## 3 SSC-MVDR Algorithm for DOA Estimation  
According to the analysis in the previous section, we already know that in the ideal case, that is, when there is no steering vector error, the steering vector model of the linear array can be expressed by Equation (2). Through further analysis, we can also find that the direction error, which is one of the unfavorable factors in practical applications, only changes the DOA of the desired signal without changing the steering vector model. However, other unfavorable factors, including random wavefront fluctuations, hydrophone position errors, and hydrophone amplitude phase errors, will affect the representation of the steering vector model. In this paper, it is assumed that the MVDR beamformer has a certain degree of direction error, and the expression of the steering vector model is known. Specifically, the actual steering vector has the following form,  
\[\hat{\mathbf{a}}(\theta)=[1,\alpha_{1}e^{j((2\pi/\lambda)d\sin(\theta)+\phi_{1 })},\cdots,\alpha_{M-1}e^{j((M-1)(2\pi/\lambda)d\sin(\theta)+\phi_{M-1})}]^{ \mathrm{T}}, \tag{7}\]  
where \(\alpha_{i},i=1,2,\cdots,M-1\) and \(\phi_{i},i=1,2,\cdots,M-1\) are known constants, the former representing the amplitude deviation of the steering vector and the latter representing the phase deviation of the steering vector. To propose the SSC-MVDR algorithm for DOA estimation, the following two cases are specifically analyzed. First, when the direction error is not equal to zero, the MVDR beamformer appears to cancel the desired signal, that is, the beam response produces sharp nulls for the desired signal; second, when the direction error is exactly equal to zero, the beam response of the MVDR beamformer produces a main lobe of a certain width for the desired signal. Furthermore, we also assume that the approximate interval of the DOA of the desired signal is known (in fact, a coarse estimate of the DOA of the desired signal can be obtained by conventional beamformer. Even in complex uncertain ocean environment, conventional beamformer still exhibits sufficient robustness). If the direction interval of the desired signal is sufficiently narrow, we will find that in the above two cases, the beam response in the direction interval appears as two different shapes. Specifically, in the first case, the beam response in the direction interval is a curve containing a steep null; and in the second case, a relatively flat curve is obtained by the beam response in the direction interval. Inspired by the above analysis, we first define the direction interval of the desired signal as \(\Theta\), and discretely sample the direction interval to get \(L\) direction samples, that is, \(\vartheta_{i}\in\Theta,i=1,2,\cdots,L\). Then, the beam response of the MVDR beamformer on the above \(L\) direction samples is calculated. \[B(\vartheta_{i})=|\frac{\bar{\mathbf{a}}^{H}(\theta_{d})\bar{\mathbf{R}}^{-1} \bar{\mathbf{a}}(\vartheta_{i})}{\bar{\mathbf{a}}^{H}(\theta_{d})\bar{\mathbf{ R}}^{-1}\bar{\mathbf{a}}(\theta_{d})}|,\quad i=1,2,\cdots,L, \tag{8}\]  
where \(B(\vartheta_{i})\) is the beam response of the MVDR beamformer on the direction \(\vartheta_{i}\). Although the shape of the beam response curve is intuitively distinguishable, how to quickly distinguish the shape of the beam response curve in the direction interval by calculation is a major problem faced by this algorithm. In this paper, we present a unit circle mapping method, whose main idea is to uniformly map all beam response sample values in the direction interval to a unit circle,  
\[B_{m}(\vartheta_{i})=|\frac{\bar{\mathbf{a}}^{H}(\theta_{d})\bar{\mathbf{R}}^{ -1}\bar{\mathbf{a}}(\vartheta_{i})}{\bar{\mathbf{a}}^{H}(\theta_{d})\bar{ \mathbf{R}}^{-1}\bar{\mathbf{a}}(\theta_{d})}|e^{i(2\pi/L)i},\quad i=1,2, \cdots,L, \tag{9}\]  
where \(B_{m}(\vartheta_{i})\) represents the unit circle mapped value of the beam response on the direction \(\vartheta_{i}\). The essence of the unit circle mapping method is to convert a series of scalars into directional vectors.