About variable resolution spectral analysis

This technique is fully described in [1] where it is applied to speech for coding purposes. Attempts to use this technique in speech and speaker recognition are described in [2,3].

Briefly, it consists in applying a bi-linear transformation of the frequency scale defined by

$$\overline{\omega} = \arctan \left\vert \frac{(1-\alpha^{2}) \sin \omega}{(1+\alpha^{2}) \cos \omega - 2 \alpha} \right\vert$$

Spectral transformations for various values of $\alpha$ are plotted in figure  where the x-axis is the original frequency scale ($\omega$) and the y-axis, the transformed frequency scale ( $\overline{\omega}$).

Figure: Bi-linear transformation of the frequency scale