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 mentionedThe 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

Which of the following is the Fourier series representation of the sig

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

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