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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