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UNIVERSIDADE FEDERAL DO CEARÁ

DEPARTAMENTO DE ENGENHARIA DE TELEINFORMÁTICA

PROGRAMA DE PÓS-GRADUAÇÃO EM ENGENHARIA DE TELEINFORMÁTICA

FRANCISCO RAFAEL MARQUES LIMA

MAXIMIZING SPECTRAL EFFICIENCY UNDER MINIMUM SATISFACTION

CONSTRAINTS ON MULTISERVICE WIRELESS NETWORKS

FORTALEZA

2012

UNIVERSIDADE FEDERAL DO CEARÁ

DEPARTAMENTO DE ENGENHARIA DE TELEINFORMÁTICA

PROGRAMA DE PÓS-GRADUAÇÃO EM ENGENHARIA DE TELEINFORMÁTICA

Maximizing Spectral Efficiency under Minimum

Satisfaction Constraints on Multiservice Wireless

Networks

Doctor of Science Thesis

Francisco Rafael Marques Lima

Advisor

Prof. Dr. Francisco Rodrigo Porto Cavalcanti

FORTALEZA

2012

FRANCISCO RAFAEL MARQUES LIMA

MAXIMIZING SPECTRAL EFFICIENCY UNDER MINIMUM SATISFACTION

CONSTRAINTS ON MULTISERVICE WIRELESS NETWORKS

Tese apresentada à Coordenação do

Programa de Pós-graduação em Engenharia

de Teleinformática, da Universidade Federal

do Ceará, como parte dos requisitos para

obtenção do título de Doutor em Engenharia

de Teleinformática.

Área de concentração: Sinais e Sistemas

Orientador: Prof. Dr. Francisco Rodrigo

Porto Cavalcanti

FORTALEZA

2012

Contents

Acknowledgement vi

Resumo vii

Abstract viii

List of Figures ix

List of Tables xii

List of Algorithms xiii

Nomenclature xiv

1 Introduction 1

1.1 Thesis Scope and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Multiple access methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 Multiple antennas techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.3 QoS and satisfaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.4 Radio Resource Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Contributions and Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6 Scientific production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 System Modeling, General Problem and Framework for Solution 14

2.1 General System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 General Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Framework for Heuristic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Maximizing Spectral Efficiency under Minimum Satisfaction Constraints in SISO

Scenario 21

3.1 System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Characterization of the Optimal Solution . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Low-Complexity Heuristic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.5.1 Simulation assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.6 Partial Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Maximizing Spectral Efficiency under Minimum Satisfaction Constraints in SU

MIMO Scenario 38

4.1 System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Spatial Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2.1 MRT spatial filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2.2 SVD-based spatial filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2.3 ZF spatial filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3 Problem Formulation, Optimal and Heuristic Solutions . . . . . . . . . . . . . . . 40



4.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43


5 Maximizing Spectral Efficiency under Minimum Satisfaction Constraints in MU

MIMO Scenario 47

5.1 System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2 BD-ZF Spatial Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.4 Characterization of the Optimal Solution . . . . . . . . . . . . . . . . . . . . . . . . 51

5.5 Low-Complexity Heuristic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 53



5.6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57


6 Maximizing Spectral Efficiency with and without Minimum Satisfaction

Constraints in SC-FDMA Uplink Scenario 64

6.1 System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.2 Unconstrained Rate Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.2.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.2.2 Characterization of the optimal solution . . . . . . . . . . . . . . . . . . . . 66

6.2.3 Low-complexity heuristic solution . . . . . . . . . . . . . . . . . . . . . . . . 66

6.2.4 Performance evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.3 Constrained Rate Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.3.2 Characterization of the optimal solution . . . . . . . . . . . . . . . . . . . . 73

6.3.3 Low-complexity heuristic solution . . . . . . . . . . . . . . . . . . . . . . . . 75

6.3.4 Performance evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80


7 Conclusions and Future Work 89

Appendix A Pseudo Code and Computational Complexity of the Algorithms in

Chapters 3 and 4 92

A.1 Complexity of Optimal Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

A.2 Algorithm and Complexity of Proposed Heuristic Solution . . . . . . . . . . . . . . 92

Appendix B Pseudo Code and Computational Complexity of the Algorithms in

Chapter 5 98

B.1 Complexity of Optimal Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

B.2 Algorithm and Complexity of Proposed Heuristic Solution . . . . . . . . . . . . . . 98

Appendix C Pseudo Code and Computational Complexity of the Algorithms in

Chapter 6 106

C.1 Complexity of Optimal Solution to the URM Problem . . . . . . . . . . . . . . . . . 106

C.2 Algorithm and Complexity of Proposed Heuristic Solution to the URM Problem . 106

C.3 Complexity of Optimal Solution to the CRM Problem . . . . . . . . . . . . . . . . . 111

C.4 Algorithm and Complexity of Proposed Heuristic Solution to the CRM Problem . 111

Bibliography 119

Acknowledgements

This thesis would not be finished without the special help of some people. I acknowledge

the confidence of my advisor, Rodrigo Cavalcanti, in my potential since the beginning when I

was only an undergraduate student. Moreover, I could not forget the guidance, support and

countless discussions with my co-advisor Tarcísio Maciel and all members of UFC.22 and

UFC.30 projects with special thanks to Ricardo Brauner, Níbia Bezerra and Walter Freitas.

I am also grateful to Wireless Telecom Research Group (GTEL) and Ericsson Research for

the financial support and for giving me opportunity of working on several research projects

that dealt with the state of the art in wireless engineering. Thanks also to FUNCAP and CNPq

for the financial support in the first year of the Ph.D course.

Moreover, I would like to thank my parents, Garcia and Fátima, for all effort and sacrifice

that they made in order to give me opportunity to study and finish this thesis. They together

with my sister, Josétima, have always believed in my capacity and understood that doing

research implies in some restrictions.

I also would like to express all my gratitude to my friend and wife, Cibelly, for the unlimited

love and for shining my life with her smile every new day. Thanks also to my wife for our new

born son Oliver and for the comprehension and patience in the pregnancy phase. Last but

not least, I thank God for listening to me and for all the special things He has given me.

Resumo

Redes celulares entraram recentemente no competitivo mercado de provimento de serviços

de dados devido principalmente aos avanços tecnológicos da terceira geração (3G) e da

iminente quarta geração (4G). Os sistemas Long Term Evolution (LTE) e LTE-Advanced são

exemplos de redes celulares que proporcionam altas taxas de dados a seus usuários. A

necessidade de estar conectado de forma permanente e os novos e poderosos dispositivos

móveis são fortes indicadores que o mercado de banda larga móvel ainda possui potencial de

crescimento em nível global.

Este novo cenário com sofisticados dispositivos móveis permite a rápida popularização de

novas aplicações de dados móveis. Como consequência, esperamos que o tráfego nas redes

móveis tenham um aumento considerável nos próximos anos. Portanto, o provimento de

Qualidade de Serviço (do inglês, Quality of Service (QoS)) para serviços heterogêneos consiste

em um cenário desafiador para os operadores dos sistemas e indústria em um futuro próximo.

De forma a enfrentar esses desafios, algumas melhorias foram realizadas no núcleo da

rede por meio do advento da arquitetura por chaveamento por pacotes baseado em protocolo

da internet (do inglês, Internet Protocol (IP)). Na rede de acesso de rádio, tivemos como

avanços o uso de múltiplas antenas nos nós da rede e a adoção dos esquemas de múltiplo

acesso por divisão de frequências ortogonais (do inglês, Orthogonal Frequency Division Multiple

Access (OFDMA)) e múltiplo acesso por divisão de frequências com portadora única (do inglês,

Single Carrier - Frequency Division Multiple Access (SC-FDMA)) nos enlaces direto e reverso

do sistema LTE, respectivamente. Outra funcionalidade que destacamos como relevante para

enfrentar os desafios das próximas gerações de redes celulares consiste no uso de alocação

de recursos de rádio (do inglês, Radio Resource Allocation (RRA)). Algoritmos de RRA são

responsáveis pelo gerenciamento dos recursos de rádio tais como intervalos de tempo (do

inglês, time slots), canais espaciais e grupos de frequências que em geral são escassos.

Neste contexto, nós estudamos nesta tese o uso de RRA em redes celulares de forma a

melhorar a eficiência no uso dos recursos e garantir um provimento sustentável de múltiplos

serviços. Especificamente, modelamos RRA como o problema de otimização de maximização

da taxa total de transmissão sujeito a restrições de satisfação mínimas por serviço. Este

problema é estudado ao longo da tese em diferentes cenários resultantes da combinação de

diferentes esquemas de múltiplo acesso e múltiplas antenas. Como principais contribuições

temos a caracterização de soluções ótimas, propostas de heurísticas de baixa complexidade,

avaliação de desempenho por meio de simulações computacionais e por fim a análise da

complexidade dos algoritmos envolvidos.

Palavras-chave: Sistemas celulares, Qualidade de serviço, Alocação de recursos de rádio

e Satisfação.

Abstract

Cellular networks are now a new player in the competitive market of data service

provision mainly due to the technological advances of 3rd Generation (3G) and the upcoming

4th Generation (4G). Long Term Evolution (LTE) and LTE-Advanced are examples of systems

that are capable of providing high data rates to the end user. The need of being connected

anytime and anywhere and the appealing mobile devices/applications are strong indications

that the mobile broadband market has potential for further worldwide increasing.

This new scenario with sophisticated mobile terminals enables the quick popularization of

new appealing data mobile applications. As a consequence, it is expected that the traffic on

mobile networks will have a considerable increase in the next years. Therefore, the sustainable

Quality of Service (QoS) provision of heterogeneous services appears as a challenging scenario

for mobile network operators and industry in the near future.

In order to deal with this challenging scenario, improvements in the core network have

been done by means of an Internet Protocol (IP)-based packet-switched architecture. In the

radio access network, the use of Orthogonal Frequency Division Multiple Access (OFDMA)

and Single Carrier - Frequency Division Multiple Access (SC-FDMA) as the multiple access

schemes of downlink and uplink of LTE system, respectively, and the addition of multiple

antennas techniques have boosted the achieved data rates in the radio part of the networks.

Another relevant functionality that is useful to deal with the challenges of next generation

cellular networks is efficient Radio Resource Allocation (RRA). RRA algorithms interact

with multiple access schemes and multiple antenna schemes and are responsible for the

management of the scarce radio resources such as power, time slots, spatial channels and

frequency chunks.

In this context, we study in this thesis the use of RRA in cellular networks in

order to improve the resource usage efficiency and guarantee the sustainable provision

of multiple services. More specifically, we model this RRA problem as the optimization

problem of maximizing the overall data rate subject to minimum satisfaction constraints

per service. Along this thesis, we study this problem in different scenarios with different

multiple access strategies and multiple antennas schemes. As main contributions we

provide the characterization of optimal solutions, proposal of low-complexity heuristic

solutions, performance evaluation by means of computational simulations and computational

complexity analysis of the involved algorithms.

Key-words: Cellular networks, Quality of Service, Radio Resource Allocation and

Satisfaction.

List of Figures

1.1 Frequency-time resource grid in OFDMA. . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Illustration of the system modeling and RRA. . . . . . . . . . . . . . . . . . . . . . 15

2.2 Illustration of the main aspects of the CRM problem. . . . . . . . . . . . . . . . . 17

2.3 Capacity region for a two-flow example to illustrate the proposed heuristic

framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1 Flowchart of the first part of the proposed solution for the SISO case:

Unconstrained Maximization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Flowchart of the second part of the proposed solution for the SISO case:

Reallocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Outage rate versus required data rate for CRM OPT, URM OPT and the proposed

solution with one service in scenarios 1 and 2 for the SISO case. . . . . . . . . . 29


solution with two services for the SISO case. . . . . . . . . . . . . . . . . . . . . . 30


solution with three services for the SISO case. . . . . . . . . . . . . . . . . . . . . 32

3.6 Outage rate versus required data rate for CRM OPT, URM OPT and proposed

solution with four services for the SISO case. . . . . . . . . . . . . . . . . . . . . . 33

3.7 CDF of total data rate for CRM OPT, URM OPT and the proposed solution with

two services in scenario 3 for the SISO case. . . . . . . . . . . . . . . . . . . . . . 34

3.8 CDF of total data rate for CRM OPT, URM OPT and proposed solution with three

services in scenarios 9 and 12 for the SISO case. . . . . . . . . . . . . . . . . . . . 35

3.9 CDF of total data rate for CRM OPT, URM OPT and the proposed solution with

four services in scenarios 13 and 14 for the SISO case. . . . . . . . . . . . . . . . 36

4.1 Flowchart of the first part of the proposed solution for the SU MISO and SU

MIMO cases: Unconstrained Maximization. . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Flowchart of the second part of the proposed solution for the SU MISO and SU

MIMO cases: Reallocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3 Comparison of outage rate versus required data rate for CRM OPT and proposed

solutions with single and multiple antennas schemes for the SU MISO and SU

MIMO cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4 CDF of the total downlink data rate for CRM OPT, URM OPT and proposed

solution at the flows’ required rate of 2Mbps with single and multiple antenna

schemes for the SU MISO and SU MIMO cases. . . . . . . . . . . . . . . . . . . . . 45

4.5 CDF of the total downlink data rate for CRM OPT, URM OPT and proposed

solution at the flows’ required rate of 2.75Mbps with single and multiple antenna

schemes for the SU MISO and SU MIMO cases. . . . . . . . . . . . . . . . . . . . . 45

5.1 Flowchart of the first part of the proposed algorithm for the MU MIMO case:

Unconstrained Maximization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.2 Flowchart of the second part of the proposed algorithm for the MU MIMO case:

Reallocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.3 Outage rate versus required data rate with CRM OPT, URMOPT and the proposed

solution for the SU MIMO and MU MIMO antenna configurations with MT = 2

and MR = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.4 CDF of total data rate for the data rate requirements of 2Mbps and 3.25Mbps

with CRM OPT, URM OPT and proposed solution in the SU MIMO and MU MIMO

antenna configurations with MT = 2 and MR = 2. . . . . . . . . . . . . . . . . . . . 59

5.5 Outage rate versus required data rate with CRM OPT, URMOPT and the proposed

solution for the SU MIMO and MU MIMO antenna configurations with MT = 4

and MR = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.6 CDF of total data rate for the data rate requirements of 4Mbps and 6Mbps with

CRM OPT, URM OPT and the proposed solution in the SU MIMO and MU MIMO

antenna configurations with MT = 4 and MR = 4. . . . . . . . . . . . . . . . . . . . 62

6.1 Basic flowchart of the proposed algorithm for the URM problem in the uplink case. 67

6.2 Illustration of steps (1) and (2) of the proposed algorithm with 3 flows and 10

RBs for the URM problem in the uplink case. . . . . . . . . . . . . . . . . . . . . . 67

6.3 Illustration of the process for building new VRs based on the example of Figure

6.2 for the URM problem in the uplink case. . . . . . . . . . . . . . . . . . . . . . . 69

6.4 CDF of total data rate for URM OPT, Wong Alg and proposed solution considering

different number of flows and 12, 18 and 24 RBs for the URM problem in the

uplink case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.5 Average total data rate versus the number of flows for URM OPT, Wong Alg and

proposed solution considering 12, 18 and 24 RBs for the URM problem in the

uplink case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.6 Flowchart of the first part of the proposed solution for the CRM problem in the

uplink case: Unconstrained Maximization. . . . . . . . . . . . . . . . . . . . . . . . 75

6.7 Flowchart of the second part of the proposed solution for the CRM problem in

the uplink case: Reallocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.8 Illustration of the process for selecting the available resources for flow 2 based

on 4 examples of resource assignment in the first part of the proposed solution

for the CRM problem in the uplink case. We consider 10 RBs and that flows 1

and 4 are the donors and that flows 2 and 3 are receivers. . . . . . . . . . . . . . 78

6.9 Illustration of the process to generate the RB groups based on the second

available RB of the fourth example of Figure 6.8 considering i equal to 1, 2 and

3 for the CRM problem in the uplink case. . . . . . . . . . . . . . . . . . . . . . . . 79

6.10Illustration of the process to generate the RB groups based on the available RB

of the second example of Figure 6.8 considering i equal to 1, 2 and 3 for the CRM

problem in the uplink case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.11Outage rate for CRM OPT, URM OPT and the proposed solution with one service

in scenarios 1, 2 and 3 for the CRM problem in the uplink case. . . . . . . . . . . 82

6.12Outage rate for CRM OPT, URM OPT and the proposed solution with two services

in scenarios 4, 5 and 6 for the CRM problem in the uplink case. . . . . . . . . . . 83

6.13Outage rate for CRM OPT, URM OPT and the proposed solution with three

services in scenarios 7, 8 and 9 for the CRM problem in the uplink case. . . . . . 83

6.14CDF of total data rate for CRM OPT, URM OPT and proposed solution with one

service in scenarios 1 and 2 for the CRM problem in the uplink case. . . . . . . . 85

6.15CDF of total data rate for CRM OPT, URM OPT and proposed solution with two

services in scenarios 4 and 5 for the CRM problem in the uplink case. . . . . . . 86

6.16CDF of total data rate for CRM OPT, URM OPT and proposed solution for required

data rate of 20 kbps with three services in scenarios 7, 8 and 9 for the CRM


List of Tables

3.1 Main simulation parameters for the SISO case. . . . . . . . . . . . . . . . . . . . . 27

3.2 Parameters of the considered scenarios for the SISO case. . . . . . . . . . . . . . 28

4.1 Main simulation parameters for the SU MISO and SU MIMO cases. . . . . . . . . 43

5.1 Main simulation parameters considered in the performance evaluation for the

MU MIMO case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57


URM problem in the uplink case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70


CRM problem in the uplink case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.3 Parameters of the considered scenarios for the CRM problem in the uplink case. 81

A.1 Description of the main parameters used in Algorithms A.1, A.2 and A.3 for the

SISO and SU MIMO cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

B.1 Description of the main parameters used in Algorithms B.1, B.2, B.3, B.4, B.5

and B.6 for the MU MIMO case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

C.1 Description of the main parameters used in Algorithms C.1, C.2, C.3 and C.4 for

the URM problem in the uplink case. . . . . . . . . . . . . . . . . . . . . . . . . . . 107

C.2 Description of the main parameters used in Algorithms C.5, C.6, C.6, C.7, C.8

and C.9 for the CRM problem in the uplink case. . . . . . . . . . . . . . . . . . . . 112

List of Algorithms

5.1 BD-ZF with MT ≥ J ′ ·MR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.2 BD-ZF with MT < J ′ ·MR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

A.1 Initialization for the SISO and SU MIMO cases. . . . . . . . . . . . . . . . . . . . . 93

A.2 First part of the proposed solution (Unconstrained Maximization) for the SISO

and SU MIMO cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

A.3 Second part of the proposed solution (Reallocation) for the SISO and SU MIMO

cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

B.1 Initialization for the MU MIMO case. . . . . . . . . . . . . . . . . . . . . . . . . . . 100

B.2 First part of the proposed solution (Unconstrained Maximization) for the MU

MIMO case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

B.3 Second part of the proposed solution (Reallocation) for the MU MIMO case. . . . 102

B.4 Procedure 1 that is part of the second part of the proposed solution (Reallocation)

for the MU MIMO case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103





C.1 Initialization for solution to the URM problem in the uplink case. . . . . . . . . . 107

C.2 Part 1 of the proposed solution for the URM problem in the uplink case. . . . . . 108



C.5 Initialization for solution to the CRM problem in the uplink case. . . . . . . . . . 113

C.6 First part of the proposed solution (Unconstrained Maximization) for the CRM


C.7 Second part of the proposed solution (Reallocation) for the CRM problem in the

uplink case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

C.8 Procedure 1 that is part of the second part of the proposed solution (Reallocation)

for the CRM problem in the uplink case. . . . . . . . . . . . . . . . . . . . . . . . . 115

C.9 Procedure 2 that is part of the second part of the proposed solution (Reallocation)

for the CRM problem in the uplink case. . . . . . . . . . . . . . . . . . . . . . . . . 118

Nomenclature

Here we summarize the conventional notational of this thesis. Firstly, we present a list of

acronyms, followed by an overview of the notation of more general nature. We conclude with

the specific notation for this thesis.

Acronyms

The abbreviations and acronyms used throughout this thesis are listed here. The meaning

of each abbreviation or acronym is indicated once, when it first appears in the text.

3G 3rd Generation

3GPP 3rd Generation Partnership Project

4G 4th Generation

ADSL Asymmetric Digital Subscriber Line

BB Branch and Bound

BD Block Diagonalization

BER Bit Error Rate

BS Base Station

CDF Cumulative Distribution Function

CQI Channel Quality Indicator

CRM Constrained Rate Maximization

CSI Channel State Information

DFT Discrete Fourier Transform

DPC Dirty Paper Coding

FDM Frequency Division Multiplexing

FDMA Frequency Division Multiple Access

FFT Fast Fourier Transform

IID Independent and Identically Distributed

ILP Integer Linear Problem

IMT-A International Mobile Telecommunications - Advanced

IP Internet Protocol

ISI Inter Symbol Interference

LTE Long Term Evolution

LTE-A LTE - Advanced

MCS Modulation and Coding Scheme

MIMO Multiple Input Multiple Output

MISO Multiple Input Single Output

MMSE Minimum Mean Square Error

MRT Maximum Ratio Transmission

MU Multi-User

OFDM Orthogonal Frequency Division Multiplexing

OFDMA Orthogonal Frequency Division Multiple Access

PAPR Peak-to-Average Power Ratio

QoS Quality of Service

RB Resource Block

RRA Radio Resource Allocation

SC-FDMA Single Carrier - Frequency Division Multiple Access

SCM Spatial Channel Model

SDMA Space-Division Multiple Access

SINR Signal to Interference-plus-Noise Ratio

SISO Single Input Single Output

SIMO Single Input Multiple Output

SNR Signal to Noise Ratio

SU Single-User

SVD Singular Value Decomposition

TDMA Time Division Multiple Access

TTI Transmission Time Interval

UMTS Universal Mobile Telecommunications System

URM Unconstrained Rate Maximization

VR Virtual Resource

WiMAX Worldwide Interoperability for Microwave Access

ZF Zero-Forcing

ZMCSCG Zero Mean Circularly Symmetric Complex Gaussian

Notations

The following notation is used throughout this thesis. We use uppercase and lowercase

boldface to denote matrices and vectors, respectively. Plain letters are used for scalars. Other

notational conventions are summarized as follows:|A| - Cardinality of set A⋃i∈I

Ai - Set operation that represents the union of the sets Ai ∀i ∈ I

|a| - Absolute value of the scalar a

(·)T - Transpose of a vector or matrix

Ia - Identity matrix with dimension a× a

0a,b - Matrix composed of 0’s with dimension a× b

1a - Column vector of length a composed of 1’s

0a - Column vector of length a composed of 0’s

diag (· · · ) - Block diagonal matrix with the arguments in the main diagonal

argmina∈A f (a) - Value of a ∈ A that minimizes the function f (·)argmaxa∈A f (a) - Value of a ∈ A that maximizes the function f (·)min (· · · ) - Minimum value among all arguments

max (· · · ) - Maximum value among all arguments

‖ · ‖2 - Euclidean norm or 2-norm of a vector

(·)H - Hermitian of a matrix

(·)−1 - Inverse of a matrix(ab

)- Number of distinct b-element subsets of any set containing a elements (binomial

coefficient)

loga (·) - Logarithm to base a of the argument

ln (·) - Natural logarithm of the argument

O (·) - Worst case complexity order of the argument (big O notation)

Specific Notations of the Thesis

We summarize here the symbols and notations that are used in the considered system

modeling of this thesis. The variables defined in the appendices are presented in specific

tables.an,p - Assumes the value 1 if the RB n is present in the assignment pattern p

A - Matrix composed of the elements an,p

bj,n - Transmit signal vector to flow j on RB n in MIMO scenarios

bj,n - Prior-filtering received signal vector of flow j on RB n in MIMO scenarios

bj,n - Post-filtering received signal vector of flow j on RB n in MIMO scenarios

c - Number of subcarriers in an RB

cj,n - Number of streams that are transmitted to flow j on RB n in MIMO scenarios

dlj,n - lth row vector of the matrix Dj,n

Dj,n - Receive matrix employed by the flow j when receiving data on RB n in MIMO

scenarios

G - Total number of SDMA groups that can be built with J flows

G - Set with all the SDMA groups that can be built with the flows of set JGn - Set of flows that compose the SDMA group assigned to RB n

f (·) - Link adaptation function that maps SNR on transmit data rate

hDLj,n - Channel transfer function of the link between flow j and the serving BS on the

RB n in the downlink when considering single antenna transceivers

hDLj,n,a,b - Channel transfer function of the link between the ath receive antenna of flow j

and the bth transmit antenna of the serving BS on RB n in downlink

hULj,z,n - Channel transfer function of the link between flow j and the serving BS at the

zth subcarrier of RB n in uplink

Hj,n - Channel transfer matrix of flow j on RB n composed of the elements hDLj,n,a,b

J - Total number of flows

Js - Number of flows that belongs to service s

J - Flow set

Js - Set of flows that belongs to service s

ks - Required minimum number of flows that should be satisfied for service s

MR - Number of receive antennas at terminals in the downlink scenario

MT - Number of transmit antennas at BS in the downlink scenario

Mj,n - Transmit matrix employed by the serving BS when transmitting to flow j on RB

n in MIMO scenarios

nj,n - Noise vector at terminal j on RB n

N - Total number of RB

N -RB set

Np - Set of Resource Blocks (RBs) that composes the assignment pattern p

og,j - Assumes the value 1 if flow j is a member of the SDMA g

O - Binary matrix composed of the elements og,j

P - Number of RB patterns according to the number of available RBs

PDL - Total power available at the serving BS (downlink)

PUL - Total power available at the mobile terminal (uplink)

P - Set with the indices of all assignment patterns

rMU DLg,j,n - Transmit data rate of flow j when the SDMA group g is assigned the RB n in the

MU MIMO downlink scenario

rSU DLj,n - Transmit data rate of flow j when RB n is assigned to flow j in SU downlink

scenario

rULj,p - Transmit data rate of flow j when RB pattern p is assigned to flow j in SC-FDMA

uplink scenario

S - Total number of services

S - Service set

tj - Required data rate of flow j

u (x, b) - Step function at b that assumes the value 1 if x ≥ b and 0 otherwise

xMU DLg,n - Assumes value 1 if the SDMA group g is assigned to RB n and 0 otherwise in

MU MIMO downlink scenario

xSU DLj,n - Assumes value 1 if RB n is assigned to flow j and 0 otherwise in SU downlink

scenario

xULj,p - Assumes value 1 if RB pattern p is assigned to flow j and 0 otherwise in

SC-FDMA uplink scenario

XMU DL - Assignment matrix with elements xMU DLg,n

XSU DL - Assignment matrix with elements xSU DLj,n

XUL - Assignment matrix with elements xULj,p

αDLj - Joint effect of the path loss and shadowing on the link between the serving BS

and flow j in the downlink

αULj - Joint effect of the path loss and shadowing on the link between the serving BS

and flow j in the uplink

γDLj,n -

SNR of flow j when receiving in RB n in the SISO downlink scenario

γDLj,n,l -

SNR of the lth stream of flow j on RB n in downlink with multiple antennas

γUL MMSEj,p - Effective SNR using MMSE frequency equalizer when RB pattern p is assigned

to flow j in SC-FDMA uplink

γULj,z,n -

SNR of flow j on zth subcarrier of RB n in SC-FDMA uplink

νj,n - Rank of the channel matrix Hj,n

ρj - Binary selection variable that assumes the value 1 if the flow j is chosen to be

satisfied in the CRM problem(σRB

)2- Noise variance considering the bandwidth of an RB

(σsub

)2- Noise variance considering the bandwidth of a subcarrier

1

Chapter 1

Introduction

This is an introductory chapter where we present the motivation and scope of this thesis

in section 1.1. After that we present basic concepts and background about relevant topics

to this thesis in section 1.2 while the state of the art is reviewed in section 1.3. The open

problems studied in this thesis and our main contributions are depicted in sections 1.4 and

1.5, respectively. Finally, the main scientific production during the Ph.D. course are presented

in section 1.6.

1.1 Thesis Scope and Motivation

With its 3rd Generation (3G) advent, cellular networks were able to switch from the

provision of the single circuit-switched voice service to a multi-service scenario with a

wide variety of multimedia services. These networks are continuously evolving and, in

particular, we are witnessing the beginning of the commercial deployment of the Long Term

Evolution (LTE) system. Furthermore, 3rd Generation Partnership Project (3GPP) and other

standardization bodies have been working on the specifications of LTE-Advanced in order to

meet the requirements of International Mobile Telecommunications - Advanced (IMT-A) or

4th Generation (4G) [1]. The fierce competition for data service provision among wireless and

wired networks, the need of being connected anytime and anywhere and the appealing mobile

devices/applications are strong indications that the mobile broadband market has potential

for further worldwide increasing.

3G and further generations have been designed to provide high transmit data rates and

offer to smartphones, tablets and notebooks a plenty of options for mobile access. Those

sophisticated mobile terminals enable the quickly popularization of new appealing data mobile

applications. Not surprisingly, the Universal Mobile Telecommunications System (UMTS)

Forum has predicted that voice and data traffic on mobile networks will grow more than

30-fold during the decade ahead [2]. This increased mobile data traffic and the need to provide

high quality data services with sustainable Quality of Service (QoS) appear as a challenging

scenario for mobile network operators and industry.

In order to deal with that scenario, improvements in the core network have been done

by means of an Internet Protocol (IP)-based packet-switched architecture. This architecture

allows for cost-efficient deployment and efficient support of mass market usage of any

IP-based service.

Improvements have also been achieved within the radio access part of those networks.

The Orthogonal Frequency Division Multiple Access (OFDMA) is a technology based on

Orthogonal Frequency Division Multiplexing (OFDM) that allows for an efficient and flexible

1.2. Background 2

use of the available spectrum. OFDMA has been selected as the downlink multiple access

scheme of modern networks such as LTE/LTE-Advanced [3] and Worldwide Interoperability

for Microwave Access (WiMAX) [4]. Due to the high Peak-to-Average Power Ratio (PAPR) of the

OFDM signal and consequently, the need of highly linear power amplifiers at the transmitter,

LTE network adopted Single Carrier - Frequency Division Multiple Access (SC-FDMA) (also

known as Discrete Fourier Transform (DFT)-spread OFDMA) instead of OFDMA in the uplink

direction. Compared to OFDMA, SC-FDMA signals have inherently lower PAPR [5].

Other functionality that is mandatory in the new standards is the use of multiple antennas

at the transmitter and receiver, i.e., Multiple Input Multiple Output (MIMO) [6]. MIMO

techniques have been intensively studied in the last decade and have become part of most

modern communication standards such as LTE, LTE-Advanced and WiMAX. Making use of

advanced signal processing algorithms, MIMO techniques together with OFDM-based multiple

access schemes allow for the exploitation of the spatial dimension of frequency channels for

obtaining diversity and/or multiplexing gains.

The use of those new multiple access strategies with MIMO are capable of boosting the

achievable data rates in the radio access. However, in order to obtain maximal capacity,

a relevant and useful functionality to deal with the challenges of next generation cellular

networks is an efficient Radio Resource Allocation (RRA). RRA algorithms interact with

multiple access schemes and MIMO and are responsible for the management of the scarce

radio resources such as power, time slots, spatial channels and frequency chunks [7].

RRA functionalities are capable of improving the spectral efficiency by a proper assignment

of such resources. Moreover, in multiservice scenarios RRA algorithms can be used to

satisfy the QoS requirements of the connected flows1 that in general have heterogeneous

demands and different channel quality states. Specifically, we call resource assignment the

functionality responsible for assigning frequency resources to the connected flows.

1.2 Background

This section is devoted to the introduction of basic concepts that are relevant for the

remaining of this thesis. In the following sections we introduce the OFDMA and SC-FDMA

multiple access methods, multiple antennas techniques, QoS and satisfaction concepts, and

RRA.

1.2.1 Multiple access methods

As will be clear later, multiple access schemes are important when studying and designing

RRA solutions. Along this thesis we will study RRA problems in the downlink OFDMA and

uplink SC-FDMA. Both multiple access methods will be modeled in this thesis not in the

signal processing level but in the level of how the resources are organized and accessed by

mobile terminals. In the following we present a brief description of those two multiple access

methods.

1.2.1.1 OFDMA

OFDMA is a multiple access scheme based on OFDM [8]. OFDM is a transmission

technology that has been utilized in wired and wireless communications. Asymmetric Digital

Subscriber Line (ADSL) broadband access and power line communications are examples of

applications of OFDM in wired systems. In wireless systems, the OFDM technology is utilized

in IEEE 802.11 a/g, LTE, LTE - Advanced (LTE-A) and Mobile WiMAX standards.

1Although a terminal can bear multiple service flows, without loss of generality, here only one service flow isconsidered per terminal. Consequently, flow and terminal are interchangeable terms throughout this thesis.

1.2. Background 3

In OFDM, the frequency band available for transmission is divided into several subcarriers

that have narrower bandwidth than the channel coherence bandwidth, as in Frequency

Division Multiplexing (FDM) systems. However, the subcarriers in OFDM are designed to

be orthogonal among each other, which leads to higher spectral efficiency than FDM. OFDM

transceivers can be efficiently implemented using the Fast Fourier Transform (FFT), and as

consequence of the narrowband subcarriers, sophisticated equalization structures are not

needed. Besides that, as the data rate transmitted in each subcarrier is low and consequently

the modulated symbols are longer than the delay spreading, OFDM is robust against Inter

Symbol Interference (ISI). In order to effectively mitigate the effects of ISI, a guard interval

named cyclic prefix, that consists in a copy of part of the OFDM symbol, is inserted before the

OFDM symbol transmission.

OFDM systems also allow for the practical use of multiple antenna schemes (MIMO).

Although, the transmission of high data rates turns the MIMO channel to be frequency

selective, the combination of multiple antenna and OFDM technologies transforms the

frequency-selective channel into a set of parallel frequency-flat channels.

With OFDMA [9], the multiple access is achieved by the assignment of different subcarriers

or block of them to individual flows at different time periods. Therefore, OFDMA is used

jointly with Time Division Multiple Access (TDMA) multiple access. Assuming single antenna

systems, the system resources in OFDMA can be arranged in a time-frequency grid as shown

in Figure 1.1. In the frequency axis the granularity is defined by the subcarriers while in the

time axis it is defined by an OFDM symbol. In MIMO OFDMA systems the spatial dimension is

added to the resource grid leading to more flexibility in the resource assignment. A Resource

Block (RB) is defined as the minimum allocable resource that consists in a group of one or

more adjacent subcarriers in the frequency dimension and a number of consecutive OFDM

symbols. The number of subcarriers and OFDM symbols in an RB depends on the system

design and channel characteristics.

Flow 1

Flow 2

Flow 3

Time (Set of

OFDM symbols)

Frequency (Group of subcarriers)

Resource block

Figure 1.1: Frequency-time resource grid in OFDMA.

In order not to generate intracell interference in wireless systems, the same RB is not

1.2. Background 4

assigned to different flows at the same time within a cell. As it will be presented later, this

constraint renders a combinatorial component to the studied RRA problem. We define this

constraint as exclusivity constraint. Note that the exclusivity constraint is not present when

multiple antennas are used to provide multiple access to the flows, as it will be presented

later.

