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A Study on the Tone-Reservation Technique for Peak-to-Average Power

Ratio Reduction in OFDM Systems

Umer Ijaz Butt

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A Study on the Tone-Reservation Technique for Peak-to-Average Power Ratio Reduction in OFDM Systems

Copyright © 2008 Umer Ijaz Butt

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ISBN-10: 1-59942-360-X ISBN-13: 978-1-59942-360-9

A STUDY ON THE TONE-RESERVATION TECHNIQUE FOR PAPR REDUCTION IN OFDM SYSTEMS

School of Electrical & Electronics Engineering - University of Manchester

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Table of Contents

LIST OF FIGURES ............................................................................................................. 5

LIST OF TABLES ............................................................................................................... 7

GLOSSARY .......................................................................................................................... 8

LIST OF SYMBOLS ........................................................................................................... 9

ABSTRACT ........................................................................................................................ 10

DECLARATION ................................................................................................................ 11

COPYRIGHT STATEMENT ........................................................................................... 11

DEDICATION .................................................................................................................... 12

ACKNOWLEDGEMENT ................................................................................................. 13

THE AUTHOR ................................................................................................................... 14

CHAPTER 1 INTRODUCTION ................................................................................... 15 1.1 Background ................................................................................................................ 15 1.2 Problem Statement and Proposed System .................................................................. 16 1.3 Aim of the Project ...................................................................................................... 16 1.4 Report Structure ......................................................................................................... 17

CHAPTER 2 LITERATURE REVIEW ....................................................................... 18

2.1 Multi Carrier Modulation ........................................................................................... 18 2.2 Orthogonal Frequency Division Multiplexing (OFDM) ............................................ 19 2.3 Peak to Average Power Ratio (PAPR) ....................................................................... 20

2.3.1 Mathematical Model ........................................................................................... 21 2.3.2 Calculation of PAPR ........................................................................................... 22 2.3.3 Complementary Cumulative Distribution function (CCDF) .............................. 22

2.3.3.1 Mathematical Explanation ........................................................................... 22 2.3.3.2 CCDF Plotting: ............................................................................................ 23 2.3.3.3 Flow Chart [CCDF] ..................................................................................... 25 2.3.3.4 Results and Analysis .................................................................................... 26

CHAPTER 3 OVERVIEW OF PAPR REDUCTION TECHNIQUES ..................... 28

3.1 PAPR Reduction Techniques ..................................................................................... 28 3.1.1 Amplitude Clipping and Filtering ....................................................................... 29 3.1.2 Coding ................................................................................................................. 29 3.1.3 Partial Transmit Sequence (PTS) ........................................................................ 30 3.1.4 Selected Mapping Technique (SLM) .................................................................. 30 3.1.5 The Interleaving Technique ................................................................................ 30 3.1.6 The Tone Reservation Technique ....................................................................... 30 3.1.7 The Tone Injection Technique (TI) ..................................................................... 31 3.1.8 Active Constellation Extension (ACE) ............................................................... 31

3.2 Criteria to select the Appropriate PAPR Reduction Technique ................................. 31 CHAPTER 4 TONE RESERVATION ......................................................................... 32

4.1 Overview .................................................................................................................... 32



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4.2 Detail .......................................................................................................................... 32 4.3 Features ...................................................................................................................... 33 4.4 Mathematical Model .................................................................................................. 34 4.5 Problem Formulation ................................................................................................. 36

CHAPTER 5 DESIGN, EXPERIMENTAION AND RESULTS ............................... 39

5.1 Choosing the PRT set ................................................................................................. 39 5.1.1 Contiguous Reserved Tones ................................................................................ 39 5.1.2 Equally spaced tones ........................................................................................... 39 5.1.3 Random Set optimization method ....................................................................... 40

5.2 Quantity of Reserved Tones ....................................................................................... 41 5.3 Basic Tone Reservation Scheme (Algorithm #1)....................................................... 41

5.3.1 Description .......................................................................................................... 41 5.3.2 Mathematical model ............................................................................................ 41 5.3.3 How It Works ...................................................................................................... 42 5.3.4 Flowchart [Algorithm 1] ..................................................................................... 43 5.3.5 Results & Analysis .............................................................................................. 43 5.3.6 Complexity .......................................................................................................... 45 5.3.7 Conclusion .......................................................................................................... 46

