Question

The Inverse Fourier transform of the signal e^-2|ω| is ____________

(a) \(\frac{2}{π(4+t^2)}\)

(b) \(\frac{1}{2π(4+t^2)}\)

(c) \(\frac{1}{π(4+t^2)}\)

(d) \(\frac{1}{(4+t^2)}\)

The question was posed to me during an online interview.

This interesting question is from Exponential Fourier Series and Fourier Transforms topic in portion Fourier Series of Signals and Systems

Answer 1

Correct choice is (a) \(\frac{2}{π(4+t^2)}\)

Easy explanation: e^-2|ω| = e^-2ω, ω > 0 and e^2ω, ω<0

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

= \(\frac{1}{2π} \int_{-∞}^0 e^{2ω} e^{-jωt} \,dω + \frac{1}{2π} \int_0^∞ e^{-2ω} e^{-jωt} \,dω\)

= \(\frac{2}{π(4+t^2)}\).