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radio-mobiles
Raphaël Visoz
To cite this version:
Raphaël Visoz. Traitement itératif et conjoint pour les systèmes radio-mobiles. domain_other.
Télé-com ParisTech, 2002. Français. �pastel-00001292�
mobiles
R.Visoz
Soutenue le 25 Mars 2002
Claude Berrou President
Pierre Humblet
Pierre Duhamel Rapporteurs
DanielDuponteil
PhilippeCiblat Examinateurs
Joseph Boutros Directeur de These
brise le complexe du monde en fragments disjoints, fractionne les problemes, separe ce qui
This work is the result of many in uences, and meetings. The main contributors from
a technical point of view to this thesis are Antoine Berthet (chapter 3,4,6,7), Elie Bejjani
(chapter 2), Hatem Boujemaa (chapter 5), Nikolai Nefedov and Markku Pukkila (chapter
4), without their help and their enthusiasm this thesis would not exist. A special thank to
Antoine, whoisnowaverygoodfriendaswellasmy favorite fellowworker. I hopethatour
future collaborationswillbe as fruitfulas they were duringthese last threeyears.
I amverygratefultoProfessor JosephBoutroswho acceptedtosupervisethis thesisand
gaveme precious advice and orientations toimprove itsoverall content.
I amalsogratefultoBertrandPenther andHiroshi Kubowhose contributions tothe
Eu-ropean Project BroadbandRadio Access Network (BRAN) were very helpfulto understand
and master the Generalized Viterbi Algorithm (GVA) [61]. These contributions [89] [71]
were specially important to me, since I consider them as the starting point of my work on
eÆcient trellis search techniques.
I wishtothank ProfessorPierreHumbletwhoseclass notes[63]keepsbeingmyreference
work and were speciallyuseful forchapter1. I hope thatthese class notes willbepublished
onaccount of their contents.
This thesiswasmadepossiblebyArmand Levywhonot onlyacceptedthat Icarriedout
a PhD inparallel tomy professionalactivities but alsoencouraged me. I also thank Daniel
Duponteil for giving me interesting work which always lay between advanced engineering
and research.
IamverygratefultothetworeviewersPierreDuhamelandPierreHumbletwhoaccepted
toreadcarefullyandcorrectthedraftofthisthesis. IwishtothanknamelyProfessorClaude
Berrou who accepted tobethe president of the jury, Professor Joseph Boutros who wasmy
eÆcientsupervisor, DanielDuponteil,andProfessorPhilippeCiblatwhoaccepted tobethe
other members of my jury.
Last but not least, thanks to Ainhoa for always being supportive, all the more during
L'egalisation pour lessystemes monoporteuses est un vieux domaine qui para^t peu
su-jet a des innovations. Cependant, on peut se demander si la separation fonctionnelle entre
egalisation, decodage et estimation de canal est pertinente [40][37]. En e et, les
traite-ments iteratifs faisant dialoguer plusieurs entites d'une m^eme cha^ne de communication se
sont averes extr^emement fructueux dans le domainedu codage et, plus recemment, dans le
domaine de l'estimation de canal et de l'egalisation. Le sujet de cette these est, en
fonc-tion des parametres du systeme etudie (typiquement la taillede l'entrelacement, letype de
canal radio mobile), d'essayer de combiner ces trois t^aches de la facon la plus performante
(la retransmission etant traitee comme une forme de codage canal). L'approche proposee
ne s'attache pas seulement aux performances obtenues mais aussi au souci de complexite,
a n de viser des applications industrielles. Trois contextes sont particulierement etudies:
les reseaux radio haut debits du type ATM sans l, le CDMA haut debit avec faible
fac-teurd'etalement,lessystemes TDMA avances(EDGE)avecmodulationd'ordreeleveet/ou
antennes d'emission etde reception multiple. A nde conserver lecritere Maximum A
Pos-teriori(MAP),touten gardantune complexiteabordable,lestechniquesd^tesde traitement
par survivant (sur treillis reduits) sont exhaustivement decrites et mises en pratiques pour
les contextes precedemment cites. Il est notamment demontrer que la generalisation du
traitementparsurvivant,consistanttressimplementagarderplusd'un survivantparnoeud
du treillis reduit, est tres robuste a la propagation d'erreur m^eme en presence de canaux a
phase non minimale. Cette generalisation fut originellement introduite par Hashimoto [61]
sous le nom d' algorithme de Viterbi generalise (GVA), la technique elle m^eme etant dans
1 Introduction 11
2 MatchedFilterBoundforMultichannelDiversityoverFrequency-Selective
Rayleigh-Fading Mobile Channels 15
2.1 Introduction . . . 15
2.2 GeneralApproach . . . 17
2.2.1 System Model . . . 17
2.2.2 Error Probability Lower Bound . . . 20
2.2.3 Asymptotic Behavior . . . 21
2.3 UnitaryTransformation E ect In Micro-Diversity . . . 22
2.4 Polarizationvs. SpatialMicro-Diversity. . . 24
2.4.1 PolarizationDiversity. . . 24
2.4.2 K x Matrix for Space/PolarizationDual-Diversity . . . 25
2.4.3 Space/PolarizationDiversity for GSMand IS-95 . . . 26
2.4.4 Rotatingthe BaseStation Antennas . . . 27
2.5 Coded Matched FilterBound . . . 29
2.6 Conclusion . . . 30
3 Joint Equalization and decoding using the Generalized Viterbi for broad-band wireless applications. 37 3.1 Introduction . . . 37
3.2 Proposed Algorithm . . . 38
3.3 Applications . . . 40
3.3.1 Application1: Cyclic block codes, jointequalizationand decoding on various full code trellises . . . 40
3.3.2 Application 2: cyclic block codes, joint equalization and decoding on reduced code trellises . . . 41
3.4 Conclusions . . . 46
4 Iterative Equalization and Estimation for Advanced TDMA Systems 53 4.1 Introduction . . . 53
4.2 System Model . . . 54
4.2.1 Notation . . . 54
4.2.2 The equivalentdiscrete-time channel model . . . 55
4.2.3 Conventional receiver . . . 56
4.3 Low Complexity Q-ary SISO Equalizer . . . 57
4.3.1 Decision Feedback Soft-In Soft-Out (DF-SISO)equalizer . . . 57
4.3.2 DF-SISO equalizer with forward recursion . . . 60
4.3.3 Minimum-phase pre- ltering . . . 61
4.4 Iterative Receiver for EDGE . . . 62
4.4.1 Turbo detection principle . . . 62
4.4.2 Q-aryturbo-detection with DF-SISO equalizers . . . 62
4.4.3 Iterative (turbo) channel estimation . . . 64
4.4.4 Combinediterative estimation-equalization . . . 65
4.4.5 Turbo equalizationforretransmission schemes . . . 66
4.5 SimulationResults . . . 67
4.5.1 Turbo detection withoutchannel re-estimation . . . 67
4.5.2 Turbo detection and LS-based channel re-estimation. . . 68
4.5.3 application of the Turbo equalization to GSM EDGE Radio Access Network . . . 68
4.6 Conclusions . . . 69
5 Iterative Low-Complexity Receiver for High Bit Rate CDMA 81 5.1 Introduction . . . 81
5.2 System model . . . 83
5.3 Equivalentchannel Model atthe Rake receiver output. . . 83
5.4 Low-complexity iterativeReceiver . . . 85
5.4.1 Turbo Rake SISO DFSE receiver . . . 85
5.5 Channel estimation improvement . . . 85
5.5.1 MMSE channel estimates. . . 85
5.5.2 iterative(turbo)channel estimation . . . 87
6 Iterative Receivers for Bit-Interleaved Coded Modulation over Wireless
Frequency-Selective Channels 93
6.1 Introduction and motivations . . . 93
6.2 Communication model . . . 95
6.3 Iterative multilayerdata detection and channel decoding . . . 96
6.3.1 SISO jointmultilayerdata detection . . . 96
6.3.2 A generalized reduced-state SOVA-likealgorithm . . . 97
6.4 Iterative channelestimation . . . 99
6.4.1 Initialchannel estimation . . . 99
6.4.2 LS-based turbo channel estimation . . . 101
6.5 Performance analysis . . . 102
6.5.1 Optimalreceiver . . . 102
6.5.2 Reduced-complexity multilayerdata detector . . . 102
6.5.3 Reduced-complexity receiver and its applicationto GERAN . . . 103
6.6 Conclusion . . . 104
7 Iterative Decoding of Serially Concatenated Multilayered Trellis-Coded Modulations in Multipath Rayleigh Fading Environment. 111 7.1 Introduction . . . 111
7.1.1 Wireless tranmission context . . . 111
7.1.2 EÆciency and limitsof fullturbo detection . . . 112
7.1.3 ImprovingspectraleÆciencyofseriallyconcatenatedspace-timetrellis coded modulation . . . 113
7.1.4 Chapter organization . . . 114
