aspi 1043 report

7/31/2019 Aspi 1043 Report

1/171

A ALBORG U NIVERSITYINSTITUTE OFELECTRONICSYSTEMS

T ITLE :Hardware/SoftwareCo-Design of a MultipathJakes Channel Simulator foran OFDM System

P ROJECT P ERIOD :September14th , 2004 -June2nd , 2005

P ROJECT G ROUP :Antanas VeiverysVara Prasad Goluguri

S UPERVISORS :Christian RomOle OlsenPeter KochTroels Bundgaard Srensen

Number of reports printed: 11

Number of pages in report: 114

Number of pages in appendix: 42

Total number of pages: 171

A BSTRACT :The rst part of this Masters thesis de-tails the analysis of a generic OFDMsystem. An analytical OFDM systemmodel is expressed using matrix-vectornotation. Several channel models areanalyzed and added to the transceiversimulator. The extreme programmingmethodology is used and evaluated dur-ing thedevelopment of the MATLAB sim-ulator for the OFDM system model.The second part of the thesis is dedi-cated to decreasing the simulation time.The most computationally complex partof the simulator is ofoaded to an FPGA-based hardware accelerator by usinghardware/software co-design techniques.A structured analysis advocated by theRugby model is followed to dene the ac-tivities and abstraction levels. The con-cepts of co-design are presented in a lit-erature study. The xed precision per-formance of the Jakes channel simula-tor algorithm is analyzed. The algorithmis optimized in order to reduce preci-sion requirements. The simulator is writ-ten inHandel-C programming languageand compiled into hardware by usingDK Suite IDE. Finally, an analysis of perfor-mance increase is performed.

7/31/2019 Aspi 1043 Report

2/171

7/31/2019 Aspi 1043 Report

3/171

P REFACE

This thesis is submitted to Aalborg University (AAU), Institute of Electronic Sys-tems, for the Master of Science in Electronic Engineering degree with specializa-tion inApplied Signal Processing & Implementation (ASPI). It presents the workperformed from September14th 2004 to June2nd 2005. It has been conducted inco-operation withCellular Systems (CSys) division at Aalborg University.

The thesis consists of two parts:

An Analytical part that contains the analysis and simulation of anOrthogonalFrequency Division Multiplexing (OFDM) system model;

An Implementation part that applies hardware/software co-design to the radio

channel model in order to speedup the simulations.

The ASPI specialization that we have been attending is not directed to dealwith topics relating to mobile communications, so an extensive effort was foundnecessary to pursue theanalytical part of the thesis. It deals with the analysisand simulation of a generic OFDM system considering various channel effects.The approach followed to conduct this part of the thesis is based on theExtremeProgramming (XP) methodology.

Theimplementation part deals with hardware/software co-design issues relat-ing to the goal of decreasing the execution time of the simulator developed in theanalytical part. The approach followed to conduct this part of the thesis is inspiredby the Rugby model.

A list of symbols and abbreviations used in the thesis is presented at the endof the report.

MATLAB, C, Handel-C, DK Suite, GCC, RC203 Celoxica development boardandDesign Trotter are thedevelopment tools and languages used during the thesis.

A CD containing the Handel-C source code,MATLAB les and LATEX & pdf versions of the report is supplied with the report.

7/31/2019 Aspi 1043 Report

4/171

The report is addressed to the supervisors and students of ASPI specializationat Aalborg University.

We would like to thank Yannick Le Moullec, a post-doc fellow with Centrefor Embedded Systems (CISS, AAU) for our useful discussions during the imple-mentation part of the thesis. Moreover, Sren Skovgrd Christensens commentson the analytical part of the report are greatly appreciated. In particular, we wouldlike to thank our supervisor Christian Rom for helping us to get the necessarybackground about OFDM by allocating his personal time.

June 2, 2005

Antanas Veiverys Vara Prasad Goluguri

7/31/2019 Aspi 1043 Report

5/171

N OTATION

References to equations, gures, sections, tables and appendices are given in theform of Equation2.1, Figure1.1, Section1, Table 3.1 and AppendixA respec-tively. The references to literature is given in the form of [Proakis, 2001] whichconsists of the last name of the author and year of publication of the correspond-ing literature. If a reference consists of more than two authors, it is represented inthe form of [Coleri et al, 2002]. When the colored item in any reference is clickedwhile accessing the text in the soft copy version of the report, the cursor directsthe reader to the details of that reference.

A list of symbols is given in AppendixA. Page number where the symbol rstappears in an equation is given next to the symbols. Some of the symbols do notappear in more than one equation, therefore they are explained below the equationand not included in the list of symbols.

A list of abbreviations is given in AppendixB.Italized letters in equations indicate scalar values, such asa. Boldface lower-

case letters indicate vectors, for example,a . Boldface uppercase letters indicatematrices:C . A hat over a variable,h, means an estimated value of h. H rep-resents the conjugate transpose (Hermitian transpose) of the matrix, whileT represents a transpose of . a indicates the smallest integer greater than orequal toa. a indicates rounding of a to the nearest integer.

Keywords are written in italics.

7/31/2019 Aspi 1043 Report

6/171

vi

7/31/2019 Aspi 1043 Report

7/171

C ONTENTS

List of Figures xi

List of Tables xiii

1 Introduction 1

1.1 Project Background. . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Project Background. . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Project Contents. . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.2 Limitations. . . . . . . . . . . . . . . . . . . . . . . . . 51.3.3 Time Schedule. . . . . . . . . . . . . . . . . . . . . . . 5

2 OFDM Analytical Model 7

2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Extreme Programming Methodology. . . . . . . . . . . . . . . . 7

2.2.1 XP Evaluation. . . . . . . . . . . . . . . . . . . . . . . 102.3 OFDM Symbol Structure. . . . . . . . . . . . . . . . . . . . . . 122.4 Basic OFDM Transmitter. . . . . . . . . . . . . . . . . . . . . . 15

2.4.1 Cyclic Prex. . . . . . . . . . . . . . . . . . . . . . . . 172.4.2 Sub-carrier Correlation Properties. . . . . . . . . . . . . 192.4.3 Approaches to Modulation. . . . . . . . . . . . . . . . . 222.4.4 Structure of Orthogonality Matrix. . . . . . . . . . . . . 252.4.5 Orthogonality Matrix with Cyclic Prex. . . . . . . . . . 262.4.6 OFDM Transmitter Using IDFT. . . . . . . . . . . . . . 27

2.5 Channel Models. . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7/31/2019 Aspi 1043 Report

8/171

viii CONTENTS

2.6 Basic OFDM Receiver. . . . . . . . . . . . . . . . . . . . . . . 282.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Downlink Simulator 33

3.1 AWGN Channel Simulator. . . . . . . . . . . . . . . . . . . . . 343.1.1 Denitions. . . . . . . . . . . . . . . . . . . . . . . . . 343.1.2 Assumptions. . . . . . . . . . . . . . . . . . . . . . . . 363.1.3 Channel Model. . . . . . . . . . . . . . . . . . . . . . . 363.1.4 Channel Estimation. . . . . . . . . . . . . . . . . . . . . 36

3.1.5 Theoretical Performance. . . . . . . . . . . . . . . . . . 363.1.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Rayleigh Channel Simulator. . . . . . . . . . . . . . . . . . . . 38

3.2.1 Denitions. . . . . . . . . . . . . . . . . . . . . . . . . 383.2.2 Assumptions. . . . . . . . . . . . . . . . . . . . . . . . 423.2.3 Channel Model. . . . . . . . . . . . . . . . . . . . . . . 423.2.4 Channel Estimation. . . . . . . . . . . . . . . . . . . . . 453.2.5 Theoretical Performance. . . . . . . . . . . . . . . . . . 463.2.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3 Jakes Channel Model. . . . . . . . . . . . . . . . . . . . . . . . 503.3.1 Reference Model. . . . . . . . . . . . . . . . . . . . . . 503.3.2 Jakes Model. . . . . . . . . . . . . . . . . . . . . . . . 513.3.3 Jakes Multipath. . . . . . . . . . . . . . . . . . . . . . 533.3.4 Approaches for Implementation. . . . . . . . . . . . . . 543.3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.3.6 Channel Proles. . . . . . . . . . . . . . . . . . . . . . 583.3.7 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.4 Channel Estimation for OFDM. . . . . . . . . . . . . . . . . . . 633.4.1 Overview of OFDM Channel Estimation Algorithms. . . 633.4.2 Time-Domain Channel Estimation. . . . . . . . . . . . . 673.4.3 Performance of Time-Domain Channel Estimation Algo-

rithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.4.4 Adaptive Time-Domain Channel Estimation. . . . . . . . 713.4.5 Performance of the Adaptive Channel Estimation Algorithm71

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7/31/2019 Aspi 1043 Report

9/171

CONTENTS ix

4 Implementation of the OFDM Simulator 75

4.1 Hardware/Software Co-Design. . . . . . . . . . . . . . . . . . . 764.1.1 Rugby Meta Model. . . . . . . . . . . . . . . . . . . . . 764.1.2 Generic Co-Design Flow. . . . . . . . . . . . . . . . . . 764.1.3 HW/SW Co-Design Literature Survey. . . . . . . . . . . 80

4.2 Algorithm Analysis. . . . . . . . . . . . . . . . . . . . . . . . . 834.3 Platform Selection. . . . . . . . . . . . . . . . . . . . . . . . . 864.4 Methodology for Channel Model Implementation. . . . . . . . . 87

4.4.1 The DK Methodology. . . . . . . . . . . . . . . . . . . 87

4.4.2 Handel-C. . . . . . . . . . . . . . . . . . . . . . . . . . 894.4.3 Followed Methodology. . . . . . . . . . . . . . . . . . . 90

