Question 2.7. Show that for all periodic physical signals that have finite power, the coefficients of the Fourier series expansion xn tends to 0 as n→∞
I have computed |xn|2 by multiplying xn with x∗n. However, as per my calculations, |xn|2=Px i.e. power of x(t), which is a constant and independent of n. I know that this is fundamentally incorrect because |xn| should vary with n. If anyone can guide me about where I went wrong that would be very helpful. Thanks and regards.
MY SOLUTION:
Please refer to the solution from the half of the first page.
Answer
A finite power periodic signal will have the following property:
Px=1T0∫T00|x(t)|2dt<∞
where Px is the power averaged over a period of the signal. Then from Parseval's theorem, the total power can also be shown to be (for a real signal I'm assuming):
Px=|a0|2+2∞∑n=1|an|2<∞
where the an's are the continuous-time Fourier series coefficients for x(t).
From the theory of power series (calculus), it's known that in order for (2) to converge (sum being less than infinity) a necessary condition is
limn→∞|an|2=0
(3) implies therefore that, for finite power periodic signal's CTFS coefficients an also goes to zero as n goes to infinity.
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