HAL Id: pastel-00000785
https://pastel.archives-ouvertes.fr/pastel-00000785
Submitted on 22 Nov 2010
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Codage et traitement de signal avancé pour les systèmes
MIMO
Abdelkader Medles
To cite this version:
Abdelkader Medles. Codage et traitement de signal avancé pour les systèmes MIMO. domain_other.
Télécom ParisTech, 2004. Français. �pastel-00000785�
Institut Eure om
TH ESE
Presentee pour obtenir le grade de do teur
de l'E ole Nationale Superieure
des Tele ommuni ations
Spe ialite: Communi ation et Ele tronique
Abdelkader Medles
Codage et traitement de signal avan e pour
les systemes MIMO
Soutenue le 15 Avril 2004 devant le jury ompose de
Jean-Claude Bel ore President
Phillipe Loubaton Rapporteurs
Emre Telatar
Karim Abed-Meraim Examinateurs
Giuseppe Caire
Meriem Jaidane
Institut Eure om
PhD THESIS
Presented in order to obtain the degree of
do teur de l'E ole Nationale Superieure
des Tele ommuni ations
Spe iality: Communi ation and Ele troni
Abdelkader Medles
Coding and advan ed signal pro essing for
MIMO Systems
Defended on April 15, 2004 before the ommittee omposed of
Jean-Claude Bel ore President
Phillipe Loubaton Rapporteurs
Emre Telatar
Karim Abed-Meraim Examinateurs
Giuseppe Caire
Meriem Jaidane
First, I would like to thank Prof. Dirk Slo k, my thesis supervisor, for
his guidan e, en ouragements, ontinuous help and availability.
Igrateful toProf. Bel ore, fordoingme the honourofpresiding the jury
of my thesis, as well asProf. Loubaton, Prof. Telatar, Prof. Abed-Meraim,
Prof. Caire and Prof. Jaidanefor a epting tobemembers of the jury.
I alsothank InstitutEure om for givingmethe opportunity of
perform-ing my resear h invery good onditions. I spe iallythank the Mobile
Com-muni ations Department sta , with whom I had a lot of intera tions and
dis ussions.
My gratitudegoestomyparentsand myfamily. To themI dedi ate this
The use of multiple transmit and multiple re eive antennas in mobile
ommuni ations o ers a high potential to improve the bit rate and the link
quality. This an be a hieved by using a higher multiplexing rate and by
exploiting the diversity ontained in the hannel, under the onstraint of
a - eptable omplexity. The hannel knowledge availability has an important
impa t on the system design. In fa t, the Channel State Information (CSI)
at the transmitter (Tx) has an impa t on the oding, whereas the quality of
the hannelknowledge atthe re eiver (Rx) side has animpa t mainly onthe
dete tion and the hannel estimation.
The rst part of this thesis onsiders the ase of absen eof CSI atthe Tx
andperfe tattheRx. Wepropose theSpa e-TimeSpreading(STS),whi h is
aspa e-time odings hemebasedonlinearpre odingthatuseaMIMO
onvo-lutive pre lter. STSa hievesfullmultiplexingrateandisoptimizedtoexploit
maximum diversity and oding gains and to save the ergodi apa ity. STS
allows to use various re eiver stru tures of low omplexity. The Stripping
MIMO De ision Feedba k Equalizer (DFE), is a non-iterative re eiver that
dete ts the streamssu essively. Theperforman es of the Stripping are
eval-uatedin termof diversityversus multiplexingtradeo . Anothernon-iterative
re eiveristheConventionalDFEappliedtotheMIMO ase. Itdete tsjointly
symbolsfor di erentstreamsbut pro eed su essivelyin time. The third
pro-posed re eiver is an iterative one. It takes advantage of the presen e of the
binary hannel ode, anditerates between the linear equalizer and the binary
hannel de oder. Simulations are provided to evaluate its performan e.
In the se ond part we onsider hannels with partial CSI at the Tx and
perfe t CSI at RX. The partial knowledge in these ases an ome from the
de ompositionof the hannelin slow varying andfast varying parameters. It
an also be the result of the re ipro ity of the downlink and uplink physi al
ergodi apa ity.
Inthelastpart,the aseofabsen eofCSIatbothTxandRxis onsidered.
The apa ities of two hannel models, blo k fading and time sele tive, are
studied. Due to the absen e of CSI at Rx in this ase the hannel needs
to be estimated in pra ti al systems. We propose semi-blind estimators that
ombinetrainingandblindinformation. Identi ability onditionsarederived
L'utilisation d'antennes multiple a la transmission et re eption dans les
ommuni ations mobiles, o re d'importantes perspe tives pour a ro^tre le
debit et ameliorer la qualite du lien. Cela peut ^etre e e tue en utilisant un
plusimportantmultiplexagespatialetenexploitantladiversite ontenuedans
le anal, tout en gardant une omplexite a eptable. L'etat de onnaissan e
sur le anal a un impa t important sur la on eption de la haine de
trans-mission. En e et, l'information sur l'etat du anal (CSI) au transmetteur
(Tx) a un impa tsur le odage alors quelaqualite du CSI au re epteur(Rx)
a prin ipalement un impa t sur ladete tion et l'estimation du anal.
Danslapremierepartiede ettethesenousavons onsiderele asd'absen e
de CSI au Tx et un parfait CSI au Rx. On propose la dispersion
spatio-temporelle (STS), qui est un s hema de odage spatio-temporel base sur le
pre odage lineaire en utilisant un ltre multi-entree multi-sortie (MIMO).
Le STS e e tue un multiplexage de ux maximal, qui est optimise pour
ex-ploiter une diversite maximale, atteindre un bon gain odage et onserver
la apa ite ergodique. Un autre avantage du STS est qu'il permet d'utiliser
une variete de re epteurs de omplexite reduite. Le Stripping MIMO ave
egalisation a retour de de ision,est un re epteur non-iteratif qui dete te les
ux d'une maniere su essive. Les performan es du Stripping sont donnees
en terme du ompromis entre diversite et multiplexage. Un autre re epteur
non-iteratifestl'egaliseuraretourdede isionappliqueau as MIMO.Il
per-metladete tiondessymbolesdesdi erents uxd'unemaniere onjointemais
su essivement dans le temps. Le troisieme re epteur propose est iteratif.
Il pro te de la presen e d'un odage anal binaire et itere entre l'egaliseur
lineaire et le de odeur anal binaire. Des simulations sont presentees pour
evaluerles performan es.
Danslase ondepartieon onsideredes anaux ave un CSIpartiel au Tx
lade ompositiondu anal enparametreslents etrapides. Ellepeut aussi^etre
le resultat d'une re ipro ite du anal physique entre la liaison montante et
des endante. A es di erents as on presente des modeles de anauxadaptes
et on etudie la apa ite ergodique.
Dans la derniere partie on traite du as d'absen e de CSI aux Tx et Rx.
