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A Unified Theory of Data-Aided Equalization
By M. 8, MUELLER and J. SALZ
(anusrint rosie! Merch 17, 19811
A unified theory is presented for dota-aided equatication of digital
data signals passed rough noisy near dispersive channels, The
‘hoary castes the some past andor future transmitted data sm
bois are nerfectiy detected, We sae this hypotiuais to derive the
‘minimum moan:square error receiver. The optimum siructure consists
(ofee ratched filter in cascade with a transcers filler combined with
1 linear intersymbol interference cuceler whioh uses the ideally
Uetected data rymbole. The main result ix an expression for the
Uplinized mean-aquare error as a function ofthe muonber and lew
fion of the canceter coufiionts, The s/n, and the channel transfer
Fnetion. When the numberof canceler coefficients is zero, we get the
teelt kaon renlt for linear squatization. When the causal or post.
fursor eanceler approaches infinite length, we obtain the well-known,
Uteision foedback result. When buh the precursor and posteurser
toncelers become infinite, ce obtain the very Bert result possible,
nantly, the malchudilier bwznd dictated fram fundamental theoret.
teal considerations. Neither the decision feedback nor the matched
filter resutte com be achiewed in practice vince their implumentation
requires infinite memory and storage. Our theory cn fe used ta
Calvudate the rate of approach to these ideals with finite cancers.
|. INTRODUCTION
“The theory of Tinear and decion feedback equalization to mitigare
the effects of intersymbol interference (st) and noi im digital daca
transanssion ia well own. *In chi paper, the prubler of wgualization
Se car in a general framework of un 11 eanceler vides! hy past andor
future data values, This general structure is auggeated from optimal
detection Ihviry and it shown in Fig The optimal detector of digital
{ala ia he presence ofedaitive Gaussian noise and st is comprised of
‘a mulched filter and an Ist estimator whieh & used to cancel the
Sn al “sagt
Ofte
inerference * The implementation of this attuctute i often impract
‘eal because of is complexity ?*
Tn our theory we postulate that some portion of the 1st can be
perfectly syuthesized and, therefore, subtracted from the incoming
endl Another words, we replace (he op mal estiaator witha penetie
fone, The effec ofthe remainiay inlerterence then eniniized by
Tineor filter or a conventional lincar equalizer. In practical ystems,
however, perfect ellalioneaizol be achiowet therufore, our results
serve as ideal limit, The inclusion of oucasional errors in our Unwory
bas proved muthemalically iatractable so fa,
Tn Seetion I, wa determine the minimal mesn-aquare error (mse)
when. an arbitrary set of data symbols is know co the receiver. In
Section UL, the optimal necsiving Ber is derived and analyzed. ‘The
performance of the infinite Hneur equalizer, the decision feedback
‘equalizer and the infinite cancer wre oblained as special cases of Che
‘eneral rut, Section TY covers a dixcssion of mumeriea results
1 MINIMUM MEE FOR DATAAIDED EQUALIZATION .
Im Fig 1, the transmitter generates the data sequence (ay) whose
laments are assumed tobe independent identically discribuced (iid)
“daerete random variables ‘Theee discrete amplimides sequentially
modulate the pulbe pC) a x rale 1/7 ln proche Ohe Uacemitted
fienil, The pulse shipe, (0, can be viewed ws the overall impulse
xeeponso of tho transmitting itor andthe transmission channel, White
ise, (0) sade to the received signal which is then applied to the
linear receiving Ete, w'(). The output signals sampled a the symbel
ate 1/7 and combined with the output of the canceler: The liner
2024 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1981
canceler ia modeled asa transversal filer with coefficients, (eo), whese
ES, and where the set of integers S denotes the range of the
‘ancelers tape
‘For mot applentios, and for the special eases investigated lator in
this Secon, the range will contain the neighboring taps ofthe refer:
fence location but never the reference Toeation ite, and consequently,
= (By =o “ly 1, ==», Nie Tho encoder operas on the past
recnived aymbols ae, +s yan on dhe Future received aymbols
iis + nets Which a7e aasimed t0 be known to che receiver
Clenily, co yealize an operation on the future data symbols, « ime
delay of at least AvP soconds has co bo introduced.
