Z-transform

Level SpecializationDifficulty ★★★★★Concept⌖ Open in the map

What is it?

The discrete-time Laplace transform: $X(z) = \sum_n x[n]z^{-n}$ , a power series in $z^{-1}$ . Difference equations (digital filters, discrete controllers) become rational functions of $z$ ; stability means poles inside the unit circle. It is the bridge from calculus to digital signal processing.

Formulas

X(z) = \sum_{n=0}^{\infty}x[n]\,z^{-n}, \qquad x[n-1] \longleftrightarrow z^{-1}X(z), \qquad z = e^{sT}

Where it shows up in computing

Digital filters★★★★★fundamentalSignals, media and vision
Filter design and analysis (IIR/FIR, poles and zeros, frequency response on $|z| = 1$ ) is done with the Z-transform.
Control theory★★★★★frequentRobotics and control
Digital controllers running on microcontrollers are designed in the $z$ -domain.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

∿ Signals, media and vision

Digital filters★★★★★

⚙ Robotics and control

Control theory★★★★★
Control theory→Trajectory optimization and MPC★★★★★

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.