Optimal Design of Companding Schemes for PAPR

Optimal Design of Companding Schemes for PAPR Reduction in OFDM Systems

Sana Mazahir NUST201362394MCEME35013F Supervisor Dr Shahzad Amin Sheikh

Department of Electrical Engineering College of Electrical and Mechanical Engineering (CEME) National University of Sciences and Technology (NUST) Rawalpindi November 2015

Optimal Design of Companding Schemes for PAPR Reduction in OFDM Systems

Sana Mazahir NUST201362394MCEME35013F

A thesis submitted in partial fulfillment of the requirements for the degree of MS Electrical Engineering

Supervisor Dr Shahzad Amin Sheikh

Supervisor’s Signature:

Department of Electrical Engineering College of Electrical and Mechanical Engineering (CEME) National University of Sciences and Technology (NUST) Rawalpindi November 2015

Declaration

I certify that this research work entitled “Optimal Design of Companding Schemes for PAPR Reduction in OFDM Systems” is my own work. The work has not been presented elsewhere for assessment. The material that has been used from other sources it has been properly acknowledged / referred.

Signature of Student Sana Mazahir NUST201362394MCEME35013F

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Language Correctness Certificate

This thesis has been read by an English expert and is free of typing, syntax, semantic, grammatical and spelling mistakes. Thesis is also according to the format given by the university.

Signature of Student Sana Mazahir NUST201362394MCEME35013F

Signature of Supervisor

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Copyright Statement

• Copyright in text of this thesis rests with the student author. Copies (by any process) either in full, or of extracts, may be made onlyin accordance with instructions given by the author and lodged in the Library of NUST College of E&ME. Details may be obtained by the Librarian. This page must form part of any such copies made. Further copies (by any process) may not be made without the permission (in writing) of the author. • The ownership of any intellectual property rights which may be described in this thesis is vested in NUST College of E&ME, 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 College of E&ME, which will prescribe the terms and conditions of any such agreement. • Further information on the conditions under which disclosures and exploitation may take place is available from the Library of NUST College of E&ME, Rawalpindi.

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Acknowledgments

I am grateful to my thesis supervisor Dr Shahzad Amin Sheikh for his guidance during my thesis work. It is due to his excellent teaching of the relevant subjects that I was able to pursue this research with confidence. In this regard, I am grateful to all the faculty members (former/present) of DEE, CEME from whom I had the opportunity to learn over the past six years. I want to express my utmost gratitude towards Dr Osman Hasan (HoD Research and Director SAVe Lab, SEECS, NUST) with whom I had the good fortune to work alongside my MS thesis work. This thesis would surely not have the quality it has now if I had not been working under his competent supervision. I am indebted to him not only for the guidance and academic grooming that he provided during my research with him in the SAVe lab, but also for the understanding, patience and generosity with which he allowed me to spend considerable time and effort on my MS thesis work. I am also grateful to my friends Nida Ishtiaq and Amina Khalid who made my time during MS studies enjoyable. My very special thanks to my parents to whom I owe everything I have today. I am grateful to them for their unwavering support and faith in me that continues to give me the strength, confidence and motivation at every step of the way.

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Dedicated to My Beloved Parents

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Abstract

Orthogonal frequency division multiplexing (OFDM) signals suffer from the problem of high peak-to-average power ratio (PAPR) that complicates the design of analog front-end of the system. Companding is a well-known technique that reduces the PAPR using a deterministic amplitude transform, with the assumption that the OFDM signal amplitude is Rayleigh distributed. In this thesis, optimization of companding schemes from following aspects is considered: symbol period/number of sub-carriers, constellation type, bit error rate (BER) and out-of-band interference (OBI). Companding transforms do not perform optimally when number of sub-carriers per OFDM symbol is comparatively smaller and/or higher order QAM is used for modulation. In order to mitigate this type of degradation, adaptive companding is proposed. In adaptive companding, the companding transforms adapts to certain pre-determined features of the input symbol during application run-time. The scheme is found to enhance the performance of the system and increase its flexibility. In case of higher order QAM based OFDM, the Rayleigh distribution is found to be insufficient to model the true amplitude characteristics of the OFDM signal because of the random nature of OFDM symbol power. The probabilistic analysis of OFDM symbol power, in relationship with amplitude distribution has been carried out. Based on this analysis, two novel schemes, namely Adjustable-parameter Companding (APC) and Adaptive Constellation Scaling (ACS), are developed to remedy the problem. In order to achieve optimum performance from companding transforms, rules or guidelines for the design of piecewise linear companding transforms are identified by relating the compander’s and decompander’s profile and parameters with the system’s performance metrics. Based on a set of criteria developed thereof, three new companding transforms are derived. Simulations are carried out to evaluate and compare the new transforms with previous works. The proposed transforms are found to outperform the existing ones in terms of BER and OBI and are also more robust to changing channel conditions. vi

Table of Contents

Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Language Correctness Certificate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Copyright Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv 1

OFDM Systems and the PAPR 1.1 OFDM Systems . . . . . . . . 1.2 Applications of OFDM . . . . 1.3 The PAPR Problem . . . . . 1.3.1 Definition of PAPR . 1.3.2 Relationship Between

Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HPA Efficiency

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An Overview of PAPR Reduction Techniques . . . . . . 2.1 Classification of PAPR Reduction Techniques . . . . . . 2.1.1 Signal Distortion Techniques . . . . . . . . . . . 2.1.2 Multiple Signaling and Probabilistic Techniques 2.1.3 Coding Techniques . . . . . . . . . . . . . . . . 2.2 Performance Considerations . . . . . . . . . . . . . . . . 2.2.1 Distribution of PAPR . . . . . . . . . . . . . . . 2.2.2 Computational Complexity . . . . . . . . . . . . 2.2.3 Average Power . . . . . . . . . . . . . . . . . . . 2.2.4 Error Performance . . . . . . . . . . . . . . . . 2.2.5 Spectral Characteristics . . . . . . . . . . . . . . 2.2.6 Bandwidth Efficiency . . . . . . . . . . . . . . . 2.2.7 Sensitivity to Errors in Side Information . . . . 2.2.8 Effects of Oversampling . . . . . . . . . . . . . . 2.2.9 Number of Sub-carriers . . . . . . . . . . . . . . 2.2.10 Signal Constellation . . . . . . . . . . . . . . . . 2.3 Scope of the Thesis . . . . . . . . . . . . . . . . . . . . .

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Companding Transforms for PAPR Reduction . . . . . . . . . . . . . . . . . . . . . . . 3.1 OFDM System with a Companding Transform . . . . . . . . . . . . . . . . . . . . . . . 3.2 Amplitude Distribution of OFDM Signal . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Literature Review . . . . . . . . . . . . . . . . . . . Some Recent Works on Companding Schemes . . . 3.4.1 Exponential Compander (EC) . . . . . . . 3.4.2 Non-linear Compander (NLC) . . . . . . . 3.4.3 Efficient Non-linear Compander (ENLC) . 3.4.4 Piecewise Exponential Compander (PEC) 3.4.5 Piecewise Linear Compander (PLC) . . . . . . . . . .

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Motivational Study and Problem Identification 4.1 Simulation Setup . . . . . . . . . . . . . . . . . 4.1.1 Effect of Number of Sub-carriers . . . . 4.1.2 Effect of Constellation Type . . . . . . 4.1.3 Applications . . . . . . . . . . . . . . . 4.2 Problem Formulation . . . . . . . . . . . . . . .

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Adaptive Companding . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Overview of Contributions . . . . . . . . . . . . . . . . . . . . 5.2 General Concept of Adaptive Companding . . . . . . . . . . . 5.3 Formal Definitions for Adaptive Companding . . . . . . . . . 5.4 Design Methodology . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Companding Transform . . . . . . . . . . . . . . . . . 5.4.2 Problem Formulation in Adaptive Compander Design Framework . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Selection of Statistical Feature . . . . . . . . . . . . . 5.4.4 Classification of OFDM Symbols . . . . . . . . . . . 5.4.5 Conditional Probability Distributions . . . . . . . . . 5.4.6 Evaluation of Compander Parameters . . . . . . . . . 5.5 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . 5.5.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . 5.5.2 With 4-QAM based OFDM . . . . . . . . . . . . . . 5.5.3 With 16-QAM based OFDM . . . . . . . . . . . . . . 5.5.4 Side Information . . . . . . . . . . . . . . . . . . . . 5.5.5 Computational Complexity . . . . . . . . . . . . . . . 5.5.6 Summary of Performance . . . . . . . . . . . . . . . .

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Companding Schemes for OFDM Systems employing Higher Order 6.1 Difference between PSK and QAM based OFDM Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Overview of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Probabilistic Modeling of Average Power and Amplitude of OFDM Symbols . . . . . . . . . . . . . . . . . . . . 6.3.1 Distribution of Average Symbol Power . . . . . . . . . . . . . 6.3.2 Distribution of OFDM Symbol Amplitude . . . . . . . . . . . 6.4 Proposed Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Classification of OFDM Symbols . . . . . . . . . . . . . . . . 6.4.2 Adjustable-parameter Companding (APC) . . . . . . . . . . . 6.4.3 Adaptive Constellation Scaling (ACS) . . . . . . . . . . . . . . 6.4.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Simulation Setup and Parameters . . . . . . . . . . . . . . . . 6.5.2 PAPR and BER Performances of APC . . . . . . . . . . . . . 6.5.3 PAPR and BER Performances of ACS . . . . . . . . . . . . . 6.5.4 Out-of-band Interference (OBI) Levels in APC and ACS Schemes . . . . . . . . . . . . . . . . . . . . . . 6.5.5 Discussions on PAPR, BER and OBI Performances . . . . . . 6.5.6 Complexity Analysis of APC and ACS . . . . . . . . . . . . . 6.5.7 Bandwidth Efficiency . . . . . . . . . . . . . . . . . . . . . . .

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Piecewise Linear Companding Transforms . . . . . . 7.1 Related Work and Motivation . . . . . . . . . . . . . 7.2 Overview of Contributions . . . . . . . . . . . . . . . 7.3 General Design Criteria . . . . . . . . . . . . . . . . 7.3.1 Constant Average Power . . . . . . . . . . . 7.3.2 Hard Clipping for Peak Power Reduction . . 7.3.3 Error Performance Optimization . . . . . . . 7.3.4 Out-of-band Interference (OBI) Reduction . 7.3.5 Partial Decompanding . . . . . . . . . . . . 7.3.6 Scalability . . . . . . . . . . . . . . . . . . . 7.4 Design Methodology . . . . . . . . . . . . . . . . . . 7.5 Proposed Companding Transforms . . . . . . . . . . 7.5.1 Companding Transform PLC-1 . . . . . . . 7.5.2 Companding Transform PLC-2 . . . . . . . 7.5.3 Companding Transform PLC-3 . . . . . . . 7.6 Performance Evaluation of PLC-1, PLC-2 and PLC-3 7.6.1 Simulation Setup . . . . . . . . . . . . . . . 7.6.2 Simulation Results . . . . . . . . . . . . . . 7.6.3 Observations and Discussions . . . . . . . . 7.7 Performance Comparison with Previous Works . . . 7.7.1 Simulation Setup . . . . . . . . . . . . . . . 7.7.2 PAPR Reduction Performance . . . . . . . . 7.7.3 Error Performance . . . . . . . . . . . . . . 7.7.4 Performance with HPA . . . . . . . . . . . . 7.7.5 Out-of-band Interference (OBI) Levels . . . 7.7.6 Computational Complexity . . . . . . . . . .

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Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 8.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Figures 1.1 1.2 1.3 1.4 1.5

OFDM system model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High peaks in MCM signal generated by the addition of multiple sinusoids . . . Envelope of a realization of an OFDM symbol . . . . . . . . . . . . . . . . . . . CCDFs of PAPR of OFDM signals generated according to various specifications Typical input power versus output power characteristic curve for SSPA . . . . .

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Taxonomy of PAPR reduction techniques [1] . . . . . . . . . . . . . . . . . . . . . . . . .

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3.1 3.2 3.3

OFDM system model with a companding scheme . . . . . . . . . . . . . . . . . . . . . . Amplitude distribution of OFDM signal . . . . . . . . . . . . . . . . . . . . . . . . . . . Profiles of companding transforms. (a) ENLC, (b) PEC, (c) EC, (d) NLC and (e) PLC .

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4.1 4.2 4.3

CCDFs of PAPR for PLC applied on various OFDM standards . . . . . . . . . . . . . . CCDFs of PAPR for PEC applied on various OFDM standards . . . . . . . . . . . . . . CCDFs of PAPR for ENLC applied on various OFDM standards . . . . . . . . . . . . .

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5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

OFDM system model with adaptive companding . . . . . . . . . . . . . . . . . . . . . . Histograms of S of OFDM symbols (WiMAX with 4-QAM) . . . . . . . . . . . . . . . . Estimated conditional distributions. (a) Estimated CDFs and (b) estimated PDFs . . . Transforms for adaptive companders, J = 4, M = 4 . . . . . . . . . . . . . . . . . . . . . Transforms for adaptive companders, J = 8, M = 4 . . . . . . . . . . . . . . . . . . . . . Transforms for adaptive companders, J = 4, M = 16 . . . . . . . . . . . . . . . . . . . . Transforms for adaptive companders, J = 8, M = 16 . . . . . . . . . . . . . . . . . . . . CCDFs of PAPR of OFDM signals transformed by fixed and adaptive companders. OFDM signals are based on 4-QAM constellation. . . . . . . . . . . . . . . . . . . . . . . BER performance of original OFDM signal and OFDM signal transformed with fixed and adaptive companders over AWGN channel using 4-QAM modulation. . . . . . . . . PSDs of original OFDM signal and OFDM signal transformed with fixed and adaptive companders. OFDM signals are based on 4-QAM constellation. . . . . . . . . . . . . . . CCDFs of PAPR of OFDM signals transformed by fixed and adaptive companders. OFDM signals are based on 16-QAM constellation. . . . . . . . . . . . . . . . . . . . . . BER performance of original OFDM signal and OFDM signal transformed with fixed and adaptive companders in scenario (i), over AWGN channel, using 16-QAM modulation. PSDs of original OFDM signal and OFDM signal transformed with fixed and adaptive companders in scenario (i). OFDM signals are based on 16-QAM constellation. . . . . . BER performance of original OFDM signal and OFDM signal transformed with fixed and adaptive companders in scenarios (ii)-(v), over AWGN channel, using 16-QAM modulation. PSDs of original OFDM signal and OFDM signal transformed with fixed and adaptive companders in scenarios (ii)-(v). OFDM signals are based on 16-QAM constellation. . .

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5.9 5.10 5.11 5.12 5.13 5.14 5.15

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

Symbol-to-energy mapping for 16-QAM . . . . . . . . . . . . . Symbol-to-energy mapping for 64-QAM . . . . . . . . . . . . . PMFs of symbol energy |Xd |2 for (a) 16-QAM and (b) 64-QAM (a) PMF of symbol power (b) Histogram of symbol power . . . (a) PMF of symbol power (b) Histogram of symbol power . . . (a) PMF of symbol power (b) Histogram of symbol power . . . (a) PMF of symbol power (b) Histogram of symbol power . . . Comparison of proposed and simulated distributions . . . . . . Conditional PDFs for WLAN with 16-QAM . . . . . . . . . . .

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6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22 6.23 6.24 6.25 6.26 6.27 6.28 6.29

7.1 7.2

7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19 7.20 7.21 7.22

Conditional PDFs for WiMAX with 16-QAM . . . . . . . . . . . . . . . . . . . . . . . . Joint probability distribution of average power and amplitude . . . . . . . . . . . . . . . Comparison of analytical and empirical CDFs for J = 4 . . . . . . . . . . . . . . . . . . Modified system with APC and ACS schemes . . . . . . . . . . . . . . . . . . . . . . . . (a) CCDFs of PAPR (b) BER with APC+ENLC on WLAN with 16-QAM . . . . . . . . (a) CCDFs of PAPR (b) BER with APC+PLC on WLAN with 16-QAM . . . . . . . . . (a) CCDFs of PAPR (b) BER with APC+PEC on WLAN with 16-QAM . . . . . . . . (a) CCDFs of PAPR (b) BER with APC+PEC on WLAN with 64-QAM . . . . . . . . (a) CCDFs of PAPR (b) BER with APC+PLC on WLAN with 64-QAM . . . . . . . . . (a) CCDFs of PAPR (b) BER with APC+ENLC on WiMAX with 16-QAM . . . . . . . (a) CCDFs of PAPR (b) BER with APC+PEC on WiMAX with 16-QAM . . . . . . . . (a) CCDFs of PAPR (b) BER with APC+PEC, APC+PLC and APC+ENLC on WLAN with 16-QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) CCDFs of PAPR (b) BER with ACS+PLC on WLAN with 16-QAM . . . . . . . . . (a) CCDFs of PAPR (b) BER with ACS+PEC on WLAN with 16-QAM . . . . . . . . . (a) CCDFs of PAPR (b) BER with ACS+PEC on WLAN with 64-QAM . . . . . . . . . (a) CCDFs of PAPR (b) BER with ACS+ENLC on WLAN with 64-QAM . . . . . . . . (a) CCDFs of PAPR (b) BER with ACS+PLC on WLAN and WiMAX with 16-QAM . (a) CCDFs of PAPR (b) BER with APC+PLC and ACS+PLC on WiMAX with 64-QAM PSDs of signals transformed by fixed compander and APC and ACS schemes with PLC, PEC and ENLC, applied on WLAN with 16-QAM . . . . . . . . . . . . . . . . . . . . . PSDs of signals transformed by fixed compander and APC and ACS schemes with PLC and PEC, applied on WiMAX with 64-QAM . . . . . . . . . . . . . . . . . . . . . . . . .

