Proposition and performance analysis of a multiuser system with joint transmit and receive generalized spatial modulation

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UNIVERSIDADE FEDERAL DE SANTA CATARINA PROGRAMA DE PÓS

-

GRADUAÇÃO EM ENGENHARIA

ELÉTRICA

Robinson Pizzio

PROPOSITION AND PERFORMANCE ANALYSIS

OF A MULTIUSER SYSTEM WITH JOINT

TRANSMIT AND RECEIVE GENERALIZED

SPATIAL MODULATION

Florianópolis

2018

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Robinson Pizzio

PROPOSITION AND PERFORMANCE ANALYSIS OF A MULTIUSER SYSTEM WITH JOINT TRANSMIT AND

RECEIVE GENERALIZED SPATIAL MODULATION

A Dissertation submitted to the Elec-trical Engineering Graduate Program in partial fulfillment of the require-ments for the degree of Doctor of Phi-losophy in Electrical Engineering. Supervisor: Bartolomeu Ferreira Uchôa Filho

Florianópolis

2018

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Pizzio, Robinson

Proposition and performance analysis of a multiuser system with joint transmit and receive generalized spatial modulation / Robinson Pizzio ; orientador, Bartolomeu Ferreira Uchôa-Filho, 2018. 133 p.

Tese (doutorado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós

Graduação em Engenharia Elétrica, Florianópolis, 2018.

Inclui referências.

1. Engenharia Elétrica. 2. Spatial Modulation. 3. Multiuser Systems. 4. Performance Analysis. 5. MIMO Systems. I. Uchôa-Filho, Bartolomeu Ferreira. II. Universidade Federal de Santa Catarina. Programa de Pós-Graduação em Engenharia Elétrica. III. Título.

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PROPOSITION AND PERFORMANCE ANALYSIS OF A MULTIUSER SYSTEM WITH JOINT TRANSMIT AND

RECEIVE GENERALIZED SPATIAL MODULATION Robinson Pizzio

This Dissertation is hereby approved and recommended for acceptance in partial fulfillment of the requirements for the degree of “Doctor of

Philosophy in Electrical Engineering.” August 03, 2018.

Prof. Bartolomeu Ferreira Uchôa Filho, Ph.D. Supervisor

Prof. Bartolomeu Ferreira Uchôa Filho, Ph.D. Coordinator of the Electrical Engineering Graduate

Program Examining Committee:

Prof. Bartolomeu Ferreira Uchôa Filho, Ph.D. – UFSC Chair

Prof. Gustavo Fraidenraich, Dr. – UNICAMP

Prof. Didier Le Ruyet, Ph.D. – CNAN

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Não diga que a vitória está perdida. Tenha fé em Deus, tenha fé na vida. Tente outra vez!

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This work is dedicated to the three most important people in my life, the main elements that kept me strong and persever-ant to accomplish this journey:

Aline Lemos Pizzio Bernardo Lemos Pizzio Henrique Lemos Pizzio

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ACKNOWLEDGEMENTS

I would like to thank my parents João Carlos Pizzio (in memoriam) and Sonia Helena Pizzio for all the inspiration and support they gave me during my education process. They mean the world to me.

I would like also to express my gratitude to Prof. Bartolomeu Ferreira Uchôa Filho. He has been much more than a supervisor. He is a friend, a colleague, a Professor, a confidant, well, a very important person in my life. In turbulent moments, he helped me find the way. Difficulties were just opportunities to learn and keep working. I have really learned a lot from you Bart. Thank you!

Also, a big thanks to the guys at the Communications and Embed-ded Systems Laboratory (LCS). I really appreciate the coffee breaks, in which the discussions and contributions were always productive, or at least relaxing.

Finally, this Dissertation is dedicated to my family. Especially my wife Aline for her unfailing love, continuous support and countless en-couragement during these years. To my kids, Bernardo and Henrique, a huge thanks for the understanding in the moments Daddy was work-ing on the Dissertation and could not stop to play with them. Thank you for being patient with me during this long journey. To my friends, thank you for keeping me focused, persistent, and for not giving up on me.

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ABSTRACT

Increasing the energy and spectral efficiencies is an important goal for the new generation of wireless communication systems, named 5G. In this direction, many signaling techniques have been proposed in the litera-ture, and spatial modulation (SM) is being considered as a very promising candidate. SM is a relatively new paradigm for input, multiple-output (MIMO) communications systems, by which information is trans-mitted not only by the choice of a symbol from a classical signal con-stellation, such as QAM, but also by the choice of an index that indicates a subset of (active) transmit antennas through which the signal is trans-mitted. The general goal of this Dissertation is to contribute to the new generation of wireless communication systems. Specifically, we propose innovations to the SM technique. As a main contribution of this Disser-tation, we propose a multiuser communication system that makes use of generalized spatial modulation where transmit and receive antennas are activated simultaneously. To this end, we develop a precoder and a post-coder for the downlink. Results show that both energy and spectral effi-ciencies are increased in comparison with previous generalizations of the technique. The second contribution of this Dissertation is the derivation of a novel performance analysis for the proposed scheme, more specifi-cally the derivation of upper bounds for the average bit error probability. Numerical results reveal a good precision with respect to the simulated results.

Keywords: Spatial Modulation, MIMO, precoder, performance analysis, multiuser systems.

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RESUMO

Aumentar a eficiência energética e espectral é uma meta importante para a nova geração de sistemas de comunicação sem fio, chamado 5G. Nessa direção, muitas técnicas de modulação tem sido propostas na literatura, e a modulação espacial (SM) está sendo considerada uma candidata pro-missora. SM é um paradigma relativamente novo para sistemas de co-municação com múltiplas entradas, múltiplas saídas (MIMO), pelo qual a informação é transmitida não apenas pela escolha de um símbolo de uma constelação de sinal clássico, como QAM, mas também pela esco-lha de um índice que indica um subconjunto de antenas de transmissão (ativas) através das quais o sinal é transmitido. O objetivo geral desta Tese é contribuir para a nova geração de sistemas de comunicação sem fio. Especificamente, propomos inovações para a técnica SM. Como uma contribuição principal desta Tese, propomos um sistema de comunica-ção multiusuário que faz uso da modulacomunica-ção espacial generalizada onde antenas de transmissão e recepção são ativadas simultaneamente. Para este fim, desenvolvemos um pré-codificador e um pós-codificador para o downlink. Os resultados mostram que tanto a eficiência energética quanto a eficiência espectral são aumentadas em comparação com generalizações prévias da técnica. A segunda contribuição desta Tese é a derivação de uma nova análise de desempenho para o esquema proposto, mais especi-ficamente derivação de limites superiores para a probabilidade média de erro de bit. Resultados numéricos revelam boa precisão em relação aos resultados simulados.

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LIST OF FIGURES

Figure 1 – 3D diagram of information coding for SM (single use of the channel). . . 29 Figure 2 – Illustration of three MIMO transmission paradigms: (a)

spatial multiplexing (SMX), (b) space-time coding (STC), and (c) spatial modulation (SM). . . 31 Figure 3 – A detailed illustration of the MU-TR-GSM scheme. . . 38 Figure 4 – Joint PDF of the complex random vector γ.. . . 80 Figure 5 – PDF of the real part of the complex random vector γ. . 81 Figure 6 – PDF of the imaginary part of the complex random

vec-tor γ. . . 81 Figure 7 – BER comparison for Scenario 1. In the proposed scheme,

the phase shifts technique has been adopted. . . 115 Figure 8 – BER comparison for Scenario 2. The proposed scheme

has been considered both with and without the phase shifts technique. . . 116 Figure 9 – Analytical curve compared with simulations curves for

Scenario 2. . . 117 Figure 10 – Trend results when varying the size of the active

trans-mit antennas index set.. . . 119 Figure 11 – Trend results when varying the number of active

trans-mit antennas.. . . 120 Figure 12 – Trend results when varying the number of active

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LIST OF TABLES

Table 1 – Multiplicities Nk1,...,kmfor n= 4 and m = 1, 2, 3 and 4 . . 55

Table 2 – Summary of the equations from the performance analysis112 Table 3 – System parameters for the simulations . . . 114 Table 4 – System parameters for the trend analysis: varying the

size of the active transmit antennas index set. . . 118 Table 5 – System parameters for the trend analysis: varying the

number of active transmit antennas. . . 119 Table 6 – System parameters for the trend analysis: varying the

