Question

Find the inverse Fourier transform of X(ω) = e^-2ω u(ω).

(a) \(\frac{1}{2π(2+jt)}\)

(b) \(\frac{1}{2π(2-jt)}\)

(c) \(\frac{1}{2(2+jt)}\)

(d) \(\frac{1}{π(2+jt)}\)

I had been asked this question in examination.

Origin of the question is Inverse Fourier Transform topic in section Fourier Transform of Signals and Systems

Answer 1

The correct option is (b) \(\frac{1}{2π(2-jt)}\)

To explain: We know that  x(t) =  \(\frac{1}{2π} \int_{-∞}^∞ X(ω) e^{jωt} \,dω\)

x(t) = \(\frac{1}{2π} \int_{-∞}^∞ e^{-2ω} \,u(ω) e^{jωt} \,dω =  \frac{1}{2π} \int_{-∞}^∞ e^{-2ω} e^{jωt} \, dω =  \frac{1}{2π(2-jt)}\).