La apa ite de deux modeles de anaux, evanes ent par blo et sele tif en
temps, est etudiees. A ause de l'absen e de CSI au Rx le anal doit ^etre
estime dans les systemes utilises en pratique. On propose des estimateurs
semi-aveugle qui ombinent l'information de la sequen e d'apprentissage et
ellede lapartieaveugle. Les onditionsd'identi abilite sontobtenuesetdes
A knowledgments . . . i
Abstra t . . . iii
Resume . . . v
Listof Figures. . . xiii
Listof Tables . . . xvii
A ronyms . . . xix
Notations . . . xxi
1 Introdu tion 1 1.1 General MIMO Channel Model . . . 2
1.1.1 Rayleigh FlatFading MIMO Channel Model . . . 5
1.1.2 Separable Spatial Channel Model (Partial CSIat Tx) 6 1.1.3 Frequen y Sele tive Rayleigh Fading MIMO Channel Model . . . 6
1.2 Capa ity of MIMO Channel . . . 7
1.2.1 Flat Channel with Perfe t CSI . . . 7
1.2.2 Ergodi Capa ity (Imperfe t CSI atTx ) . . . 7
1.2.3 Outage Capa ity (Imperfe t CSI atTx) . . . 9
1.2.4 Asymptoti Behavior inBlo k Transmission . . . 10
1.3 Conventional Multi-Antenna Re eive Diversity . . . 11
1.4 Spa e-Time Coding for MIMO System . . . 12
1.5 Diversity and Multiplexing asDe ned by Zheng& Tse . . . . 16
1.6 Thesis Overview and Outline . . . 17
1.6.1 Part One: Absen e of CSI at Tx (and Perfe t CSI at Rx). . . 18
1.6.2 Part Two: Partial CSIat Tx (and Perfe t CSI atRx) . 20
I Absen e of CSI at TX 23
2 Linear Convolutive Spa e-Time Pre oding 25
2.1 Introdu tion . . . 26
2.2 Linear Pre ltering Approa h . . . 27
2.3 Capa ity . . . 29
2.4 Mat hed Filter Bound and Diversity . . . 31
2.5 PairwiseError ProbabilityP e . . . 31
2.5.1 Choi eof Q . . . 33
2.5.2 OptimalityforQAMConstellationsinthe CaseN tx =2 k 35 2.5.3 Cir ularConvolution . . . 37
2.5.4 Frequen y Sele tive Channel Case . . . 38
2.6 ML Re eption . . . 40
2.7 Con lusion . . . 40
3 Non-Iterative Rx: Design Alternatives 43 3.1 Introdu tion . . . 44
3.2 Stripping MIMO DFE (Su essive Interferen e Can ellation) Re eiver . . . 45
3.2.1 Stripping MMSEZF DFE Rx Design . . . 47
3.2.2 Stripping MMSEDFE Rx Design . . . 48
3.2.3 MatrixSpe tral Fa torizationConsiderations . . . 49
3.2.4 Stripping DFEand V-BLAST . . . 50
3.2.5 Pra ti alImplementation of SICRe eiver . . . 50
3.3 SIC Re eiverPro essing and Capa ity Issues . . . 51
3.3.1 Stripping MMSEDFE Rx . . . 51
3.3.2 Stripping MMSEZF DFE Rx . . . 53
3.4 Diversity vs. Multiplexing Tradeo . . . 54
3.4.1 Optimal Tradeo Curve for the Frequen y Sele tive Channel . . . 54
3.4.2 Tradeo Curve forthe SICRx . . . 56
3.5 Conventional MIMO DFERe eiver . . . 60
3.5.1 Conventional MMSE MIMO DFERx . . . 60
3.5.2 Conventional MMSE ZF MIMO DFERx . . . 63
3.5.3 De oding Strategy . . . 64
3.6 Stripping vs. Conventional MIMO DFE . . . 65
3.7 Diversityvs. MultiplexingTradeo oftheConventionalMIMO DFE anOpen Problem . . . 65
3.8 Con lusion . . . 66
3.A OutageCapa ityBehaviorofSIMOFrequen ySele tive Chan-nel . . . 67
3.B Proof of Lemma 1 . . . 71
3.C Proof of Theorem 1 . . . 75
3.D Proof of Theorem 2 . . . 76
3.E Proof of Theorem 3 . . . 78
3.F Proof of Theorem 5 . . . 80
4 Iterative Rx 83 4.1 Introdu tion . . . 84
4.2 CombiningLinear Pre oding and Binary Channel Coding . . . 84
4.2.1 En oding . . . 85
4.2.2 Iterative De oding . . . 87
4.2.3 ComplexityComparison with Threading . . . 89
4.3 Multi-Blo kTime Diversity . . . 89
4.4 Performan e Analysis . . . 92
4.4.1 Comparison of Threading and STS . . . 92
4.4.2 Use of Walsh Hadamard (WH) matrix as Pre oding matrix . . . 93
4.5 Con lusion . . . 93
II Partial CSI at TX 97 5 On MIMO Capa ity with Partial CSI at Tx 99 5.1 Introdu tion . . . 100
5.2 Channel Models and Assumptions . . . 100
5.2.1 Pathwise Channel Model . . . 101
5.2.2 Channel Models for Limited Re ipro ity . . . 101
5.3 Results for Pathwise Channel Model . . . 102
5.3.1 LowSNR . . . 102
5.3.2 High SNR . . . 104
5.3.3 Water llingSolutionfortheChannel Covarian eMatrix105 5.3.4 Optimal Solution . . . 105
5.3.5 Solutionfor Separable Spatial Channel Model . . . 105
5.4 Results for Channel Models with LimitedRe ipro ity . . . 106
5.5.1 Pathwise Model . . . 109
5.5.2 LimitedRe ipro ity. . . 110
5.6 Con lusion . . . 110
III Absen e of CSI at RX 115 6 Mutual Information without CSI at Rx 117 6.1 Introdu tion . . . 118
6.2 Mutual Information De omposition . . . 119
6.2.1 GeneralFlat Fading Model. . . 119
6.2.2 MI De omposition . . . 119
6.3 Asymptoti Behavior oftheCapa ity forBlo kFadingChannel121 6.3.1 Channel Estimationfor Blo k FadingModel . . . 123
6.4 Capa ity Behaviorand Bounds for Time Sele tive Channel . . 125
6.4.1 Case of Di erentialEn oding . . . 127
6.4.2 GeneralCase . . . 130
6.5 Correlated MIMO Channel Model . . . 132
6.6 Observations. . . 133
6.7 Con lusion . . . 134
6.A Appendix A . . . 135
6.B Appendix B . . . 136
7 Semi-Blind Estimation for MIMO Channels 139 7.1 Introdu tion . . . 140
7.2 MIMO Flat Channel . . . 141
7.2.1 Maximum LikelihoodChannel Estimator . . . 142
7.2.2 InformationMatrix Issues . . . 143
7.2.3 GaussianSemi-BlindApproa h . . . 144
7.2.4 Deterministi Semi-BlindApproa h . . . 146
7.3 MIMO Frequen ySele tive Channel . . . 146
7.3.1 MIMO LinearPredi tion . . . 148
7.3.2 Deterministi Semi-BlindApproa h . . . 150
7.3.3 GaussianSemi-BlindApproa h . . . 152
7.3.4 Augmented Training-Sequen e Part . . . 154
7.4 Performan e Analysis . . . 155
7.4.1 Flat hannel ase . . . 155
7.5 Con lusion . . . 157
7.A Proof of Theorem 1 . . . 158
7.B Proof of Theorem 2 . . . 159
General Con lusion 167 Resume detaille en fran ais 171 7.3 Introdu tion . . . 172
7.3.1 Modeles des Canaux MIMO . . . 172
7.3.2 Capa itedu Canal . . . 173
7.3.3 Codage Spatio-Temporel pour des Systemes MIMO . . 174
7.3.4 Diversiteet Multiplexage omme De nis par Zheng & Tse . . . 175
7.4 Pre odage Spatio-TemporelLineaire etConvolutif . . . 177
7.4.1 Re epteur ML . . . 178
7.5 Rx Non-iteratif: Alternatifde Con eption . . . 178
7.5.1 Stripping MIMO DFE . . . 179
7.5.2 CompromisDiversite-MultiplexageduStrippingMIMO DFE . . . 180
7.5.3 MIMO DFE Conventionnel . . . 184
7.6 Re epteur Iteratif . . . 185
7.6.1 Codage . . . 185
7.6.2 De odage Iteratif . . . 185
7.6.3 Analyse de Performan e . . . 186
7.7 Modeles de Canal . . . 189
7.7.1 Modele aChemins . . . 189
7.7.2 Modele aRe ipro ite Limitee . . . 189
7.8 Resultats. . . 190
7.8.1 Modele aChemins . . . 190
7.8.2 Modele aRe ipro ite Limitee . . . 191
7.9 Resultats des Simulations . . . 191
7.10 Information Mutuelleen Absen e de CSI auRx . . . 193
7.10.1 De omposition de l'InformationMutuelle . . . 193
7.10.2 ComportementAsymptotiquedesCanauxEvanes ants par Blo . . . 194
7.11 EstimationSemi-Aveugle des Canaux MIMO . . . 194
7.11.1 Canal MIMO plat . . . 194
1.1 MIMO hannel model. . . 3
1.2 PEP vs. SNR . . . 15
1.3 Diversity vs. multiplexingoptimal tradeo . . . 17
2.1 General ST oding setup.. . . 27
2.2 Two hannel oding, interleaving, symbol mapping and de-multiplexingalternatives. . . 29
2.3 STS of one stream . . . 34
2.4 STS of all streams . . . 34
3.1 Stripping MIMO DFE re eiver. . . 45
3.2 Diversity vs. multiplexing optimal tradeo for frequen y se-le tive hannel with N tx N rx . . . 56
3.3 Diversityvs. multiplexingtradeo ofdi erents hemes. N tx = N rx =4,L=1. . . 59
3.4 Convetional MIMO DFE re eiver . . . 60
4.1 En oder for spa e-time spreading. . . 86
4.2 Serial-to-parallel onverted spa e-timeblo kbeforepre oding for N s =4.. . . 86
4.3 Iterative de oder with interferen e an ellation. . . 86
4.4 Blo k interleaver for F=2. . . 90
4.5 STS/Threading for (N tx ;N rx )=(2;2),L=1,F=1;2;4. . . 94 4.6 STS/Threading for (N tx ;N rx )=(2;2),L=2,F=1;2;4. . . 94 4.7 STS/Threading for (N tx ;N rx )=(4;4),L=1,F=1;2;4. . . 95 4.8 STS/Threading for (N tx ;N rx )=(8;8),L=1,F=1;2;4. . . 95 4.9 STS (N s =1;2;4)vs. Threading (QPSK, BPSK) and STOD for (N tx ;N rx )=(4;2), L=1 and F =1. . . 96
4.10 Two hoi e of the pre oding matrix: optimized Q vs. Walsh
Hadamard for L=1 and F =1. . . 96
5.1 Result for N tx =N rx =4, L p =2. . . 111 5.2 Result for N tx =N rx =4, L p =10. . . 112
5.3 Results for limited re ipro ity,N rx =N tx =4, Model1. . . 112
5.4 Results for limited re ipro ity,N rx =N tx =4, Model2. . . 113
5.5 Results for limited re ipro ity,N rx =N tx =4, Model3. . . 113
6.1 E e tive SNR for time sele tive hannel . . . 130
6.2 Capa ity behavior of time sele tive hannel . . . 131
7.1 NormalizedMSEvsN TS : at hannel,N tx =2;N rx =4; N B = 400, SNR= 10dB. . . 161
7.2 NormalizedMSEvsSNR: at hannel,N tx =2;N rx =4; N B = 400, N TS =4. . . 161 7.3 NormalizedMSEvsN TS : at hannel,N tx =4;N rx =4; N B = 400, SNR= 10dB. . . 162
7.4 NormalizedMSEvsSNR: at hannel,N tx =4;N rx =4; N B = 400, N TS =4. . . 162 7.5 NormalizedMSEvsN TS : at hannel,N tx =4;N rx =2; N B = 400, SNR= 10dB. . . 163
7.6 NormalizedMSEvsSNR: at hannel,N tx =4;N rx =2; N B = 400, N TS =4. . . 163
7.7 Normalized semi-blind hannelestimation MSE, s enario1. . . 164
7.8 Normalized semi-blind hannelestimation MSE, s enario2. . . 164
7.9 Normalized semi-blind hannelestimation MSE, s enario3. . . 165
7.10 Normalized semi-blind hannelestimation MSE, s enario4. . . 165
7.11 Normalized semi-blind hannelestimation MSE, s enario5. . . 166
7.12 Compromis diversite-multiplexage . . . 176
7.13 S hema General de Transmission. . . 177
7.14 Stripping MIMODFE. . . 179
7.15 Compromisdiversite-multiplexaged'un analsele tifenfrequen e182 7.16 Compromis diversite-multiplexage de di erentes te hniques. N tx =N rx =4, L=1. . . 183
7.17 MIMO DFE Conventionnel. . . 184
7.18 Stru ture de l'en odeur. . . 185
7.20 STS/Threading pour (N tx ;N rx )=(2;2),L=2,F=1;2;4. . . . 187 7.21 STS/Threading pour (N tx ;N rx )=(4;4),L=1,F=1;2;4. . . . 188
7.22 Modele a hemins, N
tx =N rx =4,L p =2. . . 191
7.23 Modele are ipro ite limitee, N
rx =N tx =4. . . 192 7.24 MSE vsN TS : anal plat,N tx =2;N rx =4; N B =400, SNR= 10 dB. . . 197
4.1 Blo kdiversityforsomepopularrate1/2binary onvolutional
odes mapped onto BPSK and QPSK(with Gray labeling).. . 90
7.1 Channel lengths, estimated lengths, training and blind data
lengths fordi erent s enarios where N
tx
Here are the main a ronyms used in this do ument. The meaning of an
a ronym isusually indi ated on e, when it rst o urs in the text.
