# Math Help - Hilbert transform problem

1. ## Hilbert transform problem

ok , my bad , here is the post again ..

i was trying to come with an easier way of finding the Hilbert transform of a function $\ x(t)$ , and here is what i did :

starting with Fourier transform of x :

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

now

$\ e^ {\ -i2\pi ft} = \frac{1}{2\pi i}\oint\frac{e^ {\ -i2\pi
fz}}{z-t} \ dz$

where the contour encloses t .

"this is done by cauchy's integral formula"

=>

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

now

$\pi\hat{x}(z)=\int^{\infty}_{-\infty}\frac{x(t)}{z-t}\ dt$

where

$\hat{x}(z)$ is the Hilbert transform of x .

=>

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

"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 $\hat{x}(s)$, by choosing the suitable contour(s) , and here is where i'm stuck !! so any help is appreciated .

2. You are going to have to type these out on your own or find a better way to copy and paste them. There is a good intro thread to learn basic LaTeX here.

-Dan

3. i think it's readable now !!

4. Originally Posted by mmzaj
ok , my bad , here is the post again ..

i was trying to come with an easier way of finding the Hilbert transform of a function $\ x(t)$ , and here is what i did :

starting with Fourier transform of x :

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

now

$\ e^ {\ -i2\pi ft} = \frac{1}{2\pi i}\oint\frac{e^ {\ -i2\pi
fz}}{z-t} \ dz$

where the contour encloses t .

"this is done by cauchy's integral formula"

=>

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

now

$\pi\hat{x}(z)=\int^{\infty}_{-\infty}\frac{x(t)}{z-t}\ dt$

where

$\hat{x}(z)$ is the Hilbert transform of x .

=>

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

"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 $\hat{x}(s)$, by choosing the suitable contour(s) , and here is where i'm stuck !! so any help is appreciated .
Here's an easier way:

1. You know that $\hat{x}(z) = x(z) * \frac{1}{\pi z}$ where * denotes the convolution operator, right?

2. $FT\left[ \frac{1}{\pi z} \right] = - i \, \text{sgn} (f)$ where sgn is the signum function (see PlanetMath: signum function).

3. From the multiplication to convolution theorem: $FT\left[ \hat{x}(z) \right] = - i \, X(f) \, \text{sgn} (f)$.

4. Therefore: $\hat{x}(z) = -i \, FT^{-1} \left[X(f) \, \text{sgn} (f) \right]$.

5. thanx a lot , that really helped .