sistemas mimo em canais correlacionados e sob treinamento dessincronizado · 2014. 11. 17. ·...

Centro de Tecnologia e Urbanismo

Departamento de Engenharia Eletrica

Ricardo Tadashi Kobayashi

Sistemas MIMO em CanaisCorrelacionados e sob Treinamento

Dessincronizado

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, PR2014


Sistemas MIMO em Canais

Correlacionados e sob Treinamento

Dessincronizado

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: Sistemas de Telecomunicacoes

Orientador:

Prof. Dr. Taufik Abrao

Londrina, PR2014

Ficha Catalografica

Kobayashi, Ricardo TadashiSistemas MIMO em Canais Correlacionados e sob Treinamento Des-

sincronizado. Londrina, PR, 2014. 77 p.

Monografia (Trabalho de Conclusao de Curso) – UniversidadeEstadual de Londrina, PR. Departamento de Engenharia Eletrica.

1.Sistemas com Multiplas Antenas. 2.MIMO Massivo 3.Di-versidade Espacial. 4.Sequencias de espalhamento. 5.Es-timativa de canais. Departamento de Engenharia Eletrica


Sistemas MIMO em CanaisCorrelacionados e sob Treinamento

Dessincronizado

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: Sistemas de Telecomunicacoes

Comissao Examinadora

Prof. Dr. Taufik AbraoDepto. de Engenharia Eletrica

Universidade Estadual de LondrinaOrientador

Prof. Msc. Jaime Laelson JacobDepto. de Engenharia Eletrica

Universidade Estadual de Londrina

Prof. Msc. Decio Luiz Gazzoni FilhoDepto. de Engenharia Eletrica

Universidade Estadual de Londrina

3 de novembro de 2014

”Imagination creates reality.” Richard Wagner.

Agradecimentos

Agradeco aos meus pais e familiares pelo incentivo dado ao longo do curso.

Tambem agradeco a todos meus amigos, colegas e professores que ajudaram de

forma direta e indireta neste trabalho e na minha formacao.

Ao prof. Dr. Taufik Abrao pelo orientacao, paciencia e pelas inumeras horas

dedicadas a diversas discussoes, sem as quais esse trabalho nao chegaria a sua

conclusao.

Por fim, expresso minha gratidao ao CNPQ e a UEL pelo auxılio financeiro.

Resumo

Neste trabalho, foram analisados sistemas de multiplas antenas em dois cenarios:unicelular sob canais correlacionados e multicelular cooperativo. No primeirocenario, o interesse reside na eficacia de diferentes tecnicas de deteccao MIMOpresentes na literatura, pois ao operar sob forte correlacao entre antenas, sistemasMIMO tem seu desempenho prejudicado, alem de sua complexidade aumentada.Neste cenario, foi estudado e analisado o comportamento, em termos de taxa deerro e complexidade, de diversas tecnicas de deteccao que englobam, por exem-plo, algoritmos de busca, cancelamento sucessivo de interferencia, reducao trelicae reducao de lista. Atraves deste estudo, foi possıvel classificar diferentes metodosde deteccao MIMO de acordo com seu compromisso desempenho-complexidade,possibilitando verificar a viabilidade pratica de tais solucoes em condicoes maisrealistas. Para o cenario multicelular, o foco se voltou para o estudo da estimativade canal de sistemas MIMO multicelular operando sob cooperacao entre celulase usuarios quase estaticos. Apesar da estimativa de canal ser popularmente feitaatraves de sequencias ortogonais, este trabalho fez o uso de sequencias pseudoa-leatorias binarias (Gold e Kasami) e quaternarias (famılia α) de baixa correlacaocruzada. O objetivo do uso de sequencias nao ortogonais e ampliar o numero deusuarios a serem servidos por celulas ou melhorar o desempenho na estimativado canal, sem comprometer a qualidade de servicos. Alem disso, o estudo desistemas MIMO em cenario multicelular contemplou o uso de grandes arranjosde antenas, que tem se mostrado muito promissor e benefico tanto no quesitodesempenho quanto eficacia. Portanto, de um modo geral, este trabalho acabapor explorar alguns dos principais desafios encontrados em sistemas MIMO mas-sivo: a complexidade inerente ao grande numero de antenas, a qual traz canaiscorrelacionados, e a contaminacao de piloto, presente na estimativa do canal.

Abstract

In this work, it was studied multiple antennas systems operating in two sce-narios: unicellular under correlated channels and multicellular with cooperation.In the first scenario, the interest resides on the efficiency of different MIMO detec-tion techniques found on the literature, because with highly correlated antennas,MIMO systems have their performance reduced, besides of their complexity in-creased. In this scenario, it was studied and analyzed the behavior, in terms oferror rate and complexity, of different detection techniques which included, forexample, search algorithms, successive-interference-cancellation, lattice-reductionand list-reduction. Through this study, it was possible to classify different MIMOdetection techniques according to their performance-complexity trade-off, en-abling to verify the practical viability of such solutions in more realistic conditions.For the multicellular scenario, the attention was turned to the channel estima-tion of multicellular MIMO systems operating with cooperation between cellsand quasi-static users. Despite of the channel estimation be popularly proceededthrough orthogonal sequences, this work used binary pseudo-random (Gold andKasami) and quaternary (α) sequences with low cross correlation. The goal ofusing non-orthogonal sequences is to enlarge the number of users served in eachbase-station or enhance the performance of channel estimation, without compro-mising the the quality of services. Besides this, the study of MIMO systems overmulticellular scenario contemplated the use of large antennas arrays, which shownto be very promising and beneficial for both performance and efficiency. There-fore, generally speaking, this work explored some of the most recurring challengesfound in massive MIMO systems: the complexity inherent to the large numberof antennas, which produces correlated channels, and the pilot contamination,present on the channel estimation.

Sumario

Lista de Abreviaturas

Convencoes

1 Introducao 1

1.1 Deteccao em sistemas MIMO . . . . . . . . . . . . . . . . . . . . 3

1.2 MIMO Massivo Celular . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Resultados 7

3 Conclusoes 9

Anexo A -- Statistical Models for Fading in Wireless Links 11

Anexo B -- On the QAM and PSK Modulation Performance 15

Anexo C -- Performance and Complexity Analysis of Sub-optimum

MIMO Detectors Under Correlated Channel 21

Anexo D -- Efficient Near-Optimum Detectors for Large MIMO Sys-

tems under Correlated Channels 49

Anexo E -- Cooperative Multi-cellular Large MIMO over Desynch-

ronized Channel Estimation 55

Referencias 76

Lista de Abreviaturas

4G Quarta geracao de telefonia movel

5G Quinta geracao de telefonia movel

BER Bit-error-rate (taxa de erro de bit)

CL Chase-list (Lista de Chase)

CSI Channel state information (informacao do estado do canal)

ERB Estacao radio base

FDD time-division-duplex (duplexacao por divisao de frequencia)

LLL Lenstra-Lenstra-Lovaz

LR Lattice-reduction (reducao trelica)

MIMO Multiple-input multiple-output (multiplas-entradas multiplas-saıdas)

MF Matched-filter (filtro casado)

ML Maximum-likelihood (maxima verossimilhanca)

MSE Mean-squared-error (erro quadratico medio)

MMSE Minimum-mean-squared-error (mınimo erro quadratico medio)

OSIC Ordered-successive-interference-cancellation (cancelamento de interferencia

ordenado)

QAM Quadrature-amplitude-modulation (modulacao em quadratura e ampli-

tude)

SD Sphere-decoding (decodificaca esferica)

SM Spatial-modulation (modulacao espacial)

SNR Signal-to-noise ratio (relaco sinal ruıdo)

SQRD Sorted-QR-decomposition (decomposicao QR ordenada)

STBC Space-time-block-code

TDD time-division-duplex (duplexacao por divisao de tempo)

V-BLAST Vertical-Bell-laboratories-layered-space-time

ZF Zero-forcing

Convencoes

As seguintes notacoes matematicas foram adotadas neste trabalho:

• Letras minusculas em negrito denotam vetores coluna (caso nao haja nehuma

especificacao)

• Letras maiusculas em negrito denotam matrizes;

• (·)T : operador de transposicao;

• (·)∗: operador de conjugacao;

• (·)H : operador hermitiano (transposicao e conjugacao);

• (·)−1: inversao de uma matriz quadrada;

• (·)†: pseudoinversa de uma matriz;

• det(·): determinante de uma matriz quadrada;

• ‖·‖: norma de de Frobenius;

• tr(·): operador traco;

• diag(·): operador de diagonalizacao;

• vec(·): vetorizacao de uma matriz;

• b·e: operador inteiro mais proximo

• IK : matriz identidade de ordem K;

• 0N×K : matriz de tamanho N ×K formada exclusivamente por zeros;

• 1N×K : matriz de tamanho N ×K formada exclusivamente por uns;

• CN (µ, σ2): variavel aleatoria Gaussiana circular e complexa com media µ

e variancia σ2;

• O(·): ordem da complexidade de uma operacao ou algorıtimo;

• Q(·): processo de slicing de um sımbolo modulado;

• E [·]: operador esperanca estatıstica;

• C: conjunto dos numeros complexos;

• N: conjunto dos numeros naturais;

• Z: conjunto dos numeros inteiros;

• ∈: pertence ao conjunto.

Define-se aqui alguns sımbolos recorrentes neste trabalho:

• N0: densidade espectral de potencia do ruıdo;

• ES: energia de sımbolo;

• Eb/N0: energia de bit por densidade espectral de ruıdo;

• SNR: relacao sinal ruıdo;

• nT : numero de antenas no transmissor;

• nR: numero de antenas no receptor;

• M : ordem da modulacao;

• N : numero de antenas presente na estacao radio base;

• K: numero de terminais moveis em cada celula;

• L: numero de celulas do sistema;

• τ : comprimento da sequencia de treinamento;

• M: numero de sequencias encontradas em uma famılia;

• ρ: ındice de correlacao do canal.

1

1 Introducao

E incontestavel o fato das telecomunicacoes moldarem a sociedade contem-

poranea, desempenhando um papel importante em nossas vidas seja atraves da

TV, internet ou telefonia. Assim como cresce o numero de aplicacoes e a im-

portancia das telecomunicacoes, seus usuarios, bem como suas exigencias, cres-

cem em proporcao ainda maior. Isso por sua vez leva a uma crise de espectro

eletromagnetico e qualidade de servico, tornando mais desafiadora a comunicacao

atraves de sistemas sem fio. Uma das solucoes mais promissoras para esses pro-

blemas e o uso de sistemas com multiplas antenas tanto na transmissao quanto

na recepcao, conhecidos como sistemas MIMO (multiple-input multiple-output).

Nestes sistemas, como mostra a figura 1.1, os sinais transmitidos sao linearmente

combinados atraves do enlace sem fio, sendo possıvel explorar a diversidade espa-

cial. Alem disso, estes sistemas podem operar com ganho de multiplexacao (dados

distintos nas antenas de transmissao) ou diversidade (insercao de redundancia na

mensagem).

Figura 1.1: Sistema MIMO (BAI; CHOI, 2012)

Apesar dos primordios dos sistemas MIMO remontarem ao inıcio do seculo

XX, essa tecnologia ainda e considerada como o estado da arte dos sistemas de

telecomunicacoes. Tal fato se deve as inumeras vantagens, das quais se destacam

o ganho de eficiencia espectral e/ou energetica. Com isso, tais sistemas despertam

cada vez mais o interesse da comunidade cientıfica, que por sua vez acaba viabi-

lizando aplicacoes comerciais atraves de extensivas pesquisas acerca do assunto.

Das aplicacoes comerciais, as mais populares se dao nos sistemas de telefonia

movel 4G e nos padroes 802.11 (redes locais) (PAULRAJ et al., 2004).

Em (WOLNIANSKY et al., 1998), foi proposta a arquitetura V-BLAST dotada

1 Introducao 2

de multiplas antenas na transmissao e recepcao, operando em modo de multi-

plexacao, i.e. dados distintos nas antenas de transmissao. Alem da proposta, a

viabilidade de tal arquitetura foi comprovada atraves de um prototipo composto

de 8 antenas na transmissao e 12 na recepcao e operando com a modulacao 16-

QAM, o qual provou ser capaz operar de forma razoavel com eficiencia espectral

na faixa de 20 − 40 [bps/Hz]. Desta forma, este trabalho acabou definindo os

sistemas MIMO como conhecemos hoje, inspirando inumeros trabalhos sobre o

assunto.

Mais recentemente, o interesse sobre sistemas MIMO esta se voltando para

um cenario multi-celular, onde cada celula possui uma estacao-radio-base (ERB)

com multiplas antenas servindo um determinado numero de usuarios equipados

com uma unica antena, como mostra a figura 1.2. Neste contexto, o interesse se

da no uso de um numero massivo e muito maior de antenas nas ERBs do que

o numero de terminais moveis, originando o nome Massive MIMO, ou MIMO

massivo.

Figura 1.2: Sistema MIMO multi-celular (RUSEK et al., 2013)

Todo o interesse sobre sistemas MIMO massivo e proveniente de uma serie

benefıcios extremamente promissores. Um deles e a grande eficiencia energetica

(LARSSON et al., 2014), que possibilita sistemas MIMO massivo operarem com

nıveis de potencias 100 vezes menores do que os sistemas atuais, abrindo possi-

bilidades do uso de energia solar ou eolica para alimentar ERBs. Com isso, o

projeto dos sistemas se tornam mais simples, logo mais baratos, pois dispensam

o uso de um unico amplificador de altas linearidade e potencia, sendo substituıdo

por varios amplificadores de baixas potencia e custo. Alem disso, devido ao baixo

consumo de energia, o cabeamento presente nas ERBs e consideravelmente redu-

zido, barateando ainda mais a infraestrutura das ERBs. Ainda em (LARSSON et

al., 2014), e exemplificado um sistema MIMO massivo atendendo 1000 usuarios

1.1 Deteccao em sistemas MIMO 3

com uma taxa de 20 [Mbps] atraves de uma ERB com 6400 antenas. Por fim, em

(MARZETTA, 2010), e mostrado que ao se manter o numero de usuarios constante

aumentando indefinidamente o numero de antenas na ERB acaba por anular

o efeito do ruıdo aditivo de fundo, um dos principais limitantes das telecomu-

nicacoes.

Apesar, dos benefıcios proporcionados pelos sistemas MIMO massivo, existem

ainda diversos problemas nao solucionados, dos quais tres limitam majoritaria-

mente a aplicabilidade de tais sistemas. Um deles e a complexidade computacio-

nal, que cresce a medida que o numero de antenas do sistema aumenta. Outro e a

contaminacao do piloto (JOSE et al., 2011), fruto da interferencia inter-celular em

casos onde a mobilidade do usuario e indispensavel. Alem disso, o uso de grandes

arranjos de antenas acaba por introduzir a correlacao entre antenas, ja que o

agrupamento destas deve ser feito em espacos relativamente restritos. Com isso,

dificulta-se a distincao de sinais provenientes de diferentes antenas, prejudicando

o desempenho de sistemas com multiplas antenas.

1.1 Deteccao em sistemas MIMO

Como citado anteriormente, nos sistemas MIMO o receptor, equipado com

nR antenas, possui a sua disposicao uma combinacao linear dos sinais enviados

por todas as nT antenas, matematicamente definida por:

x = Hs + n (1.1)

onde x e o vetor sinal recebido, H e a matriz cujos elementos descrevem os ganhos

de canais de cada caminho percorrido pelos sinais transmitidos, s e o vetor sinal

transmitido e n e o ruıdo aditivo de fundo.

A fim de reconstruir o sinal transmitido no lado do receptor, este deve de-

sacoplar o sinal recebido mediante a informacao do canal. Para isso, existem

diversas tecnicas, das quais a deteccao de maxima-verossimilhanca testa todo o

conjunto de sinais enviados, sendo a solucao otima apesar de sua complexidade

exponencial com o numero de antenas de transmissao. Uma abordagem mais

eficiente e otima e a decodificacao-esferica (HASSIBI; VIKALO, 2005), a qual reduz

o raio de busca com base na intensidade do ruido, possuindo, no entanto, uma

complexidade computacional ainda proibitiva. Ambas solucoes sao conhecidas

por apresentarem ganhos maximos diversidade (d = nR (BIGLIERI, 2007)), sendo

que esta representa a taxa de decrescimento da probabilidade de erro em elevadas

1.1 Deteccao em sistemas MIMO 4

relacoes sinal-ruıdo, sendo definida por:

d = − limρ→∞

log|Pe|log|ρ| , (1.2)

onde Pe e a probabilidade de erro e ρ e a relacao sinal-ruıdo,

Com complexidades bastante reduzidas, os detectores lineares podem se tor-

nar bastante atrativos quanto ao seu compromisso complexidade-desempenho ao

serem combinados a outras tecnicas. A solucao linear mais simples e provida pela

equalizacao zero-forcing (KUHN, 2006), a qual se resume a resolucao do sistema

(1.1) atraves da pseudo-inversa de Moore-Penrose e desconsiderando o ruıdo. Ao

considerar as estatısticas do ruıdo e do sinal, pode-se obter a solucao MMSE (SZC-

ZECINSKI; MASSICOTTE, 2005), capaz de melhorar a estimativa da mensagem.

Aplicando o cancelamento sucessivo de interferencia e/ou a ordenacao aos

detectores lineares, ganhos no desempenho sao esperados. Alem da arquitetura

V-BLAST, em (WOLNIANSKY et al., 1998) e proposto o cancelamento sucessivo

de interferencia ordenado otimo, porem com complexidade proporcional a O(n4).

No entanto, atraves da decomposicao QR ordenada (WUBBEN et al., 2003), e

possıvel realizar o cancelamento sucessivo de interferencia e a ordenacao com

complexidades da ordem O(n3) e sem perdas no desempenho. Adicionalmente, a

reducao de lista (WATERS; BARRY, 2008) introduz a repeticao de parte da deteccao

na tentativa de corrigir estimativas erroneas. Por fim, uma das tecnicas mais

eficazes aplica a reducao trelica (MA; ZHANG, 2008), responsavel por consideraveis

ganhos de desempenho na deteccao linear, alem de apresentar diversidade total e

responder muito bem a canais correlacionados, i.e. sistemas com distanciamento

de antenas menor que λ/2 (meio comprimento de onda).

Uma alternativa a deteccao convencional ou equalizacao e o uso de tecnicas de

precodificacao. Atraves dela, os dados sao enviados de tal sorte a anular o efeito

do canal, fazendo com que o receptor receba o sinal de transmissao inalterado

pelo canal, ou seja:

x = HAs + n, (1.3)

onde HA = αI, sendo α constante. Nota-se que atraves deste tipo de transmissao,

o processamento de sinais fica concentrado no transmissor, cabendo ao receptor

somente demodular os sinais de cada antena, o que e benefico para terminais

moveis de baixa capacidade energetica e/ou de processamento.

1.2 MIMO Massivo Celular 5

1.2 MIMO Massivo Celular

Na configuracao MIMO celular, cada uma das L celulas contem um conjunto

de K usuarios dotados de uma ou mais antenas, os quais sao servidos por uma

estacao radio base com N antenas. E importante destacar que as notacoes do

numero de antenas no transmissor e receptor deve ser mais generica e abrangente,

ja que neste caso a transmissao sera feita em ambas direcoes: da ERB para os

usuarios (downlink) e dos usuarios para a ERB (uplink). Em sistemas celulares

deve-se manter o processamento concentrado nas ERBs, devido as suas capaci-

dades energeticas e de processamento. Tal exigencia pode ser obtido pelo uso de

tecnicas de precodificacao para o downlink e deteccao convencional para o uplink.