1.2.1.2 SC-FDMA

Despite the many advantages of OFDMA, it suffers from strong envelope fluctuations

resulting in a high PAPR. Signals with high PAPR place a significant burden on mobile

terminals due to the need of highly linear power amplifiers to avoid excessive signal distortion.

Motivated by that reason and other practical aspects, 3GPP has chosen SC-FDMA as the

multiple access technology for the uplink of LTE networks.

SC-FDMA, that is also known as DFT-spread OFDMA, transforms the time domain data

symbols to frequency domain by a DFT before going through OFDMA modulation. Compared

to OFDMA, SC-FDMA signals have inherently lower PAPR. On the other hand, in order

to mitigate inter-symbol interference, the base station employs adaptive frequency domain

equalization. In summary, SC-FDMA reduces the requirements on linear power amplifiers of

mobile terminals but still requires frequency domain equalization at the base station [5].

SC-FDMA imposes additional constraints on RRA compared to OFDMA. Particularly, the

frequency resource blocks assigned to a given mobile terminal for transmission should be

adjacent to each other in order to obtain benefits in terms of PAPR. This new constraint

significantly reduces the freedom in RRA compared to the OFDMA case in which this

constraint does not exist. In summary, SC-FDMA imposes the following constraints when

assigning resources:

◮ Exclusivity: The same RB cannot be shared by flows within a cell. Note that this

constraint already exists for the OFDMA case;

◮ Adjacency: The RBs assigned to the flows should be adjacent to each other in the

frequency domain. This constraint is not necessary in the OFDMA case.

1.2.2 Multiple antennas techniques

At the end of the 1990s multiple antenna techniques were theoretically shown to provide

a novel means to achieve improved performance in wireless systems [10]. The use of multiple

antennas at the transmitter and/or receiver is now part of any modern mobile communication

system.

The multipath propagation due to the interaction of the electromagnetic waves with

the environment by means of reflections, refractions, scattering and diffractions has been

considered as a degrading characteristic of wireless systems when single antennas systems

are employed. Surprisingly, the multipath propagation is of utmost importance for obtaining

the gains with the use of multiple antenna systems.

The technology of transmitting/receiving with multiple antennas can be classified

according to the number of antennas at the transmitter and/or receiver. When multiple

antennas are used at the transmitter and single antennas are employed at the receiver we have

the Multiple Input Single Output (MISO) schemes. On the other hand, when we have multiple

antennas only at the receiver we have the Single Input Multiple Output (SIMO) schemes. The

use of multiple antennas at the transmitter and receiver corresponds to the MIMO schemes.

These schemes are capable of obtaining different benefits depending on the design objectives.

1.2. Background 5

Basically, multiple antennas can be employed to obtain diversity and/or multiplexing

gains. Diversity gains can be obtained by the transmission/reception of redundant signals

representing the same information. At the receiver the transmitted signals should be

coherently combined in order to achieve gains in signal strength that have as consequence

improvements in the link reliability and error performance. Basically, the SIMO and MISO

schemes are used to provide diversity gains. Multiplexing gains can be achieved by the

simultaneously transmission of different pieces of information or data streams at the same

time and frequency resources by means of the so called spatial channels. In general, the

number of data streams that can be transmitted at the same time is limited by the minimum

between the number of antennas at the transmitter and receiver. Therefore, MIMO schemes

can be used to obtain multiplexing and diversity gains. The spatial multiplexing is an

attractive technique to improve the data rates without resorting to more frequency bandwidth.

The use of MIMO can also be classified according to the capacity of transmitting to multiple

users. In Single-User (SU) MIMO schemes, multiple antennas are used for transmitting data

to a single user within a given time-frequency resource. Therefore, the spatial dimension is not

used to multiplex different flows in the spatial domain. Multi-User (MU) MIMO schemes enable

the allocation of different spatial subchannels to different flows in the same time-frequency

resource. In this case, the spatial dimension can be used as another tool to exploit the

multiuser diversity as it was already done with time, frequency and power [11]. MU MIMO

schemes are also known as Space-Division Multiple Access (SDMA) due to its capacity to

multiplex different users similar to other multiple access technologies such as TDMA and

Frequency Division Multiple Access (FDMA).

In order to perform spatial multiplexing with SU MIMO it is necessary the transmission

of multiple interference-free streams. So as to be able to do that, multiple antennas

should be employed at the transmitter and at the receiver. In other words, the number

of interference-free data streams is limited by the minimum number of antennas at the

transmitter and receiver. However, this is not mandatory with MU MIMO since the set of

mobile terminals can be seen as a virtual receiver with multiple antennas. In the case

of single antenna receivers, if the selection of mobile terminals to be spatially multiplexed

guarantees that they are far apart from each other, improved channel correlation properties

can be obtained as compared to the case of multiple receiver antennas at the same device.

The MU MIMO capability turns the RRA problems even more challenging due to the added

degree of freedom. Furthermore, MU MIMO imposes strong requirements regarding channel

state information available at the transmitter in order to achieve performance gains. This

requirement is not essential in SU MIMO schemes and compromises the uplink capacity of

the mobile systems due to the need of channel state information feedback [12].

1.2.3 QoS and satisfaction

The popularity of fixed networks with very high data rates motivated the development

of several data services that demand exchange of different multimedia information. The

development of mobile networks has opened the possibility of having access to these services

on mobile devices. Also, the improvement of the processing power of mobile devices such

as smartphones and tablets has contributed to the popularity of data services in mobile

networks.

Differently of wired networks, in mobile networks the system operator has to deal with

limited resources such as frequency bandwidth and power, and the unpredictable nature

of the wireless channels. All these factors require from the mobile network operators the

1.3. State of the Art 6

employment of a sophisticated control of the parameters that impact on how well a service is

perceived. Parameters such as data rate, packet delay and jitter are among the most common

QoS requirements. In fact, in order to support more flows in the network, resources should be

assigned to flows so as to guarantee the minimum contracted QoS. With this, more resources

are left to support new flows.

From the operator’s point of view it is important to provide the different services with a

sustainable quality. In order to measure the quality level in which each service is provided

in the network, system operators could adopt the QoS management strategy of considering

minimum user satisfaction ratios for each service. In this way, in order to consider that a

given service is provided with acceptable quality, the operators should satisfy a minimum

percentage of the data sessions. As these minimum satisfaction ratios are defined for

each service, the system operators can establish a priority hierarchy between the provided

services [13].

1.2.4 Radio Resource Allocation

RRA is a system functionality that is responsible for allocating the available resources of

the radio access network to the connected flows. When the system bottleneck is in the radio

access instead of the core network, efficient RRA can dictate the performance of the overall

system.

Among the available resources that can be allocated in the modern cellular networks we

can mention frequency bandwidth (in terms of subcarrier or group of them in OFDMA or

SC-FDMA), time slots, power and spatial subchannels when MIMO is employed. All these

resources are limited and should comply with specific constraints. As an example, the

assignment of subcarriers or group of them should be in accordance with the considered

multiple access constraints. Another example is that the number of spatial subchannels

depends on the number of antennas at the transmitter and receiver.

Besides the constraints imposed on the resources, we also have different design targets

depending on the objective to be attained. As it will be presented in section 1.3, we can find in

the literature many objectives such as improving the spectral efficiency or assuring fairness

among the connected flows.

RRA in general rely on different information in order to achieve the design targets.

Up-to-date channel state information is of fundamental importance in order to exploit the

frequency and multiuser diversity. By the knowledge of the channel state information RRA can

take advantage of the frequency selective nature of the wireless channel (frequency diversity)

and also the different propagation channels that each individual terminal experiences

(multiuser diversity).

1.3 State of the Art

In general, RRA problems are formulated in mathematical form as optimization problems

composed of objective functions and constraints that limit the search space and feasible

solutions. In the literature we can find many RRA problems with different objectives and

constraints. Particularly, different multiple access schemes and antenna configurations

impose new constraints to the problems.

Focusing first on single antenna networks or Single Input Single Output (SISO), one of

the most well-known and first RRA problems was the unconstrained rate maximization in

the downlink of OFDMA based systems [14]. Basically, this problem consists in finding the

RRA solution that results in the most efficient use of the available resources in terms of


transmit data rate. The solution to this problem is accomplished when each subcarrier is

assigned to the flow with the best channel quality on that subcarrier and the power per

subcarrier is distributed following the water-filling policy [14, 15]. The study of this problem

provides important insights regarding the maximum spectral efficiency that can be obtained

with given limited resources. However, it is well-known that the solution to this problem does

not consider QoS aspects and usually leads to service starvation of the terminals at the cell

border. In fact, maximization of spectral efficiency and QoS fulfillment for the flows are two

contradicting goals in mobile wireless networks [16].

As presented in section 1.1, the provision of multiple services imposes minimum QoS

constraints to the connected flows. In order to address QoS aspects, other problems were

formulated, such as the margin adaptive problem which consists in minimizing the total

transmit power subject to individual data rate requirements for each flow. For this problem,

a Lagrangian-based algorithm is proposed in [17], which is able to obtain considerable power

efficiency gains but at the cost of a prohibitive computational complexity. In [18], a heuristic

and low-complexity solution to this problem is proposed considering fixed modulation types.

A further improved scheme with similar complexity is proposed in [19]. In [20], a heuristic

algorithm that determines firstly how many and secondly which subcarriers should be

assigned to each flow is proposed to solve the margin adaptive problem.

Another RRA problem is the rate adaptive problem in which the objective is to maximize

the minimum flow data rate. Note that this problem is a particular case of the weighted rate

balancing problem whose objective is to maximize the total downlink data rate in the cell

while assuring that the flows’ achievable data rates are proportional to pre-defined weights

that can be seen as fairness constraints. The solution to these problems tends to balance

the achievable data rate of the flows in the cell, i.e., achieve fairness among the data rates

allocated to the connected flows. In [21], a heuristic solution to this problem is proposed which

firstly assigns to each flow its best subcarrier, then the remaining subcarriers are assigned to

the flows with lowest data rate. The work in [22] studies the weighted rate balancing problem.

The disadvantage of the solutions of the rate adaptive and weighted rate balancing problems

is that they may penalize the terminals with better channel qualities and reduce the system

efficiency, since many resources are assigned to the flows with high weights independently of

their channel states.

In order to assure a minimum QoS to the flows, some works have studied the problem

of maximizing the overall data rate subject to minimum flow data rate constraints. In [23], a

suboptimal solution is proposed that firstly determines the amount of resources to be assigned

to each flow. Then, the resources are assigned to each flow in an opportunistic manner. Since

the Hungarian algorithm is used for resource assignment, the computational complexity may

be prohibitive [24]. In [25], this problem is formulated as an Integer Linear Problem (ILP)

and a low-complexity suboptimal solution is proposed. Computational complexity is further

reduced in [26]. For more details on these RRA problems and solutions for single antenna

systems see [7,11].

RRA problems with similar objectives as the ones presented for SISO systems were also

studied and generalized to MIMO systems. Compared to the SISO case, RRA problems with

MIMO are even more difficult to solve due to the new dimension for allocation, i.e., spatial

dimension. As stated in section 1.2.2, in the context of multiple antenna systems we can

identify two specific cases: SU MIMO and MU MIMO. In SU MIMO the spatial channels

corresponding to a given frequency resource are assigned to the same flow, i.e., there is no

frequency resource sharing among different flows. This is not the case in MU MIMO. In fact,


with this antenna scheme RRA has the freedom to assign spatial subchannels corresponding

to the same frequency resource to different flows. This is also known as SDMA.

Due to the additional complexity of RRA with multiple antennas, differently of the SISO

case, the solution to the unconstrained rate maximization problem is no more trivial. In [27]

the authors have found the achievable sum rate capacity for multi-antenna downlink channel.

The work in [28] provides strategies to achieve the sum rate capacity using non-linear

processing Dirty Paper Coding (DPC) firstly described in [29]. However, obtaining the optimal

transmission policy when employing DPC is a computationally complex non-convex problem.

Therefore, in [30] the authors use duality to transform this problem into a well-structured

convex problem for the multiple access channel.

Among the QoS aware solutions for MIMO we can mention the reference [31] that proposed

solutions for the margin adaptive problem based on DPC and the uplink-downlink duality

between multiple access channels and broadcast channels. However, the DPC complexity

imposes constraints on the practical implementation of those solutions. The works [32, 33]

provided low-complexity solutions to the margin adaptive problem by using linear processing

at the transceivers. The weighted rate balancing problem has been studied for MIMO in [34,35]

by using non-linear DPC-based techniques. Efficient solutions with affordable computational

complexity using linear transceivers were considered in [36]. Another work studied the

problem of maximizing the overall data rate subject to minimum user data rate constraints [37].

The main idea is to assign more resources to the users which can contribute most to sum

capacity.

As pointed out in section 1.2.1.2, the uplink of LTE employs SC-FDMA as the multiple

access scheme. Particularly, this multiple access method imposes a new constraint on the

RRA: resource adjacency. With the resource adjacency constraint, the frequency blocks

assigned to each flow should be adjacent to each other in order to get benefit of the low

PAPR characteristic of SC-FDMA. Furthermore, there is also the constraint that the same

Modulation and Coding Scheme (MCS) should be used in all resources assigned to a given

flow.

Differently of the works devoted to RRA for downlink OFDMA, the works addressing RRA

problems for uplink SC-FDMA are quite recent. Most of them are mainly concerned with

solving RRA problems previously studied in the downlink considering the new constraints

imposed by SC-FDMA in the uplink. Focusing firstly on the unconstrained rate maximization

problem, we have that differently of the downlink SISO scenario, the optimal solution to

this problem is not easily obtained. In [38], the authors considered the unconstrained rate

maximization or sum rate maximization problem for the SC-FDMA scenario. However, this

work ignores the system constraint of subcarriers adjacency that is necessary to assure low

PAPR. In [39] the authors considered the adjacency constraint, however, it was assumed

that each flow demanded the same number of frequency resources. In practice, flows

have different data rate requirements and channel qualities leading to different demands

regarding the number of frequency resources. In [40], a different approach is followed

by formulating the total data rate maximization problem as a pure binary-integer problem

called set partitioning for which the optimal solution can be obtained without employing an

exhaustive enumeration. Therein, the authors also proposed a low-complexity suboptimal

algorithm for the studied problem. In [41], the authors proposed a suboptimal solution to

the unconstrained rate maximization with marginal gain over the algorithm proposed in [40]

but with lower computational complexity. The savings in computational complexity were

obtained mainly by simplifying the expression used to calculate the terminal data rates when

1.4. Open Problems 9

Zero-Forcing (ZF) and Minimum Mean Square Error (MMSE) frequency domain equalizers are

employed.

As the unconstrained rate maximization problem does not consider QoS aspects, some

works have considered other RRA problems that can guarantee a better resource distribution.

In [42], a suboptimal solution is proposed to the problem of utility maximization. The

considered utility functions were the sum of flows’ data rates and the sum of the logarithm

of the flows’ data rates; the last one designed to achieve proportional fairness. Although

SC-FDMA is considered in that work, the subcarrier adjacency constraint was not taken

into account for RRA. In [43], the authors studied an RRA problem based on a general

metric that quantifies the efficiency of assigning each resource to each terminal. Then,

a search-tree based algorithm was proposed that equally divides the resources among the

flows. This simplifying assumption limits the flexibility for RRA and the potential gains on

multi-user diversity. The authors of [43] subsequently proposed another algorithm in [44]

that allows the assignment of different amounts of frequency resource to the flows in order

to achieve proportional fairness. The proposed algorithm iteratively finds the terminal and

frequency resources combination with highest metric and expand the allocated bandwidth to

that terminal by assigning the contiguous frequency resources in which the selected terminal

has the highest metric compared to all other flows. The algorithm stops when all resources

are assigned. Based on the same modeling of [43, 44] and considering the knowledge of a

metric for each terminal-resource pair, the authors in [45] proposed three different suboptimal

solutions. The algorithm that meets the best performance-complexity trade-off presents

similar ideas as those shown in [44]. In [46] that algorithm is modified to provide better

performance at the cost of higher computational complexity. Proportional fairness is also

the subject of [47], in which the authors show that the adjacency constraint is sufficient to

characterize NP-hardness. Furthermore, the authors propose suboptimal algorithms to deal

with the formulated problem and which exploit frequency correlation as in [44–46].

Other contributions with QoS-aware solutions for SC-FDMA uplink were [48–50]. In [48],

the authors consider the problem of utility maximization with constraints on the minimum

terminal data rates and maximum request delay. Heuristic solutions are proposed but the

simulation results do not include comparisons of the proposed solutions with an upper bound

or even with other related solutions. The main contribution of [49] consists in the integration

of a well-known downlink scheduling solution with a method of estimating packet delays in

uplink that is not as trivial as in downlink. Although, LTE uplink is considered, the adjacency

constraint is not modeled. Finally, the margin adaptive problem that has been extensively

studied in downlink OFDMA was considered in [50] for SC-FDMA uplink. The authors propose

a suboptimal solution with variable complexity by adaptively defining the size of the search

space composed of the possible terminal-resource assignments.

1.4 Open Problems

In section 1.3 we presented different RRA problems that were studied in some scenarios

with different antenna configurations and multiple access schemes. As shown by the

literature review, the community research is aware of the relevance of QoS fulfillment to

modern networks what shows the importance of the subject. In this thesis we are still

concerned with the QoS fulfillment, however, we add another constraint motivated by the

system operator needs.

So as to assess the quality level in which each service is provided within the network,

we assume that the system performance of multiservice networks is measured by means

1.4. Open Problems 10

of minimum satisfaction constraints for each provided service type. More specifically, they

require that a certain fraction of the connected flows of each service be satisfied with the

provided QoS [13, 51, 52]. With this in mind, we propose a new RRA problem that is the

maximization of the overall data rate subject to minimum satisfaction constraints per service.

Basically, the objective of the studied problem is to maximize the total data rate in the

system. This objective is the same as some of the problems presented in section 1.3 and

aims at guaranteeing that the system resources are used in an efficient way. The main

problem constraint consists in assuring that a pre-defined number of flows from each provided

service are satisfied with the resource distribution. Due to the relevance of the unconstrained

rate maximization problem to our work we denote hereafter this problem by the acronym

Unconstrained Rate Maximization (URM). Moreover, for the sake of simplicity we denote the

maximization of the overall data rate subject to minimum satisfaction constraints per service

problem by Constrained Rate Maximization (CRM). Although we have presented many other

problems that impose constraints to the RRA problem, the use of this acronym refers only to

the main problem studied in this thesis.

To the best of our knowledge the CRM problem was not considered in the literature so

far. Related to this new problem some engineering/scientific questions arise and need to be

answered:

i. Problem formulation and modeling: As presented in the literature review, this problem

has not been studied before as far as we know. The presented RRA problems shown in

section 1.3 were formulated as optimization problems. Therefore, the identification of the

system variables and a model for the problem in the optimization form are themselves

open research topics to be studied.

ii. Characterization of optimal solution: The knowledge of the best possible solution

(optimal) to a problem or a bound for it provides important insights to researchers and

engineers. Therefore, an open problem to be studied in this thesis is to find (if possible)

methods to obtain the optimal solution to the presented problem without resorting to

brute force methods.

iii. Existence of low-complexity solutions: In general the RRA problems that involve RB

allocation are combinatorial and hard to optimally solve. Therefore, the proposal of

low-complexity solutions is of utmost importance. Consequently, another open issue

to be studied is whether good (probably suboptimal) solutions can be found to the

considered problem.

iv. Performance metrics and evaluation: The studied problem is composed of two

contradicting goals: maximization of total data rate (problem objective) and satisfaction

guarantees (problem constraints). Consequently, new performance metrics should be

identified in order to measure how efficient are the solutions. Moreover, performance

evaluation of the involved solutions should be addressed.

v. Different scenarios: As presented in the literature review, the different RRA problems

were studied in different scenarios. The previous open problems have different

particularities depending on the considered scenario. The other open problem to be

studied in this thesis is how the following scenarios impacts on the answer of problems

i, ii, iii and iv.

(a) Downlink SISO

1.5. Contributions and Thesis Organization 11

(b) Downlink SU MIMO

(c) Downlink MU MIMO

(d) Uplink SC-FDMA

vi. Computational complexity analysis: As we are dealing with algorithms, besides the

performance evaluation by means of computational simulations, the computational

complexity is needed in order to compare different algorithms in terms of the

performance-complexity trade-off.

1.5 Contributions and Thesis Organization

In Chapter 2 we present the basic system model assumed along the thesis. As different

scenarios are considered along the thesis, in this chapter we only present the common aspects

of the modeling. Also, in this chapter we present the studied problem in a general form so as

to provide an overview without going into details that are specific of each scenario. We finish

this chapter presenting a heuristic framework to be followed in order to conceive efficient

solutions to the studied problem. The contributions of this chapter help to address the open

problems i and iii.

The scenario with single antennas and downlink OFDMA is approached in Chapter 3.

In this chapter we mathematically formulate the problem and provide a discussion about a

method to obtain the optimal solution. It is important to highlight here that few works in

the literature as shown in section 1.3 provide means to obtain or characterize the optimal

solutions or bounds to their studied problems. Due to the high computational complexity of

the involved optimal solution we propose a low-complexity heuristic solution. This chapter is

finished with a performance analysis of the solutions by means of computational simulations.

The contributions of this chapter are related to the open problems i, ii, iii, iv and v(a).

Chapter 4 is devoted to the extension of the studied problem to the multiple antennas

case with SU MIMO scenario. Firstly, the system modeling considered in the last chapter

is extended to the SU MIMO including the MIMO channel and spatial filters. Then, similar

steps considered in Chapter 3 are followed: problem formulation, discussion about optimal

solution, proposal of heuristic solutions and performance analysis. The open problems i, ii,

iii, iv and v(b) are addressed in this chapter.

In Chapter 5 we generalize even more the downlink OFDMA scenario considered in previous

chapters by considering the multiple antenna case with possibility of sharing the same

frequency resource with different flows, i.e., MU MIMO scenario. As in the previous chapter,

we extend the system modeling and notation. In this chapter we introduce the concept of

SDMA groups that impacts considerably in the problem formulation and proposal of heuristic

solutions. Then we formulate the optimization problem, discuss the methods to obtain

the optimal solution, propose efficient solutions and present a performance analysis of the

involved solutions. In this chapter we present answers to the open problems i, ii, iii, iv and

v(c).

In Chapter 6 we consider the uplink SC-FDMA scenario. As pointed out previously, the

adjacency constraint limits the RRA in this scenario. We introduce in this chapter the concept

of assignment pattern in order to ease the problem formulation. As presented in the literature

review shown in section 1.3, the URM problem is a hard problem that needs efficient heuristic

solutions. Therefore, in this chapter we firstly propose a heuristic solution to this problem

along with its performance analysis. In the following we present contributions to the CRM

problem in the uplink scenario. The methodology followed in the previous chapters is present

1.6. Scientific production 12

here: problem formulation, characterization of the optimal solution and performance analysis.

Besides the proposal of efficient solutions to the URM problem, in this chapter we present

answers to the open problems i, ii, iii, iv and v(d).

In Chapter 7 we summarize the main conclusions obtained along the thesis. Furthermore,

we point out the main research directions that can be considered as extension of the study

performed in this thesis. Finally, in the appendices we present the computational complexity

of all algorithms proposed in this thesis. This contribution addresses the open problem vi.

1.6 Scientific production

The content and contributions present in Chapter 3 were published with the following

information:

◮ Lima, F. R. M.; Maciel, T. F.; Freitas, W. C.; Cavalcanti, F. R. P., “Resource Assignment

for Rate Maximization With QoS Guarantees in Multiservice Wireless Systems”. IEEE

Transactions on Vehicular Technology, 2012.

The technical content of the Chapter 4 was published with the following information:

◮ Lima, F. R. M.; Bezerra, N. S.; dos Santos, R. B.; Maciel, T. F.; Freitas, W. C.; Cavalcanti,

F. R. P., “Maximizing Spectral Efficiency with Acceptable Service Provision in Multiple

Antennas Scenarios”. European Wireless Conference, 2012.

Three provisional United States patents of the low-complexity algorithms proposed in

Chapters 5 and 6 were filled.

At the time of writing, we are working for publishing the technical content of Chapters 5

and 6. In parallel to the work developed in the Ph.D. course that was initiated on the second

semester of 2008, the Ph.D. candidate has been working on other research projects. Although

the studied problems in these projects are not the same as the ones of the Ph.D. study, the

context is within the area of radio resource management for wireless cellular networks. The

complete list of the articles is presented in the following:

◮ Silva, J. M. B.; Lima, F. R. M.; Maciel, T. F.; Cavalcanti, F. R. P., “Distributed Resource

Allocation for Wireless Service Provision in a Competitive Scenario”. XXIX Brazilian

Telecommunications Symposium, 2011.

◮ Lima, F. R. M.; Cavalcanti, F. R. P.,; Neto, R. O., “Radio Resource Management for Churn

Rate Control in Cellular Data Operators”. IEEE Globecom 2010 Workshop on Mobile

Computing and Emerging Communication Networks, 2010.

◮ dos Santos, R. B.; Freitas, W. C.; Lima, F. R. M.; Cavalcanti, F. R. P., “Method and

Arrangement for Resource Allocation”. United States Patent, (20100150088), Publication

date: July 2010.

◮ Lima, F. R. M.; Wänstedt, S.; Cavalcanti, F. R. P.; Freitas, W. C., “Scheduling for

Improving System Capacity in Multi-service 3GPP LTE”. EURASIP Journal of Electrical

and Computer Engineering, 2010.

◮ Lucena, E. O.; Lima, F. R. M.; Freitas, W. C.; Cavalcanti, F. R. P.; “Overload Prediction

Based on Delay in OFDMA Systems”. IEEE Globecom 2010 Symposium, 2010.

◮ Freitas, W. C.; Lima, F. R. M.; dos Santos, R. B.; Neto, R. O., “Resource Allocation

in Multiuser Multicarrier Wireless Systems with Applications to LTE”. Book: Optimizing

Wireless Communication Systems, New York, Springer, 2009.

1.6. Scientific production 13

◮ Lucena, E. O.;Lima, F. R. M.; Freitas, W. C.; Cavalcanti, F. R. P.; “Congestion

Control Framework for Real-Time Services in OFDMA-based Systems”. XXVII Brazilian


◮ Lima, F. R. M.; Freitas, W. C.; Cavalcanti, F. R. P.; “Scheduling Algorithm for

Improved System Capacity of Real-Time Services in 3GPP LTE”. XXVII Brazilian


◮ da Silva, A. P.; dos Santos, R. B.; Lima, F. R. M.; Cavalcanti, F. R. P.; Freitas,

W. C., “A Resource Assignment Study on Wireless OFDMA Systems”. XXVI Brazilian


◮ Lima, F. R. M.; dos Santos, R. B.; Cavalcanti, F. R. P.; Freitas, W. C., “Radio Resource

Allocation for Maximization of User Satisfaction”. The Ninth IEEE International Workshop

on Signal Processing Advances in Wireless Communications, 2008.

14

Chapter 2

System Modeling, General Problem

and Framework for Solution

In the next chapters we study the CRM problem in different scenarios such as

downlink/uplink and single/multiple antenna(s). Therefore, in section 2.1 of this chapter

we present the common aspects regarding the system model assumed along this thesis. In

section 2.2 we present the general view of the CRM problem without detailing the aspects that

depend on the considered scenarios. Later in the next chapters, we will refine the system

model according to their specificities. In section 2.3 we introduce a heuristic approach that

was followed along this thesis to propose alternative solutions to the CRM problem.

2.1 General System Model

We consider a cellular system1 composed of a number of sectored cells. For a given sector

of a cell, there is a group of flows2 connected to cell’s Base Station (BS). In this thesis we

consider the CRM problem in different scenarios including downlink and uplink directions.

In the downlink case we consider that the system combines OFDMA and TDMA while in the

uplink case the system employs SC-FDMA and TDMA. In both cases the available resources

are arranged in a time-frequency resource grid, where the minimum allocable resource, or

RB, is defined as a group of one or more adjacent subcarriers and a number of consecutive

OFDM symbols in the time domain, which represent the Transmission Time Interval (TTI). The

flows of a same sector can be simultaneously served by the assignment of different orthogonal

frequency-time RBs and, therefore, there is no intra-cell interference among flows of the same

sector in either downlink or uplink. The only exception to this is in the MU MIMO case where

the RBs are shared and interference (intra cell) could appear. As it will become clear later,

this interference could be controlled by spatial filtering.

Although intra-cell interference can be controlled by the considered multiple access

schemes (OFDMA in downlink and SC-FDMA in the uplink), the flows might still experience

inter-cell interference from other sectors that reuse the same frequency band in the cellular

system. Especially in packet-switched systems, inter-cell interference is quite unpredictable.

The reason for this is that at each TTI the resource usage pattern defined by the resource

1Note that the contributions presented in this thesis could be applied with small modifications in other systemsthat preserve similar characteristics as the one that will be shown such as flexible resource allocation.

2As stated in Chapter 1, we consider that although a terminal can bear multiple service flows, without loss ofgenerality, here only one service flow is considered per terminal. Consequently, flow and terminal are interchangeableterms throughout this thesis.

2.1. General System Model 15

Base Station

Terminal

Flow

SelectionFrequency

resources

Frequency

resources

Resource Assignment

Data flows

Channel state

of flow 1

Channel state

of flow J

Figure 2.1: Illustration of the system modeling and RRA.

assignment at each cell of the system can change considerably due to the dynamic

traffic conditions. There are many approaches to deal with inter-cell interference in the

literature [53].

Interference management is out of the scope of this work and we assume the simplifying

assumption that the inter-cell interference is modeled as a Gaussian random variable and

that it is part of the thermal noise in the Signal to Noise Ratio (SNR) expression. We highlight

that this assumption becomes more and more valid as the sector load and the number of cells

in the system increase [54].

In this thesis we consider a snapshot optimization problem that has as output the proper

association between flows and RBs (resource assignment). In other words, at each TTI we

intend to solve an assignment problem given the current system state such as channel state,

traffic state and QoS related variables.

The sequential solution to this problem along consecutive TTIs leads to decisions on which

packets are delivered using which resources. This solution is equivalent to the decisions taken

by a time-domain packet scheduler. Although the allocation along the time is not considered

in this thesis, this could be performed by the dynamic adaptation of the input variables of the

optimization problem. More comments about this issue are drawn in Chapter 7.

In this thesis we consider the approach followed by the works [7,11]. More specifically, we

consider that the RRA is split into two parts: flow selection (or flow scheduling) and resource

assignment. The system modeling, and the flow selection and resource assignment parts are

illustrated in Figure 2.1. As illustrated in this figure, the flow selection part is responsible

for pre-selecting the flows with high priorities among all connected flows in order to compete

for resource assignment in the second part. Among the criteria that can be used for flow

selection we can mention QoS aspects such as current packet delay, average data rate and

amount of buffered data. For good references about this topic see [55,56]. In the second part,

resource assignment, the proper association among the selected flows and RBs is done based

on but not limited to the RB channel states of each selected flow as shown in Figure 2.1. The

scope of this work is limited to resource assignment strategies such as the works discussed

in section 1.3 of this thesis.

We consider that in a given TTI, J active flows were pre-selected by the flow selection

part and are candidates to get resources at the resource assignment part. We assume that

there are N available RBs. Moreover, J and N are the set of active flows and available RBs,

respectively. As we are dealing with a multiservice scenario we assume that the number of

services provided by the system operator is S and that S is the set of all services. We consider

that the set of flows from service s ∈ S is Js and that |Js| = Js, where | · | denotes the cardinality

2.2. General Problem 16

of a set. Note that⋃s∈S

Js = J and∑s∈S

Js = J .

We assume that the channel state remains constant during the transmission in a TTI

(block fading channel model). This assumption is especially true when pedestrian channel

fading models are considered where the channel coherence time is much higher than the

TTI of LTE system for example [57]. Furthermore, we assume that there is perfect Channel

State Information (CSI) at the transmitter/receiver when performing RRA in all considered

scenarios. By CSI we consider the amplitude and phase of the channel transfer function.

Incomplete or imperfect CSI is left as perspectives of this thesis (see Chapter 7). More details

about channel modeling are presented in Chapters 3, 4, 5 and 6.

In this thesis we assume that the total power available at the BS in the downlink is

PDL whereas the total power at the terminal in the uplink is PUL. We also assume that

the power allocated to each RB is fixed and equal to PDL/N for downlink and PUL/N for

uplink. Therefore, the power is not optimized in our study. Although the joint optimization of

resource assignment and power would lead to a better resource allocation, it has been revealed

that the performance can hardly be deteriorated by equal power allocation [14, 25]. In fact,

the benefit of also having adaptive power control is marginal if an adaptive rate scheme is

already implemented such as channel state-aware resource assignment. Although the power

optimization is left out of the studied problem, the use of power allocation strategies after

solving resource assignment could achieve a good performance-complexity trade-off compared

to the joint optimization of resource assignment and power allocation and compared to the

resource assignment with equal power allocation (no dynamic power allocation) [58].

Link adaptation is a system functionality that is present in most of the modern mobile

networks. Basically, the transmission parameters at the physical layer are adapted according

to the current channel state. Among the transmission parameters we can mention modulation

type, constellation size and channel coding rates. Based on the adaptation of the transmission

parameters the terminals can transmit at different data rates depending on the channel state.

Along the next chapters we assume that link adaptation is a present functionality in the

network. More details are given later.

We assume that, at the current TTI, flow j has a data rate requirement equal to tj.

It is important to mention here that long-term data rate requirements can be mapped to

instantaneous data rate requirements [13]. The minimum satisfaction constraints for each

service are represented by the parameter ks which is the minimum number of flows from

service s that should be satisfied. Along the thesis we present other relevant variables that

are specific to each scenario.

2.2 General Problem

In Figure 2.2 we present an overview of the general problem to be solved in this thesis.

In the following chapters we present the problem to be studied in mathematical terms as

optimization problems. A general optimization problem is represented by an objective function

and a set of constraints.

The variable to be optimized is the resource assignment, i.e., the proper association

between flows and RBs. Note that when multiple antennas are employed each RB can be

associated to several spatial subchannels. Firstly, on the left of Figure 2.2 we can see

the objective of the studied problem. The objective to be pursued is the total data rate

or spectral efficiency maximization. This objective function is the same objective of some

problems presented in the literature review of section 1.3 such as the URM problem. This is

an attractive objective to be targeted since assignment solutions with high total data rates are

2.2. General Problem 17

Set of all RB assignments

Multiple access constraint

Minimum satisfaction constraint

Feasible assignmentsfor the CRM problem

RBassignment

SpectralEfficiency

Assignment 1

Assignment 2

Assignment 3

Assignment 4

Figure 2.2: Illustration of the main aspects of the CRM problem.

solutions that use the available spectrum in an efficient way.

On the right of Figure 2.2, we present the space of all solutions or assignments to our

problem. Each RB assignment or solution is represented by a black circle. The feasible

solutions are the ones that obey the constraints of the problem. We have two set of constraints

in the CRM problem: multiple access and minimum satisfaction constraints.

