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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>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>

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>\mathcal{Z} \left\{ \sum_{k=1}^{n} c_k f_k(t) \right\} = \sum_{k=1}^{n} c_k F(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>.

See also

Bibliography