BSTJ 60: 10. December 1981: On the Rapid Initial Convergence of Least-Squares Equalizer Adjustment Algorithms. (Mueller, M.S.)

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On the Rapid Initial Convergence of Least-
Squares Equalizer Adjustment Algorithms

By M.S. MUELLER,
‘Monasent relied May 11,1881)

Adjustment algorithms for tronawesal equalizers derived from
least oquares cost functions are Roun to converge extremely fast
White vartous simatution results confirming this fact abotond inthe
Iiteuture theory eaplatnng the fost convergence has been tacking.
"This paper reports on oteps tire auch theury. Pur some commonly
ras start-up data sequences it was found that algebraic properties
of the sunpled signal vectors play a eitical rote in the transient
behavior of theee eleorithma; runes, suncwai egual weters are
nearly independent for a larue elas of transmission chars i
‘he absence af ncise. After N erations 18 being the number of fps
he resullingsweffriont wetar ix fount! fo be rotated to well-Enown
luilizerceefiient nctore If 4 tingle pulae te used a2 a training
fan, the 20r0 forcing eguaticer ie obtained: if a pseudo ramiom
Nolee sequenee, with a period in smbots equal (0 the rurmber of
coefictente used, the stad state sation ofthe cele equalization
fs obtained, Thus, after only NY dteratione, Due Teash squares algo
thes yk vffcinal sect which kemty asymptotically objaDn
able by gradient fechaiques

1. INTRODUCTION

Adaprive equalizers aze important buieny blocks in ude: for
distal data transmission over linear dispersive cunsels. They adup
tively mitigate the adverse cflees of intersymbol snlererenet. A
trite pansmler i Uveslarlup performance af modems ithe speed
Steonvergensf he wales aeuacment algorithm, ae overall data
(evugtapat depends oni and, consequently high convergence epeed
fs detrable

\avious dtferent equalizr structures are own at this smn the
(ellowing. me concerns on the frequent. weed tmneveraal Slee
stractaro? Many equaliser update algorithms are based on the soopest
flgeent, or grndiene technique, whieh minimizes the meup-equared
fr {rs} belween fle nual ontpat and the cranamitted data
fymbole! In particular, the slothastic approximation of the gradient
tlgorit with an ws evitorion is used frequently. The convergence
peed of thi algurillen wa analyved in Rela. and 4, Le waa found to
be dependent on Ihe wumber of cootizients uzed and to @ lessor
Alegvee, onthe eigenvalue spread of he channel autocorrelation matrix

‘Several metheds to improve the convergence sped of Une gent
gor were published in Ref. 5 to In Hof. 8, prior knowledge of
fhe transmission channel is assumed sant a Lransorealion of the
recived signal is proposed which reduoes the effect of Iaexe eigen
‘ue speed te couvengence speed, whereas in Tel 6. transfor.
‘istion of thw vorruciom vectors silding the same performance, it
propocod, Used in voniunclion with » suchastie gradient algorithon,
hese methovl edocs the convergence time tothe minimal time which
isabtaimable with an ial channel having an eigenvalue spread of one.
Tn Re 7 and A eyeic equalization was proposed as a meas to apeed
‘up the convergence time, ‘Ihe number of erin roguirid thr on
vergence of the algorithm is about dhe sitte a fir the nlorhaatin
aradient. algorithm, buc theoretically, the convergence time can be
Teduced co the time required o fill Uw riser of Che exter
Pracically, however, the eanvergonce speci limived by Une uve
computational power af the implemented algorithm,

