NoiseVarianceEstimationInMIMOOFDMtestbed

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Department of Information Technology Integrated Systems Laboratory

and Electrical Engineering

Semester Thesis

Noise Variance Estimationfor MIMO-OFDM Testbed

Dominik Bischoff

Advisors: Markus Wenk

Thomas Koch

Patrick Mchler

Winter Term 2008

expected signal

received signal


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Preface

Abstract

This semester thesis deals with the problem of estimating the noise variance (or

equivalently the signal to noise ratio SNR) in a MIMO-OFDM system. In a first part, the

properties of MIMO-OFDM systems are presented. In a next step, existing algorithms

are evaluated. Most of the applicable algorithms work in the frequency domain. Theperformance of those algorithms is therefore highly dependent on the employed channel

estimator. As it is not desirable to increase the performance of the channel estimator

in a real system due to the costs in terms of throughput, a novel algorithm working in

the time domain is developed. The only prerequisite for this novel algorithm is that

periodic short preambles are available.

The performance of this proposed algorithm is evaluated in a MIMO-OFDM simulation

environment. The performance in the simulation is near the optimum and as the short

preambles are transmitted anyway, there is no loss in throughput.

In a next step, the algorithm is implemented in VHDL and mapped on a FPGA. The

hardware costs are small compared to the area occupied by the other MIMO-OFDM

signal processing blocks.

In the last part of this thesis, some measurements were conducted with the offline and

the online testbed. In case of the offline testbed, the algorithms performs better than

the previously employed constant 30dB estimator. There is some loss in performance in

the high SNR region due to transmit noise. A proposition is made how this problem

could be solved. The measurements with the online testbed show that the frequency

offset between the transmitting and the receiving board causes a problem. A possible

solution is presented but not yet implemented.


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II

Overview

The thesis is split into the following chapters:

Task Description The official task description for this semester thesis.

Introduction The terms MIMO and OFDM are explained and several

channel models are presented.

Literature Review Already existing papers with relevant information for this

thesis are presented.

Simulations The limits of an SNR estimator are elaborated.

Algorithm Design Existing algorithms are evaluated and a novel algorithm

is developed and presented.

Implementation The implementation of the novel algorithm on a FPGA is

described.

Measurements Some measurements of the algorithm with the offline and

the online testbed are presented.

Summary, Conclusion A short summary of the thesis and an outlook are given.

And Outlook


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III

Author: Dominik Bischoff [email protected]

Advisors: Markus Wenk [email protected]

Thomas Koch [email protected]

Patrick Mchler [email protected]

Supervisors: Hubert Kaeslin [email protected]

Norbert Felber [email protected]

Professor: Wolfgang Fichtner [email protected]

Acknowledgments

I thank the Integrated Systems Laboratory (IIS) at ETH Zurich for the opportunity

to realize this project and providing all the infrastructure. Special thanks go to myadvisors for offering help whenever needed but leaving me at the same time the

freedom to follow my own ideas wherever possible. As this thesis uses a lot of previous

work done by different persons (MIMO-OFDM simulation environment, offline testbed,

online testbed), I also thank whomever was involved in developing them. I further

thank Hubert Kaeslin and Norbert Felber for the VHDL code samples from the VLSI 1

lecture that were extremely helpful while writing the hardware code. I also thank the

Communication Technology Laboratory (IKT) at ETH Zurich for allowing me to use

their measurement equipment. And finally, a special thank goes also to my family and

my friends that supported me during the whole time!


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IV


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Table of Contents

1 Task Description 1

2 Introduction 7

2.1 Why Using MIMO? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Antenna Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Spatial Multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Why Using OFDM? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 The Standard Approach . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.2 Frequency Division Multiplexing . . . . . . . . . . . . . . . . . . 9

2.2.3 Orthogonal FDM (OFDM) . . . . . . . . . . . . . . . . . . . . . . 10

2.2.4 OFDM: What Are Orthogonal Signals? . . . . . . . . . . . . . . . 10

2.2.5 OFDM: How to Find Orthogonal Signals . . . . . . . . . . . . . . 11

2.2.6 OFDM: Cyclic Prefix . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.7 OFDM: Noise Considerations . . . . . . . . . . . . . . . . . . . . 12

2.2.8 Existing Systems Using OFDM . . . . . . . . . . . . . . . . . . . . 132.3 Channel Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.2 SISO Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.3 OFDM Channel Model for C Channels (SISO) . . . . . . . . . . . 15

2.3.4 MIMO Channel Model for a 44 System . . . . . . . . . . . . . . 152.3.5 MIMO-OFDM Channel Model . . . . . . . . . . . . . . . . . . . . 16

2.3.6 The TGn Channels . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Reconstruction of the Original Data . . . . . . . . . . . . . . . . . . . . . 18

3 Literature Review 19

3.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 Aldana et al. 2000 . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.2 Athanasios et al. 2005 . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.3 Athanasios et al. 2006 . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.4 Beaulieu et al. 2000 . . . . . . . . . . . . . . . . . . . . . . . . . 21


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VI TAB LE OF CONTENTS

3.2.5 Boumard 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.6 Pauluzzi et al. 2000 . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.7 Ren et al. 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.8 Ren et al. 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.9 Schmidl et al. 1997 . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.10 Shin et al. 2001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.11 Xu et al. 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.12 Xu et al. 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.13 Ycek et al. 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Other Related Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Simulations 29

4.1 Description of the Simulation Environment . . . . . . . . . . . . . . . . . 29

4.2 Best and Worst Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3 Perfect SNR Shifted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Algorithm Design 37

5.1 Several Approaches and Why They Dont Work (...Too Well) . . . . . . . 37

5.1.1 Using Only the FFT Output . . . . . . . . . . . . . . . . . . . . . 37

5.1.2 Using the Channel Matrix . . . . . . . . . . . . . . . . . . . . . . 37

5.2 Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2.1 General Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2.2 Mathematical Formulation and Analytical Results . . . . . . . . . 40

5.2.3 Simulation of the Proposed Algorithm . . . . . . . . . . . . . . . 52

5.2.4 The Influence of the Number of Samples . . . . . . . . . . . . . . 52

5.2.5 The Mean Value of the Estimated SNR . . . . . . . . . . . . . . . 52

5.2.6 Frequency Offset . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2.7 Ignore Frequency Offset and Save Hardware Costs . . . . . . . . 57

5.2.8 Limited Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2.9 Proposed Algorithm: Further Ideas and Simulations . . . . . . . . 60

6 Implementation 63

6.1 Requirements and Limitations . . . . . . . . . . . . . . . . . . . . . . . . 63

6.2 First Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.3 Second Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.4 Final Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66


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TAB LE OF CONTENTS VII

6.4.1 SNR_EST_ENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.4.2 TOTAL_POWER_ENT . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.4.3 AVERAGE_SIGNAL_ENT . . . . . . . . . . . . . . . . . . . . . . . 72

6.4.4 FULL_CYCLE_FINISHED_ENT . . . . . . . . . . . . . . . . . . . . 72

6.4.5 NUMBER_OF_FULL_CYCLES_ENT . . . . . . . . . . . . . . . . . 72

6.4.6 INITIALIZE_ENT . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.4.7 VALID_DATA_ENT . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.4.8 CONT_AV_SIG_ENT . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.4.9 NR_DIVISION_ENT . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.4.10 Mapping Onto FPGA . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.4.11 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7 Measurements 83

7.1 Measurements With Offline Testbed . . . . . . . . . . . . . . . . . . . . . 83

7.1.1 DC Carrier Removal . . . . . . . . . . . . . . . . . . . . . . . . . 84

7.1.2 Four SNR Values Estimated but Only One Required . . . . . . . . 84

7.1.3 Scaling All Streams to Equal Noise . . . . . . . . . . . . . . . . . 87

7.1.4 Transmit Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7.2 Measurements With Online Testbed . . . . . . . . . . . . . . . . . . . . . 88

8 Summary, Conclusion and Outlook 93

8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

8.2 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Bibliography 100


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VIII TAB LE OF CONTENTS


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List of Figures

2.1 Comparison of a single carrier spectrum and a FDM spectrum. . . . . . . 10

2.2 OFDM system using twice a DFT . . . . . . . . . . . . . . . . . . . . . . 11

2.3 A standard approach for a MIMO system with 4 transmitting and 4

receiving antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 A channel model for a MIMO-ODFM system. . . . . . . . . . . . . . . . . 17

