Internship Report Investigation of Space Time coding in ...Je tiens à remercier également M. Patrick Labbé pour son aide et son encouragement qui ont con-tribué au bon déroulement

Motorola Labs - July 2, 2010Wireless Access Research - Paris

Internship ReportInvestigation of Space Time coding in IEEE 802.11n

Author : Lina Mroueh

Internship Supervisors: Stéphanie Rouquette-LéveilPatrick Labbé

Version 1.0

Motorola Labs, ENST Paris: April — Sept 2006

REMERCIEMENTS

Remerciements

J’aimerais tout d’abord exprimer ma reconnaissance à Mme Stéphanie Rouquette-Léveil, pourm’avoir permis d’effectuer ce stage de fin d’études dans les meilleures conditions.Son suivi permanent de mes travaux, son soutien, sa gentillesse et sa disponibilité tout au long de monstage m’ont incontestablement permis de progresser.

Je tiens à remercier également M. Patrick Labbé pour son aide et son encouragement qui ont con-tribué au bon déroulement de mon projet.Je tiens aussi à remercier M. Mohamed Kamoun, pour ses nombreux conseils et pour le temps qu’ilm’a consacré.

Je souhaite aussi remercier M. Marc de Courville pour m’avoir accueilli dans son équipe.

Je remercie M. Jean-Claude Belfiore, professeur à l’ENST, pour m’avoir donner l’occasion de tra-vailler avec l’équipe Radio Link Technologies à Motorola, je le remercie également pour ses conseilsquant au déroulement de mon stage. Mes remerciements s’adressent aussi à Mme Ghaya Rekaya,enseignante chercheur à l’ENST, pour son suivi de mon stage.

Mes remerciements vont également à l’ensemble des stagiaires qui m’ont soutenue et ont contribué àcréer une ambiance agréable sur le lieu de travail. Je tiens aussi à remercier tous mes collègues pourleur collaboration et leur bonne humeur.

2

CONTENTS

Contents

Executive summary 5

Acronyms, notations and definitions 6

1 About Motorola and the CRM 7

1.1 History of Motorola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Four Business divisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Motorola in Europe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Motorola Labs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Radio Link Technologies RLT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5.1 Normalization of IEEE 802.11n . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5.2 WiMix Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 Project time table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Space Time Block Coding 15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Space time coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 Design criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.2 Spatial Division Multiplexing - SDM . . . . . . . . . . . . . . . . . . . . . 17

2.2.3 Alamouti scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.4 Golden code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Space time MIMO Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 ML criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.2 ML soft decoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Simulations Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.1 Simulation Protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.2 Coded Space Time System . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Investigation of space time codes in IEEE 802.11n 27

3.1 Presentation of IEEE 802.11n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Presentation of physical layer functional blocks . . . . . . . . . . . . . . . . . . . . 27

3

CONTENTS

3.2.1 Convolutional encoder and puncturing . . . . . . . . . . . . . . . . . . . . . 28

3.2.2 Interleaver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.3 Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.4 Multiple antenna processing . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.5 OFDM modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.6 IEEE 802.11 n modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Integration of the Golden code in the IEEE 802.11 n simulator . . . . . . . . . . . . 31

3.3.1 MIMO-OFDM Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.2 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Partitioning the Golden Code 38

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Mapping by set partitioning method . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.1 Conceptual description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.2 Use of partitioning in coded modulation . . . . . . . . . . . . . . . . . . . . 39

4.3 Partitioning the Golden code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3.1 Partitioning and coding gain . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3.2 Encoding and decoding the partitions of GC in E8 . . . . . . . . . . . . . . 42

4.3.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Conclusion and perspectives 46

5.1 Difficulties in the project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Summary of contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.3 Future research and Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Reference 50

4

EXECUTIVE SUMMARY

Executive summary

IEEE 802.11n is one of the latest evolutions of the previous 802.11 standards for wireless LANs. Themain objective of this technology is to provide to the end-user modes of operation that are capableof much higher throughput than 11a/b and g, with a maximum throughput of at least 100Mbps, asmeasured at the MAC data Service Access Point (SAP). The major novelty of this version is touse Multiple Input Multiple Output (MIMO) techniques so that devices embed multiple transmitterand receiver antennas to allow an increased data throughput through spatial multiplexing and anincreased range by exploiting the spatial diversity. IEEE 802.11n is currently in the last steps of thestandardization expected by end of 2007.

In this document, we study the impact of implementing the Golden code in IEEE 802.11 n. The hugeperformance gains possible with this space time technique over a Rayleigh channel in non codedMIMO system encouraged us to integrate the Golden code into the IEEE 802.11n standard.

In the first chapter of this document, we present the state of the art of space time codes, introducingbasic notions in MIMO systems, such as space time codes and Maximum likelihood decoder. Theperformance of space time codes in this chapter are evaluated over a Rayleigh channel for non codedand coded MIMO system. The second chapter is dedicated to the assessment of the Golden codeimplemented in the overall simulator chain. From this two chapters, we observe that, the mappingof bits into symbols affects largely the performance of GC. We conclude that the Gray mapping isnot the correct mapping to use with GC. That’s why, in the third chapter, we modify the mapper bypartitioning the Golden code. This work will be completed in our future studies on the partitioningfor non coded and coded systems.

5

ACRONYMS, NOTATIONS AND DEFINITIONS

Acronyms

BER Bit Error RateLLR Log Likelihood RatioMIMO Multiple Input Multiple OutputML Maximum LikelihoodMMSE Minimum Mean Square ErrorMRC Maximum Ratio CombiningNTX Number of transmit antennasNRX Number of receive antennasPER Packet Error RateQAM Quadratic Amplitude ModulationSD Sphere DecoderSDM Spatial Division MultiplexingSNR Signal to Noise RatioSTBC Space-Time Block CodingZF Zero ForcingGC Golden code

Notations

‖.‖ Euclidean norm<(.) Real part operator=(.) Imaginary part operatorE(.) Average.∗ Conjugate.T Transpose.H Hermitian. Estimate

Definitions

- Uncoded MIMO system = Mapper + MIMO + Channel + Hard decoder.

- Coded MIMO system = Convolutional encoder + Interleaver + Mapper + MIMO + Channel +Soft decoder.

6

1. About Motorola and the CRM

1.1 History of Motorola

Originally founded as the Galvin Manufacturing Corporation in 1928, Motorola has come a longway since introducing its first product, the battery eliminator. For more than 75 years, Motorolahas proven itself a global leader in wireless, broadband and automotive communications technologiesand embedded electronic products, and has become a company recognized for its dedication to ethicalbusiness practices and pioneering role in important innovations.

2003 2004

Handset with Bluetooth

Two-way pager Cable modems Personal two-way radio

First Motorola Microprocessor

6800

First Portable cellular phone

First all-digital HDTV system

Integrated voice, data and

messaging system

Fisrt Telematics Products

First Wearable cellular phone

First rectangular color TV

picture tube

3-amp power transistor

First coaxial cable TV system

Golden View first TV under

$200

First Handheld two-way radio

First practical & Affordable car radio

First words from the moon

Handie-Talkie Radio pager

Phone with touch screen

Third Generation (3G) phones

Digital wireless phone with

Internet

Figure 1.1: 75 years of innovation

Motorola is a Fortune 100 global communications leader that provides seamless mobility productsand solutions across broadband, embedded systems and wireless networks. Seamless mobility meansyou can reach the people, things and information you need in your home, auto, workplace and allspaces in between. Seamless mobility harnesses the power of technology convergence and enablessmarter, faster, cost-effective and flexible communication. Motorola had sales of US $31.3 billion in2004.

1.2 Four Business divisions

Today, Motorola is comprised of four businesses: Connected Home Solutions, Government & Enter-prise Mobility Solutions, Mobile Devices and Networks.

7

CHAPTER 1. ABOUT MOTOROLA AND THE CRM

MOBILE DEVICES NETWORKSGOVERNMENT &

ENTERPRISE MOBILITY SOLUTIONS

CONNECTED HOME

Figure 1.2: New segmentation in 4 business units (01/01/2005)

Mobile Devices

Mobile devices offer market-changing icons of personal technology - transforming the device for-merly known as the cell phone into a universal remote control for life. A leader in multi-mode,multi-band communications products and technologies, Mobile Devices designs, manufactures, sellsand services wireless subscriber and server equipment for cellular systems, servers and software so-lutions, related software and accessory products.

Networks

Networks deliver the infrastructure, network services and software that meet the needs of operatorsworld-wide today, while providing a migration path to next-generation networks that will enable themto offer innovative, revenue-generating applications and services to their customers.

