阵列信号处理中协方差矩阵为什么重要
如下图所示为阵列探测示意图，其中目标到达角定义为信号入射方向与阵列法线之间的夹角，在窄带(点信源)和远场(平面波)的假设下，同一信号到达不同阵元存在波程差，这个波程差导致了各接收阵元的相位差，该相位差组成阵列的导向矢量和阵列流型矩阵。
图1 阵列信号探测示意图
设阵元数为\(M\)，信源数为\(N\)，第n个信源的到达角为\(\theta_n\)则其对应的导向矢量可以表示为\(\boldsymbol{a}(\theta_n)=[1,e^{-i2\pi d{\rm sin}(\theta_n)/\lambda},...,e^{-i2\pi (M-1)d{\rm sin}(\theta_N)/\lambda}]^T\)，所有信源组成的阵列流型矩阵可以表示为
\[\begin{equation}
\begin{aligned}
\boldsymbol{A}(\boldsymbol{\theta})&=[\boldsymbol{a}(\theta_0),...,\boldsymbol{a}(\theta_p),...,\boldsymbol{a}(\theta_{N-1})]\\
&=
\begin{bmatrix}
&1,&1,&...,&1\\
&e^{-i2\pi d{\rm sin}(\theta_0)/\lambda},&e^{-i2\pi d{\rm sin}(\theta_1)/\lambda},&...,&e^{-i2\pi d{\rm sin}(\theta_{N-1})/\lambda}\\
&\vdots,&\vdots,&\vdots,&\vdots\\
&e^{-i2\pi (M-1)d{\rm sin}(\theta_0)/\lambda},&e^{-i2\pi (M-1)d{\rm sin}(\theta_1)/\lambda},&...,&e^{-i2\pi (M-1)d{\rm sin}(\theta_{N-1})/\lambda}
\end{bmatrix}
\end{aligned}
\end{equation}\tag{1}
\]
从(1)可以看出，\(\boldsymbol{A}(\boldsymbol{\theta})\)是一个范德蒙矩阵，且当\(M>=N\)，即信源数小于阵列维数时，阵列流型矢量是非奇异的。
所以阵列信号的接收模型可以表示为
\[\boldsymbol{x}(t)=\boldsymbol{A}(\boldsymbol{\theta})\boldsymbol{s}(t)+\boldsymbol{n}(t)\tag{2}
\]
其中\(\boldsymbol{s}(t)=[s_0(t),...,s_{N-1}(t)]^T\)，\(\boldsymbol{n}(t)=[n_0(t),...,n_{M-1}(t)]^T\)。计算阵列接收信号的协方差矩阵如下
\[\begin{equation}
\begin{aligned}
\boldsymbol{R}_x&={\rm E}[\boldsymbol{x}(t)\boldsymbol{x}^H(t)]=\boldsymbol{A}{\rm E}[\boldsymbol{s}(t)\boldsymbol{s}^H(t)]\boldsymbol{A}^H+{\rm E}[\boldsymbol{n}(t)\boldsymbol{n}^H(t)]=\boldsymbol{A}\boldsymbol{R}_s \boldsymbol{A}^H+\sigma^2 \boldsymbol{I}
\end{aligned}
\end{equation}\tag{3}
\]
对协方差矩阵进行奇异值分解可以得到
\[\boldsymbol{R}_x=\boldsymbol{A}\boldsymbol{R}_s \boldsymbol{A}^H+\sigma^2 \boldsymbol{I}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{U}^H=[\boldsymbol{U}_s,\boldsymbol{U}_n]\begin{bmatrix}\boldsymbol{\Sigma}_s,\boldsymbol{O}\\ \boldsymbol{O},\boldsymbol{\Sigma}_n\end{bmatrix} \begin{bmatrix}\boldsymbol{U}_s^H\\\boldsymbol{U}_n^H\end{bmatrix}\tag{4}
\]
其中\(\boldsymbol{\Sigma}_s={\rm diag}(\sigma_1^2+\sigma^2,...,\sigma_{N}^2+\sigma^2)={\rm diag}(\lambda_1,...,\lambda_N)\)，\(\boldsymbol{\Sigma}_n=\sigma^2 \boldsymbol{I}\)，令\(\boldsymbol{U}=[\boldsymbol{u}_1,...,\boldsymbol{u}_N, |\boldsymbol{u}_{N+1},...,\boldsymbol{u}_M]\)，则\(\boldsymbol{U}_s=[\boldsymbol{u}_1,...,\boldsymbol{u}_N]\)，\(\boldsymbol{U}_n=[\boldsymbol{u}_{N+1},...,\boldsymbol{u}_M]\)。根据奇异值分解的性质可知，\(\boldsymbol{U}\)是一个酉矩阵，\(\lbrace\boldsymbol{u}_1,...,\boldsymbol{u}_M\rbrace\)构成了一组标准正交基，即\(\boldsymbol{u}_i^H\boldsymbol{u}_i=1\)，\(\boldsymbol{u}_i^H\boldsymbol{u}_j=0,i\neq j\)，基于此，下面来证明一些结果：
