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.
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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-
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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.
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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.
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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-
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RESUME DETAILLE DE LA THESE
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-
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RESUME DETAILLE DE LA THESE
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.
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RESUME DETAILLE DE LA THESE
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].
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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
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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.
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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
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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.
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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
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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
INTRODUCTION AND OUTLINE
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
INTRODUCTION AND OUTLINE
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
CHAPTER 1. WIRELESS CHANNEL MODEL
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
CHAPTER 1. WIRELESS CHANNEL MODEL
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
CHAPTER 1. WIRELESS CHANNEL MODEL
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
CHAPTER 1. WIRELESS CHANNEL MODEL
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
CHAPTER 1. WIRELESS CHANNEL MODEL
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. UNIFIED MATRIX FORMULATION
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
CHAPTER 1. WIRELESS CHANNEL MODEL
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
1.4. UNIFIED MATRIX FORMULATION
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
CHAPTER 1. WIRELESS CHANNEL MODEL
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
1.4. UNIFIED MATRIX FORMULATION
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
CHAPTER 1. WIRELESS CHANNEL MODEL
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
CHAPTER 1. WIRELESS CHANNEL MODEL
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
2.2. STRUCTURED CODE CONSTRUCTION: A PRIMER
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
2.2. STRUCTURED CODE CONSTRUCTION: A PRIMER
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,
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CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
2.2. STRUCTURED CODE CONSTRUCTION: A PRIMER
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
2.2. STRUCTURED CODE CONSTRUCTION: A PRIMER
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
2.3. CODE CONSTRUCTION FOR SELECTIVE FADING CHANNEL
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
2.3. CODE CONSTRUCTION FOR SELECTIVE FADING CHANNEL
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
2.3. CODE CONSTRUCTION FOR SELECTIVE FADING CHANNEL
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
2.3. CODE CONSTRUCTION FOR SELECTIVE FADING CHANNEL
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
2.3. CODE CONSTRUCTION FOR SELECTIVE FADING CHANNEL
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
2.3. CODE CONSTRUCTION FOR SELECTIVE FADING CHANNEL
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
2.3. CODE CONSTRUCTION FOR SELECTIVE FADING CHANNEL
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
- 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
2.3. CODE CONSTRUCTION FOR SELECTIVE FADING CHANNEL
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
‖y(k)−He(k)Gx(k)‖2,
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
‖y(k)−He(k)Gx(k)‖2,
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
2.6. BICM-MIMO WITH FREQUENCY SELECTIVE CHANNELS
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
2.6. BICM-MIMO WITH FREQUENCY SELECTIVE CHANNELS
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
2.6. BICM-MIMO WITH FREQUENCY SELECTIVE CHANNELS
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
CHAPTER 2. NVD CODES IN STANDARDS APPLICATIONS
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
CHAPTER 3. INTERFERENCE ALIGNMENT FOR SELECTIVE FADING CHANNELS
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
CHAPTER 3. INTERFERENCE ALIGNMENT FOR SELECTIVE FADING CHANNELS
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
CHAPTER 3. INTERFERENCE ALIGNMENT FOR SELECTIVE FADING CHANNELS
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
3.4. TIME FREQUENCY DOMAIN INTERPRETATION
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
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CHAPTER 3. INTERFERENCE ALIGNMENT FOR SELECTIVE FADING CHANNELS
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
3.4. TIME FREQUENCY DOMAIN INTERPRETATION
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
CHAPTER 3. INTERFERENCE ALIGNMENT FOR SELECTIVE FADING CHANNELS
(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.
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CHAPTER 3. INTERFERENCE ALIGNMENT FOR SELECTIVE FADING CHANNELS
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
3.5. GENERAL SPREAD REQUIREMENTS FOR INTERFERENCE ALIGNMENT
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
CHAPTER 3. INTERFERENCE ALIGNMENT FOR SELECTIVE FADING CHANNELS
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
3.5. GENERAL SPREAD REQUIREMENTS FOR INTERFERENCE ALIGNMENT
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
CHAPTER 3. INTERFERENCE ALIGNMENT FOR SELECTIVE FADING CHANNELS
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
3.5. GENERAL SPREAD REQUIREMENTS FOR INTERFERENCE ALIGNMENT
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)
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CHAPTER 3. INTERFERENCE ALIGNMENT FOR SELECTIVE FADING CHANNELS
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,
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CHAPTER 3. INTERFERENCE ALIGNMENT FOR SELECTIVE FADING CHANNELS
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
3.6. INTERFERENCE ALIGNMENT WITH LIMITED FEEDBACK
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
CHAPTER 3. INTERFERENCE ALIGNMENT FOR SELECTIVE FADING CHANNELS
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.
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3.6. INTERFERENCE ALIGNMENT WITH LIMITED FEEDBACK
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.
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CHAPTER 3. INTERFERENCE ALIGNMENT FOR SELECTIVE FADING CHANNELS
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)
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CHAPTER 4. SELECTIVE BROADCAST CHANNEL WITH LIMITED FEEDBACK
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
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CHAPTER 4. SELECTIVE BROADCAST CHANNEL WITH LIMITED FEEDBACK
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).
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CHAPTER 4. SELECTIVE BROADCAST CHANNEL WITH LIMITED FEEDBACK
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
4.4. PRECODING AT THE TRANSMITTER SIDE
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.
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CHAPTER 4. SELECTIVE BROADCAST CHANNEL WITH LIMITED FEEDBACK
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
4.4. PRECODING AT THE TRANSMITTER SIDE
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.
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CHAPTER 4. SELECTIVE BROADCAST CHANNEL WITH LIMITED FEEDBACK
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.
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CHAPTER 4. SELECTIVE BROADCAST CHANNEL WITH LIMITED FEEDBACK
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
4.5. DIGITAL FEEDBACK ON SELECTIVE BC WITH ZF PRECODER
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
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CHAPTER 4. SELECTIVE BROADCAST CHANNEL WITH LIMITED FEEDBACK
(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
4.5. DIGITAL FEEDBACK ON SELECTIVE BC WITH ZF PRECODER
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
CHAPTER 4. SELECTIVE BROADCAST CHANNEL WITH LIMITED FEEDBACK
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
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CHAPTER 4. SELECTIVE BROADCAST CHANNEL WITH LIMITED FEEDBACK
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.
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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)
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CHAPTER 4. SELECTIVE BROADCAST CHANNEL WITH LIMITED FEEDBACK
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)
].
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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 ).
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CHAPTER 4. SELECTIVE BROADCAST CHANNEL WITH LIMITED FEEDBACK
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)
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CHAPTER 4. SELECTIVE BROADCAST CHANNEL WITH LIMITED FEEDBACK
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.
Use in Golden code construction
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).
Use in Golden code construction
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.
All right reserved.
Version 1.0
The materials published in this thesis may not be translated or copied in whole or in part
without the written permission of the author. Use in connection with any form of information
storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar
methodology now known or hereafter developed is forbidden.