The set of multiple access constraints limits the assignment solutions so as to obey the

constraints of the considered multiple access schemes. The RB assignments that obey this

set of constraints are within the area in red and orange in Figure 2.2. One of the aspects to

be modeled by this set of constraints is the orthogonality of the resource assignment within

the sector, i.e., assure no intra-cell interference. Depending on the considered scenario, this

group of constraints has other meanings. As an example, when SC-FDMA is considered, this

group of constraints should assure the adjacency among assigned RBs.

The second group of constraints is related to QoS and satisfaction aspects. System

operators in general measure the system capacity based on minimum user satisfaction

constraints. More specifically, the provision of a given service by the system operator is often

considered satisfactory if a minimum percentage of the flows are considered satisfied based

on QoS requirements [13,51,52,59]. The objective of this group of constraints is that for each

service a minimum number of flows should be satisfied. In other words, a minimum number

of flows should have the minimum data rate constraints fulfilled for each provided service in

the system. The RB assignments that obey this set of constraints are within the area in yellow

and orange in Figure 2.2.

In order to solve the CRM problem we are interested in the RB assignment within the

area in orange, i.e., the feasible RB assignments or the ones that fulfill the multiple access

and minimum satisfaction constraints. In Figure 2.2 we also illustrate the achieved spectral

efficiency or total data rate of four RB assignments. RB assignments 1 and 4 are not feasible

solutions to the CRM problem. RB assignment 1 obeys the multiple access constraints but

does not comply with the minimum satisfaction constraint, whereas RB assignment 4 is in

accordance with the minimum satisfaction constraint but does not obey the multiple access

constraint. Therefore, although RB assignment 1 is an efficient assignment regarding the

2.3. Framework for Heuristic Solution 18

total data rate objective we are not interested in it. Among the feasible RB assignments 2 and

3, the best solution to the CRM problem is the one that leads to the most efficient RB usage

in terms of the total transmit data rate. Therefore, the best solution in Figure 2.2 among RB

assignments 1 to 4 is the RB assignment 2.

2.3 Framework for Heuristic Solution

As commented before, URM is a classical problem in the wireless communication area

whose solution is not suitable for multiservice networks due to its QoS-unware nature.

However, the solution to this problem is relevant since it represents the resource assignment

that provides the maximum achievable spectral efficiency. Defining the capacity region in a

multiuser scenario as the set of all achievable flows’ data rates, we have that the solution to

the URM problem is located at the boundary of that capacity region. Figure 2.2 would have

to be modified in order to represent the URM problem. The URM problem does not have the

set of minimum satisfaction constraints (yellow rectangle). In fact, the URM problem does not

take into account QoS issues and the feasible solutions are all RB assignments that obey the

multiple access constraints.

When QoS and satisfaction constraints are imposed on the RRA problem, often the users’

data rates at the boundary of the capacity region are not feasible solutions anymore, i.e., the

solutions at the boundary of the capacity region do not comply with QoS and satisfaction

constraints. This means that the feasible solutions to the QoS constrained problems present

a performance loss in the total data rate or system spectral efficiency. This performance loss

is the price paid in order to fulfill the QoS demands in multiservice networks.

As it will become clear later, optimally solving the studied RRA problem is not an easy

task. Basically, the integer nature of the RB assignment places the studied optimization

problem in the class of combinatorial problems that in general are hard to solve [11]. As

a consequence, the optimal solution cannot be found by algorithms with polynomial-time

complexity. Therefore, one of the contributions of this thesis is the proposal of heuristic

solutions. The proposed solutions to the studied problem in this thesis are based on a

heuristic framework that is depicted in the following.

The proposed heuristic framework is split into two parts: Unconstrained Maximization

and Reallocation. In the Unconstrained Maximization part an optimal (or at least good

suboptimal) RB assignment of the URM problem is found. In the Reallocation part the RB

assignment solution provided by the first part of the framework is changed in order to fulfill

the satisfaction constraints. The reasoning of the proposed framework is explained in the

following.

The idea in the Unconstrained Maximization part is to find an RB assignment that is on

(or at least near to) the boundary of the rate capacity region. As the objective of the studied

problem in this thesis is to maximize the total data rate, this solution is used as an initial RB

assignment. Note that, if the optimal solution to the URM problem is used in the first part of

the proposed framework and if this solution is capable of fulfilling the minimum satisfaction

constraints, we have that the resulting RB assignment is also an optimal solution to the CRM

problem. However, it is important to mention that the solution provided by the first part of

the heuristic framework in general does not obey the satisfaction constraints since only the

flows with best channel qualities get most of the RBs. In other words, only few flows have

their required data rates fulfilled. Moreover, in some scenarios the optimal solution to the

URM problem cannot be easily obtained.

In the Reallocation part of the proposed framework, the initial solution is changed so as to


comply with the satisfaction constraints. The satisfaction constraints are fulfilled by iteratively

exchanging the RBs among the flows. The exchange of RBs is performed between the flows

that are satisfied and the flows that are unsatisfied with the solution of the first part. It is

important to highlight that each reallocation should cause a performance loss as minimum as

possible in the total data rate given by the current assignment. Therefore, efficient methods

for reallocation should be introduced. In the following we illustrate the heuristic framework

with an example.

In Figure 2.3 we illustrate the main idea of the proposed framework by showing the

resulting transmit data rates for all possible RB assignments for a two-flow case. In this

plot, a point with coordinates (x, y) represents a specific resource assignment in which flow

1 and flow 2 get allocated data rates equal to x and y, respectively. As an example, consider

that both flows demand a data rate of 300kbps and that we intend to satisfy both flows.

Therefore, the feasible assignments are the ones within the rectangular area limited by the

interval [300,∞) in the x- and y-axes. The first part of the proposed framework chooses an

assignment that maximizes the total data rate with no concern about satisfying the data rate

requirements. In Figure 2.3, the solution found in the first part of the framework does not

satisfy the data rate requirements of the second flow (point A). Then, the second part of the

framework performs reallocations of RBs between the flows so as to find a feasible assignment

(one that satisfies both data rate requirements), and that is near to the optimal point of the

studied problem. The suboptimal assignment is represented by the point B and the optimal

solution is the point C in Figure 2.3. The intermediate solutions due to the RB reallocations

are also shown in this figure.

0 100 200 300 400 500 600 700 8000

100

200

300

400

500

600

700

800

Data rate of flow 1 [kbps]

Data

rate

of flow

2 [kbps]

possible assignments

required data rate

path due iterative reallocations

CB

A

Maximum rate withQoS guarantees

Initial point − Maximumrate without QoS

guarantees

Solution provided bysuboptimum algorithm

Feasible region

Figure 2.3: Capacity region for a two-flow example to illustrate the proposed heuristic framework.

In the following chapters we address the CRM problem in different scenarios where

heuristic solutions are conceived based on the presented framework. In summary, in order to

solve the CRM problem we adopt the following heuristic approach:

◮ Firstly, a resource assignment that is on (or near to) the boundary of the capacity region

should be found. This is an effective solution in terms of spectral efficiency that in

general does not fulfill QoS and satisfaction constraints. Some flows could get an excess


of RBs becoming over satisfied whereas other flows (maybe the majority of the flows)

could get an insufficient number of RBs to become satisfied;

◮ Secondly, RBs are exchanged between the (over) satisfied flows and the unsatisfied flows.

The exchange of RBs should be performed until the minimum satisfaction constraints

for each service is fulfilled. Furthermore, the reallocation process should preserve as

much as possible the spectral efficiency attained by the initial assignment found in the

first part.

21

Chapter 3

Maximizing Spectral Efficiency

under Minimum Satisfaction

Constraints in SISO Scenario

In this chapter we study the general CRM problem presented in Chapter 2 in the context

of downlink OFDMA networks with single antennas at the transmitter and receiver. In

section 3.1 we refine the system model to this specific scenario and mathematically formulate

the studied problem in section 3.2. Sections 3.3 and 3.4 are devoted to the study of the

optimal and heuristic solutions to this problem, respectively. Finally, performance results are

presented in section 3.5 along with partial conclusions in section 3.6.

3.1 System Modeling

In this thesis we consider the channel model in the frequency domain. In this way hDLj,n

represents the channel transfer function of the link between the serving BS and the flow

j on the RB n. Note that we considered the simplifying assumption of approximating the

channel transfer function of the RB by the channel transfer function of the mid subcarrier

that composes the block.

According to this, the SNR of the link between the serving BS and flow j on RB n is given

by

γDLj,n =

(PDL/N

)· αDL

j · |hDLj,n|2

(σRB)2 , (3.1)

where |·| represents the absolute value of the argument, αDLj represents the joint effect of the

path loss and shadowing of the link between the serving BS and flow j, and(σRB

)2is the noise

power at the receiver in the bandwidth of an RB. As commented in section 2.1 of Chapter 2,

we employ equal power allocation among RBs.

We define XSU DL as a J × N assignment matrix with elements xSU DLj,n that assume the

value 1 if the RB n ∈ N is assigned to the flow j ∈ J and 0 otherwise. As we will show in

section 3.2, some constraints should be imposed on this matrix in order to assure no intra-cell

interference within a sector.

By using link adaptation, a terminal can transmit at different data rates according to

its channel state, allocated power and perceived noise/interference. We consider that the

3.2. Problem Formulation 22

terminal j ∈ J can transmit using RB n ∈ N with the data rate rSU DLj,n given by

rSU DLj,n = f

(γDLj,n

), (3.2)

where f (·) represents the link adaptation function that maps the SNR to the transmit data

rate.

As stated in section 2.1 of Chapter 2, we consider that the system operator provides

multiple services. For the sake of clarity, we assume that the indices of the flows in xSU DLj,n ,

rSU DLj,n and in tj are sequentially disposed according to the service, i.e., the flows from j = 1 to

j = J1 are from service 1, flows from j = J1 + 1 to j = J1 + J2 are from service 2, and so on.

3.2 Problem Formulation

According to the previous considerations, the resource assignment problem can be

formulated as the following optimization problem:

maxXSU DL

∑

j∈J

∑

n∈N

rSU DLj,n · xSU DL

j,n

, (3.3a)

subject to

∑

j∈J

xSU DLj,n = 1, ∀n ∈ N , (3.3b)

xSU DLj,n ∈ {0, 1}, ∀j ∈ J and ∀n ∈ N , (3.3c)

∑

j∈Js

u

(∑

n∈N

rSU DLj,n · xSU DL

j,n , tj

)≥ ks, ∀s ∈ S, (3.3d)

where u(x, b) is a step function at the value b that assumes the value 1 if x > b and 0 otherwise.

The objective function shown in (3.3a) is the total downlink data rate transmitted by the BS.

The first two constraints (3.3b) and (3.3c) assure that an RB will not be shared by different

flows, i.e., there is no intra-cell interference. Without loss of generality we considered the

equality in constraint (3.3b) instead of the inequality∑j∈J

xSU DLj,n ≤ 1, since the objective of

maximizing the total data rate leads to a complete usage of the RBs. Finally, the constraint

(3.3d) states that a minimum number of flows should be satisfied for each service.

Problem (3.3) is a combinatorial optimization problem with a non-linear constraint

(3.3d). Hence, depending on the problem dimensions, its optimal solution has prohibitive

computational complexity [25].

3.3 Characterization of the Optimal Solution

Let’s reformulate this problem by introducing some new variables. Consider ρj as a binary

selection variable that assumes the value 1 if flow j is selected to be satisfied and 0 otherwise.

Note that ρ = [ρ1 · · · ρJ ]T . In this way, problem (3.3) can be reformulated by substituting the

constraint (3.3d) by two new constraints as follows:

maxXSU DL,ρ

∑

j∈J

∑

n∈N

rSU DLj,n · xSU DL

j,n

, (3.4a)

3.3. Characterization of the Optimal Solution 23

subject to

∑

j∈J

xSU DLj,n = 1, ∀n ∈ N , (3.4b)

xSU DLj,n ∈ {0, 1}, ∀j ∈ J and ∀n ∈ N , (3.4c)∑

n∈N

rSU DLj,n · xSU DL

j,n ≥ tj · ρj , ∀j ∈ J , (3.4d)

ρj ∈ {0, 1}, ∀j ∈ J , (3.4e)∑

j∈Js

ρj ≥ ks, ∀s ∈ S. (3.4f)

In order to write this problem in a compact form we will represent the problem variables

and inputs in vector and matrix forms. In the following we utilize vertical and horizontal

lines when defining vectors and matrices so as to ease the comprehension of their structure.

Consider that rSU DLj =

[rSU DLj,1 · · · rSU DL

j,J

]T, xSU DL

j =[xSU DLj,1 · · ·xSU DL

j,N

]T, k = [k1 · · · kS ]T

and xSU DL =[(xSU DL1

)T · · ·(xSU DLJ

)T ]T. We define the optimization variable as ySU DL =

[(xSU DL

)T |ρT]T

. Note that the vectors xSU DL and ρ can be obtained from ySU DL through

the use of the following relations: xSU DL = A1ySU DL and ρ = A2y

SU DL, with A1 = [IJN |0JN×J ]

and A2 = [0J×JN | IJ ] where Ia is an a× a identity matrix and 0a×b is a a× b matrix composed

of zeros.

The objective function (3.4a) can be written as aTA1y with a =[(rSU DL1

)T · · ·(rSU DLJ

)T ]T.

Considering that 1z is a column vector of length z composed by 1’s, the constraint (3.4b)

can be written as BA1y = 1N where B =

IN IN · · · IN︸︷︷︸

J times

.

The constraint (3.4d) is written as

CA1y ≥ EA2y =⇒ (CA1 −EA2)y ≥ 0J , (3.5)

with C = diag((

rSU DL1

)T, · · · ,

(rSU DLJ

)T)and E = diag (t1, · · · , tJ) where 0z is a column vector

of length z composed by 0’s and diag (·) is a block diagonal matrix with its argument in the

main diagonal.

Finally, the constraint (3.4f) can be written as FA2y ≥ k with F = diag(1TJ1, · · · , 1T

JS

).

Please, notice that the blocks being block-diagonally organized in C and F are vectors so

that the resulting matrices are not necessarily square matrices as often expected from block

diagonal matrices.

Therefore, arranging the expressions developed so far we have

maxySU DL

(cTySU DL

)(3.6a)

subject to

GySU DL = 1N (3.6b)

JySU DL> e, (3.6c)

ySU DL is a binary vector, (3.6d)

where c = A1Ta, G = BA1, J =

[(CA1 −EA2)

T(FA2)

T]T

and e =[0J

T kT]T

.

3.4. Low-Complexity Heuristic Solution 24

Based on the previous development we have transformed (3.3) into an ILP. This problem

can be solved by standard methods such as the Branch and Bound (BB) algorithm [60]. The

computational complexity of obtaining the optimal solution by these methods is much lower

than using brute force (complete enumeration of all possible RB assignments). Nevertheless,

the complexity of the BB method grows exponentially with the number of constraints and

variables. In problem (3.6) we have J ·N + J variables and N + J + S constraints, which may

assume large values in practical scenarios, even for small numbers of flows, RBs and services.

The resulting computational complexity is usually not affordable in practice and, therefore,

low-complexity solutions should be provided to problem (3.6).

3.4 Low-Complexity Heuristic Solution

Based on the presented framework in section 2.3 of Chapter 2 we present in this section

a low complexity algorithm for solving the presented problem. Our proposed solution is split

into two parts: Unconstrained Maximization and Reallocation. These parts are depicted in

Figures 3.1 and 3.2, respectively.

In the Unconstrained Maximization part, the basic idea is to have a good initial

assignment that is on the boundary of the capacity region. As we explain in the following,

this is achieved by sequentially solving the URM problem and disregarding flows based on

channel quality and required data rate. Firstly, we define the auxiliary (B) and available (A)

flow sets and initialize them with the set of all flows (J ). The auxiliary flow set B contains

the flows that can be disregarded without infringing satisfaction constraints, whereas the

available flow set A contains the flows that were not disregarded and will get assigned RBs in

the next part. Then we solve the maximum rate allocation with the flows from the available

flow set A. After that, we define the flows that have the data rate requirement fulfilled as

the satisfied flows and the remaining ones as the unsatisfied flows. If the minimum required

number of satisfied flows of each service is achieved, i.e., the constraint (3.3d) of problem

(3.3) is fulfilled, we have found an optimal solution. However, note that this is an uncommon

situation due to the distribution of the terminals within each sector. In general, few terminals

will get most of the available RBs.

If the satisfaction constraint for any service is not fulfilled, a flow of the auxiliary flow set

B will be disregarded. By disregarding a flow we mean that it will not receive resources at the

current TTI. The criterion to select the flow j∗ to be disregarded is given by

j∗ = argminj∈B

1

N

∑n∈N

rSU DLj,n

tj, (3.7)

where B contains the flows of the services that can still be disregarded. The argument of

the argmin (·) function consists in the ratio between the average data rate and the required

data rate of flow j that represents an estimate of the average number of RBs to satisfy it.

Therefore, the adopted criterion to disregard a flow is quite reasonable: we disregard the flow

that requires, in average, more RBs to be satisfied. The selected flow is taken out of the

available and auxiliary flow sets.

The next step is to check whether the service which the disregarded flow belongs to can

have another flow disregarded without infringing the minimum satisfaction constraint of the

considered service. If this is not possible, all the flows from this service are taken out of the

auxiliary flow set B. In this case, no flow from that service will be disregarded anymore. After

that, we redo the maximum rate allocation with the remaining flows in the available flow set.


Begin

(1) Auxiliary flow set andavailable flow set arecomposed of all flows

(2) Maximum rate allocationwith the flows fromthe available flow set

(3) From the flows of theavailable flow set define the

satisfied and unsatisfied flows

(4) Is the satisfaction constraintfulfilled for all services?

(6) Take out from the availableand auxiliary flow sets the flowwith the poorest channel qualityand the highest requirement

in the auxiliary flow set

(7) Can another flow bedisregarded from the

same service of this flow?

(8) Take out from theauxiliary flow set all

flows from this service

(9) Is the auxiliaryflow set empty?

(10) Is there any satisfied flow?

(12) Define the donor, receiverand available resource sets

(5) Optimalsolution

(11) No feasiblesolution

was found

No

No

Yes

Yes

Yes

No

Yes

No

Figure 3.1: Flowchart of the first part of the proposed solution for the SISO case: UnconstrainedMaximization.

This complete procedure is repeated until either we find an optimal solution or no flow can be

disregarded, i.e., the auxiliary flow set is empty. In the latter case, we check if at least one flow

is satisfied. If so, we define from the available flow set A and the RB set N three new sets: the

donor (D) and receiver (R) flow sets, and the available resource set (K). The donor flow set D is

composed of the satisfied flows in the available flow set A and can donate RBs to unsatisfied

flows. The receiver flow set R is composed of the unsatisfied flows from the available flow

set A that need to receive RBs from the donors to have their data rate requirements fulfilled.

Finally, the available resource set K is composed of all the RBs from the flows in the donor

flow set, i.e., the RBs that can be donated to the unsatisfied flows (receiver flows). In case

there is no satisfied flow after executing the first part, the proposed algorithm is not able to

find a feasible solution, i.e., one that complies with the minimum satisfaction constraints.

Comments about this situation are provided later in this section.

In the Reallocation part we basically switch RBs from the donors to the receivers. We

start by choosing the flow from the receiver flow set R with the worst channel quality to get

resources until its data rate requirement is fulfilled. The main purpose of this procedure is

to assign the minimum number of RBs to the flows in bad channel conditions and get them


(1) Choose the flow fromthe receiver set with theworst channel condition

(2) Choose a resource fromthe available resource set

with the highest normalizedchannel quality of theselected receiver flow

(3) Can the donor lose this RB?

(4) Reallocate the RB

(5) Update data rates of thereceiver and donor flows

(6) Is the receiver flow satisfied?

(7) Take out the selectedflow from the receiver set

(8) Is there any flowin the receiver set?

(10) Take out the selected RBfrom the available resource set

(11) Are there available RBs?

(9) Afeasiblesolution

was found

(12) Nofeasiblesolution

was found

Yes

Yes

Yes

No

No

Yes

No

No

Figure 3.2: Flowchart of the second part of the proposed solution for the SISO case: Reallocation.

satisfied and assign the remaining RBs to the flows in better channel conditions. Then, we

select a resource n∗ from the available resource set K to be assigned to the receiver flow j∗

based on the following criterion

n∗ = argmaxn∈K

rSU DLj∗,n

rSU DLj+,n

, (3.8)

where j+ is the flow from the donor flow set D that has got assigned the RB n in the first part

of the proposed solution. The argument of the argmax (·) function represents the transmit data

rate of flow j on RB n normalized by the transmit data rate of the current flow that owns RB n

(donor flow) on RB n. Note that, if the reallocation of the selected RB will leave the donor flow

unsatisfied, the RB is not reallocated. In fact, this RB is taken out of the available resource

set K and another RB is chosen.

If the donor flow can lose the selected RB, this RB is assigned to the receiver flow and

the data rates of the receiver and donor flows are updated. If after reallocation, the receiver

flow is satisfied, it is taken out of the receiver set R. Otherwise, the receiver flow will get

more resources according to (3.8). This process is repeated until either all receiver flows are

satisfied or there is no available resource to be reallocated. In the former case, the proposed

algorithm has found a feasible solution. In the latter, it was not able to find a feasible solution.

When the system is overloaded, it is possible that the proposed algorithm is not able to

find a feasible solution, as indicated in Figures 3.1 and 3.2. However, in these cases we could

proceed as follows. In the first part, when the maximum number of flows were disregarded

3.5. Performance Evaluation 27

without violating the problem constraints

(J − ∑

s∈S

ks

), and no flow is satisfied, an option is

to continue disregarding flows according to (3.7) until a satisfied flow is found. This satisfied

flow will be the donor flow in the Reallocation part. If the proposed algorithm is not able to

find a feasible solution in the Reallocation part, an option is to disregard a flow according to

(3.7), assign resources to the remaining flows according to the maximum rate allocation and

then redo the reallocation process. The idea behind these procedures is to relax the constraint

(3.3d) by sacrificing some flows in order to get a feasible solution.

3.5 Performance Evaluation

This section is devoted to the performance evaluation of the proposed algorithm. In

section 3.5.1, we present the main simulation assumptions and performance metrics used for

comparison. In section 3.5.2 we show the simulation results and discuss about computational

complexity of the involved algorithms.

3.5.1 Simulation assumptions

The main assumptions stated in sections 2.1 and 3.1 were implemented in a computational

simulator. We evaluate the resource assignment in the downlink of a hexagonal sector

belonging to a tri-sectorized cell of a cellular system. The results were obtained by performing

several independent snapshots in order to get valid results in a statistical sense. In each

snapshot, the terminals are uniformly distributed within each sector, whose BS is placed at

its corner as shown in Figure 2.1. We consider resources arranged in a time-frequency grid

with each RB composed of a group of 12 adjacent subcarriers in the frequency dimension and

14 consecutive OFDM symbols in the time dimension, following the specifications in [53].

The propagation model includes a distance-dependent path loss model, a log-normal

shadowing component and a Rayleigh-distributed fast fading component. Specifically, we

consider that the fast-fading component of the channel gain of a given terminal is independent

among RBs. This hypothesis is reasonable since in general the RBs are designed to have a

frequency bandwidth on the order of the coherence bandwidth of the channel. We assume

that the link adaptation is performed based on the report of 15 discrete Channel Quality

Indicators (CQIs) used by the LTE system [61]. The SNRs thresholds for MCS switching were

obtained by link level simulations from [62]. The main simulation parameters are summarized

in Table 3.1.

Table 3.1: Main simulation parameters for the SISO case.

Parameter Value Unit

Cell radius 334 m

Transmit power per RB 0.35 W

Number of subcarriers per RB 12 -Shadowing standard deviation 8 dB

Path loss 1 35.3 + 37.6 · log10 (d) dB

Noise spectral density 3.16 · 10−20 W/Hz

Number of snapshots 3000 -

Number of flows and services, See the description -required minimum number of satisfied of the scenarios

flows and required data rate

In order to evaluate our proposal over different conditions, in the results we present some

simulation scenarios in which the main parameters of our model are changed. The scenarios

1d is the distance between the base station and the terminal in meters.


Table 3.2: Parameters of the considered scenarios for the SISO case.

Scenario S J1 J2 J3 J4 k1 k2 k3 k4 N Required data rate

1 1 5 - - - 4 - - - 10 All flows demand the same data rate

2 1 5 - - - 5 - - - 10 All flows demand the same data rate

3 2 4 4 - - 3 3 - - 15 All flows demand the same data rate



6 2 4 4 - - 4 3 - - 15Flows from service 2 demand a datarate 125kbps higher than the flows

from the service 1

7 2 4 4 - - 4 3 - - 15Flows from service 2 demand a datarate 250kbps higher than the flows

from the service 1

8 3 3 3 3 - 3 2 2 - 15 All flows demand the same data rate



11 3 3 3 3 - 3 3 2 - 15Flows from service 3 demand a datarate 125kbps higher than the flows

from services 1 and 2

12 3 3 3 3 - 3 3 2 - 15Flows from service 3 demand a datarate 250kbps higher than the flows

from services 1 and 2

13 4 3 3 3 3 3 3 2 2 20 All flows demand the same data rate



16 4 3 3 3 3 3 3 3 2 20Flows from service 4 demand a datarate 250kbps higher than the flows

from services 1, 2 and 3

17 4 3 3 3 3 3 3 3 2 20Flows from service 4 demand a datarate 500kbps higher than the flows

from services 1, 2 and 3

are described in the Table 3.2. Basically, scenarios 1 and 2 consider the single service case,

while the others presents the multiservice case with two services in scenarios 3 to 7, three

services in scenarios 8 to 12 and four services in scenarios 13 to 17. For the scenarios with

two, three and four services we change the required minimum number of satisfied flows per

service and the required data rate2 of the flows from different services.

So as to assess the relative performance of the proposed solution we simulate the optimal

solution to the CRM problem (3.3) (CRM OPT) obtained according to section 3.3 and the

optimal solution to the URM problem (URM OPT) that consists in assigning the RBs to the

flows with best channel quality on them. The channel realizations were the same for all

simulated algorithms in order to get fair comparisons. In order to solve ILP problems we used

the IBM ILOG CPLEX Optimizer [63]. The choice of the number of flows, RBs and services is

limited by the computational complexity to obtain the optimal solution.

When the performance metrics are concerned, we consider two main ones: outage ratio and

total data rate. An outage event happens when an algorithm cannot manage to find a feasible

solution, i.e., the algorithm does not find a solution fulfilling the constraints of problem (3.3).

Note that depending on the positions of the terminals within the sector, channel gains and

2Here we refer to the relative required data rate among the flows of different services. The absolute required datarates are shown in the plots of section 3.5.2.


data rate requirements of the flows, the problem itself can be infeasible and therefore the

CRM OPT solution would have an outage event. Outage rate is defined as the ratio between

the number of snapshots with outage events and the total number of simulated snapshots.

Therefore, this performance metric shows the capability of the algorithms in finding a feasible

solution to our problem. The total data rate is the sum of the data rates obtained by all the

flows in the sector in a given snapshot. Finally, increments in the offered load are emulated

by increasing the rate requirements of the flows.

3.5.2 Results

In the following figures we denote the ith scenario as SCEi. In Figure 3.3 we show the

outage rate versus the data rate required by all flows in scenarios 1 and 2 for the algorithms

CRM OPT, URM OPT and the proposed solution. Firstly, we can see that the outage rate

increases with the data rate requirement of the flows for all algorithms, as expected. This is

a behavior present in all results concerning the outage rate performance. Another general

observation is that the URM OPT solution presents high outage rates even for low data rate

requirements. The reason for this is that it maximizes the total data rate without any QoS

guarantee. Consequently, in general, only a few flows (with best channel conditions) get most

of the system resources and become satisfied.

0 2 4 6 8 10 12

x 105

0

10

20

30

40

50

60

70

80

90

100

Required rate per flow (bits/s)

Ou

tag

e r

ate

(%

)

CRM OPT SCE1

URM OPT SCE1

Proposal SCE1

CRM OPT SCE2

URM OPT SCE2

Proposal SCE2

Figure 3.3: Outage rate versus required data rate for CRM OPT, URM OPT and the proposed solutionwith one service in scenarios 1 and 2 for the SISO case.

The relative comparison of scenarios 1 and 2 shows that all algorithms perform better in

scenario 1 than in scenario 2. In fact, in scenario 2 the problem is harder than in scenario 1

since in scenario 2 all 5 flows should be satisfied instead of 4 flows as in scenario 1. This is

a general observation in all simulated scenarios. Looking at the performance of the proposed

algorithm, we can observe that our proposed algorithm and the CRM OPT solution have similar

performance for low and medium data rate requirements. Focusing on the required data rate

where the corresponding CRM OPT solution in both scenarios has an outage rate of 10%, the

difference in outage rate between proposed solution and CRM OPT are of only 4.1% and 1.2%

in scenarios 1 and 2, respectively.

Considering scenarios with 2 services, i.e., scenarios 3 to 7, we present in Figure 3.4

the outage rate versus the required data rate by the flows for the algorithms CRM OPT,


0 2 4 6 8 10 12 14

x 105

0

10

20

30

40

50

60

70

80

90

100


Ou

tag

e r

ate

(%

)

CRM OPT SCE3

URM OPT SCE3

Proposal SCE3

CRM OPT SCE4

URM OPT SCE4

Proposal SCE4

CRM OPT SCE5

URM OPT SCE5

Proposal SCE5

(a) Scenarios 3, 4 and 5, and the impact of variable ks.

1 2 3 4 5 6 7 8 9

x 105

0

10

20

30

40

50

60

70

80

90

100

Required rate per flow from service 1 (bits/s)

Ou

tag

e r

ate

(%

)

CRM OPT SCE4

URM OPT SCE4

Proposal SCE4

CRM OPT SCE6

URM OPT SCE6

Proposal SCE6

CRM OPT SCE7

URM OPT SCE7

Proposal SCE7

(b) Scenarios 4, 6 and 7, and the impact of variable tj .

Figure 3.4: Outage rate versus required data rate for CRM OPT, URM OPT and the proposed solutionwith two services for the SISO case.

URM OPT and the proposed solution. This figure shows the outage rate performance when

two aspects of our model are changed: the required minimum number of satisfied flows,

ks, in Figure 3.4(a), and the required data rate of the flows from the service 2 relative to

service 1 in Figure 3.4(b). In Figure 3.4(a), the performance of the algorithms is improved

as the required minimum number of satisfied flows is decreased, as observed previously in

Figure 3.3. In Figure 3.4(b), we can observe the impact of the flows’ required rate on the

outage rate performance. The lower the data rate demands of the flows from service 2 are,

the better the outage rate performance of the algorithms is. When the performance of the

proposed algorithm is regarded, we can observe that it is able to keep a relatively small outage

rate difference to the CRM OPT solution in low and medium loads, as in the single-service


scenarios. When the CRM OPT solution reaches an outage rate of 10% the difference in outage

rate performance between our solution and OPT are of 7.2%, 1.7% and 1% in scenarios 3, 4

and 5, respectively.

In Figure 3.5 we show the outage rate versus the data rate required by the flows in

scenarios 8 to 12 for the algorithms CRM OPT, URM OPT and the proposed solution. In Figure

3.5(a) we show the scenarios in which the minimum number of flows that should be satisfied

is changed, whereas in Figure 3.5(b) the difference between the scenarios is the required

data rate of the flows from service 3 relative to the flows from service 1 and 2. The main

conclusions obtained from Figure 3.4 are present here. We highlight the relative performance

of the proposed algorithm and the CRM OPT solution: in the presented scenarios the proposed

algorithm is capable of maintaining a small and acceptable gap to the lower bound in outage

rate. The difference in outage rate performance when the CRM OPT solution achieves the 10%

threshold are of 2%, 1.5% and 1.2% in scenarios 8, 9 and 10, respectively.

Figure 3.6 presents the outage rate versus the data rate required by the flows in scenarios

13 to 17 for algorithms CRM OPT, URM OPT and proposed solution. The conclusions achieved

so far about the impact of required minimum number of satisfied flows and the flows’ required

data rate are confirmed in Figures 3.6(a) and 3.6(b), respectively. A general analysis of the

outage rate performance from Figure 3.3 to Figure 3.6 is that the proposed solution presented

a good performance relative to the CRM OPT solution especially at low and medium loads even

when the number of services is increased.

The outage rate performance metric shows the capability of the algorithms in finding a

feasible solution to our problem. On the other hand, in the remaining figures we show

the Cumulative Distribution Function (CDF) of the total data rate for specific data rate

requirements considered in the x-axis of the figures regarding the outage rate performance.

For a specific scenario and load, the CDFs of all algorithms are built with the samples of

the snapshots in which the proposed solution and CRM OPT were able to find a solution (no

outage). Therefore, possibly many of the samples used in the CDFs for the URM OPT solution

are in outage. The main idea to include results of the URM OPT solution is to show how the

problem constraints imposed losses in the total achievable data rate.

In Figure 3.7 we present the CDFs of the total data rate for all algorithms in scenario 3 for

the required data rate of 125kbps and 1.125Mbps. We have three comments about this figure.

Firstly, the URM OPT algorithm provides the highest total data rates in both Figures 3.7(a) and

3.7(b). This is a general observation in all remaining results that comes at the cost of higher

outage rate as shown in Figure 3.4(a). Secondly, the difference in the total data rate between

the URM OPT algorithm and CRM OPT increases with the required data rate. The total data

rate of CRM OPT is penalized when the data rate requirement is high since many RBs should

be assigned to the flows in medium and bad channel conditions. The performance loss at the

50th-percentile of total data rate of CRM OPT solution relative to the URM OPT solution are of

3.8% and 8% in Figures 3.7(a) and 3.7(b), respectively. Finally, focusing on the performance

of the proposed algorithm we can see that it performs almost optimally at the required data

rate of 125kbps. At the required data rate of 1.125Mbps, the proposed algorithm leads to a

performance loss at the 50th-percentile of only 3% compared to CRM OPT.

In Figure 3.8 we selected the scenarios 9 and 12 (S = 3) to show the impact of the variation

of the difference between the required data rate of flows from service 1 and 2, and the flows

from service 3. As expected, when the required data rate of the flows from service 3 is

250kbps higher than the required data rate of the flows from services 1 and 2 (scenario 12),

the achieved total data rates are lower than in scenario 9 where the flows from all services


1 2 3 4 5 6 7 8 9

x 105

0

10

20

30

40

50

60

70

80

90

100


Ou

tag

e r

ate

(%

)

CRM OPT SCE8

URM OPT SCE8

Proposal SCE8

CRM OPT SCE9

URM OPT SCE9

Proposal SCE9

CRM OPT SCE10

URM OPT SCE10

Proposal SCE10


1 2 3 4 5 6 7 8 9

x 105

0

10

20

30

40

50

60

70

80

90

100

Required rate per flow from services 1 and 2 (bits/s)

Ou

tag

e r

ate

(%

)

CRM OPT SCE9

URM OPT SCE9

Proposal SCE9

CRM OPT SCE11

URM OPT SCE11

Proposal SCE11

CRM OPT SCE12

URM OPT SCE12

Proposal SCE12


Figure 3.5: Outage rate versus required data rate for CRM OPT, URM OPT and the proposed solutionwith three services for the SISO case.

demand the same data rate. Comparing the proposed solution in scenarios 9 and 12 we can

see that the performance loss at the 50th-percentile of the total data rate is of 1.5% at the

required rate of 625kbps.

