estudio y caracterización de técnicas de detección...

UNIVERSIDAD AUTÓNOMA DE

SAN LUIS POTOSÍ

FACULTAD DE CIENCIAS

Estudio y Caracterización de

Técnicas de Detección de Esfera

para Sistemas de Comunicación

Personal B3G

Examen de candidatura

Doctorado en Ingeniería Electrónica

con orientación en Telecomunicaciones

presenta:

MIE. Juan Francisco Castillo León

Asesor de Tesis:

Dr. Ulises Pineda Rico

Co-Asesor de Tesis:

Dr. Enrique Stevens Navarro

San Luis Potosí, S.L.P., Octubre del 2014

Índice general

Capítulo

1. Introducción 2

2. Objetivo y Metas 4

2.1. Objetivo General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4

2.2. Metas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3. Desarrollo 5

3.1. Resumen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.2. Improved Less complexity Sphere Decoder (iLSD) . . . . . .. . . . . . . . . 5

3.3. K-best - Less Complexity Sphere Decoder (KLSD) . . . . . . .. . . . . . . . 6

3.4. Semifixed Complexity Sphere Decoder (SCSD) . . . . . . . . . .. . . . . . . 7

3.5. 5-Symbols K-best . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8

4. Conclusiones 10

4.1. Productos académicos generados . . . . . . . . . . . . . . . . . . .. . . . . . 11

4.2. Docencia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

ÍNDICE GENERAL

4.3. Estancia de investigación . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 11

5. Cronograma y plan de trabajo 13

6. Anexos 15

6.1. Anexo A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6.2. Anexo B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6.3. Anexo C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Bibliografía 18

UNIVERSIDAD AUTÓNOMA DE SAN LUIS POTOSÍ


1

Capítulo 1

Introducción

En nuestros días existe una vasta diversidad de problemas relacionados con las comuni-caciones que hacen de nuestra experiencia como usuario un gran desafío técnico. Situacionestales como: velocidad insuficiente en tasas de transferencia, altos periodos de latencia en la red,interferencia entre usuarios/celdas, condiciones hostiles en canales de comunicación físicos oinalámbricos, etc.; son retos que se nos presentan día a día yque deben ser enfrentados, median-te nuevas técnicas o tecnologías.

Los arreglos de antenas MIMO prometen satisfacer los requerimientos de las futuras redesde comunicación inalámbrica (por ejemplo, Wimax2 (802.16m), LTE-Advance, WIFI (802.11n),etc...) ofreciendo una alta ganancia en cuanto a diversidadespacial (antenas) y un incrementonotable en la tasa de transferencia. Sin embargo, existen algunos retos por solucionar. Entreellos, el más serio es el de la interferencia co-canal, el cual representa un impedimento para eldesempeño eficiente de cualquier sistema de comunicación inalámbrica.

Una de las alternativas para poder cumplir de manera satisfactoria lo anterior, es el usode las técnicas de detección apropiadas. El decodificador deMáxima Verosimilitud (MLD,Maximum-Likelihood Decoder en inglés) se presenta como unasolución óptima para este ti-po de sistemas[1]. Sin embargo existe una pequeña limitante: la complejidad. Esto lo conviertenen una opción poco viable debido a que el receptor por lo general es un dispositivo con recursosfinitos en cuanto a batería y procesamiento, por ello se requiere de técnicas de detección queminimicen el consumo de recursos sin comprometer de manera alguna la correcta recepción einterpretación de la información transmitida.

No obstante existen alternativas de muy baja complejidad, dentro de ellas se destacan lastécnicas lineales basadas en la inversa de la matriz de canal: Zero Forcing (ZF), Minimum MeanSquare Error (MMSE) y variaciones de estas ultimas basadas en la técnica de cancelacionessucesivas (ZF-SIC, MMSE-SIC)[2]. Sin embargo aunque su baja complejidad es un punto a sufavor, su desempeño en términos de Tasa de Error De Bit (BER, Bit Error Rate en inglés) espobre en comparación con MLD. Por otro lado, existen técnicas no lineales que suelen alcanzardesempeños muy cercanos a MLD (cuasi-MLD) a costa de una mayor complejidad. Algunastécnicas no lineales destacadas, son las basadas en el proceso de ramificación y poda, tal comolos algoritmos de Decodificación Esférica (SD, Sphere Decoder en inglés) [3, 4].

Introducción

La Decodificación Esférica, es un tema de estudio que cobró relevancia a principios de ladécada pasada con el surgimiento de los sistemas de multiples antenas (MIMO, Multiple Input -Multiple Output en inglés). Las técnicas SD se caracterizanpor presentar una baja complejidadmientras mantienen un desempeño cuasi-MLD, su baja complejidad se debe a que estos métodosde detección limitan del área de búsqueda, generalmente mediante restricciones basadas en lascaracterísticas del canal de comunicación [3–5].

En la actualidad, gracias al estudio de los sistemas MIMO conarreglos de antenas de grandimension (Massive MIMO), el interés en los algoritmos de detección de muy baja complejidadse ha incrementado, por lo que el estudio de los algoritmos SDy variaciones es actualmente uncampo de estudio abierto con un sin fin de retos a vencer.

Este trabajo de investigación consiste en el desarrollo de algoritmos de detección basadosen técnicas SD, que puedan ser usados como detectores eficientes en sistemas MIMO o MassiveMIMO. El presente documento muestra la evolución de la investigación, comenzando con elplanteamiento del objetivo, seguido del desarrollo de la investigación separada en periodos yfinaliza presentando las principales conclusiones obtenidas hasta la fecha, el trabajo pendiente arealizar y el cronograma.



3

Capítulo 2

Objetivo y Metas

2.1 Objetivo General

El objetivo de la presente propuesta de investigación doctoral es realizar un estudio de lastécnicas de detección del tipo decodificador de esfera, así como su correcta caracterización pa-ra así identificar las técnicas mas eficientes o adecuadas para fines de innovación y eventualimplementación.

2.2 Metas

Al llevar a cabo este proyecto de investigación se espera llegar a alcanzar las siguientesmetas:

1. Lograr contribuciones importantes al estado del arte en el área de SD en redes inalám-bricas B3G.

2. Generar publicaciones para diseminar los resultados en conferencias internacionales deprestigio y en revistas científicas con arbitraje estricto.

3. Obtener del grado de Doctor en Ingeniería Electrónica conorientación en Telecomuni-caciones.

4. Colaborar en lo posible en la generación de recursos humanos de Licenciatura y Maes-tría.

Capítulo 3

Desarrollo

3.1 Resumen

En este documento se presentan los avances obtenidos y se describe la producción acadé-mica generada en el proyecto de investigación doctoral, desde el comienzo del mismo (Enero de2012) a la fecha (Octubre de 2014). El trabajo de investigación consiste en el estudio y desarrollotécnicas de detección de símbolos para sistemas de comunicación MIMO, y hasta el momentose han desarrollado tres propuestas. Dentro de las actividades realizadas en el periodo mencio-nado, se han acreditado dos materias de Postgrado, cuatro seminarios, y se han presentado cincoavances de tesis. Además, se publicaron dos trabajos en congresos internacionales (ConferenciaIberoamericana de Ingeniería Electrónica y Ciencias Computacionales (CIIECC) en su edición2012 y 2013 cuyas memorias técnicas fueron publicadas en la revista Procedia Technology de laeditorial Elsevier.), así como un trabajo en una revista indexada (IEICE Transactions on Commu-nications, Volume and Number: Vol.E97-B,No.12,pp.-,Dec.2014.). Finalmente, se ha solicitadola beca Mixta y se realizara una estancia en la Universidad Politécnica de Valencia, España, conel fin de llevar a cabo una investigación sobre las técnicas dedetección en sistemas massiveMIMO.

3.2 Improved Less complexity Sphere Decoder (iLSD)

En los primeros meses de la investigación, se generó un escrito que fue publicado en elcongreso internacional CIIECC2012, en el que se presentó el algoritmo iLSD.

El algoritmo iLSD, está basado en la Decodificación Esférica, puede ser aplicado directa-mente a sistemas de números complejos, y a diferencia de otras técnicas de detección no requierecalcular la inversa del canal. El iLSD define un área de búsqueda de nodos candidatos, medianteel ángulo de fase de la señal recibida y un ángulo de apertura,el cual se modifica dependiendode la varianza del ruido. Esto significa que los nodos candidatos seleccionados por nivel sonelegidos/descartados de acuerdo a sus ángulos de fase. Al basarse en el ángulo de fase, este algo-ritmo presenta mejores resultados cuando se realiza la detección de datos modulados mediantela técnica PSK.

Desarrollo

Los resultados obtenidos, muestran que el algoritmo iLSD esuna técnica que requiere deuna complejidad reducida para realizar el proceso de detección, y simultáneamente obtener undesempeño cuasi-ML. Además se puede destacar que al utilizar técnicas de modulación PSK,estos resultados se acentúan. Para mayor detalle sobre estapropuesta, puede dirigirse al anexoA: “Complexity-improved Sphere Decoder for MIMO Systems using PSK Modulations”.

Basados en los resultados obtenidos, así como en el estado del arte analizado en este pe-riodo. Concluimos que aunque la reducción de la complejidaden los algoritmos SD dependeprincipalmente de la elección correcta del área de búsqueda, área que a su vez ésta es definidaa través de un parámetro variable dependiente de las condiciones del canal (sea: radio, ángulo,etc.), existe la posibilidad de adoptar un parámetro fijo independiente de las condiciones delcanal, con el cual sea posible obtener un desempeño cuasi-MLD y una baja complejidad.

3.3 K-best - Less Complexity Sphere Decoder (KLSD)

Basados en lo concluido en el periodo anterior, se estudiaron los algoritmos K-best y FSD[6, 7]. Al analizar el proceso que realizan estos algoritmosdurante la detección símbolos, seobservó que el área de búsqueda por nivel esta determinada por límites definidos mediante laobservación del desempeño obtenido y a través de las características del sistema. Por lo anterior,se consideró posible definir el área de búsqueda por anticipado y a su vez de forma independientea las condiciones del canal. En consecuencia se realizó la propuesta de un nuevo algoritmo dedetección, al cual llamamos KLSD, y fue presentado en el congreso internacional CIIECC 2013.

El algoritmo KLSD incluye dos métodos de definición de área debúsqueda (etapa de pre-deteccion): En primer lugar, se recurre al concepto de ángulo de apertura utilizado en el iLSD,pero con la diferencia de que el ángulo es una constante que semantiene de nivel a nivel inde-pendiente de las características del canal. En segundo lugar, se divide la constelación de nodoscandidatos en anillos concéntricos. Estas dos técnicas en conjunto reducen el número de nodoscandidatos visitados durante el proceso de detección de forma considerable. Además como ocu-rre en FSD, para cada nodo padre existe la posibilidad de modificar el número de nodos hijossobrevivientes en cada nivel, reduciendo la complejidad ocasionada por el tamaño del arreglo deantenas MIMO. Finalmente como análisis de concepto, con el objetivo de evitar el cálculo dela distancia euclidiana y reducir la complejidad computacional que ésta implica, se realizó unanálisis comparativo entre el uso de la normaℓ1 y ℓ2.

Los resultados demuestran que el cambio de métrica no afectael desempeño BER lograndomantener un desempeño cuasi-MLD, por lo con el objetivo de reducir la complejidad es posiblesustituir con facilidadℓ2 por ℓ1 sin afectar el desempeño. Además se ha demostrado que la etapade predetección propuesta, reduce el área de búsqueda sin lanecesidad de realizar operacionesmatemáticas para estimar un área de búsqueda. Para mayor información acerca de este algoritmoconsultar el Anexo B,“MIMO Detector for uncoded symbols based on Complex SphereDecoder and Manhattan metric”.

Finalmente en este periodo se observó que la medición de la complejidad, en cada publi-cación se realiza de manera diferente, lo que en ocasiones imposibilita la comparación entrealgoritmos de forma directa. Por ello se consideró a futuro examinar diferentes métodos de me-dición de complejidad para seleccionar o desarrollar un método eficiente para la comparaciónentre algoritmos de detección. Además, se ha llegado a la conclusión de que la complejidad delos algoritmos esféricos se concentra en gran medida en la etapa de selección de nodos candida-tos por nivel, por lo que nuestros esfuerzos se enfocaron posteriormente en su estudio y mejora.



6

Desarrollo

3.4 Semifixed Complexity Sphere Decoder (SCSD)

Para este periodo de la investigación, nuestros esfuerzos se enfocaron paralelamente entre elestudio de las técnicas de medición de complejidad, así comoen la etapa de selección de nodoscandidatos por nivel.

En [8] se describen un conjunto de técnicas de medición de complejidad, cada una de ellasbasadas en la naturaleza del problema. Dado que nuestro problema consiste en medir la comple-jidad de un algoritmo de detección basado en la técnica de ramificación y poda, los métodos demedición propicios para ello son: el conteo del número de “nodos visitados” y el conteo de las“operaciones matemáticas” (FLOPS) realizadas durante el proceso de detección.

