## 2 Method Analysis
### Noise PSD Estimation Algorithm Based on Continuous Spectrum Update  
In order to calculate the initial values \(\widetilde{Y}_{i}(0)\) and \(\widetilde{N}_{i}(0)\), the noise PSD \(\sigma_{i}^{2}(k,l)\) is required. An accurate estimation of the noise PSD substantially affects the quality of the final output signal in distributed MVDR beamformers. The matrix of noise PSD is typically estimated by the temporal averaging of part of the audio with no bird vocalization [10; 20; 21]. The premise is that the noise PSD does not change too much with time [21]. However, noises usually vary with time in nature, which makes it a great challenge to accurately estimate the noise PSD. Compared with the noise estimation algorithm based on voice activity Ddetection (VAD), the algorithm based on continuous updates is more effective. These algorithms are usually based on the minimum statistical model, which consider that the point with the smallest amplitude is the reference value of noise estimation for continuous multiple frames with the same spectral component [22; 23]. Based on this model, the noise PSD can be estimated. However, if only the current frame is searched, this will cause large residual noise when the noise level rises soon afterwards [17]. In this section, an improved noise PSD estimation algorithm is proposed to estimate the noise PSD in each node during IDMVDR. This method learned from the bidirectional search of the path searching algorithm. Firstly, the smoothing operation on the power spectrum of the signal of node \(i\) is performed:  
\[S_{i}(k,l)=\alpha^{\prime}S_{i}(k,l-1)+(1-\alpha^{\prime})\big{|}Y_{i}(k,l) \big{|}^{2}, \tag{26}\]  
where \(S_{i}(k,l)\) denotes the smoothed power spectrum, and \(\alpha^{\prime}\) is the smoothing factor. Then, the bidirectional search is performed on the smoothed power spectrum. In the same spectral component, the front and rear frames are searched to find the smallest amplitude:  
\[S_{imin1}(k,l)=\min\{S_{i}(k,l^{\prime})\},l-L_{s}+1\leq l^{\prime}\leq l, \tag{27}\]  
\[S_{imin2}(k,l)=\min\{S_{i}(k,l^{\prime})\},l\leq l^{\prime}\leq l+L_{s}-1, \tag{28}\]  
where \(S_{imin1}(k,l)\) and \(S_{imin2}(k,l)\) represent the values of forward and backward searches of node \(i\), respectively, and \(L_{s}\) is the number of searched frames. The maximums of \(S_{imin1}\) and \(S_{imin2}\) are taken as the reference values, which can ensure the reduction of noise when the noise level rises by utilizing the noise reference values of later frames:  
\[S_{imin}(k,l)=\max\{S_{imin1}(k,l),S_{imin2}(k,l)\}, \tag{29}\]  
Furthermore, the bird vocalization existence probability is calculated. Considering that the bird vocalization is more susceptible to noise interference in the low-frequency segment, different discrimination thresholds are chosen for signals with different frequency bands:  
\[I_{i}(k,l)=\left\{ \begin{array}{l}1,S_{i}(k,l)/S_{imin}(k,l)>\delta(k)\\ 0,S_{i}(k,l)/S_{imin}(k,l)\leq\delta(k)\end{array} \right., \tag{30}\]  
where \(I_{i}(k,l)\) is used to determine whether bird vocalization is present in time-frame \(l\) and frequency-bin \(k\). When the value of \(I_{i}(k,l)\) is 1, it means that bird vocalization is present, otherwise bird vocalization is absent. \(\delta(k)\) is the frequency-dependent threshold determined experimentally. The existence probability of bird vocalization is further calculated by the following equation:  
\[P_{i}(k,l)=\alpha_{s}P_{i}(k,l-1)+(1-\alpha_{s})I_{i}(k,l), \tag{31}\]  
where \(P_{i}(k,l)\) is the existence probability of bird vocalization, and \(\alpha_{s}\) is smoothing constant. To quickly adapt to the increasing noise levels, the time-frequency dependent smoothing factor is computed based on the existence probability of bird vocalization. Then, the noise estimation will be updated in each frame based on the activity detection of bird vocalization.