Editing Z-transform

{{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.

== 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]], who is better known for the [[Laplace transform]], 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]] 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.<ref name="kanasewich">
{{cite book|url=https://books.google.com/books?id=k8SSLy-FYagC&q=inauthor%3AKanasewich++poles+stability&pg=PA249|title=Time Sequence Analysis in Geophysics|author=E. R. Kanasewich|publisher=University of Alberta|year=1981|isbn=978-0-88864-074-1|pages=186, 249}}</ref><ref>{{cite book  | title = Time sequence analysis in geophysics  | edition = 3rd  | author = E. R. Kanasewich  | publisher = University of Alberta  | year = 1981  | isbn = 978-0-88864-074-1  | pages = 185–186  | url = https://books.google.com/books?id=k8SSLy-FYagC&pg=PA185}}</ref>

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.<ref>{{cite journal |last1=Ragazzini |first1=J. R. |last2=Zadeh |first2=L. A. |title=The analysis of sampled-data systems |journal=Transactions of the American Institute of Electrical Engineers, Part II: Applications and Industry |date=1952 |volume=71 |issue=5 |pages=225–234 |doi=10.1109/TAI.1952.6371274|s2cid=51674188 }}</ref><ref>{{cite book  | title = Digital control systems implementation and computational techniques  | author = Cornelius T. Leondes  | publisher = Academic Press  | year = 1996| isbn = 978-0-12-012779-5  | page = 123  | url = https://books.google.com/books?id=aQbk3uidEJoC&pg=PA123  }}</ref>

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.<ref>
{{cite book
 | title = Sampled-Data Control Systems
 | author = Eliahu Ibrahim Jury
 | publisher = John Wiley & Sons
 | year = 1958
 }}</ref>{{r|JuryBook}}

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]], where the [[sequence]] of numbers under investigation is interpreted as the [[coefficients]] in the (Laurent) expansion of an [[analytic function]]. 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.{{r|JuryBook}}

==Definition==
The Z-transform can be defined as either a ''one-sided'' or ''two-sided'' transform. (Just like we have the [[Laplace transform|one-sided Laplace transform]] and the [[two-sided Laplace transform]].)<ref name="Jackson 1996 pp. 29–54">{{cite book | last=Jackson | first=Leland B. | title=Digital Filters and Signal Processing | chapter=The z Transform | publisher=Springer US | publication-place=Boston, MA | year=1996 | isbn=978-1-4419-5153-3 | doi=10.1007/978-1-4757-2458-5_3 | pages=29–54 | quote= z transform is to discrete-time systems what the Laplace transform is to continuous-time systems. ''z'' is a complex variable. This is sometimes referred to as the two-sided ''z'' transform, with the one-sided z transform being the same except for a summation from ''n'' = 0 to infinity. The primary use of the one sided transform ... is for causal sequences, in which case the two transforms are the same anyway. We will not, therefore, make this distinction and will refer to ... as simply the z transform of ''x''(''n'').}}</ref>

=== Bilateral Z-transform ===
The ''bilateral'' or ''two-sided'' Z-transform of a discrete-time signal <math>x[n]</math> is the [[formal power series]] <math>X(z)</math> defined as:

{{Equation box 1 |title=
|indent =: |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA
|equation = <math>X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] z^{-n}</math>
}}

where <math>n</math> is an integer and <math>z</math> is, in general, a [[complex number]]. In [[Polar coordinate system#Complex numbers|polar form]], <math>z</math> may be written as:
:<math>z = A e^{j\phi} = A\cdot(\cos{\phi}+j\sin{\phi})</math>
where <math>A</math> is the magnitude of <math>z</math>, <math>j</math> is the [[imaginary unit]], and <math>\phi</math> is the ''[[complex argument]]'' (also referred to as ''angle'' or ''phase'') in [[radian]]s.

=== Unilateral Z-transform ===
Alternatively, in cases where <math>x[n]</math> is defined only for <math>n \ge 0</math>, the ''single-sided'' or ''unilateral'' Z-transform is defined as:

{{Equation box 1 |title=
|indent =: |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA
|equation = <math>X(z) = \mathcal{Z}\{x[n]\} =  \sum_{n=0}^{\infty} x[n] z^{-n}.</math>
}}

In [[signal processing]], this definition can be used to evaluate the Z-transform of the [[Finite impulse response#Frequency response|unit impulse response]] of a discrete-time [[causal system]].

An important example of the unilateral Z-transform is the [[probability-generating function]], where the component <math>x[n]</math> is the probability that a discrete random variable takes the value. The properties of Z-transforms (listed in {{Slink|2=Properties|nopage=y}}) have useful interpretations in the context of probability theory.

==Inverse Z-transform==
The ''inverse'' Z-transform is:

{{Equation box 1 |title=
|indent =: |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA
|equation = <math> x[n] = \mathcal{Z}^{-1} \{X(z) \}= \frac{1}{2 \pi j} \oint_{C} X(z) z^{n-1} dz</math>
}}

where <math>C</math> is a counterclockwise closed path encircling the origin and entirely in the [[Radius of convergence|region of convergence]] (ROC). In the case where the ROC is causal (see [[#Example 2 (causal ROC)|Example 2]]), this means the path <math>C</math> must encircle all of the poles of <math>X(z)</math>.