7.2 Serially concatenated multilayeredspace-time trellis coded modulation . . . 114
7.2.1 Communication model . . . 114
7.2.2 Iterative decoding . . . 116
7.3 Discrete-time nite-state Markov modeland associated possibly-reducedtrellis116 7.3.1 Discrete-time nite-state Markov modelfor elementary ST-TCM . . . 116
7.3.2 Combinedtrellis associated with elementary ST-TCM . . . 118
7.3.3 Multilayerdiscrete-timeMarkov modeland multilayercombined trellis 118 7.3.4 Reduced-state multilayercombined trellis. . . 119
7.4 SISO jointmultilayerdata detection and inner decoding . . . 119
7.4.1 Reduced-state trellis searchand generalizedper-survivor processing . 119 7.4.2 A generalized reduced-state SOVA-likealgorithm . . . 120
7.5 Performance analysis . . . 122
chan-7.5.2 Designof inner ST-TCM inconcatenated systems . . . 124
7.5.3 Serially concatenated multilayered ST-TCM (BLAST-likeapproach) . 125
7.6 Conclusion and future research topics . . . 126
8 Conclusions 133
A Wide Sense Stationary (WSS) Random Processes 137
B Karhunen-Loeve Expansion for Circularly Symmetric Complex Variables139
C Calculus of U
n
141
Introduction
Equalizationforsingle carrierwirelesssystems may appear at rst sightasa worn out eld,
where few innovations remain tobe done. However, one can wonder if the strict functional
splitbetweenequalization,decodingandchannelestimationthatexistsnowadaysinclassical
receiversisjudicious[40][37]. Indeed,iterativeprocessingmakingseveralentitiesofthesame
communication chain communicate was proved to be very pro table in the coding domain
and more recently in the channel estimation and equalization domains. The topic of this
thesisis tocombine thesereception basictasksinthe most eÆcientway taking intoaccount
the studied system constraints, i.e., interleaving size, wireless mobile channel type, and
channel coding. The proposed approaches not only optimize the performance but also the
complexity in order toaimat industrialapplications.
Three main contexts are studiedin this thesis:
wireless Local Area Network (LAN) with very high data rate and granularity
con-straints,
high data rate CDMA with lowspreading factor,
advanced wirelessTDMA systems with high order modulationand/orMultiple Input
Multiple Outputchannel.
In the second chapter, we analytically derive the performance of optimal receivers
over wireless mobile channels. The chosen approach is the Matched Filter Bound (being
a lower bound under idealized conditions, in particular perfect channel estimation and no
Inter-Symbol Interference) for multichannel diversity over frequency selective Rayleigh
fad-ing mobile channels. It allows a better understanding of the diversity concept, which is
important for mobile radio interface design; and to quantify, to a certain extent, the gains
characteristics, the data rate, the radio channel properties. It alsohelps to explain the
per-formance degradation due to InterSymbol Interference (ISI) and to identify sub-optimality
in the receiverdesign. This bound serves asa valuable benchmark throughout this thesis.
The third chapter deals with wireless LAN, that is to say systems that have strong
granularityconstraintsandnotimediversityatallduetoveryhighdatarates,i.e,the
chan-nel coherence time is by far larger than the permitted interleaving size (e.g., much larger
than an ATM cell length). A joint equalization and decoding approach without
interleav-ing seems to be the most appropriate one for single carrier radio interfaces. However, the
major problem lies here in the receiver complexity which increases exponentially with data
rates. This chapter is, thus, naturally oriented towards suboptimaltrellis search techniques
applied to joint equalization and decoding that enable to reduce signi cantly the receiver
trellis complexity. Finally, a new low-complexity receiver is proposed based on Generalized
Per Survivor(GPSP) Processing technique derived fromthe GeneralizedViterbi Algorithm
(GVA).This technique revealed particularlyrobust toerror propagationeven inthe case of
non-minimum phasechannels.
The fourth chapter looks into the gains brought by iterative processing or by the
"turbo principle" applied to advanced TDMA systems with high order modulation and
channel interleaving. We tried toinclude as many entities as possible into one iteration. A
low complexity trellis basedreceiver isproposed, performingiteratively channel estimation,
equalizationand decoding.
The turbo detection/equalizationscheme isoriginal inthe sense that itstands out from
theclassicalapproachesbasedonDecisionFeedbackEqualizers(DFE)orMaximumA
Poste-riori(MAP)equalizers. Here, weusedaSoft-InSoft-Out(SISO)DecisionFeedbackSequence
Estimation(DFSE) equalizer with pre ltering toturn the channelinto minimumphase.
Concerning channel re-estimation, many di erent methods can be used to tackle this
issue. Once again in order to keep the receiver complexity aslowas possible, we chose one
of the simplest: a linear approach derived from the so-called bootstrap technique (using a
LeastSquare(LS)estimator,oftenrefers asLS-basedchannelre-estimation). ACramerRao
bound is alsopresented and gives insights about its performance.
The fth chapter dealswith CDMAsystems usinglowspreadingfactor orequivalently
spreadingsequences withbad correlationproperties. Itiswellknownthat the Rakereceiver
performancedegradessigni cantlyforspreadingfactorlowerthen16entailingtheneedofan
Interference canceller atitsoutput. Fortunately, the Rakeoutput canbeformallyidenti ed
The use of a low spreading factor also degrades the performance of the conventional
channel estimatorwhich relies heavily on spreading sequence properties. This issue is also
addressed in this chapter by suggesting anew channel estimation algorithmthat takes into
account the ISI structure.
The sixth chapter aims at generalizing the previous approaches to a Multiple Input
Multiple Output (MIMO) channel for TDMA systems. A reduced-complexity trellis-based
receiver performing iteratively channel estimation, multilayer detection/equalization and
channel decoding is derived. The SISO multilayered data detector/equalizer is based on
the GPSP technique already introduced in the third chapter. Indeed this technique is well
adapted to this context since a pre lter turning every single channel into minimum phase
does not exist. The channel re-estimation is based on the generalization of the bootstrap
technique of chapter 4.
Ourapproachpresentstwo-foldadvantages. ItenablestocopewithsevereMIMOchannel
ISIand allowstousemoretransmitantennasthanreceiveantennas. Focusingonthecaseof
N transmit antennas and one receive antenna, which is particularlyinteresting for handset
mobile at the receiver end, the equivalence in terms of data rate and receiver complexity
between one 2
N
-order modulationand N paralleltransmitted BPSKis pointed out.
The seventh chapter investigates Iterative Decoding of Serially Concatenated
Multi-layered Trellis-CodedModulationsina MIMO frequency selectiveradio channel . Itcan be
viewed asanextensionofchapter3and 6inordertoincrease thespectral eÆciency andthe
Matched Filter Bound for
Multichannel Diversity over
Frequency-Selective Rayleigh-Fading
Mobile Channels.
2.1 Introduction
In modern time-divisionmultiple-access (TDMA)digital mobileradio systems, data signals
are transmitted in bursts. If the channel can beassumed time invariantfor the burst
dura-tion (slow fading channel) then the optimum L-channel diversity receiver is the Maximum
Likelihood Sequence Estimation (MLSE) receiver [69]. The performance of such a receiver
is given by averaging the error probability on all possible channel outcomes. For a given
channeloutcome theerrorprobability(inwhiteGaussiannoise)isassessed bythe minimum
Euclidiandistance between all possiblereceived sequences. Thisquantityis very diÆcultto
calculate for arbitrary dispersive channels, so that only numerical performance results can
be obtained via error trellis method [77, page 420].
