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Centro de Tecnologia e Urbanismo

Departamento de Engenharia Eletrica

Bruno Felipe Costa

Deteccao MIMO sob CanaisCorrelacionados Sujeitos a Erros de

Estimativas

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

Bruno Felipe Costa

Deteccao MIMO sob Canais

Correlacionados Sujeitos a Erros de

Estimativas

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

Costa, Bruno FelipeDeteccao MIMO sob Canais Correlacionados Sujeitos a Erros de

Estimativas. Londrina, PR, 2014. 52 p.

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

1. Sistemas de Comunicacao Sem Fio. 2. Sistemasde Multiplas Antenas. 3. Deteccao MIMO. 4. Reducaotrelica. 5. Precodificacao Departamento de Engenharia Eletrica

Bruno Felipe Costa

Deteccao MIMO sob CanaisCorrelacionados Sujeitos a Erros de

Estimativas

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

Orientador

Prof. Me. Jaime Laelson JacobDepto. de Engenharia Eletrica

Universidade Estadual de Londrina

Prof. Jose Carlos MarinelloEngenharia EletricaFaculdade Pitagoras

24 de novembro de 2014

”Nao ha assunto tao antigo que algo novo nao pode ser dito sobre ele.”

Fiodor Dostoievski.

Agradecimentos

Gostaria de agradecer ao meu orientador, professor Dr. Taufik Abrao, pela

imensuravel ajuda que tenho recebido no meu processo de formacao, por todas as

discussoes , conselhos, orientacao, paciencia e tudo que me ensinou neste tempo

que trabalhamos juntos.

Agradeco a minha famılia por ser o meu alicerce, por todos os incentivos e

apoios ao longo desse curso.

A fundacao Araucaria que durante uma grande parte de minha graduacao me

auxiliaram com bolsa de iniciacao cientifica.

Enfim, agradeco a todas as pessoas que contribuıram de forma direta e indi-

reta para a realizacao deste trabalho e minha formacao.

Resumo

Este trabalho analisa o problema da deteccao em sistemas com multiplas an-tenas no transmissor e receptor (MIMO – multiple input multiple output) sobo ponto de vista de detectores classicos em canais correlacionados e com errosde estimativas. Algumas tecnicas MIMO sao estudadas visando o aumento daeficiencia espectral e/ou aumento de desempenho dos sistemas de comunicacaosem fio (wireless). Neste trabalho, primeiramente, estuda-se o desempenho dodetector MIMO classico MMSE (minimum mean squared error) com e sem o au-xilio das tecnicas reducao trelica (LR – lattice reduction) e de cancelamento deinterferencia sucessivo ordenado (OSIC – ordered successive interference cancel-lation) em canais correlacionados, com o objetivo de quantificar o impacto doefeito da correlacao espacial entre as antenas sobre o desempenho. Em seguidaanalisa-se, levando em conta o efeito da correlacao espacial, um cenario onde oconhecimento do estado do canal (CSI – channel state information) contem errosnas estimativas. Neste contexto, utiliza-se o desempenho do detector de maximaverossimilhanca (ML – maximum likelihood) como referencia na comparacao dedesempenho dos detectores sub-otimos analisados. Adicionalmente, analisa-sea complexidade computacional dos detectores MIMO sub-otimos e analisada,em funcao do numero de operacoes de ponto flutuante por segundo (FLOPS– floating-point operations per second), com o objetivo de se determinar o melhorcompromisso desempenho-complexidade. Na ultima parte do trabalho propoe-seum novo esquema de (pre-) decodificacao com o objetivo de mitigar o efeito dacorrelacao espacial e melhorar o desempenho a partir da estimacao da correlacaodo canal MIMO.

Palavras-chave – multiplas antenas no transmissor e receptor (MIMO). ReducaoTrelica (LR). Precodificacao. Decodificacao. Canais Correlacionados. Erros deestimativas de canal.

Abstract

This work analyzes the problem of detection on communication systems withmultiple antennas at the transmitter and receiver side (MIMO – multiple inputmultiple output) from the point of view of feasible detectors in correlated chan-nels and estimation error context. Some MIMO detection techniques are studiedin order to increase the spectral efficiency and/or increase the reliability of wire-less communication systems. Firstly, in this work is studied the performanceof the classical MIMO detector, namely minimum mean squared error (MMSE)with and without the lattice reduction technique (LR) and ordered successiveinterference cancellation (OSIC) procedure under correlated channels; these as-sociation is done in order to reduce the impact of the spatial correlation effectamong antennas on the system performance. Indeed, taking into account the ef-fect of spatial correlation and scenarios where the knowledge of the channel stateinformation (CSI) contains estimation errors, the MIMO system performancedegradation has been analyzed and quantified. In this context, the performanceof maximum likelihood detector (ML) is deployed as a reference for performancecomparison with the analyzed suboptimal detectors. Additionally, the compu-tational complexity of suboptimal MIMO detectors is determined as a functionof the floating-point operations per second (FLOPS), with the purpose of estab-lishing the best performance-complexity tradeoff. In the last part of the work anew (pre-) decoding scheme has been proposed, in order to mitigate the effect ofspatial correlation while improving the performance with the knowledge of thechannel correlation estimation.

Keywords – Multiple-Input-Multiple-Output (MIMO). Lattice Reduction (LR).Precoder. Decoder. Correlated Channels. Estimation Errors.

Sumario

Lista de Abreviaturas

Convencoes

1 Introducao 1

2 Conceitos Basicos e Modelo de Sistema 5

2.1 Diversidade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Modo de Multiplexacao . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Taxa de erro de bit, BER . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Modulacao por Amplitude em Quadratura . . . . . . . . . . . . . 9

2.5 Zero forcing - ZF . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.6 Raiz Quadrada de Matriz . . . . . . . . . . . . . . . . . . . . . . 10

3 Deteccao MIMO em Canais Correlacionados e com Erros de Es-

timativas 11

3.1 Detectores MIMO em Canais Correlacionados . . . . . . . . . . . 11

3.2 Deteccao MIMO em canais correlacionados e com erros na estimativa 11

3.3 Precodificacao MIMO para canais correlacionados . . . . . . . . . 12

4 Conclusoes 13

4.1 Trabalhos Futuros . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Apendice A -- MIMO Detectors Under Correlated Channels 15

Apendice B -- MIMO Detection Under Correlated and Imperfectly

Estimated Channels 27

Apendice C -- Precoder and Decoder Design to Improve the Perfor-

mance for Correlated Channel Matrix 40

Referencias 51

Lista de Abreviaturas

ASK Amplitude Shift Keying

AWGN Additive White Gaussian Noise

BER Bit Error Rate

CSI Channel State Information

LLL Lenstra, Lenstra, and Lovasz

LR Lattice Reduction

MCS Monte-Carlo Simulation

MIMO Multiple-Input Multiple-Output

ML Maximum Likelihood

MMSE Minimum Mean Squared Error

NLOS non-line-of-sight

OSIC Ordered Successive Interference Cancellation

QAM Quadrature Amplitude Modulation

RF Radio Frequency

SER Symbol Error Rate

SIC Successive Interference Cancellation

SM Spatial Modulation

SNR Signal-to-Noise Ratio

ZF Zero-Forcing

Convencoes

Na notacao das formulas, as seguintes convencoes foram utilizadas:

• letras maiusculas em negrito denotam matrizes;

• letras minusculas em negrito denotam vetores;

• (·)H denota o operador Hermitiano.

• (·)∗ denota o operador conjugado.

• Z[i] denota o conjunto dos Gaussianos inteiros.

• E[·] denota o operador de esperanca estatıstica.

• R− denota conjunto dos numeros reais negativos.

• RM ×N denota conjunto de matrizes reais M × N .

• CM ×N denota conjunto de matrizes complexas M × N .

1

1 Introducao

Nas ultimas duas decadas, a comunicacao sem fio desenvolveu-se considera-

velmente, assim como a demanda crescente de dados por usuarios, o que produz

uma busca constante por sistemas de alta capacidade de transmissao de dados

(MESLEH et al., 2008). No entanto, a disponibilidade de espectro para servicos de

dados e limitada. Nos ultimos anos, sistemas com multiplas antenas de trans-

missao e recepcao (MIMO - Multiple Input Multiple Output) tem se apresentado

como uma boa alternativa a esse problema, tendo em vista que sistemas MIMO

podem atingir um ganho notavel de eficiencia espectral ou sao capazes de me-

lhorar a confiabilidade/desempenho de comunicacao sem fio com a implantacao

de multiplas antenas tanto no lado transmissor quanto no lado do receptor (FOS-

CHINI; GANS, 1998).

No modo de ganho por multiplexacao, fluxos de informacoes sao transmitidos

de forma paralela usando multiplas antenas com o objetivo de aumentar vazao

de dados e portanto a eficiencia espectral, ao custo de se aumentar a comple-

xidade da deteccao da informacao no lado do receptor (WUBBEN et al., 2011).

Sistemas MIMO sofrem influencias de varios efeitos que podem degradar o seu

desempenho, e consequentemente acarretam reducao de sua capacidade. Assim,

o conhecimento da informacao de estado do canal ( CSI - Channel State Informa-

tion) e de fundamental importancia para um funcionamento aceitavel do sistema,

e pode ser usado no receptor, transmissor, ou em ambos os lados, dependendo

da arquitetura MIMO escolhida. Em cenarios realistas, a CSI nao pode ser per-

feitamente estimada; portanto, a informacao disponıvel contem algum erro. O

impacto da estimacao imperfeita da CSI no desempenho do sistema MIMO, bem

como o projeto de um precodificador para sistemas MIMO com estimativa imper-

feita de canal e investigado em (ANDALIBI; NGUYEN; SALT, 2013) e (YEH; WANG;

WU, 2011).

Outro efeito importante a ser considerado em cenarios MIMO realistas e a

correlacao entre as antenas. A correlacao entre as antenas tem um grande im-

pacto na taxa de erro de bit (BER - Bit Error Rate) do sistema, especialmente

1 Introducao 2

quando estruturas lineares de deteccao MIMO sao implementadas (VALENTE; FI-

LHO; ABRAO, 2014). Alguns trabalhos importantes tem analisado a capacidade

de ganho de sistemas MIMO assumindo que o desvanecimento do canal e inde-

pendente, o que na pratica e difıcil de se obter devido as restricoes fısicas (espaco

disponıvel, espacamento entre as antenas, etc), especialmente em cenarios que

requerem um numero massivo de antenas (massive MIMO), onde e fisicamente

difıcil projetar um conjunto de antenas com espaco suficiente entre elas para ga-

rantir que nao haja correlacao espacial. Recentemente, o efeito da correlacao no

desempenho de sistemas MIMO massivo foi investigado em (FANG et al., 2013).

Entre os detectores MIMO bem estabelecidos na literatura, o detector linear

zero forcing (ZF) e conhecido por cancelar completamente a interferencia entre as

antenas (WUBBEN et al., 2004), ao custo de aumentar significativamente o ruıdo

de fundo para matrizes do canal que nao estejam bem condicionadas. Nesse

contexto, o detector que minimiza o erro quadratico medio (MMSE – Minimum

Mean Squared Error) pode ser visto como uma alternativa, pelo fato deste levar

em consideracao a potencia do ruıdo aditivo durante o processo de deteccao do

sımbolo. Alem disso, o detector com cancelamento sucessivo de interferencia (SIC

- Successive Interference Cancellation) realiza uma deteccao camada-por-camada,

usando a estrategia ZF ou MMSE em cada uma destas, e cancela a interferencia

dos sinais que ja foram detectados, utilizando os sinais reconstrucao previamente

detectados nas camadas anteriores (BoHNKE et al., 2003). Como os erros de de-

teccao nas primeiras camadas sao propagados ao longo do algoritmo, um notavel

ganho de desempenho pode ser alcancado detectando-se primeiro os sımbolos mais

confiaveis, o que caracteriza os detectores MIMO SIC ordenados (OSIC - Ordered

SIC). Todas essas tecnicas lineares sub-otimas de deteccao MIMO apresentam

um desempenho inferior ao detector de maxima verossimilhanca (ML - Maximum

Likelihood), que realiza uma busca entre todos os vetores de sımbolos candidatos

e seleciona o mais verossımil.

Melhorias adicionais no compromisso entre o desempenho e complexidade

para sistemas MIMO podem ser obtidas a partir do pre-processamento de uma

tecnica denominada reducao trelica (LR - Lattice Reduction). Deteccao auxiliada

por LR pode ser empregada com o objetivo de alcancar desempenhos ainda me-

lhores enquanto a complexidade computacional se mantem controlavel. A tecnica

LR e obtida a partir de um conceito matematico usado na resolucao de muitos

problemas envolvendo reticulados. No problema de deteccao do sinal MIMO, a

tecnica LR pode ser usada para melhorar o condicionamento da matriz do ca-

nal, possibilitando o uso de estruturas mais simples para a deteccao (WUBBEN

1 Introducao 3

et al., 2011). Em outra palavras, reducao da complexidade computacional pode

ser alcancada com a implantacao da tecnica LR enquanto o desempenho do sis-

tema se mantem aceitavel (MOSTAGI; ABRAO., 2012). Em (PARK; CHUN, 2012) e

mostrado que detectores auxiliados por LR e com SIC possuem um desempenho

melhor que os detectores lineares nos cenarios com erro na estimativa do canal,

enquanto que em (CHEN et al., 2012) e demonstrado que detectores MMSE auxi-

liados por LR operando sob canais correlacionados e com estimacao imperfeita

sao capazes de alcancar um desempenho promissor, proximo ao otimo.

Com o objetivo de mitigar o efeito da correlacao espacial em canais MIMO,

alguns trabalhos investigam tecnicas de precodificacao. Em (BAHRAMI; LE-NGOC,

2006), por exemplo, e analisado a precodificacao utilizando conhecimento parcial

da matriz de correlacao de transmissao e recepcao; para canais com desvane-

cimento Ricean, foi estudado uma precodificacao para correlacao das antenas

no receptor em (ZHANG et al., 2012). O impacto da correlacao e precodificacao

usando codificacao de espaco-tempo foi investigado em (BJORNSON; OTTERSTEN;

JORSWIECK, 2009).

A contribuicao desse Trabalho de Conclusao de Curso (TCC) esta dividida

em tres partes.

1. Quantificar o impacto no desempenho de sistemas MIMO quando a tecnica

de reducao trelica e empregada para mitigar os efeitos de correlacao do

canal. Mais precisamente, a tecnica LR e aplicada para incrementar a

confiabilidade do sistema MIMO sujeito a canais correlacionados.

