Hilbert Transform - Relationship With The Fourier Transform

Relationship With The Fourier Transform

The Hilbert transform is a multiplier operator (Duoandikoetxea 2000, Chapter 3). The symbol of H is σH(ω) = −i sgn(ω) where sgn is the signum function. Therefore:

\mathcal{F}(H(u))(\omega) = (-i\,\operatorname{sgn}(\omega))\cdot \mathcal{F}(u)(\omega)\,

where denotes the Fourier transform. Since sgn(x) = sgn(2πx), it follows that this result applies to the three common definitions of

By Euler's formula,

\sigma_H(\omega) \, \ =\ \begin{cases} \ \ i = e^{+i\pi/2}, & \mbox{for } \omega < 0\\ \ \ \ \ 0, & \mbox{for } \omega = 0\\ \ \ -i = e^{-i\pi/2}, & \mbox{for } \omega > 0. \end{cases}

Therefore H(u)(t) has the effect of shifting the phase of the negative frequency components of u(t) by +90° (π/2 radians) and the phase of the positive frequency components by −90°. And i·H(u)(t) has the effect of restoring the positive frequency components while shifting the negative frequency ones an additional +90°, resulting in their negation.

When the Hilbert transform is applied twice, the phase of the negative and positive frequency components of u(t) are respectively shifted by +180° and −180°, which are equivalent amounts. The signal is negated, i.e., H(H(u)) = −u, because:

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