# Hilbert transform

Missing image
Hilbert_transform.png
The Hilbert transform, in red, of a square wave, in blue

In mathematics, the Hilbert transform [itex]\left\{\mathcal{H}s\right\}(t)[itex] (also written [itex]\hat{s}(t)[itex]) of the real function s(t) is an integral transform defined by

[itex]

\left\{\mathcal{H}s\right\}(t) = \hat{s}(t) = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{s(\tau)}{t-\tau}\,d\tau [itex]

considering the integral as a Cauchy principal value (which avoids singularities at t = τ). The inverse Hilbert transform is:

[itex]

\left\{\mathcal{H}^{-1}\hat{s}\right\}(\tau) = -\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{\hat{s}(t)}{\tau-t}\,dt [itex]

where again, the integral is a Cauchy principal value integral.

The Hilbert transform can also be written with a convolution operator as:

[itex]

\hat{s}(t) = {1 \over \pi t} * s(t) [itex]

[itex]\hat{s}(t)[itex] can be generated from [itex]s(t)[itex] by multiplying its frequency spectrum by [itex]-i \sgn(\omega)[itex], where sgn is the signum function and i is the imaginary number. This has the effect of shifting all of its negative frequencies by +90° and all positive frequencies by −90°. Note that the Hilbert transform and the original function are both functions of the same variable.

The Hilbert transform has applications in signal processing.

### Discrete Hilbert transform

The ideal discrete Hilbert transform is in the Z-domain

[itex]

H_{HT}(e^{j\omega}) = \begin{cases} -j, & 0 \leq \omega < \pi \\ j, & -\pi \leq \omega < 0. \end{cases} [itex] Clearly, it is a phase-shifting filter, with a -90 degree phase shift in the upper half plane and +90 degree shift in the lower half plane. However, it is, in the time-domain, and unrealisable system and thus the name ideal discrete Hilbert transform. Still, the impulse response [itex]h_{HT}[n][itex] can be obtained by inverse Discrete Fourier transform which yields

[itex]

h_{HT}[n]= \begin{cases} 0, & \mbox{for }n\mbox{ even},\\ \frac2{\pi n} & \mbox{for }n\mbox{ odd} \end{cases} [itex]

## External links

##### Navigation

Academic Kids Menu

• Art and Cultures
• Art (http://www.academickids.com/encyclopedia/index.php/Art)
• Architecture (http://www.academickids.com/encyclopedia/index.php/Architecture)
• Cultures (http://www.academickids.com/encyclopedia/index.php/Cultures)
• Music (http://www.academickids.com/encyclopedia/index.php/Music)
• Musical Instruments (http://academickids.com/encyclopedia/index.php/List_of_musical_instruments)
• Biographies (http://www.academickids.com/encyclopedia/index.php/Biographies)
• Clipart (http://www.academickids.com/encyclopedia/index.php/Clipart)
• Geography (http://www.academickids.com/encyclopedia/index.php/Geography)
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Maps (http://www.academickids.com/encyclopedia/index.php/Maps)
• Flags (http://www.academickids.com/encyclopedia/index.php/Flags)
• Continents (http://www.academickids.com/encyclopedia/index.php/Continents)
• History (http://www.academickids.com/encyclopedia/index.php/History)
• Ancient Civilizations (http://www.academickids.com/encyclopedia/index.php/Ancient_Civilizations)
• Industrial Revolution (http://www.academickids.com/encyclopedia/index.php/Industrial_Revolution)
• Middle Ages (http://www.academickids.com/encyclopedia/index.php/Middle_Ages)
• Prehistory (http://www.academickids.com/encyclopedia/index.php/Prehistory)
• Renaissance (http://www.academickids.com/encyclopedia/index.php/Renaissance)
• Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
• United States (http://www.academickids.com/encyclopedia/index.php/United_States)
• Wars (http://www.academickids.com/encyclopedia/index.php/Wars)
• World History (http://www.academickids.com/encyclopedia/index.php/History_of_the_world)
• Human Body (http://www.academickids.com/encyclopedia/index.php/Human_Body)
• Mathematics (http://www.academickids.com/encyclopedia/index.php/Mathematics)
• Reference (http://www.academickids.com/encyclopedia/index.php/Reference)
• Science (http://www.academickids.com/encyclopedia/index.php/Science)
• Animals (http://www.academickids.com/encyclopedia/index.php/Animals)
• Aviation (http://www.academickids.com/encyclopedia/index.php/Aviation)
• Dinosaurs (http://www.academickids.com/encyclopedia/index.php/Dinosaurs)
• Earth (http://www.academickids.com/encyclopedia/index.php/Earth)
• Inventions (http://www.academickids.com/encyclopedia/index.php/Inventions)
• Physical Science (http://www.academickids.com/encyclopedia/index.php/Physical_Science)
• Plants (http://www.academickids.com/encyclopedia/index.php/Plants)
• Scientists (http://www.academickids.com/encyclopedia/index.php/Scientists)
• Social Studies (http://www.academickids.com/encyclopedia/index.php/Social_Studies)
• Anthropology (http://www.academickids.com/encyclopedia/index.php/Anthropology)
• Economics (http://www.academickids.com/encyclopedia/index.php/Economics)
• Government (http://www.academickids.com/encyclopedia/index.php/Government)
• Religion (http://www.academickids.com/encyclopedia/index.php/Religion)
• Holidays (http://www.academickids.com/encyclopedia/index.php/Holidays)
• Space and Astronomy
• Solar System (http://www.academickids.com/encyclopedia/index.php/Solar_System)
• Planets (http://www.academickids.com/encyclopedia/index.php/Planets)
• Sports (http://www.academickids.com/encyclopedia/index.php/Sports)
• Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
• Weather (http://www.academickids.com/encyclopedia/index.php/Weather)
• US States (http://www.academickids.com/encyclopedia/index.php/US_States)

Information

• Home Page (http://academickids.com/encyclopedia/index.php)
• Contact Us (http://www.academickids.com/encyclopedia/index.php/Contactus)

• Clip Art (http://classroomclipart.com)