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{{about||standard z-score in statistics|Standard score|Fisher z-transformation in statistics|Fisher 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 [[frequency-domain]] representation.

It can be considered as a discrete-time equivalent of the [[Laplace transform]]. This similarity is explored in the theory of [[time-scale calculus]].

== History ==
The basic idea now known as the Z-transform was known to [[Laplace]], and it was re-introduced in 1947 by [[Witold Hurewicz|W. Hurewicz]]<ref name="kanasewich">
{{cite book|url=https://books.google.com/books?id=k8SSLy-FYagC&pg=PA249&dq=inauthor:Kanasewich++poles+stability&hl=en&sa=X&ei=igLcUqeXFMmxoQTxzIHoAg&ved=0CC8Q6AEwAA#v=onepage&q=inauthor%3AKanasewich%20%20poles%20stability&f=false|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> 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 equation]]s. It was later dubbed "the z-transform" by [[John R. Ragazzini|Ragazzini]] and [[Lotfi A. Zadeh|Zadeh]] in the sampled-data control group at Columbia University in 1952.<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}}</ref><ref>{{cite book  | title = Digital control systems implementation and computational techniques  | edition =  | 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>

The modified or [[advanced Z-transform]] was later developed and popularized by [[Eliahu I. Jury|E. I. Jury]].<ref>
{{cite book
 | title = Sampled-Data Control Systems
 | author = Eliahu Ibrahim Jury
 | publisher = John Wiley & Sons
 | year = 1958
 }}</ref><ref>
{{cite book
 | title = Theory and Application of the Z-Transform Method
 | author = Eliahu Ibrahim Jury
 | publisher = Krieger Pub Co
 | year = 1973
 | isbn = 0-88275-122-0
 }}</ref>

The idea contained within the Z-transform is also known in mathematical literature as the method of [[generating function]]s which can be traced back as early as 1730 when it was introduced by [[Abraham de Moivre|de Moivre]] in conjunction with probability theory.<ref>
{{cite book
 | title = Theory and Application of the Z-Transform Method
 | author = Eliahu Ibrahim Jury
 | publisher = John Wiley & Sons
 | year = 1964
 | page = 1
 }}</ref>
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.<ref>Just like we have the [[Laplace transform|one-sided Laplace transform]] and the [[two-sided Laplace transform]].</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
|indent =
|title=
|equation = {{NumBlk|:|<math>X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] z^{-n} </math>|{{EquationRef|Eq.1}}}}
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

where <math>n</math> is an integer and <math>z</math> is, in general, a [[complex number]]:
:<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
|indent =
|title=
|equation = {{NumBlk|:|<math>X(z) = \mathcal{Z}\{x[n]\} =  \sum_{n=0}^{\infty} x[n] z^{-n}.</math>|{{EquationRef|Eq.2}}}}
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

In [[signal processing]], this definition can be used to evaluate the Z-transform of the [[Finite impulse response#Impulse 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 <math>n</math>, and the function <math>X(z)</math> is usually written as <math>X(s)</math> in terms of <math>s=z^{-1}</math>. The properties of Z-transforms (below) have useful interpretations in the context of probability theory.

<br />

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

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk|:|<math> x[n] = \mathcal{Z}^{-1} \{X(z) \}= \frac{1}{2 \pi j} \oint_{C} X(z) z^{n-1} dz</math>|{{EquationRef|Eq.3}}}}
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

where ''C'' 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 ''C'' must encircle all of the poles of <math>X(z)</math>.

A special case of this [[contour integral]] occurs when ''C'' 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:

:<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 ''n'' and a finite number of uniformly spaced ''z'' 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 ''z'' to lie on the unit circle.

==Region of convergence==
The [[Radius of convergence|region of convergence]] (ROC) is the set of points in the complex plane for which the Z-transform summation converges.

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

===Example 1 (no ROC)===
Let ''x[n]'' = (0.5)<sup>''n''</sup>.  Expanding ''x[n]'' on the interval (−∞, ∞) it becomes

:<math>x[n] = \left \{\cdots, 0.5^{-3}, 0.5^{-2}, 0.5^{-1}, 1, 0.5, 0.5^2, 0.5^3, \cdots \right \} = \left \{\cdots, 2^3, 2^2, 2, 1, 0.5, 0.5^2, 0.5^3, \cdots \right\}.</math>

Looking at the sum

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

Therefore, there are no values of ''z'' that satisfy this condition.

