Source: Wikipedia, the free encyclopedia.
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://wikimedia.org/api/rest_v1/media/math/render/svg/913589e3306b1580d16c4f2092eb498a494e0c54)
where
- T is the sampling period
- m (the "delay parameter") is a fraction of the sampling period
![{\displaystyle [0,T].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d49a2b0474d5ee6d0e1967879a5489d3978f828c)
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
If the delay parameter, m, is considered fixed then all the properties of the z-transform hold for the advanced z-transform.
Linearity
![{\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://wikimedia.org/api/rest_v1/media/math/render/svg/5c99602c0d5cd8f51d64851cc21fe54c43677cf6)
Time shift
![{\displaystyle {\mathcal {Z}}\left\{u(t-nT)f(t-nT)\right\}=z^{-n}F(z,m).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea69426441f5b29711d43ffd14c9df8e6c4d5d9b)
Damping
![{\displaystyle {\mathcal {Z}}\left\{f(t)e^{-a\,t}\right\}=e^{-a\,m}F(e^{a\,T}z,m).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bed78fc31b0f407559130b121b5a6c4826d0c8d9)
Time multiplication
![{\displaystyle {\mathcal {Z}}\left\{t^{y}f(t)\right\}=\left(-Tz{\frac {d}{dz}}+m\right)^{y}F(z,m).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d749b1bc7701279ad0cab37ecc90f91ad615ba5f)
Final value theorem
![{\displaystyle \lim _{k\to \infty }f(kT+m)=\lim _{z\to 1}(1-z^{-1})F(z,m).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/978d36f2cee234074a7c4ccba8c8c1e782fe7135)
Example
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://wikimedia.org/api/rest_v1/media/math/render/svg/7b8f06d99bcc9e89defa9a1e9891edeb18548a66)
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://wikimedia.org/api/rest_v1/media/math/render/svg/26ad638663b78516d9e3b02a83bdd36fcc11bb04)
which is clearly just the z-transform of
.
References