maximum verisimilitude frequency averaging of signals

Dan Dan StefanoiuStefanoiu

Associate ProfessorAssociate [email protected]@fh-konstanz.de

[email protected]@yahoo.com

Florin Florin IonescuIonescu

[email protected]@fh-konstanz.de

University of Applied University of Applied Sciences, Konstanz, GermanySciences, Konstanz, GermanyDepartment of Department of

MechatronicsMechatronics

M

E H & P

Maximum Verisimilitude Maximum Verisimilitude Frequency Averaging of SignalsFrequency Averaging of Signals

# On leave from# On leave from “Politehnica” University of Bucharest, Romania“Politehnica” University of Bucharest, Romania

Department of Automatic Control and Computer ScienceDepartment of Automatic Control and Computer Science

The multiconference on Computational Engineering in Systems ApplicationsThe multiconference on Computational Engineering in Systems ApplicationsJuly 9-11, 2003, Lille, July 9-11, 2003, Lille,

FRANCEFRANCE

Research developed with the support of Research developed with the support of Alexander von Humboldt FoundationAlexander von Humboldt Foundation, , GermanyGermany

www.fh-konstanz.dewww.fh-konstanz.de

www.pub.rowww.pub.ro

www.avh.dewww.avh.de

HeadlinesHeadlines

The Frequency Averaging Method (FAM)The Frequency Averaging Method (FAM)

Simulation resultsSimulation results

Noise hypotheses and Maximum VerisimilitudeNoise hypotheses and Maximum Verisimilitude

On Time Domain Synchronous Averaging (TDSA)On Time Domain Synchronous Averaging (TDSA)

ConclusionConclusion

A data/signal (pre)processing paradigmA data/signal (pre)processing paradigm

ReferencesReferences

2

Usually, it is difficult, Usually, it is difficult, if not impossible.if not impossible.

Usually, it is difficult, Usually, it is difficult, if not impossible.if not impossible.


Noisy/fractal signalNoisy/fractal signal

Is it possible to make a clear Is it possible to make a clear distinction between the util distinction between the util data and the noise?data and the noise?

0 500 1000 1500 2000 2500

-1

-0.5

0

0.5

1

Sine wave

Normalized time

Ma

gn

itu

de

Period: N = 500

0 500 1000 1500 2000 2500

-1

-0.5

0

0.5

1

Sine wave

Normalized time

Ma

gn

itu

de

Period: N = 500

0 50 100 150 200 250 300 350-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2Non stationary signal

Normalized time

Ma

gn

itu

de

0 50 100 150 200 250 300 350-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2Non stationary signal

Normalized time

Ma

gn

itu

de

0 500 1000 1500 2000 2500

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Gaussian noise.

Normalized time

Ma

gn

itu

de

0 500 1000 1500 2000 2500

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Gaussian noise.

Normalized time

Ma

gn

itu

de

0 50 100 150 200 250 300 350

-0.1

-0.05

0

0.05

0.1

Uniform noise.

Normalized time

Ma

gn

itu

de

0 50 100 150 200 250 300 350

-0.1

-0.05

0

0.05

0.1

Uniform noise.

Normalized time

Ma

gn

itu

de

&&

&&

Un-mixUn-mix

0 500 1000 1500 2000 2500-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Sine wave corrupted by Gaussian noise. SNR = 15.5928 dB.

Normalized time

Ma

gn

itu

de

Period: N = 500

* Variance: 0.697359

0 500 1000 1500 2000 2500-2

-1.5

-1

-0.5

0

0.5

1

1.5

2


Normalized time

Ma

gn

itu

de

Period: N = 500


0 50 100 150 200 250 300 350

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Non stationary signal corrupted by uniform noise. SNR = -1.30917 dB.

Normalized time

Ma

gn

itu

de

* Variance: 0.00405827

0 50 100 150 200 250 300 350

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Non stationary signal corrupted by uniform noise. SNR = -1.30917 dB.

Normalized time

Ma

gn

itu

de

* Variance: 0.00405827

Util dataUtil data CorruptiCorrupting noiseng noise

stationarystationarystationarystationary

non stationarynon stationarynon stationarynon stationary

data dominates the noisedata dominates the noisedata dominates the noisedata dominates the noise

noise dominates the data noise dominates the data noise dominates the data noise dominates the data

Un-mixUn-mix??

It might be a difficult It might be a difficult Signal Processing problem.Signal Processing problem.

It might be a difficult It might be a difficult Signal Processing problem.Signal Processing problem.

How to extract the util How to extract the util data from a noisy signal?data from a noisy signal?

It depends tremendously on It depends tremendously on definition of “util” data.definition of “util” data.

It depends tremendously on It depends tremendously on definition of “util” data.definition of “util” data.

3

Signal Signal compactioncompaction

Signal Signal compactioncompaction


4

How theHow the “util” “util” data can be defined data can be defined ?? How theHow the “util” “util” data can be defined data can be defined ??

Two properties are desirable: Two properties are desirable:

Problems:Problems:Problems:Problems:

• Partial or significant attenuation of Partial or significant attenuation of noisenoise such that the extracted signal carries almost the such that the extracted signal carries almost the same information as the genuine one. same information as the genuine one.

• The rule of combination between the The rule of combination between the deterministic datadeterministic data and the and the stochastic noisestochastic noise is usually unknown. is usually unknown.

• Partial or significant attenuation of Partial or significant attenuation of redundancyredundancy such that such that the extracted signal encodes almost the same information the extracted signal encodes almost the same information but within a smaller number of data samples. but within a smaller number of data samples.

CompressingCompressingCompressingCompressing

DenoisingDenoisingDenoisingDenoising

One uses the One uses the additive/superposition hypothesisadditive/superposition hypothesis (which can fail for (which can fail for difficult signals such as: seismic, underwater acoustic or celestial).difficult signals such as: seismic, underwater acoustic or celestial).

Noise is modeled by using the Noise is modeled by using the Theorem of Central LimitTheorem of Central Limit. .

Any acquired data are affected by a certain Any acquired data are affected by a certain amount of amount of Gaussian noiseGaussian noise, usually , usually whitewhite..

Any acquired data are affected by a certain Any acquired data are affected by a certain amount of amount of Gaussian noiseGaussian noise, usually , usually whitewhite..

• Even the combination between deterministic and Even the combination between deterministic and stochastic components is known, stochastic components is known, how to separate themhow to separate them? ? Mathematical modelsMathematical models are required. are required.Mathematical modelsMathematical models are required. are required.

ParametricParametricParametricParametric

Non Non parametricparametric

Non Non parametricparametric


5

3 classes of signal compaction models 3 classes of signal compaction models 3 classes of signal compaction models 3 classes of signal compaction models

Interpolation modelsInterpolation modelsInterpolation modelsInterpolation models

In time domainIn time domainIn time domainIn time domain

Lagrange, Laguerre, Chebischev, Gauss, splines, etc.Lagrange, Laguerre, Chebischev, Gauss, splines, etc.

Compacted data provided by Compacted data provided by re-sampling re-sampling with awith a smaller sampling rate smaller sampling rate. .

