# Adaptive Speech Separation Based on Beamforming and Frequency Domain-Independent Component Analysis
## 3 Proposed Speech Separation Algorithm
### Speech Separation According to Different Mixing Condition
#### 3.2.3 Case III: \(M^{\prime}>\text{N}\)  
This case means the number of signal components is more than the number of sound sources; in this paper, the case where the localization algorithm fails to detect some sound sources is not considered, because the energy of these sound sources are much lower than other sound sources which have been counted and localized. These undetected sound sources are taken as ambient noise and still reduce the dimension to \(N\). In this situation, the low-energy sound sources and the received noise of microphones will be compressed into the signal space by PCA, and then, ICA can be used. The processing matrix is the same with Equation (23).  
Furthermore, due to the theoretical limitation of ICA, the permutation and scaling ambiguity should be solved after iteration in Case \(M^{\prime}>1\). This permutation ambiguity can be solved by postprocessing methods such as spatial directivity and correlation between frequency bins. However, these methods will increase computation load. In the set-up of Section 4.1, the postprocessing [12] costs \(\sim\)0.2 s. Therefore, a proper initialization for iteration is found to control the convergence point so that the permutation will not be changed. In Section 4.1, the simulation results show that this approach is effective in the environment with reverberation time T60 = 0.3 s. For more adverse environments, the postprocessing is still recommended.  
First, after the dimensionality reduction using PCA, the signal at each channel can be expressed as  
\[Z(f,t)=V(f)\cdot H(f)\cdot S(f,t)=P(f)\cdot S(f,t) \tag{24}\]\[P(f)=V(f){\cdot}H(f) \tag{25}\]  
where \(H(f)\) is an approximation of the real propagation process.  
Then,  
\[\begin{array}{c}Y(f,t)=P^{-1}(f)Z(f,t)=P^{-1}(f)V(f)H(f)S(f,t)\\ =P^{-1}(f)P(f)S(f,t)=I(f)S(f,t)\end{array} \tag{26}\]  
Therefore, by setting matrix \(P^{-1}(f)\) as the initial matrix for ICA, the ICA iterative process can be performed. When \(H(f)\) is close to the real propagation process, the permutation is highly likely to remain unchanged.  
\[W_{initial}=P^{-1}(f) \tag{27}\]  
Here, the scaling ambiguity can be solved by Equation (14). Considering the calculations of the Moore-Penrose pseudoinverse, a simple method is proposed to handle with the scaling ambiguity. By using the approximation of Equation (26) and the initial matrix, the separation matrix \(P^{-1}(f)\) is replaced by \(W_{\text{ICA}}(f)\), and \(Y(f,t)\) is approximately equal to  
\[Y(f,t)=W(f){\cdot}P(f){\cdot}S(f,t)=I^{\prime}(f){\cdot}S(f,t) \tag{28}\]  
\[I^{\prime}(f)=W(f){\cdot}P(f) \tag{29}\]  
where \(\dot{I}^{\prime}(f)\) maintains the permutation but the scaling of the components has been changed. We restore the scaling of the separated signal by  
\[Y(f,t)=Y(f,t){\cdot}diag(I^{\prime}(f)) \tag{30}\]  
where/denotes element-wise division and \(diag(\cdot)\) denotes the diagonal elements.  
Because the result of the dimensionality reduction with PCA is directly related to the threshold of eigenvalue, the ratio among \(W_{[1]}\left(f\right)\), \(W_{[2]}\left(f\right)\) and \(W_{[3]}\left(f\right)\) can be adjusted with the PCA threshold. According to the simulated result in Section 4.2, the processing method is modified when \(M^{\prime}>1\), and \(W_{[2]}\left(f\right)\) is merged into \(W_{[3]}\left(f\right)\). Therefore, by sorting the second largest eigenvalue of all frequency bins, the proportion of different processing schemes can be effectively controlled. Finally, the separated signals of each sound source over all frequency bins are integrated into a complete representation in the time-frequency domain, and converted into the time domain by applying the inverse Fourier transform. Figure 3 depicts the flow of speech separation according to mixing conditions.  
Figure 3: Flow chart of speech separation according to mixing conditions.