63 64 67 68 72 72 72 73 73 73 74 74 75 75 75 76 76 76 77 77

Compression and expansion profiles of (a) ENLC, (b) PEC, (c) EC, (d) NLC and, (e) PLC 85 Relationship between compander’s profile and OBI. (a) Companding transforms with changing inflexion points, (b) PSDs of companding noise, D = T (x) − x, and, (c) PSDs of companded signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Waveforms of companding noise, D = T (x) − x, for different values of inflexion point a1 91 Transform profiles of PLC-1 (a) compander and (b) decompander . . . . . . . . . . . . . 94 Profiles of PLC-2 (a) compander and (b) decompander . . . . . . . . . . . . . . . . . . . 97 Profiles of PLC-3(a) compander and (b) decompander . . . . . . . . . . . . . . . . . . . 100 CCDFs of PAPR with PLC-1, PLC-2 and PLC-3 applied on WiMAX with 4-QAM . . . 104 BER over AWGN channel with PLC-1, PLC-2 and PLC-3 applied on WiMAX with 4-QAM105 BER over AWGN channel with PLC-1, PLC-2 and PLC-3 applied on WiMAX with 16-QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 PSDs of signal transformed by PLC-1, PLC-2 and PLC-3 applied on WiMAX with 4-QAM106 PAPR reduction performances of PLC, NLC, PEC, ENLC, EC and PLC-3 . . . . . . . . 108 BER over AWGN channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 4-QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 BER over AWGN channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 16-QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 BER over SUI-1 channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 4-QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 BER over SUI-1 channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 16-QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 BER over SUI-5 channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 4-QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 BER over SUI-5 channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 16-QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Input amplitude versus output amplitude characteristic curve of SSPA . . . . . . . . . . 113 Input power versus output power characteristic curve of SSPA . . . . . . . . . . . . . . . 113 BER with SSPA (p = 2) over AWGN channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 4-QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 BER with SSPA (p = 4) over AWGN channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 4-QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 BER with SSPA (p = 2, IBO=0.5 dB) over AWGN channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 16-QAM . . . . . . . . . . . . . . . . . . . . . . 115

xi

7.23 BER with SSPA (p = 4, IBO=0.5 dB) over AWGN channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 16-QAM . . . . . . . . . . . . . . . . . . . . . . 7.24 BER with SSPA (p = 2, IBO=1 dB) over AWGN channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 16-QAM . . . . . . . . . . . . . . . . . . . . . . 7.25 BER with SSPA (p = 4, IBO=1 dB) over AWGN channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 16-QAM . . . . . . . . . . . . . . . . . . . . . . 7.26 PSDs of signals transformed by PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 4-QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.27 PSDs of signals transformed by PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 16-QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xii

116 116 117 117 118

List of Tables 2.1

Comparison of trade-offs involved in various PAPR reduction schemes . . . . . . . . . .

9

4.1

Parameters used in WLAN and Fixed WiMAX . . . . . . . . . . . . . . . . . . . . . . .

25

5.1

Summary of Trade-offs involved in adaptive companding . . . . . . . . . . . . . . . . . .

55

7.1 7.2 7.3 7.4 7.5

Parameters for SUI-1 channel simulation . Parameters for SUI-5 channel simulation . Summary of error performance comparison Summary of OBI performance comparison Comparison of implementation complexity

xiii

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107 107 112 118 119

List of Acronyms

ACE Active Constellation Extension ACI Adjacent Channel Interference ACS Adaptive Constellation Scaling APC Adjustable-Parameter Companding ASK Amplitude Shift Keying AWGN Additive White Gaussian Noise BER Bit Error Rate BPSK Binary Phase Shift Keying CCDF Complementary Cumulative Distribution Function CDF Cumulative Distribution Function CLT Central Limit Theorem CM Cubic Metric DAC Digital-to-Analog Converter DFT Discrete Fourier Transform EC Exponential Compander ENLC Efficient Non-Linear Compander EVM Error Vector Magnitude FFT Fast Fourier Transform HPA High Power Amplifier IBO Input Back-Off ICI Inter Carrier Interference IDFT Inverse Discrete Fourier Transform IFFT Inverse Fast Fourier Transform IID Independent and Identically Distributed ISI Inter Symbol Interference LCT Linear Companding Transform LLN Law of Large Numbers LTE Long Term Evolution MCM Multi-carrier Modulation NLC Non-Linear Compander OBI Out-of-band Interference OFDM Orthogonal Frequency Division Multiplexing

xiv

PAPR Peak-to-Average Power Ratio PDF Probability Density Function PEC Piecewise Exponential Compander PLC Piecewise Linear Compander PMF Probability Mass Function PSD Power Spectral Density PSK Phase Shift Keying PTS Partial Transmit Sequences QAM Quadrature Amplitude Modulation QPSK Quaternary Phase Shift Keying RCE Relative Constellation Error RCM Raw Cubic Metric SLM Selective Mapping SNR Signal-to-Noise Ratio SSPA Solid State Power Amplifier SUI Stanford University Interim TI Tone Injection TR Tone Reservation TWTA Traveling Wave Tube Amplifier WiMAX Worldwide Interoperability for Microwave Access WLAN Wireless Local Area Network WMAN Wireless Metropolitan Area Network

xv

Chapter 1

OFDM Systems and the PAPR Problem Owing to the increased demand for fast communication of information, escalated growth of multimedia wireless applications and portability of communication devices, the requirement for technologies supporting high data transmission rates and mobility has rapidly intensified over the past decade. At the same time, the concern for the efficient utilization of available bandwidth, network resources and power resources also accrues. Orthogonal frequency division multiplexing (OFDM) has been widely accepted as the technology of choice to achieve these goals and has become an essential part of a number of prevailing and emerging commercial standards [2,3]. Consequently, development of solutions for the design issues encountered in OFDM systems has been an active area of research for over two decades. High peak-to-average power ratio (PAPR) of the signal is an imperative problem in OFDM systems as it impedes the efficient design of the analog front-end of the system. Therefore, considerable research effort has been put into the development of a practical solution for PAPR reduction. In this chapter, a brief overview of OFDM systems is given, followed by a comprehensive explanation of the PAPR problem.

1.1

OFDM Systems

OFDM is a multi-carrier modulation (MCM) scheme, in which a number of data symbols are simultaneously modulated by multiple carriers. Since individual carriers are separated in frequency, this type of modulation results in partitioning of the total bandwidth into 1

Chapter 1. OFDM Systems and the PAPR Problem multiple sub-bands, such that one sub-band is occupied by one data symbol. In contrast to frequency division multiplexing (FDM), each sub-bands in OFDM overlap with its neighboring sub-bands. This increases the spectral efficiency because more symbols can be packed in the same bandwidth. Even with overlapping bands, the symbols can be independently and uniquely decoded because of the orthogonality among the sub-carriers. The division of bandwidth into multiple, narrow bands equips the system to effectively combat the effects of frequency-selective fading channels. This is because now they can be modeled as multiple, parallel flat-fading sub-channels. Frequency domain channel estimation can be used to mitigate the effects of channel impairments, which is much simpler as compared to conventional time-domain, adaptive equalizers. The discrete-time complex baseband equivalent of the OFDM symbol, given by the inverse discrete fourier transform (IDFT) of a block of QAM modulated data symbols, is represented as follows: N −1 1 X j2πkn x[n] = √ , for 0 ≤ n ≤ N − 1 Xk exp N N k=0

(1.1)

The block diagram of a typical OFDM system is shown in Fig. 1.1. Input bit stream

Output bit stream

QAM mapping

S/P

IDFT

P/S

Add cyclic prefix

S/P

Remove cyclic prefix

D/A

Passband modulation

HPA Channel

QAM demapping

P/S

DFT

A/D

Passband demodulation

Figure 1.1. OFDM system model

In order to mitigate the inter-symbol interference (ISI) between consecutively transmitted OFDM symbols, due to the delay spread of the multi-path fading channel, a guard interval with cyclic prefix is inserted between the symbols. The cyclic prefix upholds the orthogonality among the sub-carriers, thus preventing inter-carrier interference (ICI), in the presence of multi-path fading, provided that the duration of cyclic prefix is longer than the delay spread of the channel [2].

2

Chapter 1. OFDM Systems and the PAPR Problem 4

Amplitude

freq1 freq2 freq3 freq4 Sum

high peak

3 2 1 0 −1 −2

0

0.5

1

Time Figure 1.2. High peaks in MCM signal generated by the addition of multiple sinusoids

Amplitude

1.5

1

0.5

0

0

100

200

300

400

500

600

700

800

900

1000

Samples

Figure 1.3. Envelope of a realization of an OFDM symbol

1.2

Applications of OFDM

OFDM has been standardized for high speed, fixed and mobile broadband internet access systems and digital audio/video broadcasting systems. It has been incorporated in the physical layer of standards for wireless local area networks (WLANs) like IEEE 802.11a/g for high speed transmission. Other standards employing OFDM for internet access include ETSI HIPERLAN/2, multi-media mobile access communications (MMAC), worldwide interoperability for microwave access (WiMAX based on IEEE 802.16) and long term evolution (LTE) [4,5]. Broadcast technologies using OFDM are digital audio broadcasting (DAB), digital video broadcasting-terrestrial (DVB-T) and digital video broadcasting-held (DVB-H).

1.3

The PAPR Problem

It can be observed from Eq. (1.1) that the OFDM signal is constructed by addition of a number of sub-carriers with random amplitudes and phases. Due to this addition of multiple carriers, depicted in Fig. 1.2, OFDM signal assumes a highly fluctuating, 3

Chapter 1. OFDM Systems and the PAPR Problem

noise-like envelope with large dynamic range. When instantaneous amplitudes of the sub-carriers are aligned in phase, the OFDM signal exhibits a larger instantaneous power than its average power. The envelope fluctuations and in turn the dynamic range of the signal increases as the number of sub-carriers is increased. Fig. 1.3 shows the envelope of a realization of OFDM symbol generated according to IEEE 802.16d specifications. Due to the signal’s large dynamic range, the linear range of the digital-to-analog converter (DAC) and high power amplifier (HPA) at the transmitter is required to be large, so that the signal can be amplified without distortion. But increasing the linear range of the HPA leads to reduction in its power efficiency. If the linear range is kept smaller to increase power efficiency, then the signal excursion into the HPA’s non-linear region leads to non-linear distortion. The non-linear distortion leads to increase in bit error rate (BER) and spectral spreading. Also, in case of HPA linearization or pre-distortion [6], the signal excursion must be limited to the linear region for the pre-distortion algorithm to function correctly. 0

Pr [P AP R > P AP R0 ]

10

−1

10

−2

10

WiMAX, 4-QAM WiMAX, 16-QAM WiMAX, 64-QAM WLAN, 4-QAM WLAN, 16-QAM WLAN, 64-QAM

−3

10

−4

10

3

4

5

6

7

8

9

10

11

12

P AP R0 (dB)

Figure 1.4. CCDFs of PAPR of OFDM signals generated according to various specifications

1.3.1

Definition of PAPR

PAPR is the most common metric used to quantify the dynamic range of OFDM signals. It is defined as the ratio of maximum instantaneous power to average power, within a

4


symbol duration, i.e., 

2



max |x[n]|   0≤n≤N L−1   PAPR (dB) = 10 log10   N L−1 1 P |x[n]|2 N L n=0

(1.2)

PAPR is evaluated per symbol and is characterized by its complementary cumulative distribution function (CCDF). Fig. 1.4 shows empirical CCDFs of PAPR for various OFDM standards. It can be seen that the PAPR of OFDM signal can go up to 12 dB, which is considered a very large value in the design of HPA.

1.3.2

Relationship Between HPA Efficiency and PAPR

Output Power (dB)

0

Saturation output power Saturation region

Actual operating point

−5

Linear region Ideal operating point

−10

IBO

−15

−20 −20

−15

−10

−5

0

5

10

Input Power (dB) Figure 1.5. Typical input power versus output power characteristic curve for SSPA

The non-linear distortion of HPA is modeled as amplitude/amplitude (AM/AM) distortion in the solid state power amplifier (SSPA) model and as a combination of AM/AM and amplitude/phase (AM/PM) distortions in traveling wave tube amplifier (TWTA) [1]. The typical input/output characteristics of SSPA is shown in Fig.1.5. The most efficient operating point for the HPA is at the saturation level. But due to the large dynamic range of the signal, input back-off (IBO) is required to shift the operating point to the 5


left, as shown in Fig. 1.5. Increase in IBO leads to decrease in power efficiency. IBO is given as follows [1]: IBO (dB) = Psat (dB) − Pav (dB)

(1.3)

In order to limit the signal excursion of the amplified signal within the HPA’s linear range, the IBO of HPA must be at least equal to the PAPR of the signal. Hence the high PAPR of OFDM signal imposes an upper limit on the power efficiency of the HPA. So it is required to reduce the PAPR of OFDM signals to ensure high power efficiency without introducing non-linear distortion.

6

Chapter 2

An Overview of PAPR Reduction Techniques Several techniques for PAPR reduction have been proposed in the literature. In this chapter, an overview of existing PAPR reduction schemes is given. Their merits, demerits and applicability is also discussed, thereby developing the motivation for this work.

2.1

Classification of PAPR Reduction Techniques

PAPR reduction techniques are classified into three categories, as shown in Fig. 2.1.

2.1.1

Signal Distortion Techniques

In signal distortion techniques, PAPR is reduced by distorting the OFDM signal before it passes through the HPA. The most common signal distortion techniques are: (i) Clipping and Filtering [7], (ii) Peak Windowing [8], (iii) Companding [9–19] and (iv) Peak Cancellation [20]. Signal distortion includes both in-band and out-of-band distortion, resulting in elevated BER and OBI. These techniques exploit the fact that high peaks occur only rarely, so the small amount of distortion resulting from clipping the peaks is tolerable in most applications.

7

Chapter 2. An Overview of PAPR Reduction Techniques

PAPR Reduction Techniques

Signal Distortion

Multiple Signaling and Probabilistic

Coding

Clipping and Fiitering

Selective Mapping

Linear Block Coding

Companding

Partial Transmit Sequences

Golay Sequences

Peak Windowing

Interleaving

Turbo Coding

Peak Cancellation

Tone Injection

Tone Reservation

Active Constellation Extension

Figure 2.1. Taxonomy of PAPR reduction techniques [1]

2.1.2

Multiple Signaling and Probabilistic Techniques

Probabilistic techniques involve generating multiple candidate signals containing the same information and selecting the one with lowest PAPR for transmission. Most common probabilistic techniques are: (i) Partial transmit sequences (PTS) [21],(ii) Selective mapping (SLM) [22], (iii) Tone injection (TI) [23], (iv) Tone reservation (TR) [24], (v) Active constellation extension (ACE) [25], (vi) Interleaved OFDM [26].

2.1.3

Coding Techniques

These techniques involve modifying coding schemes, that are generally used for error detection and correction, to perform the additional function of PAPR reduction, by increasing complexity. Some coding techniques are: (i) Linear block coding (LBC) [27], (ii) Golay complementary sequences (GCS) [28], (iii) Turbo coding [29].

8

Chapter 2. An Overview of PAPR Reduction Techniques Table 2.1. Comparison of trade-offs involved in various PAPR reduction schemes Increase in Increase in Increase in Data Computational Scheme Average BER OBI Rate Loss Complexity Power PTS no no yes high no SLM no no yes high no Interleaving no no yes high no TI no no no high yes TR no no yes high yes ACE no no no high yes GCS no no yes high no Clipping yes yes no low no Companding yes yes no low no

2.2

Performance Considerations

The aim of any PAPR reduction technique is to reduce PAPR while preserving other desirable attributes of the system as much as possible. The techniques referenced above differ in terms of trade-offs involved. A summary is given in Table 2.1. In [1], after a detailed survey, it has been concluded that PAPR reduction techniques need to be flexible enough to exploit most of the tolerances in the system for performance gain. Performance evaluation of an OFDM system employing various PAPR reduction techniques involves following performance metrics and considerations:

2.2.1

Distribution of PAPR

The amount of PAPR reduction achieved from any scheme is quantified by the amount of reduction in complementary cumulative distribution function (CCDF), given as follows: CCDFP AP R (P AP R0 ) = Pr[P AP R > P AP R0 ]

2.2.2

(2.1)

Computational Complexity

Computational complexity of any PAPR reduction scheme is mainly quantified by the number of floating point additions and multiplications involved. Generally, probabilistic schemes involving multiple signaling are high complexity. Low complexity schemes include deterministic, signal distortion techniques like clipping and filtering, companding and peak windowing.

