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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
and
signal processing Signal processing is an electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetis ...

signal processing
, the Z-transform converts a
discrete-time signal In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which to model variables that evolve over time. Discrete time Discrete time views values of variables as occurring at distinct, separate "points ...
, which is a
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

sequence
of
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s, into a complex
frequency-domain In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
representation. It can be considered as a discrete-time equivalent of the
Laplace transform In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable t (often time in physics, time) to a function of a complex analysis, complex variable s (co ...
. 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 Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar A scholar is a person who pursues academic and intellectual activities, particularly those that develop expertise in an area of Studying, study. A ...

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 equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s. It was later dubbed "the z-transform" by Ragazzini and
Zadeh Zadeh, also Zada, is a Persian language, Persian patronymic suffix meaning 'descendant of' or 'born of' used in names mainly in Iran and Azerbaijan. Notable people whose names contain 'Zadeh' include: *Lotfi A. Zadeh (1921–2017), mathematician ...
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 function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s which can be traced back as early as 1730 when it was introduced by
de Moivre Abraham de Moivre (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematica ...

de Moivre
in conjunction with probability theory. From a mathematical view the Z-transform can also be viewed as a
Laurent series (Holomorphic functions are analytic, analytic). 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 compl ...

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.


Bilateral Z-transform

The ''bilateral'' or ''two-sided'' Z-transform of a discrete-time signal x /math> is the
formal power series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
X(z) defined as where n is an integer and z is, in general, a
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
: :z = A e^ = A\cdot(\cos+j\sin) where A is the magnitude of z, j is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad are ...
, and \phi is the ''
complex argument In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
'' (also referred to as ''angle'' or ''phase'') in
radian The radian, denoted by the symbol \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

radian
s.


Unilateral Z-transform

Alternatively, in cases where x /math> is defined only for n \ge 0, the ''single-sided'' or ''unilateral'' Z-transform is defined as In
signal processing Signal processing is an electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetis ...

signal processing
, this definition can be used to evaluate the Z-transform of the unit impulse response of a discrete-time
causal system In control theory Control theory deals with the control of dynamical system In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in whi ...
. An important example of the unilateral Z-transform is the
probability-generating function In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in w ...
, where the component x /math> is the probability that a discrete random variable takes the value n, and the function X(z) is usually written as X(s) in terms of s=z^. The properties of Z-transforms (below) have useful interpretations in the context of probability theory.


Inverse Z-transform

The ''inverse'' Z-transform is where ''C'' is a counterclockwise closed path encircling the origin and entirely in the
region of convergence In mathematics, the radius of convergence of a power series is the radius of the largest Disk (mathematics), disk at the Power series, center of the series in which the series Convergent series, converges. It is either a non-negative real number o ...
(ROC). In the case where the ROC is causal (see
Example 2 Example may refer to: * ''exempli gratia Notes and references Notes References Sources * * * Further reading * * {{Latin phrases E ...'' (e.g.), usually read out in English as "for example" * .example, reserved as a domain nam ...
), this means the path ''C'' must encircle all of the poles of X(z). A special case of this
contour integral In the mathematical field of complex analysis of the function . Hue represents the argument, brightness the magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathemati ...
occurs when ''C'' is the unit circle. This contour can be used when the ROC includes the unit circle, which is always guaranteed when X(z) is stable, that is, when all the poles are inside the unit circle. With this contour, the inverse Z-transform simplifies to the inverse discrete-time Fourier transform, or
Fourier series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
, of the periodic values of the Z-transform around the unit circle: 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
(DTFT)—not to be confused with the
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discret ...
(DFT)—is a special case of such a Z-transform obtained by restricting ''z'' to lie on the unit circle.