5.4 SCR Gradient Tone Reservation Iterative algorithm (Algorithm #2) ........................ 47 5.4.1 Description .......................................................................................................... 47 5.4.2 Mathematical model ............................................................................................ 47 5.4.3 How It Works ...................................................................................................... 49

5.4.3.1 Initialization ................................................................................................. 49 5.4.3.2 Run time ....................................................................................................... 49

5.4.4 Flowchart [Algorithm 2] ..................................................................................... 51 5.4.5 Results & Analysis .............................................................................................. 52 5.4.6 Complexity .......................................................................................................... 54 5.4.7 Conclusion .......................................................................................................... 56

5.5 Adaptive Scaling Tone Reservation (Algorithm #3) ................................................. 56 5.5.1 Description .......................................................................................................... 56 5.5.2 Mathematical model ............................................................................................ 57 5.5.3 How It Works ...................................................................................................... 58 5.5.4 Flowchart [Algorithm 3] ..................................................................................... 59 5.5.5 Results & Analysis .............................................................................................. 60 5.5.6 Complexity .......................................................................................................... 61 5.5.7 Conclusion .......................................................................................................... 62

5.6 Gaussian Pulse Based Tone Reservation (Algorithm #4) .......................................... 63 5.6.1 Description .......................................................................................................... 63 5.6.2 Mathematical model ............................................................................................ 63 5.6.3 How It Works ...................................................................................................... 64 5.6.4 Flowchart [Algorithm 4] ..................................................................................... 66 5.6.5 Results & Analysis .............................................................................................. 67 5.6.6 Complexity .......................................................................................................... 68 5.6.7 Conclusion .......................................................................................................... 69

5.7 An Active Set Approach via Tone Reservation (Algorithm #5) ................................ 70 5.7.1 Description .......................................................................................................... 70 5.7.2 Mathematical model ............................................................................................ 70

5.7.2.1 Decent Direction .......................................................................................... 70 5.7.2.2 Next Peak for the Active Set ........................................................................ 71



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5.7.3 How It Works ...................................................................................................... 72 5.7.4 Flowchart [Algorithm 5] ..................................................................................... 73 5.7.5 Results & Analysis .............................................................................................. 74 5.7.6 Complexity .......................................................................................................... 75 5.7.7 Conclusion .......................................................................................................... 76

CHAPTER 6 PERFORMANCE COMPARISON ....................................................... 77 CHAPTER 7 CONCLUSION AND SUGGESTIONS ................................................. 82

7.1 Conclusion ................................................................................................................. 82 7.2 Suggestions for Future Work ..................................................................................... 82

REFERENCES ................................................................................................................... 83 APPENDIX ......................................................................................................................... 87

Feasibility Study Report………………………………………………………… 89



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List of Figures

Figure-‎2-1: A Simple Multicarrier Transmitter ................................................................... 19

Figure ‎2-2: OFDM signals ................................................................................................... 19

Figure ‎2-3: Basic OFDM Block ........................................................................................... 20

Figure ‎2-4: High Peaks in the OFDM Signal with N=64 .................................................... 21

Figure ‎2-5: View of different stages of a CCDF block ........................................................ 23

Figure ‎2-6: Complementary Cumulative Sum of CCDF block data .................................... 24

Figure ‎2-7: Probability Calculation of CCDF block ............................................................ 24

Figure ‎2-8: PAR CCDF for N = 1024 .................................................................................. 26

Figure ‎2-9: PAR CCDF for N = 1024 & Oversampling L=1 & L=4................................... 26

Figure ‎2-10: PAR CCDF for N = 128,256,512,1024 & Oversampling L=4 ....................... 27

Figure ‎4-1: Basic Idea of Tone Reservation ........................................................................ 33

Figure ‎4-2: Tone Reservation Scheme block diagram ......................................................... 35

Figure ‎5-1: Algorithm 1 Block Diagram .............................................................................. 42

Figure ‎5-2: PAPR CCDF Plot with N=512, R=26(5%), L=1, contiguous tone method ...... 44

Figure ‎5-3: Performance Comparison of Algo1 with different Reserved Carriers .............. 44

Figure ‎5-4: Performance Comparison of Algo1 with Contiguous and RSO method .......... 45

Figure ‎5-5: Basic Operations allotment in Algorithm # 1.................................................... 45