9


2.2.3

Average Power

In any PAPR reduction scheme, it is desirable to keep the average power of signal unchanged. In ACE and TI and in some companding transforms like µ-law companding [30] and linear companding trasnform (LCT) [9] average power is increased to reduce PAPR. In PTS, SLM and interleaving schemes, average power remains constant. Most companders [10, 13, 15, 19, 31–33] are also designed to preserve the average power of the signal.

2.2.4

Error Performance

Error performance is expressed by bit error rate (BER) or probability of bit error at various values of SNR in AWGN and fading channels. Empirical curves for BER are used to evaluate error performance. BER performance is mainly affected in signal distortion techniques like clipping and companding. Generally, probabilistic schemes do not introduce signal distortion and hence no degradation in BER. However, in the schemes with constellation reshaping like TI, if the new constellation is not gray coded [23,34] or it is more dense [35], then BER performance may degrade despite keeping the signal undistorted. In PTS and SLM where side information is required for correct recovery of signal information, errors in side information can also lead to errors and hence increased BER.

2.2.5

Spectral Characteristics

Effects of PAPR reduction schemes on spectral characteristics include changes in the useful band and guard band of the channel. In-band Distortion In-band distortion involves the changes in the amplitude and phase of data carriers. This happens due to inter-modulation noise introduced by non-linear operations like clipping, companding and peak windowing. In-band distortion leads to inter-carrier interference (ICI) or loss of orthogonality among sub-carriers. This leads to degraded error performance. Also, since the noise introduced by operations like clipping and companding is not white, this distortion may not be evenly distributed among the sub-carriers. 10


In OFDM standards like IEEE 802.11 and 802.16, error vector magnitude (EVM) and relative constellation error (RCE) are used as metrics to quantify the in-band distortion. The in-band distortion results collectively from quadrature skew, I/Q gain imbalance, phase noise, clock recovery and nonlinear distortion. The non-linear distortion may include distortion due to PAPR reduction techniques in addition to other non-linearities in the circuit components. Since these metrics are experimentally evaluated and are not just related to PAPR reduction, they are generally not used to evaluate in-band distortion. Instead, in-band distortion is manifested as increase in BER. Spectral Regrowth One of the important properties of OFDM systems is its rectangle-like spectrum. Signal distortion techniques result in leakage of signal power into guard band which effectively increases the channel bandwidth. Due to the presence of high frequency contents in the transformed signal, adjacent channel interference (ACI) is increased. With spectral spreading, spacing between adjacent channel has to be increased in order to meet a given signal-to-interference ratio (SIR). Increased side-lobe level and ACI are collectively termed as out-of-band interference (OBI). Spectral regrowth due to clipping or companding is evaluated using power spectral density (PSD) plots. Theoretically PSD is defined for analog signals. Several techniques [36] can be used to estimate PSD of a random signal from its discrete-time sequence. In this thesis, periodogram estimator has been used to find PSD. Periodogram estimate is obtained by squaring the magnitude of DFT of a windowed time sequence. 1 Rx (ejω ) , N

2 −1 N X −jωn v e , vn , xn wn n n=0

(2.2)

Since periodogram is asymptotically unbiased, long data records give better estimates. Also it is not a consistent estimator, so several estimates are averaged to approximate the true PSD. This PSD is calculated for the OFDM waveform after adding the cyclic prefix. Window wn is kept long enough to span ten OFDM symbols (including the cyclic prefix).

11


2.2.6

Bandwidth Efficiency

The amount of information contained in the channel bandwidth is quantified by bandwidth efficiency. Some PAPR reduction techniques like PTS, SLM and interleaving involve transmitting side information for correctly recovering the signal at the receiver. In TR, some sub-carriers are reserved for generating peak canceling signal. In both the cases, data rate or channel throughput is reduced.

2.2.7

Sensitivity to Errors in Side Information

When side information is required for correct recovery of symbols, errors in side information can also lead to errors. Techniques like PTS and SLM render the system highly sensitive to errors in side information. Even a single bit error in side information can lead to burst errors.

2.2.8

Effects of Oversampling

The objective of PAPR reduction is to decrease the PAPR of the analog passband signal which has to pass through HPA. However, signal processing for PAPR reduction is done on discrete-time sequence. An oversampled signal more closely approximates the analog signal and its PAPR. Without oversampling, optimistic estimates of PAPR are obtained. As a convention, OFDM signal is oversampled by a factor of 4 [37]. Oversampled signal is generated by zero insertion in frequency domain. NX L−1 1 j2πkn xn = √ Xk exp NL N L k=0

(2.3)

where Xk = [X0 , X1 , X2 , ..., X N −1 , 0, ..., 0, X N , ..., XN −1 ], N is the number of sub-carriers 2 | {z } 2 N (L−1)

and L is the oversampling factor.

12


2.2.9

Number of Sub-carriers

CCDF of PAPR is a function of number of sub-carriers [38, 39]. Theoretical CCDF of PAPR is given by CCDF (P AP R0 ) = 1 − FP AP R (P AP R0 ) = 1 − (1 − exp(−P AP R0 ))αN

(2.4)

where N is the number of sub-carriers and α is determined through simulation [39]. Also, it will be discussed in detail in Chapter 5 that number of sub-carriers has significant effect on companding transforms’ performance.

2.2.10

Signal Constellation

Many techniques, like TI and ACE, reduce the PAPR by modifying the modulation. These techniques depend upon the constellation used for modulation. It will be shown in Chapter 6 that the performance of companders also differ from constellation to constellation.

2.3

Scope of the Thesis

The focus of this thesis is the design optimization of companding schemes for PAPR reduction. Performance optimization with respect to number of sub-carriers/symbol period, constellation type, error performance, spectral regrowth, computational complexity and average power is considered. Chapters 5 and 6 deal with the performance optimization with respect to symbol period and constellation. In Chapter 7, companding transforms are designed with the objective of minimizing BER, OBI and computational complexity.

13

Chapter 3

Companding Transforms for PAPR Reduction The word companding is a combination of compressing and expanding. Companding is a well known PAPR reduction technique that involves transforming the signal using a deterministic amplitude transform. It comes under the category of signal distortion techniques. Due to their low implementation complexity and capability of efficiently trading between PAPR and BER, design of companding transforms has been attracting great attention over the last decade. Input bit stream

Output bit stream

QAM mapping

S/P

IDFT

P/S

Compander

Add cyclic prefix

Decompander


D/A and HPA Channel

QAM demapping

P/S

DFT

S/P

A/D

Figure 3.1. OFDM system model with a companding scheme

3.1

OFDM System with a Companding Transform

Companding is a post modulation operation on the OFDM signal. A typical OFDM system employing a compander/decompander pair is shown in Fig. 3.1. The compander transforms the amplitude of the time domain sequence, i.e., the output of IDFT. Due to non-linearity of companding operation, the signal is inevitably distorted. This causes increase in BER and OBI. 14

Chapter 3. Companding Transforms for PAPR Reduction

1

fA (x)

0.8

fA (x) =

2x σx2

2.5

3

exp

1

2 − σx2 x

2

0.6 0.4 0.2 0

0

0.5

1

1.5

2

3.5

4

Amplitude x Figure 3.2. Amplitude distribution of OFDM signal

Since companding is an amplitude transform, the instantaneous phase of the signal remains unchanged. A companded sample yn , obtained by transforming xn by a companding function T (.) is represented as follows: yn = T (|xn |) sgn(xn )

(3.1)

where sgn(.) denotes signum function, i.e., sgn(x) = exp(j∠x).

3.2

Amplitude Distribution of OFDM Signal

Companders are amplitude transforms that modify the distribution of signal amplitude such that PAPR of the signal is reduced. The amplitude of an OFDM signal is a random variable having Rayleigh distribution as explained below. The discrete time, oversampled, complex envelope of the OFDM signal is given as follows:

NX L−1 1 j2πkn xn = √ Xk exp NL N L k=0

(3.2)

where Xk comes from the input symbol vector [X0 , X1 , X2 , ..., X N −1 , 0, ..., 0, X N , ..., XN −1 ]. 2 | {z } 2 N (L−1)

N is the number of sub-carriers including Nd data carriers, Np pilot carriers and null carriers (guard band and DC). L is the oversampling factor. Each of the data symbols in the symbol vector is modulated using QAM constellation. Since all the data symbols are independently and randomly sampled from QAM constellation, each sample xn is a random variable generated by the weighted average of independent and identically dis15


tributed (IID) random variables. When N is large, xn can be approximated as a complex Gaussian random process by the central limit theorem (CLT) approximation. Hence, the signal amplitude |xn | is a Rayleigh random process. Its cumulative distribution function (CDF) and probability density function (PDF) are given as follows: x2 FA (x) = 1 − exp − 2 σx

2 2x x fA (x) = 2 exp − 2 σx σx

(3.3)

(3.4)

where A is the random variable associated with amplitude of xn , for 0 ≤ n ≤ N L − 1, and σx2 = E [|xn |2 ] is the average power of OFDM signal.

3.3

Literature Review

• µ-law Companding: In [30], the µ-law companding scheme, which is generally used in speech processing for quantization, was used to reduce the PAPR. The µ-law companding reduces the PAPR at the expense of increased average power. • Exponential Companding (EC): In [32], EC was proposed which modifies the amplitude distribution of the signal while maintaining the average power of the signal. • Linear Companding Transform (LCT): LCT [9] was proposed that reduces the peak power by transforming small and large amplitudes with different scales. But LCT does not maintain constant signal power and the transform does not have oneto-one correspondence with its inverse. This necessitates sending of side information for decompanding. • Several transforms have been designed using the amplitude distribution modification method [10, 12, 15, 17, 19, 33, 40]. • Non-linear Compander (NLC): In [19], NLC was proposed that only transforms larger amplitudes. The larger amplitudes are transformed such that they have uniform distribution. But this transform is a fixed function that cannot be tuned to achieve any desired PAPR. Rather only one output PAPR is possible.

16


• Efficient Non-linear Compander (ENLC): The ENLC in [15] was designed to add flexibility to the non-linear compander (NLC) in [19]. In other words, NLC is a special case of ENLC. This was done by transforming the larger amplitudes such that they have a trapezium distribution. • Trapezoidal Companding Schemes: To add flexibility in attaining different configurations, complete amplitude distribution was transformed into trapezoidal distribution, instead of uniform, in [11, 33]. • Non-linear Companding Transform (NCT) with Piecewise Linear Distribution: Amplitude distribution is transformed into a piecewise linear function in [17] to provide more flexibility. • Generalized Closed Form Family of Non-linear Companders: In [10], the distribution modification method was generalized. A generic closed form family of companding transforms was derived that can be tuned to the previously proposed transforms [15, 19] and also provides solutions in operating regions that are not possible with these special cases. • Piecewise Exponential Compander (PEC): In [12], EC was modified into PEC to restrict the companding noise to larger samples. In this way, average companding noise energy is limited to minimum. • Piecewise Linear Compander (PLC): In [13], low-complexity PLC is proposed that is designed with the objective of mitigating companding distortion for a preset amount of PAPR reduction. Like PEC, companding distortion is limited by restricting the transformation to larger samples. • Two Piecewise Compander (TPWC): TPWC is another piecewise linear compander proposed in [31]. In this case, only average power constraint is used, so the compander is not optimized with respect to companding noise. • Iterative Companding Transform and Filtering (ICTF): ICTF is proposed in [18] to reduce the OBI, in which LCT and TPWC are used to implement the scheme. The linear and piecewise linear transforms are used in order to reduce the complexity.

17


ENLC

x

(a)

(b)

T (x)

EC

T (x)

T (x)

PEC

x

(c)

NLC

PLC T (x)

T (x) (d)

x

x

(e)

x

Figure 3.3. Profiles of companding transforms. (a) ENLC, (b) PEC, (c) EC, (d) NLC and (e) PLC

• BER analysis: BER analysis of companding transforms is carried out in [41, 42], which shows that slopes in the companding transforms impose the limits on error performance.

3.4

Some Recent Works on Companding Schemes

Fig. 3.3 shows profiles of ENLC [15], PEC [12], EC [32], NLC [19] and PLC [13]. They will be used to implement the schemes in Chapters 5 and 6 and for performance comparison in Chapter 7.

3.4.1

Exponential Compander (EC)

The design objectives used in EC are: • dth power of companded amplitude is uniformly distributed in the interval [0, α]. • average signal power remains unchanged, i.e., equal to σx2 .

18


The companding transform is given as follows: 2 1/d x TEC (x) = α 1 − exp − 2 σx

(3.5)

where α is given as follows: d/2

   α=  

3.4.2

  E[|xn |2 ] # " 2 1/d   x  E α 1 − exp − 2 σx

(3.6)

Non-linear Compander (NLC)

This function only transforms larger amplitudes. √ • In the interval [0, σx / 6], amplitudes remain unchanged, √ • while amplitudes in interval (σx / 6, ∞] are transformed such that the companded √ amplitudes are uniformly distributed in the interval (σx / 6, A]. • Average signal power remains constant. The transform is given as follows:    x,

σx x≤ √ 6 TN LC (x) = √ 2 1 2 1 x σ  x   6σx − exp − , x> √ 3 2 6 σx2 6

(3.7)

Output PAPR with NLC is given as follows: PAPR = 10 log10

A2 σx2

= 4.25 dB

(3.8)

NLC can only provide a fixed PAPR at the output.

3.4.3

Efficient Non-linear Compander (ENLC)

ENLC is a generalization of NLC. This means that it has some variable parameters that can be set to obtain any PAPR at the output. This transform is further generalized in [10]. 19


ENLC also transforms only larger amplitudes. • In the interval [0, cσx ], amplitudes remain unchanged. • Amplitudes in the interval (cσx , ∞] are transformed such that the companded amplitudes have a trapezoidal distribution in the interval (cσx , A]. • Average signal power remains constant. The transform is given as follows: T EN LC (x)     x ≤ cσx x, s ! 2 = 1 4c2 2c x  2 2  , x > cσx   k kcσx − σx exp(−c ) + σ 2 + 2k exp(−c ) − exp − σ 2 x x (3.9) It can be configured by setting its three parameters. In this thesis, it is configured by setting the value of peak amplitude as follows: A = σx 10P AP Rdesired /20

(3.10)

c and k are found such that they satisfy the constant average power constraint and the companded signal has the desired distribution, as described above. So they should satisfy the following equations: σx2 =

ZA

2 TEN LC (x)fA (x)dx

(3.11)

0

σx2 − k=

3.4.4

cσ Rx 0

3 2A c x2 fA (x)dx − − 2c4 σx2 exp(−c2 ) 3σx A4 A3 c4 σx4 − cσx + 4 3 12

(3.12)

Piecewise Exponential Compander (PEC)

The idea is to limit the distortion from EC to only the larger samples so that the companding distortion is minimized and the transform can be configured to get any desired PAPR. The transform has three parameters: an inflexion point Ai , a cut-off point Ac and power of companded amplitudes d. Design criteria is given as follows: 20

Chapter 3. Companding Transforms for PAPR Reduction • the dth power of companded amplitude is uniformly distributed in the interval [Ai , Ac ]. • Average power remains constant. The PEC transform is given as follows:    x, x ≤ Ai 2 TP EC (x) = 1/d x − A2i   , x > Ai + Adc  (Adi − Adc ) exp − σx2

(3.13)

Ac is set according to the desired PAPR, i.e., Ac = σx 10P AP Rdesired /20

(3.14)

Ai and d are obtained by imposing the average power constraint. Therefore, they should satisfy the following equation: 2Ad+2 + (d + 2)σx2 Adi − (d + 2)Adc A2i + dAd+2 − (d + 2)σx2 Adc = 0 c i

3.4.5

(3.15)

Piecewise Linear Compander (PLC)

PLC is specified by three parameters: a clipping level Ac , an inflexion point Ai and a slope k. These parameters are evaluated subject to the following constraints: • Average power remains constant. • Companding noise is minimized. The PLC transform is given as follows:

TP LC

    x, x ≤ Ai    = kx + (1 − k)Ac , Ai < x ≤ Ac      Ac , x > Ac

(3.16)

Ac is determined by P AP Rpreset , Ac = σx 10P AP Rpreset /20 21

(3.17)


The average power constraint is given as follows:

σx2 =

Z∞

TP2 LC (x)fA (x)dx

(3.18)

0

A solution set containing all possible solutions of the above equation is found and the solution that yields minimum mean companding distortion is selected for the compander. The mean companding distortion is defined as follows:

σc2

= E (TP LC (x) − x)2 =

Z∞ (TP LC (x) − x)2 fA (x)dx

(3.19)

0

The main idea in the design of this compander is that the companding noise is small if only larger samples are expanded. Evaluation of PLC’s parameters Eq. (3.18) can be written as a quadratic equation with respect to k as follows: I2 (Ai , Ac ) + A2c I0 (Ai , Ac ) − 2Ac I1 (Ai , Ac ) k 2 −2A2c I0 (Ai , Ac ) + 2Ac I1 (Ai , Ac ) k + A2c I0 (Ai , ∞) + I2 (0, Ai ) − σx2 = 0

(3.20)

The integrals in the above equation are given as follows: Zxf

Zxf I0 (xi , xf ) =

fA (x)dx = xi

xi

Zxf

2 2 2 xf 2x x x exp − 2 dx = − exp − 2 + exp − i2 (3.21) 2 σx σx σx σx Zxf

2 2 xf x 2x I1 (xi , xf ) = xfA (x)dx = x 2 exp − 2 dx = −xf exp − 2 σx σx σx xi xi 2 √ √ xi πσx xf πσx xi + xi exp − 2 + erf − erf σx 2 σx 2 σx

22

(3.22)

Chapter 3. Companding Transforms for PAPR Reduction 2 Rz where erf(z) = √ exp (−t2 ) dt. π0 Zxf

Zxf

2 2 xf x 2 I2 (xi , xf ) = x fA (x)dx = exp − 2 dx = −xf exp − 2 σx σx xi xi 2 2 2 xf x xi 2 2 + xi exp − 2 + σx − exp − 2 + exp − i2 σx σx σx 2

2x x2 2 σx

(3.23)

The mean companding distortion in Eq. (3.19) can be written as follows: σc2 =(k − 1)2 I2 (Ai , Ac ) + I2 (Ac , ∞) − 2(1 − k)2 Ac I1 (Ai , Ac ) + (1 − k)

2

A2c I0 (Ai , Ac )

+

A2c I0 (Ac , ∞)

(3.24)

− 2Ac I1 (Ac , ∞)

Eqs. (3.20) and (3.24) are solved to obtain (k, Ai , Ac ) with a given P AP Rpreset using Algorithm 1. Algorithm 1 Parameters of PLC 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18:

Input P AP Rpreset and σx . Find Ac using Eq. (3.17). Set Aiarray = 0 :inc: Ac , where inc is a small increment. Initialize solution set S={}. for each element Ai in Aiarray do Solve Eq. (3.20) for k to get two solutions (k1 , k2 ). Set k = k1 or k = k2 such that 0 < k < 1. Add (k, Ai , Ac ) to S. end for 2 Initialize σc,min = ∞. for each element of S do 2 using Eq. (3.24). Find σc,max 2 2 then if σc,max is less than σc,min Accept current element of S as the final solution. 2 Set σc,min = σc2 . end if end for Output accepted solution (k, Ai , Ac ).

