Table Of Content190 IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS,VOL.1,NO.1,JANUARY2002
Equalization Concepts for EDGE
WolfgangH.Gerstacker,Member,IEEE,andRobertSchober,StudentMember,IEEE
Abstract—Inthispaper, anequalization conceptfor thenovel EDGE,additionalimportantsystemparameterssuchassymbol
radio access scheme Enhanced Data rates for GSM Evolution rateandburstdurationremainunchanged.
(EDGE)isproposedbywhichhighperformancecanbeobtained
This paper focuses on suitable equalization concepts for
at moderate computational complexity. Because high-level mod-
EDGE. For equalization, the modification of the modulation
ulation is employed in EDGE, optimum equalization as usually
performedinGlobalSystemforMobileCommunications(GSM) scheme is of high significance. Optimum equalization, i.e.,
receivers is too complex and suboptimum schemes have to be maximum-likelihood(ML)sequenceestimation(MLSE)based
considered. It is shown that delayed decision-feedback sequence on the Viterbi algorithm (VA), which is used in GSM and
estimation (DDFSE) and reduced-state sequence estimation
IS-136, would require an excessive computational complexity
(RSSE) are promising candidates. For various channel profiles,
and is not realizable with currently available digital signal
approximationsforthebiterrorrateofthesesuboptimumequal-
izationtechniquesaregivenandcomparedwithsimulationresults processors (DSPs). Therefore, alternative equalization strate-
for DDFSE. It turns out that a discrete-time prefilter creating a gies have to be employed. In this paper, the applicability of
minimum-phase overall impulse response is indispensible for a reduced-statetrellis-basedequalizerstoEDGE isinvestigated.
favorable tradeoff between performance and complexity. Addi-
Two algorithms belonging to this class of suboptimum equal-
tionally, the influence of channel estimation and of the receiver
izers are considered: delayed decision-feedback sequence
inputfilterisinvestigatedandthereasonsforperformancedegra-
dationcomparedtotheadditivewhiteGaussiannoisechannelare estimation (DDFSE), introduced by Duel-Hallen and Hee-
indicated.Finally,theoverallsystemperformanceattainablewith gard [3], [4], and reduced-state sequence estimation (RSSE),
theproposedequalizationconceptisdeterminedfortransmission proposed by Qureshi and Eyubogˇlu [5], [6] (see also [7]),
withchannelcoding.
which may be viewed as a generalization of DDFSE. Both
IndexTerms—EDGE,equalization,mobilecommunications,re- schemes are well suited for implementation because of their
duced-statetrellis-basedequalizers. highregularityincontrasttorelatedreduced-stateschemeslike
the -algorithm [8] and similar schemes like that proposed
by Foschini [9]. Also, in general, performance is significantly
I. INTRODUCTION
better than that of simpler schemes such as decision-feedback
CURRENTLY, third-generation (3G) mobile communi-
equalization(DFE)[10],[11]orlinearequalization[12].
cation systems are in the phase of standardization. Two
TheperformanceofbothschemesfortheEDGEscenariois
major approaches to 3G mobile communications have to be
analyzed theoretically and by means of computer simulations.
distinguished. The first one is the introduction of completely
Inordertoenableacomparisonwiththeoreticallimits,alsothe
new radio access schemes, such as universal mobile telecom-
performanceofMLSEisdetermined.Forthis,thederivedthe-
munications service (UMTS), which is based on wideband
oretical bit error rate (BER) expressions are essential because
code-division multiple access (WCDMA). In addition, one is
withcurrentlyavailablecomputerworkstations,evenasimula-
also interested in an evolution of the existing time-division
tionofMLSEisimpossibleforEDGE.
multiple access (TDMA) standards Global System for Mobile
In general, reduced-state equalization has only an advan-
Communications (GSM) and Industry Standard 136 toward
tageous tradeoff between performance and complexity, if the
significantly higher spectral efficiency. For this, a novel
channelimpulseresponsewhichhastobeequalizedhasamin-
commonphysical layer, EnhancedData ratesfor GSM Evolu-
imum-phasecharacteristic[4]and[5].Therefore,theinfluence
tion (EDGE), will be introduced for both TDMA schemes. A
of an additional discrete-time prefilter, which transforms the
detailed description of EDGE including features such as link
overall impulse response into its minimum-phase equivalent,
adaptation, incremental redundancy protocol, etc., is given in
on equalizer performance is analyzed. Also, the effects of
[1]and[2].EDGEimprovesspectralefficiencybyapplyingthe
differentreceiverinputfiltersareconsidered.
modulation format 8-aryphase-shift keying (8PSK) insteadof
This paper is organized as follows. In Section II, the dis-
binaryGaussianminimum-shiftkeying(GMSK),whichisused
crete-timetransmissionmodelisintroduced.DDFSEandRSSE
in GSM. In order to enable a smooth transition from GSM to
are discussed in Section III. Here, also, strategies for prefilter
calculationandapproximationsfortheBERsofbothschemes,
ManuscriptreceivedDecember6,1999;revisedJune10,2000andDecember whichturnouttobequiteaccurate,areprovided.InSectionIV,
12,2000;acceptedMay20,2001.Theeditorcoordinatingthereviewofthis detailedtheoreticalandsimulationresultsaregivenforallrele-
paperandapprovingitforpublicationisP.Driessen.Thisworkwassupported
vantchannelconditionsforEDGEandtheexpectedlossdueto
byLucentTechnologiesNetworkSystemsGmbH,Nürnberg,Germany.
