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 = nwf 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.

eg parameters illustrate the effect:-
(5.0, NA, 4.39, 5.65, 2, NA, 45, 15.0, 1.5) wf = 4.986 ~ 5, n = 1
(5.0, NA, 5.0, 5.62, 2, NA, 45, 15.0, 1.5) wf = 10.13 ~ 10
(5.0, NA, 3.77, 5.62, 10, NA, 45, 15.0, 1.5) wf = 3.396 ~ 3.4
(5.0, NA, 3.11, 5.62, 2, NA, 45, 15.0, 1.5) wf = 2.503 ~ 2.5, n = 2
Changing T1, T2 to 3.46, 5.97 or 4.08, 6.59 shows that the result does not depend on whether the part chosen starts at the zero or at the minimum or maximum. (5.0, NA, 1.85, 5.62, 3, NA, 45, 15.0, 1.5) wf = 1.666 ~ 2.5, n = 3
(5.0, NA, 2.57, 5.71, 10, NA, 45, 15.0, 1.5) wf = 2.001 ~ 2.0, n = 2.5
(6.0, NA, 0.01, 10.48, 33, NA, 45, 33.0, 1.5) wf = 0.6 ~ 0.6, n = 10
(3.0, NA, 0.02, 31.43, 25, NA, 45, 64.5, 1.5) wf = 0.2 ~ 0.2, n = 25
(7.0, NA, 0.02, 31.43, 35, NA, 45, 64.5, 1.5) wf = 0.2 ~ 0.2, n = 35
(7.0, NA, 0.01, 13.93, 30, NA, 45, 30.0, 1.5) wf = 0.451 , n = 15,16. In this case ((T2 - T1) ~ (n + ½)T) the spectrum is that of a PSK signal with carrier frequency 7rad/s and a phase change of 180° every 15.5 periods.

This effect is more familiar with the DFT but is a feature of any discrete frequency(or assumed periodic) transform.


The following gif image show how the applet should appear when enabled;-
Return to main page
Return to page index
COPYRIGHT © 2007 Cuthbert Nyack.