Transcript Wiener Filtering & Basis Functions

T-61.181 Biomedical Signal Processing
Sections 4.4 - 4.5.2
Wiener Filtering &
Basis Functions
Nov 4th 2004
Jukka Parviainen
[email protected]
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
Nov 4th 2004
T-61.181 - Biomedical Signal
Processing - Jukka Parviainen
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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
Nov 4th 2004
T-61.181 - Biomedical Signal
Processing - Jukka Parviainen
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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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T-61.181 - Biomedical Signal
Processing - Jukka Parviainen
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Wiener filter in EEG
• a priori Wiener filter:
j
S s (e )
H (e ) 
j
j
S s (e )  (1 / M )Sv (e )
j
• power spectra of signal (s) and
noise (v) are F-transforms of
correlation functions r(k)
Nov 4th 2004
T-61.181 - Biomedical Signal
Processing - Jukka Parviainen
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Interpretation of Wiener
j
S
(
e
)
s
H (e j ) 
S s (e j )  (1 / M )Sv (e j )
• if ”no noise”, then H=1
• if ”no signal”, then H=0
• for stationary processes
always 0 < H < 1
• see Fig 4.22
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T-61.181 - Biomedical Signal
Processing - Jukka Parviainen
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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 < H < 1 implies
underestimation (bias)
• bias/variance dilemma
Nov 4th 2004
T-61.181 - Biomedical Signal
Processing - Jukka Parviainen
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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: 
S xa (e j )
M
H1 
(1 
)
j
M 1
MSsa (e )
Svs (e j )
H2  1
S sa (e j )
• improvements: clipping & spectral
smoothing, see Fig 4.23
Nov 4th 2004
T-61.181 - Biomedical Signal
Processing - Jukka Parviainen
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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
Nov 4th 2004
T-61.181 - Biomedical Signal
Processing - Jukka Parviainen
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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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T-61.181 - Biomedical Signal
Processing - Jukka Parviainen
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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.:
1, k  l
  
0, k  l
T
k l
Nov 4th 2004
T-61.181 - Biomedical Signal
Processing - Jukka Parviainen
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Orthonormal basis func.
• data is modelled using a set of
weight vectors and orthonormal
basic functions
N
xi   wi ,kk  w i
k 1
• example: Fourier-series/transform
x(t ) a0  a1 cos(t )  a2 cos(2t )  ...
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T-61.181 - Biomedical Signal
Processing - Jukka Parviainen
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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)
^
si   s w i   s  xi
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T
s
T-61.181 - Biomedical Signal
Processing - Jukka Parviainen
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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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T-61.181 - Biomedical Signal
Processing - Jukka Parviainen
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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?
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T-61.181 - Biomedical Signal
Processing - Jukka Parviainen
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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
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T-61.181 - Biomedical Signal
Processing - Jukka Parviainen
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