Region of convergence

The
region of convergence In mathematics, the radius of convergence of a power series is the radius of the largest Disk (mathematics), disk at the Power series, center of the series in which the series Convergent series, converges. It is either a non-negative real number o ...
(ROC) is the set of points in the complex plane for which the Z-transform summation converges. :\mathrm = \left\


Example 1 (no ROC)

Let ''x ' = (0.5)''n''. Expanding ''x ' on the interval (−∞, ∞) it becomes :x = \left \ = \left \. Looking at the sum :\sum_^x ^ \to \infty. Therefore, there are no values of ''z'' that satisfy this condition.


Example 2 (causal ROC)

Let x = 0.5^n u (where ''u'' is the
Heaviside step function 325px, The Heaviside step function, using the half-maximum convention The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the valu ...
). Expanding ''x ' on the interval (−∞, ∞) it becomes :x = \left \. Looking at the sum :\sum_^x ^ = \sum_^0.5^nz^ = \sum_^\left(\frac\right)^n = \frac. The last equality arises from the infinite
geometric series In mathematics, a geometric series (mathematics), series is the sum of an infinite number of Summand, terms that have a constant ratio between successive terms. For example, 1/2 + 1/4 + 1/8 + 1/16 + · · ·, the series :\frac \,+\, \frac \,+\, ...
and the equality only holds if , 0.5''z''−1, < 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".


Example 3 (anti causal ROC)

Let x = -(0.5)^n u
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n-1
(where ''u'' is the
Heaviside step function 325px, The Heaviside step function, using the half-maximum convention The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the valu ...
). Expanding ''x ' on the interval (−∞, ∞) it becomes :x = \left \. Looking at the sum :\sum_^x ^ = -\sum_^0.5^nz^ = -\sum_^\left(\frac\right)^ = -\frac = -\frac = \frac. Using the infinite
geometric series In mathematics, a geometric series (mathematics), series is the sum of an infinite number of Summand, terms that have a constant ratio between successive terms. For example, 1/2 + 1/4 + 1/8 + 1/16 + · · ·, the series :\frac \,+\, \frac \,+\, ...
, again, the equality only holds if , 0.5−1''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.


Examples conclusion

Examples 2 & 3 clearly show that the Z-transform ''X(z)'' of ''x ' 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. 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, :x = 0.5^nu - 0.75^nu
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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)''n''''u'' 'n''and an anticausal term −(0.75)''n''''u'' minus;''n''−1 The
stability Stability may refer to: Mathematics *Stability theory, the study of the stability of solutions to differential equations and dynamical systems **Asymptotic stability **Linear stability **Lyapunov stability **Orbital stability **Structural 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. Let us assume we are provided a Z-transform of a system without a ROC (i.e., an ambiguous ''x '). We can determine a unique ''x ' 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 ''x ' can then be found.


Properties

{, class="wikitable" , + Properties of the z-transform ! ! Time domain ! Z-domain ! Proof ! ROC , - ! Notation , x \mathcal{Z}^{-1}\{X(z)\} , X(z)=\mathcal{Z}\{x } , , r_2<, z, , - !
Linearity Linearity is the property of a mathematical relationship (''function (mathematics), function'') that can be graph of a function, graphically represented as a straight Line (geometry), line. Linearity is closely related to ''Proportionality (math ...

Linearity
, a_1 x_1 + a_2 x_2 /math> , a_1 X_1(z) + a_2 X_2(z) , \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} , Contains ROC1 ∩ ROC2 , - ! Time expansion , x_K = \begin{cases} x & n = Kr \\ 0, & n \notin K\mathbb{Z} \end{cases} with K\mathbb{Z} := \{Kr: r \in \mathbb{Z}\} , X(z^K) , \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} , R^{\frac{1}{K , - ! Decimation , x
n
n
/math> , \frac{1}{K} \sum_{p=0}^{K-1} X\left(z^{\tfrac{1}{K \cdot e^{-i \tfrac{2\pi}{K} p}\right)
ohio-state.edu
nbsp; or &nbs
ee.ic.ac.uk
, , - ! Time delay , x -k/math> with k>0 and x : x 0\ \forall n<0 , z^{-k}X(z) , \begin{align} Z\{x -k} &= \sum_{n=0}^{\infty} x -k^{-n}\\ &= \sum_{j=-k}^{\infty} x ^{-(j+k)}&& j = n-k \\ &= \sum_{j=-k}^{\infty} x ^{-j}z^{-k} \\ &= z^{-k}\sum_{j=-k}^{\infty}x ^{-j}\\ &= z^{-k}\sum_{j=0}^{\infty}x ^{-j} && x
beta Beta (, ; uppercase , lowercase , or cursive Cursive (also known as script, among other names) is any style of penmanship Penmanship is the technique of writing Writing is a medium of human communication that involves the represen ...

