Maximum Verisimilitude Frequency Averaging of Signals

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Maximum Verisimilitude Frequency Averaging of
Signals
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The multiconference on Computational Engineering
in Systems Applications
July 9-11, 2003, Lille, FRANCE
Danstef_at_fh-konstanz.de
Ionescu_at_fh-konstanz.de
Dandusus_at_yahoo.com
University of Applied Sciences, Konstanz, Germany
www.fh-konstanz.de
Department of Mechatronics
On leave from Politehnica University of
Bucharest, Romania
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www.pub.ro
Department of Automatic Control and Computer
Science
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? Research developed with the support of
Alexander von Humboldt Foundation, Germany
www.avh.de
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Headlines

• A data/signal (pre)processing paradigm

• On Time Domain Synchronous Averaging (TDSA)

• Noise hypotheses and Maximum Verisimilitude

• The Frequency Averaging Method (FAM)

• Simulation results

• Conclusion

References
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? A data/signal (pre)processing paradigm
stationary
data dominates the noise
Corrupting noise
Noisy/fractal signal
Util data
non stationary
noise dominates the data
? It might be a difficult Signal Processing
problem.
? Usually, it is difficult, if not impossible.
It depends tremendously on definition of util
data.
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? A data/signal (pre)processing paradigm
How the util data can be defined ?
Two properties are desirable

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

Partial or significant attenuation of redundancy
such that the extracted signal encodes almost the
same information but within a smaller number of
data samples.
Denoising
Compressing
Signal compaction
Problems

The rule of combination between the deterministic
data and the stochastic noise is usually
unknown.
? One uses the additive/superposition hypothesis
(which can fail for difficult signals such as
seismic, underwater acoustic or celestial).
? Noise is modeled by using the Theorem of
Central Limit.
Parametric
Non parametric
Any acquired data are affected by a certain
amount of Gaussian noise, usually white.

Even the combination between deterministic and
stochastic components is known, how to separate
them?
Mathematical models are required.
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? A data/signal (pre)processing paradigm
3 classes of signal compaction models
Time
Frequency
Time-frequency
In time domain
Interpolation models
Lagrange, Laguerre, Chebischev, Gauss, splines,
etc.
parametric
Compacted data provided by re-sampling with a
smaller sampling rate.
Noise only weakly attenuated, because models are
usually too fitted to data.
Least Squares (LS) models
based on experimental identification recipes
parametric
Models that fit the best to the data, not
necessarily maximally.
The more complex the model the better the
compaction performance.
Typical example a time series
? Simple models are preferred in pre-processing.
General trend (deterministic)

Polynomial, degree lt 7
Seasonal component (deterministic)

Elementary harmonics
Colored noise (stochastic)

Auto-regressive
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? A data/signal (pre)processing paradigm
3 classes of signal compaction models
Time
Frequency
Time-frequency
In time domain
Averaging models
based on Time Domain Synchronous Averaging
non-parametric
Described later.
In frequency domain
Spectral smoothing models
based on spectral estimation techniques
non-parametric
Smoothing the spectrum means removing some noise.
In general, complex models and methods.
Compacted signal difficult to provide because the
spectrum looses the phase information.
Averaging models
based on Maximum Verisimilitude DFT Averaging
parametric
New
Introduced within this presentation.
In time-frequency domain
Transformation models
Short Fourier Transform, Wavelet Transform,
Wigner-Ville Transform, etc.
parametric
Suitable for non stationary data sets (with
spectrum variable in time).
Complex models and methods rather inappropriate
if only pre-processing is wanted.
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? On Time Domain Synchronous Averaging
(TDSA)
? Originates from early works in Signal
Processing, such as Welch method of spectral
estimation (1967).
? Devised by P.D. McFadden in 1987.
TDSA like introduced by McFadden
How can x be extracted from y ?
Measured data model
Exploit the known periodicity.
Idea
harmonic signal with known/measurable period
unknown noise (with null average)
Util data model
Time averaging of measured data
Comb filter
Dirac impulse
Fourier Transform
Slide the comb along the data and average only
the samples pointed by its teeth.
Comb rule
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? On Time Domain Synchronous Averaging
(TDSA)
TDSA like introduced by McFadden
? Tr must accurately be known

