On Space Time Coding Design and Multiuser Multiplexing ...

Ecole Doctorale

d’Informatique

Telecommunications

et Electronique de Paris

These

presentee pour obtenir le grade de docteur

de TELECOM ParisTech

Specialite : Electronique et Communications

Lina Mroueh

Codage Spatio-Temporel et Gain de MultiplexageMulti-utilisateurs pour les Canaux Selectifs

On Space Time Coding Design and Multiuser

Multiplexing Gain over Selective Channels

Soutenue le 13 janvier 2010 devant le jury compose de :

Dr. Olivier Rioul President

Prof. Helmut Bolcskei Rapporteurs

Prof. David Gesbert

Prof. Ezio Biglieri Examinateurs

Dr. Olivier Leveque

Prof. Jean-Claude Belfiore Directeurs de these

Dr. Stephanie Rouquette-Leveil

Dr. Ghaya Rekaya-Ben Othman

To the memory of my dad

Acknowledgements

My deep gratitude goes first to Dr. Stephanie Rouquette-Leveil, my advisor at Motorola

Labs. This work would not have been completed without her unlimited encouragement and

her continuous support. It was really a great fortune for me to start my research career under

her guidance.

I am equally indebted to Prof. Jean-Claude Belfiore at Telecom ParisTech for all his valu-

able suggestions during the development of my thesis and for providing me the opportunity

to have fruitful collaborations during my thesis with Motorola Labs, on one hand, and with

the communication theory group in ETH Zurich, on the other hand.

My deepest gratitude goes also to Prof. Helmut Bolcskei at ETH Zurich for hosting me

in his group, the Communication Theory Group (CTG) during the last year of my thesis. I

am very thankful for providing me the opportunity to work on exciting topics and for all the

time he spent guiding me along the way. This collaboration was a great opportunity for me

and I really enjoyed it.

I would like to thank very much the thesis reviewers Prof. David Gesbert at Eurecom

Nice and Prof. Helmut Bolcskei at ETH Zurich for their time devoted to carefully reading

the manuscript. The same gratitude goes to the examiners Prof. Ezio Biglieri, Dr. Olivier

Leveque from EPFL and Dr. Olivier Rioul from Telecom ParisTech who gave me the honor

for presiding over the jury.

I am very grateful to Dr. Olivier Rioul for his careful reading of the earliest version of

my manuscript and for all his detailed comments that improved significantly the quality of

the final report. I am also thankful for providing me the opportunity to do the teaching

assistance of the information theory lecture at Telecom ParisTech, and for all his pedagogical

advices.

Many thanks go to all the permanent members of the Comelec Department at Telecom

ParisTech. Special thanks go to Dr. Walid Hachem, Dr. Philippe Ciblat and to Dr. Ghaya

Rekaya-Ben Othman for all their recommendations and advices. I am also grateful to the

kind secretaries in the CTG and in the Comelec Department Claudia Zurcher, Barbara Ael-

i

lig, Zouina Sahnoune, Danielle Childz and Chantal Cadiat for their constant help.

I would like to also thank all my former colleagues at Motorola Labs. I would particularly

thank Dr. Laurent Mazet for all his brilliants ideas, Dr. Mohamed Kamoun, Dr. Sheng

Yang and Dr. Sebastien Simoens for all the discussions we had, Vivien Venerozy, Fabrice

Barbarain and Olivier Lahaye for all their humour and for all their technical help. I am also

grateful to Dr. Marc de Courville and to Dr. Jean-Noel Patillon for all their support and

especially for all the efforts they made to allow me to finish my thesis in the best conditions.

Very special thanks go to Gaoning He and to Christophe Gaie for all the moments we shared

and that made my stay enjoyable in Motorola Labs.

I am very much indebted to all my friends and my colleagues in the Comelec Department

at Telecom ParisTech and in the Communication Theory Group in ETH Zurich. I would like

to thank particularly Ali Osmane, Azadeh Ettefagh, Jatin Thukral, Graeme Pope and Eric

Bouton for their friendship and their invaluable support. I am also grateful to Dr. Guiseppe

Dirusi for his valuable comments on the first chapter of my thesis.

Very special thanks go to my mother for all her prayers that guided me along my way

and for her unlimited love that always gives me the strength to advance in life.

Of course, I am very indebted to my sister Malak, for believing in me in all circum-

stances and for her unlimited support. I am particularly very grateful to my elder brother

Kassem for giving me the opportunity to come to Paris to complete my engineering degree

in Telecom ParisTech. I feel also very indebted to my brother Youssef for his constant help

during the development of my thesis and for all his recommendations while preparing my talk.

A last thought goes to all my friends particularly Zeinab Bazzi, Jessy Asmar, Layal El

Sokhon, Roula Nakhle, Ghida Harfouche and Aurelien Quaglio. I am very thankful for all the

nice moments we spent together in the 13ieme arrondissement de Paris and while discovering

new countries.

Finally, I dedicate my thesis to my father, who unfortunately passed away few months

before I finished my engineering degree. Not only was he a devoted father, but also an

exceptional teacher with an extraordinary passion and talent in mathematics. Despite all

the difficulties we encountered due to the instability in South of Lebanon, he was always

dreaming for a better future for us and working hard for that. Without his encouragements

and his wise vision, I would have never gone that far in life.

ii

Abstract

THE next generation of wireless systems such as IEEE 802.11n, IEEE 802.16m, LTE

advanced, etc features Multiple-Input Multiple-Output(MIMO) transmission and

multiuser communications.

In a point-to-point communication, the use of multiple transmitter and receiver antennas

enables an increased data throughput through spatial multiplexing and an increased range by

exploiting the spatial diversity. The design of space time coding schemes that fully achieve

the available diversity and the multiplexing gain in a MIMO system has been extensively ad-

dressed in literature yielding to the design of the optimal family of codes called perfect space

time codes constructed from cyclic division algebra. These codes, originally designed for flat

fading channels, received a lot of attention in industry in the last few years. However, the

recent standards that use multiple antenna terminals are based on more realistic assumptions

involving the use of outer codes, and multi-taps channels. In this dissertation, we propose a

new family of split NVD parallel codes to achieve the optimal diversity multiplexing tradeoff

and we show how the codes designed from cyclic division algebra can be applied in a real

world system, and we focus on their optimality and the practical limits that can be encoun-

tered in industry.

In the multiuser context, exploiting the multiuser multiplexing gains in the network al-

lows to increase considerably the overall throughput in the network. The multiuser context

has been extensively studied in the literature for the case where channels between nodes are

flat fading. However, the flat fading channel is not accurate channel for applications that

exhibit duration and bandwidth that exceed the coherence time and coherence bandwidth of

the channel. In this case, a time-frequency selective channel model is more accurate. In this

dissertation, we study two multiuser scenarios where communications between nodes occur

on channels that exhibit memory in time and frequency.

The first scenario is the interference channel, which corresponds to the scenario where

pairs of sources and destinations want to communicate reliably over the same shared medium.

We show that for the not-so large and for the large interference network, the maximal multi-

plexing gain of can be achieved using an interference alignment scheme under certain channel

spread requirements. The second scenario corresponds to the MIMO broadcast channels,

iii

where a common source transmits data simultaneously to all the multiple antennas receivers

that do not cooperate. For this scenario, we show how the correlation between time frequency

channels can be used in a selective MIMO broadcast channel to minimize the number of bits

to be fed back to the transmitter side while conserving the maximal multiplexing gain.

iv

Resume de la These

LES nouvelles generations de reseaux sans fils tels que IEEE 802.11n, IEEE 802.16m,

LTE advanced, etc sont basees sur des techniques de transmission multi-antennes et

multi-utilisateurs.

Dans les systemes de communications point a point, l’utilisation de plusieurs antennes a

l’emission et a la reception permet non seulement d’augmenter le debit transmis, mais aussi

de garantir une meilleure qualite du signal recu. La construction des codes spatio-temporels

qui permettent d’atteindre la diversite et le gain de multiplexage optimaux dans un systeme

multi-antennes a ete traitee intensivement dans la litterature ces derniers temps, aboutissant

a la construction des codes derives de l’algebre cyclique les plus performants, dits codes par-

faits. Contrairement a l’hypothese classique de modele de canal quasi-statique non code pour

lequel les codes parfaits sont concus, les standards recents qui utilisent les systemes multi-

antennes tiennent compte des considerations de transmission pratique, dont l’utilisation de

codes correcteurs d’erreur et des canaux selectifs en temps et en frequence. Dans cette these,

on propose une nouvelle famille de code spatio-temporels pour les canaux selectifs et nous

montrons comment les codes derives de l’algebre de division cyclique peuvent etre appliques

dans un systeme reel, et nous nous focalisons sur leur optimalite et les limites pratiques qui

peuvent etre rencontrees en industrie.

Dans le contexte multi-utilisateurs, l’exploitation du gain de multiplexage multi-utilisateurs

permet d’augmenter considerablement le debit global du reseau. Le contexte multi-utilisateurs

a ete largement etudie dans la litterature pour le cas ou les canaux entre les noeuds sont con-

sideres comme quasi-statiques tout au long de la duree de transmission. Cependant, cette

hypothese ne donne pas une description precise de la propagation sur un canal reel comme en

pratique le canal est selectif en temps et en frequence. Dans cette these, nous etudions deux

scenarios multi-utilisateurs, ou la communication entre les nœuds se produit sur des canaux

qui sont selectifs en temps et en frequence.

Le premier scenario est le canal a interference, qui correspond au cas ou plusieurs paires

source-destination partagent un meme media et souhaitent communiquer d’une facon efficace.

Dans cette these, on montre que le gain maximal de multiplexage peut etre atteint en utilisant

un systeme d’alignement des interferences sous certaines conditions de propagation du canal.

v

Le deuxieme scenario correspond au canal a diffusion MIMO, ou une source de donnees

commune transmet simultanement a tous les recepteurs a antennes multiples qui ne cooperent

pas. Pour ce cas, on montre comment conserver le gain de multiplexage maximal en utilisant

connaissance partielle du canal a l’emetteur avec un nombre minimal de bits de retour.

vi

Contents

Dedication c

Acknowledgements i

Abstract iii

Resume de la These v

Table of contents ix

List of figures xii

Notation xiii

Resume Detaille de la These xv

Introduction and Outline 1

1 Wireless Channel Model 5

1.1 Linear time varying channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 General LTV channel model . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.2 WSSUS assumption and statistical channel description . . . . . . . . . 7

1.1.3 Underspread channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.4 Channel classification . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 LTV channel identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Single input single output channel . . . . . . . . . . . . . . . . . . . . 9

1.2.2 Multiple input multiple output channel . . . . . . . . . . . . . . . . . 11

1.2.3 Multiuser channel identification . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Discretized channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.1 LTI systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.2 Underspread LTV channel . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Unified matrix formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4.1 Channel characterization at each time-frequency slot . . . . . . . . . . 15

1.4.2 General channel matrix decomposition . . . . . . . . . . . . . . . . . . 17

vii

1.4.3 Frequency selective channel . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4.4 Selective underspread fading channel . . . . . . . . . . . . . . . . . . . 18

1.4.5 Extension to the MIMO case . . . . . . . . . . . . . . . . . . . . . . . 22

1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 NVD Codes in Standards Applications 25

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Structured code construction: A primer . . . . . . . . . . . . . . . . . . . . . 27

2.2.1 Diversity multiplexing tradeoff (DMT) . . . . . . . . . . . . . . . . . . 28

2.2.2 Notations and normalization convention . . . . . . . . . . . . . . . . . 30

2.2.3 Optimal code design criterion . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.4 Space time code properties with fixed rate . . . . . . . . . . . . . . . . 35

2.3 Code construction for selective fading channel . . . . . . . . . . . . . . . . . . 37

2.3.1 Selective fading channel model . . . . . . . . . . . . . . . . . . . . . . 38

2.3.2 DMT of selective fading channel . . . . . . . . . . . . . . . . . . . . . 39

2.3.3 Optimal design criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3.4 Split NVD parallel codes for selective fading channel . . . . . . . . . . 41

2.3.5 Application to the block fading channel . . . . . . . . . . . . . . . . . 48

2.3.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.3.7 Discussion and observation . . . . . . . . . . . . . . . . . . . . . . . . 52

2.4 BICM system model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.4.1 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.4.2 General pairwise error probability derivation . . . . . . . . . . . . . . 55

2.5 BICM-MIMO with flat fading channel . . . . . . . . . . . . . . . . . . . . . . 57

2.6 BICM-MIMO with frequency selective channels . . . . . . . . . . . . . . . . . 61

2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.A Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.A.1 Proof of Lemma 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.A.2 Proof of Lemma 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3 Interference Alignment for Selective Fading Channels 73

3.1 Introduction and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.2 System and channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.3 Multiplexing gain of the K-SISO interference channel . . . . . . . . . . . . . . 77

3.4 Time frequency domain interpretation . . . . . . . . . . . . . . . . . . . . . . 79

3.4.1 Interference Alignment Concept . . . . . . . . . . . . . . . . . . . . . . 79

3.4.2 Toy Example: 3 Users Interference Channel . . . . . . . . . . . . . . . 79

3.5 General spread requirements for interference alignment . . . . . . . . . . . . . 84

3.5.1 General Interference Alignement Construction . . . . . . . . . . . . . . 85

3.5.2 Channel spread requirement for CJ scheme . . . . . . . . . . . . . . . 85

3.5.3 Ozgur and Tse Construction . . . . . . . . . . . . . . . . . . . . . . . 90

3.6 Interference alignment with limited feedback . . . . . . . . . . . . . . . . . . 93

3.6.1 Random vector quantization . . . . . . . . . . . . . . . . . . . . . . . 93

3.6.2 Achieving full multiplexing gain with limited feedback . . . . . . . . . 96

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4 Selective Broadcast Channel with Limited Feedback 99

4.1 Introduction and motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.2 System and channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.2.1 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.2.2 Channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.3 Multiplexing gain for the MIMO broadcast channel . . . . . . . . . . . . . . . 103

4.4 Precoding at the transmitter side . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.4.1 Linear precoding schemes . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.4.2 Performance improvement using PFC . . . . . . . . . . . . . . . . . . 106

4.5 Digital feedback on selective BC with ZF precoder . . . . . . . . . . . . . . . 111

4.5.1 Random vector quantization . . . . . . . . . . . . . . . . . . . . . . . 111

4.5.2 Throughput analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.5.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.6 Digital feedback on selective BC with BD precoder . . . . . . . . . . . . . . . 116

4.6.1 Preliminaries on Grassmann manifolds . . . . . . . . . . . . . . . . . . 116

4.6.2 Quantization codebook design . . . . . . . . . . . . . . . . . . . . . . . 117

4.6.3 Throughput analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.7 Selective MIMO broadcast channel with analog feedback . . . . . . . . . . . . 121

4.7.1 Analog feedback scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.7.2 Relationship between the channel and its analog quantification . . . . 122

4.7.3 Zero forcing with analog feedback . . . . . . . . . . . . . . . . . . . . 123

4.7.4 Block diagonalization with analog feedback . . . . . . . . . . . . . . . 123

4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Conclusion and Perspectives 125

A Algebraic Tools 127

B Weyl-Heisenberg Sequences 129

C Beta Distribution Properties 133

References 142

About the author 143

ix

List of Figures

1.1 Relationship between the channel transfer function. . . . . . . . . . . . . . . . 6

1.2 Time frequency filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1 MIMO system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Coding across time and frequency: The total rate is transmitted only during

T slots. Each entry of τi(Ξ) is a linear combination of symbols carved from

Ad(SNR) where |Ad(SNR)| = SNRrnt . In this case, Xe,d = θdΞd. . . . . . . . 45

2.3 Coding across time and frequency: The total rate is split across the NT slots.

Each entry of τi(Ξi) is a linear combination of symbols carved from As(SNR)

where |As(SNR)| = SNRr

Nnt . In this case, Xe,s = θsΞs. . . . . . . . . . . . . 46

2.4 The optimal DMT achievable by the NVD parallel code for the 2 × 2 block

fading channel with N = 2 is d(r) = 2(2 − r)(2 − r). The split code achieves

the optimal DMT of the block fading channel d(r) = (4− r)(2− r). . . . . . . 53

2.5 NVD parallel code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.6 BICM MIMO system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.7 Coded performance of spatial division multiplexing versus Golden code with a

convolutional code and dfree = 5 . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.8 Coded performance of spatial division multiplexing versus Golden code with a

convolutional code and dfree = 10 . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.9 Asymptotical behavior of the PEP over a flat fading channel . . . . . . . . . 60

2.10 (a) Coding only on each subcarrier without outer code (b) Coding across the

blocks without outer code (c) Coding only on each subcarrier in a BICM-

MIMO system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.11 Asymptotical behavior of the PEP over a frequency selective channel . . . . . 67

2.12 Golden Code vs SDM in IEEE 802.11n context . . . . . . . . . . . . . . . . . 67

3.1 A SISO interference network with K sources and destinations nodes. . . . . . 75

3.2 Outerbound on spatial multiplexing gain . . . . . . . . . . . . . . . . . . . . . 78

3.3 The signaling scheme is a equivalent to a block of M = 17 OFDM symbols,

having Nc = 7 subcarriers each. . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.4 Interference alignment for the 3 users case: shifted OFDM symbols received

at destinations 1, 2 and 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

xi

3.5 Precoding and pre-processing on S1 → D1 . . . . . . . . . . . . . . . . . . . . 84

3.6 Random vector quantization codebook . . . . . . . . . . . . . . . . . . . . . . 94

4.1 A MIMO broadcast channel with nt transmit antennas and K users having nr

receive antennas each. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.2 The r ≥ DW∆H channel coefficients are sufficient to characterize the channel. 102

4.3 Extra protection provided by periodically flipped constellation. . . . . . . . . 108

4.4 Coded performance of PFC sphere encoder versus standard sphere encoder . 111

4.5 Capacity of a broadcast channel with nt = 6 transmit antennas and K =

3 users having nt = 2 antennas each, when Zero Forcing (ZF) precoding is

performed at the transmitter side. . . . . . . . . . . . . . . . . . . . . . . . . 115

4.6 Reduced feedback vs Straightforward Approach. . . . . . . . . . . . . . . . . 116

4.7 Each user feedbacks to the source its r ≥ DW∆H channel components on a

AWGN channel. Each coefficient is transmitted during β time slots. . . . . . 122

xii

Notation

Sets and numbers

Z Set of integers.

R Set of reals.

C Set of complex numbers

Q Set of rational numbers

|A| Cardinality of a set

Z/pZ Quotient group of pZ in Zbxc Closest integer bxc ≤ xx (mod y) Remainder on division of x by y

x∗ Conjugate of a complex number

x! Factorial of x

f(x).= xb exponential equality , limx→∞

log f(x)log x = b

≥, ≤ exponential inequality

sinc(x) sin(πx)πx

SNR Signal to Noise Ratio.

Probability and statistics

CN (0, σ2) Complex Gaussian random variable with zero mean and

variance σ2

E[x] Expectation of x

Matrices and vectors

A Matrix

v Vector

IN Identity matrix with N ×N size

det(A) Determinant of square matrix A

Tr(A) Trace of a square matrix A

rank(A) Rank of matrix A

xiii

A† Transpose-conjugate of matrix A

V[T ] Transpose of vector v

‖A‖F Frobenius norm of matrix A

‖v‖ Euclidian norm of vector v

vec(A) Vectorisation of matrix A

diag(a) Diagonal matrix whose diagonal entries are the elements of

vector ai

diag(Ai)Ni=1 Block diagonal matrix having main diagonal blocks square

matrices Ai

A⊗B Kronecker product between matrices A and B

λ(A) Eigenvalue of matrix A

R Correlation matrix between the scalar (time/ frequency/

time-frequency) channel components

r Rank of matrix R

W Eigenvectors matrix of R

σ0, . . . , σr−1 Eigenvalues of R

λi Eigenvalues of the channel matrix.

αi Eigenexponents of the channel matrix, λi.= SNR−αi .

Acronyms

NVD Non Vanishing Determinant

MIMO Multiple Input Multiple Output

OFDM Orthogonal Frequency Division Multiplexing

BICM Bit Interleaved Coded Modulation

DMT Diversity Multiplexing Tradeoff

PEP Pairwise Error Probability

DFT Discrete Fourier Transform

FFT Fast Fourier Transform

LTI Linear Time Invariant

LTV Linear Time Variant

D Duration of the signal

W Bandwidth of the signal

xiv

Resume Detaille de la These

LE defi de la prochaine generation de communication sans fil est la transmission

haut debit avec une qualite de service elevee. Les techniques multi-antennes (Mul-

tiple Input Multiple Output - MIMO) et la communication multi-utilisateurs ont

ete recemment introduits dans presque toutes les nouvelles normes. Ces deux techniques

de transmission ont ete largement etudiees dans la litterature au cours des dernieres annees

visant a ameliorer la qualite de service des systemes sans fil pour s’approcher de celle des

reseaux cables. Les resultats theoriques ont ete completes par une transition rapide vers des

produits de l’industrie. Parmi les sujets consistants, la conception de codes espace-temps dans

le systeme MIMO, et le gain de multiplexage multi-utilisateurs jouent un role primordial.

Introduction et plan de la these

L’objectif principal de cette these est de montrer comment les codes espace-temps peuvent

etre utilises dans un contexte industriel et comment extraire le gain de multiplexage spatial

multi-utilisateurs avec une connaissance totale ou partielle du canal. Dans la communication

MIMO point a point, nous montrons comment les codes concus a partir de l’algebre de di-

vision cyclique peuvent etre appliques dans un systeme reel, et nous nous focalisons sur leur

optimalite et les limites qui peuvent etre rencontrees en pratique. Ensuite, nous considerons

deux systemes multi-utilisateurs (le canal a interference et le canal de diffusion MIMO) ou

nous supposons que la communication entre les nœuds s’effectuent sur des canaux selectifs en

temps et la frequence. Bien que le canal est souvent considere comme etant quasi-statitique

dans la litterature, ce modele du canal ne donne pas une description precise de la propagation

dans les environnements sans fils, en particulier pour les applications dont la duree et la bande

passante depassent le temps et la bande de coherence du canal. Pour le canal de l’interference,

nous montrons que sous certaines conditions de propagation, le gain total de multiplexage

peut etre extrait en utilisant un systeme d’alignement des interferences. Pour le canal de

diffusion MIMO, nous montrons comment la correlation entre les canaux temps-frequence

peut etre utilisee dans un canal de diffusion MIMO selectif pour minimiser le nombre de bits

a etre renvoye a l’emetteur, tout en conservant le gain maximal de multiplexage.

xv

RESUME DETAILLE DE LA THESE

Plan de la these et contributions

Cette these est organise comme suit. Chapitre 1 introduit la modelisation des canaux sans

fil qui sera utilisee tout au long de cette these. Chapitre 2 adresse la conception des codages

espace-temps pour les canaux selectifs et l’application de ces codes espace-temps dans le

contexte du standard IEEE 802.11n. Chapitre 3 etudie les conditions de propagation du

canal selectif en temps et en frequence qui sont necessaires pour obtenir le gain de multi-

plexage en utilisant un systeme d’alignement des interferences. Chapitre 4 montre comment

la correlation entre les canaux temps-frequence peut etre utilisee dans un canal de diffusion

MIMO selectif pour minimiser le nombre de bits d’informations a etre renvoyes a l’emetteur,

tout en conservant le gain maximal de multiplexage.

Les principales contributions de cette these sont resumes ci-dessous.

1. Modelisation des canaux sans fil (Chapitre 1) Dans ce chapitre, on propose

une representation matricielle unifiee pour les canaux sans fils. Cette modelisation est

basee sur le fait que tous les modeles de canaux (systeme lineaire invariant (LTI) et

systeme lineaire variant dans le temps (LTV)) peuvent etre decomposes en canaux par-

alleles statistiquement dependants [1]. Dans ce chapitre, on propose une decomposition

polynomiale du canal qui sera utilisee tout au long de cette these. La modelisation ce

canal sous cette forme permet de montrer facilement l’impact de la correlation entre

les canaux de temps-frequence.

2. Construction des codes paralleles scindes NVD pour les canaux selectifs

(Chapitre 2) Les codes spatio-temporels parfaits, derives de l’algebre de division

cyclique sont concus a l’origine pour les canaux quasi-statiques. Lorsque le canal

est selectif dans le temps ou en frequence, nous proposons une nouvelle famille des

codes paralleles NVD scindes (Split NVD parallel code) permettant d’atteindre le

compromis diversite gain de multiplexage (DMT) propose par Coronel et Bolcskei

dans [2]. Cependant l’optimalite de ces codes vient aux depens d’une complexite

elevee au recepteur. La complexite peut etre reduite en utilisant des codes correcteurs

d’erreur ce qui est d’ailleurs le cas dans les standards industriels. Dans ce chapitre, on

demontre que lorsque le canal est selectif et en presence des codes correcteurs d’erreur,

le codage entre les differents blocs des canaux paralleles n’est pas necessaire. Dans ce

cas la, il est suffisant d’envoyer un code parfait sur chaque composante frequentielle.

3. Alignement des interference pour les canaux temps-frequence (Chapitre

3) Le canal selectif a interference avec K utilisateurs est considere dans ce chapitre.

On montre que sous certaines conditions de propagation, le gain de multiplexage gain

maximal de K/2 peut etre obtenue en utilisant le systeme d’alignement d’interference

introduit dans [3] ou [4]. Pour le cas particulier avec trois utilisateurs, on pro-

pose un systeme d’alignement d’interference en utilisant des outils arithmetique sim-

xvi


ples. L’implementation de ces systemes d’alignement d’interference en pratique est

egalement abordee. On montre dans ce chapitre que la connaissance parfaite du canal

a l’emetteur peut etre reduite a une connaissance partielle en utilisant une quan-

tification vectorielle. Le nombre de bits necessaires pour quantifier le canal tout en

conservant le gain de multiplexage total est ensuite calcule.

4. Canal de diffusion selective MIMO (chapitre 4) Nous considerons que le canal

de diffusion MIMO lorsque les canaux entre la source et la destination sont selectifs

en temps et en frequence. Nous considerons d’abord le cas ou l’emetteur connaıt

parfaitement le canal. On propose une amelioration de la technique de precodage

proposee dans [5] en utilisant des constellations periodiquement retournees (Period-

ically Flipped Constellation (PFC)). Cependant, la connaissance complete du canal

n’est pas pratique a mettre en œuvre dans des systemes reels. Dans ce chapitre, on

propose une quantification reduite du canal selectif base sur les resultats [6], [7] . On

demontre que la correlation entre les canaux temps-frequence peut etre utilisee afin

de minimiser le nombre de bits a etre renvoye a l’emetteur, tout en conservant le gain

de multiplexage maximal.

xvii


Chapitre 1: Modelisation des canaux sans fils

L’etude des systemes de communication numerique sans fil necessite essentiellement une

bonne comprehension du modele de canal sans fil. Bien que la description precise du modele

de canal sans fil est donne en terme d’ondes electromagnetiques, cela reste une approche

physique et ne peut pas etre utilise dans le systeme de communication sans fil. Pour simpli-

fier la description du modele de canal, le modele de trajets multiples illustre dans Figure 1

est largement utilise en communication numerique. Le canal peut donc etre modelisee par

un systeme lineaire variant dans le temps, qui sera presente dans ce chapitre.

Line of sight

Scattering volume

TxRx

Figure 1: Canal a trajets multiples: Le signal recu est la somme du trajet direct (Line ofsight (LOS)) ainsi que les trajets indirects dus a la reflexion, refraction, ...

Comme point de depart de cette these, ce chapitre donne une formulation matricielle

unifiee pour le canal de propagation MIMO. Une caracterisation complete de ce canal est

decrite dans [1] ou dans le chapitre 2 de [8]. On donne dans ce chapitre une vue generale sur

les systemes lineaires variants (LTV), et on s’interesse plus particulierement a la formulation

matricielle du canal qui serait utilisee tout au long de cette these. On considere le cas general

du canal selectif en temps et en frequence qui peut etre modelise comme un systeme lineaire

variant dans le temps (LTV). On rappelle les notions classiques des systemes stochastiques

LTV utilises dans la litterature, et qui seront utilisees dans cette these. Partant du modele

LTV discret, on definit une decomposition polynomiale du canal qui sera utilisee dans la suite

de cette these.

La decomposition polynomiale introduite dans ce chapitre permet de mettre en evidence

l’impact de la correlation sur le modele du canal. Elle permet aussi d’analyser separement

l’effet de la selectivite en temps et en frequence. De plus, a partir de cette decomposition le

nombre de parametres minimal permettant d’identifier le canal peut etre facilement deduit

ainsi que la condition de propagation necessaire et suffisante pour identifier le canal selectif

en temps et en frequence.

xviii


Chapitre 2: Codes NVD scindes pour les canaux selectifs

Les codes spatio-temporels parfaits derives de l’algebre de division cyclique sont principale-

ment concus pour le cas des canaux quasi-statitiques. Toutefois, les normes les plus recentes

qui utilisent plusieurs antennes des terminaux tels que IEEE 802.11n ou IEEE 802.16e, con-

siderent des hypotheses beaucoup plus realistes, dont la transmission sur des canaux selectifs

en temps et en frequence et l’utilisation de codes correcteurs d’erreur.

Ce chapitre est consacre a l’analyse des codes espace-temps derives de l’algebre de division

cyclique dans un contexte standard. On commence tout d’abord par un apercu general sur la

construction des codes spatio-temporels pour les canaux MIMO quasi-statitiques non codes

illustres dans la Figure 2. Puis, on propose pour les canaux selectifs en temps ou en frequence

channel

Space Time BlockCoding

ML decoder DemodulationModulation

Figure 2: Systeme MIMO sans codage correcteur d’erreur.

une nouvelle famille de codes scindes (Split NVD parallel code) permettant d’atteindre le

compromis diversite gain de multiplexage. Cependant, l’optimalite de ces codes vient aux

depens d’une complexite elevee au niveau du recepteur. Cette complexite accrue est due

au codage entre les blocs des differents canaux paralleles qui est essentiel pour atteindre le

compromis diversite gain de multiplexage. Le codage des symboles seulement au sein de

chaque bloc temps ou frequence sans avoir besoin de coder entre les blocs pourrait etre une

solution interessante. Cependant, cette approche n’est pas optimale que si elle est utilisee

en presence des codes correcteurs d’erreur. Nous montrons que l’utilisation de codes parfaits

sur chaque bloc est optimal dans les systemes BICM (Bit Interleaved Coded Modulation)

illustres dans la Figure 3.

Convolutional

Code CModulationInterleaver

Space Time Block

Coding STBCDeinterleaver

ML soft

Decoder

Viterbi

Decoder

Channel

Figure 3: Systeme MIMO avec codage correcteur d’erreur et entrelacement

La norme IEEE 802.11n est l’une des dernieres evolutions de la norme 802.11 pour les

reseaux locaux sans fil. L’objectif principal de cette technologie est de fournir a l’utilisateur

un debit de 100 Mbps. La grande nouveaute de cette version est l’utilisation des systemes

Multiple Input Multiple Output (MIMO) permettant ainsi d’augmenter le debit et la qualite

du signal transmis.

Pour un canal nt × nr MIMO a evanouissement quasi-statique, deux approches de con-

xix


ception de schema de codage ont ete considerees dans la litterature. La premiere approche

proposee par Tarokh et al. dans [9] consiste a reduire au minimum la probabilite d’erreur en

moyennant sur tous les canaux a evanouissement donnant lieu aux deux criteres fondamen-

taux de construction de code spatio-temporel optimaux:

- Critere de rang, la difference entre deux mots transmis doit etre une matrice de rang

plein.

- Critere de determinant, le determinant minimal du code espace-temps doit etre max-

imisee.

Bien que cette approche est plus adaptee a la distribution de Rayleigh fading, Zheng et Tse

ont propose dans [10] des criteres de conception de code optimale plus generals qui sont bases

sur la caracterisation a haut SNR des gains en terme de diversite et le multiplexage spatial

en utilisant le compromis diversite gain de multiplexage (Diversity Multiplexing Tradeoff,

DMT). Afin d’atteindre le compromis diversite gain de multiplexage, Belfiore et al. ont

introduit dans [11] le critere du non-vanishing determinant (NVD). Plus tard, Elia et al.

dans [12] ont prouve que ce critere est une condition suffisante pour atteindre le DMT en

utilisant un code a taux plein. Recemment, Oggier et al. dans [13] ont propose une famille

des codes spatio-temporels connue sous le nom de codes parfaits qui remplissent les criteres

de conception de Tarokh. En outre, il a ete montre que ces codes sont les codes les plus

performants sur le canal MIMO quasi-statique.

Contrairement au canal MIMO quasi-statique, les systemes de transmission industrielle

tiennent compte de la selectivite du canal. La premiere contribution de ce chapitre est la

construction des codes spatio-temporels pour les canaux selectifs permettant d’atteindre le

compromis diversite gain de multiplexage pour les canaux selectifs dans [2]. On considere

dans ce chapitre le cas ou le canal est selectif en temps et en frequence. Dans les deux cas, le

canal peut etre decompose en N canaux nt×nr paralleles qui sont statistiquement dependants

pour le cas du canal selectif en frequence ou statistiquement independants dans le cas du canal

selectif en temps. Le DMT optimal qui peut etre atteint est (ρM − r)(m − r) ou ρ est le

rang de la matrice de correlation qui est egal a N pour le cas des canaux selectifs en temps et

egal a la memoire du canal pour le cas des canaux selectifs en frequence, M = max(nt, nr),

m = min(nt, nr) et r est le gain de multiplexage.

La structure des split NVD parallel codes proposee dans ce chapitre est illustree dans

la Figure 4. Dans plus, on demontre qu’en utilisant cette structure on arrive a atteindre le

DMT optimale. De plus, les resultats numeriques illustres dans la Figure 5 montrent que ces

codes ont une meilleurs performance que les codes NVD paralleles proposes dans [14] et qui

permettent d’atteindre seulement le DMT de ρ(nt−r)(nr−r). Les split NVD codes proposes

ne sont autre que la concatenation de N-NVD parallel code, par contre avec une taille de

constellation ajustee afin de transmettre le meme debit qu’un code parallele tout court.

La deuxieme contribution de ce chapitre est d’etudier les codes algebriques dans un con-

xx


NT slots

Ξ0 Ξ1 ΞN−1 n = 0

τ(ΞN−2) n = 1

τN−1(Ξ0) n = N − 1τN−1(Ξ2)τN−1(Ξ1)

τ(ΞN−1) τ(Ξ0)Ξs =1√N×

Figure 4: Codes NVD paralleles scindes

10-5

10-4

10-3

10-2

10-1

100

0 5 10 15 20

PE

R

SNR(dB)

Error Probability of split code and NVD parallel code

Split code BPSK - R = 4bpcu NVD parallel code QPSK - R = 4bpcu Split code QPSK- R = 8bpcu NVD parallel code 16QAM - R = 8bpcu

Figure 5: Performance des Split NVD codes vs NVD parallel codes.

xxi


texte industriel sur un canal selectif en frequence en presence des codes correcteurs d’erreur.

En l’absence des codes correcteurs d’erreur, le codage entre les blocs (chaque bloc correspond

au code envoye sur une sous-porteuse) est obligatoire pour obtenir le gain de diversite maxi-

male comme illustree dans la Figure 6(b). Ceci est due au critere de rang qui necessite que

le code bloc diagonal soit de rang plein afin d’atteindre la diversite maximale. En utilisant

uniquement des codes parfaits comme indique dans la Figure 2.10(a), il se peut que l’un

des blocs soit egal a zero, et par la suite le critere de rang n’est plus valable. Cependant,

dans un systeme MIMO-BICM-OFDM, le codage correcteur d’erreur garantit que les blocs

errones gardent une structure de rang plein en utilisant uniquement des codes parfaits (Figure

2.10(c)).

0

0 Zero block

Non zero block

(a) (b) (c)

Erroneous uncoded codeword Erroneous coded codeword

N ×N N ×N dfree × dfree

Figure 6: (a) Sans code correcteur d’erreur et codage par bloc uniquement (b) Codage entreblocs sans code correcteur d’erreur (c) Codage par bloc dans un systeme BICM-MIMO-OFDM.

