Convolution

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A convolution between two signals, and , is an operation defined as follows:

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 , having characterized its impulse response by :

System.JPG


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

Discrete Convolution[edit]

In discrete time there is no continuous time but finite samples .

So the integral can be rewritten as a sum

To understand the convolution of finite length signals better, let's look at an example with the signals and .

[ 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 where is the length of and the length of .