2. Quantificar o impacto no desempenho de sistemas MIMO sujeitos a erros

na estimativa do canal e com correlacao espacial, comparando-os ainda ao

desempenho do sistema na ausencia de ambos os efeitos. Adicionalmente e

estudado a complexidade computacional dos detectores com o objetivo de

encontrar o melhor compromissor entre desempenho e complexidade.

3. Nesta parte do trabalho, propoe-se um projeto de precodificacao MIMO

com o objetivo de mitigar o efeito da correlacao usando amostras da matriz

do canal com o objetivo de estimar a matriz de correlacao associada as

multiplas antenas do transmissor e receptor.

Note-se que as duas primeiras partes desses trabalho analisam detectores

MMSE e MMSE auxiliados pela tecnica LR (com e sem OSIC). Adicionalmente

assume-se, nas tres partes deste trabalho, uma ordem de modulacao generica

M -QAM.

1 Introducao 4

O desenvolvimento deste Trabalho de Conclusao de Curso resultou na ela-

boracao de tres artigos tecnico-cientıficos, os quais foram anexados ao final deste

documento, na forma de Apendices A a C:

[A] MIMO Detectors Under Correlated Channels

Autores: Bruno F. Costa; Taufik Abrao

Submetido a revista Semina: Ciencias Exatas e Tecnologicas em Jun-2014

[B] MIMO Detection Under Correlated and Imperfectly Estimated Channels

Autores: Bruno F. Costa; Alex M. Mussi; Taufik Abrao

Submetido a revista Expert Systems with Applications (ESWA), Elsevier

em Set-2014.

[C] MIMO Precoding for Correlated Fading Channels

Autores: Bruno F. Costa; Taufik Abrao

Submetido a revista Journal of Circuits, Systems, and Computers, World

Scientific, em Out-2014.

Esta monografia de conclusao de curso esta dividida em quatro capıtulos.

Alem deste capıtulo introdutorio, no Capıtulo 2 e feita uma breve revisao dos

conceitos basicos e modelo de sistema MIMO adotado na formulacao e analise dos

tres problemas abordados. O Capıtulo 3 discute os tres problemas desenvolvidos,

introduzindo e contextualizando as analises contidas nos Apendices A, B e C.

Finalmente, o Capıtulo 4 traz a principais conclusoes e trabalhos futuros.

5

2 Conceitos Basicos e Modelode Sistema

Este capıtulo faz uma revisao dos principais conceitos empregados na analise

de sistemas MIMO, incluindo o conceito de diversidade, modos de operacao em

sistemas MIMO (ganho de multiplexacao × ganho de diversidade), detectores

MIMO classicos, entre outros.

2.1 Diversidade

Uma das tecnicas com um otimo potencial para mitigar os efeitos do des-

vanecimento e a diversidade que faz uma combinacao dos sinais desvanecidos,

considerando eles estatisticamente independentes em cada um dos canais entre as

antenas, essa combinacao faz com que o desvanecimento resultante seja reduzido

(GOLDSMITH, 2005). A diversidade explora o fato de que sinais estatisticamente

independentes possuem baixa probabilidade de estarem sofrendo um desvaneci-

mento profundo simultaneamente. Portanto, a ideia por tras da diversidade e

transmitir a informacao em multiplos canais estatisticamente independentes. E

possıvel transmitir a mesma informacao nos multiplos canais com o intuito de

aumentar a confiabilidade da transmissao, enquanto que com informacoes dife-

rente em cada canal se aumenta a taxa de transmissao. A tecnica conhecida como

selection combining explora a diversidade espacial ao transmitir o mesmo sinal

em multiplas antenas, separadas adequadamente de forma que os canais sejam

considerados estatisticamente independentes, e no receptor e selecionado a antena

que possua o sinal menos degradado pelos efeitos do canal. Tecnicas de diver-

sidade que mitigam o efeito de desvanecimento multipercurso sao chamada de

microdiversidade. Diversidade que mitiga o efeito do sombreamento de edifıcios

e objetos e chamado macrodiversidade.

Existem mais de uma forma de se obter diversidade, a seguir sera apresentada

algumas delas:

2.1 Diversidade 6

• Diversidade Espacial - Existem algumas formas de alcancar canais inde-

pendentes em um sistema de comunicacao sem fio. O uso de sistemas MIMO

resulta em multiplos canais independentes, desde que o espacamento entre

as multiplas antenas sejam suficiente. Esse tipo de diversidade e conhe-

cida como diversidade espacial. Perceba que o receptor com diversidade

espacial e capaz de usar caminhos independentes sem adicionar potencia

ao sinal transmitido ou aumentar largura de banda. Alem disso, ao fa-

zer uma combinacao coerente dos sinais no receptor e possıvel aumentar a

relacao sinal-ruıdo (SNR - signal-to-noise ratio), o que nao e possıvel com

apenas uma antena. Esse aumento na SNR e chamado de ganho de array.

Com uma divisao apropriada de potencia relativa as antenas transmissoras,

que leve em consideracao os ganhos do canal, esse ganho tambem pode ser

obtido no transmissor.

• Diversidade Temporal - Diversidade temporal e alcancada transmitindo

o mesmo sinal em tempos diferentes, respeitando que a diferenca de tempo

entre as transmissoes sejam maior que tempo de coerencia do canal (o in-

verso do espalhamento Doppler do canal). Assim como a diversidade espa-

cial nao e necessario aumentar a potencia de transmissao para alcancar essa

diversidade, entretanto o uso dessa diversidade reduz a taxa de transmissao.

• Diversidade de Frequencia - Usando uma banda estreita para a trans-

missao de sinais com portadoras usando frequencias diferentes, onde essas

portadoras sao separadas por uma banda de coerencia de canal, e possıvel

alcancar a diversidade por frequencia. Essa tecnica de diversidade neces-

sita de uma potencia de transmissao adicional para transmitir o sinal em

multiplas bandas de frequencia.

• Diversidade de Angulo - Restringindo o Beamwidth1 da antena recep-

tora para um angulo selecionado e possıvel alcancar essa diversidade. No

caso extremo, em que angulo e muito pequeno de forma que apenas um dos

raios de multipercurso entre no seu Beamwidth, isso ira eliminar o desva-

necimento por multipercuso. No entanto para aproveitar a vantagem dessa

tecnica e necessario um numero suficiente de antenas para atingir todas as

possıveis direcoes do sinal que ira chegar ou uma antena com uma diretivi-

dade que consiga selecionar um componente dos multipercursos disponıveis

(de preferencia o mais forte).

1Beamwidth da antena e o angulo que cobre a regiao onde a antena possui metade da potenciade pico de transmissao

2.2 Modo de Multiplexacao 7

• Diversidade de Polarizacao - A diversidade por polarizacao e alcancada

utilizando duas antenas transmissoras ou receptoras polarizadas de forma

diferente (i.e. ondas polarizadas verticalmente e horizontalmente). As duas

ondas transmitidas seguem o mesmo caminho. No entanto, devido as di-

ferentes polarizacoes o canal degrada as ondas de forma idependente, de-

pendendo do cenario a media da potencia recebida no receptor pode ser

diferente para cada polarizacao. A diversidade por polarizacao possui a

desvantagem de existir apenas duas formas de polarizacao o que limita o

seu potencial para dois ramos independentes.

2.2 Modo de Multiplexacao

Em sistemas com diversidade espacial o uso da multiplexacao espacial ( Spa-

cial Multiplexing - SM), permite aumentar a taxa de dados. A ideia dessa mul-

tiplexacao e a transmissao paralela de sinais diferentes nas multiplas antenas de

transmissao. Na multiplexacao espacial nao ha aumento na largura de banda

e nao ha aumento da potencia de transmissao, desde que seja feita uma nor-

malizacao da potencia entre as antenas de transmissao, para diversos esquemas

de deteccao e importante que o numero de antenas transmissoras seja menor ao

numero de antenas receptoras. O modelo do sistema nesse modo e representado

por:

Detector

(a)

Detector

(b)

Figura 2.1: Modelo de sistema MIMO com desvanecimento e ruıdo aditivo.(a) Modo ganho de multiplexacao, (b) Modo ganho de diversidade

A Fig. 2.1 apresenta um modelo simples de sistema MIMO, usando apenas um

demodulador para a deteccao, geralmente e usada uma tecnica de deteccao no

2.3 Taxa de erro de bit, BER 8

sinal recebido (y) e entao o sinal e demodulado. A Fig. 2.1(a) representa o

sistema MIMO no modo ganho por multiplexacao, onde cada antena transmite

um sımbolo diferente, proporcionando um aumento na taxa de transmissao e si-

multaneamente um incremento na complexidade de recepcao. Ja a Fig. 2.1(b)

representa um sistema MIMO no modo ganho por diversidade, onde as nT antenas

transmitem o mesmo sımbolo; com isso, a confiabilidade da transmissao e aumen-

tada ao custo da reducao na taxa de transmissao em relacao aquela alcancada no

modo multiplexacao. A expressao que define o sinal recebido no receptor para

ambos os casos e dada por:

y = Hs + η (2.1)

sendo H ∈ CNr ×Nt a matriz de coeficientes de desvanecimento, tambem chamada

de matriz do canal, o modulo dos elementos dessa matriz segue a distribuicao

estatıstica de Rayleigh e a fase dos elementos seguem uma distribuicao uniforme o

que representa uma comunicacao sem linha de visada (NLOS –non-line-of-sight),

s e o vetor que contem as informacoes dos sımbolos, sendo que para o modo ganho

de diversidade os elementos do vetor s sao todos identicos (s1); finalmente, η e

o vetor de ruıdo aditivo Gaussiano branco (AWGN - Additive White Gaussian

Noise) contendo amostras independentes com distribuicao estatıstica normal de

media zero e variancia σ2n.

2.3 Taxa de erro de bit, BER

A taxa de erro de bit (BER – bit error rate) e uma figura de merito utilizada

na caracterizacao e comparacao de desempenho obtido pelos diferentes detectores

MIMO. Numericamente, atraves de simulacoes computacionais, a BER pode ser

encontrada atraves da seguinte equacao:

BER =nerror

ntotal

(2.2)

sendo nerror o numero de bits detectados errados e ntotal o numero total de bits

transmitidos. O pior cenario e um canal completamente aleatorio, onde a potencia

do ruıdo de fundo e muito superior ao sinal transmitido, neste caso resulta em

uma BER de transmissao igual a 0,5. A BER pode ser analisada usando si-

mulacoes computacionais estocasticas. Neste trabalho a BER e calculada atraves

da simulacao numerica de Monte-Carlo (MCS).

2.4 Modulacao por Amplitude em Quadratura 9

2.4 Modulacao por Amplitude em Quadratura

A modulacao por amplitude em quadratura (QAM – quadrature amplitude

modulation) e um esquema de modulacao digital, composta por duas componentes

na mesma frequencia de RF, uma em fase e a outra em quadratura, i.e., com

defasagem entre tais componentes de 90◦, por isso o nome em quadratura. Cada

componente carrega informacao, sendo utilizado a modulacao por chaveamento de

amplitude (ASK - Amplitude-Shift Keying) em cada uma destas. Neste trabalho,

utilizou-se modulacao digital 4-QAM, sendo 4 possıveis sımbolos, onde cada um

dos sımbolos e mapeado por 2 bits de informacao. Neste mapeamento utilizou-se

codificacao Gray, visando diminuir a variacao de bits entre sımbolos adjacentes

(menor distancia Euclidiana). No caso 4-QAM, isso resulta em um desempenho

semelhante entre a BER e a SER.

2.5 Zero forcing - ZF

O detector ZF e conhecido por cancelar completamente a interferencia entre

as antenas, porem, no seu processo de deteccao, para algumas matrizes mal con-

dicionadas, com rank baixo, o ruıdo de fundo e amplificado. O estimador ZF e

dado por:

Wzf = (HHH)−1HH (2.3)

o sinal recebido apos passar pelo estimador:

(Wzf)y = s + (HHH)−1HHη (2.4)

E possıvel observar que o ruıdo de fundo foi aumentado pelo termo (HHH)−1HH ,

esse termo representa a matriz inversa de Moore-Penrose de H, que para matrizes

com rank deficiente resultara em uma matriz com elementos de altos valores. Esse

detector nao foi usado para gerar resultados nesse trabalho, mas o funcionamento

dele e importante para o entendimento do detector MMSE, que e muito impor-

tante neste trabalho e foi utilizado para gerar todos os resultados. O detector

MMSE utiliza a mesma ideia do ZF, entretanto, ele reduz a amplificacao do ruıdo

de fundo ao considerar um termo proporcional a potencia desse ruıdo.

2.6 Raiz Quadrada de Matriz 10

2.6 Raiz Quadrada de Matriz

A raiz quadrada de uma matriz e a operacao de matriz quadrada dos numeros

escalares estendida para matrizes. Essa funcao para matrizes e mais recorrente,

usada geralmente no contexto de matrizes definidas positivas. Existe uma vari-

edade de metodos computacionais para calcular raiz quadrada de uma matriz,

com seus diferentes criterios de estabilidade. A principal raiz quadrada de uma

matriz e descrita no teorema a seguir:

Teorema 1. Seja A ∈ Cn×n e que nao possua autovalores em R−. Existe uma

raiz quadrada unica X de A com todos os autovalores se situando no semiplano

direito aberto. A matriz X e referida como a principal raiz quadrada de A e se

escreve X = A12 . Se A e real entao A

12 e real.

A prova desse teorema pode ser encontrado em (HIGHAM, 2008), assim como

a representacao por integral da principal raiz de matriz e dada por:

A12 =

π

2A

∫ ∞

0

(t2I + A)−1dt (2.5)

A funcao raiz quadrada de matriz e utilizada nesse trabalho no modelo de cor-

relacao do canal MIMO. Neste modelo, e retirada a raiz quadrada da matriz de

correlacao (R) que pertence ao grupo das matrizes de Toeplitz, ou matrizes de

diagonais constantes, e seus elementos sao reais. A matriz que descreve o canal

correlacionado e definida por:

H =√

R G√

R (2.6)

Onde G ∈ CNr xNt e seus elementos seguem uma distribuicao Gaussiana com-

plexa independente e identicamente distribuıda (i.i.d – independent identically

distributed) com media zero e variancia unitaria. No esquema de precodificacao

proposto nesse trabalho e necessario realizar a operacao de raiz quadrada para

estimar√

R, a realizacao dessa operacao e feita de forma computacional.