W.H.GerstackeriswithLehrstuhlfürNachrichtentechnikII,UniversitätEr- imperfectchannelestimationisderived.Furthermore,theinflu-
langen-Nürnberg,D-91058Erlangen,Germany. ence of the receiver input filter and the discrete-time prefilter,
R.SchoberiswiththeDepartmentofComputerEngineering,Universityof
respectively, is analyzed. Also, statistics for the minimum Eu-
Toronto,Toronto,ONM5S3G4Canada.
PublisherItemIdentifierS1536-1276(02)00191-5. clidean distance are presented by which some insight can be
1536–1276/02$17.00©2002IEEE
GERSTACKERANDSCHOBER:EQUALIZATIONCONCEPTSFOREDGE 191
Fig.1. Blockdiagramofthetransmissionmodelunderconsideration.
gainedintothereasonsforperformancedegradationcompared By this choice of the transmit filter, the transmit spectra of
totheadditivewhiteGaussiannoise(AWGN)channel.Finally, EDGE and GSM are approximately equal. Especially, the
somesimulationresultsforcodedtransmissionwithimperfect spectralmasksofGSMarestillfulfilled[15].
channelestimationaregiven. BecauseEDGEwillusethesamefrequencybandsasGSM,
the same channel models can be used for system simulations
II. TRANSMISSIONMODEL and calculations. Thus, the channel profiles typical urban area
(TU), rural area (RA), hilly terrain (HT), equalizer test (EQ),
In the following, all signals are represented by their com-
andstatic(ST)(nochanneldispersion),whicharespecifiedin
plex-valued baseband equivalents. Apart from Section IV-E,
[16],areselectedfornumericalresults.Inordertofacilitatecal-
only uncoded transmission is considered. The block diagram
culations, it is assumed until Section IV-E that the channel is
of the system model is shown in Fig. 1. At the transmitter, in
time-invariantduringeachburst,2 whichcorrespondstolowor
eachtimestep,threebitsofinformationaremappedonto8PSK
moderatevehiclespeeds,andthereceiverhasperfectknowledge
symbols using Gray
ofthechannelimpulseresponse.
encoding.1 For transmit filtering, the linearized impulse
Forthe receiverinput filter, we assume a filter withsquare-
corresponding to GMSK with time-bandwidth product 0.3,
root Nyquist frequency response. Two special cases meeting
which is the modulation format of GSM,is employed. Hence,
this condition are the whitened matched filter (WMF), which
thetransmitfilterimpulseresponseisgivenby[13]and[14]
belongs to the class of optimum input filters [17], and the
square-root raised cosine (SRC) filter [18]. Both will be
(1) consideredfornumericalresults.
else
The(continuous-time)receivedsignalisimpairedbyAWGN,
with whichischaracterizedbythesingle-sidedpowerspectralden-
sity .Thiscorrespondstothetestcaseofreceiversensitivity;
thecasesofcochannelandadjacentchannelinterferencearenot
consideredhere.
(2)
Thediscrete-timereceivedsignal,sampledattimes atthe
else outputofthereceiverinputfilter,canbewrittenas
where sisthesymbolduration.TheGaussianshaped
(5)
frequencyimpulse ofduration isgivenby
Here, , denotethecoefficientsofthecom-
bined discrete-time impulse response (CIR) of transmit filter,
channel,andreceiverinputfilter. isthelengthoftheCIR.3
Themeanreceivedenergyperbitisgivenby
(6)
for (3)
where is the logarithm with base two, is the al-
where denotesthecomplementaryGaussianerrorintegral phabetsize,and isdefinedas
E (7)
(4)
1Actually,anadditionalrotationofthe8PSKsymbolsisperformedinEDGE 2Inprinciple,itsufficestoassumethatthechanneldoesnotchangewithin
beforetransmitfiltering[13].However,theinfluenceofthisrotationonper- thepathregisterlengthoftheVA,ifuncodedtransmissionisconsidered.
formanceisnegligible.Thus,forsimplicity,itisnottakenintoaccountforthe 3Inprinciple,theCIR isofinfinite length.For definition ofd[(cid:1)],thoseL
systemmodel. consecutivetapsareselected,whichexhibitmaximumenergy[19].