A special case of this [[contour integral]] occurs when <math>C</math> is the unit circle. This contour can be used when the ROC includes the unit circle, which is always guaranteed when <math>X(z)</math> is stable, that is, when all the poles are inside the unit circle. With this contour, the inverse Z-transform simplifies to the [[Discrete-time Fourier transform#Inverse transform|inverse discrete-time Fourier transform]], or [[Fourier series]], of the periodic values of the Z-transform around the unit circle:

{{Equation box 1 |title=
|indent =: |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA
|equation = <math> x[n] = \frac{1}{2 \pi} \int_{-\pi}^{+\pi}  X(e^{j \omega}) e^{j \omega n} d \omega.</math>
}}

The Z-transform with a finite range of <math>n</math> and a finite number of uniformly spaced <math>z</math> values can be computed efficiently via [[Bluestein's FFT algorithm]]. The [[discrete-time Fourier transform]] (DTFT)—not to be confused with the [[discrete Fourier transform]] (DFT)—is a special case of such a Z-transform obtained by restricting <math>z</math> to lie on the unit circle.

The following three methods are often used for the evaluation of the inverse -transform,

=== Direct Evaluation by Contour Integration ===
This method involves applying the [[Residue theorem|Cauchy Residue Theorem]] to evaluate the inverse Z-transform. By integrating around a closed contour in the complex plane, the residues at the poles of the Z-transform function inside the ROC are summed. This technique is particularly useful when working with functions expressed in terms of complex variables.

=== Expansion into a Series of Terms in the Variables ''z'' and ''z''{{sup|-1}} ===
In this method, the Z-transform is expanded into a power series. This approach is useful when the Z-transform function is rational, allowing for the approximation of the inverse by expanding into a series and determining the signal coefficients term by term.

=== Partial-Fraction Expansion and Table Lookup ===
This technique decomposes the Z-transform into a sum of simpler fractions, each corresponding to known Z-transform pairs. The inverse Z-transform is then determined by looking up each term in a standard table of Z-transform pairs. This method is widely used for its efficiency and simplicity, especially when the original function can be easily broken down into recognizable components.

==== Example:<ref>{{Cite book |last1=Proakis |first1=John |title=Digital Signal Processing Principles, Algorithms and Applications |last2=Manolakis |first2=Dimitris |publisher=PRENTICE-HALL INTERNATIONAL, INC. |edition=3rd}}</ref> ====
A) Determine the inverse Z-transform of the following by series expansion method,

<math display=block>X(z) = \frac{1}{1 - 1.5 z^{-1} + 0.5 z^{-2}}</math>

Solution:

Case 1:

ROC: <math>\left\vert Z \right\vert > 1</math>

Since the ROC is the exterior of a circle, <math>x(n)</math> is causal (signal existing for n≥0).

<math display=block>X(z) = {1\over 1 - {3\over 2}z^{-1} + {1\over 2}z^{-2}} = 1 + {{3\over 2}z^{-1}} + {{7\over 4}z^{-2}} + {{15\over 8}z^{-3}} + {{31\over 16}z^{-4}} +....</math>

thus,

<math display=block>\begin{align} 
  x(n) &= \left\{1 , \frac{3}{2} , \frac{7}{4} , \frac{15}{8} , \frac{31}{16} \ldots \right\} \\ 
  & \qquad\! \uparrow \\ 
\end{align}</math> (arrow indicates term at x(0)=1)

Note that in each step of long division process we eliminate lowest power term of <math>z^{-1}</math>.

Case 2:

ROC: <math>\left\vert Z \right\vert < 0.5</math>

Since the ROC is the interior of a circle, <math>x(n)</math> is anticausal (signal existing for n<0).

By performing long division we get,

<math display=block>X(z) = \frac{1}{1 - \frac{3}{2}z^{-1} + \frac{1}{2}z^{-2} } = 2z^2 + 6z^3 +14z^4 + 30z^5 + \ldots</math>

<math>\begin{align} 
  x(n) & = \{30, 14, 6, 2, 0, 0\} \\ 
  & \qquad \qquad \qquad \quad\ \ \, \uparrow\\ 
\end{align}</math> (arrow indicates term at x(0)=0)

Note that in each step of long division process we eliminate lowest power term of <math>z</math>.

''Note:''

# ''When the signal is causal, we get positive powers of <math>z</math> and when the signal is anticausal, we get negative powers of <math>z</math>.''
# ''<math>z^k</math> indicates term at <math>x(-k)</math> and <math>z^{-k}</math> indicates term at <math>x(k)</math>.''

B) Determine the inverse Z-transform of the following by series expansion method,

Eliminating negative powers if <math>z</math> and dividing by <math>z</math>,

<math display=block>\frac{X(z)}{z} = \frac{z^2}{z(z^2 - 1.5z + 0.5)} = \frac{z}{z^2 - 1.5z + 0.5}  </math>

By Partial Fraction Expansion,

<math display=block>\begin{align}
  \frac{X(z)}{z} &= \frac{z}{(z-1)(z-0.5)} = \frac{A_1}{z-0.5} + \frac{A_2}{z-1} \\[4pt]
  & A_1 = \left. \frac{(z-0.5) X(z)}{z} \right\vert_{z=0.5} = \frac{0.5}{(0.5-1)} = -1 \\[4pt]
  & A_2 = \left. \frac{(z-1) X(z)}{z} \right\vert_{z=1} = \frac{1}{1-0.5} = {2} \\[4pt]
  \frac{X(z)}{z} &= \frac{2}{z-1} - \frac{1}{z-0.5} 
\end{align}</math>

Case 1:

ROC:<math>\left\vert Z \right\vert > 1  </math>

Both the terms are causal, hence <math>x(n)</math> is causal.

<math display=block>\begin{align} 
  x(n) &= 2{(1)^n}u(n) - 1{(0.5)^n}u(n) \\ 
  &= (2-0.5^n) u(n) \\ 
\end{align}</math>

Case 2:

ROC:<math>\left\vert Z \right\vert < 0.5  </math>

Both the terms are anticausal, hence <math>x(n)</math> is anticausal.