11

3 Deteccao MIMO em CanaisCorrelacionados e com Errosde Estimativas

Neste capitulo e contextualizado os artigos realizados ao longo deste trabalho

que estao presente nos Apendices A, B e C. Em todos os trabalhos foram usados

sistemas MIMO no modo ganho de multiplexacao e modulacao 4-QAM.

3.1 Detectores MIMO em Canais Correlaciona-

dos

O trabalho desenvolvido no Apendice A, intitulado Detectores MIMO em Ca-

nais Correlacionados, teve como objetivo a analise do efeito da correlacao entre

as antenas no desempenho dos detectores MIMO. O detector MMSE com e sem

o auxılio da tecnica LR e OSIC foi analisado em canais com diferente nıveis de

correlacao. Neste trabalho foi possıvel observar o grande impacto que o efeito

da correlacao tem no desempenho dos detectores MIMO, especialmente para os

detectores com esquemas lineares (MMSE – MMSE-OSIC), entretanto, os detec-

tores auxiliados pela tecnica reducao de trelica mostraram robustez contra esse

efeito, mesmo para canais fortemente correlacionados, considerando um numero

pequeno de antenas. O detector LR-MMSE-OSIC mostrou o melhor desempenho

entre os detectores sub-otimos.

3.2 Deteccao MIMO em canais correlacionados

e com erros na estimativa

O Apendice B mostra o segundo artigo desenvolvido neste trabalho de con-

clusao de curso. Neste artigo foi realizada uma expansao do trabalho realizado

no primeiro artigo, Apendice A, o efeito do erro na estimacao do canal junta-

mente com o efeito de correlacao entre as antenas. Primeiramente e estudado o

3.3 Precodificacao MIMO para canais correlacionados 12

impacto desses efeitos no desempenho de forma individual, e posteriormente de

forma combinada. Adicionalmente e estudado a complexidade computacional dos

detectores com o objetivo de se determinar o melhor compromisso desempenho-

complexidade. Neste trabalho, e quantizado o impacto dos efeitos de correlacao

e erro na estimacao dos canais e e observado que o efeito de correlacao mostrou

uma degradacao no desempenho mais significativa e novamente o detector LR-

MMSE-OSIC mostrou um desempenho superior aos outros detectores sub-otimos.

Entretanto, a analise da complexidade computacional esse detector mostra uma

sensibilidade ao aumento do nıvel de correlacao o que leva a conclusao que para

cenarios com correlacao baixa e media esse detector alcanca o melhor compromisso

desempenho-complexidade. Para cenarios altamente correlacionados os detecto-

res auxiliados pela tecnica LR alcancaram um desempenho razoavel, porem a sua

complexidade computacional inviabiliza o seu uso na pratica.

3.3 Precodificacao MIMO para canais correla-

cionados

Com o conhecimento do grande impacto do efeito de correlacao no desempe-

nho dos detectores MIMO, atraves do Apendices A e B, no Apendice C e proposto

uma tecnica de precodificacao visando mitigar o efeito da correlacao. Essa tecnica

leva em consideracao a estimacao da matriz de correlacao do canal atraves de um

metodo estatıstico. Neste trabalho, e analisado qual a faixa de numeros de amos-

tras que proporcionam uma estimacao confiavel e qual e o impacto no desempenho

para erros na estimacao da matriz de correlacao. Os resultados numericos mos-

traram que, para uma correlacao baixa e media, o esquema de precodificacao

alcancou um desempenho proximo ao mesmo detector em um cenario sem cor-

relacao. Entretanto, para canais altamente correlacionado o esquema nao foi

capaz de desacoplar de forma eficiente o efeito de correlacao.Porem e importante

salientar que o detector utilizado juntamente com o esquema de precodificacao e

linear e sensıvel ao efeito de correlacao. O esquema de precodificacao se mostrou

robusto a pequenos erros na estimacao do nıvel de correlacao do canal, resul-

tando, para um grande numero de amostras, que o ganho no desempenho se

torna marginal se comparado ao desempenho utilizando um numero pequeno de

amostras.

13

4 Conclusoes

Neste trabalho de TCC foram apresentados resultados do estudo acerca dos

detectores MIMO. Foram estudados os detectores lineares, e os mesmos detectores

auxiliados pela tecnica LR em canais correlacionados e com erros na estimativa do

canal e por fim a ideia de pre-decodificacao do sistema para canais correlacionados

foi analisada para o detector MMSE.

O efeito da correlacao nos canais se mostraram muito significantes para o

desempenho dos sistemas MIMO equipados com detectores sub-otimos com or-

dem polinomial de complexidade computacional. Entre os detectores analisados,

a tecnica de reducao trelica se demonstrou eficiente para melhorar o desempe-

nho desses detectores sub-otimos em canais correlacionados, como tambem em

canais com estimativa imperfeita do canal. Resultados numericos e analises com

os dois efeitos no sistema mostraram que o desempenho, em funcao da BER, para

sistemas equipados com a tecnica LR possui um ganho notavel em termo de per-

formance e robustez. No entanto, os detectores MMSE e MMSE-OSIC mostraram

uma grande degradacao no desempenho para canais fortemente correlacionados,

indicando que detectores ajudados pela tecnica LR podem efetivamente ser uma

alternativa para esse cenario degradado, embora a complexidade introduzida pelo

algoritmo LLL, em cenarios com alta correlacao e alto numeros de antenas, tem

um impacto significante na complexidade computacional geral do sistema. Entre

os detectores auxiliados pela LR, o detector MIMO LR-MMSE OSIC alcancou a

menor degradacao em relacao a combinacao devastadora dos efeitos de correlacao

de canal e erro na estimativa. O resultado encontrado indica que o detector LR-

MMSE-OSIC operando em canais com baixa-media correlacao e com um numero

medio-alto de Tx-Rx antenas e capaz de alcancar o melhor compromisso entre

desempenho e complexidade entre os detectores analisados.

O projeto de pre-decodificacao baseado na informacao da correlacao do canal

se demonstrou vantajoso para melhorar o desempenho do detector MMSE, prin-

cipalmente com um numero pequeno-medio de amostras (10-50). De fato, com

um grande numero de amostras (10k) o ganho no desempenho e apenas marginal,

isso acontece porque o modelo de pre-decoficacao proposto e robusto contra pe-

4.1 Trabalhos Futuros 14

quenos erros na estimacao da correlacao do canal. Entretanto, para um numero

muito pequeno de amostras (≤ 10) a uma correlacao media de canal (ρ ≥ 0.5),

o desempenho do detector pode ser degradado de tal maneira que o resultado

pode ser pior do que sem o uso da pre-decodificacao. Experimentalmente, foi

encontrado uma boa faixa de valores entre 50-100 amostras, o que expressa um

bom compromisso entre o numero de amostras e o desempenho da BER.

4.1 Trabalhos Futuros

Um perspectiva de continuidade de trabalhos futuros inclui uma extensao do

terceiro artigo submetido, Apendice C, adicionando a tecnica de precodificao aos

outros detectores, preferencialmente os estudados no Apendice A e B, bem como

sugere-se incluir e analisar o efeito dos erros nas estimativas do canal sobre o

desempenho do sistema MIMO.

A tecnica de precodificacao amplifica o ruıdo de fundo para canais com alto

nıvel de correlacao, uma outra perspectiva para essa extensao e modificar a tecnica

de precodificacao para reduzir esse efeito de amplificacao, de forma similar aquela

realizada pelo detector MMSE quando comparada ao detector ZF.

Adicionalmente, pode-se sugerir o estudo da complexidade computacional da

tecnica de precodificacao, o que deve gerar uma interessante analise quando incor-

porada a tecnica LR. A tecnica LR alcanca ganhos de desempenho consideraveis

mesmo para cenarios altamente correlacionados, porem sua complexidade compu-

tacional e sensıvel ao nıvel de correlacao e portanto pode resultar excessiva nesses

contextos. No entanto, a tecnica de precodificacao pode prover uma reducao no

nıvel de correlacao no canal. Assim, uma combinacao das duas tecnicas possui

um grande potencial para a obtencao de um detector com otimo compromisso

desempenho-complexidade.

15

Apendice A -- MIMO Detectors Under

Correlated Channels

[ MIMO Detectors Under Correlated Channels

Detectores MIMO em Canais Correlacionados

Bruno Felipe Costa1; Taufik Abrao2

1 Aluno de Graduacao do Departamento de Engenharia Eletrica, Universidade Estadual de Londrina- DEEL-UEL; [email protected]

2 Docente do Departamento de Engenharia Eletrica da Universidade Estadual de Londrina - DEEL-UEL; [email protected]

AbstractThis contribution analyses the performance of multiple-input-multiple-output (MIMO) detectorsunder correlated channels. Two MIMO detection principles, namely minimum mean squared error(MMSE) detector – with and without ordered successive interference cancellation (OSIC) – andthe lattice reduction (LR) technique, are analysed and compared with the maximum likelihood(ML) limit under specific scenarios of interest: (a) increasing spectral efficiency, by increasingnumber of antennas; (b) increasing correlated fading channel indexes. In this context, the MLMIMO detector performance are used as reference in order to compare how the performances ofthose sub-optimal detectors are close to the optimum performance.Key words: multiple-input-multiple-output (MIMO). Lattice Reduction (LR). Minimum MeanSquared Error (MMSE). Ordered Successive Interference Cancellation (OSIC); maximum likeli-hood (ML) detection.

1 IntroductionSystems with multiple transmitting antennas and multiple receiving antennas (MIMO) present a remark-able spectral efficiency and/or are able to improve the performance and reliability of wireless communi-cation by deploying multiples antennas at both transmitter and receiver side [1]. In a spatial multiplexinggain configuration, parallel data streams are transmitted using multiple antennas in order to increase thespectral efficiency at the cost of increasing complexity for data detection at the receiver [2]. The MIMOsystem suffers influences from many effects that can degrade the performance, and consequently reduceits capacity.The knowledge of the channel state information (CSI) is of fundamental importance for ac-ceptable system operation, and it can be used at the receiver, transmitter, or both sides, depending onthe chosen MIMO architecture. Under realistic scenarios, the CSI cannot be perfectly estimated, andtherefore the information available contains errors. The impact of the imperfect CSI estimation over theMIMO precoder design and respective MIMO performance is investigated in [3].

One important effect to be considered in realistic MIMO scenarios is the channel correlation betweenantennas. Some important works have analysed the capacity gain of MIMO systems assuming indepen-dent fading channel, which are in practice difficult to obtain due to the physical constraints of spacingbetween antennas. Among well-established MIMO detectors, the linear zero forcing (ZF) is known bycompletely cancel the interference between antennas [4], at the expense of increasing significantly the

Semina: Ciencias Exatas e Tecnologicas, Londrina, v. , n. , p. 1-11,1

COSTA, B.; ABRAO, T.;

background noise for badly-conditioned channel matrix. At this point, the minimum mean squared error(MMSE) detector can be seen as a better alternative, since it takes into account the noise power duringthe symbol detection process. Besides, the successive interference cancellation (SIC) detector performsthe detection layer-by-layer, using either a ZF or MMSE strategy, and canceling the interference from thepreviously detected symbols [5]. Since errors at the detection of the first layers can be propagated alongthe algorithm, a remarkable improvement on performance can be achieved detecting the most reliableantennas first, which features the ordered SIC (OSIC) MIMO detectors [6], although, all these linearsub-optimum detection techniques present a performance clearly inferior to the maximum likelihood(ML) detector.

Further improvement in MIMO performance-complexity trade-off can be obtained with a pre-processing technique named lattice reduction (LR). Indeed, LR-aided MIMO detection can be deployedin order to achieve MIMO performance improvement while holding computational complexity manage-able. The LR is a mathematical concept deployed to solve many problems involving lattice points. Forinstance, in the MIMO signal detection problem, the LR can be used to improve the channel matrixconditioning, thus allowing the use of simpler detector structures [2]; in other words, less computationalcomplexity is necessary to maintain acceptable performance [7].

The contribution of this work is quantify the impact on the MIMO system performance when lat-tice reduction technique is deployed to mitigate the effects of channel correlation. More precisely, LRtechnique is applied to improve the MIMO detector performance under correlated channels constraintsunder generic QAM modulation order (M ) and number of antennas. Both MMSE and LR-aided MMSE(with and without ordered SIC) detectors are analysed taking into consideration (a) different levels offading channels correlation, ρ; (b) increasing number of transmit and receive antennas, NT and NR,respectively.

2 MIMO System ModelIn this contribution we consider a complex baseband linear transmission system with no line of sight(NLOS) and Nt inputs and Nr outputs corrupted by additive white Gaussian noise (AWGN). The math-ematical model of the system under investigation is

y = Hs + η (1)

where H represents an Nr ×Nt fading coefficients matrix following a Rayleigh distribution representingnon-line-of-sight (NLOS) point-to-point communication, s is a vector of data symbols and η is theindependent white noise vector samples with Gaussian distribution.

2.1 Correlated MIMO ChannelsOne important class of MIMO channel model assumes that the correlation between the transmit antennas(Tx) is independent of the correlation among receive antennas (Rx); hence, admitting a MIMO channelRayleigh flat-fading [8], one can express the fading coefficients matrix:

H =√

RH,RxG√

RH,Tx (2)

where G ∈ CNr xNt is an independent identically distributed (i.i.d.) complex Gaussian zero-mean unitvariance elements. The correlation matrices RH,Tx ∈ RNt xNt and RH,Rx ∈ RNr xNr denote correlation

2Semina: Ciencias Exatas e Tecnologicas, Londrina, v. , n. , p. 1-11,

MIMO DETECTORS UNDER CORRELATED CHANNELS

observed among the transmitter antennas and receiver antennas, respectively. Assuming in this work thatthe Tx and Rx antennas are equally separated, equal numbers of antennas and equal correlation matrixRH,Rx = RH,Tx = RH. Hence, the matrix RH, can be written as:

RH =

1 ρ ρ4 · · · ρ(nT −1)2

ρ 1 ρ · · · ...ρ4 ρ 1 · · · ρ4

......

... . . . ρ

ρ(nT −1)2 · · · ρ4 ρ 1

. (3)

where ρ is the normalized correlation index. Note that a totally uncorrelated scenario means ρ = 0, whilea fully correlated scenario implies ρ = 1.

3 Conventional MIMO DetectorsIn the sequel, a sort of classical MIMO detectors found in the literature are revisited, including theminimum mean squared error (MMSE) criterion, successive interference cancelation method, QR de-composition, as well as lattice reduction (LR) aided detection.