192 IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS,VOL.1,NO.1,JANUARY2002
whereE denotesexpectation.Furthermore,asignalconstel- thereliabilityinformationprovidedbyasoft-outputRSSEare
lation with unit variance has been assumed for (6). The com- giveninSectionIV-E.
plex-valued zero-mean noise is Gaussian and white. This ForDDFSE,onlythefirst , tapsoftheCIRare
holds for the optimum WMF [17] and also for an SRC filter usedfordefinitionofatrellisdiagramwith states,
becausetheSRCfilterautocorrelationfunctionfulfillsthefirst whereasfortheremainingtapsper-survivorprocessing[25]is
Nyquistcriterion,i.e.,itequalszerofor , , performedformetriccalculations.For and ,the
[18]. limitingcasesofDFEandMLSE,respectively,result.
In front of reduced-state equalization, a prefilter is option- WithRSSE,anevenfinertradeoffbetweenperformanceand
ally inserted. The coefficients of the overall impulse response computationalcomplexityispossible.Foreachtapdelay ,
including the prefilter are denoted by , . ,thesignalconstellationispartitionedinto sub-
It should be noted that ideal allpass prefiltering does neither setsusingUngerboecksetpartitioningandtrellisstates are
changethechannellengthnorthechannelenergy, .For defined bythe concatenationof the respectivesubsetnumbers
convenience,theoverallCIRisalsorepresentedbythevector ,
(8) (9)
Inordertocreateawell-definedtrellis,
Theestimated8PSKsymbolsaredenotedby ,where
hastobevalid[5].ThenumberofstatesoftheRSSE
isthedecisiondelayofthesequenceestimator,whichisgiven
trellisisgivenby
bythepathregisterlengthoftheVA.Finally,estimatedbitsare
producedbyaninversemapping.
(10)
III. DDFSEANDRSSE
For metric calculations, again per-survivor processing is ap-
A. CharacteristicsofDDFSEandRSSE
plied.Inthespecialcase ,
EDGEreceiversshouldbeabletocopewiththesamechannel
,DDFSEresults.
conditionsasGSMreceivers.Thus,equalizershavetobetested
Here,forthefirst tapdelays,eachsubsetcontainsonly
for the GSM profiles mentioned in Section II. For the profiles
onesignalpoint,whereasforeachofthelast delays,only
HTandEQ,thediscrete-timeCIRhasaneffectivelengthof
onesubsetexists,whichisidenticaltothesignalalphabet.
,whereasfortheremainingprofiles,fewertapsaresufficient
foranaccuratecharacterization(e.g., forTU).Thelength B. PrefilteringforDDFSE/RSSE
of the CIR together with the modulation format has a direct
Inordertoobtainhighperformance,aminimum-phaseCIR
impactonthefeasibilityofequalizationstrategies.
is essential for DDFSE and RSSE [4], [5]. Therefore, a dis-
In GSM, MLSE using the VA [17], which is the optimum
crete-timeprefilter,whichideallyhasanallpasscharacteristic,
equalizer in terms of sequence error probability, is widely
shouldbeintroducedinfrontofequalizationinordertotrans-
employed [20], [21]. The computational complexity of MLSE
formtheCIRintoitsminimum-phaseequivalent.Whereasthe
is directly related to the number of states of the underlying
optimumfilterisrecursive,forapracticalimplementation,afi-
trellisdiagram,whichisgivenby .Hence,MLSE
nite impulse response (FIR) approximation is of interest. For
has amaximum complexityof states forGSM.
this,e.g.,theminimummean-squarederror(MMSE)DFEfeed-
In contrast to that, complexity of MLSE is prohibitively high
forward filter is suited [26]–[28]. In the limit case of infinite
for EDGE. Even for TU, a full-state VA would require
filter order and infinite signal-to-noise ratio, the MMSE-DFE
states, which is currently far too complex for a practical
feedforward filter tends to the optimum prefilter. Hence, for
implementation.Thus,suboptimumequalizerconceptshaveto
filter orders that are not too small, a reasonable approxima-
be considered for EDGE. In this paper, reduced-state trellis-
tioncanbeexpected.TheMMSE-DFEfilterscanbecalculated
based equalizers are discussed. Among the schemes in this
viathefastCholeskyfactorization[27]–[29].Recently,aneven
class, DDFSE [3], [4] and RSSE [5]–[7] seem to be the most
moreefficientalgorithmforFIRprefiltercalculation basedon
promising candidates because of their high regularity and
alinearprediction(LP)approachhasbeenproposed[30],[31].
good performance. Also, an extension of DDFSE and RSSE
Here, the FIR prefilter of length is realized as a cas-
for generation of additional soft output is possible, which is
cadeoftwoFIRfilters
highly desirable for subsequent channel decoding [20]–[22].
