1. Wiener Filtering & Basis Functions Nov 4th 2004 Jukka Parviainen parvi@hut.fi T-61.181 Biomedical Signal Processing Sections 4.4 - 4.5.2

2. Nov 4th 2004T-61.181 - Biomedical Signal Processing - Jukka Parviainen 2 Outline a posteriori Wiener filter (Sec 4.4) –removing noise by linear filtering in optimal (mean-square error) way –improving ensemble averaging single-trial analysis using basis functions (Sec 4.5) –only one or few evoked potentials –e.g. Fourier analysis

3. Nov 4th 2004T-61.181 - Biomedical Signal Processing - Jukka Parviainen 3 Wiener - example in 2D model x = f(s)+v, where f(.) is a linear blurring effect (in the example) target: find an estimate s’ = g(x) an inverse filter to blurring value of SNR can be controlled Matlab example: ipexdeconvwnr

4. Nov 4th 2004T-61.181 - Biomedical Signal Processing - Jukka Parviainen 4 Part I - Wiener in EEG improving ensemble averages by incorporating correlation information, similar to weights earlier in Sec. 4.3 model: x_i(n) = s(n) + v_i(n) ensemble average of M records target: good s’(n) from x_i(n)

5. Nov 4th 2004T-61.181 - Biomedical Signal Processing - Jukka Parviainen 5 Wiener filter in EEG a priori Wiener filter: power spectra of signal (s) and noise (v) are F-transforms of correlation functions r(k)

6. Nov 4th 2004T-61.181 - Biomedical Signal Processing - Jukka Parviainen 6 Interpretation of Wiener if ”no noise”, then H=1 if ”no signal”, then H=0 for stationary processes always 0 < H < 1 see Fig 4.22

7. Nov 4th 2004T-61.181 - Biomedical Signal Processing - Jukka Parviainen 7 Wiener in theory design H(z), so that mean-square error E[(s(n)-s’(n))^2] minimized Wiener-Hopf equations of noncausal IIR filter lead to H(e j ) filter gain 0 < H < 1 implies underestimation (bias) bias/variance dilemma

8. Nov 4th 2004T-61.181 - Biomedical Signal Processing - Jukka Parviainen 8 A posteriori Wiener filters time-invariant a posteriori filtering estimates for signal and noise spectra from data afterwards two estimates in the book: improvements: clipping & spectral smoothing, see Fig 4.23

9. Nov 4th 2004T-61.181 - Biomedical Signal Processing - Jukka Parviainen 9 Limitations of APWFs contradictionary results due to modalities: BAEP+VEP ok, SEP not bad results with low SNRs, see Fig 4.24 APWF supposes stationary signals if/when not, time-varying Wiener filters developed

10. Nov 4th 2004T-61.181 - Biomedical Signal Processing - Jukka Parviainen 10 APWF - What was learnt? authors: ”serious limitations”, ”important to be aware of possible pitfalls”, especially when ”the assumpition of stationarity is incorporated into a signal model”

11. Nov 4th 2004T-61.181 - Biomedical Signal Processing - Jukka Parviainen 11 Part II - Basis functions often no repititions of EPs available or possible therefore no averaging etc. prior information incorporated in the model mutually orthonormal basis func.:

12. Nov 4th 2004T-61.181 - Biomedical Signal Processing - Jukka Parviainen 12 Orthonormal basis func. data is modelled using a set of weight vectors and orthonormal basic functions example: Fourier-series/transform

13. Nov 4th 2004T-61.181 - Biomedical Signal Processing - Jukka Parviainen 13 Lowpass modelling basis functions divided to two sets, ”truncating” the model  s are to be saved, size N x K  v are to be ignored (regarded as high-freq. noise), size N x (N-K)

14. Nov 4th 2004T-61.181 - Biomedical Signal Processing - Jukka Parviainen 14 Demo: Fourier-series http://www.jhu.edu/~signals/ rapid changes - high frequency value K? transients cannot be modelled nicely using cosines/sines

15. Nov 4th 2004T-61.181 - Biomedical Signal Processing - Jukka Parviainen 15 Summary I: Wiener originally by Wiener in 40’s with evoked potentials in 60’s and 70’s by Walker and Doyle lots of research in 70’s and 80’s (time-varying filtering by de Weerd) probably a baseline technique?

16. Nov 4th 2004T-61.181 - Biomedical Signal Processing - Jukka Parviainen 16 Summary II: Basis f. signal can be modelled using as a sum of products of weight vectors and basis functions high-frequency components considered as noise to be continued in the following presentation