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 time. It takes the form

[math]\displaystyle{ 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]\displaystyle{ [0, T]. }[/math]

It is also known as the modified z-transform.

The advanced z-transform is widely applied, for example to accurately model processing delays in digital control.

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]\displaystyle{ \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]\displaystyle{ \mathcal{Z} \left\{ u(t - n T)f(t - n T) \right\} = z^{-n} F(z, m). }[/math]

Damping

[math]\displaystyle{ \mathcal{Z} \left\{ f(t) e^{-a\, t} \right\} = e^{-a\, m} F(e^{a\, T} z, m). }[/math]

Time multiplication

[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ f(t) = \cos(\omega t) }[/math]:

[math]\displaystyle{ \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]\displaystyle{ m=0 }[/math] then [math]\displaystyle{ F(z, m) }[/math] reduces to the transform

[math]\displaystyle{ 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]\displaystyle{ f(t) }[/math].

References

• Jury, Eliahu Ibraham (1973). Theory and Application of the z-Transform Method. Krieger. ISBN 0-88275-122-0. OCLC 836240.