AWGN Additive White GaussianNoise
BCJR Bahl, Co ke, Jelinek and Raviv (algorithm)
BER Bit Error Rate
BPSK Binary Phase ShiftKeying
CDMA Code-DivisionMultiple A ess
CRB Cramer-Rao Bound
CSI Channel State Information
DA Data-Aided
DFE De ision Feedba k Equalizer
DFT Dis rete Fourier Transform
DSB Deterministi Semiblind
DSBA Deterministi SemiblindAugmented
FIR FiniteImpulse Response
IC Interferen e Can eler or Interferen e Can ellation
IDFT Inverse Dis rete FourierTransform
ISI Inter-Symbol Interferen e
i if and only if
i.i.d. independent & identi allydistributed
GSB GaussianSemiblind
GSBA GaussianSemiblindAugmented
QPSK Quarternary Phase ShiftKeying
LB Lower Bound
LL LogLikelihood
MAP Maximum A Posteriori
MFB Mat hed Filter Bound
MI Mutual Information
MIMO Multiple Input Multiple Output
MISO Multiple Input Single Output
ML Maximum Likelihood
MMSE Minimum Mean Square Error
MRC Maximum Ratio Combining
MSE Mean Square Error
PIC ParallelInterferen e Can ellation
PSP Per-Survivor-Pro essing
pdf probability density fun tion
psdf powerspe tral density fun tion
Rx Re eiver
SB Singleton Bound
SIC Su essive Interferen e Can ellation
SIMO Single Input Multiple Output
SINR Signal toNoise Plus Interferen e Ratio
SNR Signal-to-Noise Ratio
ST Spa e-Time
STC Spa e-Time Code/Coding
STOD Spa e Time OrthogonalDesign
STS Spa e Time Spreading
SVD Singular Value De omposition
S/P Serial toParallel onversion
sIsO soft-Input soft-Output
TS Training Sequen e
Tx Transmitter
UB Upper Bound
UDL Upper DiagonalLower
UMTS Universal Mobile Tele ommuni ation System
vs. versus
w.p. with probability
w.r.t. with respe t to
Here is alist ofthe main notations and symbolsused in this do ument. We
have tried to keep onsistent notations throughout the do ument, but some
symbolshave di erentde nitions dependingonwhen theyo urinthe text.
a S alar variable
a Ve tor variable
A Matrixvariable
a(t); a( ) Continous-time fun tion of the temporalvariable t or
a(n) Dis rete-timefun tion a(n)=a(nT) fora given T
a n
Dis rete-timefun tion a
n
=a(nT) for agiven T
j =
p
1 The unity imaginarynumber
<(x) Realpart of x
=(x) Imaginary part of x
D Diagonalmatrix
a(z) z-transformof a(n)
() Complex onjugate () T Transpose () t Blo k transpose () H Hermitiantranspose A y (z)=A H (1=z ) I m
mm Dimensional Identity matrix
Krone ker produ t
Efg Expe tation operator
detfAg Determinantof the matrix A
trfg Tra e operator
diagfAg Diagonalmatrix of the diagonalelements of the matrix A
:
= Exponentialequality
ve (M ) M writtenin ave tor form
N tx
N rx
Number of re eive antennas
N s Number of streams P X X(X H X) 1 X H
: Proje tion onthe olumnspa e of X
E b
Energy per informationbit
E b
=N 0
Signal tonoise ratio per informationbit
N 0
Onesided noisepowerspe traldensity ofthe AWGN hannel
I(X;Y) Mutual informationbetween randomvariables X and Y
P(A) Probability of event A.
O(:) Of the order of x.
Æ ij
Krone kerdelta
ln(:) Natural logarithm
h(X) Entropy mesure of the sto hasti variable X
Ntx M i=1 X i blo k-diagfX 1 ;:::;X Ntx g
Introdu tion
In this hapter we introdu e the prin iple of multiple transmit antennasand
multiple re eiveantennas used in wireless ommuni ation, whi h an be seen
more generally as Multiple Input Multiple Output (MIMO) system. We rst
de ne the general hannel model, and spe ify the apa ity for the di erent
asesof hannel knowledge. We present some basi s on the design of
spa e-time odes in the ase of absen e of hannel knowledge at the transmitter
(Tx). We introdu e then some notions on the diversity vs. multiplexing of
Zheng &Tse and on lude this hapter by an overviewof the di erent parts
Sin etheintrodu tion,independently,ofspatialmultiplexingbyA.Paulraj
inaStanfordUniversitypatentand by Fos hini[1℄atBellLabs in 1994,the
use of multiple transmit and multiple re eive antennas has be ome the
fo- us of a lot of works. The reason behind this big interest of the s ienti
ommunity is related to the ability of MIMO systems to o era new spatial
dimension,otherthanthe timeand frequen y dimensions,thatin reases the
ergodi (average) apa ity of the hannel by a multiplying fa tor equal to
the minimumbetween the numberof transmitvs. re eiveantennas (N
tx vs.
N rx
) [2℄, and allows to lower the outage probability by the ontribution of
N tx
:N rx
diversity omponents orresponding to all the hannel oeÆ ients.
UnlikeSISO at hannel,MIMO at hannel(with absen e of hannelstate
information at the transmitter) su ers from interferen e between di erent
transmit antennas. The re ent attempts to exploit this high potential for
wireless ommuni ation have to make a ompromise in order to handle an
in reaseinrateandtakeadvantageoftheavailablediversityto ombatfading
and destru tive interferen e whithin a eptable omplexity limits.
This hapter is an introdu tion to the general framework. We present
the general MIMO linear model in wireless ommuni ation, and detail the
underlyingmodels for spe i situations. The apa ities of su h as hannels
are alsointrodu ed. Wepresent the lassi al multi-antennare eive diversity
and the spa e-time oding forMIMO systems. The diversity and
multiplex-ing as de ned by Zheng & Tse are also introdu ed. Finally we provide an
overview of this thesis.
1.1 General MIMO Channel Model
Considerlinear digitalmodulationoveralinear hannelwith additive
Gaus-sian noise. The number of antennas is N
tx
at the transmitter side and N
rx
atthe re eiver as represented in g. 1.1. We assume a linear time invariant
model for the e e t of the hannel on the transmitted signals. The re eived
signal atRx antenna i an be writtenin baseband as
y i (t) = Ntx X n=1 +1 X l = 1 x n (l)h in (t lT p ) + v i (t); (1.1) wherethe x n
(l)arethe transmittedsymbolsfromtheantennasour en,T
p is
the ommon symbol period, h
in
...
...
