Question

Find the inverse Fourier transform of X(ω) = \(\frac{1+3(jω)}{(3+jω)^2}\).

(a) 3e^-3t u(t) + 8e^-3t u(t)

(b) 3te^-3t u(t) – 8e^-8t u(t)

(c) 3e^-3t u(t) + 8te^8t u(t)

(d) 3e^-3t u(t) – 8te^-3t u(t)

I had been asked this question in an internship interview.

I want to ask this question from Inverse Fourier Transform topic in portion Fourier Transform of Signals and Systems

Answer 1

Correct option is (d) 3e^-3t u(t) – 8te^-3t u(t)

To explain: Given X(ω) = \(\frac{1+3(jω)}{(3+jω)^2} = \frac{A}{3+jω} + \frac{B}{(3+jω)^2} = \frac{3}{3+jω} – \frac{8}{(3+jω)^2}\)

Applying inverse Fourier transform, we get

x(t) = 3e^-3t u(t) – 8te^-3t u(t).