VAD Based on Kernel Smoothed Function of EGARCH Models

Abstract

An algorithm for a voice activity detector (VAD) is proposed. It is based on the exponential generalized autoregressive conditional heteroscedasticity (EGARCH) filter for generalized hyperbolic (GH), Gaussian random variables, adaptive threshold values and autocorrelation coefficients. EGARCH models are a new variation of GARCH models used especially in economic time series. A speech signal is assumed to have a GH because GH has heavier tails than the Gaussian distribution (GD) covering other heavy tailed distributions like hyperbolic, skewed $t$, variance gamma (VG), inverse Gaussian (NIG), Cauchy, Student’s $t$ and Laplace distributions. The distribution of noise signal is assumed to be uncorrelated (white noise), but in general, that is not necessary. In the proposed method, heteroscedasticity is modeled by EGARCH. A kernel smoothed function of conditional variances and autocorrelations generate the soft detection vector. Finally, hard detection is the result of comparing the soft detection vector with an adaptive threshold value. The simulation results show that the proposed VAD is able to operate down to $-5$ dB.

Appendix

1.1 Proof of (6)

Note that we can write

$$\begin{aligned} f_X \left(x\right)&= \int ^{\infty }_0 f_W \left(w\right) f_{X \mid W = w} \left( x \left|w\right.\right) dw \\&= \int \limits ^{\infty }_0 \frac{ 1}{ \sqrt{2\pi w}\eta } \exp \left\{ -\frac{ 1}{ 2} \frac{ \left(x-\mu -w\gamma \right)^2}{ w\eta ^2} \right\} f_W \left(w\right)dw \\&= \int \limits ^{\infty }_0 \frac{ \exp \left\{ \frac{ \gamma \left(x-\mu \right)}{ \eta ^2} \right\} }{ \sqrt{2\pi w}\eta } \exp \left\{ \frac{ \left(x-\mu \right)^2}{ 2w \eta ^2}\right\} \exp \left\{ -\frac{ \gamma ^2 \eta ^{-2}}{ 2 / w} \right\} f_W \left(w\right)dw, \end{aligned}$$

where $f_W (w)$ denotes the pdf of $W$. Substituting the form for $f_W (\cdot )$ from (5), we obtain

$$\begin{aligned} f_X \left(x\right)&= \frac{ 1}{ 2} \frac{ \psi ^{\lambda } \left(\sqrt{\chi \psi }\right)^{-\lambda } \exp \left\{ \frac{ \gamma \left(x-\mu \right)}{ \eta ^2}\right\} }{ \sqrt{2\pi }\eta k_{\lambda } \left(\sqrt{\chi \psi }\right)} \nonumber \\&\quad \times \int \limits ^{\infty }_0 w^{\lambda - \frac{ 1}{ 2} - 1} \exp \left\{ -\frac{ \eta ^{-2} \left(x-\mu \right)^2+\chi }{ 2w} - \frac{ \gamma ^2 {\eta }^{-2}+\psi }{ 2/w} \right\} dw. \end{aligned}$$

(15)

Transforming $w$ to

$$\begin{aligned} y=w \frac{ \sqrt{\gamma ^2 \eta ^{-2}+\psi }}{ \sqrt{\eta ^{-2} \left(x-\mu \right)^2+\chi }} \end{aligned}$$

reduces (15) to

$$\begin{aligned} f_X \left(x\right)&= \frac{ \left(\sqrt{\chi \psi }\right)^{-\lambda } {\psi }^{\lambda } \left[ \psi + \left(\frac{ \gamma }{ \eta }\right)^2 \right]^{\frac{1}{2}-\lambda }}{ \sqrt{2\pi {\eta }^2} K_{\lambda } \left(\sqrt{\chi \psi }\right)} \frac{ \exp \left\{ \frac{ \gamma \left(x-\mu \right)}{ \eta ^2} \right\} }{ \left[ \sqrt{\left( \eta ^{-2} \left(x-\mu \right)^2 + \chi \right) \left( \gamma ^2 \eta ^{-2} + \psi \right)} \right]^{\frac{ 1}{ 2}-\lambda }} \\&\quad \times \int \limits ^{\infty }_0 \frac{1}{2} y^{\lambda - \frac{ 1}{ 2} - 1} \exp \left\{ -\frac{ 1}{ 2} \sqrt{\left( \eta ^{-2} \left(x-\mu \right)^2 + \chi \right) \left( \gamma ^2 \eta ^{-2}+\psi \right)} \left[ \frac{ 1}{ y}+y\right]\right\} dy. \end{aligned}$$

