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.
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COPYRIGHT © 2012 Cuthbert Nyack.