4.1 Perfect and constant SNR estimation (FDMLE channel estimator) . . . . 31

4.2 Perfect and constant SNR estimation (ideal channel estimator) . . . . . . 32

4.3 Ideal SNR estimator with offset . . . . . . . . . . . . . . . . . . . . . . . 34

4.4 Ideal SNR estimator with offset - zoomed version. . . . . . . . . . . . . . 35

5.1 Ren2008 and an adapted EVM algorithm . . . . . . . . . . . . . . . . . . 39

5.2 Mean SNR values for different M. . . . . . . . . . . . . . . . . . . . . . 51

5.3 Simulated BER for the proposed algorithm with M = 9 . . . . . . . . . . 53

5.4 Simulated BER for the proposed algorithm with changing M . . . . . . . 545.5 Simulated BER for the proposed algorithm with changing M - zoomed

version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.6 Estimated mean SNR values for the proposed algorithm. . . . . . . . . . 56

5.7 Proposed algorithm with a frequency offset. . . . . . . . . . . . . . . . . 58

5.8 Proposed algorithm using absolute value of input signal. . . . . . . . . . 59

5.9 Proposed algorithm using limited precision. . . . . . . . . . . . . . . . . 61

6.1 Second approach, datapath of estimated signal power . . . . . . . . . 67

6.2 SNR_EST_ENT - top level design entity. . . . . . . . . . . . . . . . . . . . 69

6.3 TOTAL_POWER_ENT - calculating the power of a stream of data. . . . . 71

6.4 AVERAGE_SIGNAL_ENT - averaging all samples that belong together. . 73

6.5 NUMBER_OF_FULL_CYCLES_ENT and FULL_CYCLE_FINISHED_ENT -

counting subsignals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.6 INITIALIZE_ENT - initializes the rest of the circuit as soon as the AGC

freezes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.7 VALID_DATA_ENT - monitors the state of the arriving samples. . . . . . . 76


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X LIST OF FIGURES

6.8 CONT_AV_SIG_ENT - control for the estimated signal datapath. . . . . 77

6.9 A numerical example for the digital Non-Restoring division algorithm. . 78

6.10 NR_DIVISION_ENT - the division entity. . . . . . . . . . . . . . . . . . . 80

6.11 Overview over all signals for the final estimator entity. . . . . . . . . . . 81

7.1 A picture of the MIMO-OFDM testbed with 4 antennas. . . . . . . . . . . 83

7.2 Measurement of the BER with the offline testbed. . . . . . . . . . . . . . 85

7.3 Estimated SNR values with offline testbed compared to expected SNR

values. The expected values were approximated by taking the best

performing curves from Fig. 7.2 for each output setting. . . . . . . . . . 86

7.4 A transmit noise model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7.5 Estimated SNR for several transmit SNR values . . . . . . . . . . . . . . 897.6 Simulation showing the BER for several estimators with 30dB transmit

SNR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.7 Frequency offset compensation in online testbed. . . . . . . . . . . . . . 92


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List of Tables

6.1 SNR estimation block input and output signals. . . . . . . . . . . . . . . 64

6.2 Approximate hardware costs for approach 1. . . . . . . . . . . . . . . . . 65

6.3 Approximate hardware costs for approach 2. . . . . . . . . . . . . . . . . 66

6.4 Overview over the hardware costs for the implementation of the SNR-

estimator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82


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1 Task Description

Institut fr Integrierte Systeme

Integrated Systems Laboratory

Semester Thesis at the Departement of

Information Technology and Electrical Engineering

Autumn Term 2008

Dominik Bischoff

Noise Estimation

for MIMO-OFDM Testbed

Advisors: Markus Wenk, ETZ J69.2, Tel. 632 57 27, [email protected]

Thomas Koch, ETZ J69.2, Tel. 632 54 33, [email protected]

Patrick Mchler, ETZ J69.2, Tel. 632 65 69, [email protected]

Handout: September 15, 2008

Due: December 19, 2008

Three copies of the written report are to be turned in. All copies remain property of the

Integrated Systems Laboratory.


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2 1 TAS K DESCRIPTION

1 Project Description

In wireless systems, knowledge of the noise variance or the signal-to-noise ratio (SNR) helpsto improve performance. Especially, preprocessing and detection stage in the receiver benefitfrom the knowledge of the noise variance. So far, the multi-user MIMO-OFDM testbed devel-oped at the Integrated Systems Laboratory (IIS) in close collaboration with the CommunicationTechnology Laboratory (CTL) lacks such a noise estimator. Currently, the MMSE receiver im-plemented in the testbed uses a constant noise variance to carry out the MMSE algorithm.Fig. 1shows the noise estimation block in a MIMO-OFDM system.

Transmitter

y = Hs+ n

Receiver

MIMOdetection

(e.g. MMSE)

Channelestimation

Noiseestimation

VGAinterface

Figure 1: Overview of a MIMO system highlighting the channel estimation and the noise esti-mation blocks.

2 Noise Estimation

The estimation of the noise variance or the SNR can be carried out in time or frequency domain.A good overview of SNR estimation in OFDM systems is given in [5]. Several frequency-domainalgorithms were presented in the open literature in the last few years, e.g., estimators based ontwo training symbols (preamble) [2, 4] or for different noise statistics [6]. Most of the publishedestimators work in frequency domain.

3 Goals

The main goal of this thesis is the analysis and implementation of a noise estimation block forthe MIMO-OFDM testbed in order to improve the MIMO detection stage in the testbed. Thefollowing tasks should be accomplished during this project:

Evaluation and analysis of noise variance estimation algorithms (time domain, frequencydomain) in order to understand the impact on the error rate performance.

Integration of a noise estimator into the testbed.

2


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4 Milestones

The following milestones should be achieved during this semester thesis. However, some mile-stones can be added or skipped, depending on the projects status. The tentative calendar inFig. 2 shows all milestones.

1. Establish a project plan.

2. Get familiar with the noise estimation parameters, the literature on noise variance andSNR estimation algorithms, and the Matlab simulation environment.

3. Implementation and evaluation of different noise variance estimation algorithms in Matlab

and on the offline testbed.

4. VHDL implementation of a noise variance estimation block on the Virtex-4 FPGA.

5. BER measurements by using the PropSim channel emulator to verify the proper operationof the implemented algorithm.

6. Write the final report

5 General Recommendations

The following are some recommendations for this semester thesis:

While coding VHDL, use the IIS standard coding style [3] documented by the DesignZentrum (DZ) website [1].

VHDL coding is greatly simplified and accelerated using the Emacs editor and its famousand widely adopted VHDL mode. This Emacs installation at the institute supports amongother powerful features VHDL syntax highlighting, signal and component declaration andinstantiation, code beautifying, and automated sensitivity list updates based on the VHDLstandard. Since most assistants at the IIS are quite familiar with this editor, they can readand evaluate your VHDL code (and help to solve problems) much faster. Please consultthe corresponding FAQ under the following link:

http://www.dz.ee.ethz.ch/support/ic/emacs/index.en.html

6 Project Realization

6.1 Project Plan

Within the first week of the project you will be asked to prepare a project plan. This planshould identify the tasks to be performed during the project and set deadlines for those tasks.The prepared plan will be a topic of discussion of the first weeks meeting between the studentsand the advisors. Note that the project plan should be updated constantly depending on theprojects status.

3


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5

Literature study andsimulation environment

Matlab and offline testbedalgorithm analysis and evaluation

Implementation of a noiseestimator block in VHDL

Measurements and verification

September October November December

Documentation

Tasks

1

2

6

3

4

5

Figure 2: Tentative Calendar

[4] GuanLiang Ren, YiLin Chang, and HuiNing Zhang. SNR estimation algorithm based onthe preamble for wireless OFDM systems. Science in China Series F: Information Sciences,51(7):965974, Jul. 2008.

[5] He Shousheng and M. Torkelson. Effective SNR estimation in OFDM system simulation.IEEE GLOBECOM 98, 2:945950, 1998.

[6] T. Yzek and Arslan H. MMSE noise power and SNR estimation for OFDM systems. IEEESarnoff Conference, Princeton, March 2006.

Zurich, September 15 Prof. Dr. Wolfgang Fichtner

The thesis will not be accepted without returning the keys!

5


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6 1 TAS K DESCRIPTION


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2 Introduction

2.1 Why Using MIMO?

MIMO stands for Multiple-Input Multiple-Output. In communication systems, this

usually means that several transmitting and receiving antennas are employed.