<math>\begin{align} 
  x(n) &= -2{(1)^n}u(-n-1) - (-1{(0.5)^n}u(-n-1) ) \\ 
  &= (0.5^n-2) u(-n-1) \\
\end{align}</math>

Case 3:

ROC:<math>0.5 < \left\vert Z \right\vert < 1</math>

One of the terms is causal (p=0.5 provides the causal part) and other is anticausal (p=1 provides the anticausal part), hence <math>x(n)</math> is both sided.

<math>\begin{align} 
  x(n) &= -2{(1)^n}u(-n-1) - 1{(0.5)^n}u(n) \\ 
  &= -2u(-n-1) - 0.5^n u(n) \\ 
\end{align}</math>

==Region of convergence==
{{See also|Pole–zero_plot#Discrete-time systems}}
The [[Radius of convergence|region of convergence]] (ROC) is the set of points in the complex plane for which the Z-transform summation [[Convergent series|converges]] (i.e. doesn't blow up in magnitude to infinity):

:<math>\mathrm{ROC} = \left\{ z : \left|\sum_{n=-\infty}^{\infty}x[n]z^{-n}\right| < \infty \right\} </math>

===Example 1 (no ROC)===
Let <math>x[n] = (.5)^n\ . </math> Expanding <math>x[n]</math> on the interval <math>(-\infty, \infty)</math> it becomes

:<math>x[n] = \left \{\dots, (.5)^{-3}, (.5)^{-2}, (.5)^{-1}, 1, (.5), (.5)^2, (.5)^3, \dots \right \} = \left \{\dots, 2^3, 2^2, 2, 1, (.5), (.5)^2, (.5)^3, \dots \right\}.</math>

Looking at the sum

:<math>\sum_{n=-\infty}^{\infty}x[n]z^{-n} \to \infty.</math>

Therefore, there are no values of <math>z</math> that satisfy this condition.

===Example 2 (causal ROC)===
[[Image:Region of convergence 0.5 causal.svg|thumb|250px|ROC (blue), {{pipe}}''z''{{pipe}}&nbsp;=&nbsp;.5 (dashed black circle), and the unit circle (dotted grey circle).]]

Let <math>x[n] = (.5)^n \, u[n] </math> (where <math>u</math> is the [[Heaviside step function]]). Expanding <math>x[n]</math> on the interval <math>(-\infty, \infty)</math> it becomes

:<math>x[n] = \left \{\dots, 0, 0, 0, 1, (.5), (.5)^2, (.5)^3, \dots \right \}.</math>

Looking at the sum

:<math>\sum_{n=-\infty}^{\infty}x[n]z^{-n} = \sum_{n=0}^{\infty}(.5)^nz^{-n} = \sum_{n=0}^{\infty}\left(\frac{.5}{z}\right)^n = \frac{1}{1 - (.5)z^{-1}}.</math>

The last equality arises from the infinite [[geometric series]] and the equality only holds if <math>|(.5)z^{-1}| < 1 ,</math> which can be rewritten in terms of <math>z</math> as <math>|z| > (.5).</math> Thus, the ROC is <math>|z| > (.5).</math> In this case the ROC is the complex plane with a disc of radius 0.5 at the origin "punched out".{{clear}}

===Example 3 (anti causal ROC)===
[[Image:Region of convergence 0.5 anticausal.svg|thumb|250px|ROC (blue), {{pipe}}''z''{{pipe}}&nbsp;=&nbsp;.5 (dashed black circle), and the unit circle (dotted grey circle).]]

Let <math>x[n] = -(.5)^n \, u[-n-1] </math> (where <math>u</math> is the [[Heaviside step function]]). Expanding <math>x[n]</math> on the interval <math>(-\infty, \infty)</math> it becomes

:<math>x[n] = \left \{ \dots, -(.5)^{-3}, -(.5)^{-2}, -(.5)^{-1}, 0, 0, 0, 0, \dots \right \}.</math>

Looking at the sum

:<math>\begin{align}
\sum_{n=-\infty}^{\infty}x[n] \, z^{-n} &= -\sum_{n=-\infty}^{-1}(.5)^n \, z^{-n} \\
&= -\sum_{m=1}^{\infty}\left(\frac{z}{.5}\right)^{m} \\
&= -\frac{(.5)^{-1}z}{1 - (.5)^{-1}z} \\
&= -\frac{1}{(.5)z^{-1}-1} \\
&= \frac{1}{1 - (.5)z^{-1}} \\
\end{align}</math>

and using the infinite [[geometric series]] again, the equality only holds if <math>|(.5)^{-1} z| < 1</math> which can be rewritten in terms of <math>z</math> as <math>|z| < (.5).</math> Thus, the ROC is <math>|z| < (.5).</math> In this case the ROC is a disc centered at the origin and of radius 0.5.

What differentiates this example from the previous example is ''only'' the ROC. This is intentional to demonstrate that the transform result alone is insufficient.
{{Clear}}

===Examples conclusion===
Examples 2 & 3 clearly show that the Z-transform <math>X(z)</math> of <math>x[n]</math> is unique when and only when specifying the ROC. Creating the [[pole–zero plot]] for the causal and anticausal case show that the ROC for either case does not include the pole that is at 0.5. This extends to cases with multiple poles: the ROC will ''never'' contain poles.

In example 2, the causal system yields a ROC that includes <math>|z| = \infty</math> while the anticausal system in example 3 yields an ROC that includes <math>|z| = 0 .</math>

[[Image:Region of convergence 0.5 0.75 mixed-causal.svg|thumb|250px|ROC shown as a blue ring 0.5&nbsp;<&nbsp;{{pipe}}''z''{{pipe}}&nbsp;<&nbsp;0.75]]
In systems with multiple poles it is possible to have a ROC that includes neither <math>|z| = \infty</math> nor <math>|z| = 0 .</math> The ROC creates a circular band. For example,

:<math>x[n] = (.5)^n \, u[n] - (.75)^n \, u[-n-1]</math>

has poles at 0.5 and 0.75. The ROC will be 0.5 < {{abs|''z''}} < 0.75, which includes neither the origin nor infinity. Such a system is called a mixed-causality system as it contains a causal term <math>(.5)^n \, u[n]</math> and an anticausal term <math>-(.75)^n \, u[-n-1] .</math>

The [[Control theory#Stability|stability]] of a system can also be determined by knowing the ROC alone. If the ROC contains the unit circle (i.e., {{abs|''z''}} = 1) then the system is stable. In the above systems the causal system (Example 2) is stable because {{abs|''z''}} > 0.5 contains the unit circle.

