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Estimation of Evoked Potential by Wiener Filtration
69 ESTDI4T.ION OF EVOKED POni:NTUL BY VIElIER nLTlU.T.ION
V. Albreoht, T. Xn.titute o~ Phy.io~ogy, Prague, Czecho.~ovakia
Radi~-Vei••
Czeoho.~ovak
Aoademy of Soienoe.,
'lbe Viener ~i~tration o~ a .tationary .igna~ from. a .i~ture oompo.ed of the .isua~ p~us a .tationary noi.e ha. an app~ioation when e~traoting rea~ evoked potenti.~ from po.t.timulation EEG reoord.. 'lbe Vi.ner e~tr.otion b.oame a oo~titive .ethod with the .imp~e .veraging. Both methods, i. e. Viener fi~tr.tion and .veraging, are oompared with respeot to .~ucidade re~.tion. between resu~t. obtained by the•• Sum-eql
:u
i.
we~~
known that the Viener
fi~tration
of a .t.tionary
.isua~ whioh i . additive~y di.turbed by an independent .tationary noise .upp~ie. the estimate the mean square error of whioh is • • •_~~ a. po•• ib~e. 'lberefore the use of this method
i. very oomprehensive in the ·rea~·
evoked
potenti.~.
prob~em
of e.timation of the
One may .t.te that this method, origi-
nated by V.1ter /9/, ha. got great acceptanoe /2,3,5,6,7/. On the other hand, the Viener fi~ter bring. more oomp1icat.d data prooe.sing than th.~1a•• ica~· averaging doe•• Furthermor., the aocuracy of the r ••ult. obt.ined by both, Vi.ner fi~tration
interva~.
fo~~owing
It i.
and averaging, d.pends on the interstimu1ation
u.ed. To di.oov.r this depend.nce we use the mod.1 o~ .voked potenti~ generation.
u.ua~~y
suppo.ed that the reoorded EEG .ignaJ., "e .ha11
denote it ~(t), con.i.t. of two .tati.tio~1:y indep.ndent component.1 the prop.r .vok.d potentia1 r(t) and the ~pontaneous EEG activity n(t), i. e. (1) ~(t) . For the purpo••
= o~
r(t) + n(t) • the Viener fi1tr.tion "e suppo.e that both
r(t) and n(t) are .t.tionary prooe•••• with z.ro e~ot.tion•• 'lbe .~.ri••nta1 objeot ie·N-U. . . . . timulated by the ..... • timu1us. 'lb. EEG .isua~ (1) i . reoorded during the time T
70 after .ti.uJ.ation. We .ha~~ denote ~ the t1llle duration betwea the end ot the j-th reoord and the app~ioation ot the j+~-th .tu.u1ue. 'lbe ezper1lllent i. u._~1y arranced in .uoh a way . that al.1. the interva1.. 9" are _ch ~oager than the t1llle duration ot the reoord T. 'lbu. the j-th record oan be written in the :torm j_~ Zj(t) = r(t) + n(t + (j-~)T +;E:0k).
(2)
k=~
'lbe prooe••
(:l)
N
~
~(t) = 7
2:
Zj(t)
j=~
j-~
N
~
= r(t) + N
E
n(t +
j=1.
(j-~)T +
E
k=~
9 k)
i. oa1.~ed average evoked potentia~. Xt :to~1.ow. :tro• • tationary a ••u.ption. that a1.1 prooe••e. Zj(t), r(t) and n(t) have a oertain .peotra~ repreaentation (we auppo.e diaor.te ti•• haVing in .ind the .amp1.ing tor. o:t these prooe.aes), i. e• ."
=
J.-a~'tz". (t:I),,),
-~r
'
r,( 10)
= J.-U.tZ~(d)'),
n(t)
=
('.-alto z" (d") -f1
(aee e. s. /4/~ ,where Zz is a stooha.tio proo.ss with zero expeotations and indeP~ndent 1nore. .nts :tor which E
(I Zz/d" >.11.1-
:tz /
~
),
where t ( ~) b the speotral. density o:t Z (10) aDd B 18 the operato:jo:t enae.b1.e averaglDc (ezpeotationj.'lbe . . .e oan be repeated·:tor the re. . inias zr aDd zn prooe••e. ao that we have
71 .
vh.r. ~r ( ~ ) and 1'n ( ,.) are the .pectral. del181.ti•• 01' r (t) and net) r ••p.ctiv.1y. 7h. proc••••• zr and zn are .tat1..tica11y iDdependent 1.n cOI18.quenc. 01' the 1.nd.pend.nc. 01' r(t) end n( t). V1.th r ••p.ct to (2)v. can imaediat.1y vrit. r " i·. u (lil) '·U Z,.(dl) + "-'').!(i-1)r·.~e..]'Z,,(t1IA). • Zaj
f ..