Assim como em sistemas MIMO ponto a ponto, sistemas MIMO celulares

tem seu desempenho melhorado a medida que o numero de antenas no receptor

e transmissor e aumentado, ja que e ampliado o numero de graus de liberdade

proporcionado pelo enlace sem fio. No entanto, maiores ganhos sao observados a

medida que o numero de usuarios se torna muito menor que o numero de antenas

na ERB, i.e. K << N . Um exemplo de tal ganho pode ser encontrado em

(LARSSON et al., 2014), que descreve que ao se adotar K << N , onde cada usuario

e equipado com uma antena, a potencia de transmissao se torna inversamente

proporcional ao quadrado do numero de antenas na ERB, i.e. ∝ N−2.

Ao se adotar um numero de antenas na ERB muito superior ao numero de

usuarios, K << N , sendo N → ∞, a performance dos sistemas MIMO pode

ser consideravelmente melhorada. Isso se deve ao fato do canal se tornar extre-

mamente bem condicionado, alem de parametros que antes eram considerados

aleatorios passarem a ser determinısticos. Um exemplo disso e o canal MIMO

H ∈ CN×K modelado por uma distribuicao Rayleigh, onde hij ∼ CN (0, 1), apre-

sentar a ortogonalidade assintotica:

limN→∞

HHH

N= IK , (1.4)

que alem de melhorar o desempenho dos sistemas MIMO pode simplificar o pro-

cessamento.

Apesar de sistemas MIMO massivo (N →∞) nao serem afetados pelo ruıdo

aditivo de fundo, em cenarios multi-celulares a interferencia inter-celular ainda

permanece, sendo o principal limitante para a aplicabilidade de tais sistemas.

Para evidenciar tal problema, considere o modelo de transmissao uplink, onde o

1.2 MIMO Massivo Celular 6

sinal recebido na j−esima ERB e dado por:

Xulj =

L∑

`=1

Hj`Sul` + Nul

j , (1.5)

onde Sul` e o sinal enviado pelos usuarios na `−esima celula, Hj` e a matriz do

canal e Nulj e o ruıdo na j−esima ERB. Note que as linhas de Sul` , Nul

j e Xulj repre-

sentam a dimensao temporal (informacao em cada perıodo de sımbolo), enquanto

as colunas representam a dimensao espacial (informacao em cada antena). Nesta

equacao, a interferencia inter-celular pode ser notada de forma imediata ja que

o sinal recebido em cada ERB e uma combinacao dos sinais enviados por todos

os usuarios, sendo que somente os sinais que transitam entre usuarios e ERBs da

mesma celula sao necessarios para a reconstrucao da mensagem.

Um dos grandes desafios impostos pela interferencia inter-celular e a conta-

minacao do piloto, a qual limita a precisao da estimativa de canal, que por sua vez

desempenha um papel importante na deteccao e precodificacao. O treinamento

do canal pode ser feito atraves de sequencias ortogonais P`, onde P`PH` = IK ,

que pode ser descrito modificando a equacao (1.5):

Yj =L∑

`=1

Hj`P` + Nj. (1.6)

Devido ao numero de usuarios, a limitacoes de processamento e do tempo de

coerencia do canal (intervalo no qual o canal permanece inalterado) deve-se fazer

o reuso de sequencias em diferentes celulas. Desta forma, a estimativa de canal

entre os usuarios e a ERB da j−esima celula e obtida atraves de:

Hjj = YjPH =

L∑

`=1

Hj` + NjPH . (1.7)

Isso demonstra que a estimativa do canal da celula de interesse e corrompida

pelo ruıdo aditivo, alem de apresentar vestıgios de canais de outras celulas. Tal

fenomeno e conhecido como contaminacao de piloto e pode inviabilizar a recu-

peracao da mensagem no transmissor, dada a pobre estimativa de canal.

7

2 Resultados

Os resultados deste trabalho serao apresentados na forma de artigos gerados

ao longo desta atividade. Tais artigos fazem um estudo aprofundado de temas

contemporaneos ligados aos sistemas MIMO ponto a ponto e sistemas MIMO

massivo. Logo abaixo, estes artigos, disponıveis em anexo, sao listados em ordem

cronologica:

[A] Statistical Channel Models in Wireless Links

Relatorio Tecnico

Autores: Ricardo Tadashi Kobayashi, Fernando Ciriaco, Taufik Abrao

[B] On the QAM and PSK Performance

Relatorio Tecnico


[C] Efficient Near-Optimum Detectors for Large MIMO Systems under Corre-

lated Channels


Trabalho original submetido a revista Wireless Personal Communications.

[D] Performance and Complexity Analysis of Sub-optimum MIMO Detectors

Under Correlated Channel


Apresentado no simposio ITS 2014 (International Telecommunications Sym-

posium).

[E] Cooperative Multi-cellular Large MIMO over Desynchronized Channel Es-

timation


Trabalho original submetido a revista Transactions on Emerging Telecom-

munications Systems

Em [A], e apresentado um estudo preliminar dos principais modelos estatısticos

para enlaces sem fio, primordiais para a avaliacao de qualquer tipo de sistema

2 Resultados 8

de comunicacao via radio. Ja em [B], foram estudadas as modulacoes QAM e

PSK, sendo a primeira predominantemente aplicada neste trabalho devido a sua

performance em termos de taxa de erro. Na sequencia, [C] e [D] descrevem o funci-

onamento de diversos detectores MIMO, alem de analisar o desempenho (taxa de

erro de bit) e a complexidade (contagem de operacoes aritmeticas) destes sob ca-

nais correlacionados, sendo possıvel determinar sua viabilidade pratica. Por fim,

em [E] foram analisados sistemas MIMO em cenario multi-celular cooperativo

operando com estimativas imperfeitas de canal, realizadas atraves de sequencias

ortogonais e pseudoaleatorias, sujeitas ao dessincronismo entre receptor e trans-

missor. Neste contexto, o interesse do dessincronismo se deve ao fato de cada tipo

sequencia apresentar caracterısticas proprias de correlacao, levando a diferentes

desempenhos na estimativa de canal, podendo influenciar no desempenho final

do sistema.

9

3 Conclusoes

Ao longo deste trabalho foram estudados diversos problemas recorrentes aos

sistemas MIMO, dentre eles a contaminacao de piloto, a correlacao entre antenas

e a complexidade, sendo todos eles limitantes para a aplicacao desse tipo de

sistema.

Na primeira parte do trabalho, procurou-se determinar o melhor compromisso

complexidade-desempenho para a deteccao MIMO em canais correlacionados. De

um modo geral, a combinacao do equalizador MMSE com tecnicas de ordenacao,

LR e SIC mostrou atingir desempenho quase otimo para sistemas com poucas

antenas. No entanto, o desempenho deste detector se distancia da solucao otima

(SD, ou ML) a medida que o numero de antenas cresce, mas ainda preserva o

melhor compromisso desempenho-complexidade entre os detectores lineares. Adi-

cionalmente, notou-se que a abordagem de ordenacao adotada neste trabalho e

ineficaz para sistemas com um grande numero de antenas operando sob forte cor-

relacao, fruto instabilidade numerica da tecnica escolhida. Neste tipo de situacao,

e recomendavel desconsiderar a ordenacao, ja que o metodo coberto por este tra-

balho se mostrou ineficaz e outras abordagens, como a V-BLAST, levariam a um

aumento excessivo da complexidade da deteccao.

Para a segunda parte, foi estudada a estimativa de canal atraves de diferentes

sequencias de treinamento considerando desincronismo entre transmissor e recep-

tor. Em [D], foi derivada uma expressao para o MSE da estimativa de canal,

que possibilitou constatar a superioridade dos codigos Gold para a estimativa de

canais sob treinamento dessincronizado. Isso se deve a boa ocorrencia de valores

baixos de correlacao cruzada, os quais sao mais recorrentes se comparados com

as sequencias de Kasami, Hadamard e quaternarias. Apesar de possuir valores de

correlacao cruzada duas vezes menores do que os codigos Gold, as sequencias de

Kasami mostraram-se ineficazes devido a maior ocorrencia dos altos valores de

correlacao cruzada, sendo tal observacao valida para as sequencias quaternarias.

No entanto, sob sincronismo perfeito, as sequencias de Hadamard oferecem a

melhor estimativa, ja que elas sao estritamente ortogonais entre si. Por fim,

foi visto que a cooperacao entre celulas, combinada com o uso de sequencias

3 Conclusoes 10

de treinamento longas, limitam a mobilidade dos usuarios mas podem mitigar

a contaminacao do piloto, possibilitando altas eficiencias espectral e energetica

em cenarios multicelulares para sistemas com um grande numero de antenas, i.e.

sistemas MIMO massivo.

11

Anexo A -- Statistical Models for Fading

in Wireless Links

1

Statistical Models for Fading in Wireless LinksRicardo Tadashi Kobayashi, Fernando Ciriaco and Taufik Abrão

Abstract—In this work, it will be briefly presented statisticalmodels capable of describing an electromagnetic channel link,which is primordial for wireless systems analysis. It will beintroduced basic models for both small and large fading, whichmay be applied together in order to perform a more realistic andprecise statistical description of the wireless channel.

I. INTRODUCTION

Despite the propagation of electromagnetic waves is anwell known subject, solving Maxwell’s equation with spaceboundaries is quite impractical for wireless systems analy-sis. Electromagnetic waves may suffer attenuation, reflection,refraction and diffraction until they get to their destination,through different paths, causing constructive or destructiveinterference. Therefore, statistical models must characterizeproperly the behavior of the wireless channel for a feasiblestudy on wireless systems.

First, this work presents the large scale fading, modeled bythe Fris’ equation and the shadowing. In the sequence, thesmall scale fading is characterized by the Rayleigh channelusing Clarke’s model to incorporate the Doppler effect. Hencethe following sections will briefly discuss some basic channelmodels which may be deployed for many simulation scenariosin order to determine the performance of different wirelesssystems.

II. LARGE SCALE FADING [1]

Large scale fading is referred as the attenuation mainlycaused by space loss, where the signal travels over a largearea, and diffraction around large objects.

A. Path Loss(Path attenuation)

The path loss is the power attenuation of a electromagneticwave traveling through the space. The most simple model usesFris equation, which considers free space propagation withunobstructed clear path between the transmission and receivingside:

hPL(d) =

(λ

4πd

)2

. (1)

Notice that the attenuation is dramatically affected with thedistance and the frequency of the wave. Even though, this isthe least severe model, and models like Okumura and 10-raysmay be more suitable for urban areas, considering multi-pathpropagation and the height of both transmission and receptionterminals.

Fig. 1 depicts the path loss using the Fris. equation for asine wave with fc = 880 [MHz]:

100

101

102

103

−90

−80

−70

−60

−50

−40

−30

−20

d [m]

gain

[db]

Figure 1. Fris model for path loss

B. Shadowing

Another aspect to be considered is the path obstructioncaused by large objects, e.g. hills and large buildings. Whenan object obstructs a signal, it may suffer attenuation oramplification, leading to a fluctuation on the path loss curve.Since it takes a relatively large time for mobile terminal toovercome a obstruction, the shadowing term is considered asslow fading. This behavior is usually modeled with a log-normal distribution, which is an extension to Fris Free SpaceModel:

hSH(d) = hPL(d0) + 10n log

(d

d0

)− x [dB] (2)

where d0 is the distance reference, n is the path loss exponentand x is the shadowing dispersion in dB (modeled by a zeromean normal distribution with variance σ2).

By using the log-normal definition, fig. 2 introduces adispersion compared to fig. 1.

100

101

102

103

−100

−90

−80

−70

−60

−50

−40

−30

−20

Propagation lossLog normal dispersion

Figure 2. Fris model for path loss with a 8 [dB] dispersion

2

III. SMALL SCALE MULTI-PATH FADING [1]

When wireless communications are established, multiplesignal copies with random delays travel through the channelcausing constructive or destructive interference on the receiver.These factors can lead to rapid fluctuations on the amplitude,phase or even the signal delay, which is usually referred asthe scale fading.

A. Doppler Effect

Due to the mobility of the users, the frequency ofsent/received signal on the terminal device or on the basestation may experience a frequency change known as theDoppler effect. The Doppler frequency, or the frequency shiftobserved when the source and/or the observer is moving, isgiven by:

fD = fc ·v

c, (3)

where fc is carrier frequency, v is the relative speed betweenthe signal source and observer and c is the speed of light.

Considering a contemporaneous scenario on telecommuni-cations, with a user traveling in a high speed of 180 [km/h]communicating with a base station in a frequency of 5 [GHz],the maximum frequency shift can be evaluated with (3):

fD = 5 109180/3.6

3 108= 833.33 [Hz]

B. Coherence time [2]

The concern on the Doppler effect comes from the coher-ence time, which is the interval of time the channel remainsunchanged. This parameter is expressed as:

∆tc =0.423

fD. (4)

Hence, if the Doppler frequency is high the coherence timeis small. Besides, if the symbol period is too long, comparedto the coherence time, the symbol may experience differentfading levels. In this scenario, the channel is said to be fastfaded. In contrast, a slow fading is observed when the symboltime is shorter than the coherence time.

C. Rayleigh’s Channel

Links with no line of sight can be modeled using theRayleigh distribution, easily obtained by combining two nor-mal distributions, with zero mean and 1/2 variance, in bothphase and quadrature dimensions, compactly represented by:

h(t) ∼ CN (0, 1/2). (5)

In this distribution, the channel gain has a Rayleigh distribu-tion and the phase shift has a uniform distribution. In otherwords, the probability density function of the channel fadingis given by:

f|h(t)|(x) = 2x · e−x2

f h(t)(x) =1

2π

∏( x2π

) (6)

D. Clarke’s Model [3]

In order to consider the frequency shift spread, the Clarke’smodel modifies the Rayleigh. This model consider M propa-gation paths which may have a maximum frequency shift offD:

h(t) =1√M

M∑

k=1

exp [j(2πfD cos (αk) + φk)], (7)

where αk and φk are uniformly distributed between π and −π, i.e. αk, φk ∼ u[−π, π).

Notice that this model still preserves the statistical distribu-tion, as long as the number of multiple paths is high enough.

In fig. 3 and 4 its presented the histogram for the small-scale-fading using Clarke’s model, which confirms the as-sumption that Rayleigh and the uniform distributions arepreserved, for gain and phase respectively.

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Rayleigh Distribution (Theoretical)Clarke‘s Model (Simulated)

Figure 3. Clarke’s channel model histogram for gain, M = 40, fD =10 [Hz]

−3 −2 −1 0 1 2 30

0.05

0.1

0.15

0.2

Rayleigh Distribution (Theoretical)Clarke‘s Model (Simulated)

Figure 4. Clarke’s channel model histogram for phase, M = 40, fD =10 [Hz]

Also, through fig. 5 the coherence time can be approx-imately determined, as the channel does not present large

3

variations in a period ∆tc. For example, one can observe thatmost intervals where the gain remains at approximately at thesame level last about 0.04 ∼ 0.05 [s], i.e, a 10 [Hz] Dopplereffect.

0 0.2 0.4 0.6 0.8 1−6

−4

−2

0

2

4

t [s]

Gai

n [d

B]

0 0.2 0.4 0.6 0.8 1−4

−2

0

2

4

t [s]

Pha

se [r

ad]

Figure 5. Clarke’s Model gain, M = 40, fD = 10 [Hz]

IV. CONCLUSION

This work introduced basics on wireless channel modeling,covering the free-space path loss model, log-normal path lossmodel and Clarke’s model for fast fading. Large scale fadingproduces a massive and quasi static power loss, on the otherhand small scale fading may introduce large attenuation butwith a much more fast transition. By combining the large andsmall scale, we can derive more realistic scenarios for sim-ulation and performance analysis on wireless communicationsystems.

REFERENCES

[1] A. Goldsmith, Wireless Communications. Cambridge University Press,2005.

[2] J. Hampton, Introduction to MIMO Communications. CambridgeUniversity Press, 2013.

[3] C. Xiao, Y. Zheng, and N. Beaulieu, “Novel Sum-of-Sinusoids SimulationModels for Rayleigh and Rician Fading Channels,” Wireless Communi-cations, IEEE Transactions on, vol. 5, no. 12, pp. 3667–3679, December2006.

15

Anexo B -- On the QAM and PSK

Modulation Performance

1

On QAM and PSK Modulation PerformanceRicardo Tadashi Kobayashi, Fernando Ciriaco and Taufik Abrão

Abstract

This work will briefly compare M -QAM and M -PSK modulations through error rate simulation, signal char-acteristics and analytic analysis on the distance of adjacent symbols.

I. INTRODUCTION

Digital systems are well known to outperform analog systems as they became cheaper, faster, more efficientand far more flexible. For communication the use of digital systems is primordial since they can provide high-data rates, error correction mechanisms, higher noise immunity, multiple access and better security and privacyproperties. From this perspective, this work will study the performance of M-ary versions of the QAM and PSKmodulation, both capable of multiplexing log2(M) bits per symbol.

Basically, this work will describe both QAM and PSK modulations, giving more attention to the averageconstellation energy and the adjacent symbols distance. After that, it is calculated the BER gap between QAMand PSK modulations of same order. Finally, simulations results will validate previous calculations.

II. QUADRATURE-AMPLITUDE-MODULATION (QAM)

The M−QAM set of symbols has basically both real and imaginary parts inside the odd integer set, excludingthe zero, which is described by the following equation:

SQAM = {a+ jb | a, b ∈ {−√M + 1,−

√M + 3, . . . ,

√M − 1}}. (1)

Despite providing good spectral efficiency, the drawback of this modulation can be observed in figure 1 asM−QAM modulated signals do not posses a constant envelope and also present abrupt amplitude and phasevariation. These variations may cause noise on the the transmitting power amplifier as the output signal may varyabruptly.

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

Qua

drat

ure

In phase

(a) Constellation

0 0.05 0.1 0.15 0.2 0.25 0.3

−8

−6

−4

−2

0

2

4

6

8

Time[s]

Mud

ulat

ed s

igna

l [V

]

−3−j5 −5−j3 1+j1 3+j7 3+j1

(b) Band-pass signal with symbols marked in the constellation graph

Figure 1. 64-QAM constellation and band-pass signal

2

A. Average constellation energy

Consider now a M−QAM signal sQAM, i.e. sQAM ∈ SQAM, where all symbols are independent and identicallydistributed. In this case, the average power of sQAM is proportional to the sum of all squared symbols on theconsidered set:

ESQAM = E[|sQAM|2

]

=1

M

M∑

k=1

|SQAM[k]|2. (2)

Considering that SQAM[k] = x[k] + jy[k], the fact that both x and y are odd integers and the symmetry of theproblem:

ESQAM =4

M

√M

2−1∑

m=0

√M

2−1∑

n=0

|x[m]|2 + |y[n]|2. (3)

For the sake of organization, let β =√M/2− 1:

ESQAM =4

M

β∑

m=0

β∑

n=0

(2m+ 1)2 + (2n+ 1)2

=4

M

[(β + 1)

β∑

m=0

(2m+ 1)2 + (β + 1)

β∑

n=0

(2n+ 1)2

]

=8

M(β + 1)

β∑

m=0

(2m+ 1)2

=4√M

β∑

m=0

4m2 + 4m+ 1

(4)

Finally, using the summation formulas found in the appendix:

ESQAM =4√M

{4

[β(β + 1)(2β + 1)

6

]+ 4

[β(β + 1)

2

]+ [β + 1]

}

=2

3

[4(β2 + 2β) + 3

]

=2

3(M − 1)

(5)

B. Adjacent symbols distance

In order to compare the performance of different modulations, let us evaluate the minimum distance betweenadjacent symbols. First consider sQAM/

√ESQAM as a transmitted M−QAM modulated symbol with normalized

power, where sQAM ∈ SQAM. The power normalization is an important step for a fair comparison between differentmodulation types. In this case its straightforward that the minimum distance between adjacent symbols is given by:

dQAM = 2

√2

3

1

M − 1(6)

This is a necessary metric since additive noise may introduce errors on the symbol estimation, i.e. if symbolsare very close to each other, small amounts of noise may cause an erroneous estimation.

