Inverse Z-transform
The inverse Z-transform is
where is a counterclockwise closed path encircling the origin and entirely in the region of convergence (ROC). In the case where the ROC is causal (see Example 2), this means the path must encircle all of the poles of .
A special case of this contour integral occurs when is the unit circle (and can be used when the ROC includes the unit circle which is always guaranteed when is stable, i.e. all the poles are within the unit circle). The inverse Z-transform simplifies to the inverse discrete-time Fourier transform:
The Z-transform with a finite range of n and a finite number of uniformly spaced z values can be computed efficiently via Bluestein's FFT algorithm. The discrete-time Fourier transform (DTFT) (not to be confused with the discrete Fourier transform (DFT)) is a special case of such a Z-transform obtained by restricting z to lie on the unit circle.
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Famous quotes containing the word inverse:
“Yet time and space are but inverse measures of the force of the soul. The spirit sports with time.”
—Ralph Waldo Emerson (1803–1882)