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

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

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

$Z\left[\sum _{k=1}^{m}c_{k}f_{k}(t)\right]=\sum _{k=1}^{m}c_{k}F(z,m).$

Time shift

$Z\left[u(t-nT)f(t-nT)\right]=z^{-n}F(z,m).$

Damping

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

Time multiplication

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

Final value theorem

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

Example

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

$F(z,m)=Z\left[\cos \left(\omega \left(kT+m\right)\right)\right]$

$F(z,m)=Z\left[\cos(\omega kT)\cos(\omega m)-\sin(\omega kT)\sin(\omega m)\right]$

$F(z,m)=\cos(\omega m)Z\left[\cos(\omega kT)\right]-\sin(\omega m)Z\left[\sin(\omega kT)\right]$

$F(z,m)=\cos(\omega m){\frac {z\left(z-\cos(\omega T)\right)}{z^{2}-2z\cos(\omega T)+1}}-\sin(\omega m){\frac {z\sin(\omega T)}{z^{2}-2z\cos(\omega T)+1}}$

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

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

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

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

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.