Question

Find the Fourier transform of \(\frac{j}{πt}\).

(a) sinc(ω)

(b) sa(ω)

(c) δ(ω)

(d) sgn(ω)

I had been asked this question in unit test.

My question is taken from Properties of Fourier Transforms topic in portion Fourier Transform of Signals and Systems

Answer 1

Correct option is (d) sgn(ω)

For explanation I would say: Let  x(t) = sgn(t)

The Fourier transform of sgn(t)  is X(ω) = F[sgn(t)] = \(\frac{2}{jω}\)

Replacing ω with t

–> X(t) = \(\frac{2}{jt}\)

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

F\(\Big[\frac{2}{jt}\Big]\) = 2πsgn(-ω) = -2πsgn(ω)

\(\frac{2}{jt}\) ↔ -2πsgn(ω)

\(\frac{2}{πt}\) ↔ sgn(ω).