in the least-squar
|
||||
\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
|
||||
=
|
||||
@@ -812,7 +949,9 @@ where $(\cdot)^{\dagger}$ denotes the Moore--Penrose pseudoinverse. Let $\hat{\k
|
||||
\qquad k=1,\dots,K,
|
||||
\label{eq:st_esprit_mu_nu_estimation}
|
||||
\end{equation}
|
||||
%
|
||||
where $\angle(\cdot)$ denotes the phase angle of a complex number. Finally, the \gls{doa} estimate and, if needed, the corresponding physical Doppler conversion are given by
|
||||
%
|
||||
\begin{equation}
|
||||
\hat{\theta}_k
|
||||
=
|
||||
|
||||
Reference in New Issue
Block a user