Least Squares (LS) modelsLeast Squares (LS) modelsLeast Squares (LS) modelsLeast Squares (LS) models based on experimental identification recipesbased on experimental identification recipes

Noise only weakly attenuated, because models are usually Noise only weakly attenuated, because models are usually too fitted to datatoo fitted to data. .

Models that Models that fit the best to the datafit the best to the data, not necessarily maximally. , not necessarily maximally. The more complex the model the better the compaction performance. The more complex the model the better the compaction performance.

parametricparametricparametricparametric


0 50 100 150 200 250 300 350 400 450 500

0.85

0.9

0.95

1

1.05

1.1

1.15

USD - EURO currency (starting with January 10, 2002)

Days

1 U

SD

= *

EU

RO

1.06454

0 50 100 150 200 250 300 350 400 450 500

0.85

0.9

0.95

1

1.05

1.1

1.15

USD - EURO currency (starting with January 10, 2002)

Days

1 U

SD

= *

EU

RO

1.06454

Typical example: a time seriesTypical example: a time seriesTypical example: a time seriesTypical example: a time series

General trend General trend (deterministic)(deterministic)General trend General trend (deterministic)(deterministic)

• Polynomial, degree < 7Polynomial, degree < 7

Seasonal component Seasonal component (deterministic)(deterministic)

Seasonal component Seasonal component (deterministic)(deterministic)

• Elementary harmonicsElementary harmonics

TimeTimeTimeTime FrequencyFrequencyFrequencyFrequency Time-frequencyTime-frequencyTime-frequencyTime-frequency

• Auto-regressiveAuto-regressive

Colored noise Colored noise (stochastic)(stochastic)

Colored noise Colored noise (stochastic)(stochastic)

Simple models Simple models are preferred in are preferred in pre-processing. pre-processing.

Simple models Simple models are preferred in are preferred in pre-processing. pre-processing.

Spectral smoothing modelsSpectral smoothing modelsSpectral smoothing modelsSpectral smoothing models


6

3 classes of signal compaction models 3 classes of signal compaction models 3 classes of signal compaction models 3 classes of signal compaction models

Averaging modelsAveraging modelsAveraging modelsAveraging models based on based on Time Domain Synchronous AveragingTime Domain Synchronous Averaging

Described later. Described later.

based on spectral estimation techniquesbased on spectral estimation techniques

Smoothing the spectrum means Smoothing the spectrum means removing some noiseremoving some noise. . In general, In general, complex models and methodscomplex models and methods..

non-parametricnon-parametricnon-parametricnon-parametric

TimeTimeTimeTime FrequencyFrequencyFrequencyFrequency Time-frequencyTime-frequencyTime-frequencyTime-frequency

In time domainIn time domainIn time domainIn time domain

In frequency domainIn frequency domainIn frequency domainIn frequency domain

non-parametricnon-parametricnon-parametricnon-parametric

Compacted signal difficult to provide because Compacted signal difficult to provide because the spectrum looses the phase informationthe spectrum looses the phase information..

Averaging modelsAveraging modelsAveraging modelsAveraging models based on based on Maximum Verisimilitude DFT AveragingMaximum Verisimilitude DFT Averaging parametricparametricparametricparametric

Introduced within this presentation. Introduced within this presentation. NewNewNewNew

Transformation modelsTransformation modelsTransformation modelsTransformation models Short Fourier Transform, Wavelet Transform, Short Fourier Transform, Wavelet Transform, Wigner-Ville Transform, etc.Wigner-Ville Transform, etc.

Suitable for Suitable for non stationary data setsnon stationary data sets (with spectrum variable in time). (with spectrum variable in time).

In time-frequency domainIn time-frequency domainIn time-frequency domainIn time-frequency domain


Complex models and methods rather Complex models and methods rather inappropriate if only pre-processing is wantedinappropriate if only pre-processing is wanted. .

harmonic signal with harmonic signal with known/measurable periodknown/measurable period


7

Originates from early works in Signal Processing, Originates from early works in Signal Processing, such as such as Welch method of spectral estimationWelch method of spectral estimation (1967)(1967)..

Devised by Devised by P.D. McFaddenP.D. McFadden in in 19871987. .

)()()( tvtxty )()()( tvtxty t t

1

0

)(1

)(N

nrN nTty

Nta

1

0

)(1

)(N

nrN nTty

Nta

yca NN yca NN

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

Spectra of comb filters

Frequency [Hz]

Sp

ect

ral

po

we

r

Rotation frequency: 0.1 Hz

N = 10N = 25

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1


Frequency [Hz]

Sp

ect

ral

po

we

r


N = 10N = 25

TDSA like introduced by McFaddenTDSA like introduced by McFaddenTDSA like introduced by McFaddenTDSA like introduced by McFadden

Measured data modelMeasured data modelMeasured data modelMeasured data model

rTrTunknown noiseunknown noise

(with null average)(with null average)

How can How can xx be extracted from be extracted from yy ?? How can How can xx be extracted from be extracted from yy ??

IdeaIdeaIdeaIdea Exploit the known Exploit the known periodicity.periodicity.

Exploit the known Exploit the known periodicity.periodicity.

Util data modelUtil data modelUtil data modelUtil data model

t t

Time averaging of measured dataTime averaging of measured dataTime averaging of measured dataTime averaging of measured data

Comb Comb filterfilter

Comb Comb filterfilter

1

00 )(

1)(

N

nr

def

N nTtN

tc

1

00 )(

1)(

N

nr

def

N nTtN

tc t t

Dirac impulseDirac impulse

Fourier TransformFourier Transform

NN number of periods (the bigger, the better)number of periods (the bigger, the better)

Comb ruleComb ruleComb ruleComb rule Slide the comb along the data and average only the Slide the comb along the data and average only the samples pointed by its teeth. samples pointed by its teeth.

Slide the comb along the data and average only the Slide the comb along the data and average only the samples pointed by its teeth. samples pointed by its teeth.

TDSA is simple TDSA is simple and appealing and appealing

for applicationsfor applications

TDSA is simple TDSA is simple and appealing and appealing

for applicationsfor applications

window extracting only window extracting only NN samples samples from measured datafrom measured data

DrawbacksDrawbacks DrawbacksDrawbacks


8

k

s

def

kTtts )()( 0

k

s

def

kTtts )()( 0

)()()()()(

)(1

0

tycwsnTtyN

twtsta NN

N

nr

Ndef

N

)()()()()(

)(1

0

tycwsnTtyN

twtsta NN

N

nr

Ndef

N

TDSA like introduced by McFaddenTDSA like introduced by McFaddenTDSA like introduced by McFaddenTDSA like introduced by McFadden

t t

impulses train impulses train (ideal comb, uniform)(ideal comb, uniform)

TTrr must accurately be knownmust accurately be known

aaNN is not necessarily periodic, though is not necessarily periodic, though xx should be periodic should be periodic

ImprovedImproved model of util datamodel of util data

ImprovedImproved model of util datamodel of util data

Util data denoised and restricted to one periodUtil data denoised and restricted to one periodUtil data denoised and restricted to one periodUtil data denoised and restricted to one period

Signal compactedSignal compactedSignal compactedSignal compacted

DrawbacksDrawbacks DrawbacksDrawbacks the synchronization signal must accurately be known/acquiredthe synchronization signal must accurately be known/acquired

the method is impractical for asynchronous signals (not necessarily periodic)the method is impractical for asynchronous signals (not necessarily periodic)

k

k

def

ttts )()( 0

k

k

def

ttts )()( 0

)()()()()(

)(1

0

tycwsttyN

twtsta NN

N

nn

Ndef

N

)()()()()(

)(1

0

tycwsttyN

twtsta NN

N

nn

Ndef

N

t t

synchronization signalsynchronization signal (ideal comb, non necessarily uniform)(ideal comb, non necessarily uniform)

GeneralizedGeneralized model of util datamodel of util data

GeneralizedGeneralized model of util datamodel of util data

localization instants of comb teethlocalization instants of comb teeth

The The comb rulecomb rule works identically works identically. . The The comb rulecomb rule works identically works identically. .