III. PHASE-SHIFT-KEYING

For the M−PSK modulation scheme, symbols belong to the following set:

SPSK = {ej 2π

Mk|k ∈ {1, 2, ...,M}}, (7)

i.e. all the symbols lie on an unitary radius circle.From eq. (7) and fig. 2, PSK modulated symbols present a constant envelope. Therefore, PSK modulated signals

suffers less from abrupt variations than QAM signals.

3

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Qua

drat

ure

In phase

(a) Constellation

0 0.05 0.1 0.15 0.2 0.25 0.3

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time[s]

Mud

ulat

ed s

igna

l [V

]

0.9+j0.4 −0.7−j0.7 0−j1 0.7+j0.7 0+j1

(b) Band-pass signal with symbols marked in the constellation graph

Figure 2. 16-PSK constellation and band-pass signal

A. Average constellation energy

Due to the positioning of the PSK symbols, the calculation of the average constellation energy is quite straight-forward:

ESPSK = 1 (8)

B. Adjacent symbols distance

As the previous section, we must evaluate again the distance between adjacent symbols for a PSK modulatedsignal with normalized power. Since the average power of the constellation energy is unitary, there is no need ofadditional factors to normalize the power, hence the modulated PSK symbol with normalized power is given bysPSK ∈ SPSK. which can be easily derived and is given by:

dPSK = 2 sin( πM

)(9)

IV. PERFORMANCE COMPARISON

This section will compare the performance between PSK and QAM modulation considering baseband signals.First, it will be calculated the theoretical BER gap between these two modulation schemes and then simulationswill prove our assumptions.

For the performance gap between QAM and PSK modulation, we first evaluate the rate of the minimum distanceof adjacent symbols for both modulation:

dQAM

dPSK=

1

sin(π/M)

√3

2

1

M − 1. (10)

The previous equation describes the additional amount of space that noise can take place without corrupting QAMsymbols. Since decision boundary for QAM symbols are larger, error rate should be better on the same ratio.Therefore the performance gap between the considered modulation is expected to be:

δ(M) = 10 log

∣∣∣∣dQAM

dPSK

∣∣∣∣2

= 20 log

∣∣∣∣∣1

sin(π/M)

√3

2

1

M − 1

∣∣∣∣∣ . (11)

In order to validate this result let us test both modulation with the same order M = 16 over a AWGN channel.In this case, δ(16) = 4.1953 [dB] is the expected performance gap between QAM and PSK modulations of orderM = 16, i.e. 16−PSK requires extra 4 [dB] of power to achieve the 16−QAM error rate performance. Fig 3 shows

4

0 2 4 6 8 10 12 14 16 18

10−5

10−4

10−3

10−2

10−1

SNR [dB]

BE

R

16−QAM16−PSK

Figure 3. Simulated BER performance for single antenna 16−QAM and 16−PSK systems under AWGN channel

that the performance gap between QAM and PSK modulation schemes is actually 4 [dB] as predicted, validatingour assumptions.

It is also noteworthy that as the performance gap δ is proportional to the modulation order. For example, nowfor M = 64, the expected performance predicted by (11) is δ(64) = 9.9516, which can be validated by the graphin 4.

0 5 10 15 20 25 3010

−5

10−4

10−3

10−2

10−1

SNR [dB]

BE

R

64−QAM64−PSK

Figure 4. Simulated BER performance for single antenna 64−QAM and 64−PSK systems under AWGN channel

A more efficient approach to verify the performance gap is shown in 5, as it consists on the plot of equation (11).Despite figure 11 shows very large modulation orders, in practice, they are not deployed due to their performanceand demodulation complexity.

5

102

103

104

105

106

5

10

15

20

25

30

35

40

45

50

δ(M

)

M

Figure 5. Performance gap between QAM and PSK for different modulation orders

V. CONCLUSION

Throughout this work, it was showed that both QAM and PSK modulation schemes can both provide high spectralefficiency for high modulation orders as they are capable of multiplexing log2 |M | bits per symbol. However, it wasproven that QAM modulated symbols have better immunity to additive noise, leading to lower error rates. Howeverband-pass signals for QAM modulated signals may present abrupt variations, not found in PSK modulation, whichintroduce noise from output power amplifiers.

VI. APPENDIX

In the sequence, it is listed the summation formulas used in this work, used to derive eq. (5).n∑

k=0

1 = n+ 1

n∑

k=0

k =n(n+ 1)

2

n∑

k=0

k2 =n(n+ 1)(2n+ 1)

6

REFERENCES

[1] A. Goldsmith, Wireless Communications. Cambridge University Press, 2005.[2] L. Couch, Digital and analog communication systems. Macmillan, 1983.[3] I. Horikawa, T. Murase, and Y. Saito, “Design and performances of a 200 mbit/s 16 qam digital radio system,” Communications, IEEE

Transactions on, vol. 27, no. 12, pp. 1953–1958, Dec 1979.

21

Anexo C -- Performance and Complexity

Analysis of Sub-optimum MIMO

Detectors Under Correlated Channel

Wireless Personal Communications manuscript No.(will be inserted by the editor)

Efficient Near-Optimum Detectors for Large MIMOSystems under Correlated Channels

Ricardo Tadashi Kobayashi · FernandoCiriaco · Taufik Abrão

the date of receipt and acceptance should be inserted later

Abstract Recently, high spectral and energy efficiencies multiple antennaswireless systems under scattered environments have attracted an increasing in-terest due to their intrinsic benefits. This work focuses on the analysis of MIMOequalizers, improved MIMO detection techniques and their combinations, al-lowing a good balance between complexity and performance in a Rayleighchannel environment. Primarily, the MIMO linear equalizers combined withdetection techniques such as ordering (via sorted QR decomposition, SQRD),successive interference cancellation (SIC), list reduction and lattice reduction(LR) were investigated. An important aspect invariably present in practicalsystems take into account in the analysis has been the channel correlation ef-fect, which under certain realistic conditions could result in a strong negativeimpact on the MIMO system performance. The goal of this paper consistsin constructing a framework on sub-optimum MIMO detection techniques,pointing out a MIMO detection architecture able to attain low or moderatecomplexity, suitable performance and full diversity.

Keywords Lattice-reduction · Channel correlation · MIMO · Zero-forcing (ZF), Minimum-mean-squared-error (MMSE) · ordered successive-interference-cancellation (OSIC) · Chase-list · Sphere-decoder (SD).

R. T. KobayashiDept. of Electrical EngineeringState University of Londrina, BrazilE-mail: [email protected]

F. CiriacoDept. of Electrical EngineeringState University of Londrina, BrazilE-mail: [email protected]

T. AbrãoDept. of Electrical EngineeringState University of Londrina, BrazilE-mail: [email protected]

2 Ricardo Tadashi Kobayashi et al.

1 Introduction

The use of multiple antennas in transmission and reception is a way to real-ize higher spectral efficiency and/or reliable mobile communication systems,offering improved quality of service (QoS) through the wireless channel [4].The pioneering work [24] proposed a V-BLAST architecture capable of pro-viding spatial multiplexing gain and high data rate, motivating countless worksaround multiple antenna systems. Improvements obtained through multiple-input-multiple-output (MIMO) systems may be on transmit energy efficiency,data rate and/or symbol error rate, being determined by the antenna configu-ration and transmission-detection techniques as well. Balancing these improve-ments with the available resources in the system is a project necessity, sinceenergy and spectrum are increasingly scarce resources. As a result, it is neces-sary to formulate solutions in order to obtain improved performance under lowor moderate complexity constraints. Generally, these solutions brings an incre-ment in communication capacity, or alternatively, reducing the system powerconsumption, the radiated power, size and weight of the wireless system, whichare desirable characteristics, especially in mobile terminals (MTs). Hence, thegoal of this work consists in pointing out and analyzing MIMO architecturesand detectors with low or moderate complexity holding suitable performanceunder full diversity condition. Thus, linear MIMO equalizers combined withsub-optimal detection techniques such as ordering (via SQRD), interferencecancellation (SIC), list and lattice reduction (LR) were carefully analyzed interms of performance-complexity trade-off.

Another aspect to be considered in this work is the fading channel correla-tion; since the physical dimensions required for the mobile devices is increas-ingly restricted, the distance between antennas in the same MT is reducedas well. Indeed, under MIMO systems operating over UHF frequency range,the distance between antennas required in order to achieve the uncorrelatedchannel condition is not very small; hence, practical and versatile MIMO com-munication systems accommodating a large number of antennas and exploit-ing the maximal diversity gain (or alternatively maximal multiplexing gain)is challenging. A MIMO channel correlation scenario causes deleterious effectson the achievable data rate and on performance as well [14], requiring addi-tional power to comply with target quality of service (QoS) indexes. This canbe a concern, since the use of receivers capable of operating in high SNR andlow phase noise present a high cost [18]. Therefore, efficient MIMO detectorsable to operate under suitable bit error rate (BER) performance and powertransmission constraints are of paramount interest.

More recently, massive MIMO systems proved to be a promising technologyfor 5G systems [3], mainly for its highest spectral and energy efficiency, butmainly for being immune to additive noise for very large arrays [19]. However,large arrays bring two main problems to be addressed in this work: correla-tion between antennas and the signal processing complexity. The first comesfrom the fact that large antennas arrays must be accommodate in relativelysmall areas, causing correlated channels. On the other hand, the increase on

Title Suppressed Due to Excessive Length 3

the processing complexity is straightforward, since large arrays require morehardware and larger number of operations due to the large number of anten-nas. Therefore, analyses on MIMO processing techniques is very importantsince practical high efficient communications systems must meet their ownlimitations.

From the point-of-view of information detection, it is well known thatMIMO systems sends data through different antennas, traveling through dif-ferent paths, so the received signal is a combination of every transmit antennassignals (in each receive antenna), from which the original message is recovered.The MIMO detection problem consists primarily in decoupling the transmittedsignal from a received bandpass signal sample, while the processing load is con-centrated almost on the receive side, if no coding is applied on the transmitterside. Since the mobile communication systems do not possess high processingcapability or abundant energy resources, due to battery size and weight con-straints, the design of efficient MIMO detectors is of paramount importanceand an ongoing research topic.

It’s well known that the maximum likelihood (ML) detection principle pro-vides the best performance in terms of bit error rate, but its computationalcomplexity makes this detector impractical. On the other hand, the spheredecoder (SD) is an approach representing the state-of-the-art on MIMO de-tection; SD results in a lower complexity than the ML for small amounts ofnoise but still remains quite complex under low SNR regime, and is furtherdependent on the arrangement of the MIMO system. Besides, there are alsothe classical linear MIMO detectors, such as zero-forcing (ZF) and minimummean squared error (MMSE)-based MIMO detector, which when combinedwith ordered successive interference cancellation (OSIC) [24, 25, 27] or OSICwith repetition [22, 23] are able to exhibit lower complexity than the ML oreven SD detectors, though some degradation in symbol error performance.Furthermore, the lattice reduction (LR) [21] technique aided linear MIMO de-tectors provides full diversity exploitation, improving greatly the performanceand, in some cases, achieving near ML-performance. The complexity of lattice-reduction is known to be polynomial in time, but correlated channels impactnegatively on the MIMO system complexity, as the columns of the channelmatrix become more and more likely. Even with lower complexity than ML,the LR-aided MIMO detectors can still show undesirable complexity with alarge number of antennas and high correlation between them. Hence, it’s oneof the challenges for the applicability of large-MIMO systems.

Notation: (·)T , (·)H , (·)† and (·)−1 denote the matrix transpose, Hermitian,pseudo-inverse and inverse, respectively. Boldface lowercase letters representsvectors, while boldface uppercase letters denote matrices. I and 0 denote theidentity and an all zero matrix, respectively. Finally, tilde superscript (·) rep-resents a symbol vector estimation and hat (·) superscript represents a symbolestimation after a slicer, which quantizes an estimated symbol to its nearestconstellation point. This process will also be denoted as Q(·) = (·).


2 System Model

The transmission process of a point-to-point MIMO system composed by nT

transmit antennas and nR receive antennas can be compactly described by:

x = Hs + n. (1)

Herein we have assumed overdetermined MIMO systems, i.e., nR ≥ nT , op-erating in spatial multiplexing mode. In other words, the detected symbol isobtained by solving a system of nT unknowns with, at least, nT equations.Also, no precoding techniques were deployed.

Eq. (1) represents a transmission of nT modulated symbols, snT ×1, througha channel whose gain is denoted by HnR×nT

and additive noise nnR×1. Eachelement of H represents the channel gain in its respective path and these gainswill be assumed known at the receiver side. After passing through the channel,the symbols are combined forming the received vector, xnT ×1. Besides, a pre-processing block is responsible for the modulation and coding, if applied, whilethe MIMO detector recovers the sent data from the received signal, corruptedby background noise and inter-antenna interference.

It’s assumed that the noise vector has a circularly-symmetric complexGaussian distribution, n ∼ CN (0, N0I), with variance N0. Alternatively, thenoise can be statistically represented by its covariance matrix E

[nnH

]=

N0InR .The transmitted symbols belong to the set S, which depends on the modu-

lation order M , deployed on the transmitter side. Since each antenna’s symbolis independent and the power is divided between the antennas, the covari-ance matrix of the transmit symbols is given by E

[ssH

]= ES

nTInT

, where ES

represents the average symbol energy.Due to its spectral efficiency and performance trade-off, in this work the

M -QAM modulated symbols has been deployed; M -QAM symbols are denotedby a complex number, which real and imaginary parts are odd and limited to±(

√M − 1), i.e. [1, 10]:

S = {a + jb | a, b ∈ {−√

M + 1, −√

M + 3, . . . ,√

M − 1}}.

For this modulation, the average symbol energy is given by:

ES =2(M − 1)

3(2)

Also, it will be assumed Gray coded symbols, where adjacent symbols presentonly one bit difference, minimizing the bit error.

In some MIMO detection procedures, both the noise power and the trans-mit energy parameters, or the power-signal-to-power noise ratio (SNR) mustbe estimated at the receiver side for a suitable detection procedure, as shownin the next sections.


2.1 Spatially Correlated MIMO Rayleigh-Fading Channels

The MIMO fading channel is properly modelled by flat Rayleigh distributions.Additionally, correlation between antennas is considered through Kronecker’scorrelation model [2, 28]:

H =√

RH,RxG√

RH,Tx. (3)

where G(nR × nT ) is composed by independent, identically distributed com-plex Gaussian elements, gij ∼ CN (0, 1), the matrices RH,Rx(nR × nR) andRH,Tx(nT × nT ) represent the spatial channel correlation observed in the re-ceiver and transmitter side, respectively. The elements of these two matricesare given, in terms of normalized correlation index ρ, by:

{rH,Rx ij = ρ(i−j)2

rH,Tx ij = ρ(i−j)2 .(4)

Hence, if nT = nR both channel correlation matrices can be written as:

RH =

1 ρ ρ4 · · · ρ(n−1)2

ρ 1 ρ · · ·...

ρ4 ρ 1 · · · ρ4

......

.... . . ρ

ρ(n−1)2 · · · ρ4 · · · 1

. (5)

3 MIMO Detectors

This section revisits the most common MIMO detection techniques availablein the literature, including the maximum-likelihood (ML), sphere decoder,zero-forcing (ZF) and minimum-mean-squared-error (MMSE). Additionally,we provide a brief discussion on the use of less conventional techniques forMIMO detection, such as ordering, interference-cancellation, list-reduction andlattice-based. It is of great importance the knowledge on each detector proce-dure, especially for complexity evaluation and BER performance analysis.

3.1 Maximum-Likelihood (ML)

Despite its complexity, the maximum likelihood detector provides the optimalBER performance. ML detection is performed by an exhaustive search for theclosest symbol and signal reconstruction regarding the signal observation at thereceiver. Considering a M−ary modulation order with nT transmit antennas,each one transmitting a distinct symbol at each time-slot system, the numberof symbols combination is simply MnT . For example, the ML detector for


16−QAM and nT = 8 transmit antennas must test a list of over ≈ 4 billioncandidate-symbols.

Considering s a candidate-vector from the SnT set, of size MnT , the candidate-vector presenting the lowest distance from the received signal, and thereforethe lowest error, can be expressed by:

s = argmins∈SnT

∥x − Hs∥2. (6)

3.2 Sphere Decoder (SD)

In order to reduce the complexity of the ML detector, the sphere-decoder [12]searches for only the candidates contained in a sphere of radius d:

d2 < ∥x − Hs∥2 (7)

which, of course, is dependent on the signal-to-noise-ratio. If the radius is settoo high, the SD complexity tends to the ML one. On the other hand, if thesearch radius is small enough, there will be no candidates into the hypersphere.

Aiming to perform a sphere detection, eq. (1) is rewritten applying the QRdecomposition on the channel matrix. Through this decomposition, we obtainan orthogonal matrix Q, where I = QHQ, and an upper triangular matrix R,both with convenient properties for the detection procedure:

y = QHx

= QHQRs + QHn

= Rs + n′.

(8)

Since Q is orthogonal, the statistics on the noise, n′, remain unchanged and nonoise enhancement is expected. Also, R matrix is upper triangular, enablingestimating antennas’ signal independently. With these facts, eq. (7) becomes:

d2 < ∥y − Rs∥2 (9)

Considering R = [r1 r2 r3 · · · rnT]T , the noise norm is given by:

∥n′∥2 = ∥y − Rs∥2 =

nT∑

k=1

|yk − rks|2 (10)

Indeed, eq. (10) shows that the noise norm is the sum of each of its layer’snorm. Therefore, the noise norm can be updated as the symbols are tested ineach layer, which benefits the radius criteria test by avoiding the evaluationof the estimated noise norm for every symbol combination.

The structure of the detection problem in (9) allows a tree search thatbegins from the last antenna’s symbol to the first one, where the candidate-symbols are tested recursively and independently, differently from ML. It isnoteworthy that this process still obeys the radius constraint defined in (10).By the end of the SD detection procedure, the most likely symbol-vector insidethe sphere of radius d is taken as the solution.


3.3 Zero-Forcing (ZF)

The zero-forcing MIMO detector basically solves the linear system problemposed by (1) ignoring the additive noise. The solution proposed for this detec-tor uses Moore-Penrose pseudo-inverse matrix [15]:

s = H†x = s + H†n. (11)

Therefore, the equalization matrix is given by the pseudo-inverse of the channelmatrix:

WZF = (HHH)−1HH . (12)

Despite its low complexity, when H is ill conditioned, it is also near singular,causing noise enhancement [15], given by H†n. Hence, suitable performancecannot be expected in this situation.