+
environment
Scattering
Linear model
y 1 x 1 x 2 x N tx y 2 y N rx H v x yfromtransmitter n tore eiver antenna i. Assumingthe fx n
(l)g and fv
i (t)g
tobe jointly (wide-sense) stationary, the pro esses fy
i
(t)gare jointly
(wide-sense) y lostationary with period T
p
. If fy
i
(t)g is sampled with period T
p ,
thesampledpro essis(wide-sense)stationary. Samplinginthiswayleadsto
anequivalentdis rete-timerepresentation. WeassumetheMIMO hannelto
be ausaland niteimpulseresponse (FIR).In parti ular,aftersamplingwe
assumethe impulseresponse to beof length L. The dis rete-timeRx signal
an be represented inve tor form as
y k = Ntx X n=1 L 1 X l =0 h n (l)x n (k l)+v k = L 1 X l =0 H l x k l +v k ; (1.2) where H l is N rx N tx with (H l ) in = h in (l) for i = 1;2;:::;N rx ; n = 1;2;:::;N tx and y k = 2 6 6 6 4 y 1 (k) y 2 (k) . . . y N rx (k) 3 7 7 7 5 ;v k = 2 6 6 6 4 v 1 (k) v 2 (k) . . . v N rx (k) 3 7 7 7 5 ;h n (l)= 2 6 6 6 4 h 1n (l) h 2n (l) . . . h N rx n (l) 3 7 7 7 5 : (1.3)
ThenoiseisassumedtofollowawhiteGaussiandistribution,v
k CN (0; 2 v I N rx ),
thetransmittedsignalis hara terizedbyitspowerspe traldensity fun tion
(psdf) S xx (z) with 1 2j H dz z trS xx (z) =N tx 2 x , j = p
1, and the
signal-to-noise ratio is SNR = N tx 2 x 2 v = N tx
. The notation tr(:) denotes the tra e
fun tion.
Letqdenotethetimeshiftoperator: (q
1 x) k =x k 1 ,andH(q)= L 1 X l =0 H l q l .
The re eived signal is then
y k =H(q)x k +v k ; (1.4)
and the psdf of the re eived signal is
S yy (z)=H(z)S x x (z)H y (z)+ 2 v I; (1.5)
where H(z)is z-transformof the hannel impulseresponse.
We ould alsoobtainmultiple hannelsinthe dis retetimedomainby
over-sampling the ontinuous-time re eived signals w.r.t. the symbol rate (inthe
ase of the presen e of ex ess bandwidth),the antennas ould alsobe
gener-alized sensors (polarization, 4 EM omponents [3, 4, 5℄), and in the ase of
passband transmission of real symbol onstellations we an alsodouble the
numberof hannelsbytaking thein-phaseand in-quadrature omponentsof
the re eived signal.
Weassumeherethe hanneltobe onstantforthe durationof oneframe(or
blo k), however the hannel an varyfrom blo k to blo k, this is the reason
why this modelis alledblo kfading hannelmodel,whi histhe most
om-mon hannelmodel used inpapers dealing with wirelessMIMO systems.
Two lassesof hannelsaretobedistinguished, at hannels(L=1)and
fre-quen y sele tive hannels(L>1). The aseL>1isalso alledInter-Symbol
Interferen e (ISI) hannel. In most of the papers on MIMO the hannel is
assumed tobe at, the impulseresponse being simplythen H=H
0 .
In the di erent parts of this thesis report we willdeal with di erent
on g-uration of the Channel State Information(CSI): presen e at Rx vs. Tx and
absen e vs. perfe t or partial.
In the absen e of CSI at Tx we still assume the knowledge of the average
power ofthe hannel EtrfH
H Hg (by normalization=N rx :N tx ) [2℄, inorder
to be able to onstru t a statisti al model. This model is hosen to be a
maximum entropy modelto re e t the absen e of CSI, itleads to Gaussian
i.i.d. omponents and orresponds hen e to the Rayleigh at fadingMIMO
hannelmodel.
1.1.1 Rayleigh Flat Fading MIMO Channel Model
This model has i.i.d. entered Gaussian omponents (H)
mk CN (0;1) for 1mN rx and 1k N tx . We denote p =min(N rx ;N tx ) and q =max(N rx ;N tx ). Let USU H = H H H
be the eigenve tor de omposition of H
H H, where S = diagfs 1 ;:::;s p g
where the eigenvalues are organized inin reasing order(s
1 s 2 :::s p ), and U=[u 1 ;:::;u p ℄: N tx
p ontains the eigenve tors.
In[2℄,TelatarhasshownthatUisuniformlydistributedovertheGrassmann
pdf p(s 1 ;:::;s p )=K 1 q;p p Y i=1 s q p i Y i<j (s i s j ) 2 e P i si ; (1.6) whereK q;p is anormalizing onstant.
1.1.2 Separable Spatial Channel Model (Partial CSI
at Tx)
Inthe ase of partialCSIat the Tx, the MIMO hannelis oftenmodeled as
aspatially separable hannel model. In this model the hannel isgiven by
H= 1=2 1 W 1=2 2 ; (1.7) where W is a N rx N tx
random matrix of i.i.d. omplex ir ular Gaussian
elements withmean 0 and varian e 1.
Thematrix
1
isthe re eivearray ovarian ematrixand
2
isthe transmit
array ovarian ematrix: EfH
H Hg=trf 1 g 2 ; EfHH H g=trf 2 g 1 . For 1 =I N rx ; 2 =I N tx
we re over the Rayleigh at fadingMIMO hannel
model. Infa t,theseparable hannelmodelis onstru tedasageneralization
of the Rayleigh MIMO hannel model used in ase of no CSI at the Tx to
the ase when the se ond order statisti s of the hannel are present at the
Tx.
1.1.3 Frequen y Sele tiveRayleigh Fading MIMO
Chan-nel Model
We onstru t this modelas a generalizationof the RayleighMIMO hannel
model to the ase of frequen y sele tivity, it in orporates the power delay
pro leknowledgefor nitedelayspreadL. ForthismodelH
l
; l =0;:::;L
1 are independent, and ea h H
l
has i.i.d Gaussian omponent (H
l ) mk CN(0; 2 l ) for 1m N rx and 1 k N tx . 2 l ; 0 l L 1; re e t the
1.2 Capa ity of MIMO Channel
1.2.1 Flat Channel with Perfe t CSI
Fora at hannelwithperfe tCSIatthe Txandthe Rx, the apa ity isthe
maximum a hievable rate under the power onstraint, and is a hieved by a
frequen y at and zero-mean input
C(H)= max trfS xx gP lndet (I Nrx + 1 2 v HS xx H H ); (1.8)
where P is the power onstraint limit, S
xx
(z) = S
xx
is now simply the
ovarian e and the solution is obtained by water llingon the eigenvalues of
H H
H (spatialwater lling) [2℄. The apa ity is given in nats/se ond/Hz.
In the ase of a MIMO frequen y sele tive hannel the solution is a
water- lling overthe two dimensions spa eand frequen y, the solutioninthis ase
is not trivial.
1.2.2 Ergodi Capa ity (Imperfe t CSI at Tx )
In the aseof imperfe tCSI(noorpartialCSI)atthe Txandperfe t CSIat
Rx the hannel has apriori distribution. The instantaneous apa ity is now
a sto hasti quantity, a ommonmeasure is the average orergodi apa ity
C = max 1 2j H dz z trfS xx gP E H 1 2j I dz z lndet (I N rx + 1 2 v H(z)S x x (z)H y (z)); (1.9)
where the expe tation E
H
is here w.r.t. the hannel, and
1 2j H dz z g(z)= R 1 2 1 2 g(e j2f
)df istheintegralovertheunit ir leforafun tion
g(:).
The ergodi apa ity is hen e the average of the instantaneous apa ity, it
takes its full sense if we are able to en ode the information to transmit
over several hannel realizations (independent blo ks). Finally it be omes
a hievable if the transmitted odewords experien e an in nite number of
hannelrealizations.
Inthe aseofRayleigh atfadingMIMO hannelmodel,Telatarhasshownin
[2℄ that the ergodi apa ity isa hieved by awhite input(S
xx (z)= 2 x I N tx , 2 x = P N tx ) C= E H lndet (I Nrx +HH H ): (1.10)
In fa tin this ase the hannel distribution isinvariantby any permutation
oftheTxantennas,theyplay symmetri alrolesandhen ethereisno
prefer-able dire tionof transmission, yieldingto aspatially white input.
High SNR :Asymptoti allyfor high SNRit is shown in[2℄ that
C=O(min(N
tx
;N
rx
)ln),orequivalentlythattheMIMO hannelperforms
asymptoti allyas wellas an equivalent number of parallel AWGN hannels
equaltotherankofH. Theseasymptoti performan esoftheergodi
apa -ity arethe same asthe oneof the apa ity foraMIMO hannelwith perfe t
CSIat Tx.
The ergodi apa ity formula an be generalized to the frequen y sele tive
ase, the following theorem givesthe desiredresult.
