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Centro de Tecnologia e Urbanismo
Departamento de Engenharia Eletrica
Jose Carlos Marinello Filho
Otimizacao por Colonia de FormigasAplicavel aos Sistemas de
Telecomunicacoes
Monografia apresentada ao curso de
Engenharia Eletrica da Universidade
Estadual de Londrina, como parte dos
requisitos necessarios para a conclusao
do curso de Engenharia Eletrica.
Londrina, PR2012
Jose Carlos Marinello Filho
Otimizacao por Colonia de Formigas
Aplicavel aos Sistemas de
Telecomunicacoes
Monografia apresentada ao curso de Engenharia
Eletrica da Universidade Estadual de Londrina,
como parte dos requisitos necessarios para a
conclusao do curso de Engenharia Eletrica.
Area: Modelagem e Simulacao de Sistemas deTelecomunicacoes
Orientador:
Prof. Dr. Taufik Abrao
Londrina, PR2012
Ficha Catalografica
Marinello Filho, Jose CarlosOtimizacao por Colonia de Formigas Aplicavel aos Sistemas de Te-
lecomunicacoes. Londrina, PR, 2012. 41 p.
Monografia (Graduacao) – Universidade Estadualde Londrina, PR. Departamento de Engenharia Eletrica.
1. Sistemas de Telecomunicacoes. 2. Otimizacao por Coloniade Formigas. 3. Sistemas de Multiplo Acesso. 4. Sistemas deMultiplas Antenas. I. Universidade Estadual de Londrina. Departa-mento de Engenharia Eletrica. Departamento de Engenharia Eletrica. II. Otimizacao por Colonia de Formigas Aplicavel aos Sistemas deTelecomunicacoes.
Jose Carlos Marinello Filho
Otimizacao por Colonia de FormigasAplicavel aos Sistemas de
Telecomunicacoes
Monografia apresentada ao curso de Engenharia
Eletrica da Universidade Estadual de Londrina,
como parte dos requisitos necessarios para a
conclusao do curso de Engenharia Eletrica.
Area: Modelagem e Simulacao de Sistemas deTelecomunicacoes
Comissao Examinadora
Prof. Dr. Taufik AbraoDepto. de Engenharia Eletrica
Orientador
Prof. Msc. Jaime Laelson JacobDepto. de Engenharia Eletrica
Prof. Dr. Bruno Augusto AngelicoUniversidade Tecnologica Federal do Parana
16 de novembro de 2012
“Quando ouvires os aplausos do triunfo, que ressoem tambem aos teus ouvidos
os risos que provocaste com os teus fracassos.”
Sao Josemaria Escriva
Agradecimentos
Agradeco primeiramente a Deus pela bencao de ter chegado ate aqui; ao meu
orientador Taufik Abrao, por tudo que me ensinou nos ultimos anos; a minha
famılia, por todo o apoio e suporte que me deram para que conseguisse alcancar
meus objetivos; a minha namorada Luciana, pelo carinho e companheirismo que
tem me proporcionado desde que nos conhecemos; e a todos os meus amigos e
companheiros de curso que muito me ajudaram ao longo de minha formacao.
Resumo
Este trabalho faz um estudo e a caracterizacao da tecnica Otimizacao por Coloniade Formigas (ACO - Ant Colony Optimization), de forma a aplica-la aos sistemasde telecomunicacoes. Sao considerados tres problemas classicos em telecomu-nicacoes: Deteccao Multiusuario em Sistemas DS/CDMA (Direct Sequence/CodeDivision Multiple Access); Deteccao em Sistemas de Mutiplas Antenas (MIMO- Multiple Input Multiple Output); e Transmissao Multiusuario para Sistemasde Comunicacao MIMO. A analise de desempenho da tecnica aplicada a cadaproblema e feita por meio de simulacao computacional, sendo o canal modeladocom desvanecimento plano em frequencia (Rayleigh Flat), com o auxılio da pla-taforma MatLab, empregando-se a tecnica de simulacao Monte Carlo (MCS -Monte Carlo Simulation). Em geral, os resultados mostram a tecnica ACO comouma ferramenta de otimizacao bem versatil, podendo ser aplicada a uma classemuito ampla de problemas: tanto aqueles de natureza combinatoria (variaveisdiscretas), como aqueles de natureza contınua, utilizando-se para isso de umacomplexidade computacional viavel.
Abstract
In this work, a study and a characterization of the Ant Colony Optimization(ACO) technique is made, in order to apply it to the telecommunications sys-tems. Three classical problems on telecommunications are investigated: Multiu-ser Detection on DS/CDMA (Direct Sequence/Code Division Multiple Access)systems; Detection on multiple antennas (MIMO - Multiple Input Multiple Out-put) systems; and Multiuser Transmission for MIMO Communications Systems.The performance analysis of the technique applied for each issue is taken throughcomputational simulations, being the channel shaped with a flat frequency fading(Rayleigh Flat), with the aid of the MatLab platform, by using the Monte CarloSimulation technique. In general, the results show the ACO technique as a ver-satile optimization tool, which may be applied in a wide range of problems: boththose with a combinatorial nature (discrete variables), as the continuous ones,using for it a feasible computational complexity.
Sumario
Lista de Figuras
Lista de Abreviaturas
Convencoes
1 Introducao 1
1.1 Motivacao: Deteccao Multiusuario em Sistemas Multiplo Acesso
por Divisao de Codigo (Code Division Multiple Access) (CDMA) . 3
1.2 Motivacao: Deteccao em Sistemas Multiplas Antenas Transmisso-
ras e Receptoras (Multiple Input Multiple Output) (MIMO) . . . . 4
1.3 Motivacao: Transmissao Multiusuario (MuT) . . . . . . . . . . . . 5
2 Conclusao 8
I Detector Multiusuario ACO para DS/CDMA 10
II Detector ACO para sistemas MIMO 20
IIITransmissor Multiusuario ACO 31
Referencias 39
Lista de Figuras
1.1 Comportamento das formigas na natureza, ilustrando sua capaci-
dade de otimizar percursos. . . . . . . . . . . . . . . . . . . . . . 2
1.2 Configuracao de um sistema MIMO utilizando multiplexacao es-
pacial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Configuracao MuT MIMO, em que N antenas na Estacao Base
(Base Station) (BS) se comunicam com K unidades moveis com
uma unica antena cada. . . . . . . . . . . . . . . . . . . . . . . . 6
Lista de Abreviaturas
ACO Otimizacao por Colonia de Formigas (Ant Colony Optimization)
BER Taxa de Erro de Bit (Bit Error Rate)
BS Estacao Base (Base Station)
CDMA Multiplo Acesso por Divisao de Codigo (Code Division Multiple Access)
CIR Resposta Impulsiva do Canal (Channel Impulse Response)
DS/CDMA Direct Sequence/Code Division Multiple Access
EP Programacao Evolucionaria (Evolutionary Programming)
FDD Frequency Division Duplex
GA Algoritmo Genetico (Genetic Algorithm)
LR Reducao Trelica (Lattice Reduction)
LS Busca Local (Local Search)
LTE Long Term Evolution
LTE-A Long Term Evolution-Advanced
MACO Otimizacao por Colonia de Formigas Modificada (Modified Ant Colony
Optimization)
MC/CDMA Multi Carrier/Code Division Multiple Access
MIMO Multiplas Antenas Transmissoras e Receptoras (Multiple Input Multiple
Output)
ML Maxima Verossimilhanca (Maximum Likelihood)
MMSE Mınimo Erro Quadratico Medio (Minimum Mean Squared Error)
MS Estacao Movel (Mobile Station)
MuD Deteccao Multiusuario (Multiuser Detection)
MuT Transmissao Multiusuario (Multiuser Transmission)
PDF Funcao Densidade de Probabilidades (Probability Density Function)
PSO Otimizacao por Nuvem de Partıculas (Particle Swarm Optimization)
SD Detector Esferico (Spheric Detector)
SNR Relacao Sinal Ruıdo (Signal to Noise Ratio)
SQP Sequential Quadratic Programming
TDD Time Division Duplex
TMF Transmitter Matched Filtering
TSP Problema do Caixeiro Viajante (Travelling Salesman Problem)
V-BLAST Vertical Bell Laboratories Space-Time
ZF Zero Forcing
Convencoes
Na notacao das formulas, as seguintes convencoes foram utilizadas:
• letras maiusculas sao matrizes, exemplo: P, H;
• letras minusculas sao vetores, exemplo: x, z;
• letras em italico sao escalares, exemplo: a, β;
• vi e o i-esimo elemento do vetor v;
• Aij e o elemento da i-esima linha e j-esima coluna da matriz A;
• diag(a1, a2, . . . , ak) e uma matriz diagonal com elementos a1, a2, . . . , ak;
• Ik e uma matriz identidade de ordem k;
• {·}T e o operador matriz transposta;
• {·}H e o operador matriz conjugada transposta;
• {·}−1 e o operador matriz inversa;
• {·}† e o operador matriz pseudo-inversa;
• ℜ{·} e o operador parte real;
• ℑ{·} e o operador parte imaginaria;
• ⌈·⌋ e o operador inteiro mais proximo;
• sign(·) e o operador sinal;
• U(x, y) e um processo aleatorio com distribuicao uniforme entre as variaveis
x e y;
• N (m,σ) e um processo aleatorio com distribuicao normal de media m e
desvio padrao σ;
• Q(x) e a funcao Q Gaussiana;
• ∈ significa pertence ao conjunto;
Palavras em italico sao utilizadas para identificar termos de lıngua inglesa
nao traduzidos.
1
1 Introducao
Devido ao grande avanco tecnologico na area de telecomunicacoes, muito tem sido
pesquisado e desenvolvido nos ultimos anos a respeito de comunicacao multimıdia
movel. Esse tema pode ser considerado um dos grandes avancos cientıficos da
humanidade, uma vez que permite uma ampla variedade de servicos, como trafego
de voz, vıdeo e dados, aliados a mobilidade de uma conexao “wireless”, de forma
a libertar o usuario do par de fios. Dentro do contexto da comunicacao digital,
falar sobre comunicacao multimıdia movel equivale a falar sobre elevadas taxas
de transmissao de dados, e os requisitos de qualidade de servico cada vez mais
rigorosos exigem sofisticadas tecnicas de transmissao e recepcao de dados.
Esse mesmo avanco tecnologico, que traz consigo diversas inovacoes e varie-
dade na oferta de servicos, traz tambem uma limitacao de recursos disponıveis
cada vez mais aparente. O fato de diversos servicos e usuarios terem de comparti-
lhar a atmosfera como canal de comunicacao ocasiona um espectro de frequencias
disponıvel cada vez mais escasso. Alem disso, a necessidade de economizar energia
juntamente com o interesse em obter dispositivos cada vez mais leves e compactos
torna a potencia disponıvel cada vez menor.
Fica claro entao o grande desafio do atual cenario das telecomunicacoes: de-
senvolver tecnicas eficientes para a transmissao e recepcao de elevadas taxas de
dados, respeitando para isso as limitacoes de energia e processamento dos dispo-
sitivos utilizados.
Dentro da terceira geracao dos sistemas de comunicacao moveis, a tecno-
logia CDMA, nas suas formas Direct Sequence/Code Division Multiple Access
(DS/CDMA) e Multi Carrier/Code Division Multiple Access (MC/CDMA), por-
tadora unica e portadoras multiplas, respectivamente, e conhecida por suportar
um numero relativamente grande de usuarios, que compartilham simultaneamente
a mesma banda de frequencias, o que se torna possıvel devido ao espalhamento
espectral (ABRAO, 2001; YANG, 2009).
Ja dentro da quarta geracao de sistemas de comunicacao moveis, a multi-
1 Introducao 2
Figura 1.1: Comportamento das formigas na natureza, ilustrando suacapacidade de otimizar percursos.
plexacao espacial MIMO pode ser utilizada como forma de atender os requisitos
das tecnologias Long Term Evolution (LTE) e Long Term Evolution-Advanced
(LTE-A) (AUBERT, 2011). Quando o cenario em questao e composto por multiplas
antenas na BS e uma unica antena em cada Estacao Movel (Mobile Station) (MS),
e interessante que o maximo possıvel do processamento necessario para uma co-
municacao eficiente seja deslocado para a BS, pois esta possui mais autonomia
energetica e maior capacidade de processamento. Nesse sentido, quando a co-
municacao se da da MS para a BS (Uplink), sao aplicadas tecnicas de Deteccao
Multiusuario (Multiuser Detection) (MuD) (LIU; YANG; HANZO, 2009). Ja as
tecnicas de Transmissao Multiusuario (Multiuser Transmission) (MuT) (YAO;
CHEN; HANZO, 2012) sao aplicaveis quando a comunicacao se da da BS para a
MS (Downlink).
A tecnica Otimizacao por Colonia de Formigas (Ant Colony Optimization)
(ACO) (DORIGO; BIRATTARI; STUTZLE, 2006) foi proposta originalmente para
problemas de natureza combinatoria, como o Problema do Caixeiro Viajante
(Travelling Salesman Problem) (TSP), porem foi estendida para problemas de
natureza contınua em (SOCHA; DORIGO, 2008). Ela e baseada no comportamento
das formigas em seu estado natural, que, em busca de alimento, possuem a capa-
cidade de explorar o territorio a sua volta, encontrar os caminhos que levam as
melhores fontes de comida, e deixar rastros que irao auxiliar na busca realizada
por outras formigas (figura 1.1). Varios trabalhos ja foram publicados aplicando
essa tecnica a diversos problemas em telecomunicacoes, como deteccao em sis-
temas CDMA (XU; YANG; HANZO, 2007) (MARINELLO; SOUZA; ABRAO, 2012),
deteccao em sistemas MIMO (KHURSHID; IRTEZA; KHAN, 2010) (LAIN; CHEN,
2010), alocacao de recursos em redes sem fio (ZHAO et al., 2010) (MARQUES et al.,
2012), entre outros.
1.1 Motivacao: Deteccao Multiusuario em Sistemas CDMA 3
As principais caracterısticas que diferem o ACO discreto do contınuo estao na
forma como o algoritmo e adaptado ao problema. Na versao discreta do algoritmo,
os nos do problema podem ser mapeados na forma de tabelas, uma vez que ha
um numero finito de valores que podem ser assumidos a cada dimensao. Assim
as variaveis do algoritmo sao estruturadas na forma de matrizes, tais como a
quantidade de ferormonio, a informacao heurıstica e as probabilidades. O mesmo
ja nao acontece em sua versao contınua, pois cada dimensao do problema pode
assumir qualquer valor real dentro de determinado intervalo. Dessa forma, os
ferormonios passam a ser representados como um arquivo, que ira guardar um
certo numero de solucoes encontradas pelo algoritmo, em ordem crescente de
suas tendencias em relacao a solucao otima. Tambem as probabilidades, que
antes na forma de tabelas podiam ser vistas como em uma Funcao Densidade de
Probabilidades (Probability Density Function) (PDF) discreta, agora devem ser
organizadas na forma de uma PDF contınua, o que e feito em (SOCHA; DORIGO,
2008) por meio de uma combinacao adequada de funcoes Gaussianas.
