Hilbert spectral analysis is a signal analysis method applying the

Hilbert transform In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the func ...

to compute the

instantaneous frequency Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions. The instantaneous phase (also known as local phase or simply phase) of a ''comple ...

of signals according to :

\omega=\frac.\,

After performing the

on each signal, we can express the data in the following form: :

X(t)=\sum_^{n}a_j(t)\exp\left(i\int\omega_j(t)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

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 The Hilbert spectrum (sometimes referred to as the Hilbert amplitude 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 comp ...

. Hilbert spectral analysis method is an important part of

Hilbert–Huang transform The Hilbert–Huang transform (HHT) is a way to decompose a signal into so-called intrinsic mode functions (IMF) along with a trend, and obtain instantaneous frequency data. It is designed to work well for data that is nonstationary and nonline ...

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 Signal processing