Fourier Transform Autocorrelation and Noise.

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.
One application in which autocorrelation can be used is in the reduction of noise in a signal in cases where the noise is uncorrelated.
Fn = 1 in the Applet below shows a sinusoid in red and a noisy sinusoid in green. Autocorrelation of the red signal is shown in pink and that of the green signal is in cyan. 'N' can be used to change the noise level.
Fn = 2 to 7 show special cases of Fn = 1.
In Fn = 8 pink shows the autocorrelation of the red signal and the crosscorrelation of the red and green signal is in yellow.
Fn = 9 to 10 show special cases of Fn = 8.
In Fn = 11 is similar to Fn = 1 but with 2 sinusoids with frequencies w and w1.
Fn = 12 to 17 show special cases of Fn = 11.
In Fn = 18 is similar to Fn = 8 .
Fn = 19 to 20 show special cases of Fn = 18.
In Fn = 21 is similar to Fn = 1 but with a DSB signal.
Fn = 22 to 25 show special cases of Fn = 21.
In Fn = 26 is similar to Fn = 8 but with a DSB signal.
Fn = 27 to 28 show special cases of Fn = 26.
In Fn = 29 is similar to Fn = 1 but with a Phase Modulated signal.
Fn = 30 to 36 show special cases of Fn = 29.
In Fn = 37 is similar to Fn = 8 but with a Phase Modulated signal.
Fn = 38 to 39 show special cases of Fn = 37.
In Fn = 40 is similar to Fn = 1 but with a Square wave.
Fn = 41 to 44 show special cases of Fn = 40.
In Fn = 45 is similar to Fn = 8 but with a Square wave.
Fn = 46 to 49 show special cases of Fn = 45.
In Fn = 50 is similar to Fn = 1 but with a Triangle wave.
Fn = 51 to 54 show special cases of Fn = 50.
In Fn = 55 is similar to Fn = 8 but with a Triangle wave.
Fn = 56 to 59 show special cases of Fn = 55.
In Fn = 60 is similar to Fn = 1 but with a Rectangular Pulse.
Fn = 61 to 67 show special cases of Fn = 60.
In Fn = 68 is similar to Fn = 8 but with a Rectangular Pulse.
Fn = 69 to 74 show special cases of Fn = 68.
In Fn = 75 is similar to Fn = 1 but with a Pulse whose shape can be changed by 'a' and 'b'.
Fn = 76 to 84 show special cases of Fn = 75.
In Fn = 85 is similar to Fn = 8 but with a Pulse whose shape can be changed by 'a' and 'b'.
Fn = 86 to 87 show special cases of Fn = 85.