Wiener Filtering

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Wiener Filtering Basis Functions
T-61.181 Biomedical Signal Processing Sections
4.4 - 4.5.2

• Nov 4th 2004
• Jukka Parviainen
• parvi_at_hut.fi

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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

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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

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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)

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Wiener filter in EEG

• a priori Wiener filter

• power spectra of signal (s) and noise (v) are
  F-transforms of correlation functions r(k)

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Interpretation of Wiener

• if no noise, then H1
• if no signal, then H0
• for stationary processes always 0 lt H lt 1
• see Fig 4.22

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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 lt H lt 1 implies underestimation
  (bias)
• bias/variance dilemma

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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

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Limitations of APWFs

• contradictionary results due to modalities
  BAEPVEP ok, SEP not
• bad results with low SNRs, see Fig 4.24
• APWF supposes stationary signals
• if/when not, time-varying Wiener filters developed

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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

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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.

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Orthonormal basis func.

• data is modelled using a set of weight vectors
  and orthonormal basic functions
• example Fourier-series/transform

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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)

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Demo Fourier-series

• http//www.jhu.edu/signals/
• rapid changes - high frequency
• value K?
• transients cannot be modelled nicely using
  cosines/sines

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Summary I Wiener

• originally by Wiener in 40s
• with evoked potentials in 60s and 70s by Walker
  and Doyle
• lots of research in 70s and 80s (time-varying
  filtering by de Weerd)
• probably a baseline technique?

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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