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

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 t with the value in each bin designate

Hilbert Energy Spectrum

a

d as

at the propert) (

time, t, and the proper instantaneous

frequency, .

The is defined as the amplitude

density distribution in a time-frequency space divided into equ

Hilbert A

al size

b

mplitude

ins of

Spectru

m

t wi

th the value in each bin designated as

at the proper time, t, and the proper instantaneous

frequency,

a(t)

.

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.

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

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 / t .

In Hilbert spectral analysis, the regular Nyquist frequency

would be lower than the lower than the Fourier counterpart

at 1/ 4 t . But, there could be instantaneous valu

es 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

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

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

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

Example

Uniformly distributed white noise

Data

Data : IMF

Fourier Spectra

Fourier Spectra

Hilbert Spectra : Various F-Resolutions

Hilbert Spectra : Various T-Resolutions

Hilbert Amplitude Spectra : Various F-Resolutions

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

0 100 200 300 400 500 600 700 800 900 1000

IMF

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 Spectra

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 20 50 100 300 500 600 800]/1000

1000T

f

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

Example

Earthquake data

Earthquake data E921

IMF EEMD2(3,0.2,100)

0 2000 4000 6000 8000 10000 12000 14000

IMFs E921 : EEMD2(3, 0.1,10)

Time : second*200

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

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

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.