or ageneral st Sthe output signal 2, =(7) ean, thus, be writen
B= Brite J ent ”
“Where = r(AT) i the overall imp reponse evaluated a = KT,
not [winter @
sand where f= E(KT) Le.
soho
To facilitate meeling of various types of linear modulation schemes,
the data equence, the noise, and all impulse responses are assumed to
bbe complex valued. In general, p() will bo the preenvelope of the
passband transmission system with zqpect tox carrier frequency, Thin
fotation his become exlremely aweful and eeonomical i hit fed?
Spocifenly, it permite a unified presentation of baseband and pase-
bead myatem
"The output signa, after slicing or quantiing is uaually taken to
he an estimate ofthe transmitted date symbol ay, Our goal now is to
bcain a receiving filter, uf), and canceer tape, (c,) 50 thatthe mse,
x0"), “i
fa a minimum. To determine (he optimal caneeller coetMciente, (eu),
se diferentiae eg () With respect to cy, m'© 5, and set the result
Wer FI te—n]=0, for mes, o
‘where
DOATA-AIDED EQUALIZATION 2005
Eat} = ots, )
and 54a the Kronecker delta. The immediate conclusion froma ey. (5)
‘that form eS
o
Ansertng this into eg. (2), we wet
sogjnaces [_tontnr — nde ®
He in
‘Thun far, our approach is perfectly obvious. By knowing the data
symbols forall integers & © 8 tx posible to eynthesle tho routing
st associated with these symbols andl subtract it from the current
tgnal sample xy If Ue set 5 conan all integers #-<n, we Use al the
‘lrondy-ecided-upon data symbols (available atthe receiver without
Alay) to aynthesie the postcursor 1s. This is precisely what is dane
in decision feedback equalization. Ifthe eet 8 contains ll the intogers
fexeept the one associated with the present instant, all 18 el
nated, But this, of cours, requires infinite delay. In practice, the set S
wil be finite and our main concern will be to determine how it
inuenoee the mae.
‘We now proceed to optimize the reeciving filter, wf) fora given set
1. Inceting 9, (8) nto 6g (0) and uring 9 () the remleig mae can
bbe wxpremed te
ceea[gnrencaeiee
where
Estee + 2)°) = of840) 0)
snd where
oor flonan an
We remark that more general nose covariances can be included, but
‘the calculations become more curabersome wichout yielding additional
insights,
"To obtain the optimum w(t), let
(0) = wnt) 40) a2
ret] wtrioar nde, as)
‘2026 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1881
where un) ia the optimum impulse resyonse of the receiving filter
tnd wera che ti, aze the samples of the optimized overall impulee
esponse, I follows immedintely from oq, (2) chat
nats +a | MrpUAT = Ade 04)
‘When
a5)
{a caleulaed ftom eq. (0), we obtain an equation for the optimum
soceiving Mer:
xt)
(at E Taper ae, us
bere
: an
“The interpretation of eg (16) is standard: the optimum receiving leer
is comprised of matched fer p(~#)" in cascada with a transvereal
‘Alter having tape only at those locetions where the eanecer has none.