3.4 Minimum Mean-Squared-Error (MMSE)

By taking into account the noise and signal statistics, the MMSE detector isable to reduce the impact of the additive noise on the detection, improvingoverall MIMO performance. In the MMSE approach, the minimization of thesymbol error can be obtained by solving the following optimization problem[20]:

WMMSE = argminW

E[∥s − Wx∥2]. (13)

Solving (13), the equalization matrix for this detector is obtained as:

WMMSE =

(HHH +

N0

ESI

)−1

HH . (14)

Hence, the solution for MMSE detector is:

s =

(HHH +

N0

ESI

)−1

HHx. (15)

Alternatively, MMSE detection can also be performed as:

s = H†x

= s + H†n(16)

where the extended channel matrix and the received vector are given respec-tively by:

H =

H√

N0

ESInT

, x =

[x

0nT ×1

].

Despite being more complex than the approach given by (15), the extendedmatrix model is required for successive interference cancellation (SIC) and isalso recommended for lattice-reduction due to performance improvements [26].


3.5 Vertical Bell Laboratories Layered Space-Time (V-BLAST)

The V-BLAST architecture, proposed in [24], is a MIMO detection architecturewhich performs ordered successive interference cancelation (OSIC) through alinear detector (ZF or MMSE). Under this detection strategy, BER reductionis expected, as well as computational complexity increment.

Initially, the algorithm evaluates the equalization matrix, ZF or MMSE.Unlike the channel matrix, the equalization matrix will be represented as nT

row vectors: W = [w1 w2 · · · wnT ]T .The detection order follows the minimization of ∥wki∥, with symbol detec-

tion obtained by:sk = wkix. (17)

After passing by a slicer, the interference is reconstructed and cancelled:

x := x − skihki . (18)

where ski is the sliced version of the symbol detected applying (17).Since the kith antenna’s symbol is detected and its interference canceled,

the kith column of H is no longer needed, so it is zeroed. Due to this process,the channel matrix changes, so the equalization matrix needs to be evaluatedagain. This process is repeated until all symbols are detected. The whole V-BLAST detection procedure is summarized in the pseudo-code 1.

Algorithm 1 V-BLAST CoreInput: H, x and nT

Output: s1: for i = 1 to nT do2: W = H†

3: ki = argminj /∈{ki−1,ki−2,...}

∥wj∥2

4: ski= wki

x5: ski

= Q(ski)

6: x := x − skihki

7: null the kith column of H8: end for

The presented algorithm uses the ZF equalizer, but the MMSE equalizercan also be used by evaluating the pseudo-inverse of the extended channelmatrix H.

One of V-BLAST detection’s weakness is its high complexity, since it re-quires an update on the equalization matrix for every symbol interferencecancellation. When compared with other OSIC detectors, V-BLAST detectionrequires nT pseudo-inverse evaluations, while the sorted QR decompositionapproach, as discussed in section 3.7, requires only one matrix decomposition.In [27], it’s shown that the performance gap between these detectors is verylow, but it can disappear via post ordering. Therefore, this work will focus onthe SQRD approach.


3.6 Successive Interference Cancellation (SIC)

The successive interference cancellation can be obtained using the QR decom-position in the channel matrix H, as explained in section 3.2. Noticing for theZF approach the QR decomposition procedure is applied on the H matrix,while for the MMSE detector to the matrix H.

The process represented in (9) prepares the system’s solution, consideringR is upper triangular. Hence, the linear system is solved upwards by:

si =

yi

rii, i = nT

1

rii

(yi −

nT∑

k=i+1

riksk

), i = nT − 1, . . . , 3, 2, 1.

(19)

Noticing that every symbol must pass by the slicing step before feed for-ward with the interference cancellation, which is applied in the next symboldetection. Indeed, the slicing step must be implemented in order to perform aproper interference cancellation and improve the system performance.

3.7 Sorted QR Decomposition (SQRD)

Further performance improvements on the SIC procedure can be achievedthrough a proper ordering [27], which avoids error propagation during inter-ference cancellation computation. The ordering criteria is the minimization ofthe columns’ norm of Q, which makes the detection be proceeded from thestrongest to the weakest symbol.

The form of the decomposition is:

HΠ = QR (20)

where Π is a permutation matrix, used to reorder the symbols after applyingSIC detection. Note that the detection is followed as a conventional SIC, asdescribed in (19). However, at the end of the detection process the reorderingis carried out by multiplying the symbol vector by the permutation matrix.

The sorted QR decomposition algorithm is represented by the pseudo-codein Algorithm 2. If lines 2 and 3 of the algorithm are ignored, it will perform aconventional QR decomposition with the Gram-Schimidt approach. Since these


lines are not high complexity operations, the cost for ordering is practicallynegligible.

Algorithm 2 Sorted QR decompositionInput: Q = H,R = 0,Π = InT

Output: Q,R1: for i = 1 to nT do2: k = argmin

j=i to nT

∥qj∥2

3: exchange columns i and k in Q, R and Π4: rii = |qi|5: qi = qi/rii

6: for j = i + 1 : nT do7: rij = qH

i qj

8: qj = qj − rijqi

9: end for10: end for

Notice the QR decomposition on the extended matrix channel requiresminor modifications on the algorithm, which are found in [25].

Applying this decomposition ensures suboptimal but near BLAST order-ing performance. In order to achieve V-BLAST performance, a post sortingalgorithm can be applied, as discussed in [25]. In our analysis, the use of thisdecomposition will be referred as ordered successive interference cancellation(OSIC).

3.8 Chase List Algorithm

Performance improvements on the previous discussed MIMO detectors can beobtained, with manageable complexity, through the Chase list (CL) algorithm.The key feature for its success is repetition, since in CL algorithm the symboldetection is a recurrent procedure. Fig.1 depicts a generic diagram for theChase list MIMO detection.

hnT

s(1)

b Subdetector 1

x

x(1)

s(1)nT

hnT

s(2)

x(2)

s(2)nT Subdetector 2

x

b

s(q)

Subdetector qhnT

x(q)

s(q)nT b

x

b

b

b

x

argmink=1 to q

‖Hs(k) − x‖

det

ecti

on

and

list

crea

tion

Last

ante

nna’s

sym

bol

s

Fig. 1 Block diagram of Chase list detector.


Basically, in the CL algorithm the q best candidates are first selected aimingto detect one symbol; then interference cancellation is performed, and the re-maining symbols are finally detected (sub-detection). This process is repeatedfor all the candidates and then the best set is chosen using the Euclideannorm criterion, just like in the ML detector, but with a reduced candidateslist. Notice also that q is limited to M .

To elaborate further, initially, the last antenna’s symbol is detected (noordering criterion), which can be proceeded with the previously discussed de-tectors. From this, the q best candidates for the last antenna’s symbol, namelyC = {c

(1)nT , c

(2)nT , c

(3)nT , ..., c

(q)nT }, are classified by using the Euclidean distance cri-

terion. Next, the interference cancellation and the remaining symbol detectionare carried out. Then SIC is performed for each symbol candidate, from i = 1to q, ignoring the noise effect:

x(i) = x − s(i)nT hnT . (21)

After the interference cancellation step, the remaining symbols are detectedusing the chosen MIMO sub-detector. Hence, with all the q symbols vector-candidates detected, the solution for the Chase list detection is simply givenby solving the problem:

s = argmink=1 to q

∥Hs(k) − x∥2. (22)

For the sake of simplicity, the Chase list was presented using linear MIMOdetection techniques; however, the use of SIC-based detectors is desirable,since SIC’s first symbol detection requires only one division and interferencecancellation is inherit to the process.

A Pseudo-code for the CL-MIMO detector which applies the SIC approachis presented in Algorithm 3. In this Chase list algorithm a zero forcing SIC(ZF-SIC) detector has been deployed, but the use of other techniques, suchas ordering combined with linear detector, can be applied successfully. In thissense, the use of MMSE equalizer is also possible by carrying out the QRdecomposition on the extended channel matrix. Other Chase list approachescan be found in [22,23].

As mentioned, ordering procedure can be included in CL-MIMO detectionand the SQRD makes the strongest symbol to be detected first, as it becomesthe last symbol of the s vector, as also described in Fig. 1. In order to applyOSIC on the Chase list, its only needed two modifications in the Algorithm 3:

a) SQRD is applied instead of the conventional one (line 2);b) Received symbol must be multiplied by the permutation matrix Π (after

line 15).

3.9 Lattice Reduction (LR) aided MIMO detector

One important issue found in the previous detectors is their narrow deci-sion boundaries, which makes the detection more sensitive even under small


amounts of noise. In order to avoid this problem, detection can be done inanother domain followed by proper conversion, which is obtained through thelattice reduction technique [8, 26].

Algorithm 3 Chase List CoreInput: H, x, q and nT

Output: s1: e = ∞2: Evaluate the QR decomposition of H3: y := QHx4: snT = ynT /rnT nT

5: Sort the q closest constellation symbols of snT

C = {c(1)nT

, c(2)nT

, c(3)nT

, ..., c(q)nT

}6: for i = 1 to q do7: s

(i)nT

= c(i)nT

8: y(i) = y − s(i)nT

hnT

9: for j = nT − 1 to 1 do

10: s(i)j = Q

1

rjj

y

(i)j −

nT∑

k=j+1

rjk sk

11: end for12: if ∥Hs(i) − x∥2 < e then13: e := ∥Hs(i) − x∥2

14: a = i15: end if16: end for17: s = s(a)

The lattice reduction can be implemented through the LLL algorithm,proposed by Lenstra-Lenstra-Lovás in [16]. Basically, the LLL algorithm de-composes the MIMO channel matrix into a new base:

H = HT, (23)

where H is a base with better properties, in terms of near-orthogonality, thanthe original H, while T is an unitary matrix, which presents two features:det(|T|) = ±1, and T ∈ {ZnR×nT + jZnR×nT }. As the new matrix H has bet-ter properties, the decision boundaries are enlarged and the noise amplificationeffect is reduced.

Since the detection is done in the LR domain, it’s desirable to rewrite thesystem model in terms of LR symbols:

x = (HT)(T−1s) + n

= Hz + n.(24)

Under this modified system model, the LR symbols z can be detected usingany linear MIMO equalization technique discussed previously. Herein, it will


be presented the zero-forcing equalization approach:

z = H†x

= z + H†n.(25)

As revealed by (24) and (25), noise also corrupts the symbols on the LRdomain. Hence, a proper decision must be made on the LR domain symbols,z, including shifting and scaling operations, as follows:

z = 2

⌊z − βT−11nT

2

⌉+ βT−11. (26)

where ⌊·⌉ is the round operator, 1nT represents a column vector of ones andβ is a constant determined by the modulation format and order. Since thetransmission scheme deployed in this work uses exclusively QAM modulation,we need to set β = 1 + i.

In the last step of the LR-aided MIMO detection, the symbols must beconverted from the LR domain to the original signal space:

s = Tz. (27)

3.10 LR-aided Chase-list

In order to deploy jointly lattice reduction technique and Chase-list MIMOdetection, herein SIC structure will be used similarly as proposed in [1, 5].Firstly, the QR decomposition, or even the SQRD, is evaluated; after that, itis multiplied by the received signal, like (8), which leads to:

y = QHx

y = Rs + n[y1

ynR

]=

[R1 R2

0 rnRnT

][s1

snT

]+

[n1

nnR

] , (28)

where R1 is the composed by the columns 1 to nT − 1 and rows 1 to nR − 1of R; R2 is formed by the last column and (nR − 1)th first rows of R; 0 is azero row vector with nT − 1 elements; rnRnT

is the last element of R.On the sequence, from eq. (19), the last estimated symbol becomes:

snT= ynT /rnT nT

(29)

After the nT antennas’ symbols have been sliced, the interference cancellationis proceeded:

y1 := y1 − snTR2 (30)

Assuming a successful interference cancellation, the system defined in (28) isreduced to:

y1 = R1s1 + n1 (31)


Hence, with this new reduced system description, it is possible to apply theLR-aided method discussed previously in order to detect the remaining nT −1symbols. Notice that the lattice reduction in eq. (31) is required only once,since the interference cancellation procedure does not change the system. As-suming that the Chase-list is deployed, the signal processing operations definedby (29) to (31) are repeated q times, with the q best candidates for snT .

4 Performance Comparison

In this section, the simulated BER performance of various MIMO detectorsdiscussed previously have been compared. For a fair comparison between dif-ferent MIMO transmission systems with different modulation order and num-ber of antennas (even SISO systems), it will be considered a normalized SNREb

N0= SNR

log2 M , and transmit power constraint, with power equally distributedamong the nT antennas. Each MIMO detector performance was evaluated con-sidering three modulation formats and antenna arrangements:

a) 64−QAM-4 × 4; b) 16−QAM-8 × 8; c) 4−QAM-20 × 20.

Furthermore, three antenna correlation scenarios has been considered:a) no correlation ρ = 0;b) medium correlation ρ = 0.5;c) strong correlation ρ = 0.9.Finally, it was considered perfect channel-state-information (CSI) available atthe receiver side.

Fig. 2 depicts the first evaluated arrangement for the BER performancecomparison, with 64−QAM modulation format and 4×4 antennas. In low SNRregime, the nine analysed MIMO detectors provide very similar performance.On the other hand, the LR-aided MIMO detectors (LR-ZF, LR-MMSE, LR-MMSE-OSIC and LR-CL-MMSE-OSIC) are able to achieve full diversity, aswell as the ML and SD detectors. Recall that a system achieves full diversitywhen a 3 [dB] increasing on the transmission power implies in a BER reductionfactor of 2nT in high SNR regime [2].

When small antennas array are applied, the LR-aided detector presents anear ML/SD performance. The difference between their performance in highSNR scenarios results in a small gap, but the performance curves are parallel,implying in same diversity order.

The ZF and MMSE MIMO detectors tend to present the same performancein high SNR scenarios, for the MMSE solution is approximately equal to theZF one, as described in (11) and (15). Besides, Fig. 2 confirms that interferencecancellation, lattice-reduction techniques and combination of them can greatlyimprove the MIMO detection performance.

Regarding the antenna correlation effect on performance, still in Fig. 2, onecan conclude that as the correlation index grows the BER performance dete-riorates, specially when ρ ≥ 0.5. In high correlation scenarios non-LR-aided


0 5 10 15 20 25 30

10−3

10−2

10−1

BE

R

Eb/N

0 [dB]

ZFMMSEMMSE−OSICCL−MMSE−OSICLR−ZFLR−MMSELR−MMSEOSICLR−CL−MMSE−OSICSD

(a) ρ = 0

0 5 10 15 20 25

10−3

10−2

10−1

BE

R

Eb/N

0 [dB]


(b) ρ = 0.5

0 5 10 15 20 25 30 35 40

10−3

10−2

10−1

BE

R

Eb/N

0 [dB]


(c) ρ = 0.9

Fig. 2 BER Performance for 64−QAM and 4x4 antennas, different correlation scenarios


MIMO detectors require a very high SNR to reach reasonable BER perfor-mance. This SNR requirement for high correlation is a concerning issue, whichcan be expressed in terms of undesirable low energy-efficient systems. Indeed,only LR-based and SD MIMO detectors are able to achieve simultaneouslyfull diversity and transmission energy efficiency systems under high antennacorrelation, i.e., ρ = 0.9 in Fig. 2.(c).

As the number of antennas increases, the gap between SD/ML and otherMIMO detectors BER also grow, as depicted in Fig. 3 (8 × 8 antennas) andFig. 4 (20 × 20 antennas). In these arrangements, it is easy to see that in lowSNR non-LR-aided detectors present better performance, but in high SNRscenarios this behavior is inverted as LR-aided detectors present full diver-sity. Hence, given an array configuration with increasing number of antennas,the MIMO detectors combining lattice-reduction, interference cancellation andlist-reduction techniques exhibit a better performance than the correspondingdetector in previous array with small number of antennas.

It is worth to notice that for large array arrangements, in Fig. 4 withnT = nR = 20 antennas, the BER performance presents a different trend. Asthe correlation index grows, it takes much more power for the LR-aided MIMOdetectors to outperform the non-LR ones. However, when the correlation indexis very high, ρ = 0.9, a new trend is established. First, the ordered interference-cancellation is not able to deal with high correlation in high SNR. In thesecases, it is recommended not using ordering in SIC detection, to avoid theinversion of the slope of the BER curve. Also, MMSE lost the diversity gain,while the LR-MMSE’s BER decreased and increased as the SNR increased.For this array configuration the ZF-based detectors is unable to deal withhigh correlation scenarios (ρ ≥ 0.9); this class of MIMO detectors fails indecoupling the high-correlated inter-antennas interference. Besides, the SD-MIMO detector under high correlated channels has demonstrated an extremelyexcessive computational cost.

4.1 BER Inversion Effect in OSIC Detectors

As seem through previous BER performance results, OSIC detectors withSQRD approach are not effective under high correlated MIMO channel sce-narios while operating with a large number of antennas. This happens mainlybecause of the method by which the sorted QR decomposition proposed by [25]is a modification on the Gram-Schimidt approach. Despite being useful con-ceptually, the Gram-Schimidt method is numerically unstable. Hence, otheravailable algorithms based on re-orthogonalization, such as Householder re-flections, are able to mitigate this numerical instability.

In a QR decomposition, despite the matrix Q be referred as an orthogonalmatrix, it is not strictly orthogonal in practice. Therefore, a residual error ξ,where ∥I − QHQ∥2 = ξ2 ≥ 0, is always present, but it is particularly largerwhen QR decomposition is obtained through Gram-Schimidt algorithm [9].Although ξ is very small in most cases, rank deficiency or the dimension of the


0 5 10 15 20

10−4

10−3

10−2

10−1

BE

R

Eb/N

0 [dB]


(a) ρ = 0

0 5 10 15 20 25

10−4

10−3

10−2

10−1

BE

R

Eb/N

0 [dB]


(b) ρ = 0.5

0 5 10 15 20 25 30 35 40

10−5

10−4

10−3

10−2

10−1

BE

R

Eb/N

0 [dB]


(c) ρ = 0.9



0 2 4 6 8 10 12 14 16

10−4

10−3

10−2

10−1

BE

R

Eb/N

0 [dB]


(a) ρ = 0

0 5 10 15 20

10−3

10−2

10−1

BE

R

Eb/N

0 [dB]


(b) ρ = 0.5

0 5 10 15 20 25 30 35 40

10−3

10−2

10−1

BE

R

Eb/N

0 [dB]

ZFMMSEMMSE−OSICCL−MMSE−OSICLR−ZFLR−MMSELR−MMSEOSICLR−CL−MMSE−OSICSDCL−LR−MMSESIC

(c) ρ = 0.9



matrix to be decomposed can increase ξ enough to forbid a proper solution ofa linear system via QR decomposition. For further reading, recall to [6, 9].

