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://cdn.statically.io/img/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://cdn.statically.io/img/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.
If the delay parameter, m, is considered fixed then all the properties of the z-transform hold for the advanced z-transform.
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Consider the following example where :
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If then reduces to the transform
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which is clearly just the z-transform of .