Advanced z-transform

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


  • T is the sampling period
  • m (the "delay parameter") is a fraction of the sampling period

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.


Time shift


Time multiplication

Final value theorem


Consider the following example where :

If then reduces to the transform

which is clearly just the z-transform of .


    • Eliahu Ibraham Jury, Theory and Application of the z-Transform Method, Krieger Pub Co, 1973. ISBN 0-88275-122-0.
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