Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems(1)

7/28/2019 Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems(1)

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I EEE TRANSACTIONS ONOMMUNICATIONS, VOL. COM-28,O. 11 , NOVEMBER 1980 1867Self-Recovering Equalization and Carrier Tracking inTwo-Dimensional Data Communication Systems

DOMINIQUE N. GODARD, MEM BER, EEE

Absrr acrxonventional equalization and carrierecoveryalgorithms orminimizingmean-squareerror in digitalcommuni-cationsystemsgenerally equire an initial rainingperiodduringwhichaknowndata equence is transmittedandproperly yn-chronized at the receiver.

Thisaperolvesheeneralroblemfdaptivehannelequalizationwithout esorting oaknown rainingsequenceor oconditions of limited distortion. The criterion for equalizer adaptationis the minimization of a new class of nonconvex cost functions whichare shown tocharacterize ntersymbol nterference ndependently ofcarrierphaseandof hedata ymbol onstellation usedn thetransmission system. Equalizer convergence does not require carrierrecovery, so thatcarrierphase rackingcan be carried out at heequalizer utputn ecision-directedmode. heonvergencepropertiesof heself-recoveringalgorithmsareanalyzedmathe-matically and confirmed by computer simulation.

A I . INTRODUCTIONPPLICA TION of digital processing techniques n the de-sign of data communications equipment, and particularlythe advent in recent years of microprocessor-based modems,resulted in improved reliabil ity and performance of communi-cations systems. In addition to the execution of usual trans-mitter and receiver tasks, the high flexibil ity of microproces-sor-based modems enables them to provide a variety of func-tions such as self-diagnostics, the gathering of information online quality, or automatic switching to and from full and fall-back speeds, and thus to contribute to network management.The computing power available alsomakespractical the im-plementation of recent advances in signal theory. These signi-ficant features of microprocessor modems are of great interestwith he growing use of multipoint networks or computercommunications applications, where trends to data throughputenhancement give rise to newproblems,particularly n hefield of automatic equalization.Typically,daptivequalizers needn initialrainingperiod in which a particular data sequence, known and avail -able in proper synchronism at the receiver, is transmitted. In amultipoint network, the basic architecture of which is shownin Fig. 1, the problem of fast startup equalization is of para-mount mportance.Thecontrolstation usually operates incarrier-on mode, and tributary terminals are allowed to trans-mit only when polled by the control modem. Messages from

Paperapproved by the Editor or Communication Theory of theIEEE Communications Society for publication after presentation at theNinthAnnualCommunicationTheoryWorkshop, TrabucoCanyon,CA,April 1979. Manuscriptreceived July 20, 1979;revised June 23,1980.The author s with the IBM Centre d'Etudes et Recherches,-06610,La Gaude, France.

Contr ollDdeP l7- 7 TCPU - ONTROL -UNI T TX ~--X ITX RX Tr ibutaryXX Mode msI t t t

DTE DTE n a t a T er m n alEqui pmentsFig. 1. Typicalmultipoint network.

tributary ocontrolstationoften being short, he effectivedata throughput is, to a large degree, dependent on the startutime of thecontrolmodem which, ateach eturn messagemustadapt o he particular channeland ransmitter romwhich data are received. There is therefore considerable interest nequalizer adjustment algorithms that converge muchfaster thanheonventional estimated-gradientlgorithmA second problem peculiar tomultipoint networks is thaof retraininga tri butary receiver which, because ofdrasticchanges in channel characteristics or simply because it was nopowered-on during initial network synchronization, s not ablto recognize data and pol ling messages. Since ines are sharedthecontrolmodem has to nterruptdata transmission aninitiate anewsynchronizing procedure generally causing altributaries to retrain [4]. I t is clear that, particularly for arg

