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

<math>Z(\{x(n)\}) = X(z) = \sum_{n=-\infty}^{\infty}x(n)z^{-n}</math>

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

<math>Z(\{x(n)\}) = X(z) = \sum_{n=0}^{\infty}x(n)z^{-n}</math>

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.

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:

<math>x(n)=\frac{1}{2\pi i}\oint_CX(z)z^{n-1}dz</math>

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 (http://mathworld.wolfram.com/Z-Transform.html)