beta
= 0, \beta < 0\\ &= z^{-k}X(z)\end{align} , ROC, except ''z'' = 0 if ''k'' > 0 and ''z'' = ∞ if ''k'' < 0 , - ! Time advance , x +k/math> with k>0 , Bilateral Z-transform: z^kX(z) Unilateral Z-transform: z^kX(z)-z^k\sum^{k-1}_{n=0}x ^{-n} , , , - ! First difference backward , x - x -1/math> with ''x'' 'n''0 for ''n''<0 , (1-z^{-1})X(z) , , Contains the intersection of ROC of ''X1(z)'' and ''z'' ≠ 0 , - ! First difference forward , x +1- x /math> , (z-1)X(z)-zx /math> , , , - ! Time reversal , x
n
n
/math> , X(z^{-1}) , \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} , \tfrac{1}{r_1}<, z, <\tfrac{1}{r_2} , - ! Scaling in the z-domain , a^n x /math> , X(a^{-1}z) , \begin{align}\mathcal{Z} \left \{a^n x \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} , , a, r_2 < , z, < , a, r_1 , - !
Complex conjugation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

Complex conjugation
, x^* /math> , X^*(z^*) , \begin{align} \mathcal{Z} \{x^*(n)\} &= \sum_{n=-\infty}^{\infty} x^*(n)z^{-n}\\ &= \sum_{n=-\infty}^{\infty} \left (n)(z^*)^{-n} \right *\\ &= \left \sum_{n=-\infty}^{\infty} x(n)(z^*)^{-n}\right *\\ &= X^*(z^*) \end{align} , , - !
Real part In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this ...
, \operatorname{Re}\{x } , \tfrac{1}{2}\left (z)+X^*(z^*) \right/math> , , , - !
Imaginary part In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this ...
, \operatorname{Im}\{x } , \tfrac{1}{2j}\left (z)-X^*(z^*) \right/math> , , , - ! Differentiation , nx /math> , -z \frac{dX(z)}{dz} , \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} , ROC, if X(z) is rational; ROC possibly excluding the boundary, if X(z) is irrational , - !
Convolution In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
, x_1 * x_2 /math> , X_1(z)X_2(z) , \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 ^{-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} , Contains ROC1 ∩ ROC2 , - !
Cross-correlation In signal processing Signal processing is an electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electron ...

Cross-correlation
, r_{x_1,x_2}=x_1^*
n
n
* x_2 /math> , R_{x_1,x_2}(z)=X_1^*(\tfrac{1}{z^*})X_2(z) , , Contains the intersection of ROC of X_1(\tfrac{1}{z^*}) and X_2(z) , - ! Accumulation , \sum_{k=-\infty}^{n} x /math> , \frac{1}{1-z^{-1X(z) , \begin{align} \sum_{n=-\infty}^{\infty}\sum_{k=-\infty}^{n} x z^{-n}&=\sum_{n=-\infty}^{\infty}(x \cdots + x \inftyz^{-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} , , - !
Multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

Multiplication
, x_1 _2 /math> , \frac{1}{j2\pi}\oint_C X_1(v)X_2(\tfrac{z}{v})v^{-1}\mathrm{d}v , , At least r_{1l}r_{2l}<, z, , - Parseval's theorem :\sum_{n=-\infty}^{\infty} x_1 ^*_2 \quad = \quad \frac{1}{j2\pi}\oint_C X_1(v)X^*_2(\tfrac{1}{v^*})v^{-1}\mathrm{d}v Initial value theorem: If ''x'' 'n''is causal, then :x \lim_{z\to \infty}X(z).
Final value theoremIn mathematical analysis, the final value theorem (FVT) is one of several similar theorems used to relate frequency domain expressions to the time domain behavior as time approaches infinity. Mathematically, if f(t) in continuous time has (unilateral ...
: If the poles of (''z''−1)''X''(''z'') are inside the unit circle, then :x infty\lim_{z\to 1}(z-1)X(z).


