The Phase Vocoder

• A classic vocoder, as previously discussed, uses bandpass filters of fixed frequency for both its analysis and resynthesis phases. During the analysis phase of the system, only a channel gain is computed (and phase information is ignored).

• The DFT can also be viewed as an $N$ channel filterbank with fixed center frequencies (the $\hat{\omega}_k$ values).

• However, for each frequency component, the DFT computes both a gain and a phase. The phase data can provide information about the deviation of a sinusoidal component from its channel center frequency.

• For example, consider the sinusoidal signal
    $$x(t) = A \cos( \omega t ).$$ (6)

• The argument of the cosine function is a constantly changing value in the range 0 to $2 \pi$ (or $-\pi$ to $\pi$). If the frequency is fixed, the rate of change (or the derivative), is constant. If the frequency is changing, this will be reflected in the time derivative of the phase.

• In other words, the instantaneous frequency of the signal is directly proportional to the derivative of its phase argument.

• The “phase vocoder” takes advantage of this fact to allow natural sounding frequency-domain manipulations, such as time-stretching/compression and pitch shifting, to be performed.

• In order to compute the instantaneous frequency, the complex FFT bin data must be converted to a polar representation to get a phase value.

• The phase then should be “unwrapped”. That is, the phase is typically constrained to the range $-\pi$ to $\pi$. For these calculations, it is necessary to compute the actual phase without such constraints.

• The phase is then “differentiated” using a differencing operation:
    $$F_k[m] = \frac{\Omega_k[m] - \Omega_k[m-1]}{2 \pi R T},$$ (7)
  where $m$ is a frame index, $R$ is the frame overlap in samples, and $T$ is the sample period.

• Finally, the center frequency for a given bin $k$ should be added to this result to get the actual channel frequency.

• A simple phase vocoder implementation is provided in the Pd audio example patch I07.phase.vocoder in the documentation browser of plugdata.

• A Matlab phase vocoder example is provided by Dan Ellis.