On the Marginal Hilbert Spectrum
Outline
•Definitions of the Hilbert Spectrum (HS) andthe Marginal Hilbert Spectrum (MHS).
•Computation of MHS
•The relation between MHS and FourierSpectrum
•MHS with different frequency resolutions
•Examples
Hilbert Spectrum
Definition of Hilbert Spectra
can be amplitude or the square of amplitude (energy).
d ω
d t
Schematic of Hilbert Spectrum
Hilbert Spectra
Definition of the Marginal Hilbert Spectrum
Computing Hilbert Spectrum
Marginal Spectrum
Hilbert and Marginal Spectra
Some Properties
MHS and Fourier Spectra
1/T
MHS with different Resolutions
Observations
•Note that N/T is actually the sampling rate, so theconservation from Fourier to Hilbert is simply twicethe sampling rate, if we use the full frequency rangeto the Nyquist limit.
•If we use any zoom, the additional factor is anadditional
Some Properties
•The spectral density depends on the bin size that is onboth temporal and frequency resolutions.
•For marginal Frequency spectrum, the temporalresolution is implicit.
•For instantaneous energy density, the frequencyresolution is implicit.
•Frequency assumes instantaneous value, not mean; itis not limited by the Nyquist.
•We can zoom the spectrum to any temporal andfrequency location.
Fourier vs. Hilbert Spectra
•Adaptive basis, Data Driven
•Time-frequency spectrum
•Physical meaningful frequency at both the highand low frequency ranges
•Resolution of the frequency adjustable
•Zoom capability
•Marginal spectra for frequency and time.
Example
Uniformly distributed white noise
STD = 0.2
Data : White Noise STD = 0.2
Fourier Power Spectrum
IMF
Hilbert Marginal and Fourier Spectra
Non-zero mean data : DC components
Factor = 1
Effects on Frequency Resolution MHS
Normalized MHS
[ 10 20 50 100
300 500 600 800]/1000
Effect Frequency Resolution : bin size
Normalized
Data
Data : IMF
Fourier Spectra
Fourier Spectra
Hilbert Spectra : Various F-Resolutions
Hilbert Amplitude Spectra : Various F-Resolutions
Example
Earthquake data
Earthquake data E921
IMF EEMD2(3,0.2,100)
IMF EEMD2(3,0.1,10)
IMF EEMD2(3,0,1)
Different Frequency Resolutions
VS Fourier and Normalization
MHS and Fourier at full resolutions
MHS and Fourier Normalized
MHS Smoothed and Normalized
MHS Different Frequency Resolutions
MHS Different Resolutions Normalized
MHS EMD and EEMD
Zoom
MHS 100 Ensemble
MHS 100 Ensemble
MH Amplitude Spectrum
10 Ensemble
Poor normalization
Fourier and Hilbert Marginal Spectra
Normalized
Effect of Filter size : Fourier
Hilbert Spectrum
Hilbert Spectrum
Hilbert Spectrum
Hilbert Spectrum
Marginal Spectra
Normalized
Zoom Effects
Normalized
Effect of bin size
Normalized
Effects of bin size and zoom
Normalized
Example
Delta-Function
Influence of the resolution of frequency on the Hilbert-Huang spectrum
[ 10 20 50 100
300 500 600 800]/1000
Effects of Frequency Resolution
Fourier Energy Spectrum
Summary
•Hilbert spectra are time-frequencypresentations.
•The marginal spectra could have variousresolutions and zoom capability.
•Hilbert marginal spectra could be smoothedwithout losing resolution drastically.
•Another marginal Hilbert quantity is the time-energy distribution.
Summary
•For long time, the Hilbert Marginal Spectrum was notdefined absolutely.
•The energy and amplitude spectra were not clearlycompared; they are totally different spectra.
•Clear conversion factor are given to makecomparisons between MHS and Fourier easily.
•Conversion factor also was provided for MHS withdifferent Frequency resolutions.
•In most cases the MHS in energy is very similar toFourier in case the data are from stationary andlinear processes, for the temporal has beenintegrated out.