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CONTENTS

1 INTRODUCTION . . . 25 1.1 Dissertation Contributions . . . 26 1.2 Dissertation Organization . . . 27

2 FUNDAMENTALS OF SPATIAL MODULATION 29

2.1 Spatial Modulation . . . 29 2.2 Evolution and State of the Art of Spatial Modulation 33 2.2.1 Spatial Modulation (SM): Activation of Single

Trans-mit Antenna . . . 33 2.2.2 Generalized Spatial Modulation (GSM): Activation

of Multiple Transmit Antennas . . . 34 2.2.3 Generalized Spatial Modulation in Multiuser MIMO

Systems . . . 34 2.2.4 Receive Spatial Modulation (R-SM): Receive

An-tenna Activation. . . 35

2.2.4.1 Receive Spatial Modulation in Multiuser MIMO Systems . 35

2.3 Summary . . . 36

3 MULTIUSER TRANSMIT AND RECEIVE

GEN-ERALIZED SPATIAL MODULATION . . . 37 3.1 Introduction . . . 37 3.2 System model and the proposed scheme . . . 37 3.3 Preprocessing and Postprocessing Matrices

Devel-opment . . . 40 3.4 Signal processing in the receiver . . . 43 3.5 Improving Common SM Signals Separation Through

Phase Shifts . . . 44 3.6 Summary . . . 46

4 PERFORMANCE ANALYSIS OF MU-TR-GSM . 47

4.1 Introduction . . . 47 4.1.1 Pairwise Error Probability: The Seven Cases . . . 48

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ues of a wishart matrix . . . 49 4.2.1 Joint and marginal PDFs of the eigenvalues of a Wishart

matrix . . . 50 4.2.2 Derivation of the Joint Moment Generating Function 52 4.3 The first case of the pairwise and average pairwise

error probabilities. . . 55 4.3.1 Statistics of the first case . . . 57 4.3.2 Average Pairwise Error Probability for the first case 61 4.4 The second case of the pairwise error probability . 62 4.4.1 Statistics of the second case . . . 66 4.5 The third case of the pairwise error probability . . 70 4.5.1 Statistics of the third case . . . 72 4.6 The fourth case of the pairwise error probability . 74 4.6.1 Statistics of the fourth case . . . 78 4.7 The fifth case of the pairwise error probability . . . 89 4.7.1 Statistics of the fifth case . . . 92 4.8 The sixth case of the pairwise error probability . . 95 4.8.1 Statistics of the sixth case. . . 99 4.9 The seventh case of the pairwise error probability . 102 4.9.1 Statistics of the seventh case . . . 105 4.10 Summary of the seven cases . . . 108 4.11 Summary . . . 111 5 RESULTS . . . 113 5.1 Introduction . . . 113 5.2 Simulation results . . . 113 5.3 Trend Results of the MU-TR-GSM . . . 117 5.4 Summary . . . 121

6 FINAL DISCUSSIONS AND FUTURE WORKS . 123

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LIST OF ABBREVIATIONS AND ACRONYMS

SISO Single input single output. . . 25 MIMO Multiple input multiple output. . . 25 IM Index Modulation. . . 25 SM Spatial modulation. . . 26 QAM Quadrature Amplitude Modulation. . . 26 RF Radio frequency. . . 26 MU-TR-GSM Multiuser Transmit and Receive Generalized Spatial

Modulation. . . 26 BER Bit error rate. . . 26 MU Multiuser. . . 26 VANETs Vehicle ad hoc network. . . 27 ICI Interchannel interference. . . 29 3D Tridimensional. . . 29 QPSK Quadrature phase-shift keying. . . 29 SMX Spatial modulation. . . 30 bpcu Bits per channel use. . . 30 STC Space-time coding. . . 30 OSTBC Orthogonal space-time block code. . . 32 SSK Space shift keying. . . 33 GSM Generalized spatial modulation. . . 34 MU-GSM GSM multiuser. . . 34 BS Base station. . . 34 R-SM Receive spatial modulation. . . 35 PSM Transmitter preprocessing aided spatial modulation. 35

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GPSM Generalized precoding aided spatial modulation. . . . 35 CSIT Channel state information at the transmitter. . . 35 APM Amplitude and phase modulation. . . 35 MU-R-GSM Generalized receive spatial modulation in multiuser

MIMO systems. . . 36 SVD Singular value decomposition. . . 40 ML Maximum likelihood. . . 42 ABEP Average bit error probability. . . 47 PEP Pairwise error probability. . . 48 JMGF Joint moment generating function. . . 48 RMT Random matrix theory. . . 49 WCS Wireless communications systems. . . 49 PDF Probability density function. . . 50 CLT Central Limit Theorem. . . 80 SNR Signal-to-noise ratio. . . 114

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23

LIST OF SYMBOLS

M Number of symbols in a constellation. . . 29 Nt Number of available transmit antennas. . . 29 b.c Floor function. . . 30 N_t0 Number of active transmit antennas. . . 34 Nr Number of available receive antennas. . . 35 N_r0 Number of activated receive antennas. . . 35 K Total number of users. . . 37 Nr0(k) Number of active receive antennas at the k-th user’s unit. . . 37 Nr(k) Number of receive antennas available at the k-th user’s unit 37 sk Active receive antenna subset index. . . 38 s Spatially-modulated information vector. . . 38 k Specific user. . . 38 [·]T _{Transpose of a vector or matrix}_{. . . 38} sall Active transmit antenna subset index. . . 38 x Information vector. . . 38 H Channel gain matrix. . . 39 H(k) Channel gain submatrix associated with User k. . . 39 h(k)_N_r_,N_t Channel gain between transmit antenna Nt and receive

an-tenna Nrof User k. . . 39 H(k)sall Channel gain matrix formed by the Nt0columns of H(k)

as-sociated with the common spatial information sall. . . 39 H(k)sk,sall Channel gain matrix formed by the N

0(k)

r rows of H(k)sall

asso-ciated with the spatial information sk. . . 39 Ps,sall Preprocessing matrix. . . 40

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¯x Preprocessed data. . . 41 y(k) Received signal vector. . . 41 n(k) Noise vector. . . 42 Q(k)

e

sk,esall Postprocessing matrix. . . 42

k·k2 Frobenius norm of a vector/matrix. . . 42 G(µ, σ2_{)Gaussian random variable with mean µ and variance σ}2_{. . . 57}

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1 INTRODUCTION

Transmission of information is a need that precedes the begin-nings of society. From ancient times techniques such as the emission of sounds (such as beating drums, for instance), and even a smoke signal were already used as a way to convey some kind of information in a long distance communication.

With the evolution of humanity, new concepts and techniques, especially in the technological sphere, have emerged. As a result, communication systems were increasingly improved from rudimentary stages, which required exclusively wireline communications, to much more elaborate stages, such as the single input single output (SISO) wireless systems and coming to what we know today as multiple input multiple output (MIMO) wireless communications systems.

The use of multiple antennas in wireless systems has received wide attention from the research and industrial communities in recent years for providing high spectral efficiencies [1]. However, classic MIMO systems such as those discussed in [1] are far from meeting the yearnings and needs of the world population.

Studies from CISCO [2] showed that global mobile data traf-fic grew 63 percent in 2016. Growth rates varied widely by region, with Latin America having 66 percent. For 2021, CISCO estimates a sevenfold increase over 2016 for the overall mobile data traffic. In ad-dition, the world scenario with regard to the environment, scarcity and the preservation of natural resources, has increasingly demanded that electronic systems in general consume less energy. So, with respect to telecommunication systems, there is a paradigm change from spectral to energy efficiency and, eventually, a compromise between both.

In order to meet these needs, Index Modulation (IM) techniques, which consider innovative ways to convey information compared to tra-ditional communication systems, come out as competitive candidates for next-generation wireless networks due to the attractive advantages they offer in terms of spectral and energy efficiencies [3]. Many at-tempts have been made to explore the potential of IM schemes, but

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only after the introduction of Spatial Modulation (SM) concept, ini-tially proposed in [4,5], but formally developed by Mesleh in [6], a new wave of alternative digital modulation schemes has started.

