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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
![{\displaystyle F(z,m)=\sum _{k=0}^{\infty }f(kT+m)z^{-k}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85ata1yjBBaNaQzAePztGQnji2oNvAaDm5aArEzjC4otw5zgsQoNs0)
where
- T is the sampling period
- m (the "delay parameter") is a fraction of the sampling period
![{\displaystyle [0,T].}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CzjBFaAeQzjK0njrBntnCagsPyto3yjK5ots0yjBCaNC3ygo4aDFD)
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.
Properties[edit]
If the delay parameter, m, is considered fixed then all the properties of the z-transform hold for the advanced z-transform.
Linearity[edit]
![{\displaystyle {\mathcal {Z}}\left\{\sum _{k=1}^{n}c_{k}f_{k}(t)\right\}=\sum _{k=1}^{n}c_{k}F_{k}(z,m).}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81oNC5zDmOoNlCzqzCygo1aqw2zjG1aqzDaDhAnts0oNwNzDK3o2o2)
Time shift[edit]
![{\displaystyle {\mathcal {Z}}\left\{u(t-nT)f(t-nT)\right\}=z^{-n}F(z,m).}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Boto5zje2zjwPnDrEaDC3athCzjzAnAwPzga5ngo4ntnDzgw1njBE)
Damping[edit]
![{\displaystyle {\mathcal {Z}}\left\{f(t)e^{-a\,t}\right\}=e^{-a\,m}F(e^{a\,T}z,m).}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Enqw3ygnDaNhEago0ajK1ztCPaNlEatePoDrFzAa0yje2njlDygw5)
Time multiplication[edit]
![{\displaystyle {\mathcal {Z}}\left\{t^{y}f(t)\right\}=\left(-Tz{\frac {d}{dz}}+m\right)^{y}F(z,m).}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CzNw5oDhEoNK3ajiOzNBFnjlDoqeNz2rDoNCQnDCPoqw2atrEotrA)
Final value theorem[edit]
![{\displaystyle \lim _{k\to \infty }f(kT+m)=\lim _{z\to 1}(1-z^{-1})F(z,m).}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85zNFCaNnAaAzBnteNzjm3zgi3oNvDo2dFyga4oNhBzNGOnAs3ata1)
Example[edit]
Consider the following example where
:
![{\displaystyle {\begin{aligned}F(z,m)&={\mathcal {Z}}\left\{\cos \left(\omega \left(kT+m\right)\right)\right\}\\&={\mathcal {Z}}\left\{\cos(\omega kT)\cos(\omega m)-\sin(\omega kT)\sin(\omega m)\right\}\\&=\cos(\omega m){\mathcal {Z}}\left\{\cos(\omega kT)\right\}-\sin(\omega m){\mathcal {Z}}\left\{\sin(\omega kT)\right\}\\&=\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}}\\&={\frac {z^{2}\cos(\omega m)-z\cos(\omega (T-m))}{z^{2}-2z\cos(\omega T)+1}}.\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83oDFAajnCytBEo2a5ntG5ngrAotBFaqs5yjCPnqvBoDi4ztw4oto2)
If
then
reduces to the transform
![{\displaystyle F(z,0)={\frac {z^{2}-z\cos(\omega T)}{z^{2}-2z\cos(\omega T)+1}},}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OzAhCzDa4zDoNoDK4zti2njBBa2eQaAi4a2dCnja2nAzDathEoDm0)
which is clearly just the z-transform of
.
References[edit]