Let us assume we are provided a Z-transform of a system without a ROC (i.e., an ambiguous <math>x[n]</math>). We can determine a unique <math>x[n]</math> provided we desire the following:
* Stability
* Causality

For stability the ROC must contain the unit circle. If we need a causal system then the ROC must contain infinity and the system function will be a right-sided sequence. If we need an anticausal system then the ROC must contain the origin and the system function will be a left-sided sequence. If we need both stability and causality, all the poles of the system function must be inside the unit circle.

The unique <math>x[n]</math> can then be found.

==Properties==
{| class="wikitable"
 |+ '''Properties of the z-transform'''
 !
Property	
! Time domain
	! Z-domain
	! Proof
 	! ROC
 
|-
!Definition of Z-transform
|<math>x[n]</math>
|<math>X(z)</math>
|<math>X(z)=\mathcal{Z}\{x[n]\}</math> (definition of the z-transform)
<math>x[n]=\mathcal{Z}^{-1}\{X(z)\}</math> (definition of the inverse z-transform)
|<math>r_2<|z|<r_1</math>
 |-
 ! [[Linearity]]
	| <math>a_1 x_1[n] + a_2 x_2[n]</math>
	| <math>a_1 X_1(z) + a_2 X_2(z)</math>
	| <math>\begin{align}X(z) &= \sum_{n=-\infty}^{\infty} (a_1x_1[n]+a_2x_2[n])z^{-n} \\
         &= a_1\sum_{n=-\infty}^{\infty} x_1[n] \, z^{-n} + a_2\sum_{n=-\infty}^{\infty}x_2[n] \, z^{-n} \\
         &= a_1X_1(z) + a_2X_2(z) \end{align} </math>
	| Contains ROC<sub>1</sub> ∩ ROC<sub>2</sub>
 |-
 ! [[Upsampling|Time expansion]]
	| <math>x_K[n] = \begin{cases} x[r], & n = Kr \\ 0, & n \notin K\mathbb{Z} \end{cases}</math>
with <math>K\mathbb{Z} := \{Kr: r \in \mathbb{Z}\}</math>
	| <math>X(z^K)</math>
	| <math>\begin{align} X_K(z) &=\sum_{n=-\infty}^{\infty} x_K[n]z^{-n} \\
&= \sum_{r=-\infty}^{\infty}x[r]z^{-rK}\\
&= \sum_{r=-\infty}^{\infty}x[r](z^{K})^{-r}\\
&= X(z^{K}) \end{align}</math>
	| <math>R^{\frac{1}{K}}</math>
 |-
 ! [[Downsampling|Decimation]]
	| <math>x[Kn]</math>
	| <math>\frac{1}{K} \sum_{p=0}^{K-1} X\left(z^{\tfrac{1}{K}} \cdot e^{-i \tfrac{2\pi}{K} p}\right)</math>
	| [http://www2.ece.ohio-state.edu/~schniter/ee700/handouts/multirate.pdf ohio-state.edu] or [http://www.ee.ic.ac.uk/hp/staff/dmb/courses/DSPDF/01100_Multirate.pdf ee.ic.ac.uk]
	| 
 |-
 ! Time delay
	| <math>x[n-k]</math>
with <math>k>0</math> and <math>x : x[n]=0\ \forall \, n<0</math>
	| <math>z^{-k}X(z)</math>
	| <math>\begin{align} \mathcal{Z}\{x[n-k]\} &= \sum_{n=0}^{\infty} x[n-k]z^{-n}\\
&= \sum_{j=-k}^{\infty} x[j]z^{-(j+k)}&& j = n-k \\
&= \sum_{j=-k}^{\infty} x[j]z^{-j}z^{-k} \\
&= z^{-k}\sum_{j=-k}^{\infty}x[j]z^{-j}\\
&= z^{-k}\sum_{j=0}^{\infty}x[j]z^{-j} && x[\beta] = 0,  \beta < 0\\
&= z^{-k}X(z)\end{align} </math>
	| ROC, except <math>z{=}0</math> if <math>k > 0</math> and <math>z {=} \infty</math> if <math>k < 0</math>
 |-
 ! Time advance
	| <math>x[n+k]</math>
with <math>k>0</math>
	| Bilateral Z-transform:
<math display="block">z^kX(z)</math>
Unilateral Z-transform:<ref>{{cite book |last1=Bolzern |first1=Paolo |last2=Scattolini |first2=Riccardo |last3=Schiavoni |first3=Nicola |title=Fondamenti di Controlli Automatici |language=it |publisher=MC Graw Hill Education |isbn=978-88-386-6882-1|year=2015 }}</ref>
<math display="block">z^k \, X(z)-z^k\sum^{k-1}_{n=0}x[n] \, z^{-n}</math>
	| 
	|
 |-
 ! First difference backward
	|  <math>x[n] - x[n-1]</math>
with <math>x[n]{=}0 </math> for <math>n < 0 </math>
	| <math> (1-z^{-1}) \, X(z)</math>
	| 
	| Contains the intersection of ROC of <math>X_1(z)</math> and <math>z \neq 0</math>
 |-
 ! First difference forward
|  <math>x[n+1] - x[n]</math>
	| <math> (z-1) \, X(z)-z \, x[0]</math>
	| 
	|
 |-
 ! Time reversal
	| <math>x[-n]</math>
	| <math>X(z^{-1})</math>
	| <math>\begin{align} \mathcal{Z}\{x(-n)\} &= \sum_{n=-\infty}^{\infty} x[-n]z^{-n} \\
&= \sum_{m=-\infty}^{\infty} x[m]z^{m}\\
&= \sum_{m=-\infty}^{\infty} x[m]{(z^{-1})}^{-m}\\
&= X(z^{-1}) \\
\end{align} </math>
	| <math>\tfrac{1}{r_1}<|z|<\tfrac{1}{r_2}</math>
 |-
 ! Scaling in the z-domain
	| <math>a^n x[n]</math>
	| <math>X(a^{-1}z)</math>
	| <math>\begin{align}\mathcal{Z} \left \{a^n x[n] \right \} &=  \sum_{n=-\infty}^{\infty} a^{n}x[n]z^{-n} \\
&= \sum_{n=-\infty}^{\infty} x[n](a^{-1}z)^{-n} \\
&= X(a^{-1}z)
\end{align} </math>
	| <math>|a|r_2 < |z|< |a|r_1</math> 
 |-