In Figure 3.9 we show the CDFs of the total data rate in the scenarios 13 and 14 (S = 4)

to evaluate the impact of the required minimum number of satisfied flows in the total data

rate that consists in the objective of the studied optimization problem. As expected, when

we increase the required number of satisfied flows the achievable spectral efficiency in the

system decreases for the CRM OPT solution and the proposed algorithm. Moreover, comparing

Figures 3.9(a) and 3.9(b) we can see that the performance loss is higher as the required data

rate increases.


1 2 3 4 5 6 7 8 9

x 105

0

10

20

30

40

50

60

70

80

90

100


Ou

tag

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ate

(%

)

CRM OPT SCE13

URM OPT SCE13

Proposal SCE13

CRM OPT SCE14

URM OPT SCE14

Proposal SCE14

CRM OPT SCE15

URM OPT SCE15

Proposal SCE15


1 2 3 4 5 6 7 8

x 105

0

10

20

30

40

50

60

70

80

90

100

Required rate per flow from services 1, 2 and 3 (bits/s)

Ou

tag

e r

ate

(%

)

CRM OPT SCE14

URM OPT SCE14

Proposal SCE14

CRM OPT SCE16

URM OPT SCE16

Proposal SCE16

CRM OPT SCE17

URM OPT SCE17

Proposal SCE17


Figure 3.6: Outage rate versus required data rate for CRM OPT, URM OPT and proposed solution withfour services for the SISO case.

In summary, from the joint analysis of the results in Figures 3.3 to 3.6 and Figures

3.7 to 3.9 we can see that our proposed solution performs near optimally considering the

problem objective and constraints in low and medium load conditions. As we are dealing

with algorithms that should be employed in real-time systems, it is worthwhile to analyze the

computational complexity of the CRM OPT and the proposed solution.

In Appendix A we derive the worst case computational complexity of the proposed and

the CRM OPT solutions. The computational complexity of the CRM OPT solution when

obtained by the BB algorithm is O(2JN

). The complexity of the proposed solution is

O(N(J −∑s∈S ks

) (J +

∑s∈S ks

)). As we can see, the complexity of the proposed solution is

much lower than the one used to obtain the optimal solution. By analyzing the computational


0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

x 107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Total sector data rate (bits/s)

CD

F

CRM OPT

URM OPT

Proposal

(a) Data rate requirement of 125kbps.

0.8 0.9 1 1.1 1.2 1.3 1.4

x 107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CD

F

CRM OPT

URM OPT

Proposal

(b) Data rate requirement of 1.125Mbps.

Figure 3.7: CDF of total data rate for CRM OPT, URM OPT and the proposed solution with two servicesin scenario 3 for the SISO case.

complexity and performance of the proposed algorithm we conclude that it leads to a good

performance-complexity trade-off when compared to the strategy used to obtain the optimal

solution.


1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4

x 107

0

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1


CD

F

CRM OPT SCE9

URM OPT SCE9

Proposal SCE9

CRM OPT SCE12

URM OPT SCE12

Proposal SCE12

(a) Data rate requirement of 375kbps for flows of services 1 and 2.

0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4

x 107

0

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0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CD

F

CRM OPT SCE9

URM OPT SCE9

Proposal SCE9

CRM OPT SCE12

URM OPT SCE12

Proposal SCE12

(b) Data rate requirement of 625kbps for flows of service 1 and 2.

Figure 3.8: CDF of total data rate for CRM OPT, URM OPT and proposed solution with three services inscenarios 9 and 12 for the SISO case.


0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CD

F

CRM OPT SCE13

URM OPT SCE13

Proposal SCE13

CRM OPT SCE14

URM OPT SCE14

Proposal SCE14

(a) Data rate requirement of 250kbps.

0.8 1 1.2 1.4 1.6 1.8 2

x 107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CD

F

CRM OPT SCE13

URM OPT SCE13

Proposal SCE13

CRM OPT SCE14

URM OPT SCE14

Proposal SCE14

(b) Data rate requirement of 750kbps.

Figure 3.9: CDF of total data rate for CRM OPT, URM OPT and the proposed solution with four servicesin scenarios 13 and 14 for the SISO case.

3.6. Partial Conclusions 37

3.6 Partial Conclusions

In this chapter, we studied the CRM problem in the OFDMA downlink scenario with single

antenna transceivers. The CRM problem presented in Chapter 2 was refined and formulated

as a non-linear optimization problem. We managed to convert this problem to ILP. Therefore,

the optimal solution to the CRM problem can be obtained by BB-based algorithms instead

of applying the brute force method (complete enumeration). Then, we proposed a heuristic

solution to the CRM problem according to the framework presented in Chapter 2. The

proposed solution is able to closely follow the performance of the optimal solution in low

and moderated loads. Based on the joint analysis of the performance results and the worst

case computational complexity we can see that the proposed algorithm is suitable for practical

situations where computational processing is limited.

38

Chapter 4



Constraints in SU MIMO Scenario

In this chapter we study the general problem presented in Chapter 2 in the context of

downlink OFDMA networks with multiple antennas at the transmitter and/or receiver. More

specifically, we consider in this chapter the SU cases including SU MISO and SU MIMO. In

section 4.1 we refine the system model to this specific scenario. In section 4.2 we present

spatial filters considered in this chapter. Comments about the problem formulation, and

optimal and heuristic solutions are drawn in section 4.3. Finally, performance results are

presented in section 4.4 along with partial conclusions in section 4.5.

4.1 System Modeling

We extend here the definition of the channel model presented in Chapter 3 from single

antennas to multiple antennas. We consider a MIMO channel with MT transmit antennas

at the BS and MR receive antennas at the terminals and Hj,n is an MR × MT matrix whose

elements hDLj,n,a,b consist in the channel transfer function between the ath receiver antenna of

terminal j and the bth transmit antenna of the base station on RB n. Note that we considered

the simplifying assumption of approximating the channel transfer function of the RB by the

channel transfer function of the mid subcarrier that composes the block. We assume that the

channel coefficients remain constant during the period of a TTI. Also, perfect CSI is assumed

at both transmitter and receiver.

We consider SU access per RB so that there is no spatial resource sharing among different

flows. Each flow can be divided in many streams (depending on the channel and antenna

configuration) and transmitted through different spatial subchannels. Moreover, in this

chapter we assume the use of linear spatial filtering at the transmitter and receiver. Differently

of non-linear schemes, linear filtering is simpler by using only linear matrix operations in

signal processing which leads to easier hardware implementation and also can achieve a good

performance-complexity trade-off [32].

Assume hereafter that the RB n was assigned to the flow j†. Before transmission, the

signals to be transmitted on RB n to flow j† are filtered by an MT × cj†,n transmit matrix Mj†,n

and, at the receiver by a cj†,n × MR receiver filter Dj†,n, where cj†,n is the number of signals

transmitted to flow j† on RB n, cj†,n ≤ min(MT ,MR, νj†,n

), and νj†,n is the rank of the channel

4.2. Spatial Filters 39

matrix Hj†,n. Therefore, the input-output relation for the MIMO channel is given by

bj†,n = Dj†,nbj†,n = αDLj Dj†,nHj†,nMj†,nbj†,n +Dj†,nnj†,n, (4.1)

where bj†,n and bj†,n are the cj†,n×1 prior-filtering and the post-filtering received signal vector

of flow j† on RB n, bj†,n is the cj†,n × 1 transmit signal vector to flow j† on RB n and nj†,n is

the MR × 1 white Zero Mean Circularly Symmetric Complex Gaussian (ZMCSCG) noise vector

at terminal j† on RB n.

In section 4.2 we will show the spatial filters considered in this work. All the considered

spatial filters in this thesis are capable of cancelling the inter-stream interference. Also,

the considered spatial filtering schemes consider unitary matrices as transmit and receive

filters. Therefore, the filtered noise still preserves the original characteristics (ZMCSCG noise).

Using linear spatial filters and considering a link adaptation scheme that allows a terminal to

transmit at different data rates according to the SNR we have that the possible transmit data

rate of flow j on RB n is

rSU DLj†,n =

cj†,n∑

l=1

f

(‖αDL

j dlj†,n

Hj†,nMj†,nbj†,n‖22‖dl

j†,nnj†,n‖22

), (4.2)

where ‖ · ‖2 denotes the 2-norm of a vector and dlj†,n

consists in the lth row of the receive

matrix Dj†,n. Moreover, we assume the power allocated to each stream has been suitably

incorporated into Mj†,n. The power is equally distributed among the RBs as justified in the

Chapter 2.

As presented in Chapter 3, we define XSU DL as a J × N assignment matrix with elements

xSU DLj,n that assume the value 1 if the RB n ∈ N is assigned to the flow j ∈ J and 0 otherwise.

We consider that the indices of the flows in xSU DLj,n , rSU DL

j,n and in tj are sequentially disposed

according to the service, i.e., the flows from j = 1 to j = J1 are from service 1, flows from

j = J1 + 1 to j = J1 + J2 are from service 2, and so on.

4.2 Spatial Filters

In this section we describe the considered spatial filters in this chapter: Maximum Ratio

Transmission (MRT), Singular Value Decomposition (SVD) and Zero-Forcing (ZF).

4.2.1 MRT spatial filter

This spatial filter is used in SU MISO antenna configuration and intends to maximize the

transmit SNR [64]. The transmit and receiver filters are

Mj†,n =HH

j†,n∥∥Hj†,n

∥∥2

, and Dj†,n = 1, (4.3)

where (·)H is the conjugate transpose matrix operation.

4.2.2 SVD-based spatial filter

In the MIMO antenna configuration it is possible to transmit multiple streams on the

available spatial subchannels. The transmit and receive spatial filters are obtained as follows

Mj†,n = Vj†,n, and Dj†,n = UHj†,n, (4.4)

where Vj†,n and Uj†,n are the unitary right and left singular vector matrices of Hj†,n. These

two matrices are obtained from the SVD decomposition of Hj†,n = Uj†,nΣj†,nVHj†,n

, in which

4.3. Problem Formulation, Optimal and Heuristic Solutions 40

Σj†,n is the diagonal matrix of singular values of Hj†,n. Applying SVD spatial filtering to a

MIMO channel transforms it into a set of decoupled equivalent SISO channels that do not

interfere with each other [64].

4.2.3 ZF spatial filter

This spatial filter also allows the simultaneously transmission of multiples streams on the

same RB [64]. Each column of the transmit filter Mj†,n corresponds to the respective column

of the pseudo-inverse HHj†,n

(Hj†,nH

Hj†,n

)−1

of Hj†,n normalized so as to have unitary norm. The

receive spatial filter Dj†,n is defined as Dj†,n = IMR. ZF spatial filter implements the channel

matrix pseudo-inversion and is designed to decorrelate all the transmit signals such that the

signal at every receiver output is free of interference.

4.3 Problem Formulation, Optimal and Heuristic Solutions

The main change when switching from the SISO to the SU MISO antenna configuration is

that higher SNRs can be achieved. Consequently, by using link adaptation high transmit data

rates can be achieved. With the use of SU MIMO, multiple streams can be transmitted in the

same RB which also allows higher transmit data rates than the SISO antenna configuration.

Therefore, in general the values of rSU DLj,n for the SU MIMO or SU MISO cases are greater than

or equal to the values for the SISO case considering the same system configuration. However,

this modification does not change the structure of the CRM problem presented in Chapter 3

for the SISO case. For the sake of clarity we restate the problem in the following

maxXSU DL

∑

j∈J

∑

n∈N

rSU DLj,n · xSU DL

j,n

, (4.5a)

subject to

∑

j∈J

xSU DLj,n = 1, ∀n ∈ N , (4.5b)

xSU DLj,n ∈ {0, 1}, ∀j ∈ J and ∀n ∈ N , (4.5c)

∑

j∈Js

u

(∑

n∈N

rSU DLj,n · xSU DL

j,n , tj

)≥ ks, ∀s ∈ S. (4.5d)

Before solving problem (4.5), we have to specify the considered spatial filter according to

section 4.2 and apply equation (4.2) in order to calculate the possible transmit data rate for

each flow and RB association.

As commented on Chapter 3, problem (4.5) is a combinatorial optimization problem with

a non-linear constraint (4.5d) whose optimal solution can have prohibitive computational

complexity. Nevertheless, this problem can be turned into an ILP problem following the same

mathematical manipulations presented in section 3.3 of Chapter 3. This problem can be

solved by standard methods such as the BB algorithm [60].

The proposed heuristic solution in section 3.4 of Chapter 3 is also valid in this

new scenario. The only difference is that in the initialization of the first part of the

algorithm (Unconstrained Maximization) the possible transmit data rates rSU DLj,n should be

now calculated according to equation (4.2) considering the specific spatial filtering scheme.

The flowcharts of the first and second part of the proposed solution are reproduced in Figures

4.1 and 4.2 for the sake of clarity.


Begin

(1) Auxiliary flow set andavailable flow set arecomposed of all flows

(2) Maximum rate allocationwith the flows fromthe available flow set

(3) From the flows of theavailable flow set define the

satisfied and unsatisfied flows


(6) Take out from the availableand auxiliary flow sets the flowwith the poorest channel qualityand the highest requirement

in the auxiliary flow set

(7) Can another flow bedisregarded from the

same service of this flow?

(8) Take out from theauxiliary flow set all

flows from this service

(9) Is the auxiliaryflow set empty?



(5) Optimalsolution

(11) No feasiblesolution

was found

No

No

Yes

Yes

Yes

No

Yes

No

Figure 4.1: Flowchart of the first part of the proposed solution for the SU MISO and SU MIMO cases:Unconstrained Maximization.


(1) Choose the flow fromthe receiver set with theworst channel condition

(2) Choose a resource fromthe available resource set

with the highest normalizedchannel quality of theselected receiver flow

(3) Can the donor lose this RB?



(6) Is the receiver flow satisfied?

(7) Take out the selectedflow from the receiver set

(8) Is there any flowin the receiver set?



(9) Afeasiblesolution

was found

(12) Nofeasiblesolution

was found

Yes

Yes

Yes

No

No

Yes

No

No

Figure 4.2: Flowchart of the second part of the proposed solution for the SU MISO and SU MIMO cases:Reallocation.



This section is devoted to the performance evaluation of the proposed algorithm in the SU

MIMO and SU MISO antenna configurations. In section 4.4.1, we present the main simulation

assumptions and performance metrics used for comparison. In section 4.4.2 we show the

simulation results and discuss about computational complexity of the involved algorithms.



simulator. The MIMO channel model is the classical Independent and Identically Distributed

(IID) [65]. It is important to mention that there are other MIMO channel models that take

into account correlation between transmit/receiver antennas such as Spatial Channel Model

(SCM) [66]. Basically, this means that there is no correlation between the channel transfer

function of the links of each transmit antenna. The simulation methodology, RB composition

and link adaptation parameters are the same as the one considered in section 3.5.1 of Chapter

3. The main simulation parameters are summarized in Table 4.1.

Table 4.1: Main simulation parameters for the SU MISO and SU MIMO cases.


Cell radius 334 m


Number of subcarriers per RB 12 -Number of RBs 15 -

Shadowing standard deviation 8 dB

Path loss 1 35.3 + 37.6 · log10 (d) dB



Antenna configurations MR ×MT

1× 1 (SISO) -2× 1 (MISO) -

2× 2 (SU MIMO) -

MIMO channel model Classical IID -

Number of services 2 -

Number of flows per service 3 -Required minimum number of 2 -

satisfied flows per service

So as to perform qualitative comparisons we simulate the optimal solution to the CRM

problem, CRM OPT, obtained according to section 4.3 and the URM OPT allocation that

consists in assigning the RBs to the flows with highest possible transmit data rate on them.

In order to solve ILP problems we used the IBM ILOG CPLEX Optimizer [63]. The channel

realizations were the same for all simulated algorithms in order to get fair comparisons. The

choice of the number of flows, RBs and services is limited by the computational complexity to

obtain the optimal solution. The same performance metrics of Chapter 3 are considered here:

outage rate and total data rate.

4.4.2 Results

In Figure 4.3 we present the outage rate for the CRM OPT and our proposed solution

for the SISO, SU MISO with MRT filtering and SU MIMO with ZF and SVD spatial filtering

schemes. The results concerning the URM OPT solution were omitted here due to the poor

performance (very high outage rates). This result is expected since the URM OPT algorithm

aims at maximizing the total data rate with no concern about QoS issues.



0.5 1 1.5 2 2.5 3 3.5 4

x 106

0

5

10

15

20

25


Ou

tag

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ate

(%

)

CRM OPT SISO

Proposal SISO

CRM OPT MISO

Proposal MISO

CRM OPT ZF

Proposal ZF

CRM OPT SVD

Proposal SVD

Figure 4.3: Comparison of outage rate versus required data rate for CRM OPT and proposed solutionswith single and multiple antennas schemes for the SU MISO and SU MIMO cases.

Independent of the used spatial filtering scheme, we can see in Figure 4.3 that our

proposed solution is capable of keeping a small gap to the optimal solution at low and medium

required rates. When the required data rates are increased the difference in performance

between our proposal and the optimal solution also increases. However, at these high required

data rates the outage rate of the optimal solution becomes excessively high indicating that

these are not practical offered loads.

We can also observe in Figure 4.3 the potential gains obtained in outage rates due to the

use of multiple antennas at transmitter and/or receiver. At the outage rate threshold of 5%

our proposal with MRT, ZF and SVD spatial filters achieve gains in required data rate of 26%,

57% and 72% over the SISO schemes, respectively. These gains come from the exploitation of

spatial diversity for MRT resulting in better SNRs and spatial multiplexing in ZF and SVD by

allowing parallel transmission in different spatial subchannels.

Although the outage rate is an important figure of merit in our problem it should not be

analyzed alone. We show in Figures 4.4 and 4.5 the CDF of the total downlink data rate at

the required data rate of 2Mbps and 2.75Mbps, respectively. The CDFs of all algorithms were

built with the samples of the snapshots in which our proposed and CRM OPT solutions were

able to find a solution (no outage). Therefore, possibly many of the samples used in the CDFs

for URM OPT solution are in outage. The main motivation to include results of the URM OPT

solution is to show how the problem constraints impose losses in the total achievable data

rate. Comparing the difference in total data rate between CRM OPT and URM OPT solutions

in the required data rate of 2Mbps and 2.75Mbps we can see that the higher the required

data rate of the flows, the larger is the penalty in the maximum feasible data rates attained

by the CRM OPT solution. This is a consequence of the maximization of data rate versus QoS

fulfillment dilemma [16].

As observed in the outage rate performance, our proposed solution is also capable of

performing almost optimally for the required data rates and scenarios presented in Figures

4.4 and 4.5. This result together with the analysis of Figure 4.3 and the lower computational

complexity compared to the optimal solution indicate that our proposal presents a good


0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

x 107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Total downlink data rate (bits/s)

CD

F

CRM OPT SISO

URM OPT SISO

Proposal SISO

CRM OPT MISO

URM OPT MISO

Proposal MISO

CRM OPT SVD

URM OPT SVD

Proposal SVD

Figure 4.4: CDF of the total downlink data rate for CRM OPT, URM OPT and proposed solution at theflows’ required rate of 2Mbps with single and multiple antenna schemes for the SU MISOand SU MIMO cases.

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

x 107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Total downlink data rate (bits/s)

CD

F

CRM OPT SISO

URM OPT SISO

Proposal SISO

CRM OPT MISO

URM OPT MISO

Proposal MISO

CRM OPT SVD

URM OPT SVD

Proposal SVD

Figure 4.5: CDF of the total downlink data rate for CRM OPT, URM OPT and proposed solution at theflows’ required rate of 2.75Mbps with single and multiple antenna schemes for the SU MISOand SU MIMO cases.

performance-complexity trade-off for the studied problem. Furthermore, the use of multiple

antennas also renders gains in the total data rate. Specifically, at high percentiles of the total

data rate, the use of SVD spatial filtering withMR = MT = 2 can lead to potential gains of 100%

compared to SISO scheme. It is important to highlight that if imperfect CSI is modeled these

gains tend to decrease. The CDFs of the solutions with ZF spatial filtering were similar to the

ones with SVD spatial filtering and were omitted for the sake of better graphic visualization.



In this chapter, we studied the CRM problem in the OFDMA downlink scenario when

multiple antennas are used at the transmitter and/or receiver with SU access. Although,

the addition of multiple antennas changes some aspects of the considered system modeling,

we have shown in this chapter that the contributions of Chapter 3 are valid here with small

modifications. More specifically, the optimal solution to the CRM problem can be obtained by

ILP solvers and the heuristic solution proposed can be employed in this scenario.

According to the simulation results presented in this chapter we can see that the proposed

solution achieves a good performance - computational complexity trade-off. Furthermore, we

can see that the use of multiple antennas has potential to achieve lower outage rates and

higher transmit data rates compared to the SISO antennas configuration.

47

Chapter 5



Constraints in MU MIMO Scenario

In this chapter we study the general problem presented in chapter 2 in the context of

downlink OFDMA networks with multiple antennas at the transmitter and receiver. The

difference between this chapter and Chapter 4 is that here we extend the studied problem

to the Multi-User (MU) case. This chapter is organized as follows. In section 5.1 we refine the

systemmodel to this specific scenario and present the considered spatial filtering scheme used

for the MU MIMO scenario in section 5.2. Then, in section 5.3 we mathematically formulate

the CRM problem. Sections 5.4 and 5.5 are devoted to the study of the optimal and heuristic

solutions to this problem, respectively. Finally, performance results are presented in section

5.6 along with partial conclusions in section 5.7.

5.1 System Modeling

We consider a MIMO channel with MT transmit antennas at the BS and MR receive

antennas at the terminals and Hj,n is an MR×MT matrix whose elements hDLj,n,a,b consist in the

channel transfer function between the ath receiver antenna of terminal j and the bth transmit

antenna of the base station on RB n. Note that we considered the simplifying assumption

of approximating the channel transfer function of the RB by the channel transfer function

of the mid subcarrier that composes the block. We assume perfect CSI at both transmitter

and receiver. Furthermore, we assume that the channel coefficients remain constant during

resource allocation in a TTI.

As we are dealing with MU MIMO, multiple terminals can be spatially multiplexed in

the same RB, i.e., a group of terminals can use different spatial subchannels on the same

RB. The maximum number of orthogonal spatial subchannels that can be used per RB

in the considered sector is min (J ·MR,MT ). In general J · MR > MT , since the number of

terminals in the system can be considerably large while the number of transmit antennas

available at the BS is physically limited by the dimensions of the BS. Therefore, the maximum

number of orthogonal spatial subchannels in this case is limited by the number of transmit

antennas, MT .

A set of flows spatially multiplexed in a given RB is called an SDMA group. We define G as

a set with the indices of all SDMA groups that can be built. The number of possible groups

5.1. System Modeling 48

or the cardinality of the set G is G. More details about SDMA groups are presented in section

5.2.

The maximum sum rate in the broadcast channel can be achieved by using DPC at the

BS [29]. The key idea of DPC is to pre-cancel interference at the transmitter assuming perfect

knowledge of channel state and transmitted signals. DPC, while theoretically optimal, is an

information-theoretic concept that has been proven to be difficult to implement in practice.

With this in mind, we consider linear algorithms for spatial filtering because they are less

complex than those based on DPC techniques. Moreover, the optimization problems in MU

MIMO become more tractable and simplified from a mathematical point of view when linear

processing is considered.

Consider that Gn consists in a set with the flows’ indices of the selected SDMA group

assigned to RB n. Before transmission, the signals or streams intended to terminal j ∈ Gn are

filtered by a transmit matrix Mj,n with dimension MT × cj,n and, at the receiver, the signals

are filtered by a receiver matrix Dj,n with dimension cj,n × MR, where cj,n is the number of

transmitted signals or streams to terminal j on RB n, and cj,n ≤ min (MT ,MR, νj,n), and νj,n

is the rank of the channel matrix Hj,n. Therefore, the input-output relation for the MIMO

channel considering terminal j on RB n is given by

bj,n = Dj,nbj,n = αDLj Dj,nHj,nMj,nbj,n + αDL

j Dj,nHj,n

∑

i∈Gn,i6=j

Mi,nbi,n +Dj,nnj,n. (5.1)

Note that the variables in equation (5.1) were already defined in Chapter 4. Each flow

can be divided into many streams (depending on the channel and antenna configuration) and

transmitted through different spatial subchannels. Without loss of generality the transmit and

receive filters used in this work are obtained according to the Block Diagonalization (BD)-ZF

precoder specified in section 5.2. Considering that the flow j is part of the SDMA group

that is associated with RB n, j ∈ Gn, the Signal to Interference-plus-Noise Ratio (SINR), γDLj,n,l,

experienced by the lth stream of flow j on RB n is given by

γDLj,n,l =

‖αDLj dl

j,nHj,nMj,nbj,n‖22‖αDL

j dlj,nHj,n

∑i∈Gn,i6=j Mi,nbi,n + dl

j,nnj,n‖22. (5.2)

Moreover, we assume that the power allocated to each stream has been suitably

incorporated into Mj,n. Consider that the BS employs a link adaptation functionality that

selects the best MCS according to the channel state of a spatial subchannel. Note that the

choice of the best MCS depends on the channel transfer function, transmit and receive filters

that should be known prior to transmission. Assuming that the chosen SDMA group for RB n,

Gn, corresponds to the gth SDMA group in G, the total transmit data rate of flow j that belongs

to the gth ∈ G SDMA group is given by

rMU DLg,j,n =

cj,n∑

l=1

f(γDLj,n,l

), (5.3)

where f (·) represents the link adaptation function that maps the SINR to the transmit data

rate of a stream.

We define XMU DL as a G ×N assignment matrix with elements xMU DLg,n assuming the value

1 if the RB n ∈ N is assigned to the SDMA group g ∈ G and 0 otherwise. Let O be a binary

matrix whose element og,j assumes the value 1 if the jth flow is a member of the gth SDMA

5.2. BD-ZF Spatial Filtering 49

group and 0 otherwise.

We assume that the indices of the flows in rMU DLg,j,n , og,j and in tj are sequentially disposed

according to the service, i.e., the flows from j = 1 to j = J1 are from service 1, flows from

j = J1 + 1 to j = J1 + J2 are from service 2, and so on.

5.2 BD-ZF Spatial Filtering

In this section we describe the BD-ZF spatial filtering. This is a precoding scheme designed

for MU scenarios [67] where the interference of streams from different flows is cancelled.

Without loss of generality, in this section we omit the RB index, n, in some variables.

Therefore, we assume that the following derivation is for a given RB. Before transmission,

the signals or streams intended to flow j ∈ J are filtered by its transmit matrix Mj with

dimension MT × cj, where cj denotes the number of streams transmitted to flow j. At the

receiver the signals are filtered by a receiver filter Dj with dimension cj × MR. Note that

cj ≤ min (MT ,MR, νj), where νj is the rank of the channel matrix of flow j, Hj.

The main idea of the BD-ZF scheme is to provide transmit and receive filters that enable to

simultaneously transmit to multiple flows in the same frequency resource (MU MIMO) without

multiuser interference [67]. It is a generalization of ZF precoding scheme for multi-antenna

users in which multi-user interference is eliminated by projecting the signal of each terminal

onto the kernel of the joint null space of the channels of all other terminals sharing the same

resource. After this, the MU MIMO channel matrix becomes block diagonal thus decoupling

it into several SU MIMO channels. The remaining interference between the streams of the

same terminal can be mitigated by using SU MIMO methods based on SVD spatial filtering

(see Section 4.2.2).

Assume that the number of flows to be multiplexed is J ′. In this thesis, we consider two

particular cases of BD-ZF precoder, one in which MT ≥ J ′ ·MR, i.e., all the flows get assigned

MR spatial subchannels, and another in which MT < J ′ · MR and therefore some flows get

assigned less than MR spatial subchannels.

In the following, we briefly describe the steps of Algorithm 5.1, which presents the BD-ZF

for the case in which MT ≥ J ′ ·MR.

Algorithm 5.1 BD-ZF with MT ≥ J ′ ·MR.

1: Define Hj =[HT

1 · · ·HTj−1H

Tj+1 · · ·H

TJ′

]T, ∀j .

2: Compute V(0)j from the SVD of Hj = UjΣj

[V

(1)j V

(0)j

]H, ∀j

3: Compute V(1)j from the SVD of HjV

(0)j = Uj

[Σj 0

0 0

] [V

(1)j V

(0)j

]H, ∀j

4: Define Mj = V(0)j V

(1)j , ∀j

5: Define Dj = UHj , ∀j

In Algorithm 5.1, we define for each flow j the channel matrix Hj which stacks the channel

matrices of all other flows except flow j. Then, by performing an SVD on Hj, we find a basis

V(0)j for the right null space of Hj, i.e., the null space of the channels of all other users

together. Note that by right multiplying Hj by V(0)j the signals sent through the equivalent

channel HjV(0)j will fall into the null space of the channels of the other users. The matrix V

(0)j

is the first element used to define Mj for BD-ZF. Now taking an SVD of HjV(0)j , we determine

a basis for the range space of HjV(0)j , which is given by V

(1)j . This is the second element used

to define Mj as Mj = V(0)j V

(1)j . Observe that, for each flow j, one has that HjMk = 0MR,ck if

j 6= k, which allows to decouple the MU MIMO channel into a set of J ′ SU MIMO channels.

In the following, we describe Algorithm 5.2 which copes with case in which MT < J ′ ·MR.

5.2. BD-ZF Spatial Filtering 50

Algorithm 5.2 BD-ZF with MT < J ′ ·MR.1: Compute the SVD of Hj ← UjΣjV

Hj , ∀j. Note that Σj is a diagonal matrix with the singular values σj,l in the

main diagonal. Assume that ∀j σj,1 ≥ σj,2 ≥ · · ·2: Total number of streams ϑ← min (MT , J ′ ·MR)

3: Initialize the number of streams to be transmitted to flow j, ςj ←

⌈ϑ

J ′

⌉∀j

4: Auxiliary set E ← {1, · · · , J ′}

5: while(∑

∀j ςj > ϑ)do

6: j∗ ← argminj∈E

(σj,ςj

)

7: E ← E\j∗

8: ςj∗ ← ςj∗ − 19: end while10: Zj consists in the first ςj columns of Uj , ∀j

11: Equivalent channel of flow j Hj ← ZHj Hj

12: Hj ←[H

T

1 · · ·HT

j−1HT

j+1 · · ·HT

J

]T∀j

13: Compute V(0)j , the right null space of Hj , ∀j

14: Compute the SVD of HjV(0)j = Uj

[Σj 0

0 0

] [V

(1)j V

(0)j

]H, ∀j

15: Define Wj the first ςj columns of UHj , ∀j

16: Transmit filter corresponding to flow j: Mj ← V(0)j

V(1)j

, ∀j

17: Receive filter of flow j: Dj ← WjZHj , ∀j

In Algorithm 5.2, we first determine the number of streams that each flow j should receive.

Our choice was to equally distribute the available streams and to assign the remaining

streams to the flows with best channel qualities (higher singular values). This is represented

in steps from 1 to 9 in Algorithm 5.2. In the first step of the algorithm, we calculate the

SVD for each flow channel matrix Hj. The matrix Σj is a diagonal matrix with the elements

σj,l in the main diagonal, i.e., σj,l corresponds to the lth element of the main diagonal of

matrix Σj. Furthermore, we assume that the elements σj,l are sorted in the descending order

along the main diagonal. In step 2 we calculate the total number of possible streams in the

system, which is done by choosing the minimum values between the number of transmit

antennas (MT ) and the number of flows in the system multiplied by the number of receive

antennas (J ′ · MR). Particularly, the total number of streams is MT since we are considering

that MT < J ′ · MR. In step 3 we initialize the number of streams to be transmitted to flow

j, ςj, with the possible number of streams in the system (calculated in step 2) divided by

the total number of flows. This value is rounded to the nearest bigger integer. In the next

step, we define an auxiliary set containing all the flows. From steps 5 to 9, the worst spatial

subchannel of each flow is removed until the total number of streams is achieved. In step

10 we compute Zj that will be used to calculate the receive filter. The matrix Zj consists in

the first ςj columns of Uj (defined in step 1). The remaining of the algorithm is equivalent

to Algorithm 5.1 with the difference that we consider an equivalent channel matrix instead

of the original one as presented in step 11. This equivalent channel matrix is obtained by

the product of a guess of which receive filter the receiver would use and the original channel

matrix. Using the equivalent channel matrix for each flow we can guarantee the conditions

to use Algorithm 5.1. For more details regarding these two approaches for BD, please refer

to [67].

As it was presented in this section, we have that for a given SDMA group we try to equally

divide the spatial subchannels among the flows. If the number of streams is not multiple of

the number of flows in an SDMA group we assign the remaining spatial subchannels to the

flows with stronger channels. According to this, the possible SDMA groups that can be built

with flows 1, 2 and 3 in a system with MT = 2 and MR = 2 are {1}, {2}, {3}, {1,2}, {1,3} and {2,3}.

5.3. Problem Formulation 51

The set G contains the indices of all SDMA groups that can be built. In the previous example,

the set G would have 6 indices. The number of possible groups or the cardinality of the set Gis given by

|G| = G =

MT∑

m=1

(J

m

)=

MT∑

m=1

J !

m! (J −m)!, (5.4)

where

(a

b

)is the number of distinct b-element subsets of any set containing a elements

(binomial coefficient).

5.3 Problem Formulation

According to the previous considerations, the resource assignment problem can be

formulated as the following optimization problem:

maxXMU DL

∑

g∈G

∑

n∈N

∑

j∈J

xMU DLg,n og,j r

MU DLg,j,n

, (5.5a)

subject to

∑

g∈G

xMU DLg,n = 1, ∀n ∈ N , (5.5b)

xMU DLg,n ∈ {0, 1}, ∀g ∈ G and ∀n ∈ N , (5.5c)

∑

j∈Js

u

∑

g∈G

∑

n∈N

xMU DLg,n og,j r

MU DLg,j,n , tj

≥ ks, ∀s ∈ S. (5.5d)

The objective function shown in (5.5a) is the total downlink data rate transmitted by the

BS. The first two constraints (5.5b) and (5.5c) assure that an RB will not be shared by different

SDMA groups. Finally, the constraint (5.5d) states that a minimum number of flows should

be satisfied for each service.

This problem owns the same nature of the problems (3.3) and (4.5) presented in Chapters 3

and 4. Hence, depending on the problem dimensions, its optimal solution can have prohibitive

computational complexity.

5.4 Characterization of the Optimal Solution

Let’s reformulate this problem by introducing some new variables. Note that some vectors

and matrices are redefined here without compromising the understanding since they are used

in a different context. Consider ρj as a binary selection variable that assumes the value 1 if

flow j is selected to be satisfied and 0 otherwise. Note that ρ = [ρ1 · · · ρJ ]T . In this way, problem

(5.5) can be reformulated by substituting the constraint (5.5d) by two new constraints as

follows:

maxXMU DL,ρ

∑

g∈G

∑

n∈N

∑

j∈J

xMU DLg,n og,j r

MU DLg,j,n

, (5.6a)

5.4. Characterization of the Optimal Solution 52

subject to

∑

g∈G

xMU DLg,n = 1, ∀n ∈ N , (5.6b)

xMU DLg,n ∈ {0, 1}, ∀g ∈ G and ∀n ∈ N , (5.6c)∑

g∈G

∑

n∈N

xMU DLg,n og,j r

MU DLg,j,n ≥ ρj tj , ∀j ∈ J , (5.6d)

ρj ∈ {0, 1}, ∀j ∈ J , (5.6e)∑

j∈Js

ρj ≥ ks, ∀s ∈ S. (5.6f)

In order to write this problem in a compact form we will represent the problem variables

and inputs in vector and matrix forms. In the following we utilize vertical and horizontal

lines when defining vectors and matrices so as to ease the comprehension of their structure.