(1) \(\boldsymbol{U}_s^H \boldsymbol{U}_s=\boldsymbol{I},\boldsymbol{U}_n^H \boldsymbol{U}_n=\boldsymbol{I}\)
证明：
\[\begin{equation}
\begin{aligned}
\boldsymbol{U}_s^H \boldsymbol{U}_s&=\begin{bmatrix}\boldsymbol{u}_1^H\\ \vdots\\ \boldsymbol{u}_N^H\end{bmatrix} [\boldsymbol{u}_1,...,\boldsymbol{u}_N]=\begin{bmatrix}&\boldsymbol{u}_1^H \boldsymbol{u}_1,&...,&\boldsymbol{u}_1^H \boldsymbol{u}_N \\ &\vdots,&...,&\vdots\\ &\boldsymbol{u}_N^H \boldsymbol{u}_1,&...,&\boldsymbol{u}_N^H \boldsymbol{u}_N\end{bmatrix}=\boldsymbol{I}\\
\boldsymbol{U}_n^H \boldsymbol{U}_n&=\begin{bmatrix}\boldsymbol{u}_{N+1}^H\\ \vdots\\ \boldsymbol{u}_M^H\end{bmatrix} [\boldsymbol{u}_{N+1},...,\boldsymbol{u}_M]=\begin{bmatrix}&\boldsymbol{u}_{N+1}^H \boldsymbol{u}_{N+1},&...,&\boldsymbol{u}_{N+1}^H \boldsymbol{u}_M \\ &\vdots,&...,&\vdots\\ &\boldsymbol{u}_M^H \boldsymbol{u}_{N+1},&...,&\boldsymbol{u}_M^H \boldsymbol{u}_M\end{bmatrix}=\boldsymbol{I}\\
\end{aligned}
\end{equation}
\]
(2) \(\boldsymbol{U}_s \boldsymbol{U}_s^H+\boldsymbol{U}_n \boldsymbol{U}_n^H=\boldsymbol{I}\)
证明：
\[\begin{equation}
\begin{aligned}
\boldsymbol{U}_s \boldsymbol{U}_s^H&=[\boldsymbol{u}_1,...,\boldsymbol{u}_N]\begin{bmatrix}\boldsymbol{u}_1^H\\ \vdots\\ \boldsymbol{u}_N^H\end{bmatrix}=\sum_{i=1}^N \boldsymbol{u}_i \boldsymbol{u}_i^H\\
\boldsymbol{U}_n \boldsymbol{U}_n^H&=[\boldsymbol{u}_{N+1},...,\boldsymbol{u}_M]\begin{bmatrix}\boldsymbol{u}_{N+1}^H\\ \vdots\\ \boldsymbol{u}_M^H\end{bmatrix}=\sum_{i=N+1}^M \boldsymbol{u}_i \boldsymbol{u}_i^H\\
&\boldsymbol{U}_s \boldsymbol{U}_s^H+\boldsymbol{U}_n \boldsymbol{U}_n^H=\sum_{i=1}^M \boldsymbol{u}_i \boldsymbol{u}_i^H
\end{aligned}
\end{equation}
\]
而\(\boldsymbol{u}_i \boldsymbol{u}_i^H=\boldsymbol{C},C_{i,j}=\begin{cases}0,i\neq j\\ 1,i=j\end{cases}\)，所以，显然有\(\boldsymbol{U}_s \boldsymbol{U}_s^H+\boldsymbol{U}_n \boldsymbol{U}_n^H=\boldsymbol{I}\)
(3)\(\boldsymbol{U}_s \boldsymbol{U}_s^H=\boldsymbol{U}_s (\boldsymbol{U}_s^H \boldsymbol{U}_s)^{-1}\boldsymbol{U}_s^H\)，\(\boldsymbol{U}_n \boldsymbol{U}_n^H=\boldsymbol{U}_n (\boldsymbol{U}_n^H \boldsymbol{U}_n)^{-1}\boldsymbol{U}_n^H=\boldsymbol{I}-\boldsymbol{U}_s (\boldsymbol{U}_s^H \boldsymbol{U}_s)^{-1}\boldsymbol{U}_s^H\)