• Drawbacks

? aN is not necessarily periodic, though x
should be periodic
Improved model of util data
window extracting only N samples from measured
data
impulses train (ideal comb, uniform)
Util data denoised and restricted to one period
Signal compacted
Generalized model of util data
TDSA is simple and appealing for applications
? The comb rule works identically.
localization instants of comb teeth
synchronization signal (ideal comb, non
necessarily uniform)
? the synchronization signal must accurately be
known/acquired

• Drawbacks

? the method is impractical for asynchronous
signals (not necessarily periodic)
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? On Time Domain Synchronous Averaging
(TDSA)
Frequency effects of TDSA
? McFadden gave a frequency interpretation of
TDSA by using the Continuous Fourier Transform
after extending its definition to a train of
impulses.
But
more naturally is to operate with discrete
signals (as the measured data are) and the
Discrete Fourier Transform (DFT).
Direct
Inverse
DFTN
General case
? sampling period
Measured data
? number of samples per period
Synchronization signal
? unit impulse
Util data model
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comb teeth localization instants
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? On Time Domain Synchronous Averaging
(TDSA)
Frequency effects of TDSA
Theorem 1
Algorithm
...
Step 1. Segment the data into N successive frames
with Ks samples each, starting from each
synchronization impulse.
Frame 3
Frame 2
Frame 1
Step 2. Compute the DFT of order Ks for each
frame.
Step 3. Average the DFTs by using some harmonic
weights.
? Frames may overlap. They do not overlap for
uniform synchronization.

• Interpretation

If the main harmonic of signal has a constant
period (Ks), their (N) DFTs are quite similar and
thus, by averaging them, a noise reduction is
expected.
The synchronization signal plays a crucial role
in averaging. Inappropriate synchronization leads
to noise amplification.
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? Noise hypotheses and Maximum Verisimilitude
? Measured data not necessarily periodic.
Measured data model
compacted signal with support
unknown noise (hypotheses follow)
? Noise not necessarily additive.
Hypotheses
H1 The DFTN of signal y is affected by a set
of M complex valued and additive sub-band
noises Vm with finite supports included into
corresponding sub-bands.
? Noises are orthogonal each other.
? Here is the DFT model of measured data
How can the deterministic models Am be
extracted from Y ?
? Here are the probability densities of noises
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? Noise hypotheses and Maximum Verisimilitude
Idea
Use the Maximum Verisimilitude Method (MVM).
Util data parametric model
Example polynomial
? Linear, for simplicity.
(measured) data vector of length pm
parameters vector (of length pm )
MVM optimization problems
data segment in sub-band m
stability domain of model
density of conditional probability between data
and parameters
? Parameters should be set such that the measured
data occur with maximum probability, i.e. with
maximum verisimilitude.
Example pm0
simple averages of DFT data in sub-band m
Theorem 2 (that solves the optimization problems)
? The nice properties of LS estimation are thus
inherited.
? convergence
? accuracy of estimation (improves with K )
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? The Frequency Averaging Method (FAM)
How the MVM estimations can be employed to
construct the compacted signal ?
General solution
? Simple concatenation of MVM estimates gives the
DFT estimation of denoised signal.

The resulted spectrum keeps the appearance of
original spectrum, but is smoother.
? Compression is achieved by interpolation of MVM
estimates in a smaller number of spectral lines,
say LltK .
Example interpolation of polynomial model
Frequency Averaging Algorithm
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? The Frequency Averaging Method (FAM)

• Advantages of FAM

No synchronization signal is required.

