The Auto-correlation and the Cross-correlation.

Proceeding with our work on the Fourier transform, let us define two important functions, the Auto-correlation and the Cross-correlation.

Auto Correlation

You have seen that for a Periodic signal y(t), has Fourier series coefficients that the modulus square of the Fourier series coefficients of y(t).

Lets look at an equivalent situation with aperiodic signals, i.e:

Assume that

then

Notice that

Since

We have,

Using the dual of the convolution theorem,

The auto-correlation of x(t), denoted by is defined as:

Its Spectrum is the modulus square of the spectrum of x(t).

It can also be interpreted as the projection of x(t) on its own shifted version, shifted back by an interval ‘t'.

It can be shown that ( note that is nothing but the energy in the signal x(t) )