Convolution

A convolution between two signals, $x(t)$ and $y(t)$, is an operation defined as follows:

$x(t)*y(t)=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}x(\tau )y(t-\tau )d\tau$

The process of convolution is very useful in the time domain analysis of systems, because we can fully describe a system by its impulse response. Let's consider the following system which operates on an input as $O\{\}$, having characterized its impulse response by $o(t)$:

$x(t)⟶\begin{array}{c}O\{\}\end{array}⟶y(t)$

$y(t)=O[x(t)]$

$y(t)=x(t)*o(t)$

Put into other words, the output of a system in an instant $t$ can be written as a linear combination of past and future instants of the input and its impulse response:

$y(t)=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}x(\tau )o(t-\tau )d\tau$

Discrete Convolution

In discrete time there is no continuous time $x(t)$ but finite samples $x[n]$.

So the integral can be rewritten as a sum:

$(x*y)[m]=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}x[m-n]*y[n]$

To understand the convolution of finite length signals better, let's look at an example with the signals $x=[1,2,3]$ and $y=[6,9]$.

[ 1] * [6 9] = ?

[ 6 12 18  0]    // [1 2 3] * 6
[ 0  9 18 27]    // [1 2 3] * 9
-------------
[ 6 21 36 27]    // sum of the above

Note that the length of the output signal has the length $N+M-1$ where $M$ is the length of $x$ and $N$ the length of $y$.