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

Transcription

1 A Study on the Tone-Reservation Technique for Peak-to-Average Power Ratio Reduction in OFDM Systems Umer Ijaz Butt DISSERTATION.COM Boca Raton

2 A Study on the Tone-Reservation Technique for Peak-to-Average Power Ratio Reduction in OFDM Systems Copyright 2008 Umer Ijaz Butt All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without written permission from the publisher. Dissertation.com Boca Raton, Florida USA 2010 ISBN-10: X ISBN-13:

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4 A STUDY ON THE TONE-RESERVATION TECHNIQUE FOR PAPR REDUCTION IN OFDM SYSTEMS Table of Contents LIST OF FIGURES... 5 LIST OF TABLES... 7 GLOSSARY... 8 LIST OF SYMBOLS... 9 ABSTRACT DECLARATION COPYRIGHT STATEMENT DEDICATION ACKNOWLEDGEMENT THE AUTHOR CHAPTER 1 INTRODUCTION Background Problem Statement and Proposed System Aim of the Project Report Structure CHAPTER 2 LITERATURE REVIEW Multi Carrier Modulation Orthogonal Frequency Division Multiplexing (OFDM) Peak to Average Power Ratio (PAPR) Mathematical Model Calculation of PAPR Complementary Cumulative Distribution function (CCDF) Mathematical Explanation CCDF Plotting: Flow Chart [CCDF] Results and Analysis CHAPTER 3 OVERVIEW OF PAPR REDUCTION TECHNIQUES PAPR Reduction Techniques Amplitude Clipping and Filtering Coding Partial Transmit Sequence (PTS) Selected Mapping Technique (SLM) The Interleaving Technique The Tone Reservation Technique The Tone Injection Technique (TI) Active Constellation Extension (ACE) Criteria to select the Appropriate PAPR Reduction Technique CHAPTER 4 TONE RESERVATION Overview

5 A STUDY ON THE TONE-RESERVATION TECHNIQUE FOR PAPR REDUCTION IN OFDM SYSTEMS 4.2 Detail Features Mathematical Model Problem Formulation CHAPTER 5 DESIGN, EXPERIMENTAION AND RESULTS Choosing the PRT set Contiguous Reserved Tones Equally spaced tones Random Set optimization method Quantity of Reserved Tones Basic Tone Reservation Scheme (Algorithm #1) Description Mathematical model How It Works Flowchart [Algorithm 1] Results & Analysis Complexity Conclusion SCR Gradient Tone Reservation Iterative algorithm (Algorithm #2) Description Mathematical model How It Works Initialization Run time Flowchart [Algorithm 2] Results & Analysis Complexity Conclusion Adaptive Scaling Tone Reservation (Algorithm #3) Description Mathematical model How It Works Flowchart [Algorithm 3] Results & Analysis Complexity Conclusion Gaussian Pulse Based Tone Reservation (Algorithm #4) Description Mathematical model How It Works Flowchart [Algorithm 4] Results & Analysis Complexity Conclusion An Active Set Approach via Tone Reservation (Algorithm #5) Description Mathematical model Decent Direction Next Peak for the Active Set

6 A STUDY ON THE TONE-RESERVATION TECHNIQUE FOR PAPR REDUCTION IN OFDM SYSTEMS How It Works Flowchart [Algorithm 5] Results & Analysis Complexity Conclusion...76 CHAPTER 6 PERFORMANCE COMPARISON...77 CHAPTER 7 CONCLUSION AND SUGGESTIONS Conclusion Suggestions for Future Work...82 REFERENCES...83 APPENDIX...87 Feasibility Study Report 89 4