Nodos visitados:Son el conjunto compuesto por la expansión de un nodo hijo a partirde un nodo padre, y el cálculo del costo asociado a ese nodo hijo. El conteo de nodos vi-sitados se hace mediante el análisis del código programado,incrementando el contadorcada vez que se realiza la expansión de un nodo.

FLOPS: Son el número de operaciones de punto flotante ejecutadas porsegundo. Lamedición se realiza mediante la funciónflops(0), incluida en las versiones anteriores aMATLAB 6.0. Esta función “ estima” y no “ cuenta” el número de operaciones aritmé-ticas, puesto que contarlas de modo exacto tiene un costo prohibitivo [9].

En [6] y [7], se asegura que cuando el área de búsqueda es definida de forma anticipada, yesta área limita a un número fijo la cantidad de nodos visitados, la implementación en hardwarees mucho mas sencilla, en comparación con algoritmos que definen el área de forma variable.Dado lo anterior y siguiendo con la tendencia de limitar el área de búsqueda, se realizo la pro-puesta de un nuevo algoritmo llamado SCSD, el cual se basa en la técnica de detección FSDy a diferencia de otras técnicas, SCSD presenta una complejidad variable pero limitada a unacomplejidad máxima. Por otro lado a diferencia de técnicas como la descrita en [10] (AlgoritmoASP, Adaptive Tree Search Algorithm), nuestra propuesta limita el área de búsqueda de formaindependiente de las características del canal de comunicación.

El algoritmo propuesto se divide en las siguientes etapas:

Etapa de ordenamiento de canal:El ordenamiento de canal es parte de la etapa depreprocesamiento. Se aplica justo antes o en conjunto con ladescomposiciónQR. Sufunción es ordenar las columnas de canal de comunicaciónH, de tal forma que la señalcon la la menor probabilidad de detección correcta sea la primera en ser detectada [11].

Etapa de Predeteccion:En la etapa de predeteccion se aplican las técnicas que ayudana limitar el área de búsqueda, en este algoritmo se incluyen:la division de la constela-ción de símbolos en anillos concéntricos, y la limitación del número de nodos sobre-vivientes por nodo padre. El número de nodos y anillos en cadanivel es dependienteen gran medida del ordenamiento de canal. En nuestro caso para lograr un desempe-ño cuasi-MLD se requiere de una configuración de nodos sobrevivientes por nivel de[1, 1, 2, 16] para16-QAM, y [1, 1, 3, 64] para64-QAM. Mientras que el número deanillos requerido es de[2, 2, 2, 3] para16-QAM y [2, 3, 4, 9] para64-QAM.

Terminación Temprana: La terminación temprana es una técnica que reduce el núme-ro de nodos candidatos expandidos, mediante la comparaciónde los “mejores costos”completos o parciales respecto a las ramas restantes, y descartando las ramas con uncosto superior al “mejor costo”[12].



7

Desarrollo

En los resultados se puede observar que el desempeño BER obtenido es cuasi-MLD, con unacomplejidad notablemente inferior a algoritmos como el SD ,K-best , FSD y ASP, especialmentecuando la SNR es inferior a17 dBs. Esto confirma que la definición del área de búsqueda es unaetapa muy importante dentro del proceso de detección, por loque mejorar esta etapa traerácomo consecuencia una mayor reducción de complejidad. Paramayor información consultarel Anexo C,“A Semifixed Complexity Sphere Decoder for Uncoded Symbols for WirelessCommunications”.

3.5 5-Symbols K-best

En esta fase de la investigación, hemos estado trabajando enuna nueva técnica a la cualllamamos provisionalmente:“5-Symbols K-best”. Esta técnica nace de la observación de lastendencias de incremento de complejidad computacional, donde es claro que la complejidad seincrementa principalmente por dos factores: el orden de la modulación y el número de antenasen el arreglo MIMO.

Con el fin de reducir la complejidad nuestra propuesta utiliza dos técnicas descritas a conti-nuación:

Detección Kbest:Es una técnica de detección se caracteriza por retener únicamente losK nodos candidatos con el menor costo asociado, en cada nivel del árbol de búsqueda.Esta característica es favorable cuando se trabaja con arreglos MIMO de gran tamaño,debido a la posibilidad de controlar los nodos sobrevivientes por nivel.

Reducción del orden de la modulación:Se basa en la creación de una subconstelacionde únicamente5 nodos candidatos para cada nodo padre independientemente del ordende la modulación originalmente utilizada. Esta técnica ayuda a mantener una compleji-dad reducida independiente del orden de la modulación utilizada (64-QAM, 256-QAM,etc.).

Para lograr la reducción del orden de la modulación, se sigueel siguiente procedimiento:

1. A partir del nivelk − 1 y hastak = 1 para cada nodo padre, se deben estimar nodoscandidatos auxiliares mediante:

xk = yres/Rk,k,

dondeyres = yk +∑M

j=k Rk,jxj .

2. Calcular los vecinos más cercanos axk, haciendo uso de:

xk,i = Q(xk) + Ci i = 1, . . . , 4nv,

dondeC corresponde a un valor que genera valores de la constelaciónΩ, 4nv es elnúmero de vecinos que se desea obtener ynv define el nivel en que se encuentran losvecinos, por ejemplo:nv = 1 vecinos de primer nivel,nv = 2 vecinos de primer ysegundo nivel.



8

Desarrollo

En los resultados preliminares hemos observado un desempeño cuasi-MLD, con una com-plejidad notablemente reducida respecto al algoritmo de detección K-best. Además podemosdestacar que aun no existen comparaciones de desempeño y complejidad respecto a otros algo-ritmos de detección, pero los resultados obtenidos nos dan la confianza de lograr resultados quesean publicables.



9

Capítulo 4

Conclusiones

Como parte del proyecto de investigación hasta el momento sehan realizado tres propues-tas, en las cuales se resuelve el problema de la detección en sistemas de comunicación MIMO.La primera técnica de detección (iKLSD) se basa en la reducción del área de búsqueda, al defi-nir ésta mediante un ángulo de apertura y la division de la constelación en anillos concéntricos,este algoritmo es capaz de obtener un desempeño cuasi-ML conuna complejidad reducida (conmejor funcionamiento para modulaciones PSK). Nuestra segunda propuesta (KLSD) presentauna mayor reducción de complejidad, debido a que en este algoritmo se agrega la limitacióndel número de nodos sobrevivientes por nodo padre, tal como lo realiza FSD, manteniendo undesempeño cuasi-ML. Nuestra tercera propuesta, el algoritmo SCSD a diferencia de las pro-puestas anteriores, define el área de búsqueda sin depender del SNR o las condiciones del canal.Además para reducir el número de operaciones matemáticas a realizar, se implementa una etapallamada terminación temprana, que sumada al ordenamiento de canal, permiten un mayor aho-rro de recursos computacionales, sin afectar el desempeño.Actualmente se encuentra en estudiouna cuarta propuesta basada en el algoritmo K-best y en la técnica de optimización conocida co-mo búsqueda de patrones, esta propuesta define el área de búsqueda tomando en consideraciónlos nodos parciales detectados con anterioridad. Su característica primordial es la reducción delorden de la modulación de los datos, dado que solo requiere del análisis de un máximo de5nodos candidatos por nodo padre. De estas propuestas se desprenden dos trabajos publicados encongresos internacionales y un artículo aceptado para su publicación.

A futuro se planea la hibridación de algoritmos de detecciónesférica con métodos de bús-queda directa para obtener un algoritmo de detección de muy baja complejidad que sea com-patible con los sistemas Massive MIMO. Finalmente se pretende buscar aplicaciones de losalgoritmos estudiados en otras campos de estudio, donde se requiera resolver el problema delvector mas cercano.

Conclusiones

4.1 Productos académicos generados

Como se ha mencionado previamente, al día de hoy se han publicado los siguientes trabajos:

1. A Semifixed Complexity Sphere Decoder for Uncoded Symbols for Wireless Com-munications, J. F. Castillo-Leon, M.A. Cardenas-Juarez, U. Pineda-Rico and E. Stevens-Navarro, IEICE Transactions on Communications, Vol.E97-B,No.12,pp.-,Dec. 2014.

2. MIMO detector for Uncoded Symbols Based on Complex Sphere Decoder andManhattan Metric , Juan Francisco Castillo-León, Ulises Pineda-Rico and EnriqueStevens-Navarro, ISSN: 2212-0173, Procedia Technology, Elsevier, Volume 7, pp. 54-60, April 2013.

3. Complexity-Improved Sphere Decoder for MIMO Systems UsingPSK Modula-tions, Juan Francisco Castillo-León, Ulises Pineda-Rico and Enrique Stevens-Navarro,Procedia Technology, Elsevier, ISSN: 2212-0173, Volume 3,May 2012.

4.2 Docencia

Dentro de las labores de investigación, se contempló la impartición de un curso para estu-diantes de la carrera de Ingeniero en Telecomunicaciones, el cual tuvo como objetivo compartirconocimientos, experiencias y a su vez fomentar el interés de los alumnos por participar en pro-yectos donde puedan poner en práctica su aprendizaje teórico.

Curso: “Comunicaciones Digitales”.Carrera: Ing. en Telecomunicaciones.Periodo: Agosto - Diciembre 2013.

4.3 Estancia de investigación

La investigación sobre los sistemas de comunicación Massive MIMO, esta tomando un fuer-te auge, siendo considerada como una de las técnicas claves para los próximos estándares decomunicación 5G. Los métodos de detección para sistemas Massive MIMO, son en la actualidadun tema de estudio primordial con un sin fin de retos a vencer, por ello nuestra estancia se en-focara en el estudio de este tema, con el objetivo de desarrollar algoritmos de detección de muybaja complejidad.

Dentro de la literatura actual, existen varias vias posibles para afrontar el problema de detec-ción de señal en sistemas massive MIMO. Nosotros trabajaremos en dos de estas vias: Obtenerdetectores “hard” con coste computacional pequeño y que se escale de forma razonable conel número de antenas. Segunda, utilizar métodos de detección “soft”, relativamente sencillos yeficientes, combinados con los decodificadores iterativos.

El desempeño BER de los métodos aplicados a la detección parasistemas Massive MI-MO, varía según el método de decodificación, tamaño de arreglo de antenas y el orden de lamodulación. Se plantea para cada método de detección estudiado, analizar su rendimiento bajodiferentes órdenes de modulación y tamaños de arreglo de antenas.

Plan de trabajo



11

Conclusiones

1. Selección de métodos/estrategias de detección “hard” mas prometedoras.

2. Selección de métodos de detección “soft” adecuados para massive MIMO.

3. Colaboración en la implementación de un simulador massive MIMO multiusuario.

Asesor:Dr. Victor Manuel Garcia Mollá.Lugar: Universidad Politécnica de Valencia, España.Periodo de tiempo:Octubre 2014 - Octubre 2015.



12

Capítulo 5

Cronograma y plan de trabajo

A continuación se muestra el cronograma de las actividades realizadas durante el proyectode investigación, donde las actividades que se consideran completadas se encuentran marcadascon líneas diagonales.

Figura 5.1: Cronograma de actividades.

Durante el periodo comprendido entre octubre 2014 a febrerode 2016, de acuerdo al plan

Cronograma y plan de trabajo

de trabajo se llevarán a cabo las siguientes actividades:

1. Revisión del trabajo llevado a cabo hasta ahora; homogeneización de estilo de pro-gramación de las diversas funciones, optimización de código. Selección de los méto-dos/estrategias más prometedores. Análisis de la influencia del tamaño de la constela-ción. Duración estimada: 4 meses.

2. Estudio de métodos iterativos de decodificación (BCJR, LPDC, etc.) y de métodos dedetección “soft” adecuados para massive MIMO (ZF-SIC,MMSE-PIC). Análisis de lainfluencia del tamaño de la constelación. Duración estimada: 4 meses.

3. Implementación, en colaboración con otros doctorandos del grupo, de un simulador deun nodo completo de transmisión massive MIMO multiusuario:Duración estimada: 4meses.

Estas actividades se incluyen dentro del cronograma y estándirectamente relacionadas con:el estudio de las técnicas de detección en sistemas Massive MIMO, las técnicas multiusuario.Además de manera simultánea se está realizando la escriturade un nuevo artículo y se iniciaráen breve la redacción del documento de tesis.

Finalmente, al término de poco mas de cinco semestres de haber comenzado el doctorado,se considera un avance general aproximado de 60 %.



