# Effective Dereverberation with a Lower Complexity at Presence of the Noise
## 2 Signal Model
### Complex Gaussian Mixture Model  
Reviewing MVDR beamforming based on the CGMM [24], the enhanced speech signal by the beamforming can be expressed as follows:  
\[x_{\text{b}}(l)=\mathbf{w}_{\text{b}}^{H}\mathbf{y}(l) \tag{3}\]  
where superscript \"\(H\)\" denotes the conjugate transpose, \(\mathbf{w}_{\text{b}}\) is the vector of the filter coefficients for minimizing the output power of the beamforming under the constraint \(\mathbf{w}_{\text{b}}^{H}\mathbf{g}\) = 1. Here, \(\mathbf{g}\) is the steering vector of the desired speech signal and the subscript \"\(b\)\" indicates the beamforming. Considering the sparsity of the speech signal in the time-frequency domain [25; 26], the observed signal can be classified into three cases, i.e., the mixture of speech, noise, and reverberation; the mixture of speech and reverberation; and noise-only. Thus, the observation signals of the microphones can be written as follows [24]:  
\[\mathbf{y}(l)=\mathbf{g}^{(d)}x_{\text{b}}^{(d)}(l) \tag{4}\]  
where the superscript \"\(d\)\" denotes the scenarios of the interference such as reverberation with noise (in this case, \(d\) is denoted as \(\mathbf{x}+\mathbf{r}+\mathbf{n}\)), reverberation-only (in this case, \(d\) is denoted as \(\mathbf{x}+\mathbf{r}\)), and noise-only (in this case, \(d\) is denoted as \(\mathbf{n}\)). The complex Gaussian distribution of the random variable \(x\) with the mean \(\mu\) and variance \(\sigma^{2}\) is generally expressed as follows [24]:  
\[\text{N}(x;\mu,\sigma^{2})=\frac{1}{\pi\sigma^{2}}\text{e}^{-\frac{|x-\mu|^{2} }{\sigma^{2}}} \tag{5}\]  
Assuming that \(x_{\text{b}}^{(d)}(l)\) follows a complex Gaussian distribution \(x_{\text{b}}^{(d)}(l)\sim\text{N}(0,\phi(l)^{(d)})\), namely, its mean and variance are zero and \(\phi(l)^{(d)}\), respectively. Here, \(\phi(l)^{(d)}\) also indicatesthe power spectral density of \(x_{\rm b}^{(d)}(l)\). Thus, at each time-frequency bin, the multichannel microphone signals obey the following complex Gaussian mixture distribution [24]:  
\[{\bf y}(l)\left|{\rm d}\ \sim{\rm N}\Big{(}0,\phi(l)^{(d)}{\rm R}^{(d)}\Big{)}\right. \tag{6}\]  
where \({\rm R}^{(d)}={\rm g}^{(d)}\ ({\rm g}^{(d)})^{H}\). Thus, the multichannel microphone signals can be described by the CGMM in the case of reverberation with noise or noise-only, in which the maximum likelihood estimation of the CGMM can be obtained by the expectation maximization (EM) algorithm [24]. The Q function in the EM algorithm is defined as follows [24]:  
\[{\rm Q}(\Theta)=\sum\limits_{l}\sum\limits_{d}\lambda(l)^{(d)}\log{\rm N} \Big{(}{\bf y}(l);0,\phi(l)^{(d)}{\rm R}^{(d)}\Big{)} \tag{7}\]  
where \(\lambda(l)^{(d)}\) denotes the mask or probability of the different scenarios, which can be obtained as follows:  
\[\lambda(l)^{(d)}={\rm N}\Big{(}{\bf y}(l);0,\phi(l)^{(d)}{\rm R}^{(d)}\Big{)} /\sum\limits_{d}{\rm N}\Big{(}{\bf y}(l);0,\phi(l)^{(d)}{\rm R}^{(d)}\Big{)} \tag{8}\]  
In order to achieve the online MVDR beamforming, the speech signal is divided into a sequence of batches. Each batch contains a limited number of the speech frames. The number of the frames can be changed. Let \(c\in\{1\), \(2\),..., \(C\}\) denote the batch index and let \(l_{c}\) denote the time index within the \(c\)th batch, where \(C\) is the number of the batches.