# Effective Dereverberation with a Lower Complexity at Presence of the Noise
## 3 Proposed Dereverberation Method
### Low-Complexity Algorithm Based on the Kronecker Product  
The Kalman filter has the disadvantage of high computational complexity. By observing update Equations (31)-(37) of the Kalman filter, it can be found that the calculation of the Kalman filter involves the multiplication and summation of many matrices with high dimension data, so the dimension reduction in the matrix is the key for reducing the calculation complexity. In the observation Equation (22), the stacked microphone signal \(\mathbf{t}\) (\(l\)) is a vector with \(M\) (\(L-D\)) elements, \(\mathbf{w}_{\rm r}\) (\(l\)) is a vector including \(M\) (\(L-D\)) linear prediction coefficients. According to the method proposed in [32], \(\mathbf{w}_{\rm r}\) (\(l\)) can be expressed as the form of the Kronecker product of two subfilters, where two low-dimension subfilters can be estimated separately. Considering a signal model such as Equation (21), we have  
\[x_{\rm e}(l)=\mathbf{w}_{2}^{H}(l)\mathbf{y}(l)-\big{[}\mathbf{w}_{1}(l) \otimes\mathbf{w}_{2}(l)\big{]}^{H}\mathbf{t}(l) \tag{38}\]  
where \(x_{\rm e}\) (\(l\)) is the desired signal, symbol \"\(\otimes\)\" represents the Kronecker product, \(\hat{x}(l)=\mathbf{w}_{2}^{H}(l)\mathbf{y}(l)\), \(\mathbf{w}_{\rm r}(l)=\mathbf{w}_{1}(l)\otimes\mathbf{w}_{2}(l)\), \(\mathbf{w}_{1}\) (\(l\)), and \(\mathbf{w}_{2}\) (\(l\)) can be regarded as a (\(L-D\))-order interframe filter and a \(M\)-order spatial filter, respectively. In fact, \(\mathbf{w}_{1}\) (\(l\)) represents the filter for processing the consecutive frames, and the main purpose of \(\mathbf{w}_{1}\) (\(l\)) is to predict the reverberation components from the past \(L-D\) frames. \(\mathbf{w}_{2}\) (\(l\)) represents the filter that processes signal from all channels. Thus, \(\mathbf{w}_{2}\) (\(l\)) can be regarded as the coefficients of the spatial filter. By using the Kronecker product, the original optimization problem becomes two optimal problems of the subfilters. According to the characteristic of the Kronecker product [32], we can obtain  
\[\mathbf{w}_{1}(l)\otimes\mathbf{w}_{2}(l)=[\mathbf{I}\otimes\mathbf{w}_{2}(l)] \mathbf{w}_{1}(l)=\mathbf{W}_{2}(l)\mathbf{w}_{1}(l) \tag{39}\]  
where \(\mathbf{I}\) is a \((L-D)\times(L-D)\) identity matrix and \(\mathbf{W}_{2}\) (\(l\)) = \(\mathbf{I}\otimes\mathbf{w}_{2}\) (\(l\)) is a \(M\left(L-D\right)\times(L-D)\) matrix. Substituting (39) into (38), we have  
\[x_{\mathrm{e}}(l)=\mathbf{w}_{2}^{H}(l)\mathbf{y}(l)-\mathbf{w}_{1}^{H}(l) \mathbf{W}_{2}^{H}(l)\mathbf{t}(l)=\tilde{x}(l)-\mathbf{w}_{1}^{H}(l)\mathbf{ t}_{\mathbf{W}_{2}}(l) \tag{40}\]  
where \(\mathbf{t}_{\mathbf{W}_{2}}(l)=\mathbf{W}_{2}^{H}(l)\mathbf{t}(l)\). Since \(\mathbf{w}_{2}(l)\) is a spatial filter, \(\mathbf{w}_{2}(l)\) in (40) can be replaced by the CGMM-MVDR filter, i.e.,  
\[x_{\mathrm{e}}(l)=\mathbf{w}_{\mathrm{b}}^{H}\mathbf{y}(l)-\mathbf{w}_{1}^{H} (l)\mathbf{W}^{H}\mathbf{t}(l)=x_{\mathrm{b}}(l)-\mathbf{w}_{1}^{H}(l) \mathbf{t}_{\mathrm{W}}(l) \tag{41}\]  
where \(\mathbf{w}_{\mathrm{b}}\) = \(\mathbf{w}_{2}\) (\(l\)), \(\mathbf{W}\) = (\(\mathbf{I}\otimes\mathbf{w}_{\mathrm{b}}\)), \(\mathbf{t}_{w}\) (\(l\)) = \(\mathbf{W}^{H}\) (\(l\)) \(\mathbf{t}\) (\(l\)). It is worth noting that \(\mathbf{w}_{\mathrm{b}}\) is different at each frequency bin of all the frames. \(\mathbf{t}_{w}\) (\(l\)) can be regarded as the speech signal that was processed by the spatial filter in advance. After such replacement, two optimal problems of the subfilters become a single optimal problem of the interframe filter that can be estimated by the Kalman filter.