Consider that og = [og,1 · · · og,J ]T , rMU DLg,n =

[rMU DLg,1,n · · · rMU DL

g,J,n

]T, xMU DL

g =[xMU DLg,1 · · ·xMU DL

g,N

]T,

k = [k1 · · · kS ]T and xMU DL =[(xMU DL1

)T · · ·(xMU DLG

)T ]T. We define the optimization variable as

yMU DL =[(xMU DL

)T |ρT]T

. Note that the vectors xMU DL and ρ can be obtained from yMU DL

through the use of the following relations: xMU DL = A1yMU DL and ρ = A2y

MU DL, with A1 =

[IGN |0GN×J ] and A2 = [0J×GN | IJ ].The objective function (5.6a) can be written as aTA1y

MU DL with

a =[o1

T rMU DL1,1 · · · o1

T rMU DL1,N |o2

T rMU DL2,1 · · · o2

T rMU DL2,N | · · · |oG

T rMU DLG,1 · · · oG

T rMU DLG,N

]T, (5.7)

The constraint (5.6b) can be written as BA1yMU DL = 1N where

B =

IN · · · IN︸︷︷︸

G times

T

, (5.8)

The constraint (5.6d) is written as

CA1y ≥ EA2y =⇒ (CA1 −EA2)y ≥ 0J , (5.9)

with

C =

o1,1 rMU DL1,1,1 · · · o1,1 rMU DL

1,1,N o2,1 rMU DL2,1,1 · · · o2,1 rMU DL

2,1,N · · · oG,1 rMU DLG,1,1 · · · oG,1 r

MU DLG,1,N

o1,2 rMU DL1,2,1 · · · o1,2 rMU DL

1,2,N o2,2 rMU DL2,2,1 · · · o2,2 rMU DL

2,2,N · · · oG,2 rMU DLG,2,1 · · · oG,2 r

MU DLG,2,N

.... . .

......

. . ....

. . ....

. . ....

o1,J rMU DL1,J,1 · · · o1,J rMU DL

1,J,N o2,J rMU DL2,J,1 · · · o2,J rMU DL

2,J,N · · · oG,J rMU DLG,J,1 · · · oG,J r

MU DLG,J,N

, (5.10)

and E = diag (t1, · · · , tJ).Finally, the constraint (5.6f) can be written as FA2y ≥ k with F = diag

(1J1

T , · · · ,1JS

T).

Please, notice that the blocks being block-diagonally organized in F are vectors so that the

resulting matrix is not necessarily a square matrix as often expected from block diagonal

matrices.


Therefore, arranging the expressions developed so far we have

maxyMU DL

(cTyMU DL

), (5.11a)

subject to

GyMU DL = 1N , (5.11b)

JyMU DL> e, (5.11c)

yMU DL is a binary vector, (5.11d)

where c = A1Ta, G = BA1, J =

[(CA1 −EA2)

T(FA2)

T]T

and e =[0J

T kT]T

.

Based on the previous development we have transformed (5.5) into a linear integer (binary)

optimization problem. This problem can be solved by standard methods such as the BB

algorithm [60]. The computational complexity of obtaining the optimal solution by these

methods is much lower than using brute force (complete enumeration of all possible RB

assignments). Nevertheless, the complexity of the BB method grows exponentially with the

number of constraints and variables. In problem (5.11) we have

N ·MT∑

m=1

(J

m

)+ J = N

MT∑

m=1

J !

m! (J −m)!+ J

variables and N + J + S constraints, which may assume large values even for small numbers

of flows, transmit and receive antennas, RBs and services.

5.5 Low-Complexity Heuristic Solution

We propose a low complexity heuristic algorithm following the proposed framework of

section 2.3 of Chapter 2 that is split into two parts: Unconstrained Maximization and

Reallocation. Flowcharts describing Unconstrained Maximization and Reallocation parts are

shown in Figures 5.1 and 5.2, respectively.

Before initializing our proposed algorithm we consider that the achievable data rates of all

flows on all resources when belonging to any possible SDMA group is known, i.e., rMU DLg,j,n is

known ∀g ∈ G, ∀j ∈ J and ∀n ∈ N . In the Unconstrained Maximization part, the basic idea

is to have a good initial assignment that is on the boundary of the capacity region. Firstly,

we define the auxiliary (B) and available (A) flow sets and initialize them with the set of all

flows (J ). Then we solve the maximum rate allocation with the flows from the available flow

set A. Basically in this part we assign the RBs to the SDMA groups with highest data rate.

After that, we define the flows that have the data rate requirement fulfilled as the satisfied

flows and the remaining ones as the unsatisfied flows. If the minimum required number of

satisfied flows of each service is achieved, i.e., the constraint (5.5d) of problem (5.5) is fulfilled,

we have found an optimal solution. However, note that this is an uncommon situation due to

the distribution of the terminals within the cell. In general, few terminals will get most of the

available RBs.

In case the satisfaction constraint for any service is not fulfilled, a flow of the auxiliary flow

set B will be disregarded. By disregarding a flow we mean that it will not receive resources at


Begin

(1) Auxiliary flow set and availableflow set are composed of all flows

(2) Maximum rate allocation withthe flows from the available flow set

(3) From the flows of the available flow setdefine the satisfied and unsatisfied flows


(6) Take out from the available andauxiliary flow sets the flow with thepoorest channel quality and higherrequirement in the auxiliary flow set

(7) Can another flow be disregardedfrom the same service of this flow?

(8) Take out from the auxiliaryflow set all flows from this service

(9) Is the auxiliary flow set empty?



(5) Optimal solution

(11) No feasiblesolution was found

No

No

Yes

Yes

Yes

No

Yes

No

Figure 5.1: Flowchart of the first part of the proposed algorithm for the MU MIMO case: UnconstrainedMaximization.

the current TTI. The criterion to select the flow j∗ to be disregarded is given by

j∗ = argminj∈B

(1

G ·N∑g∈G

∑n∈N

rMU DLg,j,n

)

tj, (5.12)

where B contains the flows of the services that still can be disregarded. The adopted criterion

to disregard a flow is quite reasonable: we disregard the flow that requires, in average, more

RBs to be satisfied. The selected flow is taken out of the available and auxiliary flow sets.

The next step is to check whether the service of the disregarded flow can have another flow

disregarded without infringing the minimum satisfaction constraint of the considered service.

If this is not possible, all the flows from this service are taken out of the auxiliary flow set B.In this case, no flow from that service will be disregarded anymore. After that, we redo the

maximum rate allocation with the remaining flows in the available flow set A. This complete

procedure is repeated until either we find a feasible solution or no flow can be disregarded,

i.e., the auxiliary flow set B is empty. In the latter case, we check if at least one flow is

satisfied. If so we define from the available flow set A and the RB set N three new sets: the

donor (D) and receiver (R) flow sets, and the available resource set (K). The donor flow set

D is composed of the satisfied flows in the available flow set A and can donate/share RBs


to/with the unsatisfied flows. The receiver flow set R is composed of the unsatisfied flows

from the available flow set A and need to receive RBs from the donors to have fulfilled their

rate requirements. Finally, the available resource set K is composed by all the RBs that were

exclusively assigned to the flows from the donor flow set. These RBs can be donated/shared

to/with the unsatisfied flows (receiver flows). In case there is no satisfied flow after executing

the first part, the proposed algorithm is not able to find a feasible solution, i.e., that comply

with the minimum satisfaction constraints. Some comments about this issue are provided

later in this section.

In the Reallocation part we basically switch RBs between SDMA groups in order to satisfy

the flows from the receiver set. This is equivalent to adding or taking out flows from the SDMA

group defined in the Unconstrained Maximization part.

We start by choosing the flow from the receiver flow set (R) with the worst channel quality

to get resources until its data rate requirement is fulfilled. The main motivation with this

procedure is to assign the minimum number of RBs to get satisfied the flows in bad channel

conditions and assign the remaining RBs to the flows with better channel quality. After

that we should identify the SDMA groups and RBs pairs that are candidate to be chosen in

the reallocation procedure. The candidate SDMA groups are all possible combinations from

the set composed of the selected flow, and flows from the receiver and donor flow sets that

necessarily contain the selected flow. The RBs are the ones in the available resource set and

the RBs that were assigned to SDMA groups in the first part of our algorithm that contains

the selected flow and do not contain other receiver flows.

We consider that the candidate SDMA groups/RBs pairs (z, n) are disposed in the candidate

SDMA group/RBs set F . In order to find the best RB and the SDMA group for reallocation we

need to define a reallocation metric.

The reasoning to define the reallocation metric is to reallocate an RB to the SDMA group

that best reduces the distance between the current data rates and requirements of the flows

in the receiver flow set, while not causing a high sum rate loss. This can be achieved by the

following metric

ϕMU DLg,n =

∑j∈R

∣∣∣tj − (tj + rMU DLg,j,n − rMU DL

g′,j,n )∣∣∣

∑j∈R

∣∣tj − tj∣∣ .

Φcur

Φnewg,n

, (5.13)

where tj consists in the data rate of flow j according to the current resource assignment, Φcur

is the sum rate according to the current resource assignment while Φnewg,n is the sum rate when

the SDMA group g is chosen in the RB n without modifying the assignment on the other RBs.

We consider that g′ is the index of the SDMA group that was chosen to RB n in the first part

of the proposed solution. Also, rMU DLg′,j,n is the data rate of flow j on RB n when the SDMA group

g′ is chosen. Note that Φnewg,n ≤ Φcur since we begin with the maximum rate solution in the first

part of our proposal. The pair SDMA group and RB chosen in the reallocation process is the

one that minimizes the metric ϕMU DLg,n , i.e.,

(g∗, n∗) = arg min∀(g,n)∈F

ϕMU DLg,n , (5.14)

where the chosen pair is the SDMA group g∗ and RB n∗.

The next step is to check if the reallocation would lead any donor flow to unsatisfaction. If

so, the reallocation is not performed and the chosen pair SDMA group/RB is not available for

reallocation. Otherwise, the reallocation is performed and the receiver and donor flows data

rates are updated. Then, the algorithm checks if any receiver flow has become satisfied after


(1) Choose the flow from the receiverflow set with worst channel condition

(2) Define the set of candidateSDMA groups/RBs, F

(3) Calculate ϕMU DLg,n ∀(g, n) ∈ F

(according to equation (5.13)) andchoose the SDMA group g∗ andthe RB n∗ that minimizes ϕMU DL

g,n

(4) Can the donor(s) in the previouslySDMA group assigned to RB n lose

this RB without becoming unsatisfied?

(5) Take out the pair(g∗, n∗) from thecandidate SDMAgroup/RBs set F



(8) Are there new satisfied receivers?

(9) Take out the new satisfiedreceiver flow(s) from the receiver set

(10) Is there any flow in the receiver set?



(11) A feasiblesolution was found


Yes

Yes

Yes

No

No

Yes

No

No

Figure 5.2: Flowchart of the second part of the proposed algorithm for the MU MIMO case: Reallocation.

reallocation. If so, these flows are taken out of the receiver flow set R. The algorithm ends

with a feasible solution when there is no flow in the receiver flow set, i.e., all receiver flows

have become satisfied in the reallocation process. An outage event is reached when still exist

flows in the receiver flow set and there is no SDMA group and RB pair for reallocation.

When the system is overloaded, it is possible that the proposed solution is not able to

find a feasible solution as indicated in Figures 5.1 and 5.2. However, in this case we could

proceed as follows. In the first part, when the maximum number of flows were disregarded

without violating the problem constraints

(J − ∑

s∈S

ks

), and no flow is satisfied, an option is


flow will be the donor flow in the Reallocation part. If the proposed solution is not able to find

a feasible solution in the Reallocation part, an option is to disregard a flow according to (5.12),

assign resources to the remaining flows according to the maximum rate allocation and then

redo the reallocation process. The idea behind these procedures is to relax the constraint

(5.5d) by sacrificing some flows in order to get a feasible solution.


This section is devoted to the performance evaluation of the proposed algorithm. In

section 5.6.1, we present the main simulation assumptions and performance metrics used for


Table 5.1: Main simulation parameters considered in the performance evaluation for the MU MIMOcase.


Cell radius 334 m


Number of subcarriers per RBs 12 -

Number of RBs 10 -Shadowing standard deviation 8 dB

Path loss 1 35.3 + 37.6 · log10 (d) dB



Antenna configurations MR ×MT

2× 2 -4× 4 -

MIMO channel model Classical IID -

Number of services 2 -Number of flows per service 3 -

Required minimum number of 2 -satisfied flows per service -

comparison. In section 5.6.2 we show the simulation results and discuss the computational

complexity of the involved algorithms.



simulator. The MIMO channel model is the classical IID [65]. We simulate the MU MIMO

scenario with the following antenna configurations: MT = 2 and MR = 2, and MT = 4 and

MR = 4. We also simulate the SU MIMO scenario in order to perform relative comparisons with

the MU MIMO scenario. The simulation methodology, RB composition and link adaptation

parameters are the same as one considered in section 3.5.1 of Chapter 3. The main simulation

parameters are summarized in Table 5.1.

In order to perform qualitative comparisons in the MU MIMO scenario we simulate the

optimal solution to the CRM problem, CRM OPT, obtained according to section 5.4 and the

optimal solution to the URM problem, URM OPT allocation, that consists in assigning the

RBs to the SDMA groups with best channel quality on them. Furthermore, we simulate the

algorithms for the SU MIMO scenario obtained according to Chapter 4. So as to solve ILP

problems we used the IBM ILOG CPLEX Optimizer [63]. In order to get fair comparisons,

the channel realizations were the same for all simulated algorithms for a given antenna

configuration. The same performance metrics considered in Chapters 3 and 4 are considered

here: outage ratio and total data rate.

5.6.2 Results

In Figure 5.3 we show the outage rate versus the data rate required by all flows with

CRM OPT, URM OPT and the proposed algorithm for the MU MIMO and SU MIMO antenna

configurations with two antennas at the transmitter and at the receiver. Firstly, we can see

that the outage rate increases with the data rate requirement of the flows for all algorithms

and antenna configurations, as expected. Another general observation is that the URM OPT

solution presents high outage rates even for low data rate requirements for both antenna

configurations. The reason for this is that this algorithm maximizes the total data rate without

any QoS guarantee. Consequently, in general only few flows (with best channel conditions)



2 2.5 3 3.5

x 106

0

10

20

30

40

50

60

70

80

90

100

Required rate per user (bits/s)

Ou

tag

e r

ate

(%

)

MU MIMO CRM OPT

MU MIMO URM OPT

MU MIMO Proposal

SU MIMO CRM OPT

SU MIMO URM OPT

SU MIMO Proposal

Figure 5.3: Outage rate versus required data rate with CRM OPT, URM OPT and the proposed solutionfor the SU MIMO and MU MIMO antenna configurations with MT = 2 and MR = 2.

get most of the system resources and become satisfied.

Focusing on the MU MIMO case, we can see that the proposed solution is capable of

maintaining a small difference in the outage rate performance compared with the lower bound

provided by the CRM OPT solution. When the CRM OPT solution for the MU case has an

outage rate of 10% the difference to the proposed solution with MU MIMO is of only 6%.

The reason for this is the intelligent reallocation process performed over the maximum rate

solution in the second part of our proposed solution.

In Figure 5.3 we can also see the performance gain obtained by the MU MIMO case over the

SU MIMO one. For the CRM OPT solution considering the outage rate threshold of 10% the

difference in outage rate between MU MIMO and SU MIMO is of 12% while for the proposed

algorithm the difference is of 10%. One first reason for this gain comes from the exploitation

of the MU diversity that enables the simultaneous use of different spatial subchannels by

flows with highly orthogonal channels. Furthermore, the MU MIMO case has increased

resource granularity since a flow can get assigned only one spatial subchannel while the

other subchannels can be assigned to other flows on the same RB.


feasible solution to our problem. On the other hand, in Figure 5.4 we show the CDF of the

total data rate for the data rate requirements of 2Mbps and 3.5Mbps in the MU MIMO and SU

MIMO configurations with two antennas at the transmitter and at the receivers. In order to

plot the CDFs, we considered only the snapshots in which the proposed algorithm and CRM

OPT managed to find a feasible solution. In general, we can see that the URM OPT algorithm

provides higher total data rates for both antenna configurations. This comes at the cost of

a higher outage rate as seen in Figure 5.3. The total data rate of the optimal solution for

both antenna configurations is penalized when the data rate requirement is high since many

RBs and spatial subchannels should be assigned to the flows in medium and bad channel

conditions.

Focusing on the MU MIMO case, we can see that the performance loss of the proposed

algorithm at the 50th-percentile compared to the CRM OPT in Figures 5.4(a) and 5.4(b) are


0.8 1 1.2 1.4 1.6 1.8 2

x 107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CD

F

MU MIMO CRM OPT

MU MIMO URM OPT

MU MIMO Proposal

SU MIMO CRM OPT

SU MIMO URM OPT

SU MIMO Proposal

(a) Data rate requirement of 2Mbps.

1.3 1.4 1.5 1.6 1.7 1.8 1.9

x 107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CD

F

MU MIMO CRM OPT

MU MIMO URM OPT

MU MIMO Proposal

SU MIMO CRM OPT

SU MIMO URM OPT

SU MIMO Proposal

(b) Data rate requirement of 3.25Mbps.

Figure 5.4: CDF of total data rate for the data rate requirements of 2Mbps and 3.25Mbps with CRM OPT,URM OPT and proposed solution in the SU MIMO and MU MIMO antenna configurationswith MT = 2 and MR = 2.


4 4.5 5 5.5 6 6.5

x 106

0

10

20

30

40

50

60

70

80

90

100

Required rate per user (bits/s)

Ou

tag

e r

ate

(%

)

MU MIMO CRM OPT

MU MIMO URM OPT

MU MIMO Proposal

SU MIMO CRM OPT

SU MIMO URM OPT

SU MIMO Proposal

Figure 5.5: Outage rate versus required data rate with CRM OPT, URM OPT and the proposed solutionfor the SU MIMO and MU MIMO antenna configurations with MT = 4 and MR = 4.

of only 1.4% and 2.2%, respectively. Therefore, besides keeping a small gap to the optimal

solution in the outage rate performance, the proposed algorithm also achieves a quasi-optimal

sum rate capacity. As it can be seen in section 5.5, the criterion to reallocate the SDMA groups

takes into account the possible loss in the sum rate capacity adequately.

In Figure 5.4 we can confirm the advantage of the MU MIMO over the SU MIMO. For

the same algorithm, MU MIMO renders higher sum rate capacities over SU MIMO. For the

CRM OPT solution the performance gain at the 50th-percentile when switching from SU MIMO

to MU MIMO is of 5.8% and 5.7% in the data rate requirements of 2Mbps and 3.25Mbps,

respectively. For the proposed algorithm the performance gain is of 5.9% and 5.3% in the

data rate requirements of 2Mbps and 3.25Mbps, respectively.

In Figure 5.5 we present the outage rate versus the data rate required by all flows

with CRM OPT, URM OPT and the proposed algorithm for the MU MIMO and SU MIMO

antenna configurations with four antennas at the transmitter and at the receivers. The

main conclusions presented for the scenario with two antennas at the transmitter and at

the receivers are valid here:

◮ URM OPT presents high outage rates even for low data rate requirements;

◮ In the MU MIMO antenna configuration, the proposed solution is capable of maintaining

a small difference in the outage rate performance compared with the lower bound

provided by the CRM OPT solution. When the CRM OPT solution (MU MIMO CRM OPT)

has an outage rate of 10% the difference to the proposed solution in the MU MIMO

context is of only 5%;

◮ The MU MIMO antenna configuration provides performance gains for the studied

algorithms in the outage rate metric. For the CRM OPT solution considering the outage

rate threshold of 10% the difference in outage rate is of 15% while for the proposed

algorithm the difference is of 13%.

In Figure 5.6 we show the CDF of the total data rate for the data rate requirements of

4Mbps and 6Mbps in the MU MIMO and SU MIMO configurations with four antennas at the


transmitter and at the receivers. Here again the main conclusions from the analyses of Figure

5.3 are obtained:

◮ URM OPT algorithm provides higher total data rates for both antenna configurations;

◮ Small performance loss of the proposed solution compared to the optimal solution in the

MU MIMO antenna configuration: the performance loss of the proposed algorithm at the

50th-percentile compared to the optimal solution in Figures 5.6(a) and 5.6(b) are of only

2.2% and 3.3%, respectively;

◮ Performance gain of MU MIMO over SU MIMO: for the optimal solution the performance

gain at the 50th-percentile when switching from SU MIMO to MU MIMO is of 11% and

10.9% in the data rate requirements of 4Mbps and 6Mbps, respectively. For the proposed

algorithm the performance gain is of 9.9% and 8.6% in the data rate requirements of

4Mbps and 6Mbps, respectively.

Comparing Figures 5.3 and 5.4 with Figures 5.5 and 5.6 we can confirm the performance

gains incurred due to the use of more antennas at the transmitter and receiver. Firstly,

from Figures 5.3 and 5.5 we can see that with the same available power and bandwidth the

scenario with four antennas at both transmitter and receiver is capable of achieving lower

outage rates for the same data rate requirements in both MU MIMO and SU MIMO antenna

configurations. Considering the CDFs of the total data rate in Figures 5.4 and 5.6 we can also

see that the maximum achievable total data rate in the scenario with four antennas at the

transmitter and receiver is twice the scenario with two antennas. This gain comes from the

higher multiplexing capacity when more antennas are used at the transmitter and receiver.

It is important to highlight that in case of imperfect CSI the obtained gains due to the use of

multiple antennas are decreased.

In Appendix B we calculate the worst-case computational complexity of CRM OPT and

proposed solutions. The CRM OPT solution has complexity equal to O(2GN

). On the other

hand, the complexity of the proposed solution is given by O(GN

(∑s∈S ks

)2+ 2J

G′

∑s∈S ks

N2

)

where G′ is the number of SDMA group that can be composed with MT transmit antennas and∑

s∈S ks flows. As we can see, the computational complexity of the CRM OPT solution increases

exponentially with the input variables. The complexity of the proposed solution is higher than

the one proposed for the SISO and SU MIMO case. However, it is important to say that the

problem to be solved in the MU MIMO case involves new aspects such as the assignment

of spatial subchannels to different flows. The complexity of the proposed algorithm in the

MU MIMO scenario depends on the terms G and G′ that cannot increase indefinitely because

of the limited number of antennas at the BS, MT . As it will be commented in Chapter 7,

some strategies could be followed to decrease the complexity of the proposed solution in this

scenario.


1.5 2 2.5 3 3.5 4

x 107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CD

F

MU MIMO CRM OPT

MU MIMO URM OPT

MU MIMO Proposal

SU MIMO CRM OPT

SU MIMO URM OPT

SU MIMO Proposal

(a) Data rate requirement of 4Mbps.

2.4 2.6 2.8 3 3.2 3.4 3.6 3.8

x 107

0

0.1

0.2

0.3

0.4

0.5

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1


CD

F

MU MIMO CRM OPT

MU MIMO URM OPT

MU MIMO Proposal

SU MIMO CRM OPT

SU MIMO URM OPT

SU MIMO Proposal

(b) Data rate requirement of 6Mbps.

Figure 5.6: CDF of total data rate for the data rate requirements of 4Mbps and 6Mbps with CRM OPT,URM OPT and the proposed solution in the SU MIMO and MU MIMO antenna configurationswith MT = 4 and MR = 4.



In this chapter, we studied the CRM problem in the OFDMA downlink scenario when

multiple antennas are used at the transmitter and receiver with MU access. In order to

tackle the CRM problem in this new scenario we had to relax the constraint that a given RB

could not be shared by different terminals within a cell. Then, we introduced the concept of

SDMA group that consists in a group of flows that access the same RB to receive data.

In the problem formulation, instead of defining which flow should be associated with a

given RB, we considered the problem of which SDMA group should be associated with an

RB. Despite the different modeling, we managed to convert the formulated problem from a

non-linear integer optimization problem into ILP by introducing a new variable. This problem

can be solved by standard solvers as described in previous chapters.

Due to the high computational complexity to obtain the optimal solution we proposed a

heuristic solution to the CRM problem following the framework presented in Chapter 2. By

the performance results presented in this chapter we could see that the proposed solution

presents a manageable performance loss compared to the optimal solution for low and

moderated loads in terms of outage rate and total data rate. The computational complexity of

the proposed solution is higher than the one of the proposed solution to the SU case which is

expected due to the harder problem to be solved.

Finally, the presented results have shown the potential of the MU MIMO scenario in

improving outage rates and total data rates. It is important to keep in mind that MU MIMO is

more dependent on CSI than the other schemes. As we assumed perfect CSI at the transmitter

and receiver, the presented results can be seen as upper bounds in performance.

64

Chapter 6


with and without Minimum

Satisfaction Constraints in

SC-FDMA Uplink Scenario

In this chapter we focus on RRA in the uplink SC-FDMA scenario. As commented in

Chapter 1, the optimal solution to the URM problem can be only obtained by computationally

complex algorithms. Therefore, in this chapter, besides studying the CRM problem we also

consider the URM problem. The remaining of this chapter is organized as follows. In section

6.1, we present the system modeling. Sections 6.2 and 6.3 are devoted to the study of the URM

problem and of the CRM problem, respectively. In each of these sections, we mathematically

formulate the optimization problems, show an approach to optimally solve the problems,

propose low-complexity heuristic solutions and evaluate their performance by means of

computational simulations. Finally, in section 6.4, we present the partial conclusions of

the studies conducted in this chapter.

6.1 System Modeling

As commented in section 1.2.1.2 of Chapter 1, SC-FDMA imposes two main constraints on

RRA:

◮ Exclusivity: The same RB cannot be shared by flows within a cell. Note that this

constraint is not valid when SDMA is considered. Also, notice that this constraint already

exists for the downlink OFDMA case;

◮ Adjacency: The RBs assigned to each flow should be adjacent to each other in the

frequency domain. This constraint is not necessary in the OFDMA case.

Due to the adjacency constraint, the number of possible assignment patterns, P , depends

on N and is given by [40]

P =1

2N2 +

1

2N + 1. (6.1)

We assume that the set with the indices of all possible assignment patterns is given by

P = {1, · · · , P}. The adjacency constraint on RRA can be modeled by an N × P binary matrix

6.1. System Modeling 65

A composed of the elements an,p with n ∈ N and p ∈ P that assumes the value 1 if the nth RB

is included in the pth assignment pattern and 0 otherwise. As an example, for N = 4 we have

P = 11 possible assignment patterns that are represented by

A =

0 1 0 0 0 1 0 0 1 0 10 0 1 0 0 1 1 0 1 1 10 0 0 1 0 0 1 1 1 1 10 0 0 0 1 0 0 1 0 1 1

. (6.2)

We define XUL as a J × P assignment matrix with elements xULj,p that assume the value 1 if

the assignment pattern p ∈ P is assigned to the flow j ∈ J and 0 otherwise. As we will show

in sections 6.2.1 and 6.3.1, some constraints should be imposed on this matrix in order to

assure exclusivity and adjacency of RBs.

We consider the channel model in the frequency domain. We define the variable hULj,z,n as the

channel transfer function experienced by the jth flow at the zth subcarrier of the nth RB. Note

that we defined here the channel transfer function with the granularity of subcarriers instead

of RBs as it was done in previous chapters. As it will be made clear later, this is necessary for

the modeling of the frequency domain equalization needed in SC-FDMA systems. The SNR,

γULj,z,n, experienced by the flow j at the zth subcarrier of the RB n is given by

γULj,z,n =

(PUL

c ·N

)· αUL

j · |hULj,z,n|2

(σsub)2 , (6.3)

where αULj represents the joint effect of the path loss and shadowing of the link between flow j

and the serving BS,(σsub

)2is the noise power at the receiver in the bandwidth of a subcarrier

and c is the number of subcarrier in an RB.

A frequency domain equalizer should be used together with SC-FDMA in order to mitigate

ISI. In this work we assume that a MMSE equalizer is used, and from [68] the SNR of data

delivered by a set of RBs with MMSE equalization can be written as

γUL MMSEj,p =

1

1

c · |Np|∑

n∈Np

c∑z=1

γULj,z,n

γULj,z,n + 1

− 1

−1

, (6.4)

where γUL MMSEj,p is the effective SNR experienced by the data transmitted by flow j with the RBs

contained on the assignment pattern p, and Np is the set of RBs that compose the assignment

pattern p.

By using link adaptation, a terminal can transmit at different data rates according to its

channel state, allocated power and perceived noise/interference. We assume that there is a

link adaptation function f (·) that maps the experienced SNR to the transmit data rate. This

function can also model the use of discrete MCSs employed by modern wireless networks.

According to this model, the transmit data rate of flow j when using the assignment pattern p

is given by

rULj,p = f

(γUL MMSEj,p

). (6.5)

6.2. Unconstrained Rate Maximization 66

6.2 Unconstrained Rate Maximization

According to the system modeling presented in section 6.1 we study in this section the

URM problem. In the following we formulate the optimization problem to be solved in section

6.2.1, characterize the optimal solution in section 6.2.2, propose a low-complexity heuristic

solution in section 6.2.3 and evaluate the system performance in section 6.2.4.

6.2.1 Problem formulation

According to the definitions in section 6.1, the URM problem to be solved at each TTI can

be formulated as

maxXUL

∑

j∈J

∑

p∈P

rULj,p · xUL

j,p

, (6.6a)

subject to

∑

j∈J

∑

p∈P

an,p · xULj,p = 1, ∀n ∈ N , (6.6b)

∑

p∈P

xULj,p = 1, ∀j ∈ J , (6.6c)

xULj,p ∈ {0, 1}, ∀j ∈ J and ∀p ∈ P . (6.6d)

The objective function shown in (6.6a) is the total uplink data rate transmitted by the flows.

Constraints (6.6b) and (6.6d) assure that RBs are not reused within the cell, while constraint

(6.6c) guarantees that only one assignment pattern is chosen by each flow.

6.2.2 Characterization of the optimal solution

Problem (6.6) is an ILP that in general cannot be solved optimally with polynomial-time

complexity. According to [47], the simple constraint of resource adjacency is sufficient to

make the problem NP-hard. However, the optimal solution to this kind of problem can be

found by standard numerical methods such as the BB algorithm [60]. The computational

complexity of obtaining the optimal solution by these methods is much lower than using brute

force (complete enumeration of all possible RB assignments). Nevertheless, the complexity of

the BB method grows exponentially with the number of constraints and variables. In problem

(6.6) we have J · P variables and J +N constraints, which may assume large values even for

small numbers of flows and RBs.

6.2.3 Low-complexity heuristic solution

As pointed out in section 6.2.2, the computational complexity to solve the formulated

assignment problem is significant even with the use of standard techniques to solve ILP

problems. Therefore, we propose a low-complexity heuristic algorithm based on simple

heuristics. Figure 6.1 presents the simplified flowchart of the proposed algorithm.

The first step of the algorithm consists in finding the flow with best channel quality (or

SNR) on each RB. This assignment corresponds to the solution to the URM problem when

the network employs OFDMA as the multiple access method. Therefore, this solution in

general does not consider the adjacency constraint and is not a feasible solution to the studied

problem in SC-FDMA uplink scenario.

In step (2) we introduce the concept of Virtual Resource (VR). A VR consists in a set of


Begin

(1) Find the RB assignment thatsolves the OFDMA URM problem

(2) Build virtual resources basedon RB assignments of step (1)

(3) Does the current assignmentobeys the adjacency constraint?

(5) Generate combinations of virtualresources and calculate the metricassociated with each combination

(6) Choose virtual resourcecombination with higher metric value

(7) Update virtual resources accordingto the chosen combination in step (6)

(4) Feasiblesolution

No

Yes

Figure 6.1: Basic flowchart of the proposed algorithm for the URM problem in the uplink case.

contiguous RBs assigned to a given flow. Note that the VR belongs to the flow that owns the

contiguous RBs. Based on the RB assignment on step (1), we build the VRs and associate

them with the flows. Figure 6.2 illustrates steps (1) and (2) with an example where we have 3

flows and 10 RBs. In this example we can see that the RB assignment after the step (1) is not

in accordance with the adjacency constraint. Nevertheless, there are some contiguous RBs

that were assigned to the same flow, e.g., RBs 2 and 3, and 7 to 9 assigned to flow 1, and RBs

5 and 6 assigned to flow 3. As illustrated in Figure 6.2, those contiguous RBs assigned to the

same flow are redefined as VR after step (2) of the proposed solution.

11

11111

222

222

3

33

RB assignment after step (1)

VR assignment after step (2)

RB assignedto flow 1

VR assignedto flow 1

Figure 6.2: Illustration of steps (1) and (2) of the proposed algorithm with 3 flows and 10 RBs for theURM problem in the uplink case.


In step (3) of the proposed algorithm we evaluate if the current RB or VR assignment

complies with the adjacency constraint. Basically, this constraint is fulfilled if the number

of assigned VR to each flow is lower than or equal to one. In the example of Figure 6.2 we

can see that this constraint is not fulfilled. If the adjacency constraint is fulfilled we have a

feasible solution, otherwise we need to generate new VR combinations based on the current

VRs.

In step (5) we generate new VRs by combining the current ones and calculate an efficiency

metric in order to evaluate which combination or new VR would be more beneficial to the

system. In order to generate the new VRs combinations we use the following rules on all

current VRs:

◮ Rule 1: Consider a given assigned pair (flow j, VR v) and that v′ is the lowest VR index

greater than v that belongs to flow j, and v′′ is the highest VR index lower than v that

belongs to flow j. Based on VR v, two new VRs can be build. The first VR combination is

composed of all VRs between VRs v and v′ (including the VRs v and v′). The second VR

combination is composed of all the VRs between VRs v′′ and v (including the VRs v′′ and

v). If for a given assigned pair (flow j, VR v) only v′ or v′′ exists only one VR combination

can be build.

◮ Rule 2: If for a given assigned pair (flow j, VR v) flow j does not have any other VR with

index lower than v, a new VR combination is composed of the VRs v − 1 and v. Note that

case VR v = 1 we cannot build a new VR combination;

◮ Rule 3: If for a given assigned pair (flow j, VR v) flow j does not have any other VR with

index higher than v, a new VR is composed of the current VR v and v + 1. Note that case

VR v is the last VR we cannot build a new VR combination.

In Figure 6.3 we illustrate the new VRs that can be generated based on the example of

Figure 6.2. Each new possible VR is highlighted by dotted lines. The second combination for

flow 1, and first and second combinations for flow 2 are examples of the application of rule

1. Rule 2 is applied on the first combination for flow 1 and the first combination for flow 3.

Finally, rule 3 was applied on the third combination for flow 1 and second combination for

flow 3.