4.5 Channel Model Implementation. . . . . . . . . . . . . . . . . . 924.5.1 Partitioning and Allocation. . . . . . . . . . . . . . . . . 924.5.2 Hardware/Software Implementation Testing. . . . . . . . 934.5.3 Fixed-pointMATLAB . . . . . . . . . . . . . . . . . . . . 944.5.4 Jakes Model Optimization. . . . . . . . . . . . . . . . . 954.5.5 Handel-C Implementation. . . . . . . . . . . . . . . . . 98

4.6 Design Options and Decisions. . . . . . . . . . . . . . . . . . . 1084.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5 Conclusions 111

5.1 The Analytical Part. . . . . . . . . . . . . . . . . . . . . . . . . 1115.2 The Implementation Part. . . . . . . . . . . . . . . . . . . . . . 1135.3 Future Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Bibliography 115

A List of Symbols 125

B List of Abbreviations 131

C Sequential Handel-C Code 135

C.1 Source Code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135C.2 EDIF Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139C.3 Trace and Route Results. . . . . . . . . . . . . . . . . . . . . . 140

7/31/2019 Aspi 1043 Report

10/171

x CONTENTS

D Parallel Handel-C Code 141

D.1 Source Code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141D.2 EDIF Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146D.3 Trace and Route Results. . . . . . . . . . . . . . . . . . . . . . 146

E Lyrtech SignalMaster 149

E.1 System Requirements. . . . . . . . . . . . . . . . . . . . . . . . 149E.2 Platform Architecture. . . . . . . . . . . . . . . . . . . . . . . . 150E.3 Components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

F Celoxica RC203 Development Platform 153

F.1 System Requirements. . . . . . . . . . . . . . . . . . . . . . . . 153F.2 Platform Architecture. . . . . . . . . . . . . . . . . . . . . . . . 153F.3 Components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

G Comparison of FPGAs 157

7/31/2019 Aspi 1043 Report

11/171

L IST OF F IGURES

1.1 Digital communication system. . . . . . . . . . . . . . . . . . . 2

1.2 TheA3 framework . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Gantt chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 Comparison of OFDM and FDM spectra. . . . . . . . . . . . . . 82.2 Extreme Programming (XP) project ow. . . . . . . . . . . . . . 102.3 OFDM symbol structure (baseband). . . . . . . . . . . . . . . . 132.4 Single-carrier and multi-carrier transmission. . . . . . . . . . . . 132.5 Structure of a basic OFDM transmitter/receiver. . . . . . . . . . 14

2.6 Structure of a basic OFDM transmitter. . . . . . . . . . . . . . . 152.7 Quadrature Phase Shift Keying (QPSK). . . . . . . . . . . . . . 162.8 Inter-symbol interference and cyclic prex. . . . . . . . . . . . . 182.9 Cyclic prex insertion. . . . . . . . . . . . . . . . . . . . . . . . 192.10 Auto-correlation of orthogonality functions. . . . . . . . . . . . 212.11 Maximum cross-correlation of orthogonality functions. . . . . . 222.12 Cross-correlation of orthogonality functions. . . . . . . . . . . . 232.13 Sub-carrier modulation methods. . . . . . . . . . . . . . . . . . 24

2.14 IFFT-based OFDM transmitter. . . . . . . . . . . . . . . . . . . 282.15 Basic OFDM receiver. . . . . . . . . . . . . . . . . . . . . . . . 29

3.1 BER performance in AWGN channel. . . . . . . . . . . . . . . . 373.2 OFDM system simulator with a slow fading channel. . . . . . . . 383.3 Multipath fading channel. . . . . . . . . . . . . . . . . . . . . . 393.4 Impulse response. . . . . . . . . . . . . . . . . . . . . . . . . . 393.5 Slow fading Rayleigh channel BER performance. . . . . . . . . 47

7/31/2019 Aspi 1043 Report

12/171

xii LIST OF FIGURES

3.6 Multipath BER performance with xed SNR. . . . . . . . . . . . 483.7 Rayleigh channel BER performance,T u = 100s . . . . . . . . . 493.8 Rayleigh channel BER performance,T u = 30s . . . . . . . . . . 493.9 Jakes envelope. . . . . . . . . . . . . . . . . . . . . . . . . . . 553.10 Jakes output waveform correlation. . . . . . . . . . . . . . . . . 573.11 Jakes fast-fading channel simulation results. . . . . . . . . . . . 613.12 Pilot schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.13 Structure of a time-domain channel estimator. . . . . . . . . . . 683.14 Channel estimation BER performance at30km/h . . . . . . . . . 693.15 Channel estimation BER performance at60km/h . . . . . . . . . 703.16 Channel estimation BER performance at120km/h . . . . . . . . 703.17 Structure of an adaptive time-domain channel estimator. . . . . . 71

4.1 Rugby meta model. . . . . . . . . . . . . . . . . . . . . . . . . 774.2 Domains of the Rugby model. . . . . . . . . . . . . . . . . . . . 774.3 Generic co-design ow. . . . . . . . . . . . . . . . . . . . . . . 784.4 Manual OFDM simulator complexity estimates. . . . . . . . . . 85

4.5 Design ow with DK Suite. . . . . . . . . . . . . . . . . . . . . 884.6 Activities of the FPGA implementation. . . . . . . . . . . . . . 914.7 Block diagram of the simulator. . . . . . . . . . . . . . . . . . . 934.8 Block diagram of HW/SW implementation test setup. . . . . . . 944.9 Fixed-point simulation results. . . . . . . . . . . . . . . . . . . 994.10 Channel simulator algorithm. . . . . . . . . . . . . . . . . . . . 1004.11 Optimized Jakes algorithm. . . . . . . . . . . . . . . . . . . . . 1014.12 Data Flow Graph of the Jakes simulator. . . . . . . . . . . . . . 103

4.13 Resource-constrained schedule. . . . . . . . . . . . . . . . . . . 1044.14 Time-constrained schedule. . . . . . . . . . . . . . . . . . . . . 1054.15 Options and decisions in the co-design process. . . . . . . . . . 108

E.1 SignalMaster block diagram. . . . . . . . . . . . . . . . . . . . 150

F.1 RC203 block diagram. . . . . . . . . . . . . . . . . . . . . . . . 154

7/31/2019 Aspi 1043 Report

13/171

L IST OF TABLES

3.1 WiMAX parameters. . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Cross-correlation coefcients of Jakes model path inphase com-ponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3 Cross-correlationcoefcients of Jakes modelpath quadrature com-ponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4 ITU Vehicular A channel prole. . . . . . . . . . . . . . . . . . 583.5 ITU Vehicular B channel prole. . . . . . . . . . . . . . . . . . 583.6 ITU Pedestrian B channel prole. . . . . . . . . . . . . . . . . . 583.7 Resampled ITU Vehicular A channel prole. . . . . . . . . . . . 60

3.8 Resampled ITU Vehicular B channel prole. . . . . . . . . . . . 603.9 Resampled ITU Pedestrian B channel prole. . . . . . . . . . . . 60

4.1 Costs of FPGA implementation. . . . . . . . . . . . . . . . . . . 1064.2 Speedup estimation results. . . . . . . . . . . . . . . . . . . . . 107

C.1 Resource usage of sequential Handel-C program. . . . . . . . . . 140C.2 Minimum FPGA clock period for sequential program implemen-

tation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

D.1 Resource usage of parallel Handel-C program. . . . . . . . . . . 147D.2 Minimum FPGA clock period for parallel program implementation147

G.1 FPGA comparison. . . . . . . . . . . . . . . . . . . . . . . . . 157

7/31/2019 Aspi 1043 Report

14/171

7/31/2019 Aspi 1043 Report

15/171

C HAPTER 1

I NTRODUCTION

1.1 Project Background

The history of Orthogonal Frequency Division Multiplexing (OFDM) dates backto 1957 [Nee, 2005], but it was used in a standard for the rst time in 1995. Thisstandard was developed by the European Telecommunications Standards Insti-tute (ETSI) for Digital Audio Broadcasting (DAB). The long application list of OFDM includes Asynchronous Digital Subscriber Line (ADSL), Wireless LAN(IEEE 802.11), Digital Video Broadcast (DVB), WiMAX (IEEE 802.16), and oth-ers. With time, the use of OFDM has also been increasing. OFDM is consideredto be the most prospective candidate for use in4th generation mobile communica-tions, which makes it a good candidate for research. The basic idea of OFDM is touse a large number of parallel narrow-band sub-carriers instead of a single wide-band carrier to transmit information. This way of transmitting information helpsin handling the effects of multipath propagation efciently. The transmission isefcient because of the spectral overlap of narrowband sub-carriers. OFDM sup-ports various modulations techniques like BPSK, QAM, QPSK etc.

1.2 Project Background

The OFDM system has numerous topics which are interesting but it is unman-ageable for a Masters thesis to deal with all of those topics. To make a realistictime schedule and a task list, the scope of the project is dened in this chapter. Ageneralizeddigital communication system , shown in Figure1.1, helps to frame atask list that not only includes the parts of the system that have to be focused onbut also those that have to be neglected.

7/31/2019 Aspi 1043 Report

16/171

2 Introduction

Figure 1.1: Digital communication system

The messages producedby a sourceare converted into binary form by the inputtransducer. The number of binary bits should be as few as possible as the source

output needs to be free from redundancy. The process of efciently converting theoutput of a digital source into a sequence of binary digits is calledsource encod-ing [Proakis, Salehi, 2002]. The output of the source encoder is fed to thechannelencoder . The channel encoder adds some redundancy into the binary informationsequence to overcome the negative effects of the channel during the transmis-sion of the signal. Thedigital modulator modulates the binary sequence from thechannel encoder to be transmitted using an arbitrary transmission scheme. At thereceiver of the digital communication system, thedigital demodulator processesthe transmitted waveform that has been affected by thechannel and decides onthe transmitted data. The remaining blocks, namelychannel decoder , source de-coder andoutput transducer in Figure1.1correspond to their counterparts at thetransmitter.