Government and Enterprise Mobility Solutions

A leading provider of integrated radio communications and information solutions, with more than65 years of experience in meeting the mission-critical requirements of public safety, government andenterprise customers worldwide. It also designs, manufactures and sells automotive and industrialelectronics systems that enable automated roadside assistance, navigation and advanced safety fea-tures for automobiles.

Connected Home Solutions

Connected Home Solutions provide a scalable, integrated end-to-end system for the delivery of broad-band services that keeps consumers informed, entertained and connected.

1.3 Motorola in Europe

At the end of the year 2004, Europe totalized 19% of the overall Motorola sales (estimated to $31.3Billion).

8

1.3. MOTOROLA IN EUROPE

Figure 1.3: Sales by region, end of year 2004

Seven countries take part of these sales. These are shown below:

UK2400 employees$1,550 m

France1000 employes$580 m

Germany2600 employees$1,490 m

Italy350 employees$808 m

Israel2700 employes$530 m

Spain & Portugal380 employees$637 m

Figure 1.4: Headcount and sales by country, end of 2004

Motorola in France

Motorola has more than 35 years of ongoing presence in France. The first activity in France startedin Toulouse in 1967, and is now present in Paris and Angers.

- Paris-Saclay

– Motorola France Headquarters (The 4 business units are represented)

– Research Lab: Focus on Wireless LAN, spectrum engineering, TV on mobile, and beyond3G technologies

– 300 employees

- Rennes 1995

9

CHAPTER 1. ABOUT MOTOROLA AND THE CRM

– I-mode reserch and developement center.

– 150 employees.

- Toulouse 1967

– Design Center focusing on future software and hardware platforms of next generationscell-phones.

– World Center of Excellence for GSM / GPRS / Edge platforms

– 500 employees

1.4 Motorola Labs

Motorola Labs is working on technologies that will make it simpler for us to communicate with ourfriends and family, easier for us to communicate with our products and enable our products to com-municate with each other and configure themselves to individual preferences and the environment.

Motorola Labs is also developing intelligent networks that can configure and heal themselves for con-sistently superb performance and works closely with the company’s product businesses to transformthe promise of technology into real solutions.

Today, Motorola Labs has a strong, global team of scientists, engineers and technicians focused ondiscovering and developing new materials, technologies, architectures, algorithms and processes forsmarter devices and systems. Motorola Labs have developed a portfolio of significant, highly coor-dinated, strategic research programs and leadership technology in areas such as communication onnetworks from 2.5 to 4G and ad hoc networks, multimedia, security and privacy, microminiaturiza-tion, human experience, VoIP and more.

" Brazil" China/HK" France" Germany" India" Ireland" Israel" Japan" Malaysia" Mexico" Singapore" United Kingdom and Northern Ireland"United States

" Brazil" China/HK" France" Germany" India" Ireland" Israel" Japan" Malaysia" Mexico" Singapore" United Kingdom and Northern Ireland"United States

Figure 1.5: Motorola Labs in the world

10

MY INTERNSHIP IN MOTOROLA

The key research programs that Motorola Labs focuses on are listed below:

Broadband To and In the Home Broadband System NetworksBroadband Wireless Design and Manufacturing TechnologiesEnvironmentally Friendly Products Human ExperienceImaging Technologies Internet ResearchMicrominiaturization MultimediaPortable Energy Security and PrivacyNeuRFon Software Defined RadioSoftware Productivity Wireless Local Area and Personal Area Networks

Motorola Labs consists of a global team of nearly 1,000 people working in more than 20 researchCenters of Excellence around the world. The Centers of Excellence are focused on seven major areasof expertise:

- Embedded Systems and Physical Sciences Research Center

- Human Interaction Research Center

- Networks and Systems Research Center

- Applications, Content and Services Research Center

- Physical Realization Research Center

- The Motorola Advanced Technology Center (MATC)

- Wireless Access Research Center

Motorola Labs in Paris

Motorola Labs was Founded in 1996 in Saclay (91) with only 4 engineers, the Research Center ofMotorola (CRM) today has more than 60 researchers divided into 4 teams, including 2 laborato-ries. Its mission is to prepare Motorola with the recently technological ruptures: local area networkswireless of the Wi-Fi type, mobile communication systems beyond the 3G and mobile Internet. TheCRM cooperates with more than 30 universities in 8 countries, realize the multi-field projects, andcollaborate in many European research programs.

In particular, the CRM is composed of 3 labs:

- Seamless Radio-access Lab (SRL) develops:

1. Hybrid systems integrating cellular technologies, WIFI and numerical television, radioresource sharing which guarantees an inter-working between systems 2G and 3G to offera service integrated into the mobile users, reconfiguration.

2. Architectures for rising generation of WAN (Wide Area Network) in order to make net-works of them broadband: improvements of the physical layer, layer 2 and integratedcircuits.

- Edge Mobile Networking Lab(EMNL) develops technologies of networks allowing a com-pletely transparent mobility through all the mobile networks.

- Personalization and Knowledge Lab(PKL) offers an increasingly simple access to the servicesof the network thanks to systems multi-agents and to the knowledge management.

11


My internship in Motorola Labs

1.5 Radio Link Technologies RLT

I have done my internship, from the 3th of April to the 29th of September, with the Radio Link Tech-nologies RLT Team under the supervision of Mme Stéphanie Rouquette-Léveil and Mr. PatrickLabbé.

Radio Link Technology Marc de Courville

IEEE 802.11n Project

WIMIX Project

Stéphanie Rouquette-Léveil Patrick Labbé

Lina Mroueh

Laurent Mazet

Véronique Buzenac-Settineri

Mohamad Kammoun

Figure 1.6: Radio Link Technology Team

Inside WARCoE, the mission of the Short Range project is to create value by integrating Wireless Per-sonal Area Networks (WPAN)/Wireless Local Area Networks (WLAN) into innovative Short RangeWireless Systems and develop enabling technologies for all aspects of short range systems from sys-tem/medium access control to baseband and from simulation to prototyping.

1.5.1 Normalization of IEEE 802.11n

Within that framework, Motorola Labs participates to the High Throughput next generation WirelessLAN (IEEE 802.11n). The objective is to drive the 802.11n standard specification in order to get abenefit from spatial diversity (MIMO) not only for increasing the peak data rate but also to ensurelarger range of operation (full home coverage). The motivation is to better address handsets speci-ficities (2009 forecast: 1/3 of handsets will support WiFi: 200 million devices annually compared to70 million WiFi enabled PCs) and CE equipment (advanced settings). The IEEE802.11n target is todeliver 100Mbps on top of MAC Service Access Protocol. For that purpose Motorola has contributedto the following key enablers during the whole standardization process:

- built in support for asymmetric TX/RX antenna configurations to accommodate various ter-minal sizes (AP/Laptop/PC/PDA/Phone) offering a scalable and evolutionary solution usingSTBC

12

1.6. PROJECT TIME TABLE

- support heterogeneous traffic: increase overall peak data rate without jeopardizing lower datarates modes.

- grant range extension for limited outdoor operation (hotspot) and full home coverage.

- focus on a proven and simple solution: consider robust yet low complexity open-loop STBCMIMO PHY techniques.

1.5.2 WiMix Project

WiMix is a research project contributing to the design of a next generation Broadband WirelessAccess system. The focus of WiMix is on techniques allowing the combined self-optimization ofmacro-diversity, relaying and space-time-frequency resource allocation. The goal is to provide muchmore capacity and throughput to the user by increasing spectrum efficiency while keeping infrastruc-ture/terminal costs affordable.

1.6 Project time table

The aim of my internship in Motorla Labs was to integrate the Golden code into the existing versionof IEEE 802.11n in order to improve the performances. The Unexpected results obtained with Goldencode in IEEE802.11n context, enlight new problems. In the second part of my internship, I tried toresolve the emerging problem by modifying some blocks in the simulator such as the mapper.

Evolution of the project

First Period (Early April - Middle May): Initial Phase.

1. Get a general idea about the project.

2. Read the documents related to 802.11n and to design of space time codes.

3. Implementation in Matlab of a simple block transmission chain in order to compare ML de-coders (Exhaustive research, Sphere Decoder SD, Schnorr Euchner SE) in term of accuracyand speed.

Second Period (Middle May - Middle July): Implementations

1. Implementation in Matlab of non coded MIMO transmission chain over a Rayleigh channel.

2. Analysis of the existing version of IEEE802.11n simulator.

3. Integration of the Golden code into the simulator.

4. Tests and analysis of the results.

5. Participation to the redaction of a technical document: ’Maximum Likelihood Decoding As-sessment for IEEE802.11n’

Third Period (Middle July - August): Changing the mapper.

1. Documentation on lattice theory and partitioning.

13


2. Implementation in Matlab of a BICM(Bit Interleaved Coded Modulation)MIMO transmissionchain with both classical and modified mapper.