9

Frequency effects of TDSAFrequency effects of TDSAFrequency effects of TDSAFrequency effects of TDSA

McFadden gave a frequency interpretation of TDSA by using the McFadden gave a frequency interpretation of TDSA by using the Continuous Fourier TransformContinuous Fourier Transform after extending its definition to a train of impulses. after extending its definition to a train of impulses.

But: But: But: But: more naturally is to operate with discrete signals (as the measured data are) more naturally is to operate with discrete signals (as the measured data are) and the and the Discrete Fourier TransformDiscrete Fourier Transform (DFT). (DFT).more naturally is to operate with discrete signals (as the measured data are) more naturally is to operate with discrete signals (as the measured data are) and the and the Discrete Fourier TransformDiscrete Fourier Transform (DFT). (DFT).

1

0

][][N

n

nkN

def

wnykY

1

0

][][N

n

nkN

def

wnykY

k k

1

0

][1

][N

k

knNwkY

Nny

1

0

][1

][N

k

knNwkY

Nny

n n

DirectDirect InverseInverse

N

jndefnN ew

2

N

jndefnN ew

2

n n

General caseGeneral caseGeneral caseGeneral case

)(][ s

def

nTyny )(][ s

def

nTyny

1

0

][1

][N

mm

def

N KnyN

na

1

0

][1

][N

mm

def

N KnyN

na

Measured dataMeasured dataMeasured dataMeasured data

n n

sTsT sampling periodsampling period

Synchronization signalSynchronization signalSynchronization signalSynchronization signal

1

00 ][][

N

mm

def

Knns

1

00 ][][

N

mm

def

Knns

number of samples per periodnumber of samples per period

00 unit impulseunit impulse

comb teeth localization instantscomb teeth localization instants

n n

Util data modelUtil data modelUtil data modelUtil data model

1,0 sNn 1,0 sNn

ss

nn

1NK 1NK00 K 00 K1K1K 2K2K mKmK...... ......

11

DFTDFTNNDFTDFTNN


10

Frequency effects of TDSAFrequency effects of TDSAFrequency effects of TDSAFrequency effects of TDSA

The DFT of average signal Na is expressed as an weighted average of DFTs applied on initial

data y , for one single harmonic period sK . More specifically:

sn KjpKN

nnN epY

NpA /2

1

0

][1

][

, 1,0 sKp , where

1

/2][][sn

n

s

KK

Kq

Kjqpdef

n eqypY , 1,0 sKp .

The DFT of average signal Na is expressed as an weighted average of DFTs applied on initial

data y , for one single harmonic period sK . More specifically:

sn KjpKN

nnN epY

NpA /2

1

0

][1

][

, 1,0 sKp , where

1

/2][][sn

n

s

KK

Kq

Kjqpdef

n eqypY , 1,0 sKp .

Theorem 1Theorem 1Theorem 1Theorem 1

InterpretationInterpretation InterpretationInterpretation

Step 1.Step 1. Segment the data intoSegment the data into NN successive framessuccessive frames withwith KKss

samples each, starting from each synchronization impulse. samples each, starting from each synchronization impulse.

Step 1.Step 1. Segment the data intoSegment the data into NN successive framessuccessive frames withwith KKss

samples each, starting from each synchronization impulse. samples each, starting from each synchronization impulse.

Step 2.Step 2. Compute the DFT of orderCompute the DFT of order KKss for each framefor each frame.. Step 2.Step 2. Compute the DFT of orderCompute the DFT of order KKss for each framefor each frame..

Step 3.Step 3. Average the DFTs by using some harmonic weights.Average the DFTs by using some harmonic weights. Step 3.Step 3. Average the DFTs by using some harmonic weights.Average the DFTs by using some harmonic weights.

Frames may overlap. Frames may overlap. They do not overlap for They do not overlap for uniform synchronization.uniform synchronization.

Frames may overlap. Frames may overlap. They do not overlap for They do not overlap for uniform synchronization.uniform synchronization.

AlgorithmAlgorithmAlgorithmAlgorithm

If the main harmonic of signal has a constant period (If the main harmonic of signal has a constant period (KKss), their (), their (NN) DFTs are ) DFTs are

quite similarquite similar and thus, by averaging them, a and thus, by averaging them, a noise reduction is expectednoise reduction is expected. .

If the main harmonic of signal has a constant period (If the main harmonic of signal has a constant period (KKss), their (), their (NN) DFTs are ) DFTs are

quite similarquite similar and thus, by averaging them, a and thus, by averaging them, a noise reduction is expectednoise reduction is expected. .

ss

nn

1NK 1NK00 K 00 K1K1K

2K2KmKmK...... ......

11

The The synchronization signalsynchronization signal plays a crucial role in averaging. plays a crucial role in averaging. Inappropriate synchronization leads to Inappropriate synchronization leads to noise amplificationnoise amplification. .

The The synchronization signalsynchronization signal plays a crucial role in averaging. plays a crucial role in averaging. Inappropriate synchronization leads to Inappropriate synchronization leads to noise amplificationnoise amplification. .

| Frame 1| Frame 1| Frame 2| Frame 2

| Frame 3| Frame 3| | ......

0

M

M

2

M

3

|DFT|DFTNN||

......1100 22 M-1M-1

......M

M )1(


11

compacted signalcompacted signal with with

support support

][&][][ nvnxny ][&][][ nvnxny 1,0 Nn 1,0 Nn

Measured data modelMeasured data modelMeasured data modelMeasured data model

1,0 P 1,0 P

unknown noiseunknown noise (hypotheses follow)(hypotheses follow)

Noise not necessarily Noise not necessarily additive. additive.

Noise not necessarily Noise not necessarily additive. additive.

NP NP

HH11 The DFTThe DFTNN of signal of signal yy is affected by a set of is affected by a set of

MM complex valued and additive sub-band complex valued and additive sub-band

noises noises VVmm with finite supports included into with finite supports included into

corresponding sub-bands. corresponding sub-bands.

HH11 The DFTThe DFTNN of signal of signal yy is affected by a set of is affected by a set of

MM complex valued and additive sub-band complex valued and additive sub-band

noises noises VVmm with finite supports included into with finite supports included into

corresponding sub-bands. corresponding sub-bands.

