## 4 Speech Presence Probability-Based Noise Statistics Estimation
### A Priori SAP Estimation  
To obtain an accurate _a posteriori_ SPP that allows for robust tracking of the noise statistics, the _a priori_ SAP is a key parameter. Methods on single-channel noise tracking, as the fixed _a priori_ SNR algorithm of Reference [31] or the minima controlled recursive averaging (MCRA) algorithm [32], estimate this SAP in terms of the _a priori_ SNR. The MCRA framework was extended to multi-channel speech signals in Reference [4]. This is the SPP estimator that we used in our previous works [15; 27]. A major drawback of the MCRA scheme is the lack of robustness in case of a time-varying SNR, which makes noise changes to be detected as speech presence. More recently, the authors of [5] proposed to use spatial information in the SAP estimation, specifically the coherent-to-diffuse ratio (CDR). Alternatively, the power level difference (PLD) between microphones is used in Reference [17] to update the noise statistics, as it is a good indicator of speech presence in CT conditions. In this work, we propose a novel SAP estimator for dual-microphone smartphones that combines the CDR spatial information with the PLD between the reference and the secondary microphones. First, an SCM \(\boldsymbol{\Phi}_{Y}(t,f)\) is estimated using a rectangular window of eight past frames as in Reference [5]. Then, we calculate (1) the power spectral density (PSD) ratio between microphones as  
\[\widehat{\sigma}_{Y_{21}}(t,f)=\frac{\tilde{\boldsymbol{\phi}}_{Y_{22}}(t,f)}{ \tilde{\boldsymbol{\phi}}_{Y_{11}}(t,f)}, \tag{51}\]  
where \(\tilde{\boldsymbol{\phi}}_{Y_{ij}}\) is an estimate of \(E\left[Y_{i}Y_{j}^{*}\right]\), and (2) the short-term complex coherence between microphones,  
\[\tilde{\boldsymbol{\psi}}_{Y_{12}}(t,f)=\frac{\tilde{\boldsymbol{\phi}}_{Y_{12 }}(t,f)}{\sqrt{\tilde{\boldsymbol{\phi}}_{Y_{11}}(t,f)\tilde{\boldsymbol{\phi }}_{Y_{22}}(t,f)}}. \tag{52}\]Both terms are needed by the SAP estimators. The PLD-based _a priori_ SAP is based on the PLD noise estimator for CT conditions described in Reference [17]. In that work, a normalized difference of the noisy speech PSDs was defined as  
\[\Delta\widehat{\mathcal{G}}_{\text{nPLD}}(t,f)=\frac{\tilde{\mathcal{G}}_{Y_{11} }(t,f)-\tilde{\mathcal{G}}_{Y_{22}}(t,f)}{\tilde{\mathcal{G}}_{Y_{11}}(t,f)+ \tilde{\mathcal{G}}_{Y_{22}}(t,f)}=\frac{1-\widehat{\mathcal{G}}_{Y_{21}}(t,f) }{1+\widehat{\mathcal{G}}_{Y_{21}}(t,f)}. \tag{53}\]  
The noise statistics are updated by this parameter, as it gives information about speech presence. That is, assuming similar noise power at both microphones and that speech is more attenuated at the secondary microphone with respect to the primary one, \(\Delta\widehat{\mathcal{G}}_{\text{nPLD}}(t,f)\) is close to one when speech is present and tends to zero otherwise. Thus, an _a priori_ SAP based on this indicator can be estimated as  
\[q_{\text{PLD}}(t,f)=1-\Delta\widehat{\mathcal{G}}_{\text{nPLD}}(t,f)=\frac{2 \widehat{\mathcal{G}}_{Y_{21}}(t,f)}{1+\widehat{\mathcal{G}}_{Y_{21}}(t,f)}, \tag{54}\]  
where \(q_{\text{PLD}}(t,f)\) is upper-bounded by 1. This estimator is valid for CT conditions, but it can also be useful in FT conditions for frequencies where speech at the secondary microphone is attenuated. On the other hand, the CDR is another good indicator of speech presence [33], which is defined as  
\[\Gamma_{Y_{12}}(t,f)=\frac{\psi_{\text{diff}}(f)-\psi_{Y_{12}}(t,f)}{\psi_{Y_{ 12}}(t,f)-\psi_{X_{12}}(t,f)}, \tag{55}\]  
where \(\psi_{X_{12}}(t,f)\) is the clean speech short-term complex coherence (defined as in (52)) and  
\[\psi_{\text{diff}}(f)=\text{sinc}\left(\frac{2\pi fF_{s}d_{\text{mic}}}{Kc_{v} }\right) \tag{56}\]  
is the diffuse noise field complex coherence, with \(F_{s}\) the sampling frequency, \(K\) the number of frequency bins, \(d_{\text{mic}}\) the distance between microphones and \(c_{v}\) the speed of sound.