In Ref, 9, Godard wal the eyunline adjuscmene problem 3s. an
cextimation of w slaliomary slave veior in Gaussian noise—claaical
‘Kalman filtering problem. This recultedin anew, powerfal and rapidly
converging equalizer adjustment algorithn. While this algoritheu was
Tamiliae in the area ofaboehastic approsiration thovry. "il was never
_anpied to equalizers prior to Godan wrk 1s shown By compet
‘Simulaions’ that tia coefficient adjustment slkurhin converges rane
iderably fester then the stochastic gradical agrithea and virtually
independetny ofthe chancel wl. Alter buon ieratiana, where 17
iB the numberof cotfcitsof the equalier, the ise of the enaalined
inal ia generally clove ly the cinaal obtainable, Theis an improve-
fen hy factor of thnes to ten? compared withthe performance
of the alachastic gradiene algorithm. The eauot improvement faclor
‘Sevens ou the charuiel involved and on the madlationseheme sed
‘Godard showed farther that under cereaia mel osarmptions, Ove
cvtraa nae converges asymptotically, asthe inverse of the number uf

“Mure rly, vovions methods were published!" chat veduce che
‘rnjmtationsl eompleicy inthe implemencation of the Godard also
‘us, "Theor nsthode exploit the face thar only one now element,

2346 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1901
hick may be s vector, introduced in the vector of the reecived
Signo each eration, ‘They avoid tho procastiog of large matrices
‘which are required inthe Godard algoritom, Acuordingy, the aor
‘of arithmetic aperations ean be reduced. For the Godard alyorilhrr
Usine ges sntvaticaly, wile for che algerthme described in Rete.
12 to 1d they row proportionally to che number of equalivarvuufi-
‘enlame considerable reduction for long equalizers. In Ros. 15 und 18
al Tse sorcalled leasc-quanen algorithms are extended to include
Fackionally spaced, complex equalizer siminrey, The In -xquaris
Tate algorthen was alsa extended to « devsion feedback exualver
Ret (7 hnveatigations negarding tho implemcatation are reported in,
Ref 18

"The abjecive of this paper i to provide insight, on & fundamental
level, ino the rapid initial convergence of the lous-equares algorithms
relative co the svochuaie yrulient gorithm. Godard's approach ia
probabilistic and meds winning the enewble mae aa rapidly a
‘Poanibe, He assumed tat al she involved random vaviables have joint
Gaussian distribotions. For cis ase, che Kalman Sltering technique
ives the fies possible ennvergence.Alehough there aatumpeions are
fot generally satisfied in daca transmission applications, a very pow
ful ulgrilve emerged,

Tv Ret 19, a general equivalence between Kalman ftcring and leus-
squares estimation tecniquee was exhibited, Rur tha probe stand,

‘thatthe algorithm obtained, while noc optimal in a pric

bilistic sense iv che stlution of Ueternielic leastaquaree problem,
‘Thi foe wat stated for equalizers in Tee. 12. The Godard algorithm,
te well asthe ones proposed in Its. 12 to 17, minimize de sua of Ure
‘iwared equalizer utp emors unde the condition thatthe eeticont
‘ecige remaina constant from the start of the sexsin tn the cuerent
time, Note that chis paicuer cost fanetion is ao. the mse, Therefore,
ie dose tot explain dreary why the nrtual mae canerges 20 rapa

‘One ebvious nawin for Ue fat. coivergence of the least<qaeres
algorithms i tht ull Uwe information available from the seas of the
‘quale adaptation ie stored and exploited in che update provedure,
‘ile the stachastic gradient algrithm relics mestly on curent infor
testian. ly Het. 11, tho Gedard algorithm was interpreted as a so
tchaade gradient algerithen, where hs ene corcetions ave crane
formed by an extimuiy of the inverse of the channel sulororeelation
rmabix. "The fast inkial convergence was attributed chere to “set:
frtbogonalizalon af the paler adjustmenes, However, this ean
nly aecount for plo hr ig ape. Tenet explain the fact that
the least-squares algerthme converge conaiderablv fazer than the
lvehustie gradion® algorthans under the best conditions. when an
ial channel ia involved a, equivalemy, wacn she channel autocor

AOKPTWE EOUALIZERS 2947
relation mati is known exactly

Here, we investigate the inicial convergence of the deterministic
Teast -equare algorithon from an algebraio poine of view and offer
ltemative incerpecacions fr the fet initial convergence, In Selon
1, che problem is statod and certain algcbraie propertis ofthe cont
‘monly utilized peoudo random startup sequences are derived. In
Section LL we relate the cvlficint wert Unt results after cera
‘ons (being the number of celficieals) wo the sera forcing equalizer
‘and to tho cooficonts esulling with eyele equalization. ‘The influence
ofthe added aioe i uke inl nent in Seton TY.

