Editing Z-transform (section)

{{Short description|Mathematical transform which converts signals from the time domain to the frequency domain}}
{{distinguish|Fisher z-transformation}}

In [[mathematics]] and [[signal processing]], the '''Z-transform''' converts a [[discrete-time signal]], which is a [[sequence]] of [[real number|real]] or [[complex number]]s, into a complex valued [[frequency-domain]] (the '''z-domain''' or '''z-plane''') representation.<ref name="Mandal 2020 pp. 157–195">{{cite book | last=Mandal | first=Jyotsna Kumar | title=Reversible Steganography and Authentication via Transform Encoding | chapter=''Z''-Transform-Based Reversible Encoding | series=Studies in Computational Intelligence | publisher=Springer Singapore | publication-place=Singapore | year=2020 | volume=901 | isbn=978-981-15-4396-8 | issn=1860-949X | doi=10.1007/978-981-15-4397-5_7 | pages=157–195 | s2cid=226413693 | quote=Z is a complex variable. Z-transform converts the discrete spatial domain signal into complex frequency domain representation. Z-transform is derived from the Laplace transform.}}</ref><ref name="Lynn 1986 pp. 225–272">{{cite book | last=Lynn | first=Paul A. | title=Electronic Signals and Systems | chapter=The Laplace Transform and the ''z''-transform | publisher=Macmillan Education UK | publication-place=London | year=1986 | isbn=978-0-333-39164-8 | doi=10.1007/978-1-349-18461-3_6 | pages=225–272|quote=Laplace Transform and the ''z''-transform are closely related to the Fourier Transform. ''z''-transform is especially suitable for dealing with discrete signals and systems. It offers a more compact and convenient notation than the discrete-time Fourier Transform.}}</ref><ref name="JuryBook">{{Cite book |last=Jury |first=Eliahu Ibrahim |title=Theory and application of the z-transform method |publisher=John Wiley & Sons |year=1964 |location=New York |pages=XIII, 330 s. |language=en}}</ref>

It can be considered a discrete-time equivalent of the [[Laplace transform]] (the ''s-domain'' or ''s-plane'').<ref name="Palani pp. 921–1055">{{cite book | last=Palani | first=S. | title=Signals and Systems | chapter=The ''z''-Transform Analysis of Discrete Time Signals and Systems | publisher=Springer International Publishing | publication-place=Cham | date=2021-08-26 | isbn=978-3-030-75741-0 | doi=10.1007/978-3-030-75742-7_9 | pages=921–1055 | s2cid=238692483 | quote=''z''-transform is the discrete counterpart of Laplace transform. ''z''-transform converts difference equations of discrete time systems to algebraic equations which simplifies the discrete time system analysis. Laplace transform and ''z''-transform are common except that Laplace transform deals with continuous time signals and systems.}}</ref> 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]]. 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]]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.