1.1 Motivacao: Deteccao Multiusuario em Sis-
temas CDMA
Como ja mencionado, a principal caracterıstica que difere os sistemas CDMA
dos demais sistemas de multiplo acesso e o espalhamento espectral, ou seja, a
multiplexagem nao ocorre por divisao do tempo, espaco ou frequencia, e sim por
meio de um codigo atribuıdo a cada usuario, tambem chamado de sequencia de
espalhamento. Os diferentes usuarios do CDMA podem ser distinguidos por meio
de sua exclusiva sequencia de espalhamento. Assumindo que a correlacao cruzada
entre os codigos de espalhamento seja suficientemente baixa, o detector possuira
assim informacao suficiente para identificar os diferentes usuarios (YANG, 2009).
Com o advento da deteccao multiusuario em (VERDU, 1998), proposta como
alternativa a deteccao convencional, foi mostrado que as informacoes dos usuarios
ativos no sistema podem ser utilizadas conjuntamente a fim de atenuar/eliminar
a interferencia entre si, melhorando a deteccao de cada usuario individualmente e
aproximando a capacidade do sistema a capacidade do canal (ABRAO, 2001). As-
sim, enquanto na deteccao convencional os usuarios ativos sao vistos como ruıdo
adicional, a deteccao multiusuario utiliza as informacoes destes como forma de
combater a interferencia de multiplo acesso, resultando numa consideravel me-
lhoria de desempenho e capacidade (VERDU, 1986). O detector otimo proposto,
ao realizar uma busca exaustiva a cada deteccao, possui aumento de complexi-
1.2 Motivacao: Deteccao em Sistemas MIMO 4
dade exponencial em relacao ao numero de usuarios do sistema. Isso motivou
o desenvolvimento de diversos detectores com abordagens heurısticas, os quais
apresentam desempenho quase otimo ao custo de uma complexidade computaci-
onal notavelmente reduzida. Dentre eles se destacam o Algoritmo Genetico (Ge-
netic Algorithm) (GA) (ERGUN; HACIOGLU, 2000), a Programacao Evolucionaria
(Evolutionary Programming) (EP) (CIRIACO; ABRAO; JESZENSKY, 2004), o Busca
Local (Local Search) (LS) (LIM; VENKATESH, 2003), a Otimizacao por Nuvem de
Partıculas (Particle Swarm Optimization) (PSO) (ABRAO et al., 2010), entre ou-
tros.
Na Parte I, e mostrado um artigo publicado na revista “Expert Systems with
Applications”, o qual faz um estudo, caracterizacao e propoe uma metodologia de
otimizacao de parametros para o Detector Multiusuario baseado na tecnica ACO.
Uma versao resumida do trabalho tambem foi publicada no Simposio Brasileiro
de Telecomunicacoes (SBrT) de 2012.
1.2 Motivacao: Deteccao em Sistemas MIMO
Sistemas MIMO sao amplamente conhecidos por possibilitarem uma consideravel
melhoria na eficiencia espectral de sistemas de comunicacao sem fio (WOLNI-
ANSKY et al., 1998). Na configuracao de multiplexacao espacial, e permitido um
significante aumento na taxa de transmissao de dados para uma mesma potencia
e banda utilizadas (TSE; VISWANATH, 2010), ao se explorar a diversidade de ca-
minhos entre antenas transmissoras e receptoras (figura 1.2).
Os detectores lineares, tais como o Zero Forcing (ZF) e o Mınimo Erro
Quadratico Medio (Minimum Mean Squared Error) (MMSE), sao conhecidos por
apresentarem uma baixa complexidade computacional, porem apresentam um de-
sempenho bem inferior ao Maxima Verossimilhanca (Maximum Likelihood) (ML).
Por outro lado, o Detector Esferico (Spheric Detector) (SD) (BARBERO; THOMP-
SON, 2008) e conhecido por apresentar um desempenho quase otimo, mas com
uma complexidade computacional, para regioes de baixa Relacao Sinal Ruıdo
(Signal to Noise Ratio) (SNR), da mesma ordem da do ML.
Como uma forma de melhorar o desempenho dos detectores lineares para
canais mal condicionados (correlacionados), sem aumentar demasiadamente sua
complexidade computacional, foram propostos detectores baseados na tecnica de
Reducao Trelica (Lattice Reduction) (LR) (WUBBEN et al., 2011), (WUBBEN et al.,
2004). Essa tecnica transforma o canal parcialmente correlacionado em um equi-
1.3 Motivacao: Transmissao Multiusuario (MuT) 5
Figura 1.2: Configuracao de um sistema MIMO utilizando multiplexacaoespacial.
valente, cuja matriz apresenta melhores condicoes de ortogonalidade. De posse
dessa matriz de canal quase ortogonal no receptor, a deteccao ocorre utilizando-se
um detector linear de baixa complexidade.
Um simples detector MIMO baseado em ACO foi proposto em (KHURSHID;
IRTEZA; KHAN, 2010), o qual combinava a tecnica com os detectores ZF e Vertical
Bell Laboratories Space-Time (V-BLAST). Entretanto, os resultados numericos
mostraram um desempenho distante do ML para os detectores propostos em
regioes de media e alta SNR. Em (LAIN; CHEN, 2010), um diferente metodo de
deteccao baseado em ACO foi proposto, mas uma vez que seu desempenho nao foi
satisfatorio, foi introduzido um detector Otimizacao por Colonia de Formigas Mo-
dificada (Modified Ant Colony Optimization) (MACO) como alternativa para me-
lhorar o compromisso desempenho-complexidade em uma deteccao MIMO quase
otima.
Na Parte II, e proposto um detector que, ao combinar as tecnicas LR e ACO,
ofereca uma alternativa eficiente para os detectores MIMO existentes. O artigo
mostrado foi submetido para o evento IEEE Word Communications and Networ-
king Conference (WCNC) 2013, e uma versao estendida esta sendo elaborada
analisando o detector em canais correlacionados.
1.3 Motivacao: Transmissao Multiusuario (MuT)
Em sistemas de comunicacao sem fio, e muito comum que as unidades moveis
tenham uma unica antena, pois, dada a frequencia de operacao e as dimensoes do
dispositivo, implementa-lo com multiplas antenas se torna inviavel. Adicional-
mente, sao convenientes terminais com alta eficiencia energetica, cujas baterias
apresentem uma autonomia satisfatoria para o usuario, e que assim se tornam
incapazes de executar tecnicas sofisticadas de deteccao multiusuario. Com o
objetivo de lidar com essas limitacoes das MS’s, sem prejudicar o desempenho
1.3 Motivacao: Transmissao Multiusuario (MuT) 6
α α–1
α–1
α–1
α
α
P
XK
nK
n2
n1h1H
h2
hk
yK
y2
y1
X2
User 1
User 2
User KN
2
1X1
Figura 1.3: Configuracao MuT MIMO, em que N antenas na BS secomunicam com K unidades moveis com uma unica antena cada.
do sistema, as tecnicas de Transmissao Multiusuario foram propostas (VOJCIC;
JANG, 1998).
A ideia central desse esquema e precodificar os dados a serem transmitidos,
de forma que o receptor, equipado com um filtro casado convencional, possa fazer
a deteccao simplesmente quantizando o sinal recebido, com a menor Taxa de Erro
de Bit (Bit Error Rate) (BER) possıvel (figura 1.3). Para isso e necessario que o
transmissor possua certas informacoes, tais como a Resposta Impulsiva do Canal
(Channel Impulse Response) (CIR) e a SNR no receptor, alem de obviamente
ter conhecimento dos dados a serem transmitidos. Para sistemas Time Division
Duplex (TDD), a BS ja possui as informacoes do canal provenientes do Uplink,
as quais serao as mesmas para o Downlink uma vez que utilizam a mesma banda
de frequencias. O mesmo ja nao ocorre para sistemas Frequency Division Duplex
(FDD), e assim essas informacoes devem ser fornecidas da MS para a BS por
feedback.
As primeiras tecnicas desenvolvidas foram as lineares Transmitter Matched
Filtering (TMF) e ZF, as quais apresentam baixa complexidade, mas um baixo
ganho de desempenho. Ja a tecnica MuT baseada na aproximacao MMSE apre-
senta um desempenho melhor em relacao as anteriores, uma vez que leva em
conta a variancia do ruıdo juntamente com a interferencia multiusuario. Porem,
tais tecnicas nao levam em conta a informacao a ser transmitida na obtencao da
matriz de precodificacao, e assim minimizam a BER apenas indiretamente.
O transmissor linear otimo foi desenvolvido baseado no criterio de mini-
mizacao da BER (MBER) (HABENDORF; FETTWEIS, 2007). Devido a potencia
de transmissao fixa, o problema e formulado como a otimizacao de uma funcao
nao linear com restricoes quadraticas, podendo assim ser utilizado o algoritmo
de otimizacao nao linear Sequential Quadratic Programming (SQP). Porem, seu
custo computacional o torna inviavel para implementacao em sistemas de altas
taxas de transmissao de dados. Com o objetivo de obter um desempenho quase
otimo, ao custo de uma complexidade computacional aceitavel, foi proposto em
1.3 Motivacao: Transmissao Multiusuario (MuT) 7
(YAO et al., 2009) um transmissor em que a busca pela matriz de precodificacao e
feita pela tecnica PSO.
Na Parte III, e proposto um transmissor em que a busca pela matriz de
precodificacao e feita pela tecnica ACO. O trabalho em questao ainda nao esta
concluıdo, pois o algoritmo proposto ainda esta sendo otimizado em busca de
melhores resultados.
8
2 Conclusao
Concluımos que o algoritmo ACO e uma ferramenta de otimizacao muito efici-
ente e versatil, pois pode ser aplicado a uma variedade muito grande de proble-
mas, e os resultados alcancados sao em geral muito competitivos em relacao as
outras tecnicas de otimizacao. Este trabalho mostrou que a aplicabilidade do
algoritmo dentro das telecomunicacoes nao e diferente. Ao considerar tres pro-
blemas classicos da area, os resultados mostraram a consistencia do algoritmo e
sua viabilidade.
Dentro do problema da deteccao multiusuario em DS/CDMA, foi proposta
uma metodologia para a otimizacao dos parametros do algoritmo, e depois mos-
trado o seu excelente desempenho com os parametros devidamente otimizados.
Foi analisada a robustez do detector a variacoes das condicoes do canal, como ve-
locidade relativa entre MS e BS, controle de potencia dos usuarios, carregamento
do sistema, e os resultados mostraram que o mesmo pode operar eficientemente
sem necessidade de alteracoes significativas, nos diferentes cenarios analisados.
A complexidade do detector proposto se mostrou muito vantajosa em relacao ao
detector otimo, enquanto os resultados de ambos foram muito semelhantes.
Para a deteccao em sistemas MIMO, foi proposto um detector que combina o
algoritmo ACO com a promissora tecnica LR, de forma que a busca do algoritmo
fosse executada em um domınio melhor condicionado pela tecnica. Novamente foi
feita a otimizacao de parametros do algoritmo, e os resultados obtidos mostraram
o detector como uma eficiente alternativa a elevada complexidade do SD e ao
desempenho insatisfatorio dos lineares ZF e MMSE, para diferentes numeros de
antenas transmissoras e receptoras e diferentes ordens de modulacao. O detector
proposto tambem se mostrou muito vantajoso em relacao aos outros detectores
baseados em ACO para multiplas antenas presentes na literatura.
Para o caso da Transmissao Multiusuario em sistemas MIMO, foi proposto
um transmissor em que a busca pela matriz de precodificacao que minimizasse a
probabilidade de erro no receptor fosse efetuada pela tecnica ACO contınua. O
problema foi abordado a partir de duas aproximacoes: a primeira utilizando-se
2 Conclusao 9
uma funcao penalti e calculando os fatores por meio da teoria de otimizacao con-
vexa, fazendo-se estimativas; e a segunda na qual o calculo da probabilidade de
erro levava em consideracao o fator de normalizacao da potencia de transmissao,
ao obter uma medida de ruıdo equivalente. Apenas a segunda abordagem gerou
resultados coerentes. Entao o transmissor foi comparado com um outro descrito
na literatura, o qual e baseado na tecnica PSO. Os resultados mostraram que,
embora o transmissor proposto tenha apresentado um ganho significativo de de-
sempenho em relacao aos lineares, ele ainda nao foi capaz de superar o PSO-MuT
em termos do compromisso desempenho×complexidade. O transmissor proposto
ainda esta sendo aprimorado com o objetivo de obter um melhor desempenho
com menor custo computacional.
10
Parte I
Detector Multiusuario ACO para
DS/CDMA
Expert Systems with Applications 39 (2012) 12876–12884
Contents lists available at SciVerse ScienceDirect
Expert Systems with Applications
journal homepage: www.elsevier .com/locate /eswa
Ant colony input parameters optimization for multiuser detection in DS/CDMAsystems
José Carlos Marinello Filho, Reginaldo Nunes de Souza, Taufik Abrão ⇑Department of Electrical Engineering, State University of Londrina, Londrina, PR, Brazil
a r t i c l e i n f o a b s t r a c t
Keywords:Ant colony intelligenceMultiuser detectionInput parameters optimizationComputational complexityMultiple access communication networksDS-CDMA
0957-4174/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.eswa.2012.05.005
⇑ Corresponding author.E-mail addresses: [email protected] (J.C. Ma
gmail.com (R.N. de Souza), [email protected], [email protected]: http://www.uel.br/pessoal/taufik (T. Abrão).
1 The number of users by the processing gain ratio,
In this work a simple and efficient methodology for tuning the input parameters applied to the ant colonyoptimization multiuser detection (ACO-MuD) in direct sequence code division multiple access (DS-CDMA) is proposed. The motivation in using a heuristic approach is due to the nature of the NP complex-ity posed by the wireless multiuser detection optimization problem. The challenge is to obtain suitabledata detection performance in solving the associated hard complexity problem in a polynomial time. Pre-vious results indicated that the application of heuristic search algorithms in several wireless optimizationproblems have been achieved excellent performance-complexity tradeoffs. Regarding different systemoperation and channels scenarios, a complete input parameters optimization procedure for the ACO-MuD is provided herein, which represents the major contribution of this work. The performance of theACO-MuD is analyzed via Monte-Carlo simulations. Simulation results show that, after convergence,the performance reached by the ACO-MuD is much better than the conventional detector, and somewhatclose to the single user bound (SuB). Flat Rayleigh channels is initially considered, but the input param-eter optimization methodology is straightforward applied to selective fading channels scenarios, as wellas to joint time-spatial wireless channels diversities.
