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# Amplitude modulation spectrum detection for speech

Speech signals have a periodic envelope which can be used for speech separation. The choice of amplitude modulation spectrum (AMS) for speech separation is motivated by biological observations that variations in modulation frequency provides a long term window for characterizing speech information, especially in noisy and reverberant environments.
Suppose the received signal at the microphone is given as:

$y[n]= s[n] + \nu[n]$

where $s[n]$ is the desired speech signal and $\nu[n]$ is i.i.d zero mean Gaussian noise with variance $\sigma_{\nu^2}$. The short term spectrum of each frame is computed such that:

$y(\omega)= s(\omega)+ \nu(\omega)$

The negative frequency components of $y(\omega)$ are set to zero to synthesize a signal $\hat{y}(\omega)$ where:

$\hat{y}(\omega)= \begin{cases}y(\omega) & \omega < \frac{F_s}{2}\\0 &\text{otherwise}\end{cases}$

The inverse short term spectrum of $\hat{y}(\omega)$ is the taken, denoted $\hat{y}[n]$. The AMS is then twice the amplitude of $\hat{y}[n]$. A sample performance of this algorithm is shown in Figure 1 below:

Figure 1: Speech amplitude modulation detection

The AMS signal can also be used to detect the pitch of speech. However, this approach is very sensitive to noise.

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