14

Capítulo 6

Anexos

6.1 Anexo A

Procedia Technology 3 ( 2012 ) 52 – 60

2212-0173 © 2012 Published by Elsevier Ltd.doi:10.1016/j.protcy.2012.03.006

Complexity-Improved Sphere Decoder for MIMO SystemsUsing PSK Modulations

Juan Francisco Castillo Leon∗, Ulises Pineda Rico, and Enrique Stevens-Navarro

Facultad de Ciencias, Universidad Autonoma de San Luis Potosı, Av. Salvador Nava Mtz. S/N, Zona Universitaria S.L.P., Mexico

Abstract

It has been shown in recent years that the iterative decoding techniques, improve performance (e.g., bit error rate) ofvarious digital communication systems. Techniques of Multiple-Input Multiple-Output (MIMO) are a key technologyto promote and achieve high-speed wireless communications. They demand only a low complexity system for detectionsince a high CPU processing involves more energy consumption and therefore less flexibility in mobility terms. Thesphere decoding (SD) technique has been proposed as an efficient algorithm to solve this problem. SD is known to bean algorithm of polynomial complexity that has become a powerful tool to achieve a high performance close to thatgiven by the maximum likelihood (ML) (which is considered ideal), but with less complexity. This paper proposes amodification to the SD technique in order to reduce its complexity named: improved Less Complexity SD (iLSD).

c© 2012 Published by Elsevier Ltd.

Keywords: Sphere Decoding, Wireless Communications, Multi-Antenna Systems, Maximum Likelihood, ZeroForcing, Complexity.

1. Introduction

Wireless communications have captured the attention and imagination of the world and it has becomeone of the largest and fastest growing segments in the area of telecommunications. The main reasons arethe desire for mobility and access to the Internet without requiring any physical connection (such as wires).At the same time, various technologies and systems have been proposed and rapidly deployed worldwideto provide wireless communications services. The success of wireless communications has been primarilyassociated with a steady increase in system capacity, attractive communications services, and better Qualityof Service (QoS). However, bandwidth is limited and expensive and therefore, the continuation of this trendshould use new technologies to provide greater spectral efficiency and reliability for the coming years. [1].

Traditionally, the antenna systems are formed by one transmitter (Tx) and one receiver (Rx), e.g. sys-tems Single-Input Single-Output (SISO). The work of Foschini [2] and Telatar [3] shows that by increasingthe number of antennas on both sides of the communication system, substantially increases the capacity ofthe channel in terms of the number of bits that can be transmitted, something unthinkable in SISO systems.This increased capacity is associated with a wealth of dispersion in the environment which allows the trans-mission of information by independent paths. Due to its advantages over traditional systems, the MIMO

∗Corresponding authorEmail address: [email protected] (Juan Francisco Castillo Leon)

Available online at www.sciencedirect.com

53 Juan Francisco Castillo León et al. / Procedia Technology 3 ( 2012 ) 52 – 60

Fig. 1. Assuming a single transmitter, independent symbols are chosen from a constellation A1 formed by elements modulated in: a)8-PSK, b) 16 QAM.

communication systems have emerged as a key technology while at the same time have been proposed asextensions of existing wireless communication standards such as IEEE 802.11, 3GPP and IEEE 802.16.

The inability to accurately estimate the communication channel, results in the need of a detection stage inorder to ensure a successful information recovery. The detection methods can be optimum (i.e., often involv-ing more complexity) or suboptimal (i.e, heuristic) which involve lower levels of computational complexity.The detector Maximum likelihood (ML) in general requires joint detection of an entire block of symbols[4]. Zero Forcing Detector (ZF) uses the inverse of the channel to remove the effects of it. Additionally, de-spite the fact that the main advantage of ZF is the speed and hence the complexity, its performance remainssignificantly below compared with the ML detector. On the other hand, sphere decoding (SD) techniquesoffers a performance close to ML with the advantage of exploring only a set of possible outcomes within aradius r. Thus, reducing the number of operations performed and therefore reducing also the computationalcomplexity used at the detection stage [5].

In this paper, we propose a modification to the SD technique which aims to improve its complexitythrough the prevention of errors caused by QR decomposition. Our proposed modification, named improvedLess Complexity SD (iLSD), changes the way that the area of search is mapped and defining it with a phaseangle. Also, performance and complexity comparisons are executed against other SD algorithms usingMonte Carlo simulations in order to show the advantages of the proposed technique. The rest of the paperis organized as follows. Section 2 describes the model for a MIMO system and also presents an overviewof detectors. Section 3 introduces our proposed method iLDS. Section 4 presents the performance andcomplexity comparisons and finally Section 5 concludes the paper.

2. System Model and Overview of Detectors

The model consists of a MIMO system with N transmitters and M receiving antennas, the received signalvector y = [y1, y2, ..., yM]T , with dimension M is given by:

y = Hs + n (1)

where H denotes the channel matrix M×N, s = [s1, s2, ..., sN]T is the transmitted signal vector of dimensionN, n is a complex noise vector of dimension M and variance σ2. Finally, (·)H represents the conjugatetranspose. Inputs of s are chosen independently from a constellation AN , as shown in Figure 1. For thescope of this paper, we assume that the number of receivers equals the number of transmitters (M = N) andthe channel H is modeled as a Rayleigh flat fading channel. For sake of simplicity, the estimation of thechannel is considered ideal and it remains constant for each data transmission performed [6].


Fig. 2. Interpretation of ML. Fig. 3. Idea behind SD.

2.1. Zero Forcing (ZF)

For a channel H, the Zero Forcing technique calculates the inverse in order to cancel the interferencecaused by the channel on the received vector y, see (2). Ideally, when performing this process, the channeleffects are completely canceled.

s = H†yAN (2)

where H† denotes the Moore-Penrose pseudoinverse of H, H† = HH(HHH)−1 and ·AN denotes the opera-tion of bringing each element of the vector obtained at the nearest element in the set of vectors AN [7].

For the case of systems with an ill-conditioned H (i.e., unbalanced-eigenvalues), the ZF detector onlyworks well in the region where the signal-to-noise ratio (SNR) is high, e.g. when there are little interferenceof the noise and thus the transmitted signal y was barely disturbed, making of ZF a non-viable choice forhigh performance purposes although an easy option to implement in real systems.

2.2. Minimum Mean Square Error (MMSE)

The Minimum Mean Square Error (MMSE) detector, tries to find a coefficient which minimizes thecriterion,

E = [Wy − s][Wy − s]H (3)

solving,

W = HH(HHH + N0I)−1 (4)

See that, when the noise term N0 is zero, the MMSE detector reduces to Zero Forcing detector. The MMSEestimate provides the following solution:

s = WyAN (5)

2.3. Maximum Likelihood (ML)

It is based on the method of least squares and it finds the minimum Euclidean distance of each elementof the vector received analyzing all existing solutions as depicted in Figure 2. Thus, we have that the MLalgorithm can be represented as,

s = arg minx‖y −Hx]‖2 (6)

where y is the received vector and H is the channel, x = [x1, x2, ..., xN]T is the potential data vector thathas been sent and s = [s1, s2, ..., sN]T is the received data vector, according to the algorithm, which in atheoretical way was sent.

Considering the nature of (6), s represents the optimum solution with the inconvenience of having ex-plored all possible solutions making it not feasible for implementation purposes due its exponential com-plexity directly related to the MIMO size array.


2.4. Sphere Decoder (SD)

In [8] was proposed an efficient strategy for enumerating all the lattice points within a sphere with acertain radius. Although on its worst case the complexity is exponential, this strategy has been widelyused in close lattice point search problems due to its efficiency in many useful scenarios. This enumerationstrategy was first introduced in digital communications by Viterbo and Biglieri [9] but finally in [5, 10],Hassibi and Vikalo presented the General Sphere Decoder (GSD) as a low complexity tool to find the least-squares solution and hence as a low complexity detector for MIMO systems. The basic premise of the SDis quite simple: it comes to find the point Hx located within a sphere with radius r centered at y, reducingthe search space and therefore the calculations to be performed. It is clear that the closest point within theradius of the sphere is also the closest point within the full mesh, as shown in Figure 3. However, we haveto guarantee that the point of Hx is within the sphere of radius r and this condition is true if and only if,

r2 ≥ ‖y −Hx‖2. (7)

Therefore, the radius r represents a crucial challenge for SD detectors since it is the only warranty for findinga solution to (6) at the expense of a lesser number of operations compared with ML.

A modification to this algorithm was proposed in [11] and [12] and is called Low Complexity SphereDecoder (LSD). LSD offers similar performance as GSD but with the convenience of utilizing even lesscomplexity than GSD. One of the main challenges of the SD algorithms is the fact that they were designedonly for real valued systems. However, there is an extension for making them usable in complex-valuedapplications. Thus, we can apply the algorithms LSD and GSD to the complex system (1) only when thereal and imaginary components of y, H and x can be decoupled to create a system of real equations withtwice the dimension of the original system [13]. For example, we have for (6) the following equivalentreal-valued system,

xreal = [(x) (x)]T (8)

yreal = [(y) (y)]T (9)

Hreal =

[(H) − (H)(H) (H)

](10)

where (·) represents the imaginary part and(·) represents the real part of a vector or scalar. Notice thatwhen duplicating the dimensions of the symbol vectors, the complexity also doubles.

2.4.1. Complex Sphere Decoder (CSD)The main advantage of this method is that it does not need the separation of real and imaginary com-

ponents of y, H and x. Therefore, it does not duplicates the system dimensions and hence the complexity.For achieving this, CSD uses the Cholesky factorization finding an upper triangular matrix U with real andpositive elements on its diagonal such that UHU = HHH [13]. Then, (7) can be written as follows,

r2 ≥ (x − x)HUHU(x − x) (11)

where x = (HHH)−1HHy is the initial estimate of s. If we evaluate k = M in (11) we obtain

ruM,M

≥

∣∣∣∣∣∣xM − xM

∣∣∣∣∣∣ (12)

This inequality limits the search to the constellation points contained in a complex radio disc ruM,M

cen-tered in xM . These points are easy to find when the constellation is shaped like a complex ring as in PSKmodulations. Figure 4 shows graphically the search arc and the circle generated by the constellation 8-PSK.

Let xM = rceiθM the M element from the vector of possible symbols transmitted and xM = rceiθM theM element from the estimated vector, where θM and θM are the phase angles of xM and xM respectively.


Fig. 4. 8-PSK constellation and the search angle.

Also, rc > 0 and rc > 0 are elements of the vectors rc and rc with the radius of the circle formed by theconstellation PSK. Once defined those elements we can develop (12) to obtain

cos(θM − θM) ≥1

2rcrc

[r2

c + r2c −

r2

u2M,M

]= η (13)

it is observed that if η > 1, the search out the arc contains no PSK constellation points. On the other hand,when η < −1, the search inside the arc includes the constellation points. Thus, for −1 ≤ η ≤ 1, the arc isdescribed by

|θM − θM | ≤ cos−1 η (14)

and therefore the search range can be defined as⌈θM − cos−1 η

⌉≤ θM ≤

⌊θM + cos−1 η

⌋(15)

warranting on this way an effective search within the arc denoted by (15). Summarizing the previous state-ments, in Algorithm 1 is shown the pseudo-code of the CSD algorithm including brief comments about theimplementation of this detection technique.

Algorithm 1. CSD. Require: Input: R, x = H†y, η.Ensure: Output: Estimated vector s.

1.(Initialize) k = M.2.(Bounds) UBk = θk+cos−1(η), LBk = θk−cos−1(η). Build a couple of constellations

(Constellation A1), the first Aθ, with phase angles and the second AS D, with

the elements of modulation that are within the limits. Lower Level NIk = 1,upper level NS k = LC, where LC is the number of elements of the constellation

Aθ.3.If NIk ≤ NS k, (Increase NIk) xk = AS D

NIk, NIk = NIk + 1 and go to 5; Otherwise go

to 4.4.(Increase k) k = k + 1; If k = M + 1, go to 7; Otherwise, go to 3.5.(Decrease k) If k = 1, go to 6; Otherwise k = k − 1, go to 2.6.Solution found. Save x in XN and go to 3.7.calculate

s = arg minx‖R(H†y − x)‖2 (16)

where x ∈ XN and XN ⊆ AN. Terminate the algorithm.

end


3. Improved Less Complexity Sphere Decoder (iLSD)

The improved Less Complexity Sphere Decoder (iLSD) is an algorithm based on CSD and LSD andis proposed to avoid duplication of items when using PSK modulations and then reducing notoriously thecomplexity. Among the advantages over its counterparts, it is important to highlight that iLSD does not needto calculate the inverse of the channel using the factorization QR instead and it just requires the knowledge ofthe phase angle of the received data in order to open an angle η (which defines the search area). Additionally,it also implements a correction algorithm that helps to improve the performance. Those characteristicsensure that iLSD uses less operations than its counterparts without compromise the its performance.