2.1.1 Antenna Arrays

A special case of MIMO systems are antenna arrays that have been in use for a long

time: Several antennas can be used with a specific phase and amplitude setting to

transmit the same signal. This setup produces a higher gain in a certain direction and

is called beamforming. It also increases the diversityof the channel: If there is negative

interference of the signal transmitted from one of the antennas at the receiver, then

there is a high probability that at least one signal transmitted from another antenna of

the array is decodable. Using antenna arrays does neither increase the used bandwidth

nor does it decrease the throughput of data [1].

2.1.2 Spatial Multiplexing

The main difficulty today is that users demand higher data rates for their applications

whereas the usable spectrum is limited (both technically and by regulations). This is

due to the increase in the popularity of mobile applications as for example cell phonesor wireless internet access. Wireless systems do not provide the option of just adding

an additional cable as in wire or fibreoptics based systems. Therefore, the spectral

efficiency needs to be increased in order to enable a higher throughput. But customers

do not only want fast data access - this access also needs to be reliable (QOS - quality of

service) [1].

MIMO systems seem to be able to solve that problem at least temporarily. Instead of

just transmitting one single signal over the air from the transmitter to the receiver (as


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8 2 INTRODUCTION

done in most systems today), several independent signals are sent over the common

channel air by using multiple antennas for transmitting and receiving. The idea seems

fairly trivial - but sending signals in the same frequency band over a common channel is

generally not possible. This is because the signals interfere with each other and cannot

be easily decoded at the receiver [2].

In most applications, every signal sent from a transmitting antenna reaches the re-

ceiving antenna over multiple paths. This phenomenon called multipath propagation

is produced by electromagnetic waves that are reflected off walls and other objects.

The signal arriving at the receiver is therefore generally a superposition of scaled and

delayed versions of the original signal. Multipath propagation is generally considered

as a nuisance as it distorts the signal and common systems try to circumvent it byestablishing a line of sight (LOS) connection [2].

Instead of seeing multipath propagation as a factor that decreases the system perfor-

mance, clever approaches use it as an advantage in MIMO systems. One can imagine

the following setup:

transmitter using antennas T1 and T2

receiver using antennas R1 and R2

T1 and T2 transmit different signals

both are placed inside a building - assuming no LOS for simplicity

A signal sent from T1 and received at R1 follows a different path compared to the signal

sent from T2 and received at R2. The same is true for the signals from T1 to R2 and

from T2 to R1. If one assumes that the different paths are known at the receiver, clever

calculations can remove the effect of the superposition and decode both streams. In

that case, the data rate would have been doubled without using additional spectrum.

Due to the spatial distribution of the antennas, the reliability of the link should be

increased at the same time [2].

The critical question is how to know what those different paths look like - or in other

words: How to find the channel matrix? This is generally done in a training phase

where known signals are sent by the transmitter. This does of course decrease the

overall throughput as no real information is transmitted during that phase. This loss is

generally smaller than the additional capacity gained by using a second stream.


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2.2 WHY USING OFDM? 9

This procedure is called spatial multiplexing as the criterion used to distinguish the

different streams is the space each stream has to travel through. The number of streams

is theoretically limited by the smaller number of antennas on either the transmitter or

the receiver side. As a tradeoff between detection complexity and additional throughput,

a practical upper limit seems to be four spatial streams to be used at the same time.

There is further a problem to position a high number of antennas in a wireless system.

Most of todays commercially available systems therefore only use two spatial streams.

It is further possible to use additional antennas on either the receiver or the transmitter

side to increase the diversity gain [2].

To use spatial multiplexing in outdoor systems where one has a direct LOS, other tricks

have to be used. One possibility is to use special antennas with a 90 degree shiftedpolarization [2].

2.2 Why Using OFDM?

2.2.1 The Standard Approach

The standard approach to modulate information onto a carrier is by varying the

frequency, the phase or the amplitude. As the data rate increases, the time a single

symbol (one or several bits) is on air is decreased. In case of impulse noise or other

short period noise with high energy, it is likely that a symbol gets distorted to such a

high extent that it cannot be recovered. The shorter the period in which the symbol is

available, the higher is the probability that the symbol is fully destroyed by bursts of

noise [3].

2.2.2 Frequency Division Multiplexing

To solve this problem, one can use frequency division multiplexing (FDM). Instead

of using a single carrier that occupies the whole available frequency band, several

subcarriers are employed within the available frequency band. The data stream is

distributed over all available subcarriers. This increases the symbol period and therefore


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10 2 INTRODUCTION

decreases susceptibility to noise bursts. It also adds additional immunity to narrow-

banded noise , as such noise only affects several of the subcarriers and not the entire

signal [3].

FDM comes at the cost of a lower data rate as a guard interval has to be inserted between

the different subcarriers and therefore a part of the available frequency spectrum is

wasted. FDM also adds some complexity to the hardware by using several streams. At

the same time it also removes some of the complexity by slowing down the bit rate of

each subcarrier [3].

(a) Single carrier spectrum (b) FMD spectrum

Figure 2.1: Comparison of a single carrier spectrum and a FDM spectrum [3].

2.2.3 Orthogonal FDM (OFDM)

If one can choose a set of subcarriers that are orthogonal to each other, then there is

no need to use a guard interval to separate the subcarriers. This would increase the

spectral efficiency of the system [3].

2.2.4 OFDM: What Are Orthogonal Signals?

Two signals u(t) and v(t) are said to be orthogonal to each other iff:

< u, v >=

u(t) v(t) dt =

0 , ifu = v,const , ifu = v.


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2.2 WHY USING OFDM? 11

2.2.5 OFDM: How to Find Orthogonal Signals

There are several possible ways to create orthogonal signals. The solution presented

here uses the Discrete Fourier Transform (DFT). A hardware efficient implementation

of the DFT is the Fast Fourier Transform (FFT). All the sinusoids of the DFT form an

orthogonal basis. If a time discrete signal is transformed with the DFT, it is essentially

correlated with those base sinusoids. Furthermore, the DFT is invertible. Using the

inverse DFT (or the inverse FFT - IFFT), the original signal can be reconstructed [3].

The mathematical backgrounds are well described in [4]. A sample system is shown in

Fig. 2.2. The basic ideas behind that system are: A whole collection of source symbols

(complex) are considered to be in the frequency domain. They get translated by theIDFT into the time domain. Those discrete samples are transformed into a continuous

signal that can be transmitted over the channel. The receiver samples the signal and

transforms it back into the frequency domain by the use of the DFT. If there is no noise

present and the channel is perfect, the symbols at the receiver are the same as the ones

that were transmitted.

As the base functions of the DFT overlap each other without interfering, the spectral

efficiency of the signal is a lot higher than in the case of a simple FDM and approaches

the case of the single carrier system.

Figure 2.2: OFDM system using twice a DFT [4]. Note: Instead of using a IDFT and aDFT (or a IFFT and a FFT), one can use two DFT. This is because the DFT and the IDFT

are very similar. In that case, several adaptions need to be done to the datapath of the

transmitter!


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12 2 INTRODUCTION

2.2.6 OFDM: Cyclic Prefix

In a multipath channel, several delayed versions of the original signal appear at the

receiver. One speaks ofintersymbol interference (ISI) if a consecutive OFDM-symbol gets

distorted by the previous one. In a general case, only the first few samples of the signal

get distorted. The problem can be solved by waiting a specific time between transmitting

two consecutive symbols. This guard interval (in time domain) is depending on the

channel [3].

The other problem is that a single OFDM symbol can interfere with itself. This is called

intrasymbol interference. The reason is the following: A convolution in time domain is

equivalent to a multiplication in the frequency domain iff the signal is either periodicor infinitely long. Both is not fulfilled for a standard OFDM system [3].

The solution is to make the OFDM symbol appear periodic. This is done by using a

cyclic prefix (CP): The last few samples of the signal are copied at the beginning of the

signal where originally the guard interval would be. This cyclic prefix only contains

redundant data and can therefore be discarded at the receiver - so there is no problem

with ISI [3].

Using a cyclic prefix leads to a significant simplification of the receiver: Instead of

having to remove a convolution in time (between the signal and the channel), it is only

necessary to remove a multiplication in frequency domain [3].