Theorem1: Fora hannelwithafrequen ysele tiveRayleighfadingmodel
(se tion1.1.3) the ergodi apa ity is
C = E H 1 2j I dz z lndet (I Nrx +H(z)H y (z)); (1.11) where= P Ntx 2 v
Proof : Byapplying the resultof the at hannel ase toea h frequen y,we
on ludethat forea hfrequen y the psdfisspatiallywhite, hen eS
xx (z)= s xx (z)I Ntx , with 1 2j H dz z s xx (z) P N tx
. On the other hand, for every z
1 on
the unit ir le H
l z
l
1
; l = 0;:::;L 1 have the same joint distribution as
H l ; l =0;:::;L 1, and by onsequen e H(zz 1 ) asH(z),then E H lndet (I Nrx + s xx (z) 2 v H(z)H y (z)) = E H lndet(I Nrx + s xx (z) 2 v H(zz 1 )H y (zz 1 )) 8z 1 withjz 1 j =1 = 1 2j H dz 1 z1 E H lndet(I N rx + s xx (z) 2 v H(zz 1 )H y (zz 1 )); (1.12)
and the ergodi apa ity
1 2j H dz z E H lndet (I N rx + s xx (z) 2 v H(z)H y (z)) = 1 2j H dz z 1 2j H dz 1 z 1 E H lndet (I Nrx + s xx (z) 2 v H(zz 1 )H y (zz 1 )) = 1 2j H dz2 z 2 1 2j H dz z E H lndet (I Nrx + s xx (z) 2 v H(z 2 )H y (z 2 )) 1 2j H dz 2 z2 E H lndet(I N rx + 1 2j H dz z s xx (z) 2 v H(z 2 )H y (z 2 )) 1 2j H dz 2 z2 E H lndet(I N rx +H(z 2 )H y (z 2 )); (1.13)
inthe se ondequalityweoperateavariable hangezz 1
!z
2
,whereas inthe
third inequality we exploit the on avity of the lndet fun tion over the set
of positivede nite matri es. The nal inequality shows that
E H 1 2j H dz z lndet (I Nrx +H(z)H y
(z)) is an upper bound on the apa ity,
however this bound is a hieved for S
xx (z)= 2 x I N tx .
1.2.3 Outage Capa ity (Imperfe t CSI at Tx)
Fora hannelwithimperfe tCSIattheTx,andinthe asewheretheen oder
sees a limited number of hannel realizations (for instan e we onsider the
ase of one hannel realization) the ergodi apa ity makes no sense, the
outage apa ity is then moreappropriate. In fa t,for agiven SNR and rate
R , it expresses the probability that the transmitted rate is larger then the
instantaneous apa ity (transmissionfailure)
P out (R ) = P(C(H)<R ) = P( 1 2j H dz z lndet(I N rx +H(z)S xx (z)H y (z))<R ); (1.14) where S xx
(z) isnormalized here tohave (
1 2j H dz z trS xx (z)=N tx ).
For agiven level (0 1)of the outage, the outage apa ity is de ned
as C out ( )=(P out ) 1 ( ); (1.15) the hoi eofS x x
(z)istheonethatmaximizesC
out
,paperslike[6,7,8℄
stud-ied this problem. They show that for separable spatial at hannel model
with well onditionned
2
, and forlowoutage(small )the transmitter
op-timalinput olor tends tobewhite S
xx (z)= 2 x I N tx .
An other importantresult for at hannelis shown in [9℄, where for
asymp-toti ally high SNR, Rayleigh at fading hannel and for any onstant and
nite rate R we have
P out (R )=O Ntx:Nrx : (1.16)
TheexponentoftheSNRN
tx :N
rx
is alledthediversitygainandisrelatedto
the diversity gain thatarise inthespa e-time ode design(se tion1.4). The
diversitygain orrespondstotheasymptoti slopeoftheerrorprobability(or
outageprobability),andisequaltothetotalnumberofindependentdiversity
1.2.4 Asymptoti Behavior in Blo k Transmission
Letusassumethat we transmitoverablo kof lengthT, there eived signal
an then be writtenas Y=A T (H)X+V; (1.17) whereY =[y T 1 ;y T 2 ;:::;y T T+L 1 ℄ T , X=[x T 1 ;x T 2 ;:::;x T T ℄ T and V=[v T 1 ;v T 2 ;:::;v T T+L 1 ℄ T . A T (H)isaN rx (T+L 1)N tx T blo ktoeplitz
matrix with rst blo k olumn [H
T 0 ;H T 1 ;:::;H T L 1 ℄ T . A T (H)= 2 6 6 6 6 6 6 6 6 6 6 6 4 H 0 0 ::: 0 H 1 H 0 . . . . . . . . . H 1 . . . 0 H L 1 . . . . . . H 0 0 H L 1 . . . H 1 . . . . . . . . . . . . 0 ::: 0 H L 1 3 7 7 7 7 7 7 7 7 7 7 7 5 : (1.18)
Theinstantaneous apa ity (perinputsymbol)ofthe blo ktransmissionfor
whiteinput is C T (H)= 1 T lndet (I NrxT +A T (H)(A T (H)) H ): (1.19)
The following lemma gives an important result on the limit of C
T
(H) for
large T.
Lemma 1: The limitof the apa ity C
T (H) is lim T!+1 C T (H)=C(H)= 1 2j I dz z lndet (I+H(z)H y (z)): (1.20)
Proof: This is ageneralization of the SISO ase shown in [10, 11℄, the proof
issimilar.
Corollary 1: C
T
(H) onverges in distributionto C(H).
Proof: The limit in eq.(1.20) implies that C
T
(H) onverges almost surely
toC(H), and hen e C
T
The prin ipal onsequen e of Corollary 1 is that asymptoti ally for large
blo klengthT >> L,C
T
(H)havethesame distributionasthe oneofC(H),
then all the stated results for the ontinuous apa ity C(H) apply
asymp-toti ally to C
T (H).
1.3 Conventional Multi-Antenna Re eive
Di-versity
The lassi alMulti-antennadiversityte hniqueiswell-knowntoimprovethe
quality of the wireless link [12, 13, 14, 15℄. The hannel in these ases is
SIMO (N
tx
=1),for at hannelthe re eived signal iswritten
y k =hx k +v k ; (1.21) where h= 2 6 6 6 4 h 1 h 2 . . . h N rx 3 7 7 7 5 ; (1.22)
and the transmittedsignal x
k
isa s alar now.
By applying aMaximum Ratio Combining (MRC) re eiver
y k = h H jjhjj y k = jjhjjx k +v k ; (1.23) where jjhjj = q P N rx i=1 jh i j 2 and v k = h H jjhjj v k
follows a zero-mean Gaussian
distribution with varian e
2
v .
TheresultisaSISO fading hannelwithinstantaneous apa ityC(jjhjj
2
)=
ln(1+jjhjj
2
),a hieved for Gaussianwhite input.
Theinstantaneous apa ityC(jjhjj
2
)isanin reasingandnotbounded
fun -tion of the e e tive SNR:jjhjj
2
. Then for a given R >0 there exists >0
that veri es =C
1
(R ) and su h that
P out (R )=P(C(jjhjj 2 )<R )=P(jjhjj 2 < ): (1.24)
The quality of the wireless link is related to the e e tive SNR (jjhjj 2 ) dis-tribution,P(jjhjj 2 <
)isa measureof thisquality,and the aboveequalities
showthe tight relationbetween the outage apa ity and this measure.
jjhjj 2
has aChi-square pdf with2N
rx
degrees offreedom, we an then
evalu-ate P
out
(R ;) analyti ally. However, we an alsoapply the Cherno bound
to derive an upper bound, this method is simpler and applies for the ase
wherethe omponentsof h are stillGaussianbut not i.i.d.
P(jjhjj 2 < )= E1 fjjhjj 2 g<0 ; (1.25) and 1 fjjhjj 2 < g is upperbounded by e (jjhjj 2 )
,for any >0, then
P(jjhjj 2 < ) min >0 Ee (jjhjj 2 ) = min >0 () = min >0 e R e jjhjj 2 f(h)dh; (1.26) wheref(H)isthepdf ofh,=jjhjj 2 and ()= Ee isthe
har-a teristi fun tion of . The se ond inequality orresponds to the Cherno
upper bound.
The hannel omponentsareassumedtobeGaussianwith ovarian eC
h h =
UDU H
and rank r(= number of diversity sour es). D is an rr positive
de nitediagonalmatrix ontainingthe eigenvalues andU is N
rx
runitary
matrix of the eigenve tors (U
H U =I r ). The integralis 1 det(D) Z e h H UU H h e h H UD 1 U H h dh = det(I r +D 1 ) 1 det (D) ; (1.27)
wherewe use equalityjjhjj
2
=h
H UU
H
h. Bytaking =1the upperbound
is nally P(jjhjj 2 < ) e det ( D+Ir) = O( r ) : (1.28)
This shows that the MRC exploits all the available sour es of diversity in a
SIMO hannel.
1.4 Spa e-Time Coding for MIMO System
The MIMO hannel apa ity as we have seen in se tion 1.2, shows a high
wire-less link. The main problem is how to deal with the inter-antennas
inter-feren e, the rst response was given in [1℄. The proposed s heme, alled
V-BLAST, transmitsindependentstreamsonthe di erentantennasand use
Su essive Interferen e Can ellation (SIC) at the re eiver side. Even if this
s heme allows toa hievea highdata rate,itis farfromexploitingallthe
di-versityadvantageoftheMIMO hannel. In[16℄Tarokh etalproposedtouse
new odes, alled later Spa e-Time Codes (STC), that ombine the hannel
ode with the multiple transmitantenna aspe t, and they also introdu eda
new design riteria forthere onstru tion.
