Copyright © Philip M. Parker, INSEAD. Terms of Use.
| Domain | Definition |
Electrical Engineering | Laplace transform of a periodically sampled signal with change of variable Z = e power Ts, where T is the sampling period. Source: European Union. (references) |
Source: compiled by the editor from various references; see credits. | |
(From Wikipedia, the free Encyclopedia)
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)})
- Linearity. The Z-transform of the linear combination of two signals is the linear combination of the individual Z-transforms.
Z({x(n-k)}) = z-kZ({x(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)}*{y(n)}) = Z({x(n)})Z({y(n)})
- Convolution. The Z-transform of the convolution of two sequences is the product of the individual Z-transforms.
Z({nx(n)}) = -z dZ({x(n)})/dz
- Differentiation .
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
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Z-transform."
| Language | Translations for "Z-TRANSFORM"; alternative meanings/domain in parentheses. | |
Finnish | Z-muunnos. (various references) | |
French | transformée en Z. (various references) | |
German | Z-Transformation. (various references) | |
Greek | μετασχηματισμός Ζ. (various references) | |
Italian | trasformata Z. (various references) | |
Portuguese | transformada em Z. (various references) | |
Spanish | transformada en Z. (various references) | |
Swedish | Z-transform. (various references) | |
| Source: compiled by the editor from various translation references. | ||
Scrabble® Enable2K-Verified Anagrams | |
| Words within the letters "a-f-m-n-o-r-r-s-t-z" | |
-1 letter: transform. | |
-2 letters: formants. | |
-3 letters: fantoms, formant, formats, marrons, matrons, mortars, transom. | |
-4 letters: armors, fantom, forams, format, fronts, manors, marron, matron, matzos, mortar, ramson, ransom, razors, romans, rostra, sartor, stroma, tomans, tronas. | |
-5 letters: amort, armor, arson, atoms, azons, fanos, farms, faros, fatso, foams, fonts, foram, forms, forts, frats, frons, front, frost, manor, manos, marts, mason, matzo. | |
| Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro. | |
| 1. Translations: Modern 2. Anagrams 3. Bibliography |
Copyright © Philip M. Parker, INSEAD. Terms of Use.