‘This structure be shown in Fig 2
"To oblain our canal reat, an expromion forthe aptimal me, we
rmuleipty eg. (16) by Tut” and incoprae from co +, This yields
wth the aid of eg), (11), and (18) the result
ees ob = UY) 3
‘The exalc davernnnation of Us isthe subject of dhe next aocton
bets es
ae
\, THE OPTIMAL FILTER
‘Yo determine the simple wslwes, (Up), of the optimal overall
impulee response, defiee the sutocorsation function ofthe channel
impulze response 38
reef nat onicite
sts fate ten
wa 5 nam ”
After mmltiplying a (18) hy Fla? — 0) and inlngring frm te
‘ea we we he fllowring aprtem of linear equations im {Ti},
UpBiy= Ry = ZU» for allem, en
ca
“Zo dalarmrw tho uplinareesving Situs, we only et Us know Ui
for me #5, From eu. (21) we exliacr the equalions nevessury (©
determine Us and partition them as follows
TMs + R= Re C4 for m0,
E,Uilly =U W)Ry for mes a)
e "
where we defined
My = Rt Sido, a
and whore Js th wt $ sumer by m = 0
"Note thatthe indies of the unkrnowma and the indices ofthe right
hand sides uf ey. [23) have gape of the same size and at the sume
Tnentions (See the lefinition of7 above) Thus the ac of equations in
(23) fot in standord form and che solurion techniques not obvious
In Appendix A we develop a technique to solve this infnce act of
‘auions with Give yup involves the polation ofa special infinite
Sec of equationa wichout gaps. Te compensate for the gaps, we Mug
‘he orginal set of aquaciona, Specially, wo add for me Gite
numberof equations tothe infinite set such thatthe solution vanishes
for m ©. From Appendix a, we delarmine thot che optimum mse
Ioscomen
Mm
eet 125)
‘2028 THE BELL SYSTEM TECIINIGAL JOURNAL, NOVEMBER 1081
where If is determined from the following st of equations:
DY MalnBle® Bie NMG, for RET 8)
and where Afi the invorted sequence of Ma, Le, it satiation
YMGM A= Be for all @
In the following section, we investigate the minimal mse for some
special cases As mentioned initaly, we use the realistic assumption
thar the eet_§ sontoina the neighboring lueations (Ry === —Iy
Ly My) Then Fm {A --~ Ne] und the coefficient matrix in eg
(Gi) ie a inte Tooplita macrx, The solution af e9. (26) and, thus, He
fs unique andi guaranteed to wrist when Rs) + No is hounded away
from ero and infinity.” ‘These conditions are very mill and are
satinfid in most cae of practical interest,
2.1 Inte login equator
For Ni = Ne = 0 the solo includes only the zero integer and all
canceler coefficients vanish, Consequently, 69. (25) degenerates to 0
sale equation
HoMa? ~ 1 Nut? es)
Solving tia or Hanne inertng it into eg. (28), e obtain the standard
result forthe opsimom linear equalizer,”
ade
aa) (3)
‘where we exprenoed Af in term ofits Fourier transform,
et [de
1» the set is ininte the caneelereublracts
‘ll eh 5. Pauation (26) inthis ease yields
EM We + Nad = Sie for all ‘an
‘Comparing this with eq. (27) gives the result
My = Ha + Nobuo (2
From 96.2) and (23) i fll that fy = Re forall m. Thus, the
DATA-AIDED EQUALIZATION 2020
pur ms ar hi eas ia
not
Se ORT
‘This recognised as the mtchedflter bound forthe optimal detec
tion of «known signal in noise and in the absence of 5.
3.9 Oneaided cancel ofintnit tags
For N,= 0 and N= che cancer performs as an Weal decision
feedback oguaiza of nfie length, Ty Appoaix the mse derived
forthe more general eve 8; 70, i. for a cision fedback ecualzer
vith a limited numberof noneausel tape, The reel is
Ne
Zen
(33)
fe oy
‘where the coefficients? are determined rom che fellowing equation
ZMiMese Me for all, 95)
Here, (Mi) i tho coun “root” of tho twa sided sequence (Mf). Tt
ani Bi = D for Re O and Mz = (B=)* for k= 0. TL shown
mia
oe
and when this formula is inserted into eq, (24) we got the wall-nown
Temult forthe decision feedback equalizer,
ie ee
seer
‘Unfortunately, chee is no similar simple expression for [MG [*, for
60; therefore, we ate forced to numerically factor the ewo-sided
‘equence {Ms} into ies causal and anticausal root
\W, DISCUSSION OF NUMERICAL RESULTS
In this section, the minimal mie of data-aided equalization is eval
‘usted nomerically for certain channels und for various sels of eanceler
tape, We will exhibit and discus the behaviar ofthe M956 Cyt, NG,
as. function of Nand Ns for typical telephone channels. As a point
of rference, note the following easily proved inequalities
(0,0) el, 2) = el,
‘2000 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1981
In the following, we exatine three types of ennoeless
(i) Starting fro the infinite legs Linear equalizer whose mse is
‘cl, 0), we increate (he number of causal eanceler taps, Lc, we
texaenne eu, Ne) for Ne = QO, 0+ 1.
