Hilbert spectrum

[Image:Hilbertspectrum.png|thumb|400px|
Hilbert Spectrum of a frequency modulated waveform on the form given by.]
The Hilbert spectrum, named after David Hilbert, 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 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 or empirical mode decomposition ;
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, 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.

Definition

For a given signal decomposed to
where is the number of intrinsic mode functions that consists of and
The instantaneous angle frequency is then defined as
From this, we can define the Hilbert Spectrum for as
The Hilbert Spectrum of is then given by

Marginal Hilbert Spectrum

A two dimensional representation of a Hilbert Spectrum, called Marginal Hilbert Spectrum, is defined as
where is the length of the sampled signal. The Marginal Hilbert Spectrum show the total energy that each frequency value contribute with.

Applications

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.