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Fourier Transform Autocorrelation and Power Spectrum

Cuthbert Nyack

For a signal extending from -inf to + inf, the Autocorrelation is defined
by the following expression. The integral is evaluated for increasing
values of T and the result is averaged. If the average is not taken then
the Autocorrelation would tend to infinity.
If the signal is of finite duration, then averaging it would yield
a result of zero. Instead the following definition of the
Autocorrelation function is used.
Since the Autocorrelation function is even, then the following
definition can also be used.
The Fourier Transform of the Autocorrelation Function is the
Power Spectrum, So the Autocorrelation function and Power Spectrum
form a Fourier pair below. The power spectrum removes the phase
information from the Fourier Transform.
For RANDOM SIGNALS the autocorrelation -
Power Spectrum pair is the most useful representation. Most
spectrum analysers will display either the power spectrum or the
magnitude of the transform. In either case, the phase is not
displayed.
With the angular frequency replaced by the cyclic frequency, the
pair becomes.
Since the Autocorrelation function is even, then the following
definition for the pair can be used.

The applet below can be used to compare the autocorrelation
obtained in the time and frequency domains. In the frequency
domain, the autocorrelation is obtained by taking the inverse
Fourier transform of the power spectrum.

Fn changes the function, a and b changes the shape of the function
and wr changes the limit of the
integration used to obtain the autocorrelation in the
frequency domain.

Eg below shows the autocorrelation in the time and frequency
domains of a function when Fn = 1.

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COPYRIGHT © 1996, 2010 Cuthbert Nyack.