Fourier Series of part of 2 Sinusoids.
Cuthbert Nyack
If the Fourier Series is taken of a part of a signal consisting of 2 sinusoids, will the spectrum show 2 lines at the sinusoid frequencies?
As is the case for one sinusoid, 2 lines are obtained only if both
sinusoid frequencies are an integral multiple of the fundamental
frequency wf = 2p/T of the Fourier series. (T is the duration of the
part of the signal used to calculate the Fourier Series)
The effect is illustrated by the applet below.
Fn = 1 shows the fourier series for variable
w1, w2,
T1 and T2.
Fn = 2 to 8 show cases where the interval used contains
an integral number of periods of both
w1 and w2.
In these cases the Fourier Spectrum consists of 2 lines at the
frequency of the 2 sinusoids.
Fn = 9 and 10 show cases where the interval (T2 - T1) contains
an integral number of periods of w1
and an odd number of half periods of w2.
In these cases the spectrum contains a line at
w1 but not at w2.
Fn = 11 and 12 show cases where the interval (T2 - T1) contains
an integral number of periods of w2
and an odd number of half periods of w1.
In these cases the spectrum contains a line at
w2 but not at w1.
Fn = 13 and 14 show cases where the interval (T2 - T1) contains
an odd number of half periods of w1
and w2.
In these cases the spectrum does not contain a line at
w1 or at w2.
Fn = 15 shows a case where the sinusoid frequencies are
on either side of one of the harmonic frequencies in the
Fourier Series. Instead of containing 2 lines at the sinusoid frequencies, the spectrum has 1 large component and several smaller
components.
Fn = 16 and 17 show what happens in the case Fn = 15 as the
frequency separation of the 2 lines increases.
Fn = 18 shows a case where (T2 - T1) contains an integral
number of periods of both w1
and w2. Fn = 19 to 23 shows the number
of periods of w1 decreasing by 1/2 from
~9 to ~ 8.5.
Fn = 24 to 34 shows the number
of periods of w2 increasing by 1 from ~11 to ~12.
Fn = 35 to 39 shows the number
of periods of w1 decreasing by 1/2
from 8.5 to end at 8.
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COPYRIGHT © 2007, 2012 Cuthbert Nyack.