• Je
-.
-r
fe-a.!.
o
-"
.
7hi.. 1'orlllula y1.e1de the r.1at1.on b.tve.n .pectral. d.n.1.tie.
1'r ( ~ ), f n ( .,.) and ~ '"
(~) = 1'r
1'",
J
(
~)
,.1. ~
+
(4)
l'
"'J
the ••_ (5)
Z
j..
if" -l)T +.~ 9,./ . l'n ( 1-) I
£0"]12. 1e -i 1lej-.) T. •••
but By
(1-). Ve have "
(~)
= l'
r
1
:a
-
and therefore
(~).
+ l'
(')
n
vay ve get the spectr.l r.pr••ent.tioD o~ ~(t)
~(t) '"
J"e-ilfz.'N
(cl1-)
-lr
.,.
•
f • .•
-.
vMoh 1.. • cOl18equence
It.
/.
z,.la~
o~
)
+
( :}). 7h1.. l' orlllu1a i-.>U.. that '"
,jo.
j.. ~ -i"[(j .• )r+rl\J,~f'.(?o) ( ,. ) + / N~e. .••• ". Jot
vllere ~~ (1.) 18 the .pectral c1endty o~ tile ~ (t). For the .ake 01' abbr••iat1.on v. eha1l denote
(7)
J~",(1o>J1.
I~
N
;-f
.~e·"~!(J·o")T·t~·ll2.. 3"
72 As
Ihw(?)\~i~, ve oan summarize the .eohani.sm of averag:lng
as
fo~~ovs:·
of
~:iDear fi~ter
Tbe averag:lng of EEG
s:lgna~s
presents some kiDd
that passes the frequenoy oomponents of r(t,
vithout any attenuation whereas the frequenoy oomponents of n( t) are suppressed proport:lona~~y to
l:z. •
\!Ix("')
It fo~~ows from Ruohi::lD's paper /8/that a ooncrete form of
I !Ix (
")
:iDterva~s
I z.
stantia~~y
by Wa~ter
depends very stroDg~y on the ohoioe of the
between
stim~ations.
Th:ls function oan differ sub.
from the oonstant va~ue ~/N, vhich was proposed
/9/, even though ve do not work with
m~ation :iDterva~s. No . .tter how the
equa~
:iDtersti
SJ's are ohosen, it il
abays \ !Ix ( 0 ~~, henoe the di1"ferenoe of ~ ~ >I1-rom ~/N must be at ~east considered :iD oertain ne:lshbourhood of ~ = (
I (
Tbe frequenoy tran_fer funotion of the Yiener
fi~t.r
for
estimation r(t) from average evoked potential (,) has the form f
(8)
r ( 1-)
= -----------f,.(~) + 1!Ix(Q11fn (?)
(see e. g.
/4i).
Bovever, we know neither f r ( "), nor f n ( ?). spe~tra~ densities oan be obtained fro.
Estimators of these
soJ.ution of the equations
where
~
=N
and
So~v:lag
(11)
these equations for f r ('1.) and l'n ( 1) ve p t _
'" f
f(?)-f;v()
-..
n ( ,.) .. - - - - - - - -
~
- l!lx (,,) \:z.
73 fiN( ~) -
'\
(12)
Ibx(
~)\2. i( ~)
=
f r ( ~)
l-Ibx(~)ll
Substitution of these estimators into (8) and several manipu- . lations yield
(13)
whioh ie aD estimator of the Yiener frequency transfer funotion based on i ( ~) and l'- ( ~ )'. 'lbe Yiener fil.tered 3u.mate of r(t) fro. iJr(t) in the fr~uenoy doaaiD has the fora (l.4 )
Xx(
where ~) is the Fourier tranafora of AEP XX(t). '!be inverse '\ Fourier transfora of Y ( 1- ). ve denote i t w ( e). then yiel4a H H the Yiener esttaate of r(t) in tt.e doaaiD. '!be rel.ation between the average eVOked potential. and the Yieuer eattaate can be judged fro. the e%pressiona oODoerDill4r the errora of these estt.ators. '!be _an square error of the averaeevoked potential. is given by (l.5)
E-'H
=
E
(~(t) - r(t»2 J =
-r
SIbx(
-.
