Digital filters

Level AdvancedDifficulty ★★★★★Application⌖ Open in the map

What is it?

Difference equations that keep some frequencies and remove others. FIR filters are convolutions; IIR filters feed back previous outputs and are analysed with the Z-transform (stable iff poles inside the unit circle).

Formulas

y[n] = \sum_{k=0}^{M}b_k\,x[n-k] - \sum_{k=1}^{N}a_k\,y[n-k] \quad\Longleftrightarrow\quad H(z) = \frac{\sum_k b_k z^{-k}}{1 + \sum_k a_k z^{-k}}

The mathematics behind it

Convolution★★★★★fundamental
An FIR filter is a convolution of the signal with its impulse response.
Z-transform★★★★★fundamental
Filter design and analysis (IIR/FIR, poles and zeros, frequency response on $|z| = 1$ ) is done with the Z-transform.
Rational functions★★★★★frequent
An IIR filter's transfer function $H(z)$ is rational; its poles decide stability.
Partial fractions★★★★★frequent
Inverse Z-transforms and parallel filter realizations use partial fractions of $H(z)$ .
First-order linear equations★★★★★frequent
The exponential moving average $y_k = (1-\alpha)y_{k-1} + \alpha x_k$ is a discretized first-order linear ODE (an RC low-pass).

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