===Example 2 (causal ROC)===
[[Image:Region of convergence 0.5 causal.svg|thumb|250px|ROC shown in blue, the unit circle as a dotted grey circle and the circle &#124;''z''&#124;&nbsp;=&nbsp;0.5 is shown as a dashed black circle]]

Let <math>x[n] = 0.5^n u[n]\ </math> (where ''u'' is the [[Heaviside step function]]).  Expanding ''x[n]'' on the interval (−∞, ∞) it becomes

:<math>x[n] = \left \{\cdots, 0, 0, 0, 1, 0.5, 0.5^2, 0.5^3, \cdots \right \}.</math>

Looking at the sum

:<math>\sum_{n=-\infty}^{\infty}x[n]z^{-n} = \sum_{n=0}^{\infty}0.5^nz^{-n} = \sum_{n=0}^{\infty}\left(\frac{0.5}{z}\right)^n = \frac{1}{1 - 0.5z^{-1}}.</math>

The last equality arises from the infinite [[geometric series]] and the equality only holds if |0.5''z''<sup>−1</sup>| < 1 which can be rewritten in terms of ''z'' as |''z''| > 0.5.  Thus, the ROC is |''z''| > 0.5.  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 shown in blue, the unit circle as a dotted grey circle and the circle &#124;''z''&#124;&nbsp;=&nbsp;0.5 is shown as a dashed black circle]]

Let <math>x[n] = -(0.5)^n u[-n-1]\ </math> (where ''u'' is the [[Heaviside step function]]).  Expanding ''x[n]'' on the interval (−∞, ∞) it becomes

:<math>x[n] = \left \{ \cdots, -(0.5)^{-3}, -(0.5)^{-2}, -(0.5)^{-1}, 0, 0, 0, 0, \cdots \right \}.</math>

Looking at the sum

:<math>\sum_{n=-\infty}^{\infty}x[n]z^{-n} = -\sum_{n=-\infty}^{-1}0.5^nz^{-n} = -\sum_{m=1}^{\infty}\left(\frac{z}{0.5}\right)^{m} = -\frac{0.5^{-1}z}{1 - 0.5^{-1}z} = -\frac{1}{0.5z^{-1}-1} = \frac{1}{1 - 0.5z^{-1}}.</math>

Using the infinite [[geometric series]], again, the equality only holds if |0.5<sup>−1</sup>''z''| < 1 which can be rewritten in terms of ''z'' as |''z''| < 0.5. Thus, the ROC is |''z''| < 0.5. 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 ''X(z)'' of ''x[n]'' 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 an ROC that includes |''z''| = ∞ while the anticausal system in example 3 yields an ROC that includes |''z''| = 0.

[[Image:Region of convergence 0.5 0.75 mixed-causal.svg|thumb|250px|ROC shown as a blue ring 0.5&nbsp;<&nbsp;&#124;''z''&#124;&nbsp;<&nbsp;0.75]]
In systems with multiple poles it is possible to have a ROC that includes neither |''z''| = ∞ nor |''z''| = 0. The ROC creates a circular band. For example,

:<math>x[n] = 0.5^nu[n] - 0.75^nu[-n-1]</math>

has poles at 0.5 and 0.75. The ROC will be 0.5 < |''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 (0.5)<sup>''n''</sup>''u''[''n''] and an anticausal term &minus;(0.75)<sup>''n''</sup>''u''[&minus;''n''&minus;1].

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., |''z''| = 1) then the system is stable.  In the above systems the causal system (Example 2) is stable because |''z''| > 0.5 contains the unit circle.

If you are provided a Z-transform of a system without a ROC (i.e., an ambiguous ''x[n]'') you can determine a unique ''x[n]'' provided you desire the following:
* Stability
* Causality

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

The unique ''x[n]'' can then be found.