or heavily loaded multipoint systems, ilata throughput is increased andnetworkmonitoring is made easier by givintributary receivers the capability to acheve complete adapta-tionwithout he ooperation of the ontrol tation,andtherefore without disrupting normal data transmission to otheterminals. The purpose of this aper is the design of processinalgorithms which will allow for receiver synchronization without requiring the transmission of a known training sequence.While the subject of fast startupequalization is a welcovered topi c (see for instance the references given in [2] an.[6]),the communications literature is very poor wi th respecto he problem of self-recovering equalizationand carrietracking which is dealt with in this paper. As a matter of facwe are only awareof one paper [5] where thisproblem considered n the context of amplitude-modulated data transmission systems. In [SI , the multilevel signal is treated by thequalizer as a binary signal, that is i ts polarity, the remaininsignal being considered as random noise. This approach cannobe readily extended to combined amplitude andhase modulation systems, n particular since the problem of carrier phaserecovery is superimposed to that of equalization.

[11-[31*

0090-6778/80/1100-1867$00.750 1980 IEEE


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1868 IEEE TRANSACTIONSON COMMUNICATIONS, VOL. COM-28,NO. 11, NOVEMBER In this paper, we shall place ourselves in the context of two-dimensional modulation schemes generally used in high-speedvoice-band modems, and which we describe in Section 11. Anew variety of cost functions for equalizer adjustment, inde-pendent of carrier phase and of the symbol constellation en-coding thedata is presented and discussed in Section 111.Equalization algorithms, requiring the same computing poweras the estimated-gradient algorithm minimizing the equali zed

mean-squared error, are given in Section IV , and their conver-gence properties are addressed n Section V. The cost unc-tions to be minimized are shown to be nonconvex, but meansfor ci rcumventing this difficulty are suggested. Finally, com-puter simulations, conducted in the presence of noise andsevere distortions, confirm theeffectiveness of the approach.11. TWO-DIMENSIONAL MODUL ATION SCHEME

We considera synchronous double-sideband quadratureamplitude modulated data transmission system of the generalform shown in Fig. 2. The binary message to be transmitted isusually scrambled and converted by some coding law into datasymbols {a,} taken from a two-dimensional constellation, andthe two components are transmitted by amplitude-modulatingtwo quadrature carrier waves. Such a modulation scheme canbe reated inaconcise mannerby combining n-phase andquadrature components into complex-valued signals. Denotingby go(t) the baseband real signal element, the symbol intervalT ,and the carrier frequencyf o, the transmitted signal is of theformu ( t ) =Re ango(t- T) exp @nfot. (1)n

Assuming a dispersive transmission medium wi th additive noisew(t),the receiver input signal can be expressed asx ( t ) =Rex ang(t- T) expj(2nfot +cp(t))+w(t), (2)n

where g(t) is a generally complex baseband signal element [7]and cp(t) s a time-varying phase shift due to frequency offsetand phase jitter.At the receiver, the concept f carrier tracking fter equaliza-tion is employed. This is motivated by the fact that a properdesign of the carrier tracking loop allows removal of relativelyhigh-frequency phase jitter [8].The real-valued signalx ( t ) firstenters a phase splitter whose complex transfer function r(t)is usually matched to the transmitted ignal element, i.e.,r(t)=gd-t) exp 2nfo t, (3)

and, for thesake of simplicity,we shall assume that demodula-tion by a ocal carrier with frequency f o is carried out beforeequalization, so that the equalizer has essentially to process acomplex baseband signal of the general form~ ( t ) anh(t- T) expjdt) +Mt), (4)

f l

where h(t) is the overall baseband equivalent impulse response

IIex p j Zn f , t

IIex p- j Z nf nT

Fig. 2. Twmdimensional transmission system.