Forthis, the principles ofreduced-state equalizationand max-
(11)
imum-a-posteriori symbol-by-symbol estimation (MAPSSE)
have to be combined, which is possible in a straightforward where is a discrete-time matched filter, matched to the
way [20], [21]. Two well-known MAPSSE schemes are the (estimated) channel impulse response, and is a pre-
Bahl–Cocke–Jelinek–Raviv (BCJR) algorithm [23], per- diction-error filter of length designed for (approximately)
forming a forward and a backward recursion (suited for block whitening the output noise of . For calculation
transmission), and the algorithm proposed by Lee [24], per- of , only the autocorrelation sequence of has to
formingexclusivelyaforwardrecursion(suitedforcontinuous be known, which is proportional to the filter autocorrelation
transmission). Simulation results for channel decoding using sequence of the discrete-time channel impulse response and
GERSTACKERANDSCHOBER:EQUALIZATIONCONCEPTSFOREDGE 193
thus can be determined in a straightforward way. Then, the Here, denotes the joint probability density function(pdf)
coefficients of result from the Yule-Walker equations of the coefficients of the overall CIR. In order to separate the
[31], which can be solved via the Levinson-Durbin algorithm influence of fading and intersymbol interference (ISI) on per-
[32]. It can be shown that the transfer function tends to formance,thenormalizedminimumsquaredEuclideandistance
the transfer function of the ideal allpass filter for . [39],isintroduced,withanormalizationfactor ,
For moderate finite filter orders, a reasonable approximation where isthemeanreceivedenergyperbit
results [31]. foragivenCIR withenergy
Thetotalnumberofrequiredoperationsforthisapproachfor
prefiltercomputationis .Itshould (15)
be noted that the complexity of the MMSE-DFE approach is
given by and, thus,
is also quadratic in the filter lengths, but the constants are With(6)( isvalid),(13),and(15),weobtainforthe
quite large [29]. This is in contrast to the LP approach, which quotientunderthesquarerootin(14)
seems to be preferable for a practical implementation. Using
the LP approach, FIR prefilters with lengths up to
can be calculated in a practical receiver without violating the
limitationsimposedbyareal-timeenvironment.
(16)
C. PerformanceAnalysis
Introducing(16)in(14)yields
For a fixed CIR , the BER of MLSE, DDFSE, and RSSE
canbeapproximatedby[4],[5],and[17]4
(17)
(12)
In(17),foragiven ,theinfluenceofISIonBERisrepresented
bythenormalizedsquaredEuclideandistance andthat
Here, isdefinedin(4)and istheminimumEu- offadingbytheenergyratio .Thepdf’sof
clideandistancebeingafunctionoftheoverallCIR(including and fordifferentchannelprofileswillbeinvestigated
prefilter, if applied), the modulation format, and the selected inSectionIV-D.
equalizationalgorithm. isaconstantfactor,whichiscloseto ForthenumericalresultsofSectionIV,themultidimensional
unity in most cases for the given channel profiles, modulation integral in (17) has been evaluated by Monte Carlo integra-
format, and mapping; therefore, may be well approximated tion[40],i.e., in(17)hasbeenaveragedoverasufficient
byone. isthevarianceofinphaseandquadraturecompo- number of random channel realizations drawn from an en-
nent of semblewithpdf ,6 resultingin
(13) (18)
can be calculated using the Dijkstra algorithm
Here, and denotetheenergyandthenormal-
[33]–[35], which performs a path search inthe state transition
ized minimum squared Euclidean distance of the overall CIR
diagram of the equalizer and is highly efficient for minimum-
correspondingtothe thchannelrealization,respectively.It
phaseCIRs.5 Itshouldbenotedthattheminimumphaseprop-
turnedoutthat issufficientforreliableresults.
erty is no restriction in the case of MLSE because
of MLSE does not depend on the channel phase, i.e., the CIR
can be transformed into its minimum-phase equivalent before IV. NUMERICALRESULTS
distancecomputationwithoutlossofgenerality;inthecaseof A. BERforDifferentEqualizationSchemesandChannel
DDFSE or RSSE, a minimum-phase CIR is required by any Profiles
meansforhighperformance(SectionIV-A).
InFigs.2–6,BERsareshownforallrelevantchannelprofiles
Inourcaseofrandom,time-invariantchannels,theBERex-
and equalization algorithms, assuming an SRC receiver input
pressionof(12)hastobeaveraged
filter with rolloff factor . The influence of different
receiver input filters on performance will be examined later.
WeconsiderallpossibleDDFSEschemesincludingthespecial
(14)
casesofDFEandMLSE,andthreeRSSEschemes,whichare
characterizedby:
4Inthecitedreferences,anexpressionoftheformof(12)isgivenonlyforthe
symbolerrorrate(SER).Nevertheless,itisstraightforwardtoshowthat(12)is 1) , i.e.,
alsovalidforBER,withdifferentconstantsCthanforSER. (denotedasRSSE2);
5AnadditionalimportantattributeoftheDijkstraalgorithmisthataftercon-
vergence the exactminimum distanceis available and not only an estimate, 6Pleasenotethathereanexplicitknowledgeofthepdfisnotrequired.Only
whichisincontrasttosomerelatedschemes[36]–[38]. thechannelmodelhastobespecified.
