This is machine translation

Translated by Microsoft
Mouseover text to see original. Click the button below to return to the English version of the page.

Note: This page has been translated by MathWorks. Please click here
To view all translated materials including this page, select Japan from the country navigator on the bottom of this page.


Chirp Z-transform


y = czt(x,m,w,a)
y = czt(x)


y = czt(x,m,w,a) returns the chirp Z-transform of signal x. The chirp Z-transform is the Z-transform of x along a spiral contour defined by w and a. m is a scalar that specifies the length of the transform, w is the ratio between points along the z-plane spiral contour of interest, and scalar a is the complex starting point on that contour. The contour, a spiral or “chirp” in the z-plane, is given by

z = a*(w.^-(0:m-1))

y = czt(x) uses the following default values:

  • m = length(x)

  • w = exp(-j*2*pi/m)

  • a = 1

With these defaults, czt returns the Z-transform of x at m equally spaced points around the unit circle. This is equivalent to the discrete Fourier transform of x, or fft(x). The empty matrix [] specifies the default value for a parameter.

If x is a matrix, czt(x,m,w,a) transforms the columns of x.


collapse all

Create a random vector, x, of length 1013. Compute its DFT using czt.

rng default
x = randn(1013,1);
y = czt(x);

Use czt to zoom in on a narrow-band section of a filter's frequency response.

Design a 30th-order lowpass FIR filter using the window method. Specify a sample rate of 1 kHz and a cutoff frequency of 125 Hz. Use a rectangular window. Find the transfer function of the filter.

fs = 1000;
d = designfilt('lowpassfir','FilterOrder',30,'CutoffFrequency',125, ...
h = tf(d);

Compute the DFT and the CZT of the filter. Restrict the frequency range of the CZT to the band between 100 and 150 Hz. Generate 1024 samples in each case.

m = 1024;
y = fft(h,m);

f1 = 100;
f2 = 150;
w = exp(-j*2*pi*(f2-f1)/(m*fs));
a = exp(j*2*pi*f1/fs);
z = czt(h,m,w,a);

Plot the transforms. Zoom in on the area of interest.

fn = (0:m-1)'/m;
fy = fs*fn;
fz = (f2-f1)*fn + f1;

axis([f1 f2 0 1.2])

axis([f1 f2 0 1.2])
xlabel('Frequency (Hz)')


If m, w, or a is not a scalar, czt gives the following error message:

Inputs M, W, and A must be scalars.


czt uses the next power-of-2 length FFT to perform a fast convolution when computing the z-transform on a specified chirp contour [1].


[1] Rabiner, Lawrence R., and Bernard Gold. Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975, pp. 393–399.

See Also


Introduced before R2006a

Was this topic helpful?