From TheBestLinks.com
The Z-transform converts a discrete time domain signal, which is 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.
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)})
- 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.
Z-transform with a finite range of n and a finite number of uniformly-spaced z values can be computed efficiently via Bluestein's FFT algorithm. The discrete Fourier transform is a special case of such a Z-transform obtained by restricting z to lie on the unit circle.
See also: Formal power series
External links
de:Z-Transformation
Related links
Top visited
0 of
0 links
[no links posted yet]
>> place link >>
Discussion
Last posted
0 of
0 messages
[no messages posted yet]
>> post message >>
Watch
You can
add this article to your own "watchlist" and receive e-mail notification about all changes in this page.