3. Tests and results analysis.

Fourth Period (September): Internship report.

14

2. Space Time Block Coding

2.1 Introduction

In this chapter, we will introduce the basic notions of MIMO systems. We will consider in thischapter two different cases, a simple MIMO system over a Rayleigh channel that we will call Noncoded MIMO system, and a Binary Interleaved Coded Modulation BICM MIMO system, which wecall Coded MIMO system.

The first section would be dedicated to present the design criterion of Space Time coding ST and topresent 3 well known codes Spatial Division Multiplexing SDM, Alamouti code and Golden codeGC. In the second section, we present ML receiver in both cases coded system and non coded system.In the last section, we compute the performance of GC versus SDM performance for the 2 cases ofMIMO system.

2.2 Space time coding

Consider a Multiple Input Multiple Output (MIMO) system of nt transmit and nr receive antennas.We will assume that the transmission is done over a quasi-static rayleigh channel unknown to thetransmitter.

Figure 2.1: Transmission Block Diagram

The received signal is given by

Ynr×T = HXnr×T + Znr×T (2.1)

where

- T is the number of codewords.

- Z is the white complex Gaussian additive noise with zero mean and variance σ2 per complexcomponent.

- H is the nr × nt transfer matrix of the channel with entries hjk represent the fading betweentransmitter j and receiver k.

Many space time block code (STBC) were suggested such as Alamouti, Spatial Division Multiplexing(SDM), Golden code in order to improve the error performance. The design of these codes shouldverify the criterion described in next section.

15

CHAPTER 2. SPACE TIME BLOCK CODING

2.2.1 Design criterion

The work on the design criteria of ST codes to extract full diversity and to maximize coding gain arepresented in [6].

The design of the STBC according to [6] requires to maximize the diversity and to increase the gaincoding. Maximization of diversity order and coding gain are derived from the minimization of theupper bound of the error probability.

Let us assume that a codeword Si is transmitted. The probability that the receiver mistakes the trans-mitted codewod Si for another codeword Sj , given knowledge of channel realization at the receiver,is referred to the pairwise probability (PEP), and is given by

PEP = P (Si → Sj |H) = Q

√Es ‖H(Xi −Xj)‖2

σ2nt

(2.2)

In high SNR regime, the PEP is upper bounded by :

PEP = P (Si → Sj |H) ≤

(i=1∏r

λi

)−nr (1Es4σ2

)rnr(2.3)

Where :

- λi, i = 1...n, are the eigen values of the matrix (Xi −Xj)H(Xi −Xj)

- r = rank(Xi −Xj).

Equation (2.3) leads us to the two well-known criteria for ST construction namely "rank criterion"and "determinant criterion".

Rank criterion

The rank criterion optimizes the spatial diversity extracted by a ST code. In equation (2.3), the STcode extracts d = rnr order diversity with r = rank(Xi −Xj). In order to achieve the full spatialdiversity gain of ntnr, Xi −Xj should be full ranked, i.e r = nt.

Determinant criterion

The determinant criterion optimizes the coding gain. From equation (2.3), it is clear that the coding

gain depends on the term(∏i=1

r λi

)−nr. Hence for high coding gain, we should maximize δmin,

whereδmin = min

Xi 6=Xjdet∣∣(Xi −Xj)(Xi −Xj)

H∣∣ (2.4)

The diversity vs Multiplexing gain tradeoff

The maximal spatial multiplexing gain is the other criterion to achieve in a MIMO system.

16

2.2. SPACE TIME CODING

Definition 1 Spatial multiplexing gain

The spatial multiplexing gain is the number of symbols sent per channel use. The maximal spatialmultiplexing gain is equal to the number of degrees of freedom and is equal to rmax = min(nt, nr).

Note that this gain is not necessarily maximized when the diversity gain is maximized. Zheng andTse in [7], show that there is a fundamental tradeoff between these two gains.

For a MIMO system, with nt = 2 , nr ≥ 2 and T = 2, this tradeoff could be achieved if the code has anon vanishing determinant i.e a determinant that did not vanish when the spectral efficency increases.This implies that the coding gain should be lower bounded by a positive value.

2.2.2 Spatial Division Multiplexing - SDM

The SDM is simply the uncoded MIMO scheme. It is also known as VBLAST code. In a 2×2 MIMOconfiguration, the ST codeword may be expressed:

X =

[s1 s3 . . .s2 s4 . . .

](2.5)

The SDM code is a linear scheme. It does not achieve full diversity since r = rank(Xi −Xj) = 1.It doesn’t respect the determinant criterion since it is not a non vanishing determinant code. Finally,The SDM is a full rate code since it allows to send 2 symbols per channel use.

2.2.3 Alamouti scheme

The Alamouti scheme transmits symbol s1 and s2 from antennas 1 and 2 respectively, during thefirst period, followed by −s∗2 and s∗1 from antennas 1 and 2 respectively during the following symbolperiod. Hence the transmitted ST codeword may be expressed by:

X =

[s1 −s∗2s2 s∗1

](2.6)

Properties of Alamouti code

- The scheme is linear.

- The Alamouti scheme achieves full diversity ( r = rank(Xi −Xj) = 2).

- It has a non vanishing determinant (minimum euclidian norm |s1|2 + |s2|2 6= 0).

- It is very simple to decode due to its orthogonal structure.

- The Alamouti is not a full rate code (1 symbol per channel use).

2.2.4 Golden code

In [1], Rekaya and Belfiore propose to construct a code family in order to achieve full diversity, fullspatial multiplexing gain and also give excellent performance at low signal to noise ratio. In thissection, we will give a summary about how this code was constructed (see [4] for further details).

17


Algebraic Tools

In this section, we will present the algebraic tools used in the construction of Golden Code GC.

Definition 2 Field of rational Numbers - Algebraic Number - Number field

- The field of rational complex Q(i) is defined by Q(i) = {x+ iy, x, y ∈ Q}.

- θ is an algebraic number of degree n on Q(i), if θ is a root of a minimal polynomial of degreen with coefficients in Q(i).

- The number field K on Q(i) is defined by

K = Q(i, θ) ={∑n−1

i=0 aiθi, ai ∈ Q(i)}

Use in Golden code construction

For the construction of GC, the number field Q(i,√

5) is used.√

5 is an algebraic number where its minimal polynomial is X2 − 5 . The number field Q(i,√

5) isgiven by

Q(i,√

5) ={a0 + a1

√5, a0, a1 ∈ Q(i)

}.

Definition 3 The ring of integers is the subset of all the integers in a number field K = Q(i, θ). Thissubset is denoted by OK.The ring of integers could also be defined as the ring of numbers whose minimal polynomial is Xn +∑n−1

i=0 aiXi with ai ∈ Z[i].


In the previous example, the field number Q(i,√

5) has been introduced. We will define now thecorresponding ring of integers OK. This ring is not generated by (1,

√5), but by (1, 1+

√5

2 ).

1+√5

2 is also an algebraic number where the minimal polynomial is θ2 − θ − 1 = 0.

Thus, OK ={a0 + a1

1+√5

2 , a0, a1 ∈ Z(i)}

1+√5

2 is called the golden number θ, and θ the other root of the minimal polynomial is the conjugateof θ.

Definition 4 Conjugates - Norm in K

- The conjugates of an element x ∈ K are the roots of minimal polynomial, given by :σ1(x) = a+ bθσ2(x) = a− bθ

- The norm of an element of K is the product of all its conjugates.

NK/Q(x) =n∏i=1

σi(x) ∈ Q(i). (2.7)

If x ∈ Z(i), then NK/Q(x) ∈ Z(i).

18

2.2. SPACE TIME CODING


We propose here to find the norm of x ∈ K ⊂ Q(i,√

5), i.e x = a+ bθ, a, b ∈ Q(i).

we have :σ1(x) = a+ bθ

σ2(x) = a− bθ Thus, the norm of NK/Q(x) = σ1(x)σ2(x) = a2 − 2b2.

The norm of θ is therefore, NK/Q(θ) = −1.

Definition 5 The infinite lattice code is defined as

C∞ =

{[x1 x3x2 x4

]=

[a+ bθ c+ dθ

γ(c+ dθ

)a+ bθ

]: a, b, c, d ∈ Z(i), γ ∈ C, θ

}(2.8)

The infinite code C∞ is a linear code 1. That’s why, the minimum determinant of C∞ given byδmin = minX1 6=X2 |det(X1 −X2)|2 could be reduced to δmin = min0 6=X∈C∞ |det(X)|2.

The finite code is obtained from the infinite code by limiting the information symbols to a, b, c, d ∈S ⊂ Z(i) where S is the QAM constellation.


