# Effective Dereverberation with a Lower Complexity at Presence of the Noise
## 3 Proposed Dereverberation Method
### Application of Kalman Filter  
The linear prediction coefficients can be estimated effectively by the Kalman filter [28; 29; 30]. In the ISCLP [12], the Kalman filter used for estimating the linear prediction coefficients has shown good results for dereverberation. Here, we define the observation equation of the Kalman filter [12] as follows:  
\[x_{\text{b}}^{*}(l)=\mathbf{t}^{H}(l)\mathbf{w}_{\text{r}}(l)+x_{\text{e}}^{*} (l) \tag{22}\]  
where the superscript \"*\" indicates the conjugation operation. The vector \(\mathbf{w}_{\text{r}}\) (\(l\)) is the state vector with zero mean, and its related covariance matrix is expressed as follows:  
\[\mathbf{\Phi}_{\text{w}}(l)=\text{E}\Big{[}\mathbf{w}_{\text{r}}(l)\mathbf{w} _{\text{r}}{}^{H}(l)\Big{]} \tag{23}\]  
and \(x_{\text{e}}{}^{*}\) (\(l\)) in Equation (22) is the desired signal; it is also known as the measurement error. Similarly, the measurement error can also be regarded as a Gaussian process with zero mean, as defined by Equation (6), and its power spectral density is \(\phi_{\text{xe}}(l)=\text{E}[x_{\text{e}}(l)x_{\text{e}}^{*}(l)]\). The change in \(\mathbf{w}_{\text{r}}\) (\(l\)) means the change in the RIR or the change in the microphone positions, so \(\mathbf{w}_{\text{r}}\) (\(l\)) needs to be adjusted by the state equation. Considering the first-order Markov process, the state equation can be defined as follows [14]:  
\[\mathbf{w}_{\text{r}}(l)=\mathbf{A}^{H}(l)\mathbf{w}_{\text{r}}(l-1)+\mathbf{ v}_{\text{w}}(l) \tag{24}\]  
where \(\mathbf{A}\) (\(l\)) is the state transition matrix; it indicates the prediction result of the state vectors from the previous frames to the current frame. The zero-mean Gaussian model with the following covariance matrix can be used to model the process noise \(\mathbf{v}_{\text{w}}\) (\(l\)):  
\[\mathbf{\Phi}_{\text{v}}(l)=\text{E}\Big{[}\mathbf{v}_{\text{w}}(l)\mathbf{v }_{\text{w}}^{H}(l)\Big{]} \tag{25}\]  
According to Equations (22) and (24), the purpose of the Kalman filter is to find the optimal solution for the linear prediction coefficients based on the minimum mean square error (MMSE). Usually, \(\mathbf{A}\) (\(l\)) and \(\mathbf{v}_{\text{w}}\) (\(l\)) will be adjusted at the beginning of the algorithm [31]. In the Kalman filtering process, the essence of the algorithm is to determine the required parameters step by step through two stages. In the first stage, the state vector \(\mathbf{w}_{\text{r}}\) (\(l\)) of the next moment is predicted through the state equation. This prediction is called a priori time update. In the second stage, the prediction result of the state vector in the first stage is weighted with the so-called Kalman gain that is calculated by the observation equation combined with the state vector predicted in the first stage. The expected optimal state vector \(\mathbf{w}_{\text{r}}\) (\(l\)) can be obtained by the iterations of these two stages. Let \(\hat{\mathbf{w}}_{\text{r}}\) (\(l\)) be the estimated result given by the state Equation (24) and \(\hat{\mathbf{w}}_{\text{r}}{}^{+}\) (\(l\)) be the modified result from the observation Equation (22); their corresponding errors are obtained as follows:  
\[\mathbf{e}(l)=\mathbf{w}_{\text{r}}(l)-\hat{\mathbf{w}}_{\text{r}}(l) \tag{26}\]  
\[\mathbf{e}^{+}(l)=\mathbf{w}_{\text{r}}(l)-\hat{\mathbf{w}}_{\text{r}}^{+}(l) \tag{27}\]  
the covariance matrices of \(\mathbf{e}(l)\) and \(\mathbf{e}^{+}(l)\) are given as follows:  
\[\mathbf{\Phi}_{e}(l)=\text{E}\Big{[}\mathbf{e}(l)\mathbf{e}^{\text{H}}(l) \Big{]} \tag{28}\]  
\[\mathbf{\Phi}_{e}^{+}(l)=\text{E}\Big{[}\mathbf{e}^{+}(l)\mathbf{e}^{+\text{H} }(l)\Big{]} \tag{29}\]It is worth noting that these two covariance matrices of the errors can be regarded as the evolution in the same frame, that is, \(\mathbf{\Phi}_{\rm e}^{+}(l)\) is a further estimation of \(\mathbf{\Phi}_{\rm e}(l)\).