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)