Only digital modulator, radio channel and digital demodulator in Figure1.1are investigated in this thesis.

7/31/2019 Aspi 1043 Report

17/171

1.3 Project Contents 3

1.3 Project Contents

The project consists of two major parts:

Analytical part: A generic OFDM system model is studied and modeled. Sev-eral channel models and channel estimation techniques are analyzed.Ex-treme Programming (XP) methodology (explained in Section2.2) is fol-lowed for implementing the OFDM system simulator inMATLAB;

Implementation part: The execution time of theMATLAB simulator developedin the Analytical part is optimized. The simulator is partitioned, allocatedand implemented on a chosen heterogeneous platform. Several software/ hardware co-design techniques and tools are investigated. They are mappedto theRugby meta model (see Section4.1.1).

The framework illustrated in Figure1.2relates the analytical part and the im-plementation part. This framework consists of three domains, namely Applica-tion, Algorithm, and Architecture. Using the rst letters of each of the three do-mains, the framework is symbolically calledA3. It states that there is more thanone way to express the given application through algorithms. In this step variousradio channel modeling algorithms are evaluated. The algorithm that providesthe most realistic channel model is selected. Also, there is a one-to-many map-ping from algorithm to architecture meaning that it is possible to implement thefunctionality in more than one type of architecture.

1.3.1 Goals

Several goals are set up for the Analytical and Implementation parts of the thesis.The goals also dene the time schedule.

Analytical Part

Analysis and design of a downlink OFDM system model; Framing of a generic OFDM system. The components of the system arewritten in simple mathematical notation in order to be able to investigate

them independently;

Study of the OFDM transceiver performance in various radio channel mod-els in the increasing order of real-life resemblance;

7/31/2019 Aspi 1043 Report

18/171

4 Introduction

Figure 1.2: The A3 framework

Study of channel estimation algorithms at the receiver side; Development of the OFDM transceiver model inMATLAB;

Verication of theMATLAB system model against theoretical performancefrom literature.

Implementation Part

Decreasing the execution time of the simulator; Design space exploration for the algorithm implementation;

Use of a hardware/software co-design methodology based on the Rugbymeta-model;

Implementation of the OFDM simulator on a chosen heterogeneous archi-tecture;

7/31/2019 Aspi 1043 Report

19/171

1.3 Project Contents 5

1.3.2 Limitations

Each and every aspect of the system can not be analyzed during the duration of the thesis, therefore, the following limitations are set:

Analytical Part

Only the downlink scenario is considered; Source information processing at the transmitter and receiver is not consid-ered (see Figure1.1); Source encoding at the transmitter is not considered; Channel encoding at the transmitter is not considered; It is assumed that the transmitter and the receiver have identical clocks,they are perfectly synchronized at the start of the rst OFDM symbol path

arrival, i.e., synchronization issues are not analyzed;

Channel impulse response is assumed to be sample-spaced.

Implementation Part

Hardware platform selection is limited to items available at the departmentduring selection.

1.3.3 Time Schedule

The tasks of the project are listed in the Gantt chart shown in Figure1.3. Asalready stated, the thesis is divided into the Analytical and the Implementationparts, with an evaluation between. The time frame of the rst part is allocated alonger time period because of study courses during the 9th semester.

7/31/2019 Aspi 1043 Report

20/171

6 Introduction

Figure 1.3: Gantt chart

7/31/2019 Aspi 1043 Report

21/171

7/31/2019 Aspi 1043 Report

22/171

8 OFDM Analytical Model

(a) OFDM spectrum (b) FDM spectrum

Figure 2.1: Comparison of OFDM and FDM spectra

repeatable process for developing software, it is in fact a methodology, although alightweight one [Brewer, 2001].

XP consists of several core rules and practices:

Planning Game: at the beginning of each small iteration, a list of desired featuresfor the system is set. The required effort is estimated together with the effortthat a team can produce in a given time interval (the iteration). Each featureis written out as aUser Story , which gives the feature a name, and describesin broad strokes what is required. User stories are typically written on smallpaper cards [Brewer, 2001]. They are used to create time estimates for therelease planning meeting. They arealso used instead of a large requirementsdocument [Wells, 2004].

Small Releases: the project is started with the smallest useful feature set. Re-leases are produced early and often, adding a few features each time[Brewer, 2001].

Continuous Testing: before programmers add a feature, they write a test for it.When the suite runs, the job is done. Tests in XP come in two basic avors.

1. Unit Tests are automated tests written by the developers to test func-tionality as they write it. Each unit test typically tests only a singleclass, or a small cluster of classes [Brewer, 2001]

7/31/2019 Aspi 1043 Report

23/171

2.2 Extreme Programming Methodology 9

2. Acceptance Tests (also known as Functional Tests) are specied by thecustomer to test that the overall system is functioning as specied. Ac-ceptance tests typically test the entire system, or some large chunk of it. When all the acceptance tests pass for a given user story, that storyis considered complete. At the very least, an acceptance test couldconsist of a script of user interface actions and expected results thata human can run. Ideally acceptance tests should be automated, ei-ther using the unit testing framework, or a separate acceptance testingframework [Brewer, 2001].

System Metaphor: each project has an organizing metaphor, which provides an

easy way to remember naming convention, such as class, variable or lenames [Brewer, 2001].

Simple Design: according to [Jeffries, 2001], XP teams build software to a sim-ple design. They start simple, and through programmer testing and designimprovement, they keep it that way. An XP team keeps the design exactlysuited for the current functionality of the system. There is no wasted mo-tion, and the software is always ready for whats next.

Refactoring: any duplicate code generated in a coding session is re-factored.Refactoring can be done with condence as the functionality remains un-

changed because it is veried by tests. [Brewer, 2001].Pair Programming: all production code is written by two programmers sitting

at one machine. This practice ensures that all production code is reviewedby at least one other programmer, and results in better design, better testing,and better code [Jeffries, 2001].

Collective Code Ownership: any developer in the team is expected to be ableto work on any part of the code base at any time [Brewer, 2001].

On-site Customer: all the phases of an XP project require communication with

the customer, preferably face to face, on the site. Its best to simply assignone or more customers to the development team [Wells, 2004].

Coding Standards the code is kept consistent and easy for the entire team toread and re-factor [Wells, 2004].

The phases of a XP project are shown in Figure2.2.The development begins with requirements written as User Stories. Release

planning is performed by rst sorting the user stories by their estimated risk and

7/31/2019 Aspi 1043 Report

24/171


Figure 2.2: Extreme Programming (XP) project ow

value. The spike solutions might be developed to solve or explore critical prob-lems. Ideally those programs will be thrown away once every developer gets clearidea about the problem [Sakthivel, 2004]. Each user story is assigned a velocity(speed at which the task can be performed). Then, a release plan is made. It in-cludes several user stories to be implemented for the next release. The release dateis also determined by the picked user stories [Wikipedia, 2004].

When release planning is done, the team proceeds with implementation. XPpractices are followed. Unit tests together with Acceptance Tests from the cus-tomer are used to nd bugs. When the acceptance tests are passed, the smallrelease is nished. It gives feedback to next release planning about the projectvelocity - i.e., what speed can be considered realistic for the next tasks.

2.2.1 XP Evaluation

We applied the XP practices during the analytical part of the project. Extreme Pro-gramming gives advice for both programming and overall project management.However, we did not apply all the rules related to programming. This happenedbecause pure programming did not take a considerable time as compared to theother activities.

The XP practices seemed highly suitable for our needs at the beginning of the project. We organized the analytical part in four major iterations (AWGN,Rayleigh fading, Jakes channel models and channel estimation algorithm). Eachiteration took from 2 to 4 weeks to complete (see Gantt chart in Figure1.3). Listsof required features and activities were written for the iterations, although we didnot use small cards to write User Stories as it did not seem necessary.

7/31/2019 Aspi 1043 Report

25/171

2.2 Extreme Programming Methodology 11

Small weekly intermediate releases of the simulators were produced. Each of them added one or more features while preserving the previous functionalities andcondently moved us towards the nal simulator. We found that the approach of performing a number of small iterations is very useful because it:

forces "on-the-y" verication of each feature that adds up to the nal re-sult; allows easier returning to earlier versions after a failure; gives condence by seeing some results at once instead of keeping the

whole system in development state. In the latter case, there is a possibilityto overlook some critical design aws. In comparison, producing a smallrelease involves only addition of more functionality to an existing system.It might require renement of the system, but this is relatively safe becausethere are automatic tests from the previous iterations. The tests are repeatedin order to verify the success of the renement.

The acceptance tests were performed mainly by means of comparing theBit Error Rate (BER) curves produced by the simulators with theoretical BER curves.As this project is of non-commercial kind, we considered everyone connected to it

to be customers: the team and supervisors. Frequent discussions with supervisorswere held.The XP practices that are related to the programming process were not con-

sidered to be very important in this project. As already mentioned, it was mostlybecause of the relatively low amount of programming as compared to analysis.During the project we committed ourselves to a common coding style in both thesource code and documentation. Thepair programming paradigm proved its ben-ets at the start of the project when we had very little knowledge of OFDM. Bytrying to put our understanding into code together we were able to complementeach others ideas. However, as knowledge increased, the need of pair program-

ming decreased. We believe that pair programming is needed when:

the amount of code is large - it would be hard for one person to remembermany details of it; many external interfaces (built-in functions, etc.) are used - the "co-pilot"helps to ensure that they are used with proper semantics; and the most important - when programming is the main means of trans-forming the ideas and requirements into a product.