Figure ‎5-6: Complexity Analysis of Algorithm # 1 on different Iterations ......................... 46

Figure ‎5-7: PAPR CCDF Plot with N=512, R=26(5%), L=4, RSO method, A=0.75 ......... 52

Figure ‎5-8: PAPR CCDF Plot with N=512, R=26(5%), L=4, RSO, A=0.85 ...................... 52

Figure ‎5-9: PAPR CCDF Plot with N=512, R=26(5%), L=4, RSO method, A=0.85 ......... 53

Figure ‎5-10 Performance Comparison of Algo2 with Contiguous and RSO method.......... 53

Figure ‎5-11: Basic Operations allotment in the Initialization Section of Algorithm # 2 ..... 54

Figure ‎5-12: Basic Operations allotment in the Running Section of Algorithm #2 ............ 55

Figure ‎5-13: Complexity Analysis of Algorithm # 2 on different Iterations ....................... 55

Figure ‎5-14: PAPR CCDF Plot with N=512, R=26(5%), L=4, RSO .................................. 60

Figure ‎5-15: Performance Comparison of Algo3 with different Reserved Carriers ............ 60

Figure ‎5-16 Performance Comparison of Algo3 with Contiguous and RSO method.......... 61


Figure ‎5-18: Basic Operations allotment in the Running Section of Algorithm # 3 ........... 62


Figure ‎5-20: Gaussian Pulse Generated on R=16 and L=4 .................................................. 64



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Figure ‎5-21: Block Diagram of Gaussian Pulse Based Tone Reservation ......................... 64

Figure ‎5-22: PAPR CCDF Plot with N=256, R=16, L=4, Contiguous Tone method ......... 67






Figure ‎5-28: PAPR CCDF Plot , N=512,D=246,R=265,L=4,Contiguous Tone method .... 74

Figure ‎5-29: PAPR CCDF Plot , N=512,D=246,R=265,L=4,Contiguous Tone method .... 74




Figure ‎6-1 Performance Comparion between algo#1 and algo#2 ........................................ 77

Figure ‎6-2 Performance Comparion between algo#3 and algo#2 ........................................ 78

Figure ‎6-3 Performance Comparion between algo#4, algo2 and algo#3 ............................. 79

Figure ‎6-4: PAPR CCDF Plot of Active Set Approach ...................................................... 80



7

List of Tables

Table ‎6-1: Complexity Comparison of All the 5 algorithms on the basis of 3 iterations .... 80 Table 6-2: Complexity of All the 5 algorithms with best results ......................................... 81



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GLOSSARY ACE Active Constellation Extension

AS TR Adaptive Scaling Tone Reservation

BPSK Binary Phase Shift Keying

CCDF Complementary Cumulative Distribution function

CDF Cumulative Distributed Function

DAB Digital Audio Broadcasting

DMT Discrete Multi-tone Modulation

DVB Digital Video Broadcasting

FFT Fast Fourier Transform

IDFT Inverse Discrete Fourier Transform

IEEE Institute of Electrical and Electronic Engineering

IFFT Inverse Fast Fourier Transform

ISI Inter Symbol Interference

MCM Multi Carrier Modulation

OFDM Orthogonal Frequency Division Multiplexing

PAPR Peak to Average Power Ratio

PCS Peak Cancellation Signal

PDF Probability Density Function

PRC Peak Reduction Carrier

PRT Peak Reduction Tone

PTS Partial Transmit Sequence

QAM Quadrature Amplitude Modulation

QPSK Quadrature Phase Shift Keying

RSO Random Set Optimization

RF Radio Frequency

SCR Signal to Clipping Noise Power Ratio

SCM Single Carrier Modulation

SLM Selected Mapping

SNR Signal to Noise Ratio

TI Tone Injection

TR Tone Reservation

WLAN Wireless Local Area Networks



9

LIST OF SYMBOLS A Threshold Level

D Number of Carriers for Data

L Over Sampling factor

N Total number of Sub-Carriers

R Number of Reserved Carriers



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ABSTRACT

Multicarrier Modulation (MCM) has significant features over Single Carrier

Modulation (SCM). Orthogonal Frequency Division Multiplexing (OFDM) is a special

form of Multicarrier modulation with its key feature of Orthogonality. Along with the

several advantages of MCM or specifically OFDM, it has some drawbacks too in which

High Peak to Average Power ratio is the dominant one. There are different techniques

available to lower that PAPR. Every technique has its own benefit as well as shortcoming.

