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:

Read more about this topic:  Hilbert Transform

Famous quotes containing the words relationship with, relationship and/or transform:

    “Christianity as an organized religion has not always had a harmonious relationship with the family. Unlike Judaism, it kept almost no rituals that took place in private homes. The esteem that monasticism and priestly celibacy enjoyed implied a denigration of marriage and parenthood.”
    —Beatrice Gottlieb, U.S. historian. The Family in the Western World from the Black Death to the Industrial Age, ch. 12, Oxford University Press (1993)

    “Some [adolescent] girls are depressed because they have lost their warm, open relationship with their parents. They have loved and been loved by people whom they now must betray to fit into peer culture. Furthermore, they are discouraged by peers from expressing sadness at the loss of family relationships—even to say they are sad is to admit weakness and dependency.”
    —Mary Pipher (20th century)

    “The lullaby is the spell whereby the mother attempts to transform herself back from an ogre to a saint.”
    —James Fenton (b. 1949)