# An Efficient Multistage Approach for Blind Source Separation of Noisy Convolutive Speech Mixture
## 3 Proposed Multistage BSS Approach
Fundamentally, the IVA method [31] uses a multivariate super-Gaussian distribution source prior to model the different frequency bin inter-frequency dependencies, which are expressed as,  
\[q(s_{i})\propto\exp\left(-\sqrt{\sum_{k=1}^{K}\lvert\frac{\hat{s}_{i}(k)}{ \sigma_{i}(k)}\rvert^{2}}\right), \tag{12}\]  
where \(\sigma_{i}(k)\) represents the standard deviation of \(i\)-th source at \(k\)-th frequency block. Using Equation (11) to determine the score function of Equation (12), we obtain  
\[\varphi^{(k)}(\hat{s}_{i}(1),\cdots,\hat{s}_{i}(K))=\frac{\hat{s}_{i}(k)}{ \sqrt{\sum_{k=1}^{K}\lvert\hat{s}_{i}(k)\rvert^{2}}}. \tag{13}\]  
Equation (13) shows the non-linear score function of the fundamental IVA algorithm and is used for inter-frequency dependencies between source signals. However, the non-linear score function is not unique and is strongly dependent on the source prior. Therefore, we can use different source priors to exploit higher-order statistics. The generalized multivariate Gaussian can also be used as a source prior distribution to retain inter-frequency dependencies between different frequency blocks. Due to its heavy tails, it exploits higher-order statistical properties between the source signal and can be expressed as [32]  
\[q(s_{i})\propto\exp\left(-\sqrt[3]{(s_{i}-\mu_{i})^{\intercal}\Sigma_{i}^{-1} (s_{i}-\mu_{i})}\right). \tag{14}\]  
We assume the mean \(\mu_{i}=0\) and covariance \(\Sigma_{i}\) equal to identity. Then, using Equation (11) for Equation (14), the score function will be \[\varphi^{(k)}(\xi_{i}(1),\cdots,\xi_{i}(K))=\frac{2\xi_{i}(k)}{3\sqrt[3]{(\sum_{k=1 }^{K}|\xi_{i}(k)|^{2})^{2}}}. \tag{15}\]  
In a noisy real-time environment, the non-stationary nature of the observed convolutive mixture contains high- as well as low-energy components. Hence, it is difficult for a single source prior to model the statistical properties of a non-stationary convolutive mixture. Therefore, a hybrid model is proposed containing multivariate generalized Gaussian and super-Gaussian source priors. The hybrid source prior model can better model low and higher amplitudes [34]. The super-Gaussian source prior function models low-energy amplitude, and the high-energy amplitude is modeled by a multivariate generalized Gaussian source prior. The weights between these source priors in the hybrid source prior model are adopted based on the energy of a noisy convolutive mixture. The hybrid model can be expressed as  
\[q(s_{i})= \begin{cases}f_{GGD}&\text{if}\,\phi\geq 0.5\\ f_{SGD}&\text{if}\,\phi<0.5\end{cases} . \tag{16}\]  
\(f_{GGD}\) is the multivariate generalized Gaussian source prior distribution and \(f_{SGD}\) is the super-Gaussian source prior distribution. The non-linear hybrid score function is mathematically written as  
\[\varphi^{(k)}(\xi_{i}(1),\cdots,\xi_{i}(K))= \begin{cases}\left(\frac{2\xi_{i}( k)}{3\sqrt[3]{(\sum_{k=1}^{K}|\xi_{i}(k)|^{2})^{2}}}\right);\phi\geq 0.5\\ \left(\frac{\xi_{i}(k)}{\sqrt[3]{\sum_{k=1}^{K}|\xi_{i}(k)|^{2}}}\right)&; \phi<0.5\end{cases}  \tag{17}\]  
where \(\phi=[0,1]\) is the weighting parameter, which depends on the normalized energy of the received noisy convolutive mixture. The weights of the non-linear score functions and \(\phi\) are adjusted by the normalized energy of the mixture at every frequency block.