As explained before, ordered SIC using MMSE equalization can be obtainedthrough the QR decomposition on the extended channel matrix, therefore:

H =

[H√

N0/ESInT

]= QR =

[Q1

Q2

]R. (32)

It is straightforward that:

Q2R =√

N0/ESInT(33)

and:QHH = QH

1 H +√

N0/ESQH2

= R.(34)

For SIC detection, it must be derived an upper triangular system, using (33),(34) and (1):

QH1 x = QH

1 Hs + QH1 n

=(R −

√N0/ESQH

2

)s + QH

1 n.(35)

Therfore, it must be considered the following approximation:

QH1 H = R, (36)

which is reasonable for high SNR, also leading to:

QH1 Q1 = InT (37)

At this point, it can be observed that H is a full rank matrix even if His highly correlated, and in the worst case even if H is singular. However, forhigh values of N0/ES and extremely high correlated channels, the extendedchannel matrix H tends to exhibit orthogonality deficiency and

√N0/ESInT ≈

0nT. When this condition is combined with the low numerical stability of the

sorted QR decomposition, which is based on the Gram-Schimidt method, theerror ξ may not be negligible; therefore, QH

1 Q1 = I, which doesn’t meet therequirements for an effective ordered SIC detection.

From this analysis, it can be concluded that ordered SIC via MMSE equal-ization becomes ineffective under high correlated channel scenarios and largearrays due to the characteristics of its sorted QR decomposition. From thisperspective, V-BLAST would be recommended to perform ordered interferencecancellation if complexity was not a concern, since V-BLAST should performone pseudo-inverse to detect each layer, resulting in a complexity of O(n4

T ).Therefore, the ordering procedure should be avoided for systems operatingunder these circumstances. In order to sustain these claims, Fig. 5 depictsthe BER performance for large array (20 antennas) under highly correlatedchannels (ρ = 0.9), where ordered SIC was obtained through a) V-BLAST,


0 5 10 15 20 25 30 35 4010

−4

10−3

10−2

10−1

100

Eb/N

0

BE

R

V−BLASTSQRD

Fig. 5 BER performance for MMSE-V-BLAST and MMSE-OSIC(via SQRD) consideringa 20 × 20 4−QAM system under high correlated channels (ρ = 0.9).

and alternatively via b) SQRD procedure. Since V-BLAST is not affected bynumerical unstability, it can perform ordered interference cancellation withoutexperiencing BER performance inversion effect in high SNR region.

5 Complexity Analysis

It is of paramount importance to analyze the computational complexity ofthose MIMO detectors, since communication under mobility conditions im-pose restrictions on signal processing capability and power consumption. Bycombining complexity and BER performance analysis, it is possible to estab-lish the best trade-off among those sub-optimum MIMO detectors.

From this perspective, this section discusses the overall complexity of thosesub-optimum MIMO detectors. The overall complexity has been measured asthe number of flops needed to perform the complete MIMO detection pro-cess for each MIMO detector presented previously. Three and one flop(s)for complex product and sum, respectively, have been considered in the flopcounting [25]. Besides, with the necessary modifications, matrix operationsflop counting were based in [11]. Also, the complexity on the sorted QR de-composition found in [25, 27], as well as the lower bound for the number ofnodes visited by the SD in [13] have been considered in the following analysis.


5.1 LLL Complexity

Despite of its BER performance, there are certain scenarios where LR-aideddetectors may present a growing complexity. Numerical simulation results re-vealed not only a matrix size dependence on LLL’s complexity, but also acorrelation index dependence. The dependence between complexity and ma-trix size is of course due to the number of operations evaluated on the matrix.On the other hand, we have found that an increasing correlation index leadsto a near singular matrix, which makes difficult for LLL to find an orthogonalbasis, increasing the computational complexity.

Unfortunately, LLL’s complexity cannot be easily evaluated considering allits variables dependence. In order to circumvent this difficult, herein the LLL’scomplexity has been determined numerically through a surface fitting proce-dure on the flop count of the LLL algorithm. We have conducted numericalexperiments aiming to determine the LLL algorithm complexity dependenceon the antenna correlation index and array dimension; as a result, it was es-tablished a flop count procedure for lattice reduction of square matrices takinginto account increasing array size and correlation indexes. Besides, the LLLalgorithm applied in these numerical simulations uses the complex approachwith factor δ = 0.75, as suggested in [17]. In order to obtain a better andrealistic prediction, the mean over 500 LLL computation flops count has beencalculated for each MIMO system and channel configuration, including corre-lation index and matrix size.

The best surface fitting, which can suitably describe the flop count on LLLalgorithm has been numerically found as:

flll(nT , ρ) = (aebρ + c)n3T (38)

where a = 5.018×10−4, b = 13.48 and c = 8.396. Remembering that this fittingis valid only for nT = nR arrays. As illustration, Fig. 6 depicts the collecteddata (identified by the “+” marker) and the corresponding surface fitting givenby eq. (38). It is worth noting that the LLL computational complexity costincreases substantially under medium-high correlation index (ρ ≥ 0.5) andlarge-array configurations (nT = nR ≥ 15).

5.2 Complexity of the Analysed MIMO Detectors

Several MIMO detectors complexity has been analysed due to the varioustechniques combination possibilities. Table 1 summarises the overall complex-ity related to the most relevant combination of sub-optimum MIMO detectors.The complexity of the ML-MIMO detector is included as reference.

From Table 1, notice that SIC-based MIMO detectors are capable to offerlower complexities, since they don’t require pseudo-inverse evaluation. Be-sides, the aggregation of ordering procedure is preferable since it requires only2n2

T − 2nT flops, according to [25], while providing substantial performanceimprovement. The cost for using the Chase list may be low, but only when


Fig. 6 LR flop count dependence on antenna correlation index and number of antennas.

Table 1 MIMO Detectors Complexity

MIMO Detector Number of flopsZF 4n3

T + 8nRn2T − n2

T + 3nRnT − nR + nT

ZF-SIC 4nRn2T + 15nRnT /2 + 3n2

T /2

ZF-OSIC 4nRn2T + 15nRnT /2 + 7n2

T /2 − 2nT

CL-ZF-OSIC 4nRn2T + (4q + 15/2)nRnT + 4n2

R + (2q + 3)n2T /2

+(2q − 1)nR + (9q − 3)nT /2 + 5M/2

LR-ZF 8n3T + 8nRn2

T + 7n2T + 3nRnT − nR + 5nT + fLLL(nT , ρ)

LR-ZF-OSIC 4n3T + 12n2

T + 4nRnT + 11nT /2 + fLLL

CL-LR-MMSE-OSIC 4n3T /3 + 4nRn2

T + (4q − 9/2)n2T + (8q + 15/2)nRnT

+2qnR(7/2 − 3q/2)nT /2 + +5M/2 + fLLL(nT − 1, ρ)

MMSE 12n3T + 8nRn2

T + 2n2T + 3nRnT − nR

MMSE-SIC 4n3T /3 + 4nRn2

T + 16n2T /3 + 6nT nR + 22nT /3

MMSE-OSIC 4n3T /3 + 4nRn2

T + 19n2T /3 + 6nT nR + 16nT /3

CL-MMSE-OSIC 4n3T /3 + 4nRn2

T + (2q + 13/3)n2T + (4q + 6)nT nR

+2qnR + (9q/2 + 25/6)nT + 5M/2

LR-MMSE 16n3T + 8nRn2

T + 10n2T + 3nRnT − nR + 4nT + fLLL

LR-MMSE-OSIC 4n3T + 16n2

T + 4nRnT + 11nT /2 + fLLL(nT , ρ)


T + (10q − 11/3)n2T + (8q + 6)nRnT

+2qnR + (55/6 − 37q/2)nT + 5M/2 + fLLL(nT − 1, ρ)

ML (4nRnT + 2nR)MnT

SD [13] 4n3T + 7n2

T − nT /2 + (2nT + 2)MηnT − 1/M − 1,where η = 1/2 [c2(M2 − 1)/6N0 + 1]−1

and c2 = E[∥hi∥2

], ∀i ∈ [1, nT ]


the number of repetitions q is lower than the number of transmit antennas; ofcourse, holding the number of repetitions q much low impacts negatively on theCL-based MIMO detector performance. On the other hand, the LR-aided de-tectors show reasonable low complexity under low to medium correlation indexscenarios, while full diversity is held, which makes it a promising sub-optimumMIMO transmitting scheme. Finally, ML has a prohibitive exponential com-plexity in any practical system configuration, while SD-based MIMO detectorspresent decreasing complexity as the SNR increases, but still highly complexunder low-SNRs regime.

Since the OSIC procedure represents a reasonable difference in terms of thefinal detection complexity, Fig. 7 provide a further graphically flops-complexitycomparison, among the different detectors grouped as: a) linear MIMO detec-tors, and b) OSIC-based linear MIMO aided by lattice reduction and Chaselist procedure as well. Indeed, in Fig. 7.(b) it has been considered the deploy-ment of the ordered successive interference cancellation and a Chase list lengthequal to the constellation size, in this case M = q = 64.

5.3 Impact of Real-valued System Model Representation on the Complexity

The MIMO problem defined by eq. (1) can be rewritten in order to deal onlywith real-valued entries:

xr = Hrsr (39)

where the real matrix channel is given by:

Hr =

[ℜ(H) −ℑ(H)

ℑ(H) ℜ(H)

](40)

with the received and transmitted symbol vectors as:

xr =

[ℜ(x)

ℑ(x)

], sr =

[ℜ(s)

ℑ(s)

].

With this modification, the new system model representation has doubledits size, but all its elements are real-valued entries; mathematically, it meansxr ∈ R2nR×1, sr ∈ R2nT ×1 and Hr ∈ R2nR×2nT .

Despite the real model representation be considerably widespread in theliterature, its use may lead to a higher computational complexity, speciallyfor matrix arithmetic operations. Real arithmetic operations may requires lessflops than complex ones, but the real system size is larger than the complexone. According to [17] and [8], the impact of complex-valued system modelrepresentation on the LLL algorithm complexity is a half the complexity of thereal-valued representation. In terms of arithmetic operations, Table 2 showsthe cost comparison between complex-valued and real-valued representation.

Under MIMO system configurations discussed previously, the use of real-valued model may not be advantageous, since the complexity increases due to


0

0.2

0.4

0.6

102030405060708090100

1

2

3

4

5

x 107

nT

ρ

Flo

ps

ZFMMSELR−ZFLR−MMSESD(ρ=0)

(a) Linear MIMO complexity

0

0.2

0.4

0.6

102030405060708090100

1

2

3

4

5

x 107

nT

ρ

Flo

ps

MMSE−OSICCL−MMSE−OSICLR−MMSE−OSICLR−CL−MMSE−OSICSD(ρ=0)

(b) OSIC-based MIMO detectors

Fig. 7 Complexity in terms of flops for MIMO detectors; 64−QAM and Eb/N0 = 22 [dB].

Table 2 Comparison in terms of number of operations between complex-valued and real-valued MIMO system model representation, nR = nT

Operation Real addition Real product Total arith. operationsHrsr 4n2

T − 2nT 4n2T 8n2

T − 2nT

Hs 2n2T − 2nT 4n2

T 6n2T − 2nT

HTr Hr 8n3

T − 4n2T 8n3

T 16n3T − 4n2

T

HHH 2n3T − 2n2

T 4n3T 6n3

T − 2n2T


the real-valued system representation size. On the other hand, the use of thereal-valued model reduce the constellation size. For example, the constellationsize of a M−QAM modulation turns to a 2×

√M−PAM (in-phase and quadra-

ture representation) when the real model is applied, but it does not lead tocomplexity reduction. Furthermore, the use of the real-valued system modelrepresentation brings a small BER improvements to the SIC-based MIMOdetectors because operations such as detection and interference cancellationof both real and imaginary parts are carried out independently, leading to amore reliable detection, as demonstrated in [7]. However, this improvementrepresenting a small performance BER gap is low for the price paid by thecomplexity growth.

6 Conclusions

In this work the use of lattice reduction technique has been demonstrated auseful tool to improve the BER performance of linear sub-optimum MIMOdetectors. On the other hand, the list reduction technique was capable toachieve reasonable MIMO system performances over different array dimensionsand correlation indexes, but did not able to achieve full diversity.

Since the use of the lattice reduction technique was able to provide largeimprovements on the BER performance, its combination with list reduction didnot leave space for further BER reduction, even under combined large-MIMOand high antenna correlation condition. Furthermore, LR-aided MIMO detec-tors have demonstrated a noticeable complexity growth under high antennacorrelation scenarios, since the channel matrix becomes near-singular, gettingharder and harder for the LLL algorithm establish an orthogonal signal basis.

It is worth to note that ordering procedure has demonstrated ineffective inlarge array systems operating under high correlated channel scenarios. In theextreme correlation and large arrays situation analyzed (ρ = 0.9 and 20 × 20antennas), the LR-MMSE-OSIC and MMSE-OSIC MIMO detectors were notable to perform the symbol detection accordingly in medium- and high-SNRsregime.

Finally, the comprehensive complexity analysis of dozen MIMO detectorscarried out in this paper has demonstrated that LR-aided MIMO detectorsoperating under large number of antennas and low-medium antenna corre-lation indexes present the best complexity-performance tradeoffs under thesub-optimum MIMO detection perspective.

Acknowledgment

This work was supported in part by the National Council for Scientific andTechnological Development (CNPq) of Brazil under Grants 202340/2011-2,303426/2009-8 and in part by State University of Londrina – Paraná StateGovernment (UEL), scholarship PROIC-2013-14.


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49

Anexo D -- Efficient Near-Optimum

Detectors for Large MIMO Systems

under Correlated Channels

Performance and Complexity Analysis of Sub-optimum

MIMO Detectors Under Correlated Channel

Ricardo Tadashi Kobayashi, Fernando Ciriaco and Taufik Abrão

Abstract—The interest in multiple antennas systems comesfrom its high spectral and/or energy efficiency under scatteredenvironments, being an ongoing research topic in wireless com-munication systems. In this work, different MIMO equalizers, de-tection techniques (ordering, interference cancellation, lattice re-duction, list reduction) and its combination have been analyzed inorder to find the best complexity-performance trade-off topology.Also, it has been considered the correlation between antennas,which is invariably occurs in realistic scenarios and deterioratesthe performance of MIMO systems. From this perspective, thegoal of this paper is to point out a MIMO architecture withreasonable complexity, while holding suitable performance andfull diversity under correlated channel scenarios. We have foundthat combining lattice reduction (LR) technique and ChaseList (CL) detection enables slight BER improvements, but at arelatively high cost complexity. Furthermore, ordering has beenshown to be ineffective at high correlated and SNR scenarioswith large number of antennas, where no ordering is preferable.

I. INTRODUCTION AND SYSTEM MODEL

One of the greatest challenges for MIMO applicability is

the design of low complexity detection architectures, specially

for large or massive MIMO systems. A very important point

in these designs is the antenna correlation. Indeed, since

MIMO systems use multiple antennas in transmission and

reception, their signal may be correlated, leading to remarkable

performance deterioration. Distance between antennas is one

of the causes of channel correlation and its is invariably

present in practical MIMO systems due to the miniaturization

of mobile terminal (MT) tendency, which leads to smaller

distances between antennas and channel correlation. These

facts reveal the importance of considering the fade channel

correlation for more realistic analysis on MIMO systems.

It is well known that the Maximum-Likelihood and Sphere-

decoder [1] provide optimal detection performance, however

at a very high computational cost. With lower performance,

linear MIMO detectors got an increasing interest due to

their notable complexity reduction. For instance, Zero-Forcing

and MMSE [2] equalizers can be combined with successive

interference-cancellation (SIC), ordering SIC [3], list principle

[4] and lattice reduction [5] in order to exploit full diversity

and achieve near-ML performance at a reasonable reduced

complexity. Aiming to elect the best complexity-performance

trade-off, this work analyses the complexity versus perfor-

mance trade-off by the dozen MIMO detectors, by considering

distinct antennas arrays configuration under different antenna

correlation indexes and signal-to-noise-rate (SNR).

Notation: (·)H , (·)T , (·)† and (·)−1 denote the matrix

transpose, Hermitian, pseudo-inverse and inverse, respectively.

Boldface lowercase letter represents vectors, while boldface

uppercase letter denotes matrices. I and 0 denotes the identity

and an all zero matrix. Finally, tilde superscript (·) represents

a symbol vector estimation and hat (·) superscript represents

a symbol estimation after the slicer1.

The MIMO system composed by nT transmit antennas and

nR receive antennas can be described by:

x = Hs + n. (1)

Herein we have assumed overdetermined MIMO systems, i.e.,

nR ≥ nT . Eq. (1) represents a transmission of nT modulated

symbols, snT ×1, through a channel which gain is denoted

by HnR×nTand additive noise nnR×1. Each element of H

represents the channel gain in its respectively path, assumed to

be known at the receiver side. After passing through the chan-

nel, the symbols are combined forming the received vector,

xnR×1. A pre-processing block in the receiver is responsible

for the demodulation and decoding, if applied, while the

MIMO detector recovers the sent data from the received signal

corrupted by background noise and inter-antenna interference.

Besides, it’s assumed that the noise vector has a circularly-

symmetric complex Gaussian distribution, n ∼ CN (0, σ2nI),

with variance σ2n, also represented by E

[nnH

]= N0I.

Due to its spectral efficiency and performance trade-off, in

this work the M -QAM modulated symbols has been deployed,

denoted by a complex number in the set S = {a + jb | a, b ∈{−

√M + 1, −

√M + 3, . . . ,

√M − 1}}. Also, Gray coded

symbols have been assumed, where adjacent symbols present

only one bit of difference, minimizing the bit error. Since

each antenna’s symbol is independent the covariance matrix

of the transmit symbols is given by E[ssH

]= ESI, where

ES represents the average symbol energy.

The magnitude of MIMO fading channel is properly and sta-

tistically modeled by flat Rayleigh distributions. Besides, the

antennas correlation effect, can be modeled using Kronecker

product form [6]; as a result:

H =√

RH,RxG√

RH,Tx. (2)

where G(nR × nT ) has i.i.d. complex Gaussian elements,

G ∼ CN (0, I), RH,Rx(nR × nR) and RH,Tx(nT × nT )are the correlation observed on the receiver and transmitter,

respectively. The elements of these two last matrices are given,

in terms of normalized correlation index ρ, by:

rH,Rx ij = rH,Tx ij = ρ(i−j)2 (3)

II. MIMO DETECTORS

This section revisits the most widespread MIMO detectors,

including the maximum likelihood (ML), Sphere decoder, zero

forcing (ZF) and minimum mean squared error (MMSE).

Additionally, we have provide a brief discussion of MIMO

1A slicer quantizes an estimated symbol to its nearest constellation point.

detection techniques, such as ordering, interference cancel-

lation, list and lattice reduction. It is of great importance

the knowledge on each detector’s procedures, specially for

complexity evaluation and BER performance simulation.

A. Maximum Likelihood (ML)

The maximum likelihood detector provides the lowest BER

performance of all MIMO detectors, but at an exceedingly

high complexity. For a M−ary modulation and nT transmit

antennas system, the number of symbols set combination

is simply MnT , leading to a exponential complexity. ML

detection performs a search of the closest symbol combination

sent to the receiver, therefore its solution is given by:

s = argmins∈SnT

‖x − Hs‖2. (4)

B. Sphere Decoder (SD)

In order to reduce the complexity of the ML detector, the

sphere decoder [1] searches only the candidates contained in

a hypersphere of radius d:

d2 < ‖x − Hs‖2 (5)

which of course is dependent to the signal-to-noise ratio

(SNR). If the radius is set too large the complexity enclose

to the ML one, while if too low no candidates are found in

the hypersphere.

Aiming to perform its detection, eq. (1) is rewritten with the

QR decomposition on the channel matrix H = QR, where Qorthogonal 2 and R is upper triangular. When (1) is multiplied

by QH , a new system description is created:

y = QHx = QHQRs + QHn = Rs + n′. (6)

Since Q is orthogonal, the statistics on the noise, n′, remains

unchanged and no noise enhancement is expected. With these

facts and considering R = [r1 r2 r3 · · · rnT]T , eq. (5)

becomes:

d2 < ‖y − Rs‖2 =

nT∑

k=1

|yk − rks|2 (7)

With Eq. (7), there is no need of evaluating the estimated

noise norm for every symbol combination, for the noise norm

can be updated as the symbols layers are detected. Based in (7)

a tree search for the closest transmitted set is made considering

the radius constraint, resulting on the SD solution.