HypothesesHypothesesHypothesesHypotheses

Split the discrete spectrum of Split the discrete spectrum of yy into into

MM non overlapped sub-bands with non overlapped sub-bands with equal bandwidth and set .equal bandwidth and set .

Split the discrete spectrum of Split the discrete spectrum of yy into into

MM non overlapped sub-bands with non overlapped sub-bands with equal bandwidth and set .equal bandwidth and set .MKN MKN

KK

0V 1V 2V 1MV

Noises are Noises are orthogonal each otherorthogonal each other. .

HH22 The noises The noises VVmm are white Gaussian with null are white Gaussian with null

mean and unknown variances . mean and unknown variances .

HH22 The noises The noises VVmm are white Gaussian with null are white Gaussian with null

mean and unknown variances . mean and unknown variances . 2m

Here is the Here is the DFT model of measured dataDFT model of measured data: :

][][][ kVkAkmKY mm ][][][ kVkAkmKY mm

1,0 Mm 1,0 Mm 1,0 Kk 1,0 Kk

Here are the Here are the probability probability densities of noisesdensities of noises: : 1,0 Mm 1,0 Mm

1,0 Kk 1,0 Kk

How can the deterministic models How can the deterministic models AAmm be extracted from be extracted from YY ??

How can the deterministic models How can the deterministic models AAmm be extracted from be extracted from YY ??

Measured data not Measured data not necessarily periodic. necessarily periodic.

Measured data not Measured data not necessarily periodic. necessarily periodic.

stability domain stability domain of modelof model

With hypotheses H1 and H2, the MVM estimation is identical to Least Squares estimation, i.e.:

1

0

11

0

][][1

][][1ˆ

K

km

K

k

Tmmm kmKYk

Kkk

K , 1,0 Mm

1

0

22 ˆ][][

1ˆK

km

Tmm kkmKY

K , 1,0 Mm

With hypotheses H1 and H2, the MVM estimation is identical to Least Squares estimation, i.e.:

1

0

11

0

][][1

][][1ˆ

K

km

K

k

Tmmm kmKYk

Kkk

K , 1,0 Mm

1

0

22 ˆ][][

1ˆK

km

Tmm kkmKY

K , 1,0 Mm


12

IdeaIdeaIdeaIdea Use the Use the Maximum Verisimilitude MethodMaximum Verisimilitude Method (MVM). (MVM).Use the Use the Maximum Verisimilitude MethodMaximum Verisimilitude Method (MVM). (MVM).

mTmm kkA ][][ mTmm kkA ][][

m

m

ppmmmm kkkA ,1,0,][ m

m

ppmmmm kkkA ,1,0,][

]1[][ mpTm kkk ]1[][ mpTm kkk

][ ,1,0, mpmmmTm ][ ,1,0, mpmmmTm

)|(maxargˆmmm Y

mm

pS

)|(maxargˆmmm Y

mm

pS

Util data parametric modelUtil data parametric modelUtil data parametric modelUtil data parametric model1,0 Mm 1,0 Mm

1,0 Kk 1,0 Kk

(measured) data vector(measured) data vector

of length of length ppmm

parameters vectorparameters vector

(of length (of length ppmm ))

Linear, for Linear, for simplicity. simplicity.

Linear, for Linear, for simplicity. simplicity.

Example: polynomialExample: polynomialExample: polynomialExample: polynomial

MVM MVM optimization optimization

problemsproblems

MVM MVM optimization optimization

problemsproblems

parameters vector extended parameters vector extended with unknown variancewith unknown variance

2m

1,0]}[{

Kk

def

m kmKYY 1,0]}[{

Kk

def

m kmKYYdata segment in data segment in

sub-band sub-band mm density of conditional density of conditional probability between probability between data and parametersdata and parameters

Parameters should be set such Parameters should be set such that the measured data occur that the measured data occur withwith maximum probabilitymaximum probability, i.e. , i.e. withwith maximum verisimilitudemaximum verisimilitude..

Parameters should be set such Parameters should be set such that the measured data occur that the measured data occur withwith maximum probabilitymaximum probability, i.e. , i.e. withwith maximum verisimilitudemaximum verisimilitude..

1,0 Mm 1,0 Mm

Theorem 2 Theorem 2 (that solves the optimization problems)(that solves the optimization problems)Theorem 2 Theorem 2 (that solves the optimization problems)(that solves the optimization problems)

Example: Example: ppmm=0=0Example: Example: ppmm=0=0

1

0

][1ˆ

K

qm qmKY

KA

1

0

][1ˆ

K

qm qmKY

KA

1,0 Mm 1,0 Mm

simple averages simple averages of DFT data in of DFT data in sub-band sub-band mm

The nice The nice properties of LS properties of LS estimation are estimation are thus inherited. thus inherited.

The nice The nice properties of LS properties of LS estimation are estimation are thus inherited. thus inherited.

convergenceconvergence

accuracy of estimation accuracy of estimation (improves with (improves with K K ))

m

m

m

pp

pmmmm lL

Kl

L

KlA

1

1

1

1][ ,1,0, m

m

m

pp

pmmmm lL

Kl

L

KlA

1

1

1

1][ ,1,0,


13

1,0 Mm 1,0 Mm 1,0 Kk 1,0 Kk

1,0 Mm 1,0 Mm

1,0 Ll 1,0 Ll

How the MVM estimations can be employed to construct the compacted signal How the MVM estimations can be employed to construct the compacted signal ?? How the MVM estimations can be employed to construct the compacted signal How the MVM estimations can be employed to construct the compacted signal ??

General solutionGeneral solutionGeneral solutionGeneral solution ][ˆ][ˆ kAkmKX m ][ˆ][ˆ kAkmKX m Simple Simple concatenationconcatenation of MVM estimates of MVM estimates gives the DFT estimation of gives the DFT estimation of denoised signaldenoised signal..

CompressionCompression is achieved by is achieved by interpolationinterpolation of MVM of MVM estimates in a smaller number of spectral lines, say estimates in a smaller number of spectral lines, say L<K L<K . . NMKMLP

def

NMKMLPdef

Example: interpolation of polynomial modelExample: interpolation of polynomial modelExample: interpolation of polynomial modelExample: interpolation of polynomial model

1. Compute the frequency data )(yDFTY N .

2. Use MVM to estimate the deterministic models 1,0

}{ MmmA .

3. Perform the interpolation of each model 1,0

}{ MmmA in L equally spaced

spectral lines, with KL .4. Construct the PDFT of compacted signal by concatenation (where MLP ).

5. Apply the inverse PDFT to estimate the time values of compacted signal x̂ on

support 1,0 P .

1. Compute the frequency data )(yDFTY N .

2. Use MVM to estimate the deterministic models 1,0

}{ MmmA .

3. Perform the interpolation of each model 1,0

}{ MmmA in L equally spaced

spectral lines, with KL .4. Construct the PDFT of compacted signal by concatenation (where MLP ).

5. Apply the inverse PDFT to estimate the time values of compacted signal x̂ on

support 1,0 P .