A golden codeword is a finite lattice codeword where x1, x2, x3, x4 are restricted I, where I = αOK.Thus, CI ⊂ C∞ is defined by :

CI =

{[x1 x3x2 x4

]=

1√5

[α(a+ bθ) α(c+ dθ)

γ.α(c+ dθ

)α(a+ bθ)

]}(2.9)

where a, b, c, d ∈ Z(i), γ ∈ C , α ∈ OK, α = conj(α).

Construction of Golden code

In the previous subsection, the structure of the Golden code has been defined. α and γ have beenchosen in [4], in order to satisfy the space Time design criterion (rank and determinant criterion).This implies to have a full rank matrix X and to maximize δminC(I).

det(XX∈CI) =1

5NK/Q(α).

(NK/Q(a+ bθ)− γNK/Q(c+ dθ)

)(2.10)

- γ computation :

1. From eq. (2.10), we deduce that γ should not be a norm of elements in K in order toguarantee non zero determinants.

2. To guaranty that the same energy is transmitted for each antenna at each use limits ourchoice to |γ| = 1, i.e γ = ±1,±i.

Thus, γ = i , since it is proved that it is not a norm for K = Q(i,√

5)

- α computation:In [4], it has been proved that the ideal I, generated by α = 1 + i− iθ, maximize δmin(CI).

Note that, for γ = i, the minimum module of NK/Q(a+ bθ)− γNK/Q(c+ dθ) is 1 (take a = 1,b = c = d = 0 ).

We conclude that det(XX∈CI) = 15NK/Q(α) = 2+i

5 .

And thus, δmin(CI) =∣∣2+i

5

∣∣2 = 15

1If X1, X2 ∈ C∞, then X1 +X2 ∈ C∞.

19


The Golden Codeword is therefore :

CI =

{[x1 x3x2 x4

]=

1√5

[α(a+ bθ) α(c+ dθ)

i.α(c+ dθ

)α(a+ bθ)

]: a, b, c, d ∈ Z(i)

}(2.11)

where : θ = 1+√5

2 , θ = 1−√5

2 , α = 1 + i− iθ, α = 1 + i− iθ.

Properties of the golden code

The Golden code properties could be summarized as followed :

- The GC achieve full spatial multiplexing gain by allowing to send 2 symbols pcu.

- The GC has full diversity gain since γ has been chosen in order to satisfy the rank criterion, i.eto have det(X) 6= 0 , and thus rank(X) = 2.

- The GC achieves the trade-off diversity-multiplexing gain. The GC is a NVD code sinceδmin(CI) = 1

5 independently of spectral efficiency.

- The Golden code codeword in eq.(2.11) could be written in a reduced form in order to simplifythe ML decoding.

vecX =

x1x2x3x4

= G

abcd

=

α αθ 0 0

0 0 iα iαθ0 0 α αθ

α αθ 0 0

abcd

(2.12)

- At each use the same energy is transmitted for each antenna. Note that G is a rotation matrix(detG = 1).

2.3 Space time MIMO Receivers

2.3.1 ML criterion

The ML criterion for the MIMO system consists in finding x that minimize ‖y −H.x‖ for all x ∈Cnt . i.e,

x = argx∈Cnt(min

∥∥y −H.xT∥∥)The ML criterion is equivalent to the well known "closest point problem" in lattice theory. The SphereDecoder is one of the algorithm proposed to resolve this problem.

2.3.2 ML soft decoder

The hard decision on bits presented on the previous section, is not widely used in industrial simulatorssuch as the 802.11n simulator. In these cases, the information is coded at the transmitter, and the MLreceiver should be a soft output ML receiver.

We will consider in this section, that the information binary elements are encoded by a convolutionalcode of rateRc. Then, The coded bits are interleaved and fed to a 2m QAM mapper. Symbols are afterthat transmitted on a multiple antenna channel with nt transmit antennas and nr receive antennas. Thesystem model is illustrated in fig 2.2.

20

2.3. SPACE TIME MIMO RECEIVERS

ModulationInterleaverConvolutional

Code C+ Puncturing

ViterbiDecoder

Space Time Block Coding STBC

Deinterleaver&

Depuncturing

ML soft DecoderModulationInterleaver

Convolutional Code C

+ Puncturing

ViterbiDecoder

Space Time Block Coding STBC

Deinterleaver&

Depuncturing

ML soft Decoder

Figure 2.2: Bit Interleaved Coded Modulation (BICM)

At the reception, the probability that the codeword x = xc is :

P (x = xc|y) =

(1√2π

)nr 1√det(R)

exp−xTc R

−1xc2 (2.13)

where R is the correlation matrix of the noise vector Z.

Thus, the ML decoder generates for each bits two metrics computed as :

metrics(bi = 0) = log

∑x=(b0...bn)bi=0

exp−|y−H.x|2

σ2

(2.14)

metrics(bi = 1) = log

∑x=(b0...bn)bi=1

exp−|y−H.x|2

σ2

(2.15)

To reduce the computation of these metrics the so-called maxLogMAP approximation is commonlyused. The previous metrics can be approximated by:

metrics(bi = b) ≈ minx/bi=b

|y −H.x|2

σ2(2.16)

Finally, the Viterbi decodes the information, using the Log Likelihood Ratio (LLR) given by:

LLR(bi = 0) = log

[p(bi = 0)

p(bi = 1)

](2.17)

LLR(bi = 1) = log

[p(bi = 1)

p(bi = 0)

](2.18)

These LLRs are deduced directly from the metrics as shown below and are then used by the Viterbito decode the convolutional code.

LLR(bi = 0) = metrics(bi = 0)−metrics(bi = 1) (2.19)

LLR(bi = 1) = metrics(bi = 1)−metrics(bi = 0) (2.20)

21


Comparison examples

A typical simulation of the overall 802.11n system has been run in order to demonstrate the perfor-mance of each detector. The results presented in Figure 2.3 were obtained for a 2 × 2 MIMO SDMconfiguration with a QPSK and rate 1/2 convolutional code which corresponds to a throughput of26Mb/s on the top of the PHY layer. The results show an advantage for the ML algorithm of 3.23dBover MMSE, and 2.17dB over ZF at a PER equal to 0.01.

Figure 2.3: Comparison between ML, MMSE and ZF with nt = nr = 2 using a QPSK constellationwith a code rate equal to 1/2.

22

2.4. SIMULATIONS RESULTS

2.4 Simulations Results

2.4.1 Simulation Protocols

In this section, the performance of the Golden code is reported in two main MIMO configurations ofinterest:

- Non coded MIMO system (fig.2.1).

- Coded MIMO system(fig.2.2)

We assume that in both case that the transmission is done over a quasi-static rayleigh channelunknown to the transmitter.

The performance has been simulated in terms of packet error rate (PER) versus SNR, for a packetlength of 1000-bits . In the following, SNR gain IS GIVENto a PER of 10−2.

Non coded MIMO System

The block diagram of the system is depicted in (fig.2.1). The SD described in section ?? is used todecode information at the receiver.

Effect of the rate

Fig.2.4.1 reports the performance of the Golden code versus the SDM in a 2 × 2 MIMO configura-tion (nt = 2 and nr = 2) for the following modulations QPSK, 16QAM, 64QAM. The GC gainsrespectively 4.04 dB, 3.9 dB and 3.32 dB over SDM.

Figure 2.4: SDM vs the Golden code for a 2× 2 configuration for different q-QAM constellation

23


The significative gain of GC versus SDM is related to gain in term of coding as well as in diversitygain. The GC has full diversity gain since γ has been chosen in order to have det(X) 6= 0, howeverSDM did not achieve full diversity. In addition, SDM is not a NVD code, whereas the GC has aδmin(CI) = 1

5 independent of spectral efficiency.

Effect of the configuration

Figure 2.5 shows the performance of the GC for different MIMO configurations: 2 × 2, 2 × 3 and2× 4.

By adding antennas at the receiver, the spatial diversity (d = nrnt) of the system increases, and thusthe error rate decreases. The 2×4 configuration gains respectively 7.07 dB and 2.65 dB over the 2×2and 2 × 3 ones. The gain due in spatial diversity leads to a non constant gain in SNR that increaseswith SNR.

Figure 2.5: Comparison of differents configuration: 2× 2, 2× 3, 2× 4 for a QPSK constellation

24

2.4. SIMULATIONS RESULTS

2.4.2 Coded Space Time System

Let’s consider now, the BICM system model illustrated in fig (2.2). The ML soft decoder describedin (2.3.2) is used to decode the information.

In our simulation, the encoders used are the [133,171] and [5,7] convolutional code (CC), with re-spective dmin = 10 and 5. In the following, we will denote the [133 171] encoder by CC10, and [5 7]encoder by CC5.