Data can be periodic or not.
If data are periodic
? it is not necessary to know the main period
? if the period known, the number of
interpolation spectral lines (L) can be set
accordingly, to increase the accuracy
? if the period is poorly estimated, the
compacted signal will just lie inside a support
that is not divisible by the period

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

• Drawbacks of FAM

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

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

Parameters N, M and K should be set as a result
of a trade-off.
? On one hand accuracy is bigger for a bigger
number of spectral lines per sub-band (K).
? On the other hand the original spectrum is
better imitated by the compacted one if the
number of sub-bands is bigger (M), i.e. if the
number of spectral lines per sub-band is smaller,
given the number of samples (N).
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? The Frequency Averaging Method (FAM)
Consequences of FAM
A procedure for SNR estimation
Theorem 3 (simple frequency average models)
? Same comb rule.
TDSA
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? Simulation results
Toy example a sine wave sunk into Gaussian noise
SNR ? 6 dB (? 33 noise)
Original signal and spectrum
SNR ? 4.7 dB
Compacted signal and spectrum for M333
SNR ? 7.1 dB
Compacted signal and spectrum for M71
? The SNR is not necessarily increasing for
compacted signal.
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? Simulation results
Variation of SNR with the compacted support
length of noisy sine wave
? The trade-off between the data length and the
number of sub-bands is important.
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? Simulation results
Noisy vibration (a) and spectra (b) provided by a
bearing in service (B3850609)
Estimated SNR ? 3.27 dB
? variable rotation period due to a load and an
incipient defect
Estimated SNR ? 10.53 dB
? rotation period poorly estimated
About 4 full rotations
? The spectral appearance of compacted signal is
similar to the original one.
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? Simulation results
Filtered vibration (a) and spectra (b) provided
by a bearing in service (B3850609)
Estimated SNR ? 5.72 dB
Estimated SNR ? 12.38 dB
? this is an asynchronous signal
About 4 full rotations of unfiltered vibration
? The spectrum of compacted signal still keeps
the appearance of the original.
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? Conclusion
How to extract the util data from a noisy signal?
For example, with the help of Time Domain
Synchronous Averaging, whenever a synchronization
signal accompanies the measured data.
The Frequency Averaging Method based on maximum
verisimilitude can be employed whenever the
synchronization signal is missing or poorly
estimated.
Is it possible to make a clear distinction
between the util data and the noise?
Usually not, but it depends on how the concept of
util data is defined.
There always is a part of noise treated as util
data and a part of util data removed together
with noises.
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References
1. Cohen L. ? Time-Frequency Analysis, Prentice
Hall, New Jersey, USA, 1995. 2. Ionescu F.,
Arotaritei D. ? Fault Diagnosis of Bearings by
Using Analysis of Vibrations and Neuro-Fuzzy
Classification, Proceedings of ISMA23
Conference, Leuven, Belgium, September 16-18,
1998. 3. McFadden P.D. ? A Revised Model for
the Extraction of Periodic Waveforms by Time
Domain Averaging, Mechanical Systems and Signal
Processing, Vol. 1, No. 1, pp. 83-95, 1987.
4. McFadden P.D. ? Interpolation Techniques for
Time Domain Averaging of Gear Vibration,
Mechanical Systems and Signal Processing, Vol. 3,
No. 1, pp. 87-97, 1989. 5. Oppenheim A.V.,
Schafer R. ? Digital Signal Processing, Prentice
Hall, New York, USA, 1985. 6. Proakis J.G.,
Manolakis D.G. ? Digital Signal Processing.
Principles, Algorithms and Applications.,
Prentice Hall, New Jersey, USA, 1996.
7. Söderström T., Stoica P. ? System
Identification, Prentice Hall, London, UK, 1989.
8. Welch P.D. ? The Use of Fast Fourier
Transform for the Estimation of Power Spectra A
Method Based on Time Averaging over Short
Modified Periodograms, IEEE Transactions on Audio
and Electroacoustics, Vol. AU-15, pp. 70-73, June
1967.
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Thank you!
Dan Stefanoiu
Florin Ionescu

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