 ! [[Complex conjugation]]
	| <math>x^*[n]</math>
	| <math>X^*(z^*)</math>
	| <math>\begin{align} \mathcal{Z} \{x^*(n)\} &= \sum_{n=-\infty}^{\infty} x^*[n]z^{-n}\\
&= \sum_{n=-\infty}^{\infty} \left [x[n](z^*)^{-n} \right ]^*\\
&= \left [ \sum_{n=-\infty}^{\infty} x[n](z^*)^{-n}\right ]^*\\
&= X^*(z^*)
\end{align} </math>
	|
 |-
 ! [[Real part]]
	| <math>\operatorname{Re}\{x[n]\}</math>
	| <math>\tfrac{1}{2}\left[X(z)+X^*(z^*) \right]</math>
	|
	|
 |-
 ! [[Imaginary part]]
	| <math>\operatorname{Im}\{x[n]\}</math>
	| <math>\tfrac{1}{2j}\left[X(z)-X^*(z^*) \right]</math>
	|
	| 
 |-
 ! [[Differentiation (calculus)|Differentiation]] in the z-domain
	| <math>n \, x[n]</math>
	| <math> -z \frac{dX(z)}{dz}</math>
	| <math>\begin{align} \mathcal{Z}\{n \, x(n)\} &= \sum_{n=-\infty}^{\infty} n \, x[n]z^{-n}\\
&= z \sum_{n=-\infty}^{\infty} n \, x[n]z^{-n-1}\\
&= -z \sum_{n=-\infty}^{\infty} x[n](-n \, z^{-n-1})\\
&= -z \sum_{n=-\infty}^{\infty} x[n]\frac{d}{dz}(z^{-n}) \\
&= -z \frac{dX(z)}{dz}
\end{align} </math>
	| ROC, if <math>X(z)</math> is rational;
ROC possibly excluding the boundary, if <math>X(z)</math> is irrational<ref name = forouzan>{{cite journal  | journal = Electronics Letters| title = Region of convergence of derivative of Z transform  | author = A. R. Forouzan  | volume = 52  | issue = 8  | pages = 617–619  | year = 2016| doi = 10.1049/el.2016.0189| bibcode = 2016ElL....52..617F | s2cid = 124802942 }}</ref>
 |-
 ! [[Convolution]]
	| <math>x_1[n] * x_2[n]</math>
	| <math>X_1(z) \, X_2(z)</math>
	| <math>\begin{align} \mathcal{Z}\{x_1(n)*x_2(n)\} &= \mathcal{Z} \left \{\sum_{l=-\infty}^{\infty} x_1[l]x_2[n-l] \right \} \\
                                   &= \sum_{n=-\infty}^{\infty} \left [\sum_{l=-\infty}^{\infty} x_1[l]x_2[n-l] \right ]z^{-n}\\
                                   &=\sum_{l=-\infty}^{\infty} x_1[l] \left [\sum_{n=-\infty}^{\infty} x_2[n-l]z^{-n} \right ]\\
                                   &= \left [\sum_{l=-\infty}^{\infty} x_1(l)z^{-l} \right ] \! \!\left [\sum_{n=-\infty}^{\infty} x_2[n]z^{-n} \right ] \\
                                   &=X_1(z)X_2(z)
\end{align} </math>
	| Contains ROC<sub>1</sub> ∩ ROC<sub>2</sub>
 |-
 ! [[Cross-correlation]]
	| <math>r_{x_1,x_2}=x_1^*[-n] * x_2[n]</math>
	| <math>R_{x_1,x_2}(z)=X_1^*(\tfrac{1}{z^*})X_2(z)</math>
	|
	| Contains the intersection of ROC of <math>X_1(\tfrac{1}{z^*})</math> and <math>X_2(z)</math>
 |-
 ! Accumulation
	|<math>\sum_{k=-\infty}^{n} x[k]</math>
	|<math> \frac{1}{1-z^{-1}}X(z)</math>
	|<math>\begin{align}
\sum_{n=-\infty}^{\infty}\sum_{k=-\infty}^{n} x[k] z^{-n}&=\sum_{n=-\infty}^{\infty}(x[n]+\cdots)z^{-n}\\
        &=X(z) \left (1+z^{-1}+z^{-2}+\cdots \right )\\
        &=X(z) \sum_{j=0}^{\infty}z^{-j} \\
        &=X(z) \frac{1}{1-z^{-1}}\end{align}</math>
	|
 |-
 ! [[Multiplication]]
	|  <math>x_1[n] \, x_2[n]</math>
	| <math>\frac{1}{j2\pi}\oint_C X_1(v)X_2(\tfrac{z}{v})v^{-1}\mathrm{d}v</math>
	|
	| At least <math>r_{1l}r_{2l}<|z|<r_{1u}r_{2u}</math> |-
|}

