BSTJ 50: 8. October 1971: Some Considerations on the Application of the Volterra Representation or Nonlinear Networks to Adaptive Echo Cancellers. (Thomas, E.J.)

3.2 MB·English

Some Considerations on the Application
of the Volterra Representation of Nonlinear
Networks to Adaptive Echo Caneellers

By FJ. THOMAS,
Mawes ered Oe L810)

{In this popes we wpa th Foster representation of mac
‘othe echo contest prem und propane a generalied lap eck cnczler
lebich ammpensais fr nonbvear so pata, We prove Dat the prepooat
ec canator emg wn ues te echo bs ora ont ally te sent
sther amplicons forte agen

‘The Voliers fmesioneh cypecset lian of a nonlinear system is
csenlisution of the wall Kagwn ransu'ubion integral se for linear
Fesbea, The valiivy lives "or tar Volterra tule) eae suleie
lly wale Ta be satiny -tnas prt apaliestions

Te Cis papoe we apply tie Folin mpmesentation 19 The ere cer
rol pian end propeco 4 sqsraliae!alapeie ee eater whe
Soumsstee fp oalzcor elsy vate, We preve thn the proposes

(cares ler eouvens uni reser t32 ee ko ath the eecnoe
(swat, We also suggrsh ier apolieatons Jor ibe aise echo
soe presence

For linear sveare i i well Sonn that Eis impulse rons ae
bletely cetermisce the inal walt geletonsbip The eunpue sige,
DUD) ts fuseionally mle] bv 838 input, 4), by ae ronson
Enea)

vio [att =a ©

Uehete A(t isthe system Smpuleeaeeporse. ITs Iman Gemenstrsted
by other inveslignims™ lace nemcneas system, wine mutoua do
2708 tite meus sveruae sectsiene sucusat, Geran 1nd

rob depend on he fase ps, obey a mre general functional vel
‘sonship

wo =m + [raat = tn

4 fh [lve ceaet mint dra

wide.

afte

"Ths ie an estesion of Che fiiae inersei repreaentation of
smemiorvless acne even aad srovides for theater ta have
rrtmory. I i appseable 9 all roainearavatens woe ous
pend or che remote ons la soeta'aly sil exten. Bor example
5s sot applone ta iyeornie wn willing rates. The ene
Ufa) sin fet olin’ Uy Voller aud aa sells Veltonra faze-
tional The els, ns}, 0% apneraly eal Valter
Kernel

or tie spetomm Cok ws cer cnarema wth one a" asa tie
following:

(6) he sano ing rexpons, Meni
‘Toe spe istoue e Fanblte t
fi) Tie ape isle a Fal,

Pon [tbs snaPes
Bowsse uf Te ty teat 2) he fer

Ene, °

ato [Fo hind Ete eek

We will now generalize the acaptive echo eanceller proposal by
Sendhi” t9 anos the eahe of a nonlinear acke pati for which the
Vilerrs representation ‘= applicable
ssaenion kona canons 209

Sineo the Voltorm Kesnals are eguare legal: we may repsesent
them hy am inaensional yoneralzee Fourier svi *

gos

PP) Bled,

where (TWEIJ* is an orionneina! aot sorplete ia Ube £2 apace. The
oasis of (6) ae gen

coe eee een)

PtoOhted By
We will ume thatthe high! ules nowtineariy is of order N Then
substi (3) into (81 we obtain

EEE

a

Paleaett = ade

[rate — ab Eadnaett

eoreed it 8, be eovesen talon a shorthand not

etre
tian, Ww

sot

wo fe ne 005
To (Becomes

wo EEGs * aw

‘Naw ood tne syste sows, a0 Hig. 2, TL contains A subst
Mesignated by the ila names, Teter having hes of early
Dthonormal imu sgens |1',()} and the xe of cutpims |

7 ERB plete (0 ar th pln npn fre a
2800 rp uuu evoren sacusieaL Joya, ocivoLe TL
saris gona eoyenttem sn

tue to the input zi} ane womans to ell tee subeystorn, Schsystom
piducer the vuljnl of ty hrsy ten ene st a Hime, Subsystem 2
froduce il geeks siietons of the produce of the fltervatte
fale ero at fine fe, we ed subayetem MT prods al psi
tomlin ofthe itr enepecs taken at Ui
Se Every onal bap of every subsystem Eee umeinted wilh it two
Truliplior aod wn integer euunected in sho eowfgnontion shew in
Big. L for ony throe ys, Ober qnonties paxtaast to tho discussion
bole one ala dein ie ix. 1

"We wil num shaw thas the tap quive, Gel, f the ayeun ooueerge
to the xenemliaed Fouszer ealiernse Cj. af (1) and, coneequeat
Usk #25 emnvenges 19 p19 thuv che residual clu, (3, vanities

firs ao

Raff = 0, — Os aa
by snesection, Uke pation ymvoming the gai of

a

the

cage hpi bate
MEO wage, mE Ede wt a9}
Fo (125 2 tat
Hinge. Aas
vs as
aw
SRE py gg CO '
a RE a
Aplin 14 ud 161 te abi
ado sett
a EE laldmema ate

a5
Suraing aver all the Laps hie Uap ester, and over the
sulgtors, pele

SS ym KE EROmm a)

[Note tat2 the rightchard ee of (17) de always meguive or xo.