1, THE OPTIMAL EQUALIZER COEFFICIENTS
Tint [a(4)] denote the complex data sequence which is trnsmittod
st asigoaling ate of L/overa channel with sampled impule responce
Mad) =f, The senplen e027) ofthe recived signal, can be exprented
Bint) =F atkibline M— HIT + aT, a
sine xin) denotes the channel noise, and the equalizer length
Nn 2M 4 Liean odd number
Let x(n) dante the complex vector of the past
samples
sy “IOP NPL Se NA DTH,
land Tet eff} denote the complex eaetfcient veetar of the adaptive
‘equilzer. Then the equalizer ourpuc s(n} st time nZ) using che
oelicen! veetor (can be writen athe sealer predoct

vesived signal

yin = eckitxo, e

‘The * denotes conjugate transposed veetos (mntrices| or conjugate
complex scalars,

24 The mearaquare eterion
For the mean squntearicsvion, the equalizer coefficients are to he
hen each Ht the equalizer output you) approximates the tank
tnited data value ain} a3 lovely as possible, The oprimutn coeticene

‘nies the me
et) ~ ELfelt*xtny = oimif w

‘share £f-] denotes ensemble averazing over tho tequence (an!) The
‘octor ey mich minimizes eq (de given by

ee = A o

249 THE BELL GYSTEW TECHNICAL JOURNAL DECEMBER 1981
where

Aa #lsioint") o
‘is the autocorrelation matrix ofthe chanel, wd
= Blrindate)*] o

fs the crose-cormelation vector between the signal veetor and the
stanmited data value. Tn eu (), vi he coniciens vector which,
fon the average, will perform better than ans’ other. Ret. 2 indicates
that the elution of eq, (5) ext i the whale squave ofthe transfer
fimetion and the spectral density of Une aise have ro ra

“The least-squares algorithent”™ winnie the following cost func
tion

ao) = 5 Jelnitead) — atk) ©

i.e, the equalizer coelficien. veer ele minimizes the sum of error
fanvarer which resulted if e(a) wax used feom the beginning of tho
franamiseion to the pawn! instant 7. Viually, this ie performed
iteratively, ie, en) ie caleulated recursively for m= L-- 2. Note
that form», im invarnnl ania, ergodic daca sequences [a(n)]
tnd nose, the esl funtion approaches aremptaticaly the ensurable
tee, eln}. Alo, the soluion elm) converges thes (0

Ditlereatiting 9.1 work apcet toe} and setcing the derivative
to ravn yalde che following #0 of Towa equations for che coefficient
vector en)

E xtewiarreim) = § tha o

Althoogh the lesst-aguazys algorithms lo nol niet o minimize che
tse, fe was observed!” Uha she me

ein = 1) = Leta = Hein) ~ laf an,
which resale ifthe equiliter's coir vectors stemming frem the
previous ieretion at {a — 1iT is used, converges roughly when 1

reaches
“To find reasores far this interesting penperts, we examine closely in
‘the falling, he sation of eq, (0) altar exnelly A! ierations and
Serive relations betmeen the cocticient veevor c(N) sal sume wel
known muslizercoeficient veetore numcly, Uhe zero forcing and the
eyeic equates
Th he nselens case we show thut the vecirs<(7}, «++ 26) are

ADAPTIVE FOUALIZERS 2549,
linearly independent of each other for the daca sequences that are
‘usually used for equalizer start-up. Under chis hypothesis of linesry
Independent signal vectra xin) it can only be thatthe linear combi-
ation