23

Chapter 4

Motivational Study and Problem Identification In this chapter, a comparative study of companding transforms is conducted by evaluating some selected companders with respect to changing OFDM specifications. Generally, the companding transforms in previous works have been designed independent of system parameters, including number of sub-carriers, symbol period and constellation type. But it was found that the performance of transforms is affected by change in these parameters. The objective of the study in this chapter is to understand and assess the effects of changing OFDM specifications on companders’ performance.

4.1

Simulation Setup

The piecewise linear compander (PLC) [13], explained in Sub-section 3.4.5, the piecewise exponential compander (PEC) [12], explained in Sub-section 3.4.4 and the efficient nonlinear compander (ENLC) [15], explained in Sub-section 3.4.3, are simulated for OFDM specifications given in Table 4.1. Figs. 4.1, 4.2 and 4.3 show PAPR reduction performance evaluation for all the cases considered. Wireless local area network (WLAN), based on IEEE 802.11a and fixed worldwide interoperability for microwave access (WiMAX), based on IEEE 802.16d, are used to simulate the effect of changing number of sub-carriers. Both standards are simulated for 4, 16 and 64-QAM. It can be seen that similar trends are obtained for all the three companders.

24

Chapter 4. Motivational Study and Problem Identification Table 4.1. Parameters used in WLAN and Fixed WiMAX

Standard

N

Nd

Np

Modulation

WLAN (IEEE 802.11a/g)

64

48

4

4-QAM, 16-QAM, 64-QAM

Fixed WiMAX (IEEE 802.16d)

256

192

8

4-QAM, 16-QAM, 64-QAM

0

10

Ideal WiMAX, 4-QAM WiMAX, 16-QAM WiMAX, 64-QAM WLAN, 4-QAM WLAN, 16-QAM WLAN, 64-QAM

Pr[P AP R > P AP R0 ]

−1

10

PLC with P AP Rpreset =4.5 dB

−2

10

−3

10

−4

10

4

4.5

5

5.5

6

6.5

P AP R0 (dB) Figure 4.1. CCDFs of PAPR for PLC applied on various OFDM standards 0

10



−1

10

−2

PEC with Ac = σ x 104/20

10

−3

10

−4

10

3.5

4

4.5

5

5.5

6

P AP R0 (dB) Figure 4.2. CCDFs of PAPR for PEC applied on various OFDM standards

25

Chapter 4. Motivational Study and Problem Identification 0

10



−1

10

−2

10

ENLC with A = σ x 104.5/20 c = 0.465

−3

10

−4

10

4

4.5

5

5.5

6

6.5

P AP R0 (dB) Figure 4.3. CCDFs of PAPR for ENLC applied on various OFDM standards

4.1.1

Effect of Number of Sub-carriers

In order to evaluate the effect of number of sub-carriers, performance with WLAN (N = 64) and WiMAX (N = 256) is compared for all three transforms. Observations It can be seen that the performance degradation is larger in case of WLAN than that in WiMAX. With PLC, PAPR with WLAN at CCDF = 10−4 is 0.3 dB higher than the ideal value whereas with WiMAX, PAPR CCDF = 10−4 is 0.7 dB higher than the ideal value. Similar trends are observed in case of PEC and ENLC. Discussions In Section 3.2, it was shown that the amplitude of an OFDM signal has Rayleigh distribution by the central limit theorem (CLT) approximation. • CLT approximation is more accurate when larger number of random variables are added together. Hence, the Rayleigh distribution approximation is more valid for WiMAX as compared to WLAN. • According to Eq.(1.2), PAPR is evaluated per symbol. This means that the amplitude distribution within individual symbol duration must converge to the theoretical 26

Chapter 4. Motivational Study and Problem Identification

Rayleigh distribution. But Rayleigh distribution approximation describes the average or long-term distribution of an OFDM signal. If the number of samples per symbol is large, the local amplitude distribution, i.e., within one symbol duration, better conforms to the Rayleigh distribution. When number of sub-carriers ia larger, symbol duration and in turn the number of samples per symbol are larger. So the local distribution in WiMAX varies less around the average as compared to that in WLAN. Hence the companders perform better in case of WiMAX as compared to in WLAN. • In most existing works, OFDM with larger number of sub-carriers is used for performance evaluation of companders. Mostly WiMAX or DVB-T are used as in simulations. So the effect of number of sub-carriers is not as readily discernible from the CCDFs of PAPR as it is found to be in the simulation results in this section. Due to the above-discussed reasons, the companding transforms, that are designed with the Rayleigh distribution assumption, do not perform optimally when smaller number of sub-carriers are used.

4.1.2

Effect of Constellation Type

Observations Figs. 4.1, 4.2 and 4.3 show that, as compared to 4-QAM, variance of PAPR around the ideal value increases in case of 16 and 64-QAM. For fewer sub-carriers, i.e., in WLAN, the degradation is greater than that for larger number of sub-carriers, i.e., in WiMAX. For instance, with PLC (Fig. 4.1), PAPR at CCDF= 10−3 increases by 0.6 dB in WLAN whereas in WiMAX, it increases by 0.35 dB, when 4-QAM is replaced with 16-QAM. Similar trends are observed with PEC and ENLC. Discussions These observations can be explained by noting the fact that all these transforms are designed by assuming Rayleigh PDF, given in Eq. (3.4), for signal amplitude. The Rayleigh distribution is specified by a constant average power parameter σx2 .

27


• When the OFDM signal is constructed using 16 or 64-QAM, the average power does not remain equal to σx2 for all symbols; rather it becomes a random variable (explained in Chapter 6). • The average power of an OFDM symbol is gievn as follows:

S=

N L−1 N L−1 1 X 1 X |xn |2 = |Xk |2 N L n=0 N L k=0

(4.1)

The parameter of Rayleigh distribution used to model amplitude distribution is the expected value of S, i.e., σx2 = E[S]. The variance of S is not taken into account in conventional compander design. • Consequently, the amplitude distribution within an OFDM symbol duration exhibits deviation from the long-term average, which is more pronounced in case of fewer data carriers per symbol. This is a manifestation of the law of large numbers (LLN), which predicts that the average symbol power will be closer to its expected value if the number of superimposed carriers is larger. When sub-carriers per symbol are too few to constitute a large enough data set, the variance of average symbol power increases. This eventually leads to larger deviation of PAPR of the compander’s output from its desired/ideal value. • For all the companders proposed in the literature, PAPR reduction performance is only presented for 4-QAM based OFDM signals, so this problem apparent in Figs. 4.1, 4.2 and 4.3 cannot be identified from the simulation results found in existing works.

4.1.3

Applications

• There are several prevailing and emerging wireless communication technologies and standards that specify 4-QAM, 16-QAM and 64-QAM as modulation schemes in OFDM systems for various data rates. Applications/standards include – IEEE 802.11a/g for wireless local area network (WLAN) [43] – IEEE 802.16 for fixed and mobile worldwide interoperability for microwave access (WiMAX) [44] 28


– the European ETSI HIPERLAN/2 [43] – long term evolution (LTE) [45] – digital video broadcasting terrestrial (DVB-T) [46] Hence, 4, 16 and 64-QAM will be used for design and evaluation throughout this thesis. • WLAN and HIPERLAN/2 have similar OFDM parameters [43]. WiMAX and LTE specifications accommodate both small and large number of sub-carriers, ranging from 128 to 2048 to cover a wide range of bandwidths [44, 45]. • Moreover, it has been shown in [12,13,15,19] that increase in BER due to companding noise is also larger with 16-QAM than that with 4-QAM. Thus, with conventional companding, there are several practical applications in which significant degradation can be expected in the overall performance of the system due to the above-discussed problem.

4.2

Problem Formulation

• It is required to design companders that incorporate the deviation from Rayleigh distribution from symbol to symbol. This can be done by enabling the compander adapt to the changing amplitude distribution from one symbol to another. In Chapter 5, adaptive companding is proposed to solve this problem. • For OFDM signal constructed from higher order QAM, the stochastic nature of average symbol power should also be taken into account, in addition to the amplitude distribution. The stochastic modeling of average symbol power and its relationship with amplitude is presented in Chapter 6. Using the developed model, two schemes, adjustable parameter companding (APC) and adaptive constellation scaling (ACS), are proposed to cater for the randomness of symbol power.

29

Chapter 5

Adaptive Companding As discussed in Chapter 4, OFDM signal amplitude, on average, is Rayleigh distributed but the distribution can vary significantly from symbol to symbol, especially when constellation size increases. In this chapter, a novel adaptive companding scheme is proposed along with its design methodology, that aims at optimizing the compander performance by accommodating this variation in its design. This is achieved by designing compander parameters separately for statistically dissimilar symbols in OFDM waveform and making the compander select from these parameters during run-time according to the features of input symbols.

5.1

Overview of Contributions

The main idea in the adaptive companding scheme, presented in this chapter, is to make the compander select its parameters from a pre-determined set according to the statistical feature(s) of the input OFDM symbol. The selection of parameters is done in such a way that the change in average power and overall amplitude distribution from symbol to symbol can be accounted for in the companding transformation. • This is demonstrated to have greater control over the output signal characteristics and enhanced PAPR reduction capability while keeping the mean signal distortion same as in case of the corresponding deterministic or fixed compander. Having greater control over the signal characteristics is desirable because it can ensure higher efficiency of non-linear components in the transmitter, including HPA and digital-to-analog converter (DAC).

30

Chapter 5. Adaptive Companding

• The design framework used in this scheme makes it realizable to exploit the system’s tolerance for data rate loss and surplus computational, memory and power resources in the system for PAPR reduction. Hence the proposed adaptive companding scheme has the ability to expand the trade-space over bandwidth efficiency, power requirements and computational complexity in addition to PAPR reduction performance, error performance and out-of-band radiation levels, thereby integrating the flexibilities of probabilistic and deterministic schemes. This means that the system’s adaptability to input and channel conditions will be enhanced because now it can have more possible regions of operation as compared to those viable for systems with other PAPR reduction schemes.

5.2

General Concept of Adaptive Companding

In general, companders are deterministic functions designed to modify OFDM signal amplitude after modulation operation. The transforms are designed by assuming that the amplitude of the signal is a Rayleigh random process. However, the Rayleigh process approximation applies for large data sets, which in this case is the complete OFDM waveform comprising of sufficiently large number of symbols. In other word, Rayleigh distribution is the average distribution of OFDM signals. The sample amplitude distribution within one symbol duration does not always conform well enough to the theoretical distribution. For comparatively smaller number of sub-carriers and hence fewer samples per symbol period, the deviation of sample distribution from the average Rayleigh distribution is more pronounced. This deviation becomes even larger when higher order QAM is used for modulation instead of 4-QAM or PSK. This means that the companders designed using Rayleigh PDF are, in effect, designed for an average OFDM symbol. The deviation of amplitude statistics of individual OFDM symbols from the Rayleigh distribution is not taken into account. In the compander output, this deviation is manifested as increased variances of signal attributes, like average signal power and PAPR, around their desired or preset values. The variances imply that the compander under-performs for some symbols and over-performs for others. In other words, PAPR is reduced below the ideal value for some symbols while it is larger than the desired value in others

31


Main Idea of Adaptive Companding The idea of proposed adaptive companding scheme is based on the premise that by introducing input data dependencies in compander parameter evaluation or selection, the deviation of sample distribution from the average Rayleigh distribution can be accounted for. Hence, the deviation of output signal characteristics from design specifications can also be reduced. In this way, the compander performance would be closer to optimal; instead of over-performing or under-performing, it will be able to perform as required. This is because it will now adjust according to the amplitude distribution of individual symbols. As a result, PAPR and average symbol power will vary less around the ideal value. Smaller variance of signal power along with reduced PAPR will facilitate smaller required input back-off (IBO) and hence improved power efficiency of HPA. Moreover, the mean signal distortion remains same as in case of the corresponding fixed compander. Basic Design Approach The design approach for the proposed adaptive companding scheme is elaborated as follows: The OFDM waveform (ignoring the cyclic prefix) can be thought of as a set of symbols. This set contains all possible OFDM symbols generated according to a specific standard. In this set, the amplitude statistics of some symbols resemble more closely than others. This set is divided into subsets in such a way that within each subset, the variation of amplitude statistics affecting the compander performance is smaller as compared to that in the complete set. The compander parameters are then calculated for each of these subsets separately and independently. Since the amplitude statistics are now incorporated in the compander design, the output of the compander can be expected to exhibit better concordance with the original design objectives.

5.3

Formal Definitions for Adaptive Companding

Statistical Feature The OFDM symbols will be classified into subsets based on a statistical feature that quantifies the difference among amplitude distributions of individual OFDM symbols. Let S denote the random variable associated with the statistical measure of signal amplitude,

32


on the basis of which the OFDM waveform is divided into J subsets. Definition of the statistical measure S depends upon the form of compander function, such that change in S is causative of change in compander output from its desired value. Also, it should be such a measure of signal amplitude that differs from one symbol to another but is assumed to be a constant according to the theoretical Rayleigh distribution Criteria for the Classification of OFDM Symbols We define J disjoint, consecutive intervals of S: [s0 , s1 ], (s1 , s2 ], ..., (sJ−1 , sJ ]

(5.1)

where s0 < s1 < s2 < ... < sJ−1 < sJ s0 = min(S), sJ = max(S) and s1 , s2 , . . . , sJ−1 are selected such that the variance of S is equal within all intervals given above, Var [S | s0 ≤ S ≤ s1 ] = Var [S | s1 < S ≤ s2 ] = ... = Var [S | sJ−1 < S ≤ sJ ]

(5.2)

≤ Var(S) For J ≥ 2, S in each interval varies over a smaller range than max (S)−min (S). Thus, variance of S within each interval will also be smaller, i.e., Var[S | sj−1 < S ≤ sj ] Ac    x,       Ai , −1 TP LC (x) = x − (1 − k)Ac    ,   k    A , c

(5.8)

x ≤ Ai Ai < x ≤ kAi + (1 − k)Ac

(5.9)

kAi + (1 − k)Ac ≤ Ac x > Ac

Ac is determined by setting a preset value of PAPR such that Ac = σx 10P AP Rpreset /20 and Ai and k are determined according to following design objectives and constraints: average power remains constant and companding distortion is minimized. The mathematical 36

Chapter 5. Adaptive Companding formulation for these conditions is given in Sub-section 3.4.5. D = (T (x) − x)2 is used as the measure of distortion. Mean distortion is given by σc2 = E[(T (x) − x)2 ] and is shown to be related to receiver’s signal-to-noise ratio (SNR) in [13].

5.4.2

Problem Formulation in Adaptive Compander Design Framework

In a conventional deterministic compander, the parameters Ac , k and Ai in Eq. (5.8) are constant. In adaptive companding, they are selected from a set according to the value of statistical feature S. Ac depends upon preset PAPR specification and average signal power, so it will remain constant for all OFDM symbols in case of PSK and 4-QAM. However, it may be varied for other QAM constellations.

