the Hilbert transform
/ HILL-bairt /
On the boundary of a holomorphic function, the real and imaginary parts are not free — knowing one determines the other. The Hilbert transform is the explicit operation that produces the imaginary part of such a boundary function from its real part (up to a constant). Equivalently, it is the operation that takes a real signal and produces its quadrature companion — the same signal phase-shifted by 90 degrees at every frequency — which is the bread and butter of communications, where it builds the analytic signal and lets you read off instantaneous amplitude and phase.
On the real line the Hilbert transform of a function u is defined by a principal-value integral: (H u)(x) = (1 / pi) times the principal value of (integral over all t of u(t) / (x - t) dt). The principal value is essential because the kernel 1 / (x - t) blows up at t = x; you cut out a symmetric sliver around the singularity and let it shrink, and the two infinite halves cancel. In the frequency domain the action is wonderfully simple: H multiplies the positive-frequency components by -i and the negative ones by +i — that is, it phase-shifts every sinusoid by minus 90 degrees. The deep link to complex analysis: if you take a real function u on the real axis and harmonically extend it into the upper half-plane, its harmonic conjugate (the imaginary part of the holomorphic function whose real part is u) has boundary values exactly H u. So the Hilbert transform IS 'pass to the harmonic conjugate, then look at the boundary.'
This makes the Hilbert transform the boundary face of holomorphy, and it underlies the Kramers-Kronig relations in optics (the real and imaginary parts of a medium's response — its refraction and absorption — are Hilbert transforms of each other because the response is causal and hence the boundary value of a function holomorphic in a half-plane). It is the gateway to Hardy spaces, which are precisely the spaces of boundary functions of holomorphic functions. The honest subtleties: the defining integral needs the principal value to exist at all; H is not local — changing u near one point changes H u everywhere; and H squared equals minus the identity (applying it twice and negating recovers the original), so it is a kind of 90-degree rotation, not an averaging.
The Hilbert transform of cos(omega x) is sin(omega x), and of sin(omega x) is -cos(omega x), for omega > 0 — each sinusoid is simply shifted by minus 90 degrees. Pairing a signal u with its transform H u into u + i (H u) builds the analytic signal, whose modulus is the instantaneous amplitude (the envelope) and whose argument grows at the instantaneous frequency.
H phase-shifts every frequency by -90 degrees; u + i(H u) is the analytic signal.
The Hilbert transform of a constant is zero, which is why recovering a holomorphic function from its real part on a boundary determines it only up to an additive imaginary constant — exactly the same ambiguity as recovering a function from its harmonic conjugate.