5 xieporey ~0, ay
IDR) = Ofor all A=
that

1v. "Uae, i follow from eg 8) and (1)

XUNEIN atk Red ray

“Thi means that e() equalizes the frst V received signal vectors
ideally. All errors sze zero and accordingly 2(N) ~ 0. Inthe apecial
‘eave where the channel transfer Fane uf Une all pole wpe and of
order N— 1, s(a) wil rennin zero for n> Nand a(n) = e(N).
‘Therefore, che optinal velco veto bs Foun exactly wht =
Although this is not the case generally, e(3) sll has interesting

1M, ALGEBRAIC PROPERTIES OF SIGNAL VECTORS AND PARTICULAR,
EQUALIZER VECTORS WITHOUT NOIGE
Using a notation which i sll 1 he ae si i Ra, 4, 8
possible to expres the Nilimensional vector xi) ia dhe nobel case
a follows

xh) = Batti, ua)
hy vk oshy
where B= [ote ott on
Ley eae

and

aay 5)

sah sal,

sate

% Lema, where Esa lnet tho sum
ct part of length
en oletanay

neq (14), Big tacionary
of the channel memory plus (N ~ 1). The e
N= 2M + 1 in shown in oy (14). Too (15)
‘Ledimensional vector.

“The veecosx(1} «++ x(¥) are linearly independent, f they span the
dimensional space. hin is equivalent to the N x .V mate

f2()] 02H] e+ [20801 = BAC]

la) a

having rank N, This can only be true if both matsices on che sight.
hand side have rune N. This is a necessary bue not « sufficient
ssndition, Although eg (16) haw # very parteulay form, nanccly Too
lit it iano eagy to find sufficient conditions for linear independence,
We, therefore, investigne vo intresting epecal cases involving par-
ticular star-up data scenes; nsmnely, 9 aequence consisting of just
‘single pale, andthe other a period peeudo random noise sequence,
‘Before parsing ths line f attack, we need additional facts about dhe
smacrixB.

For # to have rank N, the rom vectors of B must be linearly
independent, A noveseary and sufficient condition is Gram’ criterion:
{Nvectons by ++ dare linearly independent if and only if
bth bite
Ber, Dithe

at #0, an

In
‘We identity 8 asthe fre vow of 1 and in general "as the nth row
of B. Consequently, the matric in eg, (17 the mntocorelation mati
St che channel, Gram'seriterion then reuires that the autocorrelation
teatrix be nonsingular. Tie is the mame enpdition as the one required
{ar the existence of a solution af eq () in the noiselees case, Thus,
‘whenever an optimal coefficient. veceor (in the mean-square wense)
fexin, chen the matrix in fll yank

yb.

2.4 Single vating poe
Consider mow the ranamicsion of a single pulse at k~ 1-4. Them,
neering ane. (13) to (18) into og, (12) yields che following set of
fquations for the equalize eelficent vetor after iterations
Foss yes hiv |

a 0
foes fee hy t= | a8)
0

Bow few he
0

este tye areas Sar
Eieenira
amr on

‘he resulting equaliner minimizes the peak distortion of the overall
‘Shannel! and Getshgorin’s enterion guarantoes & unique solution of

ADAPTIVE FOUALIZERS 2951
6, (18). This sa wuficont but not w necessary condition.

