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# Z-transform

The Z-transform converts a discrete time domain signal, which is basically a sequence of real numbers, into a complex frequency domain representation.

## 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.

An important example of the unilateral Z-transform is the probability generating function, where the component x(n) is the probability that a discrete random variable takes the value n, and the function X(z) is usually written as X(s), in terms of s = z-1. The properties of Z-transforms (below) have useful interpretations in the context of probability theory.

## Properties

Z({a1x1(n)+a2x2(n)}) = a1Z({x1(n)}) + a2Z({x2(n)})

• Shift. Time-shifting the signal by a distance of k to the right results in multiplying the Z-transform by z-k.
Z({x(n-k)}) = z-kZ({x(n)})

Z({x(n)}*{y(n)}) = Z({x(n)})Z({y(n)})

• Differentiation .
Z({nx(n)}) = -z dZ({x(n)})/dz

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.