## 2 System Overview  
The proposed enhancement system for dual-microphone smartphones is depicted in Figure 1. The microphones capture the noisy speech signals \(y_{m}(n)\), where \(m\) indicates the microphone index (\(m=1,2\)). We assume an additive noise distortion model which in the short-time Fourier transform (STFT) domain can be expressed as  
\[Y_{m}(t,f)=X_{m}(t,f)+N_{m}(t,f), \tag{1}\]  
where \(Y_{m}(t,f)\), \(X_{m}(t,f)\) and \(N_{m}(t,f)\) represent, respectively, noisy speech, clean speech and noise signal STFTs, \(t\) is the frame index and \(f\) the frequency bin. The two channel components can be stacked in a vector,  
\[\mathbf{Y}(t,f)=\left[Y_{1}(t,f)\quad,\quad Y_{2}(t,f)\right]^{\top}, \tag{2}\]  
where \([\cdot]^{\top}\) indicates matrix transposition. Similarly, we define vectors \(\mathbf{X}(t,f)\) and \(\mathbf{N}(t,f)\). In the following, we will consider that each frequency component can be processed independently from the others, which is commonly referred to as narrowband approximation [6].  
As shown in Figure 1, the beamforming block requires two previous procedures. First, a speech presence probability (SPP)-based algorithm is employed for the estimation of the noisy speech and noise spatial correlation matrices (SCM), \(\mathbf{\Phi}_{Y}(t,f)\) and \(\mathbf{\Phi}_{N}(t,f)\), respectively, and the SPP, \(p_{x}(t,f)\)  
Figure 1: Overview of the dual-microphone enhancement system for smartphones.  
Then, an extended Kalman filter (eKF)-based estimator obtains the relative transfer function (RTF) between the secondary microphone (\(m=2\)) and the reference microphone (\(m=1\)), defined as  
\[A_{21}(f,t)=\frac{X_{2}(f,t)}{X_{1}(f,t)}. \tag{3}\]  
Once the SCM matrices and RTF have been computed, the dual-channel noisy speech vector \(\mathbf{Y}(t,f)\) is processed using a Minimum Variance Distortionless Response (MVDR) beamformer, which applies spatial filtering yielding an output signal \(Z(t,f)\) defined as  
\[Z(t,f)=\mathbf{F}^{H}(t,f)\mathbf{Y}(t,f), \tag{4}\]  
where \((\cdot)^{H}\) indicates a Hermitian transpose. The coefficients of the beamformer, \(\mathbf{F}(t,f)\), are estimated as [2],  
\[\mathbf{F}(t,f)=\frac{\mathbf{\Phi}_{N}^{-1}(t,f)\mathbf{d}(t,f)}{\mathbf{d}^ {H}(t,f)\mathbf{\Phi}_{N}^{-1}(t,f)\mathbf{d}(t,f)}, \tag{5}\]  
where  
\[\mathbf{d}(t,f)=\begin{bmatrix}1&,&A_{21}(t,f)\end{bmatrix}^{\top} \tag{6}\]  
is the steering vector normalized to the reference channel.  
Finally, the speech signal at the beamformer output is enhanced by a single-channel postfilter for additional noise reduction. A spectral gain \(G(t,f)\) is obtained using the power spectral density (PSD) of the speech at the reference microphone \(\phi_{x_{1}}(t,f)\), the PSD of the residual noise \(\phi_{v}(t,f)\) and the SPP \(p_{x}(t,f)\). The above single-channel statistics (\(\phi_{x_{1}}(t,f)\) and \(\phi_{v}(t,f)\)) are estimated from the noisy speech and noise SCMs and the RTF. The gain function is further processed by a musical noise reduction algorithm [28] and finally applied to the beamformer output, thus obtaining  
\[\widehat{X}_{1}(t,f)=G(t,f)Z(t,f), \tag{7}\]  
where \(\widehat{X}_{1}(t,f)\) is the estimate of the clean speech signal at the reference microphone. In the following sections, the description of the different parts of the system is addressed.