� 2012 Elsevier Ltd. All rights reserved.
1. Introduction
In the direct sequence/code division multiple access (DS/CDMA)technology, all the users share the entire frequency band availableat the same time. This is possible due the spreading sequence withshort chip period, which is used in order to spread the user infor-mation along all the available bandwidth spectrum, as well asserves as an identification code for each user, providing some levelof multiple access interference (MAI) immunity. The application ofsequences with low cross correlation allows to support a consider-ably number of users simultaneously, as well as the possibility ofoperation in the asynchronous configuration mode, meting therequirements of wireless mobile communication uplink. However,as the system loading1 increases, the utilization of sophisticateddetectors become necessary, such as multi user detection (MuD)(Verdú, 1998), in such a way to obtain a reasonable separationamong the several user’ signals, each one under an intense multipleaccess interference level generated by K � 1 interfering users. Thebest performance is achieved by the optimum multiuser detector
ll rights reserved.
rinello Filho), reginaldouel@org (T. Abrão).
L = K/N.
(OMuD) or maximum likelihood (ML) detector, which complexitygrows exponentially with the number of users, Oð2KÞ (Verdú, 1998).
After the Verdú’s revolutionary work, a wide number of subop-timal MuD approaches have been proposed in an attempt to gethigh performance multiuser receivers with low complexity. Amongthe suboptimal MuDs, the linear multiuser detectors, such asdecorrelator (Verdú, 1986) and Minimum Mean Square Error(MMSE) (Poor & Verdu, 1997), as well as the nonlinear MuDs, suchas interference cancelers (Patel & Holtzman, 1994), zero-forcing(Duel-Hallen, 1995), among others have been widely discussed inliterature in the past two decades. More recently, a new class ofmultiuser detection approach has been successfully proposed withincreasing gain in complexity-performance trade-off: the heuris-tic-based multiuser detectors (Ciriaco, Abrpo, & Jeszensky, 2006;Ergun & Hacioglu, 2000; Hijazi & Natarajan, 2004; Lim & Venk-atesh, 2003).
In the last decade, proposals based on heuristic methods havebeen reported to solve the MuD problem, getting performanceclose to the ML performance with polynomial computational com-plexity (Abrão et al., 2009; Ergun & Hacioglu, 2000). The use ofheuristic search algorithms is motivated by the fact that optimiza-tion problems related to wireless communication systems resultsin non-polynomial (NP-hard) problems, e.g., MuD optimizationproblem (Verdú, 1989). So, from a practical point-of-view, the chal-lenge is to obtain satisfactory results for these high computational
2 Adopted without loss of generality as normalized.
J.C. Marinello Filho et al. / Expert Systems with Applications 39 (2012) 12876–12884 12877
complexity problems in a polynomial time. This requirementcomes from the fact that detection stage must be provided in ashort time under a limited computational resources in order tomet the real time wireless communication applications and ser-vices. Meting this requirement the wireless communication systemis able to be implemented on modern digital signal processorsplatforms.
Furthermore, the input parameters optimization of the heuris-tic-based algorithms is of paramount importance in order to obtainreliable results. Specifically on MuD optimization problem, inAbrão, Oliveira, Angélico, and Jeszensky (2010) a detailed studyabout the input parameters of the particle swarm heuristic algo-rithm applied to DS/CDMA multiuser detection problem has beenconducted. Hence, present work aims to develop an input parame-ters analysis for the ant colony optimization (ACO) heuristic-basedalgorithm applied to DS/CDMA multiuser detection problem.
Previous results shows the heuristic based algorithms applica-tion in several wireless communications optimization problemshas achieved great success in the performance-complexity tradeoff.In the multiuser detection context, the heuristic based algorithms(Heur-MuD) most commonly used includes the evolutionary pro-gramming (EP) based algorithms, specially the genetic algorithms(GA) (Ciriaco et al., 2006; Ergun & Hacioglu, 2000), particle swarmoptimization (PSO) (de Oliveira, Ciriaco, Abrpo, & Jeszensky, 2006;Zhao, Long, & Wang, 2006), ant colony optimization (ACO) (Xu,Yang, & Hanzo, 2007) and the local search method (LS) (Lim &Venkatesh, 2003; Oliveira, Abrão, Ciriaco, & Jeszensky, 2009).
The first algorithm using the ACO heuristic approximation wasproposed in 1991 by Colorni, Dorigo, and Maniezzo (1991), andsince that many variant algorithms were described in the litera-ture. The ACO intelligence has achieved great success in solvingcombinatorial optimization problems that had arisen in manyareas. For example in power electronics field, variants of this heu-ristic method, such as elitist ant system (EAS), rank-based ant sys-tem (ASrank) and max–min ant system (MMAS) just to name a few,have been deployed, for instance, in reactive power controls (Abb-asy & Hosseini, 2007). Recently, this ant behavior-based techniquehas been widely applied to multiple access multiuser detection(Hijazi & Natarajan, 2004; Lai & Lain, 2005; Xu et al., 2007; Zhao,Wu, Zhao, & Quan, 2010).
The computational complexity of DS/CDMA ACO multiuserdetection was analyzed in Hijazi and Natarajan (2004), noting thatwith a few iterations the ACO-MuD algorithm was able to reach thenear-optimal performance spending only a small fraction (�5%) ofcomputational effort necessary to perform an exhaustive search. InHijazi and Natarajan (2005), a similar performance is reached con-sidering different received powers for the users. Also, the lowercomputational complexity for the ACO algorithm regarding to ge-netic algorithm has been evidenced. Besides, under the same chan-nel and system operation conditions the authors have been showsthe the worse performance of the GA-MuD regarding that obtainedwith ACO-MuD.
In a more complex system and environment, Lain and Lai (2007)have explored the joint receiver diversity and ACO multiuserdetection techniques. In that work, authors have been shown thenear-far robustness effect achieved by ACO algorithm, differentlyfrom GA algorithm, which performance is degraded under severeNFR condition. Furthermore, Xu et al. (2007) analyzes the ACO-MuD applied to multi carrier DS/CDMA systems (MC-DS/CDMA).ACO-MuD in this context is able to reach the optimal performance,regardless the adopted number of carriers. Under this promisingperformance, the authors have carried out complexity analysisand verified that the ACO-MuD complexity seems the conventionaldetector, even under a system loading of 100%.
An heuristic ACO-based multiuser detector for space–timeblock coding (STBC) systems with receiver diversity was proposed
in Hongwu (2009). This STBC ACO-MuD applies the Pareto-optimally (PO) concept in the pheromone updating step, selectingonly the ants with the best results in that iteration. Numerical re-sults have indicated a very close performance to the optimal one;importantly, authors have demonstrated a very low computationalcomplexity for the STBC ACO-MuD. Furthermore, the proposedACO-MuD presented a performance much better than GA-MuD atthe same operation system and channel conditions; also, the STBCACO-MuD does not present the bit error rate saturation (BER-floor),a degradation performance effect that occurring at high SNRregion.
This work is divided in seven parts. Besides this introductory sec-tion, the adopted system model is described in Section 2. In Section3, the ACO-based heuristic algorithm applied to MuD detection isminutely exploited, while in Section 4, an input parameters optimi-zation methodology was proposed for the algorithm, consideringdifferent channel conditions. An analysis on the number of itera-tions (Golub & Loan, 1996) computed by the ACO-MuD algorithmis carried out in Section 5. The numerical results for the performanceof ACO-MuD detector are presented in Section 6. Finally, the mainconclusions of this work are offered in Section 7.
2. System model
In a DS/CDMA system, deploying BPSK modulation under non-line-of-sight (NLOS) fading channels, the time continuous base-band signal that arrives at the receiver can be described as:
rðtÞ ¼XK
k¼1
Akbkskðt � skÞ � hkðtÞ þ vðtÞ; ð1Þ
where K is the number of active users in the system; t 2 [0,Tb] andTb is the bit period2; Ak is the transmitted signal amplitude of the kth
user, given by Ak ¼ffiffiffiffiEkTb
q, where Ek is the bit energy and Pk the power
of the signal received by the kth user; bk 2 {±1} is the kth user’stransmitted bit information, assumed independent and equiprobabledistributed; hk(t) is the channel impulse response for the kth user,and v(t) is the time continuous additive white Gaussian noise(AWGN), representing the thermal noise and other noise sourcesuncorrelated to the transmitted signals, with bilateral power densityN0/2. The spreading sequence, sk, assigned to the k-th user is repre-sented by
skðtÞ ¼XN�1
i¼0
zðiÞk puðt � iTcÞ; ð2Þ
where zk is the chip vector with elements zðiÞk 2 f�1g and chip per-iod Tc; pu(.) represents the rectangular pulse with unitary amplitudein [0,Tc interval.
For a synchronous system (sk = 0) and NLOS non-selective fre-quency channel, the channel coefficients can be described as:
hkðtÞ ¼ ckðtÞdðtÞ ¼ bkej/k dðtÞ; ð3Þ
where ck(t) is the time continuous channel complex coefficients forthe kth user; bk(t) is the module of ck with a Rayleigh stocastic dis-tribution and /k(t) the phase of ck with an Uniform distribution inthe interval [0,2p].
Multiplying the received signal by the spreading sequence ofthe interest user (matched filter to this sequence), the conventionaldetector (CD) provides the information de-spreading. In this wayand using matrix notation, the output of the matched filter bank(MFB) is given by
y ¼ RCAbþ v; ð4Þ
12878 J.C. Marinello Filho et al. / Expert Systems with Applications 39 (2012) 12876–12884
where y is the K � 1 output vector, R is the K � K correlation matrix,C = diag(c1,c2, . . .,ck) the K � K channel coefficients diagonal matrix,A is the diagonal matrix of received amplitudes, and b is the K � 1vector containing one information bit for each user. v is the sam-pled AWGN K � 1 vector with bilateral power spectral density N0/2.
At the MFB output follows the hard decisor which takes deci-sion according to the signal polarity:
bcd ¼ sgnðyÞ; ð5Þ
where the modified signum function sgn(�) given by
sgnðxÞ ¼�1 if x < 0;þ1 if x P 0:
�
The conventional detector for DS/CDMA uplink receiver (MFB)considers the MAI as a additional background noise, being not ableto separate multiple access interference (MAI) from the interestsignal. On the other hand, the multiuser detectors (MuD) takesadvantage of MAI as a way to takes its performance closer to theoptimal. In Verdú (1998), it was shown that the optimal multiuserdetector (OMuD), or maximum likelihood (ML) detector, calculatesthe cost function of all the possible candidate-solutions, and returnas the optimal solution the argument of the higher value found.The cost function can be expressed as:
f ð#Þ ¼ R 2yT CHA#� #T CARACH#
n oð6Þ
where Rð:Þ is the real operator and # the K � 1 information bits can-didate vector. Consequently, the estimated transmitted bit vectorfor the K users is defined as:
bopt ¼ arg max#2f�1gK
f ð#Þ: ð7Þ
Since the optimal detector (ML) calculates the cost function forall the possible solutions, it is immediate that its performancegrows exponentially with the users number K, because the numberof possible combinations is given by 2K.
From the computational complexity viewpoint, a remarkablereduction in the number of computed operations can be obtainedcalculating previously F 1 ¼ 2yT CHA, and F 2 ¼ CARACH in (6), sincethese terms stay fixed along the algorithm iterations, and thereisn’t reason to be re-calculated at each iteration. This way, an effi-cient-running cost function can be deployed:
f ð#Þ ¼ R F 1#� #TF 2#� �
: ð8Þ
3. Heuristic ACO-MuD
Heuristic algorithms are iterative guided search methods insubspaces that presents fast convergence to global optimum, eventhat convergence in 100% of cases is not guaranteed (Hijazi & Nat-arajan, 2004). They are very efficient in obtaining almost optimumperformances, but with a computational cost very lower thanexhaustive search methods. In MuD context, the optimal solution(OMuD) is reached with the maximum likelihood (ML) approach,which performs an exhaustive search among all the candidate vec-tors in (7).
The ant colony optimization is based on the foraging behavior ofthe ant colony in nature. In search of food, the ants of a colony arescattered randomly in their neighborhood. When an ant is success-ful in it search for food, it come back nest and leaves pheromonesin the way. This pheromone will induce the other ants to take thissame way in the search for food, further strengthening the phero-mone trail. If the food at the end of a certain way runs out and theants stops taking it, this pheromone will be evaporated.
For BPSK signaling, uplink receiver side and just one antenna atthe base-station (BS) receiver and each of K users’ transmitter, i.e.,
from the interest user receiver viewpoint at BS, we have a single-input–single output (SISO) communication system, with K � 1interfering users. So, the multiuser detection problem at the BS re-ceiver side is constituted by 2K possible candidate solutions in (7).This solutions are seen by the algorithm as all the possibles vector-candidates (or trails) that the ants can travel. The quality of eachtrail is evaluated by the cost function, defined in Eq. (6). The algo-rithm steps aiming to find a solution that maximizes (6), or analo-gously, find the fastest trail for the ants until the food.
The MFB outputs serve as initial information to the ants. So,from (4) the log-likelihood function (LLF) for the kth user can bedefined:
Lkð�1Þ ¼ 2R �AðkÞyðkÞf g �AðkÞ2Rðk; kÞ; ð9Þ
where AðkÞ ¼ Aðk; kÞCðk; kÞ is the kth signal received amplitude,including the channel effects (fading, path loss and shadowing).
The desirability function is defined using the LLF function:
Dkð�1Þ ¼ 1þ e�Lkð�1Þ: ð10Þ
From the desirability function, the intrinsic affinity function isdefined, which influences the trail decision of each ant along thealgorithm iterations:
gkð�1Þ ¼ Dkðþ1Þ þDkð�1ÞDkð�1Þ : ð11Þ
The signals at the matched filter bank output are assumed asinitial information. It is then necessary to take into considerationthat the decision taken by the ants be influenced by the paths ta-ken previously, which resulted in better results. This way, the solu-tion found by the algorithm will evolve along the iterations.
In order to quantify this evolution, the 2 � K pheromone table P
is created, in which the first row refers to the probability of posi-tive bits, and the second row refers to the probability of the nega-tive bits. Its elements are initialized with probability k. Along theiterations, this table is being filled according to the quality of thepaths taken by each ant and a tendency, measured in terms ofincreasing probability of that specific bit be 1 (positive bit) or 0(negative bit).
The first way to update this table takes into account the pathschosen by each ant in that iteration, and how successful thesechooses were (measured by the cost function evaluation). A pher-omone amount which is equivalent to the cost function valueregarding the path taken by the ant is multiplied by the c coeffi-cient, and incrementally accumulated at the respective positionsin the P matrix:
P ¼ P þ c � f ðtrailðmÞÞ � T ðtrailðmÞÞ ð12Þ
where trail(m) is the path taken by the mth ant in a given iterationand T ðtrailðmÞÞ is a 2 � K filled with 1 in the positions related to thepath taken by the ant and 0 in the others.