'''[[Parseval's theorem]]'''
:<math>\sum_{n=-\infty}^{\infty} x_1[n]x^*_2[n] \quad = \quad \frac{1}{j2\pi}\oint_C X_1(v)X^*_2(\tfrac{1}{v^*})v^{-1}\mathrm{d}v</math>

'''[[Initial value theorem]]''': If <math>x[n]</math> is causal, then
:<math>x[0]=\lim_{z\to \infty}X(z).</math>

'''[[Final value theorem]]''': If the poles of <math>(z - 1) X(z)</math> are inside the unit circle, then
:<math>x[\infty]=\lim_{z\to 1}(z-1)X(z).</math>

==Table of common Z-transform pairs==
Here:
:<math>u : n \mapsto u[n] = \begin{cases} 1, & n \ge 0 \\ 0, & n < 0 \end{cases}</math>
is the [[Heaviside step function|unit (or Heaviside) step function]] and
:<math>\delta : n \mapsto \delta[n] = \begin{cases} 1, & n = 0 \\ 0, & n \ne 0 \end{cases}</math>
is the [[Kronecker delta#Digital signal processing|discrete-time unit impulse function]] (cf [[Dirac delta function]] which is a continuous-time version). The two functions are chosen together so that the unit step function is the accumulation (running total) of the unit impulse function.

{| class="wikitable"
|-
! !! Signal, <math>x[n]</math> !! Z-transform, <math>X(z)</math> !! ROC
|-
|  1 || <math>\delta[n]</math> || 1 || all ''z''
|-
|  2 || <math>\delta[n-n_0]</math> || <math> z^{-n_0}</math> || <math> z \neq 0</math>
|-
|  3 || <math>u[n] \,</math> || <math> \frac{1}{1-z^{-1} }</math> || <math>|z| > 1</math>
|-
|  4 ||<math>  -u[-n-1]</math> ||  <math> \frac{1}{1 - z^{-1}}</math> ||<math>|z| < 1</math>
|-
|  5 ||<math> n u[n]</math> || <math> \frac{z^{-1}}{( 1-z^{-1} )^2}</math> || <math>|z| > 1</math>
|-
|  6 ||<math> - n u[-n-1] \,</math> || <math> \frac{z^{-1} }{ (1 - z^{-1})^2 }</math> ||<math> |z| < 1</math>
|-
|  7 ||<math>n^2 u[n]</math> || <math>  \frac{ z^{-1} (1 + z^{-1} )}{(1 - z^{-1})^3} </math> || <math>|z| > 1\,</math>
|-
|  8 ||<math> - n^2 u[-n - 1] \,</math> || <math>  \frac{ z^{-1} (1 + z^{-1} )}{(1 - z^{-1})^3} </math> || <math>|z| < 1\,</math>
|-
|  9 ||<math>n^3 u[n]</math> || <math> \frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4} </math> || <math>|z| > 1\,</math>
|-
| 10 ||<math>- n^3 u[-n -1]</math> || <math> \frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4} </math> || <math>|z| < 1\,</math>
|-
| 11 ||<math>a^n u[n]</math> ||  <math> \frac{1}{1-a z^{-1}}</math> ||<math> |z| > |a|</math>
|-
| 12 ||<math>-a^n u[-n-1]</math> ||  <math> \frac{1}{1-a z^{-1}}</math> ||<math>|z| < |a|</math>
|-
| 13 ||<math>n a^n u[n]</math> ||  <math> \frac{az^{-1} }{ (1-a z^{-1})^2 }</math> || <math>|z| > |a|</math>
|-
| 14 ||<math>-n a^n u[-n-1]</math> || <math> \frac{az^{-1} }{ (1-a z^{-1})^2 }</math> ||<math> |z| < |a|</math>
|-
| 15 ||<math>n^2 a^n u[n]</math> || <math> \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} </math> || <math>|z| > |a|</math>
|-
| 16 ||<math>- n^2 a^n u[-n -1]</math> || <math> \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} </math> || <math>|z| < |a|</math>
|-
| 17 ||<math> \left(\begin{array}{c} n + m - 1 \\ m - 1 \end{array} \right) a^n u[n]</math> [[Permutation#k-permutations of n|<sup>[14]</sup>]]|| <math> \frac{1}{(1-a z^{-1})^m} </math>, for positive integer <math>m</math><ref name=forouzan/> || <math>|z| > |a|</math>
|-
| 18 ||<math> (-1)^m \left(\begin{array}{c} -n - 1 \\ m - 1 \end{array} \right) a^n u[-n -m]</math> || <math> \frac{1}{(1-a z^{-1})^m} </math>, for positive integer <math>m</math><ref name=forouzan/> || <math>|z| < |a|</math>
|-
| 19 ||<math>\cos(\omega_0 n) u[n]</math> || <math> \frac{ 1-z^{-1} \cos(\omega_0)}{ 1-2z^{-1}\cos(\omega_0)+ z^{-2}}</math> ||<math> |z| >1</math>
|-
| 20 ||<math>\sin(\omega_0 n) u[n]</math> || <math> \frac{ z^{-1} \sin(\omega_0)}{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }</math> ||<math> |z| >1</math>
|-
| 21 ||<math>a^n \cos(\omega_0 n) u[n]</math>||<math>\frac{1-a z^{-1} \cos( \omega_0)}{1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2}}</math>||<math>|z|>|a|</math>
|-
| 22 ||<math>a^n \sin(\omega_0 n) u[n]</math>||<math> \frac{ az^{-1} \sin(\omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }</math> ||<math>|z|>|a|</math>
|}

