Advanced z-transform

The advanced Z-transform is an extension to the Z-transform for incorporation of ideal delays that are not multiples of the sampling time.

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

where

• T is the sampling time
• m (the "delay paramter") is a fraction of the sampling time $[0,T)$

The advanced Z-transform is useful in many ways. One of which is to accuractly model 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

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

Dampening

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

Time multiplication

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

Final value theorem

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

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

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

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

See also

• Z-transform