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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

<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



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