Question

What is the equation of the Fourier series coefficient ck of an non-periodic signal?

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

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

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

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

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

I'd like to ask this question from Frequency Analysis of Continuous Time Signal topic in section Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Correct option is (b) \(\frac{1}{T_p} \int_{-\infty}^∞ x(t)e^{-j2πkF_0 t} dt\)

Easy explanation: We know that, for an periodic signal, the Fourier series coefficient is

ck=\(\frac{1}{T_p} \int_{-T_p/2}^{T_p/2} x(t)e^{-j2πkF_0 t} dt\)

If we consider a signal x(t) as non-periodic, it is true that x(t)=0 for |t|>Tp/2. Consequently, the limits on the integral in the above equation can be replaced by -∞ to ∞. Hence,

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