Z-Transform
About This Series
The Z-transform converts a discrete-time signal into a function in the complex number domain (the $z$-plane). Just as the Laplace transform plays a central role in continuous-time systems, the Z-transform is the foundation for the analysis and design of discrete-time systems.
This series begins with the definition of the Z-transform and progresses through its basic properties, the inverse transform, discrete-time system analysis, and advanced theory using complex-analytic methods.
Definition of the Z-Transform (bilateral)
$$\mathcal{Z}\{x[n]\} = X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}$$At the introductory and basic levels, we primarily use the unilateral Z-transform, which applies to causal signals ($n \geq 0$):
$$X(z) = \sum_{n=0}^{\infty} x[n] z^{-n}$$Study by Level
Learning Path
Relationship with the Laplace Transform
The Z-transform can be viewed as the discrete-time counterpart of the continuous-time Laplace transform. When a continuous signal is sampled with period $T$, $$z = e^{sT}$$ This relationship allows continuous-time system design techniques to be applied to discrete-time systems. See Intermediate Chapter 6 for details.
For Those Interested in Signal Processing Applications
To study the Z-transform from a signal processing perspective, see Signal Processing → Z-Transform Applications.
- Intermediate: Block diagrams, filter structures, frequency-selective filters, quantization
- Advanced: IIR/FIR digital filter design, multirate signal processing, adaptive filters
Related Topics
- Laplace Transform — the continuous-time counterpart
- Fourier Analysis — the foundation of frequency-domain analysis
- Complex Analysis — the theoretical basis of the $z$-plane
- Z-Transform Applications (Signal Processing) — digital filter design