## 2 Method Analysis
### Centralized Beamformer (MVDR Algorithm)  
In the audio enhancement of the WASN with FC, other nodes needed to broadcast their signals to the FC, so that the centralized beamformer could be obtained at FC. We considered a WASN consisting of \(N_{d}\) randomly connected nodes. Each node was assumed to consist of only one microphone. Here, only one signal source (bird audio) was considered. The \(N_{d}\times 1\) vector \(\mathbf{Y}(k,l)\) contains the STDFT coefficients of \(N_{d}\) collected signals under certain a time-frame \(l\) and frequency-bin \(k\), which can be shown as follows:  
\[\mathbf{Y}(k,l)=\mathbf{H}(k)\cdot s(k,l)+\mathbf{n}(k,l)=\mathbf{X}(k,l)+ \mathbf{n}(k,l), \tag{1}\]  
where the \(s(k,l)\) is the STDFT coefficient of bird audio signal, and \(N_{d}\times 1\) vector \(\mathbf{n}(k,l)\) denotes the noise STDFT coefficients. The \(\mathbf{H}(k)\) is a \(N_{d}\times 1\) vector which denotes the ATFs from the bird to each microphone. If we omit the time and frequency indices for brevity, Equation (1) can be simplified as follows:  
\[\mathbf{Y}=\mathbf{H}\cdot s+\mathbf{n}=\mathbf{X}+\mathbf{n}, \tag{2}\]  
where vector \(\mathbf{H}\) consists of \(N_{d}\) rows, which is given by:  
\[\mathbf{H}= \begin{bmatrix}H_{1}\cdots H_{N_{d}}\end{bmatrix} ^{\mathsf{T}}, \tag{3}\]  
Figure 1: The improved distributed minimum variance distortionless response (IDMVDR) beamforming method based on the local average consensus algorithm for bird audio enhancement in wireless acoustic sensor networks (WASNs). ATF is used to describe the relationship between the source signal and collected signal of microphone. ATF estimation is relatively complicated [14; 15; 16], and was not the core concern here. Considering the absence of echoes and reverberation in forest areas, the ATF in reference [9] was adopted directly in this paper, which is defined as:  
\[\mathbf{H}(k)=\left[\frac{1}{\sqrt{4\Pi}q_{1}}e^{-2j\Pi q_{1}v_{f}/c},...,\ \frac{1}{\sqrt{4\Pi}q_{N_{d}}}e^{-2j\Pi q_{1}v_{f}/c}\right]^{ \mathrm{T}}, \tag{4}\]  
when \(\mathrm{j}=\sqrt{-1}\), \(v_{f}=k\times f_{s}/K\) is the continuous frequency (in Hz) corresponding to the frequency bin \(k\in\{0,\cdots,K/2\}\) (\(K\) denotes the STDFT length of one time frame), and \(f_{s}\) is the sampling frequency. \(q_{i}\) denotes the distance between the target bird and the node \(i\). The purpose of the MVDR algorithm is to keep the output signal distortionless and minimize the noise power. The MVDR algorithm is a special form of linearly constrained minimum variance (LCMV) when the source number is limited to one. With the MVDR algorithm, the STDFT coefficient of desired bird audio can be estimated by applying a complex weight to the vector \(\mathbf{Y}\) with noisy STDFT coefficients. That is,  
\[Z=\mathbf{W}^{H}\mathbf{Y}=\mathbf{W}^{H}\mathbf{H}\mathbf{s}+\mathbf{W}^{H} \mathbf{n}, \tag{5}\]  
where \(Z\) denotes the desired bird audio STDFT coefficients, \(\mathbf{W}\) is a weight vector with filter coefficients, and \(\left(\cdot\right)^{H}\) indicates Hermetian transposition of a matrix. According to the bird audio distortionless constraint, to minimize the contribution of interference to the output \(Z\), the optimal weight vector \(\mathbf{W}\) is the solution to the following optimization problem:  
\[\left\{\begin{array}{c}\mathbf{W}=arg\ min\mathbf{W}^{H}\mathbf{R}_{\gamma \gamma}\mathbf{W}\\  
Assumed that the noise coefficient of each node was spatially uncorrelated with PSD \(\sigma_{i}^{2}\), then the noise correlation matrix PSD \(\mathbf{R}_{nm}\) can be described by the following equation [15]:  
\[\mathbf{R}_{nm}=\mathbb{E}\Big{[}\mathbf{n}\mathbf{n}^{H}\Big{]}=diag\Big{\{}R_ {N_{1}N_{1}},R_{N_{2}N_{2}},\cdots,R_{N_{d}N_{d}}\Big{\}}=diag\Big{\{}\sigma_{1 }^{2},\sigma_{2}^{2},\cdots,\sigma_{N_{d}}^{2}\Big{\}}, \tag{13}\]  
\(\mathbf{R}_{nm}\) will be estimated by the continuous spectrum update algorithm, which is described in Section 2.4.