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 1 / ( π t ) {\displaystyle 1/(\pi t)} (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.
Definition The Hilbert transform of u can be thought of as the convolution of u(t) with the function h(t) = 1/πt, known as the Cauchy kernel. Because 1/t is not integrable across t = 0, the integral defining the convolution does not always converge. Instead, the Hilbert transform is defined using the Cauchy principal value (denoted here by p.v.). Explicitly, the Hilbert transform of a function (or signal) u(t) is given by
H ( u ) ( t ) = 1 π p . v . ∫ − ∞ + ∞ u ( τ ) t − τ d τ , {\displaystyle \operatorname {H} (u)(t)={\frac {1}{\pi }}\,\operatorname {p.v.} \int _{-\infty }^{+\infty }{\frac {u(\tau )}{t-\tau }}\,\mathrm {d} \tau ,}
provided this integral exists as a principal value. This is precisely the convolution of u with the tempered distribution p.v. 1/πt. Alternatively, by changing variables, the principal-value integral can be written explicitly as
H ( u ) ( t ) = 2 π lim ε → 0 ∫ ε ∞ u ( t − τ ) − u ( t + τ ) 2 τ d τ . {\displaystyle \operatorname {H} (u)(t)={\frac {2}{\pi }}\,\lim _{\varepsilon \to 0}\int _{\varepsilon }^{\infty }{\frac {u(t-\tau )-u(t+\tau )}{2\tau }}\,\mathrm {d} \tau .}
When the Hilbert transform is applied twice in succession to a function u, the result is
H ( H ( u ) ) ( t ) = − u ( t ) , {\displaystyle \operatorname {H} {\bigl (}\operatorname {H} (u){\bigr )}(t)=-u(t),}
provided the integrals defining both iterations converge in a suitable sense. In particular, the inverse transform is
− H {\displaystyle -\operatorname {H} } . This fact can most easily be seen by considering the effect of the Hilbert transform on the Fourier transform of u(t) (see § Relationship with the Fourier transform below). For an analytic function in the upper half-plane, the Hilbert transform describes the relationship between the real part and the imaginary part of the boundary values. That is, if f(z) is analytic in the upper half complex plane {z : Im{z} > 0}, and u(t) = Re{f (t + 0·i)}, then Im{f(t + 0·i)} = H(u)(t) up to an additive constant, provided this Hilbert transform exists.
Notation In signal processing the Hilbert transform of u(t) is commonly denoted by u ^ ( t ) {\displaystyle {\hat {u}}(t)} . However, in mathematics, this notation is already extensively used to denote the Fourier transform of u(t). Occasionally, the Hilbert transform may be denoted by u ~ ( t ) {\displaystyle {\tilde {u}}(t)} . Furthermore, many sources define the Hilbert transform as the negative of the one defined here.
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