an u ( t ) is complex fdtered noise. Although we considtapped delay-line equalizer havinga tap spacing equal to Tanalyses which wil l be developed also apply to the cafractionally spaced equalizers [9].Not shown on he receiver block-diagramof Fig. 2necessary in actualmodems,are heautomatic gain co(AGC), timing recovery ci rcuit, and descrambler. Timingtrol and AGC do not requireknowledgeof the ransmidata [lo] and descramblersusually reelf-synchronizing.Therefore, as far as receiver self-adaptation is concernedsufficient to concentrateonlyonequalizationand ctracking problems.Using complex vector notation, the equalizer output z, at time t=nT can be writtenas

where yn s the vector of tap-output signals and c , thegain vector at timenT ,both being N-dimensional. Througthe paper, a prime (I)denotes the transpose of a vector. equalizer output sample is rotatedbyanestimated cphaseCpnd presented to thedecision circuit.Conventionally, the criterion for adjusting c , and&minimization of the mean-squared error

where E indicates expectation over all possible noise andsequences. Derivation of (6) with respect toc and (p leathe classical stochastic gradient algorithms

hc and A being positive real, possibly time-varying step-sizparameters, and the superscript *denoting complex conjAt this point, several remarks can be made.1) Equation (8) describes theoperation ofa first-phase-locked loop. In the presence of frequency offset, ond-order oop is necessary to lock with a zero-mean ststate phase error.2) As s obvious from (5) and (6), f a combinationminimizes the mean-squared error,hen ny ombination(c exp i L , 5 +$) is also optimum. L ater we shall take atage of thistap otationproperty ofpassbandequalize

3) I t is important to note that (7) and (8) show a coubetween equalizer updatingand carrier tracking loopsdecision-directed approach (a, replaced n (7) and (8receivers decisions 5,) is employed for receiver training,


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GODARD: TWO-DIMENSIONAL DATA COMMUNICATION SYSTEM

cessful adaptation requires that c and (p be simultaneouslyclose enough to their optimum values.At data rates of 9600 or 12000 bits/s, even without as-suming severe channel distortions, the probability of symbolerror in unequalized transmission systems is very close to 1. I tis generally observed that decision-directed ecovery fails toconverge when theerrorprobability is in theorder of 0.1,except in the case of pure phase modulation.This exception has not been well understood so far, and wethrow some light on this subject in the next section.

111. COST FUNCTIONS FOR EQUALIZER ADA PTAT IONThe primary goal of our self-recovering equalizationandcarrier tracking technique will beto reduce the effects of chan-nel distortions so that he receivers decisions become safeenough to use the conventionaldecision-directedgradientalgorithms. Intersymbol nterference (ISI) being onactualchannels of much greater importance than noise, i t will be as-sumed throughout the theoretical analyses that the noise termin (4) can be neglected.Inhe suppressed-carrier transmissionystems we aredealing with, carrier phase recovery from he received datasignal requires that he system be equalized. Furthermore,even in the absence of phase jitter and frequency offset, thereceivers initial decisions are not safe enough estimates of thetransmitted symbols to allow equalizer adaptation. Thegeneralproblem of self-recovering equalization can therefore be statedas follows: find a cost function that characterizes the amountof intersymbol interference at the equalizer output independ-ently of the data symbol constellation and of carrier phase.Our criterion will be the minimization of functionscalled dispersion of orderp (p integer>O, defined by.D(P) =E( I Z ~ I P ~ p ) ~ , (9)

with the R , being positive real constants which we shall dis-cuss later.The rationale for such a choice will become clear by com-paring the dispersion functions with cost functionsG(P)=E( 12, IP- Ian P)2, (10)

for which only independence f rom carrier phase is achieved.Let {sk} be the samples at the rate 1/THz of the overalltransmission system impulse response, including the equalizer.The equalizer output signal is then of the general formzn = an-kSk expi$,, (11)k

where J /,s the phase shift due to frequency offset andphasejitter. The eye patterns observed for pure phase modulationand combined amplitude and phase modulation examples, nthe case where only one IS1 term is nonzero, are pictured inFig. 3. The transmitted symbols are marked by circles and re-ceived points by dots. It is clear that there exists no IS1termable to produce only phase errors whichare not seen by(lo), and therefore that minimizing G(P), .e., equalizing onlythe amplitude of z,, wilI lead to a small mean-squared error.