Thus, for sake of simplicity and correct illustration of iLSD, lets start from one of the main challengesof the LSD algorithm, the radius

r2 ≥ ‖y′ − Rx‖2 (17)

and transforming similarly to what was done in (13), we obtain

r2 ≥ ‖y′M − RM,M xM‖22= r2

y′M+ r2

RM,M xM− 2ry′M

rRM,M xM cos(θy′M − θRM,M xM ) (18)

which produces

cos(θy′M − θRM,M xM ) ≥1

2ry′MrRM,M xM

[r2

y′M+ r2

RM,M xM− r2

]= η (19)

where, θy′M and RM,M xM represent the phase angles of y′M and RM,M xM respectively. In a similar manner, ry′Mand rRM,M xM represent the radio generated by M-PSK modulation. Moreover if η > 1, then the search diskdoes not contain any constellation point A1 (constellation M-PSK). If η < −1, then search the disc includesthe entire constellation. For −1 ≤ η ≤ 1, the arc is described by

|θy′M − θRM,M xM | ≤ cos−1 η (20)

therefore, the search interval can be defined as⌈θRM,M xM − cos−1 η

⌉A1

≤ θy′M ≤

⌊θRM,M xM + cos−1 η

⌋A1

(21)

On the other hand, most of the SD detectors use decompositions as QR or Cholesky for the reduction ofelements involved in the calculations, but at the same time corrupt the structure of the mesh causing errorsin the detection [5] and therefore tend to increase the probability of error. To solve this problem, we haveproposed a correction algorithm, see Algorithm 2, which using data from the component R from the QRfactorization is able to predict an error and adjust the search angle as required.

Algorithm 2. Correction Algorithm Require: Q, R, y′ = QHy, η.Ensure: η.

1.Compute for all x j ∈ A1

dist = (y′k − Rk,kx j)2 (22)

2.Identify the two closest nodes to y′k and calculate

di f = |dist1 − dist2| (23)

3.If di f ≤ Pr, Assign new opening angle (η) otherwise continue with the value of

η before assigned.

end


Thus, we summarize the iLSD algorithm, including the correction, in Algorithm 3 which essentiallydoes not use the inverse of the channel and easily can be used in complex modulation schemes, e.g. highdata rate digital modulations as M-PSK.

Algorithm 3. iLSD Require: Q, R, y′ = QHy, η.Ensure: Estimated vector s.

1.(Initialize) k = M, y′k|k+1 = y′k.2.θy′k is the phase angle of y′k|k+1. Correction Algorithm

(Bounds) UBk = θyk + cos−1(η), LBk = θyk − cos−1(η) lower limit LBk and upper limit

UBk, build a constellation (AS D) using the elements of the L-PSK modulation

(constellation A1) that are within the limits. Lower level NIk = 1, upper level

NS k = LC, where LC is the number of elements in the constellation AS D.

3.If NIk ≤ NS k, (Increase NIk) xk = AS DNIk

, NIk = NIk + 1 and go to 5; Otherwise, go

to 4.

4.(Increase k) k = k + 1; If k = M + 1, go to 7; Otherwise, go to 3.5.(Decrease k) If k = 1, go to 6; Otherwise k = k − 1, y′k|k+1 = y′k −

∑Mj=k+1 Rk, jx j.

6.Solution found. Save x in XN and go to 3.7.Calculate

s = arg minx‖QHy − Rx‖2 (24)

where x ∈ XN and XN ⊆ AN . Terminate the algorithm.

end

4. Results

The scenario consist of a MIMO system with Tx=2 and Rx=2 and, as mentioned before, a Rayleighflat fading channel. For the complexity and performance analysis shown in this section, we establish theboundaries using linear (case of ZF and MMSE) and non-linear (case of ML) detectors. The former, de-fines the lowest complexity reference and the latter, the maximum performance achievable. The idea behindthis limits, is to establish a better judgment about the trade-off of performance/complexity given by the SDalgorithms primordially analyzed (CSD and iLSD) since linear detectors always offer simplicity in imple-mentation but offering low performance while the case of ML is totally opposite, optimum performance atthe highest cost of complexity. Additionally, we have to remember that in the case of SD (compared withML) the complexity is reduced and it gets similar performance (far from ZF and closer to ML).

In Figure 5 and Figure 6 is shown the performance results about all detection methods we have studiedthorough this paper for BPSK and 8-PSK modulations, respectively. Notice that in both modulations the SDrelated algorithms have a performance very close to that one given by the ML detector. The search methodproposed for LSD and iLSD is not affected at all by the modulation order as in the case of the non-linearmethods and compared with ML, CSD and iLSD quasi-match the performance in both modulation schemes.

The interesting part and main contribution of this paper comes when we have a look to the complexityinvolved in the detection methods we are interested most, CSD, iLSD, and ML. For this test, the complexityhas been measured in number of nodes visited when searching the solution of the general problem given in(6). For example, in the case of ML the algorithm searches along all possible solutions (i.e., nodes), whileCSD and iLSD search only in the nodes delimited by η. In this case, the effectiveness of η determine thesuccess and performance of the CSD and iLSD while saving valuable operations when solving (6).

Thus, Figure 7 and Figure 8 depict the number of nodes visited by ML, CSD, and iLSD. The point tohighlight here is, from performance tests it is shown that CSD and iLSD are effective as ML but in com-plexity used CSD and iLSD are far below from ML. Clearly, we have found a good trade-off of complexity


0 2 4 6 8 10 12 14 16 18 20

10−3

10−2

10−1

SNR [dB]

BE

R

ZFMMSEMLCSDiLSD

Fig. 5. BER performance analysis with a Rayleigh fadingchannel, Tx=Rx=2 and BPSK modulation.

10 11 12 13 14 15 16 17 18 19 20 2110

−3

10−2

10−1

SNR [dB]

BE

R

ZFMMSEMLCSDiLSD

Fig. 6. BER performance analysis with a Rayleigh fadingchannel, Tx=Rx=2 and 8-PSK modulation.

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

SNR [dB]

Nodes V

isited

MLCSDiLSD

Fig. 7. Complexity analysis in terms of visited nodes andSNR. Tx=Rx=2 and BPSK modulation.

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60

70

SNR [dB]

No

de

s V

isite

d

MLCSDiLSD

Fig. 8. Complexity analysis in terms of visited nodes andSNR. Tx=Rx=2 and 8-PSK modulation.


and performance. For example (case of Figure 7), while the CSD detector visit on average 2 nodes to obtaina performance comparable to ML, the iLSD detector is able to achieve the same performance visiting onaverage 1.2 nodes while ML visits 4. Additionally, despite the fact of increasing the universe of potentialsolutions when using high order modulations as 8-PSK, the trend remains the same as shown in Figure 8.

5. Conclusions

In this paper, we have presented a computationally efficient method to solve the problem of least squares,with a result in performance similar to the ML method. The proposed method have a far lower complexitythan the ML algorithm while keeping quite similar performance.

Our proposed method can be applied to MIMO wireless communications as a detector and due its lowcomplexity behavior it can increase the mobility of the receiver (e.g., less CPU processing, more battery life,etc.). This is because most of the SD algorithms double its complexity when used in higher order modulationschemes, e.g. high data rate digital modulation as M-PSK. On the other hand, the proposed scheme iLSD,same as CDS, do not require such complex to real conversion and therefore reduces the complexity to atleast the half of traditional methods. Moreover, although the iLSD algorithm has a similar structure to CSDand both have similar performance, the complexity makes the difference. This is because, thanks to thecorrection algorithm implemented in iLSD, the complexity is used in a more efficient way because theirangle of search is generally smaller.

Acknowledgements

This work was supported by Programa de Mejoramiento del Profesorado (PROMEP) from SEP undergrant PROMEP/103.5/10/7746 and UASLP in Mexico.

References

[1] T. Rappaport, Wireless Communications: Principles and Practice, New Jersey: Prentice-Hall, 1996.[2] G. J. Foschini, M. J. Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless

Personal Communications 6 (1998) 311–335.[3] I. E. Telatar, Capacity of multiple antenna gaussian channels, European Transactions on Communications 10 (1999) 585–595.[4] J. Jalden, B. Ottersten, On the complexity of sphere decoding in digital communications, IEEE Transactions on Signal Processing

53 (2005) 1474 – 1484.[5] B. Hassibi, H. Vikalo, On the sphere-decoding algorithm i. expected complexity, IEEE Transactions on Signal Procesing 53 (8)

(2005) 2806–2818.[6] U. Pineda Rico, Link Optimisation for Mimo Communication Systems, LAP Lambert, 2010.[7] M. Grotschel, L. Lovasz, A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer Verlag, 1993.[8] U. Fincke, M. Pohst, Improved methods for calculating vectors of short length in a lattice, including a complexity analysis,

Mathematics of Computation 44 (170) (1985) 463–471.[9] M. Damen, On maximum likelihood detection and the search for the closest lattice point, IEEE Transactions on Information

Theory 49 (2003) 2389 – 2402.[10] H. Vikalo, B. Hassibi, On sphere decoding algorithm ii. generalizations, second-order statistics, and applications to communica-

tions, IEEE Transactions on Signal Processing 53 (2005) 2819–2834.[11] J. Castillo Leon, U. Pineda Rico, E. Stevens-Navarro, R. Aguilar-Gonzalez, Complexity improved sphere decoder for highly

correlated and los channels, Research in Computing Science, Advances in Computer Science and Electronic Systems 52 (2011)308–316.

[12] J. Castillo Leon, Decodificador esferico de complejidad reducida para sistemas de comunicacion mimo, Master’s thesis, Univer-sidad Autonoma de San Luis Potosi (2012).

[13] B. Hochwald, S. Brink, Achieving near capacity on a multipleantenna channel, Bell Laboratories.

Anexos

6.2 Anexo B



16

Procedia Technology 7 ( 2013 ) 54 – 60

2212-0173 © 2013 The Authors. Published by Elsevier Ltd.Selection and peer-review under responsibility of CIIECC 2013doi: 10.1016/j.protcy.2013.04.007

2013 Iberoamerican Conference on Electronics Engineering and Computer Science

MIMO Detector for Uncoded Symbols based on ComplexSphere Decoder and Manhattan Metric

Juan Francisco Castillo-Leon∗, Ulises Pineda-Rico, Enrique Stevens-Navarro

Facultad de Ciencias, Universidad Autonoma de San Luis Potosı, Av. Salvador Nava s/n, Zona Universitaria, 78290, San Luis Potosı,

Mexico

Abstract

For communication standards with high transfer rates (WiMAX, WiFi, LTE), which use MIMO (Multiple Input - Mul-

tiple Output) systems, detectors with reduced computational complexity that achieve a good Bit Error Rate (BER)

performance are of great interest. In the literature, it has been recognized the maximum likelihood (ML) detector as the

optimum, but this algorithm experiences an exponential complexity making it an impractical alternative for implemen-

tation. However, there are also alternatives such as the K-best and sphere decoder (SD) which can reach a quasi-ML

performance with lower computational complexity. This paper presents a variation of the SD algorithm based on the

Complex Sphere Decoder and K-best algorithm, named as KLSD, which limits the number of searching points during

the predetection and has a performance similar to that given by the algorithm Fixed Complexity Sphere Decoder (FSD)

without channel ordering. Furthermore, for the calculation of the weights of each candidate node is proposed to replace

the use of the Euclidean distance by the Manhattan Metric to reduce the number calculation performed. When compar-

ing performance and complexity against others algorithms, it can be seen that a similar performance without increasing

its complexity. Additionally, the results show that the change of metric, does not affect the performance of the proposed

algorithm, so it is considered a feasible complexity reduction scheme.

c© 2011 Published by Elsevier Ltd.

Keywords: Sphere Decoder, K-best Decoder, Maximum Likelihood, Multi-Antenna Systems, Wireless

Communications.

1. Introduction

In recent years the use of MIMO (Multiple Input - Multiple Output) systems has become a constant in

the communication systems of last generation, as Wimax or LTE, this is due to theoretical analysis presented

in [1], [2], this shows that a significant increase in capacity could be achieved under certain conditions of

use of multiple antennas at the transmitter and receiver. The increase in capacity can be exploited to increase

the transfer rate in MIMO systems using Spatial Multiplexing techniques [3].

∗Corresponding author. Tel. + 52 (444) 8262491 ext. 2954

Email addresses: [email protected] (Juan Francisco Castillo-Leon ), u [email protected] (Ulises

Pineda-Rico), [email protected] (Enrique Stevens-Navarro)

Available online at www.sciencedirect.com

© 2013 The Authors. Published by Elsevier Ltd.Selection and peer-review under responsibility of CIIECC 2013

55 Juan Francisco Castillo-León et al. / Procedia Technology 7 ( 2013 ) 54 – 60

Also, it is widely known that in terms of bit error rate (BER) the detector of Maximum Likelihood (ML)

compared to traditional linear detectors as Zero Forcing (ZF) and Minimum Medium Square Error (MMSE).

Additionally, its superior performance makes it be considered as optimal, however in terms of complexity

this detector has as big drawback its complexity. The complexity of the ML detector increases exponentially

with respect to the number of antennas in the MIMO array [4].