Spa e-Time Code Design Criteria
TheSTC designappliesfornon-adaptives heme,non-adaptiveinthe SNR,
the rate isthen onstant and the numberof odewords is nite.
We onsiderthetransmissionofthe odedsymbolsforadurationofT symbol
periods over a at MIMO hannel. The spa e-time ode an be then
rep-resented in N tx T matrix form: C= 1 x [x 1 ;x 2 ;:::;x T ℄. The a umulated re eived signal is Y = x HC+V; (1.29) where Y=[y 1 ;y 2 ;:::;y T ℄ and V=[v 1 ;v 2 ;:::;v T ℄ are N rx T matri es.
We onsider a Rayleigh at fading i.i.d. MIMO hannel, and a Maximum
Likelihood(ML)re eiver. The hannelisunknownattheTx(no CSIatTx)
and perfe tly known at the Rx (perfe t CSI at Rx). For a transmitted C,
the unionbound gives usan upper bound onthe errorprobability
P(error=C) X C 0 2fC fCgg P(C !C 0 ); (1.30) whereP(C!C 0
)isthePairwiseErrorProbability(PEP) ortheprobability
of de iding erroneously C
0
fortransmitted C. C is the ode book.
The STC design tries to minimize the error probability by minimizing the
PEP.
The ML de isionrule is given by
^ C=argmin C2C jjY x HCjj F ; (1.31) where jjMjj 2 F = trfM H
M g is the Frobenius norm of M. Then the
ondi-tional PEP of Cand C
0
is given by
P(C!C
0
whereÆ isthe de isionmetri di eren e Æ = jjY x HC 0 jj 2 F jjY x HCjj 2 F = jj x H(C C 0 )+Vjj 2 F jjVjj 2 F = 2 x jjH(C C 0 )jj 2 F 2 x <trfV H H(C C 0 )g: (1.33) Letd=jjH(C C 0 )jj 2 F and a=2 x <trfV H H(C C 0 )gN (0;2 2 x 2 v d). <
denotes thereal partand =willdenotetheimaginarypart. Byapplyingthe
Cherno bound weget
P(C !C 0 =H)min s>0 E(e sÆ =H): (1.34) E (e sÆ
=H) is the hara teristi fun tion of Æ and equals e
(s s 2 2 v ) 2 x d , the
minimum of whi h is a hieved for s =
1
2 2
v
. The Cherno upper bound is
then P(C!C 0 =H)e d 4 : (1.35)
ThePEP isthe average of P(C !C
0
=H) overthe hannel distribution. Let
UU be the eigenve tor de omposition of the hermitianof C C
0 , then d = trfH(C C 0 )(C C 0 ) H H H g = trfHUU H H g = trfH 0 H 0 H g; (1.36) whereH 0
=HU has the same distributionas H. We an write then
P(C! C 0 ) = EP(C!C 0 =H) E exp 4 P Ntx l =1 P Nrx k=1 l jh 0 kl j 2 = Q N tx l =1 Q N rx k=1 E exp 4 l jh 0 kl j 2 = Q Ntx l =1 1+ 4 l Nrx : (1.37)
Whi h onstitutes the desired Cherno bound onthe PEP.
The eigenvalues
l
;l = 1;:::;N
tx
are sorted in de reasing order. Let r be
the number of the non-zero values (= rank of C C
0
). Less-tight upper
bound is thengiven by
P(C !C 0 ) 4 r:N rx r Y l =1 l ! Nrx : (1.38)
Diversity Gain Coding Gain SNR(dB) Error Probability (logarithmi s ale) Figure 1.2: PEPvs. SNR
For high SNR this bound be omes tight
P(C!C 0 ) 4 r:Nrx r Y l =1 l ! Nrx : (1.39)
Diversity Gain : Thetotal diversity orderisthe exponentof the SNRand
is given by r:N
rx
. Over the all odebook the diversity is given by r
min :N rx , where r min
isthe minimumrankof overallthe possible ode words
di er-en es in C. r
min :N
rx
is alledthe diversity gain.
Coding Gain : Moreover the PEP is a de reasing fun tion of the produ t
Q r
l =1
l
,this lastone shouldbemadeaslarge aspossibleforthe errorevents
with rankr min . min Q rmin l =1 l
is alled the oding gain. Forr
min
=N
tx
, is
full rank and
Q N tx l =1 l =det (C C 0 )(C C 0 ) H .
In a plot that represents the PEP as fun tion of the SNR in a logarithmi
orresponds to the diversity gain, where the oding gain orresponds to the
position of the line(see g. 1.2).
1.5 Diversity and Multiplexing as De ned by
Zheng & Tse
In [9℄ Zheng and Tse give a new de nition of the diversity and spatial
mul-tiplexing that onsiders adaptive SNR s hemes. In fa t, a s heme C() is a
family of odes of blo k length T(one for ea h SNR level), that supports a
bitrate R ().
This s heme is to a hieve spatial multiplexing r and diversity gain d if the
data rate veri es
lim
!1 R ()
ln()
=r; (1.40)
and the average error probability veri es
lim !1 lnP e () ln() = d: (1.41)
Forblo klengthT !+1theerrorprobabilitybe omestheoutage apa ity.
For ea h r, d
(r) is de ned to be the supremum of the diversity advantage
a hievedovers hemes. Themaximaldiversitygainisde nedbyd
max
=d
(0)
and the maximal spatialmultiplexinggain is r
max
=supfr:d
(r)>0g.
Wewilluse the spe ial symbol
:
=todenote theexponentialequality,i.e.,we
write f() : = b to denote lim !1 lnf() ln() =b: (1.42)
Zheng & Tse onsidered a Rayleigh at MIMO hannel model, and using
resultsonthe distribution ofthe eigenvalues (se tion1.1.1) they showed the
following results.
Optimal Tradeo Curve : Assume T N
rx +N tx 1. The optimal tradeo urve d
(r)is given by the pie ewise-linear fun tion onne ting the
points(k;d (k)),k =0;1;:::;p, where d (k)=(p k)(q k): (1.43)
(2;(N rx 2)(N tx 2)) (0;N rx N tx ) (1;(N rx 1)(N tx 1)) (r;(N rx r)(N tx r)) (min(N rx ;N tx );0) Div ersit y Gain: d ( r )
Spatial Multiplexing Gain: r
Figure 1.3: Diversity vs. multiplexingoptimaltradeo
Re all that p = minfN
rx ;N tx g; q = maxfN rx ;N tx g. In parti ular d max = N tx :N rx and r max =min(N rx ;N tx ).
1.6 Thesis Overview and Outline
This thesis is omposed of three parts. The rst one deals with absen e of
hannelstateinformationatthetransmitter(andperfe tCSIatthere eiver),
the se ond with partial CSI at the Tx (perfe t CSI at the Rx) and the last
one onsiders the absen e of CSI at both Tx and Rx. A brief overview of
the generalframeworkof this thesisand of ea hpart isgiven inthis se tion.
Anabstra tandanintrodu tionisprovidedatthebeginningofea h hapter.
TheCSIavailabilityisafundamentallinkbetweentheseveral haptersofthis
some ases, pra ti al solutions are proposed and analyzed; in other ases,
theoreti alperforman es and fundamentallimitsare investigated. Ourwork
deals with di erent \hot" topi s in MIMO: Spa e-Time Coding (STC) and
de oding, MIMO hannelmodeling, apa itystudies and MIMO hannel
es-timation. Other works that were done during the thesis period in lude the
use of multipleantennas inthe UMTS standard [17, 18,19,20℄, the hannel
inthese asesiseitherSIMOorMISO(withN
tx
=2). Beingoutofthefo us
of this thesis, these works are not reported here.
1.6.1 Part One: Absen e of CSI at Tx (and Perfe t
CSI at Rx)
This aseisthemostpopularintheliteratureandhasbeenthe rstonetobe
onsiderede.g. [1,16,21,2℄. TheTxhas noknowledgeofthe hannel,whi h
is modeled as Rayleigh fading MIMO hannel, either at (subse tion 1.1.1)
or frequen y sele tive (subse tion 1.1.3). At the Rx side, the hannel is
as-sumed to be perfe tly known; in pra ti e this means that there is enough
trainingtoprovide a good hannelestimate.
In hapter 2,we propose a new STC s heme that is based on linear
pre od-ing using a MIMO pre lter to exploit diversity. The inputs of the pre lter
are alledstreams, and ea h symbol fromany streamgets spread overspa e
(antennas) and time by the a tion of the pre lter. This s heme is
onse-quently named Spa e-Time Spreading (STS). STS is shown to preserve the
ergodi apa ity and to a hieve full rate and full diversity. The
optimiza-tion of the MIMO pre lter isdone inorder to maximizethe Mat hed Filter
Bound (MFB) and the oding gain. This s heme was the rst to show su h
good properties
1
. In addition we show that this s heme an easily be
gen-eralized to the ase of a frequen y sele tive hannel, preserving the above
properties, and being therefore is highly attra tive. The optimal de oder
of the STS s heme is the ML de oder. However, this dete tor has a high
numeri al omplexity and o ers no possibility for ombination with binary
hannel de oding. This problemis handled in hapter 3and 4.