Frequency Averaging AlgorithmFrequency Averaging AlgorithmFrequency Averaging AlgorithmFrequency Averaging Algorithm

• The resulted spectrum keeps the appearance The resulted spectrum keeps the appearance of original spectrum, but is smoother. of original spectrum, but is smoother.

)loglog( 322 MKPPNN )loglog( 322 MKPPNN operationsoperations


14

Advantages of FAMAdvantages of FAM Advantages of FAMAdvantages of FAM

• No No synchronization signalsynchronization signal is required. is required.

• Data can be Data can be periodic or notperiodic or not. .

• Non uniform splittingNon uniform splitting of signal bandwidth of signal bandwidth can lead to better resultscan lead to better results, especially when the signal , especially when the signal energy is concentrated only inside certain sub-bands. energy is concentrated only inside certain sub-bands.

Drawbacks of FAMDrawbacks of FAM Drawbacks of FAMDrawbacks of FAM

• More complex than TDSAMore complex than TDSA, though , though the complexity can be controlled by the userthe complexity can be controlled by the user. .

• Good accuracy is obtained for Good accuracy is obtained for data sets which are large enoughdata sets which are large enough. . This is the price paid for the absence of synchronization signal.This is the price paid for the absence of synchronization signal.

• Parameters Parameters NN, , MM and and KK should be set should be set as a result of a trade-offas a result of a trade-off. . On one hand: accuracy is bigger for a bigger number of spectral lines per sub-band (On one hand: accuracy is bigger for a bigger number of spectral lines per sub-band (KK). ).

On the other hand: the original spectrum is better “imitated” by the compacted one if On the other hand: the original spectrum is better “imitated” by the compacted one if the number of sub-bands is bigger (the number of sub-bands is bigger (MM), i.e. if the number of spectral lines per sub-band ), i.e. if the number of spectral lines per sub-band

is smaller, given the number of samples (is smaller, given the number of samples (NN). ).

it is not necessary to know the main period;it is not necessary to know the main period;If data are periodic: If data are periodic:

if the period known, the number of interpolation spectral lines (if the period known, the number of interpolation spectral lines (LL) ) can be set accordingly, to increase the accuracy;can be set accordingly, to increase the accuracy;

if the period is poorly estimated, the compacted signal will just lie if the period is poorly estimated, the compacted signal will just lie inside a support that is not divisible by the period;inside a support that is not divisible by the period;


15

Consequences of FAMConsequences of FAMConsequences of FAMConsequences of FAM

1. Construct the estimated DFT of noise (by using the orthogonality of sub-band noises, as consequence of hypothesis H1):

][ˆ][][ˆ][ˆ kAkmKYkVkmKV mm

def

, 1,0 Mm , 1,0 Kk .

2. Estimate a realization of noise v̂ , by applying the inverse NDFT on V̂ .

3. Estimate the noise variance:

1

0

22 ˆ][ˆ1ˆ

N

n

vnvN

, where

1

0

][ˆ1ˆ

N

n

def

nvN

v is the noise average.

4. Estimate the variance of util signal vyudef

ˆˆ :

1

0

22ˆ

ˆ][ˆ1 N

nu unu

N , where

1

0

][ˆ1ˆ

N

n

def

nuN

u is the signal average.

5. Compute the estimated SNR: 2

2ˆ

ˆˆ

uRNS .

1. Construct the estimated DFT of noise (by using the orthogonality of sub-band noises, as consequence of hypothesis H1):

][ˆ][][ˆ][ˆ kAkmKYkVkmKV mm

def

, 1,0 Mm , 1,0 Kk .

2. Estimate a realization of noise v̂ , by applying the inverse NDFT on V̂ .

3. Estimate the noise variance:

1

0

22 ˆ][ˆ1ˆ

N

n

vnvN

, where

1

0

][ˆ1ˆ

N

n

def

nvN

v is the noise average.

4. Estimate the variance of util signal vyudef

ˆˆ :

1

0

22ˆ

ˆ][ˆ1 N

nu unu

N , where

1

0

][ˆ1ˆ

N

n

def

nuN

u is the signal average.

5. Compute the estimated SNR: 2

2ˆ

ˆˆ

uRNS .

A procedure for SNR estimationA procedure for SNR estimationA procedure for SNR estimationA procedure for SNR estimation

In case of simple average models, the compacted signal can be estimated as follows:]0[]0[ˆ yx , if 0m .

1

0 1

][1][ˆ

K

kmN

kK

mM

ww

mkMy

K

wmx , if 1,1 Mm .

In case of simple average models, the compacted signal can be estimated as follows:]0[]0[ˆ yx , if 0m .

1

0 1

][1][ˆ

K

kmN

kK

mM

ww

mkMy

K

wmx , if 1,1 Mm .

Theorem 3 Theorem 3 (simple frequency average models)(simple frequency average models)Theorem 3 Theorem 3 (simple frequency average models)(simple frequency average models)

TDSATDSATDSATDSA

1

0

][1

][K

k

def

K mkMyK

ma

1

0

][1

][K

k

def

K mkMyK

ma Same comb Same comb rule. rule.

Same comb Same comb rule. rule.

)1)(58( MK )1)(58( MK

operations operations

(no interpolation (no interpolation is necessary)is necessary)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-300

-200

-100

0

Noisy sine wave spectrum

Normalized frequency

Ma

gnitu

de [

dB]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-300

-200

-100

0

Spectrum of compacted (frequency averaged) noisy sine wave


Ma

gnitu

de [

dB]

0 50 100 150 200 250 300 350 400 450 500-3

-2

-1

0

1

2


Normalized time

Ma

gnitu

de

Period: N = 500


0 50 100 150 200 250 300 350 400 450 500-3

-2

-1

0

1

2

Compacted (frequency averaged) noisy sine wave. SNR = 7.10313 dB.

Normalized time

Ma

gnitu

de

Period: M = 71


0 50 100 150 200 250 300 350 400 450 500-3

-2

-1

0

1

2


Normalized time

Ma

gnitu

de

Period: N = 500


0 50 100 150 200 250 300 350 400 450 500-3

-2

-1

0

1

2


Normalized time

Ma

gnitu

de

Period: M = 333


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-300

-200

-100

0



Ma

gnitu

de [

dB]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-300

-200

-100

0



Ma

gnitu

de [

dB]

(a)

(b)

(c)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-300

-200

-100

0



Ma

gnitu

de [

dB]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-300

-200

-100

0



Ma

gnitu

de [

dB]

0 50 100 150 200 250 300 350 400 450 500-3

-2

-1

0

1

2


Normalized time

Ma

gnitu

de

Period: N = 500


0 50 100 150 200 250 300 350 400 450 500-3

-2

-1

0

1

2


Normalized time

Ma

gnitu

de

Period: M = 71


0 50 100 150 200 250 300 350 400 450 500-3

-2

-1

0

1

2


Normalized time

Ma

gnitu

de

Period: N = 500


0 50 100 150 200 250 300 350 400 450 500-3

-2

-1

0

1

2


Normalized time

Ma

gnitu

de

Period: M = 333


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-300

-200

-100

0



Ma

gnitu

de [

dB]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-300

-200

-100

0



Ma

gnitu

de [

dB]

