Frequency modulation detection for speech

Frequency modulation (FM) as a computational auditory scene analysis feature refers to the phoneme changes in audio streams. This is accomplished by using a two dimensional cochleagram to corresponding to the FM pattern that we want to pick up. This detection using the cochleagram is done by convolution with a time-frequency kernel.
Consider a two dimensional time-frequency zero mean Gaussian kernel defined as:

$G(\tau, \omega, \sigma_\tau , \sigma_\omega) = \frac{1}{2 \pi \sigma_\tau \sigma_\omega} \exp{\left(-\frac{\tau^2}{2\sigma_{\tau}^2} -\frac{\omega^2}{2\sigma_{\omega}^2}\right)}$

To observe a frequency change the Laplacian of the Gaussian filter is used. The Laplacian filter is:

$L(\tau, \omega, \sigma_\tau , \sigma_\omega) = \left(\frac{1}{\sigma_\omega^2} - \frac{\omega^2}{\sigma_\omega^4}\right) G(\tau, \omega, \sigma_\tau , \sigma_\omega)$

The time-frequency variances are chosen such that the Laplacian corresponds to a receptive fields in the human auditory system. Typically, the time variance is chosen to be bigger than the frequency variance. A sample Laplacian filter for FM is shown in Figure 1 below:

Figure 1: Laplacian filter for frequency change detection

FM is can be used as a feature for speech recognition systems by finding the pitch corresponding to frequency changes.

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Frequency modulation detection for speech