2.2.7 OFDM: Noise Considerations

The most common noise source in a wireless system is thermal noise - usually manifest-

ing itself as Additive White Gaussian Noise (AWGN). As the noise spectrum is uniform

in the frequency domain, this kind of noise has the same impairment on the overallsystem as it has in a single carrier system [3].

Another common type of noise is impulse noise. This type of broadband noise is generally

only present during a short period. As described before, the OFDM system performs

better under impulse noise than a single carrier system [3].

Colored noise is difficult to handle as it doesnt have a constant spectrum as AWGN. A

simple solution for high noise environments is to lower the data rate [ 3].


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2.3 CHANNEL MODELS 13

If there are other systems present, carrier interference can occur. An OFDM system can

handle that by disabling the affected subcarriers [3].

Another type of imperfection emerges from the local oscillator. There are two effects thathave to be considered: Phase noise (sometimes called phase jitter) and the frequency

offset. Phase noise originates from the fact that the oscillator frequency changes

randomly within a small range. The same argument in the frequency domain is that

the oscillator does not produce a single peak but rather a smeared out peak. Phase

noise affects every subcarrier. As the spectral width of a subcarrier is smaller than in

a single carrier system, phase noise affects OFDM systems more severly than single

carrier systems [3].

The frequency offset of an oscillator can be understood as the average frequency of theoscillator. This frequency is generally slightly different from the expected frequency.

Clock quality, temperature and other effects are generally responsible for this offset. A

solution to this problem is to introduce pilot subcarriers for synchronization. It has to be

noted that introducing pilot subcarriers affects the maximum data rate negatively [3].

2.2.8 Existing Systems Using OFDM

Two of the most prominent systems using OFDM are ADSL (Asynchronous Digital Sub-

scriber Loop) and DVB-T (Digital Video Broadcast - Terrestrial). The first is used for high

speed internet connections and the second for the European digital television [3].

A system that uses both MIMO and OFDM will be the next generation Wireless LAN

(WLAN 802.11n). The final specifications are not yet available - but there are already

existing devices on the market based on a draft (e.g. [5]). Those new devices promise

a significantly higher data rate than previous generations.

2.3 Channel Models

2.3.1 Notation

The following notation will be used:

x(t) signal leaving the transmitter (time domain)


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14 2 INTRODUCTION

X signal vector in frequency domain: Input of IFFT

y(t) signal reaching the receiver (time domain)

Y received signal vector in frequency domain: Output of FFT

h(t) channel impulse response (time domain)

H channel response matrix in frequency domain

n(t) additive noise (time domain)

N noise vector in frequency domain

T total number of transmitting antennas

number of the transmitting antenna

R total number of receiving antennas

r number of the receiving antenna

C total number of subcarriers

c number of the subcarriers

2.3.2 SISO Channel Model

The simplest possible system is a SISO (Single-Input, Single-Output) system. In time

domain, it can be written as:

y(t) = x(t) h(t) + n(t)

This is equivalent to the following notation in the frequency domain:

Y(f) = X(f) H(f) + N(f)


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2.3.3 OFDM Channel Model for C Channels (SISO)

An OFDM system can be represented by the following model in frequency domain:

[yc=1] = [Hc=1] [xc=1] + [nc=1][yc=2] = [Hc=2] [xc=2] + [nc=2][yc=3] = [Hc=3] [xc=3] + [nc=3]

... =... ... + ...

[yc=C] YC1

= [Hc=C] [xc=C] XC1

+ [nc=C] NC1

Each line corresponds to one of the orthogonal tones.

2.3.4 MIMO Channel Model for a 44 System

To simplify the notation for a MIMO system with T transmitting and R receiving

antennas, it is assumed that T = R = 4. Such a general setup is shown in Fig. 2.3. It

is straight forward to change the number of transmitting or receiving antennas. The

T1

T2

T3

T4

R1

R2

R3

R4

Transmitter Receiver

h11

h21

h31

h41

h44

Figure 2.3: A standard approach for a MIMO system with 4 transmitting and 4 receiving

antennas.


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16 2 INTRODUCTION

system shown in Fig. 2.3 can be written in the frequency domain the following way:

yr=1

yr=2

yr=3

yr=4

YR1

=

hr=1,=1 hr=1,=2 hr=1,=3 hr=1,=4

hr=2,=1 hr=2,=2 hr=2,=3 hr=2,=4

hr=3,=1 hr=3,=2 hr=3,=3 hr=3,=4

hr=4,=1 hr=4,=2 hr=4,=3 hr=4,=4

HRT

x=1

x=2

x=3

x=4

XT1

+

nr=1

nr=2

nr=3

nr=4

NR1

It can be assumed that only one antenna is transmitting and all the others are not

sending any signal at all. In that case, the equation simplifies to:

yr=1

yr=2

yr=3

yr=4

YR1

=

hr=1,=1 hr=1,=2 hr=1,=3 hr=1,=4

hr=2,=1 hr=2,=2 hr=2,=3 hr=2,=4

hr=3,=1 hr=3,=2 hr=3,=3 hr=3,=4

hr=4,=1 hr=4,=2 hr=4,=3 hr=4,=4

HRT

x=1

0

0

0

XT1

+

nr=1

nr=2

nr=3

nr=4

NR1

This can be further simplified to:

yr=1

yr=2

yr=3

yr=4

YR1

= hr=1,=1

hr=2,=1

hr=3,=1

hr=4,=1

HR1

x=1 + nr=1

nr=2

nr=3

nr=4

NR1

2.3.5 MIMO-OFDM Channel Model

As can be seen from the SISO OFDM channel model, the different OFDM subchannels

can be treated separately. This allows to formulate a simple model for a MIMO-OFDM

system: The whole system can be seen as a stack of C different MIMO systems. A

graphic showing such a system is presented in Fig. 2.4


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Y

Rx1

= H

RxT

X

Tx1

.N

Rx1

+

subchannel 1subchannel 2

subchannel 3

subchannel 4

Rx1xC RxTxC Tx1xC Rx1xC

subchannel C

Figure 2.4: A channel model for a MIMO-ODFM system.

2.3.6 The TGn Channels

In 2004, the Task Group N (TGn)1 published a set of channel models applicable

to indoor MIMO WLAN systems. The model[s] can be used for both 2 GHz and

5GHz frequency band[s.]. There are six different channel models: A, B, C, D, E and

F. Model A is an optional model and should not be used for system performance

comparisons [6].

The following steps are taken for models B to F2:

Start with delay profiles of models B-F.

Manually identify clusters in each of the five models.

Extend clusters so that they overlap, determine tap powers (see Appendix A).

Assume PAS [power angular spectrum] shape of each cluster and correspondingtaps (Laplacian).

Assign AS [angular spread] to each cluster and corresponding taps.

Assign mean AoA [angle of arrival] (AoD [angle of departure]) to each clusterand corresponding taps.

Assume antenna configuration.

Calculate correlation matrices for each tap.1IEEE P802.11 - TASK GROUP N;http://www.ieee802.org/11/Reports/tgn_update.htm

2quoted directly from [6] to show the complexity of the models
http://www.ieee802.org/11/Reports/tgn_update.htmhttp://www.ieee802.org/11/Reports/tgn_update.htm


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18 2 INTRODUCTION

The TGn also calculated the mean capacity in bits per second per Hz for all models. The

results show that model C has the lowest capacity of all proposed models. This suggests

that channel C is the most challenging of the channel models. This is the reason that

TGn C is used for the simulations in this thesis.

2.4 Reconstruction of the Original Data

The MIMO-OFDM channel model suggests that if the exact channel matrix and the

exact noise vector were known, the original data could be reconstructed perfectly. It

is obvious that in any real system with a limited amount of training data, one cannot

perfectly estimate neither the channel matrix nor the noise vector. The limitation of

available training data is justified by the loss of throughput by increasing the amount

of training data and the fact that any real wireless channel is time-varying. These

imperfections can cause errors in the detected symbols. By improving the performance

of the receiver, the amount of errors can be minimized. This thesis deals with the

estimation of the noise variance (or equivalently the SNR) as the noise variance is an

important parameter for decoding the received signals.


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3 Literature Review

3.1 Method

This part of the thesis presents a selection of papers that might be relevant to the topic

of interest. The papers are sorted alphabetically by the family name of the author.

As the methods and parameters used for simulation vary highly between the different

papers, numerical comparisons of algorithms are omitted in this section.