==Relationship to Fourier series and Fourier transform==
{{further|Discrete-time Fourier transform#Relationship to the Z-transform}}

For values of <math>z</math> in the region <math>|z| {=} 1</math>, known as the [[unit circle]], we can express the transform as a function of a single real variable <math>\omega</math> by defining <math>z {=} e^{j \omega}.</math> And the bi-lateral transform reduces to a [[Fourier series]]:

{{Equation box 1 |title =
|indent= |cellpadding= 0 |border= 0 |background colour=white
|equation={{NumBlk|:|
<math>\sum_{n=-\infty}^{\infty} x[n]\ z^{-n} = \sum_{n=-\infty}^{\infty} x[n]\ e^{-j\omega n},</math>  
|{{EquationRef|Eq.1}} }} }}

which is also known as the [[discrete-time Fourier transform]] (DTFT) of the <math>x[n]</math> sequence. This <math>2\pi</math>-periodic function is the [[periodic summation]] of a [[continuous Fourier transform|Fourier transform]], which makes it a widely used analysis tool. To understand this, let <math>X(f)</math> be the Fourier transform of any function, <math>x(t)</math>, whose samples at some interval <math>T</math> equal the <math>x[n]</math> sequence. Then the DTFT of the <math>x[n]</math> sequence can be written as follows.

{{Equation box 1 |title =
|indent= |cellpadding= 0 |border= 0 |background colour=white
|equation={{NumBlk|:|
<math>\underbrace{\sum_{n=-\infty}^{\infty} \overbrace{x(nT)}^{x[n]}\ e^{-j 2\pi f nT}
}_{\text{DTFT}} = \frac{1}{T}\sum_{k=-\infty}^{\infty} X(f-k/T),</math>  
|{{EquationRef|Eq.2}} }} }}

where <math>T</math> has units of seconds, <math>f</math> has units of [[hertz]]. Comparison of the two series reveals that <math> \omega {=} 2\pi fT</math> is a [[Normalized frequency (digital signal processing)#Alternative normalizations|normalized frequency]] with unit of ''radian per sample''. The value <math>\omega{=}2\pi</math> corresponds to <math display="inline"> f {=} \frac{1}{T}</math>. And now, with the substitution <math display="inline"> f{=}\frac{\omega }{2\pi T},</math> {{EquationNote|Eq.1}} can be expressed in terms of <math>X( \tfrac{\omega - 2\pi k}{2\pi T} )</math> (a Fourier transform):

{{Equation box 1 |title =
|indent= |cellpadding= 0 |border= 0 |background colour=white
|equation={{NumBlk|:|
<math>
\sum_{n=-\infty}^{\infty} x[n]\ e^{-j\omega n} = \frac{1}{T}\sum_{k=-\infty}^{\infty} \underbrace{X\left(\tfrac{\omega}{2\pi T} - \tfrac{k}{T}\right)}_{X\left(\frac{\omega - 2\pi k}{2\pi T}\right)}.
</math>  
|{{EquationRef|Eq.3}} }} }}

As parameter ''T'' changes, the individual terms of {{EquationNote|Eq.2}} move farther apart or closer together along the ''f''-axis. In {{EquationNote|Eq.3}} however, the centers remain 2{{pi}} apart, while their widths expand or contract. When sequence <math>x(nT)</math> represents the [[impulse response]] of an [[LTI system]], these functions are also known as its [[frequency response]]. When the <math>x(nT)</math> sequence is periodic, its DTFT is divergent at one or more harmonic frequencies, and zero at all other frequencies.  This is often represented by the use of amplitude-variant [[Dirac delta]] functions at the harmonic frequencies. Due to periodicity, there are only a finite number of unique amplitudes, which are readily computed by the much simpler [[discrete Fourier transform]] (DFT). (See ''{{slink|Discrete-time Fourier transform|Periodic data}}''.)

==Relationship to Laplace transform==
{{further|Laplace transform#Z-transform}}

===Bilinear transform===
{{Main|Bilinear transform}}
The '''bilinear transform''' can be used to convert continuous-time filters (represented in the Laplace domain) into discrete-time filters (represented in the Z-domain), and vice versa. The following substitution is used:
:<math>s =\frac{2}{T} \frac{(z-1)}{(z+1)}</math>
to convert some function <math>H(s)</math> in the Laplace domain to a function <math>H(z)</math> in the Z-domain ([[Bilinear transform|Tustin transformation]]), or
:<math>z =e^{sT}\approx \frac{1+sT/2}{1-sT/2}</math>
from the Z-domain to the Laplace domain. Through the bilinear transformation, the complex ''s''-plane (of the Laplace transform) is mapped to the complex z-plane (of the z-transform). While this mapping is (necessarily) nonlinear, it is useful in that it maps the entire <math>j\omega</math> axis of the ''s''-plane onto the [[unit circle]] in the z-plane. As such, the Fourier transform (which is the Laplace transform evaluated on the <math>j\omega</math> axis) becomes the discrete-time Fourier transform. This assumes that the Fourier transform exists; i.e., that the <math>j\omega</math> axis is in the region of convergence of the Laplace transform.

=== Starred transform ===
{{Main|Starred transform}}
Given a one-sided Z-transform <math>X(z)</math> of a time-sampled function, the corresponding '''starred transform''' produces a Laplace transform and restores the dependence on <math>T</math> (the sampling parameter):
:<math>\bigg. X^*(s) = X(z)\bigg|_{\displaystyle z = e^{sT}}</math>

The inverse Laplace transform is a mathematical abstraction known as an ''impulse-sampled'' function.

