BSTJ 50: 3. March 1971: Performance of an Adaptive Echo Canceller Operating in a Noisy, Linear, Time-Invariant Environment. (Rosenberger, J.R.; Thomas, E.J.)

12 MB·English

Performance of an Adaptive Echo
Canceller Operating in a Noisy,
Linear, Time-Invariant Environment

By J. R, ROSENBERGER and E, J. THOMAS

(asec env Oster 8 60)

In this paper we derive equations deaoibing the performance of
tnarions adantive ceho eancelir confcereCions operand x Steer,
time-inourent avuina ment, We relate the parameters iw thre equa
tions £9 mocowable environmental foetrs, suse ta efecto per
formance, and verity the teste empirically.

The ponera, the performance of av wrko eancatter camnct be exactly
sadisted for speech inputs, Thorefore, ths derived equations osswme
tC aletionany constant pecneeranvtom pt. However st ahonen that
vente tained ca this wagner yove acfud etouoten of Be par
Jormeance to be expected with spesch inpute, The similares ond
‘iferonces of the reoute for x constant power roxdow iepyt and
speech input are isin fa date

"The echo problem in the telephone nate ig enneed by te inter
tion of the following rw farcors: [The itpedance rsmotchos
thay eink yori unchione cance reer of inion: elestriow)
‘waves. (3) The existence of «biedinectional tramsneson mili per
holt the seflcte signl to ceach che talker ae ero. (8) Tine dela

‘usw ee Bite peopagatien tne o egal takes the echo abnaying,
Finariesly tke problem, hes cen alley ieveasng tral oe,
Talenring hybrids, epplsing four-wire ciceule whew yrecteal, and
raving ese apponers®

To suppress are used when the echo deny exon about 45 ms
An esho eunptoace fe 8 volee-oeratosdecoe whieh sills Lange
Toss i the weko path, aa shown i Fiz. Ts, This Jose Moke ie eo
Mlectiveiy but alee turd to leek xpeeeh from the necrend ustamer
786 nueamus evsome serene. sem, MAREE HR

Fie 1B days ba
jar atT Sea meses fo

risen he wishes to inenrape the fret eewien ‘Thi sinvtion ie
Frmsen as somz-a.sing, Danieg dogals blu ib we moseeciy
estore the commection ta fall lapiex. Some open wuiation alle
‘hopring) an esse ove ing thew duhle-asinp eile. Th
he hoon shows Pare these dozrtone socom sietesengs clue
ings tho eo delay inoeasen

‘The performance of reka suppressors en synehno wales vit
‘ute ss then eaietyng eve tothe very lms dey ef ee eet
[A new apyuosel fo the eek pauls, elle mast vlan emneelloe

te bora sggestes as possitle g-ermtive, Th teh eat

fw qvsn ation af the ea sical is autem
unl sara stom thy aetaal eee «it wo opaimmont to ype
Tribe oper, ae cia ie to analyze the operation of an taptive
vee cunelen thine necivartant savrgamicn Oe Ine 8
tive ie wae sight te the oesuatens hie Met performance
ni ee ndervltesiase, Da eathed of prschtaign ies follows
Tu Sicline Tre disene the stsnonncenta elo ae ag arm
nue st we sive nyatns describ the operation uf rian
fio eaneeller rofigusations in « Ene uw-invaniant enronment,
Binor ewer rae te ehuvatoige an eshgeaneaer Soe =pece i=
Tite, she erived egttione scar sttinry xenon ipa. Lor
ver, in Soeton ULL we Saterort fie epiions sad eho empsteally
Usk (ae mols ehnanre for asetal etinaes of
tov he expected wits apes pile

‘The eve pats thae we wl conser are avered ta he Bnew ime
invaianssnnely not teeessuriy ul Lite gl atbersise gener
TPs 2 not to inp Cha all wl el ats ro be v0 character
In inet tingevsing eric ons have born lerrved end sthees ae
fusjuetad 0 being signideontly roar Theor ceviucons owe ae

is assune aere nay seen i sesone perfume nia
thoes, Reforntes 8 aod seb (ert sas pestormanes hep te
firemen’ i either soniaear op time-resest

Alpen) na esse, savng firs with bandwidth, 2, deter
twinod by te sophie. Tan be wsed we al oh ats 0
Towel fer anvridtae ara let ae Wide ae thot af te mp
fghel. (0), en Po V2, tis san eam Ie sa for eva is
Bhie Aoi of az analog came ye. We wl sesne shat Hee are
Cla hn ince to fie