'1 )12 f n (
~ )d~
and the lDean aquare error of the Yiener fil.tar ia atated in the foll.owlDc foraula T f r ( -:l ) (l.6) E (CwH(t) - r(t» 2} = -r fr(~) + lhx(~ffn(~)
f
·fhx( ,,) I 2 f n ( 1)
41.
74 For siven .~otr.1 denaitie. o~ the re.ponae III1d the noi.e, both error. depend on the n_ber o~ .tiau1i repetition Jl' and on the inter.tUlu1ation interva1. na84. (i). Xt the inter.timn1ation interva1. are obo.en acoordins to a .tztaten .uob that 1.iJa
Jl' ~
then
BY.
=
B-'JJ
=
0
0 and 00naequent1y the
~~erenoe
E-'JJ-BY• •a-
niabe. with srow!Ds Ji, i. e. co-.pared to averasins, the sreater
accuraoy f4 the Yiener ~i1ter e2;pire. with srowins na-ber o~ .timu1i repetition~ (ii). Jiow we 01111 a.k whether the error o~ the Yiener ~i1ter OaD vaniab un1e.. the error o~ the a.erase doe•• :It i. apparent that the error o~ t¥ Yiener ~i1ter i. zero U ~r(") • ~n(~) = 0 ~Or a11 ~, i. e. U the .et o~ ~requen. oie. ~or whiob ~r ( ~ ) > 0 and the .et where ~D ( ,,) > 0 are JaUtua11y eJt01uaive. ']hi. oa.e 01111 be exa-p1Uied by the .i-.p1e.t periodioa1 _ode1. Let us .uppo.e that the .pontaneou. BBG aotivity i. a .tationary prooe•• with a periodioa1 oovariaDoe ~UI1otion.
~(~ ) = 0;:-00. ~t. o~ the averase i. siveD by the
'1ben the error
1
B.A..._
-.=
2
N
~ormu1a
..... 1
N-I&
cr.:~(- + - 2 ~I:" "Ji Ji 11., .. 4
00. ),,(kT +
E
Bp»
/#ai
2r
~ro_
whiob it ~011ow. that U i • •_ inte. .r, then
'I' + 9 i
= -i--:r-n
' where -i
Bowever, U .e 01111 .Uppo.e that the rea1 evOked potentia1 ha. a .peotra1 de~ity ~or which ~r( ~) > 0, ~C C1.<) ~ .. ) and ~r ( ') • 0 eb';where III1d U ~n" ('1 11 ~a) then the. Yiener fllter CIll1 .upp1y 00naiderab1y better re.n1t. thIIIl the averasins. 'Ibi. _ode1 o~ the .i-.p1e periodica1 nol.e i-.p11e. that the individua1 evoked potentia1. JtJ(t) are not independent and thus the averase evoked potentia1 i. not a soed e.timate o~ the rea1
75 evoked aotivity. (iii). FiDa.1.~y we examine the oase when the errors ot the both eatimates need not vanish with growing number of at:L-uli repeti. tion. Namely,
(~7)
~A", ~ E~ ~
r
i SIf•.,().) I&r.. (A)d.\ 1',.
r,. < ,,,,.1"',,
~ '""/£r,,
". f £A",
-
f
f,. (~)-"
f ,,,.(~)Jlf,,(,,) -
F;.r,-)",,,
f',. ~ 11,.. ''''.. '!hus :Lt the interstilDulation interva~a are ohoaen in 8uoh a way that the ~t integr~ vanishes with growing N, then the Viener fi~ter has asyaptotic~ly zero error :Lt and o~y :Lt the average is an estimate with aayaptotic~ly zero error. Let us elllphasize that in the majority of real e:EP8ri_nt~ data, the spectr~ density of the real evoked potential fulfils the condit:1.on that f ( 1 ) ~ f ( ,,)', i. e. the signal to noise r n ratio is less than 1. In this case it may happen that the averaging attenuates considerably o~y the noise frequenoies for whioh the real evoked potentia~ is negligible, in other words it may happen that ~ess the error of the averaging converges to zero, the integral
S
Ih.. (~)Ia.r,,(,,) -f,.(")d"
f r < Ih","r"
.
does apd consequent~y to (~7) the Viener fi~ter wi~l supply asyaptotioa~ly the same error as the averaging technique. Summarizing these considerations we ooncluc1e
that the Viener
fi~ter supp~ies
a better estimate than the averaging. Yet we IaUSt try to use such interstilll~tion intervals which guarantee that the average is a8yaptotic~ly errorless. But in this case the Viener fi~ter gives considerably better results than averaging
o~y
for small nlmber ot stimuli repetition, sinoe the
greater the rate of the convergence E"N.... O, the sooner the ditference E"N - EVN vanishes.
