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 the, 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)

    “Every man is in a state of conflict, owing to his attempt to reconcile himself and his relationship with life to his conception of harmony. This conflict makes his soul a battlefield, where the forces that wish this reconciliation fight those that do not and reject the alternative solutions they offer. Works of art are attempts to fight out this conflict in the imaginative world.”
    —Rebecca West (1892–1983)

    “It is possible to make friends with our children—but probably not while they are children.... Friendship is a relationship of mutual dependence-interdependence. A family is a relationship in which some of the participants are dependent on others. It is the job of parents to provide for their children. It is not appropriate for adults to enter into parenthood recognizing they have made a decision to accept dependents and then try to pretend that their children are not dependent on them.”
    —Donald C. Medeiros (20th century)

    “But I must needs take my petulance, contrasting it with my accustomed morning hopefulness, as a sign of the ageing of appetite, of a decay in the very capacity of enjoyment. We need some imaginative stimulus, some not impossible ideal which may shape vague hope, and transform it into effective desire, to carry us year after year, without disgust, through the routine- work which is so large a part of life.”
    —Walter Pater (1839–1894)