Spatial modulation is a relatively new paradigm for MIMO com-munication systems in which information is transmitted not only by choosing a symbol from a classical signal constellation, such as Quadra-ture Amplitude Modulation (QAM), but also by choosing an index in-dicating anactive subset of the transmit antennas through which the signal is transmitted. In addition to increasing spectral efficiency, spa-tial modulation has reduced complexity since it requires only one RF chain, rather than various RF chains (one for each antenna) required by other MIMO techniques such as spatial multiplexing [7] and space-time coding [8]. In recent years, SM has been studied and analyzed in sev-eral ways by various researchers around the world for being considered as a promising MIMO technique with great potential to be adopted in future wireless communication systems.

1.1 DISSERTATION CONTRIBUTIONS

In this Dissertation, a novel Spatial Modulation technique, called Multiuser Transmit and Receive Generalized Spatial Modulation (MU-TR-GSM), is presented [9,10]. Also, a performance analysis in terms of bit error rate (BER) is developed.

Unlike traditional SM techniques, for MU-TR-GSM in addition to the information transmitted by choosing an index that indicates an active subset of transmit antennas, there is also information trans-mitted by choosing an index that indicates an active subset of receive antennas. Besides, this new approach considers a multiuser (MU) sce-nario, where we have the possibility of sending information to several users in individual and common ways. The individual information con-sists of data from traditional digital modulations and from the spatial data associated with the choice of the set of active receive antennas. The common information, that is, the one shared among all users, con-sists of the spatial data generated by the choice of an active transmit

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1.2. Dissertation Organization 27

antennas set. A typical scenario the presented scheme fits in is a vehicle ad hoc network (VANET) [11], where cars receive independent infor-mation as well as common inforinfor-mation such as advertisement, weather report, traffic ahead information, etc. To the best of our knowledge, the presented multiuser MIMO scheme is the first one to use transmit and receive SM simultaneously.

1.2 DISSERTATION ORGANIZATION

We have divided this dissertation into six chapters. In Chapter

1, we introduce the idea of the method and the motivations for its development. A review of the SM literature is presented in Chapter2. The details of our proposed SM technique, MU-TR-GSM, is presented in Chapter3. In Chapter4, the performance analysis in terms of BER is developed. Results are presented in Chapter5. In Chapter6, some final discussions and future works are presented.

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2 FUNDAMENTALS OF SPATIAL MODULATION

2.1 SPATIAL MODULATION

Spatial modulation is a relatively new paradigm for MIMO com-munications systems [12, 13, 14]. It aims to increase the spectral ef-ficiency of systems with a single antenna while avoiding the problem of interchannel interference (ICI) [15]. In its original design, the infor-mation bits are divided, in the transmitter, into two sequences. One is mapped into aM-ary classical signal constellation (QAM, for instance), and the other one is used to select (or activate) one out of theNt

trans-mit antennas available. The signal is then transtrans-mitted through the activated antenna, whose index is used to convey extra information.

In Figure 1, the mechanism of information coding for SM is il-lustrated consideringNt = M = 4. The diagram presented in Figure1

Spatial Constellation 11 (Tx3)

10 (Tx2)

01 (Tx1)

00 (Tx0)

Signal Constellation for Tx3 Im Re (11)01 (11)00 (11)11 (11)10 . . . 11 10 . . .

Signal Constellation for Tx1 Signal Constellation for Tx0 Im

Im

Re

Figure 1 – 3D diagram of information coding for SM (single use of the channel).

is known as Spatial Constellation. In a single use of the channel, the bit stream to be coded is1110. The first two bits, 11, determine the antenna to be activated (e.g., Tx3). In turn, the two remaining bits,10, determine the QPSK symbol to be transmitted through the activated

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antenna.

The result is that a single transmission in the same bandwidth of traditional MIMO systems conveys_blog₂(M)c + blog2(Nt)cbits,

im-proving spectral efficiency. Another benefit is the reduced complexity, as a single transmit RF chain is required. Consequently, a twofold im-provement in the system energy efficiency is perceived. First, because of the use of only one RF chain instead of Nt. And second, because

of the spatial constellation bits which are transmitted without extra energy cost. Those bits will be namedgreen bits.

Detection of the antenna index (i.e., the spatial component of SM signals) relies on the uncorrelated statistics of the different channel fading gains. As the signal emitted by each transmit antenna will cross different paths to the receiver, it will undergo different propagation conditions. This is one of the main foundations of SM [12]. Thus, in order to detect the noisy signal received, the receiver must know the channel gains of all possible links between the transmit antennas and the receive antennas. This channel knowledge can be acquired by prior estimation.

In Figure2, we illustrate the concept of SM contrasting with the two most well-known MIMO techniques, namely, spatial multiplexing [7] and space-time coding [8].

In the case of spatial modulation (SMX), illustrated in Figure

2(a), two QAM symbols are transmitted simultaneously through a cou-ple of transmit antennas in a single use of the channel. In this situation, for an arbitrary number of transmit antennas (Nt), and a modulation

withMsymbols, the system rate isRSMX= Ntlog2(M)bits per channel

use (bpcu). For space-time coding (STC), which is illustrated in Figure

2(b), two QAM symbols are initially coded and then transmitted simul-taneously through a pair of transmit antennas in two uses of the chan-nel. In this way, the rate reached by such a system isRSTC= Rclog2(M)

bpcu, whereRc= NM/Ncu≤ 1is the STC rate, andNM is the number

of information symbols transmitted in Ncu uses of the channel. If we

use the Alamouti code, as presented in Figure2(b), then Rc= 1. In

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2.1. Spatial Modulation 31

symbol is effectively transmitted, while the other symbol is virtually

transmitted by the identification of the transmit antenna index in each use of the channel. That is, in the SM-MIMO the information sym-bols are modulated into two information units: one through aM-ary classical constellation; and the other through the index of the transmit antenna used in each use of the channel. Therefore, the achieved rate by the SM-MIMO system isRSM= log2(M) + log2(Nt) bpcu.

The literature presents many advantages of SM-MIMO when compared to traditional MIMO systems. However, these advantages lead to some necessary compromises and compensations. For instance, letting Nt = 2 and M= 4, we have 4, 2, and 3 bpcu for SMX, STC,

and SM-MIMO, respectively. Nonetheless, it should be considered that SMX and STC use2and4times more energy, respectively, than SM-MIMO. Another observation is that SMX requires two receive antennas

bits b1, b2 Spatial Multiplexing x1 x2 xi= 2bi− 1 i = 1, 2 bits b1, b2 Space-Time Coding x1 x2 xi= 2bi− 1 i = 1, 2 −x∗ 2 x∗ 1 bits b1, b2 Spatial Modulation x1 x1= 2b1− 1 bit b2= 0 or b2= 1 0 1 (a) (b) (c)

Figure 2 – Illustration of three MIMO transmission paradigms: (a) spa-tial multiplexing (SMX), (b) space-time coding (STC), and (c) spatial modulation (SM).

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in its operation. Di Renzo et al., in [13], present several important points, of which we highlight:

• Advantages:

– High throughput. Due to the spatial constellation,

SM-MIMO has spectral efficiency higher than single-antenna sys-tems and OSTBC-MIMO transmission.

– Simpler receivers. As a single transmit antenna is activated,

SM-MIMO will not be affected by ICI, which simplifies the detector [16].

– Simpler transmitters. With the use of only one transmit

antenna, only one RF chain is required, which makes the transmitter cheaper.

• Disadvantages:

– To explore the concept of SM, at least two antennas are required in the transmitter.

– If the communication links between receiver and transmit-ter are not sufficiently different, the SM paradigm may not generate adequate performance.

– The receiver needs full knowledge of the channel, which can generate complexity restrictions for channel estimation. Since its introduction, SM has been studied, generalized and an-alyzed in several ways by several researchers around the world. For example, a general approach for obtaining the error performance of SM was developed by Di Renzo and Haas in [17]. Because of its many advantages over other MIMO techniques, SM has become an important research topic and, as pointed out in [14], it has been recognized as a promising MIMO technique to be adopted in future wireless communi-cation systems, including those that adopt a large number of antennas, known as large-scale MIMO systems or more commonly called Massive MIMO [18]. In the next section, we will briefly discuss some of the

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2.2. Evolution and State of the Art of Spatial Modulation 33

important works that marked the evolution of SM and which are, at the same time, relevant to this dissertation.

