# Introduction

Notes

# Content

## Convolution - Definition

Let us define the function
to
equal the **convolution** between two functions
and
expressed
as

I will sometimes use subscripts to
represent variables where this is convenient and clear. This convolution is
defined as the integral

where
t
represents a “dummy variable” for integration. It might be thought
that this convolution could also be expressed in discrete variable as

However beware! Most computational
software dislikes negative indices and requires that each vector index starts at
*n* = 0. To accommodate this offset we need to add a small modification
e.g.

There
are still 2·N+1 discrete values for the summation and the offset (N+1) is a
small detail to reference the “0” position in the vector. Now *f*_{n},
*h*_{n} and *g*_{n} have
indices ranging from n=0 to
n=2·N+1 whereas the previous range of index was -N < n < N.
However we still need to prevent the index of h exceeding
its boundaries so the actual discrete domain formula becomes

Alternatively

Interestingly the discrete convolution
can also be interpreted as a matrix-vector operation i.e.

If the matrix H is invertible, then we should be able to
predict f from g (where g could be thought of as the output from a system H and
f is the input)

## Interesting Derivations

Convolution reveals a number of surprising consequences - despite its
seemingly simple but obscure proposition; so let's use this proposition to
predict a few outcomes.

### Convolution With A Step Function

# Summary

© Ian Scott 2009