Hilbert transform

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 ). The Hilbert transform has a particularly simple representation in the frequency domain: It imparts a phase shift of ±90° ( radians) to every frequency component of a function, the sign of the shift depending on the sign of the frequency (see ). 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. The Hilbert transform of u can be thought of as the convolution of u(t) with the function h(t) = 1/ pi t, known as the Cauchy kernel. Because 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 provided this integral exists as a principal value. This is precisely the convolution of u with the tempered distribution p.v. 1/pi t. Alternatively, by changing variables, the principal value integral can be written explicitly as When the Hilbert transform is applied twice in succession to a function u, the result is: provided the integrals defining both iterations converge in a suitable sense. In particular, the inverse transform is This fact can most easily be seen by considering the effect of the Hilbert transform on the Fourier transform of u(t) (see , 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.