194 IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS,VOL.1,NO.1,JANUARY2002
Fig.2. BERversus10log (E(cid:22) =N )fordifferentequalizationschemesfor Fig. 3. BER versus 10log (E(cid:22) =N ) for different equalization schemes
HTprofile(withandwithoutallpassprefiltering).“—”:DDFSE(DFE–MLSE, for TU profile (with allpass prefiltering). “—”: DDFSE (DFE–MLSE, BER
BERimprovesasK 2 f1;...;7gincreases)(theory).“–”:RSSE(RSSE2, improvesasK2f1;...;7gincreases)(theory).“–”:RSSE(RSSE2,RSSE4,
RSSE4,RSSE2(cid:2)2)(theory).“(cid:14)–(cid:14)”:DDFSE(K =1)(simulation).“ – ”: RSSE2(cid:2)2)(theory).“(cid:14)–(cid:14)”:DDFSE(K=1)(simulation).“ – ”:DDFSE
DDFSE(K=2)(simulation).“r–r”:DDFSE(K=3)(simulation). (K =2)(simulation).“r–r”:DDFSE(K =3)(simulation).
2) ,i.e.,
(denotedasRSSE2 2);
3) , i.e.,
(denotedasRSSE4).
Mainly, calculated results are presented. For the DDFSE
schemes , a comparison to simulation results is
madeinordertoverifyourtheoreticalconsiderations.
The BERs for HT (Fig. 2) show that an allpass prefilter is
mandatory for reduced-state schemes with moderate numbers
of states. Without prefilter, a severe performance degradation
occurs.7 Similarobservationshavealsobeenmadein[41]and
[42],whereithasbeenshownviaanalyticallydeterminedden-
sitiesofzerosofthe transformofCIR thatfortheprofiles
HT,TU,andEQ,onaverage,asignificantpercentageofchannel
zerosliesoutsidetheunitcircle.
It should be noted that for the case of no prefiltering, only
simulationresultscouldbeobtainedbecausetheDijkstraalgo-
rithmdidnotconvergeformanychannelrealizationswithnon-
Fig. 4. BER versus 10log (E(cid:22) =N ) for different equalization schemes
minimum-phaseCIR.Withprefiltering,fastconvergencecould for EQ profile (with allpass prefiltering). “—”: DDFSE (DFE–MLSE, BER
beobservedforallconsideredchannelprofiles. improvesasK2f1;...;7gincreases)(theory).“–”:RSSE(RSSE2,RSSE4,
Becauseofthe poorperformanceofreduced-stateequaliza- RSSE2(cid:2)2)(theory).“(cid:14)–(cid:14)”:DDFSE(K=1)(simulation).“ – ”:DDFSE
(K =2)(simulation).“r–r”:DDFSE(K =3)(simulation).
tion without prefiltering, this case is not considered in the se-
quel.Foreachofthefollowingresults,idealallpassfilteringhas
been assumed. According to Fig. 2, with prefiltering, the loss especiallyforsmall andisnottakenintoaccountintheBER
of DDFSE ( ) compared to MLSE is limited to 1.6 dB approximationof(17).
for HT at ; for and , only slight AmongtheconsideredRSSEschemes,RSSE2 2achieves
improvements can be obtained compared to . Hence, the best performance for HT, which is essentially identical to
seems to provide the best tradeoff between that of DDFSE with , whereas RSSE2 and
performanceandcomplexityforDDFSE. RSSE4performasgoodasDDFSEwith .Thus,
Obviously, calculated and simulated results for DDFSE in afurthercomplexityreductionandperformanceimprovementis
Fig.2areofthesameorder.Themaximumdifferenceisabout possiblebyRSSEcomparedtoDDFSE .Forcompar-
1.3dB,mainlycausedbyerrorpropagation,whichplaysarole ison,theBERofthestatic(ST)channelequalizedbyMLSEis
alsoincludedinFigs.2–5.
Figs.3and4showBERsfortheprofilesTUandEQ,respec-
7ThisisalsovalidforTUandEQ,ashasbeenprovedbyfurthersimulations
notshowninthispaper. tively. Simulated and calculated BERs agree even better than
GERSTACKERANDSCHOBER:EQUALIZATIONCONCEPTSFOREDGE 195
Fig. 5. BER versus 10log (E(cid:22) =N ) for different equalization schemes Fig.7. BERversus10log (E(cid:22) =N )forDDFSE(K = 2)withadditional
for RA profile (with allpass prefiltering). “—”: DDFSE (DFE–MLSE, BER allpassfilteringforallprofiles.“—”:SRCreceiverinputfilterwith(cid:11) = 0:3.
improvesasK 2f1;...;7gincreases)(theory).“–”:RSSE(RSSE2,RSSE4, “–”:WMF.
RSSE2(cid:2)2)(theory).“(cid:14)–(cid:14)”:DDFSE(K=1)(simulation).“ – ”:DDFSE
(K =2)(simulation).“r–r”:DDFSE(K =3)(simulation).
thesimulatedBERsforDFEapproachthetheoreticalcurvefor
large . This is due to the decreasing influence of error
propagation.