A much simpler measure of performance can be obtained by neglecting the e ect of
Inter-Symbol-Interference (ISI)and deriving the BitError Probability (BEP) whena single
symbol is transmitted over a perfectly known channel. In this way, an absolute BEP
per-formance lower bound isobtained, which iscommonly known as the Matched Filter Bound
(MFB).Surprisingly,theMFBgivesinmanycasesthesameperformancesastheerrortreillis
method,meaningthat withinamultiplicativeconstant the errorprobabilityof the MLSEis
essentially thesame asthe MFBeven inpresence ofISI [77, page 448]. ForDirect-Sequence
bounds under idealized conditions, in particular perfect channel estimation and no ISI) are
very attractive because they are more easily derived in analytical form, and also because
these bounds are generally found to be tight enough (in comparison with MLSE optimum
receiver or RAKE receiver) to make them serve as valuable benchmarks for system design
and evaluation.
Inthischapter,amatched lterboundoftheerrorprobabilityforanL-branchesdiversity
system that uses any linear modulation over multipath frequency-selective fading channels
is derived. On the receiver side, it is assumed that L branches of diversity are obtained
by the use of L distinct antennas. The lower bound is based on the principles of matched
lterandMaximalRatioCombining(MRC).Performance boundshavealreadybeenderived
for several systems in the past [63], [4], [5], [98], [127],[66], [100], [78], [129], [35], [46]. For
example,in [98] the problemhas been formulatedinits most generalform,but only forthe
case of ideal linear equalization, while [66] treats the case of a single multipath Rayleigh
fadingchannel with independent taps only.
In [35] the matched lter bound is calculated in the frequency domain for L
uncorre-lateddiversity branches viaacontinuous KarhunenLoevetransformationwhichallowsboth
continuously dispersive channels and discrete multipath channels to be taken into account.
Moreover [35] considers colored noise but having the same power spectral density on all
branches. Finally, in [46], the matched lter bound results for uncorrelated L diversity
branches are extended toTrellisCoded Modulationwith perfect interleaving.
Ourformulationisthe rstonewhichaccountsatthesametimeforthefollowingaspects:
pulse shape, channel taps correlation, any number of diversity branches, power mismatch
of the di erent branches (especially useful for polarization diversity), envelope correlation
between the signals on di erent branches, and white noise with di erent power spectral
density on each branch. Moreover, the presented matched lter bound derivation is the
most straightforward as it relies on a unique Karhunen-Loeve expansion. In comparison,
the approachconsideredin[78]iscomposedof twosuccessive transformationsasitneedsan
extra Cholesky decomposition totake intoaccount any kind of correlation. In this respect,
it is shown here that the compactness of our analytical formulation allows us to get more
insightin important issues relatedto antennadiversity, such aspolarizationdiversity gain.
In Section2.2,the consideredsystem modelispresented and thetheoreticalerror
proba-bilitylowerboundusingKarhunen-Loeveexpansionisderived, togetherwiththeasymptotic
expression of the BER for BPSK. This result, already known for at fading channels [101],
is thusshown toapply for the more general case of frequency-selective fadingchannels.
In Section 2.3, with the use of our model, any unitary transformation on taps of equal
diversity as it is shown that rotating the base station antennas represents a special case
of unitary transformation. Thus, the widespread belief of added diversity gain for slanted
antennas at the receiver is proven to be wrong from the matched lter bound point of
view. In addition,by extendingthework ofVaughan[117], thesame resultwas numerically
con rmed. Polarizationdiversity gain is invariant vs. antennasrotation.
In Section 2.4, polarization diversity is overviewed as there is a renewed interest for
this kind of diversity, especially in mobile radio systems. Additionally,with the use of our
theoretical model, the spatial and polarization dual-diversity gains for Global System for
Mobile communication (GSM) [82] and Interim Standard 95 (IS-95 downlink) [114] mobile
systems are comparedin various cellularenvironments.
Finally,in section 2.5the coded Matched FilterBound ispresented.
2.2 General Approach
In this sectionwe givea detailed presentation of the system modeland the computationof
the matched lter error probability bound.
2.2.1 System Model
Theconsideredbase-bandsystemmodelisdepictedinFig.2.1. Itconsistsofaonedimensional
(complexorreal)linearmodulationwithapulseshape lterg(t)atthetransmitterside. The
transmitted signal passes through L time-varyingmultipathchannels h
i
(t; ) (i=1;:::;L),
assumed tobe perfectlyknown, and is received by L di erent antennas corresponding to L
diversity branches. Each channelis modeled by a discreteK
i
-taps time varying response
h i (t; )= K i X j=1 c ij (t)Æ( ij ); (2.1) where eachc ij
(t) isa zero mean complex Gaussianrandomvariable(Rayleighfading). The
additivewhiteGaussiannoise(AWGN)ofthei-thbranchn
i
(t)has apowerspectraldensity
N i
. TheLdi erentnoisesignalsareassumedtobeuncorrelated. Inthesequel, c
ij
andc
ij (t)
will be indistinctly used as the channel is assumed to be invariant for a symbol duration.
The channel taps c
ij
of the Lbranches can bearranged inthe followingvector form
x=[c 11 :::c 1K 1 c 21 :::c 2K 2 :::c L1 :::c LK L ] T : (2.2)
The covariance matrix K
x
of the vector x is given by K
x
= E(xx
y
), where y denotes the
conjugated transpose operator. It is important to note that, unlike previously published
is considered. This allows to treat, in the same framework, the most general case of tap
correlation withinone branch aswell asbetween di erentbranches.
Asthematched lterboundisconsidered,onlyonesymbolistransmittedandreceived[5].
Inthiscase,forthei-thdiversitybranchthematched lterforeachtransmittedsymbol|say
a|isinfactmatchedtoM
i
(t; )=g( )h
i
(t; )where*denotes theconvolutionoperator.
TheoptimalwaytocombinetheLbranchesistoperformMaximalRatioCombining(MRC).
Therefore, asthe noisesignalsofthebrancheshavedi erent at powerspectraldensitiesN
i ,
any branch i must beweighted by a factor 1=N
i
,[101]. Then the outputs of the L matched
ltersare sampledatthesymbolrate (perfectsynchronizationissupposed). Wethenobtain
the samples y i = a N i kM i (t; )k 2 +b i ; (2.3) whereb i
isthe Gaussiannoisesampleofthe i-thbranchoutputand kM
i (t; )k 2 isthetotal energy of M i (t; )given by kM i (t; )k 2 = Z d " K i X j=1 c ij (t) ? g ? ( ij ) #" K i X k=1 c ik (t)g( ik ) # : (2.4)
De ning the autocorrelation functionof g(t)as
g ( )= Z g ? (t )g(t)dt; (2.5) y i
can nowbe written
y i = a N i K i X j=1 K i X k=1 c ? ij g ( ij ik )c ik +b i : (2.6) Let c i =[c i1 ;c i2 ;:::;c iKi ] T , then we have y i =ac i y A i c i +b i ; (2.7) where A i is aK i K i
matrix with elements A
i jk = g ( ij ik )=N i . The L samples y i
are addedtogether togive the nal decision variable u
u= L X i=1 y i =az+b (2.8) with z = L X i=1 c i y A i c i (2.9) and b= L X b i : (2.10)
Recalling (2.2), itis easilyobserved that (2.9) can be expressed as
z =x
y
Hx (2.11)
where the matrix H isde ned as
H= 0 B B B B @ A 1 0 0 0 0 A 2 0 0 . . . . . . . . . . . . . . . 0 0 0 A L 1 C C C C A : (2.12)
Consequently, the nal decision variable is equal tothe transmitted symbola scaled by
the coeÆcient z and addedtoa Gaussiannoise sampleb. Theinstantaneoussignal-to-noise
ratio (SNR)of the combiner output (i.e. the variable u),for given values of c
i , is equal to = jE(u)j 2 E(ju E(u)j 2 ) (2.13)
Let us verify that the receiver reallyimplements MRC combining. As b is a zero mean
Gaussiannoise, and assuming that E(jaj
2 )=1 we have jE(u)j 2 = L X i=1 kM i (t; )k 2 N i ! 2 : (2.14)
The noise sample b
i
at the output of i-th branch is expressed as
b i = 1 N i Z d n i ( )M i (t; ): (2.15)
As a resultthe varianceof the decisionvariable u can be written
E(ju E(u)j 2 )= 1 N i N j L X i=1 L X j=1 Z Z dudvE[n i (u)n j (v)]M i (t;u)M j (t;v): (2.16)
Usingthe properties of the noise signals, we have
E(n i (u)n j (v))=N i Æ(u v)Æ ij : (2.17)
Thuswecan write
E(ju E(u)j 2 )= L X i=1 kM i (t; )k 2 N i (2.18)
Equations (2.13),(2.14) and (2.18) yield
= L X kM i (t; )k 2 N i (2.19)
Finally,using (2.3) and (2.8), one can easilyverify that =z = L X i=1 i (2.20) where i =k M i (t; ) k 2 =N i = E i s =N i
is the instantaneous signal-to-noise ratio of the i-th
branch (E
i
s
is the instantaneous received symbol energy for the i-th branch). Thus, adding
the samplesisequivalenttoperformingMRCof thediversitybranches [101]. Therefore, the
combination of ideal matched lters and MRC leads to the best theoretical performance,
meaning that it corresponds to the lowest bound of the error probability that a real life
receivercan attain. However, as underlinedbefore, this bound isvery closeto the optimum
receiverperformance.