Chapitre 3: Alignement des interferences pour les canaux selectifs

Le canal a interference illustre dans la Figure 7 decrit le milieu partage entre K paires de

sources et de destinations qui partagent les memes ressources et souhaitent communiquer en

utilisant les ressources de la facon la plus efficace possible. Les approches traditionnelles de

gestion d’interference sont principalement basees sur la transmission utilisant des ressources

orthogonaux (TDMA, OFDMA, ...) et souffrent par la suite de l’absence des degres de

liberte dans le systeme. Recemment des approches plus developpees fondees sur le principe

d’alignement des interferences au recepteur permettent d’extraire tous les degres de liberte par

utilisateur. Cependant, le schema d’alignement des interferences (IA) propose par Cadambe

et Jafar dans [3] depend de facon critique sur l’hypothese que tous les canaux du reseau sont

selectif en temps. Ceci a ete plus tard etendu au cas du canal selectif en frequence par Grokop

et Tse [15].

xxii


S1

S2

SK

DK

D2

D1

Figure 7: Canal SISO a interference.

En general, les communications reelles s’effectuent sur des canaux qui sont a la fois selectifs

en temps et en frequence. Dans ce chapitre, on montre que sous certaines conditions de prop-

agation canal, l’IA permet d’extraire tous les degres disponibles sur un canal selectif en

temps-frequence. La mise en œuvre pratique des IA est egalement traitee, nous montrons

que le gain multiplexage optimal peut etre aussi obtenu en utilisant uniquement une connais-

sance partielle du canal. Le canal a l’emetteur peut dans ce cas etre reconstruit en utilisant

une quantification vectorielle pour laquelle on determine le nombre de bits minimal necessaire

pour atteindre le gain de multiplexage optimal.

Recemment, beaucoup d’efforts ont ete investi pour caracteriser la region de capacite du

canal a interference, e.g., [16, 17] aboutissant uniquement a une borne sur la region de la

capacite sans avoir une caracterisation exacte de cette region de capacite. Cependant, les

resultats preliminaires de Host Madsen et Nosratinia dans [18] ont montre que le gain de

multiplexage maximal qu’on peut atteindre a haut SNR est egal a K/2.

Cadambe et Jafar dans [3] ont propose un schema innovateur base sur l’alignement des

interferences permettant d’atteindre le gain de multiplexage maximale de K/2. L’impact ma-

jeur de l’utilisation de cette strategie est le faite que chaque utilisateur serait capable d’utiliser

la moitie des ressources partagees sans aucune interference des autres utilisateurs. Cepen-

dant, le schema propose par Cadambe et Jafar [3] depend de facon critique sur l’hypothese

que tous les canaux sont selectifs uniquement dans le temps.

Dans ce chapitre, on montre que sous certaines conditions de propagation canal, l’IA per-

met d’extraire tous les degres de liberte disponibles sur un canal selectif en temps-frequence.

La mise en œuvre pratique des IA est egalement traitee, nous montrons que le gain multi-

plexage optimal peut etre aussi obtenu en utilisant uniquement une connaissance partielle du

xxiii


2 0 1

1 2 0

0 21

12 0

0 12

2 0 1

0 1 2

interference

interference

interference

Destination 2

Destination 3

Precoded data at sources

1 2 0

0 1 2

2 0 1

Destination 1

0 1 2

2 0 1

5

2

2

4

1

3

4

5

2

Nc

1

Figure 8: Alignement des interferences aux recepteurs.

canal. Le canal a l’emetteur peut dans ce cas etre reconstruit en utilisant une quantification

vectorielle pour laquelle on determine le nombre de bits minimal necessaire pour atteindre le

gain de multiplexage optimal. Nos resultats sont bases sur la decomposition polynomiale du

canal selectif en temps et en frequence propose dans le Chapitre 1.

Afin de donner un exemple concret sur l’interpretation de l’alignement des interferences

dans le domaine temps-frequence, on considere l’exemple dans la Figure 8. On considere le

cas d’un canal a interference avec 3 utilisateurs. Chaque source souhaite envoyer 3 symboles

OFDM contenant chacun Nc symboles sur un total de 7 slots. La premiere remarque qu’on

peut faire est que si on arrive a decaler la position du symbole OFDM de tel sorte a etre recu

sans interference au recepteur, tous les sous porteuses sont aussi recues sans interference. Le

but du schema d’alignement d’interference est de trouver la position des symboles OFDM

a l’emetteur de telle sorte que les symboles interferents a chaque recepteur soient recus sur

le meme slot. Dans ce chapitre, on decrit un algorithme simple permettant de faire ce type

d’alignement au recepteur.

En resume, dans ce chapitre on montre que les schemas d’alignement des interferences

proposes dans la litterature CJ dans [3] et OT dans [19] permettent d’extraire le gain total de

multiplexage totale si le channel spread du canal est de l’ordre deK−8 etK−4, respectivement.

xxiv


Cette condition sur la propagation du canal peut etre facilement verifiee dans les systemes

pratiques comme le channel spread est de l’ordre de 10−2 pour les communications a l’interieur

des batiments et de l’ordre de 10−7 pour les canaux mobiles.

Chapitre 4: Canal de diffusion MIMO selectif

Le canal a diffusion illustre dans la Figure 9 modelise le cas ou une source de donnees MIMO

transmet simultanement des donnees a plusieurs recepteurs MIMO qui ne cooperent pas.

Dans les systemes MIMO point a point, il est bien connu des resultats de Telatar dans [20]

que le gain de multiplexage spatial ne depend pas de la connaissance du canal a cote emetteur.

Contrairement au cas mono-utilisateur, la region de capacite du canal de diffusion (Broadcast

Channel, BC) depend largement de la connaissance du canal a l’emetteur.

H[2]

S

nt

1

D1

nr

1

nr

1

D2

1

nr

DK

H[K]

H[1]

Figure 9: Canal de diffusion MIMO

Lorsque le canal est connu completement a l’emetteur, la region de capacite de ce canal

a ete caracterisee dans [21]. En plus, il a ete demontre que le Dirty Paper coding technique

(DPC) permet d’atteindre la region de capacite maximale. En depit de son optimalite, cette

technique n’est pas possible pour etre mise en œuvre dans un systeme pratique, car elle

apporte une grande complexite a l’emetteur et aux recepteurs. Les systemes lineaires de

precodage comme l’inversion du canal a l’emetteur dans [22] et la diagonalisation du canal

par bloc dans [23] sont beaucoup moins complexes a utiliser que le DPC et permettent aussi

d’atteindre le gain de multiplexage maximale comme demontre dans [24]. Partant du schema

de precodage propose par Peel dans [5], on propose une amelioration intuitive en utilisant

des schemas de constellation retournes periodiquement qu’on appelle Periodically Flipped

Constellation (PFC) permettant ainsi d’ameliorer les performances en termes de probabilite

xxv


d’ereur.

L’hypothese de la connaissance complete du canal (Full CSIT) n’est pas generalement

interessante a etre mise en oeuvre car elle necessite un grand nombre de bits de retour. Une

solution plus realiste a ete etudiee par Jindal pour le cas des utilisateurs avec une seule

antenne dans [6] et, qui a ete plus tard etendu au cas MIMO dans [7]. Il a ete demontre

que le gain de multiplexage maximal peut etre atteint en utilisant une connaissance partielle

du canal avec quantification du canal et un codebook dont la taille est proportionnelle a la

puissance du signal transmis en dB.

La plupart des resultats mentionnes ci-dessus adresse le cas ou les canaux entre la source et

les destination sont supposes etre a quasi-statique. Cependant, en realite les communications

se produisent generalement sur des canaux qui sont selectifs en temps et en frequence. Dans

ce chapitre, nous analysons le cas ou les liens sont selectifs en temps et en frequence. Pour se

faire, on se base sur les resultats de Dirusi et al. dans [1] qui montrent que lorsqu’on transmet

et recoit sur des sequences Weyl-Heisenberg, le canal peut etre decompose en des canaux

temps-frequences paralleles et statistiquement dependants. Comme les canaux sont correles,

on demontre dans ce chapitre comment cette correlation entre les canaux peut etre utilisee

pour reduire le nombre de feedback bits necessaire pour reconstruire le canal a l’emetteur, et

qui permettent de conserver le gain de multiplexage maximal.

xxvi


Conclusion et perspectives

Motive par les derniers standards MIMO et multi-utilisateurs (telles que la norme IEEE

802.11n, IEEE 802.16m, LTE Advanced, ...), deux problemes majeurs sont abordes dans

cette these: la conception des codes espace-temps pour les canaux selectifs, et l’exploitation

du gain de multiplexage maximal dans un systeme multi-utilisateurs.

Les codes derives de l’algebre cyclique ont ete analyses en premier temps dans cette these.

Pour le canal selectif en temps ou en frequence, on a propose une nouvelle famille de codes

spatio-temporels, dite Split NVD parallel code permettant ainsi d’atteindre le compromis di-

versite gain de multiplexage. La mise en œuvre pratique de codes algebriques pour les canaux

selectifs en frequence a ete aussi abordee. On a montre que dans un systeme MIMO-OFDM,

il suffit de combiner les codes correcteurs d’erreur et d’envoyer un code parfait sur chaque

sous-porteuse afin d’atteindre la diversite maximale.

Dans le contexte multi-utilisateurs, la communication sur les canaux selectifs en temps

et en frequence a ete traitee. Pour ce but, on a propose une modelisation matricielle de ce

type des canaux qui a ete utilisee tout au long de cette these. Cette modelisation matricielle

repose en principe sur le fait que les canaux temps frequence peuvent etre decomposes en des

canaux paralleles correles quand on transmet et recoit sur des sequence Weyl-Heisenberg.

L’objectif principal de l’etude des systemes multi-utilisateurs est de montrer comment

exploiter le gain de multiplexage multi-utilisateurs lorsque les canaux entre sources et desti-

nations sont selectifs en temps et en frequence. Le premier systeme qui a ete considere dans

cette these est le canal a interference. Nous avons montre que sous certaines conditions de

propagation du canal, l’alignement des interferences permet d’atteindre le gain de multiplex-

age maximal. Le second systeme est le canal de diffusion MIMO. Pour cette chaıne, il est

bien connu que le gain de multiplexage maximal depend critiquement de facon critique de

la connaissance du canal a l’emetteur. Pour le canal selectif en temps et en frequence, nous

avons montre comment la correlation entre les canaux temps-frequence peut etre utilisee pour

reduire au minimum le nombre de bits necessaire pour quantifier le canal et le gain maximal

de multiplexage. Le taux de perte en capacite du a cette quantification a ete egalement evalue.

Comme perspectives pour les travaux futurs, nous proposons les directions suivantes:

- Conception d’entrelaceur pour les systemes MIMO BICM: Dans le Chapitre 2 de cette

these, la conception d’entrelaceurs n’a pas ete abordee. Il est clair d’apres les ex-

pressions PEP que la probabilite d’erreur peut etre reduite au minimum lorsque les

bits errones appartiennent a differentes sous-porteuses. Ainsi, il serait interessant de

concevoir un entrelaceur permettant de maximiser le parametre D, qui est un facteur

limitant de l’ordre de la diversite. Dans la litterature, la conception d’entrelaceurs a ete

xxvii


traitee par Gresset dans [25]. Des efforts supplementaires dans cette direction peuvent

etre investi en vue d’ameliorer la performance dans un systeme BICM-MIMO-OFDM.

- Conception des codes espace-temps pour le canal selectif en temps et en frequence: Le

DMT optimal de ce canal a ete calcule dans [2]. Dans ce cas, le split code parallele NVD

ne peut pas etre applique comme le canal varie dans les deux dimensions temporelles

et frequentielles. Un schema optimal a ete propose dans [2] base sur la conception d’un

precodeur adapte aux statistiques du canal et la conception d’un code independant de

la statistique du canal. La construction de tels codes en utilisant les codes cycliques

est une piste interessante pour nos futurs travaux.

- Effet des interferences inter-symboles sur le modele du canal: Tout au long de cette

these, l’effet des interferences entre symboles et entre sous-porteuses a ete neglige dans

la modelisation du canal. La sensibilite de la capacite a la modelisation du canal a

ete etudiee par Durisi et al. dans [26]. Il sera interessant pour les systemes multi-

utilisateurs d’etudier l’effet de la sensibilite de la modelisation du canal sur le gain de

multiplexage dans les systemes multi-utilisateurs avec canaux selectifs en temps et en

frequence.

- Canal de diffusion MIMO, algorithme de selection: Dans le Chapitre 4, nous avons

considere le cas de canal de diffusion MIMO ou l’on suppose que les utilisateurs sont

selectionnes au hasard, sans tenir compte de la qualite de service et l’equite entre les

utilisateurs. Des algorithmes de selection qui maximisent la diversite multi-utilisateurs

ont ete introduits dans les travaux de Yoo dans [27], et les travaux de Gesbert et

al. dans [28, 29] pour les canaux quasi-statiques. Pour le cas des canaux selectifs

en frequence, un algorithme d’ordonnancement iteratif qui minimise le nombre de bits

renvoye a l’emetteur a ete propose dans [30]. L’extension de ces algorithmes iteratifs au

cas du canal selectif en temps et en frequence completerait bien les resultats existants

dans la litterature.

xxviii

Introduction and Outline

THE challenge of the next generation of wireless communication is to offer at the

receiver side a high data rate with a high quality of service. The multiple-input

multiple-output (MIMO) transmission and the multiuser communication have been

recently introduced in almost all new standards. These two techniques of transmission have

been extensively studied in the literature over the last few years aiming to boost the quality

of service of wireless systems close to the one of wireline systems. Theoretical advances were

complemented by a rapid transition to industry product. Among the consistent topics, the

space time coding design in MIMO system and the multiuser multiplexing gain played a

prominent role.

The main purpose of this dissertation is to show how space time coding can be used in

an industrial context and how to extract the multiuser spatial multiplexing gain with full or

partial channel state information. In the point-to-point MIMO communication, we show how

codes designed from cyclic division algebra can be applied in a real world system, and we focus

on their optimality and the practical limits that can be encountered in industry. Then, we

consider two multiuser systems (the interference channel and the MIMO broadcast channel)

where we assume that communication between nodes occurs on channels that exhibit memory

in time and frequency. Although the flat fading channel has been extensively studied in the

literature, it often fails to give an accurate channel model especially for applications that

exhibit duration and bandwidth that exceed the coherence time and coherence bandwidth of

the channel. For the interference channel, we show that under certain channel spread restric-

tion, the full multiplexing gain can be extracted using an interference alignment scheme. For

the MIMO broadcast channel, we show how the correlation between time-frequency channels

can be used in a selective MIMO broadcast channel to minimize the number of bits to be fed

back to the transmitter side while conserving the maximal multiplexing gain.

Outline and Contributions

This dissertation is organized as follows. Chapter 1 introduces the wireless models that will

be used throughout this thesis. Chapter 2 addresses the space time coding design in a stan-

dard context. Chapter 3 studies the time-frequency channel requirements in an interference

1

INTRODUCTION AND OUTLINE

channel needed to achieve the full multiplexing gain using interference alignment schemes.

Chapter 4 shows how the correlation between time-frequency channels can be used in a se-

lective MIMO broadcast channel to minimize the number of feedback bits to the transmitter

side while conserving the maximal multiplexing gain.

The main contributions of this thesis are summarized in the following.

1. Wireless channel model (Chapter 1) We provide a unified matrix-algebraic frame-

work for the wireless channel model. This framework is based on the description of

the channel in [1] and the fact that all channel models (linear time invariant(LTI),

linear time variant(LTV)) can be well approximated by parallel correlated (in time,

frequency or time-frequency) MIMO channels. A useful channel modeling form, which

we called polynomial channel decomposition is proposed and is used throughout this

thesis. Modeling that channel under this form allows to deal and to show easily the

impact of correlation between time-frequency channels.

2. NVD parallel codes for standard applications (Chapter 2) Perfect space

time codes [13] constructed from cyclic division algebra are originally designed over

the quasi-static fading channel, and verify the fundamental property to have a non-

vanishing determinant(NVD). When the channel is selective either in time or in fre-

quency, we propose a new family of split NVD parallel code to achieve the diversity

multiplexing tradeoff (DMT) proposed by Coronel and Bolcskei in IEEE ISIT 2007.

Decoding these codes requires a large complexity order at the receiver side, and is not

often feasible. We show that coding only across the frequency tones without requiring

to code across all the tones is optimal if it is used in a bit interleaved coded modula-

tion system. Moreover, feasible decoder based on the use of low complexity ML soft

decoder can be used on each tone to decode data.

3. Interference alignment for time-frequency selective channels (Chapter 3)

For the K-user interference time-frequency selective channel, we show that under

certain channel spread requirements the maximal multiplexing gain of K/2 can be

achieved using an interference alignment scheme [3] or [4] for the not-so large and large

networks. For the three users case, an interference alignment scheme is proposed using

some simple arithmetic tools. The practical implementation of interference alignment

scheme has been also addressed. We show that using a random vector quantization

scheme with an adequate number of bits that scales as SNR, perfect knowledge of

selective fading channel can be relaxed to a quantized channel knowledge at all nodes,

while conserving the full multiplexing gain that can be achieved will full CSI.

4. Selective MIMO broadcast channel (Chapter 4) We consider the MIMO broad-

cast channel when channels between source and destinations are selective in time and

2


frequency. We consider first the case where full channel state information is known

at the transmitter side, and review the precoding schemes that were proposed in the

literature. Then, we propose an intuitive improvement of the vector perturbation

scheme [5] based on the use of periodically flipped constellations. The assumption of

full channel knowledge at the transmitter side requires a large amount of feedback,

and is therefore not practical to implement in real systems. A more feasible solution

with finite rate feedback (analog and digital feedback) originally proposed in [6], [7]

is applied to the selective case, where the minimal number of feedback bits required

to achieve the full multiplexing gain is derived. We show that the correlation between

time-frequency channels can be used in order to minimize the number of bits to be

fed back to the transmitter side while conserving the maximal multiplexing gain.

3


4

Chapter 1

Wireless Channel Model

STUDYING digital wireless communication system essentially requires a good under-

standing of the wireless channel model. While the precise description of the wireless

channel model is given in terms of electromagnetic waves, this remains a physical

approach and cannot be used in wireless communication system. To simplify the description

of the channel model, the multipath approximation is widely used in digital communication

systems. Channels can be therefore modeled as random Gaussian linear time varying (LTV)

system, which will be presented in this chapter.

As a starting point of this thesis, this chapter gives a unified matrix-algebraic framework

for the MIMO propagation channel that will be used in the subsequent chapters. A complete

characterization of this channel is described in [1] or in chapter 2 of [8]. Here, we give a

general view on the LTV channel, and we focus mainly on the channel matrix formulation

that will be used throughout this thesis. For this purpose, we consider the general case of

the fading selective channel that can be modeled as a linear time varying (LTV) system.

We present first the continuous LTV channel model, and recall the classical fading notions

used in the literature that will be used in this thesis. Then, we describe the corresponding

discretized input output (I/O) relation. Based on this discrete I/O relation, a channel matrix

formulation for the SISO case which we call polynomial channel decomposition is defined.

Then, this formulation is generalized to the MIMO case.

1.1 Linear time varying channel

When mutipath approximation is used, the received signal is the sum of all multipath com-

ponents and the line of sight (LOS) channel. Each path induces a variation of the signal

strength (Doppler effect) and delay shift at the receiver side. Generally, the number of paths

5

CHAPTER 1. WIRELESS CHANNEL MODEL

is very high, which makes logical to model the multipath effect by a linear time varying sys-

tem.

In this section, the continuous model and the statistical description of the LTV system are

first defined. Based on this definition, we recall the classical fading notions widely used in

literature.

1.1.1 General LTV channel model

A wireless channel is generally described by a linear operator H that maps an input signal

s(t) into an output signal r(t), related by the following noise-free relationship

r(t) = (Hs)(t) =

∫t′kH(t, t′)s(t′)dt′, (1.1)

where kH(t, t′) is the kernel of the linear operator H that can be also interpreted as the channel

response at time t to a Dirac impulse at time t′, where τ = t−t′ is called the channel delay. In

wireless communication literature, the notation time varying impulse response h(t, τ) defined

as kH(t, t− τ) = h(t, τ) is more used than the kernel one.

For an LTV system, the channel can be also characterized by two other functions. The

first is the delay-Doppler spreading function SH(ν, τ) defined as Fourier transform (t → ν)

of h(t, τ) where ν denotes the Doppler spread caused by the movement of the transmitters,

receivers and scatterers. The second is the time varying transfer function LH(t, f) defined as

the Fourier transform (τ → f) of h(t, τ). The relationship between these system functions is

given by,

LH(t, f) =

∫τh(t, τ)e−j2πfτdτ, (1.2)

SH(ν, τ) =

∫νh(t, τ)e−j2πνtdt, (1.3)

LH(t, f) =

∫τ

∫νSH(ν, τ)ej2π(νt−τf)dνdτ. (1.4)

and is summarized in Figure 1.1.

h(t, τ)

Ft→ν Fτ→f

SH(ν, τ) Fν→t,τ→f

LH(t, f)

Figure 1.1: Relationship between the channel transfer function.

6

1.1. LINEAR TIME VARYING CHANNEL

Using these functions, the received signal can be expressed as

r(t) =

∫τh(t, τ)s(t− τ)dτ =

∫t′kH(t, t′)s(t′)dt′, (1.5)

=

∫ν

∫τSH(ν, τ)s(t− τ)ej2πνtdνdτ, (1.6)

=

∫fLH(t, f)S(f)ej2πftdf, (1.7)

where S(f) is the Fourier transform of the transmitted signal s(t).

1.1.2 WSSUS assumption and statistical channel description

In digital communications, the linear operator H is random and LTV channel models are

generally studied under the wide sense stationary and uncorrelated scattering (WSSUS) as-

sumption. This property consists in assuming that the random channel H is wide sense

stationary in time t and uncorrelated in scattering (delay) τ , which means that

E[h(t, τ)h(t′, τ ′)] = Rh(t− t′, τ)δ(τ − τ ′).

The WSSUS property implies that the time varying-transfer function LH(t, f) is wide-sense

stationary in both time and frequency, and the spreading function SH(ν, τ) is uncorrelated

in delay τ and in Doppler ν, i.e.,

E[LH(t, f)L∗H(t′, f ′)] = RH(t− t′, f − f ′), (1.8)

E[SH(ν, τ)S∗H(ν ′, τ ′)] = CH(ν, τ)δ(τ − τ ′)δ(ν − ν ′). (1.9)

The scattering function CH(ν, τ) is the 2-D Fourier transform of the time-frequency correlation

function RH(∆t,∆f) such that

RH(∆t,∆f) =

∫τ

∫νCH(ν, τ)ej2πν∆te−j2πτ∆fdτdν. (1.10)

1.1.3 Underspread channel

As a consequence of the limited velocity of transmitter, receiver and scatters in the propa-

gation environment, the maximum Doppler shift is limited to ν0. We also assume that the

maximum delay is bounded by −τ0 and +τ0.

The assumption of limited Doppler shift and delay implies that the scattering function

CH(ν, τ) is supported on a rectangle of spread ∆H = 4τ0ν0, such that

CH(ν, τ) = 0 for (ν, τ) /∈ [−ν0,+ν0]× [−τ0,+τ0]. (1.11)

For almost all radio channels, the time taken for channel to change significantly (1/ν0) is

7


much longer than the delay spread τ0, i.e.,

4τ0ν0 1. (1.12)

Channels satisfying these characteristics are called underspread channels and are mainly

addressed in Chapters 3 and 4 of this thesis. For this type of channels, we know from the

results of Kailath in [31] that as ∆H ≤ 1, then the channel can be identified at the receiver

side.

1.1.4 Channel classification

Usually, channel variations can be analyzed as function of two parameters: the time coherence

and the bandwidth coherence which are defined as following.

Definition 1.1 The coherence time Tc corresponds to the width of |RH(∆t, 0)| and represents

the interval of time over which LH(t, f) changes significantly as a function of t.

The coherence bandwidth Bc is defined as the width of |RH(0,∆f)| and represents the interval

of time over which LH(t, f) changes significantly as a function of f .

Depending on the environment(Tc, Bc) and the application requirements parameterized by

its duration D and its bandwidth W , channels are often categorized into 4 classes in wireless

communication literature, given as following:

Flat fading channel: If W Bc, and D Tc, then LH(t, f) = k, and the channel model

in (1.7) reduces to r(t) = k s(t).

Frequency selective channel: If W > Bc and D Tc, then h(t, τ) = h(τ), and the

channel reduces to a linear time invariant(LTI) system model where the I/O relation is given

by r(t) = (h ∗ s)(t).

Time selective channel: If W Bc and D > Tc, then h(t, τ) = h(t), and the channel

refers in this case to a linear frequency invariant (LFI) system model where the I/O relation

is given by r(t) = h(t)s(t).

Selective fading channel: If W > Bc and D > Tc, then the channel is selective in time

and frequency and the channel model is defined as in eqs.(1.5),(1.6),(1.7).

1.2 LTV channel identification

The goal of the linear time varying channel identification (or measurement) is to obtain a

complete knowledge of the channel’s operator based on a finite number of observations. The

landmark paper of Thomas Kailath in [31] was the first paper to analyze the identification

problem for the LTV channels characterized by a scattering function supported by a rectangle

defined in (1.11).

8

1.2. LTV CHANNEL IDENTIFICATION

1.2.1 Single input single output channel

Theorem 1.1 (SISO identifiability [31], [32]) For a linear time varying channel LTV

characterized by a scattering function supported by a rectangle as defined in (1.11), the un-

derspread condition is a sufficient and necessary condition for channel operator identification,

i.e,

H is identifiable if and only if ∆H ≤ 1.

Proof: The rigorous proof of this theorem based on the Gabor analysis can be found

in [32]. In the following, we just discuss the necessity of the underspread condition in the

channel reconstruction scheme based on a finite number of observations at the receiver side.

The essence of the following development can be found in [33] and in [31].

The channel’s reconstruction scheme in function of a finite number of observations is detailed

in [31] or in [33] and is depicted in Figure 1.2. As shown in Figure 1.2, the input signal s(t)

LH(t, f)

Hi(f) h0(t)

s(t)

LH(t, f)

r(t)−W/2 +W/2 −D/2 +D/2

Figure 1.2: Time frequency filtering

is filtered through a frequency limiting filter such that

S(f) = S(f) rect(f,W ).

By limiting the output observation to [−D2 ,

D2 ], then,

r(t) = rect(t,D)r(t) =

∫f

rect(t,D)LH(t, f) rect(f,W )S(f)ej2πftdf.

The effective time varying transfer function

LH(t, f) = rect(t,D)LH(t, f) rect(f,W ),

is therefore limited in time and frequency. The 2-D Shannon sampling theorem can be then

applied to SH(ν, τ) (the 2-D inverse Fourier transform of a LH(t, f)) such that,

SH(ν, τ) =1

DW

∑p

∑q

SH

( pD,q

W

)sinc

(D(ν − p

D

))sinc

(W(τ − q

W

)). (1.13)

9


The corresponding I/O relation can be deduced by replacing SH(ν, τ) by its value in (1.6),

this implies

r(t) =1

D2W

∑p

∑q

SH

( pD,q

W

)sB(t− q

W

)ej2π

pDt, |t| ≤ D/2, (1.14)

where sB(t) = s(t) ? sinc(Wt) is the equivalent input signal with limited bandwidth, and can

be written using the 1-D Shannon theorem as,

sB(t) =1

W

∑k

sB

[ kW

]sinc

(W(t− k

W

)).

The effective delay Doppler spreading function SH(ν, τ) is related to the delay Doppler spread-

ing function SH(ν, τ) by,

SH(ν, τ) = sinc(τW ) ? SH(ν, τ) ? sinc(νD). (1.15)

If SH(ν, τ) is compactly supported by the rectangle [−τ0, τ0] × [−ν0, ν0], this does not mean

necessarily that SH(ν, τ) is limited. However, most of the volume of SH(ν, τ) is supported by

[−ν0 − 1D , ν0 + 1

D ] × [−τ0 − 1W , τ0 + 1

W ]. Then, the channel can be characterized by a finite

number of parameters SH(ν, τ) equal to1

r = (2p0 + 1)(2q0 + 1),

where p0 = bν0Dc and q0 = bτ0W c.

The (2p0 + 1)(2q0 + 1) coefficients SH(ν, τ) can be computed from (1.14) by computing r(t)

at t = nW , such that,

r[ nW

]=

1

(DW )2

∑p

∑q

SH

( pD,q

W

)sB

[n− qW

]ej2π

pDW

n, −DW2

+ 1 ≤ n ≤ DW

2− 1,

The above system is feasible if the number of unknowns is less than the maximal number of

independent equations, i.e.,

(2p0 + 1)(2q0 + 1) ≤ DW − 1 ≤ DW.

This implies that in order to identify the channel, its spread should satisfy the following

condition,

4τ0ν0 ≤ 1.

1We assume that the effective delay Doppler spreading function is a continuous function and is equal tozero on the compact boundary, so that SH(ν, τ) 6= 0 if (ν, τ) ∈ [−ν0, ν0]× [−τ0, τ0]

10

1.2. LTV CHANNEL IDENTIFICATION

The effective time varying transfer function LH(t, f) = FTν→t,τ→fSH(ν, τ) can be therefore

approximated by replacing SH(ν, τ) by its value in (1.13) as,

LH(t, f) =1

DW

[∑p

∑q

SH

( pD,q

W

)e−j2π

qWfej2π

pDt]

rect(f,W ) rect(t,D)

For t ∈ [−D2 ,

D2 ] and f ∈ [−W

2 ,W2 ], the channel time varying function can be written such

that,

LH(t, f) ≈ 1

DW

p0∑p=−p0

q0∑q=−q0

SH

( pD,q

W

)e−j2π( q

Wf− p

Dt). (1.16)

Remark 1.1 (Accurate channel measurement) Note that the expression in (1.16) is

not very accurate, as it highly depends on how much the energy of the sinc function in (1.15)

is concentrated in the principal lobe. A more accurate value of LH(t, f) can be obtained by

taking into account a finite number α of side lobes, by neglecting only side lobes which are at

least 3dB weaker than the principal lobe. In this case, the equivalent delay Doppler spreading

function is non equal to zero in,[− ν0 −

α

D, ν0 +

α

D

]×[− ν0 −

α

W, ν0 +

α

W

].

The choice of α defines the number of parameters needed to characterize the channel.

In the rest of this thesis, the number of these parameters is denoted by r = (2(p0 + α) +

1)(2(q0 +α) + 1), and we let pr = p0 +α and qr = q0 +α. This choice of channels parameter

requires also the underspread condition to resolve the unknown channel parameters.

In this case,

LH(t, f) ≈ 1

DW

pr∑p=−pr

qr∑q=−qr

SH

( pD,q

W

)e−j2π( q

Wf− p

Dt). (1.17)

1.2.2 Multiple input multiple output channel

Theorem 1.2 (MIMO identifiability [34]) The nt × nr MIMO channel operator char-

acterized by a rectangle scattering function at each entry can be identified by one vector of

input signals if at each receiving antennas the sum of the area of the spreading supports of

the subchannels leading to the receiving antennas is less than 1.

Proof: The rigorous proof of this theorem can be found in [34]. The reconstruction

scheme from the input vectors can be immediately deduced from the SISO case. If we assume

that all the scattering function are supported by the same rectangle [−τ0, τ0]× [−ν0, ν0], then,

each receive antennas should find nt(2p0 + 1)(2q0 + 1) channel unknown parameters based on

DW observations. The linear system is feasible if ∆H ≤ 1/nt.

11


1.2.3 Multiuser channel identification

For the multiuser systems with K sources and K destinations having nt transmit antennas

and nr receive antennnas, each destination should identifies all the nt × nr MIMO channels

operator. This implies that, ∆H ≤ 1Knt

.

1.3 Discretized channel model

The continuous system model given in Sections 1.1 and 1.2 is very difficult to use as basis

for information theory studies. That’s why, the continuous-time channel should be converted

into a discrete-time channel. For linear time invariant (LTI) and linear frequency invariant

(LFI) systems with limited signal-bandwidth, the discrete I/O relation follows straightfor-

wardly from the application of Shannon sampling theorem. However, for the case of general

underspread LTV channel, the discrete I/O relation is more complicated to derive and re-

quires an approximate diagonalization of the underspread fading channels. In this section,

we briefly review the cyclic signal model in LTI system that will be addressed in Chapter 2

and we describe the signaling on approximate eigenfunction for the LTV channel case (For

more details, we refer the reader to [1] and references therein).

1.3.1 LTI systems

Assuming that the signal is band-limited, then the effective channel is also band-limited

in frequency and the discrete I/O relation can be easily obtained by simply applying the

Shannon sampling theorem to the effective channel. The discrete I/O noise free relation is

then given by the linear convolution, such that

r[n] =∑m

h[m]s[n−m].

When using a cyclic prefix OFDM system with a cyclic prefix that exceeds the channel delay

spread (approximatively equal to 1/Bc), then the linear convolution is replaced by a cyclic

convolution. If we assume that the delay spread admits L samples on the delay domain,

i.e., the channel has L-taps components in the delay domain drawn independently from a

continuous distribution such that E[h[l]h∗[l′]

]= σ2

l δ(l − l′), then the discrete I/O relation

reduces tor[0]

r[1]...

r[N − 1]

=

h0 0 . . . 0 hL−1 hL−2 . . . h1

h1 h0 0 . . . 0 hL−1 . . . h2

...

0 . . . 0 hL−1 hL−2 . . . h1 h0

︸︷︷︸

C

s[0]

s[1]...

s[N − 1]

. (1.18)

The matrix C is a circulant matrix, and its eigen-value decomposition is given by the following

lemma.

12

1.3. DISCRETIZED CHANNEL MODEL

Lemma 1.1 (Eigen-value decomposition of a circulant matrix) The circulant matrix

C has eigen-vectors

fm =1√N

[1, e−j2πm/N , . . . , e−j2π(N−1)m/N

],

and corresponding eigen-values

λm =L−1∑k=0

h[k]e−j2πimk/N ,

where m = 0 . . . N − 1. The matrix C can be expressed in the form C = FΛF†, where F is

the N ×N FFT matrix.

Note that λm 6= 0 almost surely as the taps h[l] are drawn independently from a continuous

distribution. That’s why Λ has a full rank of N almost surely.

Using the DFT at the transmitter and receiver side, the matrix channel model can be

converted into N parallel channel, such that

yn = hnxn + zn, n = 0 . . . N − 1, (1.19)

where zn is the additive channel noise and hn =L−1∑k=0

h[k]e−j2πnk/N . The channel coefficients

hn are correlated such that

rH(n− n′) = E[hnh

∗n′]

=

L−1∑l=0

σ2l e−j2πl(n−n′)/N .

Note that the coefficients rH(i) satisfy the following property rH(i) = rH(i−N).

1.3.2 Underspread LTV channel

The underspread assumption is very relevant as most of mobile radio channels are under-

spread. Moreover, this assumption allows to approximately diagonalize the underspread

fading channel. The discrete I/O relation can be obtained by transmitting and receiving on

an orthonormal Weyl-Heisenberg (WH) sets. This set is obtained by translating in time and

modulating in frequency a prototype g(t). In the following, this set is denoted as

(g(t), T, F ) =gm,l(t) = g(t−mT )ej2πlF t

. (1.20)

where m, l ∈ Z, T and F are the grid parameter of WH set. The triple g(t), T, F are chosen

such that g(t) has unit energy and that gm,n(t) are orthonormal, i.e.,

⟨gm,l(t), gk,p(t)

⟩=

∫tgm,l(t)gk,p(t)dt = δm,kδl,p.

13


Moreover, gm,l(t) should be at the same time well localized in time and frequency. These

properties require that TF > 1 to be satisfied (for more details on Weyl Heisenberg sets

construction, please refer to appendix B).

