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
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
Time shift
Damping
Time multiplication
Final value theorem
Example
Consider the following example where f(t) = cos(ωt)
If m = 0 then F(z,m) reduces to the Z-transform
which is clearly just the Z-transform of f(t).
See also
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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![Z \left[ u(t - n T)f(t - n T) \right] = z^{-n} F(z, m).](http://content.answers.com/main/content/wp/en/math/5/7/b/57b2dbd0e3797acbc907c85cd84aa004.png)
![Z \left[ f(t) e^{-a\, t} \right] = e^{-a\, m} F(e^{a\, T} z, m).](http://content.answers.com/main/content/wp/en/math/e/9/8/e98323e66f3ba24a331dbe58924429e4.png)
![Z \left[ t^y f(t) \right] = \left(-T z \frac{d}{dz} + m \right)^y F(z, m).](http://content.answers.com/main/content/wp/en/math/e/0/9/e097db03f2ebed1f08a9a8a07695dbf6.png)

![F(z, m) = Z \left[\cos \left(\omega \left(k T + m \right) \right) \right]](http://content.answers.com/main/content/wp/en/math/a/9/1/a91d892860da01cc9c9d81728c4dc29e.png)
![F(z, m) = Z \left[\cos (\omega k T) \cos (\omega m) - \sin (\omega k T) \sin (\omega m) \right]](http://content.answers.com/main/content/wp/en/math/d/d/f/ddf7f9c2298ea7e2182f5230d869075d.png)
![F(z, m) = \cos(\omega m) Z \left[ \cos (\omega k T) \right] - \sin (\omega m) Z \left[ \sin (\omega k T) \right]](http://content.answers.com/main/content/wp/en/math/e/c/0/ec065fcd6d868f82b11506303f546922.png)






