In
mathematics and
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, the Z-transform converts a
discrete-time signal, which is a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
or
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, into a complex
frequency-domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a si ...
(z-domain or z-plane) representation.
It can be considered as a discrete-time equivalent of the
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
(s-domain).
This similarity is explored in the theory of
time-scale calculus In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying ...
.
Whereas the
continuous-time Fourier transform is evaluated on the Laplace s-domain's imaginary line, the
discrete-time Fourier transform is evaluated over the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
of the z-domain. What is roughly the s-domain's left
half-plane, is now the inside of the complex unit circle; what is the z-domain's outside of the unit circle, roughly corresponds to the right half-plane of the s-domain.
One of the means of designing
digital filter
In signal processing, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, t ...
s is to take analog designs, subject them to a bilinear transform which maps them from the s-domain to the z-domain, and then produce the digital filter by inspection, manipulation, or numerical approximation. Such methods tend not to be accurate except in the vicinity of the complex unity, i.e. at low frequencies.
History
The basic idea now known as the Z-transform was known to
Laplace, and it was re-introduced in 1947 by
W. Hurewicz[
] and others as a way to treat sampled-data control systems used with radar. It gives a tractable way to solve linear, constant-coefficient
difference equations. It was later dubbed "the z-transform" by
Ragazzini Ragazzini is an Italian surname. Notable people with the surname include:
* Giovanni Battista Ragazzini ( 1520– 1591), Italian painter
*John R. Ragazzini
John Ralph Ragazzini (January 3, 1912 – November 22, 1988) was an American electri ...
and
Zadeh in the sampled-data control group at Columbia University in 1952.
The modified or
advanced Z-transform was later developed and popularized by
E. I. Jury.
The idea contained within the Z-transform is also known in mathematical literature as the method of
generating functions which can be traced back as early as 1730 when it was introduced by
de Moivre in conjunction with probability theory.
From a mathematical view the Z-transform can also be viewed as a
Laurent series where one views the sequence of numbers under consideration as the (Laurent) expansion of an analytic function.
Definition
The Z-transform can be defined as either a ''one-sided'' or ''two-sided'' transform. (Just like we have the
one-sided Laplace transform and the
two-sided Laplace transform
In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment generating function. Two-sided Laplace transforms are closely related to the Fourier transform, the Mellin t ...
.)
Bilateral Z-transform
The ''bilateral'' or ''two-sided'' Z-transform of a discrete-time signal