Editing Z-transform (section)

==Definition==
The Z-transform can be defined as either a ''one-sided'' or ''two-sided'' transform. (Just like we have the [[Laplace transform|one-sided Laplace transform]] and the [[two-sided Laplace transform]].)<ref name="Jackson 1996 pp. 29–54">{{cite book | last=Jackson | first=Leland B. | title=Digital Filters and Signal Processing | chapter=The z Transform | publisher=Springer US | publication-place=Boston, MA | year=1996 | isbn=978-1-4419-5153-3 | doi=10.1007/978-1-4757-2458-5_3 | pages=29–54 | quote= z transform is to discrete-time systems what the Laplace transform is to continuous-time systems. ''z'' is a complex variable. This is sometimes referred to as the two-sided ''z'' transform, with the one-sided z transform being the same except for a summation from ''n'' = 0 to infinity. The primary use of the one sided transform ... is for causal sequences, in which case the two transforms are the same anyway. We will not, therefore, make this distinction and will refer to ... as simply the z transform of ''x''(''n'').}}</ref>

=== Bilateral Z-transform ===
The ''bilateral'' or ''two-sided'' Z-transform of a discrete-time signal <math>x[n]</math> is the [[formal power series]] <math>X(z)</math> 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=-\infty}^{\infty} x[n] z^{-n}</math>
}}

where <math>n</math> is an integer and <math>z</math> is, in general, a [[complex number]]. In [[Polar coordinate system#Complex numbers|polar form]], <math>z</math> may be written as:
:<math>z = A e^{j\phi} = A\cdot(\cos{\phi}+j\sin{\phi})</math>
where <math>A</math> is the magnitude of <math>z</math>, <math>j</math> is the [[imaginary unit]], and <math>\phi</math> is the ''[[complex argument]]'' (also referred to as ''angle'' or ''phase'') in [[radian]]s.

=== 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.