In hapter 3, we propose two low omplexity non-iterative re eiver
strate-gies. Non-MLre eivers haveingeneralanimpa tonthediversityand oding
1
gains, and even onthe hannel oding setup. The Stripping MIMODe ision
Feedba k Equalizer (DFE) is a Su essive Interferen e Can ellation (SIC)
re eiver for the di erent streams. In fa t, the streams are pro essed
su - essively, ea h stream is equalized, de oded, and then its ontribution gets
subtra ted from the re eived signal before pro essing the next stream. The
Stripping DFE re eiver is shown to be a very performant re eiver for SNR
adaptive s hemes, espe ially for the diversity vs. multiplexing tradeo . In
fa t, we generalize in this hapter the diversity vs. multiplexing tradeo of
Zheng &Tse tothe frequen y sele tive hannelwith nitedelayspread. We
show also that the Stripping DFE allows to rea h an important portion of
thistradeo . Toa hievetheoptimaldiversityvs. multiplexingtradeo ofthe
Stripping DFE,the study suggests touse stream-dependent rates andhen e
di erent onstellation sizes. However, for the non-adaptive SNR ase, the
streamsexperien edi erentdiversity and odinggains,andthe binary
han-nel oding is stream dependent and should ompensate this non-symmetry.
Another re eiver strategy is the Conventional MIMO DFE. In this ase the
streamsare de oded jointly andina ausalmanner. Thepast dete ted
sym-bols ontribution are subtra ted using feedba k. To avoid performan e loss
due to error propagation, Per Survivor Pro essing (PSP) an be used. The
onventional MIMO DFE re eiver a hieves the same diversity and oding
gain for all streams.
In hapter 4, we propose an iterative turbo re eiver strategy. This re eiver
is a turbo dete tor as it iterates between a Parallel Interferen e
Can ella-tion (PIC) linearestimatorofthe streams(pre lter- hannel as ade, seen as
an inner ode), and a binary hannel de oder (binary hannel oder, seen
as an outer ode). This te hnique has the same omplexity as the turbo
re eiver for the bit interleaved te hnique applied to MIMO systems or its
variant (Threading), and shows an important gain in performan e over this
last te hnique. This re eiver is also very adapted to exploit multi-blo k
di-versity ifpresent and isvery exible, inparti ularforthe ase of less re eive
antennas than transmit antennas(N
rx
<N
tx ).
1.6.2 Part Two: Partial CSI at Tx (and Perfe t CSI
at Rx)
Inthis part the hannel knowledgeat the Txside is basedonfeedba k from
the Rx, orisdue tore ipro ity ofthe uplink and the downlink if both share
thesame arrierfrequen y. However, this knowledgeisoftenpartialbe ause
of the delay in the feedba k and the limited (or absen e of) alibration of
antennas. To study these ases, an adapted hannel model will be
formu-lated in hapter 5. In the rst ase, we will use the pathwise hannel model
a ounting forthe de omposition of the hannelintoalong term part (path
array responses, knownfromthefeedba k) andtheunknownshorttermpart
( omplex path gains). In the se ond ase, the limited re ipro ity will be
re e ted in the hannel model by introdu inga random s alar per antenna.
For ea h of these partial CSI hannel models, we will analyze the MIMO
ergodi apa ity, and show that our results present important onsequen es
for the Txdesign.
1.6.3 Part Three: Absen e of CSI at Rx (and none at
Tx)
Inthis asebothTxandRxhavenoCSI.Wewilltaketheapproa hinwhi h
the hannel will be estimated at the Rx. To this end, we will assume that
some training/pilotsymbols are present, but not ne essarilyenough tohave
ahigh quality estimate.
In hapter 6, will be onsidered two popular fading hannel models. The
rstoneistheblo k-fadingmodelandthese ondisthetimesele tivemodel.
We willshow thatthe average MutualInformation (MI)overa transmission
burst anbede omposedintosymbolpositiondependent ontributions. The
MI omponent at a ertain symbol position optimally ombines semi-blind
informationatthatsymbolposition(exploitingperfe t ausalinputre overy
up to that position ombined with blind information from the rest (future)
of the burst). We will alsoanalyze the asymptoti regimefor whi h we will
beabletoformulateoptimal hannelestimates, and toevaluatethe apa ity
loss with respe t to the ase of perfe t CSI at the Rx. Asymptoti ally, the
de reaseinMIinvolvesFisherinformationmatri es orrespondingto ertain
The mutual information de omposition suggests to ombine the training
and blind parts to estimate the hannel. Chapter 7 proposes semi-blind
approa hes whi h exploit the information from the two parts. These
te h-niques havea omplexitynot (immensely)mu hhigherthanthatof training
basedte hniques. Forthe at hannel ase, thepresented te hniquea hieves
the Cramer-RaoBound. In the frequen y sele tive hannel ase, we willuse
aquadrati semi-blind riterionthat ombinesatrainingbasedleast-squares
Linear Convolutive Spa e-Time
Pre oding
Theuseofmultipletransmitandre eiveantennasallowstotransmitmultiple
signal streams in parallel and hen e to in rease apa ity. In this hapter we
introdu e a simple onvolutive linear pre oding s heme that we all
Spa e-TimeSpreading (STS). This s hemespreads thetransmitted symbols in time
and spa e, involving spatial spreading and delay diversity. Su h linear
pre- oding allows to attain full diversity without loss in ergodi apa ity. We
show thatthegeneralizationof STS tothefrequen ysele tive MIMO hannel
2.1 Introdu tion
The use of MIMO systems o ersa new spatial dimension, and in reases the
ergodi apa ity of the hannel by amultipli ativefa tor equals to the rank
ofthe hannel(subse tion1.2.2). Itlowerstheoutageprobabilitybythe
on-tribution of N
tx :N
rx
diversity omponents orresponding to all the hannel
oeÆ ients in at hannel ase. However, the MIMO hannel (with
imper-fe tCSIatTx)su ersfrominterferen ebetweendi erenttransmitantennas.
The re ent attempts that tryto exploitthis highpotentialfor wireless
om-muni ationhavetomakea ompromiseinordertohandleanin reaseinrate.
They needtotakeadvantage ofthe availablediversity to ombatfadingand
destru tiveinterferen e whilekeepingana eptable omplexity. Trellis ode
introdu ed by Tarokh et al [16℄ takes a SISO-like solution using a binary
hannel oding designed tomap dire tly ontransmit antennas and adapted
tothe use of the ML de oder (Viterbi de oder). The binary hannel ode in
this te hnique has to be very powerful to be able to exploit multi antenna
diversity,timediversityand tohandlehighbitrateleadingtoafundamental
limit on performan e under the onstraint of omplexity. New approa hes
then appeared aiming to exploit diversity of the hannel by linear
transfor-mationofsymbols. In this ategoryspa e-time blo k odes fromOrthogonal
Design [21℄ transform the MIMO hannel in one SISO oeÆ ient that
ap-tures all the diversity and by then use the binary hannel odes designed
for Gaussian hannel. This te hnique, even if it su eeds to exploit all the
available diversity, isfar froma hieving the potentialmultiplexingrate.
More re ent s hemes [22, 23, 24℄ based on onstellation rotating, su eeds
toexploit all the diversity without a lossof rate but need to perform a ML
de oding leading to an exponential (in N
tx
and the onstellation size)
om-plexity that limitsits use. The present hapter presents the STS te hnique
that wehaveintrodu ed rst in[25℄, this s hemeis basedon Linear
Pre od-ing with a MIMO paraunitary lter, and allows to exploit all the available
diversity. As we will see in hapter 3 this s heme an be ombined with
onventionaland non- onventionallow omplexityre eivers. In hapter4we
showhow STS an be ombined with the binary Channel Code (CC) touse
a turbo dete tor and to exploit the multi-blo k diversity if present. These
properties allowtoavoid the ML dete tor omplexity and make the s heme
very attra tive.
Approa hes proposed so far deal with at hannel, however this is far tobe
environments leads to frequen y sele tive hannel with nite delay spread.
The STS approa h applies as well for frequen y sele tive hannel as for at
hannel. It leads to fulldiversity [26℄, and is then the rst te hnique that it
isshown toexploitfulldiversity inthefrequen y sele tive ase(withinTime
Division Multiple A ess ontext).
In mostof this hapterwe assumethe hanneltobe at, however thestated
results are validfor sele tive hannels aswellunless denoted.
We begin this hapterbypresenting the linearpre lteringapproa h. We
then introdu e the design riteria: apa ity, mat hed lter bound and
pair-wise error probability, in order to spe ify our STS s heme. We study the
in uen e of the ir ular onvolution and the frequen y sele tivity of the
hannel. We end the hapter by a dis ussion on the ML dete tor for the
STS s heme.
Results presented in this hapter were published in [25, 26,27, 28℄.
2.2 Linear Pre ltering Approa h
coding
& mapping
channel
+
demapper
& channel
decoder
T(z) H R (z) RX v k x k a k b k MIMO hannel TX N s N tx b x k N rx y kFigure2.1: General ST oding setup.