Forallprofiles,asignificantlossinperformancecomparedto
theSTchannelcanbeobserved.Becauseinherentpathdiversity
is maximum for EQ and minimum for RA, these two profiles
correspond to best (behind the reference case ST) and worst
equalizerperformance,respectively(Fig.7).
B. InfluenceofImperfectChannelEstimation
FortheresultsofSectionIV-A,theavailabilityoferror-free
channel estimates at the receiver has been assumed. In a real
system, DDFSE/RSSE has torely onnoisy channel estimates.
For channel estimation, a standard ML approach is usually
employed [43]–[45]. In GSM/EDGE, the specified training
sequences guarantee an optimal noise suppression of ML
channel estimation, if the length of the estimated CIR is
limited to and is valid. Then, the estimated
channelcoefficientsaregivenby
Fig. 6. BER versus 10log (E(cid:22) =N ) for different equalization schemes (19)
for ST profile (with allpass prefiltering). “—”: DDFSE (DFE–MLSE, BER
improvesasK 2f1;...;7gincreases)(theory).“–”:RSSE(RSSE2,RSSE4,
RSSE2(cid:2)2)(theory).“(cid:14)–(cid:14)”:DDFSE(K=1)(simulation).“ – ”:DDFSE wheretheestimationerrors areGaussianandmutuallyun-
(K =2)(simulation).“r–r”:DDFSE(K =3)(simulation). correlatedandhaveequalvariances
for HT and DDFSE seems to be again the most fa- (20)
vorablechoiceamongtheDDFSEschemes.AsforHT,RSSE2
and RSSE4 perform as good as DDFSE , while per- where denotesthelengthofthetrainingsequence,whichis
formance of DDFSE can be attained in principle by forGSM/EDGE.
RSSE2 2. Theinfluenceofimperfectchannelestimationonthequality
TheBERsforRA(Fig.5)showessentiallyalineardecrease ofsequenceestimationhasbeenanalyzedtheoreticallyin[46].
over becausetheRAprofileresemblesaflat An asymptotic upper bound on the BER of sequence estima-
RiceanfadingchannelwithsmallRiceanfactor.Withallconsid- tionispresentedin[46],accordingtowhichchannelestimation
eredequalizationschemesexceptforDFE,aperformancesim- errorsare equivalenttolossesinpowerefficiencyofsequence
ilar to that of MLSE can be attained. This holds also for the estimation.Ifthesequenceestimatorrequires for
ST channel Fig. 6. Furthermore, it is obvious from Fig. 6 that a given target BER in case of ideal channel knowledge, then
196 IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS,VOL.1,NO.1,JANUARY2002
in case of ML channel estimation using optimum training se-
quences
(21)
isnecessaryforthetargetBER[46,eq.10].For ,
(22)
i.e., imperfect channel estimation results in an equivalent loss
in power efficiency of dB. For , ML channel es-
timationresultsincorrelatedchannelestimationerrrors.Inthis
case,theperformancelossduetochannelestimationcannotbe
determinedinastraightforwardway.
Afterinitialchannelestimation,ingeneral,channeltracking
is necessary during sequence estimation in order to cope with
thetimevariationsoftheCIR.Nevertheless,anadditionalper-
formancelossduetothetimevariationsoftheCIRisinevitable Fig.8. BERversus10log (E(cid:22) =N )forDDFSE(K = 2)withadditional
andhastobedeterminedviasimulationsSectionIV-E.Ingen- allpassfilteringforTUprofileanddifferentreceiverinputfilters(WMFand
SRCwithdifferent(cid:11)).
eral,anerrorflooroccursfornonidealchanneltracking[47].
C. InfluenceofReceiverInputFilter
Inthefollowing,theinfluenceofthereceiverinputfilteron
performance is examined. First, the performance of DDFSE
with an SRC filter and with a WMF, in-
dividuallydesignedforeachoftherandomchannelrealizations
are compared. According to Fig. 7, differences are only small
foreachofthefiveprofiles.ForTU,adeviationofabout0.7dB
canbeobservedat andforEQ,0.4dBat
. For ST, RA, and HT, differences are even smaller. Fur-
ther investigations showed that the gap between both cases is
maximum (but still moderate) for DFE and decreases with in-
creasingnumberofstatesofthetrellis-basedequalizer.Thisis
becausetheWMF8 concentratesenergybetterinthefirsttapsof
theCIRthanthecombinationofSRCanddiscrete-timeallpass
filter; concentration of energy is especially relevant for equal-
izerswithfewstates.
Hence,afixedSRCfilterseemstobeanear-optimumchoice,
while complexity is significantly lower than for the optimum
WMF,whichhastobecontinuouslyadapted.Itshouldbenoted Fig. 9. Pdf’s of d (hhh)=d for MLSE and different channel
profiles.
that the frequency response of an optimum WMF often has
a low-pass-like characteristic with a 3-dB cutoff frequency at
.ThisjustifiestheapplicationofanSRCfilter.An- remainingprofilesandequalizers.Becauseafilterwith
otherfrequentlyusedsuboptimumreceiverinputfilterconsists is more difficult to implement in practice, is recom-
ofthecascadeofamatchedfilter,whichismatchedonlytothe mended.
transmitfilterandanoisewhiteningfilter.Forthepresentappli-
cation,weprefertheSRCfilterbecauseitissimplertoimple- D. StatisticalDistributionofMinimumEuclideanDistance
mentandoffersanear-optimumperformance.
