Some comments on the use of Wiener filtering for the estimation of evoked potentials

533

Electroencephalography and Clinical Neurophysiology, 1975, 38:533-534 C) Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

TECHNICAL SOME THE

CONTRIBUTION

COMMENTS ESTIMATION

ON OF

THE

USE

EVOKED

OF

WIENER

FILTERING

FOR

POTENTIALS

D. J. DOYLE Department of Electrical Engineering, The University of British Columbia, Vancouver, B.C. (Canada) (Accepted for publication: December 6, 1974/

It has been observed by some researchers (Brazier 1967 ; Bennett 1968) that the technique of ensemble averaging (conventional averaging) for the estimation of evoked potentials has the disadvantage that relatively large sample sizes (N ~50-100) are needed in order to reduce the noise contaminating the evoked potential to an unobjectional level. In an attempt to develop an estimation technique with noise rejection characteristics superior to that of ensemble averaging, Walter (1969) has published a filtering method based on the Wiener theory (Wiener 1949). This filtering method seeks to minimize the mean square estimation error. Subsequent to the publication of this method, at least one researcher (Nogawa et al. 1973) has made use of Waiter's filtering technique for the estimation of the visual evoked potential (VEP). In Nogawa's paper it was observed that obvious differences between the Wiener filtered VEPs and the ensemble averaged VEPs could be seen to exist. Specifically, the Wiener filtered VEPs displayed a very smooth biphasic pattern and varied little from subject to subject, while the ensemble averaged VEPs were considerably less smooth. In addition, Nogawa's results showed that, on the basis of power spectral analysis, the Wiener filtered VEPs displayed very little spectral energy beyond 6 c/see while the ensemble averaged VEPs showed significant power well beyond that point. In explaining the origin of these anomalies, Nogawa suggests that "such differences might arise from the survival of background EEGs and various instrumental artifacts". It is the purpose of this note to show that the above mentioned anomalies are not specifically due to residual EEG activity or to instrumental artifacts, but rather were caused by an incorrect formulation of the Wiener filtering algorithm. Specifically, let 4)~ (oJ) be the power spectrum of the ensemble averaged EEGs and let ~,~ (~o) be the ensemble average of all the individual power spectra. Then the Wiener filter for use on the average VEP (based on N averages) should be given by H(0)) =

q),, ((o) i

where

(i)

N

1

~°.(0)t = ~,x(,Ol- 4's,(0)1 rather than by N N-I

1 4)xi((o) -- ~

H(0)) =

4)x,(0)) (2)

~xx(0))

as stated by both Walter and Nogawa. In fact, (2) is the correct Wiener filter for any individual VEP masked in noise, but (1) is the correct filter for the average VEP. The proof for this is as follows. Let cPn,(0)) be the power spectrum of the noise present in the unaveraged VEPs and let tPss(0)) be the power spectrum of the "true" (noiseless) VE P. Then prior to averaging the power spectrum for the noisy VEPs is given by

• x,((o) = ~,,(0))+ ~..(~o)

(3)

However, after N averages we get (Walter 1969, p. 67)

• .(0)) = ~,(~) + 1 ,.~ (0))

(4)

Consequently the effect of averaging the VEPs is to reduce the power spectrum of the contaminating noise from 4).. (0)) to 4~'n.(e))=(1/N)~.(0) ). Now the Wiener filter for the general case of a signal mixed in noise is (Lathi 1971, p. 259)

H(0))

~s,(0)) ~,,(0))+ ~°.((o)

(5)

Thus, since prior to averaging the VEPs we have (Walter 1969, p. 68) N 4),,(0)) = ~ - ~

4),~(0)) - ~

i

4~,,(0))

~,..((o) = ~..(~,)- ~,,(~o) then replacement of the term 4)..(0)) by 1 (]bnn(0))

(6)

(7)

534

D . J . DOYt.~.

in (7) gives us this result for the power spectra of the noise present in the averaged VEP: I

~'on(~)) = ~ (q'~(~O)-- q'~('"l)

(8)

Finally, substitution of (8) and (6) into (5) gives the result stated in ( 1). DISCUSSION Conceptually, we can consider the Wiener filter to be a filter which accentuates those frequency components of the input signal where the desired signal is strong and the noise is weak and suppressing those frequency components of the input signal in the range where the desired signal is weak and the noise is strong. Since in the occipital area of the head the ongoing EEG is usually strongest in the region beyond about 6 c/sec (Nogawa et al. 1973) the general effect of using 4~n°(~O)rather than (1/N)q~.~(eo) in the Wiener filter would be to over-accentuate those components of the VEP below 6 c/sec and remove effectively all components of the VEP beyond 6 c/sec. This explains the origin of the anomalies present in Nogawa's results. SUMMARY It is shown that a previous formulation of the Wiener filter for use in the estimation of evoked potentials (Walter 1969) is incorrect. The origin of anomalies thereby produced is discussed.

RESUMt-~ DISCUSSION S U R L ' U S A G E DU FILTRE I)E W I E N E R P O U R L'EST1MATION DES POTENTIELS EVOQUES Nous demontrons que le fittre de Wiener con~:u par Walter (1969) pour estimer les potentiels 6voqu6s est incorrect, Nous discutons l'origine des anomalies r6sultantes. The author would like m acknowledge helpful discussions with A. Davis of the Kinsman Laboratory, Division of Neurological Sciences, and T. Ulrych of the Department of Geophysics and Astronomy at the University of British Columbia.

REFERENCES BENNEFI, J. R. A studr tff the human evoked potenthd. M.A.Sc. Thesis, Department of Electrical Engineering. The University of British Columbia, December, 1968. BRAZIER, M. A. B. Varieties of computer analysis of electrophysiological potentials. Eleetroeneeph. elin. Neurophysiol.. 1967, Suppl. 26:1 8. LArHI, B. P. An introduction to random signals and communication theoo'. International Textbook Company, Scranton, Pa., 1971. NOGAWA, T.. KATAYAMA,K.. TABArA, Y., KAWAHARA, "I. and OHSHIO, T. Visual evoked potentials estimated by "'Wiener filtering". Electroenceph. clin. Neurophysiol.. 1973, 35:375 378. WALrER, D. O. A posteriori "Wiener filtering" of average evoked responses. Eleetroeneeph. clin. Neurophysiol.. 1969, Suppl. 27:61 70. WIENER, N. Extrapolation, interpolation and smoothin.q ~/ stationary time series. John Wiley and Sons, New York, ! 949.