maximum verisimilitude frequency averaging of signals
DESCRIPTION
. Dan Stefanoiu Associate Professor. Florin Ionescu Professor. . . The multiconference on Computational Engineering in Systems Applications. July 9-11, 2003, Lille, FRANCE. Maximum Verisimilitude Frequency Averaging of Signals. [email protected]. [email protected]. - PowerPoint PPT PresentationTRANSCRIPT
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.
A data/signal (pre)processing paradigmA data/signal (pre)processing paradigm
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
Sine wave corrupted by Gaussian noise. SNR = 15.5928 dB.
Normalized time
Ma
gn
itu
de
Period: N = 500
* Variance: 0.697359
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
A data/signal (pre)processing paradigmA data/signal (pre)processing paradigm
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
A data/signal (pre)processing paradigmA data/signal (pre)processing paradigm
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
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
A data/signal (pre)processing paradigmA data/signal (pre)processing paradigm
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
parametricparametricparametricparametric
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
On Time Domain Synchronous Averaging (TDSA)On Time Domain Synchronous Averaging (TDSA)
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
Spectra of comb filters
Frequency [Hz]
Sp
ect
ral
po
we
r
Rotation frequency: 0.1 Hz
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
On Time Domain Synchronous Averaging (TDSA)On Time Domain Synchronous Averaging (TDSA)
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. .
On Time Domain Synchronous Averaging (TDSA)On Time Domain Synchronous Averaging (TDSA)
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
On Time Domain Synchronous Averaging (TDSA)On Time Domain Synchronous Averaging (TDSA)
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(
Noise hypotheses and Maximum VerisimilitudeNoise hypotheses and Maximum Verisimilitude
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
Noise hypotheses and Maximum VerisimilitudeNoise hypotheses and Maximum Verisimilitude
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,
The Frequency Averaging Method (FAM)The Frequency Averaging Method (FAM)
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
The Frequency Averaging Method (FAM)The Frequency Averaging Method (FAM)
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;
The Frequency Averaging Method (FAM)The Frequency Averaging Method (FAM)
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
Normalized frequency
Ma
gnitu
de [
dB]
0 50 100 150 200 250 300 350 400 450 500-3
-2
-1
0
1
2
Sine wave corrupted by Gaussian noise. SNR = 6.0206 dB.
Normalized time
Ma
gnitu
de
Period: N = 500
* Variance: 0.751142
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
* Variance: 0.665964
0 50 100 150 200 250 300 350 400 450 500-3
-2
-1
0
1
2
Sine wave corrupted by Gaussian noise. SNR = 6.0206 dB.
Normalized time
Ma
gnitu
de
Period: N = 500
* Variance: 0.751142
0 50 100 150 200 250 300 350 400 450 500-3
-2
-1
0
1
2
Compacted (frequency averaged) noisy sine wave. SNR = 4.73477 dB.
Normalized time
Ma
gnitu
de
Period: M = 333
* Variance: 0.295541
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
Normalized frequency
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
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
Normalized frequency
Ma
gnitu
de [
dB]
0 50 100 150 200 250 300 350 400 450 500-3
-2
-1
0
1
2
Sine wave corrupted by Gaussian noise. SNR = 6.0206 dB.
Normalized time
Ma
gnitu
de
Period: N = 500
* Variance: 0.751142
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
* Variance: 0.665964
0 50 100 150 200 250 300 350 400 450 500-3
-2
-1
0
1
2
Sine wave corrupted by Gaussian noise. SNR = 6.0206 dB.
Normalized time
Ma
gnitu
de
Period: N = 500
* Variance: 0.751142
0 50 100 150 200 250 300 350 400 450 500-3
-2
-1
0
1
2
Compacted (frequency averaged) noisy sine wave. SNR = 4.73477 dB.
Normalized time
Ma
gnitu
de
Period: M = 333
* Variance: 0.295541
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
Normalized frequency
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)
Simulation resultsSimulation results
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
Simulation resultsSimulation results
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
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
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
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)
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
Simulation resultsSimulation results
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
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
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 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 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]
(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
Simulation resultsSimulation results
Estimated SNR Estimated SNR 5.72 dB 5.72 dBEstimated SNR Estimated SNR 5.72 dB 5.72 dB
Estimated SNR Estimated SNR 12.38 12.38 dBdB
Estimated SNR Estimated SNR 12.38 12.38 dBdB
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
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
Spectra of comb filters
Frequency [Hz]
Sp
ect
ral
po
we
r
Rotation frequency: 0.1 Hz
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
Noisy sine wave spectrum
Normalized frequency
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
Spectrum of compacted (frequency averaged) noisy sine wave
Normalized frequency
Mag
nitu
de [
dB
]
0 50 100 150 200 250 300 350 400 450 500-3
-2
-1
0
1
2
Sine wave corrupted by Gaussian noise. SNR = 6.0206 dB.
Normalized time
Mag
nitu
de
Period: N = 500
* Variance: 0.751142
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
Mag
nitu
de
Period: M = 71
* Variance: 0.665964
0 50 100 150 200 250 300 350 400 450 500-3
-2
-1
0
1
2
Sine wave corrupted by Gaussian noise. SNR = 6.0206 dB.
Normalized time
Mag
nitu
de
Period: N = 500
* Variance: 0.751142
0 50 100 150 200 250 300 350 400 450 500-3
-2
-1
0
1
2
Compacted (frequency averaged) noisy sine wave. SNR = 4.73477 dB.
Normalized time
Mag
nitu
de
Period: M = 333
* Variance: 0.295541
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
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
Spectrum of compacted (frequency averaged) noisy sine wave
Normalized frequency
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
Noisy sine wave spectrum
Normalized frequency
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
Spectrum of compacted (frequency averaged) noisy sine wave
Normalized frequency
Mag
nitu
de [
dB
]
0 50 100 150 200 250 300 350 400 450 500-3
-2
-1
0
1
2
Sine wave corrupted by Gaussian noise. SNR = 6.0206 dB.
Normalized time
Mag
nitu
de
Period: N = 500
* Variance: 0.751142
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
Mag
nitu
de
Period: M = 71
* Variance: 0.665964
0 50 100 150 200 250 300 350 400 450 500-3
-2
-1
0
1
2
Sine wave corrupted by Gaussian noise. SNR = 6.0206 dB.
Normalized time
Mag
nitu
de
Period: N = 500
* Variance: 0.751142
0 50 100 150 200 250 300 350 400 450 500-3
-2
-1
0
1
2
Compacted (frequency averaged) noisy sine wave. SNR = 4.73477 dB.
Normalized time
Mag
nitu
de
Period: M = 333
* Variance: 0.295541
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
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
Spectrum of compacted (frequency averaged) noisy sine wave
Normalized frequency
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
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
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 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 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]
(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