76 Rt{'Etp9"
1. Ubr.cht. V•• Rac1i1-V.:l.••• T.: B:l.o1. Cybernet:l.c•• ~. 43 (1976) 2. Ba.ar. E•• Gonder. A•• ~ze.~. C•• Ung&rn. P.: B:l.o1. Cyb.rnet:l.c•• 32. 137 (1975) 3. Doy1•• D.J.: B1.otroeDo.ph. 01:i.D. Neuropb;y.:l.o1. J§. 533 (1975) 4. IlaDDaD.E.J.: Mu1t:1p1. T:l._ hr:l.••• Nev York: J. V:l.1ey (1970) 5. liocav•• T•• KatayaJlB. 1:•• Tabat•• Y•• O.h:l.o. T•• Xavahara. T.: J. 1:aD••:l. M.d. Sohoo1 Supp1• .ll. "9 (1973 a) 6. liogav•• T•• Kat.y.... X•• Tab.t•• Y•• O.h:l.o. T•• Xavahar•• T.: J. x.n.a:1 Med. Sohoo1. Supp1• .ll. 33 (1973 b) 7. Hocava. T•• Kat.~. 1:•• Tab.t•• Y•• Kavahar•• 1:•• O.h:l.o. T.: E1.otro.no.ph. o1:i.D. H.uropby.:l.o1. Jl•. 375 (1973 0) 8. Ruohk:1D. D.S.: :rUE Trane. B:1o-Med. Bag. Y. 87 (1965) 9. V.1ter. D.O.: B1eotroeDo.ph. c1:i.D. H.uropby.:l.o1. Supp1. 3.2. 61 (1969)
V. Albreoht, T. Xn.titute o~ Phy.io~ogy, Prague, Czecho.~ovakia
Radi~-Vei••
Czeoho.~ovak
Aoademy of Soienoe.,
'lbe Viener ~i~tration o~ a .tationary .igna~ from. a .i~ture oompo.ed of the .isua~ p~us a .tationary noi.e ha. an app~ioation when e~traoting rea~ evoked potenti.~ from po.t.timulation EEG reoord.. 'lbe Vi.ner e~tr.otion b.oame a oo~titive .ethod with the .imp~e .veraging. Both methods, i. e. Viener fi~tr.tion and .veraging, are oompared with respeot to .~ucidade re~.tion. between resu~t. obtained by the•• Sum-eql
:u
i.
we~~
known that the Viener
fi~tration
of a .t.tionary
.isua~ whioh i . additive~y di.turbed by an independent .tationary noise .upp~ie. the estimate the mean square error of whioh is • • •_~~ a. po•• ib~e. 'lberefore the use of this method
i. very oomprehensive in the ·rea~·
evoked
potenti.~.
prob~em
of e.timation of the
One may .t.te that this method, origi-
nated by V.1ter /9/, ha. got great acceptanoe /2,3,5,6,7/. On the other hand, the Viener fi~ter bring. more oomp1icat.d data prooe.sing than th.~1a•• ica~· averaging doe•• Furthermor., the aocuracy of the r ••ult. obt.ined by both, Vi.ner fi~tration
interva~.
fo~~owing
It i.
and averaging, d.pends on the interstimu1ation
u.ed. To di.oov.r this depend.nce we use the mod.1 o~ .voked potenti~ generation.
