# An Efficient Multistage Approach for Blind Source Separation of Noisy Convolutive Speech Mixture
## 3 Proposed Multistage BSS Approach  
This section presents the proposed multistage approach for BSS and speech enhancement in a noisy reverberant environment. The multistage method comprises the BSS stage and de-noising stage using MMSE filtering as shown in Figures 1 and 2, respectively. The proposed scheme evaluates different combinations of the BSS hybrid model and the de-noising MMSE method. In the first model (Figure 1), the observed convolutive mixture speech signal is first processed by the BSS stage with a hybrid source prior model for the extraction of estimated speech signals from the reverberant mixture. The de-noising module processes the resultant noisy extracted speech signals where the noisy elements in the separated speech signals are suppressed to improve the quality of the estimated signals. In the second model (Figure 2), the received reverberant observed speech mixture is de-noised by the MMSE filtering method in the first stage. In the second stage, the enhanced convolutive speech mixture is processed by the BSS stage with a hybrid source prior model to extract the de-noised estimated source speech signal from the enhanced reverberant mixture. Multivariate generalized Gaussian and super-Gaussian source priors are combined into the hybrid source prior model in the BSS stage. The generalized Gaussian model exploits higher-order statistical properties while multivariate super-Gaussian models use other related information in the hybrid source prior model approach. The weights of the source priors in the hybrid model are adopted following the energy components of the received convolutive mixture [34]. In the de-noising stage, the MMSE filtering method is used to suppress the noisy component in the received convolutive mixture signal. The hybrid source prior model provides a better separation performance and preserves the frequency dependencies between different frequency blocks for the IVA algorithm. Instead of using a single source prior distribution, a combination of multivariate gen  
Figure 1: First model. eralized Gaussian and super-Gaussian models is used for source priors for the IVA to preserve the frequency dependencies. By using the Kullback-Leibler (KL) divergence cost function to preserve the dependencies within the source speech signal while removing the dependencies among different source signals [40]. Mathematically the non-linear cost function for the hybrid model can be written as [31]  
\[ \begin{split}& C=KL(P(\hat{s}_{1},\cdots,\hat{s}_{N})||\prod_{i=1}^{ N}q(\hat{s}_{i}))\\ &=\text{const}-\sum_{k=1}^{K}\log\lvert\det(W(k))\rvert-\sum_{i=1 }^{N}E\log q(\hat{s}_{i}),\end{split}  \tag{9}\]  
where \(W(k)\) is the \(k\)-th separating matrix and \(q(\hat{s}_{i})\)is the source prior of \(i\)-th estimated source signal. The multivariate cost function in (9) is minimized by the gradient descent algorithm to remove the dependencies among different source signals and mathematically can be expressed as [31]  
\[ \begin{split}\Delta w_{ij}(k)&=-\frac{\partial C}{ \partial w_{ij}(k)}\\ &=\sum_{l=1}^{N}(I_{il}-E\varphi^{k}(\hat{s}_{i}^{(1)},\cdots,\hat {s}_{i}^{(K)})\hat{s}_{l}^{(k)})w_{lj}^{(k)},\end{split}  \tag{10}\]  
where \(I\) is the identity matrix and \(\varphi^{(k)}(. )\) is the non-linear score function which can be mathematically expressed as [34]  
\[\varphi^{k}\Big{(}\hat{s}_{i}^{(1)},\cdots,\hat{s}_{i}^{(K)}\Big{)}=-\frac{ \partial\log q(\hat{s}_{i}^{(1)},\cdots,\hat{s}_{i}^{(K)})}{\partial\hat{s}_{ i}^{k}} \tag{11}\]  
The non-linear score function retains the dependency between different frequency bins, which is the main theme of the IVA algorithm and plays a vital role in the separation process.