3.1 MMSE MIMO DetectionIn order to reduce the impact of fading and background noise, the MMSE detector employs a linearfilter that can take into account the channel matrix and the noise. The MMSE filter can be found byminimizing the mean-square error (MSE) as [9]:

Wmmse = arg minW

E[∥s − WHy∥2]

Wmmse =(E[yyH ]

)−1 E[ysH ]

Wmmse = H(HHH + N0

EsINt

)−1

(4)

The resulting estimated symbol vector can be written as:

smmse = WHmmsey (5)

3.2 MMSE-SIC MIMO DetectionThe MMSE-SIC MIMO detector is performed from on the decomposition of the channel matrix H andassuming that H is square or tall, where Nt ≤ Nr. Hence, applying for instance the QR factorization onthe channel matrix H:

H = QR (6)

where Q is a Nr × Nr unitary matrix and R is a Nr × Nt upper triangular matrix. Hence, multiplyingQH by the receive signal y we can write:

x = QHyx = Rs + QHη

(7)



where QHη is a zero-mean complex Gaussian random vector. Since QHη and η have the same statisticalproperties, η can be used to denoted QHη.The channel correlation matrix can be either square or tall. Initially, assuming a square Nt × Nt matrixH, we have:

x = Rs + η ⇒

x1

x2...

xNt

=

r1,1 r1,2 · · · r1,Nt

0 r2,2 · · · r2,Nt

...... . . . ...

0 0 · · · rNt,Nt

s1

s2...

sNt

+

η1

η2...

ηNt

(8)

where the xk and ηk denote the k th element of x and η, respectively. Thus, we have

xNt = rNt,NtsNt + ηNt

xNt−1 = rNt−1,NtsNt + rNt−1,Nt−1sNt−1 + ηNt−1...

(9)

Now assuming that the Matrix H is tall (Nt ≤ Nr) we have.

x = Rs + η ⇒

x1

x2...

xNt

xNt+1...

xNr

=

r1,1 r1,2 · · · r1,Nt

0 r2,2 · · · r2,Nt

...... . . . ...

0 0 · · · rNt,Nt

0 0 · · · 0...

......

...0 0 · · · 0

s1

s2...

sNt

+

η1

η2...

ηNt

ηNt+1...

ηNr

(10)Therefore, we have:

xNr = ηNr

...xNt+1 = ηNt+1

xNt = rNt,NtsNt + ηNt

xNt−1 = rNt−1,NtsNt + rNt−1,Nt−1sNt−1 + ηNt−1...

(11)

Since the the received signals {xNt+1, xNt+1, · · · , xNr} do not have any useful information, we cansimply ignore them. So (9) and (11) become the same.

3.2.1 The process of detection in SIC

Firstly, sNt can be detected from xNt as follows:

sNt =xNt

rNt,Nt= sNt +

ηNt

rNt,Nt(12)

Then, the contribution of sNt is canceled in detecting sNt−1 from xNt−1.This sequential detection proce-dure is terminated till all the data symbols of s are detected. The mth symbol of s ,sm can be detected



after canceling Nt − m data symbols as:

um = xm −Nt∑

q=m+1

rm,qsq, m ∈ {1, 2, · · · , Nt − 1} (13)

Using the following modifications the background noise are taken into consideration and then the meansquare error is minimized [9], the modifications are the following definitions:

• Extended channel matrix as Hex =[HT

√N0

EsI]T

;

• Extended receive signal as yex =[yT 0T

]T ;

• Extended noise AWGN as ηex =[ηT −

√N0

EssT]T

;

Therefore the receive signal can be written as:

yex = Hexs + ηex (14)

and the vector of symbols can be found by:

smmse = (Hex)−1yex (15)

3.3 Sorted QR Decomposition (SQRD)

Further performance improvement on the SIC technique can be achieved through a properly ordering[10], [6], which avoids error propagation in interference cancellation. The ordering criterium is theminimization of the H columns norm, which makes the detection be proceeded from the least corruptedsymbol to the most. The form of the decomposition is:

HP = QR (16)

Where the matrix P is a permutation matrix, used to reorder the symbols after applying the SIC de-tection, as seen previously, by multiplying it and the estimated symbol. Therefore, the application ofdecomposition SQRD instead of the QR decomposition changes the detector SIC into OSIC.

4 Lattice Reduction Based detection

The lattice (basis) reduction or LR was elaborated to transform a regular basis to a nearly orthogonalone. Choosing the channel matrix H as a basis for a lattice, the MIMO problem can be treated as alattice decoding problem. The lattice concept is explored in the sequel; after that, the MIMO LR-aideddetector is discussed in details.



4.1 LatticesLet L be a 2 x 2 matrix and u = [u1 u2]

T a 2 x 1 vector. A lattice ΛL is the set of points:

ΛL = {Lu |u1, u2 ∈ Z[i]} (17)

where Z[i] is the set of Gaussian integers. The set of Gaussian integers is the complex numbers α = a+biwhose components a and b are both integers [11]. L is called a generator matrix for the lattice ΛL. Theminimum distance of ΛL is defined as:

d2min(ΛL) = min

u =v∥ L(u − v) ∥2 (18)

where u and v are Gaussian integer vectors. From the definition of ΛL, there are infinitely differentbases in a lattice and they all span the same lattice ΛL. Assume that L′ is another basis for ΛL. So it ispossible relate the two bases by L′ = LZ, where Z is a unimodular matrix; therefore, Z has Gaussianintegers entries and det(Z) ∈ {±1, ±i}. From the definition of d2

min(ΛL) follows that:

d2min(ΛLZ) = d2

min(ΛL) (19)

A matrix U ∈ Mn is said to be unitary if UHU = I, where Mn is the set of matrices n x n [12].

4.2 MIMO system with LatticeConsider a basis B consisting of M real-valued linearly independent basis vectors which is given by

B = {b1,b2, ...,bM}. (20)

Since a lattice can be generated from a integer linear combination of a basis, with B, we can have alattice defined by

Λ ={u|u =

M∑

m=1

bmzm, zm ∈ Z[i]}

. (21)

So adopting H as a basis and s as an vector with Gaussian entries, the y becomes a vector in the latticegenerated by the basis H.

4.3 Lattice Reduction Based MIMO DetectionSince a lattice can be generated by different basis or channel matrices, with the goal to reduce the noiseand interference between multiple signals, it is convenient to find a matrix whose column vectors arenearly orthogonal to generate the same lattice. So the LR can be applied to improve the performance ofthe detection, theses methods are regarded as the LR-based detection for MIMO systems. In order to usethis technique the original constellation must be defined in terms of consecutive integers lattice. Thissymbols is represented by x.Consider two basis H and G that span the same lattice, it is shown that

H = GU, (22)

where U and T = U−1 are unimodular matrix. Then the receive signal can be rewritten as

y = Hx + ηy = HTT−1x + η

y = Gz + η(23)



wherez = Ux = T−1x. (24)

Since the received signal can be treated as the lattice points spanned by the basis, a system with a reducebasis are developed , where conventional low complexity detectors are able to be carried out to detect z.

4.3.1 Linear Detection

The LR-based linear detectors are carried out to detect z as

z = WHy, (25)

Where for the LR-based MMSE detector the linear filter is described by WH = (GHG +N0

EsU−HU−1)−1GH [9].

4.3.2 Shift and Scale Method

After the detection of z, it is necessary some operations to estimate the correct symbols (s). The solutionto this quantization problem is to use a combination of shifting and scaling operations to ensure that boththe original (x) and reduced constellation (z) are defined in terms of consecutive integer lattices [13]. Inthis scheme the conventional symbols are related to a consecutive integer lattice by:

xi = αsi + β (26)

where si is a complex integer , α is a scalar and β a complex offset. The symbol vector can be representas

x = αs + β1 (27)

where s is a vector of symbols in the complex integer lattice and 1 is a vector of ones. From (24) we canwrite

T−1x = T−1(αs + β1)z = αs + βρ

(28)

Where s = T−1s and ρ = T−11, in the normalized lattice:

z′ =

(2

α

)z = 2s + β′ρ (29)

Where β′ =(

2βα

)and has the value 1 + j for all M-QAM constellations. Isolating the s we have

s =1

2(z′ − β′ρ) (30)

and symbol estimated in the reduced lattice should be based on quantization in the consecutive integerlattice

ˆs =

⌈1

2(z′ − β′ρ)

⌋(31)

Where ⌈·⌋ represent rounding to the nearest integer. From (29) the estimate in the reduced lattice is

z′ = 2

⌈1

2(s′ − β′ρ)

⌋+ β′ρ (32)

This combinations of operations lead to a correct solution to the quantization problem.



5 Numeric ResultsIn this section, the BER versus SNR under perfect channel estimation is analysed; further realistic MIMOdetection performance analyses has been conducted considering. Along this section, numerical Monte-Carlo simulations (MCS) results are depicted and examined. Two different system and channel config-urations have been considered: a) Non-correlated channels; b) Different levels of channel correlation(ρ > 0).

Fig. 1 depicts the bit error rate (BER) performance of different detectors for a 4-QAM MIMO systemwith Nt = Nr = 2 and Nt = Nr = 4 antennas with perfect channel estimation under channels withno correlation (ρ = 0) respectively. In Fig. 1.a, the performance of MMSE-OSIC detector shows aslightly improvement compared with the MMSE detector, although this detector also presents slightlyhigher complexity; however, the performance of theses detectors, due to the noise enhancement, is poorin comparison to ML. In contrast, the LR-aided MIMO detectors clearly outperform the others MIMOdetectors, presenting higher order of diversity than the linear detectors, remarkable gain on the perfor-mance is achieved in the high SNR region, indicating that the LR technique is robust against the fadingeffect specially in this region, although the performance in low SNR region has proved worse, speciallywith the increase of the numbers of antennas. The LR-based MMSE-OSIC detector achieves near-MLperformance specially in the medium and high SNR regions, where the additive noise is manageable andnegligible, respectively.

Fig. 2 and 3 show the bit error rate (BER) performance of the detectors for a 4-QAM MIMO systemwith Nt = Nr = 2 and Nt = Nr = 4 antennas, respectively, and perfect channel estimation, butunder three different levels of channel correlation, ρ = [0 0.2 0.7 0.9].The performance of the MIMOdetectors under channels weakly correlated (ρ = 0.2) results very close to the uncorrelated channelcondition, although with the increase of the levels of correlation implies in an increasing degradationin the BER performance. The MIMO detection with the aid of the lattice technique has demonstratedto be robust against interference between antennas, preserving the same order of diversity. In stronglycorrelated channels with 2 × 2 antennas, the detectors ML and LR-aided MMSE-OSIC in high SNRregion present very similar performance; the same behavior happens with MMSE and MMSE-OSICdetectors. Under same channel and system scenario but with 4 × 4 antennas, one can note that withthe increase of the level of correlation provokes a larger degradation in the performance, specially forthe MMSE and MMSE-OSIC detectors, which under strongly correlated channels present a really poorperformance, decreasing your order of diversity. As a conclusion, the ML-MIMO performance for anynumber of antennas presents a slightly better performance than the LR-based MMSE-OSIC detector, butwith the same diversity order, indicated by the slope of BER performance curve, determined in high SNRregion.

6 ConclusionThe effects of the correlated channels have been demonstrated itself very significant on the MIMO per-formance. Among the analysed MIMO detectors, the LR-aided technique has been proved useful in orderto improve the performance of several MIMO detectors under correlated channel estimation constrains.Indeed, our numerical results and analysis of correlated channel effects over the BER performance ofMIMO system equipped with different detectors have indicated notable gains in performance and in ro-bustness of those versions of MIMO detectors aided by lattice reduction. The MMSE and MMSE-OSICshowed a really poor performance in channels strongly correlated indicating that detectors aided by lat-



0 5 10 15 20 2510

−4

10−3

10−2

10−1

SNR (dB)

BE

R

(a)

MMSE

MMSE−OSIC

LR−MMSE

LR−MMSE−OSIC

ML

0 5 10 15 20 2510

−4

10−3

10−2

10−1

SNR (dB)

BE

R

(b)

MMSE

MMSE−OSIC

LR−MMSE

LR−MMSE−OSIC

ML

Figure 1: BER performance for the MIMO detectors under 4−QAM , perfect channel estimation anduncorrelated channels for a) 2 × 2 antennas; b) 4 × 4 antennas.

tice reduction technique can be effectively an alternative to this scenario, regarding your characteristicto be robust against the effect of correlation. Among them, the LR-MMSE MIMO OSIC detector hasachieved the smaller degradation regarding the devastating effect of the MIMO channel correlation.

References

[1] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environmentwhen using multiple antennas,” Wireless Personal Communications, vol. 6, pp. 311–335, 1998.

[2] D. Wubben, D. Seethaler, J. Jalden, and G. Matz, “Lattice reduction - a survey with applications inwireless communications,” IEEE Signal Processing Magazine, vol. 28, no. 3, pp. 70–91, 2011.

[3] Z. Andalibi, H. H. Nguyen, and J. E. Salt, “Precoder design for bicm-mimo systems under channelestimation error.” Wireless Personal Communications, vol. 72, no. 4, pp. 2823–2835. [Online].Available: http://dblp.uni-trier.de/db/journals/wpc/wpc72.html



0 10 20

10−3

10−2

10−1

SNR (dB)

ρ= 0

0 10 20

10−3

10−2

10−1

SNR (dB)

ρ= 0.2

0 10 20 30

10−3

10−2

10−1

SNR (dB)

ρ= 0.7

0 20 40

10−3

10−2

10−1

SNR (dB)

ρ= 0.9

MMSE

MMSE−OSIC

LR−MMSE

LR−MMSE−OSIC

ML

MMSE

MMSE −OSIC

LR−MMSE

LR−MMSE−OSIC

ML

MMSE

MMSE−OSIC

LR−MMSE

LR−MMSE−OSIC

ML

MMSE

MMSE−OSIC

LR−MMSE

LR−MMSE−OSIC

ML

Figure 2: BER performance for the MIMO detectors under correlated channels with 2 × 2 antennas.

[4] D. Wubben, R. Bohnke, V. Kuhn, and K.-D. Kammeyer, “Near-maximum-likelihood detection ofmimo systems using mmse-based lattice reduction,” in IEEE International Conference on Commu-nications, vol. 2, june 2004, pp. 798 – 802 Vol.2.

[5] R. Bohnke, D. Wubben, V. Kuhn, and K.-D. Kammeyer, “Reduced complexity mmse detection forblast architectures.” in GLOBECOM. IEEE, pp. 2258–2262.

[6] D. Wubben, R. Bohnke, V. Kuhn, and K.-D. Kammeyer, “MMSE Extension of V-BLAST basedon Sorted QR Decomposition,” in IEEE Semiannual Vehicular Technology Conference (VTC2003-Fall), Orlando, Florida, USA, Oct. 2003.