7 A STUDY ON THE TONE-RESERVATION TECHNIQUE FOR PAPR REDUCTION IN OFDM SYSTEMS List of Figures Figure- 2-1: A Simple Multicarrier Transmitter Figure 2-2: OFDM signals Figure 2-3: Basic OFDM Block Figure 2-4: High Peaks in the OFDM Signal with N= Figure 2-5: View of different stages of a CCDF block Figure 2-6: Complementary Cumulative Sum of CCDF block data Figure 2-7: Probability Calculation of CCDF block Figure 2-8: PAR CCDF for N = Figure 2-9: PAR CCDF for N = 1024 & Oversampling L=1 & L= Figure 2-10: PAR CCDF for N = 128,256,512,1024 & Oversampling L= Figure 4-1: Basic Idea of Tone Reservation Figure 4-2: Tone Reservation Scheme block diagram Figure 5-1: Algorithm 1 Block Diagram Figure 5-2: PAPR CCDF Plot with N=512, R=26(5%), L=1, contiguous tone method Figure 5-3: Performance Comparison of Algo1 with different Reserved Carriers Figure 5-4: Performance Comparison of Algo1 with Contiguous and RSO method Figure 5-5: Basic Operations allotment in Algorithm # Figure 5-6: Complexity Analysis of Algorithm # 1 on different Iterations Figure 5-7: PAPR CCDF Plot with N=512, R=26(5%), L=4, RSO method, A= Figure 5-8: PAPR CCDF Plot with N=512, R=26(5%), L=4, RSO, A= Figure 5-9: PAPR CCDF Plot with N=512, R=26(5%), L=4, RSO method, A= Figure 5-10 Performance Comparison of Algo2 with Contiguous and RSO method Figure 5-11: Basic Operations allotment in the Initialization Section of Algorithm # Figure 5-12: Basic Operations allotment in the Running Section of Algorithm # Figure 5-13: Complexity Analysis of Algorithm # 2 on different Iterations Figure 5-14: PAPR CCDF Plot with N=512, R=26(5%), L=4, RSO Figure 5-15: Performance Comparison of Algo3 with different Reserved Carriers Figure 5-16 Performance Comparison of Algo3 with Contiguous and RSO method Figure 5-17: Basic Operations allotment in the Initialization Section of Algorithm # Figure 5-18: Basic Operations allotment in the Running Section of Algorithm # Figure 5-19: Complexity Analysis of Algorithm # 3 on different Iterations Figure 5-20: Gaussian Pulse Generated on R=16 and L=

8 A STUDY ON THE TONE-RESERVATION TECHNIQUE FOR PAPR REDUCTION IN OFDM SYSTEMS Figure 5-21: Block Diagram of Gaussian Pulse Based Tone Reservation Figure 5-22: PAPR CCDF Plot with N=256, R=16, L=4, Contiguous Tone method Figure 5-23: PAPR CCDF Plot with N=256, R=26, L=4, Contiguous Tone method Figure 5-24: PAPR CCDF Plot with N=512, R=16, L=4, Contiguous Tone method Figure 5-25: Basic Operations allotment in the Initialization Section of Algorithm # Figure 5-26: Basic Operations allotment in the Running Section of Algorithm # Figure 5-27: Complexity Analysis of Algorithm # 4 on different Iterations Figure 5-28: PAPR CCDF Plot, N=512,D=246,R=265,L=4,Contiguous Tone method Figure 5-29: PAPR CCDF Plot, N=512,D=246,R=265,L=4,Contiguous Tone method Figure 5-30: Basic Operations allotment in the Initialization Section of Algorithm # Figure 5-31: Basic Operations allotment in the Running Section of Algorithm # Figure 5-32: Complexity Analysis of Algorithm # 5 on different Iterations Figure 6-1 Performance Comparion between algo#1 and algo# Figure 6-2 Performance Comparion between algo#3 and algo# Figure 6-3 Performance Comparion between algo#4, algo2 and algo# Figure 6-4: PAPR CCDF Plot of Active Set Approach

9 A STUDY ON THE TONE-RESERVATION TECHNIQUE FOR PAPR REDUCTION IN OFDM SYSTEMS List of Tables Table 6-1: Complexity Comparison of All the 5 algorithms on the basis of 3 iterations Table 6-2: Complexity of All the 5 algorithms with best results

