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Hilbert transform problem .

  1. May 15, 2008 #1
    i was trying to come with an easier way of finding the Hilbert transform of a function [tex]\ x(t)[/tex] , and here is what i did :

    starting with Fourier transform of x :

    [tex]X(f)=\int^{\infty}_{-\infty}\ x(t) \ e^ {\ -i2\pi ft} \ dt[/tex]

    now

    [tex]\ e^ {\ -i2\pi ft} = \frac{1}{2\pi i}\oint\frac{e^ {\ -i2\pi
    fz}}{z-t} \ dz[/tex]

    where the contour encloses t .

    "this is done by cauchy's integral formula"

    =>

    [tex]X(f) = \frac{1}{2\pi i}\int^{\infty}_{-\infty}\ x(t) \oint\frac{e^ {\ -i2\pi fz}}{z-t} \ dz \
    dt = \frac{1}{2\pi i} \oint\ e^ {\ -i2\pi fz}\int^{\infty}_{-\infty}\frac{x(t)}{z-t}\ dt \ dz[/tex]

    now

    [tex] \pi\hat{x}(z)=\int^{\infty}_{-\infty}\frac{x(t)}{z-t}\ dt[/tex]

    where

    [tex]\hat{x}(z)[/tex] is the Hilbert transform of x .

    =>

    [tex]\ X(f)= \frac{1}{2i}\oint\hat{x}(s)\ e^ {\ -i2\pi fs}\ ds[/tex]

    "z is replaced with s for conventional reasons"

    now the program is to relate the last integral to the inverse laplace (or fourier !!) transform of [tex]\hat{x}(s)[/tex] , by choosing the suitable contour(s) , and here is where i'm stuck !! so any help is appreciated .
     
    Last edited: May 15, 2008
  2. jcsd
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