Fig. 9 shows pdf’s for the ratio
Next,theinfluenceoftherollofffactor isanalyzed.Fig.8
for MLSE and the profiles RA, TU, HT, and EQ. For RA,
showsBERcurvesfordifferentrollofffactors ;theresultsare
the pdf reveals a sharp peak at 0.93. This can be verified,
validforTUandDDFSE .Obviously,BERdecreases
taking into account that RA in principle is identical to a flat
with decreasing . The curves for and are
fading scenario. Hence, is essentially
almostidentical.9 Similarresultscouldalsobeobtainedforthe
given by . Via the Dijkstra algorithm,
8Here,thephaseoftheWMFisselectedinthatwaythataminimum-phase could be obtained, whereas for the AWGN
overallCIRresults.
channel and 8PSK, is valid [18]. This
9Itshouldbementionedthatalsoforsmallrollofffactorssuchas(cid:11)=0:1,
resultsinaratioofapproximately0.93,i.e.,anasymptoticloss
L=7tapsforthediscrete-timeCIRaresufficientinordertohaveonlynegli-
gibletapsfork (cid:21)Landk <0,respectively. of0.32 dB.
GERSTACKERANDSCHOBER:EQUALIZATIONCONCEPTSFOREDGE 197
For TU and EQ, the normalizedEuclidean distance often is
identical to that of the AWGN channel.10 In contrast, the pdf
forHThaspeaksat1andat0.93.Thiscanbeexplainedasfol-
lows.ForHT,pathswithlargedelayshaveonlysmallpowers(in
contrasttoEQ),i.e.,theiramplitudesareoftenclosetozero.For
thiscase,CIRssimilartotheRAcaseappear,causingthepeakat
0.93.Ontheotherhand,ifthesepathsarenotnegligible,theCIR
hasaquiterandomshape,whichimplicatesthepeakat1.0.
The pdf’s of the energy (normalized to ) are de-
pictedinFig.10forallfourprofiles.Itisclearlyvisiblethatdi-
versityincreasesintheorderRA,TU,HT,andEQ.ForEQ,very
smallenergiesarequiterare,whereastheyoccurfrequentlyfor
RA,whereanapproximatelyexponentialdistributionresults.
Finally,thepdfoftheproduct
is shown in Fig. 11, which primarily is of interest for per-
formanceevaluation(17).Obviously,thedistributionsarevery
similar to those for . This fact together with the re-
sultsforthenormalizedsquaredEuclideandistanceaccordingto
Fig.10. Pdf’sofE (hhh)=E(cid:22) fordifferentchannelprofiles.
Fig.9indicatethatperformanceismainlyinfluencedbyfading,
notbyISI,ifproperequalizationschemesareapplied.
E. SystemPerformanceIncludingChannelEstimationand
ChannelCoding
Inthefollowing,theproposedequalizationconceptisapplied
toanEnhancedGeneralPacketRadioService(EGPRS)trans-
mission,whichispartoftheEDGEstandard.Ninemodulation
andcodingschemes(MCSs)arespecifiedforEGPRS[48].Four
schemes (MCS-1–MCS-4) employ GMSK modulation, which
isalsoappliedinGSM.Forgoodchannelconditions,ahigher
throughput can be achieved with MCS-5–MCS-9, which are
basedon8PSKmodulationandcharacterizedbyindividualcode
rates.Each8PSKschemeemploysthesameconvolutionalcode
withcoderate1/3.Inordertoobtainthedesiredcoderate,punc-
turingisapplied.Forthefollowingsimulationresults,weselect
MCS-5andMCS-7,correspondingtoadatacoderate11 of0.37
and 0.76, respectively. Each encoded block is interleaved and
assignedtotwoorfourbursts,dependingontheselectedMCS
(MCS-5, MCS-7: 4bursts). Each burstconsists of2 58data Fig.11. Pdf’sofE (hhh)=E(cid:22) (cid:1)d (hhh)=d forMLSEanddifferent
symbols,26trainingsymbols,whicharelocatedinthemiddle channelprofiles.
of the burst (midamble), and three tail symbols at either end.
Theinterleavingtables ofthe EDGEstandard havebeen care- to (approximately) transform the CIR into its minimum-phase
fullyoptimized.Usingdifferenttablesmayentailasignificant equivalent,afixedFIRprefilteriscalculatedforeachburstvia
performancedegradation. the LP approach [30], [31], which uses the estimated channel
For receiver input filtering, an SRC filter with rolloff factor impulseresponse.Fortrackingofthetime-variantdiscrete-time
hasbeenselectedSectionIV-C.Becauseadditionalre- overallchannelimpulseresponse,theleast-mean-square(LMS)
liabilityinformationishighlybeneficialforchanneldecoding,a algorithm[50]isemployed.