The second way to update this table takes into account the bestpath found by the ACO-MuD algorithm until that moment, namedherein #best. Similar to the adopted procedure in the first updatestage, now, a pheromone amount which is equivalent to the costfunction of #best is multiplied by a coefficient r and deposited atthe respective positions of P:
P ¼ P þ r � f #bestð Þ � T #bestð Þ ð13Þ
Aiming to scape from possibles local optima (maxima), at eachnew iteration i, the pheromone table is multiplied by a coefficient(1 � e), being e the pheromone evaporation rate:
Piþ1 ¼ ð1� eÞ �Pi: ð14Þ
Hence, an excessive amount of pheromone is avoided to be accumu-lated over any possible trail.
J.C. Marinello Filho et al. / Expert Systems with Applications 39 (2012) 12876–12884 12879
Once factors, which influence the path choice of the ants alongthe iterations, have been defined, it is possible to define the bitchoice probability:
Pkð�1Þ ¼ Pkð�1Þ½ �a gkð�1Þ½ �b
Pkðþ1Þ½ �a gkðþ1Þ½ �b þ Pkð�1Þ½ �a gkð�1Þ½ �b; ð15Þ
where a and b parameters provide more or less importance(weighting factors) to the pheromone amount and the initial infor-mation, respectively. Note that a is related to the algorithm conver-gence speed, while b is related to the reliability that can be assignedto the MFB output, which must be set a low value in hostile condi-tions of channel and/or system loading (L > 0.5).
Algorithm 1. ACO-MuD
(1) Initialization: Niter, bcd, P = k, dim. 2 � K.(2) Calculate terms F 1 e F 2
(3) Initial Solution:
#best = bcd; fbest = f(#best)(4) ‘‘A priori’’ information mapping:
Lkð�1Þ ¼ 2Rf�AðkÞyðkÞg � AðkÞ2Rðk; kÞ Dkð�1Þ ¼ 1þ e�Lkð�1Þgkð�1Þ ¼ Dkðþ1ÞþDkð�1ÞDkð�1Þ
(5) For i = 1, . . . ,Niter:
Calculate probabilities Pk(±1) according to (15); Paths taken by each ant, m = 1, . . . ,M:– path taken trail(m) is chosen according to theprobabilities calculated above.
– The path chosen by each ant f(trail(m)) is mea-sured by (8)
– If any path is better than fbest, #best and fbest areupdated,
Pheromone Table update.– Pheromone accumulation is done according to
the path chosen by each ant (‘‘massiveincremental’’):
P ¼ P þ c � f ðtrailðmÞÞ � T ðtrailðmÞÞ
– Pheromone accumulation is done according tothe best path found, #best:
P ¼ P þ r � f ð#bestÞ � T ð#bestÞ
– Pheromone evaporation rate:
P ¼ ð1� eÞ �P
(6) When Niter is completed, the algorithm returns #best
At each iteration, the choice of a certain bit related to each anttrail will be taken from the probability defined in (15). If some trailis more successfully than #best, the best-candidate solution isupdated.
After the algorithm performs a specified number of iterationsNiter, the solution found by the algorithm is returned by the vector#best. The ACO-MuD pseudo-code is described in Algorithm 1.
4. ACO-MuD input parameters optimization
Essentially, there are four input parameters in the ACO-MuDalgorithm, a, b, c and r; the values assigned to these parameterscan dramatically affect algorithm’s performance.
The parameter a is related to the weight given to the informa-tion registered in the pheromone table during the probability cal-
culation. As a grows, more and more ants choose to take the betterpath identified in the table (higher probability value). Thus, thealgorithm’s convergence speed is improved, because the ants tendto choose the same way quickly. This affects the convergence timeand, as a consequence, the algorithm’s complexity.
The parameter b is related to the a priori information during theprobability calculation. b increasing implies in more ants followingthe initial solution trend, i.e., choosing the solution given by theMFB outputs y, in the MuD context. However, if the initial informa-tion is not reliable, i.e., in multiuser scenarios which SNR is lowand/or system loading is high (L P 0.7), high values of b could in-duce the ants to choose a mistaken path, increasing the system’sBER.
On other hand, the parameters c and r are related to thepheromone accumulation according to the quality of paths takenby each ant and the best path found so far #best, respectively.According the study about the ACO algorithm parameters ap-plied to the traveling salesman problem done in Dorigo, Mani-ezzo, and Colorni (1996), the ACO algorithm performance isnot affected by the values assigned to c and r parameters. In-deed, our simulation results described in the next subsections,related to these two input parameters optimization applied tothe MuD problem, offer support to the conclusions found in Dor-igo et al. (1996).
The evaporation rate e parameter is responsible to avoid anexcessive pheromone accumulation over a specific path, avoidinga mistaken convergence of the algorithm over some local optima.Hence, this parameter optimization also is quite relevant in orderto achieve high successful convergence rate performances, mainlywhen the objective function presents many local optima, such asthe ACO-MuD function, Eq. (6), and specially under specific systemoperation situations, i.e., high loading and hostile channel condi-tions. However, in the current study the parameter e assumes value0.5, and its importance related to the algorithm’s performanceimprovement will be analyzed in the next work.
Next, it is carried out a complete analysis optimization on thefour input parameters of ACO algorithm, specifically applied to theMuD problem considering the reverse link of DS/CDMA systems un-der flat frequency fading channels and different mobility conditionsfor the mobile terminals. Monte-Carlo simulation method is de-ployed in order to determine the optimum values of the ACO-MuDinput parameters. A 20dB SNR, Gold spreading sequences withlength (processing gain) 31 and system loading L% ¼ 100 � K
N ¼100% have been adopted. For the others ACO-MuD parameters,the following values have been assumed: initial pheromone proba-bility, k = 0.01; population = 30 ants, and Niter = 20 iterations. Table1 presents the values of the system and channel parametersdeployed in the ACO-MuD input parameter optimization analysis.
Note that the performances reached by the ACO-MuD, in termsof bit error rate (BER), were compared to the performance reachedwhen there are only one active user in the system, namely SuB(single-user-bound), since due to the full system loading (K = 31)the optimum multiuser detector (OMuD) performance calculationbecomes computationally impossible.
The optimization is made starting from presetting initial valuesfor the four main parameters, for instance, a = 1, b = 1, r = 8 andc = 1. Keeping three parameters fixed and ranging the fourth, a firstset of curves for the ACO-MuD input parameter optimization couldbe obtained. Then, the four optimized parameters at this first stepof optimization are updated. Hence, a second set of curves for theoptimized input parameters could be obtained, now in a narrowervalues range, being the optimized values of the first step the mid-dle of the values range. The values obtained at this second optimi-zation step are then assumed as optima for the ACO-MuDalgorithm at that channel condition and system operation point.
Table 1System and channel parameters used in the ACO-MuD analysis.
Parameter Adopted values
DS/CDMA systemProcessing gain N = 31Spreading sequence Gold 31Modulation BPSKNumber of users K = 31Signal–noise ratio SNR 2[5;30] dBSystem loading L ¼ K
N ¼ 1Near-far effect NFR = 0 dBMobility U½0; Vmax� ½km=h�
Vmax = [60;120;240]
Channel
Fading channel Flat Rayleigh
ACO-MuD algorithmPheromone initial prob. k = 0.01Evaporation rate e = 0.5Population M = 30 antsIterations number Niter = 20
Input parameters optimization – ACO-MuDPheromone amount a 2 [0;3]Initial information b 2 [0;15]Pheromone accumulation due #best r 2 [1;13]Pheromone accumulation due trails c 2 [1;7]
12880 J.C. Marinello Filho et al. / Expert Systems with Applications 39 (2012) 12876–12884
4.1. ACO-MuD performance under low speed vehicular channels
Fig. 1(a) shows the first performance analysis ranging theparameters according the methodology described above. For theparameters a and b, it could be observed an optimum value trend,given by a = 0.6, b = 6. For the parameters r and c, one can see thatthere were not performance degradation throughout their respec-tive range values. Hence, intermediate values have been assumedgiven by r = 5 and c = 3. Then, this set values has been deployedas the basis for the second optimization step for the parametersa and b. Results in Fig. 1(b) is taken considering a narrower rangecentered on the respective optimum value obtained from the firstoptimization step.
Finally, the optimum input parameter values for the ACO-MuDoperating on non-selective fading channels with mobile units mov-ing with uniformly distributed speeds in the range v U½0;60�km=h and system loading of L% = 100% were obtained, as shown atthe first line of Table 2 (Vmax = 60 km/h).
4.2. ACO-MuD performance under high and ultra-high speed vehicularchannels
Fig. 2 put in perspective the final results for the input parametervalues optimization under high (Vmax = 120 km/h) and ultra-high(Vmax = 240 km/h) speed vehicular non-selective Rayleigh channels.
Under higher mobility, and at same system loading L = 1, onecan see from Fig. 2 the similar results regarding the optimal inputparameters values obtained by the two-step optimization proce-dure for the low-speed vehicular channels, as described in the Sec-tion 4.1.
According to the methodology described above, under highmobility the optimal values for the a and b parameters in the firstoptimization step were changed to a = 0.6 and b = 7.5. Similarly tothe low mobility case, for r and c parameters it was assumed r = 5and c = 3. This set was used in the second optimization stage of theparameters, considering a narrower ranges centered in each opti-mum value found in the first step.
The second line of Table 2 summarizes the optimum ACO-MuDinput parameters values set for mobile units moving with uni-formly distributed speeds in the range v U½0; 120� km=h underflat frequency fading channels and system loading of 100%.
Furthermore, Fig. 2 shows too the final ACO input parametersoptimization values for ultra-high speed vehicular channels. Again,it was observed significant performance changes only for theparameters a and b. After obtained the first set of optimized valuesfor these parameters, a = 0.6, b = 3 and assuming fixed values forr = 5 and c = 3, this set serves as the basis for the second step opti-mization for a and b, but in a narrower range centered respectivelyin each one of the optimum values obtained at the first step.
Last line of Table 2 shows the optimum values set for the ACO-MuD input parameters under flat frequency fading channels withmobile units velocities uniformly distributed in the range v U½0;240� km=h.
Analyzing the optimized values on different mobility situations,one can conclude from Fig. 2 that there were not significant differ-ences for distinct channel mobility conditions, from low to ultra-high speed vehicular channels. The parameters a and b, whichare related to the convergence speed and the a priori informationreliability, respectively, are more influential to the algorithm’s per-formance (convergence), while the parameters r and c proved tobe less sensible, corroborating the analysis carried out in Dorigoet al. (1996) for a general purpose discrete ACO algorithm.
Furthermore, the input parameters optimization proceduredeveloped herein has indicated the ACO-MuD efficiency in achiev-ing the quasi-optimum BER performance with no necessity of a sig-nificant adjustments in the algorithm’s input parameters values,i.e., the algorithm is able to achieve the near-optimum BER perfor-mance under a relatively wide ranges for the four input parame-ters. Besides, the ACO-MuD input parameter optimization resultshave revealed that the parameters r and c are virtually immuneto the multiple access channel mobility, also indicating that thealgorithm is able to operate robustly under different Rayleighchannels mobility conditions and full system loading.
5. Computational complexity
In this section, the computational complexity of the conven-tional (CD), ACO-MuD, and OMuD DS/CDMA detectors have beencompared. In this analysis the number of necessary number ofoperation for each detector is presented. In order to obtain thenumber of operation in each algorithm, the respective pseudo-codehas been analyzed, counting the number of sums and multiplica-tions (Golub & Loan, 1996) performed. For the ACO-MuD algo-rithm, the number of operations is calculated considering thetotal number of iterations until the convergence, Niter. In order tosimplify and compare the algorithms, only the number of multipli-cations and divisions were considered, since the sum and subtrac-tion operations has computational time negligible. The Table 3summarizes the number of operations for each algorithm.
6. Numeric results for MuD problem with optimized ACO inputparameters
In order to prove the ACO-MuD algorithm robustness and effi-ciency, in this section the performance of the heuristic detector withand without optimized input parameters is compared, regarding thenumber of iterations (convergence speed) and signal-to-noise ratio(SNR). Fig. 3 shows the converge velocity of the ACO-MuD underSNR = 20 dB, Vmax = 120 km/h and with and without optimized in-put parameters. One can see that with optimized input parametersvalues found in Section 4, the optimized ACO-MuD with parametersa = 0.4, b = 4, r = 5 and c = 3 achieves the BER performance bound(maximum likelihood optimum detector, i.e., performance veryclose to SuB) after five iterations. Thus, for the optimized ACO-MuD, Niter = 5 iterations are enough for the complete convergence
Fig. 1. ACO-MuD input parameters optimization: Vmax = 60 km/h and SNR = 20 dB.
Table 2Optimized values for the ACO-MuD input parameters considering different system’smobility conditions.
Parameters a b r c
Vmax = 60 km/h 0.6 7 5 3Vmax = 120 km/h 0.4 7 5 3Vmax = 240 km/h 0.6 4 5 3
J.C. Marinello Filho et al. / Expert Systems with Applications 39 (2012) 12876–12884 12881
under non-selective Rayleigh SISO DS/CDMA channels and full sys-tem loading (L = 100%) condition.
In order to confirm the convergence performances related toACO-MuD input parameters values, as found in Figs. 3 and 4 presentsthe ACO-MuD BER performance for a wide range of SNR 2 [0;25]. Asone can immediately conclude, the best ACO-MuD performance(curve with marker –M–) is achievable under optimized inputparameters; besides, under with the optimized input parameters
values, the ACO-MuD is able to achieve the OMuD performance forall SNR values ranging [0;25] dB.
Note that, for this scenario, while the OMuD needs 231 costfunction calculations (cfc), resulting in over 2 billions of cost func-tion calculations, on the other hand the ACO-MuD with optimizedparameters evaluates a number of cfc given by the product of theants population M and the algorithm iterations Niter. With the val-ues assumed in the simulations, this complexity is of the order ofC ¼ M � Niter ¼ 150 [cfc]. Hence, the optimized ACO-MuD is ableto find solutions very close to those obtained by the OMuD, butwith only a fraction of the cfc, i.e., �1.4 � 107 times lower thanthe number needed by OMuD.
The complexity in terms of number of operations needs forACO-MuD (with optimized input parameters) convergence is ex-plored in Fig. 5, indicating the marginal increasing in the necessarynumber of operation as a function of the increasing number of ac-tive users. Besides, for comparison purpose, the CD and OMuD
0 1 2 360120240
10−2
α Vmax
BER
2 4 6 8 10 1260120
240
10−2
σVmax
BER
2 4 660120
240
10−2
γVmaxBE
R
0 5 10 1560120240
10−2
βVmax
BER MF
ACOSuB
Fig. 2. ACO-MuD input parameters optimization. SNR = 20 dB.
