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Notes › ENGR 2341: Signals and Systems Lecture 21

z-Transform

calendar_today April 17, 2025 article 81 words access_time 1 min replay Modified April 17, 2025

Table of Contents

$z$-Transform

$z$-Transform

Transforms a discrete-time sequence into the $z$-domain

$$X(z) = \sum_{k=0}^{N} x[k] z^{-k}$$

Inverse $z$-Transform

Transforms from the $z$-domain to a discrete-time sequence
Neither inverse nor regular $z$-transforms are practical to evaluate by hand (except with some limited summations), so use a Z-transform table (course handout) instead

$$x[n] = \frac{1}{j 2\pi} \int X(z) z^{n-1} \mathrm{d}z$$

Partial Fraction Decomposition With MATLAB

When working in the $z$-domain to compute inverse transform, use residuez instead of normal residue from Laplace Transform

References

Z-transform (course handout)

Sources

Z-transform
Z-transform table (course handout)

Graph

mindmap ↗