Definition

Section: Filtering

Definition

A filter is a linear time invariant system. It can be written as a convolution.

Impulse response

Let a filter with an impulse response $h$, then $$ \begin{aligned} y[t] & = (h*x)[t]\\ y[t] & = \sum_{n=-\infty}^{+\infty} h[n]x[t-n] \end{aligned} $$

For finite sequences (digital signals), of size $N$ two possibilities:

zeros-padding: $h[t]=0$ and $x[t]=0$ for all $t\notin\lbrace 0,\ldots,N-1 \rbrace$
circular convolution: $h$ and $x$ are supposed to be periodic (different results if the period is chosen to be $N$ or $2N$ !)