Glossary term
Z-Transform
A mathematical transform used to analyse discrete-time signals, difference equations, and sampled control systems.
Definition
methodThe z-transform maps a discrete-time sequence into a complex-domain function for analysing sampled signals and systems.
The z-transform is the discrete-time counterpart to transform methods used for continuous-time systems. It converts difference equations and impulse responses into algebraic expressions, making poles, zeros, stability, convolution, filters, and digital control dynamics easier to analyse.
The z-transform represents a discrete-time sequence x[n] as a function of a complex variable z:
This representation turns many difference-equation operations into algebraic operations. It is central to digital filters, sampled-data control, discrete transfer functions, stability analysis, and signal-processing algorithms.
Engineering use
In digital control and signal processing, poles and zeros in the z-plane describe dynamics of a discrete-time system. A pole inside the unit circle indicates stable decay for common causal linear time-invariant systems; poles near the unit circle indicate slow decay or lightly damped behaviour. The unit circle corresponds to steady-state sinusoidal frequency response when the region of convergence permits evaluation there.
The z-transform is also used to connect continuous-time models to sampled implementations. Discretization method, sampling period, zero-order hold behaviour, computational delay, quantization, and anti-alias filtering all influence the final digital system.
Sampling connection
The sampling period sets the scale of the z-plane representation. Continuous-time poles map into discrete-time poles according to the chosen discretization and hold assumptions, so stability and bandwidth must be interpreted with the sample rate in mind. In real implementations, controller execution time, zero-order hold output, sensor filtering, and quantization can add dynamics that are not present in the algebraic model.
Common mistakes
A common mistake is treating the z-transform as a direct copy of the Laplace transform with z substituted for s. Sampling changes the mapping, aliases continuous frequencies, and introduces discrete-time stability criteria. Another mistake is ignoring the region of convergence, which is necessary to distinguish sequences that share the same algebraic expression. A strong analysis states sampling period, causality assumption, region of convergence, pole-zero locations, discretization method, delay treatment, and frequency range of validity.