In recent years, several researchers have dedicated their efforts to improve or create detection algorithms

for MIMO systems. Among them, stands out the algorithms known as K-best and sphere decoder (SD),

they are able to obtain a similar performance to ML with a reduced complexity [5], [6], [7]. The SD

is based on the choice of the candidate nodes contained within a radius r, it also reduce the number of

required operations for detection using the decomposition of the channel matrix (QR, Cholesky), generating

a triangular matrix that eases the process of solve the system of simultaneous equations that represents the

detection process. An important feature is that the SD and K-best algorithms perform the detection level by

level, where the levels are defined by the number of antennas at the receiver. For example, a system with 4

receiving antennas is a system of 4 levels [4]. The enumeration process in the SD algorithms is an important

part to reduce complexity. The process of selecting the candidate nodes can increase or decrease the number

of operations performed, the number of nodes chosen and the propagated error from one to another level.

The main forms of enumeration are: Schnorr - Euchner [8] and Fincke - Pohst [9]. Thus, an alternative to

reduce complexity is complexity from the enumeration process is, setting a limit on the number of nodes to

choose as candidates as in algorithms K-best [5], [6],[7].

In this paper we propose three modifications to the ILSD (Improved Complexity Sphere Decoder) algo-

rithm [10]:

1. Independence on the noise parameter for the selection of candidate nodes, as occurs in algorithms

such as the K-best and FSD.

2. Substitution of the Euclidean distance by the Manhattan metric to reduce complexity.

3. Reduction of the search space in the predetection.

The rest of paper is organized in the following sections. Section II presents the model of the MIMO

communication system. The KLSD strategy is described in Section III, along with a comparison with other

SD proposals of the literature. Section IV introduces the performance results, and in Section V, we present

the conclusions and future work.

2. System Model

Consider a MIMO communication system with M antennas transmitting and N receiving antennas. The

transmitter sends symbols chosen from a constellation AL ∈ CM , which is defined by the existing symbols

in a L-QAM modulation. The received signal vector is given by:

y = Hs + n (1)

where H is a matrix of M × N complex values representing the channel, and hi, j is the complex transfer

function of the transmitter j to receiver i, s = [s1, s2, . . . , sM]T is the vector of transmitted symbols with

E[s] = 0 and E[sHs] = (1/M)IM , where (·)H denotes the conjugate transpose operation, n = [n1, n2, . . . , nN]T

is the vector of Gaussian White Noise with a variance σ2 and y = [y1, y2, . . . , yN]T is the vector of received

symbols. In addition, we will assume that the transmission is organized in bursts of symbol duration T

(T >> 1), where H is constant during the burst but changes randomly from one burst to the next. Also it is

assumed that the channel estimation is ideal, so the channel is perfectly known by the receiver, and modeled

as a flat fading Rayleigh channel, with a E[|hi, j|2] = 1, ∀i, j. [11].

3. K - Less Complexity Sphere Decoder (KLSD)

In recent years SD algorithms emerge as a viable alternative for use in the new technologies of high-

speed wireless communications. Based on the system model we define the ML estimate of the transmitted

vector sML as shown below


Fig. 1. Idea behind the algorithm SD, for a constellation A16 (16-QAM), distorted by the channel H.

sML = arg minx∈AL‖ y −Hx ‖2

2(2)

this type of detection algorithm is practically impossible to implement in MIMO communication systems

that utilize high-order modulation (e.g. 16-QAM, 64-QAM).

[4] and [12] presents the General form of the Sphere Decoder (GSD), based on the Fincke - Pohst

enumeration, as a tool for achieving the detection at low complexity in MIMO wireless communication

systems. The basic idea of the GSD algorithm is to search only those grid points that are within a sphere

centered at the given vector y and radius r as exemplified in Figure 1.

The received signal without noise (point represented by vector Hx in the complex space CM) is within

the sphere of radius r if and only if.

r2 ≥ ‖y −Hx‖22

(3)

where y ∈ CN is the received vector and x ∈ CM is the possible vector transmitted. Additionally, make the

decomposition QR of the channel matrix H ∈ CM×N , i.e.

H = QR (4)

where R is an upper triangular matrix of dimensions N × N with a the diagonal elements of all positive and

Q is an orthogonal matrix of dimensions M × N.

Considering that in this paper we focus on a MIMO system, where the number of transmitters and

receivers is the same (M = N, this simplifies the mathematical representation of the system). The condition

set by (3) for a system, where M = N can therefore can be written as

r2 ≥ ‖y −QRx‖22= ‖QHy − Rx‖2

2(5)

Defining y′ = QHy, (3) can be written as

r2 ≥ ‖y′ − Rx‖22

(6)

considering that R is an upper triangular matrix, the inequality established by (8) can be re-written as

r2 ≥ (y′M − RM,M xM)2 + (y′M−1 − RM−1,M xM − RM−1,M−1xM−1)2 + . . . (7)

where y′M is the M-th element of vector y′, RM,M represents the element of the position (M,M) of the matrix

R and xM is the M-th element of vector x. It can be seen that for an antenna array of dimension M = 1, (7)

depends only of xM , for an antenna array of dimension M = 2, (7) depends only of xM y xM−1,and so on.

From inequality established by (7), it follows that a necessary condition for Hx is within the sphere, is that

r2 ≥ (y′M − RM,M xM)2, or what is the same, that the component xM , belongs to the interval [4],

⌈− r + y′M

⌉≤ RM,M xM ≤

⌊r + y′M

⌋(8)

where · denotes the ceiling function, and · the floor function.

The inequality (8) restricts the search to the constellation points L-QAM contents in a interval search.


Fig. 2. Division in concentric rings of a constellation symbols, with 16-QAM modulation.

Let y′M = aMeiθM y RM,M xM = bMeiθM , where θM and θM are phase angles of y′M y RM,M xM respectively,

which form the vectors θ y θ, also aM > 0 and bM > 0 are elements of the vectors a = [a1, a2, . . . , aM]T and

b = [b1, b2, . . . , bM]T , these vectors contain the radius of the circles formed by the L-QAM constellation

symbols. To generate the search range that ensures the best results, we define now the variable η, this may

depend on the values of the noise variance or a fixed value. Once stated these elements, we can develop (8)

to obtain

⌈θM − cos−1 η

⌉≤ θM ≤

⌊θM + cos−1 η

⌋(9)

9 shows that if η > 1, then search the disc contains no point of L-QAM constellation. If η < −1, then search

disk includes the entire constellation.

3.1. KLSD AlgorithmKLSD algorithm looks forward to take advantage of the detection obtained by the algorithm iLSD [13].

In this algorithm the search area is defined by an opening angle and a stage of rings selection, see Figure 2.

This operation is simple, the algorithm has to calculate the distance of y′M to the origin and compare it to the

radius generated by RM,M xi [10].

The KLSD algorithm also uses a stage K-best, in order to minimize the number of nodes visited. After

defining the search area by the opening angle and the selection of rings, it choose the best K points level by

level. The best points are elected by the minimum euclidian distance between the received vector y′M and

the possible vector have sent RM,M xi.

dK−best = ||y′M − RM,M xi||2 where i = 1, 2, 3, 4 . . . , L (10)

3.2. Stage of PredetectionIn the SD algorithms to reduce the number of candidate nodes is performed a process of predetection.

In one embodiment of the algorithm, the Euclidean distance ‖y′M −RM,M si‖2 , needs to be calculated for each

point in the constellation AL. Therefore, to make the predetection for a 4× 4 MIMO with signals modulated

16-QAM and assuming that there is a distribution of points selected K = [1 1 1 16] for each level, it must be

calculated 16 distances in the first level and 256 for the remaining levels. In total 784 distances should be

calculate. This part of the algorithm increases enormously the computational complexity.

This can be generalized for different distributions of points, as shown below

NoDE = L + K(1)L + K(1)K(2)L + K(1)K(2)K(3)L + ... + K(1)K(2)...K(M)L (11)

where K(·) represents an element of the vector K and NoDE indicates the number of Euclidean distances

calculated.

In order to reduce the number of calculations, we always remove farthest ring from the received point

perspective. If a node is located at a distance far away, it is logical that is somewhere within the best


K nodes and therefore can be discarded without performing any calculations. The algorithm KLSD, define

their search area through rings, for the case of a 16-QAM modulation will be 3 rings. By having this division

into rings, you may discard one or two rings easily and therefore exclude from 4 to 12 points without having

to calculate its distance.

3.3. Change metric norm-2 to norm-1

In this paper, we use the norm-1, to reduce the number of mathematical calculations performed. By

replacing the Euclidean distance (norm-2) with a Manhattan distance (norm-1), the ML calculation in (2)

can be modified as

sML = arg minx∈AL‖y −Hx‖2

1(12)

where ‖z‖1=∑M

i=1 |zi|, the same way (8) can be substituted by

r2 ≥ |y′ − Rx|21

(13)

then, taking into consideration that R is an upper triangular matrix, established by inequality (8) can again

be written as

r2 ≥ |y′M − RM,M xM | + |y′M−1 − RM−1,M xM − RM−1,M−1xM−1| + . . . (14)

As shown, the operations performed depends only of summations, with the exception of the term r2 ,

there is no squared term.

4. Results

In recent studies [14] [15], has been shown that it is possible to find a quasi-ML solution, without the

need to define the search area. This brings the advantage that not required noise estimate in the system and

hence this stage is not implemented in hardware, results in one less stage in the detection hardware.

The complexity of the algorithms has been selected based on the paper [5], which shows that a detector

with a deviation of 7dB in performance to an error rate of 10−5, can be transformed into a detector with

quasi-ML performance, adding a channel ordering stage, described and implemented by Barbero in [5].

In Figure 3 shows the ML detector performance, which can be considered as the upper limit for the

studied detectors. We plotted the FSD detector without channel ordering as a reference for analyzing the

performance of the algorithm ILSD proposed in [13].

The ILSD algorithm has a correction phase stage, which helps to reduce the error by adjusting the search

area on the first level, with a C = 0.1 approximately 10% of detections 16 points are selected, the angle of

opening was defined by α = 0.8 equals approximately 36 degrees of opening. The performance of the

KLSD and ILSD algorithms is similar. They are in the proper range to be considered candidates for ML

performance, when implemented the channel ordering.

In the performance comparison analysis using different metrics, it was observed that the difference

between using the Euclidean metric y the Manhattan metric is minimal. In the Figure 4 shows a small BER

improvement by using the Manhattan metric.

In matter of complexity there are great differences between one method and another, in Figure 5 em-

phasizes that the lower limit FSD [1 1 1 16], requires only 16 nodes and maintains a fixed complexity. The

algorithms that define the number of nodes visited in advance, have the same performance but with reduced

complexity.

The algorithms of complexity fixed visit only 32 nodes and even the performance of algorithms that

perform a choice of search area, in this case with a search area fixed (iLSD) need to visit a minimum of 130

nodes and a maximum of 280 nodes to match the performance of the algorithms KLSD and FSD.


0 5 10 15 20 25 30

10−4

10−3

10−2

10−1

100

SNR

BE

RN = M = 4, 16 QAM

19 20 21

10−4

10−3

MLFSD [1 1 1 16] No orderingKLSD [1 1 2 16] ILSD C = 0.1 α = 0.8

Fig. 3. Performance of 4× 4 MIMO system, over a Rayleigh

channel, with a metric norm-2

0 5 10 15 20 25 30

10−4

10−3

10−2

10−1

100

SNR

BE

R

N = M = 4, 16 QAM

19 20 21

10−4

10−3

MLFSD [1 1 1 16] No orderingKLSD [1 1 2 16] Metric 2KLSD [1 1 2 16] Metric 1

Fig. 4. Performance comparison, KLSD (norm-1) vs KLSD

(norm-2)

5. Conclusions and future work

In this paper we have presented the algorithm called KLSD, its performance is similar to the KSD/FSD

algorithms without channel ordering and its complexity is not dependent on the noise level. The algorithm

examines a fixed number of nodes and replaces the Euclidean distance calculation by the Manhattan distance

in order to reduce the number of mathematical operations at the detection stage. Additionally, this algorithm

limits the predetection area for discarding in advance the mesh points without compute the distances.

Among the key features KLSD presents we have: its fixed complexity, the use of Manhattan metric

and the limitation of the predetection area. Therefore, KLSD can be considered a valid alternative for

implementation and integration in future wireless communication systems.

Finally, the main lines of work in progress for KLSD consider: the study of the concept of channel

ordering, the optimization of the predetection stage and the detection of signals with high-order modulations

(e.g. 64 QAM).

References

[1] I. E. Telatar, Capacity of multiple antenna gaussian channels, European Transactions on Communications 10 (1999) 585–595.

[2] G. J. Foschini, Layered space-time architecture for wireless communication in a fading environment when using multi-element

antennas, Journal Bell Labs Techn. 6 (1996) 41–59.

[3] P. W. Wolniansky, G. J. Foschini, G. D. Golden, R. A. Valenzuela.

[4] B. Hassibi, H. Vikalo, On the sphere-decoding algorithm I. expected complexity, IEEE Transactions on Signal Procesing 53 (8)

(2005) 2806–2818.

[5] L. G. Barbero, J. S. Thompson, Fixing the complexity of the sphere decoder for MIMO detection, IEEE Transactions on Wireless

communications 7 (6) (2008) 2131–2142.