7/31/2019 Aspi 1043 Report

26/171


The pair programming did not work as expected because we had to do plenty of analysis before starting to write the code (i.e., it was always clear what exactlyhas to be implemented during the next small step). Moreover, there was a slightdifference in programming skills of the team members. The task of improving theprogramming skills was considered on an individual basis in order not to ham-per the development of the working model. Therefore, it was decided to have aworking model as fast as possible.

To conclude, Extreme Programming is a relatively new andcertainly good wayto develop software systems. We used its principles during the part of the projectwhen OFDM was studied and several simulators were written inMATLAB. XPshowed its benets as a way to manage the work ow. However, its rules of programming were not very strictly followed as this project is not concentrated onsoftware development.

2.3 OFDM Symbol Structure

The structure of a baseband OFDM symbol is shown in Figure2.3. A basebandsignal implies that its spectral content extends from (or near) dc up to some nitevalue, usually less than a few megahertz, [Sklar, 2001].

An OFDM symbol is transmitted byN s orthogonalsub-carriers . The sys-tem bandwidthF 0 is equally divided among the sub-carriers. Theinter-carrier frequency (difference between frequencies of two subsequent sub-carriers) is

f =F 0N s

(2.1)

The kth sub-carrier transmits a wave of k periods during theuseful OFDM symbol transmission time T u . Its frequency is equal to

f k =kT u

= k f k = 0 , 1, . . . , N s 1 (2.2)

7/31/2019 Aspi 1043 Report

27/171

7/31/2019 Aspi 1043 Report

28/171

7/31/2019 Aspi 1043 Report

29/171

2.4 Basic OFDM Transmitter 15

2.4 Basic OFDM Transmitter

Figure2.6 depicts the structure of a basic OFDM transmitter. At a particularOFDM symbol transmission intervalm, a serial bit stream of transmitted datab is transformed into complex valued encoded symbolsd by using a mappingfunctionF map based on an arbitrary modulation scheme:

d m = F map (b m ) (2.5)= dm (0), dm (1), dm (2), . . . , d m (N s 1) (2.6)

Figure 2.6: Structure of a basic OFDM transmitter

The number of bits ind depends on the chosenF map .Quadrature Phase Shift Keying (QPSK) modulation is used in the thesis be-

cause of its popularity in the literature. The signal constellation for the QPSKmodulation is shown in Figure2.7. Two bits of the vectorb result in one modu-lated symbol of d when QPSK schemeis used. Theamplitudefor all combinationsof two bits of b is same, but signal phase is different. Gray coding is used to mapthe transmitted bits into data symbols. It gives one bit error when a symbol erroroccurs, i.e., when a received data symbol is decided to belong to a neighboringconstellation point.

After encoding, the elements ind are placed in parallel by serial-to-paralleltransformation, they may be multiplied by a weighting matrix . is a diagonalmatrix of size[N s N s ] that allows setting different values of power to differentelements in an OFDM symbol as explained in [Sandell, Edfors, 1996].

7/31/2019 Aspi 1043 Report

30/171


Figure 2.7: Quadrature Phase Shift Keying (QPSK)

=

0 0 00 1 0... ... . .. ...0 0 N s 1

(2.7)

a m =

d T m (2.8)

The symbol vectora (size[N s 1]) contains encoded weighted symbols to betransmitted by usingN s sub-carriers:

a m =

am (0)am (1)

...am (N s 1)

(2.9)

As the encoded symbols are complex, the real part of each encoded symbol ismodulated using a cosine harmonic and the imaginary part is modulated using asine harmonic. The reason that sine and cosine harmonics are used is that they are

7/31/2019 Aspi 1043 Report

31/171


orthogonal to each other and thus faithful demodulation is possible at the receiver.

2.4.1 Cyclic Prex

In an OFDM system, sub-carriers passing through a time dispersive channel loosetheir orthogonality because of inter-carrier interference (ICI) and inter-symbol in-terference (ISI).

ISI, in the context of the thesis is dened as the crosstalk between signalswithin the same sub-channel of consecutive FFT frames, which are separated intime by the signalling interval. ICI is the crosstalk between adjacent sub-channels

or frequency bands of the same FFT frames.The problem of ISI is illustrated in Figure2.8(a). Portions of the transmittedOFDM symbols overlap because of multipath, causing constructive or destructiveinterference, therefore, changing the spectral contents of the received symbol (ISIis one of possible causes of ICI). The system performance degrades because of the interference.

To overcome inter-symbol interference, aCyclic Prex (CP) is added to OFDMsymbols. It is a copyof the lastN g samples of the OFDM symbol that is prependedto the transmitted symbol and removed at the receiver before demodulation (Fig-ure2.8(b)). When a cyclic prex long enough to accommodate the maximum path

delay is added, the overlapping parts are rejected at the receiver. The receivedOFDM symbol contains no ISI during its useful periodN u .The benet of the cyclic prex is twofold. First, it avoids ISI by acting as a

guard space between symbols. Second, the cyclic prex also converts the linearconvolution between the transmitted symbol and channel impulse response intoa cyclic convolution . Cyclic convolution in time domain translates into a scalarmultiplication in frequency domain [Mitra, 1998, pp. 140], so the sub-carriersremain orthogonal and there is no ICI, [Engels, 2002], [Sramek, 2003].

Although BER performance is preserved, the spectral efciency decreases be-cause of enlarged symbol time. As the cyclic prex does not carry any new infor-mation, it should be set to a minimum length. The minimum length of the cyclicprex that cancels ISI completely is equal toN g = N ds . The multipath delayspreadN ds is the relative delay of the last path as compared to the rst arrivingpath in samples, or the length of the channel impulse response (in samples) minusone.

The insertion of cyclic prex is explained in Figure2.9, the OFDM symbolbeing sent is extended with a cyclic prex of lengthN g by copying lastN g samplesto the beginning of the symbol. Therefore, the total symbol length becomesN gu =N u + N g samples.

7/31/2019 Aspi 1043 Report

32/171

7/31/2019 Aspi 1043 Report

33/171


Figure 2.9: Cyclic prex insertion

The important principle of OFDM is the orthogonality of its sub-carriers.Cyclic Prex is a technique that combats the negative effects of multipath environ-ment on the transmitted signal. Therefore, a further insight about the sub-carrierorthogonality and Cyclic Prex is given in the next section.

2.4.2 Sub-carrier Correlation Properties

An orthogonal waveform can be represented, [Engels, 2002, pp. 35], as:

k (t) =1

T u e 2f k t (t [0, T u ])

0 otherwise(2.10)

k (t ) k th orthogonality function;

Each base function has an integer number of cycles in useful OFDM symboltime and therefore, is orthogonal to the other functions [IEC, 2004]. The cross-correlation between two arbitrary orthogonality functionsk (t) andk (t) whenchannel delay is less than the length of cyclic prex is calculated according to[Engels, 2002] as:

7/31/2019 Aspi 1043 Report

34/171


T u

0k (t) k (t)dt = (2.11)

= T u0 ( 1T u e 2f k t ) ( 1T u e 2f k t ) dt (2.12)=

1T u T u0 e 2 (f k f k )t dt (2.13)

whenk = k , f k = f k

T u0 k (t) k (t)dt = (2.14)=

1T u T u0 dt (e0 = 1) (2.15)

=1

T u1 T u 1 0 (2.16)

= 1 (2.17)

whenk = k ,

T u0 k (t) k (t)dt = (2.18)=

1

T u T u

0e 2 (f k f k )dt (2.19)

=1

T u1

2(f k f k )e 2 (f k f k )T u 1 (2.20)

= 0 (2.21)

7/31/2019 Aspi 1043 Report

35/171


(a) Auto-correlation sequence of 2 (b) Auto-correlation sequence of 3

Figure 2.10: Auto-correlation of orthogonality functions

Upon summarizing,

T s

0k (t) k (t)dt =

0 (k = k )1 (k = k ) (2.22)

Figure2.10shows how the auto-correlation of the base functions2 and 3varies when the delay varies from2N gu to 2N gu in unit increments. It canbe seen from the gure that the correlation value is one while the delay valueis between0 and N g , i.e., the orthogonality holds when the cyclic period lengthN g is greater than the maximum excess delayN ds as the cyclic prex cancels theeffects of delay. When the maximum excess delay exceedsN g, the correlation

starts decreasing.Cross-correlation sequences between different base functionsk and k are

shown in Figure2.12. It is seen that the cross-correlation sequence does not de-pend on any particular values of k and k but rather on the difference betweenthem. Figure2.12(a)and 2.12(b)correspond to the cross-correlation sequenceswhen the difference between sub-carrier indices is one. The cross-correlation se-quence has the same absolute values. The same applies to Figure2.12(c)and2.12(d), but the maximum absolute value is lower.

Figure2.11is used to investigate how the maximum absolute cross-correlation

7/31/2019 Aspi 1043 Report

36/171


Figure 2.11: Maximum values of base function 4 cross-correlation with other base func-tions

value of different pairs of orthogonality functions depends on the difference be-tween their indices. The orthogonality function4 is correlated withk , k =0 . . . N s 1. For each plot point in Figure2.11 a cross-correlation sequence be-tween4 and k is calculated with a varying time delay ( = N gu . . . N gu ). Asequence of length2N gu 1 is obtained for each plot point in Figure2.11. Themaximum absolute value of the sequence (the maximum cross-correlation coef-cient between4 andk at any delay ) is shown in the plot. It is observed inthe gure that an orthogonality function has the highest correlation with itself, butthe correlation decreases as the difference in the number of cycles in the functionsincreases.

