# Convolution

Consider an LTI system*H*:[

*Integers*→

*Reals*] → [

*Integers*→

*Reals*] with impulse response

*h*:

*Integers*→

*Reals*. Recall that we can describe any input as a sum of weighted delta functions,

By linearity, the output must be a sum of the responses of the system
to the individual terms, *x*(*k*)δ
(*n* − *k*). The response to an individual
term is *x*(*k*)* h*(*n* −
*k*), so the output must be

This summation is called the **convolution sum**. It shows how to
obtain an output of an LTI system given only the input and the impulse
response. It works for any input, so it tells us that impulse response
fully characterizes an LTI system.

We can change variables in the summation, letting *m* = *n*
− *k*, to get an equivalent form,

The continuous-time version is similar, albeit more mathematically subtle.
Given an LTI system *H*:[*Reals* →
*Reals*] → [*Reals* →
*Reals*] with impulse response *h*: *Reals* →
*Reals*, and given an input *x*: *Reals* →
*Reals*, the output is a function *y*: *Reals* →
*Reals* where for all *t* ∈
*Reals*,

This is called the **convolution integral**. The summation has become
an integral, but otherwise, the form looks very similar. Again, by a change
of variables, we write the equivalent form