(k, Ai , Ac ) =

   (k1 , Ai1 , Ac1 ),      (k2 , Ai2 , Ac2 ),

S(xn ) ∈ [s0 , s1 ] S(xn ) ∈ (s1 , s2 ]

(5.10)

   : :      (k , A , A ), S(x ) ∈ (s , s ] J iJ cJ J−1 J n For j th interval of S, (kj , Aij , Acj ) are calculated using the modified form of design equations given in Sub-section 3.4.5. Modified Design Equations • Clipping Level: Clipping level is defined for symbols in each subset separately. 2 2 is given as follows: , where σxj Since P AP Rpreset = 10 log10 A2cj /σxj

2 σxj

Z∞ =

x2 fA (x | sj−1 < S ≤ sj ) dx

(5.11)

0

Acj is given as follows: Acj = σxj 10P AP Rpreset /20

(5.12)

• Companding Distortion: Companding distortion is defined for each of the J subsets individually instead of for the whole waveform as in Sub-section 3.4.5. Eq.

37


(3.24) is modified as follows: 2 σcj =(kj − 1)2 I2j (Aij , Acj ) + I2j (Acj , ∞) − 2(1 − kj )2 Acj I1j (Aij , Acj )

(5.13)

+ (1 − kj )2 A2cj I0j (Aij , Acj ) + A2cj I0j (Acj , ∞) − 2Acj I1j (Acj , ∞) where I0j , I1j , I2j are given as follows: Zb Inj (a, b) =

xn fA (x | sj−1 < S ≤ sj ) dx

(5.14)

a

Estimator for the integral is given Sub-section 5.4.5. • Average Power Constraint: The constraint on average power is also defined for each subset individually instead of assuming same distribution for all symbols as in Sub-section 3.4.5. Eq. (3.20) is modified as follows: I2j (Aij , Ac ) + A2c I0j (Aij , Ac ) − 2Ac I1j (Aij , Acj ) kj2 −2A2cj I0j (Aij , Acj ) + 2Acj I1j (Aij , Acj ) kj + 2 Acj I0j (Aij , ∞) + I2j (0, Aij ) − σx2 = 0

(5.15)

2 Now (kj , Aij , Acj ) are such that the companding distortion σcj is minimized while keeping 2 unchanged for a given value of P AP Rpreset . Alternatively, some other average power σxj

constraint on average power can also be specified and companding distortion can be minimized for that average power. Eqs. (5.12), (5.13) and (5.15) represent J sets of design equations each of which will be solved separately and independently for j = 1, 2, ..., J using the Algorithm 1 and the estimations given in Sub-section 5.4.5. Configuration Control in Adaptive Companding In addition to optimizing the compander performance for the given objectives and constraints, the incorporation of new parameters in the problem framework provides additional degrees of freedom in specifying the objectives and constraints as compared to the fixed companders. For example, the constraint on average power can differ from subset to subset. Similarly, clipping level can also be varied. Consequently, more types of trade-offs

38


are possible. In fixed companders, PAPR reduction can be improved only by increasing signal distortion but in this case, it is possible to realize a number of operating points for the system with different levels of reduced PAPR at same signal distortion level, while the trade-offs for PAPR reduction are shifted to other, possibly more tolerant system attributes. In this way, most of the expendable resources in the system can be exploited for performance configuration. This enhances the system in terms of flexibility, efficiency and adaptability to input and channel conditions.

5.4.3

Selection of Statistical Feature

The statistical feature S used to select compander parameter should be such a measure on signal amplitude that the compander operation is directly affected by it, which means that this feature is being used in the calculation of compander parameters. For 16-QAM based OFDM For constellations with changing symbol energy like 16-QAM, fixed compander assumes 2 remains constant and equal to the long-term average power of that average power σxj 2 is not equal in the OFDM waveform. However, when considering individual symbols, σxj

all symbols. Also average power is the feature that affects the compander performance because it is being used in the evaluation of its parameters as shown in Eqs. (5.12), (5.13) and (5.15). This case will be analytically modeled in Chapter 6. In adaptive companding design framework, the average symbol power is used as the feature to adapt.

S = P oweravg

N L−1 N L−1 1 X 1 X 2 = |xn | = |Xk |2 N L n=0 N L k=0

(5.16)

2 2 So now instead of using same σxj = σx2 in all symbols, σxj can be set equal to the average

symbol power in j th subset.

2 σxj =

Z∞

x2 fA (x | sj−1 < S ≤ sj ) dx = I2j (0, ∞)

0

39

(5.17)


For 4-QAM based OFDM In PSK and 4-QAM based OFDM signals, average power remains constant for all symbols. So the input feature must be based on how the overall amplitude distribution affects the compander performance. In the compander function under consideration, peak power is reduced by clipping operation and the power deficiency created due to clipping is compensated by expanding the smaller amplitudes. Since clipping level is set by Eq. 2 remains constant for all j = (5.12), it will remain constant because average power σxj

1, 2, ..., J. But the exact amount of power deficiency due to clipping changes from symbol to symbol and hence the requirement of power compensation also changes. The change in amount of power deficiency is due to the fact that the number of samples being clipped and their amplitudes are not exactly similar in all symbols. If compander parameters are evaluated while incorporating this change, its output will be closer to optimal, which makes it pertinent to choose S as the difference between the powers of original and clipped signals. Hence the statistic S is defined as follows: S = P owerorig − P owerclipped =

X

A2 − A2c

(5.18)

A∈{|xn |} A>Ac

where {|xn |} = {|x0 |, |x1 |, . . . , |xN L−1 |}.

5.4.4

Classification of OFDM Symbols

Intervals on S are calculated according to Eqs. (5.1) and (5.2) using simulated data comprising of 200000 realizations. Simulation parameters are same as given in Table 4.1 for Fixed WiMAX. If J is a power of two, then the intervals of S are found by successively bisecting maximum variance intervals using Algorithm 2. Classification Algorithm The feature S is a random variable which can be either discrete or continuous. Its distribution can either be symmetric or skewed. The histogram of S used for 4-QAM, given in Eq. (5.18), are shown in Fig. 5.2. It can be seen that the power deficiency due to clipping can significantly vary from symbol to symbol, which leads to the degradation in the compander’s performance. 40


5000

P AP Rpreset = 4.5 dB S = P ower orig − P ower clipped

Frequency

4000 3000 2000 1000 0

5

10

15

20

25

30

35

40

45

50

Statistical Feature S 5000

P AP Rpreset = 4 dB S = P ower orig − P ower clipped

Frequency

4000 3000 2000 1000 0 10

15

20

25

30

35

40

45

50

55

60

Statistical Feature S Figure 5.2. Histograms of S of OFDM symbols (WiMAX with 4-QAM)

The analytical modeling of S used for 4-QAM is not possible due to the correlation properties of the signal envelope. The average power feature used for 16-QAM is modeled in the next chapter. However, for adaptive companding, a generic approach is used that can be applied for both constellation types. The evaluation of s0 , s1 , ..., sJ is done using the statistic S calculated from simulated OFDM symbols. The working of the algorithm is explained as follows: The entire range of S, i.e., max(S)−min(S), is divided into two such that the variances of S in the two new intervals are approximately equal. The variances are sample variances evaluated using the input data set. Subsequently, the new interval with larger variance of S is selected and divided into two new intervals satisfying the same constraint on variances of S. Hence, in each iteration, the current interval with maximum variance of S is divided into two, such that the two new intervals are approximately equal in variance of S. The process continues until J intervals are obtained.

41

Chapter 5. Adaptive Companding Algorithm 2 Classification of OFDM Symbols 1: Input data set Sdata , comprising of realizations of S. 2: Input number of subsets J. 3: Update Sdata = Sdata − O, where O = {x | x ∈ Sdata ∧ (x < m0 ∨ x > m1 )}. m0 , m1 are such that Pr[S ˆ< m0 ], Pr[S ˆ> m1 ] < ρ(= 10−5 ). 4: Set s0 = min(S), sJ = max(S). 5: Initialize array Sarr = [s0 , sJ ]. 6: Initialize variables u = m0 , v = m1 . 7: Find w ∈ (u, v], such that it minimizes |Var(S | u < S ≤ w) − Var(S | w < S ≤ v)|, where the conditional variances are evaluated as follows: Var(S | a < S ≤ b) =Var(datasetab ), where datasetab = {x | x ∈ Sdata ∧ a < x ≤ b}. 8: Append w to Sarr . 9: Sort Sarr in ascending order. 10: Find k ∈ {1, 2, 3, ..., length(Sarr ) − 1}, such that Var(S | sk < S ≤ sk+1 ) is maximum. 11: Update u = sk , v = sk+1 . 12: If length(Sarr ) < J + 1, repeat steps 7-11; Otherwise output Sarr = [s0 , s1 , s2 , ..., sJ ] and terminate.

5.4.5

Conditional Probability Distributions

In order to evaluate the integrals in Eqs. (5.11), (5.15) and (5.13), the conditional probability distributions are required. For each of the J subsets found according to Eq. (6.14), conditional CDFs are estimated using simulated data as follows: Number of samples with A ≤ x in j th subset FÂ (x | sj−1 < S ≤ sj ) = Total number of samples in j th subset

(5.19)

where x varies in small increments from zero to maximum amplitude present in the complete data set and s0 , s1 , ..., sJ are outputs of Algorithm 2. PDFs can be estimated using numerical gradient of respective CDFs. Fig. 5.3 shows the estimated curves for J = 4 for 4-QAM modulation using the feature S, given in Eq. (5.18). Since CDF curves have smaller approximation error, they are used to estimate the integrals in Eqs. (5.11), (5.15) and (5.13) as follows: Zb

Zb

n

x fA (x | sj−1 < S ≤ sj ) dx =

Inj (a, b) = a

= xn FA (x | sj−1 < S ≤ sj ) |ba −

a b Z

xn

d FA (x | sj−1 < S ≤ sj ) dx dx

nxn−1 FA (x | sj−1 < S ≤ sj ) dx

a

(5.20)

≈ b FÂ (b | sj−1 < S ≤ sj ) − an FÂ (a | sj−1 < S ≤ sj ) n

Zb − |a

nxn−1 FÂ (x | sj−1 < S ≤ sj ) dx {z

}

Numerical integration using Trapezoidal Rule

42

Chapter 5. Adaptive Companding 1

FA (x) FÂ (x | s0 ≤ S ≤ s1 ) FÂ (x | s1 < S ≤ s2 )

0.8

CDF

0.6

FÂ (x | s2 < S ≤ s3 ) FÂ (x | s3 < S ≤ s4 )

0.4 0.2 0

(a)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.5

2

PDF

fÂ (x | s2 < S ≤ s3 ) fÂ (x | s3 < S ≤ s4 )

0.5

(b)

1.8

fA (x) fÂ (x | s0 ≤ S ≤ s1 ) fÂ (x | s1 < S ≤ s2 )

1

0

1.6

Amplitude x

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Amplitude x

Figure 5.3. Estimated conditional distributions. (a) Estimated CDFs and (b) estimated PDFs

where a and b are limits of integration, n = 0, 1, 2 and j = 1, 2, ..., J. Limits at infinity in Eqs. (5.11), (5.15) and (5.13) will be replaced by maximum amplitude for which FÂ (.) is available. Now the J sets of design equations can be solved for (kj , Aij , Acj ) using the optimization algorithm in Sub-section 3.4.5 along with the integral estimator in Eq. (5.20). Fig. 5.4 shows the compander functions designed using the conditional distributions shown in Fig. 5.3. Figs. 5.5–5.7 show compander functions designed for other values of J with 4 and 16-QAM constellations.

5.4.6

Evaluation of Compander Parameters

The design methodology is summed up in Algorithm 3. Algorithm 3 Design of Adaptive Compander 1: 2: 3: 4: 5: 6: 7:

Input data set containing OFDM symbols OF DM Symbols. Input number of subsets J. Evaluate S for the symbols in OF DM Symbols to generate data set Sdata . Run Algorithm 2 to get s0 , s1 , ..., sJ . Classify OFDM symbols according to Eq. (6.14). Estimate conditional CDFs using Eq. (5.19) for each subset. For each j = 1, 2, ..., J, run Algorithm 1 using the estimated integrals in Eq. (5.20) and estimated CDFs in Eq. (5.19), to get (kj , Aij , Acj ).

43


1

Constant clipping level

T (x)

0.8

0.6

0.4

0.2

0

T (x) = x T1 (x) T2 (x) T3 (x) T4 (x)

Varying slopes

Varying inflexion points

0

0.2

0.4

0.6

0.8

1

1.2

Amplitude x

Figure 5.4. Transforms for adaptive companders, J = 4, M = 4 1.2

T (x)

1

0.8

Constant clipping level

0.6

T1 (x) T2 (x) T3 (x) T4 (x) T5 (x) T6 (x) T7 (x) T8 (x)

0.4

Varying slopes

0.2


0

0

0.2

0.4

0.6

0.8

1

1.2

Amplitude x

Figure 5.5. Transforms for adaptive companders, J = 8, M = 4

5.5

Performance Evaluation

The performance of the proposed companding scheme in terms of PAPR reduction, BER, power spectral density (PSD), computational complexity and amount of required side information is evaluated using simulated OFDM symbols. The scheme is simulated in a number of different scenarios to demonstrate its configurability. 44


3

2.5

2

T (x)

Varying clipping levels 1.5

1

Constant slopes

0.5

0

T1 (x) T2 (x) T3 (x) T4 (x)


0

0.5

1

1.5

2

2.5

3

Amplitude x

Figure 5.6. Transforms for adaptive companders, J = 4, M = 16 3

2.5

2

T (x)

Varying clipping levels T1 (x) T2 (x) T3 (x) T4 (x) T5 (x) T6 (x) T7 (x) T8 (x)

1.5

1

Constant slopes

0.5

0


0

0.5

1

1.5

2

2.5

3

Amplitude x

Figure 5.7. Transforms for adaptive companders, J = 8, M = 16

5.5.1

Simulation Setup

The OFDM symbols are simulated according to physical layer specifications given in IEEE 802.16d standard used in Fixed Worldwide Interoperability for Microwave Access (WiMAX). Oversampling factor L is 4. Parameters for the standard are given in Table 45


4.1.

5.5.2

With 4-QAM based OFDM

Figs. 5.8–5.10 show performance evaluation for OFDM system with 4-QAM modulation. PAPR Reduction Performance In Fig. 5.8, PAPR reduction performance evaluation is presented for P AP Rpreset =4 dB, 4.5 dB and 5 dB. For each case, fixed and adaptive versions of the compander are simulated. Statistical feature S is same as given in Eq. (5.18). It can be clearly seen that as the number of subsets J increases from 2 to 32, realized PAPR values come closer to their respective preset values. At CCDF = 10−4 improvement of approximately 0.25 dB for J = 32 is observed as compared to fixed compander. 0

10

P AP Rpreset = 4 dB

Fixed Adap. Adap. Adap. Adap. Adap.

P AP Rpreset = 4.5 dB


−1

10

J J J J J

=2 =4 =8 = 16 = 32

−2

10

P AP Rpreset = 5 dB −3

10

−4

10

3.8

4

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

P AP R0 (dB) Figure 5.8. CCDFs of PAPR of OFDM signals transformed by fixed and adaptive companders. OFDM signals are based on 4-QAM constellation.

Error Performance BER performance over AWGN channel for all the cases is shown in Fig. 5.9. For each value of P AP Rpreset , error performance remains similar for fixed and adaptive companders. Six coinciding BER curves, one for fixed compander and five for adaptive companders with J =2,4,8,16 and 32, are obtained in each case. Hence PAPR is reduced without increasing BER, which is not possible in deterministic companding. 46

Chapter 5. Adaptive Companding −1

10

Six coinciding curves (Fixed and Adap. with J = 2, 4, 8, 16, 32) Six coinciding curves (Fixed and Adap. with J = 2, 4, 8, 16, 32)

−2

10

Six coinciding curves (Fixed and Adap. with J = 2, 4, 8, 16, 32)

−3

BER

10

−4

10

Original OFDM Comp. Fixed. P AP Rpreset = 4 dB Comp. Fixed. P AP Rpreset = 4.5 dB Comp. Fixed. P AP Rpreset = 5 dB

−5

10

−6

10

0

2

4

6

8

10

12

Eb /N0 (dB)

Power Spectral Density (dB/rad/sample)

Figure 5.9. BER performance of original OFDM signal and OFDM signal transformed with fixed and adaptive companders over AWGN channel using 4-QAM modulation.

Two coinciding curves (Fixed and Adap.) −10 for P AP Rpreset = 5dB

Original OFDM Comp. Fixed. Comp. Adap. J = 32 Comp. Fixed Comp. Adap. J = 32 Comp. Fixed Comp. Adap. J = 32

−5

−15 −20

Two coinciding curves (Fixed and Adap.) for P AP Rpreset = 4.5dB Two coinciding curves (Fixed and Adap.) for P AP Rpreset = 4dB

−25 −30 −35 −40 −45 −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Normalized Frequency (xπ rad/sample) Figure 5.10. PSDs of original OFDM signal and OFDM signal transformed with fixed and adaptive companders. OFDM signals are based on 4-QAM constellation.

OBI Performance In Fig. 5.10, average PSDs, calculated using the periodogram estimator, are shown to be coincident for fixed compander and the best case (with respect to PAPR reduction performance/data rate loss trade-off) of adaptive (J = 32) for all the three values of preset PAPR values. The observations of BER and PSD show that the average companding distortion remains same in fixed and adaptive companders, as predicted by Eq. (5.7).