2.2.2 Error Probability Lower Bound
As the autocorrelation function
g
(t) has an Hermitian symmetry and is non-negative
def-inite, it follows that matrix A
i
is Hermitian non-negative de nite. By extension, matrix
H is also Hermitian non-negative de nite, and of dimension KK where K =
P L i=1 K i . Therefore, p
H exists so that z can be writtenz =v
y
v, where v=
p
Hx.
Knowing that x is circularly symmetric (i.e., E(xx
T
) E(x)E(x
T
) = 0, T denoting
the transpose operator) and that its covariance matrix is real (see Appendix A), v is also
circularly symmetric as it is obtained by a linear transformation of x. Using the results of
Appendix B, vcan be writtenas
v= K X i=1 q i u i ; (2.21) where q i
are complex circularlysymmetric random variables with variances the eigenvalues
of K v =E(vv y ), and u i
are complex mutually orthonormal vectors. As a result, z can be
writtenin the simple form
z = K X i=1 jq i j 2 ; (2.22) wherejq i j 2
are independentchi-square randomvariableswithmeansthe eigenvalues
i (i= 1;:::;K) of K v = p HK x p H or equivalently of HK x , K x = E(xx y
) being the covariance
matrix of x. Note that the matrix H takes into account the pulse shape characteristics in
combinationwith the channel delay spread, whereas the matrix K
x
combines the e ects of
the power pro lesof the Lchannels, the correlations(between taps of one branchas wellas
taps of di erentbranches), and the powermismatchamong the L branches.
In case ofunequal eigenvalues, the probability distribution of z isgiven by
p z (z)= K X i i e z i (2.23)
where i
are the residues (
i = Q k6=i i i k ).
More generally,someeigenvaluesmaybeequal,whichisthecaseforinstancewhen
treat-ingindependentdiversitybrancheshavingthesamechannelpro le. Thiscasewasexamined
extensively in many papers such as [100]. Alternatively, one can arti cially separate the
equal eigenvalues by a very small amount. In this way, the distribution (2.23) can stillbe
used and yields resultsvery close to the exact approach.
Weknowfrom(2.20) thatthe instantaneoussignal-to-noiseratioof the decisionvariable
u is = z. Moreover, using (2.22), the mean signal-to-noise ratio when considering the
channel variationsissimply =z=
P
i
i
. Thenthe averageerror probabilityversus can
be obtained as P e ( )= Z 1 0 dz p z (z)P 0 (z) (2.24) where P 0
( ) isthe error probabilityof the chosen modulationinAWGN channel.
For BPSK modulation, (2.24) can be solved analytically, so the matched lter bound
error probability can be expressed as [66], [100], [78]
P e ( )= 1 2 K X i=1 i 1 r +1= i (2.25) where i = i = P i
are the normalizedeigenvalues.
One can check that in a at Rayleigh fading channel (K = 1) the BER reduces to the
well known equation [92]
P e ( )= 1 2 1 r +1 : (2.26) 2.2.3 Asymptotic Behavior
It iswellknown, inthecase of K branches diversity with at Rayleighfading, thatfor large
signal-to-noiseratio ,theBERbehavesas
K
. TheorderofdiversitythusissaidtobeK.
Moreover, when the signals on the branches are correlated or have di erent energy levels,
the BER will still have the same asymptotic slope but su ers a degradation in SNR given
by the amountof signalcorrelation and/or energy mismatch[101].
Our goal is to extend this notion to the general case of frequency-selective Rayleigh
fadingchannels. The asymptotic error probability of BPSK for large SNR can be obtained
by developing (2.25) inthe following manner
P e ( )= 1 2 1 X n=1 ( 1)C n 1 2 U n 1 n (2.27) where U n = K X i i n (2.28)
and C k q = q(q 1)(q k+1) k! (2.29)
It is demonstrated in[14] (see Appendix C) that
8 > < > : U n = 1 n =0 U n = 0 n =1;:::;K 1 U n = ( 1) K+1 Q K i=1 i n =K (2.30)
Thus, for suÆciently large , (2.27)can be wellapproximated by its rst non-zero term
P e ( )' ( 1) K 2 C K 1 2 K Q K i=1 i (2.31)
orin a moresimple form
P e ( )' C K 2K 1 Q K i=1 (4 i ) (2.32) Note that Q K i=1 i is proportional to det (HK x
). Of course, even in the presence of
multipath, the asymptotic order of diversity remains equal to the total number of paths
K (of the L branches), regardless of their relative time delays and power levels. More
interesting, the product
Q K
i=1
i
is suÆcient todetermine the asymptotic SNR degradation
caused by the pulse shape autocorrelation (via H), the taps correlations, and the power
mismatch between branches (via K
x
). The product of normalized eigenvalues appearing in
(2.31) con rms the intuition that for diversity, it is better to have many small eigenvalues
than afew large ones. The highest diversity gain isobtained when
Q i ismaximized,that is for i =1=K ( P K k=1 k
=1), in which case (2.31) reduces tothe well known asymptotic
error probability forK independent equal energy diversity branches [92]
P e ( )= K 4 K C K 2K 1 (2.33)
Unfortunately some very small eigenvalues often appear in the Karhunen-Loeve
expan-sion, making this asymptoticlimitvalidonlyfor extremely high SNRvalues, far away from
the range observed in practice (10 to 30dB).
2.3 Unitary Transformation E ect In Micro-Diversity
Micro-diversitymeansthattheantennaspacingissmallenoughtoconsiderthatthechannels
on the L branches have the same delay pro le, i.e. K
j =K 0 and ij = j . This is actually
the case for all multichannel diversity systems with antennas located in the same site (few
the diversity branch number) willbe all identical, i.e. A i
=A. It isalso assumed that the
noise signals have the same spectral density, which is the case for polarization and spatial
diversity, most of the time.
In this section, it is shown that applying the same unitary matrix transformation on
each of the K
0
L-taps vectors (each vector contains the taps of the same index) leaves the
diversity gain unchanged.
