Hilbert spectrum

Hilbert spectrum

The Hilbert spectrum (sometimes referred to as the "Hilbert amplitude spectrum") is a statistical tool that can help in distinguishing among a mixture of moving signals. The spectrum itself is decomposed into its component sources using independent component analysis. The separation of the combined effects of unidentified sources (blind signal separation) has applications in climatology, seismology, and biomedical imaging.

Conceptual summary

The Hilbert spectrum is computed by way of a 2-step process consisting of:
* Preprocessing a signal separate it into intrinsic mode functions using a mathematical decomposition such as singular value decomposition (SVD);
* Applying the Hilbert transform to the results of the above step to obtain the instantaneous frequency spectrum of each of the components.

The Hilbert transform defines the imaginary part of the function to make it an analytic function (sometimes referred to as a progressive function), "i.e." a function whose signal strength is zero for all frequency components less than zero.

With the Hilbert transform, the singular vectors give instantaneous frequencies that are functions of time, so that the result is an energy distribution over time and frequency.

The result is an ability to capture time-frequency localization to make the concept of instantaneous frequency and time relevant (the concept of instantaneous frequency is otherwise abstract or difficult to define for all but monocomponent signals).

Applications of the Hilbert spectrum

The Hilbert spectrum has many practical applications. One example application pioneered by Professor Richard Cobbold, is the use of the Hilbert spectrum for the analysis of blood flow by pulse Doppler ultrasound. Other applications of the Hilbert spectrum include analysis of climatic features, water waves, and the like.

References

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


Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • Hilbert spectral analysis — is a signal analysis method applying the Hilbert transform to compute the instantaneous frequency of signals according to :omega=frac{d heta(t)}{dt}.,After performing the Hilbert transform on each signal, we can express the data in the following… …   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

  • 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

  • Spectrum of a C*-algebra — The spectrum of a C* algebra or dual of a C* algebra A, denoted Â, is the set of unitary equivalence classes of irreducible * representations of A. A * representation π of A on a Hilbert space H is irreducible if, and only if, there is no closed… …   Wikipedia

  • Spectrum (functional analysis) — In functional analysis, the concept of the spectrum of a bounded operator is a generalisation of the concept of eigenvalues for matrices. Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if… …   Wikipedia

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

  • Decomposition of spectrum (functional analysis) — In mathematics, especially functional analysis, the spectrum of an operator generalizes the notion of eigenvalues. Given an operator, it is sometimes useful to break up the spectrum into various parts. This article discusses a few examples of… …   Wikipedia

  • Compact operator on Hilbert space — In functional analysis, compact operators on Hilbert spaces are a direct extension of matrices: in the Hilbert spaces, they are precisely the closure of finite rank operators in the uniform operator topology. As such, results from matrix theory… …   Wikipedia

  • Rigged Hilbert space — In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square integrable aspects of functional analysis. Such spaces were introduced to study… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”