In the example of Figure 6.3 we have 7 new VR combinations that should be evaluated

regarding their contribution to the objective of maximizing the total data rate. In order to

measure the contribution of a new VR combination to the spectral efficiency we calculate the

effective SNR based on equation (6.4) of the subcarriers contained in the new VR. Therefore, if

the new VR belongs to flow j∗ and is composed of the RBs from n′ to n′′ the metric associated

with this new VR is given by

ϕURM ULa =

1

1

c · (n′′ − n′ + 1)

n′′∑n=n′

c∑z=1

γULj∗,z,n

γULj∗,z,n + 1

− 1

−1

, (6.7)

where ϕURM ULa is the metric of the ath VR combination.

Once the metrics associated with each new VR combination are calculated we proceed

to step (6) where we choose the new VR combination with highest metric. Then, the VR

assignment is updated according to this choice and the feasibility test is carried out again.

This procedure is repeated until a feasible solution is found. As in each iteration the number


1

11

1

1

111

1111

111

11

222

22

22222

2222

22

2

22

222

2

33

33

3

3

3

3

VR assignment after step (2)

New VRsfor flow 1

New VRsfor flow 2

New VRsfor flow 3

New VRscombination

Rule 1

Rule 1

Rule 1

Rule 2

Rule 2

Rule 3

Rule 3

Figure 6.3: Illustration of the process for building new VRs based on the example of Figure 6.2 for theURM problem in the uplink case.

of VRs decreases we can guarantee that the proposed algorithm always converges to a feasible

solution.

6.2.4 Performance evaluation

This section is devoted to the performance evaluation of the proposed solution presented

in section 6.2.3. In section 6.2.4.1 we present the main simulation assumptions and

performance metrics used for comparison, while in section 6.2.4.2 we show the simulation

results and draw preliminary conclusions.

6.2.4.1 Simulation assumptions


simulator. We evaluate the uplink resource assignment in a sector of a tri-sectorized cellular

system. We consider SC-FDMA with uplink resources arranged in a time-frequency grid. The

RB is composed of a group of 12 adjacent subcarriers in the frequency dimension and is 1 ms

long in the time dimension. We assume that the link adaptation is performed based on the

upper bound Shannon capacity [69]. In this way, the transmit data rate of flow j when using

the assignment pattern p is given by

rULj,p =

B · |Np|N

log2

(1 +

γUL MMSEj,p

Γ

), (6.8)

where B is the frequency bandwidth of an RB considered as 300 kHz and Γ is the SNR gap

given by

Γ = − ln (5 · BER)

1.5, (6.9)

where BER is the required Bit Error Rate (BER). In this work we assume BER = 10−4.

Different numbers of RBs are considered and the transmit power per RB was chosen as


Table 6.1: Main simulation parameters considered in the performance evaluation for the URM problemin the uplink case.


Cell radius 334 m


Number of subcarriers per RB 12 -

Number of RBs 12, 18 and 24 -


Path loss 1 35.3 + 37.6 · log10 (d) dB



BER for capacity gap (BER) 10−4 -

Number of flows 6, 7, 8, 9, 10, 11 and 12 -

0.1 W. Note that in general the power available at the mobile terminal is lower than the one

available at the BS. The simulation methodology and channel model are the same as the one

considered in section 3.5.1 of Chapter 3. The main simulation parameters are summarized in

Table 6.1.

In order to perform qualitative comparisons, besides our proposed algorithm, we simulate

the optimal solution to the URM problem presented in equation (6.6) obtained according to

section 6.2.2 (identified in the plots as URM OPT) and the heuristic algorithm proposed in [40]

to solve the URM problem (identified as Wong Alg in the plots). Basically, Wong Alg is a greedy

heuristic algorithm that iteratively assigns an RB to the flow that leads to the highest increase

in the objective function (total data rate maximization). In order to solve ILP problems we

used the IBM ILOG CPLEX Optimizer [63]. The channel realizations were the same for all

simulated algorithms in order to get fair comparisons. The choice of the number of flows and

RBs are limited by the computational complexity to obtain the optimal solution. When the

performance metrics are concerned, we consider the total data rate as the sum of the data

rates obtained by all the flows in the sector in a given snapshot.

6.2.4.2 Results

In Figure 6.4 we present the CDF of the total data rate for the solutions URM OPT, Wong

Alg and our proposed solution for different number of RBs and flows. In Figure 6.4(a) we

present the CDF for 6 flows while in Figure 6.4(b) we show the results for 12 flows. Firstly, we

can see in both figures that the total data rate for all algorithms is improved as a result of the

higher number of RBs or bandwidth. Another general observation is that for the same number

of RB, the total data rates are increased due to the multi-user diversity, i.e., the higher the

number of flows the higher is the probability of flows with good channel qualities.

The gains in the total data rate due to the higher bandwidth and multi-user diversity can

be better visualized in average terms in Figure 6.5 where we present the average total data

rate versus the number of flows for the URM OPT, Wong Alg and the proposed solution for

different number of RBs. Another comment can be depicted regarding the performance of

the heuristic solutions Wong Alg and our proposed one relative to the optimal solution in

Figure 6.5. The performance loss of the Wong Alg relative to the URM OPT are within the

ranges [9.2%; 12.7%], [14.2%; 15.1%] and [19.0%; 23.8%] for the scenarios with 12, 18 and

24 RBs, respectively. On the other hand, the performance loss of our proposed algorithm for



0 0.5 1 1.5 2 2.5 3

x 107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CD

F

URM OPT − 12 RBs

Wong Alg − 12 RBs

Proposal − 12 RBs

URM OPT − 18 RBs

Wong Alg − 18 RBs

Proposal − 18 RBs

URM OPT − 24 RBs

Wong Alg − 24 RBs

Proposal − 24 RBs

(a) Number of flows equal to 6 (J = 6).

0 0.5 1 1.5 2 2.5 3 3.5 4

x 107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CD

F

URM OPT − 12 RBs

Wong Alg − 12 RBs

Proposal − 12 RBs

URM OPT − 18 RBs

Wong Alg − 18 RBs

Proposal − 18 RBs

URM OPT − 24 RBs

Wong Alg − 24 RBs

Proposal − 24 RBs

(b) Number of flows equal to 12 (J = 12).

Figure 6.4: CDF of total data rate for URM OPT, Wong Alg and proposed solution considering differentnumber of flows and 12, 18 and 24 RBs for the URM problem in the uplink case.

the presented configurations of number of flows and RBs is not higher than 3.3%.

In Appendix C we present the worst-case computational complexity of the URM OPT and

proposed solutions. The complexity of the URM solution increases exponentially, O(2JP

),

while the complexity of the proposed solution is polynomial given by O(3JN +

265

2N2

).

According to the presented results and computational complexity analysis, we can see that

the proposed solution is feasible for the practical use in the resource assignment task due to

its good performance-complexity trade-off.


6 8 10 120.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

7

Number of users or flows

Avera

ge tota

l secto

r data

rate

(bits/s

)

URM OPT − 12 RBs

Wong Alg − 12 RBs

Proposal − 12 RBs

URM OPT − 18 RBs

Wong Alg − 18 RBs

Proposal − 18 RBs

URM OPT − 24 RBs

Wong Alg − 24 RBs

Proposal − 24 RBs

Figure 6.5: Average total data rate versus the number of flows for URM OPT, Wong Alg and proposedsolution considering 12, 18 and 24 RBs for the URM problem in the uplink case.

6.3. Constrained Rate Maximization 73

6.3 Constrained Rate Maximization

This section is devoted to the study of the CRM optimization problem. In the following

we formulate the optimization problem to be solved in section 6.3.1, characterize the optimal

solution in section 6.3.2, propose a low-complexity heuristic solution in section 6.3.3 and

evaluate the system performance in section 6.3.4.

6.3.1 Problem formulation

According to the previous considerations, the CRM problem to the SC-FDMA uplink

scenario can be formulated as the following optimization problem:

maxXUL

∑

j∈J

∑

p∈P

rULj,p · xUL

j,p

, (6.10a)

subject to

∑

j∈J

∑

p∈P

an,p · xULj,p = 1, ∀n ∈ N , (6.10b)

∑

p∈P

xULj,p = 1, ∀j ∈ J , (6.10c)

xULj,p ∈ {0, 1}, ∀j ∈ J and ∀p ∈ P , (6.10d)

∑

j∈Js

u

∑

p∈P

rULj,p · xUL

j,p , tj

> ks, ∀s ∈ S. (6.10e)

The objective function shown in (6.10a) is the total uplink data rate transmitted by the

flows. Constraints (6.10b) and (6.10d) assure that RBs are not reused within the cell, while

constraints (6.10c) and (6.10d) guarantee that only one assignment pattern is chosen by each

flow. Finally, constraint (6.10e) states that a minimum number of flows should be satisfied

for each service.

6.3.2 Characterization of the optimal solution

Problem (6.10) is a combinatorial optimization problem with a non-linear constraint

(6.10e). According to [47], the simple constraint of resource adjacency is sufficient to make

the problem NP-hard.

Let’s reformulate this problem by introducing some new variables. Consider ρj as a binary

selection variable that assumes the value 1 if flow j is selected to be satisfied and 0 otherwise.

In this way, problem (6.10) can be reformulated by substituting the constraint (6.10e) by two

new constraints as follows:

maxXUL,ρ

∑

j∈J

∑

p∈P

rULj,p · xUL

j,p

, (6.11a)


subject to

∑

j∈J

∑

p∈P

an,p · xULj,p = 1, ∀n ∈ N , (6.11b)

∑

p∈P

xULj,p = 1, ∀j ∈ J , (6.11c)

xULj,p ∈ {0, 1}, ∀j ∈ J and ∀p ∈ P , (6.11d)∑

p∈P

rULj,p · xUL

j,p > ρj · tj , ∀j ∈ J , (6.11e)

∑

j∈Js

ρj > ks, ∀s ∈ S. (6.11f)

In order to write this problem in a compact form we will represent the problem

variables and inputs in vector and matrix forms. In the following we utilize vertical

and horizontal lines when defining vectors and matrices so as to ease the comprehension

of their structure. Consider that rULj =

[rULj,1 · · · rUL

j,P

]T, k = [k1 · · · kS ]T , xUL =

[xUL1,1 · · · xUL

1,P | xUL2,1 · · · xUL

2,P | · · · | xULJ,1 · · · xUL

J,P

]Tand ρ = [ρ1 · · · ρJ ]T . We define the optimization

variable as yUL =[(xUL

)T |ρT]T

. Note that the vectors xUL and ρ can be obtained from yUL

through the use of the following relations: xUL = A1yUL and ρ = A2y

UL, with A1 = [IJP |0JP×J ]

and A2 = [0J×JP | IJ ].The objective function (6.11a) can be written as aTA1y

UL where a =[(rUL1

)T (rUL2

)T · · ·(rULJ

)T ]T. The constraint (6.11b) can be written as BA1y

UL = 1N where

B = [A A · · · A︸︷︷︸J times

]. (6.12)

The second constraint (6.11c) can be written as CA1yUL = 1J where C = diag

(1TP , · · · , 1T

P

).

The constraint (6.11e) is rewritten as

EA1yUL

> FA2yUL =⇒ (EA1 − FA2)y

UL> 0J , (6.13)

with E = diag(rT1 , · · · , rTJ

)and F = diag (t1, · · · , tJ). Finally, the last constraint (6.11f) can

be stated as GA2yUL > k where G = diag

(1TJ1, · · · , 1T

JS

). Please, notice that the blocks being

block-diagonally organized in C, E, and G are vectors so that the resulting matrices are not

necessarily square matrices as often expected from block diagonal matrices.

Finally, arranging the expressions developed so far we have

maxyUL

(cTyUL

), (6.14a)

subject to

JyUL = 1N+J , (6.14b)

LyUL> e, (6.14c)

yUL is a binary vector, (6.14d)

where c = AT1 a, J =

[(BA1)

T(CA1)

T]T

, L =[(EA1 − FA2)

T(GA2)

T]T

and e =[0TJ kT

]T.

Based on the previous development we have transformed (6.10) into a linear integer (binary)


optimization problem. This problem can be solved by standard methods such as the BB

algorithm [60]. The computational complexity of the BB method grows exponentially with

the number of constraints and variables. In problem (6.14) we have J · P + J variables and

2J +N + S constraints, which may assume large values even for small numbers of flows, RBs

and services.

6.3.3 Low-complexity heuristic solution

As pointed out in the section 6.3.2, the computational complexity to solve the formulated

assignment problem is significant even with the use of standard techniques to solve ILP

problems. Therefore, we propose a low-complexity algorithm based on simple heuristics. The

proposed algorithm is split into two parts: Unconstrained Maximization and Reallocation.

The flowcharts of the first and second parts are depicted in Figures 6.6 and 6.7, respectively.

Begin

(1) Auxiliary flow set and availableflow set are composed of all flows

(2) Maximum rate allocationproposed in section 6.2.3 with theflows from the available flow set

(3) From the flows of the available flow setdefine the satisfied and unsatisfied flows


(6) Take out from the available andauxiliary flow sets the flow with thepoorest channel quality and highestrequirement in the auxiliary flow set

(7) Can another flow be disregardedfrom the same service of this flow?

(8) Take out from the auxiliaryflow set all flows from this service

(9) Is the auxiliary flow set empty?


(12) Define the donor and receiver flow sets

(5) Feasible solution


No

No

Yes

Yes

Yes

Yes

No

Figure 6.6: Flowchart of the first part of the proposed solution for the CRM problem in the uplink case:Unconstrained Maximization.

In the Unconstrained Maximization part, the basic idea is to have a good initial

assignment that is on (or at least near to) the boundary of the capacity region. As we explain

in the following, this is achieved by sequentially solving the URM problem and disregarding

flows based on channel quality and required data rate. Firstly, in step (1) we define the

auxiliary (B) and available (A) flow sets and initialize them with the set of all flows (J ). The

auxiliary flow set B contains the flows that can be disregarded without infringing satisfaction


constraints, whereas the available flow set A contains the flows that were not disregarded

and will get assigned RBs in the next part. In step (2) we run the proposed solution to the

URM presented in section 6.2.3 with the flows from the available flow set A. It is important to

highlight that although this algorithm is in most of the cases suboptimal, it provides a good

performance-complexity trade-off.

After that, in step (3) we define the flows that have the data rate requirement fulfilled as

the satisfied flows and the remaining ones as the unsatisfied flows. In step (4) we check if the

minimum required number of satisfied flows of each service is achieved, i.e., the constraint

(6.10e) of problem (6.10) is fulfilled. If so, we have found a feasible solution to the studied

problem. However, note that this is an uncommon situation due to the distribution of the

terminals within the sector. In general, few terminals will get most of the available RBs. If the

satisfaction constraint for any service is not fulfilled, a flow of the auxiliary flow set B will be

disregarded in step (6). By disregarding a flow we mean that it will not receive resources at

the current TTI. The criterion to select the flow j∗ to be disregarded is given by

j∗ = argminj∈B

1

c ·N∑

n∈N

c∑z=1

γULj,z,n

tj, (6.15)

where B contains the flows of the services that can still be disregarded. The adopted criterion

to disregard a flow is quite reasonable: we disregard the flow that requires, in average, more

RBs to be satisfied. The selected flow is taken out of the available and auxiliary flow sets.

In step (7) we check whether the service from which the disregarded flow belongs can

have another flow disregarded without infringing the minimum satisfaction constraint of

the considered service. If this is not possible, all the flows from this service are taken

out of the auxiliary flow set B in step (8). In this case, no flow from that service will be

disregarded anymore. If the auxiliary flow set is not empty (step (9)), i.e., there are still

flows to be disregarded without infringing the satisfaction constraints, we redo the maximum

rate allocation with the remaining flows in the available flow set. This complete procedure

is repeated until either we find a feasible solution or no flow can be disregarded, i.e., the

auxiliary flow set is empty. In the latter case, we check if at least one flow is satisfied in step

(10). If so, in step (12) we define from the available flow set A two new sets: the donor (D) and

receiver (R) flow sets. The donor flow set D is composed of the satisfied flows in the available

flow set A and can donate RBs to unsatisfied flows. The receiver flow set R is composed of

the unsatisfied flows from the available flow set A that need to receive RBs from the donors

to have their data rate requirements fulfilled. In case there is no satisfied flow after executing

the first part, our proposed algorithm is not able to find a feasible solution (step (11)), i.e., one

that complies with the minimum satisfaction constraints. Comments about this situation are

provided later in this section.

In the Reallocation part presented in Figure 6.7 we basically switch RBs from the donors

to the receiver flows. We start in step (1) by choosing the flow from the receiver flow set R with

the worst average channel quality to get resources until its data rate requirement is fulfilled.

The main purpose of this procedure is to assign the minimum number of RBs to the flows

in bad channel conditions and get them satisfied while the other RBs remain assigned to the

flows in better channel conditions. In step (2), we identify the RBs that can be reassigned to

the selected receiver flow. We have two possibilities here:

i. Receiver flow has got an RB or a block of RBs in the first part of the algorithm: Assume

that the selected flow has got assigned the block of RBs from n′ to n′′ with n′ ≤ n′′ and


(1) Choose the flow fromthe receiver set with

worst channel condition

(2) Identify the availableRBs for reallocation

(3) Are there availableRBs for reallocation?

(4) i = 1

(5) Generate the possibleRB combinations of sizei for each available RB

(6) Is there anypossible combination?

(8) Calculate the associatedmetric with each combination

(9) Choose the combinationwith highest metric

(10) Would the donor beunsatisfied with the reassignment?

(13) Would the receiver besatisfied with the reassignment?

(15) Update assignment,receiver and donor sets, andreceiver and donor data rates

(16) Is there any receiverin the receiver flow set?

(17) Feasiblesolution found

(14) i = i + 1

(11) Delete the metriccorresponding to thechosen combination

(12) Is there anyremaining combination?

(7) No solution found

Yes

No

No

Yes

Yes

No

Yes

Yes

No

YesNo

No

Figure 6.7: Flowchart of the second part of the proposed solution for the CRM problem in the uplinkcase: Reallocation.

n′ 6= 1 and n′′ 6= N . In this case, the RBs that are available for reassignment are n′−1 and

n′′+1. Note that if n′ = 1 or RB n′− 1 belongs to another receiver flow, the RB n′− 1 is not

available for reallocation. Similarly, if n′′ = N or RB n′′ + 1 belongs to another receiver

flow, the RB n′′ + 1 is not available for reallocation.

ii. Receiver flow has not got any RB in the first part: In this case, the available RBs for

reallocation are the first and the last RBs of the blocks assigned to each donor flow in

the first part of the proposed solution.

The reasoning considered to define the available RBs is as follows. In the case i, as the

receiver flow has got an RB or a block of RBs and so as not to break the adjacency constraint,

the available resources are the ones adjacent to the RBs already assigned to the receiver flow.

In the case ii, the choice of the available RBs was defined in order not to break the adjacency

constraint on the RBs already assigned to the donor flows. Therefore, the available RBs are


the ones at the corner of the assigned block to each donor flow. In Figure 6.8 we illustrate

four hypothetical assignments after the first part of our proposed solution. In each example

we show the available RBs when flow 2 is chosen to get RBs in the second part of our solution.

Consider that flows 1 and 4 are satisfied (donor flows) and that flows 2 and 3 are unsatisfied

(receiver flows). The first three examples illustrate the case i, i.e., flow 2 has got at least

one RB in the part 1 of our proposed solution, while in example 4 we illustrate the case ii.

Note that in the second example RB 7 is not available for reassignment because it belongs to

another receiver flow.

111 11 1

111 1

1111

1111

22

22

22

3

3

3

444 4

444

444

44 4

Example 1

Example 2

Example 3

Example 4

RB assigned toflow 1 in part 1 Donor flows: 1 and 4

Receiver flows: 2 and 3

Selected receiver flow: 2

Available RB forreassignment to flow 2

Figure 6.8: Illustration of the process for selecting the available resources for flow 2 based on 4examples of resource assignment in the first part of the proposed solution for the CRMproblem in the uplink case. We consider 10 RBs and that flows 1 and 4 are the donors andthat flows 2 and 3 are receivers.

In step (3) we check if at least one RB is available for reassignment. If there is no available

RB the algorithm is not able to find a feasible solution. In step (4) we initialize the variable i

and in step (5) we generate all possible contiguous RB groups of size i based on each available

RB identified on step (2). The use of variable i will be clarified later. For i = 1 the possible

contiguous RB groups are the available RBs themselves. For i > 1 we have more possibilities

for each available RB. Basically, for each available RB we generate the contiguous RB blocks

that (1) includes the available RB and (2) that do not include any RB that belongs to receiver

flows.

In Figure 6.9 we illustrate the possible RBs groups that can be composed based on the

second available RB of the fourth example presented in Figure 6.8 with i equal to 1, 2 and 3.

Moreover, in Figure 6.10 we show the possible RB groups for the available RB in the second

example of Figure 6.8 for different values of i.

In step (6) we verify if at least one feasible RB group exists, i.e., an RB group that includes

the available RB and that does not include any RB that belongs to receiver flows. Then, in

step (8) we calculate the efficiency of each generated RB group in step (5) for each available

RB identified in step (2). Considering that j∗ is the index of the chosen receiver in step (1) and

that the RB group is composed of the RBs from n′ to n′′, the efficiency metric associated with


11 11 1

1 11 1

11 1

11 11 1

1 11 1

11 11 1

111 11 1

2 2

22 2

222

2 2

22

2

2 44

444

4444

444

4444

444

444 4

4

RB group with i = 1

RB groups with i = 2


Considered

available RB

Figure 6.9: Illustration of the process to generate the RB groups based on the second available RB ofthe fourth example of Figure 6.8 considering i equal to 1, 2 and 3 for the CRM problem inthe uplink case.

this RB group is given by

ϕCRM ULa =

1

1

c · (n′′ − n′ + 1)

n′′∑n=n′

c∑z=1

γULj∗,z,n

γULj∗,z,n + 1

− 1

−1

, (6.16)

where ϕCRM ULa is the efficiency metric of the ath RB group. This efficiency metric is equal to

the effective SNR of the subcarriers that compose the RB group (see equation (6.4)).

After calculating the efficiency metric for each generated RB group we choose, in step (9),

the RB group with highest efficiency metric. However, before reassigning the selected RB

group to the receiver flow we check in step (10) if the reassignment of the chosen RB group

would leave the donor flow(s) unsatisfied. If so, the selected RB group is discarded and the

next RB group with highest efficiency is selected. If the reassignment of the selected RB

group does not lead the donor flows to unsatisfaction, we check if the receiver flow would be

satisfied with the reassignment (step (13)). In case the receiver flow would not be satisfied we

increment the variable i, i.e., we increase the number of RBs in the RB groups. By increasing

the number of RBs in the RB groups, we increase the potential data rate of the receiver flow


1

11

111

1111

22 222

22 22

22 2

22

3

3

3

3

444

444

444

444

RB group with i = 1



Considered

available RB

Figure 6.10: Illustration of the process to generate the RB groups based on the available RB of thesecond example of Figure 6.8 considering i equal to 1, 2 and 3 for the CRM problem in theuplink case.

and the possibility of being satisfied.

Once we choose an RB group whose reassignment does not affect the satisfaction state

of the donor flows, and satisfy the selected receiver flow, we execute the reassignment and

update the flows’ data rates and the flow sets in step (15). The algorithm finds a feasible

solution when all the flows in the receiver flow set are satisfied.

When the system is overloaded, it is possible that our proposed solution is not able to find

a feasible solution, as indicated in Figures 6.6 and 6.7. However, in these cases we could

proceed as follows. In the first part, when the maximum number of flows were disregarded(J − ∑

s∈S

ks

)without violating the satisfaction constraint, and no flow is satisfied, an option is


flow will be the donor flow in the Reallocation part. If our proposed solution is not able to

find a feasible solution in the Reallocation part, an option is to disregard a flow according to

(6.15), repeat part 1 and then redo part 2. The idea behind these procedures is to relax the

constraint (6.10e) by sacrificing some flows in order to get a feasible solution.

6.3.4 Performance evaluation

This section is devoted to the performance evaluation of the proposed solution presented

in section 6.3.3. In section 6.3.4.1 we present the main simulation assumptions and

performance metrics used for comparison, while in section 6.3.4.2 we show the simulation

results and draw preliminary conclusions.

6.3.4.1 Simulation assumptions

The main assumptions stated in section 6.1 were implemented in a computational

simulator with the same characteristics as the one presented in section 6.2.4.1. The main

simulation parameters are summarized in Table 6.2.



Table 6.2: Main simulation parameters considered in the performance evaluation for the CRM problemin the uplink case.


Cell radius 334 m


Number of subcarriers per RB 12 -

Number of RBs 25 -


Path loss 2 35.3 + 37.6 · log10 d dB



BER for capacity gap (BER) 10−4 -

Number of flows See the description of the scenarios -

Number of services See the description of the scenarios -

Required minimum number of satisfied flows See the description of the scenarios -

Required data rate of the flows From 10kbps to 100kbps -

In order to analyze our proposal under different conditions, in the results we present some

simulation scenarios in which the main parameters of our model are changed. Basically, in

those scenarios we change the number of services (S), the number of flows per service (Js), and

the required minimum number of satisfied flows per service (ks). The scenarios are described

in Table 6.3.

Table 6.3: Parameters of the considered scenarios for the CRM problem in the uplink case.

Scenario S J1 J2 J3 k1 k2 k3

1 1 8 - - 6 - -

2 1 8 - - 7 - -

3 1 8 - - 8 - -

4 2 4 4 - 3 3 -

5 2 4 4 - 4 3 -

6 2 4 4 - 4 4 -

7 3 3 3 3 3 2 2

8 3 3 3 3 3 3 2

9 3 3 3 3 3 3 3

So as to perform qualitative comparisons, besides our proposed algorithm, we simulate

the optimal solution to the CRM problem obtained according to section 6.3.2 (identified in the

plots as CRM OPT) and the optimal solution to the URM problem (identified as URM OPT in

the plots). In order to solve ILP problems we used the IBM ILOG CPLEX Optimizer [63].

The channel realizations were the same for all simulated algorithms in order to get fair

comparisons. The choice of the number of flows, RBs and services are limited by the

computational complexity to obtain the optimal solutions. The same performance metrics

considered in Chapters 3, 4 and 5 are considered here: outage ratio and total data rate.

6.3.4.2 Results

In the following figures we denote the ith scenario as SCEi. In Figure 6.11 we show the

outage rate versus the data rate required by all flows in scenarios 1, 2 and 3 (only one service)

for the algorithms CRM OPT, URM OPT and the proposed solution.


0 1 2 3 4 5 6 7 8 9 10

x 104

0

10

20

30

40

50

60

70

80

90

100


Ou

tag

e r

ate

(%

)

CRM OPT SCE1

URM OPT SCE1

Proposal SCE1

CRM OPT SCE2

URM OPT SCE2

Proposal SCE2

CRM OPT SCE3

URM OPT SCE3

Proposal SCE3

Figure 6.11: Outage rate for CRM OPT, URM OPT and the proposed solution with one service inscenarios 1, 2 and 3 for the CRM problem in the uplink case.

Firstly, we can see that the outage rate increases with the data rate requirement of the

flows for all algorithms, as expected. This is a behavior present in all results concerning the

outage rate performance. Moreover, we can see that the outage rate for the same algorithm

increases from scenarios 1 to 3. The reason for this is the variation of the minimum number

of flows that should be satisfied, ks. While in the first scenario, 6 out of 8 flows should get

assigned RBs in order to become satisfied, in the third scenario, all 8 flows should have their

data rate requirements fulfilled. Another general observation that will be present in the other

outage rate plots is that the URM OPT solution presents high outage rates even for low data

rate requirements. The reason for this is that it maximizes the total data rate without any

QoS guarantee. Consequently, only few flows (with best channel conditions) get most of the

system resources and become satisfied.

In Figures 6.12 and 6.13 we present the outage rate versus the data rate required by all

flows for the algorithms CRM OPT, URM OPT and proposed solution in scenarios 4, 5 and 6

(two services), and 7, 8 and 9 (three services), respectively.

As in the outage rates presented in Figure 6.11, we can see in Figures 6.12 and 6.13

that the URM OPT solution presents high outage rates and poor resource distribution.

Furthermore, in those figures we can see again that the outage rate increases as the

requirements on the minimum number of satisfied flows augments. Another important

observation that can be drawn from the observation of Figures 6.11 and 6.12 is the impact of

the number of services on the outage rates. In scenarios 2 and 5 the total number of flows that

should be satisfied is the same: 7 flows. However, observing Figures 6.11 and 6.12 we can

see that the achieved outage rates for CRM OPT solution are not the same. The configuration

of those two scenarios is similar with only one difference: in scenario 5 the flows are split

into two services while all flows belongs to the same service in scenario 2. This interesting

observation shows the additional complexity of solving the studied problem as the number of

services increases.

The last comments about the outage rate presented in Figures 6.11, 6.12 and 6.13 are

concerned with the relative performance of the proposed solution and CRM OPT. We can


0 1 2 3 4 5 6 7 8 9 10

x 104

0

10

20

30

40

50

60

70

80

90

100


Ou

tag

e r

ate

(%

)

CRM OPT SCE4

URM OPT SCE4

Proposal SCE4

CRM OPT SCE5

URM OPT SCE5

Proposal SCE5

CRM OPT SCE6

URM OPT SCE6

Proposal SCE6

Figure 6.12: Outage rate for CRM OPT, URM OPT and the proposed solution with two services inscenarios 4, 5 and 6 for the CRM problem in the uplink case.

0 1 2 3 4 5 6 7 8 9 10

x 104

0

10

20

30

40

50

60

70

80

90

100


Ou

tag

e r

ate

(%

)

CRM OPT SCE7

URM OPT SCE7

Proposal SCE7

CRM OPT SCE8

URM OPT SCE8

Proposal SCE8

CRM OPT SCE9

URM OPT SCE9

Proposal SCE9

Figure 6.13: Outage rate for CRM OPT, URM OPT and the proposed solution with three services inscenarios 7, 8 and 9 for the CRM problem in the uplink case.

observe that the proposed solution is able to keep a relatively small outage rate difference

to the CRM OPT solution in low and medium loads in single and multiservice scenarios.

Focusing on the required data rate where the corresponding CRM OPT solution has an outage

rate of 10%, we can see that the difference in outage rate between our proposed solution

and the CRM OPT solution in Figure 6.13 are of approximately 2% in scenarios 7, 8 and 9,

respectively.


feasible solution to our problem. On the other hand, in the remaining figures we show the

CDF of the total data rate for specific data rate requirements considered in the x-axis of the

figures regarding the outage rate performance. For a specific scenario and load, the CDFs


of all algorithms are built with the samples of the snapshots in which our proposed and the

CRM OPT solutions were able to find a solution (no outage). Therefore, possibly many of the

samples used in the CDFs for URM OPT are in outage. The main idea to include results of

those solutions is to show how the problem constraints of the studied problem (6.10) impose

losses in the total achievable data rate.

In Figure 6.14 we present the CDFs of the total data rate for CRM OPT, URM OPT and

the proposed solution in the single service scenarios 1 and 2 for the required data rate of

10kbps and 50kbps. In Figure 6.15 we present the CDFs of the total data rate for CRM OPT,

URM OPT and the proposed solution in the two-services scenarios 4 and 5 for the required

data rate of 10kbps and 40kbps. Notice that the difference between scenarios 1 and 2 is the

required minimum number of satisfied flows per service (ks). Therefore, the performance of

the solution URM OPT is the same in those scenarios since those solutions do not depend on

the ks parameter. The same comment is valid for scenarios 4 and 5.

Firstly, we can see that the URM OPT solution provides higher total data rates in both

Figures 6.14 and 6.15. This is a general observation that comes at the cost of higher outage

rate as shown in Figures 6.11 and 6.12. Other observation that can be drawn is related to the

impact of the required data rate on the achieved total data rate of the CRM OPT and proposed

solutions. In order to illustrate that, the performance loss of CRM OPT related to the URM

OPT at the 50th-percentile of the total data rate is of 10 % at the required data rate of 10 kbps

in scenario 2. When the required data rate is of 50 kbps the performance loss of CRM OPT

increases to 23 %. This aspect shows that in order to satisfy increased required data rates,

the most efficient RB assignments become infeasible and, therefore, the spectral efficiency of

the system is compromised.

Other aspect that impacts on the achieved total data rate is characterized in Figure 6.16

where we present the CDFs of the total data rate for CRM OPT, URM OPT and the proposed

solution in the three-services scenarios 7, 8 and 9 for the required data rate of 20 kbps.

The difference between scenarios 7, 8 and 9 is the required minimum number of satisfied

flows (ks). We can see that ks also impacts on the achieved total data rate. As observed in

previous chapters, the achieved total data rate increases as the required number of satisfied

flows decreases. Compared to scenario 9, the performance gain at the 50th-percentile of the

total data rate of CRM OPT solution are of 9 % and 17 % in scenarios 8 and 7, respectively.

Therefore, as the required data rate, the required minimum number of satisfied flows also

limits the choice of efficient RB assignments and, consequently, decreases the achieved

spectral efficiency.

Focusing on the performance of the proposed solution we can draw some comments. At

low required data rates, i.e., required data rates that leads to small outage rates (< 10%),

the proposed solution is able to maintain a reasonable performance gap to the CRM OPT

solution. As an example, the performance loss at the 50th-percentile of the total data rate of

the proposed solution compared to CRM OPT is of 17% in scenario 1 for the required data

rate of 10 kbps. For required data rates that leads to higher outage rates the performance

loss is increased. As an example, for the required data rate of 50 kbps, the performance loss

at the 50th-percentile of the total data rate for the proposed solution compared to CRM OPT is

of 22 % in scenario 1. It is important to mention here that the increased performance losses

at high loads are not critical since these loads are not of interest for practical purposes due to

the high outage rates.

In Appendix C we present the worst-case computational complexity of the algorithms CRM

OPT and the proposed solutions. The CRM OPT solution presents exponential worst-case


0 1 2 3 4 5 6

x 107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CD

F

CRM OPT SCE1

Proposal SCE1

CRM OPT SCE2

Proposal SCE2

URM OPT SCE1 and SCE2

(a) Data rate requirement of 10 kbps.

0 1 2 3 4 5 6

x 107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CD

F

CRM OPT SCE1

Proposal SCE1

CRM OPT SCE2

Proposal SCE2


(b) Data rate requirement of 50 kbps.

Figure 6.14: CDF of total data rate for CRM OPT, URM OPT and proposed solution with one service inscenarios 1 and 2 for the CRM problem in the uplink case.

computational complexity: O(2JP

). On the other hand, the proposed solution presents

polynomial-time worst-case computational complexity given by O((∑

s∈S ks) (

N −∑s∈S ks)3)

.

As a general analysis of the outage rates and total data rates we can see that our proposed

solution achieves a good performance-complexity trade-off compared to the CRM OPT

solution.


0 1 2 3 4 5 6

x 107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CD

F

CRM OPT SCE4

Proposal SCE4

CRM OPT SCE5

Proposal SCE5


(a) Data rate requirement of 10 kbps.

0 1 2 3 4 5 6

x 107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CD

F

CRM OPT SCE4

Proposal SCE4

CRM OPT SCE5

Proposal SCE5


(b) Data rate requirement of 40 kbps.

Figure 6.15: CDF of total data rate for CRM OPT, URM OPT and proposed solution with two services inscenarios 4 and 5 for the CRM problem in the uplink case.


