## 3 Comparison and Discussion
### Convergence Analysis  
Different network topologies of WASNs will cause different convergence and enhancement performances. To evaluate the performance of WASNs of different sizes, four WASNs with \(N_{d}\) = 5, 10, 15, and 20 were simulated. All the nodes were distributed in a \(50\times 50\) m site and the heights were 1 m. Each node consisted of one microphone. The topologies of four simulated WASNs were randomly generated. A bird vocalization source was fixed at a point 5 m high. The noise source was randomly distributed. The topologies of the four simulated WASNs are shown in Figure 3a-d, respectively. In these figures, yellow dots are the node position of WASNs, dark blue dots are the position of the bird vocalization source, blue lines denote the communication connections between nodes. In the simulation experiments, bird audio of a large-billed crow with relatively high signal-to-noise ratio (SNR) was selected as the clean bird vocalization. Its waveform is shown in Figure 4.  
In this section, we only verify the convergence rate of IDMVDR. Before the iterative update of the node value, the estimation of the noise PSD was completed, so the convergence rate of the distributed algorithm had nothing to do with the noise type. A white Gaussian signal was chosen as the only noise source. Each node collected noisy bird audio signals with a sampling frequency of 44.1 kHz. To guarantee the same conditions for comparison experiments, the input SNR of node 1 was set to 1 dB. The audio signal was non-stationary, which should be framed and windowed before audio enhancement. All nodes processed the signals frame by frame, with a frame length of 706 (16 ms), 50% overlap, and Hanning window. During the noise PSD estimation, the first and second frame signals were first initialized through setting \(\sigma_{i}^{2}(k,l)=S_{i}(k,l)=S_{imin}(k,l)=\left|Y_{i}(k,l)\right|\), \(P_{i}(k,l)=0\). Then, the noise PSD estimation was performed, beginning with the second frame, by equations from Equation (26) to Equation (34) sequentially, where \(\alpha^{\prime}=0.8\) (in Equation (26)), \(L_{s}=95\) (in Equations (28) and (29)), \(\alpha_{s}=0.2\) (in Equation (31)), \(\alpha_{d}=0.95\) (in Equation (32)) and \(c=4\) (in Equation (34)). Considering the characteristics of different kinds of bird vocalization are different, relatively appropriate decision thresholds for the bird vocalizations were selected. After statistical analysis, we found that most of the bird vocalizations were in medium and low frequencies. Through lots of experiments with different kinds of birds, the decision threshold of the medium and low frequency segment was set to 2, which makes it easier to confirm that the bird vocalization exists and reduce  
Figure 4: Signal of clean bird vocalization. Figure 3: The topologies of four WASNs with different node numbers: (**a**) 5 nodes; (**b**) 10 nodes; (**c**) 15 nodes; (**d**) 20 nodes. the distortion of bird vocalization in medium and low frequency. The upper limit of the medium frequency was selected as 16.61 kHz, and the corresponding frequency-bin \(k\) was 266 and 440. For the high frequency segment, the decision threshold was set to 5. Thus, the value of \(\delta(k)\) could be set as follows:  
\[\delta(k)=\left\{ \begin{array}{c}5,\ 266<k<440\\ 2,\ \mathrm{other}\end{array} \right., \tag{38}\]  
The performance of convergence is the main concern in this section, and only the MSE is introduced herein as the measurement of differences between source signal and estimated signal, which is shown as follows [16]:  
\[\mathrm{MSE}=\frac{1}{LK}\sum\
olimits_{l=1}^{L}\sum\
olimits_{k=1}^{K} \left|\hat{Z}_{i}(k,l)-S(k,l)\right|^{2}, \tag{39}\]  
where \(K\) denotes the number of frequency bins, \(L\) is the number of time-frames, \(\hat{Z}_{i}(k,l)\) and \(S(k,l)\) denote the STDFT coefficients of the beamformer output and the desired signal at frequency-bin index \(k\) and time-frame index \(l\), respectively. During the iteration process of IDMVDR, the MSE changes.