Convolution

From Wikiversity

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

$x(t)*y(t)=\int_{-\infty}^{+\infty} x(\tau)y(t-\tau)d\tau$

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 $o (t)$ :

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:

$y(t)=\int_{-\infty}^{+\infty} x(\tau)o(t-\tau)d\tau$

[edit] 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

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

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 $y$ .

Convolution

From Wikiversity

[edit] Discrete Convolution

Views

Personal tools

Search

Navigation

Community

Toolbox