Editing Advanced z-transform

In [[mathematics]] and [[signal processing]], the '''advanced z-transform''' is an extension of the [[z-transform]], to incorporate ideal delays that are not multiples of the [[sampling rate|sampling time]]. The advanced z-transform is widely applied, for example, to accurately model processing delays in [[digital control]]. It is also known as the '''modified z-transform'''. 

It takes the form

:<math>F(z, m) = \sum_{k=0}^{\infty} f(k T + m)z^{-k}</math>

where
* ''T'' is the sampling period
* ''m'' (the "delay parameter") is a fraction of the sampling period <math>[0, T].</math>

==Properties==
If the delay parameter, ''m'', is considered fixed then all the properties of the z-transform hold for the advanced z-transform.

===Linearity===
:<math>\mathcal{Z} \left\{ \sum_{k=1}^{n} c_k f_k(t) \right\} = \sum_{k=1}^{n} c_k F_k(z, m).</math>

===Time shift===
:<math>\mathcal{Z} \left\{ u(t - n T)f(t - n T) \right\} = z^{-n} F(z, m).</math>

===Damping===
:<math>\mathcal{Z} \left\{ f(t) e^{-a\, t} \right\} = e^{-a\, m} F(e^{a\, T} z, m).</math>

===Time multiplication===
:<math>\mathcal{Z} \left\{ t^y f(t) \right\} = \left(-T z \frac{d}{dz} + m \right)^y F(z, m).</math>

===Final value theorem===
:<math>\lim_{k \to \infty} f(k T + m) = \lim_{z \to 1} (1-z^{-1})F(z, m).</math>

==Example==
Consider the following example where <math>f(t) = \cos(\omega t)</math>:

:<math>\begin{align}
F(z, m) & = \mathcal{Z} \left\{ \cos \left(\omega \left(k T + m \right) \right) \right\} \\
        & = \mathcal{Z} \left\{ \cos (\omega k T) \cos (\omega m) - \sin (\omega k T) \sin (\omega m) \right\} \\
        & = \cos(\omega m) \mathcal{Z} \left\{ \cos (\omega k T) \right\} - \sin (\omega m) \mathcal{Z} \left\{ \sin (\omega k T) \right\} \\
        & = \cos(\omega m) \frac{z \left(z - \cos (\omega T) \right)}{z^2 - 2z \cos(\omega T) + 1} - \sin(\omega m) \frac{z \sin(\omega T)}{z^2 - 2z \cos(\omega T) + 1} \\
        & = \frac{z^2 \cos(\omega m) - z \cos(\omega(T - m))}{z^2 - 2z \cos(\omega T) + 1}.
\end{align}</math>

If <math>m=0</math> then <math>F(z, m)</math> reduces to the transform

:<math>F(z, 0) = \frac{z^2 - z \cos(\omega T)}{z^2 - 2z \cos(\omega T) + 1},</math>

which is clearly just the ''z''-transform of <math>f(t)</math>.

==References==
{{reflist}}
*{{cite book |author-link=Eliahu Ibraham Jury |first=Eliahu Ibraham |last=Jury |title=Theory and Application of the z-Transform Method |publisher=Krieger |date=1973 |isbn=0-88275-122-0 |oclc=836240}}

{{DSP}}

[[Category:Transforms]]