..0

1869

I t should be noted here that, for purephase modulation, chan-nel .distortions can simply be equalized by constraining t h eequalizer output signal to have constant magnitude. This isachieved by the decision-directedgradientalgorithm even inthe presence of decision errors, which explains why ts con-vergence is generally observed.The minimization of G(p ) eads to the minimizationof IS1in a sense which we now specify. Formathematicalcon-venience, we shall consider p = 2. The datasymbolcon-stellation is assumed to have symmetriesso that

Data symbols being stationary and uncorrelatedEan*am =Elan l 2 6nm, (13)

onehas from (1 l), using (12) and (13)

wherexkindicates summation with deletion of thek=0 term.written asI t follows from (14) and (15) that, for p =2, (10) may be


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showing that G() has a minimum when I so 1 is close tounity and I S1 terms {sk},k f 0, have small magnitude.The same type of computation can be carried out for thedispersion of order 2, theevaluation of which does not requirethe knowledge of the transmitted data sequence. One obtains

- la, I t: ISk1 +R2. (17 )kIn the next section, it s demonstrated that R, mustbe chosenequal to El a, 14/Ela, 1 I t is then easy to show that (17)may be wri tten under the particular form

k

kComparing (16) and (18), it is seen that, apart from an addi-tive constant, d 2 ) and .o) have very similar expressions, pro-vided that hedata ymbolconstell ation is wch hat hequantity

4(~I a, 2 y I s0 l 2 - 2 E I a , 14is positive when I so 1 is close to unity.It may then be concluded that the dispersion has at least alocal minimum to which corresponds a small mean-squarederror, defined in the absence of noise by

Theexistence of other minima will bestudied indetail inSection V. Now we present equalizer adjustment algorithmsand show that, for an infinite lengthqualizer, perfect equaliz-ation is one steady-state solutionof the adaptation process.IV. SELF-REXOVERING EQUAL IZATION ALGORITHMSEqualizer tap-gains are adjusted according to the classicalsteepest descent algorithm

In order to take the derivative of (9) with respect to c , one

must assume that the equalizer gains are not identicallyI t can easily be shown thata- YnC =Yn*YnC IYnC I - 1,ac

from which we obtain

AS is usually done when minimizing the mean-squaredone can drop the expectation term in (21) and transforinto the stochastic approximationlgorithmC,+1 =c , -A~Y,*z, I Z Iz I -Rp),

where X, is a positive and small enough step-size parameNow we can define the values of heconstants should be noted that, from21), changing R, into &(awill result in changing the steady-state solutions c of (2suming that they exist, into ol/Pc. The value ofR, thencontrols equalizeramplification. Naturall y, we requirtap-gain increments (21) be zero when perfect equalizaachieved. This condition will define the constantsR,.L imiting ourselves to the case where the phase shift(4) is of the form

where cpo is a constant and Af the frequency offset, thetem is perfectly equalized when the equalizer output sgiven byz, =a, exp ($ +2nAfnn,

$ being any constant phase shift, owing to the tap-roproperty of passband equalizers.Using (4) for expressing the components ofy,, subs(24) i nto (21) and noting that data symbols are uncorrthe gradient of the dispersion of order p with respectzero for R, given by

Fromhedefinition of the dispersion andromequalizer adaptation does not require arrierecovery.therefore, convergence to ideal ap-gainsettings is obi.e., when (24) becomes a good approximation tothe eqoutput signal, carrier tracking can be carried out n thesion-directed mode, provided that the oop gain X, inlarge enough tohandle frequency offsets at most equalHz according toCCITT recommendations V27 and V 2phase ambiguity in the receivers decisions inherent