C. Zero Forcing (ZF)

The zero forcing MIMO detector basically solves the linear

system problem posed by (1) ignoring the additive noise. The

solution proposed for this detector uses Moore-Penrose pseudo

inverse [7]. This way, the estimated received symbol vector can

be evaluated by:

s = H†x = s + H†n. (8)

Despite of its low complexity, ill conditioned channel ma-

trices leads to noise term amplification [7], term H†n; hence,

poor performance is expected.

2Recall that an orthogonal matrix Q has the property I = QHQ.

D. Minimum Mean Squared Error (MMSE)

The ZF solution can be improved by taking into account

the noise variance [2], leading to the MMSE solution, which

minimizes the error:

s =

(HHH +

N0

ESI

)−1

HHx. (9)

Alternatively, the MMSE detection can also be performed as:

s = H†x = s + H†n (10)

where the extended channel matrix and the received vector are

given respectively by:

H =

[H√

N0/ESInT

], x =

[x

0nT ×1

].

E. Successive Interference Cancellation (SIC) and ordering

The successive interference cancellation can be obtained by

QR decomposing the channel matrix (ZF approach) or the

extended channel matrix (MMSE approach). After executing

eq. (7), the upper triangular system can solved upwards by:

si =

yi

rii, i = nT

1

rii

(yi −

nT∑

k=i+1

riksk

), i = nT − 1, . . . , 1.

(11)

In (11) every symbol must be sliced before the interference

cancellation. This step must be respected in order to perform

a properly interference cancellation and improve the perfor-

mance.

Further improvement on the SIC technique can be achieved

through a properly ordering [3], which avoids error propaga-

tion in interference cancellation. The symbol vector is ordered

and detected from the weakest to the strongest symbol, last

element of the vector. This ordering is obtained trough the

sorted QR decomposition:

HΠ = QR. (12)

where the matrix Π is a permutation matrix, used to reorder

the symbols, after applying the SIC detection, by multiplying

it and the estimated received symbol vector.

Applying this decomposition provides near V-BLAST [8]

ordering performance with lower complexity, but V-BLAST

performance can be obtained with a post sorting algorithm

[3].

F. Chase List (CL) Algorithm

The Chase list detection [4] improves symbol error perfor-

mance through detection repetition, at reasonable low com-

plexity. Summarily, the Chase list classifies the q best candi-

dates for the last antenna’s symbol, removes its interference,

detects the remaining symbols deploying one of thhe detectors

discussed previously and finally chooses the best set of q ones,

which is taken as solution.

Obviously, as the number of repetitions q increases, the BER

performance reduces while the complexity increases, since

more vector candidates are tested. Also, it is noticed that the

hnT

s(1)b Subdetector 1

x

x(1)s(1)nT

hnT

s(2)x(2)s(2)nT Subdetector 2

x

b

s(q)

Subdetector qhnT

x(q)s(q)nT b

x

b

b

b

xargmink=1 to q

‖Hs(k) − x‖

dete

ctio

nan

dlis

tcr

eati

onLa

stan

tenn

a’s

sym

bol

s

Figure 1. Block diagram of Chase list detector.

number of repetitions is restrained to the modulation order,

i.e. q ≤ M .

Another aspect to be taken into account is that the SIC

based detectors are much more preferable than non-SIC ones.

Advantages of SIC detection for Chase list include: better

BER performance, inherent interference cancellation and only

one division required for the last symbol detection. Besides,

ordering can also be employed aiming to achieve further

performance improvements.

G. Lattice-Reduction (LR) aided MIMO detector

One of the problems found on the previous detectors is their

narrow decision boundaries, which makes the detection more

sensitive even with small amounts of noise. This problem can

be avoided by detecting the symbols in another domain and

returning to the original one trough the lattice reduction [5].

The lattice reduction can be obtained through the LLL

algorithm [9], by decomposing the MIMO channel matrix:

H = HT, (13)

where H is a basis with better orthogonality properties and Tis a unitary matrix3. As the new matrix H has better properties,

the decision boundaries are larger and the noise amplification

is smaller. Since the detection is done in LR domain, it’s

desirable to rewrite the system model in terms of LR symbols:

x = (HT)(T−1s) + n = Hz + n. (14)

Under this modified system model, the LR symbols z can be

detected using any linear equalization and technique discussed

previously. The ZF equalization is presented in the sequence:

z = H†x = z + H†n. (15)

Since the noise also corrupts the symbols in the LR domain,

as viewed in (14) and (15), they must be properly sliced with

shifting and scaling operations, as follows:

z = 2

⌊z − βT−11

2

⌉+ βT−11. (16)

where 1 represents a column vector of ones and β is a

constant determined by the applied modulation. The trans-

mission scheme deployed in this work uses exclusively QAM

modulation, therefore β = 1 + i.

3An unitary matrix has the following properties |T| = {±1, ±j} andT ∈

{ZnR×nT + jZnR×nT

}

The detection is completed by converting the symbols to

the original domain:

s = Tz. (17)

In order to combine lattice reduction with the Chase list, an

additional QR decomposition should be evaluated, as proposed

in [10].

III. PERFORMANCE AND COMPLEXITY COMPARISON

The simulated BER performance of various MIMO de-

tectors discussed previously have been compared. For a fair

comparison between different transmission systems , it will

be considered a normalized SNR Eb

N0= SNR

log2 M , and transmit

power constraint, with power equally distributed among the

nT antennas. Each MIMO detector performance was evaluated

considering two modulation format and antennas arrangement:

a) 16−QAM-8 × 8; b) 4−QAM-20 × 20. Furthermore, three

antenna correlation scenarios has been considered: a) no

correlation ρ = 0; b) medium correlation ρ = 0.5; and c)

strong correlation ρ = 0.9.

As the number of antennas grows, the gap between SD/ML

and other MIMO detector’s BER also grow, as depicted in

Fig. 2 (8×8 antennas) and Fig. 3 (20×20 antennas). In these

arrangements, it is also easy to see that in low SNR non-LR-

aided detectors present better performance, but in high SNR

this behavior is inverted as LR-aided detectors present full

diversity4. Hence, given an array configuration with increasing

number of antennas, the MIMO detectors combining LR,

OSIC and CL techniques exhibit a better performance than the

correspondent detector in previous array with small number of

antennas.

For large array arrangements, in Fig. 3 with nT = nR = 20antennas the BER performance present different trend. As the

correlation index grows, it takes much more power for the

LR-aided MIMO detectors to outperform the non-LR ones.

However, when the correlation index is very high, ρ = 0.9,

a new trend is established. First, the ordered interference

cancellation is not able to deal with reordering in high SNR,

where no ordering is preferable. Also, MMSE lost its diversity

gain, while the LR-MMSE’s BER increased and decreased as

the SNR increased. For this array configuration the ZF-based

detectors is unable to deal with high correlation scenarios (ρ ≥0.9) in decoupling the inter-antennas interference. Besides,

SD-MIMO detector under this configuration has presented an

extremely exceeding computational cost.

It is of paramount importance analyze the computational

complexity of those MIMO detectors, since mobility imposes

restrictions on signal processing and power consumption.

By combining complexity and BER performance analyses,

it is possible to establish the best trade-off among those

sub-optimum MIMO detectors. In the following, the overall

complexity of those sub-optimum MIMO detectors will be

discussed. Three and one flop(s) for complex sum and product,

respectively, have been considered in the flop counting [3].

4Recall that a system achieves full diversity when a 3 [dB] increasing onthe transmission power implies in a BER reduction factor of 2nT [6].

0 5 10 15 20

10−4

10−3

10−2

10−1

ρ=0

BE

R

ZF

MMSE

MMSE−OSIC

CL−MMSE−OSIC

LR−ZF

LR−MMSE

LR−MMSEOSIC

LR−CL−MMSE−OSIC

SD

0 5 10 15 20 25

10−4

10−3

10−2

10−1

ρ=0.5

Eb/N

0 [dB]

0 5 10 15 20 25 30 35 40

10−5

10−4

10−3

10−2

10−1

ρ=0.9

Figure 2. BER Performance for 16−QAM and 8x8 antennas, different correlation scenarios

0 2 4 6 8 10 12 14 16

10−4

10−3

10−2

10−1

ρ=0

BE

R

ZF

MMSE

MMSE−OSIC

CL−MMSE−OSIC

LR−ZF

LR−MMSE

LR−MMSEOSIC


SD

0 5 10 15 20 25

10−3

10−2

10−1

ρ=0.5

Eb/N

0 [dB]

0 5 10 15 20 25 30 35 40

10−4

10−3

10−2

10−1

ρ=0.9

LR−CL−MMSESIC

Figure 3. BER Performance for 4−QAM and 20x20 antennas, different correlation scenarios

Besides, matrix operations flop counting were based in [11].

Also, the complexity on the sorted QR decomposition found

in [3] and the lower bound for the number of visited nodes by

the SD in [1]. Finally, the LLL complexity will be discussed

in the sequence.

A. LLL Complexity

Despite of its BER performance, LR-aided detectors may

present a growing complexity in certain scenarios. Numerical

simulation results revealed not only a matrix size dependence

on LLL’s complexity, but also a correlation index dependence.

These happen due to the size of the system and the difficult of

finding a orthogonal basis on ill conditioned channel matrix.

Unfortunately, LLL’s complexity cannot be easily evalu-

ated considering all its variables dependence. In order to

circumvent this difficult, herein the LLL’s complexity has been

determined numerically through a surface fitting procedure on

the flop count of the LLL algorithm. The best surface fitting

which can suitably describe the flop count on LLL algorithm

has been found as:

fLLL(nT , ρ) = (aebρ + c)n3T (18)

where a = 5.018 × 10−4, b = 13.48 and c = 8.396Remembering that this equation is valid only for nT = nR

arrays. Fig. 4 depicts the collected data ("+" marker) and

the corresponding surface fitting given by (18). It is worth

noting that the LLL computational complexity cost increases

substantially under medium-high correlation index (ρ ≥ 0.5)

and large-array configurations (nT = nR ≥ 15).

Figure 4. LR flop count dependence on antenna correlation index and numberof antennas.

B. Complexity of MIMO Detectors

The overall complexity of the most relevant MIMO detec-

tors presented in this work is summarized in Table I, which is

Table IMIMO DETECTORS COMPLEXITY

MIMO Detector Number of flops

ZF 4n3T + 8nRn2

T − n2T + 3nRnT − nR + nT

ZF-SIC 4nRn2T + 15nRnT /2 + 3n2

T /2ZF-OSIC 4nRn2

T + 15nRnT /2 + 7n2T /2 − 2nT

CL-ZF-OSIC 4nRn2T + (4q + 15/2)nRnT + 4n2

R + (2q + 3)n2T /2 + (2q − 1)nR + (9q − 3)nT /2 + 5M/2

LR-ZF 8n3T + 8nRn2

T + 7n2T + 3nRnT − nR + 5nT + fLLL(nT , ρ)

LR-ZF-OSIC 4n3T + 12n2

T + 4nRnT + 11nT /2 + fLLL


T + (4q − 9/2)n2T + (8q + 15/2)nRnT + 2qnR(7/2 − 3q/2)nT /2 + +5M/2 + fLLL(nT − 1, ρ)

MMSE 12n3T + 8nRn2

T + 2n2T + 3nRnT − nR

MMSE-SIC 4n3T /3 + 4nRn2

T + 16n2T /3 + 6nT nR + 22nT /3

MMSE-OSIC 4n3T /3 + 4nRn2

T + 19n2T /3 + 6nT nR + 16nT /3

CL-MMSE-OSIC 4n3T /3 + 4nRn2

T + (2q + 13/3)n2T + (4q + 6)nT nR + 2qnR + (9q/2 + 25/6)nT + 5M/2

LR-MMSE 16n3T + 8nRn2

T + 10n2T + 3nRnT − nR + 4nT + fLLL

LR-MMSE-OSIC 4n3T + 16n2

T + 4nRnT + 11nT /2 + fLLL(nT , ρ)CL-LR-MMSE-OSIC 16n3

T /3 + 4nRn2T + (10q − 11/3)n2

T + (8q + 6)nRnT + 2qnR + (55/6 − 37q/2)nT + 5M/2 + fLLL(nT − 1, ρ)ML (4nRnT + 2nR)MnT

SD [1] 4n3T + 7n2

T − nT /2 + (2nT + 2)MηnT −1M−1

, where η = 12

[c2(M2−1)

6N0+ 1

]−1and c2 = E

[‖hi‖2

], ∀i ∈ [1, nT ]

based on all considerations previously set.

From Table I, it can be observed that SIC-based MIMO

detectors are capable to offer lower complexities, since they

don’t require pseudo-inverse evaluation. Besides, the aggrega-

tion of ordering procedure is preferable since it requires only

2n2T − 2nT flops [3], while providing substantial performance

improvement. The cost for using the Chase list may be low,

but only when the number of repetitions q is lower than the

number of transmit antennas. On the other hand, the LR-

aided detectors present reasonable low complexity under low

to medium correlation index scenarios, while full diversity

is held, which makes it a promising sub-optimum MIMO

transmitting scheme. Finally, ML has a prohibitive exponential

complexity in any practical system configuration, while SD-

based MIMO detectors present decreasing complexity as the

SNR increases, but still highly complex under low-SNRs

regime. Since OSIC presents a better performance-complexity

trade off, their complexity is graphically compared in Fig. 5.

0

0.2

0.4

0.6

20406080100

1

2

3

4

5

x 107

nT

ρ

Flo

ps

MMSE−OSIC

CL−MMSE−OSIC

LR−MMSE−OSIC


SD(ρ=0)

Figure 5. Flop count for OSIC based MIMO detectors; 64−QAM andEb/N0 = 22 [dB].

IV. CONCLUSIONS

Throughout this work, it was highlighted the promising

features of LR technique. Indeed, among fourteen analyzed

MIMO detectors, we have shown that the best linear sub-

optimum MIMO detector is obtained with LR technique,

achieving full diversity at a reasonable cost complexity. How-

ever, this fact does not stand for high correlated scenarios

with large antennas arrays, where the complexity grows sub-

stantially. On the other hand, Chase List can bring reasonable

performance at low complexity, but cannot achieve full diver-

sity. The combination of CL and LR enables slight BER im-

provements, but at a relatively high cost. Furthermore, ordering

has been shown to be ineffective at high correlated channel

and SNR scenarios with large arrays, where no ordering is

preferable.

REFERENCES

[1] J. Jalden and B. Ottersten, “On the complexity of sphere decodingin digital communications,” IEEE Transactions on Signal Processing,vol. 53, no. 4, pp. 1474–1484, April 2005.

[2] L. Szczecinski and D. Massicotte, “Low complexity adaptation of mimommse receivers, implementation aspects,” in Global Telecommunications

Conference, 2005. GLOBECOM ’05. IEEE, vol. 4, Dec 2005, pp. 6 pp.–2332.

[3] D. Wubben, R. Bohnke, V. Kuhn, and K. D. Kammeyer, “Mmseextension of v-blast based on sorted qr decomposition,” in Vehicular

Technology Conference, 2003. VTC 2003-Fall. 2003 IEEE 58th, vol. 1,2003, pp. 508–512 Vol.1.

[4] D. Waters and J. Barry, “The chase family of detection algorithms formultiple-input multiple-output channels,” IEEE Transactions on Signal

Processing, vol. 56, no. 2, pp. 739–747, 2008.[5] X. Ma and W. Zhang, “Performance analysis for mimo systems with

lattice-reduction aided linear equalization,” IEEE Transactions on Com-

munications, vol. 56, no. 2, pp. 309–318, 2008.[6] E. Biglieri, MIMO Wireless Communications. Cambridge University

Press, 2007.[7] V. Kühn, Wireless Communications over MIMO Channels - Applications

to CDMA and Multiple Antenna Systems. Chichester, England: JohnWiley & Sons Ltd., Jun. 2006.

[8] P. Wolniansky, G. Foschini, G. Golden, and R. Valenzuela, “V-blast: anarchitecture for realizing very high data rates over the rich-scatteringwireless channel,” in International Symposium on Signals, Systems, and

Electronics, 1998. ISSSE 98. 1998 URSI, 1998, pp. 295–300.[9] H. Lenstra, A. Lenstra, and L. Lovász, “Factoring polynomials with

rational coefficients.” Mathematische Annalen, vol. 261, pp. 515–534,1982.

[10] J. Choi and H. Nguyen, “Sic-based detection with list and lattice reduc-tion for mimo channels,” IEEE Transactions on Vehicular Technology,vol. 58, no. 7, pp. 3786–3790, 2009.

[11] G. H. Golub and C. F. Van Loan, Matrix Computations (3rd Ed.).Baltimore, MD, USA: Johns Hopkins University Press, 1996.

55

Anexo E -- Cooperative Multi-cellular

Large MIMO over Desynchronized

Channel Estimation

For Peer Review

EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONS

Euro. Trans. Telecomms. 00: 1–20 (0000)

Published online in Wiley InterScience

(www.interscience.wiley.com) DOI: 10.1002/ett.0000

Cooperative Multi-cellular Large MIMO over Desynchronized Channel

Estimation

Ricardo Tadashi Kobayashi1, Fernando Ciriaco1 and Taufik Abrão1∗

1 State University of Londrina, Electrical Engineering Department (DEEL-UEL), Parana, Brazil.

SUMMARY

The aim of this work is to study multi-cell MIMO systems equipped with a very large number of antennas

in the base-station (BS) with low-complexity signal processing capabilities. Cooperation between BSs in a

multi-cell scenarios is analysed in order to reduce the well known pilot contamination effect, while the

effects of asynchronism between users and BSs in training phase using different pilot sequences, e.g.

Walsh-Hadamard, Gold and Kasami sequences on the massive MIMO system performance is examined.

As low complexity processing is deployed, the asynchronism in training phase may introduce errors on

the channel estimation process, adding its effect to the pilot contamination, resulting in an aggressive

scenario. Numerical results using the BER as an end-line measure of performance in the downlink (DL),

have been corroborated our finding. It was also developed an error formula for channel estimation subject

to desynchronization, which agrees with the BER performance results. Through these two performance

measurements, it was possible to observe that, in some situations, pseudo random sequences may be

preferable to orthogonal ones, mainly due to their out-of-phase cross-correlation properties.

Keywords – Massive MIMO, precoding, channel training, desynchronization, pseudo-random sequences

Copyright © 0000 AEIT

1. Introduction

Multiple-input-multiple-output (MIMO) systems are well

known for offering spectral and energy efficiency by

∗Correspondence to: E-mail: [email protected]

deploying multiple antennas in both transmit and receive

side, being very effective in highly scattered environments.

Despite of being studied for over two decades, MIMO

systems is still a contemporary research topic, which is

finding application in 4G and forthcoming 5G systems,


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where current comercial MIMO systems deploys up to 8

antennas at the base-stations (BS) and fewer antennas at

the mobile-terminals (MTs). Over this time, there was a

paradigm shift, which led the researches’ focus from the

point-to-point MIMO systems to the Massive multi-cell

MIMO systems, which is a candidate technology for 5G

systems [1].

Classical point-to-point MIMO is a reasonable matured

subject, which was largely discussed last decade. Therefore

there is a considerable number of techniques for MIMO

communications, including different schemes, precodings

and symbol detectors. The most popular schemes are

the Space-Time-Block-Codes (e.g. Alamouti [2]) offering

diversity gain, the BLAST scheme[3] and Spatial

Modulation[4] offering multiplexing gains. Since high

order STBCs are not straightforward and BLAST scheme

can offer higher data rates, this one became the most

popular among MIMO transmission schemes aiming to

realize large multiplexing gains. Furthermore, for symbol

detection, a vast number of solutions can be found in

the literature, where some solutions, such as ordering

[5] and lattice-reduction [6], or even their combination,

can achieve near-optimum performance, provided by

the Sphere-Decoding [7] but with a extremely reduced

complexity regarding the Maximum-likelihood solution,

depending on the system configuration, e.g., number

of transmit antennas and modulation order. The dual

approach for conventional detection, i.e. precoding, may

be preferable for concentrating the processing on the

transmitter, but is normally less effective as the set of

processing operations is limited.

