The Phase Vocoder

The steps normally taken by a ``phase vocoder'' to measure instantaneous amplitude and frequency for each bin of the current STFT frame are as follows (extending the four steps of the previous section):

5. Convert each FFT bin $\tilde{x}_m^\prime (e^{j\omega_k })$ from rectangular to polar form to get the magnitude and phase in each FFT bin, and differentiate the unwrapped phase to obtain instantaneous frequency:

$$A_k(m) \mathrel{\stackrel{\mathrm{\Delta}}{=}} \left\vert\tilde{x}_m^\prime (e^{j\omega_k })\right\vert$$ (1)

$$\Theta _k(m) \mathrel{\stackrel{\mathrm{\Delta}}{=}} \angle\tilde{x}_m^\prime (e^{j\omega_k }) \qquad \hbox{(radians)}$$ (2)

$$F_k(m) \mathrel{\stackrel{\mathrm{\Delta}}{=}} \frac{\Theta _k(m) - \Theta _k(m-1)}{2\pi R T} \qquad \hbox{(Hz)}$$ (3)