Question

Find the inverse Fourier transform of \(X(ω) = \frac{6+4(jω)}{(jω)^2 + 6(jω) + 8}\).

(a) e^-2t u(t) – 5e^-4t u(t)

(b) e^-2t u(t) + 5e^-4t u(t)

(c) -e^-2t u(t) – 5e^-4t u(t)

(d) -e^-2t u(t) + 5e^-4t u(t)

This question was addressed to me by my college professor while I was bunking the class.

My query is from Inverse Fourier Transform in portion Fourier Transform of Signals and Systems

Answer 1

Right choice is (d) -e^-2t u(t) + 5e^-4t u(t)

To explain I would say: \(X(ω) = \frac{6+4(jω)}{(jω)^2+6(jω)+8} = \frac{A}{jω+2} + \frac{B}{jω+4} = -\frac{1}{jω+2} + \frac{5}{jω+4}\)

Applying inverse Fourier transform, we get

x(t) = -e^-2t u(t) + 5e^-4t u(t).