Question

If the fourier series coefficients of a signal are periodic then the signal must be

Answer 1

Answer:

The reason why Fourier series coefficients of continuous functions are generally not periodic is because of the continuity of the function. The discrete Fourier series coefficients are periodic because the analyzed signals are discrete. Note the duality relationship of the Fourier transform

periodicity⟺discreteness

Periodicity in one domain leads to discreteness in the other domain. E.g., a periodic function has only discrete frequency components (as shown by its Fourier series). If these discrete frequency components are periodic, then, consequently, the periodic function must be discrete, i.e., it must be non-zero only at discrete times.

In his answer, Fat32 gave an example of a periodic signal with periodic Fourier series coefficients. The most general form of such a signal is

x(t)=∑m=−∞∞bmδ(t−mTN)(1)

where the coefficients bm are essentially the discrete Fourier series coefficients of the N-periodic Fourier coefficients ak of the T-periodic signal x(t):

bm=TN∑k=0N−1akej2πkm/N(2)

x(t)=∑k=−∞∞akej2πkt/T(3)

As mentioned above, the signal x(t) is of course discrete in the sense that it is non-zero only at discrete time instances tm=mT/N. Any periodic signal x(t) with N-periodic Fourier coefficients ak must be of the form (1).