: Mathématiques: Hilbert transform

mathematical analysis > functional analysis > operator > integral transform > Hilbert transform

mathematical analysis > functional analysis > transformation > integral transform > Hilbert transform

Hilbert transform

In mathematics and signal processing, the Hilbert transform is a specific singular integral that takes a function, $u (t)$ of a real variable and produces another function of a real variable $H(u)(t)$ . The Hilbert transform is given by the Cauchy principal value of the convolution with the function ${\\displaystyle 1/(\\pi t)}$ . The Hilbert transform has a particularly simple representation in the frequency domain: It imparts a phase shift of ±90° ( $π$ /2 radians) to every frequency component of a function, the sign of the shift depending on the sign of the frequency. The Hilbert transform is important in signal processing, where it is a component of the analytic representation of a real-valued signal $u (t)$ . 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.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Hilbert_transform)

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