Table of common Z-transform pairs

Here: :u : n \mapsto u = \begin{cases} 1, & n \ge 0 \\ 0, & n < 0 \end{cases} is the unit (or Heaviside) step function and :\delta : n \mapsto \delta = \begin{cases} 1, & n = 0 \\ 0, & n \ne 0 \end{cases} is the discrete-time unit impulse function (cf
Dirac delta function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
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, x /math> !! Z-transform, X(z) !! ROC , - , 1 , , \delta /math> , , 1 , , all ''z'' , - , 2 , , \delta -n_0/math> , , z^{-n_0} , , z \neq 0 , - , 3 , , u \, , , \frac{1}{1-z^{-1} } , , , z, > 1 , - , 4 , , -u
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n-1
/math> , , \frac{1}{1 - z^{-1 , , , z, < 1 , - , 5 , , n u /math> , , \frac{z^{-1{( 1-z^{-1} )^2} , , , z, > 1 , - , 6 , , - n u
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n-1
\, , , \frac{z^{-1} }{ (1 - z^{-1})^2 } , , , z, < 1 , - , 7 , , n^2 u /math> , , \frac{ z^{-1} (1 + z^{-1} )}{(1 - z^{-1})^3} , , , z, > 1\, , - , 8 , , - n^2 u n - 1\, , , \frac{ z^{-1} (1 + z^{-1} )}{(1 - z^{-1})^3} , , , z, < 1\, , - , 9 , , n^3 u /math> , , \frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4} , , , z, > 1\, , - , 10 , , - n^3 u[-n -1] , , \frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4} , , , z, < 1\, , - , 11 , , a^n u /math> , , \frac{1}{1-a z^{-1 , , , z, > , a, , - , 12 , , -a^n u
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n-1
/math> , , \frac{1}{1-a z^{-1 , , , z, < , a, , - , 13 , , n a^n u /math> , , \frac{az^{-1} }{ (1-a z^{-1})^2 } , , , z, > , a, , - , 14 , , -n a^n u
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n-1
/math> , , \frac{az^{-1} }{ (1-a z^{-1})^2 } , , , z, < , a, , - , 15 , , n^2 a^n u /math> , , \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} , , , z, > , a, , - , 16 , , - n^2 a^n u[-n -1] , , \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} , , , z, < , a, , - , 17 , , \left(\begin{array}{c} n + m - 1 \\ m - 1 \end{array} \right) a^n u /math> , , \frac{1}{(1-a z^{-1})^m} , for positive integer m , , , z, > , a, , - , 18 , , (-1)^m \left(\begin{array}{c} -n - 1 \\ m - 1 \end{array} \right) a^n u[-n -m] , , \frac{1}{(1-a z^{-1})^m} , for positive integer m , , , z, < , a, , - , 19 , , \cos(\omega_0 n) u /math> , , \frac{ 1-z^{-1} \cos(\omega_0)}{ 1-2z^{-1}\cos(\omega_0)+ z^{-2 , , , z, >1 , - , 20 , , \sin(\omega_0 n) u /math> , , \frac{ z^{-1} \sin(\omega_0)}{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} } , , , z, >1 , - , 21 , , a^n \cos(\omega_0 n) u /math>, , \frac{1-a z^{-1} \cos( \omega_0)}{1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2, , , z, >, a, , - , 22 , , a^n \sin(\omega_0 n) u /math>, , \frac{ az^{-1} \sin(\omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} } , , , z, >, a,