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GODARD: TWO-DIMENSIONAL DATA COMMUNICATION SYSTEM 1871pressed-carrier systems wi thsymmetric signal constellationswil l be removed by differential phase encoding at the trans-mitter, so that an absolute phase reference is not necessary.The block diagram of Fig. 4shows the principle of operationof the self-recovering technique.We conclude this section by discussing the influence of p.In (22), the signal

en =Zn IZn IZn I -Rp) (26)replaces the usual error signal in the least mean-square algo-rithm; Note first that, except for pure phase modulation, E ,is not small even when equalization is perfect, and itsdynamicrange is an increasing function of p . Therefore, the selection ofthe step-size X for ensuring convergence and reasonably smalltap-gain fluctuations becomes increasingly difficult with p .Furthermore,he omputation of the equalizer-coefficientincrements in (22) in a digital implementationwith finitelength arithmetic would uffer from precision or overflowproblems for large p [121. This limits the practical applica-tions of $') to p =1 or 2. For p =1, (22) becomes

with

But choosingp =2 leads to the algorithmC,+ 1 =C , - *z,( IZ I2 -R2)

h

YW 2 z n ex p- i $ ne ci s i o n Aa+ E qu al i z e r I f Ci r c ui t1 I

Fig. 4. Structureof the self-recovering technique.

We were not successful in deriving the olutions to (29)directly n terms of c . We shall therefore consider the muchsimpler problem which, from (1l), consists of expressing thedispersion under the form

and solving

T o begin with, let us considerp =2. One has(27)

(30)Using (12) and (13)and after some manipulations,oneobtains

with =o, Vl . (31)E I~,4

R2 =------ Thisetf equationsasnnfiniteumber of solutionsElan 2 which we shall denote by S M = 0 , 1, . The generalsolution s, can be defined as fol lows: amples {sk} arewhich is remarkably simple tomplement inmicroprocessor- equal to zero, except of them. The samplesbased receiver. T he speedsof convergence ofadaptation have equal squared magnitude uM 2 definedbyalgorithms (27) and (28) wil l be compared n the computersimulation section. =EIa, I 4{E)a, l4+2(M- l)(E(a, 2 ) 2 } - 1 . (32)

V. CONVERGENCE PROPERTIESIn this section, we analyze the convexity of the dispersion.We show that the dispersion is not a convex function, but thatits absolute minimum is reached for zero IS1 at the equalizer

output. The problem raised by the existence of local minima isalso shown to be soluble by simple initialization of the equal-izer reference tap-gain. Our analysis will be limited top = 1

Note that S1 is the ideal case of zero IS1 at the equalizer out-put and that solution So, for which the equalizer coefficientsare identically zero, must be discarded. For each solutionSit is now possible to compute thevalues of the energy E, andof the dispersion DM' ) at the equalizer output. Using (32)and the statistical properties of the data ymbols, one has

and 2 for the reasons mentioned above. The equalizer will also E, =ME a, 1 4 ~a , 1 2 { ~a , 14+2 ( ~ - )(Ela, 2 ~ 2 ) - 1 ,be assumed of infinite length.Our to findheolutions (33)


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1872 IEEE TRANSACTIONSNOMMUNICATIONS, V O L . COM-28,O. 11 , NOVEMBEFrom (33) and (34), it is easy to show that, if the data symbolconstellation satisfies the condition

Elan l4


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GODARD: TWO-DIMENSIONALDATA COMMUNICATION SYSTEMS 1873

d BA t t e n u a t i o n I

1 6 !

1%8

40

ms

SO 0 170090 0(a)

ms

500700 2900(b)

Fig. 5. (a) Channel 1. Amplitude anddelay distortions. (b) Channel2.Amplitude and delay distortions.

' I .8- PHASE

: : I : :. .: : I . .16-POI NT RECTANGUL AR

* I * '

16-POINT V 2 9 32-POI NT RECTANGULARFig. 6. Data symbol constellations.

Two simulation programs have been written. The f irst one,for a given channel, calculates the sample values of the wave-form h(t),normalizes the energy E, I h(rz7') I2 to unity, anddetermines theoptimum (in the sense of minimummean-squared error) equalizercoefficients Copt and theminimumattainable mean-squared error Emin*. I t also rotates the c op tvector components so that the imaginary part of the optimumreferencecoefficient is equal ozero.The secondprogramgenerates a random data signal wi th a frequency offset of 8Hz, adds white noise to i t and simulates the equalizer adapta-tion algorithms and a second-order carrier-phase tracking loopoperating indecision-directed mode. Since, n fact, we areinterested in reducing the mean-squared error, the actual MSEis periodically calculated according to