[6] Z. Guo, P. Nilsson, Algorithm and implementation of the K-best sphere decoding for MIMO detection, IEEE Journal on Selected

Areas in Communications 24 491–503.

[7] J. W. Choi, B. Shim, A. C. Singer, Algorithm and implementation of the K-best sphere decoding for MIMO detection, IEEE

Journal on Selected Areas in Communications 58 (3) 1518–1533.

[8] C. Schnorr, M. Euchner, Lattice basis reduction: Improved practical algorithms and solving subset sum problems, Lecture Notes

in Computer Science 529 (1994) 68–85.

[9] U. Fincke, M. Pohst, Improved methods for calculating vectors of short length in a lattice, including a complexity analysis,

Mathematics of Computation 44 (170) (1985) 463–471.

[10] J. F. Castillo-Leon, Decodificador Esferico de Complejidad Reducida para Sistemas de Comunicacin MIMO, Tesis de Maestrıa

en Ingenierıa Electronica con orientacion en Telecomunicaciones, Facultad de Ciencias, UASLP, 2012.

[11] U. Pineda-Rico, Link Optimization for MIMO Communication Systems, LAP Lambert, 2010.


Fig. 5. Comparison, Nodes Vs SNR per bit, over a Rayleigh channel

[12] H. Vikalo, B. Hassibi, On sphere decoding algorithm II. generalizations, second-order statistics, and applications to communica-

tions, IEEE transactions on signal processing 53 (2005) 2819–2834.

[13] J. F. Castillo-Leon, U. Pineda-Rico, E. Stevens-Navarro, Complexity-improved sphere decoder for MIMO systems using PSK

modulations, The 2012 Iberoamerican Conference on Electronics Engineering and Computer Science 3 (2012) 52–60.

[14] S. Han, T. Cui, C. Tellambura, Improved K-best sphere detection for uncoded and coded MIMO systems, IEEE Wireless com-

munications letters, accepted for publication.

[15] B. seok Kim, H. Kim, K. Choi, An adaptive K-best algorithm without SNR estimation for MIMO systems, Proceedings of the

67th IEEE Vehicular Technology Conference, VTC Spring 2008, 11-14 May 2008, Singapore 2008.

Anexos

6.3 Anexo C



17

IEICE TRANS. ??, VOL.Exx–??, NO.xx XXXX 200x1

PAPER

A Semifixed Complexity Sphere Decoder for Uncoded Symbols forWireless Communications

Juan FRANCISCO CASTILLO-LEON †, Marco CARDENAS-JUAREZ †, Nonmembers, Ulises PINEDA RICO†,and Enrique STEVENS-NAVARRO †, Members

SUMMARYThe development of high data rate wireless communications systems

using Multiple Input - Multiple Output (MIMO) antenna techniques re-quires detectors with reduced complexity and good Bit ErrorRate (BER)performance. In this paper, we present the Semi-fixed Complexity SphereDecoder (SCSD), which executes the process of detection in MIMO sys-tems with a significantly lower computation cost than the detectors: SphereDecoder (SD), K-best, Fixed Complexity Sphere Decoder (FSD), andAdaptive Set Partitioning (ASP). Simulation results show that when theSignal-to-Noise Ratio (SNR) is less than 15 dB, the SCSD reduces thecomplexity by up to 90% respect to SD, up to 60% respect to K-best orASP, and by up to 90% respect to FSD. In the proposed algorithm, theBER performance does not show significant degradation and therefore, canbe considered as a complexity reduction scheme feasible forimplementingin MIMO detectors.key words: Sphere Decoding, Wireless Communications, Multi-AntennaSystems, Maximum Likelihood, Complexity

1. Introduction

In recent years, the use of the (Multiple Input - MultipleOutput) MIMO antenna techniques has become a constantin the communication systems of last generation, such asWiMAX and 3GPP LTE. This is due to the theoretical anal-ysis presented in [1] and [2], which demonstrates that a sig-nificant increase in capacity can be achieved under certainconditions by the use of multiple antennas at the transmitterand receiver, known as spatial multiplexing techniques. Theincrease in capacity can be exploited to increase the transferrate and/or the coverage of the system [3]. In the literature,it is widely known that in terms of Bit Error Rate (BER) theMaximum Likelihood Detector (MLD), is the best detector,since it is considered optimal. However, this detector hasa main and important drawback because the level of com-plexity increases in an exponential way with the increasingof the modulation order and/or the number of antennas inthe MIMO array [4], [5]. For the past few years, many re-searchers have focused their efforts in improving the detec-tion algorithms for MIMO systems, usually trying to reacha quasi-MLD performance with a significantly lower com-plexity. Under these assumptions, the Sphere Decoders (SD)are considered the most promising candidates for implemen-tation as detectors in practical MIMO systems [6]–[8]. Thecomplexity reduction of the SD algorithms can be achievedthrough clever modifications of the stages that comprise the

Manuscript received April 7, 2014.†The author are with Facultad de Ciencias, Universidad

Autonoma de San Luis Potosı (UASLP), San Luis Potosı, Mexico.DOI: 10.1587/trans.E0.??.1

detection process. In the preprocessing stage, it includestheprocesses of: channel ordering, lattice reduction, and de-compositions of the channel matrix. In the processing stagea search tree is performed and this is defined by heuristicand/or probabilistic techniques.

In this paper, we propose a detector with a fixed up-per bound complexity called Semifixed Complexity SphereDecoder (SCSD). The proposed detector maintains a near-optimal performance and at the same time keeps a reducedcomplexity. The SCSD is based on the K-best [7], [9]–[12],Fixed Complexity Sphere Decoder (FSD) [8] and SD [5],[13] algorithms. Nevertheless, different from SD which ex-hibits a variable complexity with a random upper bound ofcomplexity, SCSD limits the maximum upper bound to aconstant, as in the case of K-best and FSD. Moreover, incontrast to K-best and FSD, the SCSD limits the searcharea. Indeed, by limiting the search area, the complex-ity is reduced through the means of early-termination andadding a stage of pre-detection. The proposed detector re-duces the number of candidate nodes (i.e., symbols) throughthe division of the main modulation constellation in sub-constellations. Also, during the search process, the SCSDimplements the method known as early termination [14],thus stopping the analysis of a candidate vector when thecost is greater than the best cost. In the literature, otherlow complexity proposals that also limits the search areacan be found such as the Adaptive Set Partitioning (ASP)[15]. However, different to SCSD, the search area limi-tation of ASP is dependent on the Signal-to-Noise Ratio(SNR) and the communication channel characteristics. Inthis manuscript, the complexity is measured in terms of vis-ited nodes and FLOPS (FLoating-point Operations Per Sec-ond). The proposed algorithm, in comparison with SD, K-best, FSD and ASP, shows a significant reduced complexity(up to 95%) when the SNR is lower than 20 dB. This SNRrange is of interest in high data rate wireless communica-tions systems.

The rest of paper is organized as follows. Section 2describes the system model considered, Section 3 explainsthe main concepts of the MLD, SD, K-best, FSD and ASPalgorithms. Section 4 describes the proposed algorithm andSection 5 discusses the performance and complexity resultsof the SCSD for different configurations. Finally, Section 6includes the conclusions and future work.

Notation: Boldface lower-case and upper-case lettersdenote vectors and matrices, respectively.I M and0M rep-

Copyright c© 200x The Institute of Electronics, Information and Communication Engineers

2IEICE TRANS. ??, VOL.Exx–??, NO.xx XXXX 200x

resent the identity matrix and zero matrix of sizeM × M.The superscripts (·)T and (·)H denote transpose and Hermi-tian operators. The statistical expectation is denoted byE[·].R andC stands for the set of real and complex numbers, re-spectively.

2. System Model

Consider a spatially multiplexed MIMO system withN an-tennas at the transmitter andM antennas at the receiver. Thereceived signal vectory is given by:

y = Hs+ n, (1)

where H is a matrix of M × N values representing thechannel, each elementhi, j of H is the transfer functionof the transmitterj to receiveri, s = [s1, s2, . . . , sN ]T isthe vector of transmitted symbols withE[sHs] = (1/N)I N ,n = [n1, n2, . . . , nM]T is the vector of independent and iden-tically distributed complex Additive White Gaussian Noise(AWGN) with a varianceσ2 = No andy = [y1, y2, . . . , yM]T

is the vector of received symbols. In addition, we assumethat the transmission is organized in bursts of symbol dura-tion T (T >> 1), whereH is constant during the burst butchanges randomly from one burst to the next. Additionally,we assume that the channel estimation is perfectly known bythe receiver, and modeled as a flat fading Rayleigh channel,with a E[|hi, j|2] = 1,∀i, j [16]. The transmitter symbols perantenna are chosen from aΩ ∈ C constellation, defined bythe existing symbols in aL-QAM modulation.

The proposed algorithm is applicable directly to a sys-tem of equations of complex numbers. Indeed, this featurebrings the advantage of a more efficient hardware implemen-tation [8]. Additionally, in order to provide the necessarybasis for understanding how the SCSD algorithm performthe detection, we introduce the basic concepts of the detec-tion algorithms: MLD, SD, K-best, FSD and ASP.

3. Detectors

The detection algorithms studied in this paper belong tothe family of algorithms known as branching and pruning,which can be represented by a tree structure withM levels.The tree is formed by the nodes (black or white dots in thefollowing figures) representing an element of the candidatevector and the branches (lines connecting between nodes)with a cost associated to it. The tree is generated by a pro-cess that starts at levelM from a root node (parent node),from which L branches are expanded, each of them with achild node. If any of the child nodes do not satisfy the con-straints proposed by the algorithm, they are removed (i.e.,pruned). Then,k = k − 1 and goes to the next level in thetree search, where the survivor nodes become parent nodesand the above process is repeated for each level until thelowest cost candidate vector (estimated vector) is obtained.

3.1 Maximum Likelihood Detector (MLD)

Based on the system model, we define the estimated vector

Fig. 1 Search tree structure of the MLD for a 3× 3 MIMO system withmodulated symbols 4-QAM.

by the MLD as:

sMLD = arg minx∈ΩM‖y − Hx‖2, (2)

where,x ∈ ΩM is the candidate vector which probably hasbeen sent. The received signal without noise is representedby vectorHx and sMLD represents the estimated vector bythe MLD algorithm, which is considered optimal [4], [5].Figure 1 shows graphically the detection process realizedby the MLD, in a MIMO system with an antenna array of3× 3 and 4-QAM modulation.

3.2 Sphere Decoder (SD)

The motivation behind the SD algorithm is to reduce thecomplexity of the MLD [5], [13], [17]. Essentially, the ideais to search the estimated vector by SD (sS D) only amongthe candidate vectors belonging to a subset of the total can-didates, defined by a sphere centered at the given vectoryand radiusr. Hence, the received signal without noise iswithin the sphere of radiusr if and only if:

r2 ≥ ‖y − Hx‖2. (3)

To reduce the complexity, the standard preprocessingis the decomposition of the channel matrix, using theQRfactorization, and then condition set by (3), can be writtenas:

r2 ≥ ‖y −QRx‖2 = ‖QHy − Rx‖2, (4)

whereR andQ are components of the factorization of thechannel matrixH. R represents an upper triangular matrixof dimensionsM×M with diagonal elements all positive andQ represents a square orthogonal matrix with dimensionsM × N. Definingy′ = QHy, now (4) can be written as:

r2 ≥ ‖y′ − Rx‖2, (5)

and considering thatR is an upper triangular matrix, the in-equality established by (5) can be re-written as:

r2 ≥ (y′M − RM,M xM)2

+(y′M−1 − RM−1,M xM − RM−1,M−1xM−1)2 + . . . , (6)

where y′M represent an element of the vectory′ =[y′1, y

′2, . . . , y

′M]T , RM,M is the element in the position (M,M)

of the matrix triangularR, andxM is theM − th element of

FRANCISCO CASTILLO-LEON et al.: A SEMIFIXED COMPLEXITY SPHERE DECODER FOR UNCODED SYMBOLS FOR WIRELESS COMMUNICATIONS3

Fig. 2 Search tree structure of the SD detector with a radiusr for a 3× 3MIMO system with modulated symbols with 4-QAM.

x. It can be seen that the solution to (5) for an antenna arrayof dimensionM = 1 depends only ofxM and for an an-tenna array of dimensionM = 2 depends ofxM andxM−1.Therefore, (5) can be solved through the individual estima-tion of elements ofx, by building a search tree ofM levelsand choosing candidate nodes belonging toΩ constellation[18]. The SD algorithms traverse the search tree, startingfrom parent nodes and visiting just child nodes that satisfythe condition:⌈−r + y′M−1|M

RM−1,M−1

⌉

≤ xM−1 ≤⌊ r + y′M−1|M

RM−1,M−1

⌋

, (7)

where⌈·⌉ denotes the ceiling function and⌊·⌋ the floor func-tion. This algorithm is described graphically in Figure 2.It is important to note that the complexity of this detectionmethod is variable and it depends on the system conditions,such as SNR and the communication channel characteris-tics.