So each technique is suitable for different circumstances.

Tone Reservation is one of them in which a small set of subcarriers are reserved for

Peak Cancellation signal. This PCS helps to lower the PAPR of the whole signal. PAPR

reduction by tone reservation depends heavily on the quantity, locations and the values of

the PCS. Several people around the globe proposed distinct ways to implement this TR

technique. Every approach has its own benefit and is appropriate in different conditions.

Some of them have good results but with high complexity, on the other hand, some of them

have much lower complexity but the results are also not so much excellent. However,

within each approach, the appropriate selection of different parameters can make a balance

between the results and the complexity. PAPR can be observed by mapping CCDF plots.

Results can vary with and without oversampling.



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DECLARATION No portion of the work referred to in the dissertation has been submitted in support of an

application for another degree or qualification of this or any other university or other

institute of learning.

COPYRIGHT STATEMENT

i. Copyright in text of this dissertation rests with the author. Copies (by any

process) either in full, or of extracts, may be made only in accordance with

instructions given by the author. Details may be obtained from the appropriate

Graduate Office. This page must form part of any such copies made. Further

copies (by any process) of copies made in accordance with such instructions

may not be made without the permission (in writing) of the author.

ii. The ownership of any intellectual property rights which may be described in

this dissertation is vested in the University of Manchester, subject to any prior

agreement to the contrary, and may not be made available for use by third

parties without the written permission of the University, which will prescribe

the terms and conditions of any such agreement.

iii. Further information on the conditions under which disclosures and exploitation

may take place is available from the Head of the School of Electrical and

Electronic Engineering.



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DEDICATION

I would like to dedicate this whole work to my beloved parents who formed my life more than

I was aware of and gave me ethics and beliefs to guide me



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ACKNOWLEDGEMENT t is Almighty & All-powerful Allah, the most humane, the most gracious who bestowed the

fellowmen with dormant & great hidden abilities. First of all, Thanks to Almighty Allah who

incited me to equip with this dissertation in a limited span of time.

All honours to my aftermost Prophet Hazrat Muhammad (P.B.U.H), who displayed the new

passages of learning knowledge, wisdom and sense for all humanity.

Thanks to my guardians who empowered me to acquire extravagant education of this level.

I am also much pleased with my respected supervisor, Dr. Emad Alsusa, who repeatedly

directed me, guided me and helped me in each and every part of this dissertation. The

completion of this dissertation was not possible without his help.

In the end, I signify the obligation to each and every person who became the source of help for

me during the whole period of this dissertation.

Umer Ijaz Butt

I



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The Author

Umer Ijaz Butt received the BS degree in Computer Engineering with majors in

Telecommunications from University of Management and Technology Lahore, Pakistan.

The core courses in the bachelors were Digital Communication, Digital Signal Processing,

and Signal & Systems. His final year dissertation in this BS was Audio Compression Using

WAVELET Technique on TMS 320 C 6711 DSP Starter Kit. He is now doing Masters in

Communication Engineering from University of Manchester, UK. The core courses studied

here included Digital communications (Introductory and Advanced), Optical

Communications, Microwave Circuit & Network Analysis and Design, Radar System,

Digital Signal Processing (Introductory and Advanced), Antennas & Propagation, Personal

and Mobile Communications, Semiconductor Materials and Devices, MMIC Components

and Technology, Finite Mathematics and Sequences-Codes and Cryptography. His final

year dissertation in MSc is based on Tone Reservation technique for Peak-to-Average

Power Ratio reduction in OFDM systems.

INTRODUCTION Chapter 1


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CHAPTER 1

INTRODUCTION

1.1 Background

uring the last few decades man has gone through novel experiences. The

unimaginable and utopian ideas have been transferred into realities. The technological and

scientific progress has discarded the centuries old absolute ideas. The ready made and

rapid growth of man‟s aspirations and needs are met with positive reactions. Every

individual in this globe aspires to go forward in the road of life. Every young man‟s

aspiration is to be at best and one and only. The world is moving fast, faster than the

imagination of a senior citizen.

