Rational functions

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

What is it?

Quotients of polynomials $p(x)/q(x)$ . They blow up at the zeros of $q$ (poles) and are the exact form of the transfer functions of linear filters and control systems.

Formulas

R(x) = \frac{p(x)}{q(x)}, \qquad H(z) = \frac{b_0 + b_1 z^{-1} + \dots}{1 + a_1 z^{-1} + \dots}

Where it shows up in computing

Digital filters★★★★★frequentSignals, media and vision
An IIR filter's transfer function $H(z)$ is rational; its poles decide stability.
Control theory★★★★★frequentRobotics and control
Linear time-invariant systems have rational transfer functions $G(s)$ .

Where is it used?

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

⚙ Robotics and control

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

∿ Signals, media and vision

Digital filters★★★★★

What depends on it

Partial fractions

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