# Auditory Device Voice Activity Detection Based on Statistical Likelihood-Ratio Order Statistics
## 4 Proposed VAD Based on LR Order Statistics  
### Signal Sparseness Model for LRT  
The decision threshold value \(\eta\) in Equation (11) can be decomposed into an a priori fixed value \(\eta_{0}\) and an increment \(\Delta\eta\), i.e., \(\eta=\eta_{0}+\Delta\eta\), in which a larger \(\Delta\eta\) subsequently leads to a larger FRP. Thus, the \(\Delta\eta\) term plays a crucial role in the false reject robustness of the binary decision of speech activity in Equation (11). By incorporating \(\eta=\eta_{0}+\Delta\eta\) into Equation (11), we can alternatively represent this decision rule as  
\[\begin{array}{lcl}\{\log\Lambda(\ell)\}^{\prime}&\geq&\eta_{0}\ :\ H_{1}\\ <&\eta_{0}\ :\ H_{0}\end{array} \tag{12}\]where \(\{\log\Lambda(\ell)\}^{\prime}\) is the attenuated version of \(\log\Lambda(\ell)\) by \(\Delta\eta\), i.e., \(\{\log\Lambda(\ell)\}^{\prime}=\log\Lambda(\ell)-\Delta\eta\). If \(\{\log\Lambda(\ell)\}^{\prime}\) is reduced by increasing \(\Delta\eta\), the FRP increases. We argue that because of the speech and noise sparseness, \(\Delta\eta\) exists in all noisy speech samples.  
For the two hypotheses in Equation (1), it has been assumed in Equations (2)-(4) that a speech signal \(S_{k}\) is present in every frequency bin for \(H_{1}\) and a noise signal \(N_{k}\) is present in every frequency bin for \(H_{0}\) and \(H_{1}\). However, speech and most types of noise (apart from white noise) do not have their energy equally distributed over all frequency bins [28,29]. Thus, to reflect the sparseness states of speech and noise, we decompose \(H_{0}\) and \(H_{1}\) into four states according to the presence or absence of speech and noise in the \(k\) th frequency bin, as shown in Table 1: \(H_{0}^{(1)}\) or \(H_{1}^{(1)}\), \(H_{0}^{(2)}\) or \(H_{1}^{(2)}\), \(H_{0}^{(3)}\) or \(H_{1}^{(3)}\), and \(H_{0}^{(4)}\) or \(H_{1}^{(4)}\).  
In the table, the minimum value of the noise components is specified as \(\varepsilon_{k}\). The superscript \(\langle 1\rangle\) is the state in which both the speech and noise components exist at the \(k\) th frequency bin; \(\langle 2\rangle\) and \(\langle 3\rangle\) are speech-only and noise-only states, respectively, and \(\langle 4\rangle\) represents the states with neither speech nor noise. Based on this notation of the four sparseness states, the LRT in Equation (11) can be expressed as  
\[\frac{\sum_{k^{(1)}}\log\Lambda_{k}(\ell)+\sum_{k^{(2)}}\log\Lambda_{k}(\ell) +\sum_{k^{(3)}}\log\Lambda_{k}(\ell)+\sum_{k^{(4)}}\log\Lambda_{k}(\ell)}{ \widetilde{K}}\qquad\eta_{0} \tag{13}\]  
where \(\widetilde{K}=num\Big{(}k^{(1)}\Big{)}+num\Big{(}k^{(2)}\Big{)}+num\Big{(}k^{( 3)}\Big{)}+num\Big{(}k^{(4)}\Big{)}\) and \(num(\cdot)\) is the number of respective frequency bins. In Equation (13), \(\sum\limits_{k^{(3)}}\log\Lambda_{k}(\ell)\) and \(\sum\limits_{k^{(4)}}\log\Lambda_{k}(\ell)\) reduce the global LR, thus increasing the FRP. Therefore, \(\sum\limits_{k^{(3)}}\log\Lambda_{k}(\ell)\) and \(\sum\limits_{k^{(4)}}\log\Lambda_{k}(\ell)\) do not robustly estimate the global LR and are less suitable to calculate the LRT:  
\[\frac{1}{\widetilde{K}^{\prime}}\Bigg{(}\sum_{k^{(1)}}\log\Lambda_{k}(\ell)+ \sum_{k^{(2)}}\log\Lambda_{k}(\ell)\Bigg{)}\begin{array}{l}H_{1}\\ \geq\\  
In the next subsection, we will explore the use of LR order statistics for a false rejection of robust VAD.