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On the Marginal Hilbert Spectrum
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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
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Hilbert Spectrum
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Definition of Hilbert Spectra
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can be amplitude or the square of amplitude (energy). d ω d t Schematic of Hilbert Spectrum
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Hilbert Spectra
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Definition of the Marginal Hilbert Spectrum
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Computing Hilbert Spectrum
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Marginal Spectrum
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Hilbert and Marginal Spectra
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Some Properties
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MHS and Fourier Spectra 1/T
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MHS with different Resolutions
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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
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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.
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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.
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Example Uniformly distributed white noise
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STD = 0.2 Data : White Noise STD = 0.2
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Fourier Power Spectrum
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IMF
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Hilbert Marginal and Fourier Spectra Non-zero mean data : DC components
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Factor = 1 Effects on Frequency Resolution MHS
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Normalized MHS
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[ 10 20 50 100 300 500 600 800]/1000
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Effect Frequency Resolution : bin size
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Normalized
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Data
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Data : IMF
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Fourier Spectra
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Hilbert Spectra : Various F-Resolutions
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Hilbert Amplitude Spectra : Various F-Resolutions
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Example Earthquake data
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Earthquake data E921
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IMF EEMD2(3,0.2,100)
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IMF EEMD2(3,0.1,10)
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IMF EEMD2(3,0,1)
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Different Frequency Resolutions VS Fourier and Normalization
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MHS and Fourier at full resolutions
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MHS and Fourier Normalized
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MHS Smoothed and Normalized
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MHS Different Frequency Resolutions
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MHS Different Resolutions Normalized
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MHS EMD and EEMD
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Zoom
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MHS 100 Ensemble
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MH Amplitude Spectrum
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10 Ensemble Poor normalization
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Fourier and Hilbert Marginal Spectra
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Normalized
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Effect of Filter size : Fourier
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Hilbert Spectrum
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Marginal Spectra
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Normalized
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Zoom Effects
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Normalized
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Effect of bin size
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Normalized
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Effects of bin size and zoom
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Normalized
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Example Delta-Function
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Influence of the resolution of frequency on the Hilbert-Huang spectrum [ 10 20 50 100 300 500 600 800]/1000
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Effects of Frequency Resolution
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Fourier Energy Spectrum
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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.
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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.
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