Hilbert transform

Integral transform and linear operator that takes a function of a real variable and produces another function of a real variable with a 90 degree phase shift

Entityintegral transformQ685437

Instance of: integral transform

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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 (see § Definition). 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 (see § Relationship with the Fourier transform). 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.

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Named after: David Hilbert
Defining formula: \operatorname {H} (u)(t)={\frac {1}{\pi }}\,\operatorname {p.v.} \int _{-\infty }^{+\infty }{\frac {u(\tau )}{t-\tau }}\,\mathrm {d} \tau
In defining formula: \operatorname{p.v.} symbol represents: Cauchy principal value
\operatorname H symbol represents: Hilbert transform
\pi symbol represents: pi

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AIDB. (2026). Hilbert transform (Q685437). Retrieved from https://aidb.si/e/hilbert-transform-Q685437