Question

Find the Fourier transform of \(\frac{1}{a+jt}\).

(a) 2πe^aω u(ω)

(b) 2πe^aω u(-ω)

(c) 2πe^-aω u(ω)

(d) 2πe^-aω u(-ω)

The question was posed to me during a job interview.

I want to ask this question from Properties of Fourier Transforms topic in chapter Fourier Transform of Signals and Systems

Answer 1

Right choice is (b) 2πe^aω u(-ω)

The best explanation: Let X(t) = \(\frac{1}{a+jt}\)

Replacing t with ω

 X(ω) = \(\frac{1}{a+jw}\)

x(t )= e^-at u(t)

As per duality property X(t) ↔ 2πx(-ω), we have

\(F[X(t)] = F\Big[\frac{1}{a+jt}\Big]\) = 2πx(-ω) = 2πe^aω u(-ω).