2.4.3 Approaches to Modulation

In practice, the data to be transmitted as represented in Equation2.8is modulatedby using a bank of orthogonal sub-carriers represented by orthogonality matrixin Equation2.27. The modulation of data onto the sub-carriers can be performedusing three approaches as shown in Figure2.13.

1. The matrix of orthogonality functions (see Section2.4.4) of size[N u N s ] is multiplied with the transmitted data vectora (see Equation2.9) of

7/31/2019 Aspi 1043 Report

37/171

7/31/2019 Aspi 1043 Report

38/171


Figure 2.13: Three approaches for modulation and demodulation using orthogonal sub-carriers

size [N s 1]. This multiplication results in a time-domain signal of size[N u 1]. Finally, cyclic prex is inserted making the size of the signal tobe transmitted to[N gu 1]. The cyclic prex insertion can be described inmatrix form as multiplication of the transmitted signal with a matrixG cpthat has two diagonals, as shown in [Engelhart et al, 1999]:

G cp =0 N g (N F F T N g ) I N g N g

I N F F T N F F T (2.23)

sm = G cp a m (2.24)

0 zero matrix;

I identity matrix;

2. Instead of generating the orthogonal sub-carriers, an IDFT operation can beused as proposed by [Weinstein, Ebert, 1971]. The advantage of using anIFFT operation instead of IDFT is described in Section2.4.6. Cyclic prexis added to the IDFT result:

7/31/2019 Aspi 1043 Report

39/171


sm = G cp IFFT (a m ) (2.25)

3. By copyingN g trailing rows before the leading rows of , an orthogonalitymatrix already containing cyclic prex ( cp , size [N gu N s ]) is formed.Both modulation and prex insertion now can be done by one matrix multi-plication( cp a ), resulting in time-domain signal vector of size[N gu 1]:

sm = cp a m (2.26)

The orthogonality matrices and cp are described in detail in Sections2.4.4and2.4.5.

2.4.4 Structure of Orthogonality Matrix

The matrix is used for mapping a vector of transmitted data onto orthogonalsub-carriers. It is dened as:

=

0,0 0,1 0,N s 2 0,N s 11,0 1,1 1,N s 2 1,N s 1... ... i,k ... ...N u 2,0 N u 2,1 N u 2,N s 2 N u 2,N s 1N u

1,0 N u

1,1

N u

1,N s

2 N u

1,N s

1

(2.27)

The dimensions of are [N u N s ].Each column of the matrix contains the values of thekth orthogonality

function:

7/31/2019 Aspi 1043 Report

40/171


i,k = cosk i + sin k i, i = 0 , 1, . . . , N u 1 (2.28)

There is a total of N s used sub-carriers in the system, i.e.,

k = 0 , 1, . . . , N s

1 (2.29)

From Equation2.4it is known thatN s = N u .

The rst column of with frequencyf 0 (with zero cycles perN u samples)contains samples of orthogonality function, the second function has one cycle in

it and so on. There is a total of N s columns (base functions) in the matrix andeach of them hasN u samples. The real part of k is a cosine harmonic and theimaginary part is a sine harmonic.

2.4.5 Orthogonality Matrix with Cyclic Prex

The orthogonality matrix can be enhanced by including the cyclic prex intoit. In this case there is no need of having a separate operations for mapping andcyclic prex insertion.

When a cyclic prex of lengthN g is added,N g samples from the end of eachbase function are copied to their beginning of that base function, so the matrix isincreased to size[(N u + N g) N s ]. The matrix after the addition of cyclic prexis represented as:

7/31/2019 Aspi 1043 Report

41/171


cp = G cp = (2.30)

=

N g ,0 N g ,1 N g ,N s 2 N g ,N s 1N g +1 ,0 N g +1 ,1 N g +1 ,N s 2 N g +1 ,N s 1... ... ... ...2,0 2,1 2,N s 2 2,N s 11,0 1,1 1,N s 2 1,N s 10,0 0,1 0,N s 2 0,N s 11,0 1,1 1,N s 2 1,N s 1... ... i,k ... ...

N u 2,0 N u 2,1 N u 2,N s 2 N u 2,N s 1N u 1,0 N u 1,1 N u 1,N s 2 N u 1,N s 1(2.31)

It is observed from Equation2.31 that the lower sub-matrix is equal to thematrix from Equation2.27. The upper sub-matrix contains the cyclic prexparts of the orthogonality functions.

The values of kth column of cp are calculated as:

cp i,k = cosk i + sin k i, i = N g , . . . , N u 1 (2.32)

2.4.6 OFDM Transmitter Using IDFT

As described in Section2.4.3, the computational complexity of the modulator canbe reduced.

The number of complex multiplications required in a DFT operation isN F F T 2and the number of complex additions isN F F T 2 N F F T . The computational com-plexity can further be reduced using an FFT operation. The number of com-plex multiplications and complex additions in an FFT operations are of orderN F F T

2 log2(N F F T ) andN F F T log2(N F F T ), respectively. AsN F F T increases the use-fulness of FFT becomes more apparent. Thus, the modulation of the encoded

7/31/2019 Aspi 1043 Report

42/171


symbols can be performed in an efcient way by using IFFT. A representation of an OFDM transmitter which uses IFFT operation instead of independent modula-tors for each encoded signal is depicted in Figure2.14.

Figure 2.14: IFFT-based OFDM transmitter

At a particular signalling intervalm, the transmitted signal is given by:

sm = G cp a m G cp IFFT (a m ) (2.33)

2.5 Channel Models

The channel models are not discussed in this iteration of the simulator. This ap-proach is followed because the channel is decoupled from the transceiver tech-nology and the basic OFDM system is described without considering a channel.Various channel models are developed in the next chapter.

2.6 Basic OFDM Receiver

The block diagram of a basic OFDM receiver which is adopted from Figure2.5is shown in Figure2.15. The received discrete signal vectorr m of size[1N gu ]contains the transmitted OFDM symbol affected by the channel. Although noparticular channel model is considered in this chapter, an imitation of the channelis assumed in order to maintain a full view of the generic OFDM system. It doesnot distort the transmitted signal.

7/31/2019 Aspi 1043 Report

43/171

2.6 Basic OFDM Receiver 29

Figure 2.15: Basic OFDM receiver

After reception, the cyclic prex is removed because it is not a part of theuseful transmitted symbol. Then, the data symbols originally mapped onto or-thogonal sub-carriers in the transmitter (vectora ) are demodulated. The receivedtime-domain signal vectorr is transformed into a frequency-domain signal vector

z . Similar to the techniques described in Section2.4.3and shown in Figure2.13,these operations are performed at the receiver in one of the following ways:

1. Discarding rstN g OFDM samples of the received signalr m by multiplyingit with a cyclic prex removal matrixG cp . The size of vectorr is decreasedfrom [1 N gu ] to [1 N u ]. Then, the signal is converted into a parallelform by transposing the vector. Finally, the received signal vector withoutthe cyclic prex is multiplied with a Hermitian of the orthogonality matrix :

G cp =0 N g

N F F T

I N F F T N F F T (2.34)zm = H r m G cp )T (2.35)

0 zero matrix;

I identity matrix;

2. Discarding rstN g samples of the received signalr m and then performingan FFT on the result:

zm = F F T (( r m G cp )T ) (2.36)

This approach has the lowest computational complexity;

7/31/2019 Aspi 1043 Report

44/171


3. Discarding the cyclic prex and performing the FFT at the same time bymultiplying the received signalr m by a Hermitian of the orthogonality ma-trix cp that has zeros in its rstN g rows (it shall be calledHcp ):

Hcp = H G T cp (2.37)

z m = Hcp r m (2.38)

This approach has high computational complexity as compared to FFT butis written in compact matrix-vector notation. Because of this compact nota-tion, this approach is used in the report.

After the demodulation, a complex-valued data symbol vectorz m with dimen-sions[N u 1] is obtained. Depending on the channel model used, the receiveddata symbolszm may have been affected by the channel. The data symbols haveto be restored by using a channel equalization functionF eq :

y m = F eq (z m ) (2.39)

This function puts the constellations of the received symbols into their esti-mated correct positions in the complex plane by canceling the effects of multi-path radio channel. Finally, a hard decision has to be made on the data symbolsy

m. A demapping scheme, corresponding to its counterpart used at the transmitter

is applied to extract a serial bit sequence:

x m = F dec (y m ) (2.40)b m = F demap (x m ) (2.41)

7/31/2019 Aspi 1043 Report

45/171

2.7 Summary 31

2.7 Summary

In this chapter, a basic analytical model of an OFDM transceiver is explainedusing matrix vector notation. The model does not take any radio channel intoaccount.

In the next chapter, a few channel models are presented and analyzed. Thetransceiver is placed in an environment represented by the channel and a channelestimation algorithm is applied at the receiver to recover the corrupted data.

7/31/2019 Aspi 1043 Report

46/171


7/31/2019 Aspi 1043 Report

47/171

C HAPTER 3

D OWNLINK S IMULATOR

The goal of this chapter is to extend the basic OFDM transceiver from the previouschapter with radio channel models. Several channel models with an increasingorder of complexity and real environment resemblance are presented.

The channel models are created in order to enhance the knowledge of OFDMtechnology that was gained in the previous chapters. The OFDM performance inseveral theoretical environments is analyzed through simulations.

A simple Additive White Gaussian Noise (AWGN) channel simulator thatdoes not require any channel estimation is presented rst, then it is extended toa multipath slow-fading Rayleigh channel. Finally, the Rayleigh channel is ex-tended to a fast-fading multipath Jakes channel model that includes InternationalTelecommunication Union (ITU) proles.

A literature survey of various channel estimation algorithms is given. Onealgorithm is applied in order to explore the effect of the channel on the transmittedsignal.