[7] Y. M. Mostagi and T. Abrao., “Lattice-reduction-aided over guided search mimo detectors,” Inter-national Journal of Satellite Communications Policy and Management, pp. 142–154, 2012.

[8] A. V. Zelst and J. S. Hammerschmidt, “A single coefficient spatial correlation model for multiple-input multiple-output (mimo) radio channels,” in in Proc. URSI XXVIIth General Assembly, 2002,pp. 1–4.

[9] L. Bai. and J. Choi., Low complexity MIMO Detection. Springer, 2002.



0 5 10 15 20 2510

−4

10−3

10−2

10−1

SNR (dB)

BE

R

ρ= 0.2

0 10 20 3010

−4

10−3

10−2

10−1

SNR (dB)

ρ= 0.5

0 10 20 30 4010

−4

10−3

10−2

10−1

SNR (dB)

ρ= 0.7

0 20 4010

−4

10−3

10−2

10−1

SNR (dB)

ρ= 0.9

MMSE

MMSE −OSIC

LR − MMSE

LR − MMSE− OSIC

ML

MMSE

MMSE − OSIC

LR − MMSE

LR − MMSE−OSIC

ML

MMSE

MMSE−OSIC

LR−MMSE

LR − MMSE − OSIC

ML

MMSE

MMSE − OSIC

LR − MMSE


ML


[10] D. Wubben, R. Bohnke, J. Rinas, K.-D. Kammeyer, and V. Kuhn, “Efficient algorithm for decodinglayered space-time codes,” IEE Electronic Letters, vol. 37, no. 22, pp. 1348–1350, Nov 2001.

[11] G. James, Modern Engineering Mathematics, 4th ed. New York, NY, USA: Prentice Hall, 2010.

[12] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge University Press, 1985.

[13] D. Milford and M. Sandell, “Simplified quantisation in a reduced-lattice mimo decoder,” Commu-nications Letters, IEEE, vol. 15, no. 7, pp. 725–727, July 2011.


27

Apendice B -- MIMO Detection Under

Correlated and Imperfectly Estimated

Channels

MIMO Detection Under Correlated and Imperfectly Estimated Channels

Bruno Felipe Costa, Alex Miyamoto Mussi, Taufik Abrão∗

Department of Electrical Engineering, State University of Londrina, Londrina, PR, Brazil.

Abstract

This contribution analyses the performance of multiple-input-multiple-output (MIMO) detectors under correlatedchannels and imperfect coefficients channel estimation. Two MIMO detection principles, namely minimum meansquared error (MMSE) detector – with and without ordered successive interference cancellation (OSIC) – and thelattice reduction (LR) technique, are analyzed. The performance-complexity trade-off of these MIMO detectors hasbeen analysed and compared with the maximum likelihood (ML) limit under specific practical scenarios of interest:(a) high spectral efficiency; (b) channel error estimates; and (c) channel/antenna correlation; (d) combining channelerrors and correlated channels. In this context, the optimum ML MIMO detector performance is deployed as referenceaiming to evaluate the performance degradation of those sub-optimal detectors.

Keywords: multiple-input-multiple-output (MIMO), Complexity, Lattice Reduction (LR), Minimum Mean SquaredError (MMSE), Ordered Successive Interference Cancellation (OSIC), Maximum likelihood (ML)

1. Introduction

Systems with multiple transmitting antennas andmultiple receiving antennas (MIMO) present a remark-able spectral efficiency and/or are able to improve theperformance and reliability of wireless communicationby deploying multiples antennas at both transmitter andreceiver side Foschini and Gans (1998). In a spatialmultiplexing gain configuration, parallel data streamsare transmitted using multiple antennas in order to in-crease the spectral efficiency at the cost of increasingcomplexity for data detection at the receiver Wübbenet al. (2011). The MIMO system suffers influences frommany effects that can degrade the performance, and con-sequently reduce its capacity. The knowledge of thechannel state information (CSI) is of fundamental im-portance for acceptable system operation, and it can beused at the receiver, transmitter, or both sides, depend-ing on the chosen MIMO architecture. Under realisticscenarios, the CSI cannot be perfectly estimated, andtherefore the information available contains errors. Theimpact of the imperfect CSI estimation over the MIMO

∗Corresponding authorEmail addresses: [email protected] (Bruno

Felipe Costa), [email protected] (Alex Miyamoto Mussi),[email protected] http://www.uel.br/pessoal/taufik(Taufik Abrão)

precoder design and respective MIMO performance isinvestigated in Andalibi et al. (2013) and Yeh et al.(2011).

Another important effect to be considered in realis-tic MIMO scenarios is the channel correlation betweenantennas. The correlation between antennas has a greatimpact on the bit error rate (BER) performance, spe-cially when linear structures of detection are deployedValente et al. (2014). Some important works have anal-ysed the capacity gain of MIMO systems assuming in-dependent fading channel, which are in practice diffi-cult to obtain due to physical constraints (spacing be-tween antennas), or specially in scenarios with a mas-sive numbers of antennas, where it is physically difficultto design antenna array with enough spacing betweenantenna elements in order to guarantee no correlation.For instance, recently, the effect of antenna correlationon the performance of massive MIMO systems has beeninvestigated in Fang et al. (2013).

Among well-established MIMO detectors, the linearzero forcing (ZF) is known by completely cancel the in-terference between antennas Wubben et al. (2004), atthe expense of increasing significantly the backgroundnoise for badly-conditioned channel matrix. At thispoint, the minimum mean squared error (MMSE) de-tector can be seen as a better alternative, since it takesinto account the noise power during the symbol detec-

Preprint submitted to ESWA-7875 - Expert Systems With Applications November 24, 2014

tion process. Besides, the successive interference can-cellation (SIC) detector performs the detection layer-by-layer, using either a ZF or MMSE strategy, and cancel-ing the interference from the previously detected sym-bols Böhnke et al. (2003). Since errors at the detec-tion of the first layers can be propagated along the algo-rithm, a remarkable improvement on performance canbe achieved detecting the most reliable antennas first,which features the ordered SIC (OSIC) MIMO detec-tors Wübben et al. (2003), although, all these linear sub-optimum detection techniques present a performanceclearly inferior to the maximum likelihood (ML) detec-tor.

Further improvement in MIMO performance-complexity trade-off can be obtained with a pre-processing technique named lattice reduction (LR).LR-aided MIMO detection can be deployed in orderto achieve MIMO performance improvement whilecomputational complexity remains manageable. TheLR is a mathematical concept deployed to solve manyproblems involving lattice points. For instance, in theMIMO signal detection problem, the LR can be used toimprove the channel matrix conditioning, thus allowingthe use of simpler detector structures Wübben et al.(2011); in other words, reduction in computationalcomplexity can be achieved with the LR techniqueaggregation while maintain overall system performanceacceptable Mostagi and Abrão. (2012). In Park andChun (2012) it is shown that the LR-aided SIC detectorsoutperform the linear detectors on the scenario withchannel estimation error, while in Chen et al. (2012) isdemonstrated the LR-aided-MMSE detector operatingunder spatial correlated MIMO channels and imperfectCSI is able to achieve promising near-optimum MIMOperformance.

The contribution of this work consists in quantify theimpact on the MIMO system performance when latticereduction technique is deployed to mitigate the effectsof channel correlation. More precisely, LR techniqueis applied to improve the MIMO detector performanceunder correlated and imperfect channel estimates un-der generic M-QAM modulation order and number oftransmit and receive antennas. Both MMSE and LR-aided MMSE (with and without ordered SIC) detectorsare analysed taking into consideration:

a) different numbers of transmit and receive antennas,NT and NR, respectively;

b) different levels of correlation antennas ρ;c) different levels of channel coefficient error esti-

mate, ε;d) imperfect CSI knowledge under correlated chan-

nels.

2. MIMO System Model

In this contribution we consider a complex base-band linear transmission system with no line of sight(NLOS) and Nt inputs and Nr outputs corrupted by addi-tive white Gaussian noise (AWGN). The mathematicalmodel of the system under investigation is

y = Hs + η (1)

where H represents an Nr×Nt fading coefficients matrixfollowing a Rayleigh distribution representing non-line-of-sight (NLOS) point-to-point communication, s is avector of data symbols and η is the independent whitenoise vector samples with Gaussian distribution.

2.1. Correlated MIMO Channels

One important class of MIMO channel model as-sumes that the correlation between the transmit anten-nas (Tx) is independent of the correlation among re-ceive antennas (Rx); hence, admitting a MIMO channelRayleigh flat-fading Zelst and Hammerschmidt (2002),one can express the fading coefficients matrix:

H =√

RH,RxG√

RH,Tx (2)

where G ∈ CNr x Nt is an independent identically dis-tributed (i.i.d.) complex Gaussian zero-mean unit vari-ance elements. The correlation matrices RH,Tx ∈ RNt x Ntand RH,Rx ∈ RNr x Nr denote correlation observed amongthe transmitter antennas and receiver antennas, respec-tively. Assuming in this work that the Tx and Rx an-tennas are equally separated, equal numbers of anten-nas and equal correlation matrix RH,Rx = RH,Tx = RH.Hence, the matrix RH, can be written as:

RH =

1 ρ ρ4 · · · ρ(nT−1)2

ρ 1 ρ · · · ...ρ4 ρ 1 · · · ρ4

......

.... . . ρ

ρ(nT−1)2 · · · ρ4 ρ 1

, (3)

where ρ is the normalized correlation index. Note that atotally uncorrelated scenario means ρ = 0, while a fullycorrelated scenario implies ρ = 1.

2

2.2. MIMO Channel Error Estimation

The estimation of the Gaussian channel matrix in thereceiver is not perfect, therefore is necessary considerthis in the modulation of the channel. The estimatedGaussian channel matrix H can be simply modeled as:

H = H + ε (4)

where ε is the stochastic complex error, assumed i.i.d.with ε ∼ N{0, σ2

ϵ }.This estimation must be normalizedby (1+σ2

ϵ )−1 Wang et al. (2008); therefore the estimated

matrix MIMO becomes:

H =H + ε1 + σ2

ε

(5)

This normalization is necessary to not change theoriginal channel characteristics. Besides, in this mod-elling, the variance σ2

ε is assumed proportional to theSNR, where the value of ε denotes the percentage inrelation to SNR.

3. Conventional MIMO Detectors

In the sequel, classical MIMO detectors found inthe literature are revisited, including the minimummean squared error (MMSE) criterion, successive in-terference cancellation method, and QR decomposition-based MIMO detectors.

3.1. MMSE MIMO Detection

In order to reduce the impact of fading and back-ground noise, the MMSE detector employ a linear fil-ter that can take into account the channel matrix and thebackground noise effect as well. The MMSE filter canbe found by minimizing the mean-square error (MSE)as Bai. and Choi. (2002):

Wmmse = arg minW

E[∥s −WHy∥2]

Wmmse =(E[yyH]

)−1E[ysH]

Wmmse = H(HHH + N0

EsINr

)−1

(6)

where Es denotes the symbol energy, N0 is the noisespectral density, I is a Nt x Nt identity matrix and E[·]is the statistical expectation operator. The resulting es-timated symbol vector can be written as:

smmse =WHmmsey (7)

3.2. MMSE-SIC MIMO Detection

The MMSE-SIC MIMO detector is performed fromthe decomposition of the channel matrix H and assum-ing that H is square or tall, where Nt ≤ Nr. Hence, ap-plying for instance the QR factorization on the channelmatrix H:

H = QR (8)

where Q is a Nr × Nr unitary matrix and R is a Nr × Nt

upper triangular matrix. Hence, multiplying QH by thereceive signal y we can write:

x = QHyx = Rs +QHη

(9)

where η = QHη is a zero-mean complex Gaussian ran-dom vector. Notice that QHη and η share the same sta-tistical properties.

The channel correlation matrix can be either squareor tall. Initially, assuming a square Nt ×Nt matriz H, wehave:

x = Rs + η

x1x2...

xNt

=

r1,1 r1,2 · · · r1,Nt

0 r2,2 · · · r2,Nt

....... . .

...0 0 · · · rNt ,Nt

s1s2...

sNt

+

η1η2...ηNt

(10)

Thus, we can write the recurrent equations in order todetermine the soft decision xNt to x1:

xNt = rNt ,Nt sNt + ηNt

xNt−1 = rNt−1,Nt sNt + rNt−1,Nt−1sNt−1 + ηNt−1...

...

(11)

Now assuming that the matrix H is tall with Nr × Nt

dimension and Nr ≥ Nt, we have

x1x2...

xNt

xNt+1...

xNr

=

r1,1 r1,2 · · · r1,Nt

0 r2,2 · · · r2,Nt

....... . .

...0 0 · · · rNt ,Nt

0 0 · · · 0...

......

...0 0 · · · 0

s1s2...

sNt

+

η1η2...ηNt

ηNt+1...ηNr

(12)Therefore, the soft decision xNr to x1 can be recursively

3

determined as:

xNr = ηNr

...xNt+1 = ηNt+1xNt = rNt ,Nt sNt + ηNt

xNt−1 = rNt−1,Nt sNt + rNt−1,Nt−1sNt−1 + ηNt−1...

(13)

Since the received signals xNt+1, xNt+2, · · · , xNr do nothave any useful information, we can simply ignorethem. Hence, eq. (11) and (13) are equivalents.

3.2.1. Successive Interference Cancellation (SIC) StepFirstly, sNt can be detected from xNt as follows:

Let sNt =xNt

rNt ,Nt

= sNt +ηNt

rNt ,Nt

(14)

Then, the contribution of sNt is to be canceled in de-tecting sNt−1 from xNt−1.This sequential detection pro-cedure is terminated until all the data symbols of s aredetected. The mth symbol of s can be detected after can-celing M − m data symbols as:

um = xm −Nt∑

q=m+1

rm,q sq,

sm =um

rm,m, m ∈ {1, 2, · · · ,Nt − 1} (15)

Finally, the background noise can be taken into ac-count in order to minimize the mean square error Bai.and Choi. (2002).

• Extended channel matrix: Hex =

[HT

√N0Es

I]T

;

• Extended receive signal: yex =[yT 0T

]T;

• Extended noise AWGN: ηex =

[ηT −

√N0Es

sT]T

.

