# A Multi-Source Separation Approach Based on DOA Cue and DNN
## 2 Mixed Signal Dereverberation
### Cost Function  
We obtain the model of the recorded signal \(\mathbf{X}\) in Section 2.1 and use \(\mathbf{Y}=\left(Y_{k,n}\right)_{K\times T}\), \(Y_{k,n}\in\mathbb{R}_{0,+}\) to denote the spectrogram of the recorded signal with reverberation in this subsection. Considering that the NMF model represents an approximate equality relationship between \(\mathbf{S}\) and \(\mathbf{W}\mathbf{V}\), the recorded signal model \(\mathbf{X}\) is also an approximate representation for the recorded signal \(\mathbf{Y}\). There are different methods which can be used to measure the accuracy of the approximate representation, such as the Euclidean distance and so on. In this subsection, we employ the generalized \(\beta\) divergence as the measurement method to obtain the optimal matrices \(\mathbf{W}\), \(\mathbf{V}\) and \(\mathbf{M}=\left(M_{k,h}\right)_{K\times H}\), \(M_{k,h}\in\mathbb{R}_{0,+}\), so that \(\mathbf{X}\) can be used to provide an accurate approximation for \(\mathbf{Y}\)[25]. Therefore, the spectrogram of sound source signal \(\mathbf{S}\) can be obtained by the product of the optimal matrices \(\mathbf{W}\) and \(\mathbf{V}\). The generalized \(\beta\) divergence of \(\mathbf{X}\) and \(\mathbf{Y}\) is defined as:  
\[Q_{\beta}(\mathbf{Y}\parallel\mathbf{X})=\sum_{k=1}^{K}\sum_{n=1}^{T}\left(Y_ {k,n}\cdot\frac{Y_{k,n}^{\beta-1}-X_{k,n}^{\beta-1}}{\beta(\beta-1)}+X_{k,n}^{ \beta-1}\cdot\frac{X_{k,n}-Y_{k,n}}{\beta}\right) \tag{5}\]  
where \(Y_{k,n}\) is the spectrogram of the recorded signal at the \(k_{th}\) frequency point and the \(n_{th}\) frame, and \(\beta\) is the value of divergence. However, there exist many ways to construct the matrices \(\mathbf{W}\), \(\mathbf{V}\) and \(\mathbf{M}\). The penalty functions for \(\mathbf{V}\) and \(\mathbf{M}\) are introduced to narrow the range of alternative ways and obtain the optimal \(\mathbf{W}\), \(\mathbf{V}\) and \(\mathbf{M}\) more efficiently [21]. Hence, the cost function combining the generalized \(\beta\) divergence and the penalty functions is defined as follows:  
\[f(\mathbf{W},\mathbf{V},\mathbf{M})=Q_{\beta}(\mathbf{Y}\parallel\mathbf{X}) +G_{\mathrm{v}}(\mathbf{V})+G_{\mathrm{m}}(\mathbf{M}) \tag{6}\]  
where \(G_{\mathrm{v}}(\mathbf{V})\) and \(G_{\mathrm{m}}(\mathbf{M})\) are the penalty functions for \(\mathbf{V}\) and \(\mathbf{M}\), respectively; \(G_{\mathrm{v}}(\mathbf{V})\) is determined based on the TF spectrum structure of the recorded signal, whereas \(G_{\mathrm{m}}(\mathbf{M})\) is determined based on the logarithmic spectrum structure of the RIR. First, we determine the penalty function \(G_{\mathrm{v}}(\mathbf{V})\) by analyzing the characteristics of the TF spectrum of the recorded signal with reverberation. An example is shown in Figure 2, where Figure 2 is the TF spectrum of the clean sound source signal sampled at 16 kHz. The recorded signal with reverberation is simulated by Roomsim software [26]. A sound field microphone is located at the center of the simulated room, and the reverberation time (\(T_{60}\)) is 300 ms. Figure 2 shows the TF spectrum of the recorded signal (i.e., W-channel signal in B-format) with \(T_{60}=300\) ms. It can be clearly seen from Figure 2 that the harmonic structure in the TF spectrum of the clean sound source signal is more obvious, while that of the recorded signal with reverberation is fuzzier, and the formant peak is not clear. Therefore, a penalty function is added to the activation matrix \(\mathbf{V}\) to highlight the harmonic structure in the TF spectrum and enhance the sparsity of the recorded signal. Since the \(\ell_{1}\) norm can be used to represent the sparsity of a matrix, we use it to construct the penalty function for \(\mathbf{V}\). In a non-negative matrix, the \(\ell_{1}\) norm can be represented as the sum of the total elements in the matrix.