3.3 K-best Algorithm

The K-best algorithm proposed in [7], is a low complexitydetection technique, whose main characteristic is the limi-tation of the surviving nodes in each levelk by a constantK (i.e., Kk). The process of detection starts from the levelk = M and then, for each parent node in levelk, L childnodes are expanded, which involves the generation ofKkLchild nodes and therefore the calculation of same number ofassociated costs. From this child nodes, only theKk nodeswith the lowest costs are selected. Finally, in the next levelk − 1 the selected nodes become parents nodes and the pro-cedure is repeated, untilk = 1, where the candidate vectorwith the lowest cost will be selected as the estimated vectorby K-best (sK−best), this procedure is shown in Figure 3. Thecost of thei candidate vector from the root to the levelk canbe calculated by:

costk,i =M∑

j=k

(y′j −M∑

m= j

R j,mxm,i), i = 1, . . . ,Kk. (8)

3.4 Fixed Complexity Sphere Decoder (FSD)

The FSD is similar to SD and K-best algorithms. The pro-cess of detection is based in costs, it starts from levelk = M

Fig. 3 Search tree structure of the K-best detector for a 3× 3 MIMOsystem with modulated symbols 4-QAM.

and descends tok = 1. In order to reduce the error propa-gation, this algorithm includes a stage of channel ordering,where the elements of the triangular matrixR, Ri,i satisfy:

E[R2M,M] < E[R2

M−1,M−1] < . . . < E[R21,1]. (9)

If we definep as the number of child nodes extendedin the levelk (i.e., pk) from a given parent node that satisfies(5), with 1≤ pk ≤ L, we obtain from (9) that:

E[pM] ≥ E[pM−1] ≥ ... ≥ E[p1]. (10)

Using the established in (10), in order to achieve aquasi-MLD performance, FSD allocates a fixed number ofcandidate nodes (which are less than MLD). Thus, it facili-tates its implementation in hardware [8]. Figure 4 shows thesearch tree structure of the FSD.

3.5 Adaptive Set Partitioning (ASP)

The ASP algorithm proposed in [15] and depicted in Fig-ure 5, is a low complexity detection technique based in theK-best and FSD algorithms. It defines a search area bypartitioning theΩ constellation into|Ω|/pk disjoint subsets,wherepk is estimated by:

pk =

min

√2σ

d′min,k

Q−1f unc

Prtarget

4

2

, p(max)

2

, (11)

whereQ−1f unc(·) is the inverse ofQ f unc(·) which is the right

tail probability of the normalized Gaussian distribution,d′min,k = Rk,kd, d is the minimum distance between sym-bols of the transmitted constellationΩ, (·)(max) stands forthe maximum value and Prtarget is a target probability [15].At the same time in which the detection process is carried

Fig. 4 Search tree structure of the FSD detector for a 3×3 MIMO systemwith modulated symbols 4-QAM.


Fig. 5 Search tree structure of the ASP detector for a 3×3 MIMO systemwith modulated symbols 4-QAM.

out, the number of selected child nodes per level is limitedto 1≤ Kk ≤ K(max) using the constraint:

costk,i+1 > costk,1 + σ2 ln

1− Prtarget

Prtarget

. (12)

In the same manner as K-best, the candidate node with thelowest cost will be selected as the estimated vector (sAS P).

4. Semi-fixed Complexity Sphere Decoder (SCSD)

The proposed algorithm reaches a quasi-MLD performancewhile keeping low complexity levels. SCSD is based in theFSD and adds the stages of predetection and early termina-tion which functions are to limit the search area and thusreduce complexity. They are independent of the noise andthe channel characteristics.

Regarding the complexity, when achieving the best per-formance, the maximum number of visited nodes (in theworst case) is fixed and it can be computed by:

complexity= LM∑

i=1

M−1∏

j=M−i

p j+2, pM+1 = 1. (13)

4.1 Channel Preprocessing and Ordering

The channel preprocessing is the stage where the knowndata are analyzed and modified to help reducing the com-plexity of the detection. This process must be of low com-plexity and should guarantee the decrease in the overallcomplexity of the detection process [3], [8], [19].

In order to reduce the error propagation from the levelk to level k − 1, the SCSD algorithm includes the stage ofchannel ordering, as proposed in [3], [8]. This stage deter-mines the order in which the detection of the symbols mustbe performed and it is described in the Algorithm 1.

4.2 Predetection

The predetection is the process which aims to reduce thecomplexity by limiting the search area in order to reducethe number of nodes to be analyzed at each level of thetree detection. This process is done through the selectionof a subset of candidate nodes ofΩ, in which shall be con-tained the solution node. In order to reduce the search area,

SCSD dividesΩ in concentric rings, for later to select theRik concentric rings closest to an auxiliary node (defined byyres/Rk,k) and finally to save the nodes of each ring in a aux-iliary constellation namedΘ. To select theRik closest ringsit is necessary to know the existing Euclidean distance (ED)between the auxiliary node and theNrings rings:

EDi =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

yres

Rk,k

∣

∣

∣

∣

∣

∣

− di

∣

∣

∣

∣

∣

∣

∣

, i = 1, . . . ,Nrings, (14)

whereyres = y′k +∑M

j=k Rk, jx j, Nrings is the number of ringsdefined by theL-QAM modulation (Nrings = 3 for 16-QAMandNrings = 9 for 64-QAM) andd = [d1, . . . , dNrings] rep-resents the vector with the ED between the origin to theithring. The previous issue is shown graphically in the Figure6. Also is important to note that, the number of selectedrings should ensure a lower level of complexity without af-fecting the performance in terms of BER.

Algorithm 1 Channel Ordering(H)Initialize: k = M, Vk = [01, 02, . . . , 0M ]while k , 0 do

Calculate the matrixH†i = (HHi Hi)−1HH

i , whereHi = Hki+1 is thechannel matrix with theki+1 columns selected in the previous itera-tions zeroed.The columns ofH and signals must be ordered according to:

Vk = arg maxj

[H†i ] j

where, j represent the index of a row ofH†ik = k − 1

end whilereturn (V,Hord)← The vector with the original sequence of the channelV and the reordered matrixH

4.3 Early Termination

Early termination reduces the number of visited nodes dur-ing the detection process. It is based on the fact that eachcandidate vector in the search tree of the proposed algorithmhasM partial costs associated [14]. The early terminationmethod compares the cost (best cost) obtained for the firstcandidate vector that has been analyzed with the partial costof the current partial candidate vector at the levelk. If the

Fig. 6 Division in concentric rings of a constellation of symbols modu-lated with 16-QAM.


best cost has been exceeded, the analysis of this candidatevector is stopping and the process is repeated for the nextcandidate. In the contrary,sS CS D is updated, its cost is con-sidered the new best cost and the process is repeated for thenext candidate. The process continues with the next candi-date and so on until finish all the candidates.

Therefore, avoiding redundant operations causes com-plexity reduction. Nevertheless, it also turns the algorithmof fixed complexity into an algorithm with variable com-plexity, but with lower (on average) complexity.

Algorithm 2 SCSD(H, y, p)Comment: get a suitable vectorsInitialize: s = [01, 02, . . . ,0M ]T , k = M, Part − Cost = ∞, yres = 0,NoS = 1Best− Cost= [01, 02, . . .]T andp = [p1, p2, . . . , pM ]T .(V,Hord) = Channel−Ordering (H,M )(Q,R) = QR(Hord)y′ = Qy(Θ) = Predetection(Rk,k, y

′k, yresk,k+1,Ω)

(βk,:, Part − costk,:) = Best− Nodes(Rk,k, y′k, yresk,k+1, pk,Θ)

LBk = 1 , UBk = pk + 1while k ≤ M do

if LBk ≤ UBk thenxk = βk,LBk ,LBk = LBk + 1if k , 1 then

k = k − 1yresk,k+1 = yk +

∑Mj=k Rk, j+1x j

if (NoS > 0) and (Part −Costk+1,: < BestCostNoS ) thenk = k + 1

else(Θ) = Predetection(Rk,k, y

′k, yresk,k+1,Ω)

(βk,:, Part−costk,:) = Best− Nodes(Rk,k, y′k, yresk,k+1, pk,Θ)

LBk = 1 , UBk = pk + 1end if

elseSolution found, savesNoS = x and their cost (BestCostNoS ).Also doNoS = NoS + 1

end ifelse

k = k + 1if k = M + 1 then

Finish the algorithmreturn (s)← the estimated vector associated to the best cost.

end ifend if

end while

4.4 SCSD Algorithm

In order to find the estimated vector via the lowest cost ofcomplexity, the algorithm starts with the elements’ sortingof the channel matrixH and the received data vectory.Then, the predetection stage is realized to getΘ. Also, frominitial parent node, branches and child nodes with an asso-ciated cost are expanded and only a set of thepk child nodeswith the lowest costs are selected, this set is known asβ.Subsequently, the first symbol ofβ is assigned toxk = βk,LBk ,whereLBk represents the lowest index of the vectorβ atthe levelk. Thus, in the search tree, the detection level ischanged fromk to k − 1. Finally, whenk = 0 the candidate

Fig. 7 Search tree structure of the SCSD detector for a 3× 3 MIMOsystem with modulated symbols 4-QAM.

vector and its associated cost are saved (as estimated vectorand best cost) and the process continues for a levelk = k+1and repeat untilk = M + 1.

For sake of a better understanding, Algorithm 2 de-scribes in more detail the process to follow by SCSD. Essen-tially, if more candidatespk are selected, the performance iseven closer to MLD algorithm. Additionally, in Figure 7 isshown the structure of the search tree executed by SCSD.

5. Numerical Results

In order to assess the proposed SCSD algorithm, a 4× 4MIMO array is used with 16 and 64-QAM modulated sym-bols over flat fading Rayleigh channel. The performance ofthe MIMO system is evaluated in terms of BER, whereas thecomplexity is evaluated in terms of number of visited nodesand FLOPS.

5.1 Performance of Detectors

The error rate is widely used as a measure unit to evaluatethe quality of a link. Therefore, it is considered an impor-tant parameter in communication systems where the errorrate metric should be very close to zero. In this paper, theBER is used as the performance metric of the detection al-gorithms. To quantify the BER, Monte Carlo simulationsare performed using different values of SNR. Table 1 showsthe configuration of survivor nodes by level for K-best, FSD,ASP and SCSD. The SD algorithm is not included since thenumber of visited nodes depends of a radius defined by thenoise variance [5]. All of these algorithms attain a perfor-mance close to MLD. For ASP and SCSD the configurationrepresents the maximum number of survivor nodes and/orrings accordingly. Figure 8 demonstrates the performanceof the studied algorithms in terms of BER against SNR.For both, 16-QAM and 64-QAM modulated symbols, it can

Table 1 Configuration of visited nodes by level for K-best, FSD, ASP,and SCSD algorithms.

Algorithm16-QAM 64-QAM

Level [1 2 3 4] Level [1 2 3 4]K-best (K) [1 4 12 16] [1 8 24 64]FSD (p) [1 1 2 16] [1 1 3 64]

ASP (p, K) [1 16 16 16],[1 16 16 16] [1 64 64 64],[1 64 64 64]SCSD (p, Ri) [1 1 2 16],[2 2 2 3] [1 1 3 64],[2 3 4 9]


0 5 10 15 20 2510

−6

10−5

10−4

10−3

10−2

10−1

100

SNR (dB)

BE

R

SCSDFSDSDK−bestASP

16 QAM

64 QAM

Fig. 8 Performance analysis of different algorithms considering a 4× 4MIMO antenna array with 16-QAM and 64-QAM modulated symbols.

be seen that the analyzed algorithms exhibit a quasi-MLDperformance, as explained above. Furthermore, the perfor-mance comparison in terms of BER is not sufficient to showthe advantages amongst algorithms. Therefore, it is neces-sary to perform a set of experiments to measure the com-putation complexity. The configuration of visited nodes bylevel and the characteristics of the algorithms cause a differ-ent complexity effect for each case, which is reflected in thenumber of visited nodes and the number of FLOPS required.

5.2 Level of Complexity

In the next experiments, in order to carry out a comparativeanalysis of different detection algorithms, the complexity(i.e., The difficulty of performing the detection process) ismeasured in terms of visited nodes and FLOPS. The count-ing number of nodes that are visited during the detectionprocess is a valid measure of complexity since the genera-tion of child nodes from a parent node depends on the con-ditions set by the detection algorithm used. In this regard,avisited node is considered the set formed by a branch and achild node expanded from a parent node.

Moreover, for the analysis of complexity in terms ofFLOPS, operations such as addition, subtraction, multipli-cation and division are taken into account. Indeed, the built-in function f lops(·) (included in MATLAB 5.3) estimatesthe number of FLOPS used in the simulation process andgives a value to each mathematical operation as shown inTable 2.

Thus, Figure 9 a) and Figure 9 b), show the complexity

Table 2 Elementary scalar operations measured in FLOPS.

