Chirp z-Transform :: Special Topics (Signal Processing Toolbox)

Signal Processing Toolbox

Chirp z-Transform

The chirp z-transform, or CZT, computes the z-transform along spiral contours in the z-plane for an input sequence. Unlike the DFT, the CZT is not constrained to operate along the unit circle, but can evaluate the z-transform along contours described by

where A is the complex starting point, W is a complex scalar describing the complex ratio between points on the contour, and M is the length of the transform.

One possible spiral is

A = 0.8*exp(j*pi/6);
W = 0.995*exp(-j*pi*.05);
M = 91;
z = A*(W.^(-(0:M-1)));
zplane([],z.')

czt(x,M,W,A) computes the z-transform of x on these points.

An interesting and useful spiral set is m evenly spaced samples around the unit circle, parameterized by A = 1 and W = exp(-j*pi/M). The z-transform on this contour is simply the DFT, obtained by

```
y = czt(x)
```

czt may be faster than the fft function for computing the DFT of sequences with certain odd lengths, particularly long prime-length sequences.

Specialized Transforms Discrete Cosine Transform