# Voice Activity Detection Using Fuzzy Entropy and Support Vector Machine
## 2 Proposed Methodology
### Feature Extraction--Fuzzy Entropy (FuzzyEn)  
Let _s(i)_ be the sample speech sequence, where \(i=1,\,2,\,3,\,\ldots\,\), \(N\) (\(N=512\) in present case), which is reconstructed by phase-space with an embedded dimension \(m\), and the reconstructed phase-space speech vector \(S_{i}^{m}\) is given in (3),  
\[S_{i}^{m}=\left\{s\left(i\right),\,s\left(i+1\right),\,\ldots,\,s\left(i+m-1 \right)\right\}-s_{0}\left(i\right),\,i=1,2,\ldots,N-m+1 \tag{3}\]  
and is generalized by removing the baseline as in (4),  
\[s_{0}\left(i\right)=m^{-1}\sum_{j=0}^{m-1}s\left(i+j\right) \tag{4}\]  
For given vector \(S_{i}^{m}\), the similarity degree \(D_{ij}\) of its neighboring vector \(S_{j}^{m}\) through its similarity degree is defined by fuzzy function, given in (5),  
\[D_{ij}=\upmu\left(d_{ij}^{m},r\right) \tag{5}\]  
and \(d_{ij}^{m}\) is the maximum absolute difference of the scalar components of \(S_{i}^{m}\) and \(S_{j}^{m}\), given in (6),  
\[d_{ij}^{m}=d\left[S_{i}^{m},S_{j}^{m}\right]=\max_{l\,\in\left(0,m-1\right)} \left[\left(s\left(i+l\right)-s_{0}\left(i\right)\right)-\left(s\left(j+l \right)-s_{0}\left(j\right)\right)\right] \tag{6}\]  
Figure 2: Speech signals (**a**–**c**) with its additive noise [0, 5 dB] with amplitude along vertical axis and time (s) along horizontal axis; its corresponding histograms (**d**–**f**), with amplitude along horizontal axis and frequency along the vertical axis. Here \(\mu(d_{ij}^{m},r)\) is the fuzzy membership function, which is given by the exponential function, as in (7),  
\[\mu\left(d_{ij}^{m},n,r\right)=exp\left(\frac{-\left(d_{ij}^{m}\right)^{n}}{r}\right) \tag{7}\]  
where \(m\), \(n\), and \(r\) are the embedding dimension, gradient, and width of the fuzzy membership function, respectively. For each \(S_{i}^{m}\), averaging all similarity degree \(D_{ij}\) of the neighboring vectors \(S_{j}^{m}\), we get (8),  
\[\Phi_{i}^{m}=\frac{1}{\left(N-m-1\right)}\sum_{j=1\;j\
eq i}^{N-m}D_{ij}^{m} \tag{8}\]  
Now construct \(\varphi^{m}\left(r\right)\) given in (9) and \(\varphi^{m+1}\left(r\right)\) which is given in (10),  
\[\varphi^{m}\left(r\right)=\frac{1}{N-m}\sum_{i=1}^{N-m}\Phi_{i}^{m}\left(r\right) \tag{9}\]  
and  
\[\varphi^{m+1}\left(r\right)=\frac{1}{N-m}\sum_{i=1}^{N-m}\Phi_{i}^{m+1}\left(r\right) \tag{10}\]  
From this, FuzzyEn\(\left(m,r\right)\) of the speech, is defined by, given in (11),  
\[\text{FuzzyEn}\left(m,r\right)=\lim_{N\rightarrow\infty}\left[\text{ln}\varphi ^{m}-\text{ln}\varphi^{m+1}\left(r\right)\right] \tag{11}\]  
Three parameters are crucial in calculating FuzzyEn, which are to be fixed at first. The first parameter, \(m\), is the embedded dimension which focusses on the sequence length to be compared. The other two parameters are \(r\) and \(n\), which determine the width or similarity tolerance and the gradient of the exponential function, respectively. Figure 3 illustrates the impact of different selection of parameters on the exponential function. In Figure 3a, the width \(r\) in the exponential membership function is fixed at 0.2 and \(n\) is varied from 1, 2, and 5. Similarly, in Figure 3b, the exponential membership function parameter \(n\) is set to 2 and the width \(r\) varies between 0.1 and 0.3. Experimentally, the width \(r\) is optimal when multiplied with standard deviation (SD) and small values of \(n\). In this work, the embedding dimension, \(m\), is 2 and the exponential function parameters \(n\) and \(r\) are set as 2 and 0.2 times standard deviation, respectively. Generally, too large values of embedding dimension might lead to loss of useful information. Also, underestimating the similarity tolerance, \(r\), leads to higher noise sensitivity. So the selection of FuzzyEn parameter numbers are decided based on Mann-Whitney U-test. The lowest \(p\)-value was obtained for the parameter combination \(m\) as 2 and \(n\) and \(r\) as 2 and 0.2 times SD, respectively. FuzzyEn is computed for all the frames of the speech signal and these values are used as the features for the SVM classifier.