(a)

(b)

(c)

The SNR is not necessarily increasing for compacted signal. The SNR is not necessarily increasing for compacted signal. The SNR is not necessarily increasing for compacted signal. The SNR is not necessarily increasing for compacted signal. 16

Original signal and spectrumOriginal signal and spectrum Original signal and spectrumOriginal signal and spectrum

Toy example: a sine wave sunk into Gaussian noiseToy example: a sine wave sunk into Gaussian noiseToy example: a sine wave sunk into Gaussian noiseToy example: a sine wave sunk into Gaussian noise

Compacted signal and spectrum for M=333Compacted signal and spectrum for M=333 Compacted signal and spectrum for M=333Compacted signal and spectrum for M=333

Compacted signal and spectrum for M=71Compacted signal and spectrum for M=71Compacted signal and spectrum for M=71Compacted signal and spectrum for M=71

SNR SNR 6 dB ( 6 dB ( 33% 33% noise)noise)

SNR SNR 6 dB ( 6 dB ( 33% 33% noise)noise)


SNR SNR 4.7 dB 4.7 dBSNR SNR 4.7 dB 4.7 dB

SNR SNR 7.1 dB 7.1 dBSNR SNR 7.1 dB 7.1 dB

17


0 50 100 150 200 250 300 350 400 450 5004

5

6

7

8

9

10

11

12

13

Variaton of SNR for an averaged noisy sine wave

Averaged frame length

Sig

na

l-to

-No

ise

Ra

tio

[d

B]

Initial: SNR = 6.0206 dBMax: (M,SNR) = (12,13.0671)

Min: (M,SNR) = (333,4.73477)

0 50 100 150 200 250 300 350 400 450 5004

5

6

7

8

9

10

11

12

13



Sig

na

l-to

-No

ise

Ra

tio

[d

B]


Min: (M,SNR) = (333,4.73477)

0 50 100 150 200 250 300 350 400 450 5004

5

6

7

8

9

10

11

12

13



Sig

na

l-to

-No

ise

Ra

tio

[d

B]


Min: (M,SNR) = (333,4.73477)

0 50 100 150 200 250 300 350 400 450 5004

5

6

7

8

9

10

11

12

13



Sig

na

l-to

-No

ise

Ra

tio

[d

B]


Min: (M,SNR) = (333,4.73477)

Variation of SNR with the compacted support length of noisy sine waveVariation of SNR with the compacted support length of noisy sine waveVariation of SNR with the compacted support length of noisy sine waveVariation of SNR with the compacted support length of noisy sine wave

The trade-off between the data length and The trade-off between the data length and the number of sub-bands is important. the number of sub-bands is important.

The trade-off between the data length and The trade-off between the data length and the number of sub-bands is important. the number of sub-bands is important.

The spectral appearance of compacted signal is similar to the original one. The spectral appearance of compacted signal is similar to the original one. The spectral appearance of compacted signal is similar to the original one. The spectral appearance of compacted signal is similar to the original one. 18


0 5 0 1 0 0 1 5 0 2 0 0 2 5 0

- 0 . 0 5

0

0 .0 5

0 .1

A se gme nt of ra w vibra tion a cquire d from be a ring < B3 8 5 0 6 0 9 >

T ime [ms]

Acce

lerat

ion [c

m/s

2] V ibra tion le ngth: 8 0 9 . 3 5 ms > > >

* V a ria nce : 4 . 8 6 3 1 1 e - 0 0 8

* S a mpling ra te : 2 0 kHz

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0

- 0 . 0 5

0

0 .0 5

0 .1

C ompa cte d ( fre que ncy a ve ra ge d) vibra tion ( 4 full rota tions)

T ime ( ms)

Acce

lerat

ion [c

m/s

2] Le ngth: 9 0 . 1 5 ms

* V a ria nce : 6 . 0 2 6 2 e - 0 0 8

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0

- 0 . 0 5

0

0 .0 5

0 .1

A se gme nt of ra w vibra tion a cquire d from be a ring < B3 8 5 0 6 0 9 >

T ime [ms]

Acce

lerat

ion [c

m/s

2] V ibra tion le ngth: 8 0 9 . 3 5 ms > > >

* V a ria nce : 4 . 8 6 3 1 1 e - 0 0 8

* S a mpling ra te : 2 0 kHz

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0

- 0 . 0 5

0

0 .0 5

0 .1

C ompa cte d ( fre que ncy a ve ra ge d) vibra tion ( 4 full rota tions)

T ime ( ms)

Acce

lerat

ion [c

m/s

2] Le ngth: 9 0 . 1 5 ms

* V a ria nce : 6 . 0 2 6 2 e - 0 0 8

0 1 2 3 4 5 6 7 8 9

-6 0

-4 0

-2 0

0

2 0

4 0

R a w vibra tion spe ctrum

F re que ncy [kH z]

Spec

tral p

ower

[dB]

0 1 2 3 4 5 6 7 8 9

-6 0

-4 0

-2 0

0

2 0

4 0

S pe ctrum of compa cte d ( fre que ncy a ve ra ge d) ra w vibra tion

F re que ncy [kH z]

Spec

tral p

ower

[dB]

0 1 2 3 4 5 6 7 8 9

-6 0

-4 0

-2 0

0

2 0

4 0

R a w vibra tion spe ctrum

F re que ncy [kH z]

Spec

tral p

ower

[dB]

0 1 2 3 4 5 6 7 8 9

-6 0

-4 0

-2 0

0

2 0

4 0

S pe ctrum of compa cte d ( fre que ncy a ve ra ge d) ra w vibra tion

F re que ncy [kH z]

Spec

tral p

ower

[dB]

(a) (b)

Noisy vibration (a) and spectra (b) provided by a bearing in service (B3850609)Noisy vibration (a) and spectra (b) provided by a bearing in service (B3850609)Noisy vibration (a) and spectra (b) provided by a bearing in service (B3850609)Noisy vibration (a) and spectra (b) provided by a bearing in service (B3850609)

Estimated SNR Estimated SNR 3.27 dB 3.27 dBEstimated SNR Estimated SNR 3.27 dB 3.27 dB

Estimated SNR Estimated SNR 10.53 10.53 dBdB


About 4 full rotationsAbout 4 full rotations About 4 full rotationsAbout 4 full rotations

variable rotation variable rotation periodperiod due to a load due to a load and an incipient defectand an incipient defect

rotation period rotation period poorly estimatedpoorly estimated

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0

-0 .0 5

0

0 .0 5

0 .1A se gme nt of high pa ss filte re d vibra tion. Be a ring < B3 8 5 0 6 0 9 > .