Approximate underspread channel diagonalization

As we mentioned in Subsection 1.1.3, CH(ν, τ) is supported by a rectangle with a total channel

spread ∆H 1. This implies that SH(ν, τ) should also be supported by a rectangle almost

surely. For the underspread fading channel, it is easy to check that the two Nyquist conditions

(T ≤ 1/(2ν0) and F ≤ 1/(2τ0)) can be satisfied when ∆H ≤ 1 (underspread condition) and

TF > 1 required for WH sets orthonormality. The two Nyquist conditions implies a more

restrict condition on the channel spread,

∆H ≤1

TF. (1.21)

As the time varying transfer and the Doppler spreading function are related by a 2-D Fourier

transfer, the Shannon sampling in 2-D can be applied, and the samples LH(mT, nF ), taken

on a rectangle grid with T ≤ 1/(2ν0) and F ≤ 1/(2τ0) are sufficient to characterize LH(t, f).

It has been shown in [35] that the kernel of the linear operator kH(t, t′) can be then approxi-

mated by setting,

kH(t, t′) =

∞∑m=−∞

∞∑l=−∞

LH(mT, lF

)gm,l(t)g

∗m,l(t

′). (1.22)

Discrete time discrete frequency I/O relation

Using the set of WH defined in (1.20), any input signal can be written as

s(t) =M−1∑m=0

Nc−1∑l=0

x[m, l]gm,l(t), (1.23)

where D = MT is the approximate time duration of s(t) and W = NcF is its approximate

bandwidth. The projection of the received signal y(t) = r(t)+z(t), where z(t) is the additive

white noise onto the WH setgm,l(t)

where m = 0 . . .M − 1 and l = 0 . . . Nc − 1 gives

y[m, l] =⟨y(t), gm,l(t)

⟩=

⟨Hx, gm,l(t)

⟩+⟨z(t), gm,l(t)

⟩,

=∑k,p

x[k, p]⟨Hgk,p, gm,l(t)

⟩+ z[m, l], (1.24)

= LH(mT, lF )x[m, l] + z[m, l]. (1.25)

where (1.25) follows from the approximate decomposition of underspread channel in (1.22).

Note that due to the orthonormal WH set, z[m, l] are i.i.d for all (m, l) ∈ 0 . . .M − 1 ×0 . . . Nc − 1, such that z[m, l] ∈ CN (0, 1) and E

[z[m, l]z[m′, l′]

]= δm,m′δl,l′ .

14

1.4. UNIFIED MATRIX FORMULATION

Interpretation

As shown in [36], the signaling scheme in (1.25) can be interpreted as a block of size M ×Nc,

where M denotes the total number of OFDM symbols and Nc denotes the total number of

subcarriers. The couple n = (m, l) denotes the time-frequency slots such that n = 0 . . . N−1,

where N is the total number of time-frequency slots and is equal to MNc.

We denote by hn the channel coefficient where hn = LH(mT, lF ). As mentioned above due

the WSUSS assumption, channel coefficients are wide-sense stationary in time and frequency,

such that

r[m; l] = ELH((m+ k)T, (l + p)F )L∗H(kT, pF )

= RH

(mT, lF

). (1.26)

The SISO system model in (1.25) can be finally written as

yn = hnxn + zn, n = 0 . . . N − 1. (1.27)

1.4 Unified matrix formulation

The discrete I/O relation in the previous section can be interpreted as a transmission over

N time (LFI system), frequency (LTI system) and N = MNc time-frequency slots (for the

underspread LTV model). The channel coefficients over these parallel channels are correlated.

Depending on this correlation, we give in this section a formal representation of the channel

model that we will use in the subsequent chapters. We started by analyzing the SISO case,

and then we provide the extension to the MIMO case.

In this thesis, we restrict our analysis to the case of the Rayleigh fading channels, where

the coefficients hn = LH(mT, lF ) are drawn from a continuous Gaussian distribution. We

emphasize that these channels coefficients are correlated across n.

Let h = [h0, . . . , hN−1] be the N × 1 stacked channel vector that contains the N channel’s

components, and R its N × N hermitian covariance matrix such that R = E[hh†]. The

covariance channel matrix coefficients can be deduced from (1.10) and is supposed to be

known at both the transmitter and the receiver side in all the subsequents chapters.

Define r = rankR the rank of R and R = WΛW† its eigenvalue decomposition where

Λ = diagσ20, . . . , σ

2r−1, 0, . . . , 0.

1.4.1 Channel characterization at each time-frequency slot

At each time-frequency slot, the channel is characterized as shown in Lemma 1.2.

Lemma 1.2 (Scalar time-frequency slot channel) At each (m, l) time-frequency slot de-

15


noted by n, the channel hn = LH(mT, lF ) can be written such that,

hn =r−1∑i=0

wn,iσihω,i, n = 0 . . . N − 1, (1.28)

The Gaussian vector hω ∼ CN (0, Ir) contains the r i.i.d CN (0, 1) coefficients required to

identify the channel, such that

hω =[hω,0 . . . hω,r−1

][T ],

The parameters wn,i (n = 0, . . . N−1 and i = 0, . . . , r−1) denote the eigen-vectors coefficients

of matrix R and σ2i are the eigen-values of R.

Before going to the proof, we note that when transmitting and receiving on a set of

Weyl-Heisenberg sets, the rank of the covariance matrix r ≤ MNc defines the number of

parameters required to identify the channel. As shown in remark 1.1, to identify the channel

at least r ≥ (2pr + 1)(2qr + 1) ≥ DW∆H are required to be known. This implies that

∆HDW ≤MNc,

and consequently,

∆H ≤1

TF.

For the underspread fading channel, this condition is satisfied when transmitting and receiving

on a set of Weyl-Heisenberg as shown in (1.21) and is a necessary condition for channel

reconstruction.

Proof: The vector h can be written in function of its covariance matrix R such that

h = R1/2hω′ ,

where hω′ is an i.i.d CN (0, 1) vector with the same dimension as h. Using the eigen-value

decomposition of R,

h = WΛ1/2W†hω′ ,

= WΛ1/2hω, (1.29)

where hω in (1.29) is also a random Gaussian vector CN (0, 1), since W† is a unitary matrix.

It follows from (1.29) that

hk =r−1∑i=0

wk,iσihw,i, k = 0 . . . N − 1. (1.30)

16


1.4.2 General channel matrix decomposition

As we can see from (1.19) and (1.27), the channel can be decomposed into N parallel channels

where the channel components are correlated. We start first by giving a formalized decom-

position of the diagonal matrix.

The expression of the corresponding diagonal channel matrix H = diagh0, . . . , hN−1 is

given by the following lemma.

Lemma 1.3 (Channel matrix decomposition) The diagonal channel matrix H can be

decomposed in function of the eigen-vectors of R as following

H =r−1∑i=0

hiWi, (1.31)

where

Wi = diagw1,i, . . . , wN,i, (1.32)

wi,j denotes the element of the channel correlation eigen-vector matrix W and hi are i.i.d

drawn from a continuous distribution CN (0, σ2i ), with σ2

i being the eigen-value of the corre-

lation matrix.

Proof: Using the time-frequency channel expression in Lemma 1.2, the diagonal channel

matrix can be written such that,

H = diaghk =

∑r−1

i=0 w1,iσihw,i ∑r−1i=0 w2,iσihw,i

. . . ∑r−1i=0 wN,iσihw,i

,

=r−1∑i=0

σihw,i

w1,i

w2,i

. . .

wN,i

=r−1∑i=0

σihw,iWi =r−1∑i=0

hiWi,

with hi = σihw,i.

1.4.3 Frequency selective channel

Lemma 1.4 (LTI channel decomposition) For the frequency selective channel with L-

taps, the diagonal channel matrix can be written as

H =1

N

L−1∑i=0

h[i]Zi, (1.33)

17


where h[i] = hi are i.i.d drawn from a continuous distribution CN (0, σ2i ), σ

2i is the auto-

covariance of the i-th channel tap, and Z is given by

Z = diag1, ω, . . . , ωN−1 where ω = e−j2πN .

Proof: As shown in Subsection 1.3.1, for the frequency selective channel with L-taps using

a CP-OFDM, the covariance matrix is a circulant matrix with rank = L, and its eigen-value

decomposition is such that

R = F diagσ20, . . . , σ

2L−1, 0, . . . , 0F†, (1.34)

where F is the N × N FFT matrix. Lemma 1.4 follows by replacing the vectorized eigen-

vectors in Lemma 1.3 by the L vectorized FFT columns.

Note that Lemma 1.4 can be also applied to the case of time selective channel (linear frequency

invariant system) where the circulant covariance matrix can be obtained by using a basis

expansion model.

1.4.4 Selective underspread fading channel

In this case, the covariance matrix R is the autocorrelation of a 2-D discrete random process,

which is well-known as Toeplitz-Block-Toeplitz(TBT) matrix. The definition of the TBT

matrix as well as the circulant block circulant matrix (CBC) is given in the following. In

order to find the rank and the eigen-values decomposition of this matrix, we need the Theorem

1.3 that has been already proved in [37].

Basic preliminaries on Toeplitz-Block-Toeplitz matrix

Definition 1.2 (Toeplitz-Block-Toeplitz matrix(TBT)) The MNc×MNc Toeplitz-Block-

Toeplitz matrix induced by a 2-D sequence of r[m, l] is defined as follows

R =

R[0;−] R[−1;−] . . . R[1−M ;−]

R[1;−] R[0;−] . . . R[2−M ;−]...

R[M − 1;−] R[M − 2;−] . . . R[0;−]

,

where each block Nc ×Nc block given by

R[i− j,−] =

r[i− j, 0] r[i− j,−1] . . . r[i− j, 1−Nc]

r[i− j, 1] r[i− j, 0] . . . r[i− j, 2−Nc]...

r[i− j,Nc − 1] r[i− j,Nc − 2] . . . r[i− j, 0]

18


is also a Toeplitz matrix.

Definition 1.3 (Circulant-Block-Circulant matrix(CBC)) Let c[m, l] be a 2-D sequence.

The MNc ×MNc circulant-Block-Toeplitz matrix is defined as follows

C =

C[0;−] C[−1;−] . . . C[1−M ;−]

C[1;−] C[0;−] . . . C[2−M ;−]...

C[M − 1;−] C[M − 2;−] . . . C[0;−]

,

where each block Nc ×Nc block given by

C[i− j,−] =

c[i− j, 0] c[i− j,−1] . . . c[i− j, 1−Nc]

c[i− j, 1] c[i− j, 0] . . . c[i− j, 2−Nc]...

c[i− j,Nc − 1] c[i− j,Nc − 2] . . . c[i− j, 0]

is also a circulant matrix.

The eigen-value decomposition of CBC hermitian matrix can be easily derived such as

C = (F⊗G)Λ(F⊗G)†, (1.35)

where F and G are M ×M and Nc ×Nc FFT matrix and Λ = diagλk such that

λt =M−1∑m=0

Nc−1∑l=0

cm,le−j2π(mp/M+lq/Nc)

where t denotes the number of (p, q) couples such that p = 0 . . .M − 1 and q = 0 . . . Nc − 1.

Theorem 1.3 (Asymptotical equivalence [lemma 1 in [37]) ] Let r[m, l] be a 2-D se-

quence that induces a TBT matrix R and s(θ, ϕ) be the 2-D discrete Fourier transform of

r[m, l] such that

s(θ, ϕ) =∑m

∑l

r[m, l]e−j2π(mθ−lϕ), |θ|, |ϕ| ≤ 1/2.

The CBC matrix C having the t eigen-values equal to the uniformly spaced samples of s(θ, ϕ)

such that

λt = s( pM,q

Nc

), (1.36)

is asymptotically equivalent (when M →∞, and Nc →∞) to R.

19


Toeplitz-Block-Toeplitz matrix for underspread channel

For the underspread fading channel, the 2-D discrete Fourier transform of r[m, l] is such that

s(θ, ϕ) =∑m

∑l

r[m, l]e−j2π(mθ−lϕ), |θ|, |ϕ| ≤ 1/2,

=∑m

∑l

RH(mT, lF )e−j2π(mθ−lϕ).

By replacing r[m, l] by its value in (1.10), then

s(θ, ϕ) =∑m

∑l

∫τ

∫νCH(ν, τ)ej2πνmT e−j2πτlF e−j2π(mθ−lϕ)dτdν, (1.37)

=

∫τ

∫νCH(ν, τ)

∑m

ej2πmT (ν− θT

)∑l

e−j2πlF (τ− ϕF

)dτdν.

Using the Fourier series decomposition of the periodic impulse train,

s(θ, ϕ) =1

TF

∫τ

∫νCH(ν, τ)

∑m

δ(ν − θ −mT

)∑l

δ(τ − ϕ− lF

)dτdν,

=1

TF

∑m

∑l

CH

(θ −mT

,ϕ− lF

),

=1

TFCH

(θ

T,ϕ

F

), |θ|, |ϕ| ≤ 1/2. (1.38)

where (1.38) follows from the fact that (a) CH(ν, τ) is compactly supported in a rectangle

and (b) that the grid parameters of the Weyl-Heisenberg frame T and F are chosen such that

2τ0F ≤ 1 and 2ν0T ≤ 1 are satisfied.

By applying Theorem 1.3, the asymptotical CBC matrix C equivalent to R can be constructed

as

C = (F† ⊗G)Λ(F⊗G†),

where

λ(p,q)(C) = s( pM,q

Nc

)=

1

TFCH

(p

TM,q

FNc

),

and t denotes the couple (p, q).

As CH(ν, τ) is compactly supported in a rectangle [−ν0, ν0]× [−τ0, τ0]2, this implies that

(p, q) ∈ −p0 + 1, . . . , p0 − 1 × −q0 + 1, . . . , q0 − 1,

where

p0 = bν0TMc = bν0Dc and q0 = bτ0FNcc = bτ0W c.

2The scattering function is continuous and is equal to zero at the boundary.

20


Polynomial channel decomposition for large D and W

Theorem 1.4 (Polynomial underspread channel decomposition for large D and W )

For the underspread fading channel, the asymptotical (in term of N = MNc) diagonal channel

matrix can be written such that

H =∑

(p,q)∈Aλ(p,q) (Z p

M ⊗ Z qNc

), (1.39)

where

A =

(p, q) : p ∈ −p0 + 1, . . . , p0 − 1, q ∈ −q0 + 1, . . . , q0 − 1, (1.40)

p0 = bν0Dc, q0 = bτ0W c. (1.41)

The coefficients λ(p,q) are i.i.d random variable drawn from a continuous Gaussian distribu-

tion CN (0, σ2(p,q)), such that

σ2(p,q) =

1

DWCH

( pD,q

W

).

These coefficients represent the delay-Doppler spreading function SH(ν, τ) sampled at ν = pD

and τ = qW , i.e.,

λ(p,q) =1

DWSH

( pD,q

W

).

Matrix ZM is such that

ZM = diag1, ωM , . . . , ωM−1M where ωM = ej

2πM .

and ZNc is given by

ZNc = diag1, ωNc , . . . , ωNc−1Nc

where ωNc = e−j2πNc .

Proof: The theorem is a straightforward consequence of applying Lemma 1.3 to the

asymptotical TBT of the underspread fading case.

Note that this result can be also deduced by observing that when the channel bandwidth and

its duration are large, then the relationship between the delay-Doppler spreading function

SH(ν, τ) and the effective delay-doppler spreading function SH(ν, τ) in (1.15) can be reduced

to

SH(ν, τ) ≈ SH(ν, τ)

as sinc(2Wx) ≈ δ(x), when W →∞. Then, the time varying transfer function in (1.16), can

be written as

LH(t, f) ≈ 1

DW

p0−1∑p=−p0+1

q0−1∑q=−q0+1

SH

( pD,q

W

)ej2π

pDt−j2π q

Wf .

21


Using the polynomial channel decomposition, allows to analyze separately the shift in time

due to the Doppler spread and in frequency due to the delay as will be shown in Chapter 3.

1.4.5 Extension to the MIMO case

The discrete I/O relation in the previous section can be generalized to the nt × nr MIMO

case. As transmitters and receivers are co-located, channels between antennas have identical

statistics and are assumed to be statistically independent. We assume a transmission over

N time (LFI system), frequency (LTI system) and N = MNc time-frequency slots (for the

underspread LTV model).

The diagonal channel model in (1.19) and (1.27) can be generalized to the MIMO case by

(a) transmitting and receiving on the N ×N FFT eigen-vectors for the LFI case, and (b) by

transmitting and receiving on a common set3 of orthonormal Weyl-Heisenberg (WH) which

diagonalize all subchannels. Consequently, the MIMO channel model is given by

yn = Hnxn + zn, n = 0 . . . N − 1, (1.42)

where yn, xn and zn corresponds respectively to the nr×1 received signal, nt×1 transmitted

signal and nr×1 additional noise, such that zn ∼ CN (0, Inr) and Hn is a nr×nt matrix such

that its component h(i,j)n (where i = 1 . . . nr, j = 1 . . . nt) is the (a) DFT of the L-taps for

the frequency selective case, and (b) L(i,j)H (mT, lF ) for the selective channel case (note that

(m, l) corresponds to the n-th couple).

The channels H0, . . . ,HN−1 can be generated using the stack matrix H given by

H = [H0 . . . HN−1].

By observing that the channel statistical dependence can be expressed as

Eh(i,j)n h(i′,j′)

m = r(n−m)δi,i′δj,j′ , ∀i, i′ = 1, . . . nt, and j, j′ = 1, . . . nr.

where r(n−m) is the correlation between two frequency or time-frequency slots. This implies

that

H = [H0 . . . HN−1] = Hw(R1/2H ⊗ Int), (1.43)

where RH is the N × N correlation between the scalar subchannels, Hw denotes the nr ×Nnt matrix such that Hw = [Hw,0 . . .Hw,N−1] and Hw,n denotes i.i.d CN (0, 1) matrices of

dimension nt × nr.

3As all channels have the same statistical distribution, they can be diagonalized using a common WH set.Moreover, the covariance matrix between the scalar subchannels is the same for all channels between transmitand receive antennas

22

1.5. CONCLUSION

1.5 Conclusion

In this chapter, we give a unified channel matrix formulation that will be used throughout

this thesis. Starting from the modeling of the channel as a linear time varying system, we

show how to discretize the I/O relation according to the application requirements and to

the channel coherence in time and frequency. Based on this discrete I/O relation, we give

the channel matrix formulation for the SISO and for the MIMO case. For both cases, we

show that the channel is equivalent to N -time-frequency parallel channels where channels are

correlated. This channel model is used in Chapter 2 to analyze the optimality of space time

code over frequency selective channel.

For the underspread fading channel, we show that when the bandwidth and the duration of the

signal are very large, the covariance matrix can be approximated by a circulant block circulant

matrix. Based on this approximation, we define the polynomial channel’s decomposition form,

which is very useful in several applications such as interference alignment that we analyze in

Chapter 3, and for quantized feedback analysis in Chapters 3 and 4.

23


24

Chapter 2

NVD Codes in Standard

Applications: Optimality and

Practical Limits

THE optimal design of perfect space time codes constructed from cyclic division alge-

bra(CDA) on the quasi-static uncoded MIMO channel has received lots of attention

in industry over the last few years. However, the recent standards that uses mul-

tiple antennas terminals such as IEEE 802.11n or IEEE 802.16e, are based on more realistic

assumptions involving the use of outer codes, and multi-taps channels.

This chapter is devoted to the analysis of the performance of space time codes constructed

from CDA in a standard context. We start by giving a general overview on non-vanishing

determinant code construction (NVD) over flat fading MIMO channel. Then, we propose for

the selective fading channel, a new family of split NVD parallel codes constructed from CDA

to achieve the optimal DMT. Although these codes are optimal, a high decoding complexity

order is required at the receiver side, which make these codes not easy to be implemented in

a practical system. Coding symbols only across subcarriers without requiring to code across

blocks could be an interesting solution. However, this approach is not optimal if used alone in

a cyclic-prefix orthogonal frequency division multiplexing (CP-OFDM) system. We show that

using only perfect codes over each subcarrier can be made possible if used in a bit interleaved

coded modulation MIMO system. Finally, we show that using numerical results, the expected

gains of these codes cannot be reached due to the moderate range of target packet error rate

(in the range of 10−2 for a packet of 1000 bits) required by standard application.

25

CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS

2.1 Introduction

The IEEE 802.11n is one of the latest evolutions of the previous 802.11 standards for wireless

LANs. The main objective of this technology is to provide to the end-user modes of operation

that are capable of much higher throughput than 11a/b and g, with a maximum throughput

of at least 100Mbps, as measured at the MAC data Service Access Point (SAP). The major

novelty of this version is to use Multiple Input Multiple Output (MIMO) techniques so

that devices embed multiple transmitter and receiver antennas to allow an increased data

throughput through spatial multiplexing and an increased range by exploiting the spatial

diversity.

In order to exploit fully the available diversity of a quasi-static fading uncoded1 nt × nrMIMO channel, two approaches have been considered in the literature. The first approach

proposed by Tarokh et al. in [9] consists to minimize the error probability over all the fading

distribution, and yields to two fundamental criterion that should be satisfied by a space time

code to be optimal. These two criteria are as follows

- The rank criterion, stating that the difference between two distinct codewords must be

a full rank matrix.

- The determinant criterion, stating that the minimal determinant of space time code is

maximized.

While this approach is more tailored to the Rayleigh fading distribution, Zheng and Tse pro-

posed in [10] a powerful approach based on the high SNR characterization of the dual benefits

in term of diversity and spatial multiplexing using the diversity multiplexing tradeoff (DMT)

framework. In order to fully achieve the optimal diversity multiplexing tradeoff, Belfiore et

al. introduce in [11] the non-vanishing determinant criterion. Later, Elia et al. in [12] prove

that this criterion is a sufficient condition to achieve the optimal DMT using a full rate code

and for nt ≤ nr. A more general design criterion has been later proposed by Tavildar and

Viswanath in [38], where universal (and approximately) code design has been established.

These codes have the main property to achieve reliable communication over MIMO channel

realization that are not in outage.

More recently, Oggier et al. in [13] proposed a family of optimal space time codes known

as perfect space time codes that fulfill the design criteria of Tarokh. Moreover, it has been

shown that these codes are the optimal codes over quasi-static uncoded MIMO channel since

they achieve full rate and full diversity, preserve the mutual information, achieve the Diver-

sity Multiplexing Tradeoff (DMT) [10] and have a non vanishing determinant [11].

Unlike the simplified quasi-static uncoded MIMO channel, industrial transmission schemes

are based on more realistic assumptions involving the use of multi-tap channels and outer

codes such as the convolutional code. The main objective of this chapter is to show how codes

1In this chapter, the uncoded MIMO system refers to the case when no outer codes such as the convolutionalcodes are used

26

2.2. STRUCTURED CODE CONSTRUCTION: A PRIMER

designed from cyclic division algebra can be applied in an industrial context focusing on their

optimality and the practical limits that can encountered in industry. For this purpose, we

first give an extended overview about space time code design over a flat fading MIMO channel

in Section 2.2. Then, the selective fading channel is considered in Section 2.3. We propose a

new family of split NVD parallel codes to achieve the optimal DMT derived in [2]. The high

complexity required at the receiver side make these codes impractical to be implemented in

a real system. The MIMO-BICM system (bit interleaved coded modulation) based on the

IEEE 802.11n transmission scheme with non iterative maximum-likelihood (ML) decoding

algorithm (Viterbi Algorithm) is then considered. For the flat fading channel in Section 2.5,

we show that the coding gain of perfect codes can be enhanced but without any gain in

term of diversity. For the frequency selective channel in Section 2.6, we show that when

a convolutional code is used, coding data only across each subcarrier allows to achieve the

optimal diversity and to maximize the coding gain. This reduces considerably the complexity

compared to the case when no convolutional code is used and a global code is required across

all subcarriers. Finally, Section 2.7 concludes this chapter.

2.2 Structured code construction: A primer

In this section, we give a global overview on structured code construction for the nt × nrslow fading MIMO channels depicted in Figure 2.1. A more complete description on code

construction can be found in Chapter 9 of [8] or in Chapter 1 of [39]. We just focus here on

the main steps needed to have a comprehensive vision on the optimality of perfect space time

codes. The optimal diversity multiplexing tradeoff (DMT) is used as a unified framework to

compare the optimality of space time codes and is presented in Subection 2.2.1. We highlight

the normalization convention in Subection 2.2.2. Then, we recall the code design criterion in

Subection 2.2.3. Finally, we review the properties of the space time codes that will be used

throughout this chapter in Subection 2.2.4.

channel

Space Time BlockCoding

ML decoder DemodulationModulation

Figure 2.1: MIMO system

Flat fading channel model

We consider first the flat fading channel model given by

Y = θHX + z

27


where X ∈ Cnt×T is a space time code drawn from code Xp of rate R per channel use,

Y ∈ Cnr×T is the received signal, z ∈ Cnr×T ∼ CN (0, 1) is the additive noise and H ∈ Cnr×nt

is the channel matrix with i.i.d complex Gaussian CN (0, 1) entries. The scaling factor θ is

chosen to ensure the power constraint,

θ2 E[‖X‖2F

]= TSNR. (2.1)

2.2.1 Diversity multiplexing tradeoff (DMT)

The DMT is a powerful framework used to compare the space time coding schemes. In the

following, we first define the tradeoff of any space time coding schemes, and then we define the

optimal tradeoff curve which characterizes the slow fading performance limit of the channel.

DMT of the code dXp(r)

Definition 2.1 A coding scheme Xp(SNR) with data rate R bits PCU achieves a multiplexing

gain r and diversity gain d if the data rate R is such that

limSNR→∞

R(SNR)

log SNR= r,

and the average error probability Pe(SNR) with maximum likelihood-decoding is such that

−d = limSNR→∞

logPe(SNR)

log SNR.

For a given multiplexing gain r, the largest diversity supported by any coding scheme is denoted

by dXp(r).

DMT of the channel dout(r)

For the slow fading channel, when no CSIT is available at the transmitter, there is a positive

probability, that the entries of H are small. In this case, whatever the code used by the

transmitter, the decoding error probability cannot be small. Consequently, the Shannon

capacity of the i.i.d. Rayleigh slow fading MIMO channel is zero. For this case, we focus on

characterizing the ε-outage capacity, which is the largest rate of reliable communication such

that the error probability is no more than ε.

The channel outage is usually discussed for non ergodic fading channels, i.e, when the

channel matrix H is chosen randomly but is held fixed for all time. An outage event is defined

as the event that the channel cannot not support a target data rate of R at a given SNR,

and can be described as,

O =H : I(x,y|H) ≤ R

.

In high SNR regime, a convenient characterization of the tradeoff between rate and reliability

is offered by the DMT introduced by Zheng and Tse in [40].

28


Definition 2.2 Given a point-to-point MIMO system, the gains in terms of diversity gain d

−d = limSNR→∞

logPout(R,SNR)

log SNR

and spatial multiplexing gain r

r = limSNR→∞

R(SNR)

log SNR

can be simultaneously obtained. But, there is a fundamental tradeoff dout(r) between these

two gains provided by any coding scheme.

Theorem 2.1 The DMT of nt×nr Rayleigh channel is a piecewise-linear function connecting

the points(r, d(r)

), r = 0, . . . ,min(nt, nr) where

d(r) = (nt − r)(nr − r). (2.2)

Proof: For the MIMO case, the outage event occurs when the channel does not support

the data rate R = r log SNR. The outage event can be described such that,

O =

H : log det(Int +SNR HH†) ≤ r log SNR.

Let λi denotes the ordered non-zero eigen-value of HH†, i.e (λ1 ≤ . . . ≤ λq), i = 1 . . . q =

rankH = min(nt, nr) and αi be the eigen-exponents corresponding to λi such that λi =

SNR−αi , then

O = q∑i=1

log(1 + SNRλi) ≤ r log SNR,

=α :

q∑i=1

(1− αi)+ ≤ r

= O[nt,nr]α (r, SNR). (2.3)

Using the density function of the αi given in [10],

p(α1, . . . , αq).=

q∏i=1

SNR−(2i+|nt−nr|+1)αi

and by averaging over all the channels, the outage probability is

Pout(r).= SNR−dout(r),

where

dout(r) = infα∈Oα(SNR)

q∑i=1

(2i+ |nt − nr|+ 1)αi. (2.4)

The solution of the linear optimization in (2.4) yields to the DMT expression in (2.2).

29


Relation between dXp(r) and dout(r)

Theorem 2.2 For any coding scheme with rate scaling as r log SNR, the DMT of the code

dXp(r) is upper bounded by dout(r), i.e,

dXp(r) ≤ dout(r). (2.5)

Proof: The proof of this theorem is due to Lemma 5 in [40], and results as a consequence

of the Fano inequality that gives a lower bound on the probability of detection error, such

that

Pe(SNR) ≥ SNR−dout(r).

2.2.2 Notations and normalization convention

Before going to the structured code construction that achieves the DMT, we define the nota-

tions and the normalization conventions that will be used in the following. The unnormalized

space time code is defined as

X = θX, (2.6)

where X ∈ Xp(SNR) refers to the normalized space time code. Let Xi and Xj be two distinct

codewords, such that Xi,Xj ∈ θXp(SNR), and ∆X be the difference codeword matrix,

∆X = Xi −Xj . (2.7)

The eigen-values of ∆X∆X† are denoted by µ−1i , and βi corresponds to the µ−1

i eigen-

exponents, such that

µi.= SNRβi .

It can be easily deduced from the power constraint in (2.1) that µ−1i ≤ SNR. For a sake of

notation simplicity, we denote by

µ2i =

µ−1i

SNR

.= SNRβi−1 ≤ 1, (2.8)

which represents the eigen-value of 1SNR∆X∆X†.

2.2.3 Optimal code design criterion

Achieving the DMT of the channel have been widely addressed in literature. The unstructured

coding schemes such as Gaussian code in [40] and LAST codes in [41] allow to achieve

this DMT. However, the random coding arguments used in such schemes makes the explicit

construction of such codes very challenging. In the following, we focus only on the structured

code construction. The essence of the following development can be found in [12], [38], [13],

[40].

30


Sufficient condition for DMT achievability

Theorem 2.3 (Sufficient condition) A coding scheme Xp(SNR) achieves the DMT of the

channel if

limSNR→∞

k∑i=1

βi ≥ k − r, k = 1 . . . q, (2.9)

where βi are the eigen-exponents corresponding to the eigen-values of the difference codeword

matrix ∆X∆X†.

Proof: The proof of Theorem 2.3 follows from [12] and from the matching between error

region and outage region developed in [39]. A sketch of the proof is given in the following for

sake of completeness.

The error probability can be upper bounded by the pairwise error probability,

Pe(SNR) ≤ EH

ProbXi → Xj |H

,

Using the approximation of the average PEP given by the sphere bound in [12,40],

ProbXi → Xj |H .= Prob‖H(Xi −Xj)‖2F ≤ 1,

and the lower bound on the Frobenius norm as in [12] and references therein,

‖H(Xi −Xj)‖2F ≥k∑i=1

λiµ−1i , k = 1 . . . q,

where λi, µ−1i with λ1 ≥ . . . ≥ λq and µ1 ≥ . . . ≥ µnt are respectively the eigen-values of

H†H and ∆X∆X†, it follows that,

ProbXi → Xj |H ≤ Prob k∑i=1

λiµ−1i ≤ 1, k = 1 . . . q

,

≤ Prob

( k∏i=1

λi

)( k∏i=1

µ−1i

)≤ k−k, k = 1 . . . q

, (2.10)

≤ Prob k∑i=1

αi ≥k∑i=1

βi, k = 1 . . . q︸︷︷︸Eα,β(r,SNR)

,

where (2.10) follows from the arithmetic-geometry mean inequality, αi and βi are respectively

the eigen-exponents corresponding to λi and µ−1i . It can be easily observed that for the high

SNR regime, ifk∑i=1

βi ≥ k − r, , k = 1 . . . q,

31


then,

Eα,β(r, SNR) = k∑i=1

αi ≥k∑i=1

βi ≥ k − r, k = 1 . . . q,

= O[nt,nr]α (r, SNR).

It follows that,

Pe(SNR).= SNR−dXp (r) ≤ SNR−dout(r),

and hence dXp(r) ≥ dout(r).From (2.5), we know that dXp(r) ≤ dout(r). Both inequalities imply that,

dXp(r) = dout(r).

Note that the sufficient condition in (2.9) can be written in function of the eigen-values µ2i

defined in (2.8) such that,

µ21µ

22 . . . µ

2k = SNR(

∑ki=1 βi−k),

≥ SNR(k−r+ε−k) =1

2R(SNR)+o(log SNR), k = 1 . . . q.

Approximately universal code construction

The approximately universal code design provides a structured code design criterion that

achieve the DMT. This design criteria is derived from the performance of the code over the

worst-channel case that is not in outage. Universal codes achieve reliable communication

over MIMO channel realization that are not in outage. The criterion that should be satisfied

by a code to be universal are given in Theorem 2.4.

Theorem 2.4 A coding scheme Xp(SNR) is approximately universal over the MIMO channel

if and only if, for every pair of distinct codewords

µ21µ

22 . . . µ

2q ≥

c

2R(SNR)+o(log SNR), c > 0 (2.11)

where q = rankH = min(nt, nr) and µ1 ≤ . . . ≤ µq are the smallest q eigen-values of the

normalized codewords difference matrix ∆X.

Non-vanishing determinant code construction

The non-vanishing determinant(NVD) criteria is a particular form of the approximately uni-

versal condition defined in Theorem 2.4 and has been proposed separately by Elia et. al

in [12]. In the following, we first define the structure of NVD codes and show that un-

der certain conditions, codes satisfying the NVD criterion are optimal. Then, we recall the

construction of NVD codes from cyclic division algebra (CDA).

32


Definition 2.3 (NVD codes) A coding scheme Xp(SNR) is called a rate-n NVD code if

Xp(SNR) satisfies the following properties

- Each entry xi,j of the Xp(SNR) is a linear combination of symbols from A(SNR), where

A(SNR) is a universal code over the scalar channel with data rate RA(SNR) bits PCU.

The quadrature amplitude modulation (QAM) constellation such as QPSK, 16QAM,

64QAM,... or HEX constellation are usually used as scalar universal codes.

- The average number of symbols transmitted by Xp(SNR) is n symbols PCU.

- The following NVD property is satisfied

det

(∆X∆X†

SNR 2−RA(SNR)

)≥ SNR0 (2.12)

Lemma 2.1 NVD codes achieve the DMT for nt × nr MIMO configuration when nt ≤ nr,

and for full rate codes (n = nt).

Proof: Using the identity detaA = am detA, where a ∈ C and A ∈ Cm×m, the

NVD property in (2.12) can be rewritten as,

det

(∆X∆X†

SNR 2−RA(SNR)

)=

1

2−ntRA(SNR)det

(∆X∆X†

SNR

).

As Xp(SNR) transmits an average of n symbols, then

R(SNR) = RXp(SNR) = r log SNR = nRA(SNR).

It follows that,

det

(∆X∆X†

SNR 2−RA(SNR)

)= SNR

ntnrµ2

1 . . . µ2nt ≥ SNR0

where µ2i are the eigen-values defined as in (2.8).

It can be immediately deduced for a NVD code that if nt ≤ nr, then

SNR−ntnr ≤ µ2

1 . . . µ2nt ≤ µ2

1 . . . µ2k, k = 1 . . . q = nt, (2.13)

where (2.13) follows from the normalization identity µ2i ≤ 1. This implies that for a NVD

code with nt ≤ nr and a full rate n = nt, the code is approximately universal and is therefore

DMT achieving. The NVD condition can be therefore rewritten such that,

µ21 . . . µ

2nt ≥

1

2R(SNR)+o(log SNR)(2.14)

33


Scaling factor θ for NVD codes

For a non-vanishing determinant code, the average number of symbols transmitted by Xp(SNR)

is n symbols PCU can be defined as

n =1

Tlog|A| |Xp|, (2.15)

where |Xp| denotes the total number of possible codewords in Xp and |A| denotes the total

number of constellation symbols in A . Let A be the M2-QAM constellation2, such that

A =a+ ib, |a|, |b| ≤M − 1 a, b are odd

,

which is a universal code over a SISO channel, then by definition

|A| = M2.