"Reming rit) sl Uw vba gal it, ane blind oH
swe en equate recess thie ae op soners af se camped als
AU ties! = 0P whew O12, v= Silla other signa

pectin to the ee eae te die eons ed ase the
Ico sevate op pespase'e, Tae he ask a bosity we
fell vigil = wonton notation, ting Henote £0" WP. Alo Une e90
‘rlutinn operation will be enol

apo)
788 repay aver meme. youn, MAREE
thie worresrus to

[ase = nae

inthe ansog exse snl to

aittiate — 7)

4m tho dita eete, Where other diferoncs ceeus the spoeige variable
vv bee,

Whether the arho ancler is digital oe anal, ib sould be ives’
into the connection ae shewn sn Tig. Th. The easton on te Te
tae end) ig ranasted “ven een by le cameller#horne The
cusvorer om Uhe right (near en) is Ding poles By striae echo
fnecoe on the far ent 0 the eannostion

"Threw (eel Une ater performance of an eelo canoeler are
the types of sigonls une, echo paths rreuuntere, and who exmeller
fonfigivarion. ‘We will snsider those chree peinz eeparately” ‘There
ave hos ifs: signals posers. in te erhe eenrsller envinoomest:

‘The spooth ofthe ae-end customer, ealed the input simnel (0,
(Ge) The spect the nea-endvastomen: When he nearaevd evstomer

and Tarend customer spesk simulianoney, we hve double-

talking. This ennatintes an interference to the ocho canceler,
8) Tenting eit noise ie ie inhont the sh path,

‘The eaho eanceller must perform: stifeotorly when these sigue ane
tmvaeat in all pomble combinetions, Cvuit ies dented aa py
mumed to be a zoro mean random procees with vaviance ef band-
limited to Hs,

‘A block diagram of the cavecller circuit uscd is shown in Hg. 2
“The busie omponents of tis eaeelor we

(Avot of AF filter having wethorormal impulke reyponses whic
feu the fist AF menor of compe be et
(ii) A ontrol network which mstomtvally weights ond. sums the
fauspte of the AF filters wo generste an aporuiation af the eho
(6) Devies to couple whe eancllr fo the telephone pant, The AcD
sd D-A converte ae required Jor un anuloy euler operation
inva digital pleat ar viow vrai. Ths et uf A ll have imple
esponats NUE] AG), =» ani) The antpat of ens er, do
red wea) a given by
amorivn neato eAneriaRe 70

date) a
ie weight by the value ofthe top grin gut) ALS = Othe tap gan bs
set to cama Eta tly asetzne to be worl. The sum of these
wiyhted stp isthe woqwusioariee af tho ceo wd

Ig). The we awe

ate = Sato ®

swhioh $9 enbtrete! fran y(f) lo lve the eaneeed chi dencted a
(Gh. Phe ranseled esi is wise ff) is operated on by a funtion F
fa then sultplied by a positive factor AC muy be any od non
egteasng fonstien
came everest memo soravat, aco mt
Wr vit ones Ee ino oes
@ Feet) snd
[1 it L1> 0,
i in tice.

‘The reultng signal is nuliplied by eh we iniegrated to yield the
‘wale af ach
he wralag error ik aver by Ube set of ifernting egutions

ren wetsf

feel) LA

lity wan
ost = Reet
Tho digital network ia gawemed ly ‘he ark of ditevence wvationse
pai0T) = agin — T) 4 RPRMIAT — 7)
Fela = Mwai = 7), mat Ben Me
We ean wile the ers 8),

as) = G) — 9)
= Eke ethos + 0 6
vere
to = Beaton ®

‘The oodisients Gq, m= 1,2, +> are the generalized Fourier etic
tents ofthe ecko path tranrr sunelinn 47()) cued land [f]
‘relative to tho complete baie 2. They are given by the uation

mek, a

Where gfft is dhe Bounestraneorr: of Affi. The tare) i onl
‘he uneezeellable part of 2

Velo suppresion nebievca ¢ cecands after che start of oaneeller
‘operation ie defined a8

wie ®

« a
8) B1FO]

19 og

igre das sg nit.
smurave sew cavoniant mo

Masiouin_ sobievable supprosion is denoted a8 Sau an equate
Tinyse [Si We define the average siting time f Uo be Uhe time
in seconds requited. for the suppresion SIG) w each UH poreent,
Of Sue it eis

el) mw derive cquations for mosizmu achievable euparcesion
aud avon satling the forthe two eases PU] = ['] ard BL]

sgn(-], To Snviltae the desivations we deine the eolumm matrices

ocd
jer | [outs
ni =e a=] 2 |=)
ee) Laude
‘ang these matrices, we me write ef) ns
Ha) = ROWED + at ®
end ye) ae
v5) = CG) = att ao

