Question

Find the inverse Fourier transform of sgn(ω).

(a) \(\frac{1}{πt}\)

(b) \(\frac{j}{πt}\)

(c) \(\frac{j}{t}\)

(d) \(\frac{1}{t}\)

I got this question during a job interview.

This question is from Inverse Fourier Transform in chapter Fourier Transform of Signals and Systems

Answer 1

Right option is (b) \(\frac{j}{πt}\)

To explain I would say: Given the function F(ω)=sgn(ω). The Fourier transform of a Signum function is sgn(ω) = \(\frac{2}{jω}\).

Applying the duality property F(t) ↔ 2πf(-ω), we get

F(\(\frac{2}{jt}\)) = 2πsgn(-ω).

As sgn(ω) is an odd function, sgn(-ω)=-sgn(ω).

Hence, \(\frac{2}{jt}\) ↔ -sgn(ω)

Or \(\frac{2}{πt}\) ↔ sgn(ω)

Therefore, the inverse Fourier transform of sgn(ω) is \(\frac{j}{πt}\).