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:
where is the desired speech signal and is i.i.d zero mean Gaussian noise. The frequency domain representation then becomes
The long term spectral envelope, denoted is given as
It is clear that is non-causal and a such a buffer has to be used for real time implementation. The size of will impact the overall systems latency. Too big a means large latency whilst too small a will mean not enough averaging which may cut off some speech frames or result in abrupt transitions. The long term spectral estimate, denoted then is extracted from as
where is an estimate of the noise spectrum and denotes cardinality .
A detection threshold, is used and is defined as:
Here, is the expected noise floor for clean speech, is the noise floor for high noise condition. and 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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