Editing Hilbert transform (section)

== Table of selected Hilbert transforms ==
In the following table, the [[frequency]] parameter <math>\omega</math> is real.

{| class="wikitable"
|-
! Signal <br/><math>u(t)</math>
! Hilbert transform<ref group="fn">Some authors (e.g., Bracewell) use our {{math|−H}} as their definition of the forward transform. A consequence is that the right column of this table would be negated.</ref> <br/><math>\operatorname{H}(u)(t)</math>
|-
| align="center"| <math>\sin(\omega t + \varphi)</math> <ref group="fn" name="ex02">The Hilbert transform of the sin and cos functions can be defined by taking the principal value of the integral at infinity.  This definition agrees with the result of defining the Hilbert transform distributionally.</ref> || align="center" |
<math>\begin{array}{lll}
\sin\left(\omega t + \varphi - \tfrac{\pi}{2}\right)=-\cos\left(\omega t + \varphi \right), \quad \omega > 0\\
\sin\left(\omega t + \varphi + \tfrac{\pi}{2}\right)=\cos\left(\omega t + \varphi \right), \quad \omega < 0
\end{array}</math>
|-
| align="center"| <math> \cos(\omega t + \varphi) </math> <ref group="fn" name="ex02"/> || align="center" |
<math>\begin{array}{lll}
\cos\left(\omega t + \varphi - \tfrac{\pi}{2}\right)=\sin\left(\omega t + \varphi\right), \quad \omega > 0\\
\cos\left(\omega t + \varphi + \tfrac{\pi}{2}\right)=-\sin\left(\omega t + \varphi\right), \quad \omega < 0
\end{array}</math>
|-
| align="center"| <math> e^{i \omega t} </math> || align="center"|
<math>\begin{array}{lll}
e^{i\left(\omega t - \tfrac{\pi}{2}\right)}, \quad \omega > 0\\
e^{i\left(\omega t + \tfrac{\pi}{2}\right)}, \quad \omega < 0
\end{array}</math>
|-
| align="center"| <math> e^{-i \omega t} </math> || align="center"|
<math>\begin{array}{lll}
e^{-i\left(\omega t - \tfrac{\pi}{2}\right)}, \quad \omega > 0\\
e^{-i\left(\omega t + \tfrac{\pi}{2}\right)}, \quad \omega < 0
\end{array}</math>
|-
| align="center"| <math> 1 \over t^2 + 1 </math> || align="center"| <math> t \over t^2 + 1 </math>
|-
| align="center"| <math> e^{-t^2} </math> || align="center"| <math> \frac{2}{\sqrt{\pi\,}} F(t) </math><br/>(see [[Dawson function]])
|-
| align="center"| '''[[Sinc function]]''' <br /> <math> \sin(t) \over t </math> || align="center"| <math> 1 - \cos(t)\over t </math>
|-
| align="center"| '''[[Dirac delta function]]''' <br /><math> \delta(t) </math> || align="center"| <math> {1 \over \pi t} </math>
|-
| align="center"| '''[[Indicator function|Characteristic function]]''' <br /> <math> \chi_{[a,b]}(t) </math> ||  align="center"| <math>{ \frac{1}{\,\pi\,}\ln \left\vert \frac{t - a}{t - b}\right\vert }</math>
|}

'''Notes'''
<references group="fn"/>
An extensive table of Hilbert transforms is available.{{sfn|King|2009b}}
Note that the Hilbert transform of a constant is zero.