# 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.