An algorithm is suitable if the following points are satisfied:

better or equally accurate as other algorithms of similar setup and complexity

adaptable to MIMO-OFDM

well enough documented to be implementable in a reasonable amount of time

complexity of calculations within reasonable limits and therefore suitable forhardware implementation

Any additions not present in the paper and added by the author of this thesis are written

in italics.

3.2 Papers

3.2.1 Aldana et al. 2000: Accurate Noise Estimates in Multicarrier

Systems

Aldana et al. [7] presented in their work two different algorithms to estimate the noise

variance in multicarrier systems. Those algorithms would therefore be suitable for

OFDM systems. The two presented algorithms do not use any known training signals.


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20 3 LITERATURE REVIEW

The first algorithm presented is the EM (Expectation Maximization) algorithm. The

algorithm is iterative and converges only slowly. Those two facts make this algorithm

unsuitable for application in a real system.

The second algorithm is a decision directed algorithm. Similar to the previous algo-

rithm, this one is suitable for OFDM signals, operates in the frequency domain and does

not need any training data.

Nk = Yk Hk Xk

2k =1

L

Ln=1

|Nk|2

SN RQAM = |Hk|2

d2

(M2

1)6 2

Yk is the received signal of the k-th tone. Hk is the gain of subchannel k and assumed to

be known (or at least accurately guessed). Xk is the estimation of the transmitted symbol

of the k-th tone. Known training symbols might improve the quality of the estimated

SNR. M is the number of symbols (M-ary QAM) and L is the blocklength. d is the

distance between symbols. The authors come to the conclusion that their algorithm

does underestimate the true SNR and that in order to get reliable results, a look up

table (LUT) depending on the modulation scheme should be implemented.

3.2.2 Athanasios et al. 2005: SNR Estimation Algorithms in AWGN for

HiperLAN/2 Transceiver

Athanasios et al. [8] present two different algorithms for the HiperLAN/2 system that

employs OFDM. Both algorithms estimate the SNR in a 64-QAM system.

The first algorithm is called MMSE (Minimum Mean Square Error). This algorithm usestraining signals a and works in the frequency domain.

a = {a1, a2,...,aL}C = Y aH

E = |Y|2

SN R =|C|2

|a|2 E |C2|


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3.2 PAP ER S 21

The authors state that it is also possible to only use the real or the imaginary part of the

received data to reduce the complexity of the calculation, whereas the drop in precision

should be only minimal.

The second algorithm is called EVM (Error Vector Magnitude). It estimates the sent

symbols and calculates the average and the variance of them. It is not specified in detail

how those symbols should be estimated and the algorithm seems to exhibit a rather

poor performance compared to the MMSE algorithm.

3.2.3 Athanasios et al. 2006: SNR Estimation for Low Bit Rate OFDM

Systems in AWGN channels

Athanasios et al. [9] present two different algorithms for OFDM systems. The second

one is the MMSE algorithm already presented in [8].

The first algorithm is called SNV (Squared Signal to Noise Variance). Again, this

estimator needs estimates of the received symbol and the performance seems to be

inferior to the MMSE algorithm.

3.2.4 Beaulieu et al. 2000: Comparison of Four SNR Estimators for

QPSK Modulations

Beaulieu et al. [10] present four different estimators for QPSK modulations in time

domain. Xi is the in phase component and Yi is the quadrature component. Thealgorithm with the best performance is:

2 = L

Li=1

(|Xi| |Yi|)2X2i + Y

2i

1

It has to be further investigated if and how this algorithm could be used for an OFDM

system. The same algorithm is also presented in the frequency domain by Hong et

al. [11].


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3.2.5 Boumard 2003: Novel Noise Variance and SNR Estimation

Algorithm for Wireless MIMO OFDM Systems

Boumard [12] presents an algorithm to estimate the SNR in a 2x2 MIMO-OFDM system

in the frequency domain. The algorithm needs some well defined training symbols (two

per antenna - sent individually) and the results from a channel estimator. The algorithm

is able to calculate both the SNR per subcarrier and the overall SNR. The algorithm

seems to perform well as long as the channel is reasonably slow fading. It needs to be

further investigated, how this algorithm can be adapted for a 4x4 MIMO-OFDM system

with predefined training symbols. The principal challenges are the use of given training

symbols and the expansion to a 4x4 system.

3.2.6 Pauluzzi et al. 2000: A Comparison of SNR Estimation

Techniques for the AWGN Channel

Pauluzzi et al. [13] present five different SNR estimation techniques for PSK modulation

in an AWGN channel.

The first algorithm is called SSME (Split Symbol Moments Estimator) and is only valid

for BPSK modulation.

The second algorithm is the ML (Maximum Likelihood) estimator. There are two

versions of that algorithm: One that uses known training symbols and one that uses

guesses of the transmitted symbols. The data-aided version seems to perform near

the optimum and the non-data-aided performs equally well for high SNRs. To use this

algorithm, it has to be adapted to the MIMO-OFDM system as the system used by Pauluzzi

et al. is quite different.

The third algorithm is the SNVestimator that is also presented in [14] and [9].

The fourth algorithm is the M2M4 (Second- and Fourth-Order Moments) estimator.

This estimator seems to perform similar to the ML algorithm except in low SNR

environments, where it performs worse.

The fifth algorithm presented is the SVR(Signal to Variance Ratio) estimator. It per-

forms significantly worse than the ML estimator especially in high SNR environments.


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3.2 PAP ER S 23

3.2.7 Ren et al. 2005: A New SNRs Estimator for QPSK Modulations

in an AWGN Channel

Ren et al [15] present the M2M4 algorithm from [13] and an improved version of

this algorithm. The improved version seems to perform better than the original and

also better than the ML in high noise environments (SNR < 0dB). As this region is not

suitable for fast wireless communication anyway, the algorithm doesnt offer any advantage

over the ML algorithm.

3.2.8 Ren et al. 2008: SNR Estimation Algorithm Based on the

Preamble for Wireless OFDM Systems

Ren et al. [16] analyze the algorithm presented by Boumard [12] and come to the

conclusion that the performance of this algorithm depends highly on the frequency

selectivity of the channel. They propose an improved version of Boumards algorithm tosolve that problem. The authors also present several simulations that seem to confirm

that fact.

W =4

N

N1k=0

Im

Y0,k c0,k

Hk|Hk|

2S = M2 W

M2 =1

N

N

k=0 |Y0,k|2

SN Rav =S

W

SN Rsubch k =|Hk|2

W

N is the size of the IFFT/FFT. Ym,k is the m-th symbol of the k-th subcarrier after the

FFT at the receiver. cm,k is the m-th symbol on the k-th subcarrier. Hk is the channel

coefficent estimate.


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3.2.9 Schmidl et al. 1997: Robust Frequency and Timing

Synchronization for OFDM

Schmidl et al. [17] present a time domain approach for synchronizing transmitter and

receiver. As a by-product they suggest an SNR estimator working in the time domain.

This estimator works well for the SNR below 20 dB. Above this level, M(dopt) is so

close to 1 that an accurate estimate of the SNR can not be determined, but only that

the SNR is high.

3.2.10 Shin et al. 2001: Simple SNR Estimation Methods for QPSK

Modulated Short Bursts

Shin et al. [18] present two algorithms to estimate the SNR in a QPSK modulated

system.

The first algorithm is the EVM algorithm also presented by Athanasios et al. [8]. The

algorithm is rather simple and doesnt need any estimates at all (at least for the QPSKcase and not too low SNR). The authors also attribute a higher accuracy to this algorithm

than in [8].

1. check ifRe{Y} > 0 and ifIm{Y} > 0

2. for a given time period, collect the values for each of the four regions

3. estimate the SNR by: SN R = |average|2

variance

4. repeat to get an average

As this algorithm is simple to implement and independent of any other hardware. It should

also be easy to transform to the OFDM case.

The second algorithm presented is the MMSE that is also presented by Athanasios et

al. [8]. Interestingly, the MMSE algorithm is considered to be inferior to the EVM

algorithm by Shin et al., whereas Athanasios et al. come to the opposite conclusion.


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3.2 PAP ER S 25

3.2.11 Xu et al. 2005: Subspace-Based Noise Variance and SNR

Estimation for OFDM Systems

Xu et al. [19] present a subspace based algorithm for SNR estimation in OFDM

systems. The algorithm is computationally quite complex: 1) Make an eigenvector

decomposition of the correlation matrix R.