Figure (2.6) and (2.7) refer to QPSK modulation with coding rate 12 . We notice that, by concatenating

the GC with CC the significant gain obtained with Non coded MIMO system is reduced to 1.87 dBfor the CC5 case in fig.(2.6), and to 0.44 dB in CC10 case in fig.(2.7). From Figure (2.6) and (2.7), itis obvious that the GC gain is reduced when dmin increase.

From these results, we realize that the concatenation of a binary outer code such as the convolutionalcode, with spatial time coding is quite easy to do for the SDM. However, with golden code, it is notthe case. Our results prove that the labeling used for the simulations, i.e gray mapping is not thecorrect label to use with the Golden code.

Figure 2.6: The SDM vs the Golden code in a 2× 2 BICM system for a QPSK modualtion and [5 7]encoder (dmin = 5).

25


Figure 2.7: The SDM vs the Golden code in a 2× 2 BICM system for a QPSK modualtion and [133171] encoder (dmin = 10)

2.5 Conclusion

In this chapter, we present the transmission block of two different MIMO system, the Non codedMIMO system, and the coded MIMO system. We evaluate the performance of the GC versus theSDM in each system. We notice that, the Golden code has a significant gain over the SDM in thenon coded MIMO case. However when concatenated with convolutional code, in BICM case, thereis no gain for GC over SDM. From these results, we can deduce that gray labeling is not the correctlabeling to use with the Golden code. In next chapter, we will investigate how to label correctly theGC in order to extract the GC gain.

26

3. Investigation of space time codes in IEEE802.11n

3.1 Presentation of IEEE 802.11n

The objective of IEEE802.11n standardization is to achieve 100Mbps in top of MAC layer whilestill being backward compatible with IEEE802.11a/g, which results in a maximum PHY rate ofabout 130Mbps. This significant rate increase compared to other IEEE802.11 standards such asIEEE802.11a, whose maximum PHY rate is 54Mbps, is enabled by the introduction of multiple an-tennas at the Access Point and at the Mobile Terminal. These multiple antennas are used to increasethe peak data rate, but also to derive benefit from spatial diversity in order to ensure for instance alarger range of operation for full home coverage, or to better address outdoor hotspot environments.Another feature of IEEE802.11n consists in addressing handsets specificities, such as a small numberof antennas. In next sections, we present the main features of IEEE802.11n physical layer and theimpact of adding a space time coding such as Golden code.

3.2 Presentation of physical layer functional blocks

One major difference of IEEE802.1n compared to other IEEE802.11x PHY layer architectures (e.g.IEEE802.11a) is hence the introduction of multiple transmit and multiple receive antenna concepts inorder to exploit the Multiple Input Multiple Output (MIMO) channel properties. In order to supportthe various features mentioned above, it is necessary to support symmetric and asymmetric antennaconfigurations to accommodate various terminal sizes (Laptop/PC/PDA/Phone). The support of thesedifferent configurations is enabled by the definition of PHY modes based on the transmission ofa number of spatial steams, varying from one to four, that is limited by the minimum number oftransmit and receive antennas.

The Transmission block diagram for IEEE 802.11 n simulator 1 is given in fig. 3.1

The stream parser divides the output of the encoders into blocks that will be sent to different inter-leaver and mapping devices for rate increase compared to a single antenna system. Note that thenumber of convolutional encoders depends on the number of spatial streams. This spatial divisionmultiplexing operation is optionally followed by space-time block coding for range increase. Theconstellation points from one spatial stream are spread into two space-time streams using one of thespace-time block codes (STBC). Finally, spatial mapping can be optionally applied to map the result-ing space-time streams to different transmit chains. Note that this last functional block is intendedto perform spatial processing that is unknown at the receiver. The receiver will then perform the op-erations that are required for decoding of the space-time streams at the input of the spatial mappingblock and that correspond to the content of the signalling field. This is enabled by applying this spa-tial mapping block to the data, but also to the preambles, for appropriate channel estimation and datadecoding. Finally OFDM modulation is applied prior transmission of the space-time streams to the

1Reference [5]

27

CHAPTER 3. INVESTIGATION OF SPACE TIME CODES IN IEEE 802.11N

Scr

ambl

er

FEC

enc

oder

Interleaver

Interleaver

Spa

tial M

appi

ng

QAM mapping

QAMmapping IFFT

IFFT CDD

InsertGI

and Window

InsertGI and

Window

Analogand RF

Analogand RF

FEC

enc

oder

Stre

am P

arse

r

Interleaver

Interleaver QAM mapping IFFT CDD

InsertGI and

Window

Analogand RF

QAM mapping IFFT CDD

InsertGI and

Window

Analogand RF

Enc

oder

Par

ser

N SS Spatial Streams N TX Transmit chains

STB

C

CDD

CDD

CDD

N STS Space Time Streams

Figure 3.1: Transmission Block Diagram for IEEE 802.11 n

analog and RF chains.

3.2.1 Convolutional encoder and puncturing

The data are encoded using the .11a rate 1/2 convolutional coder characterized by the polynomials133 (octal) and 171 (octal) (see figure 3.2). Six tail bits are required for the termination of the code.

Figure 3.2: Convolutional encoder

Two puncturing modes have already been specified for .11a in order to provide rate 2/3 and rate 3/4coding schemes. The corresponding puncturing patterns are presented in Figure 3.3.

Additionally, the code rate of 5/6 is implemented according to the puncturing pattern illustrated inFigure 3.4.

3.2.2 Interleaver

The bit interleaver is used for frequency and spatial interleaving, as illustrated in Figure 3.5.

3.2.3 Mapping

The mapping operation is performed using one of the following constellations: BPSK, QPSK, 16QAMand 64QAM. Each constellation is the Gray coded one specified in .11a.

28

3.2. PRESENTATION OF PHYSICAL LAYER FUNCTIONAL BLOCKS

Figure 3.3: .11a puncturing patterns

3.2.4 Multiple antenna processing

Spatial Division Multiplexing modes

Spatial Division Multiplexing is used for all modes in which the number of transmit antennas is equalto the number of spatial streams.

Space-time Block Coding modes

Robust transmission modes based on Space-Time Block Coding have also been defined. They consistin transmitting a number of spatial streams which is inferior to the number of transmit antennas. Thenthey are applicable in asymmetrical antenna configurations when the number of transmit antennas issuperior to the number of receive antennas.

3.2.5 OFDM modulation

The OFDM modulation for 20MHz bandwidth is based on a 64-point IFFT, with 52 data subcarriersand 4 pilots on subcarriers ±7 and ±21.

29


Figure 3.4: Puncturing pattern to achieve a code rate of 5/6

Figure 3.5: Interleaving operation

3.2.6 IEEE 802.11 n modes

The data modes in 20MHz bandwidth are for 2 spatial streams are presented in Tables 3.1. The samemodulation and coding schemes are considered in 40MHz bandwidth, thus allowing to get twice thedata rate provided in 20MHz bandwidth.

30

3.3. INTEGRATION OF THE GOLDEN CODE IN THE IEEE 802.11 N SIMULATOR

Data rate Modul. Code Nb. cod bits Nb Data Nb Coded Bits Data Bits(Mbits/s) Rate (R) per subc. subc. Pilots per OFDM per OFDM

per ant. (NSD) (NSP ) symbol symbol(NBPSC) (NCBPS) (NDBPS)

13.0 BPSK 1/2 1 52 4 104 2726.0 QPSK 1/2 2 52 4 208 5439.0 QPSK 3/4 2 52 4 208 8152.0 16-QAM 1/2 4 52 4 216 41678.0 16-QAM 3/4 4 52 4 416 162

104.0 64-QAM 2/3 6 52 4 624 416117.0 64-QAM 3/4 6 52 4 624 468130.0 64-QAM 5/6 6 52 4 624 520

Table 3.1: EWC modes for 2 spatial stream - 20 MHz bandwidth

3.3 Integration of the Golden code in the IEEE 802.11 n simulator

In [1], it has been shown that the Golden code outperforms all the existing codes. The huge perfor-mance gains possible with this space time technique encouraged us to integrate the Golden code intothe IEEE 802.11n standard for wireless LANs.

3.3.1 MIMO-OFDM Processing

At the Transmitter

The bit stream is first coded, then divided into two spatial streams S1 and S2 in Golden code case.Each stream is then interleaved and modulated. The data symbols to be transmitted are first encodedby a space-frequency encoder into blocks of size nt × PU , where PU = 52 is the number of datasubcarriers.

