Question

The Fourier series coefficient of the signal y(t) = Even{x(t)} is ___________

(a) \(\frac{X[k]+X[-k]}{2}\)

(b) \(\frac{X[k]-X[-k]}{2}\)

(c) \(\frac{X[k]+X^* [-k]}{2}\)

(d) \(\frac{X[k]-X^* [-k]}{2}\)

I got this question by my college director while I was bunking the class.

The query is from Fourier Series Coefficients topic in section Fourier Series of Signals and Systems

Answer 1

Correct option is (a) \(\frac{X[k]+X[-k]}{2}\)

Easy explanation: even {x (t)} = \(\frac{x(t)+x(-t)}{2}\)

The Fourier series coefficient transform of x (t) is

X1[k] = \(\frac{1}{T}\) ∫ x (-t)e^-jkωt dt

= \(\frac{1}{T}\) ∫ x (α)e^jkωα dα

= X [-k]

∴ The Fourier coefficient of Even{x(t)} = Y[k] = \(\frac{X[k]+X[-k]}{2}\).