==Linear constant-coefficient difference equation==
The linear constant-coefficient difference (LCCD) equation is a representation for a linear system based on the [[Autoregressive moving average model|autoregressive moving-average]] equation:

:<math>\sum_{p=0}^{N}y[n-p]\alpha_{p} = \sum_{q=0}^{M}x[n-q]\beta_{q} .</math>

Both sides of the above equation can be divided by <math>\alpha_0</math> if it is not zero. By normalizing with <math>\alpha_0{=}1,</math> the LCCD equation can be written

:<math>y[n] = \sum_{q=0}^{M}x[n-q]\beta_{q} - \sum_{p=1}^{N}y[n-p]\alpha_{p}.</math>

This form of the LCCD equation is favorable to make it more explicit that the "current" output <math>y[n]</math> is a function of past outputs <math>y[n-p],</math> current input <math>x[n],</math> and previous inputs <math>x[n-q].</math>

===Transfer function===
Taking the Z-transform of the above equation (using linearity and time-shifting laws) yields:

:<math>Y(z) \sum_{p=0}^{N}z^{-p}\alpha_{p} = X(z) \sum_{q=0}^{M}z^{-q}\beta_{q}</math>

where <math>X(z)</math> and <math>Y(z)</math> are the z-transform of <math>x[n]</math> and <math>y[n],</math> respectively. (Notation conventions typically use capitalized letters to refer to the z-transform of a signal denoted by a corresponding lower case letter, similar to the convention used for notating Laplace transforms.)

Rearranging results in the system's [[transfer function]]:

:<math>H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{q=0}^{M}z^{-q}\beta_{q}}{\sum_{p=0}^{N}z^{-p}\alpha_{p}} = \frac{\beta_0 + z^{-1} \beta_1 + z^{-2} \beta_2 + \cdots + z^{-M} \beta_M}{\alpha_0 + z^{-1} \alpha_1 + z^{-2} \alpha_2 + \cdots + z^{-N} \alpha_N}.</math>

===Zeros and poles===
From the [[fundamental theorem of algebra]] the [[numerator]] has <math>M</math> [[root of a function|roots]] (corresponding to zeros of ''<math>H</math>'') and the [[denominator]] has <math>N</math> roots (corresponding to poles).  Rewriting the [[transfer function]] in terms of [[zeros and poles]]
:<math>H(z) = \frac{(1 - q_1 z^{-1})(1 - q_2 z^{-1})\cdots(1 - q_M z^{-1}) } { (1 - p_1 z^{-1})(1 - p_2 z^{-1})\cdots(1 - p_N z^{-1})} ,</math>
where <math>q_k</math> is the <math>k^\text{th}</math> zero and <math>p_k</math> is the <math>k^\text{th}</math> pole. The zeros and poles are commonly complex and when plotted on the complex plane (z-plane) it is called the [[pole–zero plot]].

In addition, there may also exist zeros and poles at <math>z{=}0</math> and <math>z{=}\infty.</math> If we take these poles and zeros as well as multiple-order zeros and poles into consideration, the number of zeros and poles are always equal.

By factoring the denominator, [[partial fraction]] decomposition can be used, which can then be transformed back to the time domain. Doing so would result in the [[impulse response]] and the linear constant coefficient difference equation of the system.

===Output response===
If such a system <math>H(z)</math> is driven by a signal <math>X(z)</math> then the output is <math>Y(z) = H(z)X(z).</math> By performing [[partial fraction]] decomposition on <math>Y(z)</math> and then taking the inverse Z-transform the output <math>y[n]</math> can be found. In practice, it is often useful to fractionally decompose <math>\textstyle \frac{Y(z)}{z}</math> before multiplying that quantity by <math>z</math> to generate a form of <math>Y(z)</math> which has terms with easily computable inverse Z-transforms.

==See also==
* [[Advanced Z-transform]]
* [[Bilinear transform]]
* [[Difference equation]] (recurrence relation)
* [[Convolution#Discrete convolution|Discrete convolution]]
* [[Discrete-time Fourier transform]]
* [[Finite impulse response]]
* [[Formal power series]]
* [[Generating function]]
* [[Generating function transformation]]
* [[Laplace transform]]
* [[Laurent series]]
* [[Least-squares spectral analysis]]
* [[Probability-generating function]]
* [[Star transform]]
* [[Zak transform]]
* [[Zeta function regularization]]

==References==
{{Reflist}}

==Further reading==
* Refaat El Attar, ''Lecture notes on Z-Transform'', Lulu Press, Morrisville NC, 2005. {{isbn|1-4116-1979-X}}.
* Ogata, Katsuhiko, ''Discrete Time Control Systems 2nd Ed'', Prentice-Hall Inc, 1995, 1987. {{isbn|0-13-034281-5}}.
* Alan V. Oppenheim and Ronald W. Schafer (1999). Discrete-Time Signal Processing, 2nd Edition, Prentice Hall Signal Processing Series. {{isbn|0-13-754920-2}}.

==External links==
* {{springer|title=Z-transform|id=p/z130010|mode=cs1}}
* {{cite arXiv |eprint=1409.1727 |last1=Merrikh-Bayat |first1=Farshad |title=Two Methods for Numerical Inversion of the Z-Transform |date=2014 |class=math.NA }}
* [http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/LaplaceZTable/LaplaceZFuncTable.html Z-Transform table of some common Laplace transforms]
* [http://mathworld.wolfram.com/Z-Transform.html Mathworld's entry on the Z-transform]
* [http://www.dsprelated.com/comp.dsp/keyword/Z_Transform.php Z-Transform threads in Comp.DSP]
* [https://www.youtube.com/watch?v=4PV6ikgBShw A graphic of the relationship between Laplace transform s-plane to Z-plane of the Z transform]
* [https://www.youtube.com/watch?v=B4IyRw1zvvA A video-based explanation of the Z-Transform for engineers]
* [https://www.allaboutcircuits.com/technical-articles/what-is-the-z-transform/ What is the z-Transform?]

{{DSP}}
{{Authority control}}

[[Category:Transforms]]
[[Category:Laplace transforms]]