WiMAX is one of the latest applications in which OFDM is applied. Theparameters specied in WiMAX (IEEE 802.16) standard are used for testing thesimulator.

WiMAX is a high-throughput broadband wireless technology that providesconnections over long distances. It is intended to be used for a number of ap-plications, including "last mile" broadband connections, hotspots and high-speedenterprise connectivity for business [Intel, 2004].

Various parameters regarding WiMAX specication are given in[Yaghoobi, 2004, pp. 203]. The parameters that are relevant to this project areshown in Table3.1.

7/31/2019 Aspi 1043 Report

48/171

34 Downlink Simulator

Parameter ValueSampling frequency (F 0) 20MHzSample time (1F 0 ) 50nsFFT size (N F F T = N s ) 2048Sub-carrier spacing ( f = F 0N s ) 9.76kHzUseful symbol time (T u = 1 f ) 102.4sGuard time (T g = T u8 ) 12.8sOFDM symbol length (T gu = T u + T g) 115.2s

Table 3.1: WiMAX parameters

3.1 Additive White Gaussian Noise Channel Simu-lator

The noise characteristics - white, additive and Gaussian are most often used tomodel noise in a communication system. Since zero-mean Gaussian noise is com-pletely characterized by its variance, this model is particularly simple to use in thedetection of signals, [Sklar, 2001, pp.33]. In order to seek analytical tractability,the problem of analyzing a complex channel model is simplied by conditioningit with an elementary Additive White Gaussian Noise (AWGN) effected channel.Unconditioning is performed in order to approach reality in further models.

3.1.1 Denitions

A random signal with known statistical properties of amplitude, distribution, andspectral density having a frequency spectrum that is continuous and uniform overa specied frequency band is calledwhite noise , [Haykin, 2001]. For a speciedbandwidth consisting of a continuous frequency spectrum, the total power in thespecied bandwidth divided by the specied bandwidth is termed asspectral den-sity, [Haykin, 2001]. It is usually expressed in Watt per Hertz,[Telecom Glossary, 2000]. The power spectral density of white noise is indepen-dent of the operating frequency and is given according to [Haykin, 2001] as:

7/31/2019 Aspi 1043 Report

49/171

7/31/2019 Aspi 1043 Report

50/171


3.1.2 Assumptions

The simulator uses AWGN as the only source of degradation to the trans-mitted signal;

All the sub-carriers would have experienced the same attenuation when theyreach the receiver;

The transmitter and the receiver are perfectly synchronized.

3.1.3 Channel Model

Since the noise in this iteration of the OFDM simulator is a Gaussian processand the samples are uncorrelated, the noise samples are also independent. Suchchannel is called memoryless channel because the AWGN affects the transmittedsymbol independently. The term additive means that the noise is simply addedto the transmitted signal and that there are no multiplicative mechanisms involved,[Sklar, 2001, pp. 33].

3.1.4 Channel Estimation

No channel estimation is considered for the AWGN channel simulator. The chan-nel estimator is transparent in the sense that it does not make any effect on thesignal received by it.

3.1.5 Theoretical Performance

One of the most important metrics of performance of digital communication sys-tems is a plot of the bit-error probability (P b) versus bN 0 , [Sklar, 2001]. The datareceived at the receiver is corrupted by noise while transmission. This results in anerror while estimating the transmitted data. The smaller the requiredbN 0 , the moreefcient is the detection process for a given probability of error, [Sklar, 2001].

The theoretical BER curves are generated according to the formulae given in[Proakis, 2001, pp.256]. The practical curves are plotted against the theoreticalcurves as described in [Proakis, 2001, pp. 260].

7/31/2019 Aspi 1043 Report

51/171

3.1 AWGN Channel Simulator 37

P b = Q 2b

N 0(3.4)

bN 0 signal-to-noise ratio per bit;

3.1.6 Results

The simulated BER curve corresponds to the theoretical curve as shown in Fig-ure 3.1. The basic OFDM transceiver simulator produces theoretically expectedresults.

Figure 3.1: BER performance in AWGN channel

7/31/2019 Aspi 1043 Report

52/171


3.2 Rayleigh Channel Simulator

A slow-fading Rayleigh channel model is chosen to be the extension to the simpleAWGN channel model in the previous section. The process of extending is whathas been referred to as unconditioning ([Jeruchim et al, 2000, pp. 16]) in Sec-tion3.1. The Rayleigh channel is selected because it is commonly used to describethestatistical time varying multipathcomponent of a channel [Hara, Prasad, 2003].This model introduces a multipath channel and a channel estimator (Figure3.2).This system model is more realistic compared to the AWGN channel model be-cause of the simulated multipath effects that are inevitable in real environments.

Figure 3.2: Block diagram of the OFDM system simulator with a slow fading channel

3.2.1 Denitions

According to [Hara, Prasad, 2003, pp. 13-18], multipath fading is due to multipathreections of a transmitted wave by local scatterers such as houses, buildings andman-made structures, or natural objects such as forest surrounding a mobile unit.Figure3.3shows a typical multipath fading channel withL paths.

Before dealing further with multipath channels, it is necessary to have an un-derstanding about the parameters of multipath channels and various types of fad-ing. The channel can be characterized as a function of channel impulse response(CIR)h (t, ). The termt represents the time variations occurring because of themovement of the receiver (assuming that the transmitter does not move). The

7/31/2019 Aspi 1043 Report

53/171

7/31/2019 Aspi 1043 Report

54/171


delay prole is useful in obtaining excess delay spread. It is calculated by takingthe spatial average of h (t, ) 2. The spatial average is calculated by averagingthe values of h (t, ) 2 at same i for different values of t.

A channel that passes all spectral components with approximately equal gainand linear phase is a at channel.Coherence bandwidth , ( f )c, is a statisticalmeasure of the range of frequencies over which the channel can be consideredat. Coherence bandwidth is also described as the range of frequencies overwhich two frequency components have a strong potential for correlation in ampli-tude. Two sub-carriers that have their frequency separation greater than( f )c areaffected differently by the channel.

Delay spread and coherence bandwidth describe the time ( ) dispersive natureof the channel. The time(t) varying nature of the channel is given byDoppler spread and coherence time . When a pure sinusoidal frequencyf is transmitted,the received signal spectrum, called the Doppler spectrum, will have componentsin the range of (f f d) to (f + f d), wheref d is the Doppler shift depending onthe velocity of the receiver and the angle between the transmitter and the receiver[Rappaport, 1996, pp. 141]. The effects of Doppler spread are negligible at thereceiver if the baseband transmitted signal bandwidth is greater than the Dopplerspread.

Coherence time , ( t)c is the time duration over which two received signals

have a strong correlation. The channel will change during the transmission of baseband message if the time duration of the baseband signal is greater than( t)c,thus causing distortion at the receiver.

Different transmitted signals undergo different types of fading depending onthe relation between the signal parameters like bandwidth, symbol period, etc.and the channel parameters like delay spread, Doppler spread, etc. There are fourpossible effects, that are exhibited depending on the nature of the transmitted sig-nal the channel, and the velocity of the mobile unit. The multipath delay spreadleads to time dispersion and frequency selective fading. The Doppler spread leadsto frequency dispersion and time selective fading. These two propagation mecha-

nisms are independent of one another.Time dispersion due to multipath causes at fading and frequency selective

fading. The time duration of the transmitted signal in a at fading channel islarger than the multipath time delay spread of the channel. Flat fading channelsare sometimes referred to as amplitude varying channels, frequency non-selectivechannels or narrowband channels, since the bandwidth of the transmitted signalis narrow compared to the bandwidth of the at fading channel. The instanta-neous amplitude distribution of at fading channels is commonly considered tobe Rayleigh distributed [Rappaport, 1996, pp. 169]. Thus, a Rayleigh at fad-

7/31/2019 Aspi 1043 Report

55/171

3.2 Rayleigh Channel Simulator 41

ing channel model assumes that the channel induces an amplitude which varies intime (t) according to the Rayleigh distribution.

The PDF of Rayleigh distribution envelope and phase is given by

p() =

2re

2

2 2r , ( > 0) (3.5)

p() =1

2, (0 < 2) (3.6)

p() andp() are statistically independent.

Frequency selective fading occurs on the received signal if the channel pos-sesses a constant gain and linear phase response over a bandwidth that is smallerthan the bandwidth of transmitted signal. In frequency selective fading, the mul-

tipath delay spread of the channel impulse response is greater than the time du-ration of the transmitted signal and the received signal contains multiple versionsof the transmitted signal. These multiple versions are attenuated and delayed intime and make the received signal distorted. Certain frequency components inthe received signal spectrum have greater gains than others in the frequency do-main. Frequency selective fading channels are also called wideband channels[Rappaport, 1996, pp. 169] since the bandwidth of the transmitted signal is widerthan the bandwidth of the channel impulse response.

If the channel impulse response changes rapidly within the symbol duration

i.e., the coherence time of the channel is smaller than the time duration of thetransmitted symbol, the channel is a fast fading channel. On the contrary and if the channel response changes at a slower rate than the transmitted symbol, thechannel is aslow fading channel . The channel is decided to be fast or slow fadingby the bandwidth of the transmitted signal and the velocity of the mobile unit thatgives the Doppler spread.

The received signal after passing through the multipath fading channel isr (t)and is described by Equation3.7.

7/31/2019 Aspi 1043 Report

56/171


r (t) =L1

l=0

pl(t)e 2f k l (t ) (3.7)

=L1

l=0

l(t) (3.8)

l (t ) complex-valued stochastic process;

3.2.2 Assumptions

The following assumptions are made for this system model:

The channel is slow fading, frequency non-selective;

There is a perfect synchronization between the transmitter and the receiver;

Perfect channel estimation exists.