u.ua~~y
suppo.ed that the reoorded EEG .ignaJ., "e .ha11
denote it ~(t), con.i.t. of two .tati.tio~1:y indep.ndent component.1 the prop.r .vok.d potentia1 r(t) and the ~pontaneous EEG activity n(t), i. e. (1) ~(t) . For the purpo••
= o~
r(t) + n(t) • the Viener fi1tr.tion "e suppo.e that both
r(t) and n(t) are .t.tionary prooe•••• with z.ro e~ot.tion•• 'lbe .~.ri••nta1 objeot ie·N-U. . . . . timulated by the ..... • timu1us. 'lb. EEG .isua~ (1) i . reoorded during the time T
70 after .ti.uJ.ation. We .ha~~ denote ~ the t1llle duration betwea the end ot the j-th reoord and the app~ioation ot the j+~-th .tu.u1ue. 'lbe ezper1lllent i. u._~1y arranced in .uoh a way . that al.1. the interva1.. 9" are _ch ~oager than the t1llle duration ot the reoord T. 'lbu. the j-th record oan be written in the :torm j_~ Zj(t) = r(t) + n(t + (j-~)T +;E:0k).
(2)
k=~
'lbe prooe••
(:l)
N
~
~(t) = 7
2:
Zj(t)
j=~
j-~
N
~
= r(t) + N
E
n(t +
j=1.
(j-~)T +
E
k=~
9 k)
i. oa1.~ed average evoked potentia~. Xt :to~1.ow. :tro• • tationary a ••u.ption. that a1.1 prooe••e. Zj(t), r(t) and n(t) have a oertain .peotra~ repreaentation (we auppo.e diaor.te ti•• haVing in .ind the .amp1.ing tor. o:t these prooe.aes), i. e• ."
=
J.-a~'tz". (t:I),,),
-~r
'
r,( 10)
= J.-U.tZ~(d)'),
n(t)
=
('.-alto z" (d") -f1
(aee e. s. /4/~ ,where Zz is a stooha.tio proo.ss with zero expeotations and indeP~ndent 1nore. .nts :tor which E
(I Zz/d" >.11.1-
:tz /
~
),
where t ( ~) b the speotral. density o:t Z (10) aDd B 18 the operato:jo:t enae.b1.e averaglDc (ezpeotationj.'lbe . . .e oan be repeated·:tor the re. . inias zr aDd zn prooe••e. ao that we have
71 .
vh.r. ~r ( ~ ) and 1'n ( ,.) are the .pectral. del181.ti•• 01' r (t) and net) r ••p.ctiv.1y. 7h. proc••••• zr and zn are .tat1..tica11y iDdependent 1.n cOI18.quenc. 01' the 1.nd.pend.nc. 01' r(t) end n( t). V1.th r ••p.ct to (2)v. can imaediat.1y vrit. r " i·. u (lil) '·U Z,.(dl) + "-'').!(i-1)r·.~e..]'Z,,(t1IA). • Zaj
f ..
• Je
-.
-r
fe-a.!.
o
-"
.
7hi.. 1'orlllula y1.e1de the r.1at1.on b.tve.n .pectral. d.n.1.tie.
1'r ( ~ ), f n ( .,.) and ~ '"
(~) = 1'r
1'",
J
(
~)
,.1. ~
+
(4)
l'
"'J
the ••_ (5)
Z
j..
if" -l)T +.~ 9,./ . l'n ( 1-) I
£0"]12. 1e -i 1lej-.) T. •••
but By
(1-). Ve have "
(~)
= l'
r
1
:a
-
and therefore
(~).
+ l'
(')
n
vay ve get the spectr.l r.pr••ent.tioD o~ ~(t)
~(t) '"
J"e-ilfz.'N
(cl1-)
-lr
.,.
•
f • .•
-.
vMoh 1.. • cOl18equence
It.
/.
z,.la~
o~
)
+
( :}). 7h1.. l' orlllu1a i-.>U.. that '"
,jo.
j.. ~ -i"[(j .• )r+rl\J,~f'.(?o) ( ,. ) + / N~e. .••• ". Jot
vllere ~~ (1.) 18 the .pectral c1endty o~ tile ~ (t). For the .ake 01' abbr••iat1.on v. eha1l denote
(7)
J~",(1o>J1.
I~
N
;-f
.~e·"~!(J·o")T·t~·ll2.. 3"
72 As
Ihw(?)\~i~, ve oan summarize the .eohani.sm of averag:lng
as
fo~~ovs:·
of
~:iDear fi~ter
Tbe averag:lng of EEG
s:lgna~s
presents some kiDd
that passes the frequenoy oomponents of r(t,
vithout any attenuation whereas the frequenoy oomponents of n( t) are suppressed proport:lona~~y to
l:z. •
\!Ix("')
It fo~~ows from Ruohi::lD's paper /8/that a ooncrete form of
I !Ix (
")
:iDterva~s
I z.
stantia~~y
by Wa~ter
depends very stroDg~y on the ohoioe of the
between
stim~ations.
