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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
Hilbert Spectrum
n n
k k kk 1 k 1
tn
k kk 1 0
2n n2 2
k k i jk 1 k 1
After EMD, we should have
x(t) = c ( t ) = a ( t ) cos ( t )
a ( t ) cos ( )d
x (t) = c ( t ) = c ( t ) 2 c c
n
2 2k k
k 1
a ( t ) cos ( t )
Definition of Hilbert Spectra
2
The is defined as the energy density
distribution in a time-frequency space divided into equal size
bins of with the value in each bin designa
Hilbert Energy
ted as
Spectrum
t
a (t at th p) r pee o r
and the
The is defined as the amplitude
density distribution
Hilbert
in a ti
Amplitude Spe
me-frequency s
time, t, proper instantaneous
frequen
pace divided into equal size
bins of
cy, .
ct
i
rum
t w
th the value in each bin designated as
at the proper time, t, and the proper instantaneous
frequency,
a(t)
.
00.
10.
20.
30.
40.
50
0.050.1
0.150.2
0.25
Fre
quency
: 1/
T
Power Spectrum Density: L2 T
f
t
jiS , can be amplitude or the square of amplitude (energy).
jiS ,d ω
d t
Schematic of Hilbert Spectrum
Hilbert SpectraCurrently, the spectrum is not defined in terms of density.
The value is simply the energy value at the particuler bin.
To define the spectra in terms of density would facilitate
comparison with the F
i , j
2i , j
ourier spectra which is defined in terms
of density.
Therefore, the value in each bin should be
a for amplitude spectra
t
a for energy spectra
t
Definition of the Marginal Hilbert Spectrum
T N
ii 10
Given the Hilbert Spectrum as H( ,t), the Marginal Spectrum
is defined simply as
h( ) = H( ,t) dt = H( , t ) .
Simple as it seems, the actual computation and evaluation
is more involv
ed. The main reason is that, with the adaptive
basis, we do not have the rigid limitation on frequency resolution
dictated by the total data length and the uncertainty principle.
The freedom on our choice of frequency-time resolution; however,
makes the marginal frequency evaluation much more complicated.
We need to define it rigorously for detailed comparisons with
other forms of spectrum.
Computing Hilbert Spectrum
0 0 0 0 1
0 0 1
In the time-frequency space,
time is designated as : t , t + t, t +2 t, ..... t i t , .... t .
frequency is designated as : , , ...., +j , ... , .
Here the values and sizes of all th
0 1
e variables could be selected
to fit our need subjecting only to the following restrictions:
1. t and t have to reside within th interval of data span, [0, T].
2. t cannot be smaller than the sampl ing step.
Marginal Spectrum
In the Fourier Spectrum, the time scale is out of the
formulation. The frequency scales are limited by the
sampling rate, t, and total data length, T:
Frequency resolution : = 1/T.
Nyquist frequency
(the highest frequency, 2 points) = 2 / t .
In Hilbert spectral analysis, the regular Nyquist frequency
would be lower than the Fourier counterpart at 1/ t (4 points).
But, there could be instantaneo
us values higher than this
limit. The frequency values are continuous and there is
absolutely no limitation to its range of values.
Hilbert and Marginal Spectra
i 0
j 0 i , j
2j i i , j i , j
j i
Let us designate the values at an arbitrary bin at t t i t ,
= + j as S . The the Hilbert spectrum is
H( , t ) = S = a for all the i and j.
The Marginal spectrum,
1
t
j
n2
j i , j i , ji 1 i
n2 2
i , j j i , jj 1 j j i
2
h( ), is
h( ) = S = a .
The total energy , E, is
E = S = h( ) = a = 2x ;
therefore, the energy is not averaged x as in Fouri
1 t t
t
1 t
er
.
Some Properties
2 2h j i , j i , j
j j i j i
2N N2 2
n nn 1 n 1
N2 2i , j n
i j n 1
The total energy is
h S a 2 x .