47


5.5.3

With 16-QAM based OFDM

Figs. 5.11–5.15 show performance evaluation with 16-QAM constellation. Simulation Setup As discussed in Sub-section 5.4.2, the incorporation of new variables in the design framework makes it realizable to obtain many different operating conditions for the system using the same companding transform. This type of flexibility is important in PAPR reduction schemes because it allows manageable trade-offs in many different situations and the same technique can meet performance constraints in many applications. It has been shown in [1] that none of the existing PAPR reduction techniques can provide good trade-offs in all situations. The proposed adaptive companding scheme has the capability that it allows the distribution of trade-offs for PAPR reduction among error performance (BER), spectral spreading (PSD), computational complexity, data rate loss and average power, which is not possible in any conventional PAPR reduction scheme [1, 37]. In order to illustrate the all-encompassing trade-space and enhanced flexibility in obtaining various operating conditions for the system with the proposed design framework, companders are designed and evaluated in five different scenarios. 2 = σx2 in Eq. • Scenario (i): S = P oweravg , Acj = Ac = σx 10P AP Rpreset /20 , σxj

(5.15), where σx2 is the long-term average power of the OFDM waveform. These conditions aim at providing such a power scaling that all symbols can be transmitted at same power level σx2 . Since average power is same, clipping level will also be constant, according to Eq. (5.12). Figs. 5.11–5.13 show evaluation for this case with J = 2, 4, 8, 16, 32. 2 • Scenario (ii): S = P oweravg , Acj = σxj 10P AP Rpreset /20 . σxj in Eqs. (5.11) and

(5.15) are calculated using Eq. (5.17). In this case, compander is designed to maintain the average power of individual symbols. Figs. 5.11, 5.14 and 5.15 show results for this case with J = 16. Additional power scaling will be needed at the transmitter if it is required to maintain constant transmission power, but BER will be improved due to the elimination of unnecessary clipping as observed in Fig. 5.14. • Scenario (iii): S = P oweravg . For j = 1, 2, ..., J/2, Acj = σx 10P AP Rpreset /20 and 2 2 σxj = σx2 . For every j = J/2 + 1, J/2 + 2, ..., J, Acj = σxj 10P AP Rpreset /20 and σxj in

48


Eqs. (5.12) and (5.15) are calculated using Eq. (5.17). In this case, the power of 2 those symbols in which σxj ≤ σx2 is increased to overall average power σx2 and the 2 > σx2 , power is kept unchanged. According to these conditions, symbols in which σxj 2 σxj is set in Eqs. (5.12) and (5.15). This is done to increase the SNR for low power

symbols by a small increase in average power resulting in overall reduction in BER as shown in Fig. 5.14. Increase in average power results in increase in both in-band and out-of-band power which means that the side-lobe level is increased as shown in Fig. 5.15. Results for this case are shown in Figs. 5.11, 5.14 and 5.15 with J = 16. • Scenario (iv): Two features are used in classification. S1 = P oweravg , S2 = P owerorig − P owerclipped . OFDM waveform is divided into J1 subsets based on S1 . Each of the J1 subsets is further divided into J2 subsets based on S2 . Acj1 = σxj1 10P AP Rpreset /20 , for j1 = 1, 2, ..., J1 . Clipping level varies in each of J1 subsets but in each of the J2 subsets of every j1th subset, it remains constant. Figs. 5.11, 5.14 and 5.15 show results for this case with J1 = 8, J2 = 4 and J1 = 8, J2 = 8. 2 2 2 . + ∆σxj = σxj • Scenario (v): S = P oweravg . Acj = σxj 10P AP Rpreset /20 and σxj

Overall average power is increased to raise SNR and hence reduce BER. Also increase in average power results in raised side-lobe level as shown in Fig. 5.15, in addition to increased in-band power. Figs. 5.11, 5.14 and 5.15 show results for this case with J = 16. The changes in amplitude distribution will be incorporated by using the specified S in conditional distribution, given in Eqs. (5.13) and (5.15), in all five cases. In scenario (iv), conditions on S1 and S2 are simultaneously imposed. The conditional distributions are estimated as described in Sub-section 5.4.5. PAPR Reduction Performance In Fig. 5.11, PAPR reduction performance comparison of fixed and adaptive companders for P AP Rpreset =4.5 dB is presented. The results of CCDF of PAPR obtained for scenarios (i), (ii), (iii) and (v) were found to be approximately similar. Only those for (i) are shown in Fig. 5.11. This shows that PAPR reduction performance depends upon J and in turn the extent of similarity of feature S within a subset. Improvement of approximately 0.31

49


10

Coincident curves obtained in (i), (ii), (iii) and (v)

Pr[ PAPR > PAPR0 ]

−1

10

P AP Rpreset = 4.5 dB Fixed Adap. Adap. Adap. Adap. Adap. Adap. Adap.

−2

10

−3

10

J =2 J =4 J =8 J = 16 J = 32 J1 = 8, J2 = 4 J1 = 8, J2 = 8

(iv)

−4

10

4

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5

5.1

5.2

5.3

P AP R0 (dB) Figure 5.11. CCDFs of PAPR of OFDM signals transformed by fixed and adaptive companders. OFDM signals are based on 16-QAM constellation.

dB at CCDF = 10−4 is achieved with J = 16 and J = 32 as compared to the fixed compander. It can be seen that as J increases in (i), (ii), (iii) and (v), the CCDF approaches the CCDF of 4-QAM with fixed compander in Fig. 5.8. This means that since within each subset, variation of average power is very small, the amplitude statistics within the subsets approach those of PSK or 4-QAM. Hence the best PAPR reduction performance achievable, using average power in Eq. (5.16) as feature, is the performance of 4-QAM with fixed compander. In order to further reduce PAPR, a second feature S2 is required to incorporate the overall distribution variation, as done in case of 4-QAM. Further improvement of 0.12 dB is obtained with J1 = 8, J2 = 8, as shown in Fig. 5.11. BER and OBI Performances in Scenario (i) Figs. 5.12 and 5.13 show BER and PSD for companders designed according to conditions in scenario (i). Performance is similar to that of fixed one for J =2,4,8,16 and 32. BER and PSD remain unchanged irrespective of J, as predicted by Eq. (5.7). BER and OBI Performances in Scenario (ii) Figs. 5.14 and 5.15 show BER and PSD curves, respectively, for companders design in scenario (ii) for the best case (with respect to PAPR reduction/side information trade-

50


10

Six coinciding curves for Fixed and Adap. with J = 2, 4, 8, 16, 32

−1

10

−2

10

P AP Rp r eset = 4.5 dB

BER

−3

10

Original OFDM Fixed Comp. Adap. J Comp. Adap. J Comp. Adap. J Comp. Adap. J Comp. Adap. J

−4

10

−5

10

−6

10

=2 =4 =8 = 16 = 32

−7

10

0

2

4

6

8

10

12

14

16

18

20

Eb /N0 (dB) Figure 5.12. BER performance of original OFDM signal and OFDM signal transformed with fixed and adaptive companders in scenario (i), over AWGN channel, using 16-QAM modulation.


5

Original OFDM Comp. Fixed Comp. Adap. J Comp. Adap. J Comp. Adap. J Comp. Adap. J Comp. Adap. J

0 −5 −10 −15

Coinciding curves for Fixed and Adap. Companders with J = 2, 4, 8, 16, 32

−20

=2 =4 =8 = 16 = 32


−25 −30 −35 −40 −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Normalized Frequency (xπ rad/sample) Figure 5.13. PSDs of original OFDM signal and OFDM signal transformed with fixed and adaptive companders in scenario (i). OFDM signals are based on 16-QAM constellation.

off), i.e., J = 16. The improvement in performance is almost negligible for J = 32, due to which J = 16 is considered as the best case. PSD remains unchanged while BER at high SNR is slightly reduced which may be explained by the fact that unnecessary clipping is being avoided. So the performance floor at high SNR, due to clipping noise, has been lowered. From the observations in scenario (i) and (ii) and results in case of 4-QAM, it can be concluded that BER predominantly depends on clipping level. This is because the 51


10

−1

10

−2

10


BER

−3

10

Original OFDM Comp. Fixed Comp. Adap. (ii) J = 16 Comp. Adap. (iii) J = 16 Comp. Adap. (iv) J1 = 8, J2 = 8 Comp. Adap. (v) J = 16, 2 ∆σ xj = 0.4 dB

−4

10

−5

10

−6

10

−7

10

0

2

4

6

8

10

12

14

16

18

20

Eb /N0 (dB) Figure 5.14. BER performance of original OFDM signal and OFDM signal transformed with fixed and adaptive companders in scenarios (ii)-(v), over AWGN channel, using 16-QAM modulation.


5

Original OFDM Comp. Fixed Comp. Adap. (ii) J = 16 Comp. Adap. (iii) J = 16 Comp. Adap. (vi) J1 = 8, J2 = 8 Comp. Adap. (v) J = 16, 2 ∆σ xj = 0.4 dB

0 −5 −10 −15 −20

Coinciding curves for Fixed, Adap.(ii) and (iv)


−25 −30 −35 −40 −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Normalized Frequency (xπ rad/sample) Figure 5.15. PSDs of original OFDM signal and OFDM signal transformed with fixed and adaptive companders in scenarios (ii)-(v). OFDM signals are based on 16-QAM constellation.

companding noise due to the clipping operation is larger in amplitude as compared to noise due to expansion. As a result, it has greater impact on the BER performance. PSD remains unchanged in scenarios (i) and (ii). This shows that it depends upon average noise power, i.e., E[(T (x) − x)2 ]. This can be explained by considering the companded signal as a sum of original signal and companding noise. According to Eq. (5.7), average distortion remains constant in fixed and adaptive companders. This property was also verified through simulations by computing the distortion in all J subsets and

52


comparing the average with the theoretical expected value of D. BER and OBI Performances in Scenario (iii) Figs. 5.14, 5.15 show BER and PSD, respectively, for companders design in scenario (iii) for the best case (with respect to PAPR reduction/side information trade-off), i.e., J = 16. In this case, half of the symbols are transformed such that their power is increased to σx2 . Increase in average power improves error performance. For other half, power is 2 maintained at σxj and clipping level is changed. Overall average power is increases slightly,

which is manifested as raising of in-band and out-of-band power levels. BER and OBI Performances in Scenario (iv) Figs. 5.14, 5.15 show BER and PSD, respectively, for companders design in scenario (iv) for the best case (with respect to PAPR reduction/side information trade-off), i.e., J1 = 8, J2 = 8. PSD remains unchanged as overall average distortion is constant irrespective of J1 and J2 . BER at high SNR is slightly reduced which may be explained by the fact that unnecessary clipping is being avoided. So the performance floor at high SNR, due to clipping noise, has been lowered. BER and OBI Performances in Scenario (v) Figs. 5.14, 5.15 show BER and PSD, respectively, for companders design in scenario (v) for the best case (with respect to PAPR reduction/side information trade-off), i.e., J = 16. BER is reduced both due to overall increase in signal power and elimination of redundant clipping operation. In case of PSD, both in-band and out-of-band energy is increased due to the increase in overall average power. Number of Parameters The compander is specified by three parameters. So for every adaptation of the compander, we need to store these parameters. In cases when clipping level remains unchanged,

53


only one value of Ac is required to be stored, whereas J values of Ai and k are required. So the number of parameters needed to be stored in memory is 2J + 1. It was found that the slope k remains same for all subsets when clipping level is changed according to average power. Therefore the number of parameters needed to be stored in memory still remains 2J + 1 because now we need one value of k and J values of Ai and Ac . In case of scenarios (iv), J1 values of Ac are needed and J1 J2 values of Ai and k are required to be pre-calculated and stored. In Table 5.1, number of parameters is indicated as memory requirement in each of the case considered in simulation.

5.5.4

Side Information

The adaptive compander is not a deterministic transform. Its parameters change according to input symbols. Therefore, it is required to communicate the parameters to the receiver. Since there are some specific, finite number of adaptations of the compander, the parameters values can be encoded in a small number of bits of side information. In case of 4-QAM and scenarios (i), (ii), (iii) and (v) in 16-QAM, dlog2 Je bits of side information are required. In scenario (iv), dlog2 J1 J2 e bits are required to be sent with each symbol.

5.5.5


For the two features used in the simulations, computational complexity is given below. Amplitude calculations and companding function arithmetic are common in fixed and adaptive companders, for which complexity analysis and comparison is presented in [13]. Complexities of various companding transforms are also compared in Chapter 7. Approximately 1.32N L floating point multiplications and additions are required for fixed companding with P AP Rpreset =4.5 dB [13]. Additional complexity at the transmitter, involved in introducing adaptive behavior in the proposed scheme, is given below. • For feature in Eq. (5.16), S1 = P oweravg : Average power of the input symbol can be calculated with low complexity from the frequency domain representation using QAM values. Out of N L values in the sum, only Nd are data bearing subcarriers while rest are pilot and null sub-carriers. Hence Nd (192 in Fixed WiMAX) floating

54


point additions and one multiplication per symbol will be required if symbol to energy mappings for the constellation are pre-calculated as a look-up table (LUT). • For feature in Eq. (5.18), S2 = P owerorig −P owerclipped : On average, N L

R∞

fA (x)dx

Ac

(0.06N L for P AP Rpreset = 4.5 dB) floating point multiplications and additions are required per symbol. Additionally, J comparisons per symbol will be required to enable the adaptive compander select its parameters according to the value of calculated feature. In Table 5.1, {(S1 ) and {(S2 ) denote complexities of S1 and S2 , respectively, and represent the required number of arithmetic operations as given above.

5.5.6

Summary of Performance

The results presented clearly demonstrate that with the proposed scheme, it is possible to cover a wide range of operating conditions for the system with reduced PAPR without introducing added distortion; a feat not achievable with deterministic companders. The trade-offs are shifted to added computational complexity required for calculating the input feature, additional memory required to store parameter values and requirement of side information for decompanding. Table 5.1. Summary of Trade-offs involved in adaptive companding 16-QAM 4-QAM (i) (ii) (iii) (iv) Best PAPR at CCDF=10−4 BER OBI Side Information Complexity Memory requirement Average power

(v)

4.55 dB

4.8 dB

4.8 dB

4.8 dB

4.67 dB

4.8 dB

unchanged unchanged

unchanged unchanged

reduced unchanged

reduced increased

reduced increased

dlog2 Je

dlog2 Je

dlog2 Je

dlog2 Je

{(S2 )

{(S1 )

{(S1 )

{(S1 )

reduced unchanged dlog2 Je, J = J1 J2 {(S1 ) +{(S2 )

2J + 1

2J + 1

2J + 1

2J + 2

J1 + 2J1 J2

2J + 1

unchanged

unchanged

unchanged

increased

unchanged

increased

55

dlog2 Je {(S1 )

Chapter 6

Companding Schemes for OFDM Systems employing Higher Order QAM In Chapter 4, it was established that the PAPR reduction performance of the compander is affected by constellation type. For higher order QAM, the performance degrades as compared to that for PSK or 4-QAM. In this chapter, the probability distribution of amplitude in OFDM signals employing higher order QAM is analytically modeled. The derived probability distributions are then used in the design of two novel schemes for OFDM systems employing higher order QAM.

6.1

Difference between PSK and QAM based OFDM Systems

In Chapter 3, it was explained that the amplitude of OFDM signal is assumed to be Rayleigh distributed. The Rayleigh distribution is specified by an average power parameter σx2 . In PSK based OFDM systems, the average power of all OFDM symbols is constant. But in OFDM systems employing higher order QAM, like 16 or 64-QAM, this probabilistic model is insufficient to model the true characteristics of the signal. This is due to the stochastic nature of average symbol power, which alters the complete amplitude distribution from one OFDM symbol to another. Since the Rayleigh distribution is specified by a constant average power parameter, so the stochastic nature of symbol 56

Chapter 6. Companding Schemes for OFDM Systems employing Higher Order QAM

power renders it insufficient to approximate the true amplitude distribution of an OFDM symbol. In conventional companding schemes, all the symbols are transformed using the same deterministic function. Owing to the fact that the prevalent randomness of symbol power and its relationship with the symbol’s amplitude distribution is neglected in these schemes, the PAPR reduction performance significantly degrades when higher order QAM constellations are employed as modulation schemes, instead of 4-QAM or PSK. It should be noted here, that the Rayleigh distribution, with constant parameter, is an effective approximation of the average or long-term signal amplitude statistics, i.e., when a large number of OFDM symbols are collectively considered. But since PAPR is always evaluated symbol by symbol [1], so, for the design optimization of companding schemes, it is logical to consider the signal’s amplitude distribution correspondingly.

6.2


In this chapter, two novel schemes, namely adjustable-parameter companding (APC) and adaptive constellation scaling (ACS). The main idea is to enable the compander comply with the changing amplitude distribution from one symbol to another. The analytical probabilistic model of average symbol power and its relationship with the amplitude distribution is developed. The derived probability distributions are then utilized in the design of the proposed schemes, in such a way that symbol power variations are accommodated in the companding operation. Simulations are carried out to demonstrate that the proposed schemes outperform the conventional ones with regard to PAPR, BER and OBI. Particularly, the proposed schemes are capable of jointly reducing the PAPR and BER, while keeping OBI level unaffected, which is not possible in conventional signal distortion techniques. Like adaptive companding, these schemes also facilitate a more even distribution of trade-offs for PAPR reduction as compared to that realizable in traditional techniques, thereby increasing the system’s adaptability. Furthermore, the proposed schemes can be conveniently and effectively applied to enhance the performance of any companding transform.