Let U be a unitary transformation matrix of dimension LL (i.e. U
y
U=I, where I
denotes the identity matrix), and t
i =[c 1i c 2i c Li ] T
be a vector of the L taps of the same
delay i . t i 0 =Ut i (2.34)
By notingfrom (2.2) that the taps inx are arranged ona branch by branch basis, the new
taps y can be deduced from xby anequivalentLK
0
LK
0
unitary transformation
y= x (2.35)
where isa block matrix of the form
= 2 6 6 6 6 4 11 12 1L 21 22 2L . . . . . . . . . L1 L2 LL 3 7 7 7 7 5 (2.36) and ij are K 0 K 0
diagonalmatrices relatedto the transformation U by
ij
=U
ij
I: (2.37)
By substitutingy as given in(2.35) into(2.11), we easilyshow that z becomes
z =x
y y
H x: (2.38)
Finally,referring to (2.12),H is ablock matrix of the form
H= 2 6 6 6 6 4 A 0 0 0 A 0 . . . . . . . . . . . . 0 0 A 3 7 7 7 7 5 (2.39)
It can be easilyveri ed that H and commute. Using the fact that is a unitary
trans-formationwe nallyget
z =x
y
Hx: (2.40)
the sameindex. InSection2.4.4wepointout thefactthat antennarotationforpolarization
diversityisaparticularcaseofsuchunitarytransformation,andthusthatitdoesnotprovide
any additionalgain.
2.4 Polarization vs. Spatial Micro-Diversity
In this section, the case of dual-diversity (L =2) for both polarizationand space diversity
techniquesisconsidered. Itisassumed thatthe AWGN hasthe samepowerspectraldensity
for the 2 branches (this is the case for almost all pratical systems). The subject of space
diversity has been widely studied for the past several decades [64], [101]. In general, the
main disadvantage ofspace diversity isthe existenceof anon-negligiblecorrelation between
the di erent branches especially when the multipath angular spread of the channel is very
narrow[1], [2]. However, the localmean power is generallythe same on the two branches.
2.4.1 Polarization Diversity
Although polarization diversity has been well known for over 20 years [76], [117], [64],
[101],[1], [2], space diversity schemes have been preferred as polarization diversity su ers
from astrong imbalance between the localmean powers received on itstwobranches. This
imbalance is commonly referred to ascross-polar-discrimination(XPD, denoted by
here-after) and is the ratio of the received vertical and horizontal polarizationpower. However,
two main arguments can explain the increasing popularity of polarization diversity
nowa-days. First of all,the miniaturization of base stations makes the antenna spacing required
by space diversity both costly and inconvenient. Secondly, polarizationdiversity is very
at-tractive with handheld portablesas their moving antennas are onthe average closer to the
horizontal,which decreases the XPD.
The nature of polarization diversity relies on the elementary processes responsible for
the depolarization of electromagnetic waves. Three di erent processes are responsible for
depolarization: scattering from rough surfaces, di raction [124], and Fresnel re ection. As
the later has the greatest impact inmobile channels, we willbrie y present it.
Two Fresnel re ection coeÆcients R
?
and R
k
are de ned inreference tolocalaxes.
the orthogonal polarization(?) isde ned by the electric eld component included in
the plane de ned by the normal tothe obstacle and the propagation vector,
In general, the propagation occurs mainly in the horizontal plane, either in microcells
(where the antenna is well under roof tops) or in macrocells [64]. On the other hand,
the scatterers are mainly vertical (especially in urban environments). In Fig. 2.2 it is
obviousthat thepolarizationparalleltotheobstacleisless attenuated,thereforethevertical
polarizationis strongly favored atthe expense of the horizontalone. This explains why the
verticalpolarizationremainsthestrongestevenwhentheportablemobileantennaisinclined
at70
0
fromthe vertical [79].
Moreover, the phase di erence introduced by Fresnel coeÆcients between vertical and
horizontalpolarization(Fig. 2.2) ensures that the received horizontaland vertical polarized
signals are merelyuncorrelated [76]. Indeed,experimentalresults show anenvelope
correla-tion around 0.2[76], [79]. It is wellknown that a correlationcoeÆcient under 0.5has small
impact ondiversity [64],[101], [1].
2.4.2 K
x
Matrix for Space/Polarization Dual-Diversity
As shown in Section 2.2, the matched lter bound performance is given by the eigenvalues
of the matrix HK
x
. The exact structure of matrix K
x
for both space/polarization
dual-diversity is K x = " C 1 C 2 # : (2.41)
Assuming that the taps of each channel are uncorrelated, C
1 , C 2 and are K 0 K 0
diagonalmatrices with diagonalelements C
1i =E(jc 1i j 2 ),C 2i =E(jc 2i j 2 )= 2 i E(jc 1i j 2 ),and i =E(c 1i c ? 2i )= i i E(jc 1i j 2
)respectively. The coeÆcient
i 2
denotes the power mismatch
between the i-th taps of the 2 channels, whereas
i
represents the correlation coeÆcient of
those same taps.
Obviously, (2.41) covers the most general situation where power mismatch and
correla-tion vary from one tap to another. Unfortunately, no such ne channel measurements are
available, neither for space nor for polarization diversity. However, when the number of
scatterersisrelativelyhigh andallthechanneltaps areRayleighfading, itissafetoconsider
these coeÆcients asconstants |say and |for all the K
0 taps.
Space diversity presents no power mismatch between branches, thus
2
= 1. Inversely,
polarization diversity exhibits power mismatch between its two branches |as previously
highlighted| given interms of XPD (dB).This means that
2
=10
=10
forpolarization
diversity.
Finally,notethatonlytheenvelopecorrelation
env
isavailableexperimentally. However,
it is shown in [101] that is very close to
p
env
. From now on, all results will be given in
terms of
2
2.4.3 Space/Polarization Diversity for GSM and IS-95
The GMSK modulation used in GSM is well approximated by binary linear modulation
with the rst pulse shape ofLaurentdevelopment[75]. However theMinimum-ShiftKeying
(MSK) approximation is practically suÆcient. MSK can be modeled as a linear OQPSK
(O set Quadrature Phase-Shift Keying) with a pulse shape lter g(t) = cos (
t 2T b ) of time duration2T b , whereT b
isthe bitduration[88]. Indeed, inthe rangeof the commondelay
spreadofmobileradiochannels, theautocorrelationfunctionofthe rstLaurentpulseshape
is very close to that of the MSK.
On theotherhand,the IS-95system usesaRaised-Cosinepulseshapeg(t)with aroll-o
factor =0:33 and a binary modulation indownlink, at a chip rate of 1.2288 Mcps. Note
thatwedonot consider powercontrolfortheIS-95downlink. Ideally,itisassumed thatthe
spreadingsequences ensureaperfectDiracÆ( )autocorrelationfunction,andthatinterfering
users can be modeled by additive Gaussian noise (an acceptable approximation except for
very low number of interferers). In this case, the performances are uniquely determined by
the autocorrelation function
g
( ) of the pulse shape,which isknown tobe[92]
g ( )= sin( =T c ) =T c cos( =T c ) 1 4 2 2 =T 2 c (2.42)
Although many authors stilluse the cumulativeprobability distribution of the received
energy(i.e. theprobabilityoftheenergybeinggreaterthanagiventhreshold)toevaluatethe
diversity gain [76], [117], we prefer tocompare the polarizationand space diversity schemes
on the basis of the gain obtained for a given biterror probability. In fact, the later gain is
much more relevant than the former in the case of digital systems [101]. Inthis respect, we
consider that for uncoded bits (class II bits of the GSM frame) a BER equal to510
3
is
enough to ensure a good speech quality. The same value will be considered in the case of
IS-95in order tocompare the two systems.
It is emphasized here that the relevant antenna diversity gain G
2
is the one obtained at
510
3
BER afterexcludingtheinherentmultipathdiversitygain G
1
duetothe multipath
channel (as depicted inFig. 2.3).
Experimentalmeasurements showed that the averagevalueof XPDinurban and
subur-ban environments is between 1 and 10 dB with an average value of 6 dB [79], [76], and in
rural environments (e.g. Hilly Terrain) that XPD is very high, ranging from 10 to 18 dB.
This can be understood by the fact that a very small amount of energy is transposed from
one polarization to another due to the lack of scatterers in rural areas (scatterers are
re-sponsible for depolarizationvia Fresnel coeÆcient, see Section2.4.1). Polarizationdiversity
isthenuninterestingforthis typeofchannel. In thefollowing,atypicalenvelopecorrelation
GSM, and inFigs. 2.6and 2.7 for IS-95. As areference, the diversity gain for a at fading
(1tap) Rayleighchannelisplotted ineach gure. The antennadiversity gainis the highest
for this channel, which has no intrinsic multipathdiversity.