Therefore the receive signal can be written as:

yex = Hexs + ηex (16)

and the vector of symbols can be found by:

smmse = (Hex)−1yex (17)

3.3. Sorted QR Decomposition (SQRD)Further performance improvement on the MIMO

MMSE-SIC technique can be achieved through a prop-erly ordering Wübben et al. (2001), Wübben et al.(2003), which avoid error propagation in the interfer-ence cancellation step. The ordering criterion is theminimization of the H columns norm, which makes thedetection be proceeded from the least noise corruptedsymbol to the most. The form of the decomposition issimply:

HP = QR (18)

where matrix P is a permutation matrix, used to reorderthe symbols after applying the SIC detection, eq. (15),by multiplying it and the estimated symbol. There-fore the application of decomposition SQRD insteadof the QR decomposition change the detector SIC intoOSIC.The sorted QR decomposition is summarized bythe pseudo-code in Algorithm 1.

Algorithm 1 Sorted QR decompositionRequire: Q = H,R = 0,P = InT

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

j=i to nT

∥q j∥2

3: exchange columns i and k in Q, R and P4: ri j = |qi|5: qi = qi/rii

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

i q j

8: q j = q j − ri jqi

9: end for10: end for

4. Lattice Reduction Based detection

The lattice (basis) reduction or LR was elaborated totransform a regular basis to a nearly orthogonal one.Choosing the channel matrix H as a basis for a lattice,the MIMO problem can be treated as a lattice decodingproblem. The lattice concept is explored in the sequel;after that, the MIMO LR-aided detector is discussed indetails.

4.1. LatticesLet L be a 2 x 2 matrix and u = [u1 u2]T a 2 x 1

vector. A lattice ΛL is the set of points:

ΛL = {Lu | u1, u2 ∈ Z[i]} (19)

where Z[i] is the set of Gaussian integers. The set ofGaussian integers is the complex numbers α = a + bi

4

whose components a and b are both integers James(2010). L is called a generator matrix for the latticeΛL. The minimum distance of ΛL is defined as:

d2min(ΛL) = min

u,v∥ L(u − v) ∥2 (20)

where u and v are Gaussian integer vectors. From thedefinition of ΛL are infinitely different bases in a latticeand they all span the same lattice ΛL. Assume that L′is another basis for ΛL. So it is possible relate the twobases by L′ = LZ, where Z is a unimodular matrix;therefore, Z has Gaussian integers entries and det(Z) ∈{±1,±i}. From the definition of d2

min(ΛL) follows that:

d2min(ΛLZ) = d2

min(ΛL) (21)

A matrix U ∈ Mn is said to be unitary if UHU = I, whereMn is the set of matrices n×n Horn and Johnson (1985).

4.2. MIMO System with LatticeLet the basis B consisting of M real-valued linearly

independent basis vectors given by

B = {b1,b2, ..., bM}. (22)

Since a lattice can be generated from an integer linearcombination of a basis, with B, we can have a latticedefined by

Λ ={u|u =

M∑

m=1

bmzm, zm ∈ Z[i]}. (23)

Hence, adopting H as a basis and s to produce an integerlinear combination of the basis, the y becomes a vectorin the lattice generated by the basis H.

4.3. The Lenstra - Lenstra - Lovász (LLL) AlgorithmA well-know and powerful algorithm for reduction is

the LLL algorithm, proposed by Lenstra, Lenstra andLovász in 1982 Lenstra and Lenstra (1982). A basis Awhich can be decomposed by a QR decomposition asA = QR is named reduced LLL with parameter δ ( 1

4 <δ ≤ 1) if the following inequalities hold:

|rl,k | ≤ 12|rl,l| for 1 ≤ l < k ≤ m,

andδr2

k−1,k−1 ≤ r2k,k + r2

k−1,k for k = 2, ...,m. (24)

If only the first inequality in (24) is fulfilled, the basisis called size reduced. The parameter δ influences thequality of the reduced basis. In this paper, we will as-sume δ = 3

4 as proposed in Lenstra and Lenstra (1982).In this work, the adopted LLL algorithm is based onWübben et al. (2004) and represented by the pseudo-code in Algorithm 2.

4.3.1. Computational Complexity of LLL AlgorithmThe complexity of LLL algorithm depends on the ma-

trix size and also the correlation index, ρ. In Kobayashiet al. (2014) the complexity of this algorithm was evalu-ated by numerical experiment, considering Nt = Nr = Nand δ = 3

4 . The function that describe the complexity innumbers of flops has been obtained by fitting, and givenby:

fLLL(N, ρ) = (aebρ + c)N3 (25)

where a = 5.018 × 10−4, b = 13.48 and c = 8.396. It isworth noting that the computational complexity cost forthe LLL algorithm increases substantially under large-array configurations and medium-high correlation index(ρ ≥ 0.5).

Algorithm 2 The LLL algorithmRequire: Q,R,PEnsure: Q, R,T

1: Initialization: Q := Q, R := R,T := P2: k = 23: while k ≤ m do4: for l = k-1,...,1 do5: µ = ⌈R(l, k)/R(l, l)⌋.6: if µ , 0 then7: R(1 : l, k) := R(1 : l, k) − µR(1 : l, l)8: T(:, k) := T(:, k) − µT(:, l)9: end if

10: end for11: if δR(k−1, k−1)2 > R(k, k)2 + R(k−1, k)2 then12: Swap columns k − 1 and k in R and T13: Calculate Givens rotation matrix Θ such that

elements R(k, k − 1) becomes zero:

Θ =

[α β−β α

]with α = R(k−1,k−1)

∥R(k−1:k,k−1)∥ and

β = R(k,k−1)∥R(k−1:k,k−1)∥

14: R(k−1 : k, k−1 : m) := ΘR(k−1 : k, k−1 : m)15: Q(:, k − 1 : k) := Q(:, k − 1 : k)ΘT

16: k := max{k − 1, 2}17: else18: k := k + 119: end if20: end while

4.4. Lattice Reduction Based MIMO Detection

Since a lattice can be generated by different basis orchannel matrices, with the goal of reducing the noiseand interference between multiple signals, it is conve-nient to find a matrix whose column are nearly orthog-onal to generate the same lattice. Hence, LR technique

5

can be applied to improve the MIMO detection perfor-mance; these methods are regarded as the LR-based de-tection for MIMO systems.

In order to deploy LR technique, the original constel-lation must be defined in terms of consecutive integerslattice. Consider two basis H and G that span the samelattice; it also shown that

H = GU, (26)

where U and T = U−1 are a unimodular matrices. As aconsequence the received signal in (1) can be rewrittenas:

y = Hs + η = HTT−1s + η = Gz + η, (27)

where z = T−1s = Us. (28)

Since the received signal can be treated as the latticepoints spanned by the basis, lattice-based MIMO sys-tem detection can be developed aiming to reduce thecomplexity of conventional detectors. Hence, under lat-tice transform, z is initially detected; after that, the orig-inal symbols s can be obtained applying a combinationof shifting and scaling operations, eq. (31), as describedin the sequel, in order to correctly estimate z and thende-mapping deploying (28).

4.4.1. LR-aided Linear DetectorsThe LR-based MIMO linear detection can be carried

out initially detecting z by applying linear filtering inthe received signal vector; disregarding the backgroundnoise we have:

z =WHy, (29)

where for the LR-based MMSE detector the linear filteris described by

WH =

(GHG +

N0

EsU−HU−1

)−1

GH (30)

4.4.2. Shift and Scale MethodAfter computing soft decisions on z it is necessary

to estimate the correct symbols. The solution to thisquantization problem is to use a combination of shiftingand scaling operations to ensure that both the original sand the reduced constellation z are defined in terms ofconsecutive integer lattices Milford and Sandell (2011).The shifting and scaling operations is given by:

z = 2⌈12

(z − β′ι)⌋+ β′ι (31)

where ⌈·⌋ represent rounding to the nearest integer.Hence, the original symbols s can be obtained applying

(31) and then de-mapping in (28). The steps to achieveshifting and scaling equation (31) is explained in the fol-lowing.

In the LR-aided scheme the constellation symbols arerelated to a consecutive integer lattice x by:

si = αxi + β (32)

where si is a complex integer , α is a scalar and β acomplex offset. The symbol vector can be represent as

s = αx + β1 (33)

where s is a vector of symbols in the complex integerlattice and 1 is a vector of ones. Another equivalentrepresentation for s is normalizing by 2

α:

s = 2x + β′1 (34)

where β′ =(

2βα

), assuming the value 1 + j for all M-

QAM constellations. From (28) we can write

T−1s = T−1(2x + β′1)z = 2x + β′ι (35)

where x = T−1x and the row-sum vector ι = T−11. Iso-lating the x we have

x =12

(z − β′ι) (36)

and symbol estimate in the reduced lattice should bebased on quantization in the consecutive integer latticebasis

x =⌈

12

(z − β′ι)⌋

(37)

Finally, from (35) the estimate in the reduced lattice fol-lows:

z = 2⌈12

(z − β′ι)⌋+ β′ι (38)

The core of LR-aided MMSE MIMO detection pro-cess is described by the pseudo-code shown in Algo-rithm 3.

Algorithm 3 LR-MMSE MIMO detectorRequire: H, yEnsure: s

1: [G,T] = LLL(H)2: WH =

(GHG + N0

EsTHT

)−1GH

3: z =WHy

4: shifting and scaling: z = 2⌈

12 (z − β′ι)

⌋+ β′ι

5: symbol estimation: s = Tz

6

5. Numeric Results

In this section performance in terms of BER ver-sus SNR under perfect and channel error estimatesis analysed; further realistic MIMO detection perfor-mance analysis has been conducted considering scenar-ios with correlated channels. Along this section, nu-merical Monte-Carlo simulations (MCS) results are de-picted and examined. Different system and channel con-figuration scenarios have been considered:

a) perfect channel state information (CSI) knowledgeavailable at receiver side, but not at the transmitterside, and uncorrelated antennas;

b) perfect channel state information and different lev-els of antenna correlation (ρ > 0);

c) imperfect CSI knowledge under uncorrelated chan-nels;

d) imperfect CSI knowledge under correlated chan-nels.

Besides, 4-QAM modulation and Gray coding havebeen deployed across this section; hence, the analysedBER and SER in this section are equivalent.

Fig. 1 depicts the BER performance of different de-tectors for a 4-QAM MIMO system with Nt = Nr = 2and Nt = Nr = 4 antennas with perfect channel estima-tion (ε = 0) under channels with no correlation (ρ = 0)respectively. In Fig. 1.a, the performance of MMSE-OSIC detector shows a slightly improvement comparedwith the MMSE detector, although this detector alsopresents slightly higher complexity; however, the per-formance of theses detectors, due to the noise enhance-ment, is poor in comparison to ML. In contrast, theLR-aided MIMO detectors clearly outperform the oth-ers MIMO detectors, remarkable in the high SNR re-gion, indicating that the LR technique is robust againthe fading effect specially in this region, although yourperformance in low SNR region has proved worse, spe-cially with the increase of the numbers of antennas. TheLR-based MMSE-OSIC detector achieves near-ML per-formance, with the same order of diversity, specially inthe medium and high SNR regions, where the additivenoise is manageable and negligible, respectively.

Fig. 2 and 3 show the BER performance of the de-tectors for a 4-QAM MIMO system with Nt = Nr = 2and Nt = Nr = 4 antennas, respectively, and perfectchannel estimation (ε = 0), but considering three differ-ent levels of channel correlation, ρ = [0; 0.2; 0.7; 0.9].The performance of the MIMO detectors under chan-nels weakly correlated (ρ = 0.2) results very close to theuncorrelated channel condition, although with the in-crease of the levels of correlation implies in an increas-ing degradation in the BER performance. The MIMO

0 5 10 15 20 2510

−4

10−3

10−2

10−1

SNR (dB)

BE

R

(a)

MMSE

MMSE−OSIC

LR−MMSE

LR−MMSE−OSIC

ML

0 5 10 15 20 2510

−4

10−3

10−2

10−1

SNR (dB)

BE

R

(b)

MMSE

MMSE−OSIC

LR−MMSE

LR−MMSE−OSIC

ML

Figure 1: BER performance for the MIMO detectors under 4−QAM,perfect channel estimation and uncorrelated channels for a) 2 × 2 an-tennas; b) 4 × 4 antennas.

detection with the aid of lattice-reduced has demon-strated be robust against interference between antennas.In strongly correlated channels with 2 × 2 antennas, thedetectors ML and LR-aided MMSE-OSIC in high SNRregion present very similar performance, preserving thesame order of diversity; the same behavior happens withMMSE and MMSE-OSIC detectors. Under same chan-nel and system scenario but with 4 × 4 antennas, onecan note that with the increasing of the level of correla-tion provokes a larger degradation in the performance,specially for the MMSE and MMSE-OSIC detectors,which under strongly correlated channels present a re-ally poor performance where this detectors showed asmaller order of diversity.

In strongly correlated channels combined with highnumber of antennas it is noticeable that in low-mediumSNR regions the LR-MMSE-based MIMIO detectorshows a higher performance degradation regarding theMMSE or MMSE-OSIC. The reason is that the LR tech-nique applied to MIMO systems, specially under largenumber of antennas condition, makes them more sensi-tive to the noise and inter-antenna interference Valenteet al. (2014). However, this performance degradationeffect does not occurs with LR-aided MMSE-OSIC be-cause the OSIC substantially mitigates the interferenceeffect.

Finally, one can conclude that the ML-MIMO perfor-mance for any number of antennas presents a slightlybetter performance than the LR-based MMSE-OSIC de-

7

tector at cost of much higher complexity, while LR-MMSE-OSIC detector is able to offer the same diver-sity order, as indicated by the slope of BER performancecurve in high SNR region.

Fig. 4 and 5 show the BER performance of the de-tectors for a 4-QAM MIMO system with Nt = Nr = 2and Nt = Nr = 4 antennas, respectively, consideringuncorrelated channels (ρ = 0) but under different lev-els of error in the estimation of the channel coefficients,ε = [0.2; 0.5; 0.7; 0.9]. One can notice that with theincrease of CSI error levels the degradation of BER per-formance is remarkable for all analyzed MIMO detec-tors. It is noticeable that under very high level of errorin the channel estimation (ε = 0.9) the performance gapbetween the LR-based MIMO detectors and the lineardetectors is smaller indicating that LR-based detectorsare more sensitive to noise and to CSI estimation errors,however, this detectors presents higher order of diver-sity.