3.2.12 Xu et al. 2005: A Novel SNR Estimation Algorithm for OFDM

Xu et al. [20] present a broad range of algorithms. Among them are the ML, the MMSE

and the M2M4 algorithms already presented in other papers.

Based on Boumards algorithm [12], they develop a new algorithm that should perform

better in time varying channels.

RG(l) =1

J

J1j=0

y(i, j) y(i, l +j) (3.1)

SG RG(1) + RG(1) RG(2)3

(3.2)

NG =1

J

J1j=0

y(i, j) y(i, j) SG (3.3)

SN R =SG

NG(3.4)

y(i, j) is the j-th symbol on the i-th subcarrier.

3.2.13 Ycek et al. 2006: MMSE Noise Power and SNR Estimation forOFDM Systems

Ycek et al. [21] propose to use an estimator with a two dimensional filter over

time and frequency. To reduce the calculational complexity, they propose to have

a rectangular window for the filter. The authors come to the conclusion that their

approach significantly improves the SNR estimation in colored noise. The paper

continues work proposed in an earlier paper by the same authors [22]. If colored


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noise should be a problem, this algorithm could be further investigated - despite its high

computational complexity.

3.3 Other Related Papers

The following papers were somehow related to the problem but were too far away from

the actual problem to be adapted with a reasonable amount of work:

Alagha 2001: Cramer-Rao Bounds of SNR Estimates for BPSK and QPSK Modu-

lated Signals [23]

This paper presents the theoretical bounds that can be achieved by the best

possible algorithm.

Benedict et al. 1967: The Joint Estimation of Signal to Noise from the SumEnvelope [24]

This paper provides some basic theory about estimating noise in narrowband

AWGN systems.

He et al. 1998: Effective SNR Estimation in OFDM System Simulation [25]Some basic principles about using OFDM without the DFT are presented. But more

important is the following quote: Disregarding the form of distortions/interferences,

by the virtual of the central limit theorem, the noise part in eqn. (10) tends to

approach a Gaussian process, and it has been shown that ifn(t) is a Wide-Sense

Stationary (WSS) process, the noise part in eqn. (10) tends to be white. This

indicates that it might be reasonable to assume that SNR estimation has a higher

probability of success if done in frequency domain.

Further, a rather basic algorithm for SNR estimation is presented.

Jeruchim et al. 1989: Estimation of the Signal-to-Noise Ratio (SNR) in Commu-nication Simulation [26]

A very basic paper providing some estimator theory.

Kerr 1966: On Signal and Noise Level Estimation in a Coherent PCM Chan-nel [27]

A basic paper that is too far away from the actual problem to be of any direct use.


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3.3 OTHER RELATED PAP ER S 27

Trkboylari et al. 1998: An Efficient Algorithm for Estimating the Signal-to-Interference Ratio in TDMA Cellular Systems [28]

A rather complex algorithm for TDMA systems.

Wiesel et al. 2002: Data-Aided Sigal-to-Noise-Ratio Estimation in Time SelectiveFading Channels [29]

A time selective channel model is presented and a generalized class of ML detec-

tors for that model is derived.

Wiesel et al. 2002: Non-Data-Aided Signal-to-Noise-Ratio Estimation [30]A non data aided version of the ML detector is presented along with a M2M4estimator. Further, a non data aided iterative algorithm is presented.

Wiesel et al. 2006: SNR Estimation in Time-Varying Fading Channels [31]The Cramer-Rao bound (CRB) is derived for data aided SNR estimation. It is

shown that the data aided CRB is the same for time constant and time varying

channels. But this doesnt mean that all the algorithms perform equally well in

time varying channels. A generalized ML detector is derived for a polynominal-

in-time, time-varying fading channel. This algorithm is iterative. If time variation

should be found to be a problem in the real system, it would probably be worth

to consider this algorithm - even though iterative behavior usually means high

computational costs.


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4 Simulations

4.1 Description of the Simulation Environment

The simulation environment performs the following tasks for each sweep:

1. generate a data-stream in the time domain, consisting of:

2 short preambles (64 samples + 16 samples for the CP each) 2 long preambles (a total of 128 samples + 32 samples for the CP) MIMO training (320 samples - 80 per transmitting antenna) random data to transmit (64 samples + 16 samples for the CP)

2. transmit the data (apply channel matrix)

3. generate AWGN noise corresponding to the SNR setting (all channels equal

amount of noise)

4. add the generated noise to the received data

5. estimate SNR

6. configure receiver and decode data bits

7. calculate the BER

8. repeat steps 2 to 6 for all SNR steps

At the end of all sweeps, the average of the BER is calculated for each channel SNR.

It has to be noted, that a real system should send more data in order to increase

the throughput. This is not done here because the focus is on the SNR estimation.

In order to get reliable results with a reasonable amount of computation time, it is

preferred to increase the amount of sweeps rather than to increase the amount of data

per sweep. The estimated SNR is the average of the four SNRs calculated for each

receiving antenna.


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4.2 BEST AND WORST CAS ES 31

0 5 10 15 20 25 3010

3

102

101

100

SNR channel dB

BER

SNRest = SNR of channel

SNRest = 10dB

SNRest = 20dB

SNRest = 30dB

SNRest = 50dB

Figure 4.1: This simulation shows the differences between a SNR estimator using the

actual channel SNR and several constant SNR estimators. The used channel estimatoris FDMLE.


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32 4 SIMULATIONS

0 5 10 15 20 25 3010

4

103

102

101

100

SNR channel dB

BER

SNRest = SNR of channel

SNRest = 10dB

SNRest = 20dB

SNRest = 30dB

SNRest = 50dB

Figure 4.2: This simulation shows the differences between a SNR estimator using the

actual channel SNR and several constant SNR estimators. The ideal channel estimator(i.e. perfect channel knowledge) is used.


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4.3 PERFECT SNR SHIFTED 33

estimator instead of a constant (high) SNR estimator. This is equivalent to a decrease

in the BER by about a factor of two (if the channel SNR is above 5dB). The benefit is

lower if compared to the 30dB curve, but still significant. It is therefore worth investing

some time to find a good SNR estimator.

4.3 Perfect SNR Shifted

Fig. 4.1 suggests, that it is generally better to overestimate the SNR than to underes-

timate it. This is certainly true for large deviations of the actual channel SNR. The

effects of slightly over- or underestimating the channel SNR are explored3 in Fig. 4.3

and 4.4.

Fig. 4.3 and Fig. 4.4 confirm that the the channel estimator adds approximately 2dB of

noise. They also show that approximately half a decibel is lost if the estimation is in

the range of -5...+1 dB of the actual channel SNR and that around one decibel is lost

for the range -6...+2 dB channel SNR.

The second interesting result from Fig. 4.3 and Fig. 4.4 is that the loss in performance

increases quite fast for higher deviations. If one assumes -2dB to be the optimal case,

then 3dB deviation result in half a decibel of performance loss, whereas 4dB deviationlead to a full decibel of performance loss!

3SNR range: 0-30 [dB](step: 1 [dB]) number of sweeps: 20000 (seed=0..9)channel model: TGn C transmitting antennas: 4 receiving antennas: 4 number of tones: 64channel estimator: FDMLE demapper: MMSE modulation: QPSK


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34 4 SIMULATIONS

0 5 10 15 20 25 30103

102

101

100

SNR channel dB

BER

SNRest = SNR channel

SNRest = SNR channel + 6dB




SNRest = SNR channel 1dB






SNRest = 50dB

Figure 4.3: This figure shows the simulation results of an SNR estimator using theactual channel SNR with an offset of several decibels.


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4.3 PERFECT SNR SHIFTED 35

21 21.5 22 22.5 23 23.5 24

102

SNR channel dB

BER

SNRest = SNR channel











SNRest = 50dB

Figure 4.4: This figure shows the simulation results of an SNR estimator using theactual channel SNR with an offset of several decibels. Detailed version of the plot inFig. 4.3.


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36 4 SIMULATIONS


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5 Algorithm Design

5.1 Several Approaches and Why They Dont Work (...Too

Well)

5.1.1 Using Only the FFT Output

The simplest approach would be using the output of the FFT directly - without any

correction terms from the channel matrix. This does generally not produce any reliable

results, as every tone on every possible channel generally experiences a different

influence from the channel itself (phase shift and amplitude change - multiplication

with a complex channel matrix coefficient). To use an EVM-style algorithm, one would

have to apply the algorithm for every transmitter-receiver-tone combination. It would

therefore be necessary to send the same known symbol several times in series. This is

obviously not a good solution as a lot of potential channel capacity is wasted.