BCJR-typesoft-outputRSSEalgorithmasproposedin[20]and Fig. 12 shows the resulting BERs after channel decoding
[21]isusedforequalization.Astatenumberof ischosen versus for the TU3 profile, where “3” indi-
(RSSE2),whichseemstobeespeciallyinterestingforapractical catesavelocityofthemobilestationof km/h.Frequency
implementation withonly moderate complexity.Two different hopping has been used in order to obtain a high diversity
equalization processes are performed. Both start from the mi- for channel coding. For each of the four bursts contained in
damble,thefirstoneproceedsinnegativetimedirectionandthe a block, a different frequency is used. Hence, the channel
secondoneinpositivetimedirectionalso[49].Forchanneles- impulse responses of different bursts in a block are mutually
timation, the standard ML approach [43] is adopted. In order statistically independent and interleaving is highly beneficial.
At , MCS-5 performs dB better than
10ThisingeneraltendstoholdforrandomlygeneratedCIRswithapproxi-
MCS-7. On the other hand, MCS-7 is twice as efficient as
matelyevenlydistributedenergyoverthewholetimespan,asfurthercalcula-
tionshaveshown. MCS-5 in terms of information bit rate (MCS-7: 44.8 kb/s,
11Forencodingofheaderinformation,differentcoderatesareused. MCS-5: 22.4 kb/s). A comparison of Figs. 3 and 12 shows
198 IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS,VOL.1,NO.1,JANUARY2002
for km/h especially at low BERs. Also, nonideal pre-
filteringcausesaperformancelossespeciallyforlongnonmin-
imum-phaseCIRs,whichfrequentlyarisefortheEQprofile.All
the described effectscannot be avoidedin a practical receiver.
TheyexplainwhythesimulatedBERcurveforuncodedtrans-
missionofFig.13isworsethantheresultspresentedinFig.4
forwhichpracticaleffectsarenottakenintoaccount.
Hence,theresultspresentedinSectionIVcanserveasguide-
linesforthedesignofequalizationforEDGE,buttheycannot
replaceadetailedsimulationofapracticalreceiverincludingall
nonidealeffectsandcomponents.
V. CONCLUSION
In this paper, equalization for EDGE has been considered.
Because of the high-level modulation applied in EDGE, in
contrasttoGSM,MLSEcannotbeemployedforequalizationin
Fig. 12. BER after channel decoding versus 10log (E(cid:22) =N ) for RSSE2 apracticalscheme.Ashasbeendemonstratedbyanalyticaland
for TU3 profile (MCS-5 and MCS-7, frequency hopping, imperfect channel
simulation results, DDFSE or RSSE with only a few number
estimation).BERofuncodedtransmissionisalsoshownforcomparison.
of states together with prefiltering are attractive solutions. A
state number of for DDFSE and of or
forRSSEseemstobesufficientforaperformanceratherclose
to the theoretical limit given by the performance of MLSE.
Furthermore,ithasbeenshownthatareceiverinputfilterwith
an SRC frequency response performs close to the optimum
WMF,whilecomplexityissignificantlylower.Also,ithasbeen
demonstratedviapdf’softheminimumEuclideandistancethat
theperformancelosscomparedtotheAWGNchannelismainly
caused by fading, not by ISI, if well-designed trellis-based
equalizers are applied. Finally, the effects of channel coding
and channel estimation have been considered. It is shown that
the loss due to ML channel estimation is limited, however,
especiallyathighvehiclespeedsimperfecttrackingoftheCIR
causesaconsiderabledegradationwhichmaybecompensated
bypowerfulchannelcoding.
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ofErlangen-Nürnberg.From1999to2000,hewasa
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99301299.6,Feb.1999.
Zealand,sponsoredbyafellowshipfromtheGermanAcademicExchangeSer-
[31] W.H.Gerstacker,F.Obernosterer,R.Meyer,andJ.B.Huber,“Aneffi-
vice(DAAD).Hiscurrentresearchinterestsincludeequalizationandchannel
cientmethodforprefiltercomputationforreduced-stateequalization,”
estimation,blindtechniques,space-timecoding,andnoncoherentdetectional-
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technik—Grundlagen und Anwendungen (in German). Berlin, missionandMobileCommunications),UniversityofErlangen-Nürnberg.Since
Germany:Springer-Verlag,1992. May2001,hehasbeenaPostdoctoralFellowwiththeUniversityofToronto,
[37] J.M.Cioffi,“Anerroreventlengthconjecture,”inProc.Conf.Specific Toronto,ON,Canada,sponsoredbyafellowshipfromtheGermanAcademic
Problems in Communication and Computation, Palo Alto, CA, Sept. ExchangeService(DAAD).Hiscurrentresearchinterestsareintheareaofnon-
1986. coherentdetection,equalization,space-timecoding,andmultiuserdetection.