Same trend is observed in the field of communication. The minimization of time

consumed for communication from one place to another place is a facility provided to the

young man of today. Now it takes seconds or part of a second to communicate from one

place to another.

The healthy competition and urge for unearthing the unknown is gaining ground.

This led to the transformation from Analog to Digital in the field of communication, which

produces a huge change. This conversion not only increases the capacity that gave

economical benefit but also gives numerous advantages like easy regeneration of digital

signals and the use of digital signal processing techniques[1]. This advancement of

digitalizing the communication world makes it a global village.

In addition to this, the transformation from wired medium to the wireless medium

takes man into unimaginable utopia, but it is reality.

However both in wired and wireless communication, the main objective is to

transmit data from source to sink in such a way that there should be least alteration in it, as

numerous impairment factors like noise, attenuation, distortion, fading, interference etc

tried to change the data during the communication process[2].

D



16

Therefore different technical steps follow, including the encoding, modulation and

multiplexing, while transmitting a message from source to sink in order to avoid these

destruction factors.

In addition to this, utilization of bandwidth is also a big issue in any

communication system. Therefore there are different ways of utilizing it to make the

communication system more efficient. Multiplexing is one of those ways through which

large amount of data is transmitted on a communication line with a limited amount of

bandwidth. However, in multiplexing, the resource allocations are fixed. There is another

way of utilizing the bandwidth and that is Multiple Access in which the resource

allocations are assigned dynamically [5].

Whereas modulation is mainly used to synchronise the signal characteristics to

channel characteristics, this helps to reduce the noise and interference [3].

1.2 Problem Statement and Proposed System

Modulation can be classified in to two categories:

Single Carrier Modulation (SCM)

Multi Carrier Modulation (MCM)

Orthogonal Frequency Division Multiplexing (OFDM) is a type of Multicarrier

modulation. Multicarrier modulation or particularly, OFDM along with numerous

advantages, on the other hand, has some drawbacks too. High peak to Average Power

Ratio (PAPR) is one of them which is stated that when majority of sub-carriers align

themselves in phase, there will be high peaks in the signal [16].

In order to get advantage from this superb OFDM technique, the high PAPR issue

has to be resolved. Therefore several people from different places around the globe

proposed distinct techniques to overcome this problem. Tone Reservation is one of those

techniques that give remarkable results with minimum usage of the resources.

1.3 Aim of the Project

The main aim of this dissertation is to analyse this Tone Reservation technique both

theoretically and by simulation analysis. Different Tone Reservation algorithms are

discussed, simulated, analysed and compared with each other in different scenarios. Also a

brief comparison between this TR technique and other PAPR reduction techniques are

taken into account.



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1.4 Report Structure

The remaining chapters of this thesis discussed the following:

CHAPTER 2: A brief introduction to multicarrier modulation & Orthogonal Frequency

Division Multiplexing (OFDM) and a thorough description of Peak to

Average Power Ratio (PAPR) in OFDM Systems is discussed in this

chapter.

CHAPTER 3: An overview of all the PAPR reduction techniques have been taken into

account and a specific criteria has been discussed in order to select the

appropriate technique.

CHAPTER 4: A detail explanation including the mathematical clarification of Tone

Reservation has been discussed in this chapter.

CHAPTER 5: Detail of all those Tone Reservation algorithms which are taken into

account during the dissertation period are given in this chapter. A complete

introduction, working, mathematical model, flowchart, results, Complexity

and Conclusion of each algorithm has examined separately.

CHAPTER 6: An overall analysis on all the algorithms examined in the previous chapter

is discussed here with the help of several results and complexity issues.

CHAPTER 7: In this chapter, Summary and Conclusion of the whole work is discussed.

Also the future work recommended about this dissertation is given.

LITERATURE REVIEW Chapter 2


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CHAPTER 2

LITERATURE REVIEW

2.1 Multi Carrier Modulation

odulation can be classified into Single Carrier modulation (SCM) and

Multicarrier Modulation (MCM). Single carrier modulation is one in which data is

transmitted in serial by modulating one single carrier at a time [24].

On the other hand, in multicarrier modulation, the whole bandwidth is divided into

different sections called sub bands or sub carriers. Also the data stream is divided into

blocks of data symbols. These data symbols are then transmitted in parallel by modulating

onto the subcarriers. [6-9] [24].