0 1 2 3 4 5 6

x 107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


CD

F

CRM OPT SCE7

Proposal SCE7

CRM OPT SCE8

Proposal SCE8

CRM OPT SCE9

Proposal SCE9

URM OPT SCE7, SCE8 and SCE9

Figure 6.16: CDF of total data rate for CRM OPT, URM OPT and proposed solution for required datarate of 20 kbps with three services in scenarios 7, 8 and 9 for the CRM problem in theuplink case.



In this chapter, we presented our last technical contribution that is the study of the URM

and CRM problems in the SC-FDMA uplink scenario. Differently of the previous scenarios,

the optimal solution to the URM problem cannot be easily obtained. With this in mind, we

proposed a low-complexity heuristic solution to the URM problem. Our proposed solution

starts with an initial assignment based on the solution to the URM problem for OFDMA.

Although this is not a feasible solution to our studied problem, it is a good initial point in order

to obtain feasible and possibly good solutions. Based on this initial solution, we proposed

intelligent RB combinations and a metric to measure the efficiency of each combination that

should be calculated iteratively in order to find a feasible solution.

In the second part of this chapter, we studied the CRM problem. As in the previous

chapters, we managed to convert the previously non-linear integer problem to an ILP problem.

The optimal solution can then be obtained by BB-based methods. We also proposed a

low-complexity heuristic solution following the framework presented in Chapter 2.

Simulation results were presented and worst-case computational complexity was

discussed for both proposed algorithms. According to the presented scenario, we could

see that the proposed solution to the URM problem was able to achieve a maximum

performance loss of 3.3% related to the optimal solution at the cost of a polynomial worst-case

computational complexity for the set of simulated scenarios. When the proposed solution to

the CRM problem is concerned, we could see that it was able to perform closely to the optimal

solution in low and moderated loads in the outage rate criterion. The obtained total data

rates of the proposed solution to the CRM problem presented a performance gap greater

than the ones observed in previous chapters. The main reason for that is the use of a

heuristic solution in the first part of the proposed solution (Unconstrained Maximization).

The worst-case computational complexity of the proposed solution to the CRM problem is

polynomial in contrast to the complexity of the optimal solution that increases exponentially

with the input variables.

89

Chapter 7

Conclusions and Future Work

Along this thesis we dealt with Radio Resource Allocation (RRA) for wireless networks

in different scenarios. Specifically, the main problem studied is the Constrained Rate

Maximization (CRM) problem that consists in the maximization of the overall data rate subject

to minimum satisfaction constraints per service. As presented in Chapter 1, this is an

important problem to be solved in modern wireless networks where different services with

heterogeneous Quality of Service (QoS) demands are present. In Chapter 2 we presented

that problem in general terms. In general, RRA problems that involve Resource Block (RB)

assignment are hard to solve optimally, and therefore low-complexity solutions are needed.

As a result, we presented in Chapter 2 a general framework for proposal of heuristic solutions

to the CRM problem. Basically, the heuristic framework consists in firstly finding the solution

to the Unconstrained Rate Maximization (URM) problem and then reallocating the RBs among

the flows in order to get a feasible solution.

In Chapter 3 we studied the CRM problem in single antenna downlink scenario. The

formulated problem in this chapter is a non-linear integer optimization problem. We managed

to convert this problem to an Integer Linear Problem (ILP) by adding new variables to it.

Consequently, the optimal solution to the problem can be obtained by standard solvers found

in the literature. Motivated by the computational complexity of these methods, we proposed a

low-complexity heuristic solution to the problem. According to the simulation results we could

verify that the proposal is able to maintain a small performance loss related to the optimal

solution especially at low and moderated loads. We also studied the impact of the problem

variables in the performance of the involved algorithms.

In Chapter 4 we considered the addition of multiple antennas to the problem considered

in Chapter 3. Considering the Single-User (SU) access per RB, we have shown that the

problem itself is hardly changed compared to the previous chapter. Therefore, with few

modifications, all the development for Single Input Single Output (SISO) scenario is valid here.

We provided in this chapter a performance evaluation of the proposed algorithm in SISO, SU

Multiple Input Single Output (MISO) and SU Multiple Input Multiple Output (MIMO) antenna

configurations. Zero-Forcing (ZF) and Singular Value Decomposition (SVD) spatial filters were

considered for the MIMO antenna configuration. Despite the simplified assumptions regarding

the MIMO modeling, we could see that the SU MIMO configuration can provide substantial

gains compared to SISO case in the outage rate and total data rate criteria. Furthermore, as

in the SISO scenario, the proposed solution presented a competitive performance compared

to the optimal solution.

The multiple antenna scenario was generalized in Chapter 5 where we considered the

90

multiple antenna configuration with Multi-User (MU) access per RB. In this case, the CRM

problem formulation was significantly changed with the definition of the Space-Division

Multiple Access (SDMA) groups. In order to cancel out the inter-streaming interference

between different flows we considered the Block Diagonalization (BD)-ZF spatial filtering. As

in Chapter 3, the formulated problem was non-linear and integer, however, by the addition

of new variables the problem was converted to ILP. Then, we proposed an alternative

solution to the CRM problem following the heuristic framework presented in Chapter 2.

By the simulation results we could verify that the proposed solution also maintains a good

performance compared to the optimal solution in low and moderated offered loads. Also, the

gains of the MU MIMO over SU MIMO in the CRM problem are discussed.

In Chapter 6, differently of the previous chapters, we consider the uplink scenario.

The considered multiple access scheme is the Single Carrier - Frequency Division Multiple

Access (SC-FDMA) where the assigned RBs to a flow should be adjacent to each other. This

constraint turns RRA allocation even more challenging. In this chapter, besides the CRM

problem, we considered the URM problem. The latter problem is not a trivial problem in

this scenario differently of the scenarios considered in previous chapters. For the URM

problem, we proposed a heuristic solution that according to the simulated scenarios presents

a performance loss not higher than 3.3 % compared to the optimal solution. In the second part

of the chapter, we dealt with the CRM problem. As in previous chapters, we reformulated the

problem so as to be solved by standard Branch and Bound (BB)-based methods. Furthermore,

we proposed an alternative solution based on the heuristic framework presented in Chapter

2. Our proposed solution to the CRM problem has as basic step our proposed algorithm to

the URM problem. The performance results shows that small performance loss in the outage

rate criterion can be achieved in low and moderated offered loads compared to the optimal

solution to the CRM problem.

As we dealt with algorithms in this thesis, we also presented the worst-case computational

complexity of most of the algorithms in the appendices of this thesis. In general terms, we

could verify that the proposed heuristic algorithms have polynomial-time complexity while the

methods to obtain the optimal solution have exponential-time worst-case complexity in terms

of the problem inputs. Consequently, from the joint analysis of the simulation results together

with the complexity analysis we believe that we have provided important contributions to the

RRA field of wireless networks.

The work developed in this thesis opened new research directions to be investigated. In

the following we mention some of them:

◮ More detailed modeling of system aspects: Along our work we considered some

simplifying assumptions such as perfect CSI at the transmitter/receiver, Independent

and Identically Distributed (IID) channel modeling for MIMO and inter-cell interference

modeled as noise. A possible interesting study that could be developed from this thesis

is the evaluation of the impact of the modeling of these aspects in the performance

evaluation. In general terms, we expect that the relative results between the algorithms

will not change significantly, differently of the absolute performance that could be

deteriorated.

◮ Time scheduling: We followed the strategy of solving at each Transmission Time Interval

(TTI) the snapshot problem of maximizing the total data rate subject to minimum

satisfaction guarantees, assuming that the selection of the flows to the problem and

the definition of the problem inputs are done previously. However, in order to achieve

91

the traced objectives in long term in a dynamic environment some modifications are

needed. Basically this is a possible continuation of this work. In order to solve the

time scheduling problem the ideas stated in [70] could be followed. In that work,

at each TTI a snapshot optimization problem is solved where the problem input are

dynamically changed by a control loop algorithm. This strategy could be used in the

CRM problem. More specifically, in a regular basis, e.g., at each TTI, an algorithm based

on a control loop could adapt the problem inputs such as flows’ required data rates (tj )

and minimum satisfaction guarantees (ks) according to the environment changes. The

proposed algorithms in this thesis can be used to solve the snapshot problem at each

TTI.

◮ Infeasibility: As commented along the thesis, the proposed heuristic solution can finish

without a feasible solution. This can happen due to two reasons. The first one is if

the CRM problem itself is infeasible, i.e., it is not possible to satisfy the satisfaction

constraints with the given channel states and QoS requirements. The other reason is

due to the suboptimality of the proposed solution that sometimes are not able to find

a feasible solution. Along the chapters, we have drawn some comments with ideas to

proceed when a feasible solution is not found. One possible solution is to relax the

satisfaction constraints, i.e., for a given service the minimum number of satisfied flows

is decreased. Other option, is to relax the required data rates of the flows. This issue is

posed here as a perspective of this thesis.

◮ Lower computational complexity: Although the proposed heuristic solutions along this

thesis have polynomial-time worst-case computational complexity we believe that the

complexity could be reduced. In the MU MIMO and SC-FDMA uplink scenarios the

complexity order of the problem could be reduced if we restricted the feasible SDMA

groups in the MU MIMO case, and assignment contiguous patterns in the SC-FDMA

uplink. The open question in this case is whether the possible loss in performance is

worth of considering.

Appendix A

Pseudo Code and Computational

Complexity of the Algorithms in

Chapters 3 and 4

In this appendix we provide the worst-case computational complexity of the optimal and

heuristic solutions proposed in Chapters 3 and 4. In sections A.1 and A.2 we show the

complexity of the optimal and heuristic solutions, respectively.

The computational complexity considered in all appendices of this thesis is the worst-case

one that gives an upper bound on the computational resources required by an algorithm

and is represented by the asymptotic notation O (·). As in [25,26], we consider summations,

multiplications and comparisons as the most relevant and time-consuming operations.

A.1 Complexity of Optimal Solution

The optimal solution to the problem studied in Chapter 3 can be obtained by using the BB

algorithm. For an arbitrary number of integer variables l, the number of linear programming

subproblems to be solved is at least(√

2)l

[71]. Meanwhile, the number of iterations needed

to solve one linear programming problem with m constraints and l variables is approximately

2 (m+ l), and each iteration encompasses (lm−m) multiplications, (lm−m) summations, and

(l −m) comparisons [25,71]. As in problem (3.6) there are JN+J integer variables and J+N+S

constraints, the required total number of operations is

√2(JN+J)

2 (JN + 2J +N + S) (2 (JN + J) (J +N + S)− 3 (J +N + S) + JN + J) .

Retaining the term of higher order we have that the worst-case computational complexity

is O(2JN

).

A.2 Algorithm and Complexity of Proposed Heuristic Solution

In Algorithms A.1, A.2 and A.3, we show a pseudo-code of the proposed algorithm for the

SU case presented in Chapters 3 and 4. In Table A.1 we present the definition of the variables

used in those algorithms. It is worth to mention that the purpose of the description of the

proposed algorithm in this section is only the calculation of its computational complexity.

For a more complete description of the proposed algorithm the interested reader should see

section 3.4 of Chapter 3 of this thesis.

A.2. Algorithm and Complexity of Proposed Heuristic Solution 93

Table A.1: Description of the main parameters used in Algorithms A.1, A.2 and A.3 for the SISO and SUMIMO cases.

Variable Definition

J Set with the index of all flows

N Set with the index of all RBs

S Set with the index of all services

Js Set with the index of the flows from service s

R Set with the index of receiver flows (unsatisfied)

B Auxiliary flow set with the flows that can be disregarded in the

first part of the proposed algorithm

A Set with the index of the flows that were not disregarded in the

first part of the proposed algorithm (Available flow set)

As Set with the index of the flows that were not disregarded in the

first part of the proposed algorithm and belong to service s

K Set with the index of available RBs for reallocation

Nj Set of assigned RBs to flow j

rSU DLj,n Transmit data rate of flow j on the RB n

ϕj,n Normalized transmit data rate of flow j on RB n

tj Required data rate of flow j

ks Required minimum number of satisfied flows of service s

an Store the index of the flow assigned to RB n

tj Current allocated data rate to the flow j

ωs Current number of satisfied flows of service s

βs Number of flows from service s that were not disregarded

γs Flag used to indicate if a flow from service s can be disregarded

λj Used to identify the service of the flow j

rj Average transmit data rate of the flow j

fj Ratio between average transmit data rate and required data rate of flow j

q with elements qi Contain the ith flow with lowest value of fj

l with elements li Contain the ith flow with lowest value of rj

c1, c2, Auxiliary variables

flag1, flag2, flag3 and aux1

Algorithm A.1 Initialization for the SISO and SU MIMO cases.1: βs ← |Js| ∀s ∈ S2: ωs ← 0 ∀s ∈ S3: ϕj,n ← 0 ∀j ∈ J ∀n ∈ N4: γs ← 0 ∀s ∈ S

5: rj ←

∑n∈N rSU DL

j,n

N∀j ∈ J

6: fj ← rj/tj ∀j ∈ J7: Sort fj in ascending order and store the sorted indices in the vector q composed of the elements qi8: Sort rj in ascending order and store the sorted indices in the vector l composed of the elements li9: A ← J10: As ← Js ∀s ∈ S11: an ← 0 ∀n ∈ N

We begin with the Algorithm A.1 where we present some steps necessary for the

initialization of the proposed algorithm. We assume that the transmit data rate rSU DLj,n , the

mapping between flow and service λj, ks and the sets J , N , S, and Js, ∀s ∈ S, and their

respective sizes are already known before initialization. In line 5 we have the computation of

the average transmit data rate for all flows. In this operation we have J (N − 1) summations

and J multiplications. In line 6 we calculate the ratio between the average transmit data rate

and the data rate requirement which leads to J multiplications. The operations in lines 7 and

8 in Algorithm A.1 are operations of sorting the elements of a vector. An algorithm that could

be used is the MergeSort that according to [72] has worst-case complexity O (n lnn) where n is

the length of the vector.

Let’s focus on the first part of proposed solution shown in Algorithm A.2. Depending on the

system state such as channel conditions, data rate requirements and satisfaction constraints,

a feasible solution can already be found in the first part of the proposed solution. However,


as we are considering the worst-case complexity we will not consider this hypothesis. In

the worst case condition, the main loop of Algorithm A.2 will iterate J −∑s∈S ks + 1 times

and after the first part of the algorithm we have only one donor and∑

s∈S ks − 1 receiver

flows. According to this we have that the operation of searching the maximum in line 4 takes(J −∑s∈S ks + 1

) (NJ +N

∑s∈S ks − 2N

)

2comparisons. In line 6 we have N summations per

iteration in the main loop which lead to N(J −∑s∈S ks + 1

)operations. The if -sentence in

line 10 requires

(J −∑s∈S ks + 1

) (J +

∑s∈S ks

)

2comparisons. J −∑s∈S ks +1 summations are

needed in line 12. The number of comparisons in lines 20, 28 and 31 is S(J −∑s∈S ks + 1

)

each. The comparisons in lines 35 and 37 are repeated J −∑s∈S ks + 1 times. In line 40 we

have S(J −∑s∈S ks

)comparisons since in the last iteration this operation is not performed.

The number of comparisons in line 45 is

(J −∑s∈S ks

) (J − 1 +

∑s∈S ks

)

2. The number of

summations in line 48 is J −∑s∈S ks. Finally, in the worst case scenario the comparison in

line 50 will be performed only once in the last iteration.


Algorithm A.2 First part of the proposed solution (Unconstrained Maximization) for the SISOand SU MIMO cases.1: loop2: K ← N R ← ∅ tj ← 0 ∀j ∈ A Nj ← ∅ ∀j ∈ A3: for all n ∈ N do

4: j∗ ← argmaxj∈A

(rSUDLj,n

)

5: an ← j∗

6: tj∗ ← tj∗ + rSUDLj∗,n

7: Nj∗ ← Nj∗ ∪ {n}8: end for9: for all j ∈ A do10: if tj ≥ tj then11: s∗ ← λj

12: ωs∗ ← ωs∗ + 113: else14: R ← R∪ {j}

15: K ← K \ Nj

16: end if17: end for18: flag1 = 0 {Auxiliary variable that assumes the value 1 whereas flows can still be disregarded without infringing

problem constraints and 0 otherwise}19: for all s ∈ S do20: if βs > ks then21: γs ← 122: flag1 ← 123: end if24: end for25: flag2 ← 1 {Auxiliary variable that assumes the value 1 when the problem was solved and zero otherwise}26: flag3 ← 0 {Auxiliary variable that assumes the value 1 when at least one flow is satisfied and 0 otherwise}27: for all s ∈ S do28: if ωs < ks then29: flag2 ← 030: end if31: if ωs 6= 0 then32: flag3 ← 133: end if34: end for35: if flag2 = 1 then36: OPTIMAL SOLUTION FOUND37: else if flag1 = 1 then38: B ← A39: for all s ∈ S do40: if γs = 0 then41: B ← B \ As

42: end if43: ωs ← 0 γs ← 0 ∀s ∈ S44: end for45: j∗ ← argmin

j∈B(fj)

46: s∗ ← λj∗

47: As∗ ← As∗ \ {j∗}

48: βs∗ ← βs∗ − 149: A ← A \ {j∗}50: else if flag3 = 0 then51: NO FEASIBLE SOLUTION FOUND52: else53: BREAK (GO TO SECOND PART)54: end if55: end loop


In the second part of the proposed algorithm there are two main loops. The first loop is

necessary to calculate the normalized channel gain of all receivers or unsatisfied flows. As

commented before, the worst-case scenario is the one in which, after executing the first part of

the proposed algorithm, there is only one donor and∑

s∈S ks−1 receiver flows. Moreover, there

are N available RBs and the N − 1 RBs are reassigned to the receivers since the donor flow

should get at least one RB. The second loop iterates N − 1 times in the worst-case condition.

Algorithm A.3 Second part of the proposed solution (Reallocation) for the SISO and SU MIMOcases.1: c1 ← 0 {Auxiliary variable used to check when there is no available RB for reallocation}2: for all n ∈ K do3: c1 ← c1 + 14: for all j ∈ R do5: j⋆ ← an6: ϕj,n ← rSU DL

j,n /rSU DLj⋆,n

7: end for8: end for9: Take out the indices of the vector l that are not part of the set R10: c2 ← 1 {Auxiliary variable to control if all the receiver flows were satisfied in the reallocation part}11: aux1 ←

∑s∈S ks {Auxiliary variable with the number of flows in the receiver flow set}

12: loop13: j∗ ← lc214: n∗ ← argmax

n∈K

(ϕj∗,n

)

15: j+ ← an∗

16: if tj+ − rSU DLj+,n∗ ≥ tj+ then

17: an∗ ← j∗

18: tj∗ ← tj∗ + rSU DLj∗,n∗

19: tj+ ← tj+ − rSUDLj+,n∗

20: if tj∗ ≥ tj∗ then21: if c2 = aux1 then22: BREAK (FEASIBLE SOLUTION FOUND)23: else24: c2 ← c2 + 125: end if26: end if27: end if28: K ← K \ {n∗}29: c1 ← c1 − 130: if c1 = 0 then31: BREAK (NO FEASIBLE SOLUTION FOUND)32: end if33: end loop

In the following we present the number of operations in the second part of the proposed

solution shown in Algorithm A.3. In lines 3 and 6 we have N summations and N(∑

s∈S ks − 1)

multiplications, respectively. In order to perform the operation in line 9, J(∑

s∈S ks − 1)

comparisons are necessary. There are S summations in line 11. Within the second loop

of Algorithm A.3, in line 14 we have an operation of searching the maximum of a vector that

needsN (N − 1)

2comparisons. The if -sentence in line 16 requires N − 1 comparisons and

N − 1 summations. There are at most N − 1 summations in lines 18 and 19 (each), and N − 1

comparisons in line 20. The condition in the if -sentence between lines 16 and 26 will be true

at most∑

s∈S ks − 1 times. Consequently, we have∑

s∈S ks − 1 comparisons in line 21 and∑

s∈S ks summations in line 24. Finally, in lines 29 and 30 there are N − 1 summations and

comparisons, respectively.


In summary, the number of operations in the initialization of Algorithm A.1 is

JN + 2J ln J + 3J.

Assuming that θ1 = J −∑s∈S ks and θ2 = J +∑

s∈S ks, the total number of operations in

Algorithm A.2 is

(θ1θ2N

2

)+ (θ1θ2 + θ2N + 4Sθ1) +

(θ22

+7θ12

+ 3S

)+ 4.

Finally, the number of operations in the second part of proposed solution represented in

Algorithm A.3 is

(N2

2+ J

∑

s∈S

ks +N∑

s∈S

ks

)+

(13N

2+ 2

∑

s∈S

ks − J + S

)− 10.

The most significant term in the number of operations is the term of third order

found in the first part of the proposed algorithm:N(∑

s∈S ks − 1) (∑

s∈S ks + 1)

2. Therefore,

the complexity of the proposed solution in the SISO and SU MIMO scenario is

O(N(J −∑s∈S ks

) (J +

∑s∈S ks

)). It is worth to mention that this complexity could be reduced

since the maximum rate allocation performed by the for-loop between lines 3 and 8 of the

Algorithm A.2 need not to be done completely every iteration. In fact the resource assignment

changes only for the resources that were previously assigned to the disregarded flows.

Appendix B



Chapter 5


heuristic solutions proposed in Chapter 5. In sections B.1 and B.2 we show the complexity of

the optimal and heuristic solutions, respectively.

B.1 Complexity of Optimal Solution

The optimal solution to the problem studied in Chapter 5 can be obtained by using the

BB algorithm. In problem (5.11) there are GN + J variables and J + N + S constraints. The

required total number of operations is

√2(GN+J)

2 (GN + 2J +N + S) (2 (GN + J) (J +N + S)− 3 (J +N + S) +GN + J) .

Retaining the term of higher order we have that the worst-case computational complexity is

O(2GN

).

B.2 Algorithm and Complexity of Proposed Heuristic Solution

In Algorithms B.1, B.2, B.3, B.4, B.5 and B.6 we show a pseudo-code of the proposed

algorithm for the MU case presented in Chapter 5. In Table B.1 we present the definition of the

variables used in those algorithms. It is worth to mention that the purpose of the description of

the proposed algorithm in this section is only the calculation of its computational complexity.


section 5.5 of Chapter 5.

In Algorithm B.1 we show the initialization of the proposed algorithm. We assume that the

transmit data rate rMU DLg,j,n , the mapping between flow and service λj, og,n, ks and the sets J ,

N , S, G, Jg and Js, ∀s ∈ S, and their respective sizes are already known before initialization.

In line 5 we have the computation of the average transmit data rate for all flows. In this

operation we have J (GN − 1) summations and J+1 multiplications. In line 6 we calculate the

ratio between the average transmit data rate and the data rate requirement which leads to J

multiplications. The operations in lines 7 and 8 in Algorithm B.1 are operations of sorting the

elements of a vector that, as pointed out in Appendix A, have worst-case complexity O (n lnn)

B.2. Algorithm and Complexity of Proposed Heuristic Solution 99

Table B.1: Description of the main parameters used in Algorithms B.1, B.2, B.3, B.4, B.5 and B.6 forthe MU MIMO case.

Variable Definition





R Set with the index of receiver flows (unsatisfied) after the

first part of the proposed solution







D Set with the index of donor flows (satisfied)

K Set with the index of available RBs for reallocation

G Set with the index of all SDMA groups


R Set with the index of receiver flows (unsatisfied) along the second part of the proposed

solution. This set is updated along the second part of our proposed solution

D Set with the index of receiver flows that have become

satisfied along the second part of our proposed solution

G Set with the index of SDMA groups that are candidate

to be assigned in the reallocation process

K Set with the index of the RBs that are assigned to

the selected flow and are not assigned to the other receiver flows

Jg Set with the index of the flows that compose the gth SDMA group

U Set with the index of donor flows that are candidate to lose

a specific RB in the reallocation process

T Set with the index of the receiver flows that have become

satisfied with a given reallocation

G Set with the index of all SDMA groups that contains the flows that were not disregarded

rMU DLg,j,n Transmit rate of flow j when the SDMA group g is assigned to RB n

og,j Variable that assumes the value 1 if flow j belongs to the SDMA group g


tj Current allocated rate to flow j

ttempj

Temporary variable that stores the possible current allocated rate to flow j






ϕi Metric associated with the ith SDMA group / RB

for reallocation


τ1i Variable that stores the SDMA group of the ith pair SDMA group/RB for reallocation

τ2i Variable that stores the RB of the ith pair SDMA group/RB for reallocation

rj Average transmit data rate of the flow j


N Total number of RBs

G Total number of SDMA groups

q with elements qi Contain the ith flow with lowest value of fj


c1, c2, c3, Auxiliary variables

flag1, flag2, flag3,



flag10, flag11,

aux1 and aux2

where n is the length of the vector.

Now we present the number of operations of Algorithm B.2 that corresponds to the first

part of the proposed solution. Although we can get a feasible solution in the first part of the


Algorithm B.1 Initialization for the MU MIMO case.1: βs ← |Js| ∀s ∈ S2: ωs ← 0 ∀s ∈ S3: ϕj,n ← 0 ∀j ∈ J ∀n ∈ N4: γs ← 0 ∀s ∈ S

5: rj ←

∑g∈G

∑n∈N rMU DL

g,j,n

N ·G∀j ∈ J

6: fj ← rj/tj ∀j ∈ J7: Sort in the ascending order fj and store the sorted indices in the vector q composed of the elements qi8: Sort in the ascending order rj and store the sorted indices in the vector l composed of the elements li9: A ← J10: As ← Js ∀s ∈ S11: ǫ← 10−6

12: G ← G

proposed solution, in the worst case condition, the main loop of Algorithm B.2 will iterate J −∑

s∈S ks+1 times and after the first part of the algorithm we have only one donor and∑

s∈S ks−1

receiver flows. According to this, the operation of searching the maximum in line 4 takes

N (G− 1)(J −∑s∈S ks + 1

)comparisons and

GN(J −∑s∈S ks

) (J +

∑s∈S ks − 2

)

2summations.

Note that in order to ease the calculation we considered the pessimistic assumption that along

the iterations, the set G remains equal to G. In lines 7 and 8 we have MTN(J −∑s∈S ks + 1

)

summations (each) assuming that all SDMA groups have MT flows. The if -sentence in line

13 requires

(J −∑s∈S ks + 1

) (J +

∑s∈S ks

)

2comparisons. J − ∑s∈S ks + 1 summations are

needed in line 16. The number of comparisons in lines 24, 32 and 35 is S(J −∑s∈S ks + 1

)

each. The comparisons in lines 39 and 41 are repeated J −∑s∈S ks + 1 times. In line 44 we

have S(J −∑s∈S ks

)comparisons since in the last iteration this operation is not performed.

The number of comparisons in line 49 is

(J −∑s∈S ks

) (J − 1 +

∑s∈S ks

)

2. The number of

summations in line 52 is J −∑s∈S ks. The if -sentence in line 55 requires GMT

(J −∑s∈S ks

)

comparisons. Finally, in the worst case scenario the comparison in line 59 will be performed

only once in the last iteration.

As in SU case, the worst case complexity for the proposed solution to the MU scenario

assumes that after running the first part of the solution there are∑

s∈S ks − 1 receiver flows

and only one donor flow. All the RBs are assigned to the donor flow and they will be

spatially shared with the receiver flows. Therefore, the loop between lines 8 and 24 iterates

at most∑

s∈S ks − 1. The loop between lines 10 and 14 iterates N times corresponding to the

reallocations. Note that we assume that in the last iteration of this loop the donor spatially

shares its unique RB with a receiver flow.

In the following we present the number of operations of the Algorithm B.3. In line 1 we

have S summations. N summations are presented in line 3. The operation of searching for

the minimum in a vector in line 9 has

(∑s∈S ks − 2

) (∑s∈S ks − 2

)

2. In lines 15, 18 and 21 we

have∑

s∈S ks − 1 comparisons each.

The number of operations in Algorithm B.4 is presented in the following. As we consider

that in the worst-case scenario the receiver flows do not have any assigned RB in the

first part of the proposed solution, the for-loop between lines 2 and 13 is not executed.

In line 17 there are N∑MT

m=1

(∑s∈S ks

m

)comparisons. The for-loop in line 19 selects the

receiver flows that have become satisfied along the reallocation process. In order to

simplify the calculation we consider that the number of flows that is selected in this loop

is equal to∑

s∈S ks − 2. Note that this is a pessimistic estimative since this is true only

in the last iterations of the reallocation process. The number of comparisons in line 20

is N

∑MT

m=1

(∑s∈S ks

m

)∑

s∈S ks

(∑s∈S ks − 2

). The comparison in line 25 is executed N

∑MT

m=1

(∑s∈S ks

m

)∑

s∈S ks.


Algorithm B.2 First part of the proposed solution (Unconstrained Maximization) for the MUMIMO case.1: loop2: K ← N R ← ∅ tj ← 0 ∀j ∈ A Nj ← ∅ ∀j ∈ A Φ← 0 D ← ∅3: for all n ∈ N do

4: g∗ ← argmaxg∈G

(∑j∈A rMU DL

g,j,n

)

5: an ← g∗

6: for all j ∈ Jg∗ do

7: Φ← Φ+ rMU DLg∗,j,n

8: tj ← tj + rMU DLg∗,j,n

9: Nj ← Nj ∪ {n}10: end for11: end for12: for all j ∈ A do13: if tj ≥ tj then14: s∗ ← λj

15: D ← D ∪ {j}16: ωs∗ ← ωs∗ + 117: else18: R ← R∪ {j}

19: K ← K \ Nj

20: end if21: end for22: flag1 = 0 {Auxiliary variable that assumes the value 1 whereas flows can still be disregarded without infringing

problem constraints}23: for all s ∈ S do24: if βs > ks then25: γs ← 126: flag1 ← 127: end if28: end for29: flag2 ← 1 {Auxiliary variable that assumes the value 1 when the problem was solved}30: flag3 ← 0 {Auxiliary variable that assumes the value 1 when at least one flow is satisfied}31: for all s ∈ S do32: if ωs < ks then33: flag2 ← 034: end if35: if ωs 6= 0 then36: flag3 ← 137: end if38: end for39: if flag2 = 1 then40: OPTIMAL SOLUTION FOUND41: else if flag1 = 1 then42: B ← A43: for all s ∈ S do44: if γs = 0 then45: B ← B \ As

46: end if47: ωs ← 0 λs ← 0 ∀s ∈ S48: end for49: j∗ ← argmin

j∈B(fj)

50: s∗ ← λj∗

51: As∗ ← As∗ \ {j∗}52: βs∗ ← βs∗ − 153: A ← A \ {j∗}

54: for all g ∈ G do55: if j∗ ∈ Jg then

56: G← G \ {g}57: end if58: end for59: else if flag3 = 0 then60: NO FEASIBLE SOLUTION FOUND61: else62: BREAK (GO TO PART 2)63: end if64: end loop

The for-loops in lines 32 and 33 are repeated different times at each iterations of the

reallocation cycle. However, for the sake of simplicity we consider a pessimistic estimative


Algorithm B.3 Second part of the proposed solution (Reallocation) for the MU MIMO case.1: aux1 ←

∑s∈S ks {Auxiliary variable with the initial number of flows in the receiver flow set}

2: c1 ← 0 {Tracks the number of receiver flows that have become satisfied}3: c2 ← |K| {Tracks the number of available RBs for reallocation}4: flag10 ← 1 {Auxiliary variable that assumes the value 1 if there is no more RBs for reallocation and 0 otherwise}5: flag11 ← 1 {Auxiliary variable that assumes the value 0 if all receiver flows have become satisfied and 1 otherwise}

6: R ← R7: D ← ∅8: loop9: j∗ ← arg min

j∈R(lj)

10: loop11: PROCEDURE 112: PROCEDURE 213: PROCEDURE 314: end loop15: if flag6 = 1 then16: BREAK (NO FEASIBLE SOLUTION FOUND)17: end if18: if flag10 = 0 then19: BREAK (NO FEASIBLE SOLUTION FOUND)20: end if21: if flag11 = 0 then22: BREAK (FEASIBLE SOLUTION FOUND)23: end if24: end loop

that the number of SDMA groups in G and available RBs in K are

∑MT

m=1

(∑s∈S ks

m

)∑

s∈S ksand N ,

respectively. Therefore, the number of summations in lines 39 and 40 is N2J

∑MT

m=1

(∑s∈S ks

m

)∑

s∈S kseach. In order to compute the number of operations in line 42 we assume that the

number of receiver flows in set R is∑

s∈S ks − 1. In fact, the number of flows in this

set is equal to this value only in the beginning. Along the reallocation iterations the

number of flows in this set decreases. According to these considerations the number of

summations, comparisons and multiplications in line 42 are 4N2(∑

s∈S ks − 1) ∑MT

m=1

(∑s∈S ks

m

)∑

s∈S ks,

2N2(∑

s∈S ks − 1) ∑MT

m=1

(∑s∈S ks

m

)∑

s∈S ksand 3N2

(∑s∈S ks − 1

) ∑MT

m=1

(∑s∈S ks

m

)∑

s∈S ks, respectively. The last

operation in Algorithm B.4 is the one in line 43 that requires N2

∑MT

m=1

(∑s∈S ks

m

)∑

s∈S ks.

In Algorithm B.5 the loop in line 2 is executed only once per reallocation iteration since all

the RBs in the beginning of the reallocation process is assigned to only one (donor) flow. In

line 3 there is N

(N

∑MT

m=1

(∑s∈S ks

m

)∑

s∈S ks− 1

)comparisons. The number of comparisons in lines 7

and 10 is N each. The if -sentence in line 16 requires NMT comparisons. 2N summations are

executed in line 23. Finally, in lines 24, 30 and 34 there are N comparisons each.

The number of operations concerning the Algorithm B.6 is presented in the following. In

lines 2, 3, 10 and 11 we have NMT summation each. NMT

(∑s∈S ks − 1

)comparisons are

performed in line 13. We assume that the set T has MT receiver flows in the first N − 1

reallocation iterations and MT − 1 receiver flows in the last iteration since in the last iteration

the unique RB is shared with the donor flow. Consequently, in line 20 we have (N − 1)MT +

MT − 1 comparisons. In lines 23, 24 and 27 we have∑

s∈S ks − 1 operations. Finally, there are

N operations in each line 34, 35 and 39.

In summary, the number of operations in the initialization of Algorithm B.1 is

GNJ + 2J ln J + 3J + 1.


Algorithm B.4 Procedure 1 that is part of the second part of the proposed solution(Reallocation) for the MU MIMO case.1: K ← ∅2: for all n ∈ Nj∗ do3: flag4 ← 0 {Auxiliary variable that assumes the value 0 if a given RB can be considered for reallocation and 1

otherwise}4: for all j ∈ (R \ {j∗}) do

5: if n ∈ Nj then6: flag4 ← 17: BREAK8: end if9: end for10: if flag4 = 0 then11: K ← K ∪ {n}12: end if13: end for14: K ← K ∪ K15: G ← ∅16: for all g ∈ G do17: if og,j∗ = 1 then18: flag5 ← 0 {Auxiliary variable that assumes the value 0 if a given SDMA group can be considered as a

candidate for reallocation and 1 otherwise}19: for all j ∈ D do20: if og,j = 1 then21: flag5 ← 122: BREAK23: end if24: end for25: if flag5 = 0 then26: G ← G ∪ {g}27: end if28: end if29: end for30: c3 ← 131: ϕi ← 0 τ1i ← 0 τ2i ← 0 ∀i

32: for all g ∈ G do33: for all n ∈ K do34: τ1c3 ← g

35: τ2c3 ← n36: aux2 ← Φ37: g′ ← an38: for all j ∈ (R∪D) do39: aux2 ← aux2 − rMU DL

g′,j,n

40: aux2 ← aux2 + rMU DLg,j,n

41: end for

42: ϕc3 ←

∑j∈R

min(tj − tj + rMU DL

g,j,n − rMU DLg′,j,n

, ǫ)· Φ

∑j∈R

min(tj − tj , ǫ

)· aux2

43: c3 ← c3 + 144: end for45: end for


s∈S ks, the total number of operations in

Algorithm B.2 is

(GNθ1θ2

2

)+

(2MTNθ1 +GMT θ1 +GN −Nθ1 + 2MTN + θ1θ2 + 4Sθ1) +(θ22

−N +7θ12

+ 3S

)+ 4.

