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.
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Adaptive signal processing, A/D and D/A. This module provides the basic
tools for adaptive filtering and a solid mathematical framework for sampling and
quantization
Ce cours pose les bases d'un concept essentiel en ingénierie : la notion de système. Plus spécifiquement, le cours présente la théorie des systèmes linéaires invariants dans le temps (SLIT), qui sont
An introduction to methods of harmonic analysis.
Covers convergence of Fourier series, Hilbert transform, Calderon-Zygmund theory, Fourier restriction, and applications to PDE.
This course introduces modern computational electronic structure methods and their broad applications to organic chemistry. It also discusses physical organic concepts to illustrate the stability and
In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator whose kernel function K : Rn×Rn → R is singular along the diagonal x = y. Specifically, the singularity is such that |K(x, y)| is of size |x − y|−n asymptotically as |x − y| → 0. Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over |y − x| > ε as ε → 0, but in practice this is a technicality.
In Fourier analysis, a multiplier operator is a type of linear operator, or transformation of functions. These operators act on a function by altering its Fourier transform. Specifically they multiply the Fourier transform of a function by a specified function known as the multiplier or symbol. Occasionally, the term multiplier operator itself is shortened simply to multiplier. In simple terms, the multiplier reshapes the frequencies involved in any function.
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. In this method, a singularity on an integral interval is avoided by limiting the integral interval to the singularity (so the singularity is not covered by the integral). Depending on the type of singularity in the integrand f, the Cauchy principal value is defined according to the following rules: In some cases it is necessary to deal simultaneously with singularities both at a finite number b and at infinity.
Covers the basics of Density Functional Theory, challenges in DFT, and improvements in functional approximations and corrections for accurate calculations.
Situational awareness strategies are essential for the reliable and secure operation of the electric power grid which represents critical infrastructure in modern society. With the rise of converter-interfaced renewable generation and the consequent shift ...
EPFL2024
H & uuml;sler-Reiss vectors and Brown-Resnick fields are popular models in multivariate and spatial extreme-value theory, respectively, and are widely used in applications. We provide analytical formulas for the correlation between powers of the components ...
Springer2024
,
In a society that recognizes the urgency of safeguarding the environment and drastically limiting land transformations and energy-intensive activities like constructing new buildings, the protection of architectural and environmental heritage is no longer ...