Consider the 4 data symbols s1,1, s2,1, s1,53, s2,53 that belongs to the two spatial streams S1 and S2,such as following:

S1 : s1,1 s1,2 . . . s1,53 . . .S2 : s2,1 s2,2 . . . s2,53 . . .

si,j denotes the jth symbol of the spatial stream i. The Golden codeword is made from these 4 datasymbols according to :

vecX =

x1,1x2,1x1,53x1,53

=

α αθ 0 0

0 0 iα iαθ0 0 α αθ

α αθ 0 0

s1,1s1,53s2,1s2,53

(3.1)

- x1,1 and x2,1 are transmitted over antennas 1 & 2 respectively on tone #1 of the first OFDMsymbol ;

- x1,53 and x2,53 are transmitted over antennas 1 & 2 respectively on the next tone #1 of thesecond OFDM symbol.

- The Golden codeword is then transmitted within 2 consecutive OFDM symbols.

31


The channel

The implementation of the Golden code requires that the channel remains constant over the spacetime codeword. In IEEE802.11n context, the channel remains constant in time over one subcarrier ,i.e:

H[PU

(k)i

]= H

[PU

(k+1)i

]= . . . (3.2)

where :

- PU (k)i is the index of subcarrier i in the kth OFDM symbol,

- PU (k+1)i is the index of the same subcarrier i in the (k + 1)th OFDM symbol.

Channel diversity

The diversity coding techniques discussed in chap.2 extract only spatial diversity in a MIMO-OFDMsystem. However, frequency diversity may also be available if the channel is frequency selective.

In OFDM system, the tones spaced greater than the coherence band of channel experience indepen-dent fading. Let nc be the number of the coherence bandwidths within B(B is the channel Bandwidth= 20 MHZ).

nc =BcB

(3.3)

where Bc is the coherent Bandwidth. The total diversity available is then

d = ntnrnc (3.4)

when the antennas are uncorrelated. In order to extract full diversity, data must be suitably spreadacross frequency and space.

Channel models

In order to assess the performance in different channel conditions of practical interest, two differentselective channel models (for more details on these channel, see [?]) are considered:

- Channel B : Characterized by a 9-tap tapped delay line profile with 15 ns root mean square(rms).Channel B is useful benchmark for scenarios with limited frequency selectivity, like home res-idential environement (nc = 1).

- Channel D : Characterized by a 18-tap and 50 ns rms delay spread. Channel D has a significantfrequency diversity as typical of indoor office (nc = 5).

In both case, low correlation factors are experienced, then full diversity must be achievable.

At the reception

The OFDM demodulate first the received signal. The ML soft decoder described in sec.2.3.2, is usedto compute the ML metrics as detailed in equations 2.15 and 2.15. The LLR for the Viterbi decoderare deduced from the metrics in equations as presented 2.20 and 2.20.

32


3.3.2 Performance Analysis

Scenarios

The performance of the Golden code is evaluated in the IEEE context where the block transmissiondiagram is depicted in 3.6.

Bit Source

CCEncoder

Interleaver

Interleaver Mapper

Mapper

STBC Golden Code

(Optional)

OFDMModulator

OFDMModulator

CCDecoder

Deinterleaver

Deinterleaver Demapper

Demapper

Soft Output MIMO Decoder

OFDMDemodulator

OFDMDemodulator

MIMOChannel

Figure 3.6: Block transmission diagram for IEEE 802.11n

The performance has been evaluated in terms of packet error rate (PER) versus SNR, for a packetlength of 1000-bits. In the following, SNR gain will be related to a PER of 10−2.

GC in IEEE 802.11n (QPSK-1/2)

Fig. 3.7 reports the performance of Golden code versus the Alamouti code and SDM, for channelmodel D in a 2 × 2 MIMO configuration using QPSK -1/2 modulation. It can be shown from thiscomparison that the GC have nearly the same performance as SDM in 802.11n context. This loss hasalready been observed when we concatenate the GC with convolutional code in 2.4 for a Rayleighchannel model.

As we said before, there is no gain due to the fact that the concatenation of a binary outer code suchas convolutional code, with spatial time coding is quite easy to do for SDM. However, with goldencode, it is not the case. Labeling used for in IEEE 802.11 n, i.e Gray mapping is not the correct labelto use with golden code. Note that the very close performance is achieved using the simple Alamouticode.

33


Figure 3.7: Golden Code vs SDM and Alamouti code in IEEE 802.11n context for a QPSK - 1/2modulation with spectral efficiency = 2bpcu - Channel model D

Impact of Convolutional code

In Figure 3.8, we compare for the same simulations parameters (i.e QPSK-1/2, channel D) SDMwith GC in non coded case. In this simulation, we eliminate the convolutional code, and we make harddecisions over bits2. We observe that the GC gains about 8dB over SDM, and 2dB over Alamouti.This gain was also observed in a simple Rayleigh case(see 2.4).

2The value of the bit bi is equal to b where b = 0 or 1 if ln(p(bi = b)) < ln(p(bi = 1− b)).

34


Figure 3.8: Golden Code vs SDM and Alamouti code in IEEE 802.11n context with out convolutionalcode for a QPSK Modulation with spectral efficiency = 4bpcu - Channel model D

Various Considerations

In figure 3.9, we summarize the performance of GC vs SDM in 3 different cases:

- case 1: IEEE 802.11n context.

- case 2: Hard ML decoder and Viterbi decoding according to Hamming distance metrics.

- case 3: With out convolutional encoder and with hard ML decoding.

From figure 3.9, we can deduce that the Golden code gains 3dB in case 2, and 8dB in case 3. In case1, no gain is observed.

GC in IEEE 802.11n (16QAM-3/4)

The same results as above are observed for a 16QAM-3/4 modulation on channel D. It is clear fromfigure 3.10 that no gain is observed even if the rate increases.

GC in IEEE 802.11n (16QAM-3/4) - Channel B

In this case, we run our simulations for 16QAM-3/4 over channel B. We notice that we have a smallgain for the GC over SDM. This gain is similar to the gain obtained in Rayleigh case. In fact, channelB has a large coherence Bc, i.e nc = Bc

B ≈ 1, and this lead to similar results for channel B andRayleigh channels.

35


Figure 3.9: SDM vs GC in 3 different cases for QPSK 1/2 - Channel model D

Figure 3.10: Golden Code vs SDM in IEEE 802.11n context for a 16QAM - 3/4 modulation withspectral efficiency = 6bpcu - Channel model D

36


Figure 3.11: Golden Code vs SDM in IEEE 802.11n context for a 16QAM - 3/4 Modulation withspectral efficiency = 6bpcu - Channel model B

37

CHAPTER 4. PARTITIONING THE GOLDEN CODE

4. Partitioning the Golden Code

4.1 Introduction

As we see, in chap 2 and in 3, the mapping of bits into symbols affects largely the performance of GC.From these two chapters, it was clear that the Gray mapping is not the correct mapping to use withGC. This chapter will give answers about the impact of mapping on performances, and it will ask newquestions and enlight new problems. In the first section, we will present the partitioning principle.The next section will be dedicated to the presentation of Golden code partitioning.

4.2 Mapping by set partitioning method

In this section, the mapping by set partitioning principle is presented. The first subsection is dedicatedto define some algebraic notions such as quotient group, coset leaders used in partition. In the secondsubsection, the use of partitioning in coded modulation is presented. This section, is followed by twoexamples illustrating the partition in Z and in Z8.

4.2.1 Conceptual description

The figure 4.1 is an example on mapping by set partitioning for a set Lof nL elements.

L

L1 L1 + g1

L2 L2 + g2 L2 + g2 + g1L2 +g1

L3 + g3L3 L3 + g2 + g3 L3 + g1 L3+ g3 + g1 L3 + g2 + g1 L3+ g2+ g3 + g1L3 + g2

0

00

0

0

0

0

1

1 1

1 1 1 1

L

L1 L1 + g1

L2 L2 + g2 L2 + g2 + g1L2 +g1

L3 + g3L3 L3 + g2 + g3 L3 + g1 L3+ g3 + g1 L3 + g2 + g1 L3+ g2+ g3 + g1L3 + g2

0

00

0

0

0

0

1

1 1

1 1 1 1

Figure 4.1: Ungerboeck partition

In the first partitioning, L is subdivided into 2 subsets L1 and L1 + g1 of nL2 elements, such that

minimal distance between points increase. These two subsets form a partition1 of L denoted by[L|L1].

1A and B form a partition of a given set E if :

- A⋂

B = ∅.- A

⋃B = E.

38

4.2. MAPPING BY SET PARTITIONING METHOD

[L|L1] is called the quotient group and the elements of [L|L1] are called the coset leaders.In our example, [L|L1] = {bg1, b ∈ GF(2)} and 0 and g1 are the coset leaders.

For the second partitioning, L1 is subdivided into 2 subsets: L2 and L2 + g2. The other partitions ofL are deduced from L1 partitions by a simple translation of g1.Thus, [L|L2] = {b1g1 + b2g2, (b1,b2 ) ∈ GF(2)}, where b1, and b2 are the bits used to select the subset.

