Question

The Fourier series coefficient of the signal z(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) \(\frac{X[k]-X^* [-k]}{2}\)

The question was asked in homework.

My question is taken from Miscellaneous Examples on Fourier Series topic in division Fourier Series of Signals and Systems

Answer 1

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

The 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]

∴ Z[k] = \(\frac{X[k]+X^* [-k]}{2}\).