2.2 EVOLUTION AND STATE OF THE ART OF SPATIAL

MODU-LATION

In this section, we have chosen to divide the works to be reported in groups of topics. In this way, we believe that a better understanding by the reader is achieved.

2.2.1 Spatial Modulation (SM): Activation of Single Transmit

An-tenna

The first publication related to spatial modulation appeared in the 2001 Vehicular Technology Conference. The paper "Space

Modu-lation on Wireless Fading Channels" by Chau and Yu [4] presented the

idea for what we now call Space Modulation. In that work, the au-thors developed a modulation technique, which they called Space Shift Keying (SSK), whose operation relies on the distinct multipath fad-ing characteristics of the MIMO channels. The system developed in [4] was exemplified with two transmit antennas. In this case, trans-mitting a signal only through one antenna means sending the spatial bit 0. If the transmission of the signal is through the two antennas, a bit1is spatially transmitted. The detector used was the maximum likelihood detector. Therefore, the receiver must rely on the different channel characteristics in order to recover the transmitted information. A more elaborate SSK study was published in 2009 by Jeganathan et al. [19].

On top of pure spatial modulation (i.e. SSK), a digital modula-tion can be used to transmit addimodula-tional informamodula-tion. It was only in 2006 that the term Spatial Modulation was coined, by Mesleh et al. [20]. Throughout the decade, the SM system appeared in the literature as originally conceived, that is, using single transmit antenna activation, by which a QAM symbol is transmitted.

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2.2.2 Generalized Spatial Modulation (GSM): Activation of Multi-ple Transmit Antennas

For several years the academic community investigated ways to circumvent a system limitation of SM with single antenna activation, namely, the number of transmit antennas must be a power of two. This restriction was removed about 8 years ago in [21, 22, 23], while the simplicity of having only one RF chain was maintained.

In the so-called Generalized Spatial Modulation (GSM), a sub-set of theN_t0 transmit antennas (instead of a single antenna in SM) is activated and they all transmit the same point in the signal constella-tion. By associating information with the active antennas subset, the resulting number of bits per transmission is increased to_blog₂(M)c + j

log2 Nt

N_t0

k

. This means that the same spectral efficiency of SM can now be achieved with fewer antennas available. In addition, the advan-tage of single RF chain is still maintained.

2.2.3 Generalized Spatial Modulation in Multiuser MIMO Systems

Most references on SM consider peer-to-peer systems with a sin-gle user at each end. It was only recently that SM was considered as a modulation format in a multiuser MIMO context. In [24,25], focusing only on the uplink part, the authors considered a system withK users sending independent information to a base station (BS). The BS is equipped with a large number of receive antennas (under the concept of massive MIMO [18]). Each user has Nt transmit antennas and is

provided withN_t0RF chains, corresponding to the number of active an-tennas. Note that in these works multiple RF chains are required since a different constellation signal point is sent through each active an-tenna. As a result, if a classicalM-ary modulation is adopted, then, in a given transmission, each user sends _bN_t0log₂(M)c +jlog₂ Nt

Nt0 k

bits. The K users transmit simultaneously in the uplink way, and the BS performs maximum likelihood multiuser detection to recognize which subset of the transmit antennas have been activated by each user and which conventional modulation symbols have been transmitted through

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2.2. Evolution and State of the Art of Spatial Modulation 35

these antennas.

2.2.4 Receive Spatial Modulation (R-SM): Receive Antenna

Activa-tion

In another recent wave of publications [26, 27, 28, 29, 30], the so-called receive spatial modulation (R-SM), also known as transmit preprocessing aided spatial modulation (PSM), and further elaborated to generalized precoding aided spatial modulation (GPSM) has been introduced. The novel characteristic of this scheme is that a subset of the receive antennas, as opposed to transmit antennas, is activated as a means of conveying information. Of course, this can only be made possible with channel state information at the transmitter (CSIT), and through the use of precoding.

In [26], the authors developed a PSM system that makes use of the receive antenna indexes as a constellation to convey information, in addition to using traditional amplitude and phase modulation (APM). In [27], the authors consider R-SM with imperfect CSIT. They develop R-SM schemes that perform well under this more realistic scenario. A similar approach is proposed by Zhang et al. [28], who expand the idea of [26]. They focused on a performance analysis for both cases of perfect and imperfect CSIT. In addition, the authors propose a GPSM scheme aided by a reinforcement matrix which yields better performance as compared to GPSM. Recently, a BER analysis for GPSM schemes was developed by Zhang et al. [29].

2.2.4.1 Receive Spatial Modulation in Multiuser MIMO Systems

The recent work due to Humadi et al. [30] considers receive spa-tial modulation in downlink multiuser MIMO systems. A BS equipped with Nt transmit antennas produces K independent information bit

streams to be conveyed to K different users, each equipped with Nr

receive antennas. At each transmission, and for each user,_blog₂(M)c

bits are mapped into a conventional signal constellation and

j log2 NNrr0

k

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CSIT is assumed and the BS relies on precoding to activate a different subset of receive antennas at every user’s unit. For simplicity, herein we refer to such kind of generalized receive spatial modulation in mul-tiuser MIMO systems as MU-R-GSM. The ideas in [30] have motivated the development of this dissertation.

2.3 SUMMARY

This chapter has the objective of presenting, briefly, a history of the techniques of spatial modulation from its inception to the present day. Thus, even without the intention of being a complete tutorial, this presentation will be important for the understanding of certain definitions that will come in the next chapters.

From the above, we have noticed that in the great majority of the works already published in the SM area there is only one user in the system. Some works apply the concept of SM in the transmit antennas (activation of these), others in the receive antennas. However, no work uses the concept of SM to activate both transmit and receive antennas at the same time. In the following chapters, we will present the proposed system, which, to the best of our knowledge, presents itself as original and innovative.

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37

3 MULTIUSER TRANSMIT AND RECEIVE GENERALIZED SPATIAL MODULATION

3.1 INTRODUCTION

Motivated by the suitability of spatial modulation in scenarios with many transmit and receive antennas, in this dissertation we de-velop a new generalization of this idea which is detailed next. Along the way, we highlight the modifications and new developments we made to the original concept of SM and its recent evolution presented in Chap-ter2. This development leads us to instigating and promising results for wireless applications, which are described and discussed in Chapter

5.

3.2 SYSTEM MODEL AND THE PROPOSED SCHEME

We consider a multiuser MIMO downlink system where a base station wants to send information toKusers. Two types of information are considered. One is independent information targeted to each user, and the other is common information addressed to all users. In the former type, the information bit stream is split into two sequences. One is mapped into a conventionalM-QAM modulation, and the other is employed in a R-SM scheme, i.e., the bits are used to select a subset ofNr0(k)active receive antennas fromNr(k) receive antennas available at

thek-th user’s unit. Multicast information (the latter type) is conveyed through the selection of a subset ofN_t0 active transmit antennas from

Nt transmit antennas available at the BS.

It should be remarked that the concept of active receive antenna is somewhat misleading as all receive antennas are active (or are

lis-tening) at all times. What is meant by an active receive antenna in

the context of R-SM is an antenna in which the received signal is the intendedcleantransmitted signal plus noise only. In other words, the precoder is designed in such a way as to deliver to a subset of active receive antennas the intended signal free of inter-channel interference, whereas all the other receive antennas are having interference. The

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SM detector should be able to differentiate between the two reception states.

To the best of our knowledge, this is the first GSM scheme that activate a subset of transmit and receive antennas simultaneously [9].

A detailed illustration of the proposed scheme, called MU-TR-GSM, is presented in Figure3.

bits to User 1

1 N(1)r

common bits to all users

Preprocessor User 1 Nr0(1)blog2(M )c + $ log2 N(1) r N_r0(1) % j log2NtN 0t k Switch (N0 tout of Nt) N_t0 Ps,sall x1,s1 x ¯ x 1 Nt

sall(Tx antenna subset selector)

y(1) Q(1)˜s1,˜sall 1 ¯ x1 ¯ xN0t Base station through ML detector xk,sk xK,sK ˆ s1, ˆx1, ˆsall ˆ sk, ˆxk, ˆsall ˆ sK, ˆxK, ˆsall bits to User k Nr0(k)blog2(M )c + $ log2 N(k) r N0(k)_r % bits to User K N0(K)r blog2(M )c + $ log2 N(K) r Nr0(K) % 1 N(k) r User k Q(k) ˜ sk,˜sall through ML detector 1 N(K) r User K Q(K)˜sK,˜sall through ML detector H y(k) y(k) y(K)

Figure 3 – A detailed illustration of the MU-TR-GSM scheme.

