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.

Table of contents

1 Definition 2 Properties 3 External links

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

• Linearity. The Z-transform of the linear combination of two signals is the linear combination of the individual Z-transforms.

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)})

• Convolution. The Z-transform of the convolution of two sequences is the product of the individual Z-transforms.

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.

See also: Formal power series

External links

• http://mathworld.wolfram.com/Z-Transform.html