## 4 Online Computation of SSC-MVDR Algorithm  
The computational complexity of an algorithm is one of the important factors we need to consider when applying the algorithm to the actual project. In the SSC-MVDR algorithm (i.e., Equation (10)), the inverse of the sample covariance matrix and the corresponding matrix operations are the main calculation steps. In the following, the online computation process of the SSC-MVDR algorithm is given and a way to reduce the amount of calculation is provided.  
First, the iterative calculation process of the sample covariance matrix is as follows  
\[\hat{\mathbf{R}}(k+1)=\gamma\hat{\mathbf{R}}(k)+\frac{1}{k+1}\mathbf{x}(k+1) \mathbf{x}^{H}(k+1), \tag{11}\]  
where \(\hat{\mathbf{R}}(k+1)\) and \(\hat{\mathbf{R}}(k)\) represent the sample covariance matrices at times \(k\) and \(k+1\), respectively. \(\gamma\) is a constant less than 1 but very close to 1. When the array snapshots are non-stationary, the coefficient \(\gamma\) is used to ensure that the SSC-MVDR algorithm still works reliably. Using the Woodbury matrix identity, the inverse of the sample covariance matrix at time \(k+1\) can be expressed as,  
\[\hat{\mathbf{R}}^{-1}(k+1)=\gamma^{-1}(\hat{\mathbf{R}}^{-1}(k)-\frac{ \mathbf{y}(k+1)\mathbf{y}^{H}(k+1)}{(k+1)\gamma+\mathbf{x}^{H}(k+1)\mathbf{y} (k+1)}), \tag{12}\]  
where \(\mathbf{y}(k+1)\) represents the product of the inverse of the sample covariance matrix at time \(k\) and the array snapshot at time \(k+1\), that is, \(\mathbf{y}(k+1)=\hat{\mathbf{R}}^{-1}(k)\mathbf{x}(k+1)\).  
Second, the symbol \(g\) is introduced to represent the generalized inner product of the steering vectors with respect to the inverse of the sample covariance matrix,  
\[g_{k+1}(\theta,\theta_{i})=\bar{\mathbf{a}}^{H}(\theta)\hat{\mathbf{R}}^{-1}( k+1)\bar{\mathbf{a}}(\theta_{i}), \tag{13}\]  
Substituting Equation (12) into Equation (13), the generalized inner product \(g\) can be equivalently expressed as,  
\[g_{k+1}(\theta,\theta_{i})=\gamma^{-1}(g_{k}(\theta,\theta_{i})-q_{k+1}( \theta,\theta_{i})), \tag{14}\]  
where \(q_{k+1}(\theta,\theta_{i})\) is defined as  
\[q_{k+1}(\theta,\theta_{i})=\frac{\bar{\mathbf{a}}^{H}(\theta)\mathbf{y}(k+1) \mathbf{y}^{H}(k+1)\bar{\mathbf{a}}(\theta_{i})}{(k+1)\gamma+\mathbf{x}^{H}( k+1)\mathbf{y}(k+1)}. \tag{15}\]  
Finally, the pseudo spatial power spectrum at time \(k+1\) can be calculated by substituting Equation (14) into Equation (10), that is,  
\[P_{\text{SSC-MVDR}}^{k+1}(\theta)=1/|\sum_{i=1}^{L}|\frac{g_{k}(\theta,\theta _{i})-q_{k+1}(\theta,\theta_{i})}{g_{k}(\theta,\theta)-q_{k+1}(\theta,\theta) }|e^{i(2\pi/L)i}|,\quad\theta\in\Theta. \tag{16}\]  
It can be seen from Equation (16) that the calculation of the pseudo spatial power spectrum at time \(k+1\) depends on the results of two functions, which are the generalized inner product at time \(k\) and the function \(q\) at time \(k+1\), respectively. Please note that the former is known during the calculation process at time \(k+1\) (because it has been obtained in the previous calculation), and the latter is calculated as Equation (15). Since it only involves vector operations, its computational complexity is relatively small.