Fig. 6 put into perspective the joint effect of an-tenna correlation and CSI error estimation on the BERperformance degradation, considering 4-QAM MIMOsystem with Nt = Nr = 2. The linear and LR-aided MIMO BER degradations have been calculatedfor ρ = [0.2; 0.5; 0.7; 0.9] combined to ε =[0.2; 0.5; 0.7; 0.9]. In Fig. 6.a) the LR-MMSE-OSIC MIMO detector performance is analysed for dif-ferent SNR levels and correlation, while CSI errors areparametrized. As expected, depending on the level ofchannel correlation and CSI error the impact on BERperformance degradation can be remarkable. It is possi-ble to see the difference between how the effects degradethe performance, the impact by the effect of correlationhave a envelopment exponential, which indicate that forvariation on smalls values of ρ have a marginal impact,however, variations for high values of ρ has great im-pact. The effect of estimation error have a envelopmentlogarithmic, which indicate an opposite behavior, wherevariations for small values of ε has great impact and forhigh values of ε the impact is marginal.

Complementary, in Fig. 6.b) the joint effects of chan-nel correlation and CSI error estimates on the BERdegradation are analysed considering a fixed SNR =10dB. Notice that in the region of high channel error es-timation (ε = 0.9) BER performance for the all MIMOdetectors is less degraded than in high correlated chan-nel (ρ = 0.9), which indicate the effect of channel cor-relation on the BER is more devastating than the errorin the channel error estimation.

0

5

10

15

20 0

0.2

0.4

0.6

0.8

110−4

10−3

10−2

10−1

100

ρSNR (dB)

BE

R

ε= 0

ε= 0.2

ε= 0.5

ε= 0.7

ε= 0.9

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

10−3

10−2

10−1

100

ερ

BE

R

MMSE

MMSE−OSIC

LR−MMSE


ML

Figure 6: BER performance degradation under different levels of CSIestimation errors and correlation for 2 × 2 antennas. a) LR-MMSE-OSIC MIMO detector for different SNRs and correlation; b) All con-sidered MIMO detectors with SNR = 10 dB

8

0 10 20

10−3

10−2

10−1

SNR (dB)

ρ= 0

0 10 20

10−3

10−2

10−1

SNR (dB)

ρ= 0.2

0 10 20 30

10−3

10−2

10−1

SNR (dB)

ρ= 0.7

0 20 40

10−3

10−2

10−1

SNR (dB)

ρ= 0.9

MMSE

MMSE−OSIC

LR−MMSE

LR−MMSE−OSIC

ML

MMSE

MMSE −OSIC

LR−MMSE

LR−MMSE−OSIC

ML

MMSE

MMSE−OSIC

LR−MMSE

LR−MMSE−OSIC

ML

MMSE

MMSE−OSIC

LR−MMSE

LR−MMSE−OSIC

ML


0 5 10 15 20 2510

−4

10−3

10−2

10−1

SNR (dB)

BE

R

ρ= 0.2

0 10 20 3010

−4

10−3

10−2

10−1

SNR (dB)

ρ= 0.5

0 10 20 30 4010

−4

10−3

10−2

10−1

SNR (dB)

ρ= 0.7

0 20 4010

−4

10−3

10−2

10−1

SNR (dB)

ρ= 0.9

MMSE

MMSE −OSIC

LR − MMSE

LR − MMSE− OSIC

ML

MMSE

MMSE − OSIC

LR − MMSE

LR − MMSE−OSIC

ML

MMSE

MMSE−OSIC

LR−MMSE


ML

MMSE

MMSE − OSIC

LR − MMSE


ML


6. Complexity Analysis

In this section, MIMO detector complexities are com-pared in terms of floating-point operations per second

(flops). Table I summarizes the overall complexityrelated to the analysed MIMO detectors, considering

9

0 5 10 15 20 2510

−4

10−3

10−2

10−1

ε= 0.2

SNR (dB)

BE

R

0 10 2010

−4

10−3

10−2

10−1

ε= 0.5

SNR (dB)

0 10 20 30

10−4

10−3

10−2

10−1

ε= 0.7

SNR (dB)

0 10 20 30

10−4

10−3

10−2

10−1

ε= 0.9

SNR (dB)

MMSE

LR−MMSE

MMSE−OSIC

LR−MMSE−OSIC

ML

MMSE

LR−MMSE

MMSE−OSIC

LR−MMSE−OSIC

ML

MMSE

LR−MMSE

MMSE−OSIC

LR−MMSE−OSIC

ML

MMSE

LR−MMSE

MMSE−OSIC

LR−MMSE−OSIC

ML

Figure 4: BER performance for the MIMO detectors with channel errors estimation; 2 × 2 antennas.

0 10 20 30

10−4

10−3

10−2

10−1

SNR (dB)

BE

R

0 10 20 30

10−4

10−3

10−2

10−1

ε= 0.5

SNR (dB)

0 10 20 30

10−4

10−3

10−2

10−1

ε= 0.7

SNR (dB)

0 10 20 30

10−4

10−3

10−2

10−1

ε= 0.9

SNR (dB)

MMSE

LR− MMSE

MMSE − OSIC


ML

MMSE

LR − MMSE

MMSE − OSIC

LR − MSSE − OSIC

ML

MMSE

LR − MMSE

MMSE − OSIC


ML

MMSE

LR − MMSE

MMSE − OSIC


ML

Figure 5: BER performance for the MIMO detectors with channel errors estimation; 4 × 4 antennas.

Nt = Nr = N and M modulation order for the M-QAM.The complexity of the ML-MIMO detector is included

as reference. The complexity is evaluated in terms of thetotal numbers of floating-point operations (flops), which

10

one flop is defined as an addition, subtraction, multipli-cation or division between two floating points numbers,and matrix and vector operations flop count are based inGolub and Van Loan (1996). Also, the complexity onthe sorted QR decomposition can be found in Wübbenet al. (2003).

Table 1: Detectors Complexity

MIMO Detector Number of flops

MMSE 263 N3 + 4N2

MMSE-OSIC 403 N3 + 13

3 N2 + 256 N

LR-MMSE 263 N3 + 4N2 + fLLL(N, ρ)

LR-MMSE-OSIC 403 N3 + 13

3 N2 + 256 N + fLLL(N, ρ)

ML (8N2 + 2N)MN

First of all, from Table 1, it can be observed thatML has a prohibitive exponential complexity in anypractical MIMO system configuration with moderateor higher number of antennas for any channel corre-lated and estimation errors scenarios. On the otherhand, MMSE detector offers a lower complexity, withpolynomial order O(N3). This complexity is explaineddue to the fact that a matrix inversion is necessaryfor this detector. SIC-based MIMO detectors are ca-pable to provide substantial performance improvementwithout increase the order of complexity regardingthe MMSE MIMO detector; hence SIC-based MIMOtopologies are able to offer a suitable BER performance-complexity tradeoff. Furthermore, despite the samepolynomial complexity order O(N3) achieved by LR-aided MIMO detectors, these MIMO detector topolo-gies present a second complexity term that could be-come significant regard the first term, given by thefunction of fLLL, which is dependent of the level ofchannel/antennas correlation and number of antennas,as predicted by (25). In uncorrelated channel/antennasscenarios, the LR-based MIMO detectors complexityhas dominated by the first term, and this techniquecan demonstrate an attractive performance-complexitytrade-off. Besides, under low to medium correlation in-dex scenarios LR-aided MIMO technique show a rea-sonable and suitable complexity, while preserving fulldiversity, which makes it a promising near-optimumMIMO transmitting scheme.

Fig. 7 depicts a further graphically flops-complexitycomparison for the sub-optimal MIMO detectors. Inter-esting, the Tx-Rx antenna correlation has a much greatimpact over the complexity in the LR-aided detectors,specially in scenarios with a large number of antennas

0

0.2

0.4

0.6

0.8110

2030

4050

6070

8090

100

0

0.5

1

1.5

2

2.5

3

3.5

x 107

ρ

N

Flo

ps

MMSE

MMSE − OSIC

LR − MMSE


Figure 7: Flop complexities for sub-optimum MIMO detectors.

and strong correlation. Furthermore, one can confirmthe effect of channel correlation and number of antennason the substantial complexity increasing of LR-aidedMIMO detectors, mainly when ρ ≥ 0.7 and N > 50due to the increasing complexity added by the LLL al-gorithm.

7. Conclusion

The correlated channels effects have been demon-strated very significant on the MIMO system per-formance equipped with polynomial-complexity sub-optimal detectors. Among the analysed MIMO detec-tors, the LR-aided technique has been demonstratedvery useful aiming to improve the performance of thosesub-optimal MIMO detectors under antenna correla-tion, as well as under imperfect channel estimation con-strains. Indeed, numerical results and analyses for botheffects on the BER performance of LR-aided MIMOsystem equipped with different detectors have indicatednotable gains in terms of performance and robustness.However, the MMSE and MMSE-OSIC MIMO detec-tors have shown a remarkable performance degradationin strongly correlated channels, indicating, in addition,that detectors aided by lattice reduction technique caneffectively be an alternative to this degraded scenario,although the complexity introduced by the LLL algo-rithm has a significant impact on the overall computa-tional complexity under high correlation and numberof antennas scenarios. Among them, the LR-MMSE

11

MIMO OSIC detector has achieved the smaller degra-dation regarding the devastating effect of the combinedeffects of channel correlation and error estimation. Ourfinding indicates the LR-MMSE MIMO OSIC detec-tor operating under low-medium channel correlation in-dexes and medium-large number of Tx-Rx antennas isable to achieve the best performance-complexity trade-off among those analysed MIMO detectors.

Acknowledgement

This work was supported in part by the NationalCouncil for Scientific and Technological Development(CNPq) of Brazil under Grants 202340/2011-2, and inpart by Londrina State University - Paraná State Gov-ernment (UEL).

References

Andalibi, Z., Nguyen, H. H., Salt, J. E., 2013. Precoder design forbicm-mimo systems under channel estimation error. Wireless Per-sonal Communications 72 (4), 2823–2835.

Bai., L., Choi., J., 2002. Low complexity MIMO Detection. Springer.Böhnke, R., Wübben, D., Kühn, V., Kammeyer, K.-D., 2003. Reduced

complexity mmse detection for blast architectures. In: GLOBE-COM’03. IEEE, pp. 2258–2262.

Chen, R., Li, J., Li, C., Liu, W., Feb 2012. Lattice-reduction-aidedmmse precoding for correlated mimo channels and performanceanalysis. Systems Engineering and Electronics, Journal of 23 (1),16–23.

Fang, X., Fang, S., Ying, N., Cao, H., Liu, C., Dec 2013. The per-formance of massive mimo systems under correlated channel. In:Networks (ICON), 2013 19th IEEE International Conference on.pp. 1–4.

Foschini, G. J., Gans, M. J., 1998. On limits of wireless communica-tions in a fading environment when using multiple antennas. Wire-less Personal Communications 6, 311–335.

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

Horn, R. A., Johnson, C. R., 1985. Matrix Analysis. Cambridge Uni-versity Press.

James, G., 2010. Modern Engineering Mathematics, 4th Edition.Prentice Hall, New York, NY, USA.

Kobayashi, R. T., Ciriaco, F., Abrão, T., 2014. Efficient near-optimumdetectors for large mimo systems under correlated channels. Wire-less Personal Communications, 1–18Accepted.

Lenstra, A. K., Lenstra, H. W., 1982. Factoring polynomials with ra-tional coefficients. Muth. Ann, 515–534.

Milford, D., Sandell, M., July 2011. Simplified quantisation in areduced-lattice mimo decoder. Communications Letters, IEEE15 (7), 725–727.

Mostagi, Y. M., Abrão., T., 2012. Lattice-reduction-aided over guidedsearch mimo detectors. International Journal of Satellite Commu-nications Policy and Management, 142–154.

Park, J., Chun, J., June 2012. Improved lattice reduction-aided mimosuccessive interference cancellation under imperfect channel esti-mation. Signal Processing, IEEE Transactions on 60 (6), 3346–3351.

Valente, R. A., Filho, J. C. M., Abrão, T., 2014. Lr-aided mimo detec-tors under correlated and imperfectly estimated channels. WirelessPersonal Communications 77 (1), 173–196.

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Wübben, D., Böhnke, R., Kühn, V., Kammeyer, K.-D., Oct. 2003.MMSE extension of V-BLAST based on sorted QR decompo-sition. In: IEEE Semiannual Vehicular Technology Conference(VTC2003-Fall). Orlando, Florida, USA, pp. 1–5.

Wubben, D., Bohnke, R., Kuhn, V., Kammeyer, K.-D., june2004. Near-maximum-likelihood detection of mimo systems usingmmse-based lattice reduction. In: IEEE International Conferenceon Communications. Vol. 2. pp. 798 – 802 Vol.2.

Wübben, D., Böhnke, R., Rinas, J., Kammeyer, K.-D., Kühn, V., Nov2001. Efficient algorithm for decoding layered space-time codes.IEE Electronic Letters 37 (22), 1348–1350.

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Wübben, D., Böhnke, R., Kühn, V., dirk Kammeyer, K., 2004. Mmse-based lattice-reduction for near-ml detection of mimo systems. In:in ITG Workshop on Smart Antennas. pp. 18–19.

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12

40

Apendice C -- Precoder and Decoder

Design to Improve the Performance for

Correlated Channel Matrix

November 20, 2014 15:22 WSPC/INSTRUCTION FILE Preco_MIMO

Journal of Circuits, Systems, and Computersc⃝ World Scientific Publishing Company

MIMO Precoding for Correlated Fading Channels

Bruno Felipe Costa, Taufik Abrão

Department of Electrical Engineering, State University of Londrina (DEEL-UEL),Rod. Celso Garcia Cid | Pr 445 Km 380,Londrina, Paraná CEP 86.057-970, Brazil

[email protected] [email protected]

Received (Day Month Year)Revised (Day Month Year)Accepted (Day Month Year)

This contribution proposes a precoder-decoder design aiming to improve the performanceof multiple-input-multiple-output (MIMO) detectors under correlated fading channels.The MIMO detection principle namely minimum mean squared error (MMSE) is ana-lyzed under such channel condition. The proposed design deploys the channel state in-formation (CSI) to estimate the level of channel spatial correlation, namely normalizedρ index, and uses this information to improve the MIMO system performance. Further-more, the impact of the ρ estimation errors on the performance, as well the performancedegradation for different levels of correlation have been analyzed and compared with thescenario without correlation.

Keywords: multiple-input-multiple-output (MIMO); Correlated Channels; Precoding;channel error estimation.

1. Introduction

The use of multiples antennas at the transmitter and receiver sides (MIMO)in wireless communication systems has the potential of dramatically increasing thespectral efficiency and/or be able to improve the performance and reliability ofwireless communication by creating multiple uncorrelated channel paths betweentransmitter and receiver antennas [1].

