Hilbert transform

In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). This linear operator is given by convolution with the function 1 / ( π t ) {displaystyle 1/(pi t)} : H ( u ) ( t ) {displaystyle H(u)(t)} = u m ( t ) ⋅ e i ( ω t + ϕ ) {displaystyle =u_{m}(t)cdot e^{i(omega t+phi )},}   (by Euler's formula)    (Eq.1) In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). This linear operator is given by convolution with the function 1 / ( π t ) {displaystyle 1/(pi t)} : the improper integral being understood in the principal value sense. The Hilbert transform has a particularly simple representation in the frequency domain: it imparts a phase shift of 90° to every Fourier component of a function. For example, the Hilbert transform of cos ⁡ ( ω t ) {displaystyle cos(omega t)} , where ω > 0, is cos ⁡ ( ω t − π / 2 ) {displaystyle cos(omega t-pi /2)} . The Hilbert transform is important in signal processing, where it derives the analytic representation of a real-valued signal u(t). Specifically, the Hilbert transform of u is its harmonic conjugate v, a function of the real variable t such that the complex-valued function u+iv admits an extension to the complex upper half-plane satisfying the Cauchy–Riemann equations. 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/(πt), known as the Cauchy kernel. Because h(t) is not integrable, 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/πt (due to Schwartz (1950); see Pandey (1996, Chapter 3)). Alternatively, by changing variables, the principal value integral can be written explicitly (Zygmund 1968, §XVI.1) as: When the Hilbert transform is applied twice in succession to a function u, the result is negative u: provided the integrals defining both iterations converge in a suitable sense. In particular, the inverse transform is −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 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 plane 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. In signal processing the Hilbert transform of u(t) is commonly denoted by u ^ ( t ) {displaystyle {widehat {u}}(t),} (e.g., Brandwood 2003, pg 87). However, in mathematics, this notation is already extensively used to denote the Fourier transform of u(t) (e.g., Stein & Weiss 1971). Occasionally, the Hilbert transform may be denoted by u ~ ( t ) {displaystyle { ilde {u}}(t)} . Furthermore, many sources define the Hilbert transform as the negative of the one defined here (e.g., Bracewell 2000, pg 359).