---
tag: K0V6T18367
kind: concept
name: Advanced Z-transform
msc: [44A55, 93C57]
specializes: [5FY0JQGHYZ]
sources:
  - url: https://en.wikipedia.org/wiki/Advanced_z-transform
same_as:
  wikidata: Q1970283
  wikipedia: Advanced z-transform
---

## Summary

The advanced Z-transform is a variant of the [[Z-transform|5FY0JQGHYZ]] of sampled signals.
It samples a continuous-time signal at the instants $kT + m$, shifted by a part $m$ of the sampling period $T$.
It is $F(z, m) = \sum_{k \ge 0} f(kT + m) z^{-k}$, with $0 \le m \le T$.
It lets one describe delays that are not whole multiples of the sampling period, and the behaviour of a sampled system between sampling instants.
For fixed $m$ it has the usual properties of the Z-transform.

## Explanations

### L1 human

Sampling a signal at $0, T, 2T, \ldots$ only shows it at those instants.
The advanced Z-transform samples it at $m, T + m, 2T + m, \ldots$ instead, for a chosen offset $m$.
Letting $m$ run from $0$ to $T$ shows the signal at every time.
For $f(t) = e^{-at}$ one gets $F(z, m) = e^{-am} z/(z - e^{-aT})$.