Fourier Series of part of 1 Sinusoid.

Cuthbert Nyack

If a part f(t) between T1 and T2 of a sinusoid with frequency w is used to find the Fourier series of the sinusoid, would the calculation give a spectrum containing only 1 line at w?

The Fourier Series coefficients are given by:-

As the following applet shows, the "correct" result is only obtained if w = 2p/T is a multiple of the fundamental frequency 2p/(T2 - T1).

ie w = nw_f or 2p/T = n2p/(T2 - T1) or (T2 - T1) = nT.
This is a consequence of the Fourier Series calculation assuming a periodic signal. If a periodic signal is constructed by adding f(t ± n(T2 - T1)) to f(t), then the correct spectrum is obtained if the constructed periodic signal is identical to the sinusoid. In the applet below the correct result is obtained when the green f(t) and yellow f(t - (T2 - T1)) plots join smoothly at the boundary.

Fn = 1 shows the effect for T1, T2 and w.

Fn = 2 to 9 show special cases where (T2 - T1) contains an integral number of periods.
In these cases the spectrum consists of a single line at the frequency of the sinusoid.
For Fn = 7, w = 6 and the white text is showing that c(10) has magnitude 1.0000 and frequency 6.0011. (The accuracy is limited by the resolution of T1 and T2).
Fn = 10 to 13 show special cases where (T2 - T1) contains an odd number of half periods.
In these cases the spectrum does not contain a component at the frequency of the sinusoid.
For Fn = 13, w = 5.7 and the white text is showing that the largest components in the spectrum are c(6) with magnitude 0.6556 and frequency 5.2578 and c(7) with magnitude 0.61949 and frequency 6.1342.
Fn = 14 and 24 shows what happens as the number of periods increases from 7 to 8 with w kept constant at 6.0 and T2 is varied.
Fn = 25 and 35 shows what happens as the number of periods increases from 7 to 8 with T1, T2 kept constant at 0.01 and 7.34 and w is varied.

This effect is more familiar with the DFT but is a feature of any discrete frequency(or assumed periodic) transform. There is no sampling here. When sampling is introduced the spectrum becomes periodic.

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