One general observation is that antenna diversity gain is high when the multipath
di-versity gain is low and vice-versa. Note that multipath diversity gain is small either if the
channel has a low intrinsic diversity, or if its diversity has not been exploited by the pulse
shape. ForIS-95, the multipathdiversity is wellresolved by the relatively shortdurationof
the pulse g(t) which makes antenna diversity useless in practice as its gain rarely exceeds
1 dB. However, multiple antennas are always useful for interference reduction [127], beside
that of increasing the average signal to noiseratio.
Furthermore, our results show that the two branch polarizationdiversity gain is almost
equivalenttothatof spatialdiversity inallurban/suburbanenvironments. Spatialdiversity
clearlyoutperformspolarizationdiversityonlyinruralenvironments(e.g. HillyTerrainHT).
2.4.4 Rotating the Base Station Antennas
The possible improvementof polarizationdiversity gain through spatial rotationof the two
receiver's antennas is now analysed. In fact, slanted antenna polarizationdiversity is very
popular nowadays [79], [117]. A greatnumberof manufacturersclaim that it achievesmore
diversitygain(upto1.5dBextragain)thanordinaryvertical/horizontalpolarizedantennas,
themainreasonadvancedbeingtheimprovementofpowerbalancebetweenthetwodiversity
branches.
The new XPDafterrotationis rst derived together with thenew correlation coeÆcient
aftera rotationof the base stationantennas by anangle fromthe vertical (Fig. 2.8).
Let the horizontaland vertical received electric elds be respectively
E x1 =jx 1 jcos(!t+ 1 ) (2.43) E x 2 =jx 2 jcos(!t+ 2 ) (2.44) where x 1 = jx 1 je j 1 and x 2 = jx 2 je j 2
are two correlated circularly symmetric complex
Gaussianvariableswith an XPDequalto
1
and correlation coeÆcient
1 .
After rotation the electric eld received on the two rotatedantennas, E
y1 and E y2 , can be deduced from E x1 and E x2
by a rotation matrix. By linearity, the relation between the
complex envelopes is the same. As a result, antenna rotationis equivalent to a rotationof
any 2 channel taps of the same index. In a genericway, we have
" y 1 y 2 # = " cos sin sin cos #" x 1 x 2 # (2.45)
where (x 1
;x
2
) denotes the pair of initial taps, and (y
1 ;y
2
) the pair of rotated taps. This
transformation is a particular unitary transformation for L = 2 which has been shown not
tohaveanyimpactondiversitygain(Section2.3). Therefore,thewidespreadbeliefofadded
diversity gain is completelyfalse from the matched lter bound point of view. It turns out
that polarizationdiversity gain is invariant by antenna rotation.
Considering now the argument of a more balanced power distribution between the
di-versity branches, one should note that in reality there is a tradeo between the powerlevel
balance and the correlation of the two branches. After rotation, the new taps exhibit new
XPDand cross-correlationcoeÆcients
2
and
2
that are functionsof the initialones
1
,
1
and of the rotation angle .
We must evaluate 2 = E(jy 2 j 2 ) E(jy 1 j 2 ) (2.46) and 2 = E(y 1 y ? 2 ) p E(jy 1 j 2 )E(jy 2 j 2 ) (2.47)
After astraightforward development itis found
2 = f 1 ( 1 ; 1 ; ) f 2 ( 1 ; 1 ; ) (see Fig. 2.9) (2.48) 2 = (1 1 )tan + 1 p 1 (1 tan 2 ) p f 1 ( 1 ; 1 ; )f 2 ( 1 ; 1 ; ) (see Fig. 2.10) (2.49) wheref 1 ( 1 ; 1 ; )=tan 2 + 1 +2 1 p 1 tan andf 2 ( 1 ; 1 ; )=1+ 1 tan 2 2 1 tan p 1 .
Theseresultsextend thoseof[117],wherethee ect ofrotatingthe basestationantennas
was alsostudied but inthe special case of independent Rayleighfadingsignals (
1
=0).
By using (2.48) and (2.49) for various combinations of channel type, initial XPD, and
initialcorrelationcoeÆcient, toevaluatethediversitygain asbefore, itwasnoticedthatthe
gainremained unchangedforwhateverrotationangle considered. Thisresult con rmsthe
one obtained above using the property of unitary transformations.
Consequently, even ifthe reportedextragains arevalid, theyshouldnot beattributedto
diversity itself,but rathertosome imperfectionorsub-optimalityofthe consideredreceivers
which may be more sensitive to the power imbalance than to branch correlation. We
be-lieve that the measured diversity gain would be practicallythe same for vertical/horizontal
and slanted polarization if the receiver is well designed. We conclude that the question of
system performance with polarizationdiversity should not be limitedto the study of signal
propagation aspects (attenuation, spatial correlation, power mismatch), but should rather
advan-simple, and therefore far from optimal. To the best of our knowledge, no such combined
investigations are availablein the literature.
2.5 Coded Matched Filter Bound
This sectionis very inspiredby [46] where the MFB isderived fortrellis coded modulation.
Consider the transmission of trellis coded symbols a corresponding to a particular path
through the code trellis. An error event of length T of the decoded sequence
e
a is taken
to start with symbol a
1 6= a~
1
and end with symbol a
T 6= a~
T
. Along the error path, the
T T nonzero branch metrics d
2 i = ja i ~ a i j 2
are accumulated leading to the Euclidian
distance d 2 (T) = P T i=1 d 2 i
, in the case of static channels, the error event with minimal
euclidiandistance d
2
min
isdominantbutinfadingenvironmentthee ective codelengthT
min
(minimumnumberof branches with non-zero branch metrics)may be more important. Let
consider an error event of length T, the individual SNR
i
along the error path are to be
weightedbyd
2
i
and sumtoformthe e ectiveSNRwhichenablestoobtaintheperformances
afterthe decoder. The e ective SNR (at the outputof the decoder) becomes
e = T min X i=1 d 2 i i (2.50)
Onecanseethattheprobabilitydistributionof
e
iseasilyderivableonlyinthetwofollowing
cases:
the
i
are independent variables,
the
i
are completely correlated (i.e. 8i;
i
= ), this is typically the situation of
chapter 3.
These cases correspond to either nointerleaving orperfect interleaving. Unfortunately, the
reality liesalways between these twocases. This is the reason why wepreferred tofocus on
the uncoded MFB which is independent of the interleaving scheme. Moreover, simulations
Fig. 2.11showthattheuncodedMFBisveryclosetorealityforGSMandIS95systems. As
statedintheintroduction,theMFBdoesnottakeintoaccounttheISI.Ifthe Uncoded MFB
isattainedattheoutput ofthe equalizerthenitcan onlymeansthat the distributionofthe
minimum distance associated to each channel outcome, has a large peak at the Gaussian
2.6 Conclusion
Inthischapter,thematched lterandMRCboundforL-branchantennadiversityandlinear
modulation over frequency-selective Rayleigh fading multipath channels has been derived
using a novel compact approach. The comparison of the space/polarization diversity gains
forGSManddownlinkIS-95systems(binarymodulationinbothcases)showsthatthereisa
trade-o between multipathdiversity gainandantennadiversitygain. Polarizationdiversity
has alsobeen shown toprovidealmostthe samegainasspatialdiversity,especiallyinurban
environments. Moreover,withthehelpofourgeneralmodel,ithasbeenpossibletoprovefor
the rsttimethatpolarizationdiversitygainisinvariantbyrotationofthereceiverantennas,
whichisincontradictionwith generalbelief. Some considerationwerealsogiven concerning
the MFB extended to trellis coded modulation. It was nallyconcluded that the uncoded
MFB was of more practical interest since the uncoded MFB is relatively tight to existing
system performance(e.g., GSMand IS95) andisveryusefultoidentifysuboptimalityinthe
Figure2.1: System modelwith L diversity branches.
