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Z-Transform

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

 (1)

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

 (2)

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

 (3)

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

 (4)

where is the Laplace transform,

 (5) (6)

the one-sided shah function with period is given by

 (7)

and is the Kronecker delta, giving

 (8)

An alternative equivalent definition is

 (9)

where

 (10)

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) (12) (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

 (14)

translation

 (15) (16) (17) (18)

scaling

 (19)

and multiplication by powers of

 (20) (21)

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

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

 (22)

and a -transform with

 (23)

for is called a fractional Fourier transform.

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

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References

Arndt, J. "The -Transform (ZT)." Ch. 3 in "Remarks on FFT Algorithms." http://www.jjj.de/fxt/.Boxer, R. "A Note on Numerical Transform Calculus." Proc. IRE 45, 1401-1406, 1957.Boxer, R. and Thaler, S. "A Simplified Method of Solving Linear and Nonlinear Systems." Proc. IRE 44, 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. Differential and 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's Outline of Feedback and Control Systems, 2nd ed. New York: McGraw-Hill, 1995.Elaydi, S. N. An Introduction 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. Theory and 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. Finite Difference Equations. New York: Dover, 1992.Ljung, L. System Identification: Theory for the User. Prentice-Hall, 1987.Mickens, R. E. Difference Equations, 2nd ed. Princeton, NJ: Van Nostrand Reinhold, 1987.Miller, K. S. Linear Difference Equations. New York: Benjamin, 1968.Ogata, K. Discrete-Time Control Systems. Englewood Cliffs, NJ: Prentice-Hall, 1987.Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.Sedgewick, R. and Flajolet, P. An Introduction to the Analysis of Algorithms. Reading, MA: Addison-Wesley, 1996.Tsypkin, Ya. Z. Sampling System Theory. New York: Pergamon Press, 1964.Vidyasagar, M. Control System 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.

Z-Transform

Cite this as:

Weisstein, Eric W. "Z-Transform." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Z-Transform.html