The rate of the space time code R = r log SNR can be related to |Xp| by

R =1

Tlog2 |Xp| = r log SNR.

It follows that,

|Xp| = 2TR = SNRTr = |A|nT ,

and therefore,

|A| = SNRrn . (2.16)

As each entry xi,j is a linear combination of symbols sl carved from a M2-QAM constellation,

i.e

xi,j =

n∑l=1

alsl, al ∈ C and ‖a‖2 = ‖[a1 . . . an]‖2 = 1

then it can be easily checked that ,

E[|xi,j |2] = ‖a‖2 E[|s|2] ≤ M2 = |A| = SNRrn .

Using the normalization constraint in (2.1), it follows that

θ2 .= SNR1− r

n . (2.17)

NVD code construction: Perfect space time codes

Perfect space time codes are full rate codes(n = nt) constructed from cyclic division algebras

(CDA) defined as following. For the sake of clarity of presentation, some basic algebraic tools

are provided in appendix A. Let L = Q(i, θ) be a cyclic extension of degree nt on the base

field Q(i). The generator of Galois group Gal(L/Q(i)) is denoted by σ, and assume that

2The same construction can be extended to HEX constellation, leading to the same value of scaling factor.

34


Gal(L/Q(i)) = σ0, . . . , σnt−1. Let γ ∈ Q(i) be such that γ, γ2, . . . , γnt−1 are non-norm

elements in L. The CDA of degree nt is given by

C =(L/Q(i), σ, γ

).

Each element X of C is given by,

X =

x1 x2 . . . xnt

γσ(xnt) σ(x1) . . . σ(xnt−1)...

...

γσnt−1(x2) γσnt−1(x3) . . . σnt−1(x1)

(2.18)

where xi ∈ I ⊂ OL is a linear combination of symbols carved from a QAM or Hex constella-

tion, OL being the ring of the integers, and I is an properly chosen ideal that preserves the

constellation shaping. As perfect space time codes are linear codes constructed from a CDA,

then

min∆X6=0

det∆X∆X† ≥ δ,

where δ is the inverse of the discriminant of Q(θ) (refer to [13] for more details), and is

independent of the constellation size. The NVD property in (2.12) can be easily verified,

such that

det

(∆X∆X†

SNR 2−RA(SNR)

)= δnt det

( θ2

SNR1− rn

)≥ SNR0.

2.2.4 Space time code properties with fixed rate

In previous sections, we mainly focus on space time codes design when the size of the code

grows as SNR. When the rate of the code is independent of SNR, i.e R(SNR) = R, the

corresponding multiplexing gain is such that,

r = limSNR→∞

R

log SNR= 0.

In this case, minimizing the average error probability over the distribution of the fading

channel is studied instead of the outage formulation. The average PEP for the nt×nr MIMO

channel has been derived in [9]. Assuming that a maximum likelihood decoder is used and

that the energy per antenna at each time slot is equal to 1, the PEP is bounded by,

PEP ≤ c SNR−d, (2.19)

where d is the diversity given by,

d = nr rank∆X∆X†, (2.20)

35


and c is the coding gain and is equal to,

c = 4d(

det

∆X∆X†)−nr . (2.21)

Perfect space time codes

Perfect space time codes fulfill the design criteria of Tarokh et. al in [9] for a quasi static

uncoded MIMO fading channel. For this family of code, the spreading factor3 s = nt, and

the matrix codeword given in (2.18) belongs to a cyclic division algebra. The fundamental

properties of perfect space time code that will be used in the following are:

- Perfect space time codeword has full rank of nt.

- Perfect space time code has a non-vanishing determinant

minC6=0

detCC† ≥ δ(dmin

)2nt ,with δ being the inverse of the discriminant of Q(θ) (refer to [13] for more details), and

is independent of the constellation size. dmin = 2√Es

is the minimal Euclidian distance

between two symbols and Es is the energy per symbol (Es = 2, 10, 42 for QPSK,

16QAM and 64QAM respectively). This implies that the PEP in (2.19) is bounded by

PEP ≤ δ−nrEntnrs SNR−ntnr . (2.22)

- Using the vectorized notations, vecC = Gx, where G is the n2t×n2

t rotation precoding

matrix constructed from cyclic algebra such that GG† = Int .

Spatial division multiplexing

The spatial division multiplexing corresponds to the case when no space time code is used.

The spreading factor in this case is s = 1, and the codeword x is a nt × 1 vector of symbols.

For SDM schemes, properties used in the following are

- The rank of xx† is equal to 1.

- The minimum determinant of the nt × nt matrix xx† is equal to d2min, where dmin =

2√Es

is the distance between two constellation points. The PEP in (2.19) is therefore

bounded by

PEP ≤ Enrs SNR−nr , (2.23)

- In order to have consistent notation, x can be written, x = Gx, where G = Int .

Alamouti schemes

The Alamouti scheme is designed for the 2 × nr MIMO configuration with nr ≥ 1. This

scheme transmits symbol s1 and s2 from antennas 1 and 2 respectively, during the first

3or equivalently the number of time slots.

36

2.3. CODE CONSTRUCTION FOR SELECTIVE FADING CHANNEL

period, followed by −s∗2 and s∗1 from antennas 1 and 2 respectively during the following

symbol period, i.e,

X =

[s1 −s∗2s2 s∗1

]. (2.24)

The properties of this code that we will use in the following are such that,

- The scheme is linear and its rank is equal to 2.

- The minimum determinant of the nt × nt matrix XX† is equal to d4min, where dmin =

2√Es

is the distance between two constellation points. The PEP in (2.19) is therefore

bounded by

PEP ≤ Entnrs SNR−ntnr (2.25)

- It is very simple to decode due to its orthogonal structure.

- The Alamouti is not a full rate code for nr > 1, only 1 symbol is sent per channel use.

This is traduced in a high coding gain c, as shown in the following example.

Numerical example: 2 BPCU

Considering the 2 × 2 MIMO configuration, and respectively a Golden code, SDM and an

Alamouti coding scheme of 2 BPCU. For this spectral efficiency, QPSK constellation is used

for the Golden code and the SDM cases and a 16QAM should be used with the Alamouti in

order to achieve the same spectral efficiency.

The PEP of these 3 schemes is upperbounded by,

PEP ≤ 400SNR−4, Golden code (2.26)

≤ 4SNR−2, SDM (2.27)

≤ 10000SNR−4, Alamouti code. (2.28)

At very low SNR, the SDM has better coding gain then both Alamouti schemes and the

Golden code. However, for the high SNR regime, the diversity gain dominates the coding

gain. As the Alamouti scheme is not a full rate scheme, using a 16QAM constellation instead

of QPSK constellation in order to achieve the same spectral efficiency induces a loss in term

of coding gain.

2.3 Code construction for selective fading channel

While most of the above results address the case of flat fading channel, we focus in this chap-

ter on the DMT achieving coding schemes for selective fading channel. The main objective of

previously proposed coding schemes such as [42] and [43] is to achieve the optimal multipath

diversity. However, these codes are not full rate, and therefore cannot extract all the available

spatial degrees of freedom. For the frequency selective channel, the optimal DMT has been

separately derived in [44] and [45]. More recently, Coronel and Bolcskei propose an optimal

coding scheme in [2] that achieves the DMT of the channel. This construction is originally

37


tailored to the time-frequency selective channel case, but can also apply to the case of fre-

quency selective channel. That’s why, coding is only performed across subcarriers without

making use of the time component. This optimal design is separated into two simpler design

problem. The first problem consists in designing a precoder that is adapted to the channel

statistics. The second problem consists in designing a code independent of the channel statis-

tics. In our contribution, we propose a more structured alternative to achieve the optimal

DMT by extending the non-vanishing determinant criterion to the selective channel case. We

show that for this channel, a sufficient condition to achieve the diversity multiplexing tradeoff

(DMT) is to code across time and frequency using a split NVD parallel code.

We consider a particular class of the general channel model considered in [2], [45] where

the channel is selective either in time or in frequency. For this class of channels, we propose

a systematic way to achieve the optimal DMT by extending the non-vanishing determinant

criterion to the selective channel case. A new code construction based on split NVD parallel

codes is then proposed to satisfy the NVD parallel criterion. Moreover, for the block fading

channel, we provide an extension of the geometrical interpretation to show the achievability

of the optimal DMT. This result is of significant interest not only in its own right, but also

as it shows that the optimal DMT in [45] is achievable for all the classes of fading channels

including the block fading channel.

2.3.1 Selective fading channel model

The input-output relation for the class of channels considered in this chapter is given by

Y[nr×T ]n =

√SNR

ntH[nr×nt]n X[nt×T ]

n + Z[nr×T ]n , (2.29)

where n = 0, 1, . . . , N − 1 represents the sub-channel n, the sub-channel H[nr×nt]n is a nt×nr

MIMO channel that remains constant during all the duration of the transmission T , Xn

represents the transmitted signal, and Zn denotes the additive i.i.d. CN (0, I) noise. The

channels Hn are correlated across the sub-channels n = 0 . . . N − 1 according to,

H = [H0 . . . HN−1] = Hw(R1/2H ⊗ Int), (2.30)

where RH is the N×N correlation between the scalar sub-channels with rank equal to ρ ≤ N ,

Hw is an nr × Nnt matrix with i.i.d. CN (0, 1) entries. The transmitted signal satisfies the

following power constraint,N−1∑i=0

E[‖Xi‖2F

]≤ TN. (2.31)

Throughout this chapter, we set m = min(nt, nr) and M = max(nt, nr).

The input-output relation considered in (2.29) models the case when the channel is selec-

tive either in time or in frequency.

38


For the frequency selective channel, n stands for the frequency and the channel is constant

across time. In this case, N represents the total number of subcarriers and RH is a circulant

matrix. For the time selective case (or the block fading channel), the channel remains constant

during a block n of T time slots and changes in a statistically independent manner across

blocks. For this case, N represents the total number of blocks and RH = IN .

2.3.2 DMT of selective fading channel

The discussion about the outage derivation will be revisited in the last Section 2.3.5. We just

recall here the Jensen outage bound derived by Coronel et al. in [45]. We refer the interested

reader to [2] and [45] for more details.

Theorem 2.5 (Outage bound on the DMT) For a selective fading channel, the outage

probability is lower-bounded by,

Pout(r) ≥ PJ(r).= SNR−dJ (r)

where,

dJ(r) = (ρM − r)(m− r). (2.32)

Proof: The essence of the proof is due to the Jensen bound derived by Coronel and Bolcskei

in [45]. In this case,

dJ(r) = infα∈Oα(r,SNR)

m∑i=1

(2i− 1 + LM −m)αi,

where αi are the eigen-exponents of the eigen-values of the equivalent m×m Wishart matrix

with ML degrees of freedom, and Oα(r, SNR) is the outage event such that,

O[m,LM ]α (r, SNR) =

α :

k∑i=1

αi ≥ k − r, k = 1, . . . ,m = min(nt, nr).

2.3.3 Optimal design criterion

Unlike the case of time-frequency selective channel in [2], we show here that when the channel

is selective either in time or in frequency, there is no need to construct an additional precoder

adapted to the channel statistics in order to achieve the optimal DMT. The optimal code

design criterion required to achieve the optimal DMT is summarized in the following theorem.

Theorem 2.6 (Sufficient condition for DMT achievability) A coding scheme X achieves

the optimal DMT (ρM − r)(m− r), if for any two different codewords X, X ∈ Xp(SNR), the

39


eigenvalues of the block diagonal matrix DD†, where D = diag

(Xn − Xn)N−1

n=0satisfy

minX,X∈Xp(SNR)

m∏i=1

λi(DD†) ≥ 1

2R(SNR)+o(SNR). (2.33)

Proof: Let X be the transmitted codeword, X the nearest decoded codeword and

∆Xn = Xn − Xn the difference codeword matrix. The pairwise error probability of the

correlated parallel channels is upper-bounded as following,

PEP ≤ EH exp

(−SNR

4nt

N−1∑n=0

‖Hn∆Xn‖2F

),

≤ EH exp(− SNR

4ntTr(HwΘH†w

)), (2.34)

where Hw denotes the nr ×Nnt i.i.d. CN (0, 1) matrix, and

Θ = (R1/2H ⊗ Int) diag

∆Xn∆X†n

N−1

n=0(R

1/2H ⊗ Int)

is the effective codeword matrix.

Assuming that Xp(SNR) satisfies the NVD criteria, then D = diag

∆Xn

N−1

n=0is a full rank

matrix with rank equals to Nnt. The rank and the eigenvalues of the effective codeword

matrix Θ can be computed using the following lemma 2.2.

Lemma 2.2 Let A be a p× p Hermitian matrix given by,

A = B(CC†)B†,

where B is p× p matrix with rank s, C is full rank p× p matrix. Then, the matrix A has the

following properties:

a) The rank of A is equal to s, the rank of B.

b) The non zero eigenvalues λk(A) of A are lower bounded by,

λk(A) ≥ λ1(BB†)λk(CC†). (2.35)

Proof: The proof of this lemma uses the same matricial tools as [2], and is detailed in

Appendix 2.A.1.

By applying Lemma 2.2-a to Θ, it follows that,

rankΘ = rankR1/2H ⊗ Int

= rankR1/2H rankInt = ρnt.

40


By noticing that Θ is not full rank, the Frobenius norm in (2.34) has the same distribution

as TrHwΛH†w where Hw is the nr × ρnt effective channel with i.i.d. entries ∼ CN (0, 1)

and Λ is the ρnt × ρnt diagonal matrix containing the non-zero eigenvalues of the effective

codeword Θ bounded using Lemma 2.2-b such that

λi(Θ) ≥ σ2H λi

(DD†

), i = 1 . . . ρnt,

where σ2H is the smallest eigenvalue of RH.

By following the same footsteps as in [(105) and (108) in [2]], this Frobenius norm can be

bounded such that,

TrHwΛH†w ≥m∑i=1

λi(HwH†w)λm−i+1(Θ)

≥ σ2H

m∑i=1

λi(HwH†w)λm−i+1(DD†)

where Hw denotes the m× ρM Jensen channel with i.i.d. CN (0, 1) entries such that,

Hw =

[Hw,0 . . . Hw,ρ−1], if nr ≤ nt,[H†w,0 . . . H†w,ρ−1], if nr > nt.

(2.36)

The rest of the proof uses the same technique as presented in [45], [2]. It can be deduced

that if the code satisfies the NVD criteria in (2.33), then the error region event Eα(r, SNR)

for a given channel realisation α matches with the outage region O[m,ρM ]α (r, SNR) of the

equivalent m× ρM MIMO channel,

Eα(r, SNR) = k∑i=1

αi ≥ k − r, k = 1, . . . ,m,

= O[m,ρM ]α (r, SNR), (2.37)

with α being the vector containing the eigen exponents of the channel HwH†w, such that

λi(HwH†w).= SNR−αi .

2.3.4 Split NVD parallel codes for selective fading channel

In this section, we propose a new family of split NVD parallel codes to achieve the optimal

DMT of (ρM − r)(m− r). Before studying the optimality of these split NVD parallel codes,

we briefly review the structure of the NVD parallel codes in Subsections 2.3.4 and 2.3.4. An

equivalent system model for the selective channel is defined in Subection 2.3.4. Finally, the

code construction and the optimality of the split NVD parallel code is addressed in Subection

2.3.4.

41


NVD parallel scheme

Let X = diagXnN−1n=0 ∈ Xp(SNR) be the block diagonal matrix containing the transmitted

codeword Xi in (2.29), and constructed such that X = θ Ξ, where θ is a scaling factor that

depends on the structure of the code, and chosen to ensure the power constraint in (2.31).

The block diagonal matrix Ξ = diagΞiN−1i=0 is an NVD parallel code denoted by C(SNR),

and defined as follows:

Definition 2.4 (NVD parallel scheme) Let A(SNR) be an alphabet4 that is scalably dense,

such that

∀s ∈ A(SNR) ⇒ |s|2 ≤ |A(SNR)|.

Then, C(SNR) is called NVD parallel code if,

1. Each entry of Ξ is a linear combination of symbols carved from A(SNR).

2. The total number of transmitted symbols carved from A(SNR) is equal to TNnt.

3. For any pair of different codewords Ξ and Ξ ∈ C(SNR), the NVD property is satisfied

det((Ξ− Ξ)(Ξ− Ξ)†

)≥ κ > 0, (2.38)

with κ is a constant independent of SNR.

Cyclic division algebra (CDA) code structure

We recall here the most relevant concepts of the construction of the codeword matrix Ξ =

diagΞiN−1i=0 based on cyclic division algebra. We refer the reader to [14], [46] for more

details on the NVD parallel code construction. In the following, we conider,

- The field F as a Galois extension of degree N over Q(i), and that have τ as generator,

such that

Gal(F/Q(i)) = τ1, . . . , τN.

- The field K is a cyclic extension of degree nt over F, and that have σ as generator, such

that

Gal(K/F) = σ0, . . . , σnt−1.

The code Ξ is constructed by setting Ξi = τi(Ξ), i.e.,

Ξ =

τ1(Ξ)

τ2(Ξ)

· · ·τN (Ξ)

(2.39)

4We assume here without restriction that the signal constellation is a QAM constellation, i.e, A(SNR) =AQAM(SNR). This can be also extended to the case of HEX constellations.

42


where Ξ = Ξ0 belongs to the cyclic division algebra C = (K/F, σ, γ), and γ ∈ F chosen such

that γ, γ2, . . . , γnt−1 are not norms of an element of K. The matrix Ξ is defined such that

Ξ =

x0 x1 . . . xnt−1

γσ(xnt−1) σ(x0) . . . σ(xnt−2)...

...

γσnt−1(x1) γσnt−1(x2) . . . σnt−1(x0)

,

where, xi =∑Nnt

j=1 si,jωj , si,j ∈ A(SNR) and ωj ∈ K. For the NVD parallel code, the

determinant is such that,

det(

diagΞiNi=1

)=∏k

τk(det(Ξ))

= NF/Q(i)(det(Ξi)) ∈ Z[i],

and which is equal to zero if and only if all xi are zeros. It follows that for Ξ 6= 0 ,

| det(Ξ)|2 ≥ SNR0.

Example: Two transmit antenna and 2-parallel NVD scheme

In this case, the code is given by

X =

[Ξ 0

0 τ(Ξ)

](2.40)

where

Ξ =

(x1 x2

γσ(x2) σ(x1)

)

and x1, x2 ∈ OK, with OK = a+ bθ | a, b ∈ OF and θ = 1+√

52 .

Let F = Q(ζ8) be an extension of Q(i) of degree 2, with ζ8 = eiπ4 , then a = s1 + ζ8s2. The

Galois generators σ and τ are chosen such that

σ(x) = a+ bθ, x = a+ bθ

τ(a) = s1 − ζ8s2, a = s1 + ζ8s2

By choosing the xi in an ideal generated by αOK, with α = 1 + i− iθ, and knowing that

γ = ζ8 is not a norm, the matrix codeword is then given by,

Ξ =1√5

[α(s1 + s2ζ8 + s3θ + s4ζ8θ) α(s5 + s6ζ8 + s7θ + s8ζ8θ)

ζ8α(s5 + s6ζ8 + s7θ + s8ζ8θ) α(s1 + s2ζ8 + s3θ + s4ζ8θ)

](2.41)

43


Choice of θ for NVD parallel codes

The scaling factor θ that insures the power constraint in (2.31) is such that,

θ2N−1∑i=0

E[‖Ξi‖2F] ≤ TN.

Due the linearity of this code and to the use of unit transformation, each entry of x ∈ Ξ is

such that,

E[|x|2] = E[|s|2], s ∈ AQAM(SNR),

=2(|A(SNR)| − 1)

3.

This implies that,

N−1∑i=0

E[‖Ξi‖2F] = TN E[|x|2],

.= TN |A(SNR)|.

The scaling factor θ that ensures the power constraint is therefore,

θ2 .= |A(SNR)|−1. (2.42)

Using the NVD parallel criterion in (2.38) and the value of θ2 in (2.42), the eigenvalues

of the block diagonal matrix D = X− X = θ(Ξ− Ξ) for any different codewords X, X, are

such that,Nnt∏i=1

λi(DD†) =|det(Ξ− Ξ)|2|A(SNR)|Nnt ≥

1

|A(SNR)|Nnt .

Due to the power constraint in (2.31), these eigenvalues necessarily satisfy λi(DD†) ≤ 1.

Then, the NVD parallel criterion is equivalent to,

minX,X∈Xp(SNR)

m∏i=1

λi(DD†) ≥ 1

|A(SNR)|Nnt . (2.43)

Equivalent model scheme

The selective fading channel operating at a total multiplexing gain rt = Nr is equivalent in

term of its error performance to a system operating at the same multiplexing as each sub-

channel, i.e., re = rtN = r. The optimal DMT of the selective fading channel ds(rt) is related

to the equivalent scheme by ds(rt) = d(rtN

), where d(r) is the optimal DMT of the equivalent

model. The equivalent model has the following form,

Y[Nnr×NT ]e =

√SNR

ntHX[Nnt×NT ]

e + Z[Nnr×NT ]e , (2.44)

44


where H = diagHiN−1i=0 ∈ CNnr×Nnt is the channel block diagonal matrix and Xe =

[X[T ]e,0 . . . X

[T ]e,N−1][T ] is the transmitted coding scheme such that E[XeX

†e] = INT .

Split NVD parallel codes and optimality

The NVD parallel codes as put straightforwardly by Lu in [46] and Yang et al. in [14] are sub-

optimal, as the DMT achieved by these codes is only ρ(nt−r)(nr−r) < (ρM−r)(m−r). The

main idea of the new split code construction is to design a coding scheme for the equivalent

model in Subection 2.3.4 that guarantees to transmit a rate of R(SNR) over each sub-channel

and to satisfy the NVD parallel criterion in Theorem 2.6. The two possible ways of splitting

the data over the parallel channels are depicted in Figure 2.2 and Figure 2.3 .

Block diagonal NVD parallel code

The first way of splitting the data over the parallel channels has been previously studied

in [46] and is depicted in Figure 2.2. In this case, the total rate NR is transmitted during

only T slots over each sub-channel.

Ξ

τ(Ξ)

n = N − 1

n = 1

n = 0

NT slots

τN−1(Ξ)

Ξd =

Figure 2.2: Coding across time and frequency: The total rate is transmitted only during Tslots. Each entry of τi(Ξ) is a linear combination of symbols carved from Ad(SNR) where

|Ad(SNR)| = SNRrnt . In this case, Xe,d = θdΞd.

It can be easily verified that for this scheme the outage event is such that,

O1(r, SNR) = I1(x, y|H) < Nr log SNR ,

where,

I1(x, y|H) = log det(IN +

SNR

ntHH†

).

Each block τi(Ξ) contains TNnt symbols carved from a signal constellation Ad(SNR).

In order to maintain a rate of R(SNR) over each sub-channel, the size of the constellation

|Ad(SNR)| should be chosen such that,

R(SNR) = r log SNR =1

NTlog |Ad(SNR)|ntTN .

45


i.e., |Ad(SNR)| = SNRrnt . It can easily be verified that for this choice of signal constellation

size, the NVD parallel criterion in (2.43) is,

minX,X∈Xp(SNR)

m∏i=1

λi(DD†)≥ 1

2NR(SNR)+o(SNR).

Obviously, the sufficient condition in Theorem 2.6 is not satisfied in this case. The achievable

DMT by this transmission scheme is only ρ(nt−r)(nr−r) as shown in [46], and it is therefore

sub-optimal.

Split NVD parallel code

The second way we propose to split the data that guarantees to transmit a rate of R(SNR)

using a total power of SNR over each sub-channel is shown in Figure 2.3. In this case, the

total rate is split equally among all the NT slots. Each block Ξi transmits TNnt symbols

carved from a signal constellation As(SNR). The same TNnt symbols are transmitted over

blocks Ξi . . . τN−1(Ξi) but encoded differently. However, different symbols are transmitted

over two different blocks Ξi and Ξj .

RΞ0 Ξ1 ΞN−1 n = 0

τ(ΞN−2) n = 1

τN−1(Ξ0) n = N − 1τN−1(Ξ2)τN−1(Ξ1)

τ(ΞN−1) τ(Ξ0)

NT slots

Ξs =1√N×

TNnt symb

R R

Figure 2.3: Coding across time and frequency: The total rate is split across the NTslots. Each entry of τi(Ξi) is a linear combination of symbols carved from As(SNR) where

|As(SNR)| = SNRr

Nnt . In this case, Xe,s = θsΞs.

For this transmission scheme, the outage event occurs when at least one of the NVD

parallel code scheme with rate R(SNR) = r log SNR is in outage, meaning that,

O2(r, SNR) =N−1⋃s=0

Os(r, SNR),

where,

Os(r, SNR) =

1

NI2(x, y|H) < r log SNR

, ∀s,

46


and,

I2(x, y|H) = log det(IN +

SNR

NntHH†

).

Note that the normalization factor 1/N in the first side of the inequality in the outage event

Os(r, SNR) traduces the fact that N blocks are needed to decode the information of each

NVD parallel code with rate R(SNR).

Using the union bound and the inclusion bound (Os ⊆ O2), the outage probability can be

bounded as,

P(Os) ≤ P(O2) ≤N−1∑i=0

P(Os) (2.45)

Assuming that P(Os) scales as SNR−ds(r), it follows from (2.45) that at high SNR,

P(O2).= SNR−ds(r) .

= P(Os) .= P(O1),

This implies that this scheme is equivalent in term of outage to the first scheme.

In order to maintain the rate of R(SNR) over each sub-channel, the signal constellation

As(SNR) should be chosen such that,

R(SNR) = r log SNR =1

Tlog |As(SNR)|ntTN .

The size of the signal constellation for the split NVD parallel scheme is therefore reduced

compared to the block diagonal case, and

|As(SNR)| = SNRr

Nnt = |Ad(SNR)| 1N .

Due to the block diagonal channel matrix structure, it can be deduced that the split NVD

parallel code is equivalent to a concatenation of N independent parallel NVD codes, where

the symbols of each NVD parallel code are carved from a constellation As(SNR) with size

SNRr

Nnt . The system is in error if at least one of the NVD parallel codes is in error, i.e.,

ε(r, SNR) =

N−1⋃i=0

εi(r, SNR),

where ε(r, SNR) represents the event that the system is in error and εi(r, SNR) denotes the

event that the ith NVD parallel code formed by the blocks Ξi . . . τN−1(Ξi) is in error.

For each NVD parallel code with symbols carved from As(SNR), it can be easily verified by

replacing the cardinality of As(SNR) in (2.43) that the NVD parallel criterion in Theorem 2.6

is satisfied, i.e.,

minX,X∈Xp(SNR)

m∏i=1

λi(DD†) ≥ 1

2R(SNR)+o(SNR).

47


It follows from Theorem 2.6 that,

P(εi).= SNR−di(r),

where di(r) = (ρM − r)(m− r), ∀i.Using the inclusion and the union bound as for the outage analysis in (2.45), it follows that,

Pe(r, SNR) = P(ε).= SNR−d(r),

with d(r) = di(r) = (ρM − r)(m− r).The split NVD parallel codes in Figure 2.3 achieve therefore the optimal DMT of (ρM −r)(m− r).

2.3.5 Application to the block fading channel

The block fading channel is a particular case of the selective fading channel model considered

in (2.29) with covariance matrix RH = IN . The optimal DMT expression is therefore d∗(r) =

(NM − r)(m − r), which is the DMT expression of the general channel model considered

in [2], [45] applied to this particular channel setting. Obviously, this result does not match

with the corresponding result in [10], i.e., dl(r) = N(M − r)(m − r) ≤ d∗(r), ∀r. This

incoherence in results has given rise to lots of debate in literature e.g. [47]. The authors of [47]

base their arguments on a non-accurate outage probability derivation (Pout,l(r).= SNR−dl(r))

to claim that the DMT of the block fading channel cannot exceed dl(r) ≤ d∗(r). As we

will show in the following, deriving the analytical outage probability is not a straightforward

generalization of the flat fading channel and should be carefully performed. The outage

derivation we provide here is based on the geometrical argument previously used for the flat

fading channel in [10] and for the selective fading case in [2].

Geometrical interpretation

For the block fading channel, the outage probability is,

Pout(r) , Prob

log det(

I +SNR

ntHH†

)< Nr log SNR

,

where H = diagHnN−1n=0 is the block diagonal channel matrix.

In order to generalize the geometrical interpretation in [2] to the block fading channel, we

start first by finding an equivalent expression of the outage probability. For this, we consider

hij the N×1 Gaussian vector ∼ CN (0, IN ) containing the N independent channel realisations

between transmit antenna j and receive antenna i. It is well-known that the Gaussian vector

hij is identically distributed as Fhω,ij for any unitary matrix F, i.e., hij ∼ Fhω,ij ,∀i, j.In the following, we specify our result to the case where F is a N × N Fast Fourier

Transform (FFT) matrix. This means that each channel realisation is identically distributed

48


as,

h[n]ij ∼

1√N

N−1∑l=0

h[l]ij,we

−j2π lnN , n = 0 . . . N − 1.

The block diagonal matrixH is therefore identically distributed as DH, i.e.,H ∼ DH, where,

DH =1√N

N−1∑l=0

Hw,lω0l

. . .N−1∑l=0

Hw,lωN−1l

, (2.46)

with ωl = e−j2πlN and Hω,l = (h

[l]ij,ω)1≤i≤nr,1≤j≤nt .

Consequently, the mutual information is identically distributed as,

I(x,y|H) ∼ log det(

I +SNR

NntDHD†H

)= ID(SNR).

By using an FFT precoder and an FFT equalizer as in an OFDM system to transmit over

the channel DH in (2.46), the matrix DHDH† can be made unitarly equivalent to CHC†H,

where

CH =

Hw,0 Hw,1 . . . Hw,N−1

Hw,N−1 Hw,0 . . . Hw,N−2

...

Hw,1 Hw,2 . . . Hw,0

. (2.47)

Thus, the corresponding mutual information ID(SNR) can be written as,

ID(SNR) = log det

(I +

SNR

NntCHC†H

)∼ I(x,y|H).

It follows therefore that the outage probability is such that,

Pout(r) = Prob

log det(

I +SNR

NntCHC†H

)< Nr log SNR

.

Following the geometrical interpretation of the flat fading channel in [10], the typical

outage event occurs when the channel matrix CH is close to the manifold of all matrices with

rank Nr denoted by RNr, such that,

RNr = CH : rankCH = Nr.

By following the same reasoning as in [10], this requires that the d(r) components of CH

orthogonal to RNr to be collapsed, i.e., be on the order of SNR−1. The probability of this

event is Pout(r).= SNR−d(r). The number of these components is given by

d(r) = NMm− dim(RNr),

49


where dim(RNr) is the sufficient minimal number of parameters required to specify matrix

CH with rank Nr.

Dimensionality of RNr

We first note that due to the structure of CH in (2.47), the number of parameters required

to characterize a matrix CH in RNr is equal to the number of parameters required to specify

an m ×NM matrix (m = min(nt, nr) and M = max(nt, nr)) with rank r that contains the

nt first columns if nt ≤ nr, and the nr first rows if nr ≤ nt. Characterizing a matrix CH with

rank Nr reduces therefore to the problem of characterizing a matrix of dimension m×NMwith rank r that requires only NMr + (m− r)r, i.e,

dim(RNr) = NMr + (m− r)r,

where MNr is the number of independent parameters needed to identify r independents

vectors and (m − r)r parameters are needed to identify the linear dependent vectors as a

function of the r independent vectors. It can be be easily verified here that the MNr free

i.i.d. Gaussain parameters that identify the r linear independent vectors generate a block

circulant matrix with rank Nr with a probability equal to one.

It can be deduced that the optimal DMT for the class of block fading channel is,

dout(r) = NMm− dim(RNr) = (NM − r)(m− r).

and not,

dout(r) 6= NMr −N dim(Rr) = N(M − r)(m− r).

Comments on related work’s derivation

It should be finally emphasized that the number of parameters needed to describe the sub-

space RNr is therefore different than the number of parameters needed to characterize N

independent subspaces Rr separately, i.e., dim(RNr) 6= N dim(Rr) = N(Mr + (m− r)r).Note that when applying the geometrical argument to the outage definition, one may

have tendency to deduce that N(Mr + (m− r)) parameters are required to characterize the

subspace of block diagonal matrix H = diagHiN−1i=0 with rank Nr. Although, the parallel

sub-channels are statistically independent, there is an implicit dependency between the eigen-

exponents of the channels induced by the NVD parallel criterion in Theorem 2.6 and traduced

in (2.37).

Due to the NVD parallel criteria in Theorem 2.6 and using a split NVD parallel code,

the system is in outage and therefore in error if the eigen-exponents of the diagonal channel

matrix satisfies the equation in (2.37). Using the geometrical argument in [10], this error

event occurs if the m ×NM Jensen matrix is close to a rank r matrix, which requires only

NMr + (m− r)r to be characterized.

50


If the Jensen channel matrix Hw is a rank r matrix, then using the QR decompostion,

Hw = Q[m×r]R[r×NM ],

where Q is a unitary matrix, such that Q†Q = Ir and R is an upper triangular matrix. Each

m×M matrix can be therefore written as,

H0 = Q[m×r] H[r×M ]

0 , (2.48)

Hi = Q[m×r] H[r×M ]i , i = 1 . . . N − 1. (2.49)

where H0 = R([1 : r], [1 : M ]) and Hi = R([1 : r], [iM + 1 : (i + 1)M ]) are statistically

independent matrix with i.i.d. entries and rank r.

It can easily be checked from the product matrix rank property (D ∈ Ca×b,E ∈ Cb×c),

rankD+ rankE − b ≤ rankDE≤ min

rankD, rankE

, (2.50)

that the matrices Hi, i = 1 . . . N − 1 are rank r matrices, amd therefore the block diagonal

matrix is close to a rank Nr matrix.

As a consequence of NVD parallel property, it can be noticed from (2.48) and (2.49), that

the sub-channels share the left eigen vectors without violating the fact these channels are

statistically independent. The number of parameters required to specify the block diagonal

matrix is therefore less than the one required to identify N independent matrix with rank r.

2.3.6 Numerical results

In order to compare the performance of the split NVD parallel code with the classical NVD

parallel code, we consider the case of 2 parallel 2 × 2 MIMO channel, i.e. a block fading

channel with a total number of blocks equal to 2.

The structure of the NVD parallel code for this configuration is given in [14], such that

X =

(Ξ 0

0 τ(Ξ)

)(2.51)

where,

- Matrix Ξ is such that,

Ξ =

(x1 x2

γσ(x2) σ(x1)

)

- F = Q(ζ8) be an extension of Q(i) of degree 2, with ζ8 = eiπ4 , then a = s1 + ζ8s2.

- x1, x2 ∈ OK, with K = F(√

5), and OK = a+ bθ | a, b ∈ OF and θ = 1+√

52 .

51


- The Galois generators σ and τ are chosen such that

σ(x) = a+ bθ, x = a+ bθ

τ(a) = s1 − ζ8s2, a = s1 + ζ8s2

- xi in an ideal generated by αOK, with α = 1 + i− iθ.- γ = ζ8 is not a norm of an element of K.

- The matrix codeword is then given by,

Ξ =1√5

(α(s1 + s2ζ8 + s3θ + s4ζ8θ) α(s5 + s6ζ8 + s7θ + s8ζ8θ)

ζ8α(s5 + s6ζ8 + s7θ + s8ζ8θ) α(s1 + s2ζ8 + s3θ + s4ζ8θ)

)(2.52)

For the same channel model, the structure of the split NVD parallel code is such that,

X =1√2

(Ξ1 Ξ2

τ(Ξ2) τ(Ξ1)

)(2.53)

As we showed in previous section, the optimal DMT achievable by the NVD parallel is

only 2(2−r)(2−r). However, the optimal DMT achievable by the split code is (4−r)(2−r).These two DMT are depicted in Figure 2.4.