En2 = (Copt - ,)*'A(cop, E,) +Em&where A is the channel correlation matrix and E, is derivedfrom actual gains c, by rotating them so that the referencetap-coefficient is real.Denotingy ho theptimum step-size proposed byUngerboeck [I41 when thedata sequence is knownat hereceiver

hl and h2 in (27) and (28) were initiallychosen equal toX0/ 5 and X0/200, respectively, and divided by 2 each 10000iterations n he course of the convergence process. Thesechoices were found appropriate a posteriori from simulationresults. When choosing hl and h2 greater than ndicated, thesqualizer comes close to instability.Equalizer gains were initially set o zero, except C o, theinitial value of which was a variable parameter in the simula-tion program.For p = 1, reliable convergence was obtained when takingco>0.8 for channel 1and co>1.2 for channel 2 whose dis-tortions are extremely severe. For p = 2, onehad to takeco>1.3 for channel 1 and co>2 for channel 2.


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MSE ( dB)5 1

5 1MSE ( dB)

0

-5

-1 0

-1 5

- 20

L i n e 2p= SI J R=30AB

MSE ( dB)

5 .

32-POI NT RECTXI GULAR

-2 0 .

t i n e i n s yn hn l i n t er

M

5L i n e ?p=2SNR=26dB

i E

L i n e 2p=2RNR=3f l dR

T i n e i n sv mhol i n t er v al1noon ~2nnnn 3nonn 10000

(C)2nnnn 3nn

(dlFig. 7. (a) Speed of convergence-Line 1, p =.1. (b) Speed of con-vergence-L ine 2, p = 1. (c) Speed of convergence-Line 1, p =2.(d) Speed of convergence-Line 2,p =2.

In the case of the V/29constellation, for which the ratioEla, 14/(E la, 1 is the largest, condition (38) imposes tochoose co>1.4for channel 1and co>2.4for channel2.Theresults obtained by simulations are therefore n good agree-ment with theanalysis of Section V.The speed of convergence of algorithms (27) and (28) isil lustrated by heplots of Fig. 7 where each curve wasobtained by averaging five computerunswith if ferentinitializations of , noise and data sources. The convergence forp =2 appears to be faster than or p =1.

I t hould also benoted hat he equalizer coeffminimizing the dispersion functions closely approximatewhich minimize the mean-squared error.The eye is open when the MSE approaches -15 dB. Apoint, convergence couldbe speeded upby switchinequalizer into decision-directed mode. One may thereforsider that the timetoadapt the equalizer is of order of 1I t rnust be noted hatno difficulties were encouwhen updating hecarri er. phase estimate & using tceivers decisions from the beginning of equalizer adjustm


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GODARD: TWO-DIMENSIONAL DATA COMMUNICATION SYSTEMSFinally, we also tested the convergence properties of thedispersion of order 3. Convergence in that case is much slowerandequires that he step-size parameter be smaller than10-4X0. Such a small value is not practical to implement withfixed-point arithmetic.

VII. SUMMARY AND CONCLUSIONSWe have introduced in this paper a new class of cost func-

tionsand algorithms forautomatic equalization n data re-ceivers employingwo-dimensional modulation. Equalizeradaptation does not require the knowledge of the transmitteddata sequence nor carrier phase recovery and is also, apartfrom a constant multiplier in final tapgain settings, inde-pendent of thedatasymbol constellation used in the rans-mission system.The cost unctions to be minimizedare not convex, butconvergence tooptimal gains can be ensuredby employingsmall step-size parameters in adaptation oops and initializingthe equalizer reference gain to any large enough value, typi-cally in the orderof 2 when the equalizer input energy isnormalized to that of the data symbol constellation. Practi-cally, data receivers are usually equipped wi th an automaticgain control ci rcuit, so that equalizer nitiali zation should notbe a critical problem.Simulations have shown that the self-recovering algorithmsare extemely robust with respect to channel distortions.As expected, since data symbols are not known at the re-ceiver, equalizer convergence is slow, of the order of 10s fortransmission at 2400bauds over severely distorting lines. How-ever, for he purposeof etraining a tributary eceiver inmultipointnetworkswi thout disrupting normaldata rans-mission, speedof convergence is not of paramount mpor-tance. For this eason, noattempts were made o definetheoretically the step-size parameters which should be used,and to evaluate the influence of the number of taps on thespeed of convergence.