Finally, the number of operations in the second part of proposed solution represented

by Algorithms B.3, B.4, B.5 and B.6 is shown in the following. In order to simplify the

presentation we define θ3 =G′

∑s∈S ks

=

∑MT

m=1

(∑s∈S ks

m

)∑

s∈S kswhere G′ is the number of SDMA group


Algorithm B.5 Procedure 2 that is part of the second part of the proposed solution(Reallocation) for the MU MIMO case.1: flag6 ← 0 {Auxiliary variable that assumes the value 1 if there is no more pairs SDMA group / RB for reallocation}

2: loop3: i∗ ← argmin

∀i(ϕi)

4: g+ ← τ1i∗

5: n+ ← τ2i∗6: g⋆ ← an+

7: if ϕi∗ = inf then8: flag6 ← 19: BREAK10: else if g⋆ = g+ then11: ϕi∗ ← inf12: CONTINUE13: end if14: U ← ∅15: for all j ∈ Jg⋆ do16: if j ∈ D then17: U ← U ∪ {j}18: end if19: end for20: ttemp

j ← tj ∀j ∈ J

21: flag7 ← 0 {Auxiliary variable that assumes the value 1 if at least one donor will become unsatisfied if thecurrent reallocation is performed and 0 otherwise}

22: for all j ∈ U do23: ttemp

j ← ttempj − rMUDL

g⋆,j,n+ + rMU DLg+,j,n+

24: if(ttempj < tj

)then

25: ϕi∗ ← inf26: flag7 ← 127: BREAK28: end if29: end for30: if flag7 = 0 then31: BREAK32: end if33: end loop34: if flag6 = 1 then35: BREAK36: end if

that can be composed with MT transmit antennas and∑

s∈S ks flows.

(2Jθ3N

2 + 9N2

(∑

s∈S

ks − 1

)θ3

)+

(2Nθ3

∑

s∈S

ks + 2N2θ3 +NMT

∑

s∈S

ks

)+

(5NMT −Nθ3 +

(∑s∈S ks

)2

2

)+

(9∑

s∈S ks

2+ 10N + S

)− 6.

As J ≥ ∑s∈S ks, the most significant term in the number of operations is the

term of quarter order found in the first and second parts of the proposed algorithm:GN

(∑s∈S ks − 1

) (∑s∈S ks + 1

)

2+ 2J

G′

∑s∈S ks

N2.

Therefore, the complexity of the proposed solution is O(GN

(∑s∈S ks

)2+ 2J

G′

∑s∈S ks

N2

).

As in the SU case, this complexity could be reduced since the maximum rate allocation

performed by the for-loop between lines 3 and 11 of the Algorithm B.2 need not to be done

completely every iteration. In fact the resource assignment changes only for the resources

that were previously assigned to the disregarded flow.


Algorithm B.6 Procedure 3 that is part of the second part of the proposed solution(Reallocation) for the MU MIMO case.1: for all j ∈ Jg⋆ do

2: tj ← tj − rMU DLg⋆,j,n+

3: Φ← Φ− rMU DLg⋆,j,n+

4: Nj ← Nj \ {n+}5: end for6: an+ ← g+

7: T ← ∅8: flag8 ← 1 {Auxiliary variable that assumes the value 0 if at least one receiver has become satisfied and 0 otherwise}

9: for all j ∈ Jg+ do

10: tj ← tj + rMU DLg+,j,n+

11: Φ← Φ+ rMU DLg+,j,n+

12: Nj ← Nj ∪ {n+}

13: if j ∈ R then14: T ← T ∪ {j}15: flag8 ← 016: end if17: end for18: flag9 ← 1 {Auxiliary variable that assumes the value 0 if the selected receiver flow was satisfied with the

reallocation and 1 otherwise}19: for all j ∈ T do20: if

(tj − tj

)≥ 0 then

21: R ← R \ {j}

22: D ← D ∪ {j}23: c1 ← c1 + 124: if j = j∗ then25: flag9 ← 026: end if27: if c1 = aux1 then28: flag11 ← 029: BREAK30: end if31: end if32: end for33: K ← K \ {n+}34: c2 ← c2 − 135: if c2 = 0 then36: flag10 ← 037: BREAK38: end if39: if flag9 = 0 then40: BREAK41: end if

Appendix C



Chapter 6


heuristic solutions proposed on Chapter 6. In sections C.1 and C.2 we show the complexity

of the optimal and heuristic solutions of the URM problem, respectively. Furthermore, in

sections C.3 and C.4 we present the complexity of the optimal and heuristic solutions to the

CRM problem.

C.1 Complexity of Optimal Solution to the URM Problem

The optimal solution to the URM problem presented in (6.6) can be obtained by using the

BB algorithm. In that problem there are JP variables and J + N constraints. The required

total number of operations is

√2(JP )

2 (JP + J +N) (2 (JP ) (J +N)− 3 (J +N) + JP ) .


O(2JP

).

C.2 Algorithm and Complexity of Proposed Heuristic Solution to the URM

Problem

In Algorithms C.1, C.2, C.3 and C.4, we show a pseudo-code of the proposed algorithm

for the URM problem presented in Chapter 6. In Table C.1 we present the definition of the

variables used in those algorithms. It is worth to mention that the purpose of the description of

the proposed algorithm in this section is only the calculation of its computational complexity.


section 6.2.3 of Chapter 6 of this thesis.

We assume that rj,n, γULj,z,n and the sets J and N and their respective sizes are already

known before initialization. The worst-case situation of the proposed algorithm corresponds

to the case in which the maximum rate solution without adjacency constraint presented in

Algorithm C.2 returns an assignment in which N − 1 flows get only one RB and consequently

VR, and 1 flow gets two non-adjacent RBs or VRs, the first and the last RB. Note that this

C.2. Algorithm and Complexity of Proposed Heuristic Solution to the URM Problem 107

Table C.1: Description of the main parameters used in Algorithms C.1, C.2, C.3 and C.4 for the URMproblem in the uplink case.

Variable Definition




Nv Set of RBs that composes the vth Virtual Resource (VR)

Vj Set of VRs that belongs to flow j

Zi Set with the indices of the RBs that composes the ith VR combination

Zi Set with the indices of the VRs that composes the ith VR combination

γULj,z,n Signal to Noise Ratio (SNR) experienced by flow j on the zth subcarrier of the RB n

rj,n Transmit data rate of flow j when only the RB n is assigned to it

f (·) Function that returns the achieved data rate according to a specific effective SNR. See equation (6.8)

h (j, C) Function that returns the effective SNR when the set of RBs C is assigned to flow j. See equation (6.4)

tj Current allocated data rate to the flow j

νj Number of VRs that belongs to flow j


av Store the index of the flow assigned to RB v

τi Index of the flow that owns the ith VR combination

ϕi Metric associated with the ith VR combination

c Number of subcarriers that composes an RB

c1, c2, c3, Auxiliary variables

c4, c5, c6 and flag1

Algorithm C.1 Initialization for solution to the URM problem in the uplink case.

1: Nj ← ∅ ∀j ∈ J2: Vj ← ∅ ∀j ∈ J3: νj ← 0 ∀j ∈ J

4: Nv ← ∅ ∀v5: av ← 0 ∀v6: tj ← 0 ∀j ∈ J7: an ← 0 ∀n ∈ N

assignment does not comply with the adjacency constraint. Then the parts 2 and 3 presented

in Algorithms C.3 and C.4, respectively, are executed N − 2 times so that at each iteration a

new VR is composed based on the combination of two previous VRs. In this way the number

of VRs is decreased by a unit at each iteration. In the last iteration of the proposed solution

we have 1 flow with 1 VR and 1 flow with 2 VRs.

Note that the initialization of the proposed solution does not contain any summation,

multiplication or comparison. We begin with the Algorithm C.2 where we present the

maximum rate solution without adjacency constraints and where the composition of the VRs

is performed. In line 3 there are N (J − 1) comparisons. The number of comparisons in line 5

is N . In lines 6, 7 and 9 we have N − 1 operations. In line 13 we have only 1 summation and

in lines 20 and 24 we have J and 1 operations, respectively.

The function of the Algorithm C.3 is to combine the current VRs and calculate the

associated metric. Note that if we have V VRs the maximum number of combinations

is 2V − 3. The number of operations is presented in the following. In line 4 we have

J (N − 2) comparisons. The number of operations in lines 6, 7 and 8 is(N − 2) (N − 1)

2

each.(4c+ 1) (N − 2) (N − 1)

2is the number of summations and

(2c+ 10) (N − 2) (N − 1)

2the

number of multiplications in line 10. The number of operations in lines 11, 13, 14 and

15 is(N − 2) (N − 1)

2each. The number of summations and multiplications in line 17

are of(4c+ 1) (N − 2) (N − 1)

2and

(2c+ 10) (N − 2) (N − 1)

2, respectively. In line 18 we have

(N − 2) (N − 1)

2summations. In lines 20 and 23 we have J (N − 2) and 4 (N − 2) operations,


Algorithm C.2 Part 1 of the proposed solution for the URM problem in the uplink case.1: c1 ← 1 {Auxiliary variable that tracks the current number of VRs}2: for all n ∈ N do3: j∗ ← argmax

j∈J(rj,n)

4: Nj∗ ← Nj∗ ∪ {n}5: if n 6= 1 then6: if (n− 1) /∈ Nj∗ then7: c1 ← c1 + 18: Vj∗ ← Vj∗ ∪ {c1}9: νj∗ ← νj∗ + 110: end if11: else12: Vj∗ ← Vj∗ ∪ {c1}13: νj∗ ← νj∗ + 114: ac1 ← j∗

15: end if16: Nc1 ← Nc1 ∪ {n}17: end for18: flag1 ← 1 {Auxiliary variable that assumes the value 1 if the adjacency constraint is fulfilled and 0 otherwise}19: for all j ∈ J do20: if νj > 1 then21: flag1 ← 022: end if23: end for24: if flag1 = 1 then25: for all j ∈ J do

26: tj ← f(h(j, Nj

))

27: for all n ∈ Nj do28: an ← j29: end for30: end for31: FEASIBLE SOLUTION FOUND32: end if

respectively. The number of summations in lines 24 and 25 is (N − 2) each. The calculation of

the metric in line 27 requires (4c+ 1) (N − 2) summations and (2c+ 10) (N − 2) multiplications.

(N − 2) summations and 4 (N − 2) comparisons are performed in lines 28 and 29, respectively.

In lines 30 and 31 there are (N − 2) summations each. In line 33 there are (4c+ 1) (N − 2)

summations and (2c+ 10) (N − 2) multiplications. In lines 34 and 36 we have (N − 2)

and 2 (N − 2) operations, respectively.(N − 2) (N + 3)

2and

(N − 2) (N + 1)

2operations are

performed in lines 41 and 42, respectively. In lines 43 and 44 we have(N − 2) (N + 1)

2operations each. In the calculation of the metric in line 47 there are (N − 2) (2cN + 1)

summations and (N − 2) (cN + 10) multiplications. Finally, the number of operations in lines

48 and 50 are (N − 2) and 2 (N − 2), respectively.

The part 3 presented in Algorithm C.4 is executed N − 2 times until finding a feasible

solution in the worst case. The number of comparisons in line 1 is(N − 2) (2N − 2)

2. In line 2

we have 3 (N − 2) summations. The number of operations in lines 5 and 6 are(N − 2) (N + 5)

2

and(N − 2) (N + 3)

2, respectively. In line 8 there are (N − 2) comparisons whereas in line 13

we have 2 (N − 2) summations. In lines 14, 15 and 16 we have (N − 2) summations each. The

test in line 18 is executed(N − 2) (N + 1)

2times. In lines 19 and 22 we have

3 (N − 2) (N − 1)

2

summations each. The number of summations in lines 31 and 32 is(N − 2) (N + 3)

2each. The

calculation of the cardinality of the set in line 37 requires(N − 2) (N + 1)

2summations. In line

38 there areN (N − 3)

2+ 2 summations. J and N − 2 are the number of operations in lines 43

and 47, respectively. Finally, assuming that in the last iteration we have only two flows with

assigned resources, the number of summations and multiplications in line 49 are 2cN +2 and


Algorithm C.3 Part 2 of the proposed solution for the URM problem in the uplink case.

1: Zi ← ∅Zi ← ∅ τi ← 0ϕi ← 0 ∀i2: c2 ← 1 {Auxiliary variable used to control the indices of the new VR combinations}3: for all j ∈ J do4: if νj = 1 then5: v′ ← Vj6: if v′ 6= 1 then7: Zc2 ← Nv′−1 ∪ Nv′

8: Zc2 ← {v′ − 1, v′}

9: τc2 ← j10: ϕc2 ← f (h (j,Zc2))11: c2 ← c2 + 112: end if13: if v′ 6= c1 then14: Zc2 ← Nv′ ∪ Nv′+1

15: Zc2 ← {v′, v′ + 1}

16: τc2 ← j17: ϕc2 ← f (h (j,Zc2))18: c2 ← c2 + 119: end if20: else if νj > 1 then21: c3 ← 1 {Auxiliary variable that is used to check if a given VR is the leftmost or rightmost VR of a flow}22: for all v ∈ Vj do23: if (c3 = 1) and (v 6= 1) then

24: Zc2 ← Nv−1 ∪ Nv

25: Zc2 ← {v − 1, v}26: τc2 ← j27: ϕc2 ← f (h (j,Zc2))28: c2 ← c2 + 129: else if (c3 = νj) and (v 6= c1) then

30: Zc2 ← Nv ∪ Nv+1

31: Zc2 ← {v, v + 1}32: τc2 ← j33: ϕc2 ← f (h (j,Zc2))34: c2 ← c2 + 135: end if36: if c3 = 1 then37: v⋆ ← v38: else39: Zc2 ← Nv⋆

40: Zc2 ← {v⋆}

41: while v⋆ < v do42: Zc2 ← Zc2 ∪ Nv⋆+1

43: Zc2 ← Zc2 ∪ {v⋆ + 1}

44: v⋆ ← v⋆ + 145: end while46: τc2 ← j47: ϕc2 ← f (h (j,Zc2))48: c2 ← c2 + 149: end if50: c3 ← c3 + 151: end for52: end if53: end for

cN + 20, respectively.

In summary, the number of operations in Algorithm C.2 is

(NJ) + (3N + J) + (−1) .

Assuming c = 12 hereafter we have that the total number of operations in Algorithm C.3 is

(2JN + 125N2

)+ (−138N − 4J) + (−224) .


Algorithm C.4 Part 3 of the proposed solution for the URM problem in the uplink case.1: i∗ ← argmax

∀i(ϕi)

2: c4 ← c1−|Zi∗ |+1 {Auxiliary variable with the current number of VR after the selection of the best VR combination}

3: c5 ← 1 {Auxiliary variable used in the update of most of the variables related to resource assignment}4: c6 ← First element of Zi∗ {Auxiliary variable used in the update of most of the variables related to resource

assignment}5: while c6 ≤ c1 do6: if c5 = 1 then7: j⋆ ← ac68: if j⋆ 6= τi∗ then9: Vτi∗ ← Vτi∗ ∪ {c6}10: Vj⋆ ← Vj⋆ \ {c6}11: ac6 ← τi∗12: end if13: for all v ∈ {1, · · · , |Zi∗ | − 1} do

14: Nc6 ← Nc6 ∪ Nc6+v

15: j⋆ ← ac6+v

16: Vj⋆ ← Vj⋆ \ {c6 + v}17: end for18: else if c6 ≤ c4 then19: Nc6 ← Nc6+|Zi∗ |−1

20: j+ ← ac621: Vj+ ← Vj+ \ {c6}

22: j† ← ac6+|Zi∗ |−1

23: Vj† ← Vj† ∪ {c6}

24: ac6 ← j†

25: else26: Nc6 ← {∅}27: j⋆ ← ac628: Vj⋆ ← Vj⋆ \ {c6}29: ac6 ← 030: end if31: c6 ← c6 + 132: c5 ← c5 + 133: end while34: flag1 ← 135: c1 ← 036: for all j ∈ J do37: νj ← |Vj |38: c1 ← c1 + νj39: Nj ← ∅ ∀j ∈ J40: for all v ∈ Vj do

41: Nj ← Nj ∪ Nv

42: end for43: if νj > 1 then44: flag1 ← 045: end if46: end for47: if flag1 = 1 then48: for all j ∈ J do

49: tj ← f(h(j, Nj

))

50: for all n ∈ Nj do51: an ← j52: end for53: end for54: FEASIBLE SOLUTION FOUND55: else56: GO TO PART 257: end if

Finally, the number of operations in Algorithm C.4 is

(15N2

2

)+

(69

2N + J

).

The most significant term in the number of operations are the terms of second order

presented in the following: 3JN +265

2N2.

C.3. Complexity of Optimal Solution to the CRM Problem 111

Therefore, the complexity of the proposed algorithm is O(3JN +

265

2N2

). It is important

to highlight that the average computational complexity is lower when the algorithm is applied

in the wireless environment. In this scenario, due to the near-far effect, some flows get most

of the RBs in the initial phase of the algorithm decreasing the total number of iterations to get

a feasible solution.

C.3 Complexity of Optimal Solution to the CRM Problem

The optimal solution to the CRM problem presented in (6.14) can be obtained by using the

BB algorithm. In that problem there are JP + J variables and 2J + N + S constraints. The

required total number of operations is

√2(JP+J)

2 (JP + 3J +N + S) (2 (JP + J) (2J +N + S)− 3 (2J +N + S) + JP + J) .


O(2JP

).

C.4 Algorithm and Complexity of Proposed Heuristic Solution to the CRM

Problem

In Algorithms C.5, C.6, C.7, C.8 and C.9 we show a pseudo-code of the proposed algorithm

for the CRM problem in the uplink scenario presented in Chapter 6. In Table C.2 we present

the definition of the variables used in those algorithms. It is worth to mention that the

purpose of the description of the proposed algorithm in this section is only the calculation

of its computational complexity. For a more complete description of the proposed algorithm

the interested reader should see section 6.3.3 of Chapter 6.

C.4. Algorithm and Complexity of Proposed Heuristic Solution to the CRM Problem 112

Table C.2: Description of the main parameters used in Algorithms C.5, C.6, C.6, C.7, C.8 and C.9 forthe CRM problem in the uplink case.

Variable Definition





P Set with the index of all contiguous assignment patterns







D Set with the index of the donor flows (satisfied)

R Set with the index of receiver flows (unsatisfied)

K Set with the available candidate RBs for reallocation


K Set with a contiguous block of RBs that is candidate for reallocation

N ′j Auxiliary set that contains the assigned RBs to flow j

rULj,p Transmit data rate of flow j when the RB pattern p is assigned to it

f (·) Function that returns the achieved data rate according to a specific effective SNR. See equation (6.8)

h (j, C) Function that returns the effective SNR when the set of RBs C is assigned to flow j. See equation (6.4)



an Store the index of the flow assigned to resource n

a′n Auxiliary variable that contains the index of the flow that would get assigned the RB n

tj Current allocated rate to the flow j




νj Number of VRs that belongs to flow j

ϕi Metric associated with the ith VR combination

π1i First RB of the ith contiguous block of RBs candidate for reallocation

π2i Last RB of the ith contiguous block of RBs candidate for reallocation



N Total number of RBs

P Total number of assignment patterns


c1, c2, c3 Auxiliary variables

flag1, flag2, flag3

flag4, flag5, flag6

flag7, flag8, flag9

aux1 and aux2


Algorithm C.5 Initialization for solution to the CRM problem in the uplink case.1: βs ← |Js| ∀s ∈ S2: ωs ← 0 ∀s ∈ S3: γs ← 0 ∀s ∈ S

4: rj ←

∑p∈P rUL

j,p

P∀j ∈ J

5: fj ← rj/tj ∀j ∈ J6: Sort in the ascending order fj and store the sorted indices in the vector q composed of the elements qi7: Sort in the ascending order rj and store the sorted indices in the vector l composed of the elements li8: A ← J9: As ← Js ∀s ∈ S10: an ← 0 ∀n ∈ N

In Algorithm C.5 we show the initialization of the proposed algorithm. We assume that the

transmit data rate rULj,p, the mapping between flow and service λj, tj, ks and the sets J , N , S,

P and Js, ∀s ∈ S, and their respective sizes are already known before initialization. In line 4

we have the computation of the average transmit data rate for all flows. In this operation we

have J (P − 1) summations and J multiplications. In line 5 we calculate the ratio between the

average transmit data rate and the data rate requirement which leads to J multiplications.

The operations in lines 6 and 7 in Algorithm C.5 are operations of sorting the elements of a

vector that, as pointed out in Appendix A, have worst-case complexity O (n lnn) where n is the

length of the vector.

Now we present the number of operations of Algorithm C.6 that corresponds to the first

part of the proposed solution. Although we can get a feasible solution in the first part of the

proposed solution, this possibility is not considered. The worst-case condition corresponds

to the situation in which the calculation of the metric for each RB combination in line 26 of

Algorithm C.9 is repeated most often. Assuming that N >>∑

s∈S ks, the worst-case condition

is met when after J −∑s∈S ks + 1 iterations of the main loop in the first part of our proposed

solution there are only one receiver flow and∑

s∈S ks−1 donor flows. In this case, there would

be 2(∑

s∈S ks − 1)RBs to be swept in the for-sentence in line 6 of Algorithm C.9. In the end,

the single receiver flow will get a set of contiguous RBs of size N −(∑

s∈S ks − 1)while each

donor flow would stay with only one RB.

In line 3 of Algorithm C.6 we have the execution of the proposed heuristic solution to the

URM problem. The complexity of this algorithm was calculated in section C.2 of this Appendix.

Retaining the terms of higher order we have that the number of operations executed in the

worst-case situation in line 3 is(J −∑s∈S ks + 1

)(3NJ +

265N2

2

)assuming that each RB is

composed of 12 subcarriers. The test in line 5 is performed

(J −∑s∈S ks + 1

) (J +

∑s∈S ks

)

2times.

(J −∑s∈S ks + 1

) (∑s∈S ks − 1

)summations are performed in line 7. The number

of comparisons executed in lines 15, 23 and 26 is S(J −∑s∈S ks + 1

)each. In lines

30 and 32 there are(J −∑s∈S ks + 1

)comparisons each. The test in line 35 requires

S(J −∑s∈S ks

)comparisons while in line 40

(J +

∑s∈S ks − 1

) (J −∑s∈S ks

)

2comparisons are

executed. Finally, in lines 43 and 45 there are(J −∑s∈S ks

)summations and 1 comparison,

respectively.

Now we present the number of operations performed in Algorithm C.7. As there is only one

receiver flow in the second part of the proposed solution, the test in line 7 is executed only

once. In line 17 we have 2cN+(∑

s∈S ks)+1 summations and cN+10

(∑s∈S ks

)multiplications.

In lines 20 and 21 we have only one operation each.

As in the worst-case complexity case we consider that the receiver flow does not have any

RB assigned in the first part of the proposed solution, the commands between lines 3 and


Algorithm C.6 First part of the proposed solution (Unconstrained Maximization) for the CRMproblem in the uplink case.1: loop2: R← ∅ tj ← 0 ∀j ∈ A Nj ← ∅ ∀j ∈ A D ← ∅ tj ← 0 ∀j ∈ J

3: Run the proposed solution to the URM problem in uplink and update an, tj , νj and Nj

4: for all j ∈ A do5: if tj ≥ tj then6: s∗ ← λj

7: ωs∗ ← ωs∗ + 18: D ← D ∪ {j}9: else10: R ← R∪ {j}11: end if12: end for13: flag1 = 0 {Auxiliary variable that assumes the value 1 whereas flows can still be disregarded without infringing

problem constraints}14: for all s ∈ S do15: if βs > ks then16: γs ← 117: flag1 ← 118: end if19: end for20: flag2 ← 1 {Auxiliary variable that assumes the value 1 when the problem was solved}21: flag3 ← 0 {Auxiliary variable that assumes the value 1 when at least one flow is satisfied}22: for all s ∈ S do23: if ωs < ks then24: flag2 ← 025: end if26: if ωs 6= 0 then27: flag3 ← 128: end if29: end for30: if flag2 = 1 then31: FEASIBLE SOLUTION FOUND32: else if flag1 = 1 then33: B ← A34: for all s ∈ S do35: if γs = 0 then36: B ← B \ As

37: end if38: ωs ← 0 γs ← 0 ∀s ∈ S39: end for40: j∗ ← argmin

j∈B{fj}

41: s∗ ← λj∗

42: As∗ ← As∗ \ {j∗}43: βs∗ ← βs∗ − 144: A ← A \ {j∗}45: else if flag3 = 0 then46: NO FEASIBLE SOLUTION FOUND47: else48: BREAK (GO TO PART 2)49: end if50: end loop

15 are not executed. Line 2 is executed only once while the number of comparisons and

summations in line 23 are 1 and 2(∑

s∈S ks − 1)− 1, respectively.

In the following we present the number of operations in Algorithm C.9. The number of

summations and comparisons in line 8 is

2

(∑

s∈S

ks − 1

)(5

2

(N −

∑

s∈S

ks + 1

)(N −

∑

s∈S

ks + 2

)+

(N −∑s∈S ks + 1

) (N −∑s∈S ks + 2

) (2N − 2

∑s∈S ks + 3

)

3−

(N −

∑

s∈S

ks + 1

)(N −

∑

s∈S

ks + 2

)).


Algorithm C.7 Second part of the proposed solution (Reallocation) for the CRM problem inthe uplink case.1: c1 ← 0 {Auxiliary variable to control if all the receiver flows were satisfied in the reallocation part}2: aux1 ← |R| {Auxiliary variable with the number of flows in the receiver flow set}3: loop4: j∗ ← lc15: PROCEDURE 16: PROCEDURE 27: if flag4 = 0 then8: BREAK (NO FEASIBLE SOLUTION FOUND)9: end if10: for all n ∈ {π1

i∗ , · · · , π2i∗} do

11: j+ ← an12: Nj+ ← Nj+ \ {n}13: an ← j∗

14: Nj∗ ← Nj∗ ∪ {n}15: end for16: for all j ∈ (R∪D) do

17: tj ← f(h(j, Nj

))

18: end for19: R ← R \ {j∗}20: c1 ← c1 + 121: if c1 = aux1 then22: BREAK (FEASIBLE SOLUTION FOUND)23: end if24: end loop

Algorithm C.8 Procedure 1 that is part of the second part of the proposed solution(Reallocation) for the CRM problem in the uplink case.1: K ← ∅2: if νj∗ 6= 0 then

3: n′ ← First element of Nj∗

4: n′′ ← Last element of Nj∗

5: if n′ 6= 1 then6: K ← K ∪ {n′ − 1}7: end if8: if n′′ 6= N then9: K ← K ∪ {n′′ + 1}10: end if11: for all n ∈ K do12: if an /∈ D then13: K ← K \ {n}14: end if15: end for16: else17: for all j ∈ D do18: n′ ← First element of Nj

19: n′′ ← Last element of Nj

20: K ← K ∪ {n′, n′′}21: end for22: end if23: if |K| = 0 then24: BREAK (NO FEASIBLE SOLUTION FOUND)25: end if

The test in line 12 is executed 4(∑

s∈S ks − 1) (N −∑s∈S ks + 1

) (N −∑s∈S ks + 2

) (2N − 2

∑s∈S ks + 3

)

6times. The number of operations in line 17 is

2

(∑

s∈S

ks − 1

)N

(N −∑s∈S ks + 1

) (N −∑s∈S ks + 2

) (2N − 2

∑s∈S ks + 3

)

6.

The if -sentence in line 21 is performed

2

(∑

s∈S

ks − 1

) (N −∑s∈S ks + 1

) (N −∑s∈S ks + 2

) (2N − 2

∑s∈S ks + 3

)

6.


2(∑

s∈S ks − 1) (

N −∑s∈S ks + 1) (

N −∑s∈S ks + 2)comparisons are executed in line 25. The

calculation of the efficiency of each RB set to be reallocated in line 26 requires

2

(∑

s∈S

ks − 1

)((N −∑s∈S ks + 1

) (N −∑s∈S ks + 2

)

2+

c

3

(N −

∑

s∈S

ks + 1

)(N −

∑

s∈S

ks + 2

)(2N − 2

∑

s∈S

ks + 3

))

summations and

2

(∑

s∈S

ks − 1

)(c(N −∑s∈S ks + 1

) (N −∑s∈S ks + 2

) (2N − 2

∑s∈S ks + 3

)

6+

5

(N −

∑

s∈S

ks + 1

)(N −

∑

s∈S

ks + 2

))

multiplications. The update of the variable c3 at the end of line 26

requires(∑

s∈S ks − 1) (

N −∑s∈S ks + 1) (

N −∑s∈S ks + 2)

summations. In lines

30 and 36 we have N − ∑s∈S ks + 1 comparisons while in line 35 there are

(N −∑s∈S ks + 1

) ((∑s∈S ks − 1

) (N −∑s∈S ks + 2

)− 1)

operations. In line 47 we calculate

the new data rate of each donor in case the resource reallocation is executed. For

the sake of simplicity we consider the pessimistic assumption that the number of RBs

assigned to each donor flow does not decrease along the iterations which is not true.

Therefore, this operation requires(N −∑s∈S ks + 1

) (2cN +

∑s∈S ks − 1

)summations and

(N −∑s∈S ks + 1

) (cN + 10

∑s∈S ks − 10

)multiplications. In lines 48 and 53 we have

(N −∑s∈S ks + 1

) (∑s∈S ks − 1

)and N −∑s∈S ks + 1 comparisons, respectively. In line 56 we

calculate the data rate that the selected receiver would get in case the resource reallocation

is executed. This operation requires c(N −∑s∈S ks + 1

) (N −∑s∈S ks + 2

)+ N −∑s∈S ks + 1

summations and c

(N −∑s∈S ks + 1

) (N −∑s∈S ks + 2

)

2+ 10

(N −∑s∈S ks + 1

)multiplications.

Finally, the number of operations required in lines 57, 60 and 65 are N − ∑s∈S ks + 1,

N −∑s∈S ks and 2(N −∑s∈S ks + 1

), respectively.

In summary, the number of operations in the initialization of Algorithm C.5 is

JP + 2J ln J + 3J.


s∈S ks the total number of operations in

Algorithm C.6 is

(3Jθ1N +

265θ1N2

2

)+

(3NJ +

265N2

2+ θ1θ2 + θ21 + 4Sθ1

)+

(7θ12

+θ22

+ 3S

)+ (3) .

Finally, the number of operations in the second part of proposed solution represented in

Algorithms C.7, C.8 and C.9 is shown in the following. In order to simplify the presentation


we define θ4 = N −∑s∈S ks.

(2∑

s∈S

ksθ34

)+

29

∑

s∈S

ksθ24 − 2θ34 − 2θ4

(∑

s∈S

ks

)2

− 2θ4∑

s∈S

ksN

+

85

∑

s∈S

ksθ4 − 11θ24 + 34θ4N − 4

(∑

s∈S

ks

)2

− 4∑

s∈S

ksN

+

(71∑

s∈S

ks − 16θ4 +229

3N

)+

(38

3

)

The most significant term in the number of operations is the term of quarter order found

in the second part of the proposed algorithm: 2∑

s∈S ksθ34. Therefore, the complexity of the

proposed solution is O((∑

s∈S ks) (

N −∑s∈S ks)3)

.


Algorithm C.9 Procedure 2 that is part of the second part of the proposed solution(Reallocation) for the CRM problem in the uplink case.1: c2 ← 1 {Auxiliary variable with the current size of the block of adjacent RBs that are tested}2: flag4 ← 0 {Auxiliary variable that assumes the value 1 if the current selected receiver flow is satisfied with the

reallocation of RBs and 0 otherwise}3: loop4: ϕi ← 0 π1

i ← 0 π2i ← 0 ∀i

5: c3 ← 0 {Auxiliary variable that tracks the number of feasible RB patterns generated}6: for all n ∈ K do7: for all i ∈ {1, · · · , c2} do

8: K ← ∅ K ← K ∪ {n+ 1− i, · · · , n+ c2 − i}9: flag5 ← 0 {Auxiliary variable that assumes the value 1 if at least an RB index of the generated block of

RBs is invalid}10: flag6 ← 0 {Auxiliary variable that assumes the value 1 if at least one RB of the generated block of RBs

belongs to a receiver flow}11: for all n∗ ∈ K do12: if (n∗ < 1) or (n∗ > N) then13: flag5 ← 114: end if15: flag7 ← 0 {Auxiliary variable that assumes the value 1 if a specific RB belongs to a donor flow}16: for all j ∈ D do17: if n∗ ∈ Nj then18: flag7 ← 119: end if20: end for21: if flag7 = 0 then22: flag6 ← 123: end if24: end for25: if (flag5 = 0) and (flag6 = 0) then

26: ϕc3 ← f(h(j∗, K

))π1c3← First element of K π2

c3← Last element of K c3 ← c3 + 1

27: end if28: end for29: end for30: if c3 = 0 then31: BREAK (NO FEASIBLE SOLUTION FOUND)32: end if33: flag8 ← 1 {Auxiliary variable that assumes the value 0 if there is no feasible assignment pattern to choose and

1 otherwise}34: loop35: i∗ ← argmax

∀iϕi

36: if ϕi∗ = −inf then37: flag8 ← 038: BREAK39: end if40: a′n ← an ∀n ∈ N

41: N ′j ← Nj ∀j ∈ J

42: for all n ∈ {π1i∗ , · · · , π

2i∗} do

43: j+ ← an N ′j+← N ′

j+\ {n} a′n ← j∗ N ′

j∗ ← N′j∗ ∪ {n}

44: end for45: flag9 ← 0 {Auxiliary variable that assumes the value 1 if at least one donor flow would be unsatisfied if the

current assignment pattern is chosen and 0 otherwise}46: for all j ∈ D do

47: aux2 ← f(h(j,N ′

j

)){Auxiliary variable with the data rate that the current donor flow would get if the

assignment pattern is chosen}48: if aux2 < tj then49: flag9 ← 150: BREAK51: end if52: end for53: if flag9 = 1 then54: ϕi∗ ← −inf55: else56: aux2 ← f

(j∗,N ′

j∗

)

57: if aux2 ≥ tj∗ then58: flag4 ← 159: else60: c2 ← c2 + 161: end if62: BREAK63: end if64: end loop65: if (flag8 = 0) or (flag4 = 1) then66: BREAK67: end if68: end loop

119

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