For a rate per channel use equal to 4 bpcu (resp. 8 bpcu), the symbols s1, s2, . . . , s8

should be carved from a BPSK (resp. QPSK) constellation for the scheme with split code

and from a QPSK (resp. 16QAM) constellation for the scheme with NVD parallel code. One

should expect here that the gain provided by the use of a smaller size of constellation used

in the split NVD parallel code to be compensated by the normalization factor 1/√

2. Due to

the gain in DMT, this is not the case and the comparison of both schemes is in Figure 2.5.

2.3.7 Discussion and observation

Although the split NVD parallel schemes are optimal, this solution does not seem to be

eligible to be implemented in a practical system and specially in standards considerations,

where the number of FFT tones is N = 64, 256, . . .. The decoding of such a code using a

lattice decoder exhibits a high order of complexity as QAM symbols that belongs to Nnt

OFDM symbols should be decoded at once. Using a sphere decoder or a Schnorr Euchnerr

decoder with these high dimensions seems to not be feasible. Coding only symbols across the

frequency without coding across blocks could be an interesting solution in term of complexity.

However, this scheme is not optimal if used alone in a CP-OFDM system. As we will show in

next section that this can be made possible if information bits are coded using a convolutional

code. This motivates the study of bit interleaved coded modulation BICM-MIMO system.

52


Suboptimal bound, NVD parallel code

[Coronel and Bolcskei, 2007]

Optimal bound, Split NVD code

[Zheng and Tse, 2003]

d(r)

8

3

2

2 r1

Figure 2.4: The optimal DMT achievable by the NVD parallel code for the 2×2 block fadingchannel with N = 2 is d(r) = 2(2 − r)(2 − r). The split code achieves the optimal DMT ofthe block fading channel d(r) = (4− r)(2− r).

10-5

10-4

10-3

10-2

10-1

100

0 5 10 15 20

PE

R

SNR(dB)

Error Probability of split code and NVD parallel code

Split code BPSK - R = 4bpcu NVD parallel code QPSK - R = 4bpcu Split code QPSK- R = 8bpcu NVD parallel code 16QAM - R = 8bpcu

Figure 2.5: Split NVD code versus NVD parallel code for the 2×2 block fading channel withN = 2.

53


2.4 BICM system model

The performance of the optimal NVD codes will be studied in a more complete system using

a convolutional code and over multi-tap channel defined in Subection 2.4.1. We focus on the

case where the rate does not grow with SNR. For this reason, the PEP is first derived for a

general channel case in Subection 2.4.2, then we specify our result to the flat fading case in

section 2.5 and to the frequency selective case in section 2.6.

2.4.1 System model

The block diagram of the considered system based on the IEEE 802.11n transmission scheme

is depicted in Figure 2.6.

Convolutional

Code CModulationInterleaver

Space Time Block

Coding STBCDeinterleaver

ML soft

Decoder

Viterbi

Decoder

Channel

Figure 2.6: BICM MIMO system

During transmission, the binary information elements b are first encoded by a binary code

of rate Rc e.g. a convolutional code, and then interleaved by a bit interleaver which will be

denoted by π. The coded and interleaved sequence c is fed into the 2m-QAM gray mapper

and is mapped onto the signal sequence x ∈ X . The resulting symbols are coded by a space

time block code with spreading factor s and generator matrix G. The coded codewords are

finally transmitted on a multiple antenna channel H with nt transmit antennas and nr receive

antennas. Bit interleaver can be modeled as π: k′ → (k, i), where k

′denotes the original

ordering of the coded bits ck′ , k denotes the time or frequency ordering of the vectorized5

MIMO codewords x(k) where x ∈ X snt and i indicates the position of the bits ck′ in the

codeword.

At the receiver, the vectorized received signal is given by

y(k) = He(k)Gx(k) + z(k), (2.54)

where z(k) is the complex Gaussian noise z ∼ CN (0, N0Inr), G is the snt × snt rotation

precoding matrix defined as in section 2.2.4 such that GG† = Int and

He(k) = diag

H(k), . . . ,H(k)︸︷︷︸s

,

denotes the equivalent block diagonal channel.

The ML soft decoder generates for each coded bit ck,i two metrics: λick=0 and λick=1.

Theses metrics correspond to the log-MAP computed over one codeword (refer to [48] for

5In the following, vectorized notations for space time code are used instead of the matricial notations forsimplicity.

54

2.4. BICM SYSTEM MODEL

more details on λ-metrics), and are given by :

λi(ck) = log∑

x∈X ick

p(y(k)∣∣He(k),x),

= log∑

x∈X ick

exp−‖y(k)−He(k)Gx(k)‖2 .

These metrics can be approximated by

λi(ck) ≈ minx∈X ick

‖y(k)−He(k)Gx‖2, (2.55)

where X ib denotes the constellation subset

X ib = x ∈ X snt = X × . . .×X︸︷︷︸snt

: li(x) = b,

and li(x) is the ith bit of the codeword x. For low complexity algorithms, these metrics can

be computed using the list sphere decoder [49].

Then, the metrics associated to the interleaved bits are deinterleaved. Finally, the λ metrics

are used by the Viterbi decoder to decode the information bits by finding the shortest path

in the trellis according to

c = arg minc∈C

∑k′

λ(cik). (2.56)

2.4.2 General pairwise error probability derivation

In [50], the pairwise error probability of BICM-MIMO-OFDM using an orthogonal space time

code such as the Alamouti code was derived. In this section, we extend the pairwise error

probability expression of [50] to a more general case using a space-time code with a spreading

factor s. The main result of this subsection is summarized by the following theorem.

Theorem 2.7 The PEP over a general channel model is upper bounded by,

P(c→ c) ≤ EH exp(− 1

4N0

∑k′,dfree

‖H(k)C(k)‖2F), (2.57)

where dfree is the free distance of the convolutional code, C(k) denotes the nt × s codeword

matrix associated to the snt × 1 vectorized vector, and is a non zero matrix for all c 6= c and

k = 1 . . . dfree.

55


Proof: Assuming that the code sequence c is transmitted and c is detected, the PEP

given the channel knowledge can be written as

P(c→ c|H) = Prob∑

k′

minx∈X ick

‖y(k)−He(k)Gx(k)‖2

≤∑k′

minx∈X ick

‖y(k)−He(k)Gx(k)‖2. (2.58)

Let dfree be the minimum Hamming distance of the convolutional code. Error occurs when

the distance between the incorrect path associated to c and the correct one associated to c is

equal to dfree. In this case, X ick and X ick in equation (2.58) are equal for all k′

except for the

dfree distinct values of k′. Therefore, only dfree terms are different in the inequality (2.58).

Let us denote in the following x(k) and x(k) as

x(k) = arg minx∈X ick

‖y(k)−He(k)Gx(k)‖2,

x(k) = arg minx∈X

cik


where Xcik is the complementary set of Xcik .

The PEP can be written as

P(c→ c|H) = Prob ∑k′ ,dfree

‖y(k)−He(k)Gx(k)‖2

≤∑

k′ ,dfree


where∑

k′ ,dfreemeans that we only consider the dfree terms of the inequality for which x(k)

and x(k) are different. Consequently,

P(c→ c|H) = Prob ∑k′ ,dfree

ωk ≤ 0,

where

ω =∑

k′ ,dfree

ωk,

ωk =∥∥∥He(k)G

(x(k)− x(k)

)︸︷︷︸

d(k)

+z(k)∥∥∥2− ‖z(k)‖2.

Due to the bit interleaver, bits are uncorrelated, and then ω is the sum of dfree independent

Gaussian variable ωk with mean∥∥He(k)Gd(k)

∥∥2and variance 4N0

∥∥He(k)Gd(k)∥∥2

. Conse-

56

2.5. BICM-MIMO WITH FLAT FADING CHANNEL

quently, ω is a Gaussian variable, such that

N( ∑k′,dfree

‖He(k)Gd(k)‖2 , 4N0

∑k′,dfree

‖He(k)Gd(k)‖2)

For a Gaussian variable ω ∼ CN (µ, σ2), it is well known that

Probω < 0 = Q(µσ

)≤ exp

(− µ2

2σ2

).

It follows that

P (c→ c|H) ≤∏

k′ ,dfree

exp(− ‖He(k)Gd(k)‖2

8N0/2

).

Let C(k) denotes the nt × s matrix associated to the snt × 1 vectorized vector Gd(k), then

the following identity is verified

‖He(k)Gd(k)‖2 = ‖H(k)C(k)‖2F.

By averaging over all the channel, the PEP over a general channel model can be written such

that

P(c→ c) ≤ EH

[exp

(− 1

4N0

∑k′,dfree

‖H(k)C(k)‖2F)]

.

From the definition of C(ki), we can note that all matrices C(ki) (i = 1 . . . dfree) are non

zero matrix for all c 6= c. This remark follows from the definition of C(ki) in section 2.4.2.

Note that in this case d(ki) = x(ki) − x(ki) contains at least one non zero symbol, as x(ki)

and x(ki) belongs to two complementary set.

2.5 BICM-MIMO with flat fading channel

For a flat fading channel, the channel is fixed during the whole duration of transmission,

H(k) = H, ∀k, with H a Gaussian matrix with i.i.d entries hi,j ∼ CN (0, 1), with i = 1 . . . nt

and j = 1 . . . nr. The main result of this section is summarized in the following theorem.

Theorem 2.8 For a MIMO-BICM system with a flat fading channel, perfect space time code

allows to extract the full diversity order of ntnr and the PEP is upperbounded by,

PEP ≤(dfreeδ

)−nr(Es)ntnrSNR−ntnr . (2.59)

57


2 4 6 8 10 12 14 1610

-3

10-2

10-1

100

SNR (dB)

PE

R

QPSK - CC = [5 7] , 1/2 (dfree = 5)

2x2 MIMO - SDM2X2 MIMO - GC

Figure 2.7: Simulation results for a packet of 1 ko: SDM vs the Golden code in a 2×2 BICMsystem for a QPSK modulation with [5 7] encoder (dfree = 5)

2 4 6 8 10 12 14 1610

-4

10-3

10-2

10-1

100

SNR (dB)

PE

R

QPSK - CC = [133 171] , 1/2 (dfree = 10)

2x2 MIMO - SDM2x2 MIMO - GC

Figure 2.8: Simulation results for a packet of 1 ko: SDM vs the Golden code in a 2×2 BICMsystem for a QPSK modulation with [133 171] encoder (dfree = 10)

58

2.5. BICM-MIMO WITH FLAT FADING CHANNEL

Proof: When perfect space time codes are used, the above equation (2.57) in the general

PEP expression can be simplified∑k′,dfree

‖HC(k)‖2F ,∑k′,dfree

Tr

H(C(k)C(k)†

)H†,

(a)= Tr

∑k′,dfree

H(C(k)C(k)†

)H†,

= Tr

H∑k′,dfree

(C(k)C(k)†

)H†.

In the following, A denotes the nt × nt matrix such that,

A =∑k′,dfree

C(k)C(k)† = CC†,

where

C = [C(k1) . . .C(kdfree)].

Let A = UΛU† be the eigen-value decomposition of A, then∑k′,dfree

‖HC(k)‖2F = Tr

(HU)Λ(HU)†,

=

nr∑i=1

nt∑j=1

λj(A)βi,j ,

where βi,j are i.i.d random variable resulting from the multiplication of H by the unitary

matrix U and λj are the non zero eigen-values of A.

By averaging over all the βi,j variables which are Gaussian distributed as shown in [9] , the

PEP is bounded as,

P(c→ c) ≤

nt∏j=1

λj(A)

−nr ( 1

4N0

)−ntnr. (2.60)

As all matrices C(k)C(k)† are positive definite matrices, the minimum determinant can be

bounded such that

∆min = detAA† ≥dfree∑k′=1

detC(k)C(k)†.

From Theorem 2.7, we know that C(ki) matrices are non-zero matrices and therefore,

∆min ≥ dfree∆cmin = dfreeδ(dmin)2nt ,

where ∆cmin is the minimal determinant of the code and dmin is the minimal Euclidian disc-

tance between two symbols (dmin = 2√Es

). It is clear that the convolutional code over a flat

59


10 15 20 25 30

10-10

10-8

10-6

10-4

10-2

100

SNR

PE

PAsymptotical PEP behavior

Non coded GCNon coded SDMCoded GC - dfree = 10Coded SDM - dfree = 10Coded GC - dfree = 5Coded GC - dfree = 5

Intersection point

Intersectionpoint

CC-GC gain

CC-SDM gain

Figure 2.9: Asymptotical behavior of the PEP over a flat fading channel

fading channel improves the coding gain. Assuming that the energy per coded symbol at

each antenna and on eachtime slot is equal to one, the PEP is bounded by

PEP ≤(dfreeδ

)−nr(Es)ntnrSNR−ntnr . (2.61)

Notice here that the main difference between [51,52], that using a combination of convo-

lutional code and perfect space time codewords guarantees that all C(ki) are non zero and

therefore the coding gain is enhanced automatically, which is not the case in [51] and [52].

Combining convolutional code with perfect codewords can be another alternative that could

be investigated to enhance the coding instead of performing set partitioning at the mapper

as shown in [51,52].

Perfect space time codes versus SDM

When no space time code is used (SDM), the PEP can be bounded following the same steps

as for the perfect code case. This implies that,

PEP ≤(dfree

)−nr(Es)nrSNR−nr . (2.62)

In Figure 2.9, the asymptotical behavior of SDM versus coded GC is depicted for the 2×2

MIMO configuration using QPSK constellation. The coding gain of SDM is largely enhanced

compared to the Golden code coding gain specially when a convolutional code with a large

60

2.6. BICM-MIMO WITH FREQUENCY SELECTIVE CHANNELS

free distance is used. At low SNR, this coding gain dominates the high diversity gain that

can be achieved by the GC. However, at high SNR, the diversity gain became dominant.

As shown in Figure 2.7 for the 2×2 MIMO flat fading channel and a convolutional code [5 7]

- 1/2 with dfree = 5, the gain provided by the full diversity of the GC dominates the coding

gain for low SNR as well as for the high SNR order. A gain of 1.9 dB is obtained for a PER

of 10−2. However, when using a convolutional code with a large free distance, e.g [133 171]

- 1/2, with dfree = 10, as shown in Figure 2.8 the coding gain dominates the diversity gain.

The impact of the additional diversity cannot be observed at a moderate range of PER.

2.6 BICM-MIMO with frequency selective channels

We consider a cyclic prefix MIMO OFDM system with L-taps and N frequency slots per

OFDM symbols. The main result of this section is summarized in the following theorem.

Theorem 2.9 For a MIMO-BICM system with frequency selective fading channel, perfect

space time code used over subcarriers allows to extract the full diversity order of ntnr min(L,D)

and the PEP is upperbounded by,

PEP ≤(α δ)−nrdfree(Es)

ntnr min(L,D)(σ2L−1SNR

)−nrnt min(L,D), (2.63)

where D ≤ dfree denotes the number of different subcarriers on which erroneous bits are

received. The interleaver design allows to maximize the parameter D. The parameter α is a

constant that depends on the covariance matrix.

Before going to the proof, we note that if an ideal interleaver and a convolutional code

with a free distance dfree = L = D in a BICM-MIMO system are assumed, it is sufficient

to use a perfect codeword over subcarriers rather than using a global N -parallel NVD space

time code over the N carriers to extract the channel diversity, to minimize the coding gain

and to fulfill the NVD criteria.

Proof: The expression in equation (2.57) can be simplified as following∑k′,dfree

‖H(k)C(k)‖2F = TrH CC† H†

(2.64)

where H is the equivalent channel matrix over dfree frequency slots.

H =[H(f1) . . . H(fdfree

)]

(2.65)

and

CC† = diag

C(fi)C(fi)†dfree

i=1

61


Lemma 2.3 (Equivalent channel matrix) The equivalent channel matrix H can be ex-

pressed as,

H = Hw

(V ⊗ Int

), (2.66)

where Hw is the nr × Lnt i.i.d CN (0, 1) such that,

Hw =[H0 . . . HL−1

],

w = e−j2πN , and,

V =

σ0 σ0 . . . σ0

σ1wf1 σ1w

f2 . . . σ1wfD

...

σL−1w(L−1)f1 σL−1w

(L−1)f2 . . . σL−1w(L−1)fD

. (2.67)

Proof: As shown in chapter 1, (1.30), each hi,j(k) term can be written such that,

hi,j(k) =

L−1∑l=0

wk,lσlhi,jl , k = 1 . . . N, i = 1 . . . nt, j = 1 . . . nr.

where wk,l = wkl and w = e−j2πN . The corresponding channel matrix over a subcarrier k, is

therefore

H(k) =L−1∑l=0

wk,lσlHl. (2.68)

Using (2.68), it can be easily checked that the equivalent channel matrix in (2.65) can be

written [H(f1) . . . H(fdfree

)]

=[H0 . . . HL−1

](V ⊗ Int

)As in Subection 2.3, let

Θ =(VL×dfree

⊗ Int)CC†

(VL×dfree

⊗ Int)†, (2.69)

be the effective matrix codeword. In the following, D denotes the number of different sub-

carriers on which erroneous bits are received, such that D ≤ dfree. The interleaver design

allows to maximize the parameter D. The interleaver design is not addressed in this thesis.

The interested reader can refer to [25] for suboptimal interleaver design.

62


Equation (2.64) can be simplified using the eigen-value decomposition of Θ = UΛU†, to∑k′,dfree

‖H(k)C(k)‖2F = Tr

(HωU)Λ(HωU)†

=

nr∑i=1

r∑j=1

λj(Θ)βi,j

where r is the rank of Θ, βi,j are i.i.d random variable resulting from the multiplication of

H by the unitary matrix U and λj(Θ) are the non zeros eigen-values of Θ.

The rank and the eigen-values of Θ can be deduced from the following Lemma 2.4.

Lemma 2.4 Let A be the M ×M Hermitian matrix given by,

A = B(CC†)B†,

where B is M ×N matrix having D different columns such that rank r ≤ min(M,D) and C

is full rank M ×M matrix. Then, the matrix A has the following property.

a) The rank of A is equal to r, the rank of B.

b) Let λk(A) and λk(B) be respectively the decreasing ordered eigen-values of A and B,

such that

λ1(A) ≥ . . . ≥ λr(A) ≥ 0 = λr+1(A) = . . . = λM (A),

λ1(B) ≥ . . . ≥ λr(B) ≥ 0 = λr+1(B) = . . . = λM (B),

then the non zero eigen-values of A can be lower bounded by,

λk(A) ≥ λr(BB†)λk(CC†), k = 1 . . . r, If M ≥ N, (2.70)

λk(A) ≥ λr(BB†)λN−M+k(CC†), k = 1 . . . r, If N ≥M. (2.71)

Proof: The proof of Lemma 2.4 is given in appendix 2.A.2.

By applying Lemma 2.4 to Θ, it follows that

rankΘ = rankV ⊗ Int,= rankVnt.

The rank of V can be deduced by considering only the D different subcarriers in V. These

columns form a linear independent family of rank min(L,D) as the equivalent L×D matrix

can be assimilated to a submatrix of a Vamdermonde matrix, and therefore the rank is equal

63


min(L,D). This implies that,

r = rankΘ = nt min(L,D).

Using Lemma 2.4, the eigen-values are bounded such that,

λj(Θ) ≥ αλm(CC†),

where

m =

j if L ≥ dfree

dfree − L+ j if L ≤ dfree,(2.72)

and α = λr(VV† ⊗ Int). By averaging over all the Gaussian variable βi,j as described in [9],

the PEP over a frequency selective channel is bounded by

P(c→ c) ≤

nt min(L,D)∏j=1

λj(Θ)

−nr ( 1

4N0

)−nrnt min(L,D)

.

The product of the non zero-eigen-values can be lowerbounded such that,

nt min(L,D)∏j=1

λj(Θ) ≥ (dmin)2nt min(L,D)

nt min(L,D)∏j=1

µj(Θ),

≥ 4nt min(L,D)(Es)−nt min(L,D)αnt min(L,D)

nt min(L,D)∏j=1

µm(CC†),

where dmin is the minimal Euclidian distance between two symbols (dmin = 2√Es

), µm(Θ) are

the normalized eigen-values andm = f(j) as defined in equation 2.72. From the normalization

power constraint with a total energy equals to 1 per antenna at each time slot as shown in

2.8, we know that

µm(C(k)C†(k)) ≤ 1, ∀k = 1 . . . N, j = 1 . . . nt min(L,D),

where C(k) denotes the matrix codeword over a subcarrier k. As CC† is a block diagonal

matrix containing C(k)C(k)†, k = f1 . . . fdfree, then the eigen-values of CC are trivially equals

to the eigen-value of C(k)C(k)†, for all k = f1 . . . fdfree, and then,

µj(CC†) ≤ 1, ∀j = 1 . . . dfreent.

64


The product of eigen-values can be therefore bounded as,

nt min(L,D)∏j=1

λj(Θ) ≥ 4nt min(L,D)(Es)−nt min(L,D)αnt min(L,D)

ntdfree∏j=1

µj(CC†),

≥ 4nt min(L,D)(Es)−nt min(L,D)αnt min(L,D)δdfree .

Finally, the PEP can be bounded such that,

P(c→ c) ≤(α δ)−nrdfree(Es)

ntnr min(L,D)SNR−nrnt min(L,D). (2.73)

Coded versus no coded selective fading channel

For the selective fading channel, the diversity order is maximized in the pairwise error prob-

ability expression if the whole diagonal block matrix is different of zero, meaning that the

erroneous block matrix is a full rank matrix. When no outer code is used and symbols are

coded independently on each subcarriers and without coding across the blocks, then the er-

roneous codeword as shown in Figure 2.10(a) can contain at least one block equal to zero.

However, when coding across the blocks, all the blocks are different of zero as shown in Figure

2.10(b). Coding only symbols on each subcarriers can be optimal if it is used in a BICM-

MIMO-OFDM system. In this case, using convolutional code guarantees that all the blocks

in the erroneous coded codeword in Figure 2.10(c) are non equal to zero.

0

0 Zero block

Non zero block

(a) (b) (c)

Erroneous uncoded codeword Erroneous coded codeword

N ×N N ×N dfree × dfree

Figure 2.10: (a) Coding only on each subcarrier without outer code (b) Coding across theblocks without outer code (c) Coding only on each subcarrier in a BICM-MIMO system. Thedashed block denotes a non zero-block, the non dashed one refers to a zero-block.

Perfect space time code versus SDM

Similar to the case with perfect space time coding, the PEP expression can be derived for the

SDM case. When no space time coding is used, the minimal rank of the effective codeword

65


Θ in (2.69) arises when all the erroneous codewords at the dfree subcarriers are received on

the same antenna 6. This implies that, C is proportional to IN ⊗C(k), where C(k) is a non

zero SDM codeword received on an arbitrary subcarrier k, with rank equals to 1 for the SDM

case. Therefore, the rank of the effective codeword matrix is equal, to

rankΘ = rank(V ⊗ Int)(ID ⊗C(k)),= rankV ⊗C(k) = min(L,D).

The eigen-values of Θ can be computed in a similar way as in Lemma 2.4. We emphasize

here that the rank of C in Lemma 2.4 does not impact the eigen-values bound, but impacts

only the rank of A. The proof can be easily repeated for the case where C is not full rank.

The PEP for the SDM is therefore bounded such that,

P(c→ c) ≤(α)−nrdfree(Es)

nr min(L,D)SNR−nr min(L,D), (2.74)

where α depends on the covariance matrix.

Numerical results

In Figure 2.11, the asymptotical behavior of SDM versus coded GC is depicted for the 2× 2

MIMO configuration using QPSK constellation, and a convolutional code [57] − 1/2, with

dfree = 5, and a over a multi-tap channel with L = 18. It can be observed that at low SNR,

the coding gain of SDM is largely enhanced compared to the Golden code coding gain. At

low SNR, this coding gain dominates the high diversity gain that can be achieved by the GC.

Both schemes gain in diversity compared to the non coded case. At high SNR, the diversity

gain became dominant. The diversity gain can be observed at a very low PER rate (range of

10−7).

The performance of the Golden code versus SDM has been evaluated in the IEEE context

in terms of packet error rate (PER) versus SNR, for a packet length of 1000-bits. In the

following, SNR gain will be related to a PER of 10−2. The packet error rates in Figure 2.12

are evaluated over channel D using QPSK constellation. The channel D is characterized by a

50ns rms delay spread and 18 taps, and then by significant frequency diversity. In the IEEE

802.11n context, the convolutional code [133 171] with a coding rate of Rc = 1/2 is used with

dfree = 10. No additional gain is observed at a PER = 10−2. This behavior of Golden code

compared to SDM have been also observed in [53].

Practical limits of space time codes use in a standard context

Recent standards that use MIMO system such that IEEE 802.11n and IEEE 802.16e aim

to increase the throughput and the reliability of the system. However, increasing the relia-

bility comes often at the expense of increased complexity at both transmitter and receiver

6This observation is due to the structure of the effective codeword in (2.69).

66


6 8 10 12 14 16 18 20

10-25

10-20

10-15

10-10

10-5

100

SNR

PE

P

Asymptotical PEP behavior

Non coded SDM

Non coded GC

coded OFDM-GC

coded OFDM-SDM

Intersectionpoint

Figure 2.11: Asymptotical behavior of the PEP over a frequency selective channel

10-4

10-3

10-2

10-1

100

2 4 6 8 10 12 14 16 18 20 22 24

PE

R

Eb/N0(dB)

Error Probability of GC vs SDM in IEEE 802.11n

Channel D - QPSK 1/2 - SDMChannel D - QPSK 1/2 - GCChannel D - 16QAM 3/4 - SDMChannel D - 16 QAM 3/4 - GCChannel B - 16QAM 3/4 - SDMChannel B - 16QAM 3/4 - GC

Figure 2.12: Golden Code vs SDM in IEEE 802.11n context

67


side. Scarifying the complexity order can be done if promising gains at reasonable PER

range are observed. Although, the theoretical optimality of these codes is studied in the high

SNR regime, practical assumptions are more realistic, and address generally a moderate SNR

regime and moderate range of PER.

For the flat fading case, when no outer code is used, the huge gain observed by the

Golden code over all other known code make it promising to be used in such systems. How-

ever, when a complete chain as BICM-MIMO-OFDM system is considered, the situation

become considerably different. As we show in a BICM-MIMO-OFDM system, the diversity

of BICM-OFDM system can be extracted when no space time code is used. Additional diver-

sity can be provided by using perfect space time codes over each subcarriers. This additional

diversity comes at the expense of an increased lattice decoder, i.e instead of using a 2nt×2nt

ML soft decoder a 2n2t × 2n2

t is required. Moreover, the impact of this additional diversity

cannot be unfortunately observed at moderate range of PER.

2.7 Conclusion

In this chapter, we studied the performance of non-vanishing determinant code constructed

from cyclic division algebra in a standard context. We propose for the selective fading chan-

nel a new family of split NVD parallel codes to achieve the optimal DMT. One of the main

hindrance to the practical implementation of such code is the high complexity order required

at the receiver side. We show that more feasible schemes based on coding the symbols inde-

pendently across the frequencies and without coding across blocks are optimal in a MIMO

BICM system.

The PEP is derived for BICM-MIMO for the cases of flat fading and frequency selective

channels when respectively perfect codes and spatial division multiplexing schemes are used

at each subcarrier. When the channel is flat, we show that the diversity order remains the

same as the non-coded case. However, the coding gain is improved compared to the non

coded case. We noticed that the coding gain of the SDM is largely enhanced compared to the

coding gain of perfect code especially for a convolutional with a large free distance. For the

frequency selective channel case, we show that using perfect codes at each subcarrier allow

to extract the transmit diversity. We show then that the diversity order of BICM-OFDM

system can be also extracted when no space time code is used. In a practical context, such

as in IEEE 802.11n, the numerical results we provide show that this gain in diversity cannot

be observed at a moderate range of PER.

68

2.A. APPENDICES

2.A Appendices

2.A.1 Proof of Lemma 2.2

It can be easily checked from the product matrix rank property (D ∈ Cm×k,E ∈ Ck×n),

rankD+ rankE − k ≤ rankDE ≤ min

rankD, rankE, (2.75)

As CC† is a full rank matrix, this implies that,

rankB+M −M ≤ rankA ≤ rankB,

and therefore,

rankA = rankB.

Using the fact that for a square matrix M ∈ CM ,

λ(MM†) = λ(M†M), (2.76)

implies that

λk(A) = λk(C†B†BC).

Let B†B = UΛU† be the eigen-value decomposition of B†B, with Λ = [Λ 0M−r]. Then,

λk(A) = λk(C†UΛU†C), (2.77a)

= λk(Λ1/2U†CC†UΛ1/2), (2.77b)

where (2.77b) follows from using the matrix property in (2.76). Let

Ω = U†(CC†)U

and Ω be the r × r principal submatrix of Ω. Then,

λk(A) = λk(Λ1/2ΩΛ1/2), (2.77c)

= λk(Λ1/2

ΩΛ1/2

), (2.77d)

As Λ1/2

in (2.77d) is non singular matrix and Ω is Hermitian, The Ostrowski theorem in [54]

can be applied,

λk(A) ≥ λr(Λ)λk(Ω), (2.77e)

≥ λr(B†B)λk(Ω), (2.77f)

= λr(BB†)λk(CC†). (2.77g)

69


As Ω is a r× r submatrix of the Hermitian matrix Ω, (2.77f) follows from the application of

theorem 4.3.15 in [54]. Finally, (2.77g) follows from the fact that U is unitary matrix, and

therefore,

λk(Ω) = λk(CC†).

2.A.2 Proof of Lemma 2.4

To simplify the notations, we denote in the following by λ(.) the eigen-values ordered in

increasing order, and λ(.) the eigen-values ordered in a decreasing order. The rank of A can

be easily deduced as shown in the proof of Lemma 2.2, and then,

r = rankA = rankB.

Case 1 : M ≥ N

Let AN be the N ×N principal submatrix of the M ×M matrix A, such that

AN = BNCC†B†N

where BN = B([1 : N ], [1 : N ]) is N × N minor of matrix B with rank r ≤ N . Assuming

that, the increasing eigen-values of AN and A are such that,

λ1(AN ) = . . . = λN−r(AN ) ≤ λN−r+1(AN ) . . . ≤ λN (AN )

and

λ1(A) = . . . = λM−N (A) = . . . = λM−r(A) = 0 ≤ λM−r+1(A) ≤ . . . ≤ λM (A)

Then, using theorem theorem 4.3.15 in [54],

λk(AN ) ≤ λk+M−N (A), k = 1 . . . N (2.78)

The non-zero eigen-values of A can be bounded by taking k = N−r+1 . . . N in the inequality

2.78, or equivalently

λM−r+i(A) ≥ λN−r+i(AN ), i = 1 . . . r,

and in term of the decreasing order eigen-values,

λk(A) ≥ λk(AN ), k = 1 . . . r.

As AN is a square matrix, the eigen-values of AN can be lower bounded using Lemma 2.4,

and therefore,

λk(A) ≥ λk(AN ) ≥ λr(BNB†N )λk(CC†), k = 1 . . . r.

70

2.A. APPENDICES

The matrix BNB†N is a N ×N submatrix of the Hermitian matrix BB†, and therefore,

λr(BNB†N ) ≥ λr(BB†),

and therefore,

λk(A) ≥ λr(BB†)λk(CC†), k = 1 . . . r.

Case 2 : M ≤ N

In this case, A is a M ×M principal submatrix of the N ×N matrix AN with rank r, such

that

AN = BNCC†B†N

and BN is a N × N matrix constructed such that B is its M × N minor and BN has rank

N −M + r. Assuming that, the increasing eigen-values of AN and A are such that,

λ1(AN ) = . . . = λM−r(AN ) = 0 ≤ λM−r+1(AN ) . . . ≤ λN (AN ),

and

λ1(A) = . . . = λM−r(A) = 0 ≤ λM−r+1(A) . . . ≤ λM (A).

Then, using theorem 4.3.15 in [54],

λk(A) ≥ λk(AN ), k = 1 . . . r, (2.79)

and in term of the decreasing order eigen-values,

λk(A) ≥ λN−M+k(AN ), k = 1 . . . r.

As AN is a square matrix, the eigen-values of AN can be lower bounded using Lemma 2.4,

and therefore,

λk(A) ≥ λN−M+k(AN ) ≥ λr(BNB†N )λN−M+k(CC†), k = 1 . . . r.

The matrix BNB†N is a N ×N submatrix of the Hermitian matrix BB†, and therefore,

λr(BNB†N ) ≥ λr(BB†),

and therefore,

λk(A) ≥ λr(BB†)λN−M+k(CC†), k = 1 . . . r.

71


72

Chapter 3

Interference Alignment for

Selective Fading Channel

DEALING with interference in wireless adhoc networks has received lots of attention

recently. While traditional approaches based on orthogonalization suffer from the

lack of degrees of freedom in the system, more developed approaches based on

interference alignment schemes allow to extract all the available degrees of freedom per user.

The interference alignment(IA) scheme proposed by Cadambe and Jafar [3] depends critically

on the assumption that all channels in the network are time selective. This has been later

extended to the case of the frequency selective channel by Grokop and Tse [15].

Generally, real communication scenarios occur on channels that are selective both in time

and frequency. We show that under certain channel spread restrictions, the IA allows to

extract all the available degrees over a time-frequency selective fading interference channel

for both cases of finite and large scaling network. The practical implementation of IA is also

addressed; we show that using a random vector quantization scheme with an adequate number

of bits, perfect knowledge of selective fading channel can be relaxed to a quantized channel

knowledge at all nodes, while conserving the full multiplexing gain that can be achieved will

full CSI.

3.1 Introduction and motivation

In Chapter 2, the point-to-point MIMO channel has been considered. It is well known that

in this case, the joint ML (maximum likelihood) decoder is the optimal decoder that allows

to eliminate data-streams’ interference at the receiver side. However, for the case of single

73

CHAPTER 3. INTERFERENCE ALIGNMENT FOR SELECTIVE FADING CHANNELS

input single output interference channel, the situation is considerably different.

In this chapter, we shift the focus to the case of K-SISO interference channel. This sce-

nario describes the shared medium between K pairs of sources and destinations that want

to communicate reliably. Recently, lots of efforts have been investigated to characterize the

capacity region of this channel [16, 17] leading to an outer bound without giving an exact

characterization of this capacity region. However, for the high SNR analysis, the outer bound

on the sum capacity is known from the preliminary results in [18] of Host-Maden and Nos-

tarnia where it was shown that the spatial multiplexing gain is upper-bounded by K/2.

Recently, Cadambe and Jafar in [3] propose an innovative scheme called interference align-

ment for time-varying interference channel that allows to extract K/2 degrees of freedom.

The strong implication of this result is that even when more than two interfering users are

considered, the sum capacity that can be achieved per user is (1/2) log SNR, i.e., every one

take the the half of the cake, in the Cadambe Jafar terminology. The interference align-

ment(IA) scheme proposed by Cadambe and Jafar [3] depends critically on the assumption

that all channels in the network are time selective to construct precoders such that non-

desired signals are perfectly or partially aligned at the receiver side.

More recently, Grokop and Tse in [15] extend the IA to the frequency selective channel. In

both schemes, perfect channel knowledge is required at all nodes. However, the assumption of

perfect channel knowledge at all nodes, makes IA scheme very complex when implemented in

a practical system due to the large amount of required feedback. For the frequency selective

case, Thukral and Bolcskei in [55] show that the perfect knowledge assumption can be relaxed

to a partial channel knowledge at all nodes, while conserving the full multiplexing gain that

can be achieved will full CSI.

Contributions: In this chapter, we consider the case where all channels are selective in

both time and frequency. We show that under certain channel spread restrictions, the IA

allows to extract all the available degrees over a time-frequency selective fading interference

channel for both cases of finite and large scaling network. Moreover, we propose a lim-

ited feedback IA scheme based on random vector quantization (RVQ) of the selective fading

channel, that allows also to achieve the full multiplexing gain of the network. Our results

are based on the polynomial channel decomposition that we derive for the time-frequency

selective channel model when the number of time-frequency slots is very large.