0 2 4 6 8 10 12 14 16 18 20
10−2
Iterations
BER
CDSuB (BPSK)ACO α = 0.4, β = 4, σ = 5 e γ = 3ACO α = 0.07, β = 0.6, σ = 0.12 e γ = 5ACO α = 7, β = 6, σ = 12 e γ = 0.05
Fig. 3. ACO-MuD convergence performance under SNR = 20 dB, flat Rayleighchannel, L = 100% and Vmax = 120 km/h.
0 5 10 15 20 25
10−3
10−2
10−1
Eb/No [dB]
BER
CDSuB (BPSK)ACO α = 0.4, β = 4, σ = 5 e γ = 3ACO α = 7, β = 6, σ = 12 e γ = 0.05ACO α = 0.07, β = 0.6, σ = 0.12 e γ = 5
Fig. 4. BER performance for ACO-MuD under Flat Rayleigh channels, L = 100%loading, Vmax = 120 km/h, considering different values for the input parameters.
12882 J.C. Marinello Filho et al. / Expert Systems with Applications 39 (2012) 12876–12884
complexity were considered. The figure pinpoints the exponentialcomplexity behavior of the OMuD with the number of users, whichbecomes unfeasible for a high number of users, while the ACO-MuD complexity keeps itself close to the conventional detectorwith polynomial complexity order OðK3Þ, as shown in Table 3.
Comparing the ACO-MuD complexity according to the expres-sions shown in Table 3 with the previously adopted M, N, K and Niter
values, the number of operation needs by ACO-MuD results�6.8 � 106 times lower than OMuD. This difference in the complex-ity ratio between the detectors, when comparing with the previousresult taking into account only cfc, is justified by the number ofoperations analysis adopted here, which takes into account all thepre-processing performed by the ACO-MuD algorithm beyond thecost function calculation, while for OMuD the overall processingcomplexity is almost entirely dominated by the cfc step.
Deploying the same values for the input parameters used inFig. 3, the excellent achievable ACO-MuD BER performance in
terms of convergence speed is put into perspective in Fig. 6, consid-ering a wide range of system SNR operation and full system loadingL% = 100%. The number of iterations to achieve total convergenceincreases with SNR values; e.g., for SNR = 10 dB, Niter = 2, whilefor SNR = 20 dB, Niter = 5 and for SNR = 30 dB, Niter = 9. However, itis worth noting that the necessary number of iteration for totalconvergence remains small, proven the high efficiency of theACO-MuD with optimized input parameters in finding the globaloptimum under a high multiple access interference and a widerange of SNR.
The robustness to the near-far effect for the three MuD’s is com-pared in Fig. 7, which evaluates the BER performance degradation asa function of increasing system loading and NFR. Here, half of theusers was considered as the interest performance evaluation andthe another half as interfering users. It is clear that the ACO-MuDperformance degradation is solely affected by the near-far effect,been robust to the system loading increasing. However, under not
Table 3Number of operations necessary for each DS/CDMA detector.
Detector Number of operations
CD K N
ACO-MuD [3K(K2 + K + 5) + K N(2K + 1)] + Niter[M(K2 + 4K + 1) + 24K + 1]
OMuD K(3K2 + 2K + 1) + K N(2K + 1) + 2K(K2 + 2K)
1 2 3 4 5 6 7 8 9 10 11 120
1
2
3
4
5
6
7
8x 105
Number of Users, K
Num
ber o
f Ope
ratio
ns
CDACO−MuDOMuD
1 2 3 4 5 6 7 8 90
2
4
6x 104
Fig. 5. Number of operation as a function of increasing K.
0
20
40
51015202530
10−4
10−3
10−2
10−1
SNR (dB)Iterations
BER
ACOSuB
Fig. 6. ACO-MuD convergence performance under SNR 2 [5;30] dB, flat Rayleigh,L% = 100%, Vmax = 120 km/h; a = 0.4; b = 7; r = 5; c = 3.
0.40.6
0.81 −10 −5 0 5 10 15 20
10−2
10−1
NFR (dB)System Loading, L
BER
MFACO MuDSuB
Fig. 7. ACO-MuD performance under NFR 2 [�10;20] dB, flat Rayleigh, L% 2 [40;100]%, Vmax = 120 km/h, SNR = 20 dB; a = 0.4; b = 7; r = 5; c = 3.
J.C. Marinello Filho et al. / Expert Systems with Applications 39 (2012) 12876–12884 12883
so strong near-far effect, the ACO-MuD is relatively robust in therange NFR 2 [�10;10] dB for small system loadings, becoming alittle less robust under high system loadings approaching one in this
NFR range. To be more specific, in the range of NFR 2 [�10;5] dB, theACO-MuD performance is very close to the theoretic OMuD bound,and just marginally degraded in the range NFR 2 [5;10] dB.
7. Conclusions
A heuristic multiuser detector based in an ant colony optimiza-tion (ACO-MuD) suitable to perform in BPSK DS/CDMA systemsunder flat Rayleigh fading channels was proposed and successfullycharacterized. An input parameters optimization methodology forthe ACO-MuD was carried out, in order to achieve the best possibleperformance with a fixed number of iterations. The optimizedvalues proved to be robust enough, such a way to ensure a near-optimum performance for different system and channel operationsscenarios, as well as different power control situations, without theneed of significant changes in the algorithm.
Indeed, the input parameters optimization for the ACO-MuDshows that the parameters r and c are virtually immune to thechannel mobility and loading system variation, indicating thatthe algorithm is able to operate robustly under different mobilechannel coherence times; in practice, only the a and b parametersneeds to be slightingly adjusted when drastic changes in both sys-tem operation conditions and multiple access channel occur.
The computational complexity for the proposed ACO-MuDdeploying optimized input parameters, was analyzed by both thenumber of cost function calculations, as well as by the number ofoperations. ACO-MuD complexity results very low, resulting in avery small fraction of the OMuD computational complexity(�10�7) under full system loading condition, wide range of NFRand channel mobility, but with very similar performances.
Acknowledgements
This work was supported in part by CAPES and Araucaria Foun-dation (scholarships) and by the National Council for Scientific andTechnological Development (CNPq) of Brazil under Grant 303426/2009-8. The authors would like to thanks the fruitful technical dis-cussions shared with colleagues from the Telecomm and DSP Lab-oratory of Londrina State University.
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20
Parte II
Detector ACO para sistemas
MIMO
1
Lattice Reduction Aided Detector for MIMOCommunication via Ant Colony Optimization
José Carlos Marinello, Taufik Abrão
Abstract—In this work heuristic ant colony optimiza-tion (ACO) procedure is deployed in conjunction withlattice reduction (LR) technique aiming to improve theperformance-complexity tradeoff of detection schemes inMIMO communication. A hybrid LR-ACO MIMO detec-tor using the linear minimum mean squared error (MMSE)criterion as initial guess is proposed and compared withother traditional (non)linear MIMO detector, as well asheuristic MIMO detection approaches from the literaturein terms of both performance and complexity. Numericalresults show that the proposed LR-ACO outperforms thetraditional ACO-based MIMO detectors and the ACOdetector with the MMSE solution as initial guess, witha significant complexity reduction while is able to reachfull diversity degree in all antennas and modulation orderconfiguration scenarios considered.
Index Terms—Large-MIMO systems, ant colony op-timization, evolutionary computing, lattice reduction,maximum-likelihood estimation, MMSE, diversity order.
I. INTRODUCTION
Multiple-input-multiple-output (MIMO) systems areknown by providing a significant spectral efficiencyimprovement on wireless communication systems [1].Therefore, in the last decade many researches in thisfield have been done due to the new wireless systemrequirement of high data rates, imposed by the recenttelecommunications services and technologies, alliedwith the large interest on saving resources, like spectrumand power. Furthermore, exploiting the path diversitybetween the multiple-antenna transmitter and multiple-antenna receiver, another MIMO configuration allowsto improve the link reliability for a given data rate,or maximize the data rate given a certain reliabilityrequirement [2]. It’s also known that very high datarates can be achieved when using a large number ofantennas and/or modulations orders (Large MIMO), andthat on these cases the detection task becomes challenger,
Part of this work (reduced version) has been submitted to the IEEE-WCNC’13 Conference.
J. C. Marinello and T. Abrão are with the Electrical Engineer-ing Department, State University of Londrina, PR, Brazil. E-mail:[email protected]; [email protected]
since there is a considerably enhancement on the interlayer/antenna interference (ILI), and/or a lower noiserobustness, requiring on this way more sophisticated andefficient detection techniques.
Among the linear detectors widely known the zeroforcing (ZF) and the minimum mean squared error(MMSE) based techniques present low complexity andthe ability to operate under ill-conditioned channel matri-ces; however both linear detection techniques for MIMOis clearly inferior to the performance achieved by themaximum likelihood (ML) detector. Recently, the spheredecoding (SD) approach [3] has becoming an alternativeto the ML, presenting a near-optimum performance;however SD approach results in a prohibitive complexityfor implementation under low or medium SNR operationregions on real communication systems; in fact, underlow SNR operation region, the SD complexity becomesof the same order of ML complexity.
An appealing way to improve the MIMO linear de-tectors performance under correlated channels and si-multaneously resulting in a feasible complexity consistsin deploying lattice reduction (LR) technique [4], [5] .This technique transforms the partial correlated channelinto an equivalent one, with a better conditioned chan-nel matrix. Having a near-orthogonal near-uncorrelatedchannel transformation implemented, at the receiver sidethe detection can be carried out easily deploying a low-complexity linear detector scheme.
The ant colony optimization (ACO) is a techniqueoriginally proposed for combinatorial optimization prob-lems, such as the traveling salesman problem [6]. It’sinspired on the behavior of ants in nature, that, lookingfor food, having the ability of exploit the territoryaround, find the ways that lead to the best sourcesof food, and leave trails that will help other ants ontheir searches. Many works have been disseminatedapplying this technique to several combinatorial (ordiscrete) and continuous optimization problems that arisein telecommunications, such as detection in code divisionmultiple access systems (CDMA) [7] [8], detection inMIMO systems [9] [10], resource allocation on wirelessnetworks [11] [12], among others.
A simple ant colony optimization applied to MIMO
2
detection is presented in [9]. Authors combine ACOheuristic technique with the well-known zero-forcing(ZF) and V-BLAST MIMO detectors. However, nu-merical results show that the proposed ACO-ZF andACO-BLAST MIMO detectors were not able to achieveML performance for medium and high SNR regions.In [10], a different detection scheme based on ACOwas proposed, but since this technique presented a poorperformance compared with ML, a modified ACO algo-rithm is suggested herein to improve the performance-complexity trade-off in a near-optimal MIMO detection.
In this sense, this work proposes, as a alternative to theACO detector of [9] and [10], a near-optimum MIMOdetector based on ant colony optimization heuristic ap-proach, in which part of the search complexity has beenshifted for the initial detection stage, by deploying alow complexity linear detector works as a start pointin the search solution carried out in a second stage. Thisway, will be shown that, performing the search on abetter conditioned domain provided by the LR technique,it’s possible to obtain a significant improvement in theperformance × complexity tradeoff with this LR-ACOdetector for MIMO systems, even when applied to aMIMO system with a large number of antennas (Dense-MIMO systems).
The remainder of this paper is organized as follows:In Section II, the system model and notations are intro-duced; in Section III and IV, different detection schemeswith and without reduction of basis, respectively, arereviewed. In section V, the state of the art of theACO MIMO detection is viewed. The proposed detectionschemes are introduced in Section VI, and the numericalresults in VII. Concluding remarks can be found inSection VIII.
II. MIMO SYSTEM MODEL
In the adopted multiple-input-multiple-output(MIMO) system it is assumed that there are nT
transmit antennas at the transmitter side, and nR
receive antennas at the receiver. Besides, informationtransmitted symbol vector x = [x1 x2 . . . xnT
]T , wherexi denotes the transmitted symbol at the ith antenna,with xi is assumed taken a value of the squaredquadrature amplitude modulation (M -QAM) alphabet;so, the complex-valued symbol (finite) set is givenby S =
{A+
√−1 · A
}, where the real-valued finite
set A ={±1
2a; ±32a; . . . ; ±
√M−12 a
}, with
√M
representing the order modulation (per dimension)of the corresponding real-valued ASK scheme. Theparameter a =
√6/(M − 1) is used for normalizing
the power of the complex valued transmit signals to
1. Furthermore, we have assumed that the system isdetermined, i.e., nR ≥ nT .
For analysis simplicity, using matrix notation, thereceived complex-valued signal over a MIMO channelis written as:
r = Hx+ n (1)
where x denote the complex valued nT × 1 transmitsignal vector, the corresponding receive signal vectorr has dimension nR × 1; besides, n ∼ CN (0, N0I)represents the additive white Gaussian noise (AWGN)with variance σ2
n = N0, which is observed at the nR
receive antennas while the average transmit power ofeach antenna is normalized to one, i.e. E[xxH ] = InT
and E[nnH ] = InR, thus Eb/N0 = nR/(log2(M)σ2
n)holds. Furthermore, the nR×nT complex-valued channelmatrix H was assumed uncorrelated complex Gaussianfading gains with unit variance. Flat fading environmentwas assumed, where the channel matrix H is admittedconstant over the entire frame and changes independentlyfrom frame to frame (block fading channel assumption).
In order to facilitate the numerical analysis, real andimaginary part of (1) are treated separately; so, thesystem model can be rewritten as:
r = Hx+ n (2)
with the real-valued channel matrix
H =
[ℜ{H} −ℑ{H}ℑ{H} ℜ{H}
]∈ Rn×m (3)
and the real-valued vectors
r =
[ℜ{r}ℑ{r}
]; x =
[ℜ{x}ℑ{x}
]; n =
[ℜ{n}ℑ{n}
]∈ Rn
(4)where r,n ∈ Rn, x ∈ Am, m = 2nT and n = 2nR.Note that now, the information vector assumes valuesonly over the finite set of real-valued: x ∈ A2nT ,where A is the set of real-valued entries in the signalconstellation, e.g., A = a
2 · {±1,±3,±5 ± 7} in a 64-QAM signaling.
III. MIMO DETECTORS
A. ML MIMO Detector
The maximum likelihood (ML) detector (MLD) con-sists in to find the symbol in the set A2nT that maximizesthe likelihood function, i.e
xML = arg maxx∈A2nT
f(r|x) (5)
where f(r|x) denotes the likelihood function of x for agiven r. Since the noise in (2) is assumed to be circularlysymmetric complex Gaussian (CSCG), the ML detection
3
becomes:
xML = arg minx∈A2nT
||r−Hx||2 (6)
= arg minx∈A2nT
(r−Hx)T R−1n (r−Hx)
where Rn is the noise covariance matrix, which isassumed i.i.d., so we have
Rn = diag(σ2n,1, σ
2n,2, . . . , σ
2n,nR
)= N0InR
,
As shown in (6), the MLD in MIMO systems isequivalent to an exhaustive search of a combinatorialoptimization problem, which becomes prohibitive whenthe constellation order and number of antennas increasesubstantially; for example, if M = 16 and nT = 4, thenumber of candidates to be evaluated becomes astonish-ingly large M2nT ≈ 4.3 · 109.
