# Hilbert spectral analysis

Hilbert spectral analysis

Hilbert spectral analysis is a signal analysis method applying the Hilbert transform to compute the instantaneous frequency of signals according to
:$omega=frac\left\{d heta\left(t\right)\right\}\left\{dt\right\}.,$After performing the Hilbert transform on each signal, we can express the data in the following form::$X\left(t\right)=sum_\left\{j=1\right\}^\left\{n\right\}a_j\left(t\right)exp\left(iintomega_j\left(t\right)dt\right).,$

This equation gives both the amplitude and the frequency of each component as functions of time. It also enables us to represent the amplitude and the instantaneous frequency as functions of time in a three-dimensional plot, in which the amplitude can be contoured on the frequency-time plane. This frequency-time distribution of the amplitude is designated as the Hilbert amplitude spectrum, or simply Hilbert spectrum.

Hilbert spectral analysis method is an important part of Hilbert-Huang transform.

*Hilbert transform
*Hilbert-Huang transform
*Hilbert spectrum

References

*Alan V. Oppenheim and Ronald W. Schafer, "Discrete-Time Signal Processing"," Prentice-Hall Signal Processing Series, 2 ed., 1999.

*Huang, et al. "The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis." "Proc. R. Soc. Lond. A" (1998) 454, 903–995 ( [http://keck.ucsf.edu/~schenk/Huang_etal98.pdf Link] )

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Spectral analysis — may refer to:* Spectrum analysis, in physics, a method of analyzing the chemical properties of matter from bands in their optical spectrum * Spectral theory, in mathematics, a theory that extends eigenvalues and eigenvectors to linear operators… …   Wikipedia

• Hilbert-Huang transform — The Hilbert Huang Transform (HHT) is a way to decompose a signal into so called intrinsic mode functions (IMF), and obtain instantaneous frequency data. It is designed to work well for data that are nonstationary and nonlinear. In contrast to… …   Wikipedia

• analysis — /euh nal euh sis/, n., pl. analyses / seez /. 1. the separating of any material or abstract entity into its constituent elements (opposed to synthesis). 2. this process as a method of studying the nature of something or of determining its… …   Universalium

• Hilbert space — For the Hilbert space filling curve, see Hilbert curve. Hilbert spaces can be used to study the harmonics of vibrating strings. The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It… …   Wikipedia

• Spectral theory of ordinary differential equations — In mathematics, the spectral theory of ordinary differential equations is concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation Hermann Weyl… …   Wikipedia

• Spectral theorem — In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a …   Wikipedia

• Spectral theory — In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix. The name was introduced by David Hilbert in his original formulation of Hilbert space theory, which was… …   Wikipedia

• Spectral theory of compact operators — In functional analysis, compact operators are linear operators that map bounded sets to precompact ones. Compact operators acting on a Hilbert space H is the closure of finite rank operators in the uniform operator topology. In general, operators …   Wikipedia

• Hilbert–Pólya conjecture — In mathematics, the Hilbert–Pólya conjecture is a possible approach to the Riemann hypothesis, by means of spectral theory.Initial hunchesDavid Hilbert and George Pólya speculated that real number values of t such that : frac12 + it is a zero of… …   Wikipedia

• David Hilbert — Hilbert redirects here. For other uses, see Hilbert (disambiguation). David Hilbert David Hilbert (1912) Born …   Wikipedia