Outline: The rest of the chapter is organized as follows. In Section 3.2, we provide

some background materials on the time-frequency selective channel model and define the

corresponding input-output relation at each destination for the K-SISO interference channel.

The channels spread requirement for finite SISO interference channel and for the large scaling

network are derived in Section 3.5. Then, the practical implementation of IA with partial

channel knowledge at Tx side is addressed in Section 3.6. Finally, Section 3.7 concludes this

74

3.2. SYSTEM AND CHANNEL MODEL

chapter.

3.2 System and channel model

We consider the K single antenna users interference channel where K pairs of sources and

destinations (Si, Di), i = 1 . . .K coupled randomly want to communicate. Each source Si

wants to communicate an independent message to its corresponding destination Di, and

induces interference at all the other destinations Dj , j 6= i. We assume that all channels

Si → Dj in the network are both selective in time and frequency. In the rest of the section,

we first recall the time-frequency selective channel model over the SISO links Si → Dj as

described in Chapter 1. Then, the corresponding input-output relation at each destination

for the K-SISO interference channel is provided.

S1

S2

SK

DK

D2

D1

Figure 3.1: A SISO interference network with K sources and destinations nodes.

We consider the case when fading processes corresponding to all channels are characterized

by non-disjoint scattering function C[i,j]H (ν, τ), such that

C[i,j]H (ν, τ) = 0 for (ν, τ) /∈

]0,+ν

[i,j]0

]×]0,+τ

[i,j]0

].

By choosing the sampling period T , and sampling frequency F such that T ≤ 1/ν0 and

F ≤ 1/τ0 where

ν0 = max1≤i,j≤K

ν[i,j]0 , and τ0 = max

1≤i,j≤Kτ

[i,j]0 ,

channels can be diagonalized using the same Weyl-Heisenberg sets. In this case, as shown in

Chapter 1, the channel spread should satisfy ∆H ≤ 1TF . The corresponding I/O relation at

75


each destination i is given by

yi(n) = h[i,i](n)xi(n) +∑k 6=i

h[i,k](n)xk(n) + zi(n) n = 0, 1, . . . , N − 1

where n denotes the time-frequency slot (m, l), such that m = 1 . . .M and l = 1 . . . Nc where

M is the total number of OFDM symbols, Nc the total number of subcarriers and N = MNc

is the total number of time frequency slots.

The channel matrix model can be written such that

yi =K∑k=1

H[i,k]xk + zi, i = 1 . . . ,K

where

H[i,k] = diagh[i,k](0), h[i,k](1), . . . , h[i,k](N − 1)

,

h[i,k](n) = L[i,k]H (mT, lF ), n corresponds to the time-frequency slot (m, l) and yi ∈ CN×1 is

the received signal. xk denotes the precoded signal such that

xk = V[k]xk

and V[k] is the linear N×dk precoding matrix, and xk is the dk×1 data vector. The precoded

signal transmitted by each source satisfies the following power constraint,

E[|xk(n)|2] ≤ P

K, k = 1 . . .K, and, n = 0 . . . N − 1.

Lemma 3.1 (Polynomial channel decomposition) The selective fading channels H[i,j]

can be decomposed according to Theorem 1.4 developed in Chapter 1 such as,

H[i,j] =∑p,q∈A

λ[i,j](p,q)W(p,q) (3.1)

where

W(p,q) = ZpM ⊗ ZqNc , (3.2)

A =

(p, q) : p ∈ 0, . . . , p[i,j]0 − 1, q ∈ 0, . . . , q[i,j]

0 − 1, (3.3)

p[i,j]0 = bν[i,j]

0 TMc, q[i,j]0 = bτ [i,j]

0 FNcc, (3.4)

The coefficients λ[i,j](p,q) are i.i.d random variable drawn from a continuous Gaussian distribu-

tion CN (0, σ2,[i,j](p,q) ), such that

σ2,[i,j](p,q) =

1

TFNC

[i,j]H

(p

TM,q

FNc

).

76

3.3. MULTIPLEXING GAIN OF THE K-SISO INTERFERENCE CHANNEL

Matrix ZM is such that

ZM = diag1, ωM , . . . , ωM−1M where ωM = ej

2πM .

and ZNc is given by

ZNc = diag1, ωNc , . . . , ωNc−1Nc

where ωNc = e−j2πNc .

Note that if all the channels have the same scattering function, then we denote by p0 =

p[i,j]0 , and q0 = q

[i,j]0 .

Decomposing the time-frequency channel using the polynomial form in Theorem 1.4 has

two main implications:

- It can be easily noticed that the diagonal channel matrix ZpM and the (right)-cyclic

shift matrix

Sp =

[0 Ip

IM−p 0

]are similar as there exists a change basis such that Sp = F ZpN F† with F is the N ×NFFT matrix. Similarly, Zq and the (left)-cyclic shift matrix

Sq =

[0 INc−qIq 0

]

are also similar.

The first remark that can be deduced from this channel decomposition is that the

analysis in time and frequency can be done separately as the shifts in time and in

frequency are simply separated by a kronecker product.

- The second remark that can be observed is that at the transmitter side the channel can

be reconstructed using a finite number of independent parameters that do not exceed

the duration of the transmission |A| = bτ0FMcbν0TNcc ≤ MNc = N due the choice

of the WH grid parameter in (1.21). The channel spread condition due to the grid

parameter choice in (1.21)

∆H ≤1

TF,

is therefore necessary and sufficient to identify the channel.

3.3 Multiplexing gain of the K-SISO interference channel

In this section, we briefly review the outerbound on the total spatial multiplexing that has

been derived in [18] and extended later to the time varying channel in [3] for the sake of

completeness.

Theorem 3.1 (Theorem 1 in [3]) For the K-user SISO interference channel, the total

77


number of spatial multiplexing gain is upper bounded by K/2, i.e.,

r1 + r2 + . . .+ rK ≤ K/2. (3.5)

where ri denotes the multiplexing gain per user.

Proof: The proof of this upperbound is based on the assumption that (a) a reliable

coding scheme is used for the K-user interference channel, and (b) only two users i and j are

active. Then, the K-user interference channel is equivalent to a two user interference channel.

The proof can be immediately deduced from the following steps depicted in Figure 3.2.

(a) (b) (c)

(d)

Degraded BC

MAC channel

Wi

Reliable decoding of Wi

Wi

Reliable decoding of Wj

Wi

Wj

Wj

Wj

Wi

Wi

Wj

Figure 3.2: (a) Message Wi is provided to Rx j. (b) Using a reliable coding scheme, Rx idecodes successfully Wi.(c) Equivalent degraded BC. (d) MAC channel.

In (a), Receiver j is a cognitive receiver. This implies that Rx j has complete knowledge of

the channel, and therefore the dashed link can be removed between Tx i and Rx j. Assuming

a reliable coding scheme, Rx i can decode successfully message Wi in (b), and the dashed

link between Tx i and Rx i can be also removed. The noisy message Wj is received by Rx

i and j in (c). As reliable coding scheme is used, Rx j decodes successfully this message.

Assuming that IC can be made equivalent to a degraded BC by prioritizing data transmitted

to Rx i(increasing the power). Then, Rx i can decode what ever Rx j can decode and hence

Wj . Rx i is able to decode messages Wi and Wj in (d), which is a MAC scheme. Rates must

lie in the capacity region of the MAC channel from Tx i, j and Rx i This implies that the

total multiplexing gain r1 + r2 cannot exceed 1 for a SISO MAC channel. Therefore,

ri + rj = limP→∞

Ri(P ) +Rj(P )

log2 P≤ 1. (3.6)

78

3.4. TIME FREQUENCY DOMAIN INTERPRETATION

By simply adding all the inequalities in (3.6) implies

r1 + . . .+ rK ≤ K/2.

3.4 Time frequency domain interpretation

3.4.1 Interference Alignment Concept

The interference alignment scheme proposed by Cadambe and Jafar [3] extracts the full

multiplexing gain in an interference channel. It consists simply to receive all the interfering

signal in the same subspace at each destination, and the desired signal on a different subspace.

The strong implication of using such schemes is that each user will be able to communicate

during the half of the shared medium.

3.4.2 Toy Example: 3 Users Interference Channel

In order to give a toy example for interference alignment in the time-frequency domain,

we consider the three users case where channels induce only a single shift in time p[i,j] =

bτ [i,j]0 FNcc and frequency q[i,j] = bν[i,j]

0 TMc, such that

CH(ν, τ) = CH(ν0, τ0)δ(τ − τ0)δ(ν − ν0).

By applying Lemma 3.1 , this implies that,

H[i,j] = λ[i,j](Zpi,jM ⊗ Z

qi,jNc

).

At each destination Di, the received signal is given by,

yi = H[i,i]V[i]xi +∑j 6=i

H[i,j]V[j]xj + ni.

The linear precoder that insures interference alignment should be constructed such that

V[i] =(ZpiM ⊗ INc

)(F⊗G)†,

H[i,j]V[j] = λ[i,j](Zpi,j+piM ⊗ Z

qi,jNc

)(F⊗G

)†,

where F and G are respectively the M ×M and Nc ×Nc FFT matrices.

The received signal should be equalized using a preprocessor matrix P = F ⊗G, such

79


that

PH[i,j]V[j] = λ[i,j](F⊗G

)(Zpi,j+piM ⊗ Z

qi,jNc

)(F⊗G

)†= λ[i,j]

[ (F Z

pi,j+piM F†

)︸︷︷︸shift in time

⊗(G Z

qi,jNc

G†)︸︷︷︸

shift in frequency

]

The above precoding and pre-processing entails a shift in both time and frequency. The

OFDM symbols are shifted by a time shift of pi,j + pi, and subcarriers of the same OFDM

symbol are shifted by qi,j . As shown in Figure 3.3, the signaling scheme is equivalent to a

block of M OFDM symbols having Nc subcarriers each. The post and pre-processing allow

1

Nc

l

1 m M

Figure 3.3: The signaling scheme is a equivalent to a block of M = 17 OFDM symbols,having Nc = 7 subcarriers each.

to shift in time OFDM symbols by pi,j + pi. If the OFDM symbol is interference free, then

the data over the Nc subcarriers are also interference free. At destination to say D1, signals

received from S2 and S3 should overlap. However, the desired signal should not be aligned

with interference. For the selective fading channel, shift is performed in time as shown in the

following.

Time shift

The undesired signals at receivers 1, 2 and 3 are aligned if and only if the following conditions

are satisfied,

p2 + p1,2 = p3 + p1,3,

p1 + p2,1 = p3 + p2,3,

p1 + p3,1 = p2 + p3,2.

80


This implies that

p1 − p3 = p2,3 − p2,1,

p1 − p3 = p3,2 − p3,1 + p1,3 − p1,2,

The above equation system is feasible in Z/pZ, where

p = (p2,3 − p2,1)− (p3,2 − p3,1 + p1,3 − p1,2), (3.7)

which implies that

p1 ≡ p3,2 − p3,1 (mod p)

p2 ≡ 0 (mod p),

p3 ≡ p1,2 − p1,3 (mod p),

or equivalently, ∃k1, k2, k3 such that

p1 = k1p+ p1, p1 = p3,2 − p3,1, (3.8a)

p2 = k2p+ p2, p2 = 0, (3.8b)

p3 = k3p+ p3, p3 = p1,2 − p1,3, (3.8c)

where k1, k2, k3 ∈ [0,m] and m + 1 ≤ M denotes the maximal number of OFDM symbols

that can be transmitted free from interference. From (3.8a), (3.8b) and (3.8c), we can note

that the interference alignment scheme is feasible if the cyclic shift p1, p2 and p3 is applied

to the position kp (mod M). The precoding processing is performed in two steps:

- The first precoding step consists therefore in converting the position of the OFDM

symbols from k ∈ [0,m] to kp (mod M).

- The second step adds a cyclic shift in time to the new OFDM symbol’s position.

The equivalent precoder V[i] can be therefore written as,

V[i] =(ZpiM ⊗ INc

)(F⊗G)†(S⊗ INc), (3.9)

with S = (si,j)1≤i,j≤M , such that,

si,j =

1 if (i, j) =(kp (mod M) , k

), k = 0 . . .m,

0 otherwise.

The position of shifted OFDM symbols at destination Di is given by pi+pi,j (mod M). Note

that, M should be a prime number to avoid symbols to overlap. As the desired signal should

not be aligned with the interference, the direct doppler pi,i spread should satisfy the following

81


conditions independently of the delay spread,

p1,1 + p1 (mod M) 6= p2 + p1,2 (mod M)

p2,2 + p2 (mod M) 6= p1 + p2,1 (mod M)

p3,3 + p3 (mod M) 6= p2 + p3,2 (mod M)

By replacing p1, p2 and p3 by their values in (3.8a), (3.8b) and (3.8c), it follows that,

p1,1 6= k2p+ p1,2 − (k1p+ p3,2 − p3,1) (mod M)

p2,2 6= k1p+ p3,2 − p3,1 + p2,1 − k2p (mod M)

p3,3 6= k2p+ p3,2 − k3p+ p1,2 − p1,3 (mod M)

Let c1 = k2− k1, c2 = k1− k2 and c3 = k2− k3. As k1, k2, k3 ∈ [0,m], then it is to check that

c1, c2, c3 ∈ [−m,m]. Therefore,

p1,1 6= c1p+ p1,2 − p3,2 + p3,1 (mod M), (3.10)

p2,2 6= c2p+ p3,2 − p3,1 + p2,1 (mod M), (3.11)

p3,3 6= c3p+ p3,2 − p1,2 + p1,3 (mod M), (3.12)

with c1, c2, c3 ∈ [−m,m]. As each direct doppler spread should be different of 2m+ 1 values

in Z/MZ, the system in (3.10), (3.11) and (3.12) has a solution if and only 2m+ 1 < M − 1.

This implies that,

m =M − 3

2.

Then, the maximal number of symbols that can be transmitted is equal to m+ 1 = M−12 .

Full multiplexing

Each receiver can decode m + 1 OFDM free from interference and on each OFDM symbol

Nc subcarriers interference free. Therefore, the maximal total multiplexing gain that can be

achieved is such that,

r = K(m+ 1)Nc

MNc= K

(M − 1)Nc

2MNc

M→∞−−−−→ K/2.

Numerical example

In order to illustrate this, we consider the numerical example with M = 17 be the prime

number of OFDM blocks, and p1,2 = 2, p1,3 = 5, p2,1 = 4, p2,3 = 6, p3,1 = 1 and p3,2 = 3.

Therefore, p = −3 as given in (3.7). The maximal number of data that can be transmitted

in this case, is m+ 1 = M−12 = 8. For this numerical case, the interference alignment scheme

is feasible if the direct Doppler spread are chosen following the equations (3.10), (3.11) and

82


3

5

3 2 7 6 0 5 41

5 4 3 2 7 1 6 0

6 0 5 4 3 2 7 1

1 6 0 4 3 2 75

3 2 7 1 6 0 5 4

1 6 0 4 3 2 7

0 5 4 3 2 7 1 6

Destination 1

Destination 2

Destination 3

interference

interference

interference

desired signal

desired signal

desired signal

4 3 2 7 1 6 0 5

2 7 1 6 0 5 4

5 6 7 8 1615141312100 1 2 3 119

S1

S2

S3

I2

I1

I1

I3

I3

I2

4

Figure 3.4: Interference alignment for the 3 users case: shifted OFDM symbols received atdestinations 1, 2 and 3.

83


(3.12), i.e,

p1,1 = 7, p2,2 = 16, p3,3 = 13.

The precoding and pre-processing steps at the S1 → D1 are depicted in Figure 3.5. As

we discussed previously the precoding cyclic shift p1 + p1,1 (mod M) = 9 should be applied

to the position of the OFDM symbol at kp (mod M), k = 0, . . . , M−32 = 7.

0 1 2 3 4 5 6 7

(a)

60 5 4 3 2 17

(b)

0 5 463 2 17

(c)

Figure 3.5: Precoding and pre-processing on S1 → D1 (a) - Signaling scheme : Each blockdenotes an OFDM symbol with Nc subcarriers. (b) - First precoding step: Change theOFDM symbol position k to kp (mod M), with p = −3. (c) - Second precoding step +channel: Cyclic shift in time of p1,1 + p1 (mod M) = 9

The same steps should be repeated for each Si → Dj , (i,j = 1,2,3). It can be easily verified

in Figure 3.4, that the interference coming from user 2 and 3 are aligned at receiver 1, and

so on. Finally, notice that when a OFDM symbol is received interference free, all the Nc

subcarriers are also received free from interference. The maximal multiplexing that can be

achieved per user is therefore,

r =8Nc

17Nc= 0.47 ≈ 1/2.

3.5 General spread requirements for interference alignment

In this section, the 3-user SISO interference channel case is first analyzed. We assume that

all channels between node have the same statistical distribution, and the same scattering

function. For this case, we briefly review the interference alignment concept from [3]. Then,

we derive the conditions required by the selective fading channel for the general case to

achieve the full multiplexing gain of K/2.

The large wireless network is also considered. This has been initially studied in the seminal

work by Gupta and kumar in [56] where they show that the maximal total throughput in

an adhoc network cannot scale better than O(√K) when a multihop architecture is used.

In a recent work of Morgenshtern and Bolcskei in [57], it has been shown that the same

scaling of O(√K) can be achieved when dedicated relays are used to assist all sources and

84

3.5. GENERAL SPREAD REQUIREMENTS FOR INTERFERENCE ALIGNMENT

destinations communications. More recent, it has been shown in [19] that the total linear

throughput scaling O(K) can be achieved under opportunistic CSIT assumption using a

hierarchical transmission strategy. We show here that the linear scaling can be also achieved

using interference alignment schemes under certain channel spread requirements, but at the

expense of a very large bandwidth-time product.

3.5.1 General Interference Alignement Construction

For the general case, the interference alignment scheme proposed by Cadambe and Jafar (CJ)

in [3] or Ozgur and Tse (OT) in [19] for the K users extracts the total multiplexing gain

K/2. These scheme consist in designing linear precoders V[i] such that the total space at the

receiver side i is divided into two subspaces the desired space Si of dimension dS,i(n) and the

interference subspace Ii of dimension dI,i(n), as shown in (3.13),

yi = H[i,i]V[i]︸︷︷︸Signal subspace

Si of dim ds,i(n)

si +∑j 6=i

H[i,j]V[j]︸︷︷︸Interference subspace

Ii 6= Si of dim dI,i(n)

sj + zi, (3.13)

with n is an auxiliary variable and N(n) = dI,i(n) + ds,i(n) is the total number of slots. The

two subspaces Si and Ii should be disjoint. That’s why, by performing zero forcing scheme

U[i] at the receiver side i nulls out only the interference space Ii. Finally, the dimensions of

Si and Ii are chosen such that

r = limn→∞

∑i ds,i(n)

N(n)=K

2.

In the following, we recall the Cadambe-Jafar (CJ) construction, and Ozgur-Tse (OT) con-

struction. Unlike the time selective channel where coefficients are uncorrelated, we show that

the construction of these schemes is constrained by the time-frequency correlation. We prove

that these schemes can be both applied in the time-frequency domain under certain channel

spread requirements which can be easily met in practical system.

3.5.2 Channel spread requirement for CJ scheme

Cadambe and Jafar (CJ) construction

For the CJ scheme, the total duration of transmission is assumed to be be N = (n+1)Q+nQ,

and the desired signal dimensionalities that can be extracted are such that,

ds,k =

(n+ 1)Q k = 1

nQ i = 2, . . . ,K.

85


In this case, linear precoders V[i] are chosen such that

H[1,2]V[2] = H[1,3]V[3] = . . . = H[1,K]V[K], (3.14)

spanH[i,j]V[j]

⊆ span

H[i,1]V[1]

. (3.15)

Equations (3.14) and (3.15) can be written as following

V[j] = S[j]B j = 2 . . .K, (3.16)

spanT

[i]j B⊆ span

V[1]

∀i /∈ 1, j, ∀j 6= 1, (3.17)

where

B = (H[2,1])−1H[2,3]V[3],

S[j] = (H[1,j])−1H[1,3](H[2,3])−1H[2,1], j = 2, . . .K,

T[i]j = (H[i,1])−1H[i,j]S[j] i, j = 2 . . .K, j 6= i.

and T[2]3 = IN . The columns of matrix B and V[1] are chosen in the setB and V [1] respectively,

such that

B =

( ∏m,k∈S

(T

[m]k

)αmk)w : ∀αmk ∈ 0, 1, 2 . . . n− 1

, (3.18)

and

V [1] =

( ∏m,k∈S

(T

[m]k

)βmk)w : ∀βmk ∈ 0, 1, 2 . . . n

. (3.19)

where

S = m, k ∈ 2, 3, . . .K,m 6= k, (m, k) 6= (2, 3) ,

and

Q = |S| = (K − 1)(K − 2)− 1.

We can check easily that for this construction the (3.17) is satisfied. If b is a column

vector of B, then T[i]j b is necessarily a column vector of V[1]

T[i]j b = T

[i]j

( ∏m,k∈S

(T

[m]k

)αmk)w,

=∏

m,k∈S

(T

[m]k

)βmkw ∈ V [1].

where

βmk =

αmk + 1 (m, k) = (i, j),

αmk otherwise.∈ 0, . . . , n

For the time selective channel when channels coefficients are uncorrelated, it has been

86


shown in [3] that these two sets B and V [1] contain respectively nQ and (1 + n)Q distinct

columns vectors. Moreover, for the uncorrelated case, the interference subspace Ii is inde-

pendent of the signal subspace Si, and is nulled out using a zero forcing U[i] ∈ Cdi×N at each

user i, such that,

U[i] ∈ ker Ii,

where,

di = dim(

ker Ii)

= N(n)− dI,i(n) = ds,i(n).

The rate achieved at user i is lower bounded by

Ri =1

Nlog det

(IN +

P

KH[i,i]H[i,i]†),

where

H[i,i] = U[i]H[i,i]V[i]

is the effective channel between Si → Di with rank = ds,i. Let λi denote the eigen-values of

H[i,i]H[i,i]†, then

Ri =1

N

di∑j=1

log(

1 +P

Kλj

)P→∞−−−−→ di

NlogP

The total spatial multiplexing gain is therefore,

r = limP→∞

∑iRi

logP=

∑i diN

,

=(n+ 1)Q + (K − 1)nQ

(n+ 1)Qn→∞−−−→ K/2.

For large K →∞ and large n, the total multiplexing gain is such that,

r =(1 + 1

n)Q + (K − 1)

(1 + 1n)Q + 1

∼Qn +K

2 + Qn

.

The auxiliary variable n should scale as

n ∼ Q1+ε, (3.20)

where ε > 0 in order to achieve the full multiplexing gain K/2. The total multiplexing can

be therefore achieved at the expense of a large duration and bandwidth product,

DW ∼ 2TFK2K2(1+ε).

Channel spread requirement for CJ scheme

Unlike the uncorrelated case, the CJ scheme is constrained by the correlation of the channels

coefficients. This correlation can reduce the dimensionality of the desired signal if the channel

87


spread is not chosen appropriately. The main result of this subsection is summarized in the

following theorem.

Theorem 3.2 In the time-frequency domain, the total multiplexing gain K/2 can be achieved

using the CJ scheme if and only if the channel spread ∆H satisfies the following condition

1

9TFQ2n2

[(n+ 1)Q

nQ + (n+ 1)Q

]≤ ∆H ≤

1

TFK−1. (3.21)

For large K, the condition on the channel spread is reduced to,

1

18TFK−8−ε ≤ ∆H ≤

1

TFK−1,

with ε > 0.

Proof: In the time-frequency domain, the dimensionality of the desired signal space

xi = V[i]xi ∈ Si is affected by the correlation of the channels coefficients. It is clear that

the dimension of the desired signal depends on the rank of V[i], and hence on the number

of independent vectors in the sets B and V [1] in equations (3.18) and (3.19). Using the

commutativity over the diagonal channel matrix, the i-th column v[j]i of the linear precoder

matrix V[j] in equations (3.16) and (3.18), can be written as,

v[j]i = Sj

( ∏m,k∈S

(T

[m]k

)αmk)w, ∀ 2 ≤ j ≤ K,

= Sj

( ∏m,k∈S

H[m,1]H[1,k]H[2,3]

)−nC

[j]i w.

In a similar way, for j = 1, v[1]i can be written as

v[1]i =

( ∏m,k∈S

H[m,1]H[1,k]H[2,3]

)−nC

[1]i w,

where the index i depends on the

- nQ choice of the Q-αmk values in 0, 1, 2 . . . n− 1 for v[j]i , where 2 ≤ j ≤ K.

- (n+ 1)Q choice of the Q-βmk values in 0, 1, 2 . . . n for v[1]i .

and

C[j]i =

∏m,k∈S

(H[m,1]H[1,k]H[2,3]

)n−δmk(H[m,k]H[1,3]H[2,1])δm,k ,

with

δmk =

βmk if j = 1,

αmk if 2 ≤ j ≤ K.

Notice that first term in the product does not depend on i, that’s why it is sufficient to

study linear independence of c[j]i = C

[j]i w to prove the linear independence of v

[k]i . We should

88


find the conditions that should be satisfied to guarantee the linear independence of vectorsc

[j]i

i=1...dj

for each j.

As stated in Theorem 1.4, any channel matrix between user i and j can be written such

as

H[i,j] =∑

(p,q)∈Aλ

[i,j](p,q)(Z

pM ⊗ ZqNc),

where λ[i,j](p,q) are i.i.d random variables drawn from a continuous distribution.

Using the property of the Kronecker product (A ⊗ B)(C ⊗D) = AC ⊗ BD, and the fact

that Ci is the product of 3nQ polynomial, Ci can be written such that

C[j]i =

∑(p,q)∈A′

ζi,[j](p,q)(Z

pM ⊗ ZqNc),

where ζi,[j](p,q) are polynomial of variables drawn from a continuous distribution with a degree

that depends on i,

A′ =

(p, q) : p ∈ 0, . . . , p0, q ∈ 0, . . . , q0,

and p0 and q0 are given such that

p0 = min(3nQ(p0 − 1),M )− 1 ≤ 3nQ(ν0TM − 1) (3.22)

q0 = min(3nQ(q0 − 1), Nc − 1) ≤ 3nQ(τ0FM − 1). (3.23)

The cardinality of A′ is therefore,

t = |A′| = (p0 + 1)(q0 + 1) ≤ 9n2Q2TFN∆H,

where N = MNc = (n+ 1)Q +nQ is the total number of time-frequency slots and ∆H = τ0ν0

is the channel spread. Consequently, c[j]i = C

[j]i w can be written as the sum of t = |A′|

independent linear vectors, such that

c[j]i =

∑(p,q)∈A′

ζi,[j](p,q)(fp ⊗ gq), i = 1 . . . dj .

where fp and gq are respectively the p-th and q-th columns of the M ×M and Nc ×Nc FFT

matrices F† and G. The matrix C[j] can be therefore written as,

C[j] = (F† ⊗G)C[j]

89


where,

C[j] =

ζ

1,[j](0,0) . . . ζ

ds,j ,[j]

(0,0)...

...

ζ1,[j](p0,q0) . . . ζ

ds,j ,[j]

(p0,q0)

0N−t,ds,j

.

Following the same reasoning as in [3], it can be shown that any two columns of C[j] are

linearly independent as the coefficients are polynomial of variable drawn from continuous

distribution. The rank of this matrix is therefore min(t, ds,j). The dimensionality ds,j of the

desired signal can be extracted if and only if the matrix C[j] has a rank equal to ds,j . This

requires that,

t ≥ ds,j ∀1 ≤ j ≤ K.

As ds,1 ≥ ds,j for all j ≥ 2, the dominant condition is given by

t ≥ d1 = (n+ 1)Q.

and therefore,

9n2Q2FTN∆H ≥ t ≥ (n+ 1)Q.

or equivalently,

∆H ≥ ∆H,min =1

9TFQ2n2

[(n+ 1)Q

nQ + (n+ 1)Q

].

For large number of users, the auxiliary variable n should scale as Q1+ε as shown in (3.20)

and Q ∼ K2. In this case, ∆H,min scales as

∆H,min ∼ 1

18TFQ4Q2ε,

∼ 1

18TFK−8−ε′ , ε′ = 4ε > 0.

Finally, using the same reasoning as in [3], it can be shown that in the time-frequency domain

the desired subspace Si and the interference subspace Ii are independent. Therefore, when

the channel spread condition is fulfilled, the CJ scheme extracts the full multiplexing gain in

the time-frequency domain.

3.5.3 Ozgur and Tse Construction

Ozgur and Tse (OT) construction

In [4], Ozgur and Tse proposed a slightly modified interference alignment scheme of Cadambe

and Jafar (CJ) scheme previously presented. The main difference here is that a unique

precoder is used at all the the sources, such that,

V[i] = S, i = 1 . . .K.

90


Moreover, the interference can be removed using the same zero forcing preprocessor at all

destinations, such that,

U[i] = Q, i = 1 . . .K.

The input output relationship can be written such that,

yk = H[k,k]Sxk +∑k 6=l

H[k,l]Sxl + zk, k = 1 . . .K (3.24)

where the columns of the precoder matrix si can be written such that,

si =∏k 6=l

(H[k,l]

)αk,l[i]w, i = 1 . . . ds (3.25)

and αk,l[i] take values in 0, 1, . . . , n with,∑k 6=l

αk,l[i] = n. (3.26)

The dimension of the desired signal space is therefore given by,

ds =

(n+K(K − 1) + 1

K(K − 1) + 1

).

In this case, the interference space spans the space formed by the vectors,

bi =∏k 6=l

(H[k,l]

)βk,l[i]w, i = 1 . . . ds

where βk,l[i] take values in 0, 1, . . . , n+ 1, and∑k 6=l

βk,l[i] = n+ 1. (3.27)

The dimension of the space spanned by the vectors bi with i = 1 . . . dI is equal to,

dI =

(n+K(K − 1) + 2

K(K − 1) + 1

).

The number of time-frequency slot is chosen in order to have,

N(n) = dI(n) + ds(n),

The maximal multiplexing gain that can be achieved is given by,

r = limn→∞

Kds(n)

ds(n) + dI(n)= lim

n→∞K(n+ 1)

2n+ 3 +K(K − 1)= K/2. (3.28)

91


It can be easily verified that for large users case, that the auxiliary variable n should scale as

n ∼ K2+ε, (3.29)

with ε > 0 in order to achieve the full multiplexing gain.

Channel spread requirement for the OT scheme

The OT scheme is also constrained by the time-frequency correlation as this correlation

affects the dimension of the desired signal. In this case, the condition on the channel spread

is summarized by the following theorem.

Theorem 3.3 In the time-frequency domain, the total multiplexing gain K/2 can be achieved

using the OT scheme if and only if the channel spread ∆H satisfies the following condition

1

TFn2

[n+ 1

2n+ 3 +K(K − 1)

]≤ ∆H ≤

1

TFK−1. (3.30)

For large K, the condition on the channel spread is reduced to,

1

2TFK−4−ε ≤ ∆H ≤

1

TFK−1.

Proof: For the OT scheme, using the polynomial channel decomposition in Theorem

1.4, each column vector si can be written such that,

si =∏k 6=l

(H[k,l]

)αk,l[i]w,=

∏k 6=l

( ∑(p,q)∈A

λ[k,l](ZpM ⊗ ZqNc))αk,l[i]

w,

=∑

(p,q)∈A′ξ[i]p,q(Z

pM ⊗ ZqNc),

with

A′ = (p, q) : p ∈ 0, . . . , p0, q ∈ 0, . . . , q0,

and p0 and q0 are such that,

p0 = min((p0 − 1)∑k 6=l

αk,l[i],M) = min((p0 − 1)n,M),

q0 = min((q0 − 1)∑k 6=l

αk,l[i], Nc) = min((q0 − 1)n,Nc).

Using the same reasoning as in the above section,

(n(p0 − 1) + 1)(n(q0 − 1) + 1) ≥ ds,

92

3.6. INTERFERENCE ALIGNMENT WITH LIMITED FEEDBACK

This implies that,

n2∆HFTN ≥ ds

and therefore,

∆H ≥ dsn2FT (ds + dI)

,

≥ (n+ 1)

n2FT (2n+ 3 +K(K − 1)).

For a large scaling network, the auxiliary variable n should scale as K2+ε as shown

in (3.29). In this case, the channel spread should scale at least as,

∆H,min =ds

n2FT (ds + dI)∼ 1

2TFK−4−ε′ , ε′ = 2ε.

3.6 Interference alignment with limited feedback

In the previous sections, IA construction is built on the assumption of perfect channel knowl-

edge at all nodes. However, this assumption cannot be made practical for implementation

in real system due to the large amount of required feedback. In this section, we propose a

limited feedback IA scheme based on the random vector quantization of the selective fading

channel. For this, we briefly review the random vector quantization, and then we show how

this quantization can be combined with IA scheme to achieve full multiplexing gain.

We assume that each destination Di has a perfect channel knowledge of channels Sk → Di,

k = 1 . . .K and needs to quantize each of the K channel using B bits and feedbacks the total

KB bits perfectly and instantaneously to all others nodes in the network. In the following, we

assume that all channels have the same statistical characterization and the same scattering

function CH(ν, τ).

3.6.1 Random vector quantization

Based on the observation we made on the polynomial decomposition of selective fading chan-

nel in Theorem 1.4, once can notice that each node j can be able to reconstruct the diagonal

channel i→ k based only on the knowledge of |A| = p0q0 ≤ DW∆H terms, such that

H[i,k] =∑

(p,q)∈Ah

[i,k](p,q)(Z

pM ⊗ ZqNc),

A =

(p, q) : p ∈ 0, . . . , p0 − 1, q ∈ 0, . . . , q0 − 1,

93


closest vector

channel vector

The quantized channel is the

i

2

1

2B

a = sin2(θ) ≤ 2−B

|A|−1

Figure 3.6: Random vector quantization codebook

The terms D = TM and D = FNc are respectively the duration and the bandwidth of the

signal. Let h[i,k] denotes the |A| × 1 vector such that,

h[i,k] =[h

[i,k](0,0) . . . h

[i,k](p0−1,q0−1)

],

and h[i,k] be the normalized vector, such that

h[i,k] =1

‖h[i,k]‖h[i,k].

Each destination Di quantizes each channel (Sk → Di) to B bits. The quantization

depicted in Figure 3.6 is performed using a vector quantization codebook that is known at

all nodes. We assume that each node uses different codebooks for each channel to prevent

quantizing of two different channels with the same quantization vector. The quantization

codebook C consists of 2B - |A| dimensional unit norm vectors , such that C = ω1, . . . , ω2B,where B is the number of feedback bits. Each destination nodes (to say destination i) feeds

the index F [i,k] of the ω vectors that is closest (in term of its angle) to its channel vector

h[i,k], such that

F [i,k] = arg maxj=1...q

|h[i,k]†ωj |,

= arg minj=1...q

sin2(∠(h[i,k], ωj)

). (3.31)

and feeds the index back to the all others 2K − 1 nodes in the network. Each destination Di

feed backs K index F [i,k], where k = 1 . . .K of B bits each, i.e., in total KB bits.

Preliminary calculation

In order to derive the total number of degrees of freedom with limited feedback, we need the

two following lemmas. Let h = 1‖h‖h be a unit norm |A| × 1 vector and h its quantization

vector. We denote by H and H the selective fading channels that can be reconstructed from

94


h and h respectively.

Lemma 3.2 The quantization error can be bounded almost surely by

sin2(∠(h, h)

)≤ 2

− B(|A|−1) . (3.32)

As |A| ≤ DW∆H, the following inequality follows,

sin2(∠(h, h)

)≤ 2

− B(DW∆H−1) . (3.33)

Proof: As shown in (3.31), h is the closest vector(in term of its angle) to h among 2B

vectors. As shown in appendix C, the angle between two isotropic random vector ∈ C1×|A|

is beta distributed β(1,M − 1). This implies that, cos2(∠(h, h)

)is the maximum between

2B independent beta variables β(1,M − 1), and hence X = sin2(∠(h, h)

)is the minimum

between 2B independent beta variables β(M − 1, 1), where the CDF is given by,

F (x) = ProbX ≤ x = 1− (1− x|A|−1)2B .

