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

:$Z\; left\; [\; sum\_\{k=1\}^\{m\}\; c\_k\; f\_k(t)\; ight]\; =\; sum\_\{k=1\}^\{m\}\; c\_k\; F(z,\; m).$

Time shift

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

Damping

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

Time multiplication

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

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 $f(t)\; =\; cos(omega\; t)$

:$F(z,\; m)\; =\; Z\; left\; [cos\; left(omega\; left(k\; T\; +\; m\; ight)\; ight)\; ight]$:$F(z,\; m)\; =\; Z\; left\; [cos\; (omega\; k\; T)\; cos\; (omega\; m)\; -\; sin\; (omega\; k\; T)\; sin\; (omega\; m)\; ight]$:$F(z,\; m)\; =\; cos(omega\; m)\; Z\; left\; [\; cos\; (omega\; k\; T)\; ight]\; -\; sin\; (omega\; m)\; Z\; left\; [\; sin\; (omega\; k\; T)\; ight]$:$F(z,\; m)\; =\; cos(omega\; m)\; frac\{z\; left(z\; -\; cos\; (omega\; T)\; ight)\}\{z^2\; -\; 2z\; cos(omega\; T)\; +\; 1\}\; -\; sin(omega\; m)\; frac\{z\; sin(omega\; T)\}\{z^2\; -\; 2z\; cos(omega\; T)\; +\; 1\}$:$F(z,\; m)\; =\; frac\{z^2\; cos(omega\; m)\; -\; z\; cos(omega(T\; -\; m))\}\{z^2\; -\; 2z\; cos(omega\; T)\; +\; 1\}.$

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 $f(t).$

ee also

*Z-transform

Bibliography

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