Regarding the spatially-modulated information, lets_{∈ S}K×1_be

the spatial data vector whosek-th component,

sk∈ S ⊆ ( 0, . . . , N (k) r Nr0(k) − 1 ) , (3.1)

represents theactive receive antenna subset index intended to Userk. The idea is to send Nr0(k) different QAM symbols to these Nr0(k) active

receive antennas. Therefore, the BS can send up to Nr0(k)blog2(M)c +

j log₂ N (k) r Nr0(k) _k

bits to each user. Also, define sall∈ Sall⊆

n

0, . . . , Nt

Nt0 − 1 o

as the active trans-mit antenna subset index intended to all users. In this case, up to

blog2 Nt

Nt0cbits of common spatial information is sent to all users.

Letx= [xT

1, . . . , xTK]T ∈ M

∑K_k=1Nr0(k)

×1 _{be the information data}

vector, where xk∈ MN

0(k)

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3.2. System model and the proposed scheme 39

xk(n), represents the symbol from the conventional complex signal

con-stellation M (with _{|M | = M}) to be transmitted to the n-th active

receive antenna of Userk.

Assuming CSIT, transmission is accomplished by appropriate preprocessing of the data vectorxat the BS. However, unlike the sig-nal processing in MU-R-GSM [30], in the proposed scheme the ICI is only fully removed after complementary postprocessing and detection at each user’s unit, as seen in Figure 3. The development of the pre-processor and postpre-processor is presented in Section3.3.

For this, let the channel gain matrix be defined as

H=          H(1) .. . H(k) .. . H(K)          , (3.2)

where H(k) ∈ CNr(k)×Nt _{is the channel gain submatrix associated with}

Userk, H(k)=     h1,1(k) ··· h1,Nt(k) .. . hNr,1(k) ··· h (k) Nr,Nt     . (3.3)

No particular statistical assumption on Hneeds to be made at this point. However, we assume that the whole matrixH is known to the BS, and that Userkonly knows its associated submatrixH(k), but has no knowledge of any other submatrixH(ek), fork6= ek.

We also define for Userkthe matrixH(k)sall∈ C

Nr(k)×Nt0_{as formed by}

theNt0columns ofH(k)associated with the common spatial information

indexsall, and the matrixH(k)sk,sall∈ C

Nr0(k)×Nt0 _{as formed by the}_N_r0(k)_rows

of H(k)sall associated with the spatial information index sk. We assume

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following condition holds [31]: N_t0≥ K

∑

k=1 Nr0(k) (3.4)

3.3 PREPROCESSING AND POSTPROCESSING MATRICES

DEVEL-OPMENT

Having defined the system model in the previous section, we will now determine the preprocessing and postprocessing matrices to be employed into the SM scheme proposed herein. We follow the singular value decomposition (SVD) approach developed in [28,30], adapted to our generalized scheme.

As we assume that each user has knowledge only of the channel gains of the communication between the BS and itself, to determine the preprocessing and postprocessing matrices, we will take the SVD of the channel gain matrixH(k)sk,sall.

The SVD ofH(k)sk,sall is given as H(k)sk,sall = U (k) sk,sall h Λ(k)sk,sall , 0i V(k)sk,sall H (3.5)

whereU(k)sk,sall is a unitary matrix of dimensionN

0(k) r × Nr0(k),

h

Λ(k)sk,sall, 0 i

has dimensionNr0(k)× Nt0, withΛ (k) sk,sall being aN 0(k) r × Nr0(k)diagonal ma-trix, and V(k)sk,sall H

is a unitary matrix of dimensionNt0× Nt0.

An additional decomposition of (3.5)yields

H(k)sk,sall = U (k) sk,sall h Λ(k)sk,sall, 0 i   V(k)₁_,s k,sall H V(k)_2,s k,sall H   (3.6) = U(k)sk,sallΛ (k) sk,sall V(k)_1,s k,sall H , (3.7) where V(k)₁_,s k,sall H has dimensionNr0(k)× Nt0.

Givens andsall, the preprocessing matrix is given by

Ps,sall= V1,s,sallV

H

1,s,sallV1,s,sall −1

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3.3. Preprocessing and Postprocessing Matrices Development 41

whereV_1,s,s_all has dimensionNt0× ∑Kk=1Nr0(k)and is defined as

V1,s,sall= h V(1)_1,s 1,sall, . . . , V (k) 1,sk,sall, . . . , V (K) 1,sK,sall i , (3.9) and β_s,s_all = diag n β(1)s1,sall,··· ,β (k) sk,sall,··· ,β (K) sK,sall o = diag β_s(1)₁_,1,s all,··· ,β (1) s1,Nr0(1),sall ,··· , β_s(k) k,1,sall,··· ,β (k) sk,Nr0(k),sall ,··· , β_s(K) K,1,sall,··· ,β (K) sK,Nr0(K),sall , (3.10)

whereβ(k)sk,sall can be used for power control, given as

β(k)sk,sall= v u u u u t Nr0(k) trace V(k)_1,s k,sall H V(k)_1,s k,sall −1. (3.11) In our case, asV(k)_1,s

k,sall is a unitary matrix, the result of equation (3.11) is always one. We shall consider the use of β(k)sk,sall for other

purposes, as presented in Section3.5. Usually, as in [31], these betas are defined as a function of the user’s subchannel matrix since, supposedly, both transmitter and receiver know this information. However, this is not a constraint as long as the transmitter and the receiver agree on a common value.

The preprocessed data is given by

¯x= Ps,sallx, (3.12)

where¯xis aN_t0_×1vector andPs,sall, defined in(3.8), is aNt0×∑Kk=1Nr0(k)

matrix. The vector ¯x is then transmitted through the Nt0 antennas

selected according to the spatial information indexsall.

The signal vector received by Userkis given by

y(k) = H(k)sall¯x+ n

(k) _(3.13)

= H(k)sallPs,sallx+ n

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wheren(k)_{∈ C}Nr(k)×1 _{is the noise vector.}

Although the signal preprocessing done by the BS uses the ma-trixH(k)sk,sall, in the received signal model the use of the matrix H

(k) sall is

necessary because the receiver does not yet know which of its anten-nas have been activated, i.e., the information associated with the index

sk. Therefore, the receiver will have to determine from the signals

re-ceived in all of its antennas, which of these antennas were activated. This need is directly related to the concept of active receive antenna

presented earlier.

Lets now definey(k)

e sk ∈ C

N_r0_×1 _{as a subvector formed by the}_N0(k) r

elements ofy(k) associated with the spatial information indexs_ek. It is

given by y(k) e sk = H (k) e sk,sallPs,sallx+ n (k) e sk , (3.15) where n(k) e

sk is the corresponding noise subvector of n

(k)_{. Since User} _k

knowsH(k), it also knows H(k)

e

sk,esall for all esk∈ S and for all esall∈ Sall.

Thus, the decomposition in(3.5) for any tentative spatial information detection can also be done by Userk.

Assuming complete knowledge of H(k)sall in the receiver, the

opti-mum detectorg: y(k)_7→_bxk is the Maximum Likelihood (ML) detector,

given by

(ˆxk, ˆsk, ˆsall) = arg min e xk∈MN0 (k) r ×1 e sk∈S e sall∈Sall y (k) e sk − Q (k) e sk,esallexk 2 , (3.16) where Q(k) e sk,esall= U (k) e sk,esallΛ (k) e sk,esallβ (k) e sk,esall (3.17)

is the postprocessing matrix of dimensionNr0(k)× Nr0(k) through which

the ML detection is performed. The signal processing involved in the ML detection is presented in Section3.4.

