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

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

])([

)()()(

22

1

neE

knxanxne

pe

p

kkp

Prediction error filter Estimation of parameters based on

minimization of prediction error ep 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 ap

Ti=1)

pxTppe

pTpp

neE

nne

aRa

xa~

])([

)(~)(22

iaR 2~epx

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

pTpx XXR~~~

Data matrix

Data matrices in detail

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

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

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

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

Partial fraction expansion, example

)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

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

)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

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

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

1

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)

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