Operation ifc ∈ R if c ∈ Ca + b = c 1 2 FLOPSa − b = c 1 2 FLOPSa ∗ b = c 1 6 FLOPSa ÷ b = c 1 6 FLOPS

of SCSD, FSD, SD, K-best and ASP algorithms in terms ofthe number of visited nodes against SNR, for 16-QAM and64-QAM modulated symbols, respectively. In cases wherethe SNR is less than 20 dB, SCSD reduces the complexityin comparison to the others. For example, in the case of16-QAM, it can be seen in Figure 9 a) that SCSD visits ap-proximately 60% fewer nodes compared to SD, K-best andASP when SNR is 10 dB. Similarly, for 64-QAM, it canbe seen in Figure 9 b) that SCSD visit approximately 75%fewer nodes respect to K-best and ASP, whilst respect to SD,SCSD visits up to 90 % fewer nodes when SNR is 10 dB.

Figure 10 a) and Figure 10 b) show the complexity interms of FLOPS against SNR, for 16-QAM and 64-QAMmodulated symbols, respectively. Results show that theSCSD outperforms the FSD, SD, K-best and ASP algo-rithms in cases where SNR is below 10 dB for 16-QAM and15dB for 64-QAM. For example, it can be seen in Figure10 a) that the number of required FLOPS is reduced by theSCSD up to 30% compared to K-best and ASP, whereas re-spect to SD, SCSD requires similar number of FLOPS whenSNR is 10 dB. Similarly, for 64-QAM, Figure 10 b) showsthat the number of required FLOPS is reduced by the SCSDup to 50% compared to the SD and ASP, whilst respect to K-best, SCSD requires up to 90% fewer FLOPS when SNR is10 dB. It is worth to point out that, compared to the SD andASP algorithms the complexity of the SCSD increases upto 10% for 16-QAM when SNR is greater than 10 dB, andup to 20% for 64-QAM when SNR is greater than 15 dB.However, although in these cases the complexity of the SDand ASP is slightly better than that of the SCSD, in practi-cal scenarios, a robust algorithm that performs well in noisyenvironments while exhibiting a low complexity (like theSCSD) would be preferred.

The previous analysis shows that, in the low SNR re-gion (below 20 dB), SCSD reduces the complexity in termsof number of visited nodes and FLOPS. Regarding FSD, itcan be seen from previous graphics that the number of vis-

0 10 20 300

500

1000

1500

2000

2500

SNR (dB)

Nu

mb

er

of vi

site

d n

od

es

a) 16−QAM

0 10 20 300

1

2

3

4

5

6x 10

4

SNR (dB)

Nu

mb

er

of vi

site

d n

od

es

b) 64−QAM


Fig. 9 Complexity analysis of different algorithms considering a 4× 4MIMO antenna array in terms of the number of visited nodes against SNR.


0 10 20 300

1

2

3

4

5

6x 10

4

SNR (dB)

Nu

mb

er

of F

LO

PS

a) 16−QAM

0 10 20 300

2

4

6

8

10

12x 10

5

SNR (dB)

Nu

mb

er

of F

LO

PS

a) 64−QAM


Fig. 10 Complexity analysis of different algorithms considering a 4× 4MIMO antenna array in terms of FLOPS against SNR.

ited nodes and FLOPS required for the detection process isextremely high in both scenarios. This is explained by thefact that the number of survivor nodes by level is high whencompared for example to K-best, which is also a fixed com-plexity algorithm.

Table 3 shows the average cost per visited node in termsof FLOPS. This refers to the complexity required to selector exclude a node in the detection process. The SCSD al-gorithm is practically the most costly due to the requiredcomplexity to perform the reduction of search area (prede-tection). However, this is compensated by the fact that onlya small number of candidate nodes are visited during theoverall detection process.

Table 4 shows a complexity comparative analysis ofSCSD with and without stages of predetection and early ter-mination in terms of number of visited nodes and FLOPS. Inthis context, the SCSD without both stages reduces to FSDalgorithm. Also, it can be seen in the table how the com-plexity is reduced by adding individually each of the stagesto the FSD algorithm. The largest reduction of complexityof the SCSD respect to FSD is achieved mainly at the stageof early termination, obtaining a complexity reduction up to90 % independently of the modulation used. On the otherhand, in the stage of predetection, the reduction is approxi-mately 20 % for 16-QAM and near to 70% for 64-QAM. Inthe case where both stages are implemented simultaneously(i.e., SCSD) the complexity reduction is greater than 90 %for 16-QAM and 64-QAM. Indeed, a significant reductionin complexity occurs at higher order modulations, which are

Table 3 Average cost per visited node in terms of FLOPS.

Algorithm 16-QAM 64-QAMSD 17.47 5.36FSD 37.18 34.83

K-best 28.57 27.35ASP 26.07 28.50

SCSD 40.37 26.56

Table 4 Complexity reduction due to SCSD stages (early terminationand predetection) at SNR=15 dB. NVN stands for number of visited nodes.

Withoutboth

stages

With earlytermination

Withpredetection

Withboth

stagesModulation 16-QAM

NVN 1296 151 976 121FLOPS 53131 5879 42251 5536

Modulation 64-QAMNVN 28736 1523 6453 700

FLOPS 1074113 76700 404460 22728

required for future high data rate wireless communicationssystems using MIMO.

6. Conclusions and Future Work

In this paper, a new detection algorithm called SemifixedComplexity Sphere Decoder was presented. In terms ofBER, the proposed SCSD algorithm is able to achieve per-formance similar to that attained by the MLD. Moreover,its complexity level, in terms of visited nodes and FLOPS,reduces significantly in the low SNR region. In order toreduce the complexity at the detection process, the SCSDalgorithm examines a semifixed number of candidate nodes.Thus fewer mathematical calculations are needed. When thechannel characteristics are unfavorable, the number of vis-ited nodes reaches up to a maximum limit. On the contrary,when the channel characteristics are favorable, the numberof visited nodes is smaller. Furthermore, this algorithm lim-its the search area, discarding beforehand elements of theconstellation. The variability in the number of visited nodesin each detection cause a variable level of complexity, butit was shown that the averaged complexity for the SCSD issignificantly lower.

Finally, an interesting issue for future research is tostudy the performance of SCSD with coded symbols inMIMO systems. For example, in the case of turbo-MIMObased on bit-interleaved coded modulation (BICM) [20], itis necessary to extend the algorithm for finding not onlythe MLD estimation but also obtaining a set of candidatesaround the MLD estimation that can be used to calculatesoft-output information about the interleaved bits.

Acknowledgements

This work was partially supported by Consejo Nacional deCiencia y Tecnologıa (CONACYT), Fondo de Apoyo a laInvestigacin (FAI 2014) and Universidad Autonoma de SanLuis Potosı (UASLP).

References

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[5] B. Hassibi and H. Vikalo, “On the sphere-decoding algorithm I. ex-pected complexity,” IEEE Transactions on Signal Procesing, vol.53,no.8, pp.2806–2818, August 2005.

[6] I. Kanaras, A. Chorti, M.R.D. Rodrigues, and I. Darwazeh, “Afast constrained sphere decoder for ill conditioned communicationsystems,” IEEE Communications Letters, vol.14, pp.999 – 1001,November 2010.

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[9] J.W. Choi, B. Shim, and A.C. Singer, “Eficient soft-inputsoft-output tree detectionvia an improved path metric,” IEEE Journalon Selected Areas in Communications, vol.58, no.3, pp.1518–1533,March 2012.

[10] S. Han, T. Cui, and C. Tellambura, “Improved k-best sphere detec-tion for uncoded and coded mimo systems,” IEEE Wireless Com-munications Letters, vol.1, pp.472–475, October 2012.

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[13] H. Vikalo and B. Hassibi, “On sphere decoding algorithmII. gen-eralizations, second-order statistics, and applicationsto communi-cations,” IEEE Transactions on Signal Processing, vol.53,pp.2819–2834, August 2005.

[14] Guanghui Li, P. Zhang, S. Lei, C. Xiong, and D. Yang, “An earlytermination-based improved algorithm for fixed-complexity spheredecoder,” IEEE Wireless Communications and Networking Confer-ence (WCNC 2012), Shanghai, China, pp.624–629, April 2012.

[15] Kuei-Chiang Lai and Li-Wei Lin, “Low-complexity adaptive treesearch algorithm for mimo detection,” IEEE Transactions onWire-less Communications, vol.8, no.7, pp.3716–3726, July 2009.

[16] U. Pineda Rico, Link Optimisation for Mimo Communication Sys-tems, LAP Lambert, 2010.

[17] U. Fincke and M. Pohst, “Improved methods for calculating vectorsof short length in a lattice, including a complexity analysis,” Mathe-matics of Computation, vol.44, no.170, pp.463–471, April 1985.

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[19] P. Yang, Y. Xiao, Q. Tang, B. Zhou, and S. Li, “A low-complexityordered sphere decoding algorithm for spatial modulation,” IEICETransactions on Communications, vol.E95-B, pp.2494–2497, July2012.

[20] L.G. Barbero and J.S. Thompson, “Extending a fixed-complexitysphere decoder to obtain likelihood information for turbo-mimo sys-tems,” IEEE Transactions on Vehicular Technology, vol.57,no.5,pp.2804–2814, September 2008.

Juan Francisco Castillo-Leon received theM.Sc. degree in Electronic Engineering fromthe Universidad Autonoma de San Luis Potosi(UASLP), San Luis Potosi, Mexico in 2012.Currently he is a PhD student at UASLP andhis research interests are sphere decoders forMIMO systems.

Marco Cardenas-Juarez received the Ph.D.degree in Signal Processing for Communica-tions from the School of Electronics and Elec-trical Engineering, University of Leeds, UnitedKingdom, in 2012. Currently, he is an AssistantProfessor at Facultad de Ciencias of the UASLP,Mexico. His research interests are in signal pro-cessing and communications.

Ulises Pineda-Rico received the Ph.D.degree in Electrical and Electronic Engineer-ing from The University of Manchester, UnitedKingdom in 2009. Currently, he is an AssociateProfessor at Facultad de Ciencias of the UASLP,Mexico. His main research interests are MIMOwireless technologies and related. Dr. Pineda-Rico is member of the IEICE and IEEE Commu-nications Societies and member of the MexicanNational Research System since 2010.

Enrique Stevens-Navarro received thePh.D. degree in Electrical and Computer Eng.from the Univ. of British Columbia, Canada in2008. He is currently an Associate Professorat Facultad de Ciencias of the UASLP, Mexico.His research interests include the design andevaluation of protocols for wireless networks.He is member of the IEEE Communications So-ciety since 1999, member of the IEICE Commu-nications Society since 2010 and member of theMexican National Research System since 2010.

Bibliografía

[1] Mourad Barkat. Signal Detection and Estimation. Artech House, 2nd Edition, 2005.

[2] Rafael Arturo Trujillo Rasúa. Algoritmos paralelos par a la solución de problemas deoptimización discretos aplicados a la decodificación de señales. Universidad Politec-nica de Valencia, España.

[3] Babak Hassibi and Haris Vikalo. On the sphere-decoding algorithm i. expected com-plexity. IEEE Transactions on Signal Procesing, 53(8):2806–2818, Agosto 2005.

[4] H. Vikalo and B. Hassibi. On sphere decoding algorithm ii. generalizations, second-order statistics, and applications to communications. IEEE transactions on signalprocessing, 53:2819–2834, Agosto 2005.

[5] S. Han, T. Cui, and C. Tellambura. Improved k-best spheredetection for uncoded andcoded mimo systems. IEEE Wireless communications letters,Mayo 2013.

[6] L. G. Barbero and J. S. Thompson. Fixing the complexity ofthe sphere decoder formimo detection. IEEE Transactions on Wireless communications, (7).

[7] Zhan Guo and P. Nilsson. Algorithm and implementation of the k-best sphere de-coding for mimo detection. IEEE Journal on Selected Areas inCommunications,24:491–503, Marzo 2006.

[8] José A. Mañas. Análisis de algoritmos complejidad.www.lab.dit.upm.es/simlprg/material/apuntes/o/index.html. Universidad Politec-nica de Madrid, España.

[9] Javier García de Jalón. Aprenda Matlab 6.5 como si estuviera en primero. EscuelaTécnica Superior de Ingenieros Industriales Universidad Politécnica de Madrid, 2004.

[10] Kuei-Chiang Lai and Li-Wei Lin. Low-complexity adapti ve tree search algorithm formimo detection. IEEE Transactions on Wireless Communications, 8(7):3716–3726,July 2009.

BIBLIOGRAFÍA

[11] P.W.Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela. V-blast: An ar-chitecture for realizing very high data rates over the rich-scattering wireless channel.Proc. URSI International Symposium on Signals, Systems andElectronics, Atlanta),pages 295–300, Septiembre 1998.

[12] Guanghui Li, PXin Zhang, Sheng Lei, Cong Xiong, and Dacheng Yang. An earlytermination-based improved algorithm for fixed-complexity sphere decoder. IEEEWireless Communications and Networking Conference (WCNC 2012), Shanghai,China, pages 624–629, April 2012.



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