It is easy to show that for x0 = 2− B

(|A|−1) ,

F (x0) = 1−(

1−(2− B

(|A|−1))|A−1|)2B

, (3.34)

= 1−(

1− 2B2−B +O(2− B

(|A|−1) )), (3.35)

= 1−O(2−B). (3.36)

This implies that when B is large, X < x0 with high probability.

Note that a closer bound has been also developed in [55].

Lemma 3.3 The selective fading channel can be written in function of its quantized matrix

such that

H = ‖h‖√

1− aH + ‖h‖√aS, (3.37)

where a = sin2(∠(h, h)

)is the quantization error, S is the diagonal matrix reconstructed

from s ∈ C|A|×1 and s is a unit norm vector isotropically distributed in the null-space of h,

independent of a.

Proof: The proof of this lemma is based on the proof of Lemma 2 given by Jindal in [6].

As h = 1‖h‖h is a unit norm vector isotropically distributed in C|A|×1, then h can be written

as the sum of two vectors, one, h in the direction of the quantized vector and the second, s

is isotropically distributed in the null-space of h, independent of a. Then

h =√

1− ah +√as.

95


Equation (3.37) follows as a consequence of the relation between H and h as given in theo-

rem 1.4, i.e.,

H =∑i

hiWi = ‖h‖∑i

hiWi,

= ‖h‖∑i

(√1− ahi +

√asi)Wi,

= ‖h‖√

1− a∑i

hiWi + ‖h‖√a∑i

siWi,

= ‖h‖√

1− aH + ‖h‖√aS.

Note that S has a full rank of N , as there is zero probability that any one of the diagonal

elements is equal to zero.

3.6.2 Achieving full multiplexing gain with limited feedback

Theorem 3.4 Assuming a communication IA scheme over K-SISO interference channel

using an RVQ scheme, the total spatial multiplexing gain of K/2 can be achieved if the total

number of feedback bits Nf broadcast by each destination to all the sources and to all other

destinations scales as

Nf = K(DW∆H − 1

)log2 P,

where D, W are respectively the signal and the bandwidth duration and ∆H is the channel

spread.

Proof: IA scheme including the precoder matrices V [i] and the zero forcing pre-processing

matrices U[i] should be constructed using the knowledge of quantized channels matrices. The

received signal at destination i, is therefore

yi = U[i]H[i,i]V[i]xi +∑k 6=i

U[i]H[i,k]V[k]xk + U[i]zi. (3.38)

If the channel spread condition in Theorem 3.2 is satisfied, then it can be guaranteed that

rank U[i] = di, and rank V[i] = di.

Using Lemma 3.3 and the IA implications, (3.38) can be written as,

yi = U[i]H[i,i]V[i]xi + ‖h‖∑k 6=i

(√1− a U[i]H[i,k]V[k] +

√aS[i,k]V[k]

)xk + U[i]zi,

= U[i]H[i,i]V[i]xi + ‖h‖√a∑k 6=i

U[i]S[i,k]V[k]xk + U[i]zi.

96


It follows that, the rate achieved at the destination is such that

Ri =1

Nlog det

IN +P

KH[i,i]†

(Idi +

Pa

K‖h‖2

∑k 6=i

S[i,k]S[i,k]†)−1

H[i,i]

,

where

H[i,i] = U[i]H[i,i]V[i], and S[i,k] = U[i]S[i,k]V[k].

Using the following properties of matrices (A ∈ Cm×n,B ∈ Cn×p),

rank(A) + rank(B)− n ≤ rank(AB) ≤ min

rank(A), rank(B), (3.39)

and the fact that H[i,i] and S[i,k] has full rank of N , it can be easily checked that,

rank H[i,i] = rank S[i,k] = di.

Let Ap denote the di × di matrix such that

Ap =(

Idi +Pa

K‖h‖2

∑k 6=i

S[i,k]S[i,k]†)−1

.

At high SNR, the total number of degrees of freedom can be achieved if and only if H[i,i]ApH[i,i]†

has also full rank of di. This implies that Ap should have a full rank di, or equivalently all

its eigen-values λi(Ap) should be strictly positive.

As matrix S[i,k] is a full rank matrix, it follows that∑

k 6=i S[i,k]S[i,k]† has di non-zero eigen-

values µi > 0 that are independent of P . Then,

λi(Ap) =1

1 + PaK ‖h‖2µi

≥ 1

1 + 1K ‖h‖2µiP 2

− BDW∆H−1

is strictly positive if and only if B scales as (DW∆H − 1) log2 P .

Provided that B scales as (DW∆H − 1) log2 P , then Ap has a full rank, and H[i,i]ApH[i,i]†

has di non zeros eigen-values λj , j = 1 . . . di. The achievable rate Ri can be rewritten such

that

Ri =1

N

di∑j=1

log(

1 +P

Kλj

)P→∞−−−−→ di

NlogP

The total multiplexing gain that can be achieved is therefore,

r = limP→∞

K∑i=1

RilogP

=

∑Ki=1 diN

=(n+ 1)Q + (K − 1)nQ

nQ + (n+ 1)Qn→∞−−−→ K/2.

97


3.7 Conclusion

In this chapter, we consider the K-user SISO interference channel where channels between

sources and destination are time-frequency selective. Based on the polynomial channel matrix

proposed in Chapter 1, we show how interference alignment scheme can be deployed in

this context by performing adequate shift for the ODFM symbols only. We show that the

interference alignment schemes proposed in literature CJ in [3] and OT in [19] extract the

full multiplexing gain if the channel spread scales at least as K−8 and K−4 respectively. This

condition on channel spread can be easily met in practical system as the channel spread is in

the order of 10−2 for indoor channels and 10−7 for land mobile channels. Finally, we extend

the result of [55] to the time-frequency domain; We show that if the number of bits to be

fed back by each receiver to all others node scales as K(DW∆H− 1

)log2 P , the total spatial

multiplexing gain can be also achieved.

98

Chapter 4

Selective MIMO Broadcast

Channel with Limited Feedback

THE MIMO broadcast channel has recently received significant attention due to the

fact that this channel can provide MIMO spatial gains without requiring multiple

antennas at the receiver side. This channel has been studied widely in the literature

over the last few decades, where it is common to assume that channels are flat fading.

In this chapter, we analyze the case of selective MIMO Broadcast channel, where links are

selective in both time and frequency. We consider first the case where full channel state

information is assumed at the transmitter side, and review the precoding schemes that were

proposed in the literature. Then, we propose an intuitive improvement of the vector pertur-

bation scheme [5] based on the use of periodically flipped constellations. The assumption of

full channel knowledge at the transmitter side requires a large amount of feedback, and it

is therefore not practical to implement in real systems. A more feasible solution with finite

rate feedback originally proposed in [6], [7] is applied to the selective case, where the minimal

number of feedback bits required to achieve the full multiplexing is derived. We show that

the correlation between time frequency channels can be used in order to minimize the number

of feedback bits to the transmitter side while conserving the maximal multiplexing gain.

4.1 Introduction and motivations

The next generation cellular system (such as IEEE 802.16m, LTE advanced, etc) features

Multiple-Input Multiple-Output (MIMO) transmission and multi-user communications.

In the uplink channel, as known as the Multiple Access Channel (MAC), of such systems,

99

CHAPTER 4. SELECTIVE BROADCAST CHANNEL WITH LIMITED FEEDBACK

multiple mobile terminals transmit simultaneously to the base station. The latter treats the

received signal in such a way that messages from different mobile terminals are distinguish-

able. The capacity region of a multiple access [58] has been known decades ago. A capacity

achieving scheme is the Successive Interference Cancellation (SIC). This scheme has been

well studied and extends naturally in the MIMO case.

Unlike the uplink channel, little is known for the downlink channel, as known as the

Broadcast Channel (BC), until recent years. Solid progress on the capacity region of MIMO

broadcast channel has been made in [59–62], and the exact characterization of the capacity

region was found in [21]. It has been shown that the Dirty Paper Coding (DPC) achieves

the capacity region. As a dual counterpart of the SIC for the MAC, the DPC for the BC

successively removes the inter-user interference at transmitter provided that exact Channel

State Information (CSI) is available at the transmitter side.

The main hindrance to the practical implementation of the DPC is its high complexity

(see, for example, [63]) and its sensibility to the CSI, as shown by [64]. Low complexity so-

lutions come naturally to linear precoding based schemes, such as the Zero-Forcing (ZF) [22]

and block diagonalization (BD) [23] for multiple antennas users case or non-linear precoding

schemes such that vector perturbation proposed in [5]. Based on the intuitive observation

that the performance of the vector perturbation scheme at low SNR depends highly on sen-

sitivity of the modulo function to the noise perturbation, we propose in this chapter a new

non-linear precoder based on the use of a more sophisticated constellation scheme which we

call Periodically Flipped Constellation (PFC) in [65] at the encoder associated to a modified

modulo function at the receiver to perform decoding.

The above precoders require a full channel knowledge at the transmitter side, which is

not practical in real systems as it requires a large amount of feedback. A more feasible

solution with finite rate feedback has been proposed in [6], [7]. It has been shown that using

an adequate number of feedback bits that scales as SNR, the full multiplexing gain can be

also achieved using limited feedback. So far, this scenario has been essentially studied for

the case where channels between source and different destinations are assumed to be flat

fading. In this chapter, a limited feedback scheme for the selective MIMO broadcast channel

is proposed. We show that the correlation between time frequency channels can be used in

order to minimize the number of feedback bits to the transmitter side.

4.2 System and channel model

4.2.1 System model

In this chapter, we consider a K receiver multiple-antenna broadcast channel where a source

S wants to communicate with K destinations Di as shown in Figure 4.1. We assume that

all communications occur on selective fading channels. We denote in the following by N the

100

4.2. SYSTEM AND CHANNEL MODEL

number of time-frequency slots, nt the number of transmit antennas at the source and nr the

number of receive antennas at each destination Di.

H[2]

S

nt

1

D1

nr

1

nr

1

D2

1

nr

DK

H[K]

H[1]

Figure 4.1: A MIMO broadcast channel with nt transmit antennas and K users having nrreceive antennas each.

At each destination, the received signal yk(n) is given by

y[k](n) = H[k](n)x(n) + n[k](n), k = 1 . . .K, n = 0 . . . N − 1. (4.1)

where H[k](n) ∈ Cnr×nt is the channel matrix for the time frequency sot n. The vector

x(n) ∈ Cnr×1 is the transmitted signal, and n[1](n), . . . ,n[K](n) are independent complex

Gaussian noise terms with unit variance. The transmitter is subject to an average power

constraint P , such that

E[x(n)x(n)†] ≤ P. (4.2)

We assume that channels are spatially uncorrelated, that for a given time-frequency slot n,

H[k](n) has i.i.d CN (0, 1) entries. The channels corresponding to different destinations are

assumed to be statistically independent. However, channels are correlated across n for a

given destination k. For simplicity of notations, we assume that all scalar sub channels h[k]i,j

with(k = 1 . . .K, i = 1 . . . nr and j = 1 . . . nt) have the same correlation function i.e.,

E[h

[k]i,j(n)h

[k′]i′,j′(n

′)]

= Rh(n, n′)δ(k − k′)δ(i− i′)δ(j − j′). (4.3)

101


4.2.2 Channel model

As shown in Chapter 1, each scalar channel between antenna i and antenna j requires the

knowledge of r ≥ DW∆H values of the sampled delay-Doppler spreading function to be

reconstructed as shown in fig 4.2. Assuming nt antennas at the transmitter side and nr at

the receiver side, the receiver needs to compute during the whole duration of the transmission

rntnr parameters in order to reconstruct the channel at each time frequency slot. At each

time-frequency slot, the MIMO channel between the source and each destination can be

written as given in Lemma 4.1.

h[k]ω,1h

[k]ω,0 h

[k]ω,r−1User k

rTs

Figure 4.2: The r ≥ DW∆H channel coefficients are sufficient to characterize the channel.

Lemma 4.1 (Time-frequency MIMO channel matrix) The user k channel matrix H[k](n) ∈Cnr×nt at a time-frequency slot n can be written as

H[k](n) = H[k]ω Γ(n) (4.4)

where H[k]ω ∈ Cnr×rnt is a Gaussian matrix with i.i.d CN (0, 1) entries such that,

H[k]ω =

[H

[k]ω,0 H

[k]ω,1 . . . H

[k]ω,r−1

],

and Γ(n) is a rnt × nr deterministic matrix that depends only on the channel statistics, and

is such that

Γ(n) =

σ0wn,1

...

σr−1wn,r

⊗ Int .

Proof: As shown in Chapter 1, each scalar channel can be written in function of the

covariance matrix eigenvectors (please refer to Lemma 1.2) as,

h[k]i,j(n) =

r−1∑l=0

wn,lσlh[k]i,j [l],

102

4.3. MULTIPLEXING GAIN FOR THE MIMO BROADCAST CHANNEL

As channels between transmit antennas and receive antennas are not correlated, this implies

that

H[k](n) =

∑r−1

l=0 wn,lσlh[k]1,1[l] . . .

∑r−1l=0 wn,lσlh

[k]1,nt

[l]...∑r−1

l=0 wn,lσlh[k]nr,1

[l] . . .∑r−1

l=0 wn,lσlh[k]nr,nt [l]

=

r−1∑l=0

wn,lσl

h

[k]1,1[l] . . . h

[k]1,nt

[l]...

h[k]nr,1

[l] . . . h[k]nr,nt [l]

=

r−1∑l=0

wn,lσlH[k]ω,l

Consequently, the matrix form in (4.4) can be easily deduced by simple matrix manipu-

lations.

h[k]j (n) = h

[k]ω,jΓ(n), j = 1 . . . nr, k = 1 . . .K. (4.5)

where h[k]ω,j ∈ C1×rnt is a Gaussian vector with i.i.d entries.

Remark 4.1 Let v ∈ C1×nr be a unitary vector, and u = Γ(n)v. Then, it can be easily

verified that

‖u‖2 =

r−1∑i=0

σ2i |wn,i|2 = σ2

t,n

In the following, we let,

Γ(n) =1

σt,nΓ(n),

and,

σ2t = max

n=0...N−1σ2t,n.

Consequently,

‖u‖2 = ‖Γ(n)v‖2 = 1.

4.3 Multiplexing gain for the MIMO broadcast channel

The sum capacity of the broadcast channel (BC) depends largely on the availability of CSI at

the transmitter side. When perfect channel state information (CSI) is assumed at both the

BS and receivers, it is well known that the Dirty Paper Coding technique (DPC) achieves the

maximum sum capacity [21]. This technique is the most efficient strategy that allows a base

station to transmit data to multiple users at same time. In this case, the multiuser system is

equivalent to a nt×Knr MIMO system as shown in [66]. The implementation of DPC brings

high complexity to both the transmitter and the receiver. In addition, full CSI is required

at the transmitter side which is not practical in a real system. Suboptimal linear precoding

103


techniques, which we present in the following achieve a large portion of DPC capacity while

being simpler to operate than DPC [22], [5]. In both cases, the maximum multiplexing gain

that could be achieved is equal to min(nt,Knr).

On the other hand, when no CSI is available at the transmitter and the channels of all

receivers are statistically identical, then, the BC channel is degraded in any order and TDMA

is the optimal strategy. In this case, the multiuser BC channel is equivalent to a single user

M ×N MIMO channel and the maximal multiplexing gain that can be achieved is equal to

min(M,N). As we can see, there is a huge gap between the multiuser gains with and without

transmit CSI. Since lack of CSI does not lead to multiuser gains and since perfect CSIT is

not feasible, solution based on partial CSI at the transmitter side has been considered in [6]

and [7]. It has been shown that for the flat fading channel when the number of feedback bits

scale in a adequate way with the power, the maximal multiplexing gain ca be also achieved.

As the spatial multiplexing gain is addressed in this chapter, we only focus on the case where

nt ≥ Knr. For the large networks case, we assume that the K destinations are selected

among a large number of destinations in the cell. The scheduling strategies allowing to select

these destinations are not addressed in this thesis. The interested reader can refer to the

work of Yoo et al. in [27], Kountouris et al. in [29], [28] and Zakhour in [67] for more details

on iterative feedback schemes for the MIMO broadcast channel with large number of users.

4.4 Precoding at the transmitter side

As shown in [21], it is well known that the Dirty Paper Coding (DPC) achieves the capacity

region of the MIMO broadcast channel, and therefore achieves the maximal transmitted

sum capacity. However, the implementation of DPC brings high complexity to both the

transmitter and the receiver. Since the capacity-achieving dirty paper coding approach is

difficult to be implemented in a real system, many more practical downlink transmission

techniques have been proposed. Downlink linear beamforming, although suboptimal, has

been shown to achieve a large portion of DPC capacity while being simpler to operate than

DPC [22].

4.4.1 Linear precoding schemes

When linear precoding is used, the transmitted signal vector x(n) is a linear function of the

destinations’ data symbols sk(n) ∈ CN×1. Let Vk(n) denotes the precoding matrix of user

k, such as

x(n) =

K∑k=1

V[k](n)s[k](n), n = 0 . . . N − 1. (4.6)

104

4.4. PRECODING AT THE TRANSMITTER SIDE

The received signal for user k is given by,

y[k](n) = H[k](n)V[k](n)s[k](n) +∑j 6=k

H[k](n)V[j](n)s[j](n) + n[k](n), n = 0 . . . N − 1. (4.7)

where the second term represents the multi-user interference from every other user’s signal.

The precoding matrices V[k], k = 1 . . .K must satisfy conditions in order to eliminate the

multiuser interference. In the following, we consider two linear precoding schemes which

eliminate the multiuser interference when nt ≥ Knr : channel inversion and block diagonal-

ization.

Zero forcing(ZF)

Zero forcing at the transmitter side allows to cancel the multiuser and the inter-antenna

interference. The precoding matrices V[k](n) ,k = 1 . . .K are chosen to eliminate multiuser

and inter antenna interference, such that

h[k]i (n)v

[j]l (n) = 0, ∀j 6= k ∈ [1,K], ∀i, l ∈ [1, nr], (4.8)

h[k]i (n)v

[k]l (n) = 0, ∀l 6= i ∈ [1, nr], (4.9)

where vj,l(n) denotes the lth column vector of Vj(n).

The received signal is given by

y[k]i (n) = h

[k]i (n)v

[k]i (n)s

[k]i (n) + n

[k]i (n), i = 1 . . . nr, n = 0 . . . N − 1. (4.10)

The channel inversion converts the MIMO broadcast channel into KN parallel channels with

effective channel g[k]i (n) = h

[k]i (n)v

[k]i (n). Due to the isotropic nature of i.i.d Rayleigh fading,

this orthogonality constraint consumes Knr − 1 degrees of freedom at the transmitter, and

reduces the channel from the 1× nt vector h[k]i to a (nt −Knr + 1)× 1 Gaussian vector.

The effective channel norm∣∣g[k]i (n)

∣∣2 of each parallel channel is chi-squared with 2(nt−Knr+

1) degrees of freedom. If single user detection and signaling are used, the achievable rate of

user k is given by

Rk =1

N

N−1∑n=0

nr∑i=1

log2

(1 +

P

Knr

∣∣g[k]i (n)

∣∣2). (4.11)

and the total multiplexing gain that can be achieved is such that,

r1 + . . .+ rK = limP→∞

∑k Rk

log2 P= Knr = min(nt,Knr). (4.12)

Although, zero forcing scheme allows to extract all the available degrees of freedom in the

channel, this scheme suffers from significant loss in term of

- diversity at the Rx side, for the case when destinations have multiple antennas(nr ≥ 1).

105


As the zero forcing scheme eliminates also inter-antenna interference, the destination

cannot perform joint processing of data at the Rx side, which entails a loss in the

receiver diversity order. The block diagonalization scheme proposed by Choi and Murch

in [23] is a solution for this problem.

- power, for ill-conditioned channel case. As the system is subjected to a total power

constraint, the channel inversion of ill-conditioned channel reduces the signal power.

A fix to this problem is to use non-linear precoding scheme. A vector version of the

Tomlinson-Harashima precoding [68,69] scheme, also known as the vector perturbation

scheme, is proposed in [5] and is described in the following. An intuitive improvement

of the vector perturbation scheme based on the use of the so-called periodically flipped

constellation is proposed in Subsection 4.4.2.

Block diagonalization(BD)

When the block diagonalization (BD) precoding schemes is used, the precoding matrices are

chosen in order to eliminate the multiuser interference, such that

H[k](n)V[j](n) = 0 ∀j 6= k ∈ [1,K], n = 0 . . . N − 1. (4.13)

The received signal at each user side is given by

y[k](n) = H[k](n)V[k](n)s[k](n) + n[k](n), n = 0 . . . N − 1. (4.14)

The BD converts the system into K parallel MIMO channels with effective channel matrices

G[k](n) = H[k](n)V[k](n), k = 1 . . .K. In the BD case, the orthogonality consumes (K −1)nr degrees of freedom. This reduces the channel matrix which is originally nr × nt to

nr× (nt− (K− 1)nr) complex Gaussian matrix. The nr×nr equivalent Gk(n)Gk(n)† matrix

is a Wishart matrix with nt− (K − 1)nr degrees of freedom. The achievable rate of user k is

given by

Rk =1

N

N−1∑n=0

log2 det(Inr +

P

ntG[k](n)G[k]†(n)

), (4.15)

and the total multiplexing gain is such that,

r1 + . . .+ rK = limP→∞

∑k Rk

log2 P= Knr = min(nt,Knr). (4.16)

4.4.2 Improving performance using periodically flipped constellations1

As detailed in the above section, zero forcing schemes suffers from the loss in power. The

equivalent zero forcing can be written in function of the pseudo-inverse H+ of the nt ×Knr1In this subsection, the time-frequency selective channel is also considered. However, for the sake of

simplicity of notations the time-frequency index is dropped. The notation H[k] stands for the user k channelmatrix at any time frequency.

106


destinations channel stack matrix, H =[H1 H2 . . . HK

]given by,

x ,1√γ

H+s (4.17)

with s being the vector of signals intended for different destinations, H+ = H†(HH†)−1 and

γ is a scaling factor required to satisfy the power constraint in (4.2). It is assumed that s

belongs to a constellation carved from the translated lattice Λ defined by

Λ , τcZK [i] + τc1 + i

2(4.18)

and is normalized in power, i.e. E |si|2 = 1 for all i. τc is the minimum distance between two

different points in the constellation. With the ZF precoding, the equivalent channel is

yk =1√γsk + zk, ∀ k (4.19)

For ill-conditioned channel, the scaling factor γ is very large, which imply a signal attenuation

and therefore a loss in term of power and diversity.

Vector Perturbation

A fix to this problem is to use non-linear precoding scheme. A vector version of the Tomlinson-

Harashima precoding [68, 69] scheme, also known as the vector perturbation scheme, is pro-

posed in [5] and is described briefly as follows. This transmitted signal x is

x =1√γ

H+(s + p(s)

)(4.20)

with p(s) ∈ P(s) being the perturbation vector. Thus, an obvious optimal choice of p is

p∗(s) = arg minp∈P(s)

‖x‖2

= arg minp∈P(s)

‖−H+s−H+p‖2. (4.21)

Note that the naive ZF scheme is a particular case of the above scheme, which can be seen by

setting trivially P(s) = 0. Therefore, the non-linear scheme is at least as good as the linear

scheme. In [5], P(s) is set as a sub-lattice τZK [i] of the lattice τcZK [i] independent of s.

The factor τ is chosen in order to get a periodic extension of the original signal constellation.

Thus, τ/τc ∈ Z and s + p(s) belongs to a coset of τZK [i] determined by the symbol vector2

s. The received signal for each user is

yk =1√γs′k + zk, s′k ∈ τcZ[i] + τc

1 + i

2,

2For a QAM signaling, it is readily shown that the cardinality of the constellations (τ/τc)2.

107


where γ is the scaling factor required to satisfy the power constraint in (4.2), such that

The receiver tries to decide the most probable coset. For a hard detector, the closest

lattice point is first found and then is used to determine the representation s of the coset by

a mod-τZK [i] operation using the modulo function fτ (.) where fτ (y) = y −⌊y+τ/2τ

⌋τ .

Periodically Flipped Constellations

As shown above, the conventional vector perturbation scheme restricts the possible pertur-

bation vectors within the sublattice τZK [i]. In this section, we show that the performance

can be improved with another set of perturbation vectors. The motivation is shown by the

following example.

Motivating example

noise

replicated QPSK original QPSK

perturbation

detection

replicated QPSK

(a) Replicated constellation

noise

original QPSK

perturbation

flipped QPSKreplicated QPSK

detection

(b) Periodically flipped constellation

Figure 4.3: Extra protection provided by periodically flipped constellation.

For simplicity of demonstration, we consider, in this example, the special case of QPSK

modulation. Suppose that the source needs to transmit to some user k a symbol dk = −1+ i.

With the conventional vector perturbation scheme, a replicated constellation is used as an

108


infinite extension of the original constellation (cf. Fig. 4.3(a)). Let us assume that another

point (say, −5 + i) that is in the same coset turns out to minimize the transmit power and is

chosen. If the noise happens to draw the received symbol outside the constellation as shown

in Fig. 4.3(a), the receiver will make a wrong decision by searching the closest point in the

constellation to the received symbol.

The situation can be improved with a better choice of perturbation set, i.e. a better

infinite extension. The idea is shown in Figure 4.3(b). Assume that dk = 1 + i is the

information symbol. Instead of associating the information symbol with its periodically

replicated counterparts, as in the previous case, the original constellation is successively

flipped away. We call this constellation scheme the periodically flipped constellation. In

this example, the transmitter finds that −5 + i minimizes the transmit power over all the

associated points of dk = 1 + i in the PFC. Now, with the same noise as in the previous

case, the receiver can make a right decision by searching the closest point at the extended

constellation, i.e. the PFC. Thus, the overall error rate performance is improved.

Scheme definition

Mathematically, a PFC of a K-dimensional QAM constellation C can be represented bys + p(s) | s ∈ C, p(s) ∈ PPFC(s)

with the set of perturbation vectors defined by

PPFC(s) ,

p

∣∣∣∣∣∣∣∣∣∣∣

<pi∈(2τZ)∪(f(<si)+2τZ)

∀i = 1, . . . ,K,

=pi∈(2τZ)∪(f(=si)+2τZ)

∀i = 1, . . . ,K

(4.22)

where f(s) = τ−s, s ∈ R is the flip function; <x and =x represent the real and imaginary

parts of x, respectively. Since the above set is defined in a dimensional-wise manner, we can

rewrite it as

PPFC(s) ,

p

∣∣∣∣∣∣∣∣∣∣∣

<pi ∈ PPFC(<si)∀i = 1, . . . ,K,

=pi ∈ PPFC(=si)∀i = 1, . . . ,K

(4.23)

With the flipped replication, it is obvious that points at the border of the constellation

enjoy a better protection. This is due to the fact that the number of neighbours that are

at the minimum distance to any of these points is reduced by at least one3. For BPSK, the

number of neighbours of minimum distance to any symbol is 1 for PFC compared to 2 in the

conventional case. Similarly, this number is equal to 2 compared to 4 for QPSK and, to 2 or

3In the QPSK example, the number of neighbours is reduced by two for all constellation points.

109


3 compared to 4 for 16QAM.

The table 4.1 summarizes the number of neighbors of minimum distance to any symbol

in QAM constellations of different size.

M-QAM RC PFC

BPSK 2 1QPSK 4 2

16QAM 4 2 or 3

Table 4.1: Number of neighbors of minimum distance in QAM constellations.

At the receiver side, similar operation is performed as with the conventional vector per-

turbation schemes. More specifically, the closest lattice point is first found, and then is used

to determine the representation in the original coset using a modified modulo function which

corresponds to −fτ (.) when the closest point happens to be within one of the flipped constel-

lations and fτ (.) if not. Hence, detection complexity remains the same as the conventional

constellation with the PFC.

Numerical results

For illustration, we consider the case of a broadcast channel with 4 transmit antennas and 4

selected single antennas destination among a large number of destinations. In Fig. 4.4, we

compare for this antenna configuration the packet error rate for the conventional vector per-

turbation technique versus the PFC perturbation scheme when a convolutional code [133 171]

is used for a packet size of 1kB. A MIMO OFDM system, having N = 10 subcarrier is used.

On each subcarrier, the channel is considered flat.

We can see that the PFC provides a gain of 1.5dB for QPSK at PER = 10−2. Although

this technique protects the border points in the constellation, there is no gain for higher order

of modulations (16QAM and 64QAM). This loss in gain can be explained from the fact that

the power normalization factors with replicated constellation is, in average, smaller than the

one with periodically flipped constellation.

The proposed technique applies to data and pilot transmission, and is providing enhanced

protection for BPSK/QPSK preambles, signalisation fields and pilots.

Practical implementation 1: composition of sphere encoders

Since PPFC(s) can be seen as a union of 22K shifted sub-lattices 2τZK [i], finding the closest

point in PPFC(s) can be implemented by finding the closest points in each of the shifted

sub-lattices and then taking the one with minimum distance. In each shifted sub-lattice,

a standard sphere decoder can be used for the research. Therefore, the complexity of the

composition is that of the sphere decoder multiplied by a factor 22K . This exponential time

complexity becomes unacceptable for a large number of users K.

110

4.5. DIGITAL FEEDBACK ON SELECTIVE BC WITH ZF PRECODER

10−2

10−1

100

−5 0 5 10 15 20 25

SE perf for coded modulations

QPSK SE STD

QPSK SE PFC

16QAM SE STD

16QAM SE PFC

64QAM SE STD

64QAM SE PFC

Figure 4.4: Coded performance of PFC sphere encoder versus standard sphere encoder

Practical implementation 2: modified Schnorr-Euchner algorithm

A more efficient way proposed by Mazet, Yang et al. consists to modify the sphere decoder

in such a way that it can work directly on the set PPFC(s). This modification is possible

since each dimension of any point in PPFC(s) belongs to a manageable set. The main idea of

this Schnorr Euchnerr modification consists to take into account the manageable structure of

flips in the visiting order while searching for the optimal point. A more complete description

can be found in [70].

4.5 Digital feedback on selective BC with ZF precoder

4.5.1 Random vector quantization

The random vector quantization scheme has been presented previously in Section 3.6.1. When

a Zero Forcing (ZF) linear precoding scheme is used for a system with K users having nr

antennas receivers, the system is equivalent to a MIMO broadcast with nt transmit antennas

and Knr single antennas receivers.

111


Relationship between the matrix and its quantification

As it can be noticed from (4.5), it is sufficient to know h[k]ω,j ∈ C1×rnt to determine the

channel at each time frequency slot and at each antenna j = 1 . . . nr. If we let a be the

quantization error between the normalized vector h[k]ω,j and its quantified vector h

[k]ω,j as defined

in Subsection 3.6.1. Then h[k]ω,j can be written as the sum of two vectors, one, h

[k]ω,j in the

direction of the quantized vector and the second, sj is isotropically distributed in the null-

space of h[k]ω,j , independent of a as shown in [6], such that

h[k]ω,j =

1

‖h[k]ω,j‖

h[k]ω,j =

√1− a h

[k]ω,j +

√a sj .

This implies that,

h[k]j (n) = h

[k]ω,jΓ(n),

= ‖h[k]ω,j‖

(√1− a h

[k]ω,jΓ(n) +

√a sj Γ(n)

),

= ‖h[k]ω,j‖

(√1− a h

[k]j (n) +

√a σt,n sj Γ(n)

), (4.24)

with E[‖h[k]ω,j‖] = rnt and sj Γ(n) is a unitary vector.

4.5.2 Throughput analysis

Theorem 4.1 For the K-selective MIMO broadcast channel with nt transmit antennas at the

source and nr receive antennas at the destinations(nt = Knr) when a zero forcing scheme is

used, the total spatial multiplexing gain of K can be achieved using a quantization scheme if

the number of feedback bits Nf broadcast by each user scales as,

Nf = nr(rnt − 1) log2 P, (4.25)

where r ≥ DW∆H is the rank of the selective fading channel covariance matrix.At high SNR,

the rate loss incurred by the above quantization scheme is upper bounded by,

∆R ≤ nr log2

(1 + σ2

t

r(K − 1)

rnt − nr

),

Proof: Let ∆Rk = RQuant − RFull CSIT be the rate loss incurred by the quantization,

then due to the isotropic nature of the channel matrices, the rate loss can be written such

112


that,

∆Rk ≤ E[ 1

N

N−1∑n=0

nr∑j=1

log2

(1 +

P

nt

∑i 6=k‖h[k]

j (n)v[i](n)‖2)],

≤ nrN

N−1∑n=0

log2

(1 +

P

nt

∑i 6=k

E[‖h[k]

j (n)v[i](n)‖2]),

≤ nrN

N−1∑n=0

log2

(1 +

P

nt(K − 1)E

[‖h[k]

j (n)v[i](n)‖2])

(4.26)

Using the relation between the channel vector and its quantized channel vector in (4.24),

and from the ZF constraint,

h[k]j (n)v[i](n) = h

[k]ω,jΓ(n)v[i](n) = 0, i 6= k, (4.27)

it follows that,

h[k]j (n)v[i](n) = ‖h[k]

ω,j‖√a σt,n sj Γ(n)v[i](n),

where sj is a unit vector isotropically distributed in the null-space of h[k]ω,j as mentioned above

and Γ(n)v[i](n) is also a unit vector isotropically distributed in the null-space of h[k]ω,j as a

consequence of the zero forcing constraint in (4.27). Then, these two vectors are distributed

in the rnt−1 nullspace of h[k]ω,j . As shown in Appendix C, the angle between these two vectors

∈ Crnt−1 is therefore beta distributed with parameters β(1, rnt − 2), and hence,

E[‖h[k]

j (n)v[i](n)‖2]

= σ2t,n E[β(1, rnt − 2)]E[‖h[k]

ω,j‖]E[a],

=1

rnt − 1rnt E[a].

It is well know from [6] that the quantization error a corresponds the minimum angle be-

tween the channel vector and the 2B codebooks vector, and therefore it is distributed as the

minimum between 2B beta variables, and

E[a] ≤ 2− Brnt−1 .

Then, the rate loss is therefore upper bounded by,

∆Rk ≤nrN

N−1∑n=0

log2

(1 +

r(K − 1)

rnt − 1σ2t,n P2

− Brnt−1

),

The maximal multiplexing gain can be achieved, if the gap capacity between the full CSIT

and the quantized capacity are independent of P . This occurs if the number of bits scale as

113


(rnt − 1) log2 P , then ∆Rk is constant and independent of P , such that

∆Rk ≤ nrN

N−1∑n=0

log2

(1 + σ2

t,n

r(K − 1)

rnt − 1

),

≤ nr log2

(1 + σ2

t

r(K − 1)

rnt − 1

), σ2

t = maxn=0...N−1

σ2t,n.

Consequently,

R = R−∆R, (4.28)

≥ R− c, (4.29)

≥ Knr log2 P − nr log2

(1 + σ2

t

r(K − 1)

rnt − nr

), (4.30)

and therefore the maximal multiplexing gain can be achieved, but with a constant capacity

gap.

4.5.3 Numerical results

In order to illustrate the reduced qunatized scheme, we consider the MIMO broadcast chan-

nel with K = 3 destinations having nr = 2 receive antennas each and a source with nt = 6

transmit antennas. We assume that the communication occurs over a radio channels char-

acterized by the parameters in Table 4.2 (Table 2.1 in [8]). These parameters correspond to

the context of the standard IEEE 802.16 (or WIMAX).

