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.