In tho derivations whi follow we wil assume:

‘he ing sgl, 29), 2a veatiomary monn prowsns having
rectangulor power density epectrim

nay fam 8
ea

(i) The cit nae, pt), faa stationary sero mon matdon process
Dundee independent of a0;

(Gi) Bow the case FJ ~ gn [| we mill ether sssuose Ul 0) and
ate puueina ah ror ren

(9) Les nlpendenn of both 1 an

With rogaed 4o the Jost assumption, it ia clear that, since #22) sw
fuetion of x2) and 9) i cannot tly be independent of (0) and
it). Tor reson vale othe foie Tanker Ky the rat of chau
792 muam ance sven eonteay seme, eae 207
‘of R{g) wil bo mush alower than that of 2) orp) 0 thet the asump-
tion is justified fora wide ange af opeeating eonditions.

Substituting equations (@) ss (10) into aquation (8), it ean be
shown that

[nora -eoeame ne],

Limo |

‘The intgral in the denominstor of equation (11) fe defined the
echo path energy which will he denoted as y. Wo see feom enantion
(1) that 8(¢) ie masimized fore given oche bath if (Rg) 2200]
0. The suai va e

si) ~ ~10 mg [4= 22] a
Aste hs prio: ay nt be ail ome BL) 101
ny va owner tis (2) veh ae aso
Shonen ted Tok beg Tron a ai
iol tnd te fer tt uur. Wel bean ineoapenre
vent) fart

Bi = —10 log

eee 0
teh 13)

[Note chat Fig a voneeparable funotion of the environment and the
‘echo eancellr. That's io rey, to ceoulace the incompleteness factor
‘one rust naw the cho path teaonfer faneion over the baacwidth of
the init igual, che Bier aet win the echo canelle and tke nun
‘er of taps employed in the cancels.

‘To fd meakmury ciate syression nv avecage setting tin
2 mutt evainate che tara BAY) #48 | fr the digits and analog,
‘830 under the oro conditions of the function F,

24 Ronlnaton of ELR'Q)-ROO forthe Analog Canto Yield Soy a
‘Tring ie Meiko of BG) a mquation (8), we may write

4 pro.m) — a+ 0 + OTRO. 8

aa he Case FL, = [

or thi ae, we va wate equation (14) ae
aourrivn re caxenian 788

(mo ROL slot a1. 08)

2k (Re Teo + LEO.

a

) ten 0

[4 BOLO Ket ] m1
iy 2 ang [1 LOE og (AE J] a

oat, =e. 09 (~

Act =, we have
Soc = 10 log as)

For the given escunptions. the maxizum sehievwble suppecesion is

ful a furetinn of eiouie noise ard 3 Kazited only the inecraplee

Sess of te filter wel DF sate tect epeckine, Sy, wold be less

thc, she Tit by amount depnfing on che sotrelscon existing

benmecn (0) wn pl

Defining the term

xt) = ng (BBB 1
09 = Jog * (285) 9)
In: ancitog of the srngmessiom ache settang time, litnting this
fn equation (27) and rescranging 8 ge

“nl »
ana Phe Cave Fi] = v4
‘As abores Sow atid 1, may be devia ylolding the Zoloning two
eqintions:

Baw = —10 log 1 21)
wd

ye Bg pean some(li JEL) oo
TO rama syerne THEMNIOML IOTRYAL, MAREN OMT

eof eee!
(1 eee)
|) d= C04 oto!
' cr,
sh cnt nen nee an fF 0 tan equation 2)
a come neta — ve -
selene th va °

‘shove isthe antilog of the suppression at tne
ta Evaluation of HWE): Ro] Jo the Dita! Cows bv Vet
st,
Using the ésfsition of RG} cad equacion 4, we car write

Rio) Ret) — RT — T-RET — 7)
= RPL Fa — 7b yw = Ty
1 rod Tyee — marian
1 Laer

FLERE 1

+ out = Wer = Ty Fa — 7) 2

pan The Coee FL] = [1
Solving squetion @21) for cho expectation af RU Rilh it wy be
own thet Shu is elven by

sin = 10 es ile baer =f)

Ry 4 MRM at 1]

are
for = 01,2, 20+ vn wha
G1 Kise TUTE De, Meg

P= MRE — C6) | eee!
Note tnt the tus ona 28 neossney 0 yield w eanvensen! system,
We cats the exalt rnin vate ws the palo th esha th 98