In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
and
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
, 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 cal ...
of
real 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 for ...
s, into a complex valued
frequency-domain (the z-domain or z-plane) representation.
It can be considered a discrete-time equivalent of the
Laplace transform
In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a f ...
(the ''s-domain'' or ''s-plane'').
This similarity is explored in the theory of
time-scale calculus.
While the
continuous-time Fourier transform is evaluated on the s-domain's vertical axis (the imaginary axis), the
discrete-time Fourier transform is evaluated along the z-domain's
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 ...
. The s-domain's left
half-plane maps to the area inside the z-domain's unit circle, while the s-domain's right half-plane maps to the area outside of the z-domain's unit circle.
In signal processing, one of the means of designing
digital filter
In signal processing, a digital filter is a system that performs mathematical operations on a Sampling (signal processing), sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other ma ...
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 foundational concept now recognized as the Z-transform, which is a cornerstone in the analysis and design of digital control systems, was not entirely novel when it emerged in the mid-20th century. Its embryonic principles can be traced back to the work of the French mathematician
Pierre-Simon Laplace
Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
, who is better known for the
Laplace transform
In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a f ...
, a closely related mathematical technique. However, the explicit formulation and application of what we now understand as the Z-transform were significantly advanced in 1947 by
Witold Hurewicz and colleagues. Their work was motivated by the challenges presented by sampled-data control systems, which were becoming increasingly relevant in the context of
radar
Radar is a system that uses radio waves to determine the distance ('' ranging''), direction ( azimuth and elevation angles), and radial velocity of objects relative to the site. It is a radiodetermination method used to detect and track ...
technology during that period. The Z-transform provided a systematic and effective method for solving linear difference equations with constant coefficients, which are ubiquitous in the analysis of discrete-time signals and systems.
[
]
The method was further refined and gained its official nomenclature, "the Z-transform," in 1952, thanks to the efforts of
John R. Ragazzini and
Lotfi A. Zadeh, who were part of the sampled-data control group at Columbia University. Their work not only solidified the mathematical framework of the Z-transform but also expanded its application scope, particularly in the field of electrical engineering and control systems.
A notable extension, known as the modified or
advanced Z-transform, was later introduced by
Eliahu I. Jury. Jury's work extended the applicability and robustness of the Z-transform, especially in handling initial conditions and providing a more comprehensive framework for the analysis of digital control systems. This advanced formulation has played a pivotal role in the design and stability analysis of discrete-time control systems, contributing significantly to the field of digital signal processing.
Interestingly, the conceptual underpinnings of the Z-transform intersect with a broader mathematical concept known as the method of
generating functions, a powerful tool in combinatorics and probability theory. This connection was hinted at as early as 1730 by
Abraham de Moivre, a pioneering figure in the development of probability theory. De Moivre utilized generating functions to solve problems in probability, laying the groundwork for what would eventually evolve into the Z-transform. From a mathematical perspective, the Z-transform can be viewed as a specific instance of a
Laurent series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansio ...
, where the
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 cal ...
of numbers under investigation is interpreted as the
coefficients
In mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or any other type of expression. It may be a number without units, in which case it is known as a numerical factor. It may also be a ...
in the (Laurent) expansion of an
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
. This perspective not only highlights the deep mathematical roots of the Z-transform but also illustrates its versatility and broad applicability across different branches of mathematics and engineering.
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.)
Bilateral Z-transform
The ''bilateral'' or ''two-sided'' Z-transform of a discrete-time signal