Discover

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.

Contents

Properties

Linearity

Time shift

Damping

Time multiplication

Final value theorem

Example

See also

Bibliography

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 rac{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]
:
ight)}{z^2 - 2z cos(omega T) + 1} - sin(omega m) rac{z sin(omega T)}{z^2 - 2z cos(omega T) + 1}
:
If $m=0$ then $F(z,\; m)$ reduces to the Z-transform
:
which is clearly just the Z-transform of $f(t).$

See also

★ Z-transform

Bibliography

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