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Hilbert transform

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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)±iX(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),


and 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)+jx(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.