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Bilateral Z-Transform

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A bilateral Z transform is two-sided (doubly infinite) Z-Transform given by

Z^((2))[{a_n}_(n=-infty)^infty](z)=sum_(n=-infty)^infty(a_n)/(z^n)

(Zwillinger 1996; Krantz 1999, p. 214). The bilateral transform is generally less commonly used than the unilateral Z-transform, since the latter finds widespread application as a technique essentially equivalent to generating functions.

The following table summarized values of the bilateral Z-transforms for various functions. Here, delta_(n0) is the Kronecker delta, H(x) is the Heaviside step function, and Li_k(z) is the polylogarithm.

a_nZ^((2))[{a_n}_(n=-infty)^infty](z)
delta_(n0)1
(-1)^nH(n)z/(z+1)
H(n)z/(z-1)
nH(n)z/((z-1)^2)
a^nnH(n)(az)/((a-z)^2)

See also

Unilateral Z-Transform, Generating Function, Z-Transform

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References

Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, 1999.Zwillinger, D. (Ed.). "Z-Transform." §6.27 in CRC Standard Mathematical Tables and Formulae, 30th ed. Boca Raton, FL: CRC Press, pp. 543-547, 1996.

Referenced on Wolfram|Alpha

Bilateral Z-Transform

Cite this as:

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

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