(Gi) Starting from the infinie length Linear equalizer, we increase
the number of known dala symbols altemating between causal and
fponcausal ones; icy we exainine eyiN, Neb for Q = 1, + Nem
Oye 1band whore 3) ~ Ne for @ oven, Ne= Ny +1 for Q of
iit Starting from the inne length decision fodck equalizer
whose mse sO, =), we examine the bebvior when nonesusal tape
fare added ity tN) for My ~ Q =O, +=» 15
‘Rquncion (25) isused Wy determine the mae for eases () and (i, where
Fis in obtained aa che solution of eq, (26) To detormine the soquence
{Ms") forall, wo observe thatthe Fourier transform of the waquence
{Ra inrelatedto dheuveralltransterfanedion ofthe channel folly
5)
Clearly, Ha) ia periodic wth period 26/T, and itis only dependent on
the magnitude ofthe overall channel transfer Function, P(.) There-
fore, phnse dscorion in the channel haa no effect on Uhe me. This it
relleced inthe well nov far! that phate distortion can be perfectly
fcavilied ‘without nose enhancement. ‘Therefore, ihe sequence
(ti'j i obtained aa follows
at 7 eta
ee or om
Fost Fourier transfor eniques ane used to evaluate
4, (80), Numerical tact aow that i sulfces to take 4-128 samples
‘of Ra) + Nein the intorval [—»/7, x/T], The fact thatthe coefficient
‘tris 0g (26) pom ive definite and Toeplit, makes it posible to
‘bain the deeized solution, My recursively. This is done with the
Tevinson algorithm"
For cate (ii), we evaluate 9, (14, The sequence, (MZ), i obtained
scsi nat Fi ate TE) fea eh at
wate 3 meme wo
wear on 5 ne, «
DATA.AIDED EQUALIZATION 2031
ate
[stare tte “
me
‘Fast Fourier transforms are used to obtain consecutively (Fi), M(c,
land Mz. The overall channel power transfer function, |P(«), is
‘sstumed te consi of 2 rained conine shaped tranamitting Filter with
Pelative excess bandwith, a = O15,
ITW)|”
wy trlsictlar
QE] mo-mkeccnent
os[tvia(erg)Z] e-em gcoscu og
: este
and eascaded with the channel power transfer foneron |G(w)[*
igure 8 shows vo different channel power transfer functions,
Go) which are used lo dative che subwequent numerical resus,
(al we show the equivalent breebuod (ansfer function for the worst
‘channel meeting tho basio conditions of private Lines (BASICBAD).”
Part (b) howe a transfer function (CABLE) with linearly increwing
atomuation. The parameters P, uni P, indicate the attenuation at
““r/T and o = 1/7. A model for a baseband eable channel le
‘biained when Pi = Ps
‘igure 4 shown the mse as function ofthe numberof canceler tap
receiver inpu, The dotted line representa type
line, ¢ype Gi; and the solid line, type (ii), The eurves for types i)
‘and (i) stat at Ue minimal re for ue invite Lert equaliar andl
‘the curv for type (i) starts at the minimal mse of the infinite decision
feodback equalizer.
‘Asan be observed ll curves converge very rupidy to their asymp-
toes. The curve for type () indicatus thal only 3 caused cooficints
sulice 1 clove approximate the performance of an infinite devision
feedback equalizer. The curve for type (i) soggesca that a woul of &
coeficienta (3 causa and 3 anticausa) resulta in performance which
ia vary close to the optimal (the matched-filter bound). The curve for
type (ii) roaches very close to the mee obtained from the matched
filter bound with only 3 noneausal coefficients, in addition to an infinive
decison fodback equalizer. Tove results are virtually independent of
‘the channel involved.
‘2082 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1981
