Question

Which of the following is the equation for the Fourier series coefficient?

(a) \(\frac{1}{T_p} \int_0^{t_0+T_p} x(t)e^{-j2πkF_0 t} dt\)

(b) \(\frac{1}{T_p} \int_{t_0}^∞ x(t)e^{-j2πkF_0 t} dt\)

(c) \(\frac{1}{T_p} \int_{t_0}^{t_0+T_p} x(t)e^{-j2πkF_0 t} dt\)

(d) \(\frac{1}{T_p} \int_{t_0}^{t_0+T_p} x(t)e^{j2πkF_0 t} dt\)

I have been asked this question during an online exam.

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

Answer 1

Correct choice is (c) \(\frac{1}{T_p} \int_{t_0}^{t_0+T_p} x(t)e^{-j2πkF_0 t} dt\)

For explanation I would say: When we apply integration to the definition of Fourier series representation, we get

ckTp=\(\int_{t_0}^{t_0+T_p} x(t)e^{-j2πkF_0 t} dt\)

=>ck=\(\frac{1}{T_p} \int_{t_0}^{t_0+T_p} x(t)e^{-j2πkF_0 t} dt\)