## 2 Signal Self-Cancellation of MVDR Beamformer  
Assume that the sensor array is a horizontal linear array composed of \(M\) omnidirectional hydrophones. In addition, suppose that there are \(N\) far-field narrowband underwater acoustic signals impinging on the hydrophone array, and their directions of arrival are equal to \(\theta_{1},\theta_{2},...,\theta_{N}\). Let \(\mathbf{x}(k)\) be the array snapshot vector at time \(k\), then it can be expressed as  
\[\mathbf{x}(k)=[\mathbf{a}(\theta_{1}),\mathbf{a}(\theta_{2}),\cdots,\mathbf{a }(\theta_{N})]\left[ \begin{array}{c}s_{1}(k)\\ s_{2}(k)\\ \cdots\\ s_{N}(k)\end{array} \right]+\mathbf{n}(k), \tag{1}\]  
where \(s_{i}(k),i=1,2,\cdots,N\) represents the amplitude of the \(i\)th received underwater acoustic signal at time \(k\), \(\mathbf{n}(k)\) denotes the array noise vector at time \(k\), where the data element on the \(j\)th row corresponds to the recorded noise of the \(j\)th hydrophone, \(j=1,2,\cdots,M\). Besides, \(\mathbf{a}(\theta)\) is the array manifold (or steering vector ) towards direction \(\theta\), which can be formulated as  
\[\mathbf{a}(\theta)=[1,e^{i(2\pi/\lambda)d\sin(\theta)},\cdots,e^{i(M-1)(2\pi/ \lambda)d\sin(\theta)}]^{\mathrm{T}}, \tag{2}\]  
where the superscript T represents the transposition operation, \(\lambda\) denotes the signal wavelength, and \(d\) is the distance between adjacent hydrophones. Beamformer can preserve the received signal impinging from a specific direction while suppressing signals in other directions, that is, with spatial filtering capability. The MVDR beamformer achieves the above objects by solving the convex optimization problem as follows,  
\[\min_{\mathbf{w}}\mathbf{w}^{H}\mathbf{R}_{\mathbf{i+n}}\mathbf{w}\quad s.t. \quad\mathbf{w}^{H}\mathbf{a}(\theta_{d})=1, \tag{3}\]  
where \(\theta_{d}\) is the specified direction (i.e., the direction of the desired signal), \(\mathbf{w}\) denotes the weight vector of MVDR beamformer, and \(\mathbf{R}_{\mathbf{i+n}}\) represents the interference-plus-noise covariance matrix. In Equation (3), the objective function is equal to the power of the interference and noise passing through the beamformer, and the constraint guarantees that the gain of the signal in the specified direction is 1. It should be pointed out that in some practical applications, such as passive sonar detection, the array received snapshot data contains both interference, noise, and the desired signal. Therefore, in this case, it is difficult to directly obtain a covariance matrix only for interference and noise. A simple solution is to replace the interference-plus-noise covariance matrix directly with the sample covariance matrix. Therefore, the optimization problem about the MVDR beamformer needs to be re-expressed as  
\[\min_{\mathbf{w}}\mathbf{w}^{H}\mathbf{\hat{R}}\mathbf{w}\quad s.t.\quad \mathbf{w}^{H}\mathbf{a}(\theta_{d})=1, \tag{4}\]where \(\hat{\mathbf{R}}\) represents the sample covariance matrix, which can be directly calculated from several array snapshot data,  
\[\hat{\mathbf{R}}=\frac{1}{K}\sum_{k=1}^{K}\mathbf{x}(k)\mathbf{x}^{H}(k), \tag{5}\]  
where \(K\) is the number of array snapshots that can be used to calculate the sample covariance matrix. In Equation (4), the objective function is equal to the total power of the desired signal, interference, and noise passing through the beamformer. Please note that when the array snapshot number \(K\) is large enough, the sample covariance matrix is approximately equal to the theoretical signal covariance matrix (i.e., \(\mathbf{R}=\mathrm{E}\{\mathbf{x}(k)\mathbf{x}^{H}(k)\}\)). Furthermore, when the specified direction is completely equal to the DOA of the desired signal, it is easy to prove that the optimization problems in Equations (3) and (4) are equivalent. However, in the practical applications, such as underwater acoustic source localization, even if the number of available array snapshots is sufficient, there are still many unfavorable factors that make the performance of the MVDR beamformer significantly degraded.