On the Marginal Hilbert Spectrum

Outline Definitions of the Hilbert Spectrum (HS) and the Marginal Hilbert Spectrum (MHS). Computation of MHS The relation between MHS and Fourier Spectrum 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 the conservation from Fourier to Hilbert is simply twice the sampling rate, if we use the full frequency range to the Nyquist limit. If we use any zoom, the additional factor is an additional

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 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 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

[ ]/1000

Effect Frequency Resolution : bin size

Normalized

Data

Data : IMF

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

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

Example Delta-Function

Influence of the resolution of frequency on the Hilbert-Huang spectrum [ ]/1000

Effects of Frequency Resolution

Fourier Energy Spectrum

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 drastically. 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 in case the data are from stationary and linear processes, for the temporal has been integrated out.