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3.4. Signal processing in the receiver 43

3.4 SIGNAL PROCESSING IN THE RECEIVER

In order to illustrate the signal processing in the receiver, we expand the termy(k)

e sk − Q (k) e sk,esallexk in(3.16)as y(k) e sk − Q (k) e sk,esallexk= H (k) e sk,sallPs,sallx+ n (k) e sk − U (k) e sk,esallΛ (k) e sk,esallβ (k) e sk,esallexk = U(k) e sk,sallΛ (k) e sk,sall V(k)_1, e sk,sall H V1,s,sallV H 1,s,sallV1,s,sall −1 | {z } A β_s,s_allx +n(k) e sk − U (k) e sk,esallΛ (k) e sk,esallβ (k) e sk,esallexk. (3.18)

Assuming that_esk= sk, the factorA in (3.18)evaluates to

V(k)₁_,e_s k,sall H V1,s,sallV H 1,s,sallV1,s,sall −1 =h 01 I Nr0(k) 02 i , (3.19)

where01and02are all-zero matrices with dimensionsNr0(k)×∑k_i=1−1Nr0(i)

andNr0(k)× ∑K_i=k+1Nr0(i), respectively.

In this case,(3.18)can be rewritten as

y(k) e sk − Q (k) e sk,esallexk= U (k) e sk,sallΛ (k) e sk,sallβ (k) e sk,sallxk + n(k) e sk − U (k) e sk,esallΛ (k) e sk,esallβ (k) e sk,esallexk. (3.20)

Finally, assuming that _esall= sall andxek= xk, equation(3.20)is

reduced to y(k) e sk − Q (k) e sk,esallexk= n (k) e sk , (3.21)

which is Gaussian noise only.

On the other hand, if _esk6= sk, the factor A in (3.18) does not

have the form in (3.19). Instead, it becomes a matrix whose entries contain nonzero multiuser interference terms. Moreover, ifs_eall6= sallor

exk6= xk, then(3.20)is not reduced either. Therefore, we can conclude

that, if the noisen(k)

e

sk is not too high, then the minimization in(3.16)

is likely to be successful and the receiver will be able to recover all the information.

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3.5 IMPROVING COMMON SM SIGNALS SEPARATION THROUGH PHASE SHIFTS

When the number of active transmit antennas, N_t0, is close to the number of transmit antennas,Nt, or when different subsets of

ac-tive transmit antennas largely overlap, the user’s ability to detectsall

may be reduced due to the high degree of similarity between different subchannel matrices. This is specially critical whenNr0(k)= 1, in which

case the matrixU(k)sk,sall is a scalar.

Tackling this problem may require the adoption of a reduced selection of active transmit antenna subsets from all the Nt

N_t0

possible such subsets, at the expense of a reduced number of common spatial information bits to be conveyed to all users.

Alternatively, in the proposed scheme we improve the separabil-ity between common SM signals by introducing at the BS a positive phase shift in the transmitted symbols according to sall. This phase

shift may be introduced through β_s,s_all, in (3.8). It may also be made user-dependent and be introduced throughβ(k)sk,sall, in(3.10). This

gen-eralization is useful if different signal constellations are used for differ-ent users. At the User k’s unit, the same phase shifts utilized in the transmitter are applied in the ML detector.

Let Φall={φ0, . . . , φ_|S_all_|−1} be the set of phase shifts, where

|Sall| ≤ N_Nt0 t

. For simplicity, consider the case where the same signal constellation, M, is assumed for all users. In this case, for a given

sall∈ Sall, we have thatβs,sall in(3.8)is given byβs,sall= e

φ_sall_I ∑K_k=1Nr0(k)

. For a given signal constellation, M, the set Φall must be optimized

aiming to improve the detector’s performance. Next, we develop the expression for the phase shifts considering aM-PSK constellation.

We begin by assuming that the two matrices from(3.20)can be approximated as follows: U(k) e sk,sallΛ (k) e sk,sall ≈ U (k) e sk,esallΛ (k) e sk,esall. (3.22)

This is likely to occur, particularly when the number of active transmit antennas is close to the number of transmit antennas or when different

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3.5. Improving Common SM Signals Separation Through Phase Shifts 45

subsets of active transmit antennas largely overlap. Under this assump-tion, the performance of the ML detector in(3.16)can be improved if

Φall is chosen to maximize the parameterDas follows:

D= max

Φall

min

x, ˜x∈M φ_sall,φ_sall˜ ∈Φall

x_{6= ˜x or s}all6= ˜sall e jφ_sall_x_{− e}jφ_sall˜ _x_˜ 2 (3.23)

If φsall = φ for all sall∈ Sall, i.e., if the phase shifts are not

used, then D is zero. Consider that M is the M-PSK constellation and assume, without loss of generality, that x= 1 andx˜= ejθ_{, where}

θ= 2π`

M , for some`∈ {0,...,M − 1}. Assume also that the phases in Φall are given as φi = i.∆φ, for i= 0, . . . ,|Sall| − 1. Then, expanding

the squared term in(3.23) yields

e jφ_sall_x_{− e}jφ_sall˜ _x_˜ 2

= 2− 2cos[(φsall− φs˜all)− θ]. (3.24)

We can say that φsall− φs˜all = m.∆φ, for some m∈ {−|Sall| +

1, . . . , 0, . . . ,|Sall| − 1}. The minimum in(3.23)occurs whenm.∆φ gets

So, for a M-PSK signal we have shown that the phase shifts should be given by

φsall =

2π/M

|Sall|

sall, (3.27)

where sall∈ Sall={0,...,|Sall| − 1}. The effect of this phase shift

on the error performance will be shown in the simulation results of Chapter5.

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3.6 SUMMARY

In this chapter, we have introduced the system model for the proposed scheme. The mathematical development of the transmitter and receivers, as well as the solutions necessary for its operation were presented. A variation of the scheme is proposed to enhance the detec-tion of the common SM signal through a phase shift dependent on the T-SM information. It helps detection in situations where the number of active transmit antennas is close to the total number of transmit antennas available in the BS or when they overlap. In the next chap-ter, we develop the performance analysis of the proposed MU-TR-GSM scheme.

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47

4 PERFORMANCE ANALYSIS OF MU-TR-GSM

4.1 INTRODUCTION

In this chapter, we analyze the error performance of the pro-posed SM scheme, MU-TR-GSM, which was illustrated in Figure 3. This analysis is very important in order to have a better understand-ing on the behavior of the proposed method. In particular, the average bit error probability (ABEP) is theoretically analyzed. Before we start, let us summarize the scenario, the notation, and some important defi-nitions presented in Chapter3.

The symbol sk∈ S ⊆ ( 0, . . . , N (k) r Nr0(k) − 1 )

denotes the index which indicates the subset ofNr0(k) activereceive antennas for thek-th

user and xk∈ MN

0(k)

r ×1 _{denotes the} _N0(k)

r × 1 symbol vector intended

to the same user. Finally, sall∈ Sall⊆

n

0, . . . , Nt

Nt0 − 1 o

denotes the common transmit spatial information index, which indicates the subset ofN_t0 active transmit antennas. Collectively, they form the composed symbolXx(k)_k,sk,sall.

Recall from equation(3.15)that the subvectory(k)sk of the received

vectory(k) associated with index sk at userkis

y(k)sk = H (k) sk,sallPs,sallx+ n (k) sk , (4.1) wherex= [xT 1, . . . , x T K]T ∈ M ∑K_k=1Nr0(k) ×1_.

We list below some expressions from Chapter 3 which are im-portant to our analysis:

H(k)sk,sall = U (k) sk,sall h Λ(k)sk,sall, 0 i   V(k)_1,s k,sall H V(k)_2,s k,sall H   (4.2) = U(k)sk,sallΛ (k) sk,sall V(k)_1,s k,sall H , (4.3) Q(k)sk,sall = U (k) sk,sallΛ (k) sk,sallβ (k) sk,sall (4.4) Ps,sall = V1,s,sallV H 1,s,sallV1,s,sall −1 β_s,s_all (4.5)

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V_1,s,s_all = hV(1)_1,s 1,sall, . . . , V (k) 1,sk,sall, . . . , V (K) 1,sK,sall i . (4.6)

The average bit error probability seen by userkcan be analyzed by the well-known union bound technique as [17]:

P_b(k)≤ 1 |X|log2(|X|)_{X ∈X}

∑

f X ∈X f X 6=X dH(X , fX )APEPX →Xf , (4.7)

wheredH(X , fX )is the Hamming distance between the bit-words

rep-resentingX andXf, APEPX →Xf = EH n PEPX →X fH o , (4.8) PEPX →X fH

is the conditional pairwise error probability (PEP), andEH· stands for expectation overH.