Multicarrier modulation (MCM) was first used in military HF radio links in late

1950s [6] [8]. “The main objective in multicarrier transmission is to alter the serial high

data stream into multiple parallel low rate sub streams.” [13]. Due to its significant features

like the strength to multipath and use of Fast Fourier Transform (FFT) & its inverse, it

become more important and used numerously as compared to SCM [23].

Also, the difference between SCM and MCM can be seen in such a way that on the

same parameters, multicarrier modulation is capable of transmitting more data as compared

to single carrier modulation [12].

Multicarrier modulation is use in wired communication in the form of discrete

multi-tone modulation (DMT) and in wireless communication in the form of orthogonal

M



19

frequency division multiplexing (OFDM). This special form of multicarrier modulation

(OFDM) was introduced in U.S in 1970[8].

A simple multicarrier transmitter is shown in the figure below [11] :

Figure 2-1: A Simple Multicarrier Transmitter

Due to its significant supremacy over single carrier modulation, it is used in

numerous applications including digital transmission like digital audio broadcasting(DAB),

digital video broadcasting(DVB), wireless local area networks(WLAN), wireless local

loop(WLL) etc [8][4]. It is also found in IEEE standards 802.11a, 802.15.3 and 802.16a

[23].

2.2 Orthogonal Frequency Division Multiplexing (OFDM)

“OFDM was first introduced more than three decades ago” [14]. “It follows the rule

of multicarrier transmission and transmits a single stream over a number of low rate

subcarriers” [15].By definition, “OFDM is a sum of N independent signals modulated onto

sub channel of equal bandwidth with frequency separation of 1/T between adjacent

subcarriers, where T is the time duration of OFDM symbol” [34].

OFDM can be considered as a modulation technique or a multiplexing technique.

The main feature of OFDM that differentiate it from the conventional frequency division

multiplexing is the factor of Orthogonality. Means, the subcarriers are orthogonal to each

other, and this orthogonality is attain by placing the subcarriers in such a way that the

spacing between these sub carriers is equal to the reciprocal of the useful symbol period. In

this way, the subcarriers will not disturb each other and there will be no interference [19].

This can be shown in the figure illustrated below [22]:

Figure 2-2: OFDM signals



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OFDM is implemented by the Fast Fourier Transform (FFT) and its Inverse (IFFT).

Basic OFDM system works in such a way that, “it takes a data stream, divide it into N

parallel data streams, each at a rate of 1/N of the original rate. Each stream is then mapped

to a tone at a unique frequency, then all of them are combined together to sent to Inverse

Fast Fourier Transform (IFFT)”[16]. The IFFT block converts the frequency domain data

into corresponding time domain samples. The output of the IFFT is then entered into

Parallel to serial converter that converts the parallel data in to serial to generate the time

domain signals sequentially. The output is then send to the channel where the factors, like

noise, interacts. On the receiver end, the same process repeats in a reverse order. The block

diagram of a basic OFDM system is given below [17]:

Figure 2-3: Basic OFDM Block

OFDM has several advantages that dominate it from other techniques. Like, “it is

good to deal with multipath, By OFDM, single-frequency networks can be possible, which

is used in broadcasting applications”[15], “it provides better coverage and penetration, it

require less operation and installation costs, It has ultra high spectral efficiency, It has high

resistance to multi path and it minimized inter symbol interference(ISI)”[16].

2.3 Peak to Average Power Ratio (PAPR)

In addition to numerous significant advantages of multicarrier modulation and

particularly OFDM, it has a severe negative aspect too that it suffers from a problem of

High Peak to Average Power Ratio.

As in OFDM, each data symbol is sent on different carrier frequency to reduce the

ISI (Inter Symbol Interference) and also the OFDM is more spectrum efficient means it

contains more subcarriers, this is however led to the problem of Peak to Average Power

Ratio (PAPR).Or simply, it can say that there will be peak in the signal when majority of

the sub-carriers align themselves in phase [16].



21

It can be explained by Central limit theorem of probability theory, which says that

when there is a linear combination of variables in large quantity, this is considered or

approximated under Gaussian, and here in OFDM system where there are large number of

variables, the OFDM signal is considered by Gaussian distributed signal [19].

