Advanced z-transform

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title: "Advanced z-transform" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["transforms"] topic_path: "general/transforms" source: "https://en.wikipedia.org/wiki/Advanced_z-transform" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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. The advanced z-transform is widely applied, for example, to accurately model processing delays in digital control. It is also known as the modified z-transform.

It takes the form

:F(z, m) = \sum_{k=0}^{\infty} f(k T + m)z^{-k}

where

  • T is the sampling period
  • m (the "delay parameter") is a fraction of the sampling period [0, T].

Properties

If the delay parameter, m, is considered fixed then all the properties of the z-transform hold for the advanced z-transform.

Linearity

:\mathcal{Z} \left{ \sum_{k=1}^{n} c_k f_k(t) \right} = \sum_{k=1}^{n} c_k F_k(z, m).

Time shift

:\mathcal{Z} \left{ u(t - n T)f(t - n T) \right} = z^{-n} F(z, m).

Damping

:\mathcal{Z} \left{ f(t) e^{-a, t} \right} = e^{-a, m} F(e^{a, T} z, m).

Time multiplication

:\mathcal{Z} \left{ t^y f(t) \right} = \left(-T z \frac{d}{dz} + m \right)^y F(z, m).

Final value theorem

:\lim_{k \to \infty} f(k T + m) = \lim_{z \to 1} (1-z^{-1})F(z, m).

Example

Consider the following example where f(t) = \cos(\omega t):

:\begin{align} F(z, m) & = \mathcal{Z} \left{ \cos \left(\omega \left(k T + m \right) \right) \right} \ & = \mathcal{Z} \left{ \cos (\omega k T) \cos (\omega m) - \sin (\omega k T) \sin (\omega m) \right} \ & = \cos(\omega m) \mathcal{Z} \left{ \cos (\omega k T) \right} - \sin (\omega m) \mathcal{Z} \left{ \sin (\omega k T) \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{align}

If m=0 then F(z, m) reduces to the transform

:F(z, 0) = \frac{z^2 - z \cos(\omega T)}{z^2 - 2z \cos(\omega T) + 1},

which is clearly just the z-transform of f(t).

References

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