Editing Hilbert transform (section)

{{Short description|Integral transform and linear operator}}
In [[mathematics]] and [[signal processing]], the '''Hilbert transform'''  is a specific [[singular integral]] that takes a function, {{math|''u''(''t'')}} of a real variable and produces another function of a real variable {{math|H(''u'')(''t'')}}. The Hilbert transform is given by the [[Cauchy principal value]] of the [[convolution]] with the function <math>1/(\pi t)</math>  (see {{slink|#Definition}}).  The Hilbert transform has a particularly simple representation in the [[frequency domain]]: It imparts a [[phase shift]] of ±90° ({{pi}}/2&nbsp;radians) to every frequency component of a function, the sign of the shift depending on the sign of the frequency (see {{slink|#Relationship with the Fourier transform}}).  The Hilbert transform is important in signal processing, where it is a component of the [[Analytic signal|analytic representation]] of a real-valued signal {{math|''u''(''t'')}}.  The Hilbert transform was first introduced by [[David Hilbert]] in this setting, to solve a special case of the [[Riemann–Hilbert problem]] for analytic functions.