3.2.3 Channel Model

The modeled radio channel involves distortion of the transmitted signal becauseof two reasons: multipath effects and white Gaussian noise. As the channel fol-lows slow fading, the coherence time is longer than OFDM symbol timeT gu , inother words, the channel impulse response can be considered constant during oneOFDM symbol duration. An account of the impulse response characteristics havebeen given in Section3.2.1. According to [Jeruchim et al, 1992, pp. 374], theeffects of a channel on a sent signal may be described as:

r (n) =L1

l=0

pl(n) s(n l(n)) (3.9)

7/31/2019 Aspi 1043 Report

57/171


A simulated radio channel hasL paths, the rst of them appears at time delay = 0 , and each path is associated with a complex attenuation coefcientpl and atime delay l.

Tests are performed with themultipath delay spread being both smaller thancyclic prex lengthNg (no ISI expected) and longer (to investigate the effects of ISI). According to [Jeruchim et al, 1992, pp. 375], the complex low-pass impulseresponse of the multipath channel can be described as:

h ( ; n) =L1

l=0

pl(n) ( l(n)) (3.10)

The effect of channel on the transmitted signal can be expressed in matrix-vector notation as multiplication of the matrixH m with the transmitted signalvector. The matrixH m has N gu columns and each column contains a delayedchannel impulse response with lengthN ds . If the channel is approximated byslow-fading, i.e., the channel impulse responseh is considered to be constantduring one OFDM symbol duration, the channel matrixH m is written as:

H m =

hm [0] 0 0hm [1] hm [0] 0hm [2] hm [1] 0... ... ...hm [N ds 1] hm [N ds 2] 00 hm [N ds 1] 0... ... ...

0 0 hm [0]... ...0 hm [N ds 1] hm [N ds 2]0 0 hm [N ds 1]

(3.11)

7/31/2019 Aspi 1043 Report

58/171

7/31/2019 Aspi 1043 Report

59/171


r m = H m [0]sm + w m (3.15)= H m [0] a m + w m (3.16)

3.2.4 Channel Estimation

If themultipath delay spread is longer than the guard period length, but not longerthan 2N gu , the current received OFDM symbol is corrupted with one previoussymbol:

r m = H m [0]sm + H m 1[1]sm 1 + w m (3.17)

At the receiver the cyclic prex is removed from the received signal and FFTis performed. Both FFT operation and cyclic prex removal can be substituted byreceived signal multiplication by orthogonality matrixHcp , i.e., a Hermitian of and has zeros instead of the cyclic prex elements as described in Section2.6.

zm = Hcp r m (3.18)= Hcp H m [0] cp

C m [0]a m + Hcp H m 1[1] cp

C m [1]a m 1 (3.19)

+ Hcp w m (3.20)

According to [Kim et al, 1999], the variance of AWGN does not change af-ter FFT. Therefore, the noise component is written asw instead of Hcp w m inEquation3.21. The transmitted and received OFDM symbols are related by thediagonal matrixC m [0] containing the channel effects.C m [1]a m 1 is the inter-symbol interference. If the multipath delay spread is shorter than the guard periodlength,H m 1[1]contains only zeros and the ISI term can be ignored.

7/31/2019 Aspi 1043 Report

60/171


Finally, the received OFDM symbols after the FFT operation as depicted inFigure2.15can be expressed as:

zm = C m [0]a m + C m [1]a m 1 + w m (3.21)

TheC matrices contain the channel effects, therefore, successful recovery of the transmitted data symbols relies on correct estimation of C m [0]. The issues of

OFDM channel estimation are presented in Section3.4.For the purpose of verifying the Rayleigh multipath channel model, a per-fect channel estimation is assumed, i.e., the channel estimation matrixH m [0] isknown. As there is also an assumption of normalized channel delay spread be-ing less than the cyclic prex length,H m [1]contains only zeros and the sent datasymbols can be extracted:

C m [0] = Hcp H m [0] (3.22)y m = diag (C m [0])H z m (3.23)

3.2.5 Theoretical Performance

According to [Proakis, 2001, pp. 831], the expression of bit error rate for QPSKis:

P b = 121 2 2

L

1

l=0

2ll

1 2

4 22l (3.24)

= c1 + c (3.25)

where c is average received SNR per channel.

7/31/2019 Aspi 1043 Report

61/171


3.2.6 Results

Figure3.5 shows the bit error performance of the model in comparison to thetheoretical performance given by Equation3.24. The theoretical AWGN BERcurve from Figure3.1 is also included. The simulations are performed with one,two and three Rayleigh fading paths. The maximum excess delay of the channelis less than the cyclic prex length. The plot shows that the simulated BER curvescorrespond to theoretical estimates. However, the Rayleigh BER curve is differentfrom AWGN. The transceiver BER performance in a Rayleigh fading channel ispoorer than in OFDM channel, because the symbol energy becomes scattered intime when passing a multipath environment. The effect of of scattering can be

mitigated using methods that would gather the dispersed symbol energy back intoits original time interval.

Figure 3.5: Slow fading Rayleigh channel BER performance

A second simulation is performed with two paths and the SNR xed at 10dB.The delay of rst path arrival is xed at zero ( 0 = 0 ). The delay of the secondpath 1 changing from1 to 2N g OFDM samples. It can be seen in Figure3.6thatthe OFDM performance does not suffer when themaximum excess delay is lessthan cyclic prex lengthN g.

The performance of different symbol and cyclic prex lengths is compared.Figure3.7 shows the BER performance curves of the basic OFDM system with

7/31/2019 Aspi 1043 Report

62/171


Figure 3.6: BER performance with xed SNR and varying second path delay

100s symbol length. Two values,T u / 8 and0 are considered as the cyclic prexlength. Theoretical BER performance is also shown on the plot.ITU Vehicular B and ITU Pedestrian B channel proles (Tables3.8 and 3.9) are used for thissimulation (channel proles are discussed in Section3.3.6). The channel modelused for these simulations differs from the model described in Section3.2becausethe paths have xed delays and different variances.

The maximum delay spread of theITU Vehicular B channel prole is20ms .It exceeds the cyclic prex of 12ms . Figure3.7shows that the BER performanceis worse than the theoretical limit. TheITU Pedestrian B channel prole hasthe maximum delay spread of 3.7ms . It can be seen that the BER performance

deteriorates when the maximum delay spread is greater provided the same cyclicprex length.Figure3.8shows the results of a simulation with30s OFDM symbol length.

As expected, the shorter symbol length results in higher inter-symbol interferenceand bit error rate.

7/31/2019 Aspi 1043 Report

63/171


Figure 3.7: Rayleigh channel BER performance, T u = 100 s

Figure 3.8: Rayleigh channel BER performance, T u = 30 s

7/31/2019 Aspi 1043 Report

64/171


3.3 Jakes Channel Model

The channel models analyzed in the previous sections are Additive White Gaus-sian Noise (AWGN) and Rayleigh multipath channels. The AWGNchannel modelhas a drawback of not considering any multipath effects on the transmitted signal.The Rayleigh fading channel model is only suitable for simulating slow-fadingchannels because the channel impulse response is calculated for each OFDM sym-bol independently. The individual channel samples are not correlated in time. Thenext step is to introduce time correlation to the Rayleigh channel model.

Historically, the Jakes model has been used for modeling a Rayleigh fadingchannel. The Jakes simulator models the low-pass envelope of a stationary (at)frequency non-selective (see Section3.2.1) mobile fading channel under isotropicscattering conditions [Rappaport, 1996]. The condition when the transmitted en-ergy arrives equally distributed over all possible spatial angles, with uniformlydistributed phases is called an isotropic condition [ien, 2003]. An approximateanalytical model for such a channel is a zero-mean complex Gaussian noise pro-cess with uncorrelated inphase and quadrature components. Jakes model allowsan effective approximation of the desired analytical model by using a nite num-ber of low-frequency oscillators [Ptzold, Laue, 1998].

A reference model gives theoretical performance of the Jakes model and al-lows performance evaluation of the practical Jakes simulator. Next, a practicalapproach to Jakes simulator is given together with its statistical properties. Inorder to use the Jakes simulator for multipath frequency-selective channel mod-eling, a single-path case is considered and extended to multipath.

3.3.1 Reference Model

The complex low-pass Rayleigh envelope for the frequency non-selective (singlepath) Jakes reference model is, [Xiao et al, 2002]:

7/31/2019 Aspi 1043 Report

65/171

3.3 Jakes Channel Model 51

g(t) = g1(t) + g2(t) (3.26)g1(t) = 2N osc

N osc

n =1

cos(2f d tcos n + n ) (3.27)

g2(t) = 2N oscN osc

n =1

sin (2f dtcos n + n ) (3.28)

For largeN osc , the central limit theorem justies thatg1(t) andg2(t) can be ap-proximated as a Gaussian random processes assuming that andn are mutually in-dependent and uniformly distributed over[, ] for each oscillator,[Xiao et al, 2002].

The reference model gives the principle of generating a Rayleigh process byusing two banks of oscillators.

