## 2 Review of the MCRA Noise Estimator  
A noise-interfered speech signal \(y(\eta,\ v)\) can be modeled as the sum of the speech signal \(s(\eta,\ v)\) and interference noise \(d\ (\eta,\ v)\) in the frame \(\eta\) of the time domain, given as  
\[y(\eta,\ v)=s(\eta,\ v)+d(\eta,\ v) \tag{1}\]  
where \(v\) is the sample index in a frame. The noise-interfered signal \(y(\eta,\ v)\) is analyzed and transformed to the frequency domain, given as  
\[Y(\eta,\Omega)=\sum_{v=0}^{N-1}y(v+\eta M)\cdot h(v)\cdot e^{-j(2\pi/N)v \Omega} \tag{2}\]  
where \(\Omega\) and \(h\) represent the frequency bin index and analysis window, respectively. \(N\) and \(M\) are the frame size and update step in time. Let \(H_{0}(\eta,\Omega)\) and \(H_{1}(\eta,\Omega)\) indicate the hypotheses for speech-absence and speech-presence, respectively. They are presented as [3]  
\[ \begin{split} H_{0}(\eta,\Omega):Y(\eta,\Omega)=D(\eta,\Omega)\\ H_{1}(\eta,\Omega):Y(\eta,\Omega)=S(\eta,\Omega)+D(\eta,\Omega) \end{split}  \tag{3}\]  
where \(S(\eta,\Omega)\) and \(D(\eta,\Omega)\) represent the spectrum of clean speech and additive noise, respectively. Let \(\lambda_{d}(\eta,\Omega)=\left|D(\eta,\Omega)\right|^{2}\) denote the variance of the noise. The noise estimates for speech absence and presence can be obtained, given as  
\[ \begin{split} H_{0}^{\prime}(\eta,\Omega):\hat{\lambda}_{d}(\eta,\Omega)=\alpha_{d}\hat{\lambda}_{d}(\eta-1,\Omega)+(1-\alpha_{d})\cdot\left| Y(\eta,\Omega)^{2}\right|\\ H_{1}^{\prime}(\eta,\Omega):\hat{\lambda}_{d}(\eta,\Omega)= \hat{\lambda}_{d}(\eta-1,\Omega)\end{split}  \tag{4}\]  
where \(\alpha_{d}\) denotes a smoothing parameter. \(H_{0}^{\prime}\) and \(H_{1}^{\prime}\) respectively represent the hypotheses of speech absence and presence. The noise estimate given in (4) can be obtained by  
\[\hat{\lambda}_{d}(\eta,\Omega)=\hat{\lambda}_{d}(\eta-1,\Omega)\cdot p^{ \prime}(\eta,\Omega)+[\alpha_{d}\cdot\hat{\lambda}_{d}(\eta-1,\Omega)+(1- \alpha_{d})\cdot\left|Y(\eta,\Omega)\right.\left|^{2}\right|\cdot\left[1-p^{ \prime}(\eta,\Omega)\right] \tag{5}\]  
where \(p^{\prime}(\eta,\Omega)\) denotes the probability of speech presence, which can be obtained by  
\[\hat{p}^{\prime}(\eta,\Omega)=\alpha_{p}\cdot\hat{p}^{\prime}(\eta-1,\Omega)+ \left(1-\alpha_{p}\right)\cdot I(\eta,\Omega) \tag{6}\]  
where \(\alpha_{p}(\alpha_{p}=0.2)\) is a smoothing factor for speech presence probability. \(I(\eta,\Omega)\) is an indicator function for speech presence, given as  
\[I(\eta,\Omega)=\left\{ \begin{array}{l}1\,,\text{if}\ \gamma(\eta,\Omega)> \delta_{\gamma}\\ 0\,,\text{otherwise}\end{array} \right. \tag{7}\]  
where \(\delta_{\gamma}\) represents a threshold for speech presence. \(\gamma(\eta,\Omega)\) represents the ratio between the local energy of the noise-interfered signal \(P_{Local}(\eta,\Omega)\) and its estimated minimum \(P_{\min}(\eta,\Omega)\), given as  
\[\gamma(\eta,\Omega)=P_{Local}(\eta,\Omega)/P_{\min}(\eta,\Omega) \tag{8}\]  
where  
\[P_{Local}(\eta,\Omega)=\sum_{i=-\omega_{1}}^{\omega_{1}}b(i)\cdot\left|Y(\eta,\Omega-i)\right|^{2} \tag{9}\]  
The smoothed version of the local energy \(P_{Local}^{S}(\eta,\Omega)\) is computed by a first order recursive average, given as  
\[P_{Local}^{S}(\eta,\Omega)=\alpha_{s}P_{Local}^{S}(\eta-1,\Omega)+(1-\alpha_ {s})P_{Local}(\eta,\Omega) \tag{10}\]  
The minimum \(P_{\min}(\eta,\Omega)\) and a temporary variable \(P_{Imp}(\eta,\Omega)\) are initialized by \(P_{\min}(0,\Omega)=P(0,\Omega)\) and \(P_{Imp}(0,\Omega)=P(0,\Omega)\).