# Dictionary:Hilbert transform

Given an *h*(*t*) that is nonsingular at *t*=0 and that is a causal response so that *h*(*t*)=0 for *t*<0, then its Fourier transform,

*H*(ω)=

*R*(ω)±i

*X*(ω),

(where ω=angular frequency) has the special property known as the Hilbert transform, expressed by

*X*(ω)=–(1/π)

*R*(ω)

^{*}(1/ω)=–(1/π)℘ ∫

*R*(

*y*)

*dy*/(ω–

*y*),

*R*(ω)=–(1/π)

*X*(ω)*(1/ω)=–(1/π)℘ ∫

*X*(

*y*)

*dy*/(ω–

*y*),

where ℘ denotes the Cauchy principal value at discontinuities. If *H*(ω) vanishes for ω<0, its Fourier transform,

*h*(

*t*)+

*jx*(

*t*),

has *h*(*t*) and *x*(*t*) forming a **Hilbert transform pair**. *h*(*t*) and *x*(*t*) have the same amplitude spectrum but differ in phase by 90°. [*h*(*t*)+*j*′*x*(*t*)] is called the **analytic signal** belonging to *h*(*t*), and *x*(*t*) is the **quadrature signal** corresponding to *h*(*t*). Often used in *complex trace analysis* (q.v.). Named for David Hilbert (1862–1943), German mathematician.