Question

The Fourier series coefficient of the signal y(t) = \(\frac{d^2 x(t)}{dt^2}\)  is _____________

(a) \((\frac{2πk}{T})^2 X[k]\)

(b) –\((\frac{2πk}{T})^2 X[k]\)

(c) j\((\frac{2πk}{T})^2 X[k]\)

(d) -j\((\frac{2πk}{T})^2 X[k]\)

The question was posed to me in an interview for job.

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

Answer 1

Right option is (b) –\((\frac{2πk}{T})^2 X[k]\)

The explanation is: \(x (t) = ∑_{k=-∞}^∞ X[k]e^{j \frac{2π}{T} kt}\)

Now, \(\frac{dx(t)}{dt} = -j (\frac{2π}{T})k ∑_{k=-∞}^∞ X[k]e^{j \frac{2π}{T} kt}\)

And, \(\frac{d^2 x(t)}{dt^2}  = -(\frac{2π}{T})^2 k^2  ∑_{k=-∞}^∞ X[k]e^{j \frac{2π}{T} kt}\)

∴ Y[k] = – \((\frac{2πk}{T})^2 X[k]\).