## 4 Speech Presence Probability-Based Noise Statistics Estimation
### A Posteriori SPP Estimation  
The _a posteriori_ SPP \(p_{x}(t,f)\) allows us to control the updating procedure of Equation (38) for the computation of the noise statistics. Nevertheless, \(p_{x}(t,f)\) is also exploited in two additional parts of our system:  
* The estimation of the RTF presented in the previous section is only accurate in time-frequency bins where speech is present. The _a posteriori_ SPP indicates those bins where speech presence is more likely. Therefore, in our implementation we only update the eKF in those bins where \(p_{x}(t,f)>p_{\text{thr}}\), with \(p_{\text{thr}}\) being a predefined probability threshold. Otherwise, the previous values are preserved.
* The postfiltering performance can be improved if additional information about SPP is provided, as shown later in Section 5.  
The estimation of the _a posteriori_ SPP can be addressed assuming complex multivariate Gaussian distributions for the noisy speech \(\mathbf{Y}\)[4; 5], according to the two hypotheses previously formulated (see (36) and (37)). Using the Bayes' rule, the _a posteriori_ SPP at each time-frequency bin can be calculated as  
\[p_{x}=\frac{\left(1-q_{n}\right)f\left(\mathbf{Y}|H_{x}\right)}{\left(1-q_{n} \right)f\left(\mathbf{Y}|H_{x}\right)+q_{n}f\left(\mathbf{Y}|H_{n}\right)}, \tag{46}\]where \(q_{n}=P\left(H_{n}\right)\) is the _a priori_ speech absence probability (SAP) and  
\[f\left(\mathbf{Y}|H_{x}\right)=\frac{e^{-\mathbf{Y}\boldsymbol{\Phi}_{Y}^{-1} \mathbf{Y}^{H}}}{\pi^{M}\det\left[\boldsymbol{\Phi}_{Y}\right]}, \tag{47}\]  
\[f\left(\mathbf{Y}|H_{n}\right)=\frac{e^{-\mathbf{Y}\boldsymbol{\Phi}_{N}^{-1} \mathbf{Y}^{H}}}{\pi^{M}\det\left[\boldsymbol{\Phi}_{N}\right]}, \tag{48}\]  
are the likelihoods of observing the noisy speech signal under the different hypotheses, with \(\det\left[\cdot\right]\) being the matrix determinant operator and \(M\) the number of microphones. Equation (46) can be redefined using these likelihoods, yielding the following expression for the SPP,  
\[p_{x}=\left(1+\frac{q_{n}}{1-q_{n}}\frac{\det\left[\boldsymbol{\[PERSON]}_{Y} \right]}{\det\left[\boldsymbol{\Phi}_{N}\right]}\frac{e^{-\mathbf{Y} \boldsymbol{\Phi}_{N}^{-1}\mathbf{Y}^{H}}}{e^{-\mathbf{Y}\boldsymbol{\Phi}_{ Y}^{-1}\mathbf{Y}^{H}}}\right)^{-1}. \tag{49}\]  
Then, the _a posteriori_ SPP can be estimated at each frame for all frequencies according to the following two-iteration algorithm:  
* _Initialization_: Estimate the noisy SCM with a recursive updating, \[\widehat{\boldsymbol{\Phi}}_{Y}(t,f)=\bar{\alpha}\widehat{\boldsymbol{\Phi}}_ {Y}(t-1,f)+\left(1-\bar{\alpha}\right)\mathbf{Y}(t,f)\mathbf{Y}^{H}(t,f),\] (50) where \(\bar{\alpha}\) is an updating constant as in (39). Also, estimate the _a priori_ SAP \(q_{n}(t,f)\) (see Section 4.2).
* _1 st iteration_: Estimate \(p_{x}(t,f)\) using \(\widehat{\boldsymbol{\Phi}}_{N}(t-1,f)\) in (49). Then, estimate \(\widehat{\boldsymbol{\Phi}}_{N}(t,f)\) using \(p_{x}(t,f)\) in (38).
* _2 nd iteration_: Re-estimate \(p_{x}(t,f)\) using now \(\widehat{\boldsymbol{\Phi}}_{N}(t,f)\) in (49). Finally, re-estimate \(\widehat{\boldsymbol{\Phi}}_{N}(t,f)\) using \(p_{x}(t,f)\).