Thealgorithmswhich we have proposeddonot requiremore computing power than the conventionalgradient algo-rithm or minimization of he mean-squared error, whichmakes their mplementation easy, and therefore attractive, inmicroprocessor-based data receivers.REFERENCES

[ I ] K . H. Mueller and D. A. Spaulding,Cyclic equalization-A newrapidlyconvergingequalization technique for synchronous data

1875communication, Bell S ysf. Tech. J . . vol. 54, pp. 370-406, Feb.1975.D. N. Godard, Channel equalization using a Kalman fi lter for fastdata transmission, I BM J . Res. Develop.. vol. 18, pp. 267-273,May 1974..R . D. Gitlin and F. R. Magee, Self-orthogonalizing adaptiveequalizationalgorithms, IEEE Trans.Comrnun., vol. COM-25,G. D. Forney, S . U. H.Qureshi, and C. K . Miller, Multipointnetworks: Advances in modem design and control, presented atthe Nat. Telecommun. Conf.. Dallas, TX, Nov. 1976.Y . Sato, A method of self-recovering equalization for multilevelamplitude-modulationystems, IEEE Trans.omrnun., vol.COM-23. pp. 679-682, June 1975.R. W. Lucky, A survey of the communication theory literature:1968-1973, I EEE Trans. nform.Theory, vol. IT-19,pp. 725-739. Nov. 1973.L. E . Franks, Signal T heory. Englewood Cliffs, NJ: Prentice-Hall,D. D. Falconer, J ointly adaptive equalization and carrier recoveryin two-dimensional digital communication systems, BellSyst.Tech. J . , vol. 55, pp. 316-334, Mar. 1976.G. Ungerboeck, Fractional tap-spacing equalizer and conse-quencesor clock recovery in datamodems, I EEE Trans.Comrnun., vol. COM-24, pp. 856-864, Aug. 1976.D. N. Godard, Passband timing recovery in an all-digital modemreceiver, IEEE Trans. Comrnun . , vol.COM-25, pp. 517-523,May 1978.R. D. Gitlin, E. Y . Ho, and J . E. Mazo, Passband equalization ofdifferentially phase-modulated data signals, Bell Syst. Tech. J . ,vol. 52, pp. 219-238, Feb. 1973.R. D. Gitlin and S. B . Weinstein, On the required tap-weightprecision for digitally implemented, adaptive, mean-squaredequalizers, Bell Syst. Tech. J . , vol. 58 , pp. 302-321, Feb. 1979.G. J . Foschini, R. D. Gitlin, and S . B. Weinstein, On theselection of a two-dimensional signal constellation in the presenceof phase jitter and Gaussian noise, BellSyst. Tech. . , vol. 52, pp,G.Ungerboeck, Theory on thespeedof convergence in adaptativeequalizers for digital communication, I BM J . Res. Develop., vol.

pp. 666-672, July 1977.

1969, pp. 79-87.

927-965, J uly-Aug. 1973.

16. pp. 546-555, NOV. 1972.*

cover data transmissionand estimation theory.

DominiqueN Ward (80) was born in Lyon,France, on December 29, 1947. He received theMastership degree in physics from the Universityof Lyon,France, in 1971, the Elaborate Studydegree in electronics from the University of Parisin 1972, and the Ph.D. degree in electricalengineering in 1974.Since974, he has been employed byCompagnie IBM France, La Gaude Laboratory,France, where he s a member of he AdvancedTechnology Department. His present interests

I systems, digital signal processing, and detection

Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems(1)

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Transcript of Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems(1)