B. Linear Zero-Forcing MIMO Detector
In a conventional zero-forcing (ZF) detector the in-terference is completely suppressed by multiplying thereceive signal vector x with the Moore-Penrose pseudo-inverse of the channel matrix given by:
H† =(HTH
)−1HT (7)
The decision step consists of mapping each element ofthe filter output vector.
The conventional ZF detector output is given by:
xZF = H†r = H† (Hx+ n) = x+H†n (8)
i.e., the linear transformation matrix for zero-forcingis given by the Moore-Penrose pseudo-inverse matrix:WZF = H† =
(HTH
)−1HT .
C. Linear MMSE MIMO Detector
The conventional minimum mean squared error(MMSE) detector for MIMO system is another lineardetector, whose preprocessor output is given by:
xMMSE = WTMMSEr = H
(N0
EsI+HTH
)−1
r (9)
Note that both ZF and MMSE MIMO detectors canalso be derived by the following unified approach. Letus consider the extended real-valued channel matrix asfollows:
HEXT =
[H
κI2nT
](10)
where κ ≥ 0 is a constant. Hence, the pseudo-inverse ofHEXT is obtained extending (11):
H†EXT =
(κ2I2nT
+HTH)−1 [
HT κI2nT
](11)
Let take [H†
EXT
]=
(κ2I2nT
+HTH)−1
HT
then, we can re-define the linear transformation matrixfor the ZF and MMSE MIMO detectors as follows:
WT =[H†
EXT
](12)
=
{WT
ZF = H† if κ = 0
WTMMSE if κ =
√2N0
Es=
√2N0
Eb log2 M
D. Sphere Decoding
Sphere Decoding solves (6) by searching only amongthe solutions inside a hypersphere of radius d [13]:
||r−Hx||2 ≤ d2 (13)
To obtain those points without an exhaustive search,the search is made taking advantage of the triangularstructure of the R matrix, given by the QR decomposi-tion of the channel matrix H, such that H = QR, beingQ an orthogonal matrix with dimension n ×m and Ran upper triangular matrix, dimension m × m. Hence,the points inside the hypersphere can be found layer-by-layer, starting from the last row of R, from the PartialEuclidian Distances by evaluating:
||QTr−Rx||2 ≤ d2 (14)
at that layer, and summing the previous. Then, enumer-ating the points inside the hypersphere, the solution isthe one closest to the center QTr.
IV. LATTICE REDUCTION AIDED MIMO DETECTORS
The aim of lattice-reduction is to transform a givenbasis H into a new basis H with vectors of shortestlength or, equivalently, into a basis consisting of roughlyorthogonal basis vectors. Usually, H is much betterconditioned than H.
Hence, let suppose that the lattices generated by thecolumn vectors of MIMO channel H and LR-reducedchannel H matrices are the same. This implies thereexists an integer unimodular matrix T that satisfies:
H = HT (15)
Then, the received signal in (2) can be rewritten as:
r = Hx+ n
= HT−1x+ n
= Hz+ n (16)
where z = T−1x
4
A. LR-based Zero-Forcing MIMO detection
For the zero forcing (ZF) detector, received signalvector on the nr receive antennas, r, is multiplied by thereduced channel pseudo-inverse matrix H†, resulting:
zZF = H†r = H†(Hz+ n
)= z+ H†n (17)
Since in the lattice reduction procedure the originalsymbols of the QAM constellation are shifted and scaledby the matrix T, before re-mapping the vector zZF tothe original constellation, this shifting-scaling operationsmust be reverted [14], resulting in:
zZF = 2 ·⌈zLR − β′T−11
2
⌋+ β′T−11 (18)
where 1 is unitary column vector, and ⌈·⌋ is the roundingfunction for the near integer.
As shown in [4], this process is equivalent to thequantization operation (over the vector zZF) for thenearest point of the constellation T−1x, as describedby (18). After quantization operation, the vector zZF isdemapping onto the original base by multiplying to theunimodular matrix:
xZF = TzZF (19)
Hence, xZF is detected as the symbol associated to theconstellation point with the minimal distance.
B. LR-based Minimum Mean Squared Error MIMO de-tection
For the minimum mean squared error (MMSE) de-tector, the extended received signal vector rEXT, defined
as rEXT =
[r
02nT
], where 02nT
is a column vector with
zeros and length 2nT , is multiplied by the pseudo-inverseof the reduced extended channel matrix H†
EXT, resulting:
zMMSE = H†EXTrEXT (20)
Then, the quantization operation and the demappingonto the original base is made in the same way as theLR-ZF.
V. ACO MIMO DETECTION
From (6), the MIMO detection problem can be seen asa combinatorial optimization problem. Therefore, the antcolony optimization is a proper technique to be applied.In the ACO-MIMO detector proposed in [10], Nants antsseeks iteratively for better solutions accordingly to acost function. For MIMO communications systems, theoptimal solution is given by (6), so the following cost
function can be adopted in the ant colony optimizationcontext:
F (s) = ||r−Hs||2 (21)
In a convenient way, we can decompose the H ma-trix using the QR decomposition. Hence the vector ofreceiving signal is readily re-written as:
r = QHr = QH(Hx+ n) = Rx+QHn, (22)
been the cost function conveniently re-written as:
F (s) = ||r−Rs||2 (23)
Hence, the dij distance related to the "sij" path, i.e.,assumed that the jth symbol of the set A has beentransmitted at the ith antenna is calculated into the ACOalgorithm deploying the recursive relation:
dij = |ri −m∑
l=i+1
Rilsl −Riisj | (24)
where i should progressively decrease from m to 1, sj ∈A, and sl are the hard decision of the transmitted symbolsl that have been tentatively made decision [10]. Hence,si can be iteratively obtained as
si = arg minsij∈A
dij , (25)
Besides, the distances dij are then converted into theheuristic values ηij using a log-sigmoid function:
ηij =1
1 + edij(26)
Such values have influence in the probability cal-culation of the paths traced by the active ants acrossthe iterations of the algorithm; this ant paths followingprobability is computed as:
pij =[τij ]
α[ηij ]β∑
j∈M [τij ]α[ηij ]β(27)
where τij is the pheromone level over the path sij ,and the constants α and β weight the importance ofτij and ηij , respectively. These levels are a way toimplement evolution mechanism in the algorithm alongthe iterations, and are analog to the substances that realants deposit on the best paths in searching food process.The log-sigmoid converted distances and pheromonelevel are set up with a correspondent initial value, andas soon as the Nants complete their paths s
(n)k , both
variables are updated at the n + 1 iteration accordingto (24) and
τ(n+1)ij = (1 + ρ)τ
(n)ij +
Nants∑k=1
∆τkij (28)
5
respectively, where ρ is the pheromone evaporation rate(ER) and ∆τkij is given by:
∆τkij =
{F (s
(n)k ) if (i, j) ∈ s
(n)k ,
0 otherwise.(29)
At the end of the first iteration, the cost functionof the paths taken for each ant had been calculated,being the lower cost function and the associated pathassigned to the variables Fbest and sbest, respectively.These variables are updated along the iterations. At theend, sbest represents the solution given by the ACOsearch algorithm.
VI. PROPOSED DETECTION SCHEMES
Firstly, some modifications in the ACO-MIMO de-tector described above is introduced in such a waythat the search can be readily started from an initialpoint, which in fact would be the solution offered byany low-complexity linear MIMO detector; besides, thepheromone update is done on a closest way of that shownin [6]. Finally, adapting this heuristic search algorithmtaken into account the improved channel conditionsprovided by the LR technique, we can considerablyimprove the overall MIMO system performance at costof an affordable increasing in complexity, as discussedin Section VII.
A. ACO MIMO Detection with an Initial Guess
Initially, if in the eq. (24) we use s as the initialsolution given by a any low-complexity linear MIMOdetector, such as ZF or MMSE, with or without LRaiding, the heuristic information will be obtained takingadvantage of the information provided by this detector.This way, as much better the information provided bythe initial detector is (i.e., the bit-error-rate (BER) per-formance), the more reliable the final achieved heuristicinformation, and less processing (time-consuming) isneeded for the ants search and find a reliable high-qualitysolution. Hence, in this work we use the initial guess forthe ACO input as s = xMMSE, deployed in the calculationof (24).
The other modification introduced herein is related tothe pheromone updating. Since the cost function (23)should be minimized, eq. (28) and (29) may be notcompletely appropriate, since these calculations possiblyintroduce an excessive pheromone accumulation on thepaths controlled by ρ, which can provide erroneously ahigher pheromone deposition level over the paths underworse evaluation (i.e., high cost function). To correct this
effect, herein (29) is re-written as:
∆τkij =
{ϱ
F (s(n)k )
if (i, j) ∈ s(n)k ,
0 otherwise.(30)
where ϱ is a constant to be adjusted experimentally.In a similar way suggested in [6], herein we also
implemented a second pheromone deposition rule, whichtakes into account the best solution found so far by theACO algorithm:
∆τ(n)elitist =
{δ
F (s(n)best)
if (i, j) ∈ s(n)best,
0 otherwise.(31)
where δ is a constant to be adjusted experimentally.As a consequence, eq. (28) becomes:
τ(n+1)ij = (1− ρ)τ
(n)ij +
Nants∑k=1
∆τkij +∆τ(n)elitist (32)
where ∆τkij is given by (30), and ∆τ(n)elitist by (31).
B. LR-based ACO MIMO detection
One of the main issues to be solved in the ACO MIMOdetector adaptation to the reduced domain, with betterconditioned channel gain matrix, is the fact on the LRdomain, there is no fixed constellation, as discussed in[15]. Since z = T−1x, each channel matrix remains ona different reduced transformed constellation. Obviously,to calculate all these constellation points for each channelmatrix gain realizations is practically unfeasible.
In order to solve this problem, the ants’ search forthe optimal solution initially takes place on the neigh-borhood of a initial solution, namely reduced domainneighborhood (RDN) procedure [15]. In this work, theinitial solution is feed by the LR-MMSE MIMO detector.
Since every constellation can be seen as a shifted andscaled version of the integer constellation Zn [14], theprocedure starts taking the shifted and scaled version ofthe vector zMMSE, expressed by:
zMMSE =zMMSE − β′T−11
2(33)
Hence, the neighborhood is obtained as the combina-tions of the adjacent integers on each element of zMMSE,after quantization, in other words, for a given dimensioni, the neighborhood will be:
⌈zMMSEi⌋ − N − 1
2, ⌈zMMSEi
⌋ − N − 1
2+ 1, . . .
. . . , ⌈zMMSEi⌋+ N − 1
2− 1, ⌈zMMSEi
⌋+ N − 1
2
(34)
being N the number of elements per dimension to becalculated. N = 5 was adopted in [15]; however, the
6
problem was treated in a complex-values format. Since inthis work we are adopting equivalent real-values format,we have adopted N = 3. So, the j-th neighbor of⌈zMMSEi
⌋, zij , can be defined as:
zij = ⌈zMMSEi⌋ − N + 1
2+ j (35)
In a same way, eq. (24) also should be adapted to thereduced domain. Hence, it becomes [15]:
dij =∥∥∥R(zMMSE − z)
∥∥∥2 (36)
in which i should decrease from m to 1, z is a vectorformed by the elements of ⌈zMMSE⌋ on the positions i+1 . . .m, zij at the i-th position, and zeros at the positionsi− 1 . . . 1, and R is given by the QR decomposition ofthe reduced channel matrix H.
Finally, the cost function deploying the reduced LRdomain description becomes:
F (z) = ||r− Rz||2 (37)
where r = Qr.The remainder of the algorithm is constructed in the
same way as the ACO MIMO detector on the originalchannel gain matrix domain. Pseudo-codes for bothheuristic MIMO algorithms are described in Algorithms1 and 2.
Algorithm 1 ACO-MIMOInput: r, R, xMMSE.Initialization: τij ← τ0, n ← 1, sbest ← xMMSE,Fbest ←∞.
For each path sij :dij = |ri −
∑ml=i+1Rilsl −Riisj |;
ηij =1
1+edij;
endwhile n ≤ It
For each sij :Evaluate pij according to (27);
endFor each k-th ant:
Generate trails sk according to pij ;Evaluate cost function F (sk);
endUpdate Fbest and sbest;Update τij according to (30), (31) e (32);n = n+ 1;
end;Output: sbest.
Algorithm 2 LR-ACO-MIMO
Input: r, R, zMMSE, T.Initialization: τij ← τ0, n ← 1, zbest ← zMMSE,Fbest ←∞.
Evaluate zMMSE according to (33);Generate RDN according to (34);For each neighbor zij :
dij =∥∥∥R(zMMSE − z)
∥∥∥2;
ηij =1
1+edij;
endwhile n ≤ It
For each zij :Evaluate pij according to (27);
endFor each k-th ant:
Generate trails zk according to pij ;Evaluate cost function F (zk);
endUpdate Fbest and zbest;Update τij according to (30), (31) e (32);n = n+ 1;
end;Output: sbest = Tzbest.
VII. NUMERICAL RESULTS
It is well-known that the ACO algorithm performancedepends on the appropriate choice for its parameters,ensuring this way a desirable acceleration in algorithm’sconvergence [16] while simultaneously guarantees a re-duction in the computational complexity. Hence, as afirst step in this section, a procedure aiming to obtaina ACO-MIMO detector input parameter optimization iscarried out. Three different MIMO system configurationsare considered: a) 4 × 4 antennas and 64-QAM; b)8 × 8 antennas and 16-QAM; c) 20 × 20 antennasand 4-QAM modulation. After that, performance andcomplexity analyses under optimized input parametersare developed.
TABLE IACO PARAMETERS.
α [0, 1]β [0.2, 2]ϱ 1δ 3ρ 0.3
Nants 20τ0 0.01
7
A. Input Parameter Optimization
As generically analyzed in [6] and specifically in [16]considering the multiuser DS-CDMA detection problem,the parameters ϱ, δ and evaporation rate (ER) ρ presentlittle influence on the ACO algorithm performance. Thisway, after non-exhaustive tests, these parameters werechosen empirically, been adopted herein the followingvalues: ϱ = 1, δ = 3 and ρ = 0.3.
The parameters that drastically affect the ACO al-gorithm performance are α and β. As discussed in[6], [16], α is related to the importance given to thepheromone levels in the probability calculations, eq.(27), in such a way to determine the convergence speedof the algorithm, while β is related to the importancegiven to the "a priori" information in (27).
