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. The advanced z-transform is widely applied, for example, to accurately model processing delays in digital control. It is also known as the modified z-transform.

It takes the form

$F(z,m)=\sum _{k=0}^{\mathrm{\infty }}f(kT+m)z^{-k}$

where

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

Properties

If the delay parameter, m, is considered fixed then all the properties of the z-transform hold for the advanced z-transform.

Linearity

$𝒵\{\sum _{k=1}^n{c}_k{f}_k(t)\}=\sum _{k=1}^n{c}_k{F}_k(z,m).$

Time shift

$𝒵\{u(t-nT)f(t-nT)\}=z^{-n}F(z,m).$

Damping

$𝒵\{f(t)e^{-at}\}=e^{-am}F(e^{aT}z,m).$

Time multiplication

$𝒵\{t^{y}f(t)\}={(-Tz\frac{d}{dz}+m)}^{y}F(z,m).$

Final value theorem

$\underset{k\to \mathrm{\infty }}{\lim }f(kT+m)=\underset{z\to 1}{\lim }(1-z^{-1})F(z,m).$

Example

Consider the following example where $f(t)=\cos (\omega t)$:

$\begin{array}{cc}F(z,m)& =𝒵\{\cos \left(\omega (kT+m)\right)\}\\ & =𝒵\{\cos (\omega kT)\cos (\omega m)-\sin (\omega kT)\sin (\omega m)\}\\ & =\cos (\omega m)𝒵\{\cos (\omega kT)\}-\sin (\omega m)𝒵\{\sin (\omega kT)\}\\ & =\cos (\omega m)\frac{z(z-\cos (\omega T))}{z^2-2z\cos (\omega T)+1}-\sin (\omega m)\frac{z\sin (\omega T)}{z^2-2z\cos (\omega T)+1}\\ & =\frac{z^2\cos (\omega m)-z\cos (\omega (T-m))}{z^2-2z\cos (\omega T)+1}.\end{array}$

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

$F(z,0)=\frac{z^2-z\cos (\omega T)}{z^2-2z\cos (\omega T)+1},$

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

References

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