The result follows by the definition of the modified Bessel function of the third kind. $\square$

1.2 Mean and variance of the GH distribution

If $W$ is a GIG random variable then

$$\begin{aligned} \mathbb E \left( W^{\alpha }\right)&= \frac{ \chi ^{-\lambda } \left(\sqrt{\chi \psi }\right)^{\lambda }}{ k_{\lambda } \left(\sqrt{\chi \psi }\right)} \int \limits ^\infty _0 \frac{ 1}{ 2} w^{\lambda +\alpha - 1} \exp \left\{ -\frac{ 1}{ 2}\left(\chi w^{-1}+\psi w\right) \right\} dw \\&= \left( \frac{ \chi }{ \psi }\right)^{\alpha / 2} \frac{ K_{\lambda +\alpha } \left(\sqrt{\chi \psi } \right)}{ k_{\lambda } \left(\sqrt{\chi \psi }\right)} \\&\times \int \limits ^{\infty }_0 \underbrace{\frac{ \chi ^{-\left(\lambda +\alpha \right)} \left(\sqrt{\chi \psi }\right)^{\lambda +\alpha }}{ 2k_{\lambda +\alpha } \left(\sqrt{\chi \psi }\right)} w^{\lambda +\alpha -1} \exp \left\{ -\frac{ 1}{ 2} \left(\chi w^{-1}+\psi w\right) \right\} }_{f_W \left(w \right)} dw \\&= \left( \frac{ \chi }{ \psi }\right)^{\alpha / 2} \frac{ K_{\lambda +\alpha } \left(\sqrt{\chi \psi } \right)}{ k_{\lambda } \left(\sqrt{\chi \psi }\right)}. \end{aligned}$$

In addition, $\mathbb E (X) = \mu + \mathbb E (W) \gamma$ and Var $(X) = \eta ^2 \mathbb E (W) + \gamma ^2 \mathrm{Var} (W)$. $\square$

1.3 Proof of Proposition 1

Let $X\sim \ GH (\lambda , \chi , \psi , \mu , \eta ^2, \gamma )$. By (2),

$$\begin{aligned} \phi _X\left(s\right)&= \mathbb E \left( \mathbb E \left( \exp (\mathrm{i} sX) \left|W\right.\right)\right) = \mathbb E \left( \exp \left(\mathrm{i} s\left(\mu +W\gamma \right) - W s^2{\eta }^2 / 2 \right)\right)\\&= \exp (\mathrm{i} s\mu ) H\left( s^2 \eta ^2 / 2 - \mathrm{i} s\gamma \right), \end{aligned}$$

where $H (\theta ) = \mathbb E [\exp (-\theta W)]$ is the Laplace transform of a GIG random variable.

Let $Y=c+\sum ^n_{j=1} b_jX_j$ and let $X_j \sim GH (\lambda , \chi , \psi , \mu _j, \eta ^2_j, \gamma _j)$ for each $j$. Then,

$$\begin{aligned} \phi _Y\left(s\right)&= \mathbb E \left[ \exp (\mathrm{i} s\left(c+\sum ^n_{j=1} b_j X_j \right) \right] \\&= \exp (\mathrm{i} sc) \phi _X \left(s\sum ^n_{j=1}{b_j}\right) \\&= \exp \left[ \mathrm{i} s \left(\mu \sum ^n_{j=1}{b_j}+c\right) \right] H\left( s^2 \eta ^2 \sum ^n_{j=1} b^2_j / 2 - \mathrm{i} s \gamma \sum ^n_{j=1}{b_j}\right). \end{aligned}$$

The result follows. $\square$