Notice also that
x ( t ) c ( t ) c ( t )
Therefore , we have
a 2 c ( t )
E
E
MHS and Fourier Spectra
M2
ii 1
22
M M2 2 2
i i i , ji 1 i 1
By definition, the Fourier Spectrum is
1 x S( )d S( ) , where = .
T
xBut x , where N is the total number of data points.
N
Therefore,
1 x N x N S(
N) S( )
Ta
2
i j
M2i , j i
i j i 1
a S( )2N
T
1/T
MHS with different Resolutions2
j i i , j i , jj i
n2
j i , j i , ji 1 i
By definition, H( , t ) = S = a for all the i and j.
h( ) = S = a .
Therefore, for arbitrary different resolution
1
t
1
s,
t
we would have dif
n2
a j a i , j i , ji 1 i
2a j i , j
i
aa
a
a
a
fernt
values for the marginal spectra:
h ( ) = S = a
To convert them to the same scale, we should use a
1
factor: .
h ( ) =
t
1 a
2i , j j
i
= a h( ) 1
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
.a
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.
0 100 200 300 400 500 600 700 800 900 10000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
STD = 0.2
Data : White Noise STD = 0.2
10-3
10-2
10-1
100
10-6
10-5
10-4
10-3
10-2
10-1
100
101
Frequency: Hz (T-1)
Pow
er
Spectr
um
Density: L
2 TFourier Power Spectrum
10-3
10-2
10-1
100
10-5
10-4
10-3
10-2
10-1
100
101
102
FourierHHT
Hilbert Marginal and Fourier SpectraNon-zero mean data : DC components
10-3
10-2
10-1
100
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
Frequency: Hz (T-1)
Hilbert
-Huang M
arg
inal S
pectr
um
: L
2 T
1/8001/6001/5001/3001/1001/501/201/10Fourier
Factor = 1
Effects on Frequency Resolution MHS
10-3
10-2
10-1
100
10-4
10-3
10-2
10-1
100
101
102
103
Frequency: Hz (T-1)
Hilbert
-Huang M
arg
inal S
pectr
um
: L
2 T
Fourier1/8001/6001/5001/3001/1001/501/201/10
Normalized MHS
10-3
10-2
10-1
100
10-4
10-3
10-2
10-1
100
101
102
Frequency: Hz (T-1)
Hilbert
-Huang M
arg
inal S
pectr
um
: L
2 T
BSF=500 FACTOR=1BSF=250 FACTOR=1BSF=50 FACTOR=1
Effect Frequency Resolution : bin size
10-3
10-2
10-1
100
10-4
10-3
10-2
10-1
100
101
102
Frequency: Hz (T-1)
Hilbert
-Huang M
arg
inal S
pectr
um
: L
2 T
BSF=500 FACTOR=1BSF=250 FACTOR=0.5BSF=50 FACTOR=0.1
Normalized
0 2000 4000 6000 8000 10000 12000 14000
IMFs E921 : EEMD2(3, 0.1,10)
Time : second*200
IMF EEMD2(3,0.1,10)
10-2
10-1
100
101
102
10-4
10-2
100
102
104
106
Frequency : Hz
Spectr
al D
ensity
Fourier and Marginal Hilbert Spectra : Raw
Fourier:WS=14,000Hilbert: BS=7,000
Fourier and Hilbert Marginal Spectra
10-2
10-1
100
101
102
100
101
102
103
104
105
106
107
Frequency : Hz
Spectr
al D
ensity
Fourier and Marginal Hilbert Spectra : Normalized
Fourier:WS=14,000
Hilbert: BS=7,000
Normalized
10-2
10-1
100
101
102
10-6
10-4
10-2
100
102
104
Frequency : Hz
Spectr
al D
ensity
Fourier Spectra with Various Window Sizes
Hanning=14,000
Hanning=3,500
Effect of Filter size : Fourier