==Properties==
{| class="wikitable"
 |+ '''Properties of the z-transform'''
 !
 	! Time domain
	! Z-domain
	! Proof
 	! ROC
 |-
 ! Notation
	| <math>x[n]=\mathcal{Z}^{-1}\{X(z)\}</math>
	| <math>X(z)=\mathcal{Z}\{x[n]\}</math>
	|
	|<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]&nbsp; or &nbsp;[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} 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 ''z'' = 0 if ''k'' > 0 and ''z'' = ∞ if ''k'' < 0
 |-
 ! 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 |date= |title=Fondamenti di Controlli Automatici |trans-title= |url= |language=it |location= |publisher=MC Graw Hill Education |isbn=978-88-386-6882-1}}</ref>
<math display="block">z^kX(z)-z^k\sum^{k-1}_{n=0}x[n]z^{-n}</math>
	| 
	|
 |-
 ! First difference backward
	|  <math>x[n] - x[n-1]</math>
with ''x''[''n'']=0 for ''n''<0
	| <math> (1-z^{-1})X(z)</math>
	| 
	| Contains the intersection of ROC of ''X<sub>1</sub>(z)'' and ''z'' ≠ 0
 |-
 ! First difference forward
|  <math>x[n+1] - x[n]</math>
	| <math> (z-1)X(z)-zx[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
	| <math>nx[n]</math>
	| <math> -z \frac{dX(z)}{dz}</math>
	| <math>\begin{align} \mathcal{Z}\{nx(n)\} &= \sum_{n=-\infty}^{\infty} nx(n)z^{-n}\\
&= z \sum_{n=-\infty}^{\infty} nx(n)z^{-n-1}\\
&= -z \sum_{n=-\infty}^{\infty} x(n)(-nz^{-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 = Elec. Lett.  | title = Region of convergence of derivative of Z transform  | author = A. R. Forouzan  | volume = 52  | issue = 8  | publisher = IET  | pages = 617–619  | year = 2016| DOI = 10.1049/el.2016.0189}}</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 + x[-\infty])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 ''x''[''n''] is causal, then
:<math>x[0]=\lim_{z\to \infty}X(z).</math>

'''[[Final value theorem]]''': If the poles of (''z''−1)''X''(''z'') 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> || <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==
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, &omega;, by defining <math>z=e^{j \omega}</math>. And the bi-lateral transform reduces to a [[Fourier series]]:

{{NumBlk|:|<math>\sum_{n=-\infty}^{\infty} x[n]\ z^{-n} = \sum_{n=-\infty}^{\infty} x[n]\ e^{-j\omega n},</math>|{{EquationRef|Eq.4}}}}

which is also known as the [[discrete-time Fourier transform]] (DTFT) of the <math>x[n]</math> sequence.  This 2{{pi}}-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, ''T'', equal the x[''n''] sequence.  Then the DTFT of the ''x''[''n''] sequence can be written as follows.

{{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.5}}}}

When ''T'' has units of seconds, <math>\scriptstyle f</math> has units of [[hertz]].  Comparison of the two series reveals that &nbsp;<math>\scriptstyle \omega = 2\pi fT</math>&nbsp; is a [[Normalized frequency (digital signal processing)#Alternative normalizations|normalized frequency]] with units of ''radians per sample''.  The value ω=2{{pi}} corresponds to <math>\scriptstyle f = \frac{1}{T}</math> Hz.&nbsp; And now, with the substitution &nbsp;<math>\scriptstyle f = \frac{\omega }{2\pi T},</math>&nbsp; {{EquationNote|Eq.4}} can be expressed in terms of the Fourier transform, X(•)''':'''

{{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.6}}}}

As parameter T changes, the individual terms of {{EquationNote|Eq.5}} move farther apart or closer together along the f-axis.  In {{EquationNote|Eq.6}} however, the centers remain 2{{pi}} apart, while their widths expand or contract.  When sequence ''x''(''nT'') 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).&nbsp; (See [[DTFT#Periodic data|DTFT; periodic data]].)

==Relationship to Laplace 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 =\frac{2+sT}{2-sT}</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, X(z), of a time-sampled function, the corresponding '''starred transform''' produces a Laplace transform and restores the dependence on sampling parameter, T''':'''

:<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 α<sub>0</sub>, if it is not zero, normalizing α<sub>0</sub> = 1 and 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 ''y[n]'' is a function of past outputs ''y[n−p]'', current input ''x[n]'', and previous inputs ''x[n−q]''.

===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>

and rearranging results in

:<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 ''M'' [[root of a function|roots]] (corresponding to zeros of H) and the [[denominator]] has N 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 ''q<sub>k</sub>'' is the ''k''-th zero and ''p<sub>k</sub>'' is the ''k''-th 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 ''z'' = 0 and ''z'' = ∞. 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 ''H(z)'' is driven by a signal ''X(z)'' then the output is ''Y(z)'' = ''H(z)X(z)''.  By performing [[partial fraction]] decomposition on ''Y(z)'' and then taking the inverse Z-transform the output ''y[n]'' can be found.  In practice, it is often useful to fractionally decompose <math>\textstyle \frac{Y(z)}{z}</math> before multiplying that quantity by ''z'' to generate a form of ''Y(z)'' 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]]
* [[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}}
* [https://arxiv.org/abs/1409.1727 Numerical inversion of the Z-transform]
* [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 An video based explanation of the Z-Transform for engineers]

{{DSP}}

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