Definition The Z-transform of a signal x(n) is the function X(z) defined by
where n is an integer and z is a complex number.
Sometimes we are only interested in the values of the signal x(n) for non-negative values of n. If such is the case, the Z-transform is defined as
The latter is sometimes called a unilateral Z-transform and the former a bilateral or doubly infinite Z-transform. In signal processing, the latter definition is used when the signal is causal[?] in nature.
The inverse Z-transform can be computed as follows:
where C is any closed curve around the origin and lying in the region of convergence[?] of X(z).
The (unilateral) Z-transform is to discrete time domain signals what the Laplace transform is to continuous time domain signals.
The Discrete Fourier transform is a special case of the Z-transform obtained by restricting z to lie on the unit circle.
See also: Formal power series
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