T-61.181 – Biomedical Signal Processing

1. T-61.181 – Biomedical Signal Processing Chapters 3.4 - 3.5.2 14.10.2004

2. Overview • Model-based spectral estimation • Three methods in more detail • Performance and design patterns • Spectral parameters • EEG segmentation • Periodogram and AR-based approaches

3. Model-based spectral analysis • Linear stochastic model • Autoregressive (AR) model • Linear prediction

4. Prediction error filter • Estimation of parameters based on minimization of prediction error ep variance

5. 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

6. Autocorrelation/covariance method • Straightforward minimization of error variance • Linear equations solved with Lagrange multipliers (constraint apTi=1)

7. 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

8. 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

9. Data matrices in detail

10. 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

11. 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

12. Burg’s method recursion steps

13. 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

14. 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

15. 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

16. Spectral parameters • Power, peak frequency and bandwidth • Complex power spectrum • Poles have a complex conjugate pair

17. 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

18. Partial fraction expansion, example

19. Power, frequency and bandwidth

20. 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

21. EEG segmentation principles • A reference window and a test window • Dissimilarity measure • Segment boundary where dissimilarity exceeds a predefined threshold

22. 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

23. 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

24. 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

25. 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)

26. 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