Question

The trigonometric Fourier series of a periodic time function can have only ___________

(a) Only cosine terms

(b) Only sine terms

(c) Both cosine and sine terms

(d) Dc and cosine terms

The question was asked in an online interview.

This is a very interesting question from Common Fourier Transforms topic in chapter Fourier Transform of Signals and Systems

Answer 1

Right answer is (d) Dc and cosine terms

Best explanation: The Fourier series of a periodic function () is given by,

X (t) = \(∑_{n=0}^∞ a_n  \,cos⁡nωt + ∑_{n=1}^∞ b_n \,sin⁡nωt\)

Thus the series has cosine terms of all harmonics i.e., n = 0,1,2,…..

The 0th harmonic which is the DC term = a0.

So, the trigonometric Fourier series of a periodic time function can have only Dc and cosine terms.