The goal with the introduction of the input parametersoptimization procedure is to obtain a set of values thatensures an affordable and suitable algorithm performanceadjusted to deal with the MIMO detection problemunder realistic channel scenarios and a wide range oftransmit and receive antennas configurations. The nu-merical results discussed in the following corroboratethe good performance × complexity for the proposedACO-MIMO detector under a wide varieties of MIMOsystem configurations.
Varying the parameters α ∈ [0, 1] and β ∈ [0.2, 2],with steps of 0.2, all possible parameters combinationson these intervals was performed. Fig. 1 shows the bit-error-rate (BER) performance as a function of these inputparameter variations, for 4 × 4 antennas and 64-QAMmodulation format. Fig. 2 shows α and β parametersvariation for 8×8 antennas under 16-QAM. Finally, Fig.3 shows the parameters variation for 20 × 20 antennasunder 4-QAM.
00.20.40.60.81
00.5
11.5
2
10−5
10−4
αβ
BE
R
LR−MMSE MIMO 4x4LR−ACO MIMO 4x4
Fig. 1. LR-ACO input α and β parameters optimization for 4× 4under 64-QAM modulation. SNR= 26dB.
00.20.40.60.810
12
10−4
10−3
10−2
βα
BE
R
LR−MMSE MIMO 8x8LR−ACO MIMO 8x8
Fig. 2. LR-ACO input α and β parameters optimization for 8× 8,16-QAM and SNR = 18 dB.
00.20.40.60.810
0.51
1.52
10−4
10−3
10−2
βα
BE
R
LR−MMSE MIMO 20x20LR−ACO MIMO 20x20
Fig. 3. LR-ACO input α and β parameters variation for 20 × 20,4-QAM and SNR = 16 dB.
In order to obtain the best α, β-parameters combi-nation, the ratio between the bit error rate of eachcombination and the minimum BER obtained over thatspecific MIMO configuration, BERα,β
BERmin, was calculated.
Also, for each parameters combination, the mean amongthe ratio considering the three configurations presentedin Fig. 1, 2 and 3 was obtained. Finally, the optimumα, β-parameters combination was obtained as that onewith the lower mean. As a result, the best combinationobtained was α = 0.8 and β = 0.8. Hereafter, thesevalues are adopted for both proposed LR-ACO MIMOdetectors.
B. LR-ACO MIMO Performance under Optimized InputParameters
Fig. 4 shows the (a) performance and correspondent(b) convergence of the MIMO detectors at a signal-to-noise ratio (SNR) of 26 dB, considering the ACO-MIMO
8
detectors for 4 × 4 antennas and 64-QAM modulation.In a same way, Fig. 5 and 6 show the same behaviorof the MIMO detectors for 8 × 8 antennas, 16-QAM,SNR= 18 dB; and for 20 × 20 antennas, 4-QAM,SNR= 16 dB, respectively. For a notation issue, theACO proposed in [10] is called "ACO1", while theACO proposed herein which performs the search startingfrom the LR-MMSE solution in the original domain as"ACO2", and the proposed ACO herein performing thesearch in a reduced domain is denominated "LR-ACO".For comparison purpose, the sphere decoding MIMOperformance of [17], namely "SD", is included in thegraphs of this section.
0 5 10 15 20 2510
−5
10−4
10−3
10−2
Iterations
64−QAM
MMSE MIMO 4x4LR−MMSE MIMO 4x4ACO
1 MIMO 4x4
ACO2 MIMO 4x4
LR−ACO MIMO 4x4
0 3 6 9 12 15 18 21 24 2610
−5
10−4
10−3
10−2
10−1
Eb/N
0 [dB]
BE
R
α = 0.8, β = 0.8, 64−QAM
MMSE MIMO 4x4LR−MMSE MIMO 4x4ACO
1 MIMO 4x4
ACO2 MIMO 4x4
LR−ACO MIMO 4x4SD MIMO 4x4
Fig. 4. a) BER performance; b) Convergence (SNR=26dB) for theMIMO detectors considering 4 × 4 antennas, 64-QAM modulationand optimized ACO input parameters.
Comparing the BER performance results, one canconclude that while ACO1 presents a performance nearto MMSE MIMO for all antenna and modulation orderconfigurations, the ACO2-MIMO detector achieves per-formance near to LR-MMSE MIMO when the numberof antennas is not so large. Beyond these performances,it is worth noting the proposed LR-ACO MIMO detectorreaches full diversity degree1 in all number of antennaconfigurations, leading to a significant performance im-provement, although the SNR gap regarding the spheredetector (SD-MIMO) has grown with the increasingnumber of antennas.
Analyzing the convergence figures, we can see thatfor the proposed LR-ACO MIMO detector, in the worst
1Defined as the asymptotic slope of the BER curve in each numberof antennas scenario.
0 5 10 15
10−4
10−3
10−2
10−1
Eb/N
0 [dB]
BE
R
MMSE MIMO 8x8LR−MMSE MIMO 8x8ACO
1 MIMO 8x8
ACO2 MIMO 8x8
LR−ACO MIMO 8x8SD MIMO 8x8
0 10 20 30 4010
−4
10−3
10−2
Iterations
MMSE MIMO 8x8LR−MMSE MIMO 8x8ACO
1 MIMO 8x8
ACO2 MIMO 8x8
LR−ACO MIMO 8x8
Fig. 5. a) BER performance; b) Convergence (SNR=18dB) for theMIMO detectors considering 8 × 8 antennas, 16-QAM modulationand optimized ACO input parameters.
0 5 10 15
10−4
10−3
10−2
10−1
Eb/N
0 [dB]
BE
R
MMSE MIMO 20x20LR−MMSE MIMO 20x20ACO
1 MIMO 20x20
ACO2 MIMO 20x20
LR−ACO MIMO 20x20SD MIMO 20x20
0 10 20 30 40
10−3
10−2
Iterations
MMSE MIMO 20x20LR−MMSE MIMO 20x20ACO
1 MIMO 20x20
ACO2 MIMO 20x20
LR−ACO MIMO 20x20
Fig. 6. a) BER performance; b) Convergence (SNR=16dB) for theMIMO detectors considering 20× 20 antennas, 4-QAM modulationand optimized ACO input parameters.
case, I = 10 iterations are enough for the total algorithmconvergence. However, in the other ACO-based detectorsthe algorithm remains evolving slowly along all the I =40 iterations evaluated.
C. Complexity Analysis
The algorithm complexity can be evaluated in termsof the total number of floating-point operations (flops). Aflop is defined as an addition, subtraction, multiplicationor division between two floating point numbers [18]. Ta-ble II shows the number of flops for the considered ACO-
9
MIMO detectors, assuming equal numbers of antennas(nT = nR = n) and M -QAM modulation format. It isalso considered that Nants and I are the number of antsand iterations, respectively, for the ACO, as well as Nis the neighborhood size for the LR-ACO.
In another way, the complexity for the ACO-MIMOdetector is depicted on Fig. 7 when both the numberof antennas nT = nR and the modulation order Mincreasing.
TABLE IINUMBER OF OPERATIONS NECESSARY FOR EACH MIMO
DETECTOR.
Detector Number of OperationsLR-MMSE (258/3)n3 + 33n2 − 1
ACO1 (208/3)n3 + 8n2 − 4n+ 4n2√M + 8n
√M+
+ I[10n√M + (4n2 + 6n)Nants + 1
]ACO2 (466/3)n3 + 41n2 − 4n+ 4n2
√M + 8n
√M+
+ I[10n√M + (4n2 + 6n)Nants + 2n+ 1
]LR-ACO (466/3)n3 + 41n2 − 4n+ 4n2N + 8nN+
+ I[10nN + (4n2 + 6n)Nants + 2n+ 1
]
510
1520
252
4
16
64
256
0
1
2
3
4
5
x 106
FLO
PS
LR−MMSE MIMOACO
1 MIMO
ACO2 MIMO
LR−ACO MIMO
MnT = nR
Fig. 7. ACO-MIMO Detector Complexity. ACO1 and ACO2:Nants = 20, I = 40; LR-ACO: Nants = 20, I = 10.
In the same way as [15], one advantage of the LR-ACO is that its complexity doesn’t depends on themodulation order (M ) used, because the neighbourhoodsize (N ) remains the same.
One can see that the LR-ACO detector complexityremains always below the complexity of ACO1 andACO2, whereas it presented a substantial performanceimprovement, for the Large-MIMO conditions.
VIII. CONCLUSION
This work proposed a new heuristic-based detector forMIMO communications systems that is able to combinethe promising features of both diminished search spacepropitiated by ant colony optimization with the latticereduction (LR) technique; as a result the proposed LR-ACO MIMO detector is capable to achieve a substantialimprovement on BER performance, reaches full diversitydegree while expending lower computational burdenthan others ACO-based MIMO detectors considered. Thepromising performance-complexity trade-off results wereobtained taking into consideration different number ofantennas and modulation order configurations.
ACKNOWLEDGEMENT
This work was supported in part by the National Coun-cil for Scientific and Technological Development (CNPq)of Brazil under Grants 202340/2011-2, 303426/2009-8and in part by Londrina State University - Paraná StateGovernment (UEL).
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31
Parte III
Transmissor Multiusuario ACO
1
Multiuser Transmission Schemes for MIMOCommunication via Ant Colony Optimization
José Carlos Marinello & Taufik Abrão
Abstract—In this paper we use the heuristic antcolony optimization (ACO) technique to improve theperformance-complexity tradeoff of MIMO multiusertransmission (MuT) schemes. Specifically, we consider theminimum bit-error-rate in low-complexity linear MuT us-ing heuristic ACO approach (ACO MBER-LMuT MIMO).
Index Terms—MIMO systems, Multiuser Transmission,ACO, Minimum Bit-Error-Rate, performance-complexitytradeoff.
I. INTRODUCTION
In multiple-input-multiple-output (MIMO) systemsseveral transmit and receive antennas are deployed tocommunicate base-station (BS) and mobile terminals(MTs). For practical reason, other important config-uration consists to equip MTs with a single-receive-antenna. This scenario is justified by the fact that MTsusually don’t have the energy and the processing ca-pability enough to perform multiuser detection (MuD)techniques. Hence, a useful strategy to improve the sys-tem’s performance without harming the energy efficiency(EE), in terms of MTs batteries autonomy, consists toshift the processing necessary to combat the multiuserinterference from the MTs to the BS, leading to theappealing concept of multiuser transmission (MuT) [1].
When relative good estimation of the channel stateinformation are available at the transmitter side, basi-cally, two MuT techniques could be enabled: a) linearMuT (LMuT) and b) non-linear MuT (NLMuT). Be-sides LMuT, known as transmit zero forcing or transmitWiener filters, recently nonlinear minimum bit-error-rate MuT (MBER-MuT) has been proposed. The lat-ter one computes the transmit signal by minimizingthe mean BER at the detector output deploying someconstrained numerical optimization method. Nonlinearpre-equalization techniques for downlink of multiusersystems applied to the transmitter equipped with multipleantennas to single antenna (MISO) mobile receivers
J. C. Marinello and T. Abrão are with State University of Londrina,Electrical Engineering Department (DEEL-UEL), Parana, Brazil. E-mails: [email protected], [email protected]
without cooperation have been proposed in [2]; thistechnique was extended for higher-order modulationin [3]. Indeed, it was shown that due to the fixedtransmit power the problem should be formulated as aquadratically constrained optimization problem with anonlinear objective function. Thus, its can be typicallysolved by some nonlinear optimization method, such assequential quadratic programming (SQP), which is ableto reduce the average BER at the expense of an excessivecomputational burden. In order to achieve the same per-formance but with a lower complexity, a new frameworkwith an unconstrained optimization was proposed in [3],showing an interesting trade-off among performance ofthe proposed optimization framework over LMuT whilethe computational complexity is significantly reducedcompared to the constrained MBER-MuT approach.
Recently, nonlinear MuT schemes inspired on heuris-tic methods have been proposed; for example, the par-ticle swarm optimization (PSO) was deployed in MuT-MIMO optimization problems in [4] and [5]. The PSOwas invoked to solve the constrained nonlinear opti-mization problem considering the MBER-MuT metric.Authors have been shown that the PSO-aided MBERMuT scheme is able to improve system performance incomparison with the conventional MMSE-MuT schemebased on transmit Wiener filters, while imposing a sig-nificantly reduced complexity compared to the state-of-the-art SQP-based MBER-MuT design.
The ant colony optimization (ACO) is a techniqueoriginally proposed to solve combinatorial optimizationproblems, such as the traveling salesman problem [6];nevertheless this heuristic optimization tools was ex-tended for continuous problems in [7]. The technique isinspired on the behavior of ants in nature that looking forfood have the ability of exploit the territory around, find-ing paths that lead to better sources of food, and leavingtrails that will help another ants on their searches. Manyworks have been disseminated applying this technique toseveral combinatorial (or discrete) and continuous opti-mization problems that arise in telecommunications, suchas multiuser detection in code division multiple accesssystems (CDMA) [8] [9], detection in MIMO systems[10] [11], resource allocation on wireless networks [12]
2
[13], among others.This work proposes a multiuser transmitter scheme for
MIMO system in which the precoder matrix optimizationis performed by deploying heuristic ACO technique.Since the optimization problem herein belongs to thecontinuous domain, the algorithm proposed in [7] shouldbe adapted.
II. MBER-LMUT
Let’s consider the downlink transmission from a basestation (BS) to non-cooperative energy-efficient mobileterminals (MTs), each equipped with a single receiveantenna. The linear MuT-aided MIMO system is shownin Fig. 1. BS is equipped with N transmitting antennascommunicates with K low-complexity MTs. Hence, con-sidering energy-efficient downlink communication de-sign, simple MTs are unable to perform sophisticatedmultiuser detection (MuD) in order to mitigate themultiuser interference (MuI). As a solution adoptedherein, the BS performs multiuser transmission (MuT),assuming that reliable estimates for the channel stateinformation are available at transmitter side. In thiswork, we suggest deploy a MuT scheme similar to thatproposed in [4], [5], as depicted in Fig. 1. The adoptedMuT is linear since the BS deploys linear precoding topre-process the transmitted multiuser downlink signals.The model of the adopted LMuT scheme aided MIMOsystem is explained in section II-A.
α α–1
α–1
α–1
α
α
P
XK
nK
n2
n1h1H
h2
hk
yK
y2
y1
X2
User 1
User 2
User KN
2
1X1
Fig. 1. MuT-aided MIMO system with linear precoding. BS employsN Tx antennas to communicate with K mobile devices equipped withsingle-receiver-antenna [5].