In the analysis, we split the conditional PEP into seven different cases. For the expectation of the first case, we elaborate on and make use of the joint moment generating function (JMGF) of the eigenvalues of a Wishart matrix. For the other cases, we make use of numerical integration through Wolfram Mathematica [32] to evaluate the expec-tation in(4.8).

4.1.1 Pairwise Error Probability: The Seven Cases

The seven cases of the PEP are summarized below:

Case 1:Xx(k)k,sk,sall →X

(k) e

xk,sk,sall (only xkis in error) (4.9)

Case 2: →Xx(k)k,sk,esall

(only sallis in error) (4.10)

Case 3: _→X(k)

e

xk,sk,esall (both xkand sallare in error) (4.11)

Case 4: →Xx(k)k,esk,sall (only skis in error) (4.12)

Case 5: →Xx(k)k,esk,esall

(both skand sallare in error) (4.13)

Case 6: _→X(k)

e

xk,esk,sall (both xkand skare in error) (4.14)

Case 7: →X(k)

e xk,esk,esall

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4.2. Joint moment generating function of the eigenvalues of a wishart matrix 49

where_e_·represents a variable in error.

In the sections that follow, we develop the PEP expressions for each one of the seven cases. But before that, we derive an expression for the JMGF of the eigenvalues of a Wishart matrix, which proves useful for obtaining the expectation of the first case of the PEP.

4.2 JOINT MOMENT GENERATING FUNCTION OF THE

EIGEN-VALUES OF A WISHART MATRIX

Random Matrix Theory (RMT) plays a central role in the anal-ysis and design of MIMO wireless communications systems (WCS). As well known, RMT is in the core of the mathematical derivation of the channel capacity of MIMO channels, presented by Telatar [33] and Foschini and Gans [34] in the late 90’s. Since then, several WCS based on the MIMO paradigm have been designed, aiming to achieve its promised gains.

The capacity and performance analyses of WCS often yield quite complex expressions. The challenge faced by researchers is to arrive at closed-form expressions from which some insight can be gained. For instance, when deriving the error probability of a WCS, researchers often strive to seek an expression whose asymptotic value (as the signal-to-noise ratio (SNR) goes to infinity) indicates the system’s diversity gain.

An elegant approach for the performance analysis of WCS, which usually yields closed-form expressions, is based on themoment

gener-ating function(MGF) of some key random variable, such as SNR. The

MGF of a random variableX is defined as1MGFX(a) = E{eaX}, where

a is a real parameter. The so-called MGF-based approach [35] has widely been employed in the past 20 years.

Chiani et al. [36] have derived a closed-form expression for the characteristic function of the MIMO channel capacity, namely, C=

log₂(det(I + ρ/NtHHH)), whereHis the random MIMO channel gains

1 _{An alternate name for the MGF is the characteristic function, usually defined as Ψ} X(a) =

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matrix. Numata and Kuriki [37] have derived the moment generating functionE

n etr(ΘW)

o

, whereΘis a real parameter matrix,_tr(·)denotes the trace of a matrix, andWis a Wishart random matrix [38]. Many other expectations involving the Wishart random matrix, not all of them a MGF, can be found in [38, Sec. 2.1.6].

In this section, we derive a essentially closed-form expression for the joint MGF (JMGF) of the eigenvalues of a Wishart matrix. The following additional notations will be used: Tr(·)denotes the trace of a matrix,_(·)Tdenotes vector/matrix transposition,_(·)∗denotes complex conjugate, and(·)H _{denotes both transposition and complex conjugate}

(Hermitian). det(·)denotes matrix determinant,_k·k2denotes (squared) Frobenius norm of a vector/matrix,diag(v)gives a square diagonal ma-trix whose diagonal elements are the entries of vectorv, and_◦denotes component-wise vector/matrix (Hadamard) product.

4.2.1 Joint and marginal PDFs of the eigenvalues of a Wishart

ma-trix

Let H be a m× n random matrix with i.i.d. columns, each forming a circularly-symmetric jointly-Gaussian complex random vec-tor with covariance matrixΣ.

The matrix W=∆

HHH _{has a (central) Wishart distribution,}

i.e.,W_{∼ W}n(m, Σ), whose probability density function (PDF) is given

by [38, eq. (2.6)]

f(W) = π−

m(m−1) 2

[det(Σ)]n∏m_i=1(n− i)!exp−tr{Σ

−1_W_}_[det(W)]n_−m

. (4.16)

Throughout this section, we assumeΣ= I. The singular value decom-positions ofHandWare given by

H= UΛVH and W= UΛΛHUH= UΩUH, (4.17)

respectively, where Uand V are unitary matrices. In particular,U is a Haar-distributed random matrix, i.e., it is uniform on the set of all

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of Ω [38, Lemma 2.6]). The matrix Ω is a diagonal square matrix,

Ω= diag(λ₁2, . . . , λ_m2), whose entries λ_i2= ωi, for i= 1, . . . , m, are the

eigenvalues ofW.

The joint PDF of the ordered eigenvalues ofWis given by [33]

fΩ,ordered(ω1, . . . , ωm) = Km−1,ne−∑iωi

∏

i

ω_in−m

_∏

j>i

(ωi− ωj)2, (4.18)

whereω1≥ ··· ≥ ωm≥ 0, andKm,nis a normalization factor. A slightly

different expression2 is given for this PDF in [39], for which the nor-malization factor is provided. After a change of variables, it can easily be shown that the normalization factor for the PDF in(4.18)is

K_m,n−1 = π m(m−1) e Γm(n)eΓm(m) , (4.19) where e Γm(a) , πm(m−1)/2 m

∏

i=1 Γ(a− i + 1), (4.20)

andΓ(z)is the Gamma function, given by

Γ(z) =

Z ∞ 0

xz−1e−xdx. (4.21)

The joint PDF of the unordered eigenvalues is obtained dividing

(4.18)bym!, yielding fΩ(ω1, . . . , ωm) = (m!Km,n)−1e−∑iωi

_∏

i ω_in−m

_∏

j>i (ωi− ωj)2. (4.22)

The marginal PDF of (any) unordered eigenvalueλi2is [33]

f_λ2(ω) = 1 m m

∑

k=1 ϕk(ω)2ωn−me−ω, (4.23) where ϕk+1(ω) = k! (k + n− m)! 1/2 Ln_k−m(ω), k = 0, . . . , m− 1 (4.24)

andLn_k−m(x)is the Laguerre polynomial of orderk, given by

Ln_k−m(x) = 1 k!e x xm−n d k dxk(e−xx n_−m+k_). _(4.25)

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Example 1. For n= 4 and m = 2, the joint PDF of the unordered eigen-values is given by fΩ(ω1, ω2) = 1 24e −(ω1+ω2) ω12(ω1− ω2)2ω22, (4.26)

and the marginal PDF of an unordered eigenvalue is given by

f_λ2(ω) =

1

12e

−ω_ω2

(12− 6ω + ω2). (4.27)

We are interested in the joint moment generating function (JMGF) of the eigenvalues of the random matrixΩ, which is derived next.

