Wiener Filtering & Basis Functions

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

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 T-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 T-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) T-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) T-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 T-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(ej ) • filter gain 0 < H < 1 implies underestimation (bias) • bias/variance dilemma T-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 T-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 T-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” T-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.: T-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 T-61.181 - Biomedical Signal Processing - Jukka Parviainen

13. Lowpass modelling • basis functions divided to two sets, ”truncating” the model • sare to be saved, size N x K • vare to be ignored (regarded as high-freq. noise), size N x (N-K) T-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 T-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? T-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 T-61.181 - Biomedical Signal Processing - Jukka Parviainen