# T-61.181 – Biomedical Signal Processing - CIS

AI and Robotics

Nov 24, 2013 (4 years and 7 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
p

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
p
T
i=1)

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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
-

p
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Data matrix

Data matrices in detail

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R
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p
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x

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
-
valued model order

Divide the transfer function H(z) into
second
-
order transfer functions H
i
(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

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