# The differences between autocorrelation and convolution

1. ### azserendipity

10
Hi,

This is something that has appeared in a module, we've had a lab session in it but im still not sure what it is.

I don't understand the formulas given in lecture notes so I was hoping someone could explain it?

Autocorrelation
R1(τ) = ∫f(t)f(t+τ)dt = f V f

Convolution
C12(τ)= ∫ f1(t)f2(-t+τ) dt

Any help would be really appreciated!

2. ### mathman

6,474
I'll describe it in the context of probability theory.

Autocorrelation refers to a property of a stochastic process, describing how events at one time are related to events at another time. The integral you displayed is (after taking statistical average) is the auto covariance. It needs to be normalized by the variance to get the autocorrelation.

Convolution is used to get the distribution function of a sum of two independent random variables, given the distribution functions of the given random variables.

3. ### azserendipity

10

So how would you apply it in mathematical terms?

I understand what you have said it is in terms of probability theory but how would you apply it to signal theory?

Last edited: Jan 21, 2012
4. ### mathman

6,474
In this context, signal theory uses probability theory - look at the signal as a stationary Gaussian process with a spectrum. The autocorrelation is essentially the Fourier transform of the spectrum (or the inverse transform). Convolution would come into play when adding two signals.

5. ### Antiphon

Convolution is used in signal processing in the time domain. Convolution runs the impulse response backward in time against the signal to solve for the output given an arbitrary input.