# Effective Dereverberation with a Lower Complexity at Presence of the Noise
## 3 Proposed Dereverberation Method
### Initialization of Kalman Filtering  
Some parameters need to be initialized at the start of the Kalman filtering. In update Equations (31)-(37) and (49)-(55), these parameters include the state transition matrix \({\bf A}\) (\(l\)), covariance matrix \({\bf\Phi_{v}}\) (\(l\)) of the process noise \({\bf w}_{\rm w}\) (\(l\)), covariance matrix \({\bf\Phi_{\rm w_{A}}}(l)\) of the process noise \({\bf w}_{\Delta}\) (\(l\)), power spectral density \(\phi_{x_{\rm e}}(l)\) of the desired signal, and the covariance matrices \({\bf\Phi_{\rm e}}\) (\(l\)) and \({\bf\Phi_{\rm e_{1}}}(l)\) of the errors \({\bf e}\) (\(l\)) and \({\bf e_{1}}\) (\(l\)). The state transition matrix \({\bf A}(l)\) is regarded as a time-invariant function similar to the forgetting factor in the update equation [31]. The covariance matrices \({\bf\Phi_{\rm e}}\) (\(l\)) and \({\bf\Phi_{\rm e_{1}}}(l)\) gradually become convergent during the Kalman filtering. The covariance matrix \({\bf\Phi_{\rm e}}\) (\(l\)) can be initialized as a diagonal matrix, i.e., Diag \([{\bf\Phi_{\rm e}}(l)]={\bf\varphi_{\rm e}}\), where vector \({\bf\varphi_{\rm e}}\) includes \(M\) (\(L-D\)) elements that rise exponentially with every \(M\) elements [12], i.e.,  
\[{\bf\varphi_{\rm e}}=\left[{\varphi_{\rm e}^{1}1^{T}},{\varphi_{\rm e}^{2}1^{ T}},\cdots,{\varphi_{\rm e}^{(L-D)}1^{T}}\right]_{M(L-D)\times 1}^{T} \tag{56}\]  
where \(1=[1,1,\cdots,1]_{M\times 1}^{T}\), \({\varphi_{\rm e}^{1}}\), \({\varphi_{\rm e}^{2}}\),..., \({\varphi_{\rm e}^{(L-D)}}\) are \(L-D\) scalars that rise exponentially. Similarly, the covariance matrix \({\bf\Phi_{\rm e1}}\) (\(l\)) is also initialized as a diagonal matrix, i.e., Diag \([{\bf\Phi_{\rm e1}}\) (\(l\))] = \({\bf\varphi_{\rm e1}}\), where vector \({\bf\varphi_{\rm e1}}\) only includes \(L-D\) elements that drop exponentially, i.e.,  
\[{\bf\varphi_{\rm e1}}=\left[{\varphi_{\rm e1}^{1}},{\varphi_{\rm e1}^{2}}, \cdots,{\varphi_{\rm e1}^{(L-D)}}\right]^{T} \tag{57}\]  
The process noises \({\bf v_{\rm w}}\) (\(l\)) and \({\bf w_{\Delta}}\) (\(l\)) are modeled by the zero-mean Gaussian model, and their covariance matrixes \({\bf\Phi_{v}}\) (\(l\)) and \({\bf\Phi_{\rm w\Delta}}\) (\(l\)) are usually assumed to be time-invariant and initialized as follows [12]:  
\[{\bf\Phi_{v}}(l)=(1-\beta){\bf\Phi_{\rm e}}(l) \tag{58}\]  
\[{\bf\Phi_{\rm w\Delta}}(l)=(1-\beta){\bf\Phi_{\rm e_{1}}}(l) \tag{59}\]  
where \(\beta\in\) (0,1) is the forgetting factor. The weighted prediction error (WPE) algorithm also needs to initialize the PSD of the desired signal. In the WPE, the PSD of unprocessed signal is regarded as one of the desired signals to estimate the linear prediction coefficients used for obtaining the enhanced signal. Then, the PSD of the enhanced signal is used to estimate the new linear prediction coefficients. The above steps are iterated until the algorithm converges. When the WPE is used for a short-time speech signal, the frame number of the speech signal has to be reduced. In this case, the estimated PSD of the desired signal greatly degrades the performance of the algorithm because of using few frames. Therefore, a method of directly estimating the PSD of the desired signal from the reverberated speech by using the DNN was proposed [33]. In [14; 15], the PSD of the desired signal is determined by a decision-oriented method, that is, the current PSD of the desired signal is composed of the PSD of the previous frame and the PSD of the current frame; the later one is obtained by the linear prediction coefficients of the previous frame. These two PSDs are weighted by a weight factor. This method relies on the setting of the weight factor, and how to choose an appropriate weight factor is crucial. In this paper, the speech signal processed by the beamformer is utilized to initialize the PSD of the desired signal.