Th:ls function oan differ sub.
from the oonstant va~ue ~/N, vhich was proposed
/9/, even though ve do not work with
m~ation :iDterva~s. No . .tter how the
equa~
:iDtersti
SJ's are ohosen, it il
abays \ !Ix ( 0 ~~, henoe the di1"ferenoe of ~ ~ >I1-rom ~/N must be at ~east considered :iD oertain ne:lshbourhood of ~ = (
I (
Tbe frequenoy tran_fer funotion of the Yiener
fi~t.r
for
estimation r(t) from average evoked potential (,) has the form f
(8)
r ( 1-)
= -----------f,.(~) + 1!Ix(Q11fn (?)
(see e. g.
/4i).
Bovever, we know neither f r ( "), nor f n ( ?). spe~tra~ densities oan be obtained fro.
Estimators of these
soJ.ution of the equations
where
~
=N
and
So~v:lag
(11)
these equations for f r ('1.) and l'n ( 1) ve p t _
'" f
f(?)-f;v()
-..
n ( ,.) .. - - - - - - - -
~
- l!lx (,,) \:z.
73 fiN( ~) -
'\
(12)
Ibx(
~)\2. i( ~)
=
f r ( ~)
l-Ibx(~)ll
Substitution of these estimators into (8) and several manipu- . lations yield
(13)
whioh ie aD estimator of the Yiener frequency transfer funotion based on i ( ~) and l'- ( ~ )'. 'lbe Yiener fil.tered 3u.mate of r(t) fro. iJr(t) in the fr~uenoy doaaiD has the fora (l.4 )
Xx(
where ~) is the Fourier tranafora of AEP XX(t). '!be inverse '\ Fourier transfora of Y ( 1- ). ve denote i t w ( e). then yiel4a H H the Yiener esttaate of r(t) in tt.e doaaiD. '!be rel.ation between the average eVOked potential. and the Yieuer eattaate can be judged fro. the e%pressiona oODoerDill4r the errora of these estt.ators. '!be _an square error of the averaeevoked potential. is given by (l.5)
E-'H
=
E
(~(t) - r(t»2 J =
-r
SIbx(
-.
'1 )12 f n (
~ )d~
and the lDean aquare error of the Yiener fil.tar ia atated in the foll.owlDc foraula T f r ( -:l ) (l.6) E (CwH(t) - r(t» 2} = -r fr(~) + lhx(~ffn(~)
f
·fhx( ,,) I 2 f n ( 1)
41.
74 For siven .~otr.1 denaitie. o~ the re.ponae III1d the noi.e, both error. depend on the n_ber o~ .tiau1i repetition Jl' and on the inter.tUlu1ation interva1. na84. (i). Xt the inter.timn1ation interva1. are obo.en acoordins to a .tztaten .uob that 1.iJa
Jl' ~
then
BY.
=
B-'JJ
=
0
0 and 00naequent1y the
~~erenoe
E-'JJ-BY• •a-
niabe. with srow!Ds Ji, i. e. co-.pared to averasins, the sreater
accuraoy f4 the Yiener ~i1ter e2;pire. with srowins na-ber o~ .timu1i repetition~ (ii). Jiow we 01111 a.k whether the error o~ the Yiener ~i1ter OaD vaniab un1e.. the error o~ the a.erase doe•• :It i. apparent that the error o~ t¥ Yiener ~i1ter i. zero U ~r(") • ~n(~) = 0 ~Or a11 ~, i. e. U the .et o~ ~requen. oie. ~or whiob ~r ( ~ ) > 0 and the .et where ~D ( ,,) > 0 are JaUtua11y eJt01uaive. ']hi. oa.e 01111 be exa-p1Uied by the .i-.p1e.t periodioa1 _ode1. Let us .uppo.e that the .pontaneou. BBG aotivity i. a .tationary prooe•• with a periodioa1 oovariaDoe ~UI1otion.
