T-61.181 – Biomedical Signal Processing - CIS

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Nov 24, 2013 (3 years and 9 months ago)

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T
-
61.181


Biomedical Signal
Processing

Chapters 3.4
-

3.5.2


14.10.2004

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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Model
-
based spectral analysis


Linear stochastic model


Autoregressive (AR) model


Linear prediction

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Prediction error filter


Estimation of parameters based on
minimization of prediction error e
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variance

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


Straightforward minimization of error
variance




Linear equations solved with Lagrange
multipliers (constraint a
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Autocorrelation/covariance
method

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


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

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Data matrix

Data matrices in detail

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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

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

Burg’s method recursion steps

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

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

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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Spectral parameters


Power, peak frequency and bandwidth


Complex power spectrum





Poles have a complex conjugate pair





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Partial fraction expansion


Assumption of even
-
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Divide the transfer function H(z) into
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-
order transfer functions H
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(z)




No overlap between transfer functions


Partial fraction expansion,
example

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Power, frequency and
bandwidth

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

EEG segmentation principles


A reference window and a test window


Dissimilarity measure


Segment boundary where dissimilarity
exceeds a predefined threshold

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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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

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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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)

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