证明：由证明1可得\(\boldsymbol{U}_s^H \boldsymbol{U}_s=\boldsymbol{I},\boldsymbol{U}_n^H \boldsymbol{U}_n=\boldsymbol{I}\)，所以\(\boldsymbol{U}_s \boldsymbol{U}_s^H=\boldsymbol{U}_s (\boldsymbol{U}_s^H \boldsymbol{U}_s)^{-1}\boldsymbol{U}_s^H\)和\(\boldsymbol{U}_n \boldsymbol{U}_n^H=\boldsymbol{U}_n (\boldsymbol{U}_n^H \boldsymbol{U}_n)^{-1}\boldsymbol{U}_n^H\)显然是成立的，再由证明结果2，可以得到\(\boldsymbol{U}_n \boldsymbol{U}_n^H=\boldsymbol{U}_n (\boldsymbol{U}_n^H \boldsymbol{U}_n)^{-1}\boldsymbol{U}_n^H=\boldsymbol{I}-\boldsymbol{U}_s (\boldsymbol{U}_s^H \boldsymbol{U}_s)^{-1}\boldsymbol{U}_s^H\)也是成立的。
由上面的证明结果(3)可知，\(\boldsymbol{U}_s \boldsymbol{U}_s^H\)和\(\boldsymbol{U}_n \boldsymbol{U}_n^H\)分别表示在s空间和n空间上的投影算子，所以\(\boldsymbol{U}_s \boldsymbol{U}_s^H X\)，\(\boldsymbol{U}_n\boldsymbol{U}_n^H X\)分别表示将信号\(\boldsymbol{X}\)往s空间和n空间上进行投影，基于这个性质可以做很多事情。
(4)可以写成
\[\boldsymbol{R}_x=[\boldsymbol{U}_s,\boldsymbol{U}_n]\begin{bmatrix}\boldsymbol{\Sigma}_s,\boldsymbol{O}\\ \boldsymbol{O},\boldsymbol{\Sigma}_n\end{bmatrix} \begin{bmatrix}\boldsymbol{U}_s^H\\\boldsymbol{U}_n^H\end{bmatrix}=\boldsymbol{U}_s\boldsymbol{\Sigma}_s\boldsymbol{U}_s^H+\boldsymbol{U}_n\boldsymbol{\Sigma}_n\boldsymbol{U}_n^H=\sum_{i=1}^N \lambda_i \boldsymbol{u}_i \boldsymbol{u}_i^H+\sum_{i=N+1}^M \sigma^2 \boldsymbol{u}_i \boldsymbol{u}_i^H\tag{5}
\]
对\(\boldsymbol{R}_x\)求逆可以得到
\[\boldsymbol{R}_x^{-1}=\sum_{i=1}^N \frac{1}{\lambda_1} \boldsymbol{u}_i \boldsymbol{u}_i^H+\sum_{i=N+1}^M \frac{1}{\sigma^2} \boldsymbol{u}_i \boldsymbol{u}_i^H\tag{6}
\]
现在考虑杂波抑制前的场景，此时信号中的直达波和多径杂波信号远远强于目标信号和噪声信号，所以上述s空间即为杂波空间，n空间即为目标和噪声空间。由于\(\lambda_1,...,\lambda_N \gg \sigma^2\)，所以可以对(6)进行如下近似
\[\boldsymbol{R}_x^{-1}\approx \sum_{i=N+1}^M \frac{1}{\sigma^2} \boldsymbol{u}_i\boldsymbol{u}_i^H=\frac{1}{\sigma^2}\boldsymbol{U}_n\boldsymbol{U}_n^H\tag{7}
\]
结合上面证明(3)的结论可知，此时\(\boldsymbol{R}_x^{-1}\)等效于n空间的投影算子。所以利用\(\boldsymbol{R}_x^{-1}\)可以将原始信号往能量较弱的信号成分构成的子空间进行投影，可以较好地实现信号分离，正是有这样的性质，在很多空域处理算法中经常能看到\(\boldsymbol{R}_x^{-1}\)的身影。所以\(\boldsymbol{R}_x\)在空域信号处理中具有举足轻重的地位。
参考文献
[1]H. Krim and M. Viberg, “Two decades of array signal processing research: the parametric approach,” IEEE Signal Process. Mag., vol. 13, no. 4, pp. 67–94, 1996.
[2]Wan X, Yi J, Zhao Z, et al. Experimental research for CMMB-based passive radar under a multipath environment[J]. IEEE Transactions on Aerospace and Electronic Systems, 2014, 50(1): 70-85.
