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

$F(z, m) = \sum_{k=0}^{\infty} f(k T + m)z^{-k}$

where

T is the sampling period
m (the "delay parameter") is a fraction of the sampling period $[0, T).$

It is also known as the modified Z-transform.

The advanced Z-transform is widely applied, for example to model accurately 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

$Z \left[ \sum_{k=1}^{m} c_k f_k(t) \right] = \sum_{k=1}^{m} c_k F(z, m).$

Time shift

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

Damping

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

Time multiplication

$Z \left[ t^y f(t) \right] = \left(-T z \frac{d}{dz} + m \right)^y F(z, m).$

Final value theorem

$\lim_{k = \infty} f(k T + m) = \lim_{z = 1} (1-z^{-1})F(z, m).$

Example

Consider the following example where $f (t) = cos(ω t)$

$F(z, m) = Z \left[\cos \left(\omega \left(k T + m \right) \right) \right]$

$F(z, m) = Z \left[\cos (\omega k T) \cos (\omega m) - \sin (\omega k T) \sin (\omega m) \right]$

$F(z, m) = \cos(\omega m) Z \left[ \cos (\omega k T) \right] - \sin (\omega m) Z \left[ \sin (\omega k T) \right]$

$F(z, m) = \cos(\omega m) \frac{z \left(z - \cos (\omega T) \right)}{z^2 - 2z \cos(\omega T) + 1} - \sin(\omega m) \frac{z \sin(\omega T)}{z^2 - 2z \cos(\omega T) + 1}$

$F(z, m) = \frac{z^2 \cos(\omega m) - z \cos(\omega(T - m))}{z^2 - 2z \cos(\omega T) + 1}.$

If $m = 0$ then $F (z, m)$ reduces to the Z-transform

$F(z, m) = \frac{z^2 - z \cos(\omega T)}{z^2 - 2z \cos(\omega T) + 1}$

which is clearly just the Z-transform of $f (t).$

Bibliography

Eliahu Ibraham Jury, Theory and Application of the Z-Transform Method, Krieger Pub Co, 1973. ISBN 0-88275-122-0.

Digital signal processing
Theory — Discrete frequency, Nyquist–Shannon sampling theorem, estimation theory, detection theory
Sub-fields — audio signal processing \| control engineering \| digital image processing \| speech processing \| statistical signal processing
Techniques — Discrete Fourier transform (DFT) \| Discrete-time Fourier transform (DTFT) \| bilinear transform \| Z-transform, advanced Z-transform
Sampling — oversampling \| undersampling \| downsampling \| upsampling \| aliasing \| anti-aliasing filter \| sampling rate \| Nyquist rate/frequency

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