Question

Which of the following relation is correct between Fourier transform X(F) and Fourier series coefficient ck?

(a) ck=X(F0/k)

(b) ck= 1/TP (X(F0/k))

(c) ck= 1/TP(X(kF0))

(d) none of the mentioned

The question was posed to me by my college professor while I was bunking the class.

This interesting question is from Frequency Analysis of Continuous Time Signal in division Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Correct choice is (c) ck= 1/TP(X(kF0))

The best I can explain: Let us consider a signal x(t) whose Fourier transform X(F) is given as

X(F)=\(\int_{-∞}^∞ x(t)e^{-j2πF_0 t}dt\)

and the Fourier series coefficient is given as

ck=\(\frac{1}{T_p} \int_{-∞}^∞ x(t)e^{-j2πkF_0 t}dt\)

By comparing the above two equations, we get

ck=\(\frac{1}{T_p} X(kF_0)\)