# Auditory Device Voice Activity Detection Based on Statistical Likelihood-Ratio Order Statistics
## 3 Conventional Statistical LRT-Based VAD  
Assuming that speech and noise signals are additive, the detection of voice activity at the \(\ell\) th segmented frame is accomplished by deciding upon one of two hypotheses of \(H_{0}\) and \(H_{1}\):  
\[\begin{split} H_{0}:\,\text{speech absent}:\,\,\,\mathbf{X}( \ell)=\,\,\mathbf{N}(\ell)\\ H_{1}:\,\text{speech present}:\,\,\mathbf{X}(\ell)=\,\,\mathbf{N}( \ell)+\mathbf{S}(\ell)\end{split} \tag{1}\]  
where \(\mathbf{X}(\ell)\), \(\mathbf{S}(\ell)\), and \(\mathbf{N}(\ell)\) are \(\
icefrac{{K}}{{2}}+1\). dimensional vectors composed of \(k\) spectral components \((k=0,1,\ldots,\,K/2)\) of the input signal, speech, and noise, i.e., \(X_{k}(\ell),S_{k}(\ell)\), and \(N_{k}(\ell)\), respectively, such that  
\[\mathbf{X}(\ell)=[\,\,\,X_{0}(\ell),\,\,\,X_{1}(\ell),\ldots,X_{K/2}(\ell)]^{T} \tag{2}\]  
\[\mathbf{S}(\ell)=[\,\,\,S_{0}(\ell),\,\,\,S_{1}(\ell),\,\ldots,S_{K/2}(\ell)]^ {T} \tag{3}\]  
\[\mathbf{N}(\ell)=[\,\,\,N_{0}(\ell),\,\,\,N_{1}(\ell),\ldots,N_{K/2}(\ell)]^{T}. \tag{4}\]  
Figure 1: Block diagram of the voice activity detection (VAD) implemented on the auditory device filter bank.  
Assuming that \(S_{k}(\ell)\) and \(N_{k}(\ell)\) follow complex Gaussian distributions, the probability density functions conditioned on \(H_{0}\) and \(H_{1}\) are given by  
\[p(X_{k}(\ell)|H_{1})=\prod_{k=0}^{K-1}\frac{1}{\pi\big{[}\hat{\lambda}_{N,k}( \ell)+\hat{\lambda}_{S,k}(\ell)\big{]}}\exp\!\left(-\frac{|X_{k}(\ell)|^{2}}{ \hat{\lambda}_{N,k}(\ell)+\hat{\lambda}_{S,k}(\ell)}\right) \tag{5}\]  
and  
\[p(X_{k}(\ell)|H_{0})=\prod_{k=0}^{K-1}\frac{1}{\pi\big{[}\hat{\lambda}_{N,k}( \ell)\big{]}}\exp\!\left(-\frac{|X_{k}(\ell)|^{2}}{\hat{\lambda}_{N,k}(\ell)}\right) \tag{6}\]  
where \(\hat{\lambda}_{N,k}(\ell)\) and \(\hat{\lambda}_{S,k}(\ell)\) are estimate values of the noise variance \(\lambda_{N,k}(\ell)\) and speech variance \(\lambda_{S,k}(\ell)\), respectively, at the \(k\)th frequency bin. Here, the hat symbol \(\hat{\ }\) denotes an estimate value. The \(k\) th local LR \(\Lambda_{k}(\ell)\). under \(X_{k}(\ell)\) can be estimated [2,4] as the ratio between \(p(X_{k}(\ell)|H_{1})\) and \(p(X_{k}(\ell)|H_{0})\) as  
\[\begin{array}{l}\Lambda_{k}(\ell)=\frac{p(X_{k}(\ell)|H_{1})}{p(X_{k}(\ell)| H_{0})}\\ =\frac{1}{1+\hat{\xi}_{k}(\ell)}\exp\!\left(\frac{\hat{\xi}_{k}(\ell)\cdot\hat{ \xi}_{k}(\ell)}{1+\hat{\xi}_{k}(\ell)}\right)\end{array} \tag{7}\]  
where \(\hat{\xi}_{k}(\ell)\) is the a priori SNR estimate, which is estimated using a decision-directed (DD) approach in [2,3,10] as  
\[\hat{\xi}_{k}(\ell)=\alpha\cdot\hat{\xi}_{k}(\ell-1)+(1-\alpha)\cdot\max( \hat{\gamma}_{k}(\ell)-1,\ 0) \tag{8}\]  
where \(0\leq\alpha<1\) is the smoothing parameter. In Equations (7) and (8), \(\hat{\gamma}_{k}(\ell)\) is called the a posteriori SNR, which is expressed as  
\[\hat{\gamma}_{k}(\ell)=\frac{|X_{k}(\ell)|^{2}}{\hat{\lambda}_{N,k}(\ell)} \tag{9}\]  
where the noise variance estimate \(\hat{\lambda}_{N,k}(\ell)\) is obtained via a recursive procedure with a smoothing parameter \(\beta\), such that  
\[\hat{\lambda}_{N,k}(\ell)=\beta\cdot\hat{\lambda}_{N,k}(\ell-1)+(1-\beta)\cdot |X_{k}(\ell)|^{2}\text{if}H_{0}\text{ is true.} \tag{10}\]  
Then, from the averaged log value of LR \(\Lambda_{k}(\ell)\) in Equation (7), a decision rule can be established using a decision threshold \(\eta\):  
\[\log\Lambda(\ell)=\frac{1}{\widetilde{K}}\sum_{k=0}^{\widetilde{K}-1}\log \Lambda_{k}(\ell)\begin{array}{l}\geq\\ <\\ \eta\end{array}\begin{array}{l}\eta\\ :H_{1}\\ :H_{0}\end{array} \tag{11}\]  
where \(\widetilde{K}=\kappa/2+1\), and \(\Lambda(\ell)\) is referred to as the global LR in this paper.