# Dual-Mic Speech Enhancement Based on TF-GSC with Leakage Suppression and Signal Recovery
## 2 Summary of TF-GSC and Postfiltering  
Let \(x_{p}(m)\), \(n_{p}(m)\), and \(s(m)\) denote the input microphone signal and interfering signal at the \(p^{th}\) microphone and the desired speech at the source for time \(m\), respectively. With additive noise assumption, \(x_{p}(m)\) is represented as:  
\[x_{p}(m)=a_{p}(m)*s(m)+n_{p}(m),\quad p=1,2 \tag{1}\]  
where \(a_{p}(m)\) is the acoustic impulse response from the desired source to the \(p^{th}\) microphone and \(*\) denotes the convolution operation. The short-time Fourier transform (STFT) coefficients for signals \(X_{p}(n,k)\), \(N_{p}(n,k)\), and \(S(n,k)\) for frame \(n\) and frequency \(k\) are related as:  
\[\mathbf{X}(n,k)=\mathbf{A}(n,k)S(n,k)+\mathbf{N}(n,k) \tag{2}\]  
in which:  
\[ \begin{array}{l}\mathbf{X}(n,k)=[X_{1}(n,k)\quad X_{2}(n,k)]^{\mathsf{T}}\\ \mathbf{A}(n,k)=[A_{1}(n,k)\quad A_{2}(n,k)]^{\mathsf{T}}\\ \mathbf{N}(n,k)=[N_{1}(n,k)\quad N_{2}(n,k)]^{\mathsf{T}}\end{array}  \tag{3}\]  
where \(A_{p}(n,k)\) is the acoustic TF from the desired source to the \(p^{th}\) microphone for frame \(n\) and frequency \(k\). A block diagram of the TF-GSC is shown in Figure 1 in black, which mainly consists of three blocks [2; 7; 8; 18]. The FBF conserves signals from a desired direction while rejecting other signals as much as possible. The output of the FBF \(\mathbf{W}(n,k)\), \(Y_{\mathrm{FBF}}(n,k)\) is given by:  
\[Y_{\mathrm{FBF}}(n,k)=\mathbf{W}(n,k)^{H}\mathbf{X}(n,k), \tag{4}\]  
where \(\mathbf{W}(n,k)\) is constrained to satisfy \(\mathbf{W}(n,k)^{H}\mathbf{A}(n,k)=\mathcal{F}^{*}(n,k)\), in which a prespecified filter \(\mathcal{F}^{*}(n,k)\) is usually assumed to be a simple delay [9]. However, since the actual TFs are very difficult to obtain, the TF ratios are estimated instead and the optimal \(\mathbf{W}(n,k)\) is represented as [18]:  
\[\mathbf{W}(n,k)=\frac{\mathbf{H}(n,k)}{\left\|\mathbf{H}(n,k)\right\|^{2}} \tag{5}\]in which \(\mathbf{H}\) is the acoustic TF ratio vector:  
\[\mathbf{H}(n,k)=\big{[}H_{1}(n,k)\quad H_{2}(n,k)\big{)}\big{]}^{T}= \begin{bmatrix} 1&\dfrac{A_{2}(n,k)}{A_{1}(n,k)}\end{bmatrix} ^{T}. \tag{6}\]  
In the other branch, the BM \(\mathbf{B}(n,k)\) tries to block the signals from a desired direction while passing through all other signals, resulting in the noise reference \(Y_{U}\):  
\[Y_{U}(n,k)=\mathbf{B}(n,k)^{H}\mathbf{X}(n,k) \tag{7}\]  
in which:  
\[\mathbf{B}(n,k)= \begin{bmatrix}-H_{2}(n,k)^{*}\\ 1\end{bmatrix} . \tag{8}\]  
The third block, the ANC, tries to remove the components related to \(Y_{U}(n,k)\) from the FBF output \(Y_{FBF}(n,k)\) using the normalized least-mean-square (NLMS) algorithm:  
\[Y(n,k)=Y_{FBF}(n,k)-G(n,k)^{H}Y_{U}(n,k) \tag{9}\]  
where \(G(n,k)\) is the ANC filter updated by:  
\[G(n+1,k)=G(n,k)+\mu\dfrac{Y_{U}(n,k)Y_{FBF}(n,k)^{H}}{P_{est}(n,k)} \tag{10}\]  
in which \(\mu\) is a step size and the noise reference power \(P_{est}(n,k)\) is updated as:  
\[P_{est}(n,k)=\rho_{U}P_{est}(n-1,k)+(1-\rho_{U})Y_{U}(n,k)Y_{U}(n,k)^{H} \tag{11}\]  
where \(\rho_{U}\) is a smoothing factor. In order to effectively operate FBF and BM, the estimation of accurate TF ratios is essential. In [18], the TF ratios were estimated using least squares, assuming that the TF ratios are slowly changing in time compared to the desired signal and the background noise signals are stationary.