## 3 SSC-MVDR Algorithm for DOA Estimation
Specifically, the amplitude of the \(i\)-th vector is equal to the \(i\)-th beam response sample value, and the phase of the \(i\)-th vector is equal to \((2\pi/L)i\). If all beam response sample values are equal to 1, then the converted vectors are exactly on the unit circle, so the above method is called the unit circle mapping method. Next, all the unit circle mapped values are summed. In the summation process, the beam response curves of the two different shapes will correspond to significantly different results. Specifically, for a beam response curve segment containing a steep null, the magnitude of the summation result is related to the depth of the null, and the deeper the null, the greater the magnitude of the summation result. The phase of the summation result differs from the phase of the null on the unit circle by approximately 180 degrees. For the relatively flat beam response curve segment, the unit circle mapped values have almost cancelled each other during the summation process, so that the summation result is approximately equal to zero. Therefore, we define the pseudo spatial power spectrum as follows,  
\[P_{\text{SSC-MVDR}}(\theta)=1/|\sum_{i=1}^{L}|\frac{\bar{\mathbf{a}}^{H}( \theta)\bar{\mathbf{R}}^{-1}\bar{\mathbf{a}}(\vartheta_{i})}{\bar{\mathbf{a}} ^{H}(\theta)\bar{\mathbf{R}}^{-1}\bar{\mathbf{a}}(\theta)}|e^{i(2\pi/L)i}|, \quad\theta\in\Theta. \tag{10}\]  
It should be noted that the pseudo spatial power spectrum \(P_{\text{SSC-MVDR}}(\theta)\) is only defined in the direction interval \(\Theta\). When the direction \(\theta\) is exactly equal to the DOA of the desired signal, the pseudo spatial power spectrum is expected to achieve a maximum. In practical applications, to achieve the desired estimation performance of the SSC-MVDR algorithm, a reasonable direction interval should be selected. Generally, the center of the direction interval can be set as the coarse estimate of the DOA of the acoustic source, and the width of the direction interval should be set to a suitable value to ensure that the true DOA of the acoustic source always falls within the direction interval, and meanwhile, the beam response segment in the direction interval contains only the null generated by the signal self-cancellation, and does not contain other unrelated nulls. Therefore, the direction interval width is not only related to the error of the coarse estimation of the DOA, but also to the specific position of the nulls in the beam response. The implementation principle and detailed steps of the SSC-MVDR algorithm are shown in Figure 1. First, the sample covariance matrix is calculated according to the array snapshots, then the beam response of the MVDR beamformer is calculated, and the beam response on the direction interval is mapped to the unit circle, and finally the pseudo spatial power spectrum is calculated. Meanwhile, in Figure 1, examples of the beam response of the MVDR beamformer and examples of the unit circle mapping of the beam response curve segment on the direction interval are also given. Specifically, Figure 1a,b are the results when there is no signal self-cancellation. In the Figure 1a, the direction interval is indicated by two dashed lines, and the beam response curve segment in this direction interval is relatively flat. Therefore, in the Figure 1b, all the mapping vector amplitudes are approximately equal. Moreover, since the phases of the mapping vectors are uniformly distributed in the range of 0 to 360 degrees, the magnitude of the sum of the mapping vectors will be very small. Therefore, the pseudo spatial power spectrum for this case will be very large, as shown by the higher red dashed line in Figure 1e. Figure 1c,d correspond to the case where there is signal self-cancellation. In the Figure 1c, the direction interval is also indicated by two dashed lines, and the beam response curve segment in this direction interval contains a steep null. Therefore, in the Figure 1d, the magnitudes of some mapping vectors are much smaller than the amplitudes of other mapping vectors.