ROC of z-transform is indicated with circle in z-plane. ROC does not contain any poles. If x(n) is a finite duration causal sequence or right sided sequence, then the ROC is entire z-plane except at z = 0. 9.1.1 Admissible form of a z-transform Formulas for do not arise in a vacuum. In an introductory course they are expressed as linear combinations of z-transforms corresponding to elementary functions such as. In Table 9.1, we will see that the z-transform of each function in is a rational function of the complex variable. It can be shown that. In signal processing, 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 is the probability that a discrete random variable takes the value, and the function is usually written as in terms of =. Now the z-transform comes in two parts. The first part is the formula as shown above and the second part is to define a region of convergence for the z-transform. Both parts are needed for a complete z-transform as a z-transform without a ROC would not be of much help in signal processing. More on the region of convergence will be discussed below.

The (unilateral) -transform of a sequence is defined as

This definition is implemented in the Wolfram Language as ZTransform[a, n, z]. Similarly, the inverse -transform is implemented as InverseZTransform[A, z, n].

'The' -transform generally refers to the unilateral Z-transform. Unfortunately, there are a number of other conventions. Bracewell (1999) uses the term '-transform' (with a lower case ) to refer to the unilateral -transform. Girling (1987, p. 425) defines the transform in terms of samples of a continuous function. Worse yet, some authors define the -transform as the bilateral Z-transform.

In general, the inverse -transform of a sequence is not unique unless its region of convergence is specified (Zwillinger 1996, p. 545). If the -transform of a function is known analytically, the inverse -transform can be computed using the contour integral

where is a closed contour surrounding the origin of the complex plane in the domain of analyticity of (Zwillinger 1996, p. 545)

The unilateral transform is important in many applications because the generating function of a sequence of numbers is given precisely by , the -transform of in the variable (Germundsson 2000). In other words, the inverse -transform of a function gives precisely the sequence of terms in the series expansion of . So, for example, the terms of the series of are given by

Girling (1987) defines a variant of the unilateral -transform that operates on a continuous function sampled at regular intervals ,

where is the Laplace transform,

(5)

the one-sided shah function with period is given by

and is the Kronecker delta, giving

An alternative equivalent definition is

where

This definition is essentially equivalent to the usual one by taking .

The following table summarizes the -transforms for some common functions (Girling 1987, pp. 426-427; Bracewell 1999). Here, is the Kronecker delta, is the Heaviside step function, and is the polylogarithm.

1

1

The -transform of the general power function can be computed analytically as

(11)

(13)

where the are Eulerian numbers and is a polylogarithm. Amazingly, the -transforms of are therefore generators for Euler's number triangle.

The -transform satisfies a number of important properties, including linearity

translation

(15)

(17)

scaling

and multiplication by powers of

(20)

(Girling 1987, p. 425; Zwillinger 1996, p. 544).

The discrete Fourier transform is a special case of the -transform with

and a -transform with

for is called a fractional Fourier transform.

SEE ALSO:Bilateral Z-Transform, Discrete Fourier Transform, Euler's Number Triangle, Eulerian Number, Fractional Fourier Transform, Generating Function, Laplace Transform, Population Comparison, Unilateral Z-TransformREFERENCES:

Arndt, J. 'The -Transform (ZT).' Ch. 3 in 'Remarks on FFT Algorithms.' https://www.jjj.de/fxt/.

Boxer, R. 'A Note on Numerical Transform Calculus.' Proc. IRE45,1401-1406, 1957.

Boxer, R. and Thaler, S. 'A Simplified Method of Solving Linear and NonlinearSystems.' Proc. IRE44, 89-101, 1956.

Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 257-262, 1999.

Balakrishnan, V. K. Schaum's Outline of Combinatorics, including Concepts of Graph Theory. New York: McGraw-Hill, 1995.

Brand, L. Differentialand Difference Equations. New York: Wiley, 1966.

Cadzow, J. A. Discrete-Time Systems: An Introduction with Interdisciplinary Applications. Englewood Cliffs, NJ: Prentice-Hall, 1973.

DiStefano, J. J.; Stubberud, A. R.; and Williams, I. J. Schaum'sOutline of Feedback and Control Systems, 2nd ed. New York: McGraw-Hill, 1995.

Elaydi, S. N. AnIntroduction to Difference Equations, 2nd ed. New York: Springer, 1999.

Germundsson, R. 'Mathematica Version 4.' Mathematica J.7,497-524, 2000.

Girling, B. 'The Z Transform.' In CRC Standard Mathematical Tables, 28th ed (Ed. W. H. Beyer). Boca Raton, FL: CRC Press, pp. 424-428, 1987.

Graf, U. Applied Laplace Transforms and z-Transforms for Scientists and Engineers: A Computational Approach using a Mathematica Package. Basel, Switzerland: Birkhäuser, 2004.

Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.

Grimaldi, R. P. Discrete and Combinatorial Mathematics: An Applied Introduction, 4th ed. Longman, 1998.

Jury, E. I. Theoryand Applications of the Z-Transform Method. New York: Wiley, 1964.

Kelley, W. G. and Peterson, A. C. Difference Equations: An Introduction with Applications, 2nd ed. New York: Academic Press, 2001.

Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.

Levy, H. and Lessman, F. FiniteDifference Equations. New York: Dover, 1992.

Ljung, L. SystemIdentification: Theory for the User. Prentice-Hall, 1987.

Mickens, R. E. DifferenceEquations, 2nd ed. Princeton, NJ: Van Nostrand Reinhold, 1987.

Online Z Transform Calculator

Miller, K. S. LinearDifference Equations. New York: Benjamin, 1968.

Ogata, K. Discrete-TimeControl Systems. Englewood Cliffs, NJ: Prentice-Hall, 1987.

Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B.Wellesley, MA: A K Peters, 1996. https://www.cis.upenn.edu/~wilf/AeqB.html.

Find Z Transform Calculator

Sedgewick, R. and Flajolet, P. An Introduction to the Analysis of Algorithms. Reading, MA: Addison-Wesley, 1996.

Tsypkin, Ya. Z. SamplingSystem Theory. New York: Pergamon Press, 1964.

Vidyasagar, M. ControlSystem Synthesis: A Factorization Approach. Cambridge, MA: MIT Press, 1985.

Wilf, H. S. Generatingfunctionology,2nd ed. New York: Academic Press, 1994.

Zwillinger, D. (Ed.). 'Generating Functions and Transforms' and '-Transform.' §3.9.6 and 6.27 in CRC Standard Mathematical Tables and Formulae, 30th ed. Boca Raton, FL: CRC Press, pp. 231-233 and 543-547, 1996.

Referenced on Wolfram|Alpha: Z-Transform

Inverse Z Transform Matlab

CITE THIS AS:

Weisstein, Eric W. 'Z-Transform.' FromMathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Z-Transform.html

Inverse Z Transform Calculator With Boundaries

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