The chirp Z-transform (CZT) is useful in evaluating the Z-transform
along contours other than the unit circle. The chirp Z-transform is
also more efficient than the DFT algorithm for the computation of
prime-length transforms, and it is useful in computing a subset of
the DFT for a sequence. 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 *z _{ℓ}* =

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)

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