Question

The Fourier series coefficient of the signal y(t) = Re{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 had been asked this question in an online quiz.

I need to ask this question from Fourier Series Coefficients topic in section Fourier Series of Signals and Systems

Answer 1

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

Explanation: Re{x (t)} = \(\frac{x(t)+x^* (-t)}{2}\)

The Fourier coefficient of x* (t) is

X1[k] = \(\frac{1}{T}\) ∫ x* (t)e^-jkωt dt = \(X_1^*\) [-k]

Or, \(X_1^*\) [k] = \(\frac{1}{T}\) ∫ x(t)e^jkωt dt = X [-k]

So, X1[k] = \(X_1^*\) [-k]

∴ Y[k] = \(\frac{X[k]+X^* [-k]}{2}\).