Question

What is the frequency response of a Hilbert transform H(F)=?

(a) \(\begin{cases}&-j  (F>0) \\ & 0  (F=0)\\ & j  (F<0)\end{cases}\)

(b) \(\left\{\begin{matrix}-j & (F<0)\\0 & (F=0) \\j & (F>0)\end{matrix}\right. \)

(c) \(\left\{\begin{matrix}-j & (F>0)\\0 &(F=0) \\j & (F<0)\end{matrix}\right. \)

(d) \(\left\{\begin{matrix}j&(F>0)\\0 & (F=0)\\j & (F<0)\end{matrix}\right. \)

The question was asked in an internship interview.

This interesting question is from The Representation of Bandpass Signals topic in division Sampling and Reconstruction of Signals of Digital Signal Processing

Answer 1

Correct choice is (a) \(\begin{cases}&-j  (F>0) \\ & 0  (F=0)\\ & j  (F<0)\end{cases}\)

For explanation I would say: H(F) =\(\int_{-∞}^∞ h(t)e^{-j2πFt} dt\)

=\(\frac{1}{π} \int_{-∞}^∞ 1/t e^{-2πFt} dt\)

=\(\left\{\begin{matrix}-j& (F>0)\\0&(F=0) \\ j& (F<0)\end{matrix}\right.\)

We Observe that │H (F)│=1 and the phase response &odot;(F) = -1/2π for F > 0 and &odot;(F) = 1/2π for F < 0.