Editing Z-transform (section)

=== Unilateral Z-transform ===
Alternatively, in cases where <math>x[n]</math> is defined only for <math>n \ge 0</math>, the ''single-sided'' or ''unilateral'' Z-transform is defined as:

{{Equation box 1 |title=
|indent =: |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA
|equation = <math>X(z) = \mathcal{Z}\{x[n]\} =  \sum_{n=0}^{\infty} x[n] z^{-n}.</math>
}}

In [[signal processing]], this definition can be used to evaluate the Z-transform of the [[Finite impulse response#Frequency response|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 <math>x[n]</math> is the probability that a discrete random variable takes the value. The properties of Z-transforms (listed in {{Slink|2=Properties|nopage=y}}) have useful interpretations in the context of probability theory.