0
5
10
15
20
25
30
35
40
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
BEP versus average SNR
Average SNR
BEP
Rayleigh, L=1
BU, L=1
BU, L=2
χ
=1 [dB]
ρ
env
=0
BU : Bad Urban
L : number of diversity branches
G1
G2
Figure 2.3: Multipath diversity gain and antenna diversity gain. Example of GSM system
in BadUrban Channel (BU) [36].
0
2
4
6
8
10
12
14
16
18
0
1
2
3
4
5
6
7
Polarization diversity gain versus XPD with
ρ
env
= 0.2
χ
[dB]
Gain at 5e
−
3
TU : Typical Urban
BU : Bad Urban
HT : Hilly Terrain
Rayleigh
Micro−Cellular
TU
BU
HT
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
1
2
3
4
5
6
7
Spatial diversity gain versus
ρ
env
with XPD = 0
ρ
env
Gain at 5e
−
3
TU : Typical Urban
BU : Bad Urban
HT : Hilly Terrain
Rayleigh
Micro−Cellular
TU
HT
BU
Figure2.5: Spatialdiversitygain vs. envelopecorrelation inGSM,for variouschannels[36].
0
2
4
6
8
10
12
14
16
18
0
1
2
3
4
5
6
7
Polarization diversity gain versus XPD with
ρ
env
= 0.2
χ
[dB]
Gain at 5e
−
3
Rayleigh
TU
HT
Micro−Cellular
BU
TU : Typical Urban
BU : Bad Urban
HT : Hilly Terrain
Figure 2.6: Polarizationdiversity gain vs. XPD (dB) in IS-95 for various channels [36],
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
1
2
3
4
5
6
7
Spatial diversity gain versus
ρ
env
with XPD =0
ρ
env
Gain at 5e
−
3
TU : Typical Urban
BU : Bad Urban
HT : Hilly Terrain
MC : Micro−Cellular
Rayleigh
HT
BU
MC
TU
Figure2.7: Spatial diversity gain vs. envelope correlationin IS-95for various channels[36].
x
2
x
1
y
2
y
1
x
2
+
Eƪā|y
2
|
2
ƫ
Eƪā|y
1
|
2
ƫ ,ā ò
2
a
x
1
+
Eƪā|x
2
|
2
ƫ
Eƪā|x
1
|
2
ƫ ,ā ò
1
Vertical
Horizontal
0
2
4
6
8
10
12
14
16
18
0
5
10
15
20
25
30
35
40
45
0
2
4
6
8
10
12
14
16
18
20
χ
1
[dB]
α
[°]
χ
2
[dB]
Figure 2.9: New XPD 2(dB) as a function of the initial XPD
1
(dB) and the rotation
angle of the base station antennas with aninitialenvelope correlation coeÆcient of 0.2.
0
2
4
6
8
10
12
14
16
18
0
5
10
15
20
25
30
35
40
45
0
0.2
0.4
0.6
0.8
1
χ
1
[dB]
α
[°]
ρ
2
Figure 2.10: Correlation coeÆcient
2
as a function of the initial XPD
1
(dB) and the
rotationangle of the base station antennas with aninitialenvelope correlation coeÆcient
0
5
10
15
20
25
30
35
40
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
error probability typical urban
SNR db
Pe
MSK
rayleigh
inf
cossap
Figure 2.11: Comparison of the Matched Filter Bound to a Monte Carlo simulation: MSK
represents the Matched Filter Bound for GMSK transmitted through TU [36] channel and
Joint Equalization and decoding using
the Generalized Viterbi for
broadband wireless applications.
3.1 Introduction
The optimal way todecode trellis encoded signals transmitted onInterSymbol Interference
(ISI) Channels isto use the Maximum Likelihood(ML)"supertrellis",acombinationof ISI
anderror-controlcodetrellises,whosestate complexityisthe product ofboth[34].
Unfortu-nately,forfrequency selectiveradio channels, thenumberofstates oftheISI trellisincreases
exponentiallywith the bit rate, which precludes this approachfor broadband wireless radio
interfaces. Asaconsequence, alotofworkhasbeen doneonsub-optimalreceiversforTrellis
Coded Modulation(TCM) in the presence of ISI [34], [126], [38], [44]. In [34], a systematic
method is developed for lowering the state complexity of the supertrellis. An interesting
case arises whenthe receivertrellis isreduced tothe code trellis [34],[126], [38], [44], whose
complexity does not depend on bit rate. The ISI due to the channel is not taken into
ac-count in the trellis states but in the edge metric, as done in a classical Decision Feedback
Equalizer(DFE).Itfollowsthatsuchareceiver, commonlycalledParallelDecisionFeedback
Decoding (PDFD),inherentlysu ers (as the DFE)fromerror propagation,especiallyinthe
case of non minimumphase channels. Therefore, PDFD receiver needs pre- ltering toturn
the channelintominimum phase. This pre- lteringis cumbersome andincreases theoverall
receivercomplexity. Besides, error propagation stillremains.
In parallel, many e orts have been devoted to improve sub-optimal equalization
tech-niques for broadband wireless channels. Once again, the issue is the complexity of the ML
to ght against errorpropagation.
The proposed receiver combines the PDFD algorithmwith the GVA. Simulationsprove
thattheGVAmakesthePDFDreceiververyrobusttoerrorpropagation(eveninthecaseof
nonminimumphasechannels)forareasonablecomplexityincrease. Itiseven shown thatin
most cases the ML optimalperformance isattainedwith only foursurvivors per state. The
paperisorganized asfollows. Section3.2describesindetailsthe proposedalgorithmforany
error-controlcodetrellis. In Section3.3some possible applicationstogetherwith simulation
results are presented for cyclic block codes, convolutionalcodes, for static and time-varying
multipathRayleighchannels. Notably, the algorithmwas proved toperform well for simple
convolutional codes, in the contextof Broadband Radio mobilechannels.
3.2 Proposed Algorithm
The discrete time equivalent structure of the proposed communication model is shown in
Fig. 3.1. The data signals are transmitted in bursts containing N coded data symbols
andaknowntrainingsequence (locatedatthe beginningoftheburst)used bothfor channel
estimationandalgorithminitialization. Letthe estimated(symbolspaced) impulseresponse
of the convolution of the transmitter lter, the receiver lter and the radio-mobile channel,
be denoted fh l g l ;l 2[0;K 1]; (3.1)
where Kis the channel constraint length.
Note that the overall channel does not need to be minimum phase [29, page 78]. The
receiver lter should ensure however that noise samples atthe symbol rate R
s
are
uncorre-latedatitsoutput,whichisthe case forasquared-rootraised cosine lter forexample. The
output of the received lter at timinginstantt is given by
y t =h 0 x t +b t +I t ; (3.2) where x t
is the current coded data symbolto be received, b
t
is a Gaussiannoise sample,
and I
t
is the ISI contributionterm with
I t = K 1 X l =1 h l x t l (3.3)
Wede netheerror-correctingcodeCinageneralsense,seeingitasatime-variantMarkovian
process. The coded symbolsx
t
are then relatedto the incomingbinary sequence by a
time-variant relationshipof the form
x t = (u t ;u t 1;; u t L ); (3.4)
taking into account encoding and bit mapping operations, where L t
is the instantaneous
code constraintlengthattimet. Inthegeneralcase,forboth blockandconvolutionalcodes,
the coded sequences produced by C can be described by an irregular trellis T(V;E;#) of
rankN whereV andE respectivelydenotethe vertex andedgespaces and# theset ofedge
multivaluations. We alsointroduce V
t
and E
t
the vertex and edge subspaces at time index
t of complexitiesjV
t
j and jE
t
j. Withthose notations, we have
jVj= N X t=1 jV t j jEj= N X t=1 jE t j W = max 0tN jV t j: (3.5)
For time-invariant Markovian processes, such as convolutional codes, or linear cyclic block
codes, the (regular) trelliscan be brought back toa single section.
The PDFD evaluates the ML metric
= N X t=1 y t h 0 x t b I t 2 ; (3.6)
on the full C trellis, where
b I
t
is the estimated ISI evaluated, as in the DFE, by the use of
a traceback array of size in O(jVj) that saves the path leading to a given survivor [126] at
every time t<N. A pathisasuccession of edges,eachone carryingthe input bit#
(1)
=u
t ,
the output producedsymbol#
(2)
=x
t
, and the departure and arrivalvertices.
