T-61.181

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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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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 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
• Burgs method
• The actual model is the same for all methods

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Autocorrelation/covariance method

• Straightforward minimization of error variance
• Linear equations solved with Lagrange multipliers
  (constraint apTi1)

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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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Data matrix

• 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 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

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Burgs 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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Burgs 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
• Burgs 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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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 Hi(z)
• No overlap between transfer functions

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Partial fraction expansion, example
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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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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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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