Hilbert : FR=7000, TR=70, CF=100Hz
Time : second
Fre
quency : H
z
0 10 20 30 40 50 60 700
10
20
30
40
50
60
70
80
90
100
Hilbert Spectrum
Hilbert : FR=7000, TR=70, CF=10Hz
Time : second
Fre
quency : H
z
0 10 20 30 40 50 60 700
1
2
3
4
5
6
7
8
9
10
Hilbert Spectrum
Hilbert : FR=700, TR=70, CF=10Hz
Time : second
Fre
quency : H
z
0 10 20 30 40 50 60 700
1
2
3
4
5
6
7
8
9
10
Hilbert Spectrum
Hilbert : FR=350, TR=70, CF=10Hz
Time : second
Fre
quency : H
z
0 10 20 30 40 50 60 700
1
2
3
4
5
6
7
8
9
10
Hilbert Spectrum
10-2
10-1
100
101
102
100
101
102
103
104
105
106
107 Marginal Hilbert Spectra with Different Bin Sizes
Spectr
al D
ensity
Frequency : Hz
BS=7,000BS=3,500BS=700
Marginal Spectra
10-2
10-1
100
101
102
100
101
102
103
104
105
106
107 Marginal Hilbert Spectra with Different Bin Sizes Normalized
Frequency : Hz
Spectr
al D
ensity
BS=7,000BS=3,500/(2)BS=700/(10)
Normalized
10-3
10-2
10-1
100
101
102
10-2
10-1
100
101
102
103
104
105
106 Hilbert Marginal Spectra: Zoom
Frequency : Hz
Spectr
al D
ensity
CF=10Hz; BS=7,000CF=100Hz; BS=7,000
Zoom Effects
10-3
10-2
10-1
100
101
102
10-2
100
102
104
106
108Hilbert Marginal Spectra: Different Cut-off Frequency and Bin Sizes Normalized
Frequency : Hz
Spectr
al D
ensity
CF=10Hz; BS=7,000 (*10)CF=10Hz; BS=700
Normalized
10-3
10-2
10-1
100
101
10-2
10-1
100
101
102
103
104
105
106
Frequency : Hz
Spectr
al D
ensity
Hilbert Marginal Spectra: Zoom V1
CF=10Hz; BS=7,000
CF=10Hz; BS=700
Effect of bin size
10-3
10-2
10-1
100
101
10-1
100
101
102
103
104
105
106
107
Frequency : Hz
Spectr
al D
ensity
Hilbert Marginal Spectra: Zoom Normalized
CF=10Hz; BS=7,000 *(10)
CF=10Hz; BS=700
Normalized
10-2
10-1
100
101
102
10-2
100
102
104
106
Hilbert Marginal Spectra: Zoom V3
Frequency : Hz
Spectr
al D
ensity
CF=100Hz; BS=7,000CF=10Hz; BS=700CF=10Hz,BS=350
Effects of bin size and zoom
10-2
10-1
100
101
102
10-2
100
102
104
106
Hilbert Marginal Spectra: Zoom V3 Normalized
Frequency : Hz
Spectr
al D
ensity
CF=100Hz; BS=7,000CF=10Hz; BS=700CF=10Hz,BS=350 (/2)
Normalized
0 100 200 300 400 500 600 700 800 900 10000
0.5
1-function
0 200 400 600 800 1000-2-101
x 10-3
Resid
ual
-20
2
x 10-3
IMF 5
-4-202468
x 10-3
IMF 4
-0.050
0.050.1
IMF 3
-0.4-0.2
00.20.4
IMF 2
-0.4-0.2
00.20.4
IMF 1
Influence of the resolution of frequency on the Hilbert-Huang spectrum
f [ 10 20 50 100 300 500 600 800]/1000
1000T
Effects of Frequency Resolution
10-3
10-2
10-1
100
10-20
10-15
10-10
10-5
100
105
Frequency: Hz (T-1)
Hilbert
-Huang M
arg
inal S
pectr
um
: L
2 T
1/8001/6001/5001/3001/1001/501/201/10Fourier
Fourier Energy Spectrum
10-3
10-2
10-1
100
2.5
3
3.5
4
4.5
5
5.5x 10
-3
Frequency: 1/T
Pow
er
Spectr
um
Density: L
2 T
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.