## 3 Extended Kalman Filter-Based Relative Transfer Function Estimation
### A Priori RTF Statistics  
The dynamic model for the RTF vector \(\mathbf{a}_{21,t}\) presented in Equation (12) accounts for the process variability across time in terms of a white noise \(\mathbf{w}_{t}\). This variability has a twofold meaning: first, the possible temporal variations of the acoustic channel due to environment changes, head or smartphone movements, etc., and, second, the inaccuracy of the narrowband approximation of Equation (8) [6]. This is due to the fact that, especially in reverberant environments where the analysis window of the STFT is shorter than the impulse response of the acoustic channel, the transfer function is not multiplicative but convolutive. This convolutive transfer function expands in both time and frequency dimensions. Thus, nearby frames and frequencies are correlated, which violates the narrowband approximation [6]. In order to account for the variability associated to this effect, we will assume the statistical model proposed in (12) so that, although the channel could be time-invariant, the RTF, as defined in Equation (3), is time-variant.  
The eKF-based RTF estimator requires the _a priori_ statistics of the RTF. These statistics are the overall mean vector and covariance matrix of the RTF, respectively,  
\[\boldsymbol{\mu}_{a_{21}}=E\left[\mathbf{a}_{21,t}\right], \tag{32}\]  
\[\boldsymbol{\Sigma}_{a_{21}}=E\left[\left(\mathbf{a}_{21,t}-\boldsymbol{\mu} _{a_{21}}\right)\left(\mathbf{a}_{21,t}-\boldsymbol{\mu}_{a_{21}}\right)^{ \top}\right], \tag{33}\]  
and also the covariance of the RTF variability across frames,  
\[\mathbf{Q}=E\left[\mathbf{w}_{t}\mathbf{w}_{t}^{\top}\right]=E\left[\left( \mathbf{a}_{21,t}-\mathbf{a}_{21,t-1}\right)\left(\mathbf{a}_{21,t}-\mathbf{ a}_{21,t-1}\right)^{\top}\right]. \tag{34}\]  
Additionally, we can define the overall correlation matrix of the RTF as  
\[\mathbf{R}_{a_{21}}=E\left[\mathbf{a}_{21,t}\mathbf{a}_{21,t}^{\top}\right]= \boldsymbol{\Sigma}_{a_{21}}+\boldsymbol{\mu}_{a_{21}}\boldsymbol{\mu}_{a_{21 }}^{\top}. \tag{35}\]  
The previous statistics can be estimated in advance using a training set of dual-channel clean speech utterances in different acoustic environments and device positions. In order to avoid outliers which might yield useless estimates, we select, on an utterance basis, time-frequency bins where the speech power at the reference channel is large enough (higher than the maximum power at that frequency in the utterance minus 3 dB). For those bins, we estimate the RTF using (3), which yields a set of RTFs \(A_{21}^{(l)}(t,f)\) for the \(l\)-th utterance, that are later converted to RTF vectors \(\mathbf{a}_{21,t}^{(l)}\). For each utterance, a sample mean vector \(\widehat{\boldsymbol{\mu}}_{a_{21}}^{(l)}\) and a sample correlation matrix \(\widehat{\mathbf{R}}_{a_{21}}^{(l)}\) are estimated using those \(\mathbf{a}_{21,t}^{(l)}\) vectors, while a sample covariance matrix \(\widehat{\mathbf{Q}}^{(l)}\) is computed from consecutive RTF vectors. Finally, the global sample statistics \(\widehat{\boldsymbol{\mu}}_{a_{21}}\), \(\widehat{\mathbf{R}}_{a_{21}}\) and \(\widehat{\mathbf{Q}}\) are obtained by averaging the particular utterance-dependent statistics. The sample covariance \(\boldsymbol{\Sigma}_{a_{21}}\) can then be estimated using \(\widehat{\boldsymbol{\mu}}_{a_{21}}\) and \(\widehat{\mathbf{R}}_{a_{21}}\) in (35).