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

Week 11

Convolution