# Phase Vocoder

# 1.1 Parameters

These can be tuned to taste.

$M$: input audio block size (integer, eg 8192).

$\Delta I$: input audio hop size (integer, eg 19).

$N$: zero-padded block size (for DFT, integer, eg 262144).

$\Delta O$: output audio hop size (integer, eg 432).

The time dilation factor is $\frac{\Delta O}{\Delta I}$.

# 1.2 Algorithm

Take $M$ samples of input audio every $\Delta I$ samples.

Multiply by raised cosine window of length $M$ (peak amplitude $2$, mean $1$).

Zero-pad to length $N$, call it $x(n)$ where $n$ is the block index in $0, 1, 2, \ldots$.

Take the discrete Fourier transform of $x(n)$, call it $X(n)$.

For each bin, normalize the (complex-valued) ratio $\frac{X(n)}{X(n - 1)}$ to magnitude $1$, and raise it to the power $\frac{\Delta O}{\Delta I}$ (which need not be an integer). Call the result $\delta \theta(n)$. In case of division by zero or other badness, set $\delta \theta(n) = 1$.

Increment the phase of each bin by $\theta(n) = \theta(n - 1) \times \delta \theta(n)$ and normalize (just to be safe in case of rounding errors). Phase of $\theta(-1)$ is probably arbitrary but should have magnitude $1$.

Then the output Fourier transform has the input’s magnitude with the accumulated phase: $Y(n) = |X(n)| \times \theta(n)$.

Take the inverse Fourier transform of $Y(n)$, call it $y(n)$.

Multiply by raised cosine window of length $M$ (peak amplitude $2$, mean $1$).

Multiply by gain factor: $G = \frac{\Delta O}{M \times N}$. This assumes that the gain of the DFT/FFT is not normalized.

Accumulate $M$ samples spaced every $\Delta O$ samples to output audio stream (overlap-add).

# 1.3 References

• Pure-data documentation 3.audio.examples/I07.phase.vocoder.pd (Pd version 0.53).