5.1.2 Using the Channel Matrix

Every approach that employs the inverse of the channel matrix is doomed: The channel

matrix is generally not invertible. Inverting the channel matrix can be circumvented by

rewriting the algorithm or using known training signals where no tone is sent by more

than one antenna at any moment.

But not only the inversion is a problem: Using the channel matrix itself is highlyproblematic. To estimate the channel matrix in a 4x4 system, each antenna has to

transmit each tone once alone. It is then possible to fill in the channel matrix with the

values at the receiver. This results in four complete OFDM symbols that have to be sent

including their CP. Compared to other setup steps, this step is quite costly and should

therefore not be repeated - at least not in a 4x4 system.

If an algorithm - for example1 the one presented by Ren et. al. [16] - uses this estimated

1The same problem exists for Aldana et. al. [7], Athanasios et. al. [9], Boumard [12] and others.


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38 5 ALGORITHM DESIGN

channel matrix, the measured noise is zero. This is because the estimation of the

channel matrix assumed that there is no noise. If then the signal power is divided by the

noise power, the result is a high number which has nothing to do with the actual SNR.

As mentioned before, it would be possible to get a better estimate of the channel matrix

- but this is no option in a real system. It is also not desirable to have an SNR estimator

that is dependent on the performance of the channel estimator. SNR estimators that

need the channel matrix are not generally bad - some of them (e.g. the one from Ren et.

al. [16]) have a performance near the optimum for a perfect channel estimator. They

can therefore be a valid solution if an extremely accurate channel estimator is used. A

plot2 showing the performance of the Ren2008 and an adapted EVM algorithm can be

seen in Fig. 5.1.

2SNR range: 0-30 [dB](step: 1 [dB]) number of sweeps: 20000 (seed=0..9)channel model: TGn C transmitting antennas: 4 receiving antennas: 4 number of tones: 64channel estimator: FDMLE/ideal demapper: MMSE modulation: QPSK


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5.1 SEVERAL APPROACHES AND WHY THE Y DON T WOR K (...TOO WEL L) 39

0 5 10 15 20 25 3010

3

102

101

100

SNR channel dB

BER

Ren2008 perfect channel estimatorRen2008 FDMLE channel estimator

EVM ideal channel estimator

EVM FDMLE channel estimator

constant 50dB

channel SNR

Figure 5.1: This plot shows the high performance of the Ren2008 and an adapted EVMalgorithm for an ideal channel estimator. It further shows the bad performance whenusing the FDMLE channel estimator. It is not entirely clear why the Ren2008 algorithm

performs bad at low channel SNR in combination with the ideal channel estimator. Itcan further be noted that with the FDMLE channel estimator, both algorithms perform

slightly worse than the 50dB constant algorithm. This suggests that 50dB is not enoughto be the upper limit but it is close enough for the 0..30dB range.


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5.2 Proposed Algorithm

5.2.1 General Idea

The algorithm uses the short preambles transmitted in the training phase. The system

transmits a clearly defined number of short preambles (generally two or four). One

short preamble consists of a repeating signal part of 16 samples plus another 16 samples

for the CP. In the ideal case, this leads to a series of five 16-sample-signals (subsignals)

per short preamble that are identical. For four short preambles, this results theoretically

in twenty identical subsignals that can be compared to estimate the signal power and

the noise power. It has to be noted that at least the first subsignal is heavily distorteddue to the setup of filters and the automatic gain control (AGC) and therefore cannot be

used.

To estimate the SNR, an average of all available subsignals is taken. This average

signal should be nearly identical to the signal received without noise, as long as the

noise is additive and has a mean value near zero (this is the case for AWGN). Out of

this estimated subsignal, the signal power Ps,est can be calculated. Using the original

received signal, the power of the signal plus noise Ps+n can be calculated. Those results

can be used to estimate the SNR:

SNRest =Ps,est

Ps+n Ps,est =Ps,est

Pn,est

This algorithm will be denoted proposed algorithm to distinguish it from other algo-

rithms. The numbers provided are specific for the the used system but can easily be

adapted for other configurations.

5.2.2 Mathematical Formulation and Analytical Results

Original Signals

All formulas provided are written in the discrete time domain - i.e. directly after the

IDFT at the transmitter and directly before the DFT at the receiver.


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5.2 PROPOSED ALGORITHM 41

16 sample subsignal c[l] that is part of the short preamble transmitted by antenna

:

c[l] =c,l C , if l = 0...15,0 , else.

The transmitted signal s[k] can then be written as a concatenation of several instances

of the signal c[l] where m is the number of transmitted short preambles:

s[k] =m5i=0

c[k i 16]

The received signal yr[k] for receiving antenna r is then the following:

yr[k] =4

=1

(s hr,)[k] + n[k]

(s hr,)[n] =

k=s[k] hr,[k n]

=

k= s[k n] hr,[k]

n is assumed to be IID AWGN and h is the channel impulse response. Due to the

convolution, the received signal yr[k] is generally not periodic anymore.

The Received Signal Rewritten

It is shown that if the first and the last 8 samples ofy are cut away, the remaining signal

is periodic again. The important points are:

hr,[k] = 0 ifk < 0 due to the causality of the channel.

The cyclic prefix is 16 samples long and assumed to be chosen carefully to avoidISI. Therefore the impulse response hr,[k] is zero ifk 8.

h is assumed to be constant during the whole transmission (slow enough fadingchannel).


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The channel does in the worst case distort the first 8 samples of the next 16-samplesubsignal. This is done in a periodic manner.

The sum of multiple signals with the same period is periodic again.Those three facts lead to the conclusion, that if the first and the last 8 samples are cut

away, the rest of the signal is periodic again. It is easy to see that this is true for all

receiving antennas. Every receiving antenna can therefore be treated individually.

This leads to a modified received signal y[k] that can be written in the following

way:

y[k] = M1i=0

z[k i 16] + n[k]The newly introduced signal z[l] is defined as:

z[l] =

z,l C , ifl = 0...15,0 , else.

It is possible to calculate the different components z,l but it is in this case not necessary.

M is the number of available 16-sample subsignals. The noise signal n[k] is generally atruncated version of the former noise signal n[k] and can be defined (assuming AWGN)

the following way:

Re{n[k]} =nkr R so that nkr N(0,

2n2

) , if l = 0...16 M 1,0 , else (cut away).

Im{n[k]} =

nki R so that nki N(0, 2n

2) , if l = 0...16 M 1,

0 , else (cut away).

E Re{n[k]}2 + Im{n[k]}2 = 2nThe Received Signal as a Random Variable

Each sample of the received subsignal can also be interpreted as a random variable:

y[k] N

z[mod16(k)], 2n


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The Averaged Signal

In a next step, the average s[l] of all 16-sample subsignals in y[k] is calculated. If an

infinite amount of such subsignals would be available, the average is expected to be

z[l], as the noise terms cancel out according to the law of large numbers:

s[l] =1

M

M1i=0

y[l + 16 i]

=1

M[y[l] + y[l + 16] + ... + y[l + (M 1) 16]]

= z[l] +1

M

M1

i=0 n[l + 16 i]

The Averaged Signal as a Random Variable

The expectation of this average signal is calculated:

E[s[l]] = z[l] +1

M

M1i=0

E [n[l + 16 i]]

= z[l]

The following property was used:

E[X+ Y] = E[X] + E[Y]

Further, the variance of the average signal is calculated.

var(s[l]) = E

(s[l] z[l])2

= 1

M2E[(

M1

i=0

n[l + 16 i]

N(0,M2n)

)2]

=2nM

The following formulas were used:

X, Y N(0, 2) , IID


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X+ Y N(0, 2 + 2)var(Z) = 2z = E[(Z E[Z])2]

The average signal s[l] can then be written as a random variable:

s[l] N

z[l] ,2nM

This result is plausible as the mean value is as expected and the variance decreases

linearly with an increasing number of samples.

The Signal Power

In a next step, the signal power3 is calculated:

Ps = Rss[0]

=15

i=0

|s[i]|2

=15

i=0 s[i] (s[i])

In those formulas, s[i] denotes the complex conjugate and Rss denotes the autocorrela-

tion function of the signal s[i].

