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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, the Z-transform converts a
discrete-time signal In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...
, which is a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of real or
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, into a complex frequency-domain (z-domain or z-plane) representation. It can be considered as a discrete-time equivalent of the
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
(s-domain). This similarity is explored in the theory of
time-scale calculus In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying ...
. Whereas the continuous-time Fourier transform is evaluated on the Laplace s-domain's imaginary line, the
discrete-time Fourier transform In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the ...
is evaluated over the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
of the z-domain. What is roughly the s-domain's left half-plane, is now the inside of the complex unit circle; what is the z-domain's outside of the unit circle, roughly corresponds to the right half-plane of the s-domain. One of the means of designing
digital filter In signal processing, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, t ...
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 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 and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
, 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, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
s. It was later dubbed "the z-transform" by Ragazzini and Zadeh 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, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary serie ...
s which can be traced back as early as 1730 when it was introduced by
de Moivre Abraham de Moivre FRS (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He mov ...
in conjunction with probability theory. From a mathematical view the Z-transform can also be viewed as a
Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
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. (Just like we have the one-sided Laplace transform and the two-sided Laplace transform.)


Bilateral Z-transform

The ''bilateral'' or ''two-sided'' Z-transform of a discrete-time signal x /math> is the
formal power series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...
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 extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
: :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 x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
, and \phi is the '' complex argument'' (also referred to as ''angle'' or ''phase'') in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
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 subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, this definition can be used to evaluate the Z-transform of the 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 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 (ROC). In the case where the ROC is causal (see Example 2), this means the path ''C'' must encircle all of the poles of X(z). 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 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 A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
, 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 chirp Z-transform (CZT) is a generalization of the discrete Fourier transform (DFT). While the DFT samples the Z plane at uniformly-spaced points along the unit circle, the chirp Z-transform samples along spiral arcs in the Z-plane, correspon ...
. The
discrete-time Fourier transform In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the ...
(DTFT)—not to be confused with the
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
(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 (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'' 'n''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 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 value of which is zero for negative arguments and one for positive argum ...
). Expanding ''x'' 'n''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 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 n-1 (where ''u'' is the
Heaviside step function 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 value of which is zero for negative arguments and one for positive argum ...
). Expanding ''x'' 'n''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, 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 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 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'' 'n''. We can determine a unique ''x'' 'n''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 , 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/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 ; grc, βῆτα, bē̂ta or ell, βήτα, víta) is the second letter of the Greek alphabet. In the system of Greek numerals, it has a value of 2. In Modern Greek, it represents the voiced labiod ...
= 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/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, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
, 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 , \operatorname{Re}\{x } , \tfrac{1}{2}\left (z)+X^*(z^*) \right/math> , , , - ! Imaginary part , \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 (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
, 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, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
, r_{x_1,x_2}=x_1^* 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 Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
, 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 In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates ...
:\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 theorem: 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, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
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 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 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/math> , , \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 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 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/math> , , \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/math> , , \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 In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
, 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 A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
: which is also known as the
discrete-time Fourier transform In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the ...
(DTFT) of the x /math> sequence. This 2-periodic function is the periodic summation of a
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
, 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 The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that o ...
. Comparison of the two series reveals that   \omega = 2\pi fT  is a normalized frequency with units of ''radians per sample''. The value ''ω''=2 corresponds to f = \frac{1}{T}.  And now, with the substitution   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 In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (). More generally, an impulse response is the reac ...
of an
LTI system In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defi ...
, 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 samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
(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 ( 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 mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
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 equation. :\sum_{p=0}^{N}y -palpha_{p} = \sum_{q=0}^{M}x -qbeta_{q} Both sides of the above equation can be divided by ''α''0, if it is not zero, normalizing ''α''0 = 1 and the LCCD equation can be written :y = \sum_{q=0}^{M}x -qbeta_{q} - \sum_{p=1}^{N}y -palpha_{p}. 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 :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 fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
the
numerator A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
has ''M''
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(corresponding to zeros of ''H'') and the
denominator A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
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,
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decomposition can be used, which can then be transformed back to the time domain. Doing so would result in the
impulse response In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (). More generally, an impulse response is the reac ...
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 In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction a ...
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 \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 The bilinear transform (also known as Tustin's method, after Arnold Tustin) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa. The bilinear t ...
*
Difference equation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
(recurrence relation) * Discrete convolution *
Discrete-time Fourier transform In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the ...
*
Finite impulse response In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of ''finite'' duration, because it settles to zero in finite time. This is in contrast to infinite impulse ...
*
Formal power series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...
*
Generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary serie ...
* Generating function transformation *
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
*
Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
* Least-squares spectral analysis * 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

A video-based explanation of the Z-Transform for engineers

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