The novelty in the proposed algorithmconsists in combining classical PDFD and GVA,
thus keeping at each vertex the S best incoming paths instead of a single one, and storing
them ina generalized traceback array of size in O(SjVj).
Let
t
i
denote a vertex of label i, 0 i jV
t
j 1, at time t, 0 t N, and e
t 1;t
i !j
the
edge associated with transition
t 1 i ! t j . Let also M k ( t i
) be the accumulated vertex
metric (or path metric) at terminationvertex
t
i
of the survivor of rank k; 0k S 1 .
Wede ne k (e t 1;t i !j )= y t h 0 x t d I (k) t 2 (3.7)
the edge metric for the transition e
t 1;t
i !j
associated with the k
th
survivor stored at time
t 1 .
The so-called Generalized Parallel Decision-Feedback Decoder (GPDFD) can be
recur-sivelydescribed as follows
Generalized Joint Equalizer and Decoder (GPDFD)
Initialization step: Attime t
0
, initializeallthe pathmetrics toin nity except M
0 (
0
0 )
1) Path extension step: Go through the trellis section at time t and compute, for all
SjE
t
j possible extended paths,the new candidate pathmetrics
M ( t j )=M k ( t 1 i )+ k (e t 1;t i !j ); i2V t 1 ;j 2V t ;k2[ 0;S 1]; (3.8)
using the generalized traceback array .
2)Pathselectionstep: ClassifythecandidatepathmetricsM
( t j )ateachvertexj 2V t
and keep the S best ones. Simultaneously update the section t 1 of the generalized
traceback array .
3) Final step: Go up the best path from the nal all-zero state using the complete
ful lled . Read the input bits fromthe stored edges among the path.
It is to be underlined that the GPDFD comes down to the PDFD algorithmin the case of
S=1. Simulation results, hereafter, always include that simple case in order to enable the
comparison of this two algorithms.
3.3 Applications
3.3.1 Application 1: Cyclic block codes, joint equalization and
decoding on various full code trellises
Inthissection,weshowthattheproposedreceiveralsoworksforblockencodedsignals. The
TCM code trellis used by the GPDFD can be designed in several ways. The rst way aims
at optimizing the receiver decoding complexity, which is in O(jEj). The problem consists
in searching eÆcient time axis orderings, leading to reduced trellises. Optimal minimal
Kschichang-Sorokine(KS) trelliseshave been found viasimulated annealing basedheuristic
[70],[18]. The code is used in its systematic form for encoding step, but codewords are
permutedaccordingtooptimalexhibitedorderings, beforeBPSKmappingand transmission
over ISI channel.
The second approach aims at introducing a natural QPSK mapping, as done in
appli-cation 3.3.3. The receiver is then applied on sectionalized trellises. By sectionalization, we
mean the choice of a symbol alphabet at each time index. For a given code of time axis
, the sectionalization e ectively shrinks at the expense of increasing the code alphabet
and thetrellis vertices out-degrees. Forexample, binaryextended Hammingcodes of length
n = 2n
0
can be thought as quaternary codes of length n
0
by grouping pairs of consecutive
coded bitstogether. Suchanoperationsubstantiallya ectthe edgeand vertex complexities
Athirdapproachwouldaimatestablishingaconnectionbetweenblockandconvolutional
encoded signals by means of tail-biting trellises. Some results are presented in [32], where
it is shown that unwrapping a tail-biting representation of a good block code, such as the
extended Golaycode, can produce agoodconvolutionalcode.
We nally focus on a fourth approach and investigate the performance of the GPDFD
when appliedon the regulartrellis of any binary polynomialblock code of generators
g(x)= n k X i=0 g i x i (3.9)
Such a trellis is directly designed using the shift register which would perform the
non-systematic polynomial encoding operation. As explained in [115], a systematic encoding
must be realizedforthe purpose of optimizingthe nal BERon messagebits. Atreception,
the GPDFD is applied onto the regular trellis associated with the non-systematic code
version. A convolution between the non-systematic decoded message sequence
b u 0
(x) and
g(x) isperformed for recovering the nal decoded messageu(x)b which consists ofthe k last
symbolsoftheproducedcodeword. Bywayofanillustration,Fig. 3.2showstheperformance
in terms of BER of the GPDFD used for decoding a TCM made of an expurgated binary
BCH code (31;25;dmin = 4) mapped onto a simple BPSK constellation and transmitted
through the worst static 6-taps ISI channel [92]
H(z)=0:23+0:42z 1 +0:52z 2 +0:52z 3 +0:42z 4 +0:23z 5 (3.10)
whose ISI theoretical loss is 7 dB . The regular code trellis section has an overall state
complexity of
w=2
31 25
=64: (3.11)
ConsideringtheoptimalMAPdetectionperformanceofuncodedBPSKsignalsasareference,
theTCMgainprovidedbythis1-errorcorrectingcodeisweakcomparedtothegainprovided
by convolutionalTCM presented in section3.3.3. We alsoobserve that errorpropagation is
completelyeliminated for S=2.
3.3.2 Application 2: cyclic block codes, joint equalization and
de-coding on reduced code trellises
In this second application, we employ the GPDFD to decode more powerful binary BCH
codes. Even reduced by eÆcient time axis orderings, trellises of such codes are usually
Algorithm 1:
This algorithm is inspired by a procedure, rst described in [80]. Let C be an expurgated
t-correcting binary BCH code of primitive length n , of designed distance Æ = 2t+2 and
generator polynome g(x)=(1+x)m (x) | {z } e g(x) ( t Y j=2 m i j (x) ) | {z } g(x) (3.12) where m
(x) is the primitive minimalpolynome corresponding to the primitive n
th
rootof
unity . Finally,we alsointroduce the check polynome of C
h(x)= (x n 1) g(x) (3.13)
LetT bethe optimalminimaltrellisofC . Afather code
e
C ofC isacode whichcontainsall
codewords of C. Typically, expurgated Hamming codes are father codes of more powerful
expurgated BCH codes of same length. Let
e
T be the trellis of
e
C directly constructed from
the generator polynome ge(x). Such a trellis, even in its regular non-reduced form, is far
smallerthan T. Atemission,weencode the messagesequence u(x)systematicallyusingthe
generator polynome g(x). Let
c(x)=r(x)+u(x)x
n k
=u
0
(x)g(x) (3.14)
be the produced systematiccodeword where r(x) is equal to u(x)x
n k
modulo g(x). Given
the received word, the GPDFD performs joint equalization and decoding, working on the
regular trellissection
e
T,and producesin parallela listof the S best messagesequences
b e u 0 1 (x); b e u 0 2 (x); ; c f u 0 S (x); (3.15)
whichall are under the form
b e u 0 i = b u 0 i (x)g(x); (3.16)
and whichwould generate the best father codewords
b e c 1 (x); b e c 2 (x); ; b e c S (x); (3.17)
naturallyclassi edwith respect toanincreasingpath metricorder. Toexplicitlyobtainthe
list b e c 1 (x); b e c 2 (x);; b e c S
(x), each estimated message sequence
b e u 0 i is re-encoded by a simple
convolution with eg(x). A simple syndromecomputationis sequentially performedusing the
checkpolynomeh(x)oneachofthecandidatefathercodewords. The naldecisiondelivered
by algorithm 1 isthe messagesequence u(x)b corresponding tothe rst candidate codeword
in the list,for which
b
![Figure 2.5: Spatial diversity gain vs. envelope correlation in GSM, for various channels [36].](https://thumb-eu.123doks.com/thumbv2/123doknet/2971094.82547/34.918.253.668.110.464/figure-spatial-diversity-gain-envelope-correlation-various-channels.webp)
![Figure 2.7: Spatial diversity gain vs. envelope correlation in IS-95 for various channels [36].](https://thumb-eu.123doks.com/thumbv2/123doknet/2971094.82547/35.918.253.669.148.506/figure-spatial-diversity-gain-envelope-correlation-various-channels.webp)