A MIMO system is said massive when the number

of transmit and/or receive antennas is very large, say

hundreds or even thousands. These systems can be used

in a cellular multi-user (MU) configuration, where each

cell’s base station is equipped with a number of antennas,

normally larger then the number of mobile terminals (MTs)

equipped with only one antenna. In this scenario, the

interest is to fix the number of users per cell and make the

number of BS antennas as large as possible, where random

processes tend to become deterministic [8] and the noise

is averaged off [9]. One practical concern is determining

a number of BS antennas to be considered high enough to

achieve the asymptotic orthogonality [10], while ensuring

feasible computational complexity. Another discussion is

the use of time-division-duplex (TDD) systems, which

provide channel reciprocity at a higher interference cost,

over frequence-division-duplex (FDD) ones, which reduce

the inter-cell interference but require a feedback loop. A

more serious concern is the pilot contamination [11], where

the channel state information (CSI) may not be recovered

due to inter-cell interference and the length of the training

sequence, limited by the coherence time of the channel

[12], leading this application suitable only for low mobility

scenarios.

By using long sequences and avoiding sequence

reuse among different cells, pilot contamination may be

vanquished [13]. However, it would severely limit the

number of users to be served by each BS, besides of

reducing their mobility. Another aspect to be considered


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for training sequences is the synchronism between

transmitter and receiver, since uplink (UL) training step

may not achieve a perfect synchronism as obtained in

coordinated multi-cell downlink (DL) transmission. In this

sense, the correlation properties of the training sequence

sets, specially cross-correlation levels, is a key issue in

order to improve the capacity of massive MIMO systems.

Hence, the study of different binary [14, 15, 16], or even

quaternary [17] sequences, in this new channel training

context is of paramount importance in order to achieve

substantial performance and/or capacity improvement in

massive MIMO systems.

Still in multi-cellular MIMO systems, the inter-cell

interference can be mitigated via cooperation between

cells, which is obtained through coordination backhaul

links between different BSs. Through this, data and

channel aspects are shared between different BSs,

enabling interference cancellation and more precise

beamforming, for example. Nevertheless, this topology

requires centralized processing, besides of overloading

the backhaul link when a massive number of antennas is

deployed [18].

Notation: (·)H , (·)T , (·)†, (·)−1 and (·)∗ denote

the transpose, Hermitian, pseudo-inverse, inverse and

conjugate matrices, respectively. Boldface lowercase letter

represents vectors, while boldface uppercase letter denotes

matrices. IK and 0K denotes the identity and an all zero

matrices of dimension K . Also, tr(·), vec(·) and diag(·)

represent the trace, vectorization and diagonal operators,

respectively. Finally, imaginary number is denoted as ι =

√−1 and E [·] is the expected value of a random variable.

2. Point-to-Point MIMO

A point-to-point MIMO system is composed by a

transmitter and receiver equipped with nT and nR antennas,

respectively. Mathematically, this system can be modeled

by [19]:

X =

√ES

nTHS + N (1)

where the symbols S ∈ CnT×F , with E[|sij |2

]= 1, are

transmitted over a frame of length F † with transmitting

power ES. The signal is transmitted through the Rayleigh

channel H ∈ CnR×nT , where the channel coefficient hij ∼

CN (0, 1), received on the other side X ∈ CnR , is corrupted

by AWGN noise nR ∈ CnT , with nij ∼ CN (0, σ2wgn),

where σ2wgn is the power of the background noise.

Since this work will focus on the use of a large number

of antennas, simple precoding and detection techniques

are preferable, while tend to become optimal when nT

approaches to infinity.

2.1. Precoding

By adopting precoding techniques, computational process-

ing can be centralized on the transmitter side, as the signal

is transmitted in such a way that the received signal is

actually the modulated information. Therefore, the only

signal processing required on the receive side is the slicer

on each receive antenna. For this operation, we should

†Each column of the signal matrix S corresponds to one of the F time-slots.




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consider:

X =

√ES

nTHAS + N (2)

where the received and the transmitted data matrices

must have the same size, i.e. X,S ∈ CnR×F . Besides, the

precoding matrix A ∈ CnT×nR must present the following

property, HA/γ = InR , where γ is a power control

parameter which ensures that no extra transmit power is

added or removed due to the precoding.

The zero-forcing (ZF) precoding approach employs the

Moore-Penrose pseudo-inverse, and is given by:

AZF =1√γZF

HH(HHH)−1 (3)

where the factor γZF = tr(HHH)−1/nT. This factor is

of great importance for ZF precoding since the pseudo-

inverse leads to high gains case of the channel matrix is

ill conditioned [20].

Another approach, valid only for nT/nR → ∞, is the

matched-filter precoding:

AMF =1√γMF

HH , (4)

where again γMF = tr(HHH)/nT. It is an interesting

solution for large MIMO systems due to its low signal

processing requirements.

2.2. Detection

Conventional detection is the dual process of precoding,

but that is more straightforward and presents more

freedom from the signal processing perspective. The

optimal detector in terms of performance is the maximum-

likelihood (ML), which tests all possible combination

for transmitting information symbols, where the best

candidate of the signal constellation set is taken as the

solution. Despite of its performance, the ML complexity is

prohibitive as it searches in a set of MnT elements. A better

option is the sphere-decoding (SD) [21], which can achieve

the same performance with a reduced, but still high,

complexity. Within lower complexity approaches, linear

MIMO detection techniques becomes a good choice, as

they can achieve near optimum performance if combined

with other techniques, but yet lower complexity.

A couple of popular MIMO detection approaches are the

zero-forcing [22]:

SZF =

√nT

ESH†X =

√nT

ES

(HHH

)−1HHX (5)

and the MMSE equalization:

SMMSE =

√nT

ES

(HHH +

σS

σwgnInR

)−1

HHX (6)

These detectors can be improved by using different

techniques, such as ordered successive-interference-

cancellation (OSIC)[5] and lattice-reduction[23] tech-

niques. In fact, the lattice-reduction is very useful for

ill conditioned and/or high correlated MIMO channels,

besides of achieving near-optimal performance while

holding full diversity.

Finally, a special case for the ZF equalization is the

matched filter, which is a suitable approximation for large




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MIMO systems:

SMF =

√nT

ESHHX (7)

2.3. Channel Estimation

It is known that the knowledge of the CSI is paramount

for detection and precoding, as MIMO systems deploy

signal combinations over time and/or space, hence these

signals must be decoupled at the receiver. For the model

described herein, the CSI estimation is obtained on the BS

through pilots sent at the beginning of every coherence

time interval. This implies that the coherence time must be

long enough to perform channel estimation, precoding and

transmission steps in the broadcasting mode; and also long

enough to proceed the detection in the uplink transmission

mode.

For channel estimation, let us consider pilot sequence

transmission:

Y =

√ES

nTHP + N, (8)

where each column in Y and N ∈ CnR×τ represents

received signal and noise vectors from a time slot in a

frame of length τ . The channel H will be estimated through

pilot sequences with length τ , contained in each row of

matrix P ∈ CK×τ , which are sent by the nT antennas to the

nR receive antennas with equally distributed power (ES)

among the nT transmit antennas.

Notice that MIMO channel estimation problem in (8) is

similar to a conventional MIMO detection problem, but the

unknown variable is the channel matrix H instead of the

symbols P. A straightforward solution problem (8) applies

the well-known equalization based on the pseudo-inverse

channel matrix [24]:

H =

√nT

ESYP†. (9)

Case of orthogonal sequences are employed as pilots,

PPH = τInT , the processing can be reduced to:

H =1

τ

√nT

ESYPH . (10)

3. Multi-Cell Model

A Multi-cell MIMO system is composed by L cells, each

one composed by one BS equipped with N antennas,

and K mobile users equipped with only one antenna. For

uplink transmission, the users are taken as the transmit

side and the BS as the receive side, and the opposite

situation holds for downlink transmission depending on

the transmission direction. Due to this, from now on, the

notations nT and nR are dropped since communication is

established in both directions. Fig. 1 depicts a multi-cell

MIMO system with 4 cells with 1 [km] radius, each one

containing one centered BS serving 10 users uniformly

distributed.

3.1. Uplink Model

Multi-cell MIMO uplink transmission can be described as

L interfering point-to-point MIMO systems:

Xulj =

L∑

ℓ=1

HjℓSulℓ + Nul

j , (11)




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−1000 0 1000 2000 3000

−1500

−1000

−500

0

500

1000

1500

2000

2500

3000

Users − Cell 1

Users − Cell 2

Users − Cell 3

Users − Cell 3

Base Stations

Figure 1. Multi-Cell MIMO system, K = 10 users, L = 4 cells

where Xulj ∈ CN×τ and Nul

j ∈ CN×τ are the signal and

noise received matrices at the jth BS; Hjℓ ∈ CN×K is the

channel gain matrix between the jth BS and the users in

the ℓth cell; and Sulℓ ∈ CK is the signal matrix sent by the

K users inside the ℓth cell. In order to keep the notation

as simple as possible, we define Sulℓ =

√Eℓ/KSul

ℓ , where

the term√

1/K normalizes the power among all antennas,

Ej = diag(eℓ1, eℓ2, · · · , eℓK) describe the transmission

power of each antenna and Sulℓ is the actual modulated

message. As Xulj , Sul

j and Nulj are signal matrices with

F columns, each one of them represents a symbol period.

Due to its performance, M − QAM Gray-coded

symbols is a widespread modulation format, which will

be deployed in this work for Sℓ, where all elements of

the modulated message have unitary average energy, i.e.

E[vec(Sul

ℓ )vec(Sulℓ )H

]= IKF . Also, all the elements of

the noise matrix are i.i.d. circularly-symmetric-complex

Gaussian, where vec(Nulj ) ∼ CN (0NF , σwgnINF ).

In multi-cell scenarios, it must be considered the fast

fading and the pathloss channel effects, which reduce

substantially the inter-cell interference in macro-cell

environment. Hence, the columns of channel matrix Hjℓ =

[h1jℓ · · · hKjℓ] are expressed as:

hkjℓ = d−βjℓkINvjℓ. (12)

where the d−βjℓkIN term represents the pathloss, with djℓk

is the distance between jth BS and the kth user in the ℓth

cell, and β is the fading factor. On the other hand, the fast

fading term is modeled using the Rayleigh distribution, i.e.

vjℓ ∼ CN (0N , IN ).

Since the BS possesses greater signal processing

capabilities, its preferable to perform conventional MIMO

detection for uplink, which can provide better BER

performance, depending on the adopted technique. On

the other hand, DL transmission must deploy precoding

techniques, since the users possesses low processing power

and are physically separated, and hence the detection is

performed by a simple slicer.

3.2. Downlink transmission

Considering the use of TDD mode, the channel can

be considered reciprocal, i.e. Hdljℓ = (Hul

jℓ)T . This is a

desirable situation since there is no need of feed-backing

the channel-state-information (CSI) case precoding is ap-

plied, besides simplifying the DL transmission description.




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Hence, the received signal in the jth cell is given by:

Xdlj =

L∑

ℓ=1

(Hjℓ)T AℓSdl

ℓ + Ndlj , (13)

where Xdlj ∈ CK×F is the received signal matrix by the K

users inside the jth cell; (Hjℓ)T is the DL channel matrix

between the users served by the ℓth BS and the users placed

in the jth cell; and Adlℓ Sdl

ℓ ∈ CN×F is the sent signal by

the ℓth BS, where Sdlℓ =

√Eℓ/KSdl

ℓ ∈ CK×F . Finally, Ndlj

is the background additive noise with variance σwgn.

As stated previously, since there is an excess of antennas

transmitting information and a crescent need of low

complexity receivers, precoding techniques are usually

employed at the BS. Aiming to provide low-complexity

massive MIMO detectors, a zero-forcing precoding matrix

has been proposed [13]:

AℓZF =1√γZF

H∗ℓℓ(H

TℓℓH

∗ℓℓ)

−1 (14)

where the ZF factor γZF = tr(HTℓℓH

∗ℓℓ)

−1/K.

Notice that simplifications on the zero-forcing precoding

leads to the MF matrix solution suitable for massive MIMO

applications:

AℓMF =1√γMF

H∗ℓℓ, (15)

where MF factor γMF = tr(HTℓℓH

∗ℓℓ)/K

It is noteworthy that both MIMO precoding and

detection ignore the interfering links from different cells,

affecting directly the system’s performance.

3.3. Channel Estimation and Pilot Contamination

In TDD systems, pilot contamination is challenging,

affecting the CSI estimation, which greatly limits the

performance of multi-cell massive MIMO systems. The

training phase consists on sending pilot sequences from

all K users of each cell to their BS, which processes the

received signal in order to estimate the CSI.

Following eq. (11), the received signal in the channel

estimation mode Yj ∈ CN×τ is given by:

Yj =

L∑

ℓ=1

HjℓPℓ + Nj , (16)

where Pℓ =√

Eℓ/KPℓ is the pilot sequences matrix scaled

by the normalized transmission power on each antenna.

Also, the pilot sequences have a length τ , contained in each

row of Pℓ, leading to a K × τ matrix. Usually, orthogonal

sequences are used as pilot, but their set site may be limited

leading to reuse of sequences among different cells, i.e.

Pℓ = Pj , ∀j, ℓ.

From this fact, the channel estimation in massive MIMO

problem using orthogonal pilot sequences (i.e., PPH =

τIK), becomes:

Hjj =

√K

τYjE

−1/2j P

H

= Hjj +∑

ℓ 6=j

Hjℓ

√EℓE

−1j +

√K

τNjE

−1/2j P

H

(17)

Notice that (17) estimates not only the channel link of

interest, since vestiges from other links (second term

in (17)) cannot be removed. Furthermore, despite of

E[‖Hjj‖2

]> E

[‖Hjℓ‖2

], due to the pathloss effect,




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these residue is enough to cause an erroneous channel

estimation.

4. Cooperative Multi-cell MIMO

By sharing information between BS through a backhaul

net, the communication between MTs and BS can

be improved substantially if interference cancellation

procedure could be employed for symbol detection,

channel estimation and precoding purposes.

Rewriting (11) for UL transmission multi-cell scenario

in matricial form:

Xul1

...

XulL

=

H11 · · · H1L

.... . .

...

HL1 · · · HLL

Sul1

...

SulL

+

Nul1

...

NulL

, (18)

which can be compactly expressed as:

X ul = HSul + N ul. (19)

The diagonal elements (matrices Hℓℓ) of H are referred as

direct link channel, while the remaining elements are called

interfering links. As BSs are isolated and are not linked

in non-cooperative cell MIMO, for precoding, channel

estimation and symbol detection procedures, only the

main elements of H are considered, leading to intercell

interference and pilot contamination problems.

Similarly, the compact matricial DL multi-cell descrip-

tion can also be obtained:

X dl = HHASdl + N dl. (20)

Under this system description, the ZF precoding matrix

can be rewritten as:

AZF =1√γZF

H∗(HT H∗)−1, (21)

with ZF factor γZF = tr(HTℓℓH∗)−1/LK. Also, the

matched-filter precoding matrix is given by:

AMF =1√γMF

H∗, (22)

where γMF = trace(HT H∗)/LK .

4.0.1. Channel Estimation In shared channel estimation

procedure, each BS estimates only the CSI of their

own cell (Hℓℓ), i.e. only the diagonal of H on eq.

(20) are estimated and subject to interference. If P =[P

T

1 PT

2 · · · PT

L

]T

and Y are the pilot sequence and

the received signal on the BS’s, respectively, both shared

in all BS, we can reliably estimate the CSI of all cells’ link

with:

H = YP†= (HP + N )P†

. (23)

The price for the cooperation in multicell channel

estimation is the need of long sequences, implying in no

pilot reuse. Another issue is the traffic carried through

the backhaul link, which may turn out very intense for a

massive number of antennas. Finally, the signal processing

must be centralized for the sake of reliability, complexity

and efficiency.

Choosing orthogonal pilot sequences for P matrix, such

as Walsh-Hadamard (W-H) matrices, one can find the




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solution:

H =

√LK

τYE−1/2PH

, (24)

with the energy control matrix given by:

E =

E1 · · · 0K

.... . .

...

0K · · · EL

. (25)

Despite of improving the channel estimation, the length

of the pilot sequence must be at least τ ≥ KL, limiting

this solution just for low mobility applications, due to the

channel coherence time. Also, from eq. (19), it can be

observed that cooperation enable multi-cell systems to be

treated as single-cell point-to-point MIMO systems, where

pilot contamination is absent, improving link reliability.

5. Asymptotic Orthogonality

Now consider a generic channel matrix HN×K with all

elements i.i.d. with complex Gaussian distribution, i.e. flat

fading Rayleigh channel. When H gets extremely tall, or

N >> K , it tends to be very well conditioned, therefore

we have the following property [13]:

limN→∞

HHH

N= IK , (26)

which leads, for example, to the MF solution. This

tendency is highly desirable since the linear detectors with

low complexity are able to achieve superior performance,

near-optimal performance in fact, besides of showing that

random processes tend to become deterministic for large

arrays. This behavior can be statistically tested by the

asymptotic orthogonality (AO) index:

AO(N) , tr

[E

(HHH

KN

)]≈ 1 (27)

To illustrate this concept, in Fig. 2, eq. (27) is evaluated

using Monte Carlo experiments considering different

values for both K and N . For better readability, let us

consider the expression |1 − AO(N)|. In this graph, it can

be confirmed the convergence of (27) is improved as the

number of antennas N increases.

10 20 30 40 50 60 70 80 90 10010

−6

10−5

10−4

10−3

K

|1−

AO

|

N=100N=500N=1000

Figure 2. Numerical Asymptotic Orthogonality for massiveMIMO channels as a function of number of users K andparameterized by the number of transmit antennas N .

6. Orthogonal and Pseudo Random Sequences

The characterisation of sequence sets with good cross-

and auto-correlation properties is of paramount interest

in telecommunications systems, primarily for spread-

spectrum system applications. For this work, different

sequence sets will be deployed in low-complexity channel

training procedures. Despite of not being orthogonal,




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some sequence sets may present far better correlation

properties, which are very desirable for desynchronized

systems with small synchronization errors related to the

symbol period interval. In this section, a brief discussion

on the correlation properties of representative sequence

sets to be deployed later is carried out; more specifically

the Hadamard codes, Gold family, small Kasami set and

quaternary sequences are revisited.

This work will cover families with M cyclically distinct

sequences (size of the sequence set) of length τ . Hence,

if si(n) denotes a sequence, it is cyclically distinct to

another sequence sj(n) if there is no integer T for which

si(n) = sj(n). Furthermore, the sequences herein studied

are periodic, i.e., si(n) = si(nτ). Besides the parameters

M and τ , the periodic cross-correlation function among

sequences of the same set is very important, since this

measure allows the identification of sequences (or channels

or users) from their own shifted versions or other shifted

sequences. The discrete periodic correlation values of a

family is denoted as θij(n), where it is referenced as auto-

correlation if i = j and cross-correlation if i 6= j.

It is straightforward the fact that binary sequences are

composed by 0s, and 1s, i.e. b(n) ∈ {0, 1}, n = 1, . . . τ .