~(~ ) = 0;:-00. ~t. o~ the averase i. siveD by the
'1ben the error
1
B.A..._
-.=
2
N
~ormu1a
..... 1
N-I&
cr.:~(- + - 2 ~I:" "Ji Ji 11., .. 4
00. ),,(kT +
E
Bp»
/#ai
2r
~ro_
whiob it ~011ow. that U i • •_ inte. .r, then
'I' + 9 i
= -i--:r-n
' where -i
Bowever, U .e 01111 .Uppo.e that the rea1 evOked potentia1 ha. a .peotra1 de~ity ~or which ~r( ~) > 0, ~C C1.<) ~ .. ) and ~r ( ') • 0 eb';where III1d U ~n" ('1 11 ~a) then the. Yiener fllter CIll1 .upp1y 00naiderab1y better re.n1t. thIIIl the averasins. 'Ibi. _ode1 o~ the .i-.p1e periodica1 nol.e i-.p11e. that the individua1 evoked potentia1. JtJ(t) are not independent and thus the averase evoked potentia1 i. not a soed e.timate o~ the rea1
75 evoked aotivity. (iii). FiDa.1.~y we examine the oase when the errors ot the both eatimates need not vanish with growing number of at:L-uli repeti. tion. Namely,
(~7)
~A", ~ E~ ~
r
i SIf•.,().) I&r.. (A)d.\ 1',.
r,. < ,,,,.1"',,
~ '""/£r,,
". f £A",
-
f
f,. (~)-"
f ,,,.(~)Jlf,,(,,) -
F;.r,-)",,,
f',. ~ 11,.. ''''.. '!hus :Lt the interstilDulation interva~a are ohoaen in 8uoh a way that the ~t integr~ vanishes with growing N, then the Viener fi~ter has asyaptotic~ly zero error :Lt and o~y :Lt the average is an estimate with aayaptotic~ly zero error. Let us elllphasize that in the majority of real e:EP8ri_nt~ data, the spectr~ density of the real evoked potential fulfils the condit:1.on that f ( 1 ) ~ f ( ,,)', i. e. the signal to noise r n ratio is less than 1. In this case it may happen that the averaging attenuates considerably o~y the noise frequenoies for whioh the real evoked potentia~ is negligible, in other words it may happen that ~ess the error of the averaging converges to zero, the integral
S
Ih.. (~)Ia.r,,(,,) -f,.(")d"
f r < Ih","r"
.
does apd consequent~y to (~7) the Viener fi~ter wi~l supply asyaptotioa~ly the same error as the averaging technique. Summarizing these considerations we ooncluc1e
that the Viener
fi~ter supp~ies
a better estimate than the averaging. Yet we IaUSt try to use such interstilll~tion intervals which guarantee that the average is a8yaptotic~ly errorless. But in this case the Viener fi~ter gives considerably better results than averaging
o~y
for small nlmber ot stimuli repetition, sinoe the
greater the rate of the convergence E"N.... O, the sooner the ditference E"N - EVN vanishes.
76 Rt{'Etp9"
1. Ubr.cht. V•• Rac1i1-V.:l.••• T.: B:l.o1. Cybernet:l.c•• ~. 43 (1976) 2. Ba.ar. E•• Gonder. A•• ~ze.~. C•• Ung&rn. P.: B:l.o1. Cyb.rnet:l.c•• 32. 137 (1975) 3. Doy1•• D.J.: B1.otroeDo.ph. 01:i.D. Neuropb;y.:l.o1. J§. 533 (1975) 4. IlaDDaD.E.J.: Mu1t:1p1. T:l._ hr:l.••• Nev York: J. V:l.1ey (1970) 5. liocav•• T•• KatayaJlB. 1:•• Tabat•• Y•• O.h:l.o. T•• Xavahara. T.: J. 1:aD••:l. M.d. Sohoo1 Supp1• .ll. "9 (1973 a) 6. liogav•• T•• Kat.y.... X•• Tab.t•• Y•• O.h:l.o. T•• Xavahar•• T.: J. x.n.a:1 Med. Sohoo1. Supp1• .ll. 33 (1973 b) 7. Hocava. T•• Kat.~. 1:•• Tab.t•• Y•• Kavahar•• 1:•• O.h:l.o. T.: E1.otro.no.ph. o1:i.D. H.uropby.:l.o1. Jl•. 375 (1973 0) 8. Ruohk:1D. D.S.: :rUE Trane. B:1o-Med. Bag. Y. 87 (1965) 9. V.1ter. D.O.: B1eotroeDo.ph. c1:i.D. H.uropby.:l.o1. Supp1. 3.2. 61 (1969)