t-61.181 – biomedical signal processing chapters 3.4 - 3.5.2 14.10.2004
TRANSCRIPT
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T-61.181 – Biomedical Signal Processing
Chapters 3.4 - 3.5.2
14.10.2004
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Overview Model-based spectral estimation
Three methods in more detail Performance and design patterns Spectral parameters EEG segmentation
Periodogram and AR-based approaches
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H zA z a z a zpp p
p( )( ) ...
1 1
1 1
1
)()(...)1()( 1 nvpnxanxanx p
Model-based spectral analysis Linear stochastic model
Autoregressive (AR) model Linear prediction
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])([
)()()(
22
1
neE
knxanxne
pe
p
kkp
Prediction error filter Estimation of parameters based on
minimization of prediction error ep variance
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Estimation of model parameters Parameter estimation process
critical for the successful use of an AR model
Three methods presented Autocorrelation/covariance method Modified covariance method Burg’s method
The actual model is the same for all methods
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Straightforward minimization of error variance
Linear equations solved with Lagrange multipliers (constraint ap
Ti=1)
pxTppe
pTpp
neE
nne
aRa
xa~
])([
)(~)(22
iaR 2~epx
Autocorrelation/covariance method
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Levinson-Durbin recursion Recursive method for solving
parameters Exploits symmetry and Toeplitz
properties of the correlation matrix Avoids matrix inversion Parameters fully estimated at each
recursion step
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The correlation matrix can be directly estimated with data matrices
In covariance method the data matrix does not include zero padding, but the resulting matrix is not Toeplitz
In autocorrelation method the data matrix is zero-padded
pTpx XXR~~~
Data matrix
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Data matrices in detail
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iaRR 2)~( epxx
Modified covariance method Minimization of both backward and
forward error variances Parameters from forward and
backward predictors are the same Correlation matrix estimate not
Toeplitz so the forward and backward estimates will differ from each other
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Burg’s method Based on intensive use of
Levinson-Durbin recursion and minimization of forward and backward errors
Prediction error filter formed from a lattice structure
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Burg’s method recursion steps
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Performance and design parameters Choosing parameter estimation method
Two latter methods preferred over the first Modified covariance method
no line splitting might be unstable
Burg’s method guaranteed to be stable line splitting
Both methods dependant on initial phase
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Selecting model order Model order affects results significantly
A low order results in overly smooth spectrum
A high order produces spikes in spectrum Several criteria for finding model order
Akaike information criterion (AIC) Minimum description length (MDL) Also other criteria exist
Spectral peak count gives a lower limit
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Sampling rate Sampling rate influences AR
parameter estimates and model order
Higher sampling rate results in higher resolution in correlation matrix
Higher model order needed for higher sampling rate
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p
jjj
vvx
zdzdzAzA
zS
1
*1
2
1
2
)1)(1()()(
)(
*122 ii dd
Spectral parameters Power, peak frequency and
bandwidth Complex power spectrum
Poles have a complex conjugate pair
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2/
1
)()(p
ii zHzH
2/
1
2/
1
1221 )()()()()()(p
i
p
ixiivvx zSzHzHzHzHzSi
Partial fraction expansion Assumption of even-valued model
order Divide the transfer function H(z) into
second-order transfer functions Hi(z)
No overlap between transfer functions
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Partial fraction expansion, example
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)1(2
)cos2
1arccos(
))(
)(arctan(
)0(
)0(
2
'
ii
ii
ii
i
ii
x
ii
xi
r
r
r
d
d
r
PP
rPi
Power, frequency and bandwidth
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EEG segmentation Assumption of stationarity does
not hold for long time intervals Segmentation can be done
manually or with segmentation methods
Automated segmentation helpful in identifying important changes in signal
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EEG segmentation principles A reference window and a test
window Dissimilarity measure Segment boundary where
dissimilarity exceeds a predefined threshold
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Design aspects Activity should be stationary for at
least a second Transient waveforms should be eliminated
Changes should be abrupt to be detected Backtracking may be needed
Performance should be studied in theoretical terms and with simulations
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)0,0(),0(
))0,(),(()(
2
1xx
N
Nkxx
rnr
krnkrn
The periodogram approach Calculate a running periodogram
from test and reference window Dissimilarity defined as normalized
squared spectral error Can be implemented in time
domain
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The whitening approach Based on AR model Linear predictor filter “whitens” signal When the spectral characteristics
change, the output is no longer a white process
Dissimilarity defined similarly to periodogram approach The normalization factor differs
Can also be calculated in time domain
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1
0
2
3 )1)0,0(
)((
1)(
tN
k er
kne
Nn
r
t
t
r
N
k e
r
r
N
k e
t
t nr
ke
Nr
kne
Nn
1
2
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2
4 )1),0(
)((
1)1
)0,0(
)((
1)(
Dissimilarity measure for whitening approach
Dissimilarity measure asymmetric Can be improved by including a
reverse test by adding the prediction error also from reference window (clinical value not established)
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Summary Model-based spectral analysis
Stochastic modeling of the signal Is the signal an AR process?
Spectral parameters Quantitative information about the
spectrum EEG segmentation
Detect changes in signal