Transmitting data streams in parallel increase the spectral efficiency of the sys-tem; however, the complexity on the data detection at the receiver side increasesproportionally [2]. The MIMO system suffers influence from many effects that candegrade the performance, such as channel spatial correlation, errors of channelestimates and so forth; as a consequence system capacity could be reduced dra-matically. Furthermore, since the fading effect reduces the capacity of the system,using multiple antennas at both side to mitigate the fading effect could increase thereliability of the system if the same data information is transmitted simultaneouslyin all transmitting antennas.

Some knowledge of the channel state information (CSI) is necessary in order to

1


2 Bruno F. Costa, Taufik Abrão

deploy a detector on the receiver side aiming to increase the system throughput;indeed, with the CSI available at both the transmitter and receiver side increase thecomplexity, although, it is possible to deploy successfully low complexity detectorson the receiver [3] complexity

Another important effect to be considered in realistic MIMO scenarios is thechannel correlation between antennas. This effect occurs when antennas are phys-ically too close, for instance. This effect has a great impact on the bit error rate(BER) performance and in the diversity gain provided by the system. When a linearstrategy of detection is deployed the correlated channels effect decreases dramati-cally the system performance [4].

Some important works have analysed the capacity gain of MIMO systems as-suming independent fading channel; in practice it is difficult to design antennasarray respecting the physical constraints (spacing between the antennas), speciallyin scenarios with massive MIMO. For instance, recently, the effect of antenna cor-relation on the performance of massive MIMO systems has been investigated in [5].

The classical linear MIMO detector, namely zero forcing (ZF), can cancel com-pletely the interference between antennas [6], at the cost of increasing the back-ground noise for a low rank channel matrix. Another classical linear detector, theminimum mean squared error (MMSE) detector, takes into account the noise powerin the detection process, with a slightly increase in the computational complexity.

In order to mitigate effect of correlation, some works have investigated precoderas a alternative. In [7] is analyzed a precoder using partial knowledge on the transmitand receive correlation matrix. For Ricean fading channels was studied a precoderfor receive correlation [8]. The impact of correlation and precoder using space-timecoding was investigated on [9].

The contribution of this work consists in propose a precoder and decoder designto mitigate the effect of the correlation between the antennas using samples of thechannel matrix to estimate the matrix of correlation on the transmitter and receiverside.

2. MIMO System Model

In this contribution we consider a complex baseband linear transmission systemwith no line of sight (NLOS) and Nt inputs and Nr outputs (regarding as transmit-ting and receiving antennas) corrupted by additive white Gaussian noise (AWGN).The mathematical model of the system under investigation is

y = Hs + η (1)

where H ∈ CNr × Nt fading coefficients matrix with magnitudes following a Rayleighstatistical distribution representing non-line-of-sight (NLOS) point-to-point com-munication, s is a vector of data symbols and η is the independent white noisevector samples with Gaussian distribution.


MIMO Precoding for Correlated Fading Channels 3

2.1. Correlated MIMO Channels

One important class of MIMO channel model assumes that the spatial correla-tion between the transmit antennas (Tx) is independent of the spatial correlationamong receive antennas (Rx); hence, admitting a MIMO channel Rayleigh flat-fading [10], one can express the fading coefficients matrix:

H =√

RH,RxG√

RH,Tx (2)

where G ∈ CNr x Nt is an independent identically distributed (i.i.d.) complexGaussian matrix with zero-mean unit-variance elements. The correlation matri-ces RH,Tx ∈ RNt x Nt and RH,Rx ∈ RNr x Nr denote correlation observed among thetransmitter antennas and receiver antennas, respectively.

Furthermore, for simplicity of analysis, assuming in this work that the Txand Rx antenna elements are equally separated, with equal numbers of antennas(Nr = Nt = N), symmetrically arranged under equally correlation matrix elements,resulting in

RH,Rx ≡ RH,Tx ≡ R, (3)

The matrix R ∈ RN × N must be able to describe the state of correlation on thechannel matrix; remembering that the channel correlation depends of the physicaldistance between the antennas, carrier frequency, physical array layout, channelenvironment among others. For simplicity of analysis, assuming that antennas el-ements at Tx and Rx are equally spaced in a linear antenna array, the matrix R

must be symmetric, and can be describe by [10]:

R =

1 ρ ρ4 · · · ρ(nT −1)2

ρ 1 ρ · · ·...

ρ4 ρ 1 · · · ρ4

......

.... . . ρ

ρ(nT −1)2 · · · ρ4 ρ 1

N×N

, (4)

The elements of R, are the index of correlation ρ, with different exponents, that hasa growth squared, which describe how the correlation decreases with the square ofthe distance among (i, j)-antenna elements. Also, ρ represents a normalized index ofcorrelation; under totally uncorrelated scenario, ρi,j = 0, ∀i = j, and the matrix R

becomes equal to a matrix identity, while under a fully correlated scenario impliesρ = 1, ∀i = j, and R = 1N×N .

3. MMSE MIMO Detection

In order to reduce the impact of fading and background noise, the MMSE-MIMOdetector employs a linear filter that can take into account the channel matrix andthe background noise effect as well. The MMSE filter matrix can be found by



minimizing the mean-square error (MSE) as [11]:

Wmmse = arg minW

E[∥s − WHy∥2]

=(E[yyH ]

)−1 E[ysH ]

=H

(HHH +

N0

EsI

)−1

(5)

where Es denotes the symbol energy, N0 is the noise spectral density, I is a N ×N identity matrix and E[·] is the statistical expectation operator. The resultingestimated symbol vector can be readily written as:

smmse = WHmmsey (6)

4. Precoder-Decoder Design based on Channel Correlation

In this work, the proposed design metric consists in multiplying a P matrixon the transmitting (Precoder) and receiver (Decoder) signal aiming to mitigatethe deleterious effect of spatial channel correlation over the system performance. Inorder to do that, we initiate expressing the matrix P as:

P = R− 12 (7)

The domain of P changes according with the numbers of antennas and the ρ val-ues. This occurs because when the nonsingular matrix R ∈ RN × N presents realnegative eigenvalues, P has complex values entries [12]. In practical scenarios, forN ≥ 4, at least one of the eigenvalues of R tends to zero. When this occurs tendingfrom the left the matrix P ∈ CN × N , for the others case P ∈ RN × N .After the signal pass through the Precoder and Decoder its is received as:

y = PHPs + Pη = Us + Pη (8)

Then using U as a new matrix channel it is possible to use scheme of detection, forinstance, like a MMSE.

4.1. Normalizing U

In order to ensure a fair comparison, matrix U = PHP in (8) must have thesame amount of energy of H. Hence, aiming to normalize the energy on the trans-mitter side, a function ε : CN × N → R can be defined as:

ε(Z) =

√⟨ℜ(zi,j)

2⟩2

+⟨ℑ(zi,j)

2⟩2

(9)

where zi,j refers to the element of i-th row and j-th column of matrix Z, and⟨·⟩ is the average operator across i = 1, 2, . . . N and j = 1, 2, . . . N . Notice thatfunction ε is used to calculate the energy of the channel matrix H, as well as it is



deployed to normalize U. Hence, the energy of the matrix U can be normalized onthe transmitter side by:

P =√

ε(H)ε(U) · P

U = PHP

(10)

So the normalized transmitting signal can be written as:

q = Ps (11)

and the corresponding normalized receive signal:

y = HPs + η (12)

The process of decoding results in an amplification of the additive noise by P;therefore, the signal after passing through decoder is given by:

y = Py = Us + Pη (13)

5. The estimation of ρ

In order to mitigate the effect of correlation on the system, it is necessary toestimate the level of correlation between the antennas. Using the channel matrix(H) and assuming equal number of antennas, it is possible to find the correlationmatrix by [10]:

R = E[hnhHn ], for n = 1, . . . , N (14)

where hn denotes the the n-th column of H. For instance, considering the firstcolumn of H (n = 1), the matrix R can be written using Eq. (14), as explicitlywritten in eq. (15), where [ · ]∗ represents the conjugate operator.

R =

E[h∗11h11] E[h∗

12h11] E[h∗13h11] . . . E[h∗

1Nh11]

E[h∗11h12] E[h∗

12h12] E[h∗13h12] . . . E[h∗

1Nh12]

E[h∗11h13] E[h∗

12h13] E[h∗13h13] . . . E[h∗

1Nh13]...

......

. . ....

E[h∗11h1N ] E[h∗

12h1N ] E[h∗13h1N ] . . . E[h∗

1Nh1N ]

(15)

Observing the first column of the matrices in Eq. (4) and Eq. (15), the index ofcorrelation ρ can be identified as:

ρ(n−1)2 = E[h∗11h1n ] (16)

For any n ≤ N integer positive; for the case n = 2, ρ can estimated by:

ρ = E[ h∗11h12 ] (17)

Considering the level of correlation the same on the transmit and receive side, theestimation of ρ using Eq. (17) can be used for any numbers of antennas on bothsides.



5.1. Estimating the Spatial Correlation

The estimation of ρ using Eq. (17) involves the expected value operator. Toobtain the expected value of a variable, it is necessary to take in amount a significantnumber of samples and calculate the mean of these samples. In order to determinethe amount of samples necessary to obtain a reliable estimation of ρ, Fig. 1 depictsthe histogram of the ρ estimation with different numbers of samples (10000, 100 and50 samples) and considering the true value of ρ = 0.5. From these histograms onecan conclude that, with 10000 samples, the mean is very similar with the real valueof ρ, although with 100 and 50 samples the deviation is about 10% of the true valorof channel correlation; furthermore, in the three simulations, the standard deviationis similar. The estimation with less samples than 50 samples becomes unstable inrelation to your mean and standard deviation, therefore, the estimation of ρ isunreliable. The similarities presents on the histogram indicate that the estimationwith a number of channel greater than 50 samples is enough to achieve a goodperformance.

−2 −1 0 1 2 3 4 5 6 7 80

100

200

300

400

500

600

700

800

900

Standard Deviation = 0.7909

10000 Samples

Mean = 0.5015

−2 −1 0 1 2 3 40

5

10

15

20

25

30

35

40

45


Mean = 0.5477

100 Samples

−1 −0.5 0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

12

Mean = 0.5415


50 Samples

Fig. 1. Histogram of the estimation of ρ for 10000, 100 and 50 samples.

5.2. Errors in the estimation of ρ

The errors on the estimation of ρ have influence on the performance of thesystem. Hence, in order to determine the influence of these errors, a MIMO system



equipped with a MMSE detector operating with N = 2 × 2 antennas and a SNR= 10 dB was simulated while channel correlation indexes ranging ρ ∈ [0; 1[ havebeen generated; the correspondent values of the estimated correlation is identifiedas ρe in Fig. 2.

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

10−2

10−1

ρρe

BE

R

Fig. 2. Surface showing the BER performance for different values of ρ and ρe; N = 2× 2 antennasand SNR = 10 dB

Indeed, Fig. 2 shows the BER performance for different levels of ρ and ρe. Onecan observe a considerable degradation on the MIMO system performance withthe increase of ρ, the case when ρ = ρe shows the better performance; however,this performance is still sensitive with the increasing of channel correlation indexρ. In other words, for a strongly correlated channel the precoder is not able tocompletely remove the correlation effect. However, this surface showed that theprecoder is robust against small errors in the estimation of ρ.

6. Simulation Results

In this section BER performance versus SNR under perfect and channel errorestimates is analyzed numerically; further realistic MIMO detection performanceanalysis has been conducted considering scenarios with correlated channel. Hence,along this section, numerical Monte-Carlo simulations (MCS) results are depicted



and examined. MIMO system equipped with the proposed precoder-decoder wassimulated the BER performance using different numbers of samples to estimate thecorrelation index ρ, in order to find a number of samples that is able to guaranteereliable estimation of the correlation index ρ. After the precoder-decoder scheme thesignal was detected using the MMSE detector, besides, 4-QAM modulation formatand Gray coding have been deployed; hence, the BER and symbol error rate (SER)values in this section very similar.

Fig. 3 depicts the BER performance of different MMSE detector designs, numberof channel samples and different levels of correlation of the channel using a 4-QAMMIMO system equipped with N = Nt = Nr = 2 antennas. A perfect channelestimation using the MMSE detector under channels without correlation (ρ = 0)have been included for reference. The MMSE-MIMO with precoder and decoderbased on channel correlation estimation index ρe has been obtained for differentnumbers of channel samples.

0 5 10 15 20 25 30

10−3

10−2

10−1

BE

R

ρ= 0.2

0 5 10 15 20 25 30

10−3

10−2

10−1

ρ= 0.5

SNR (dB)

0 5 10 15 20 25 30

10−3

10−2

10−1

ρ= 0.7

0 5 10 15 20 25 30

10−3

10−2

10−1

ρ= 0.9

MMSE non correlated

MMSE correlated

MMSE precoder 10 samples

MMSE precoder 50 samples

MMSE precoder 10K samples

MMSE precoder perfect estimation

Fig. 3. BER performance for different levels of correlation and number of channel samples.

In Fig. 3, the performance of the detectors under weakly correlated channels(ρ = 0.2) is very similar; however, with the increasing of the correlation level inthe range 0.2 < ρ ≤ 0.5 and using few samples (10 samples), the deviation in chan-



nel correlation estimation ρe becomes progressively inadequate. As a consequence,the BER performance becomes worse than the scheme using only MMSE detectorwithout precoder.

Furthermore, it is worth to note that in all scenarios the MMSE detector with(pre-)decoder and 10k samples shows only a slightly improvement compared withthe same detector using just 50 channel samples, which uses far less samples. Finally,under strongly correlated channels (ρ = 0.9, Fig. 3) the MMSE with pre-decoderpresents a noticeable degradation in BER performance regarding performance ofthe same MMSE under uncorrelated channel. Even so, there is a remarkable gainin performance regarding the MMSE correlated case (without pre-decoder).

Indeed, under a strongly correlated channels MMSE-MIMO with pre-decoderbased on correlation information results in a better performance, despite the pre-coder was not able to completely remove the correlation effect.

7. Conclusion

The correlated channel effects have been demonstrated very significant on theMIMO system performance. The pre-decoder design approach based on channelcorrelation information has been demonstrated very useful aiming to improve theperformance of the MMSE-MIMO detectors, mainly with a small-medium numberof samples (10-50). Indeed, with a far high number of samples (10k) the performancegain is only marginal; this occurs because the proposed pre-decoder design is robustagainst small errors on the channel correlation estimation ρe.

However if the numbers of samples is very small (≤ 10) and the channel corre-lation is medium ρ < 0.5, the performance can be degraded in such a way that canresult worse that without use of pre-decoder design. Experimentally, it was founda good range of samples between 50-100 samples, which express a good trade-offbetween number of samples and BER performance.

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