Key channel and signal parameters Values

Carrier frequency fc 2.5 GHzCommunication bandwidth W 1 MHzDelay requirement D 50msDoppler spread ν0 100 HzCoherence time Tc 2.5msDelay spread τ0 1µsCoherence bandwidth Wc 500 KHzChannel spread ∆H 10−4

Table 4.2: Channel and signal parameters

In this case, it can be easily verfied that the signal duration and bandwidth are much

larger than the coherence bandwidth and the coherence time of the channel. The channel is

therefore selective in time and frequency. This channel can be approximately decomposed

into parallel time-frequency channel using Weyl-Heisenberg sequences as explained in Section

4.2, where the channel grid parameter T and F are chosen such that,

TF ≤ 1

∆H= 104.

114


We assume in the following that TF = 103. In this case, the number of time-frequency slots

is such that,

N =DW

TF= 50.

and the sufficient number of parameters required to identify the channel is,

ρ = bDW∆Hc = 10.

For this channel and signal model, the total throughput transmitted is depicted in Figure

4.5. The source reconstructs the channel using a random vector quantization (RVQ) and a

limited of feedback bits. A zero forcing precoding is performed using the quantized channel.

As it can be shown, when using a reduced number of feedback bits that scales as shown

in Theorem 4.1, the total multiplexing gain of min(nt,Knr) = 6, which is the same as the

full CSIT case. The same multiplexing gain can be observed when using the straightforward

strategy previously defined. As it can be shown in Figure 4.6, the number of feedback bits

using the reduced strategy we proposed is significantly reduced compared to the straight-

forward strategy. As for the flat fading case, when using a constant number of feedback,

the total multiplexing cannot be achieved. However, the statistical knowledge of the channel

combined with the reduced feedback gives better reconstruction of the channel.

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25 30 35 40

To

tal

Da

ta R

ate

Transmit Power (dB)

Total Data Rate in the Cell

Full Channel KnowledgeN = 250 log(SNR) bits

N = 59 log(SNR) bits N = 29 log(SNR) bits N = 100 bits Same throughput

of feedback bits

Insufficient numberto full channel knowledge

Small gap compared

Figure 4.5: Capacity of a broadcast channel with nt = 6 transmit antennas and K = 3users having nt = 2 antennas each, when Zero Forcing (ZF) precoding is performed at thetransmitter side.

115


0

1000

2000

3000

0 5 10 15 20 25 30 35 40

Fee

dbac

k bi

ts (

bits

)

SNR(dB)

Comparison of the number of feedback bits

Straightfoward Feedback (SF)Reduced Feedback (RF)

Figure 4.6: Reduced feedback vs Straightforward Approach.

4.6 Digital feedback on selective BC with BD precoder

In this section, we propose a quantization scheme for the block diagonalization when a time-

frequency selective channel is considered. Based on the observation that time-frequency

selective channel are correlated, we compute the minimal number of feedback bits required

to achieve the full multiplexing gain. These quantization codebooks are constructed over a

Grassmann manifolds which we present in Subsection 4.6.1. Then, we define the quantization

codebook in Subsection 4.6.2 and derive in Subsection 4.6.3 the minimal number of feedback

bits needed to achieve the full multiplexing gain.

4.6.1 Preliminaries on Grassmann manifolds

The Grassman manifolds are generally used for quantifying M × T dimensional matrix with

T ≥ M . Before going to the quantization codebook design over Grassmann manifold, we

start first by defining the Grassmann manifold as defined in [71].

Definition 4.1 (Stiefel manifold) The Stiefel manifold S(T,M) for T ≥M is defined as

the set of all unitary matrices M × T , i.e.,

S(T,M) =Q ∈ CM×T : QQ† = IM

.

116

4.6. DIGITAL FEEDBACK ON SELECTIVE BC WITH BD PRECODER

The real dimension of the Stiefel manifold is given by

dim(S(T,M)) = 2TM −M2.

Definition 4.2 (Grassmann manifold) The Grassmann manifold G(T,M) is defined as

the quotient space of S(T,M) with respect to the following equivalence relation on the Stiefel

manifold:

P, Q ∈ S(T,M) are equivalent if the row vectors (T -dimensional) span the same subspace,

i.e., P = UQ for some unitary M ×M matrix U. The dimensionality of the Grassmann

manifold is given by,

dim(G(T,M)) = M(T −M).

4.6.2 Quantization codebook design

As the time-frequency channel matrices H[k](n) are correlated, it is not necessary that the

receiver feeds back channel at each time-frequency slot. As it can be shown from Lemma 4.1,

it is sufficient that the transmitter knows matrix H[k]ω ∈ Cnr×rnt defined in the following to

reconstruct the channel at each time-frequency slot.

Codebook design

It can be easily deduced from Lemma 4.1 that the knowledge of H[k]ω ∈ nr × rnt is sufficient

to know the channel at each time-frequency slot. Usually, quantized matrices are chosen in

a Grassmaniann manifold G(T,M), where T > M . That’s why, the quantization problem of

selective fading channel consists in finding a quantization for H[k]†ω ∈ Crnt×nr (with rnt ≥ nr).

The quantization codebook is known at the transmitter side as well at the receivers side.

Each receiver uses a different codebook Ck of 2B unitary matrices in Crnt×nr , such that

Ck =W1, . . . ,W2B

,

where, B is the number of feedback bits allocated per user. Each user(to say user k) feeds

back the index of the W matrix that is closest in term of its chordal distance to the channel

matrix H[k]†ω ∈ Crnt×nr , i.e.,

H[k]ω = arg min

W∈Ckd2(H[k]

ω ,W), (4.31)

where d is the chordal distance between two matrices.

Each of the 2B unitary matrices are chosen independently and are uniformly distributed over

a Grassmaniann G(rnt, nr). As shown in [7] (and references therein), the distortion associated

117


with a given codebook Ck for the quantization of H[k]†ω ∈ Crnt×nr is such that

Ds = E[d2(H[k]

ω , H[k]ω )]≤ D, (4.32)

where D is equivalent when the number of bits B goes to infinity to,

DB→∞−−−−→ C 2

− Bnr(rnt−nr) . (4.33)

and C is a constant independent of B given by

C =Γ(1

g )

(g + 1)

( nr∏i=1

(rnt − i)!g! (nr − i)!

)− 1g. (4.34)

with g = nr(rnt − nr) is the dimensionality of the Grassmanian manifold.

Relationship between the matrix and its quantification

The following lemma derived in [7] is a key result that will be used in the following in order

to derive the minimal number of bits required when a block diagonalization scheme is used.

Lemma 4.2 (Lemma 1 in [7]) The quantization H ∈ CT×M of a channel matrix H ∈CT×M , with T > M can be decomposed as following,

H = HXY + SZ, (4.35)

where

- H ∈ CT×M is an orthonormal basis of the subspace spanned by the columns of H, given

by the left singular vectors decomposition of HH†, such that

HH† = HΛH†.

- X ∈ CM×M is a unitary and uniformly distributed matrix over G(T,M).

- Z ∈ CM×M is upper triangular matrix with positive diagonal elements, that satisfies

Tr(ZZ†) = d2(H, H),

E[ZZ†] =D

MIM ,

where D is the quantization distortion defined as in (4.32) and (4.33).

- Y ∈ CM×M is upper triangular matrix with positive diagonal elements, that satisfies

YY† = IN −ZZ†.

- S ∈ CT×M is an orthonormal basis for the isotropically distributed M -dimensional

plane in the T −M dimensional left null space of H.

118

4.6. DIGITAL FEEDBACK ON SELECTIVE BC WITH BD PRECODER

The matrices H, Y and X are distributed independently one of each other. Also, S and Z

are distributed independently one of each other.

As we can see from Lemma 4.1, the knowledge of H[k]ω is sufficient to reconstruct the

time-frequency user channel matrix. By applying Lemma 4.2 to matrix H[k]†(n), and its

quantized channel matrix H[k]†(n), then

σt,nΓ(n)†H[k]ω (n) = σt,nΓ(n)†H[k]†

ω X[k](n)Y[k](n) + σt,nΓ(n)†S[k](n)Z[k](n)

= H[k]†(n)X[k](n)Y[k](n) + σt,nΓ(n)†S[k](n)Z[k](n) (4.36)

4.6.3 Throughput analysis

Theorem 4.2 For the K-selective MIMO broadcast channel with nt transmit antennas at the

source and nr receive antennas at the destinations(nt ≥ Knr) when a block diagonalization

scheme is used, the total spatial multiplexing gain of Knr can be achieved using a quantization

scheme if the number of feedback bits Nf broadcast by each user scales as,

Nf = nr(rnt − 1) log2 P, (4.37)

where r ≥ DW∆H is the rank of the selective fading channel covariance matrix. At high

SNR, the rate loss incured by the above quantization scheme is upper bounded by,

∆R ≤ nr log2

(1 + σ2

t

r(K − 1)

rnt − nrC

),

where C is a constant defined as in equation (4.34).

Proof: Unlike the flat fading channel, the transmitter needs to reconstruct its channel

matrix based on the quantized matrix version using Lemma 4.1. The same footsteps as in [7]

are used, with the major difference that quantization is not made on the channel itself, but

on a related version of this channel as shown previously in Lemma 4.1.

Let ∆Rk = RQuant−RFull CSIT be the rate loss incurred by the quantization, then due to the

isotropic nature of the channel matrices, the rate loss can be written such that,

∆Rk ≤ E[ 1

N

N−1∑n=0

log2 det(Inr +

P

ntH[k](n)

(∑j 6=k

V[j](n)V[j]†(n))H[k]†(n)

)], (4.38a)

≤ E[ 1

N

N−1∑n=0

log2 det(Inr +

P

ntH[k]†(n)H[k](n)

(∑j 6=k

V[j](n)V[j]†(n)))]

,(4.38b)

119


Equation 4.38b follows from the identity log | I +AB| = log | I +BA|. By replacing H[k](n)

by its value in Lemma 4.1, then

∆Rk ≤ E[ 1

N

N−1∑n=0

log2 det(Inr +

P

ntσ2t,n Γ(n)†H[k]†

ω (n)H[k]ω (n)Γ(n)

(∑j 6=k

V[j](n)V[j]†(n)))]

,

Let H[k]ω (n)Λ[k]†(n)H

[k]†ω (n) be the svd decomposition of H

[k]†ω (n)H

[k]ω (n), then

∆Rk ≤ 1

N

N−1∑n=0

E[

log2 det(Inr +

P

ntσ2t,n Γ(n)†H[k]

ω (n)Λ[k](n)H[k]†ω (n)Γ(n)

(∑j 6=k

V[j](n)V[j]†(n)))]

,

≤ 1

N

N−1∑n=0

E[

log2 det(Inr +

P

ntσ2t,n H[k]†

ω (n)Γ(n)(∑j 6=k

V[j](n)V[j]†(n))Γ(n)†H[k]

ω (n)Λ[k](n))],

≤ log2 det(Inr +

P

nt(K − 1)σ2

t E[H[k]†ω (n)Γ(n)V[j](n)V[j]†(n)Γ(n)†H[k]

ω (n)]︸︷︷︸

(a)

E[Λ[k](n)]︸︷︷︸(b)

),

with

σ2t = max

n=0,...,N−1σ2t,n

The singular value matrix of a Wishart matrix H[k]†ω (n)H

[k]ω (n) is such that,

(b) = E[Λ[k](n)] = rnt Inr

Using the relationship in (4.36) between the channel matrix and its quantization, and the

block diagonalization implications,

H[k](n)V[j](n) = Hω[k]

(n)Γ(n)V[j](n) = 0, (4.39)

we have,

(a) = E[Z[k]†(n)

(S[k]†(n)Γ(n)V[j](n)V[j]†(n)Γ(n)†S[k](n)

)Z[k](n)

]= E

[Z[k]†(n)U(n)Z[k](n)

].

As Γ(n)V[j](n) is an orthonormal matrix in the left null-space of Hω[k]

(n) (BD requirement

in (4.39)), and S[k]†(n) is in the left null-space of Hω[k]

(n) as shown in Lemma 4.2, then

U(n) = S[k]†(n)Γ(n)V[j](n)V[j]†(n)Γ(n)†S[k](n) ∼ Beta(nr, rnt − 2nr)

distributed (please refer to appendix C for more details.). For this matrix, we have

E[Z[k]†(n)U(n)Z[k](n)

]=

nrrnt − nr

E[Z[k]†(n)Z[k](n)

].

120

4.7. SELECTIVE MIMO BROADCAST CHANNEL WITH ANALOG FEEDBACK

As shown in Lemma 4.2,

E[Z[k]†(n)Z[k](n)] =D

nrInr .

Therefore,

∆Rk ≤ nr log2

(1 + P (K − 1)σ2

t

r

rnt − nrD

), (4.40)

From the quantization codebook design in Subsection 4.6.2, the distortion error rate is such

that,

Ds ≤ D (4.41)

where D is equivalent for large B to

DB→∞−−−−→ C 2

− Bnr(rnt−nr) . (4.42)

and C is a constant independent of B. It follows that,

∆Rk ≤ nr log2

(1 + Pσ2

t

r(K − 1)

rnt − nrC 2

− Bnr(rnt−nr)

), (4.43)

It can be easily deduced that, ∆R is independent of P , if and only if B scales as nr(rnt −nr) log2 P , then

∆R ≤ nr log2

(1 + σ2

t

r(K − 1)

rnt − nrC

)= c,

where c is a constant independent of P . Consequently,

R = R−∆R (4.44)

≥ R− c (4.45)

≥ Knr log2 P − nr log2

(1 + σ2

t

r(K − 1)

rnt − nrC

)(4.46)

and therefore the maximal multiplexing gain can be achieved, but with a constant capacity

gap.

4.7 Selective MIMO broadcast channel with analog feedback

4.7.1 Analog feedback scheme

For the analog feedback case, each user should send back to the transmitter each of rntnr

channels coefficients β times in order to reconstruct the channel at each time-frequency slot.

We assume that these coefficients are sent on an unfaded uplink AWGN channel with the

same power as the downlink scheme. In this case, the number of feedback bits required for

analog feedback is,

Nf = βntnrr log2(1 + P ).

121


h[k]ω,1h

[k]ω,0 h

[k]ω,r−1User k

βTs

rβTs

Figure 4.7: Each user feedbacks to the source its r ≥ DW∆H channel components on aAWGN channel. Each coefficient is transmitted during β time slots.

4.7.2 Relationship between the channel and its analog quantification

The matrix received at the transmitter side is such that

G[k]ω =

√βPH[k]

ω + N[k]ω (4.47)

where F[k]ω is the feedback noise, with i.i.d CN (0, 1) entries.

In order to estimate H[k]ω , the MMSE estimator is used, this implies that,

H[k]ω =

√βP

1 + βPG[k]ω

At each time-frequency slot, the estimated channel matrix is such that,

H[k](n) = H[k]ω Γ(n) =

√βP

1 + βPG[k]ω Γ(n)

or equivalently,

H[k](n) = H[k](n) +1√

1 + βPF[k]ω Γ(n) (4.48)

with F[k]ω is a Gaussian matrix with i.i.d entries. At each receive antennas, the vector channel

can be expressed as,

h[k]j (n) = h

[k]j (n) +

1√1 + βP

f[k]j,ω Γ(n), j = 1 . . . nr. (4.49)

122

4.7. SELECTIVE MIMO BROADCAST CHANNEL WITH ANALOG FEEDBACK

4.7.3 Zero forcing with analog feedback

When there is no perfect CSI at the transmitter side, the rate gap between the full CSIT and

quantized CSIT is bounded as shown in (4.26). In this case,

‖h[k]j (n)v[i](n)‖2 =

σ2t,n

1 + βP‖f [k]j,ω Γ(n)v[i](n)‖2

=σ2t,n

1 + βP‖f [k]j,ω‖2 cos2

(f

[k]j,ω, Γ(n)v[i](n)

)∼

σ2t,n

1 + βPχ2

2rnrβ(1, rnr − 1) =σ2t,n

1 + βPχ2

2, (4.50)

where (4.50) follows from the fact that the angle between a unitary vector Γ(n)v[i](n) and

the Gaussian unit vector f[k]j,ω is beta distributed.

This implies that,

∆Rk ≤ nrN

N−1∑n=0

log2

(1 +

P

nt(K − 1)

σ2t,n

1 + βP

)≤ nr log2

(1 +

σ2t

nt(K − 1)

P

1 + βP

), σ2

t = maxn=0...N−1

σ2t,n,

≤ nr log2

(1 +

σ2t

βnt(K − 1)

), P →∞.

and which is independent of P , and therefore the full multiplexing gain can be achieved.

4.7.4 Block diagonalization with analog feedback

The rate gap between the quantized channel rate and the perfect CSIT is bounded as shown

in (4.38a). This implies that,

∆Rk ≤1

N

N−1∑n=0

log2 det(

Inr +P

nt(K − 1)E

[H[k](n)V[j](n)V[j]†(n)H[k]†(n)

])

Using the relationship between the channel and its quantized version and the block diag-

onalization implications, it follows that,

H[k](n)V[j](n)V[j]†(n)H[k]†(n) =σ2t,n

1 + βPF[k]ω Γ(n)V[j](n)V[j]†(n)Γ(n)†F[k]†(n)

As all the columns of Γ(n)V[j](n) have unit norms, the equivalent matrix B = F[k]ω Γ(n)V[j](n)

123


is Gaussian matrix with i.i.d CN (0, 1) entries. This implies that,

∆Rk ≤ log2

(Inr +

P

nt(K − 1)

σ2t

1 + βPInr

)≤ nr log2

(1 +

σ2t

ntβ(K − 1)

)The rate loss incured by the analog feedback with block diagonalization is the same as the

one incured by Zero Forcing. This gap is independent of P , and therefore there is no loss in

the total number of the degrees of freedom in both cases. For the analog feedback, it is clear

that the rate gap does not depend on the number of channel parameters r as it was in the

case of the digital feedback. When β → ∞, the gap between the full CSIT and the analog

quantization goes to zero.

4.8 Conclusion

In this chapter, we consider the MIMO broadcast channel, where channels between source

and destination are assumed to be selective in time and frequency. We propose an intuitive

scheme based on the use of Periodically Flipped Constellation that improves the performance

of the vector perturbation scheme especially for low constellation order. While considering

the periodically flipped constellation instead of the replicated ones in the sphere encoder

scheme, the signal detection becomes more robust to noise perturbation, and this induces a

gain of 1.5 dB as shown in numerical results.

Then, the MIMO broadcast channel with limited feedback is studied. Two feedback

schemes are considered, the digital and the analog ones. We show that as time-frequency

channels are correlated it is not necessary to do the quantization on each time-frequency

channel itself. However, it is sufficient to reconstruct the channel based on a finite number of

parameters by making use of the correlation in time and frequency while conversing the full

spatial multiplexing gain. The optimal number of feedback bits required to achieve the full

multiplexing gain is computed for both cases of digital and analog. Moreover, the rate loss

incurred by the analog and the digital feedback schemes is also derived.

124

Conclusion and Perspectives

MOTIVATED by recent MIMO and multiuser standards (such as IEEE 802.11n,

IEEE 802.16m, LTE advanced, ...), two major problems are addressed in this

dissertation: the space time coding design in a standard context and the mul-

tiuser multiplexing gain for wireless channels with time-frequency selective links.

The non-vanishing determinant space time code designed from cyclic division algebra has

been considered in this work. For the selective fading channel, we proposed a new family

of split NVD parallel codes that achieve the optimal diversity multiplexing tradeoff of this

channel. The main hindrance to the practical implementation of these codes is the high com-

plexity order required at the receiver side. A more feasible scheme based on the use of perfect

code across each subcarrier is shown to be optimal if used in a MIMO-BICM system. Finally,

we show that in a standard context the expected gain of these codes cannot be observed at

moderate PER range.

In the multiuser context, we focused on the case where communication occurs on channel

that exhibits memory in time and frequency. This dissertation provides a unified matricial

framework for modeling the time-frequency selective channel. Using the fact that trans-

mitting and receiving on a set of Weyl-Heisenberg sequences approximately decomposes the

channel into N time-frequency parallel channels which are stationnary in time and frequency,

we proposed a useful form for modeling this channel which we call the polynomial channel

decomposition. We then show how the correlation between time-frequency channels can be

used in order to identify the channel.

The main purpose of studying multiuser systems is to show how to exploit the multiuser

multiplexing gain when channel between sources and destinations is time-frequency selective.

The first system that has been addressed in this dissertation is the interference channel. We

show that under certain channel spread restriction, interference alignment scheme allows to

achieve the maximal multiplexing gain for the not-so-large and for the large wireless networks.

A time-frequency interpretation of the interference alignment scheme has been also provided

for the three users case. The second system we considered is the MIMO broadcast channel.

For this channel, it is well-known that the maximal multiplexing gain that can be achieved

depends critically on the channel knowledge at the transmitter side. For the time-frequency

125

CONCLUSION AND PERSPECTIVES

selective channel, we show how the correlation between the time-frequency channels can be

used to minimize the number of feedback bits while conserving the maximal multiplexing

gain. The rate loss incurred by this feedback scheme has been also derived.

As perspectives for future works, we point out the following directions:

- Interleaver design in a standard context: In Chapter 2 of this dissertation, the in-

terleaver design has not been addressed. It is clear from the PEP expressions that

this latter can be minimized when the number of erroneous bits fall over D different

subcarriers. The interleaver design allows to maximize the parameter D, which is a

limiting factor of the diversity order. In the literature, the design of interleaver has

been adressed by Gresset in [25]. Potential efforts to improve the performace in a

BICM-MIMO-OFDM system could be investigated in the design of the interleaver.

- Design of CDA space time codes for the time-frequency selective channel: The optimal

DMT of this channel has been derived in [2]. In this case, the NVD parallel code cannot

be applied as the time component is not constant like in the case of the frequency

selective channel. An optimal scheme has been proposed in [2] based on the design of

a precoder adapted to channel statistics and the design of a code independent of the

channel statistics. It is of interest to design another alternative to achieve the DMT of

this channel using codes constructed from cyclic division algebra.

- Effect of intersymbol and intercarrier interference on the channel model: Throughout

this dissertation, the effect of the intersymbol (ISI) and intercarrier (ICI) interference

were neglected in the channel modeling. The sensitivity of the capacity to the channel

modeling has been studied by Durisi et al. in [26]. Using the same framework as

in [26], possible study on the sensitivity of the multiuser multiplexing gain to the

channel approximation model can be carried out.

- Large MIMO broadcast channel, selection algorithm: In Chapter 4, we considered the

case of MIMO broadcast channel where we assume that users are randomly selected

without specifying any selection algorithm to maximize the quality of service or to en-

sure fairness between users. Dealing with the large and not so-large MIMO broadcast

channel has been addressed in literature mainly in the work of Yoo in [27], Kountouris

et al. in [29], [28] and Zakhour in [67] for the flat fading channel. For the case of fre-

quency selective channel, an iterative scheduling algorithm that minimizes the number

of bits fed back to the transmitter has been proposed in [30]. Iterative algorithms that

maximize the multiuser diversity can be also extended to the case of time-frequency

selective channel.

126

Appendix A

Algebraic Tools

Definition A.1 (Field of rational Numbers - Algebraic Number - Number field) The

field of rational complex Q(i) is defined by Q(i) = x+ iy, x, y ∈ Q. The algebraic number θ

of degree non Q(i) is defined as the root of a minimal polynomial of degree n with coefficients

in Q(i).

The number field K on Q(i) is defined by

K = Q(i, θ) =

n−1∑i=0

aiθi, ai ∈ Q(i)

Use in Golden code construction

For the construction of GC, the number field Q(i,√

5) is used. It can be easily shown that√5 is an algebraic number as its minimal polynomial is X2 − 5 . The number field Q(i,

√5)

is given by

Q(i,√

5) =a0 + a1

√5, a0, a1 ∈ Q(i)

.

Definition A.2 The ring of integers is the subset of all the integers in a number field K =

Q(i, θ). This subset is denoted by OK.


In the previous example, the field number Q(i,√

5) has been introduced. The corresponding

ring of integers OK is generated by (1, 1+√

52 ). It can be shown that θ = 1+

√5

2 is also an

algebraic number and its minimal polynomial is given by θ2 − θ − 1 = 0. The number

θ = 1+√

52 is called the golden number, and its conjugate θ is the other root of the minimal

polynomial. Thus, OK =a0 + a1

1+√

52 , a0, a1 ∈ Z(i)

127

APPENDIX A. ALGEBRAIC TOOLS

Definition A.3 (Conjugates - Norm in K) The conjugates of an element x ∈ K are the

roots of minimal polynomial, given by :

σ1(x) = a+ bθ

σ2(x) = a− bθ

The norm of an element of K is the product of all its conjugates.

NK/Q(x) =

n∏i=1

σi(x) ∈ Q(i). (A.1)

If x ∈ Z(i), then NK/Q(x) ∈ Z(i).


Let x ∈ K ⊂ Q(i,√

5), i.e x = a+ bθ, a, b ∈ Q(i). Then, the norm of x is such that,

NK/Q(x) = σ1(x)σ2(x) = a2 − 2b2.

128

Appendix B

Weyl-Heisenberg Sequences

This framework gives a brief introduction on the construction of the set of Weyl Heisenberg

that are used throughout this dissertation. The essence of the following development has

been reviewed in Matz et. al in [72]. A more complete description can be found in Chapter

4 and 5 in [73] or in [74].

A primer on frames

Definition B.1 (Frame) A set of signal gj(t) is called a frame of an Hilbert space H if,

A‖x‖2 ≤∑j

|〈x, gj〉|2 ≤ B‖x‖2, x(t) ∈ H,

with B ≥ A ≥ 0.

Definition B.2 (Frame operator) The linear operator T associates to each signal x(t) a

sequence of inner products 〈x, gj〉,

T : x→ 〈x, gj〉j

The frame operator is defined as,

S = T∗T,

(Sx)(t) =∑〈x, gj〉gj(t).

Definition B.3 (Dual frame) The dual frame gj(t) of a frame gj(t) is defined as,

gj(t) = (S−1gj)(t).

129

APPENDIX B. WEYL-HEISENBERG SEQUENCES

Theorem B.1 (Signal expansion) Any signal in the Hilbert space can be written in func-

tion of the frame and its dual as,

x(t) =∑j

〈x, gj(t)〉gj(t).

Weyl Heisenberg sequences in LTV systems

When linear time vaying channel is considered, we already mention in chapter 1 that the

discrete I/O relationship can be obtained by transmitting and receiving on a set of Weyl

Heisenberg.

Let gm,l(t) be a set of Weyl-Heisenberg. The input function x(t) ∈ H can be expressed as,

x(t) =∑m,l

〈x, gm,l〉gm,l(t),

where,

x[m, l] = 〈x, gm,l(t)〉 =

∫x(t)g∗m,l(t)dt.

and

gm,l(t) = g(t−mT )ej2πlF t

is a time frequency shifted version of the transmit impulse g(t).

At the receiver side, the received signal r(t) is projected on a Weyl-Heisenberg set γm,l(t).

As stated in 1, these two Weyl-Heisenberg sets gm,l(t) and γm,l(t) should be biorthgonal. In

the following, the construction of these two weyl-Heisenberg sets is described.

Definition B.4 (Weyl-Heisenberg Riesz sequence) A sequence(g(t)m,l∈N, T, F

)is called

a Weyl-Heisenerg Riesz sequence of H if there exist two constant B′ ≥ A′ > 0, such that,

A′∑m,l

|x[m, l]|2 ≤∑m,l

|〈x, gm,l〉gm,l|2 ≤ B′∑m,l

|x[m, l]|2.

with

gm,l(t) = g(t−mT )ej2πlF t

and the grid parameters T and F satisfy TF ≥ 1.

If gm,l(t) is a Weyl-Heisenerg Riesz sequence, then there exists a non unique Weyl-

Heisenerg Riesz sequence γm,l(t) orthogonal to gm,l(t), such that,

〈gm,l(t)γm′,l′(t)〉 = δl,l′δk,k′

and that allows to reconstruct the received signal. If the gj(t)j∈N are linearly independent

then the frame is not redundant and is called a Riez basis.

130

Definition B.5 (Weyl-Heisenberg frame) A sequence(g(t)m,l∈N, T , F

)is called a Weyl-

Heisenerg frame of H if there exist two constant B ≥ A > 0, such that,

A‖x‖2 ≤∑j∈N|〈x, gm,l〉|2 ≤ B‖x‖2,

with

gm,l = g(t−mT )ej2πlF t,

and the grid parameters T and F satisfy T F < 1. When A = B, the frame is said to be tight.

If the frame condition is satisfied, this means any signal s(t) can be recovered from its

frame coefficients 〈s, gm,l〉 via the frame expansion,

s(t) =∑m,l

〈s, gm,l〉γm,l(t)

with

γm,l(t) = γ(t−mT )ej2πlF t

is the dual frame of gm,l(t).

For a tight frame,

γ(t) =T F

Ag(t).

Theorem B.2 (Duality of WH Riesz sequence and WH frames) Let gm,l(t) and γm,l(t)

be WH sets with parameters grids T and F , and gm,l(t) and γm,l(t) be WH sets with param-

eters grids T = 1F and F = 1

T . The following equivalence are verified.

1. gm,l(t) is Riesz sequence if and only gm,l(t) is a frame.

2. gm,l(t) and γm,l(t) are biorthogonal Riesz sequences if and only if gm,l(t) and

γm,l(t) are dual frames.

3. gm,l(t) is an orthogonal Riesz sequence if and only if gm,l(t) is a tight frame.

131

APPENDIX B. WEYL-HEISENBERG SEQUENCES

132

Appendix C

Beta Distribution Properties

Relation between the β and the χ2- distribution

Definition C.1 (Beta distribution) The beta β(p, q) has a continuous pdf inside the in-

terval [0, 1] given by,

p(x) = B(a, b)xa−1(1− x)b−1

where

B(a, b) =(a+ b− 1)!

(a− 1)!(b− 1)!

Lemma C.1 If X and Y are two independent chi-square variables with 2a and 2b degrees

of freedom respectively, then Z = XX+Y have the beta distribution with parameter a and b,

β(a, b).

Proof: To find the distribution of Z, we let U = X and V = XX+Y . The joint distribution

of u,v is given by

p(u, v) = pX(u)pY (u1− vv

)|u|v2

Using the marginalisation, we get

p(v) =

∫ +∞

−∞pX(u)pY (u

1− vv

)|u|v2du

Applying this formula to the chi-squared distributions, we get

p(z) =Γ(a+ b)

Γ(a)Γ(b)za−1(1− z)b−1

133

APPENDIX C. BETA DISTRIBUTION PROPERTIES

Geometrical application to the beta distribution

Theorem C.1 We consider a fixed 1- space line v ∈ C1×M and a Gaussian vector h, isotrop-

ically distributed in C1×M , such that h ∼ CN (0, IM ). Let θ be the angle between these two

vectors. The distribution of cos2(θ) is β(1,M − 1)

Proof: We first pick a random point v ∈ C1×M to form the first random line v = Ov.

We may regard v as a fixed one-space line, with unitary vector v = v/‖v‖. Next, we pick a

random point h(x1, . . . , xn) in the new basis generated by v and v⊥ = ker(v). Note that the

distribution of Oh does not change since the vector is isotropically distributed in the space.

It is basically known that

cos(θ) =projh(v)

‖h‖Thus,

cos2(θ) =x2

1

x21 + . . .+ x2

n

=x

x+ y

with X ∼ χ22 and Y ∼ χ2

2(M−1), then applying lemma we get cos2(θ) ∼ β(1,M − 1).

Product of two random variables

Lemma C.2 Let X and Y be two random variables, and Z = XY . The pdf of Z is given by

pZ(z) =

∫ +∞

−∞

1

|t|pX,Y (t, z/t)dt

Proof: If we let u = x, v = xy, then joint distribution of (u, v) is such that

p(u, v) = |J(u, v)|f(x(u, v), y(u, v))

with J(u, v) = 1u . The pdf of the product v is then

p(v) =

∫ +∞

−∞p(u, v)du

Example C.1 An interesting application can be for the case of

X ∼ χ22M , Y ∼ β(1,M − 1)

In this case, Z = XY is exponentially distributed (χ22)

Proof:

pX(x) =1

(M − 1)!x(M−1)e−x

134

and

pY (y) = (M − 1)(1− y)M−2 0 ≤ y ≤ 1

By simply using the above formula, we get

p(z) =1

(M − 2)!

∫ ∞z

e−t(t− z)M−2dt

=1

(M − 2)!e−z

∫ ∞0

e−uuM−2du

= e−z

Matrix variate Beta distributions

Definition C.2 (Definition 5.2.1 [75]) A p×p random symmetric positive definite matrix

U is said to have a matrix variate beta distribution with parameters (a, b), denoted as, U ∼Beta(a, b), if its pdf is given by,

βp(a, b)−1 det(U)a−

12

(p+1) det(Ip−U)b−12

(p+1).

with

βp(a, b) =

∫ p

0ta−1(1− t)b−1dt.

Theorem C.2 (Theorem 5.3.12 and 5.3.19 in Chapter 5 [75]) Let U ∼ Beta(a, b). Then

for a constant matrix Z ∈ Cq×q of rank q ≤ p,we have,

E[Z†UZ

]=

a

a+ bE[Z†Z

]

135

APPENDIX C. BETA DISTRIBUTION PROPERTIES

136

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About the author

Lina Mroueh was born in Lebanon, on June 29, 1983. She received her engineering diploma

from Telecom ParisTech former Ecole Nationale Superieure des Telecommunications de Paris

in 2006 and her master degree from Universite Pierre et Marie Curie, France in the same

year. Since March 2006, she has been pursuing her PhD in the group of Professor Jean-

Claude Belfiore.

From 2006 to 2008, she worked with the radio link technology team in Motorola Labs as

a research engineer and participated to the IEEE 802.11n project, and to other internal and

European projects. From February 2009, she joined the communication theory group (CTG)

led by Prof. Helmut Bolcskei in ETH Zurich, Switzerland as a visitor researcher.

Publications

The content of this thesis was submitted to the following conferences and journal papers.

Journal papers

- L. Mroueh and J-C. Belfiore, ”How to achieve the optimal DMT in selective fading

channel?,” submitted to IEEE Transaction on Information Theory.

- L. Mroueh, S. Rouquette-Leveil and J-C. Belfiore, ”Application of perfect space time

codes: Optimality and practical limits,”submitted to IEEE transaction on communica-

tion.

- L. Mroueh, S. Rouquette-Leveil and J-C. Belfiore, ”Reduced feedback for selective

fading MIMO broadcast channel,” submitted to EURASIP Journal, special issue on

Wireless Communications on Recent Advances in Multiuser MIMO Systems.

143

Conference papers and Patent

Conference papers and filed patent related to topics treated in this manuscript are as follow-

ing.

- L. Mroueh and J-C Belfiore, ”How to achieve the optimal DMT of selective fading

channel?” , To appear in the Information Theory Workshop proceeding, ITW 2010,

Sept 2010.

- L. Mroueh, S. Rouquette-Leveil, G. Rekaya-Ben Othman and J-C Belfiore, ”On the

performance of the Golden code in BICM-MIMO and IEEE 802.11n cases ”, Asilomar

Conference on Signals, Systems and Computers, California, USA, November 2007.

- L. Mroueh, O. Damen, S. Rouquette Leveil, G. Rekaya-Ben Othman, and J-C. Belfiore,

“Code construction for the selective cooperating broadcast channel,” Sept 2008, invited

paper to (NWMIMO) Workshop in PIMRC 2008 conference.

- L. Mroueh, S. Rouquette Leveil, G. Rekaya-Ben Othman, and J-C. Belfiore, “DMT

achieving schemes for the isotropic fading broadcast channel,” Sept 2008, in the pro-

ceeding of the IEEE International Symposium on Personal, Indoor and Mobile Radio

Communications (PIMRC), Cannes (France).

- L. Mroueh, S. Rouquette Leveil, G. Rekaya-Ben Othman, and J-C. Belfiore, “DMT of

weighted parallel channels: Application to the broadcast channel,” July 2008, in pro-

ceeding of the International Symposium on Information Theory ISIT, Toronto, Canada.

- L. Mroueh, S. Rouquette Leveil, L. Mazet and M de Courville, ”Periodically Flipped

Constellation for MIMO broadcast channel”, US filed patent.

c© Copyright by Lina Mroueh, 2009.

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