Second-order statistics, namely auto-correlation and cross-correlation func-tions, are useful for analyzing the correlation properties of the inphase and quadra-ture components. They are given as, [Xiao et al, 2002]:

R g1 g1 ( ) = E [g1(t)g1(t + )] = J 0(2f d ) (3.29)R g2 g2 ( ) = E [g2(t)g2(t + )] = J 0(2f d ) (3.30)R g2 g1 ( ) = E [g2(t)g1(t + )] = 0 (3.31)R g1 g2 ( ) = E [g1(t)g2(t + )] = 0 (3.32)

3.3.2 Jakes Model

It is possible to reduce the computational complexity of the reference model. Ac-cording to [Xiao et al, 2002] and [Pop, Beaulieu, 2002], the number of oscillatorsin each bank can be reduced. The complex low-pass envelopeg is given by theJakes model as:

7/31/2019 Aspi 1043 Report

66/171

7/31/2019 Aspi 1043 Report

67/171


Rg1 g1 ( ) =4

N oscM +1

n =1

un 22 cos( n ) (3.42)

Rg2 g2 ( ) =4

N osc

M +1

n =1

vn 2

2 cos( n ) (3.43)Rg1 g2 ( ) = E g2(t)g1(t + )

=4N

M +1

n =1

un vn2 cos( n ) (3.44)

Rg2 g1 ( ) = Rg1 g2 ( ) (3.45)

3.3.3 Jakes Multipath

The Jakes fading model is suitable for simulating aat fading , i.e., single-pathchannel [Li, Guan, 2000]. For simulating frequency-selective channel, it has tobe extended to a multipath. The wide-sense stationary-uncorrelated scattering(WSSUS) channel is a commonly employed model for the multipath channel,[Sadowsky, Kafedziski 1998]. The WSSUS model for multipath channel includesboth the variations int and (see Section3.2.1). The time-varying nature of thechannel is modeled as a wide-sense stationary (WSS) process. The attenuationand phase shift associated with different delays are modeled with an uncorrelatedscattering assumption, [Jeruchim et al, 1992].

The extension to multipath can be achieved in a few different ways. A methodof assigning different arrival angles for the paths and applying orthogonal weight-ing functions is proposed in [Dent et al, 1993]. Another possible approach is touse the theoretical correlation function to nd a time offset after which the auto-correlation of the process reaches a negligible value. The different paths could beproduced by the same Jakes model with the time offset.

A Jakes model with random phases of the oscillators is analyzed in[Xiao et al, 2002] and is used in this project.

In this case, each path is modeled as a Jakes fading process with randomphases assigned to its low-frequency oscillators. All the paths are low-correlatedbecause of the random phases.

7/31/2019 Aspi 1043 Report

68/171


3.3.4 Approaches for Implementation

There are different approaches for the implementation of the multipath Jakeschannel model. One possibility is to pre-generate the sequences of g (one se-quence for each multipath component, or multipath can be achieved by using asingle delayed process) and reuse them later in the simulations. However, this ap-proach implies that the channel is identical throughout the simulations. Moreover,long sequences of the Jakes process require a lot of storage space. For example,the complex Jakes coefcients for one second of channel data (T s = 50ns , singlepath) occupy 320 megabytes.

It is decided to investigate the possibilities to calculate theg function for everytransmitted OFDM sample on the y.

MATLAB code proling was performed because the initial simulator code wasextremely slow. Proling showed that almost all computational complexity is con-centrated in only few lines of the code. After some optimizations, namely replac-ing for loops with vector operations and usage of sparse matrices, the executiontime of the simulation was reduced up to1500 times. Nevertheless, performing asimulation with a big amount of transmitted-received bits still stays time consum-ing. For the performance evaluation of the channel estimation algorithm describedin Section3.4, a fast-fading channel simulator is required. A50106 bit transmis-sion through a fast fading channel with 6 paths takes around 1 hour for a singlepoint of a BER curve. The rate of the simulated transmission is 7500 OFDM sam-ples per second. The PC used for this measurement has a 2.4GHz Pentium IVCPU, 1GB RAM and runs Linux 2.6 operating system.

3.3.5 Results

Figure3.9(a)shows the envelope of the Jakes model output waveform. TheRayleigh process is generated using one path of Equation3.33. It can be ob-served that the process is correlated in time. The probability density function(PDF) of the waveform envelope is shown in Figure3.9(b). The Jakes enve-lope is an approximation of Rayleigh random process. It can be observed that thePDF approaches theoretical curve asM increases. Choosing parameterM is atrade-off between computational complexity and approximation accuracy. It canbe seen from Equations3.33-3.40 that increment of M corresponds to increaseof storage requirements by 5 real numbers, adds asin and acos operation, fourmultiplications and four additions for one calculation of complex channel attenu-ation. We decided to set the value of M to 20 for further simulations as it closelyapproximates the theoretical Rayleigh PDF.

7/31/2019 Aspi 1043 Report

69/171


(a) Jakes envelope (b) Probability Density Function of Jakes enve-lope

Figure 3.9: Jakes envelope ( f d = 400 Hz, f s = 20 MHz, carrier frequency: 3 .5 GHz)

The auto and cross-correlation plots shown in Figure3.10depict the compari-son of the correlation properties of the Jakes model against the reference model.The simulated curves are obtained using Jakes single path simulator given inEquation3.33, the reference curves correspond to Equations3.29 to 3.32. Thetheoretical correlation sequences are given in Equations3.42 to 3.45. It can beobserved that the obtained correlation sequences approximate the desired correla-tion characteristics.

The auto-correlation function of inphase, or quadrature components, decreasesin time at a rate that corresponds to Doppler frequency. As the Doppler frequencyincreases, the time in which the channel is highly correlated (channel coherencetime ( t)c) decreases. This effect can be observed in Figures3.10(b)(400Hzmaximum Doppler frequency: carrier frequency is3.5GHz , speed of movement -120km/h ) and 3.10(d) (100Hz maximum Doppler frequency -3.5GHz at30km/h ). The Jakes simulator provides a sequence of complex Rayleigh dis-tributed path gains. These path gains are correlated in time in a controllable fash-ion.

According to [Rappaport, 1996], the coherence time of a channel (see Section3.2.1) corresponding to the Doppler shift of 400Hz (3.5GHz carrier frequency atthe speed of 120km/h ) is equal to:

7/31/2019 Aspi 1043 Report

70/171


( t)c = 916 f 2d 1ms (3.46)

The coherence time( t)c is also dened as the time for which the channelcorrelation decreases by3dB [Engels, 2002 (2), pp. 27]. It can be seen fromFigure3.10that Jakes path correlation decreases by3dB (the correlation coef-cient decreases by 2 times) in approximately 1 millisecond, thus the correlationcorresponds to the theoretical assumption in Equation3.46.

As the channel coherence time (( t)c = 1 ms ) is much larger than the OFDMsymbol time (N gu = 100s), the channel can be considered to beslow fading inWiMAX system. The model is capable to provide afast fading channel model aswell.

Tables3.2and3.3give thecross-correlationcoefcientsof inphase andquadra-ture components between 5 different Jakes paths. It can be seen that the differ-ent paths are low-correlated. It approaches the WSSUS condition of zero cross-correlation between paths.

Path 1 Path 2 Path 3 Path 4 Path 5Path 1 1.0000 -0.0841 -0.1458 -0.0513 -0.0364Path 2 1.0000 0.0410 -0.0670 0.0981Path 3 1.0000 -0.0237 0.1185Path 4 1.0000 0.1174Path 5 1.0000

Table 3.2: Cross-correlation coefcients of Jakes model path inphase components

Path 1 Path 2 Path 3 Path 4 Path 5Path 1 1.0000 0.0295 -0.0410 0.0269 0.0542Path 2 1.0000 0.0510 0.0409 0.1791Path 3 1.0000 -0.0663 0.1793Path 4 1.0000 -0.0393Path 5 1.0000

Table 3.3: Cross-correlation coefcients of Jakes model path quadrature components

7/31/2019 Aspi 1043 Report

71/171


(a) Inphase component auto-correlation,f d =400 Hz

(b) Quadrature component auto-correlation,f d =400 Hz

(c) Cross-correlation,f d = 400 Hz (d) Quadrature component auto-correlation,f d =100 Hz

Figure 3.10: Jakes output waveform correlation ( f s = 20 MHz, carrier frequency: 3 .5 GHz)

7/31/2019 Aspi 1043 Report

72/171


3.3.6 Channel Proles

In order to bring a channel model even closer to reality, a well-dened channelprole has to be used. A channel prole denes the average signal attenuations atcertain path delays. The ITU Vehicular A, Vehicular B and Pedestrian B channelproles are used to specify the properties of the multipath Jakes model in thisproject. The path delays and the average powers are given in Tables3.4- 3.6.

Relative delay Average relative i , ns powerAi , dB0 0.0

310 -1.0710 -9.01090 -10.01730 -15.02510 -20.0

Table 3.4: ITU Vehicular A channel prole

Relative delay Average relative i , ns powerAi , dB0 -2.5300 08900 -12.012900 -10.017100 -25.220000 -16.0

Table 3.5: ITU Vehicular B channel prole

Relative delay Average relative i , ns powerAi , dB0 0.0200 -0.9800 -4.91200 -8.02300 -7.83700 -23.9

Table 3.6: ITU Pedestrian B channel prole

7/31/2019 Aspi 1043 Report

73/171


It is seen that the delays i of the ITU Vehicular A prole are not sample-spaced according to the OFDM sampling rate (f s = 50 .0ns ), specied by WiMAXin Table3.1. The channel prole is resampled using this sampling rate in orderto fulll the assumption of the delays i being sample-spaced. The channel isresampled using a simple approach of assigning the paths to the closest OFDMsample:

i = i , i = 1 , 2, . . . , N (3.47)

N number of paths in channel prole;

Next, the mean attenuation valuesA, given in dB are converted to path vari-ances2 p. In order to be able to control the signal power in the simulator, it isdesired to have the sum of path variances equal to one, but the given proles donot have this property. Therefore, the average path powers are recalculated for allthe three channel proles by using the following equations:

Ai = 10 log102i21 , i = 1 , 2 . . . , N (3.48)

2i = 21 10A i10 (3.49)

N

i=1

2i =N

i=1

21 10A i10 = 21

N

i=1

10A

aspi 1043 report

Documents