On the Marginal Hilbert Spectrum
Outline Definitions of the Hilbert Spectra (HS) and the Marginal Hilbert Spectra (MHS). Computation of MHS The relation between MHS and Fourier Spectrum MHS with different frequency resolutions Examples
Hilbert Spectrum
Definition of Hilbert Spectra
Hilbert Spectra
Definition of the Marginal Hilbert Spectrum
can be amplitude or the square of amplitude (energy). d ω d t Schematic of Hilbert Spectrum
Computing Hilbert Spectrum
Marginal Spectrum
Hilbert and Marginal Spectra
Some Properties
MHS and Fourier Spectra
MHS with different Resolutions
Some Properties The spectral density depends on the bin size that is on both temporal and frequency resolutions. For marginal Frequency spectrum, the temporal resolution is implicit. For instantaneous energy density, the frequency resolution is not implicit. Frequency assumes instantaneous value, not mean; it is not limited by the Nyquist. We can zoom the spectrum to any temporal and frequency location.
Fourier vs. Hilbert Spectra Adaptive basis, Data Driven Time-frequency spectrum Physical meaningful frequency at both the high and low frequency ranges Resolution of the frequency adjustable Zoom capability Marginal spectra for frequency and time.
Example Delta-Function
Influence of the resolution of frequency on the Hilbert-Huang spectrum [ ]/1000
Effects of Frequency Resolution
Fourier Energy Spectrum
Example Uniformly distributed white noise
Data
Data : IMF
Fourier Spectra
Hilbert Spectra : Various F-Resolutions
Hilbert Spectra : Various T-Resolutions
Hilbert Amplitude Spectra : Various F-Resolutions
STD = 0.2 Data : White Noise STD = 0.2
Fourier Power Spectrum
IMF
Hilbert Marginal and Fourier Spectra
Factor = 1 Effects on Frequency Resolution MHS
Normalized MHS
[ ]/1000
Effect Frequency Resolution : bin size
Normalized
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
MH Amplitude Spectrum
10 Ensemble Poor normalization
Fourier and Hilbert Marginal Spectra
Normalized
Effect of Filter size : Fourier
Hilbert Spectrum
Marginal Spectra
Normalized
Zoom Effects
Normalized
Effect of bin size
Normalized
Effects of bin size and zoom
Normalized
Summary Hilbert spectra are time-frequency presentations. The marginal spectra could have various resolutions and zoom capability. Hilbert marginal spectra could be smoothed without losing resolution. Another marginal Hilbert quantity is the time- energy distribution.
Summary For long time, the Hilbert Marginal Spectrum was not defined absolutely. The energy and amplitude spectra were not clearly compared; they are totally different spectra. Clear conversion factor are given to make comparisons between MHS and Fourier easily. Conversion factor also was provided for MHS with different Frequency resolutions. In,most cases the MHS in energy is very similar to Fourier, for the temporal has been integrated out.