The figure given below clearly shows that high peaks:

Figure 2-4: High Peaks in the OFDM Signal with N=64

This figure shows that two of the samples have high peaks, and in order to transmit

a signal with these high peaks, a high signal span is required from the power amplifier. But

such types of power amplifiers are costly [19].

This high PAPR can make a cause of certain drawbacks, for example: complexity

will be increased for analog to digital and digital to analog convertors and also the

efficiency of the RF power amplifier will be degraded [15].

One way to reduce this peak power is to reduce the average power of the signal

which will defiantly reduce the peaks but by doing this, the Signal to noise ratio of the

signal at the receiver will also reduce therefore the overall performance will be degraded.

So on one side, there is a facility to have more and more carriers in an OFDM signal as

compared to FDM but on the same time it increases the PAPR.

2.3.1 Mathematical Model

Let‟s have a look on the mathematical explanation of the PAPR. The input bit

stream given to OFDM system is first mapped on a selected modulation technique Eg:

QAM,QPSK, BPSK etc. From this, a complex OFDM vector of symbols is obtained. Let‟s

this complex valued vector is a data block „X‟ consisting of complex data symbolsX0 X1 X2

…. XN-1 [18]. The complex base band representation of this multicarrier signal can be

determined by the equation 2.1 [20].



22

10;1)( /21

0

NkXN

nx e NknjN

kk

Equation (2.1)

“In the above equation, „N‟ tells the number of subcarriers in the OFDM system. Xk

is the modulated data carried by the kth sub-carrier” [20].This equation is actually

performing the Inverse Fast Fourier Transform (IFFT) through which, a time domain

symbol is obtained. It can be represented as:

x = IFFT(X)

Therefore, „x‟ represents a vector of N elements (0 to N-1) of time domain signals

and x(n) is it‟s nth element. Whereas „X‟ is a set of modulated data carried by „N‟ different

subcarriers [20].

2.3.2 Calculation of PAPR

Now the Peak to Average power ratio of this transmitted signal can be calculated

from equation given below [18] [30]:

])([

)(max10)( 2

2

1010log

nxE

nxxPAPR Nn Equation (2.2)

This is defined as the ratio between the square of the maximum power and square

of the average power of the signal. However, in order to observe the PAPR performance of

a signal, the Cumulative Distributed Function (CDF) or the Complementary Cumulative

Distribution function (CCDF) is used. Through the CCDF, it can be observed that the

PAPR of the signal exceeded a certain value.

2.3.3 Complementary Cumulative Distribution function (CCDF)

In the modern communications world, CCDF measurements prove as one of the

precious tool. The CCDF plots offer a comprehensive analysis of signal power peaks. “It is

a statistical technique that provides the amount of time, a signal spends above any given

power level” [21].

However by using these plots, a probability can be seen that a signal data block

exceeds a given threshold. These CCDF plots can be used to analyse the PAPR

performance of the signals [18].

2.3.3.1 Mathematical Explanation: In order to calculate the CCDF of a given data, the

following steps should be followed:



23

Cal. The Probability density function(PDF) of the data

Take the integral of the PDF to get the CDF (Cumulative Distributed Function)

Subtract the CDF from „1‟ to get the CCDF(as: CCDF=1-CDF)

Or equivalently, it can summarize as:

CDF = ∫PDF & CCDF = [1 – CDF]

Mathematically, it can be explained as follows [18]:

1

( ) 1 ( )

( )

1 (1 exp( ))

N

N

P PAPR z P PAPR z

F z

z

Equation (2.3)

Where: F(z) represents the CDF.

2.3.3.2 CCDF Plotting: The CCDF plot can easily be simulated in MATLAB by

following the procedure given below:

First of all, total number of subcarriers „N‟ of a multicarrier signal is selected. The

signal is then oversampled by factor „L‟ (though CCDF can also plotted of non-

oversampled signal). After this, It is transformed from frequency domain to time domain

and the PAPR of the signal is calculated by equation 2.2.

In parallel to this, a block having „z‟ number of sections is created and a range for

each section is set. All the sections contain null in initial. The calculated PAPR is checked

and that section value is incremented by 1 whose range occupies that PAPR value. A

number of multicarrier symbols are observed and corresponding section value of CCDF

block incremented. This can be further explained by the figure below:

Figure 2-5: View of different stages of a CCDF block

A Study on the Tone-Reservation Technique for Peak-to ... · A Study on the Tone-Reservation...

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