Question

Which of the following is the Fourier series representation of the signal x(t)?

(a) \(c_0+2\sum_{k=1}^{\infty}|c_k|sin(2πkF_0 t+θ_k)\)

(b) \(c_0+2\sum_{k=1}^{\infty}|c_k|cos(2πkF_0 t+θ_k)\)

(c) \(c_0+2\sum_{k=1}^{\infty}|c_k|tan(2πkF_0 t+θ_k)\)

(d) None of the mentioned

The question was posed to me in an interview for internship.

I want to ask this question from Frequency Analysis of Continuous Time Signal in chapter Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Right answer is (b) \(c_0+2\sum_{k=1}^{\infty}|c_k|cos(2πkF_0 t+θ_k)\)

To elaborate: In general, Fourier coefficients ck are complex valued. Moreover, it is easily shown that if the periodic signal is real, ck and c-k are complex conjugates. As a result

ck=|ck|e^jθkand ck=|ck|e^-jθk

Consequently, we obtain the Fourier series as x(t)=\(c_0+2\sum_{k=1}^{\infty}|c_k|cos(2πkF_0 t+θ_k)\)