4.2.2 Derivation of the Joint Moment Generating Function

Leta be am-dimensional real parameter vector. The JMGF is defined as JMGFΩ(a) = EΩ ( exp − m

∑

i=1 aiωi !) . (4.28)

In order to gain insight, we derive next the JMGF for the casen= 4

andm= 2, given by JMGFΩ(a) = (2!K2,4)−1 Z ∞ 0 Z ∞ 0 e−(1+a1)ω1−(1+a2)ω2 ω12ω22(ω1− ω2)2dω1dω2. (4.29)

This double integral breaks into three double integrals as

JMGFΩ(a) = (2K2,4)−1 Z ∞ 0 Z ∞ 0 e−(1+a1)ω1−(1+a2)ω2 ω14ω 2 2dω1dω2 − K2,4−1 Z ∞ 0 Z ∞ 0 e−(1+a1)ω1−(1+a2)ω2 ω13ω23dω1dω2 + (2K2,4)−1 Z ∞ 0 Z ∞ 0 e−(1+a1)ω1−(1+a2)ω2_ω2 1ω24dω1dω2. (4.30)

Note that each one of these three double integrals is a particular case of the general form:

Z ∞ 0 Z ∞ 0 e−(1+a1)ω1−(1+a2)ω2 ω₁k1ω₂k2dω1dω2, (4.31)

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which can be solved as [40, p. 340, Sec. 3.351, 3]: Z ∞ 0 e−(1+a1)ω1 ω₁k1dω1 Z ∞ 0 e−(1+a2)ω2 ω₂k2dω2 = k1!k2! (1 + a1)(k1+1)(1 + a2)(k2+1) . (4.32)

Therefore, the JMGF for this case is given by

JMGFΩ(a) = (2K2,4)−1 4!2! (1 + a1)5(1 + a2)3− 2(3!3!) (1 + a1)4(1 + a2)4 + 2!4! (1 + a1)3(1 + a2)5 . (4.33)

The JMGF for the generic case is given by

JMGFΩ(a) = (m!Km,n)−1 Z ∞ 0 ··· Z ∞ 0 e−∑mi=1(1+ai)ωi_P 0(ω1, . . . , ωm)dω1···dωm, (4.34)

whereP0(ω1, . . . , ωm)has the well-known (Vandermonde) discriminant

polynomial as a factor, and is defined as

j=1 kj! (1 + aj)(kj+1) , (4.37)

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where the range of(k1, . . . , km)and the multiplicities Nk1,...,km have to

be indicated.

In fact, the Nk1,...,km’s are related to the Kostkas numbers

ap-pearing in the combinatorial definition of Schur functions [41, p. 311], for which no simple formula is known in general. For small values ofn

andm, one can findNk1,...,km by inspection, upon expanding(4.35). For

large parameters, we offer Algorithm 1 below (a symbolic computation program may still be required). The nonzero multiplicities, Nk1,...,km, Algorithm 1 Specification of k1, . . . , kmand Nk1,...,km

1: Initialization: For j= 1, . . . , m, let kjvary in the range n− m ≤ kj≤

n+ m− 2. Whenever ∑m

i=1ki6= m(n − 1), Set Nk1,...,km ← 0. For all

other cases, run the rest of the algorithm to find the possibly nonzero multiplicities; 2: For i= 1, . . . , m, Set Pi(ωi+1, . . . , ωm)← 1 ki! ∂ki_P i−1(ωi, . . . , ωm) (∂ ωi)ki _ω_i= 0 ; 3: Output: Set Nk1,...,km← Pm.

forn= 4andm= 1, 2, 3 and 4, are given in Table1.

The next result will be useful in the performance analysis of interest.

Lemma 1. Let the parameter vector a be given as: a= αa0, for some

α∈ R+. Also, assume that a0i, for all i∈ {1,...,m}, can take values on a

set of nonnegative real numbers including the zero element. Then, lim

α→∞

−log(JMGFΩ(a))

log(α) = n− m + 1 (4.38)

Proof. For large α, it is easy to see that JMGFΩ(a) is dominated by terms

of the form

N

(1 + αa0

`)n−m+1

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4.3. The first case of the pairwise and average pairwise error probabilities 55

Table 1 – Multiplicities Nk1,...,kmfor n= 4 and m = 1, 2, 3 and 4

n m Nk1,...,km † 4 1 N3= 1 4 2 N4,2= 1, N3,3=−2 4 3 N5,3,1= 1, N4,4,1=−2, N5,2,2=−2, N4,3,2= 2, N3,3,3=−6 4 4 N6,4,2,0= 1, N5,5,2,0=−2, N6,3,3,0=−2, N5,4,3,0= 2, N4,4,4,0=−6, N6,4,1,1=−2, N5,5,1,1= 4, N6,3,2,1= 2, N5,4,2,1=−2, N5,3,3,1=−4, N4,4,3,1= 4, N6,2,2,2=−6, N_5,3,2,2= 4, N4,4,2,2= 4, N4,3,3,2=−6, N3,3,3,3= 24 †_N

k1,...,km= Nπ(k1,...,km), for all permutations π on the set

{k1, . . . , km} (only one permutation is shown for each case).

which are the terms in (4.37) for which a`is the only nonzero entry of the

vector a, for some`, and k`assumes its minimum value, namely, n− m.

The proof now follows immediately.

4.3 THE FIRST CASE OF THE PAIRWISE AND AVERAGE

PAIR-WISE ERROR PROBABILITIES

In the first case of the PEP for the proposed MU-TR-GSM sys-tem, we have the situation where only the transmitted symbol vector,

xk, is in error. It is presented as PEPXx(k)_k,sk,sall→ X (k) e xk,sk,sall H= Pr y (k) sk − Q (k) sk,sallxk 2 > y (k) e sk − Q (k) e sk,esall xk 2 = Pr n (k) sk 2 > y (k) sk − Q (k) sk,sallexk 2 (4.40)

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The term y (k) sk − Q (k) sk,sallexk 2 in(4.40)expands to y (k) sk − Q (k) sk,sallexk 2 = H (k) sk,sallPs,sallx+ n (k) sk − Q (k) sk,sallexk 2 = U(k)sk,sallΛ (k) sk,sall V(k)_1,s k,sall H V1,s,sallV H 1,s,sallV1,s,sall −1 β_s,s_allx +n(k)sk − U (k) sk,sallΛ (k) sk,sallβ (k) sk,sallexk 2 (4.41) From(3.19)we have y (k) sk − Q (k) sk,sallexk 2 = U (k) sk,sallΛ (k) sk,sallβ (k) sk,sall(xk−exk) + n (k) sk 2 = (xk−exk) H β(k)sk,sall H Λ(k)sk,sall H U(k)sk,sall H U(k)sk,sallΛ (k) sk,sallβ (k) sk,sall(xk−exk) + n (k) sk 2 + + 2Re n(k)sk H U(k)sk,sallΛ (k) sk,sallβ (k) sk,sall(xk−exk) = (xk−exk) H β(k)sk,sall H Λ(k)sk,sall H Λ(k)sk,sallβ (k) sk,sall (xk−exk) + 2Re n(k)sk H U(k)sk,sallΛ (k) sk,sallβ (k) sk,sall (xk−exk)} + n (k) sk 2 (4.42)

Sinceβ(k)_s_k_,s_all is the identity matrix (see page41for explanation), we have y (k) sk − Q (k) sk,sallexk 2 = Nr0(k)

∑

i=1 |xk,i−xek,i| 2 λ_s(k) k,sall,i 2 + n (k) sk 2 + 2Re n(k)sk H U(k)sk,sallΛ (k) sk,sall(xk−exk) (4.43)

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4.3.1 Statistics of the first case

Considering equation (4.40) and taking into account the result presented in equation(4.43), we have

PEPXx(k)k,sk,sall→ X (k) e x_k,sk,sall H = Pr n (k) sk 2 > n (k) sk 2 +2Re n(k)sk H U(k)sk,sallΛ (k) sk,sall(xk−exk) + Nr0(k)

∑

∑

j ω∗jEninj (4.48) Where, Eninj = ( En2 i , if i = j 0, if i6= j (4.49)

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But, fori= j, we have

En2

i = E {(Re(ni) + jIm(ni))(Re(ni) + jIm(ni))}

= En(Re(ni))2− (Im(ni))2+ 2 jRe(ni)Im(ni)

o = N0 2 − N0 2 + 0 = 0. (4.50)

In a similar way, we can show that the third term in (4.47) is equal to zero, as claimed. Defining the matrix

i (xk,i−exk,i) 2 λ_s(k) k,sall,i 2 , (4.53)

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where, in(4.52), we have used the matrix property that, for a matrix A = U(k) sk,sallΛ (k) sk,sall(xk−exk) | {z } B BH z }| { (xk−exk) H Λ(k)sk,sall U(k)sk,sall H , (4.54) we haveTr{A } = kBk2_.

Now, the expectation of the first term in equation (4.47) is

E n(k)sk n(k)sk H

= N0I. So, for the first term we have

0(k) r i |xk,i−xek,i| 2 λ_s(k) k,sall,i 2 .