Also, quaternary sequences assume four distinct values:

q(n) ∈ {0, 1, 2, 3}. In order to deploy them properly, these

values must be remapped into a unitary circles: eιπ b(n) ∈

{±1} and eι π2 q(n) ∈ {±1, ±ι}, for binary and quaternary

sequences, respectively.

6.1. Walsh-Hadamard sequences

Walsh-Hadamard (WH) sequences are known to be

orthogonal besides of being fairly easy to build, which

gives their importance in telecommunication systems.

They are composed by M sequences of length τ ,

easily build using the iterative Sylvester method [14].

Since they are orthogonal, they present zero cross-

correlation for non shifted sequences. However, their cross-

correlation values may assume very high values when

WH sequences are desyncronized, leading to θij(n) =

{0, ±4, ±8, · · · , M}, as depicted in Fig. 3.

−60 −40 −20 0 20 40 600

10

20

30

40

50

60

70

80

Occ

urre

nce

perc

enta

ge

Corelation value

−60 −50 −40 −30 −200

0.2

0.4

0.6

0.8

20 30 40 50 600

0.2

0.4

0.6

0.8

Figure 3. Discrete cross-correlation values distribution forHadamard sequences of length τ = 64.

6.2. Maximum-length-sequence or m-code

The maximum-length-sequences are binary periodic

sequences of great importance since they present excellent

correlation properties and are used to derive many other

sequences. An r order m-code of length τ = 2r − 1 is

generated by r linear feedback shift registers, which

is summarized in [25]. Their discrete auto-correlation




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assumes the following values:

θij(n) =

2r − 1, n = 0, ±τ, ±2τ, . . .

−1, otherwise

Despite of presenting low correlation values, their size set

M is very restrict, limiting its raw applicability.

6.3. Gold codes

The Gold set is a three valued cross correlation family

[15] formed by M = 2r + 1 sequences of length τ = 2r −

1, which are build from two specific m-codes, referred

as preferred pair. To generate them, it is taken first a

m-code p1(n) of length τ = 2r − 1, an then it is re-

sampled at a rate q generating the second element of

the pair p2(n) = p1(qn), which has also length 2r − 1.

By combining shifted versions of this pair, the remaining

sequences of the Gold set can be constructed.

As introduced before, the Gold family assume three

cross-correlation values [15], plus the auto-correlation

(AC):

θij(n) =

2r − 1 , in-phase AC: i = j, n = 0

−1

−2r + 1/2 − 1

2r + 1/2 − 1

(28)

for r odd and:

θij(n) =

2r − 1 , in-phase AC: i = j, n = 0

−1

−2r + 2/2 − 1

2r + 2/2 − 1

(29)

for r even. Also, the cross-correlation distribution of this

family presents just three values, and can be obtained

through expressions in [25]. The correlation distribution,

in terms of percentage occurrence, for Gold sequences of

length τ = 63 is exampled by Fig. 4. Notice that Gold sets

present relatively low correlation values {−17, 15} and

high occurrence of a low value −1.

−17 −1 150

10

20

30

40

50

60

70

80

Occ

urre

nce

perc

enta

ge

Corelation value

Figure 4. Discrete cross-correlation values distribution for Goldsequences of length τ = 63.

6.4. Small set of Kasami sequences

As Gold sequences, the small Kasami (s-Kasami) set can

be obtained through the combination of shifted m-codes

[25]. However, in this case, the s-Kasami set is obtained

by a m-code and its shorter sampled version, leading to

a family of M = 2r/2 sequences of length τ = 2r − 1.




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The s-Kasami family is a much smaller set compared to

the Gold one, however the Kasami’s cross-correlation are

almost half:

θij(n) =

2r − 1 , in-phase AC: i = j, n = 0

−1

−2r/2 − 1

2r/2 − 1

(30)

In fact, the cross-correlations values of r-order Gold

sequences are the same found in r + 1 small set of Kasami

sequences, but their distributions are quite different. From

Fig. 5 one can conclude that the s-Kasami sets present

a trade-off between M and θij(n), in which, unlike

Gold, s-Kasami sets exchange low-correlation values for

a smaller family of codes. Also, it can be seemed that

high-correlation values are more frequent on the Kasami

sequences.

−9 −1 70

10

20

30

40

50

60

Occ

urre

nce

perc

enta

ge

Corelation value

Figure 5. Discrete cross-correlation values distribution for s-Kasami sequences of length τ = 63.

6.5. Quaternary codes

Possessing more levels than binary sequences, quaternary

codes are able to present low cross-correlation values,

distributed in both in-phase and quadrature components.

In special, this work will cover the α Quaternary family,

extensively discussed in [26] and [17]. The construction

of quaternary sequences is very similar to m-codes as

they use r shift feedback register, but they use modulo

4 operations instead of modulo 2 ones. As Gold’s set, a

r-order quaternary code set is composed by M = 2r +

1 sequences of length τ = 2r − 1. The cross-correlation

distribution of these sequences can be explored in detail

in [17]; their values are summarized in the sequel:

θij(n) =

2r − 1 , in-phase AC: i = j, n = 0

−1

−1 + 2s ± ι2s

−1 − 2s ± ι2s

(31)

for odd r = 2r − 1, and:

θij(n) =

2r − 1 , in-phase AC: i = j, n = 0

−1

−1 ± 2s

−1 ± ι2s

(32)

for even r = 2r.

With a family as large as Gold’s and correlation values

comparable to the Kasami set, the quaternary sets present




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good qualities, but their distribution reveals higher cross-

correlation values being far more frequent than low ones,

as shown in Fig. 6.

−1 7 −1−8i −1+8i −90

5

10

15

20

25

30

35

Occ

urre

nce

perc

enta

ge

Corelation value

Figure 6. Cross-correlation distribution for Quaternary se-quences of τ = 63

7. Desynchronized Channel Estimation

As stated previously, uplink channel training may not

ensure synchronization, as users are physically apart along

their cells, deteriorating the channel estimation further

more.

In order to study this effect, let us consider a single cell

scenario and simplifying considerations:

Y = H [P]n + N, (33)

where all elements of H are i.i.d. CN (0, 1), P is

the training sequence, [·]n is transmitted training pilot

with desynchronization up to n samples and N is the

background noise with σ2wgn power. When transmit and

receiver side are synchronized and the pilot training

sequences are orthogonal, PPH = τIK ; hence, the

transmission and channel estimation is free from errors. On

the other hand, under synchronization errors, the channel

estimation matrix H can be written in terms of correlation

matrix R:

H =1

τYPH

=1

τH [P]

nPH +

1

τNPH

=1

τHR(n) +

1

τNPH ,

(34)

where R(n) will be referred as a correlation matrix of

discrete time n. By considering P = [p1 · · ·pK ]T :

[P]n PH =

[p1]n pH

1 · · · [p1]n pH

K

.... . .

...

[pK ]npH

1 · · · [pK ]npH

K

, and

R(n) =

Rp1p1(n) · · · Rp1pK (n)

.... . .

...

RpKp1(n) · · · RpKpK (n)

.

For simplicity, from now on let us drop (n) index.

In order to evaluate the error introduced by non

synchronized pilot transmission, the squared distance

between the channel matrix and its estimation must be

calculated:

‖H − H‖2 = tr[(H− H)(H− H)H

]

tr(HHH)− tr(HRHHH)

τ

− tr(HPNH)

τ− tr(HRHH)

τ

+tr(HRRHHH)

τ 2+

tr(HRPNH)

τ 2

− tr(NPHHH)

τ+

tr(NPHRHHH)

τ 2

+tr(NPHPNH)

τ 2.

By evaluating the expected value of the previous equation

considering that P and R are deterministic, tr(E[·]) =




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E [tr(·)] and tr(AB) = tr(BA), it is showed that:

E[‖H − H‖2

]= tr(E

[HHH

])− tr(E

[HHH

]RH)

τ

− tr(E[NHH

]P)

τ− tr(E

[HHH

]R)

τ

+tr(E

[HHH

]RRH)

τ 2+

tr(E[NHH

]RP)

τ 2

− tr(E[HHN

]PH)

τ+

tr(E[HHN

]PHRH)

τ 2

+tr(E

[NHN

]PHP)

τ 2.

Since H and N are uncorrelated random matrices,

E[HHN] = 0K . Also E[HHH

]= NIK , E

[NHN

]=

NσwgnIK and tr(PPH) = Kτ for every set of K

sequences of length τ :

E[‖H − H‖2

]= NK +

Nσ2wgn − 2Ntr(R)

τ

= +Ntr(RRH)

τ2

Finally the normalized mean-squared-error is given by:

MSEH =E

[‖H − H‖2

]

E [‖H‖2]

= 1 +σ2

wgn − 2tr [R(n)]

Kτ+

tr[R(n)R(n)H

]

Kτ2.

(35)

Equation (35) reveals that the number of BS antennas

(N ) does not affect the channel estimation performance.

Besides, if the MSE should be calculated for a

desynchronization lower than an unit of symbol period TS,

all signals are ressampled versions with rate ns. Therefore,

(35) must be modified such as τ → τns.

From (35), and remembering that for synchronized

orthogonal sequences R(0) = τIK , one can conclude that

perfect channel estimation can be achieved with minimum

MSE given by:

MSEminH

=σ2

wgn

Kτ

On the other hand, desynchronized channel estimation

procedures introduce estimation errors, mainly because of

auto- and cross-correlation terms of the selected training

sequence, i.e., tr(R) and tr(RRH) in eq. (35).

In Fig. 7, eq.(35) is tested with the estimation of a

20 × 8 MIMO channel using WH pilot sequences of length

τ = 64 and training power of 20 [dB]. For this work and

in order to simplify the analysis, the desynchronization

was modeled as a cyclic shift of n samples. The region

of practical interest of desynchronization errors in massive

MIMO channel estimation can be bounded as∣∣nTS

τ

∣∣ < 0.2.

0 0.2 0.4 0.6 0.8 110

−4

10−3

10−2

10−1

100

101

Units of TS

MS

E

SimulationTheoretical

Figure 7. Theoretical and measured MSE for a 20× 8 systemwith WH pilot sequences of length τ = 64

7.1. Correlated Massive MIMO Channels

Considering now a noiseless scenario, based on eq. (33) , a

MIMO channel matrix can be estimated through:

H =1

τYPH =

1

τHR(n), (36)




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As expected, for synchronized training using orthogonal

sequences R(0) = τIK , leading to a perfect estimation.

However, as desynchronized training sequences takes

place, R becomes more and more different from a

diagonal matrix, making the estimation less accurate.

Besides, as elements outside the diagonal of R shows

off, the estimated channel becomes correlated‡ as different

elements of the original channel H are combined to

generate the estimated channel matrix H. Elaborating

further, each different element of H is a weighted

combination of links (elements of) from H, i.e., hij =

Σk hikrkj . Hence, desynchronization introduces not only

erroneous channel estimations but also MIMO channel

correlations, resulting in a very poor and ill conditioned

estimated channel matrix.

8. Numerical Results

This section presents numerical results obtained via

Monte-Carlo simulation (MCS) for both channel estima-

tion and data error performance for DL massive MIMO

channels with training Gold and Quaternary sequences of

length τ = 63 and WH of length τ = 64. Furthermore, for

simplicity of analysis, it has been assumed that the pilot is

transmitted at the same DL transmitting rate.

8.1. Channel Training

The results presented in this section describe the MSE of

the channel matrix over desynchronized training, modeled

by right cyclic shifted sequences. The choice of using

‡Indicating correlation between transmit antennas.

cyclic shift to characterize desynchronization comes from

its simplicity, besides of enabling a link between this work

and the vast analyses on sequence available in literature,

widely discussed, for instance in [25].

For a fair comparison, Fig. 8 depicts the MSE for

channel estimation based on (35) and considering a

massive MIMO system with K = 8 users, N = 100 BS’s

antennas and deploying different sequence sets: WH, Gold,

s-Kasami and α-quaternary sequences with reasonable

long training sequence length of τ = 64, 63, 63 and 63,

respectively. It was also considered an unicellular scenario

and training signal-to-noise-ratio SNR∗ = 20 [dB]. For

this setup, WH sequences presented the best channel

estimation as they are strictly orthogonal, however this

picture changes as desynchronization gets larger. Under

this situation Gold and s-Kasami sequences have been

demonstrated to perform a better channel estimation

since their cross-correlation results lower for TS >

⌈0.05 · τ⌉ and TS > ⌈0.17 · τ⌉, respectively. Also, it can

be remarked that s-Kasami sequence set presents lower

cross-correlation values than Gold sequences, as illustrated

in Fig. 5) and 4, respectively; however, low correlation

values are less frequent, so their performance showed to

be equivalent when the asynchronism increases.

In order to support more users, a larger number

of sequences is required. To demonstrate the channel

estimation behavior when more users must be served, Fig.

9 depicts the MSE for channel estimation when the entire

set of sequences is deployed, e.g. Gold (τ = 63), WH (τ =

64) and Quaternary (τ = 63); also, the s-Kasami (τ = 8)




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0 0.2 0.4 0.6 0.8 110

−4

10−3

10−2

10−1

100

101

MS

E

Units of TS

HadamardGoldKasamiQuaternary

Figure 8. MSE for a K = 8×N = 100 system with differentpilot sequence over desynchronized training

set has not been considered since its set is much smaller

than other sequences. In this case, the channel estimation

was also proceeded in an unicellular system with SNR∗ =

20 [dB] for pilot transmission, where 64 users will be

served by 100 BS antennas.

Comparing MSE results from Fig. 8 and 9, it can

be noted that WH sequences didn’t present performance

losses when more channel estimation links are included,

while Gold and quaternary sequences become quite inaccu-

rate for channel estimation under heavy number of channel

estimation sharing the same BS’s antennas infra-structure.

It comes from the fact that cross-correlation distribution

of WH sequences remains unchanged independent from

the number of sequences deployed. On the other hand,

Gold and quaternary sequence present the opposite effect,

hence, as the number of sequences used increase, the term

tr(RRH) of equation (35) experiences a larger growth

than tr(R), deteriorating the massive MIMO channel

estimates.

0 0.2 0.4 0.6 0.8 110

−4

10−3

10−2

10−1

100

101

MS

E

Units of TS

HadamardGoldQuaternary

Figure 9. MSE for a K = 64×N = 100 system with differentpilot sequences over desynchronized training interval.

8.2. Downlink Transmission

This section presents numerical resuls for data error per-

formance of DL massive MIMO systems operating under

desynchronized training procedure. For this scenario, it

has been considered massive MIMO systems with L =

4 cooperative cells§ with radius of 1 [km], 4−QAM

modulation, a number of BS antennas in the range N ∈

[8; 700]; furthermore, the system operates with 16.6%

of desynchronism through a link with pathloss exponent

β = 1.9. For channel training, WH, Gold and Quaternary

sequences have been deployed. The uplink transmission

will be performed by zero-forcing and matched filter

precoding. Also, it will be considered power normalization

between antennas for both pilot transmission and downlink

data transmission.

§All sharing information and assuming no pilot reuse, as discussedpreviously.




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In the first setup, it is considered K = 8 users per

cell, leading to 32 users served along the 4 cells, where

pilots were sent with SNR∗ = 5 [dB] and bits were sent

with Eb/N0 = 5 [dB]. Fig. 10 summarizes the bit error

rate (BER) performance for different number of BS

antennas operating with perfectly synchronized training.

This set of performance curves shows the superiority

of the WH set, which is straightforward since the

all sequence in the set are orthogonal. Gold training

sequences present close performance to the WH training

set, while quaternary sequences leaded to the poorest

BER performance. These performance results agree with

the MSE results presented previously; hence, an accurate

channel estimation procedure leads to a better DL BER

performance.

101

102

10−6

10−5

10−4

10−3

10−2

10−1

BE

RD

L

N

ZF−HadamardZF−GoldZF−QuaternaryMF−HadamardMF−GoldMF−Quaternary

Figure 10. Downlink BER performance for a system with 4 cells,each one containing a BS with N antennas serving 8 users, withsynchronized training with SNR∗ = 5 [dB] and DL transmissionwith Eb/N0

DL = 5 [dB]

On the other hand, when desynchroniztion in training

step arises (Fig. 11), WH sequence set no longer performs

the best channel estimation, and Gold training sequences

become a better choice due to its suitable cross-correlation

properties. It is worth to note that quaternary sequences

still present poor performance since channel estimation is

not as precise as the ones performed by other sequences.

Finally, it is also noteworthy that the MF filter performance

is conditioned to the quality of the channel estimation and

the size of the MIMO system.

101

102

10−4

10−3

10−2

10−1

BE

RD

L

N


Figure 11. Downlink BER performance for a system with 4 cells,each one containing a BS with N antennas serving 8 users, withsynchronized training with SNR∗ = 5 [dB] and DL transmissionwith Eb/N0

DL = 5 [dB]

If more users must be served, more sequences should

be used and, in some cases, the channel estimation

becomes less accurate. By using the same setup as previous

simulations to serve now K = 16 users per cell, Figures

12 and 13 depict both synchronized and desynchronized

training scenarios, respectively. One can see that Gold

training set results in worse performance compared to

the WH ones in both synchronized and desynchronized

training scenarios, considering high density of users




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per number of sequences¶. Notice that it was already

predicted by eq. (35) and Fig. 9, which show that only

orthogonal training sequences, i.e.,WH codes, are able to

perform reasonable channel estimation in massive MIMO

configurations.

102

10−4

10−3

10−2

10−1

BE

RD

L

N


Figure 12. Downlink BER performance for a system with 4 cells,each one containing a BS with N antennas serving 16 users percell, with synchronized training with SNR∗ = 5 [dB] and DLtransmission with Eb/N0

DL = 5 [dB]

Notably, from both Figures 13 and 12 one can conclude

that if the massive MIMO channel is poorly estimated, the

zero-forcing precoding is not effective as its solution is

obtained through channel matrix inverse, which are very

sensitive to ill conditioned channels. It is also noteworthy

that, both Gold and Quaternary training sequences result

in very ill conditioned channels, as discussed previously,

which combined to the zero-forcing precoding, causes

an ineffective solution with impracticable bit error rate

performances.

¶This is due to the adopted system configuration: 16 users per cell, anda multi-cellular environment with four cells; hence, the number of servedusers is approximately the size of training sequence sets. Note that due tothe adopted sequence length, there is no pilot sequence reuse.

102

10−4

10−3

10−2

10−1

BE

RD

L

N


102

0.4995

0.5

0.5005

Figure 13. Downlink BER performance for a system with 4 cells,each one containing a BS with N antennas serving 16 usersper cell, with 16.67% desynchronized training sequences withSNR∗ = 5 [dB] and DL transmission with Eb/N0

DL = 5 [dB]

9. Conclusion

This work investigated the deployment of different training

sequence sets for channel estimation in multi cell massive

MIMO environment with cooperation between BS’s by

using more compact expressions. An expression for

evaluating the MSE performance of the training sequence-

based channel estimation has been proposed, which

was validated through direct MSE and downlink BER

performance simulations.

MCS numerical results have corroborated the fact

that WH training sequences present, overall, the best

performance, particularly under training configurations

with perfect synchronism. Under desynchronized training

and low number of users served per cell, Gold training

sequences have achieved the best performance for massive

MIMO channel estimation. On the other hand, s-Kasami

training sets even presenting suitable correlation properties




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and a small number of sequences available in the set,

the channel estimation performed with s-Kasami sets have

not demonstrated to greatly outperform Gold sets. Finally

quaternary sequence sets presented the poorest channel

training and estimation performance and therefore the

worst BER performance arises, mainly determined by its

cross-correlation distribution.

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