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)=
| infty | |
| \sum | |
| k=0 |
f(kT+m)z-k
where
[0,T].
It is also known as the modified z-transform.
The advanced z-transform is widely applied, for example to accurately model processing delays in digital control.
If the delay parameter, m, is considered fixed then all the properties of the z-transform hold for the advanced z-transform.
l{Z}\left\{
| n | |
| \sum | |
| k=1 |
ckfk(t)\right\}=
| n | |
| \sum | |
| k=1 |
ckFk(z,m).
l{Z}\left\{u(t-nT)f(t-nT)\right\}=z-nF(z,m).
l{Z}\left\{f(t)e-a\right\}=e-aF(eaz,m).
l{Z}\left\{tyf(t)\right\}=\left(-Tz
| d | |
| dz |
+m\right)yF(z,m).
\limkf(kT+m)=\limz(1-z-1)F(z,m).
Consider the following example where
f(t)=\cos(\omegat)
\begin{align} F(z,m)&=l{Z}\left\{\cos\left(\omega\left(kT+m\right)\right)\right\}\\ &=l{Z}\left\{\cos(\omegakT)\cos(\omegam)-\sin(\omegakT)\sin(\omegam)\right\}\\ &=\cos(\omegam)l{Z}\left\{\cos(\omegakT)\right\}-\sin(\omegam)l{Z}\left\{\sin(\omegakT)\right\}\\ &=\cos(\omegam)
| z\left(z-\cos(\omegaT)\right) | |
| z2-2z\cos(\omegaT)+1 |
-\sin(\omegam)
| z\sin(\omegaT) | |
| z2-2z\cos(\omegaT)+1 |
\\ &=
| z2\cos(\omegam)-z\cos(\omega(T-m)) | |
| z2-2z\cos(\omegaT)+1 |
. \end{align}
If
m=0
F(z,m)
F(z,0)=
| z2-z\cos(\omegaT) | |
| z2-2z\cos(\omegaT)+1 |
,
which is clearly just the z-transform of
f(t)