Question

The Fourier transform of the signal \(\frac{1}{a^2+w^2}\) is _____________

(a) \(\frac{π}{a} e^{a|ω|}\)

(b) \(\frac{π}{a} e^{-a|ω|}\)

(c) \(\frac{2}{a} e^{-a|ω|}\)

(d) \(\frac{π}{a} e^{a|ω|}\)

The question was posed to me in an online quiz.

My question comes from Common Fourier Transforms in portion Fourier Transform of Signals and Systems

Answer 1

Right answer is (b) \(\frac{π}{a} e^{-a|ω|}\)

The best explanation: Fourier transform of e^-a|t| = \(\frac{2a}{a^2+ω^2}\)

Now, e^-a|t| = e^-at, t>0 and eat, t<0

Using Duality property, \(\frac{2a}{a^2+w^2}\) ↔ 2πe^-a|ω|

Or, \(\frac{1}{a^2+w^2} \leftrightarrow \frac{π}{a} e^{-a|ω|}\).