Relationship to Fourier series and Fourier transform

For values of z in the region , z, =1, known as the unit circle, we can express the transform as a function of a single, real variable, ω, by defining z=e^{j \omega}. And the bi-lateral transform reduces to a
Fourier series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
: which is also known as the
discrete-time Fourier transform In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
(DTFT) of the x /math> sequence. This 2-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 X(f) be the Fourier transform of any function, x(t), whose samples at some interval, ''T'', equal the x 'n''sequence. Then the DTFT of the ''x'' 'n''sequence can be written as follows. = \frac{1}{T}\sum_{k=-\infty}^{\infty} X(f-k/T). , When ''T'' has units of seconds, \scriptstyle f has units of hertz. Comparison of the two series reveals that  \scriptstyle \omega = 2\pi fT  is a Normalized frequency (digital signal processing)#Alternative normalizations, normalized frequency with units of ''radians per sample''. The value ω=2 corresponds to \scriptstyle f = \frac{1}{T} Hz.  And now, with the substitution  \scriptstyle f = \frac{\omega }{2\pi T},  can be expressed in terms of the Fourier transform, X(•): As parameter T changes, the individual terms of move farther apart or closer together along the f-axis. In however, the centers remain 2 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 x(nT) 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 In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discret ...
(DFT).  (See .)


Relationship to Laplace transform


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: :s =\frac{2}{T} \frac{(z-1)}{(z+1)} to convert some function H(s) in the Laplace domain to a function H(z) in the Z-domain (Bilinear transform, Tustin transformation), or :z =e^{sT}\approx \frac{1+sT/2}{1-sT/2} 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 j\omega 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 j\omega axis) becomes the discrete-time Fourier transform. This assumes that the Fourier transform exists; i.e., that the j\omega axis is in the region of convergence of the Laplace transform.


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: :\bigg. X^*(s) = X(z)\bigg, _{\displaystyle z = e^{sT 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. :\sum_{p=0}^{N}y[n-p]\alpha_{p} = \sum_{q=0}^{M}x[n-q]\beta_{q} Both sides of the above equation can be divided by α0, if it is not zero, normalizing α0 = 1 and the LCCD equation can be written :y = \sum_{q=0}^{M}x[n-q]\beta_{q} - \sum_{p=1}^{N}y[n-p]\alpha_{p}. This form of the LCCD equation is favorable to make it more explicit that the "current" output ''y ' is a function of past outputs ''y[n−p]'', current input ''x ', and previous inputs ''x[n−q]''.


Transfer function

Taking the Z-transform of the above equation (using linearity and time-shifting laws) yields :Y(z) \sum_{p=0}^{N}z^{-p}\alpha_{p} = X(z) \sum_{q=0}^{M}z^{-q}\beta_{q} and rearranging results in :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}.


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 :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})} where ''qk'' is the ''k''-th zero and ''pk'' 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 ' can be found. In practice, it is often useful to fractionally decompose \textstyle \frac{Y(z)}{z} 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 In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable t (often time in physics, time) to a function of a complex analysis, complex variable s (co ...
*
Laurent series (Holomorphic functions are analytic, analytic). 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 compl ...

Laurent series
* Probability-generating function * Star transform * Zak transform * Zeta function regularization


References


Further reading

* Refaat El Attar, ''Lecture notes on Z-Transform'', Lulu Press, Morrisville NC, 2005. . * Ogata, Katsuhiko, ''Discrete Time Control Systems 2nd Ed'', Prentice-Hall Inc, 1995, 1987. . * Alan V. Oppenheim and Ronald W. Schafer (1999). Discrete-Time Signal Processing, 2nd Edition, Prentice Hall Signal Processing Series. .


External links

*
Numerical inversion of the Z-transform





Z-Transform threads in Comp.DSP

A graphic of the relationship between Laplace transform s-plane to Z-plane of the Z transform

An video based explanation of the Z-Transform for engineers

What is the z-Transform?
{{Authority control Transforms Laplace transforms