Similarly, for the third partition, we get:

[L|L3] = {b1g1 + b2g2 + b3g3, (b1,b2,b3 ) ∈ GF(2)}, where b1, b2 and b3 are the bits used to select thesubset.

As an example, L3 + g2 + g3 is selected by 011.

Notice that, it is not necessary to carry out the partition to the limit where each subset contains onlyone single point. In this case, b1, b2 and b3 would be used to select the subset.

4.2.2 Use of partitioning in coded modulation

In figure 4.2, we consider a block of m bits, m = k1 + k2.

BinaryEncoder

Select Subset

Select Point From the

SubsetSignal Point

K1 bits n bits

K2 bits: Uncoded bits

BinaryEncoder

Select Subset

Select Point From the

SubsetSignal Point

K1 bits n bits

K2 bits: Uncoded bits

Figure 4.2: Coded Modulation Block diagram

For the partitioning, the n bits used to select the subset should more protected than other bits. That’swhy, we encode the k1 bits into n bits. These n bits will be used to select the corresponding subsetwhile k2 bits are left uncoded and are used to select one of the 2k2 points in each subset.

The binary encoder was not integrated to our simulator yet, but this will be the object of our futureresearch. The rules of designing this binary encoder were presented by Ungerboeck.

Simple example

As as simple example, let’s take L = Z as the ring of integers. 2Z is the subset of odd integers, and2Z+1 of even integers, such as:

2Z = {. . . ,−6,−4,−2, 0, 2, 4, 6, . . .}2Z + 1 = {. . . ,−5,−3,−1, 1, 3, 5, . . .}

These 2 subsets form a partition of Z denoted by [Z|2Z], where: [Z|2Z] = {0,1}. 0 and 1 are the cosetleaders.

39


Z

2Z 2Z + 1

4Z 4Z + 2 4Z + 34Z +1

8Z +48Z 8Z +6 8Z +1 8Z +5 8Z +3 8Z +78Z +2

0

00

0

0

0

0

1

1 1

1 1 1 1

Z

2Z 2Z + 1

4Z 4Z + 2 4Z + 34Z +1

8Z +48Z 8Z +6 8Z +1 8Z +5 8Z +3 8Z +78Z +2

0

00

0

0

0

0

1

1 1

1 1 1 1

Figure 4.3: Partitioning [Z|8Z]

In figure 4.3, 2Z is partitioned by 4Z, and so on, ... Finally, Z could be written as:

Z = 8Z + [Z|2Z] + [2Z|4Z] + [4Z|8Z].

The partitioning [Z|2Z|4Z|8Z] is given in fig.4.3.

The Ungerboeck labeling would be then:

0 1 2 3 4 5 6 7-1

000 100 010 110 001 101 011 111

0 1 2 3 4 5 6 7-1

000 100 010 110 001 101 011 111

Figure 4.4: Ungerboeck labeling

Partitioning in Z8

In lattice theory, an important partition chain of order 4 for Z8 lattice is defined by:

2Z8 ⊂ L8 ⊂ E8 ⊂ D24 ⊂ Z8

The quotient group for two consecutive lattices is a group of order 4 which is generated by two binarygenerating vectors h1 and h2, such as : [Λk |Λk+1] = {b1h1 + b2h2 |b1, b2 ∈ GF (2)}. Coset leadersare therefore 0, h1, h2, h1 + h2. The lattices in the partition chain can be obtained by ConstructionA using the nested sequence of binary codes. From these binary codes, the binary generator vectorcould be deduced then, by:

[Z8|D24] :

h11 = (0, 0, 0, 0, 0, 0, 0, 1)

h12 = (0, 0, 0, 1, 0, 0, 0, 0)

[D24|E8] :

h21 = (0, 0, 0, 0, 0, 1, 0, 1)

h22 = (0, 0, 0, 0, 0, 0, 1, 1)

40

4.3. PARTITIONING THE GOLDEN CODE

[E8 | L8] :

h31 = (0, 1, 0, 1, 0, 1, 0, 1)

h32 = (0, 0, 1, 1, 0, 0, 1, 1)

[L8|2Z8]:

h41 = (0, 0, 0, 0, 1, 1, 1, 1)

h42 = (1, 1, 1, 1, 1, 1, 1, 1)

4.3 Partitioning the Golden code

In [2], Champion, Belfiore and Rekaya propose to partition of the Golden code in order to increasethe minimum determinant. It has been demonstrate that the subcodeG1 = {XB,X ∈ G} obtainedas a right principal ideal of the Golden code G where

B =

[i(1− θ) 1− θiθ iθ

]provides the minimum determinant 2δmin(for further details see [2]). Similarly, the subcode Gkobtained by GK =

{XBk, X ∈ G

}provides the minimum determinant 2kδmin.

In [3] , Hong, Viterbo and Belfiore implement this partitioning by finding that this partitioning isequivalent to the partition of Z8 presented before.

In fact, by vectorizing and separating real and imaginary parts in the expression of received signal,we find the Golden codeword correspond to the rotated Z8 point lattice. Similarly, it can be shownthat:

- The codewords of G1 correspond to D24

- The codewords of G2 correspond to E8

- The codewords of G3 correspond to L8

- The codewords of G4 correspond to 2Z8

In table 4.3, we summarize the relations between lattices and Golden subcode.

Level Golden subcode Lattice ∆min

0 G Z8 δmin1 G1 D2

4 2δmin2 G2 E8 4δmin3 G3 L8 8δmin4 G4 2Z8 16δmin

4.3.1 Partitioning and coding gain

Over a Rayleigh channel, The pairwise error is upper bounded by : (∆min)−nr(EsN0

), where

∆min = minX 6=0|det(XXT )|.

41


Thus, the asymptotic coding gain over a Rayleigh channel is given by

γs =

√∆min,1/Es,1√∆min,2/Es,2

(4.1)

Thus, by multiplying the determinant by 2, the asymptotic coding gain is equal to 3dB.

4.3.2 Encoding and decoding the partitions of GC in E8

In this subsection, we will consider a constellation point in E8. Therefore, by comparing with classi-cal mapping, i.e in Z8, the minimum squared determinant is multiplied by 4.

E8 encoder

A constellation point x ∈ E8 can be written as x = 2u + c, where u is a 8-dimensional vector withinteger components {0, 1, ...,mi − 1}, i = 1, ...8 and c is used to select one of the 16 subsets, i.e:

c = (b1, b2, b3, b4)

h31h32h41h42

The E8 lattice constellation point have integer components in {0, 1, ...2mi − 1}. For a Q-QAMconstellation, where Q is an odd power of 2, mi is computed such as :

2mi =√Q

x = 2u + c

q bits 2Z8 Encoder 8

2u

[C2 / C4 ] =

[E8/2Z8]8 bits

c4 bits

+ x = 2u + c

q bits 2Z8 Encoder 8

2u

[C2 / C4 ] =

[E8/2Z8]8 bits

c4 bits

+

Figure 4.5: E8 encoder

The Golden codeword formed by 4 Q-QAM symbols could be then addressed by a total of 8ν + 4bits. The following table gives the number of bits per codeword for the corresponding modulation.

42


Nbpcw Equivalent inModulation bits per Spectral non coded

Q-QAM codeword Efficiency caseQPSK 4 2 bpcu BPSK

16-QAM 12 6 bpcu 8-QAM64-QAM 20 10 bpcu 32-QAM

Example

For example, let’s consider a Golden codeword of 20 bits in E8 illustrated in figure 4.6. The first fourbits would be used to select the corresponding subset among the 16 other subsets (2Z8 and the otherspartitions). The other 16 bits are mapped by a Gray mapper into four symbols: The real part and theimaginary < and = are bounded by 0 and 3. ui are then either the real part or the imaginary part ofeach symbol.

c = (b1,b2, b3, b4)G2

(c1, c2, c3, c4, c5,c6, c7, c8)

Gray Mapping

u1 u2 u3 u4 u5 u6 u7 u8c = (b1,b2, b3, b4)G2

(c1, c2, c3, c4, c5,c6, c7, c8)

Gray Mapping

u1 u2 u3 u4 u5 u6 u7 u8

Figure 4.6: Example of mapping for 10 bpcu

Finally, x = 2u+ c.

Decoding in E8 lattice

Given the received point y, the lattice decoder first minimize the 16 squared Euclidean distances ineach coset

d2(j) = minu(j)∈Z8

∥∥yj − 2Huj∥∥ (4.2)

where y(j) = y −Hc(j), j = 0, 1, ..., 15, then makes the final decision as u = arg minj

(d2(j)

).