10 A STUDY ON THE TONE-RESERVATION TECHNIQUE FOR PAPR REDUCTION IN OFDM SYSTEMS GLOSSARY ACE Active Constellation Extension AS TR BPSK CCDF CDF DAB DMT DVB FFT IDFT IEEE IFFT ISI MCM OFDM PAPR PCS PDF PRC PRT PTS QAM QPSK RSO RF SCR SCM SLM SNR TI TR WLAN Adaptive Scaling Tone Reservation Binary Phase Shift Keying Complementary Cumulative Distribution function Cumulative Distributed Function Digital Audio Broadcasting Discrete Multi-tone Modulation Digital Video Broadcasting Fast Fourier Transform Inverse Discrete Fourier Transform Institute of Electrical and Electronic Engineering Inverse Fast Fourier Transform Inter Symbol Interference Multi Carrier Modulation Orthogonal Frequency Division Multiplexing Peak to Average Power Ratio Peak Cancellation Signal Probability Density Function Peak Reduction Carrier Peak Reduction Tone Partial Transmit Sequence Quadrature Amplitude Modulation Quadrature Phase Shift Keying Random Set Optimization Radio Frequency Signal to Clipping Noise Power Ratio Single Carrier Modulation Selected Mapping Signal to Noise Ratio Tone Injection Tone Reservation Wireless Local Area Networks 8

11 A STUDY ON THE TONE-RESERVATION TECHNIQUE FOR PAPR REDUCTION IN OFDM SYSTEMS LIST OF SYMBOLS A D L N R Threshold Level Number of Carriers for Data Over Sampling factor Total number of Sub-Carriers Number of Reserved Carriers 9

12 A STUDY ON THE TONE-RESERVATION TECHNIQUE FOR PAPR REDUCTION IN OFDM SYSTEMS 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. 10

13 A STUDY ON THE TONE-RESERVATION TECHNIQUE FOR PAPR REDUCTION IN OFDM SYSTEMS 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. 11

14 A STUDY ON THE TONE-RESERVATION TECHNIQUE FOR PAPR REDUCTION IN OFDM SYSTEMS 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 12

15 A STUDY ON THE TONE-RESERVATION TECHNIQUE FOR PAPR REDUCTION IN OFDM SYSTEMS ACKNOWLEDGEMENT I 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 13

16 A STUDY ON THE TONE-RESERVATION TECHNIQUE FOR PAPR REDUCTION IN OFDM SYSTEMS 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. 14

17 INTRODUCTION Chapter 1 CHAPTER 1 INTRODUCTION 1.1 Background During 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]. 15

18 INTRODUCTION Chapter 1 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. 16

19 INTRODUCTION Chapter 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. 17

20 LITERATURE REVIEW Chapter 2 CHAPTER 2 LITERATURE REVIEW 2.1 Multi Carrier Modulation Modulation 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 18

21 LITERATURE REVIEW Chapter 2 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 a, and a [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 19

22 LITERATURE REVIEW Chapter 2 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]. 20

23 LITERATURE REVIEW Chapter 2 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 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]. 21

24 LITERATURE REVIEW 1 x ( n) N Chapter 2 N 1 X e k 0 j 2 kn / N k ;0 k N 1 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] Calculation of PAPR Now the Peak to Average power ratio of this transmitted signal can be calculated from equation given below [18] [30]: max x(n) 2 PAPR ( x) 10 log 0 n N E[ x(n) ] 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 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] Mathematical Explanation: In order to calculate the CCDF of a given data, the following steps should be followed: 22

25 LITERATURE REVIEW Chapter 2 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]: P( PAPR z ) 1 P( PAPR z ) 1 F ( z) N 1 (1 exp( z )) Equation (2.3) N Where: F(z) represents the CDF 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 nonoversampled 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 23