Ta hm 0 we Ma
202 MULL Sve amULaNL JRA, oCKomR Ie7
Als mete tha

wos EB Dewee as,

‘Thos, se ‘ours & manegatve faneton, whose deivetive is eae
worgenive, aaa eunclle Leak §(0 wenee Be noninercasing. I ie
sdnctiy dastensing wherever

3, 20 general

ay

Since every tera in yi) Ie oni've, we fuer cone that

$0 fee pa, ey

Aplin (20) te (12) we se shat
Fmt 0 = eg PMN, en
‘The ne uve shut (nt Ue Lap gaa converge to the eaceraired
Pousor euufiens ut Byinyy +0", ayy Whe is wank weet out to
"The svener ean no doubt appresase hot suing (IT) for anything
teu the fink, valve cull be Mell and oud only be vobd for 5
specific rid). The svletivs ‘ur even a vey ample irpat aval may
Dot be seuesale, We will ros adlenpLeuch an ane-vais here. However,
ftom (17) e ace hah le Inst te make | the euieker the eho
vanishes. Uowaver, [ein be ciate ails range bees
uring eallcing paniod:* wl eauco divergence prblews

nslvis cms» inlees oul sign yf). When
diy pala precet equation (17)

The poe
brew ass i], meer in
frou 2e writen

Sigang thang i alo one whan bor eenicn ia n tlesbone com
cin ie pring anol
2808

4 Darien — 21K (EERO an, we.)

si |wol EE kuoews es)

a

"The mane ngumest wel Ife then ee

tin (E Dat y= em

(Of eourey, bef shis Hit Je ourked one etasstelly independent
fssurptien woud no longar Se valid, but at least we may eoneluce
hab ceaversence will tal plist wil the nee pewor becomes com
pareble io the whan seer

‘There ars severel chee diel secociated wit she enbo een-
pelle Heal that me epestenio the teb-erder Volterra kernel by em
Sp" fold “aint summation [equation i. Ubviowey, rue a ey
tam fs to ss inferentes, eons. inlide ey ance nuber of
ements tnd 6} cam ely bo noprorizated by

fo Taleo
en

wwheve tie Jane Ered ass posible for an abuts deer.

on

‘Birthermone, evo if the J's ane small the astra shown in Figs

requires lerge nuriber of sotnponents, However, one point may not
‘be obviozs It ean hr soma tba is Veltova keels are nymetrieal.*
ae rou it ry be verti thal aT the Cj. 820 equ Tor any
Dermutarion of w xpi set of rmbees fy yoo. g For example,

Can = Ce tome
Po} ee weLL erat zacusHeAL Jouma, oomowe oo

Singo tho act of gains, Gu for sy vombinution ofa spel st of
inegers #1. aze the aosul of tho mame phywieal openations, ony
fone tap ironed to eenouns forall nach tarts, For example, eonsidor
The wy tpn Haat, Eo Fig.

= [eon — 5018

t= fom

ges).

tin

‘The eonwitution of Gy, and Gy tops toward $0 is given by

90) In = Gusta + Papcasy — (Gig ~ ane — Wane

‘lou, che Gy wal iy taps may Deep yw nae ta Any wo
fof tape, Gracy for ony portntation of « apeice set of numbers,
fy vei, cae Ke rophinsl yo ingle th Ts euniernbly eae.

ho total number of tape eyed

atin, thee

Although ye tes stata he ci ennai
ane mug be aol ler ana." pale tee ey ri

Delo
481 Chmprnantir fr a Nonlinear Syne

Accom thet we wish to Hnwarize the nonlinear system shown in
Fig. 1, ase thes the zeulting cutpu, 2), ean be expreced by the
Tineueeanvovatin integral,

[Unioase = rode ea

He adaptise aystom of Fig. 1
onverge Jong enough x0 th dhe tombone of tie ot (0 (0)} ea be
fonsierod to equil the earesponding mombers of the st |C,! equi
tion (21). ARer oregon, tie moambers of tue web (Ge) any freed
‘toaero while the member ofthe sot (G,).j # Lane xed atthe values
‘dsenmnined provinnly. Ate Tort the compensated oat will atiey
saquation (25),
anaerie ge errs 206

aaah

near Syton Synievier

SSuppoee one wisest ssi the electrical shameosisties of » non
Tinea erotem hie east be Bena inthe Laboratory. le eoakd
Hho thia by mating inp ouput tape rooming ofthe ayetem and ws
these at pal lan semper simlstnn of The alate spscem of
Fig. 1A good chuiee of an inp signal would be white nko or sry
Inher easily generated browllad signs. hy alloting the simultion
to convogge ad then (xg te tap grins Gat hel final value, the
tina chacueterisse et identi, ‘Thon it en he determined
Inns Fie ie ayer wil Setar for aay eabitears mpot hy anplyng
this ine: to the computor simnJation seh te tap gue ded ar the
veslesnmained prvionss

"The Volterrn resentation ik rise cto of characterising
nonlinear apstens with memory wten the witpas oes not depend ox
the infiice past. It 58 « gonralastion of tho convelition Tnegesh
tel in Tinea analyse, an sun Fear eareopts muy be estenbed
Using une Vollara representation, we hove propane x generates,
fdaplive acho canes. expeble of renlsaar compensttion. dike
Sevancage af she proposed cea eanes ie ie eomp.es:y, Th puns
OF these wow be gry cnmnoed iF an fin meas e! mee
fring the Vellora konels 9° i ungnaw: pobwork coud be fou
"The auchor av! oter Tesearehers ate pereans working town. this
fd and cover ees have ben stoposed, ‘To date, sinfortnstely,
they’ all suger fom the complet problem,

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a echt anh ere ese Wl 6 ware

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