## 3 Extended Kalman Filter-Based Relative Transfer Function Estimation  
The proposed system requires knowledge of the RTF between the two microphones, namely \(A_{21}(f,t)\). Among other approaches, the computation of this RTF can be addressed as the estimation of a variable that changes across frames in terms of a stochastic model. Given this dynamic model, the noise statistics and the noisy observations, we proposed in [15] the tracking of the RTF using an extended Kalman filter (eKF), showing a better estimation performance in comparison with other state-of-the-art approaches. Next, we describe in detail the derivation of this eKF-based RTF estimation. First, we formulate the narrowband approximation for a given frequency of the noisy speech signal at the secondary microphone in terms of the reference microphone as  
\[Y_{2}(t)=A_{21}(t)\left(Y_{1}(t)-N_{1}(t)\right)+N_{2}(t), \tag{8}\]  
where the frequency index \(f\) has been omitted for the sake of simplicity. In addition, complex variables are expressed as stacked vectors of their real and imaginary parts. For example, given \(Y_{m}(t)=Y_{m,t}^{r}+jY_{m,t}^{i}\) with \(m=1,2\), we can define  
\[\mathbf{y}_{m,t}= \begin{bmatrix}Y_{m,t}^{r}&,&Y_{m,t}^{i}\end{bmatrix} ^{\top}, \tag{9}\]and, in a similar way,  
\[\mathbf{a}_{21,t}=\left[A_{21,t}^{r}\quad,\quad A_{21,t}^{i}\right]^{\top}, \tag{10}\]  
\[\mathbf{n}_{m,t}=\left[N_{m,t}^{r}\quad,\quad N_{m,t}^{i}\right]^{\top}. \tag{11}\]  
Using these definitions for the model variables in terms of vectors, we propose the following dynamic and observation models needed by the Kalman filter:  
* Dynamic model for the RTF \(\mathbf{a}_{21,t}\): We assume that the state vector \(\mathbf{a}_{21,t}\) is a random walk stochastic process which can be expressed as \[\mathbf{a}_{21,t}=\mathbf{a}_{21,t-1}+\mathbf{w}_{t},\] (12) where \[\mathbf{w}_{t}\sim\mathcal{N}\left(\mathbf{0},\mathbf{Q}\right)\] (13) is a zero-mean multivariate white Gaussian noise that models the variability of the RTF. A detailed discussion on this model is provided in Section 3.2. * Observation model for the noisy speech at the secondary microphone, \(\mathbf{y}_{2,t}\): It is defined using the distortion model in (8) as \[\mathbf{y}_{2,t} =\mathbf{h}\left(\mathbf{a}_{21,t},\mathbf{n}_{1,t};\mathbf{y}_ {1,t}\right)+\mathbf{n}_{2,t}\] (14) \[=\left( \begin{bmatrix}1&0\\ 0&1\end{bmatrix} \left(\gamma_{1,t}^{r}-N_{1,t}^{r}\right)+ \begin{bmatrix}0&-1 \\ 1&0\end{bmatrix} \left(\gamma_{1,t}^{i}-N_{1,t}^{i}\right)\right)\mathbf{a}_{21, t}+\mathbf{n}_{2,t},\] where the noises are assumed to be zero-mean multivariate Gaussian variables, \[ \begin{bmatrix}\mathbf{n}_{1,t}\\ \mathbf{n}_{2,t}\end{bmatrix} \sim\mathcal{N}\left(\mathbf{0}, \begin{bmatrix} \mathbf{\Sigma}_{n_{11},t}&\mathbf{\Sigma}_{n_{12},t}\\ \mathbf{\Sigma}_{n_{21},t}&\mathbf{\Sigma}_{n_{22},t}\end{bmatrix} \right),\] (15) where \(\mathbf{\Sigma}_{n_{ij},t}=E\left[\mathbf{n}_{i,t}\mathbf{n}_{j,t}^{\top}\right]\) is a noise covariance matrix whose estimation is addressed in Section 4. Additionally, we assume that there is no correlation between \(\mathbf{a}_{21,t}\) and \(\mathbf{n}_{m,t}\) while \(\mathbf{n}_{1,t}\) and \(\mathbf{n}_{2,t}\) are correlated. The model \(\mathbf{h}\) is non linear because of the product between the variables \(\mathbf{a}_{21,t}\) and \(\mathbf{n}_{1,t}\). This model depends on the observation \(\mathbf{y}_{1,t}\), which acts as a model parameter. Finally, the Kalman filter framework is applied to obtain a (recursive) minimum mean square error (MMSE) estimate of \(\mathbf{a}_{21,t}\). This is a two-step procedure that is applied frame-by-frame for all frequencies:  
1.