The Signal Power as a Random Variable

The mean and the variance of this signal are calculated. In order to do this, the

following formulas for the noncentered chi-square distribution (random variable Z) and

the expectation of a random variables are used:

Xi N(, 2i )

Z =k1i=0

Xii

23It has to be noted that the power of s is only equal to the signal power for the limes M . The

algorithm assumes that the signal power is equal to the power of s for all M > 1. This is justified bythe fact that at the end the SNR is estimated and not calculated.


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z =k1i=0

ii

2

mean(Z) = k + z

2z = var(Z) = 2 (k + 2z)E[a Xn] = a E[Xn]

var(a X) = a2 var(X)

Out of those formulas it can be seen that the power of s is noncentered chi-square

distributed. This can be written the following way:

Ps =

2

nM

15i=0

|s[i]|nM

2

:=Z

Z =15

i=0

z[i] M

n

2

=M

2n

15i=0

|z[i]|2

mean(Z) = 16 +M

2n

15

i=0 |

z[i]|2

One could argue, that this is not true, as |z[i]| is not Gaussian distributed. But this doesnot matter as the square is taken anyway. The following property holds:

|z2| = |z|2

The mean signal power is then written as:

mean(Ps) = 2

nM mean(Z)

=2nM

16 +

M

2n

15i=0

|z[i]|2

=16 2n

M+

15i=0

|z[i]|2


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This result makes sense, as it is exactly the signal power for M (many samples)or for n 0 (no noise). Next, the variance is calculated:

var(Ps) = 4n

M2 var(Z)

=4n

M2 2

16 +

2M

2n

15i=0

|z[i]|2

=32 4n

M2+

4 2nM

15i=0

|z[i]|2

As before, the variance is zero as expected for the cases M (many samples) or forn

0 (no noise). It is slightly confusing that the signal power has an influence on the

variance of the signal power. The following example helps to clarify the situation. It is

assumed that the noise power is in the range [1, 1] (not AWGN anymore). If the signalamplitude is equal to 1, then the resulting signal power is distributed in the range [0, 4].

If the signal amplitude is assumed to be 3, then the resulting signal power is distributed

in the range [4, 16]. It is therefore obvious that a higher average signal power leads to a

higher variance in the total signal power.

The Signal Plus Noise Power

In the next step, the total power is calculated.

Py = Ryy[0]

=M161

i=0

|y[i]|2

=M161

i=0

y[i] (y[i])


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The Signal Plus Noise Power as a Random Variable

As before, the total power is noncentered chi-square distributed (with the same argu-

mentation for |y[i]| as before):

Py = 2n

M161i=0

|y[i]|22n

:=Z

Z =M161

i=0

|z[mod16(i)]|n

2

=M2n

15i=0

|z[i]|2

mean(Z) = 16 M + M2n

15i=0

|z[i]|2

var(Z) = 2

16 M + 2 M2n

15i=0

|z[i]|2

This leads to the following mean power value:

mean(Py) = 2n mean(Z)

= M(162n +15

i=0

|z[i]|2)

This is the expected result, as it is the sum of the signal power and the noise power.

The variance can be calculated as:

var(Py) = 4n

var(Z)

= M 2n

32 2n + 4 15

i=0

|z[i]|2

As expected, the variance goes to zero for n 0 (no noise). It is slightly confusingto have a factor ofM in front of the variance term. But again, an example shows the

reason: Assume that the noise is in the interval [1, 1]. The signal amplitude is assumedto be 1. If only one sample is taken, the signal power is in the region [0, 4]. If two

samples are taken, the total signal power is in the region [0, 8] = 2 [0, 4].


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z =M161

i=0

|M1M

n[k]|2(M1)2n

M2

= (M 1)2n

M

16

1

i=0

|n[k]|2

E[|n2|]=2n= M 16 (M 1)mean(Z) = M2 16

var(Z) = 2z = 2 16 M(2M 1)

This leads to the following mean noise power value:

mean(n) = 16 (M 1) 2n

For M , this results in a mean value of2n per sample as expected. The variancecan be calculated as:

var(n) = 32 4n (M 1)2 (2M 1)

M3

Summary of the Mean Power Terms Normalized Per Sample

As an overview, the mean values of the different power terms are presented here -

averaged per sample:

mean(Py) = 2n +

1

16

15i=0

|z[i]|2

mean(Ps) =2nM

+1

16

15i=0

|z[i]|2

mean(Pn) = 2

n

M

1

M

Those results indicate that the following property is true:

Pn = Py Ps

The property cannot easily be proven. Numerical examples strongly indicate that the

property holds - and the mean values indicate it too. This property is important as it is

therefore needless to calculate the estimated noise signal and hardware costs can be


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saved. It also makes sense out of a physical point of view: The total power is the power

of the signal plus the power of the noise. So if from this total power the signal power is

subtracted, the remaining power is the noise power.

Calculation of the SNR

The last step is to estimate the SNR. This is done in the following way:

SN R :=Ps M

Py Ps M

It is interesting to see what the average SNR looks like:

mean SNR =M mean(Ps)

mean(Py) M mean(Ps)

= 1 +M

162n 15i=0 |z[i]|2M 1

=1 + M SN Rtrue

M 1= E[SN R]

It has to be noted that this result is not equal to the expectation of the SNR, as the

following equation is generally not true:

A, B : arbitrary random variables

E

A

B A

= E[A]E[B] E[A]

It is not easily possible to calculate the expectation value of the division of two non-

centered chi-square variables. Therefore, the approximated values of the mean SNR

were calculated for several M and various SNR, as they should show a tendency. The

results can be seen in Fig. 5.2. As expected, the results get better with a higher M and

higher channel SNR.


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0 5 10 15 20 25 305

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

SNR channel dB

SNR

SNR_

hat[dB]

Figure 5.2: This figure shows the difference between the expected SNR and the meancalculated SNR. The lowest curve is for M = 2. Each higher curve increases the valueofM by one - the highest curve is for M = 100.


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5.2.3 Simulation of the Proposed Algorithm

The proposed algorithm was tested using the simulation environment4. No nonidealities

were considered in this run. The noise was purely AWGN. The results of the simulation

are shown in Fig. 5.3. It can be seen that the algorithm performs near the optimum

for M = 9. The result of the SNR estimation is independent of the channel estimator,

whereas the BER depends on the estimated channel matrix!

5.2.4 The Influence of the Number of Samples

In a next step, the influence of the number of available subsignals M was investigated5.

Fig. 5.2 together with Fig. 4.4 suggest that the influence of the number of subsignals

should be rather small - at least for M > 4. The results can be seen in Fig. 5.4 and

Fig. 5.5.

5.2.5 The Mean Value of the Estimated SNR

As mentioned before, Fig. 5.2 only shows an approximation of the estimated SNR. The

exact curves were calculated using the simulation environment6. The results can be seen

in Fig. 5.6. Qualitatively, the curves look the same which proves that the approximation

made is quite accurate. The most obvious difference is the offset difference of around

one decibel that can be seen by comparing the two figures.

4SNR range: 0-30 [dB](step: 1 [dB]) number of sweeps: 20000 (seed=0-9)channel model: TGn C transmitting antennas: 4 receiving antennas: 4 number of tones: 64channel estimator: FDMLE demapper: MMSE modulation: QPSK




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0 5 10 15 20 25 3010

3

102

101

100

SNR channel dB

BER

proposed algorithm (M=9)

const 50dB

channel SNR

Figure 5.3: This figure shows the simulation results that were obtained using theproposed algorithm with M = 9. The simulated BER is close to the best possible BERand is as discussed already better than taking the exact channel SNR.


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0 5 10 15 20 25 3010

3

102

101

100

SNR channel dB

BER

channel SNR

constant 50dB

proposed algorith (M=9)








Figure 5.4: This figure shows the simulation results that were obtained using theproposed algorithm with different M. As expected, the performance is better for highM.


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22 22.5 23 23.5 24 24.5 25

102

SNR channel dB

BER

channel SNR

constant 50dB









Figure 5.5: This figure shows the same results as Fig. 5.4. It can be seen that the BER is

near the optimum for M > 4 and even the results with smaller M are still acceptable(losing less than 1dB in the worst case M = 2).


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0 5 10 15 20 25 305

4

3

2

1

0

1

SNR channel dB

mean(SNR

SNR_

hat)[dB]









Figure 5.6: This figure shows the simulated mean SNR values of the algorithm forseveral M.

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