57


6.3

Probabilistic Modeling of Average Power and Amplitude of OFDM Symbols

In this section, probability distribution of the average power of OFDM symbols is derived and the dependence of amplitude distribution on symbol power is modeled. This yields the theoretical framework necessary for the development of solutions proposed in Section 6.4.

6.3.1

Distribution of Average Symbol Power

Let S represent the random variable associated with the average power of an OFDM symbol. By Parseval’s theorem of discrete Fourier transform (DFT), we can write N L−1 N L−1 1 X 1 X 2 S= |xn | = |Xk |2 N L n=0 N L k=0

(6.1)

2 = d2min (M − 1)/6, Average energy of M-ary square QAM constellation is given by σSQAM

where dmin is the distance between nearest neighbors in the constellation. If σp2 is the average energy of pilot symbols, σx2 = E [S] = Let D = {d1 , d2 , ..., dNd } and P =

1 2 Nd σSQAM + Np σp2 NL

(6.2)

p1 , p2 , ..., pNp represent sets containing indexes of

data bearing sub-carriers and pilot sub-carriers, respectively. Then Xd1 , Xd2 , ..., XdNd are identical random variables. Let Xd represent any one of these random variable such that it can attain any value from a given QAM constellation. In case of 16 and 64-QAM, |Xd |2 is also a random variable, because all symbol vectors in the constellation do not have the same amplitude. This is in contrast with 4-QAM and PSK in which |Xd |2 is a constant. Symbol-to-energy mappings for 16-QAM and 64-QAM constellations are shown in Figs. 6.1 and 6.2, respectively. Using this mapping and assuming that all QAM symbols are equally likely, probability mass functions (PMF) of |Xd |2 for 16-QAM and 64-QAM are evaluated and shown in Fig. 6.3.

58


4

|X d |2 = 18 |X d |2 = 10 |X d |2 = 10

|X d |2 = 18

dmin = 2

ℑ [Xd ]

2

|X d |2 = 10

|X d |2 = 2

|X d |2 = 2

|X d |2 = 10

|X d |2 = 10

|X d |2 = 2

|X d |2 = 2

|X d |2 = 10

0

−2

|X d |2 = 18 |X d |2 = 10 |X d |2 = 10 |X d |2 = 18

−4 −4

−2

0

2

4

ℜ [Xd ]

Figure 6.1. Symbol-to-energy mapping for 16-QAM 8

2 2 2 2 |Xd |2 = 98 |Xd |2 = 74 |Xd | = 58 |Xd |2 = 50 |Xd |2 = 50 |Xd | = 58 |Xd | = 74 |Xd | = 98

6

2 2 2 2 2 2 |Xd | = 74 |Xd | = 50 |Xd | = 34 |Xd | = 26 |Xd |2 = 26 |Xd | = 34 |Xd | = 50 |Xd | = 74

4

2 2 2 |Xd |2 = 58 |Xd | = 34 |Xd |2 = 18 |Xd |2 = 10 |Xd | = 10 |Xd | = 18 |Xd |2 = 34 |Xd |2 = 58

2

2 |Xd |2 = 50 |Xd |2 = 26 |Xd | = 10 |Xd |2 = 2 |Xd |2 = 2 |Xd |2 = 10 |Xd |2 = 26 |Xd |2 = 50

ℑ[Xd ]

dmin = 2 2

0

−2

−4

2 |Xd |2 = 2 |Xd |2 = 10 |Xd | = 26 |Xd |2 = 50

2 |Xd |2 = 50 |Xd |2 = 26 |Xd |2 = 10 |Xd | = 2

2 2 2 2 2 2 2 |Xd |2 = 58 |Xd | = 34 |Xd | = 18 |Xd | = 10 |Xd | = 10 |Xd | = 18 |Xd | = 34 |Xd | = 58

2 2 2 2 2 2 |Xd |2 = 74 |Xd | = 50 |Xd | = 34 |Xd | = 26 |Xd | = 26 |Xd | = 34 |Xd |2 = 50 |Xd | = 74

−6

−8 −8

2 2 2 2 2 2 |Xd |2 = 98 |Xd | = 74 |Xd | = 58 |Xd | = 50 |Xd | = 50 |Xd | = 58 |Xd | = 74 |Xd |2 = 98

−6

−4

−2

0

2

4

6

8

ℜ[Xd ]

16-QAM d min = 2 1/2 1/4 0

(a)

p|Xd |2 (x)

p|Xd |2 (x)

Figure 6.2. Symbol-to-energy mapping for 64-QAM

2

10

x

18

1/8 1/16 0

(b)

64-QAM d min = 2

3/16

2

10

18

26

34

50

x

58

74

Figure 6.3. PMFs of symbol energy |Xd |2 for (a) 16-QAM and (b) 64-QAM

59

98


Now, S, given in Eq. (6.1), can be re-written as follows: S=

X 1 X 1 X 1 |Xk |2 + |Xk |2 + |Xk |2 N L k∈D N L k∈P NL ∪D)0 {z } | {z } | k∈(P{z | } constant

random

=0, (guard band and DC)

(6.3)

2 N 1 p 2 2 2 = |Xd1 | + |Xd2 | + ... + XdNd + σ NL NL p Let Sd = N LS − Np σp2 , so that 2 Sd = |Xd1 | + |Xd2 | + ... + XdNd 2

2

(6.4)

2 Each of |Xd1 |2 , |Xd2 |2 , ..., XdNd is distributed according to PMF p|Xd |2 (x). Since the PMF of sum of independent random variables is equal to the convolution of PMFs of individual random variables [47], so Sd has the following distribution: 2 (x) pSd (x) = p X 2 (x) ∗ p X 2 (x) ∗ ... ∗ p | d1 | | d2 | XdN d {z } |

(6.5)

Nd −1 convolutions

2 (x) = p where p X 2 (x) = p X 2 (x) = ... = p |Xd |2 (x). Now the PMF pSd (x) is | d1 | | d2 | XdN d transformed into pS (x) as follows:

Sd + Np σp2 pS (x) = Pr[S = x] = Pr =x NL

pS (x) = pSd N Lx − Np σp2

(6.6)

(6.7)

The derived PMFs and histograms of S, obtained from simulated OFDM symbols are compared in Figs. 6.4–6.7. The non-zero variance of symbol power around the mean σx2 can be clearly observed. It is the effect of this variance that is manifested in the degradation in the PAPR reduction performance observed in the simulation results presented in Chapter 4. Interestingly, it was found that, although the variance of S is larger in case of 64-QAM p than that in case of 16-QAM, but the relative standard deviation, i.e., Var(S)/σx2 , in both cases is almost the same. PAPR also represents peak power relative to average power. Hence, the variance of PAPR around the ideal value is almost identical for both constellations, as observed in Chapter 4. 60


0.08

p S (x)

WLAN, 0.04 16-QAM dmin = 2

Frequency

(a)

0 1.4

1.6

1.8

2

2.2

2.4

2.6

2.4

2.6

OFDM symbol power x

4000

WLAN, 16-QAM dmin = 2

2000

(b)

σ x2 = 2.0313

0 1.4

1.6

1.8

2

2.2

OFDM symbol power

Figure 6.4. (a) PMF of symbol power (b) Histogram of symbol power

p S (x)

0.02

Frequency

(a)

0

6

6.5

7.5

8

8.5

9

9.5

10

10.5

11

9.5

10

10.5

11


500 0

7

OFDM symbol power x

1000

(b)

σ x2 = 8.5313


0.01

6

6.5

7

7.5

8

8.5

9

OFDM symbol power


p S (x)

0.04 WiMAX,

16-QAM dmin = 2

Frequency

(a)

0 1.7

1.75

1.8

1.85

1.9

1.95

2

2.05

2.1

2.15

2.2

2.05

2.1

2.15

2.2

OFDM symbol power x

1500

WiMAX, 16-QAM 500 dmin = 2

1000

(b)

0 1.7

1.75

1.8

1.85

1.9

1.95

2

OFDM symbol power


0.01

p S (x)

WiMAX, 64-QAM dmin = 2

0

Frequency

(a)

(b)

7

7.2

7.4

7.6

7.8

8

8.2

8.4

8.6

8.8

9

9.2

9.4 9.5

8.6

8.8

9

9.2

9.4 9.5

OFDM symbol power x

300

WiMAX, 64-QAM dmin = 2 0

7

7.2

7.4

7.6

7.8

8

8.2

8.4

OFDM symbol power


61

Chapter 6. Companding Schemes for OFDM Systems employing Higher Order QAM 1

WLAN, 16-QAM

Cumulative Distribution Function

0.9 0.8 0.7

2 2 2 2 2 σx1 < σx2 < σx3 < σx4 < σx4

0.6

2 Theory, FA (x | S = σ x1 ) 2 ˆ Est., FA (x | S = σ x1 ) 2 Theory, FA (x | S = σ x2 ) 2 ˆ Est., FA (x | S = σ x2 )

0.5 0.4

2 Theory, FA (x | S = σ x3 ) 2 Est., FÂ (x | S = σ x3 )

0.3


0.2


0.1 0

0

0.5

1

1.5

2

2.5

3

3.5

Amplitude x

Figure 6.8. Comparison of proposed and simulated distributions

6.3.2

Distribution of OFDM Symbol Amplitude

Since S is found to deviate from σx2 , so Rayleigh PDF with constant parameter σx2 is not sufficient to describe the amplitude distributions of individual OFDM symbols. Rather, they can be more appropriately characterized by a family of conditional Rayleigh PDFs. The parameter of a Rayleigh distribution, belonging to the said family, can have any value for which pS (.) 6= 0. The probability that the amplitude distribution of a randomly generated OFDM symbol conforms to Rayleigh distribution with parameter x is equal to pS (x). The PDF of amplitude of a sample |xn | (represented by A) belonging to an OFDM symbol with S = y, is found by modifying the Rayleigh PDF as follows: 2 2x x fA (x | S = y) = exp − y y

(6.8)

The distributions in Eq. (6.8), for various values of S, were validated by comparing them with estimated distributions using simulated data. The comparison is shown in Fig. 6.8.

The conditional PDFs for a range of values of S are shown in Figs. 6.9 and 6.10. It can be seen that the amplitude distribution can significantly vary around the average

62

Chapter 6. Companding Schemes for OFDM Systems employing Higher Order QAM distribution fA (x | S = σx2 ). Also, this variation is larger in case of WLAN than that in WiMAX, which explains the difference between the performance of the two standards observed in Chapter 4. 0.7

! " fA x | S = σ x2

fA (x | S)

0.6


0.5 0.4

Decreasing S

Decreasing S

0.3 0.2 0.1 0

0

0.5

1

1.5

2

2.5

3

3.5

4

Amplitude x Figure 6.9. Conditional PDFs for WLAN with 16-QAM


! " fA x | S = σ x2

0.6

fA (x | S)

0.5 0.4

Decreasing S

0.3

Decreasing S

0.2 0.1 0

0

0.5

1

1.5

2

2.5

3

3.5

4

Amplitude x Figure 6.10. Conditional PDFs for WiMAX with 16-QAM

The joint PDF of amplitude and symbol power can be expressed as follows: 2 2x x fA,S (x, y) = pS (y) exp − y y

(6.9)

The joint distributions are shown in Fig. 6.11. For every possible value of S, there is a corresponding Rayleigh PDF for A. In other words, the random variable A is dependent upon S. A companding transform is designed to modify the amplitude distribution. Consequently, it will also yield closer-to-desired output if it recognizes and deals with this dependence of signal amplitude on symbol’s average power. To accomplish this objective, two novel schemes are presented in the next section, that modify the companding 63


fA,S (x, σx2 ) 0.03

fA,S (x, y)

fA,S (x, y)

0.015


0.01 0.005 0 2.2 3

0.01

y

2.4

2.2

2

1.8

1 0

fA,S (x, σx2 )

0.02

0 2.6 2


2

y

x

3 1.8

2 1.6

1.4

1 0

x

Figure 6.11. Joint probability distribution of average power and amplitude

operation to accommodate this dependence.

6.4

Proposed Solutions

It has been established that since the amplitude distribution within an OFDM symbol duration depends on the average power S of that symbol, so the companding operation should also adapt to S. This is the foundational idea for both the schemes presented in the Sub-sections 6.4.2 and 6.4.3. In order to cut down the overheads on complexity and throughput involved in these schemes, the OFDM symbols are first classified on the basis of S in Sub-section 6.4.1.

6.4.1

Classification of OFDM Symbols

Motivation Theoretically, S can attain values over a large range. For instance, in case of WLAN with 16-QAM, S can attain 105 values, ranging from 0.4063 to 3.6563. However, the probability of S having these values is of the order of 10−32 . pS (x) for this case is shown in Fig. 6.4(a) (complete range of S is not shown). Similarly in WLAN with 64-QAM, S can have 621 different values, with probabilities as small as 10−63 . This implies that it would be inefficient to force the companding operation adjust differently for every single value of S. Instead, the computational and bandwidth resources would be more costeffectively utilized if the process governing the adaptivity to S is designed by assigning appropriate weightages to average and outliers.

64


Criteria for Classification The OFDM symbols are classified in the same way as in adaptive companding. The ensemble of OFDM symbol will be divided into J subsets, such that within each subset, the variance of S of the constituent symbols is smaller as compared to that in the complete ensemble. Let [s0 , s1 ], (s1 , s2 ], ..., (sJ−1 , sJ ] be J consecutive and disjoint intervals of the symbol power S, where s0 < s1 < s2 < ... < sJ . s0 = min (S), sJ = max (S) and s1 , s2 , ..., sJ−1 are such that the variance within each interval is equal, i.e., Var [S | s0 ≤ S ≤ s1 ] = Var [S | s1 < S ≤ s2 ] = ... = Var [S | sJ−1 < S ≤ sJ ]

(6.10)

The conditional variances in Eq. (6.10), for j = 1, 2, ..., J, are evaluated as follows: 2 2 Var [S | sj−1 < S ≤ sj ] = E (S − σxj ) | sj−1 < S ≤ sj =

∞ X

2 x − σxj

2

(6.11) pS (x | sj−1 < S ≤ sj )

x=0 2 where σxj is the expected value of S within the subset j, i.e.,

2 σxj = E [S | sj−1 < S ≤ sj ] =

∞ X

x pS (x | sj−1 < S ≤ sj )

(6.12)

x=0 2 is the overall average power of the symbols belonging to the subset j. Using Hence σxj

the definition of conditional probability, the conditional PMF, used in Eqs. (6.11) and (6.12), is found as follows: Pr[S = x, sj−1 < S ≤ sj ] pS (x | sj−1 < S ≤ sj ) = Pr[sj−1 < S ≤ sj ]  pS (x)   , for sj−1 < x ≤ sj P p (x) S x∈(s ,s ] j−1 j =   0, otherwise

(6.13)

Algorithm 4 is used to evaluate s0 , s1 , ..., sJ , which is similar to Algorithm 2 used in adaptive companding. But in this case, the derived PMF is used instead of training data.

65

Chapter 6. Companding Schemes for OFDM Systems employing Higher Order QAM Algorithm 4 Evaluation of s0 , s1 , ..., sJ 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11:

Input arrays pS (x) and x, such that pS (x) 6= 0. Input number of subsets J. Set s0 = min(x), sJ = max(x). Initialize array Sarr = [s0 , sJ ]. Initialize variables u = s0 , v = s1 . Find w ∈ (u, v], such that it minimizes |Var(S | u < S ≤ w) − Var(S | w < S ≤ v)|, where the conditional variances are evaluated using Eqs. (6.11), (6.12) and (6.13). Append w to Sarr . Sort Sarr in ascending order. Find k ∈ {1, 2, 3, ..., length(Sarr ) − 1}, such that Var(S | sk < S ≤ sk+1 ) is maximum. Update u = sk , v = sk+1 . If length(Sarr ) < J + 1, repeat steps 6-10; Otherwise output Sarr = [s0 , s1 , s2 , ..., sJ ] and terminate.