A general Spa e-Time (ST) oding setup is sket hed in g. 2.1. The
in oming stream of bits gets transformed to N
s
symbol streams through a
ombinationofbinary hannel oding,interleaving,symbolmappingand
de-multiplexing. The result is a ve tor stream of symbols b
k
ontaining N
s
symbols per symbol period. The N
s
streams then get mapped linearly to
the N
tx
transmit antennas and this part of the transmission is alled linear
ST pre oding. The output is a ve tor stream of symbols a
k
ontaining N
symbols per symbol period. The linear pre oding is spatiotemporal sin e a
linearfun tionofb
k
mayappearinmultiple omponents(spa e)andmultiple
timeinstan es (time)of a
k
. The ve tor sequen e a
k
gets transmittedover a
MIMO hannelHwithN
rx
re eiveantennas,leadingtothe symbolrate
ve -torre eivedsignaly
k
aftersampling. The linearpre oding an be onsidered
tobean inner ode, while the binary hannel oding et . an be onsidered
to be an outer ode. As the number of streams is a fa tor in the overall
ode-rate,weshall allthe aseN
s
=N
tx
thefullrate ase,whileN
s
=1
or-responds tothe single rate ase. Instead of multipleantennas, more general
multiple hannels an be onsidered by oversampling, by using polarization
diversity or other EM omponent variations, by working in beam-spa e, or
by onsideringin-phaseandin-quadrature(orequivalentlyrealand omplex)
omponents. In the ase of oversampling, some ex ess bandwidth shouldbe
introdu edatthetransmitter,possiblyinvolvingspreadingwhi hwouldthen
be part of the linear pre oding.
Asweshallsee below, hannel apa ity anbeattainedby afullrate
sys-tem without pre oding (T(z)= I). In that ase, the binary hannel oding
has to be fairly intense sin e it has to spread the information ontained in
ea h transmitted bit over spa e (a ross Tx antennas) and time, as pi tured
on the left part in g. 2.2 and [29℄. The goal of introdu ing the linear
pre- odingistosimplify(possibly goingasfaraseliminating)thebinary hannel
odingpart [16℄. Inthe aseof lineardispersion odes [30℄,[31℄, transmission
is not ontinuous but pa ket-wise (blo k-wise). In that ase, a pa ket of T
ve torsymbolsa
k
(hen eaN
tx
T matrix)gets onstru tedasalinear
om-binationof xed matri es inwhi h the ombination oeÆ ients are symbols
b k
. A parti ular aseisthe Alamouti ode [32℄whi hisafulldiversity single
rate ode orresponding toblo k lengthT =N
tx
=2,N
s
=1.
The STSs heme fo uses on ontinuoustransmission,large blo ks (T >>
1),in whi h linear pre oding orresponds toMIMO pre ltering. This linear
onvolutive pre oding an be onsidered as a spe ial ase of linear
disper-sion odes (making abstra tion of the pa ket boundaries) in whi h the xed
matri es are time-shifted versions of the impulse responses of the olumns
of T(z) (see g. 2.1). Whereas in the absen e of linear pre oding, the last
operation of the en oding part is spatial demultiplexing (Serial-to-Parallel
(S/P) onversion) (see left part of g. 2.2), this S/P onversion is the rst
operationinthe aseoflinearpre oding(see therightpartof g. 2.2). After
the S/P onversion, we have a mixture of binary hannel oding,
coding &
mapping
S/P
DEMUX
S/P
DEMUX
coding &
mapping
mapping
coding &
symbol stream 1
x k b 1;k b N s ;k . . . x k b 1;k . . . b N s ;k symbolstreamN sFigure2.2: Two hannel oding, interleaving,symbol mappingand
demulti-plexing alternatives.
systems are spe ial ases of this approa h. V-BLAST is a full rate
sys-tem with T(z) = I
Ntx
whi h leads to quite limited diversity. The delay
transmit diversity used in the UMTS standard is a single rate system with
T(z) = [1 z 1 ; :::; z (N tx 1) ℄ T
whi h leads to full diversity. We would like
to introdu e a pre ltering matrix T(z) withouttaking a hit in terms of
a-pa ity,while a hievingfull diversity. The MIMOpre ltering willallowus to
apture alldiversity (spatial,and frequentialfor hannelswithdelay spread)
and willprovide some oding gain. The optionalbinary hannel oding per
stream then serves to provide additional oding gain and possibly (with
in-terleaving)to apture thetemporaldiversity(Dopplerspread)ifthereisany.
Finally, though time-invariant ltering may evoke ontinuous transmission,
thepre lteringapproa hisalsoimmediatelyappli abletoblo ktransmission
by repla ing onvolution by ir ulant onvolution.
2.3 Capa ity
Consider the MIMO at AWGN hannel
y k =Ha k +v k =HT(q)b k +v k ; (2.1)
where the noise powerspe traldensity matrix isS
vv (z)= 2 v I N rx ,q 1 b k = b k 1
. The ergodi apa itywithabsen e of CSIatthe Txand perfe tCSI
at the Rx is given by C(S aa ) = E H 1 2j H dz z lndet(I + 1 2 v HS aa (z)H H ) = E H 1 2j H dz z lndet(I + 1 2 v HT(z)S b b (z)T y (z)H H ) = E H 1 2j H dz z lndet(I +HT(z)T y (z)H H ); (2.2)
wherewe assumethat the hannel oding and interleaving per streamleads
tospatially and temporallywhite symbols: S
bb (z)= 2 b I Ntx , and = 2 b 2 v = SNR N tx
. The hannel is modeled as Rayleigh at fading, see subse tion 1.1.1.
As we have shown in se tion 1.2for su h a hannelmodel, the optimization
of the apa ity subje t to the Tx power onstraint
1 2j H dz z tr(S aa (z)) N tx 2 b
leads to the requirement of a white (and
Gaus-sian) ve tor transmission signal S
aa
(z) =
2
b
I. Combined with the
white-ness of the ve tor stream b
k
resulting fromthe hannel en oding, this leads
to the requirement for the pre lter to be paraunitary: T(z)T
y
(z) = I in
orderto avoid apa ity loss.
Motivated by the onsideration of diversity also(see below), we propose
touse the following paraunitary pre lter
T(z) = D(z)Q D(z) = diagf1; z 1 ;:::;z (Ntx 1) g; Q H Q =I; jQ ik j= 1 p N tx ; (2.3)
whereQ isa ( onstant)unitary matrix with equalmagnitude elements
(ex-ampleofthosein lude: WalshHadamardandtheDis reteFourierTransform
matri es). Note that for a hannel with delay spread, the pre lter an be
immediately adapted by repla ing the elementary delay z
1
by z
L
for a
hanneloflength(delayspread)L. Forthe atpropagation hannelH
om-binedwith the pre lterT(z) in(2.3), symbolstreamn (b
n;k
)passes through
the equivalentSIMO hannel
N tx X i=1 z (i 1) H :;i Q i;n ; (2.4)
whi h now has memory due to the delay diversity introdu edby D(z). It is
important that the di erent olumns H
:;i
of the hannel matrix get spread
out in time to get full diversity; otherwise the streams just pass through
a linear ombination of the olumns, as in V-BLAST, whi h o ers limited
diversity. The delay diversity only be omes e e tive by the introdu tion
of the mixing/rotation matrix Q, whi h has equal magnitude elements for
uniform diversity spreading. The pre lter introdu ed in [33℄ is essentially
the same as the one in eq.(2.3). However, the symbol stream b
k
in [33℄ is a
sub-sampled stream, sub-sampled by a fa tor N
tx
. As a result, the system
is single rate. The advantage in that ase though is that no interferen e
2.4 Mat hed Filter Bound and Diversity
The Mat hed Filter Bound (MFB) is the maximum attainable SNR for
symbol-wisedete tion,whentheinterferen efromallothersymbolshasbeen
removed. Hen e the multi-stream MFB equals the MFB for a given stream.
For V-BLAST (T(z)=I), the MFB for streamn is
MFB n =jjH :;n jj 2 2 ; (2.5)
hen e, diversity is limited to N
rx
. For the proposed T(z) = D(z)Q on the
other hand, stream n has MFB
MFB n = 1 N tx jjHjj 2 F ; (2.6)
hen ethisT(z)providesthesamefulldiversityN
tx N
rx
forallstreams. Larger
diversity order leadsto larger outage apa ity.
2.5 Pairwise Error Probability P
e
This se tion studies the pairwise error probability of the STS s heme. The
study will allow us to omplete the spe i ation of the pre oding matrix
T(z) and derive some interesting optimalityresults. We willalso study the
in uen e of the introdu tion of the ir ular onvolution and the e e t of
frequen y sele tive hannels onthe error probability.
The re eived signal is
y k =HT(q)b k +v k =HD(q)Qb k +v k =HD(q) k +v k ; (2.7) where k =Qb k =[ 1 (k); 2 (k); :::; N tx (k)℄ T
. We onsider now the
trans-mission of the oded symbols over a duration of T symbol periods. The
a umulatedre eived signalis then
Y=
b
HC+V; (2.8)
where Y and V are N
rx T matri esand C isN tx T. The stru ture of C is detailedbelow C= 1 b 2 6 6 6 4 1 (1) ::: ::: ::: ::: 1 (T N tx +1) 0 ::: 0 0 . . . ::: ::: ::: ::: . . . . . . . . . . . . . . . . . . ::: ::: ::: ::: . . . 0 0 ::: 0 N tx (1) ::: ::: ::: ::: N tx (T N tx +1) 3 7 7 7 5 (2.9)
