The Hilbert spectrum named after David Hilbert

David Hilbert

David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

, 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

Independent component analysis

Independent component analysis is a computational method for separating a multivariate signal into additive subcomponents supposing the mutual statistical independence of the non-Gaussian source signals...

. The separation of the combined effects of unidentified sources (blind signal separation

Blind signal separation

Blind signal separation, also known as blind source separation, is the separation of a set of signals from a set of mixed signals, without the aid of information about the source signals or the mixing process....

) has applications in climatology

Climatology

Climatology is the study of climate, scientifically defined as weather conditions averaged over a period of time, and is a branch of the atmospheric sciences...

, seismology

Seismology

Seismology is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other planet-like bodies. The field also includes studies of earthquake effects, such as tsunamis as well as diverse seismic sources such as volcanic, tectonic, oceanic,...

, and biomedical imaging

Magnetic resonance imaging

Magnetic resonance imaging , nuclear magnetic resonance imaging , or magnetic resonance tomography is a medical imaging technique used in radiology to visualize detailed internal structures...

.

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
  Singular value decomposition
  In linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....
  
   (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

Hilbert transform

In mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, u, and produces a function, H, with the same domain. The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the...

 defines the imaginary part of the function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 to make it an analytic function (sometimes referred to as a progressive function), i.e. a function whose signal strength

Signal strength

In telecommunications, particularly in radio, signal strength refers to the magnitude of the electric field at a reference point that is a significant distance from the transmitting antenna. It may also be referred to as received signal level or field strength. Typically, it is expressed in...

 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

Energy

In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

 distribution over time

Time

Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

 and frequency

Frequency

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the 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

Blood flow

Blood flow is the continuous running of blood in the cardiovascular system.The human body is made up of several processes all carrying out various functions. We have the gastrointestinal system which aids the digestion and the absorption of food...

 by pulse Doppler ultrasound

Ultrasound

Ultrasound is cyclic sound pressure with a frequency greater than the upper limit of human hearing. Ultrasound is thus not separated from "normal" sound based on differences in physical properties, only the fact that humans cannot hear it. Although this limit varies from person to person, it is...

. Other applications of the Hilbert spectrum include analysis of climatic features

Climatology

Climatology is the study of climate, scientifically defined as weather conditions averaged over a period of time, and is a branch of the atmospheric sciences...

, water waves

Ocean surface wave

In fluid dynamics, wind waves or, more precisely, wind-generated waves are surface waves that occur on the free surface of oceans, seas, lakes, rivers, and canals or even on small puddles and ponds. They usually result from the wind blowing over a vast enough stretch of fluid surface. Waves in the...

, and the like.