T ime [ms]

Acce

lerati

on [c

m/s

2] V ibra tion le ngth: 7 5 8 .1 5 ms > > >

* Va ria nce : 3 .0 7 9 3 3 e -0 0 8

* S a mpling ra te : 2 0 kH z

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0

-0 .0 5

0

0 .0 5

0 .1C ompa cte d ( fre que ncy a ve ra ge d) filte re d vibra tion (4 full rota tions)

T ime (ms)

Acce

lerati

on [c

m/s

2] Le ngth: 9 0 .1 5 ms

* Va ria nce : 2 .5 2 5 6 2 e -0 0 8

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0

-0 .0 5

0

0 .0 5


T ime [ms]

Acce

lerati

on [c

m/s


* Va ria nce : 3 .0 7 9 3 3 e -0 0 8


0 5 0 1 0 0 1 5 0 2 0 0 2 5 0

-0 .0 5

0

0 .0 5


T ime (ms)

Acce

lerati

on [c

m/s

2] Le ngth: 9 0 .1 5 ms

* Va ria nce : 2 .5 2 5 6 2 e -0 0 8

0 1 2 3 4 5 6 7 8 9

-4 0

-2 0

0

2 0

4 0

F ilte re d vibra tion spe ctrum

F re que ncy [kH z]

Spec

tral p

ower

[dB]

0 1 2 3 4 5 6 7 8 9

-4 0

-2 0

0

2 0

4 0

S pe ctrum of compa cte d ( fre que ncy a ve ra ge d) filte re d vibra tion

F re que ncy [kH z]

Spec

tral p

ower

[dB]

0 1 2 3 4 5 6 7 8 9

-4 0

-2 0

0

2 0

4 0


F re que ncy [kH z]

Spec

tral p

ower

[dB]

0 1 2 3 4 5 6 7 8 9

-4 0

-2 0

0

2 0

4 0


F re que ncy [kH z]

Spec

tral p

ower

[dB]

(a) (b)

The spectrum of compacted signal still keeps the appearance of the original. The spectrum of compacted signal still keeps the appearance of the original. The spectrum of compacted signal still keeps the appearance of the original. The spectrum of compacted signal still keeps the appearance of the original. 19


Estimated SNR Estimated SNR 5.72 dB 5.72 dBEstimated SNR Estimated SNR 5.72 dB 5.72 dB



About 4 full rotations of About 4 full rotations of unfiltered vibrationunfiltered vibration

About 4 full rotations of About 4 full rotations of unfiltered vibrationunfiltered vibration

this is an this is an asynchronous signalasynchronous signal

Filtered vibration (a) and spectra (b) provided by a bearing in service (B3850609)Filtered vibration (a) and spectra (b) provided by a bearing in service (B3850609)Filtered vibration (a) and spectra (b) provided by a bearing in service (B3850609)Filtered vibration (a) and spectra (b) provided by a bearing in service (B3850609)

The The Frequency Averaging MethodFrequency Averaging Method based on maximum verisimilitude based on maximum verisimilitude can be employed whenever the synchronization signal is missing or can be employed whenever the synchronization signal is missing or poorly estimated. poorly estimated.

The The Frequency Averaging MethodFrequency Averaging Method based on maximum verisimilitude based on maximum verisimilitude can be employed whenever the synchronization signal is missing or can be employed whenever the synchronization signal is missing or poorly estimated. poorly estimated.

Is it possible to make a Is it possible to make a clear distinction clear distinction between the util data between the util data and the noise?and the noise?

How to extract the util How to extract the util data from a noisy signal?data from a noisy signal?

Usually Usually notnot, but it depends on how , but it depends on how the concept of the concept of “util data”“util data” is defined. is defined.

Usually Usually notnot, but it depends on how , but it depends on how the concept of the concept of “util data”“util data” is defined. is defined.

For example, with the help of For example, with the help of Time Domain Synchronous AveragingTime Domain Synchronous Averaging, , whenever a whenever a synchronization signalsynchronization signal accompanies the measured data. accompanies the measured data.

For example, with the help of For example, with the help of Time Domain Synchronous AveragingTime Domain Synchronous Averaging, , whenever a whenever a synchronization signalsynchronization signal accompanies the measured data. accompanies the measured data.

ConclusionConclusion

20

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1


Frequency [Hz]

Sp

ect

ral

po

we

r


N = 10N = 25

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1


Frequency [Hz]

Sp

ect

ral

po

we

r


N = 10N = 25

ss

nn

1NK 1NK00 K 00 K1K1K 2K2K

mKmK...... ......

11

0

M

M

2

M

3

|DFT|DFTNN||

......

1100 22 M-1M-1

......M

M )1(

0V 1V 2V 1MV

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-300

-200

-100

0



Mag

nitu

de [

dB

]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-300

-200

-100

0



Mag

nitu

de [

dB

]

0 50 100 150 200 250 300 350 400 450 500-3

-2

-1

0

1

2


Normalized time

Mag

nitu

de

Period: N = 500


0 50 100 150 200 250 300 350 400 450 500-3

-2

-1

0

1

2


Normalized time

Mag

nitu

de

Period: M = 71


0 50 100 150 200 250 300 350 400 450 500-3

-2

-1

0

1

2


Normalized time

Mag

nitu

de

Period: N = 500


0 50 100 150 200 250 300 350 400 450 500-3

-2

-1

0

1

2


Normalized time

Mag

nitu

de

Period: M = 333


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-300

-200

-100

0



Mag

nitu

de [

dB

]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-300

-200

-100

0



Mag

nitu

de [

dB

]

(a)

(b)

(c)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-300

-200

-100

0



Mag

nitu

de [

dB

]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-300

-200

-100

0



Mag

nitu

de [

dB

]

0 50 100 150 200 250 300 350 400 450 500-3

-2

-1

0

1

2


Normalized time

Mag

nitu

de

Period: N = 500


0 50 100 150 200 250 300 350 400 450 500-3

-2

-1

0

1

2


Normalized time

Mag

nitu

de

Period: M = 71


0 50 100 150 200 250 300 350 400 450 500-3

-2

-1

0

1

2


Normalized time

Mag

nitu

de

Period: N = 500


0 50 100 150 200 250 300 350 400 450 500-3

-2

-1

0

1

2


Normalized time

Mag

nitu

de

Period: M = 333


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-300

-200

-100

0



Mag

nitu

de [

dB

]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-300

-200

-100

0



Mag

nitu

de [

dB

]

(a)

(b)

(c)

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0

-0 .0 5

0

0 .0 5


T ime [ms]

Acce

lerati

on [c

m/s


* Va ria nce : 3 .0 7 9 3 3 e -0 0 8


0 5 0 1 0 0 1 5 0 2 0 0 2 5 0

-0 .0 5

0

0 .0 5


T ime (ms)

Acce

lerati

on [c

m/s

2] Le ngth: 9 0 .1 5 ms

* Va ria nce : 2 .5 2 5 6 2 e -0 0 8

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0

-0 .0 5

0

0 .0 5


T ime [ms]

Acce

lerati

on [c

m/s


* Va ria nce : 3 .0 7 9 3 3 e -0 0 8


0 5 0 1 0 0 1 5 0 2 0 0 2 5 0

-0 .0 5

0

0 .0 5


T ime (ms)

Acce

lerati

on [c

m/s

2] Le ngth: 9 0 .1 5 ms

* Va ria nce : 2 .5 2 5 6 2 e -0 0 8

0 1 2 3 4 5 6 7 8 9

-4 0

-2 0

0

2 0

4 0


F re que ncy [kH z]

Spec

tral p

ower

[dB]

0 1 2 3 4 5 6 7 8 9

-4 0

-2 0

0

2 0

4 0


F re que ncy [kH z]

Spec

tral p

ower

[dB]

0 1 2 3 4 5 6 7 8 9

-4 0

-2 0

0

2 0

4 0


F re que ncy [kH z]

Spec

tral p

ower

[dB]

0 1 2 3 4 5 6 7 8 9

-4 0

-2 0

0

2 0

4 0


F re que ncy [kH z]

Spec

tral p

ower

[dB]

(a) (b)

There always is There always is a part of noise treated as util dataa part of noise treated as util data and and a part of util data removed together with noisesa part of util data removed together with noises. .

