Voice activity detectors are designed to always discriminate between features of speech and noise. The long term spectral divergence approach is used to produce a decision rule aimed at minimizing the number of decision errors. It is inherently a non-causal procedure since the decision of a frame depends on features of future temporal frames.

Suppose the received signal at the microphone is given as: $y(t)= s(t) + \nu(t)$

where $s(t)$ is the desired speech signal and $\nu(t)$ is i.i.d zero mean Gaussian noise. The frequency domain representation then becomes $y(t, \omega) = s(t,\omega) + \nu{(t,\omega)}$

The long term spectral envelope, denoted $\alpha(\omega)$ is given as $\alpha(t,\omega) = \underset{\tau}{argmax} ~~|y(t+\tau, \omega)|,~~ \tau \in [-T,T]$

It is clear that $\alpha(t,\omega)$ is non-causal and a such a buffer has to be used for real time implementation. The size of $T$ will impact the overall systems latency. Too big a $T$ means large latency whilst too small a $T$ will mean not enough averaging which may cut off some speech frames or result in abrupt transitions. The long term spectral estimate, denoted $\gamma(t)$ then is extracted from $\alpha(t,\omega)$ as $\gamma(t) = \frac{1}{|\omega|_k} \sum\limits_{w} \left( \frac{\alpha(t,\omega) }{\alpha_n(\omega) } \right)^2$

where $\alpha_n(\omega)$ is an estimate of the noise spectrum and $||_k$ denotes cardinality .
A detection threshold, $\beta_T(t)$ is used and is defined as: $\beta_T(t)= \begin{cases}\beta_0, & \text{if}\ \gamma(t) \le \gamma_0\\\frac{\beta_0 - \beta_1}{\gamma_0 - \gamma_1} \gamma(t) + \gamma_0 -\gamma_0 \frac{\gamma_0 - \gamma_1}{\gamma_0-\gamma_1}, & \text{if}\ \gamma_0 < \gamma(t) < \gamma_1 \\\beta_1, & \text{if}\ \gamma(t) \ge \gamma_1\end{cases}$

Here, $\gamma_0$ is the expected noise floor for clean speech, $\gamma_1$ is the noise floor for high noise condition. $\beta_0$ and $\beta_1$ are constants used to ensure a sigmoid like activation function. A smoothening function can be applied to both the threshold and the noise estimates to prevent spurious transitions.

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