Convolution and Autocorrelation

Cuthbert Nyack

Here the difference between autocorrelation and convolution is illustrated by considering the following function which is a unit ramp cut off at t = 1.

The Fourier Transform of the function f(t) is given below.

Autocorrelation

The autocorrelation function is defined by the following equation.

When evaluated the Autocorrelation function results in the following 2 equations.

The proceedure for calculating the autocorrelation function is illustrated by the applet below. In the applet f(t) is shown in purple, f(t + tau) in orange, the area to be integrated is shown in green and the autocorrelation in red. The green vertical line shows the value of the autocorrelation at the current value of tau.

The Fourier transform of the autocorrelation which is also the POWER SPECTRUM is given by the following equation:-

The relation between the fourier transform of the autocorrelation (POWER SPECTRUM) and the transform of f(t) is shown below.

Convolution

Convolution is defined by the equation below.

Evaluating the integral results in the following 2 equations.

The Fourier Transform of the convolution is:-

and the relation between the fourier transform of f(t) and the transform of the convolution is shown below. From the convolution theorem, the transform of the convolution of 2 functions is the product of the transform of each function.

Proceedure for evaluating the integral is shown in the applet below. Convolution is shown by the red curve. As f(t - tau) moves over f(tau) the product of the 2, which is the area to be integrated, is shown in green.

The plot below shows a graph of the real part of the spectrum of f(t) in red, the imaginary part in blue, the spectrum of the autocorrelation in green and the real and imaginary parts of the spectrum of the convolution in purple and brown respectively.

Return to main page
Return to page index