In our simulator, we compare the accuracy of decoder with exhaustive research with the one presentedabove. Both algorithms provide the same results.

4.3.3 Simulation results

The performance has been simulated in terms of packet error rate (PER) versus SNR in the uncodedMIMO case and for the coded MIMO case.

Non coded MIMO system

In figure 4.7 and 4.8, we show our simulation results over a golden codeword of 12 bits which couldbe mapped into four 16QAM symbols in E8 constellation, and into four 8QAM symbols in Z8. Weconsider an AWGN channel in figure 4.7, and a rayleigh channel in figure4.8.

43


Over the Gaussian channel, the asymptotic gain does not depend on constellation expansion. That’swhy, the asymptotic gain in AWGN

γs = 10 log10

√4δmin√δmin

∼ 3dB (4.3)

However for a Rayleigh quasi-static channel, the loss due to expansion of the constellation(∼3dB)compensates the gain due to determinant increase, i.e :

γs = 10 log10

√4δmin√δmin

− 10log10Es,partEs,class

∼ 3dB − 3dB ∼ 0dB (4.4)

Figure 4.7: Comparison of performance of GC for a partitioned 16-QAM versus classical 8-QAM(6bpcu) over AWGN channel

44


Figure 4.8: Comparison of performance of GC for a partitioned 16-QAM versus classical 8-QAM(6bpcu)over Rayleigh channel

Coded MIMO system

The same results as in uncoded MIMO case for E8 partitioning are observed for coded MIMO casefor the same reasons detailed above.

Figure 4.9: Comparison of performance of GC for a partitioned 16-QAM versus classical 8-QAM(6bpcu)over Rayleigh channel in coded MIMO case

45

CHAPTER 5. CONCLUSION AND PERSPECTIVES

5. Conclusion and perspectives

5.1 Difficulties in the project

During my internship, I found that the courses I have learnt at ENST Paris and in my master at Paris6, are very useful and helpful in the implementation and in the analysis of results. However, severaldifficulties still appear in the project.

The first difficulty is the complexity of the problem. In fact, the results obtained over a non codedMIMO system do not match with the one obtained in coded MIMO system and therefore in IEEE802.11n context. There is no expression of the error probability in Bit Interleaved Coded Modulationis not given exactly and that what create the complexity of the problem. To resolve the problem,several blocks such as the mapper, the interleaver, the channel , ... should be taken into considerations.In our case, we simplify the 802.11n problem to a problem over Rayleigh channel, and we show theimpact of modifying the mapper in this case.

The second difficulty is that there are many ’open questions’. For example : How can we prove thatif a certain method works in a particular context, it is also valid in other cases? How far do we mustinvestigate the questions that arise during the study to meet the deadline of the project?.

5.2 Summary of contribution

1. Assessment of Golden code over a Rayleigh channel in uncoded MIMO and coded MIMO caseand in IEEE 802.11n context.

2. Not finished work on the impact of partitioning the Golden code in coded MIMO case.

5.3 Future research and Perspective

In [3], Y.Hong propose to increase the ∆min,1 in order to increase the coding gain. This could bedone by adding a trellis encoder to protect bits that select a partition in Z8 as illustrated in figure 5.1designed using Ungerboeck design rules and will increase efficiently the coding gain. Performancehas been evaluated for uncoded MIMO case and provides 3dB of gain over the classical GC. Fu-ture research will analyze the impact of adding trellis in such coded MIMO system to improve theperformance.

46

5.3. FUTURE RESEARCH AND PERSPECTIVE

q1 bits

q2 bits

q3 bits

TrellisEncoder

[Cl0+l / C4]

[Cl0 / Cl0 + l ] 8 bits

8 bits

Mod 2

+2Z8

Encoder 8

2u

c1

c2

c

x = 2u + c

nc bitsq1 bits

q2 bits

q3 bits

TrellisEncoder

[Cl0+l / C4]

[Cl0 / Cl0 + l ] 8 bits

8 bits

Mod 2

+2Z8

Encoder 8

2u

c1

c2

c

x = 2u + c

nc bits

Figure 5.1: Trellis Coded Modulation

47

LIST OF FIGURES

List of Figures

1.1 75 years of innovation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 New segmentation in 4 business units (01/01/2005) . . . . . . . . . . . . . . . . . . 8

1.3 Sales by region, end of year 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Headcount and sales by country, end of 2004 . . . . . . . . . . . . . . . . . . . . . 9

1.5 Motorola Labs in the world . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6 Radio Link Technology Team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1 Transmission Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Bit Interleaved Coded Modulation (BICM) . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Comparison between ML, MMSE and ZF with nt = nr = 2 using a QPSK constel-lation with a code rate equal to 1/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 SDM vs the Golden code for a 2× 2 configuration for different q-QAM constellation 23

2.5 Comparison of differents configuration: 2× 2, 2× 3, 2× 4 for a QPSK constellation 24

2.6 The SDM vs the Golden code in a 2 × 2 BICM system for a QPSK modualtion and[5 7] encoder (dmin = 5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.7 The SDM vs the Golden code in a 2 × 2 BICM system for a QPSK modualtion and[133 171] encoder (dmin = 10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1 Transmission Block Diagram for IEEE 802.11 n . . . . . . . . . . . . . . . . . . . . 28

3.2 Convolutional encoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 .11a puncturing patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 Puncturing pattern to achieve a code rate of 5/6 . . . . . . . . . . . . . . . . . . . . 30

3.5 Interleaving operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.6 Block transmission diagram for IEEE 802.11n . . . . . . . . . . . . . . . . . . . . . 33

3.7 Golden Code vs SDM and Alamouti code in IEEE 802.11n context for a QPSK - 1/2modulation with spectral efficiency = 2bpcu - Channel model D . . . . . . . . . . . 34

3.8 Golden Code vs SDM and Alamouti code in IEEE 802.11n context with out convolu-tional code for a QPSK Modulation with spectral efficiency = 4bpcu - Channel modelD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.9 SDM vs GC in 3 different cases for QPSK 1/2 - Channel model D . . . . . . . . . . 36

3.10 Golden Code vs SDM in IEEE 802.11n context for a 16QAM - 3/4 modulation withspectral efficiency = 6bpcu - Channel model D . . . . . . . . . . . . . . . . . . . . . 36

48

LIST OF FIGURES

3.11 Golden Code vs SDM in IEEE 802.11n context for a 16QAM - 3/4 Modulation withspectral efficiency = 6bpcu - Channel model B . . . . . . . . . . . . . . . . . . . . . 37

4.1 Ungerboeck partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Coded Modulation Block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 Partitioning [Z|8Z] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.4 Ungerboeck labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.5 E8 encoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.6 Example of mapping for 10 bpcu . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.7 Comparison of performance of GC for a partitioned 16-QAM versus classical 8-QAM(6bpcu) over AWGN channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.8 Comparison of performance of GC for a partitioned 16-QAM versus classical 8-QAM(6bpcu)over Rayleigh channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.9 Comparison of performance of GC for a partitioned 16-QAM versus classical 8-QAM(6bpcu)over Rayleigh channel in coded MIMO case . . . . . . . . . . . . . . . . . 45

5.1 Trellis Coded Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

49

BIBLIOGRAPHY

Bibliography

[1] J.-C Belfiore, G.Rekaya, and E.Viterbo. The golden code: A 2 × 2 full rate space time codewith non vanishing determinants. IEEE Transactions on information theory, 51(2):1432–1436,apr 2005.

[2] D. Champion, J.-C. Belfiore, G.Rekaya, and E.Viterbo. Canadian workshop on information the-ory on information theory. 2005.

[3] Yi Hong, E.Viterbo, and J.-C. Belfiore. Golden space-time trellis coded modulation. IEEE Trans-actions on Information Theory, 2006.

[4] G. Rekaya-Ben Outhman. Nouvelles Constructions algébriques de codes spatio-temporels at-teignant le compromis multiplexage diversité. PhD thesis, ENST Paris, December 2004.

[5] Stéphanie Rouquette-Léveil, Patrick Labbé, Laurent Mazet, and Markus Muck. Technology in-vestigation and assessment for ieee802.11n. Motorola internal document.

[6] Vahid Tarokh, Nambi Seshadri, and A. R. Calderbank. Space time codes for high data ratewireless communication : performance criterion and code construction. IEEE transactions oninformation theory, 44(2):744–765, mar 1998.

[7] L. Zheng and D. Tse. Diversity and multiplexing : A fundamental tradeoff in multiple antennachannels. IEEE Transactions on Information Theory, may 2003.

50

Documents

Internship Report Investigation of Space Time coding in ...Je tiens à remercier également M. Patrick Labbé pour son aide et son encouragement qui ont con-tribué au bon déroulement