"This caample shows that che lens-squurcs oqualizer adjustment
technique viels the zero forcing equalizer in iverations [if x unique
lution of ea, (8) exist), whereas the gradient technique only attains
this asymptotically in the steady atate and practically requires a
‘multiple of NV Reraions to obtain a good approximation

‘As the time instant approaches infinity, the m equations

z(k)eln) —a(k)" and A= 1, eo
cannot be satiafied simultancously any mare. Tn this cas, et) wil be
Alotorrind fom eg, (8). Tsering eqs (18) to (05) inte. () yields
a

0
lim BB*eint = 2/1 ny
lo
‘Since A" ia proportional to A as defined in eg 6) and the right-hand
fide of eg, (21 is proportional tow as defined in on (7), withthe same
propertonality constant, i ellows that
Tim e(n) = exes ea

i.e, the lest-aquares algorithms converge in the noseles case to the
‘optimal coefficients in che mse sonst

32 Periodic peeudo random taining sequence

‘While the tachnique af sounding the channel ehrough isolated tat
lacs i technically possible, a different method has found wider Use,
I Refs 7,8, and 20, peviodic poeuda random noise sequences (Pex)
‘were propased and analyzed for equalizer taining purposes, When
{hese sequences are used, che eeuting equalizer coefficient vector in
the steady sate ie found to be different from the optinia one mhen
fandom data was used. Nevertheless, the former vector approache
the optimal solution very closely oven for short periads: In PRN of
period length P = Misused, then the vcore the sampled signal nce
periodic alto and nay be writen

2952 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1081
fiom 3 teams, (25)

ana
Ua = Jaan. + on
‘tie vows of J and the vector dh) have dimension N-

‘Aualysing eqs. (24) and (25) events that B ia a N x 1 circulant
acrix Ite well nove from Ret 7 tal the eigenvalues of cireulant
‘matrices are determined by the diner Fourier tranaform of cho fst
‘ov. Since the ie rove ofthe mentioned Ax ¥ dirulant i formed by
amples of the porieically repeated channel impulse response, itis
fonelued chat has full rank NV, provide that the diaerete Foret
‘transform of the periodically repeated channel impulee responso has
zo zero vals

"This condition is very similar to the one statod for the existence of
‘an optimal eneiient vector in the ma wens’ which, inthe nobles
tase aires thatthe absolute value of the channel transfer function
Taave no eras.

rom eq. (25) and the fact that the date saquence te perindi
iy ale +N) = aif}, ic follows dat N successive data veciars
Gk) +++. dt +N 1) form an N x N circulant matt. Arguing as
bow, ie follows in general chat chebe vectors er nearly iependent
iF he dt ofthe data sequence has no 220 values

For Panes of period fv in parccuar (eis known chat

ot (25)

aunyain =
aera

“The matrix in Grams erverion (17) is thon a elreulant with eigen-
wales

bes on

w
y

‘The determinant is (V+ 1)" 70; Uecefore, Wauccessve dete vectors
ofa rus with period are always linearly independent. This, together
‘with the fact hat B has rank W, implies thac any NV different ¥ vectors
tre linearly independent. ‘henefore, the ooeficent vector after 1V
iterations i given a8 the solution of

sUeyreiN) = at" for

tds uarantee to exist. Ln Het Tit wine shown tha forthe particular
tase of N= Pie, when the equalizer length equ the period P of
the Pax, the colution of cq (2) hrs on intresting interpretation in
‘he frequency domain: it equine Ue channel transfer function ofthe

NeL pode, ew

Lew em

ADAPTIVE EQUALIZERS 2353
periodically repeated impulse response at V equidistant points. Again,
thin solution is obtained by the Teast-squares algorithms after 1°
Werations, whereas the grade. rebnigae obaina this only asymptol-
ically as the nuzaberofLerations becomes large

TF the equalizer length is smaller than che period of the runs, ie,
1N = P no specific information on the neture of c(N) ean be obained
‘Wa, therefore, consider the solution et) after P iterations, Inserting
4, 23) in oq (9) and using 0, (27) yields

1

1
Bu-DB-eiry = alr | 5

P leer (99)
al
‘where Dis a matrix containing identical elements
1
Dungy ny
sing the fact chat all rows of have the same som.
1 ios that
Areip) 044, co
‘where 9 sa vector with identical lements
Da (3a)
and
oo
Observe that

Zor

qual the de value of the tranafer fanetion ofthe sampled channel and

"ena the th dement f

2384 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1981