A. MuT-MIMO System Model
The MuT-MIMO system model deployed herein issimilar to [5]. Information transmitted symbol vectorx = [x1 x2 . . . xK ]T , where xk denotes the transmittedsymbol to the kth MT, with xk assuming a value ofthe QPSK alphabet. The symbol set for the QPSK ordermodulation is taken as I = {±0.5± j0.5}. Let’s defineN ×K precoder matrix of the multiuser transmission:
P = [p1 p2 . . . pK ] (1)
whee pk is the precoder’s coefficient vector for prepro-cessing the kth user’s data stream.
Hence, given a total fixed transmit power PT at theBS, an appropriate scaling factor α should be used tofulfill this transmit power constraint such that:
α =
√PT
||Px||2=
√ET
Ts||Px||2(2)
where Ts is data symbol period time and ET is the totalenergy available at BS for each symbol period. Besides,this normalized transmit power constraint PT allowsa fair comparison between different pre-equalizationschemes. Besides, the instantaneous MIMO channel ma-trix
H = [h1 h2 . . . hK ] (3)
is assumed constant at each Ts period basis.Finally, the equivalent base-band received vector sig-
nal can be written as:
y = HT Px + α−1n (4)
B. Linear MuT techniques
According [14], the linear multiuser transmitter(LMuT) based on the zero forcing (ZF) technique isexpressed by:
PZF = H∗ (HTH∗)−1β (5)
where β = diag (β1 β2 . . . βK) is a normalization ma-trix necessary to maintain the transmit power fixed andbounded. An equivalent way to fulfill this transmit powerconstraint is to calculate the PZF matrix as if there wereno power constraint, i.e., without the β matrix. Then, ascaling factor αZF is calculated, which is in fact the ratioof the power available and the power that would be used:
αZF =
√PT
||PZFx||2=
√ET
Ts||PZFx||2(6)
At the receiver side, the received signal should be thenmultiplied by α−1
ZF , ensuring then the signal power fulfillsthe transmit power constraint.
In addition, the precoder matrix based on the MMSEapproach can be described by:
PMMSE = H∗ (HTH∗ + σnIK)−1
(7)
while the respective power factor constraint is
αMMSE =
√PT
||PMMSEx||2=
√ET
Ts||PMMSEx||2(8)
3
C. The Optimization Problem for the Linear MinimumBER MuT Precoder Design
Given a 4−QAM symbol vector x, the average BER1
of the in-phase component of y at the receiver as afunction of N ×K precoder matrix is expressed as:
P Ie,x(P) =
1
K
K∑k=1
Q
(sgn(R[xk])R[hT
kPx]
σn
)(9)
where Q(·) is the standard Gaussian error function.Similarly, given the symbol vector x, the average BER ofthe quadrature-phase component of y can be expressedas:
P Qe,x(P) =
1
K
K∑k=1
Q
(sgn(I[xk])I[h
TkPx]
σn
)(10)
Thus, the average BER for the specific M−QAMsymbol vector x is
Pe,x(P) =P Ie,x(P) + PQ
e,x(P)
2(11)
Finally, the minimum bit-error-rate multiuser trans-mission (MBER-MuT) design can be formulated by thefollowing constrained optimization problem:
PMBER,x = arg minP
Pe,x(P, x, H, σ)(12)
s.t. : (C.1) ||Px||2 = PT
Hence, under this optimization design approach, the biterror rate should be predicted at the transmitter sidebased on the transmit signal x, precoder matrix P, aswell as the current channel state information H shouldbe available at the transmission side. The only parametertreated statistically in the transmitter side is the additivewhite Gaussian noise (AWGN) of the receiver, which itsvariance σ2 must be estimated at the transmitter side.
As explained in [5], this constrained nonlinear opti-mization is typically solved by an iterative gradient basedalgorithm known as the sequential quadratic program-ming (SQP). The computational complexity of the SQP-based linear MBER MUT design can be found in [4].It is worth noting that the SQP-based design has a highcomputational burden. Therefore, for practical high-rateinformation systems, it may be difficult or expensive toimplement this SQP-based MBER-LMuT solution.
Hence we propose deploying heuristic ACO algorithmas an alternative to the SQP approach in order toachieve a low-complexity MBER LMuT design but withsimilar performance, aiming to achieve a much betterperformance-complexity trade-off.
1The average and exact BER expression result distinct, as well asfor different modulation order M [15].
1) Unconstrained Transformed MBER-LMuT Opti-mization Problem: The original constrained optimizationproblem in (12) is convert to an unconstrained optimiza-tion problem by introduction of a penalty function [4],[16]. In doing so, the unconstrained problem automati-cally meets the transmit power constraint. Hence, definethe cost function by:
Fx(P) = Pe,x(P) +Gx(P) (13)
where the penalty function is given by:
Gx(P) =
{0 , ||Px||2 − PT ≤ 0
λ(||Px||2 −PT
), ||Px||2 − PT > 0
(14)and the penalty factor λ > 0 should be chosen appro-priately. Hence, the original constrained MBER-LMuToptimization design (12) becomes the following uncon-strained optimization problem:
PMBER,x = arg minP
Fx(P) (15)
In [17], an analytical expression for λ is proposed,based on the fact that Fx(P) is a convex function withglobal minimum in PMBER. Hence, taking the derivativeof (13) with respect to the elements of P, setting theresult to equal to zero, and then isolating the λ factor,the expression is obtained with the aid of the powerconstraint (C.1). However, figuring out analytically thisderivative may be not a simple task to be implementedand continuously recalculated in the BS.
Another way to convey the problem in (12) un-constrained one is discussed in [3]. Since scaling thereceived signal by the factor α−1 will affect directlythe background noise amplitude, the following equivalentnoise measure σneq
has been suggested:
σneq=
√E[| 1αnk|2
]2
=
(1
α
)√E [|nk|2]
2=
σnα
(16)
Hence, the unconstrained function to be optimizedshould be written as:
PMBER,x = arg minP
Pe,x
(P, x, H, σneq
)(17)
In the Section IV, the MBER-LMuT problem will betreated by two approaches: initially according to (13),and then according to (17).
III. ACO FOR CONTINUOUS DOMAINS
As shown in [7], to adapt the original ACO algorithm[6] to continuous domains problems, it is necessary torepresent the pheromone information, which for discreteproblems was represented as a table, in the context of acontinuous problem. It can be made by using an archive
4
T that stores the Tsz best solutions in ascending orderof their respective cost function values found by thealgorithm so far.
The transition probabilities, which were previouslydefined as a discrete probability density function (PDF),also shall be adjusted, becoming a continuous PDF,represented herein by a weighted sum of several Gaus-sian PDF functions, namely Gaussian kernel, for eachproblem dimension. As a practical simplification, thesampling of this Gaussian kernel is equivalently donein two phases: first, a discrete sampling, in which it ischosen what Gaussian function will be sampled (relatedto the ℓg-th solution of T ), and then a continuous one,related to its. The probability of choice for each Gaussianfunction is given by:
Pr(ℓ) =wℓ∑Tsz
r=1wr
(18)
where wℓ is the weight of the ℓ-th solution of T , definedas:
wℓ =1
qTsz
√2π
exp
{−(ℓ− 1)2
2q2T 2sz
}(19)
where q is a parameter of the algorithm, that influencesits search diversity since q is related to the probabilityof being chosen the best solutions of the archive.
At the beginning of each iteration, a new Gaussianfunction, which subsequently will be sampled from (18),is selected. Then, on the current iteration each antfollows new path, which is done sampling the Gaussianchosen on each dimension of the problem; in this case,each element of the precoder matrix PN×K . For eachdimension, the mean of the Gaussian function is givenby the value of the element of that solution on T : thepower element pi,jℓg with the respective standard deviationgiven by:
σi,jℓ = ξ
Tsz∑e=1
|pi,je − pi,jℓg |Tsz − 1
(20)
where ξ is the parameter related with the algorithm’sconvergence speed.
Finally, at the end of each ACO algorithm iteration,having each ant chosen its path, the archive T is updated,which includes or not solutions found, in order to keepalways the same size of the solution file Tsz. Thealgorithm search finishes when all the solutions of T areequal or if the maximum number of iterations is reached,returning the first solution stored on the top of archive.
IV. NUMERICAL RESULTS
It is well known from the convex optimization theory[18] that the penalty factor λ in constrained optimizationproblems, must be of the same magnitude order than the
first term related to the objective function (in this casePe,x), and should steadily and continuously decreaseswhile the algorithm evolvies across the iterations (con-straint relaxation).
At the first approach considered, the decrement of λoccurs on a linear basis, as suggested by:
λ1 =
(Imax − i10 · Imax
)Pe,x (21)
where Imax is the pre-determined maximum number ofiteration and i is the current iteration. For a notationpurpose, hereafter we call this approach by λ1.
In the second approach analyzed, the penalty factor isdecremented while the ACO algorithm iterations evolve.
λ = 0.1
(Imax
i
)n
Pe,x (22)
where n = 1 for the second approach (λ2), and n = 2for the third approach (λ3), assuming in the last case astronger quadratic hyperbolic function behavior.
A. Performance and Convergence Analysis
Convergence comparison deploying the three penaltyfactor λi, i = 1, 2, 3 in PSO MBER-LMuT and MMSE-LMuT MIMO with 4 antennas is shown in Fig. 2, wherethe bit error rate reduction is observed in PSO-MuTapproaches when algorithm evolves; SNR = 15dB ands = 20 particles, inertia weight 0, search and velocitylimits equals to 1 were adopted following the setup of[5]. Fig. 3 brings the same convergence analysis for theACO-MuT deploying a = 10 ants, ξ = 0.85, q = 0.1,and solution archive size Tsz = 40.
0 10 20 30 40 50 60 70 80 90 100
10−2.7
10−2.6
Iterations
BE
R 4x4−MIMO, 4−QAM
MMSE−MuTPSO−MuT (λ
1)
PSO−MuT (λ2)
PSO−MuT (λ3)
Fig. 2. Convergence of the PSO-MuT for different λ approaches.SNR = 15 dB and s = 20 particles.
5
50 100 150 200 250 300
10−2
Iterations
BE
R
4x4−MIMO, 4−QAM
MMSE−MuT
ACO−MuT (λ1)
ACO−MuT (λ2)
ACO−MuT (λ3)
Fig. 3. Convergence of the ACO-MuT for different λ approaches.SNR = 15 dB, and a = 10 ants.
As one can note, the λ2 approach had resulted on abest performance for the PSO-LMuT, while for the ACO-MuT none of them worked rightly. The Fig. 4 and 5 com-pares the convergences using the λ2 approach with theconvergence by using the optimization function in (17),for the PSO-MuT and for the ACO-MuT, respectively.
0 10 20 30 40 50 60 70 80 90 100
10−2.8
10−2.7
10−2.6
10−2.5
Iterations
BE
R
4x4−MIMO, 4−QAM
MMSE−MuT
PSO−MuT (λ2)
PSO−MuT
Fig. 4. Convergence of the PSO-MuT: λ approach × eq. (17).SNR = 15 dB and s = 20 particles.
One can see from Fig. 4 and 5, the optimizationapproach deploying in (17) has resulted in a reliable con-vergence for ACO, evolving continuous and achievinga lower BER than the that one using the λ2 approach,in the case of heuristic PSO. Besides, note that fromFig. (17) the ACO-MuT (λ2) with a = 10 ants was notable to evolve even after Imax = 300 iterations. Finally,Fig. 6 compares the BER performance versus SNR withboth heuristic MBER LMuT techniques. It is clear that,
50 100 150 200 250 300
10−2.8
10−2.7
10−2.6
10−2.5
10−2.4
10−2.3
Iterations
BE
R
4x4 MIMO, 4−QAM
MMSE−MuT
ACO−MuT (λ2)
ACO−MuT
Fig. 5. Convergence of the ACO-MuT: λ approach × eq. (17).SNR = 15 dB and a = 10 ants.
for the conditions analyzed, the PSO-MuT presenteda better performance, since its BER remained alwaysbelow the one of ACO-MuT, although both transmittershave provided performance improvement regarding theconventional linear MMSE-MuT approach.
0 5 10 15 20 2510
−4
10−3
10−2
10−1
Eb/N
0 [dB]
BE
R
4x4 MIMO, 4−QAM
MMSE−MuTACO−MuTPSO−MuT
Fig. 6. BER Performance comparison, including both heuristicACO-LMuT × PSO-LMuT and the MMSE-MuT approach.
B. Complexity Analysis
Table I shows the number of operations (flops) neededfor each linear precoding transmitter technique analyzedherein. N represents the number of transmitter antennas,K is the MS’s number, s is the number of particlesfor the PSO algorithm, Tsz and a represent the archivesize and the number of ants, respectively, for the ACOalgorithm.
Fig. 7 compares the computational complexity of bothtechniques with an increase in the number of transmitter
6
TABLE INUMBER OF OPERATIONS PER ITERATION FOR EACH MUT.
MuT Number of OperationsPSO ((8N + 2NK +K + 27)K +N + 10)s+ 8ACO ((2TszN + 2N + 2NK +K + 27)K +N + 10)a
antennas and MTs, assuming N = K and 4-QAMmodulation. A number of iterations Imax = 100 andImax = 200 for the PSO-MuT and the ACO-MuT,respectively, was adopted based on the convergenceanalysis results of Fig. 4 and 5. As a result, the ACO-MuT complexity is remarkable higher than the PSO-MuT one. For the number of antennas considered in theprevious subsection (N = K = 4), while the PSO-MuTneeds nearly 7.9 · 105 flops per symbol transmission, theACO-MuT requires nearly 31.5 · 105 flops, i.e., almost 4times higher complexity.
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
3.5x 10
7
N = K
flops
PSO MuTACO MuT
PSO: 20 particles, 100 iterationsACO: 10 ants, 200 iterations
Fig. 7. Number of operations × number of antennas.
V. CONCLUSION
In this work, a linear multiuser transmitter scheme forMIMO systems applications with low-complexity MTshas been proposed and compared with other heuristic-based MuT scheme available in the literature. TheLMuT MIMO scheme uses heuristic ACO technique tosolve the MBER-MuT optimization problem. Although itpresented a significant performance improvement whencompared to the linear ZF-MuT and MMSE-MuT, theproposed multiuser transmitter ACO-based approach wasnot able to outperform the PSO-MuT proposed in [4] and[5], in terms of performance versus complexity trade-off, presenting a worth performance, however, at theexpense of a complexity almost 4 times higher thatPSO for the MIMO system conditions analyzed. In orderto turn the analysis more complete and obtain a more
promising ACO-MuT scheme, it is necessary to proceedthe input ACO parameter optimization, aiming to find theparameters combination that maximizes the algorithmBER performance and minimizes its complexity, whichremains a field for further works.
ACKNOWLEDGMENT
This work was supported in part by the National Coun-cil for Scientific and Technological Development (CNPq)of Brazil under Grants 202340/2011-2, 303426/2009-8and in part by Londrina State University - Paraná StateGovernment (UEL).
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