There always is There always is a part of noise treated as util dataa part of noise treated as util data and and a part of util data removed together with noisesa part of util data removed together with noises. .

1.1.Cohen L. Cohen L. Time-Frequency AnalysisTime-Frequency Analysis, , Prentice Hall, New Jersey, USA, Prentice Hall, New Jersey, USA, 19951995. .

2.2.Ionescu F., Arotaritei D. Ionescu F., Arotaritei D. Fault Diagnosis of Bearings by Using Analysis of Vibrations Fault Diagnosis of Bearings by Using Analysis of Vibrations and Neuro-Fuzzy Classificationand Neuro-Fuzzy Classification, , Proceedings of ISMA’23 Conference, Leuven, Belgium, Proceedings of ISMA’23 Conference, Leuven, Belgium, September 16-18, 1998September 16-18, 1998. .

3.3.McFadden P.D. McFadden P.D. A Revised Model for the Extraction of Periodic Waveforms by Time A Revised Model for the Extraction of Periodic Waveforms by Time Domain AveragingDomain Averaging, , Mechanical Systems and Signal Processing, Vol. 1, No. 1, pp. 83-95,Mechanical Systems and Signal Processing, Vol. 1, No. 1, pp. 83-95, 19871987. .

4.4.McFadden P.D. McFadden P.D. Interpolation Techniques for Time Domain Averaging of Gear Interpolation Techniques for Time Domain Averaging of Gear VibrationVibration, , Mechanical Systems and Signal Processing, Vol. 3, No. 1, pp. 87-97,Mechanical Systems and Signal Processing, Vol. 3, No. 1, pp. 87-97, 19891989. .

5.5.Oppenheim A.V., Schafer R. Oppenheim A.V., Schafer R. Digital Signal ProcessingDigital Signal Processing, , Prentice Hall, New York, Prentice Hall, New York, USA, USA, 19851985. .

6.6.Proakis J.G., Manolakis D.G. Proakis J.G., Manolakis D.G. Digital Signal Processing. Principles, Algorithms and Digital Signal Processing. Principles, Algorithms and Applications.Applications., , Prentice Hall, New Jersey, USA, Prentice Hall, New Jersey, USA, 19961996. .

7.7.Söderström T., Stoica P. Söderström T., Stoica P. System IdentificationSystem Identification, , Prentice Hall, London, UK, Prentice Hall, London, UK, 19891989. .

8.8.Welch P.D. Welch P.D. The Use of Fast Fourier Transform for the Estimation of Power Spectra: The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging over Short Modified PeriodogramsA Method Based on Time Averaging over Short Modified Periodograms , , IEEE IEEE Transactions on Audio and Electroacoustics, Vol. AU-15, pp. 70-73, Transactions on Audio and Electroacoustics, Vol. AU-15, pp. 70-73, June 1967June 1967. .

1.1.Cohen L. Cohen L. Time-Frequency AnalysisTime-Frequency Analysis, , Prentice Hall, New Jersey, USA, Prentice Hall, New Jersey, USA, 19951995. .

2.2.Ionescu F., Arotaritei D. Ionescu F., Arotaritei D. Fault Diagnosis of Bearings by Using Analysis of Vibrations Fault Diagnosis of Bearings by Using Analysis of Vibrations and Neuro-Fuzzy Classificationand Neuro-Fuzzy Classification, , Proceedings of ISMA’23 Conference, Leuven, Belgium, Proceedings of ISMA’23 Conference, Leuven, Belgium, September 16-18, 1998September 16-18, 1998. .

3.3.McFadden P.D. McFadden P.D. A Revised Model for the Extraction of Periodic Waveforms by Time A Revised Model for the Extraction of Periodic Waveforms by Time Domain AveragingDomain Averaging, , Mechanical Systems and Signal Processing, Vol. 1, No. 1, pp. 83-95,Mechanical Systems and Signal Processing, Vol. 1, No. 1, pp. 83-95, 19871987. .

4.4.McFadden P.D. McFadden P.D. Interpolation Techniques for Time Domain Averaging of Gear Interpolation Techniques for Time Domain Averaging of Gear VibrationVibration, , Mechanical Systems and Signal Processing, Vol. 3, No. 1, pp. 87-97,Mechanical Systems and Signal Processing, Vol. 3, No. 1, pp. 87-97, 19891989. .

5.5.Oppenheim A.V., Schafer R. Oppenheim A.V., Schafer R. Digital Signal ProcessingDigital Signal Processing, , Prentice Hall, New York, Prentice Hall, New York, USA, USA, 19851985. .

6.6.Proakis J.G., Manolakis D.G. Proakis J.G., Manolakis D.G. Digital Signal Processing. Principles, Algorithms and Digital Signal Processing. Principles, Algorithms and Applications.Applications., , Prentice Hall, New Jersey, USA, Prentice Hall, New Jersey, USA, 19961996. .

7.7.Söderström T., Stoica P. Söderström T., Stoica P. System IdentificationSystem Identification, , Prentice Hall, London, UK, Prentice Hall, London, UK, 19891989. .

8.8.Welch P.D. Welch P.D. The Use of Fast Fourier Transform for the Estimation of Power Spectra: The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging over Short Modified PeriodogramsA Method Based on Time Averaging over Short Modified Periodograms , , IEEE IEEE Transactions on Audio and Electroacoustics, Vol. AU-15, pp. 70-73, Transactions on Audio and Electroacoustics, Vol. AU-15, pp. 70-73, June 1967June 1967. .

ReferencesReferences

21

Bodensee & Säntis (2542 m)Bodensee & Säntis (2542 m)

(Danny’s photo gallery)(Danny’s photo gallery)

Thank you!

Thank you!Dan Dan

StefanoiuStefanoiuDan Dan

StefanoiuStefanoiu

[email protected]@yahoo.com [email protected]@fh-konstanz.de

[email protected]@yahoo.com [email protected]@fh-konstanz.de

http://www.geocities.com/dandusus/Danny.htmlhttp://www.geocities.com/dandusus/Danny.html http://www.geocities.com/dandusus/Danny.htmlhttp://www.geocities.com/dandusus/Danny.html

Florin Florin IonescuIonescuFlorin Florin

IonescuIonescu

[email protected]@fh-konstanz.de [email protected]@fh-konstanz.de

maximum verisimilitude frequency averaging of signals

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