Amplitude Distribution Within Subsets Now, the j th subset of OFDM symbols, xn = [x0 , x1 , ..., xN L−1 ], is defined as follows: {xn | S(xn ) ∈ (sj−1 , sj ]} where S(xn ) = (1/N L)

PN L−1 n=0

(6.14)

|xn |2 , i.e., the average power of symbol xn . As discussed

in Section 6.3, the amplitude distribution fA (.) of every sample belonging to an OFDM symbol is a function of the average power S of that symbol. Thus, by using the law of total probability [47] and Eq. (6.8), the amplitude distribution of a sample belonging to any symbol xn in j th subset is given by the conditional PDF as follows: fA (x | sj−1 < S ≤ sj ) =

X

fA (x | S = y) pS (y | sj−1 < S ≤ sj )

y∈(sj−1 ,sj ]

=

X y∈(sj−1 ,sj ]

2 2x x exp − pS (y | sj−1 < S ≤ sj ) y y

(6.15)

Eq. (6.15) is essentially the average of Rayleigh distributions with parameters in the range (sj−1 , sj ]. Alternatively, since the average power of all the samples in j th subset is 2 σxj , given in Eq. (6.12), and they are all Rayleigh distributed, so the conditional PDF in 2 Eq. (6.15) can also be approximated as a Rayleigh PDF with parameter σxj , i.e.,

fA (x | sj−1 < S ≤ sj ) ≈ fA x | S =

2 σxj

2x x2 = 2 exp − 2 σxj σxj

(6.16)

In Fig. 6.12, the conditional cumulative distribution functions (CDFs), obtained as derivatives of the PDFs in Eq. (6.16), are compared with the empirical CDFs obtained

66


Cumulative Distribution Function

1

WLAN, 16-QAM J =4

0.9 0.8

2 2 2 2 σx1 < σx2 < σx2 < σx3 < σx4

0.7

Est. FÂ(x | s0 ≤ S ≤ s1 )

0.6

2 Theory, F A (x | S = σx1 ) ˆ Est. F A(x | s1 < S ≤ s2 )

0.5 0.4

2 Theory, F A (x | S = σx2 ) Est. FÂ(x | s2 < S ≤ s3 )

0.3

2 Theory, F A (x | S = σx3 ) Est. FÂ(x | s3 < S ≤ s4 )

0.2 0.1 0

2 Theory, F A (x | S = σx4 )

0

0.5

1

1.5

2

2.5

3

3.5

Amplitude x Figure 6.12. Comparison of analytical and empirical CDFs for J = 4

from simulated OFDM signal. It can be seen that the conditional distributions, given in Eq. (6.16), are in good agreement with those obtained from simulated data. It should be noted that both the PDF expressions above, given in Eqs. (6.15) and (6.16), were found to be in good agreement with the simulated distributions. For simplicity, the PDFs in Eq. (6.16) will be used in the development of schemes proposed in the next sub-sections. 2 In Fig. 6.12, it can also be observed that FA (.) with different values of σxj significantly

differ from one another. Therefore, a compander designed for one of them will not perform well for a symbol with another value of S. Mechanism for Selecting Companding Adaptation In the schemes proposed in the next sub-sections, for each input OFDM symbol, S will be calculated and classified, i.e., a value of j is assigned after comparing S with s0 , s1 , ..., sJ . Then the signal is transformed by the companding adaptation that is designed for signal 2 amplitude with Rayleigh distribution specified by parameter σxj .

The modified system with the two proposed schemes is illustrated in Fig. 6.13. S can be evaluated from either frequency or time domain sequence, as given in Eq. (6.1). The number of non-zero elements in frequency domain sequence is smaller. Thus for low

67

Chapter 6. Companding Schemes for OFDM Systems employing Higher Order QAM Evaluate S Input bit stream

Select scaling factor and encode side information

QAM mapping

Output bit stream

x

QAM demapping

x Select scaling factor

S/P

P/S

Side information IDFT

DFT

Adjust/select compander parameters

P/S

S/P

Side information

D/A and HPA

Compander

Add cyclic prefix

Decompander


Adjust/select decompander parameters

Original system APC and ACS Channel

A/D

APC only ACS only

Side information

Figure 6.13. Modified system with APC and ACS schemes

complexity implementation, frequency domain sequence, comprising of complex QAM symbols will be used.

6.4.2

Adjustable-parameter Companding (APC)

Main Idea In this scheme, the compander parameters are adjusted according to the value of S of each input symbol, as illustrated in Fig. 6.13. The APC scheme is similar to adaptive companding. Design of APC parameters The scheme essentially involves designing the compander parameters by assuming the 2 , given in Eq. (6.12), if signal amplitude to be Rayleigh distributed with parameter σxj

S ∈ (sj−1 , sj ]. • The parameters for each j = 1, 2, 3, ..., J can be separately and independently calculated by substituting σx with σxj in the transform design equations and the compander can select from a parameter set during run-time. • Alternatively, if the values of parameters at power σx2 are known, then parameters 2 for symbol with power σxj can be derived as a function of these values and σx2 .

For instance, the piecewise linear compander (PLC) in [13] is specified by three parameters: clipping level Ac , inflexion point Ai and slope k, at symbol power σx2 . The parameters for APC for a symbol belonging to j th subset are found by substituting amplitude x with (σxj /σx ) x and σx with σxj in the design equations. The j th set of parameters comes out to be as follows: (Acj , Aij , kj )PLC =

σxj σxj Ac , Ai , k σx σx

68

(6.17) PLC


Similarly, for piecewise exponential compander (PEC) [12], (Acj , Aij , σxj , dj )PEC =

σxj σxj Ac , Ai , σxj , d σx σx PEC

(6.18)

For efficient non-linear compander (ENLC) [15], (Aj , cj , kj , σxj )ENLC =

σxj σx2 A, c, 2 k, σxj σx σxj

(6.19) ENLC

Mathematical Definition of the APC Scheme The APC scheme, using PLC, is given as follows: TAPC (x) = TPLC (x; Acj , Aij , kj ) , if x ∈ xn and S(xn ) ∈ (sj−1 , sj ]

(6.20)

for j = 1, 2, ..., J. TPLC (x; Acj , Aij , kj ) is the transform function for PLC with parameters (Acj , Aij , kj ). Side Information Like adaptive companding, dlog2 Je bits of side information are required in APC scheme as well.

6.4.3

Adaptive Constellation Scaling (ACS)

Main Idea In adaptive constellation scaling (ACS) scheme, the companding transform itself is deterministic. However, prior to companding, the signal is transformed in such a way that its amplitude distribution matches with that Rayleigh distribution for which the fixed compander is designed. Design of Constellation Scaling Factor Let the fixed compander be designed according to fA (x) = fA (x | S = σx2 ). The samples belonging to an OFDM symbol in j th subset, according to Eq. (6.14), have amplitude 2 distribution fA (x | S = σxj ). We need to transform the amplitude of input signal by

hj (x), such that its distribution becomes fA (x). 69


The transformation hj (x) can be simply obtained by using the distribution modification method [10, 15–17, 33]. Let the original amplitude distribution be represented 2 ) and the required distribution be FA,req (x) = by CDF FA,orig (x) = 1 − exp(−x2 /σxj

1 − exp(−x2 /σx2 ). Also, hj (x) must be monotonically increasing, so that it is uniquely invertible and compatible with the companding transform. Hence hj (x) is obtained as follows: −1 hj (x) = FA,req (FA,orig (x)) =

σx x σxj

(6.21)

The transform hj (x) is simply scaling of the signal by a constant. This is intuitively satisfying because one Rayleigh distribution is being transformed into another, which means that only the power of the signal is changed. Moreover, DFT is a linear operation which means that the same transformation can be done in frequency domain by scaling the QAM symbols. Thus, in effect, the constellation is scaled to change the symbol power. The scaling factor is selected according to the input symbol power S, as shown in Fig. 6.13. S of input symbol is compared with s0 , s1 , ..., sJ and subset index j and hence the transformation hj (.) is selected for that symbol. Mathematical Definition of the ACS Scheme The companding transformation with the proposed ACS scheme, using PLC, is expressed as follows: TACS (x) = TPLC (hj (x); Ac , Ai , k) , if x ∈ xn and S(xn ) ∈ (sj−1 , sj ]

(6.22)

for j = 1, 2, ..., J. Side Information As in APC and adaptive companding, dlog2 Je bits of side information are transmitted per symbol in ACS as well. The received signal is scaled by h−1 j (x) before demodulation and detection, where j is determined by the side information.

70


6.4.4

Discussions

Difference between the Outputs of APC and ACS schemes The main difference between the properties of output signal in APC and ACS schemes is that 2 , while its clipped • in APC, the average symbol power remains unchanged, i.e., σxj 2 peak power is scaled by a factor σxj /σx2 (by adjusting compander’s clipping level),

to achieve the desired PAPR, whereas • in ACS, clipped peak power remains constant and average power of the symbol 2 changes from σxj to σx2 , so that the signal has the desired PAPR.

ACS involves comparatively larger computational overhead due to the additional scaling operation. But it enables all symbols to be transmitted at same average and peak power level, which may enable more efficient and simpler design of the HPA.

6.5

Performance Evaluation

In this section, APC and ACS schemes are evaluated by applying them to PLC [13], PEC [12] and ENLC [15]. In order to demonstrate high fidelity of results, the proposed schemes are evaluated for different transforms with various configurations, changing modulation schemes and number of sub-carriers.

6.5.1

Simulation Setup and Parameters

OFDM symbols are generated according to the physical layer specifications given in IEEE 802.11a, used in WLAN, and IEEE 802.16d, used in Fixed WiMAX. Parameters are given in Table 4.1. Oversampling factor L is 4. APC and ACS schemes are evaluated for various values of J. Perfect synchronization, zero carrier frequency offset and ideal channel estimation are assumed at the receiver.

6.5.2

PAPR and BER Performances of APC

Figs. 6.14–6.21 show CCDF of PAPR curves for APC scheme. APC scheme is applied on PEC, PLC and ENLC for WLAN and WiMAX, with 16 and 64-QAM. In the figures, 71


’Orig.’ represents conventional companding using the respective transforms. 0

10

−1

Aj = σ xj 105/20 c = 0.568

10

Aj = σ xj 104.5/20 c = 0.465

−2

10

−1

Aj = σ xj 104.5/20 c = 0.465

−2

10

−3

10

Bit Error Rate


10

WLAN, 16-QAM Orig. ENLC APC, J = 2 APC, J = 4 APC, J = 8 Orig. ENLC APC, J = 2 APC, J = 4 APC, J = 8

WLAN, 16-QAM Original OFDM Orig. ENLC APC, J = 2 APC, J = 4 APC, J = 8 Orig. ENLC APC, J = 2 APC, J = 4 APC, J = 8

−4

10

−5

−6

10

−7

4.1

(a)

−3

10

10

−4

10

Aj = σ xj 105/20 c = 0.568

4.4

4.7

5

5.3

5.6

5.9

6.2

P AP R0 (dB)

10

(b)

0

3

6

9

12

15

18

21

24

27

Eb /N0 (dB)

Figure 6.14. (a) CCDFs of PAPR (b) BER with APC+ENLC on WLAN with 16-QAM

0

10

P AP Rpreset = 4 dB P AP Rpreset = 4.5 dB

−1

10

−2

10

Bit Error Rate


10

−2

−1

10

WLAN, 16-QAM Orig. APC, APC, APC, Orig. APC, APC, APC,

−3

10

PLC J=2 J=4 J=8 PLC J=2 J=4 J=8


−3

10

P AP Rpreset = 4 dB

WLAN, 16-QAM Original OFDM Orig. PLC APC, J = 2 APC, J = 4 APC, J = 8 Orig. PLC APC, J = 2 APC, J = 4 APC, J = 8

−4

10

−5

10

−6

10

−4

10

(a)

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

P AP R0 (dB)

5.8

0

(b)

3

6

9

12

15

18

21

24

27

Eb /N0 (dB)

Figure 6.15. (a) CCDFs of PAPR (b) BER with APC+PLC on WLAN with 16-QAM

0

10

−1

10

Acj = σ xj 104.5/20


−2

10

−2


−3

10

PEC J=2 J=4 J=8 PEC J=2 J=4 J=8

Acj = σ xj 104.5/20 −3

10

WLAN, 16-QAM Original OFDM Orig. PEC APC, J = 2 APC, J = 4 APC, J = 8 Orig. PEC APC, J = 2 APC, J = 4 APC, J = 8

−4

10

−5

10

−4

10

(a)

Acj = σ xj 104/20

10

Acj = σ xj 104/20 Bit Error Rate

−1

10

−6

3.6

3.8

4

4.2

4.4

4.6

4.8

P AP R0 (dB)

5

5.2

5.4

5.6

10

(b)

0

3

6

9

12

15

18

21

Eb /N0 (dB)

Figure 6.16. (a) CCDFs of PAPR (b) BER with APC+PEC on WLAN with 16-QAM

72

24

27


10

Acj = σ xj 105.5/20

−1

10

5/20

Acj = σ xj 10 −1

−2

10

Acj = σ xj 105.5/20 Bit Error Rate


10


−3

10

PEC J=2 J=4 J=8 PEC J=2 J=4 J=8

(a)

Acj = σ xj 105/20

WLAN, 64-QAM Original OFDM Orig. PEC APC, J = 2 APC, J = 4 APC, J = 8 Orig. PEC APC, J = 2 APC, J = 4 APC, J = 8

−3

10

−4

10

−2

10

−4

4.2 4.4 4.6 4.8

5

5.2 5.4 5.6 5.8

6

6.2 6.4 6.6 6.8

7

P AP R0 (dB)

10

(b)

0

3

6

9

12

15

18

21

24

27

Eb /N0 (dB)

Figure 6.17. (a) CCDFs of PAPR (b) BER with APC+PEC on WLAN with 64-QAM 0

10

P AP Rpreset = 5.5 dB P AP Rpreset = 5 dB

−1

10

P AP Rpreset = 5.5 dB Bit Error Rate


10

−2

−1

10

WLAN, 64-QAM

−3

10

Orig. APC, APC, APC, Orig. APC, APC, APC,

PLC J=2 J=4 J=8 PLC J=2 J=4 J=8

(a)

4.4

4.6

4.8

5

5.2

Original OFDM Orig. PLC APC, J = 2 APC, J = 4 APC, J = 8 Orig. PLC APC, J = 2 APC, J = 4 APC, J = 8

−3

−4

10 5.4

5.6

5.8

6

6.2

6.4

6.6

6.8

0

(b)

P AP R0 (dB)

P AP Rpreset = 5 dB

WLAN, 64-QAM

10

−4

10

−2

10

3

6

9

12

15

18

21

24

27

Eb /N0 (dB)

Figure 6.18. (a) CCDFs of PAPR (b) BER with APC+PLC on WLAN with 64-QAM 0

10

Aj = σ xj 104.5/20 c = 0.465 Aj = σ xj 105/20 c = 0.568

−1

−2

−3

10

WiMAX, 16-QAM Orig. APC, APC, APC, Orig. APC, APC, APC,

ENLC J=2 J=4 J=8 ENLC J=2 J=4 J=8

−2

(a)

4.2

−3

10

−4

10

WiMAX, 16-QAM −5

10

−6

10

−4

10

Original OFDM Orig. ENLC A = σx 10 4.5/20, c = 0.465 APC, J = 8, Aj = σxj 10 4.5/20, c = 0.465 Orig. ENLC A = σx 10 5/20, c = 0.568 APC, J = 8, Aj = σxj 10 5/20, c = 0.568

10

Bit Error Rate


10

10

−1

10

−7

4.4

4.6

4.8

5

P AP R0 (dB)

5.2

5.4

10

(b)

3

6

9

12

15

18

21

24

27

Eb /N0 (dB)

Figure 6.19. (a) CCDFs of PAPR (b) BER with APC+ENLC on WiMAX with 16-QAM

It can be seen that APC can be applied to any companding transform and both the PAPR and BER performances are enhanced. 73


10

Acj = σ xj 105.5/20

−1

10

Acj = σ xj 105/20 Acj = σ xj 105.5/20

−1

−2

−2

Bit Error Rate


10

WiMAX, 64-QAM

10

Orig. APC, APC, APC, Orig. APC, APC, APC,

−3

10

PEC J=2 J=4 J=8 PEC J=2 J=4 J=8

10

Acj = σ xj 105/20

WiMAX, 64-QAM Original OFDM Orig. PEC APC, J = 2 APC, J = 4 APC, J = 8 Orig. PEC APC, J = 2 APC, J = 4 APC, J = 8

−3

10

−4

10

−4

10

(a)

4.6

4.8

5

5.2

5.4

5.6

5.8

6

6.2

P AP R0 (dB)

0

(b)

3

6

9

12

15

18

21

24

27

Eb /N0 (dB)

Figure 6.20. (a) CCDFs of PAPR (b) BER with APC+PEC on WiMAX with 16-QAM 0

10

PLC P AP Rpreset = 4 dB

−2

−2

10

−3

10

WLAN, 16-QAM

Bit Error Rate


WLAN, 16-QAM Orig. APC, APC, APC, Orig. APC, APC, APC, Orig. APC, APC, APC,

PLC J=2 J=4 J=8 PEC J=2 J=4 J=8 ENLC J=2 J=4 J=8

(a)

Orig. Orig. APC, APC, APC, Orig. APC, APC, APC, Orig. APC, APC, APC,

−3

10

−4

10

−5

10

−6

10

−7

−4

10

ENLC Aj = σxj 10 5/20 c = 0.568 PEC Acj = σxj 10 4.5/20

10

PEC Acj = σxj 10 4.5/20

−1

10

−1

10

ENLC Aj = σxj 10 5/20 c = 0.568

OFDM PLC J=2 J=4 J=8 PEC J=2 J=4 J=8 ENLC J=2 J=4 J=8


10

3.1 3.3 3.5 3.7 3.9 4.1 4.3 4.5 4.7 4.9 5.1 5.3 5.5 5.7 5.9 6.1

P AP R0 (dB)

(b)

0

3

6

9

12

15

18

21

24

27

Eb /N0 (dB)

Figure 6.21. (a) CCDFs of PAPR (b) BER with APC+PEC, APC+PLC and APC+ENLC on WLAN with 16-QAM

6.5.3

PAPR and BER Performances of ACS

Figs. 6.22–6.26 show CCDF of PAPR curves for ACS scheme. ACS scheme is applied on PEC, PLC and ENLC for WLAN and WiMAX, with 16 and 64-QAM.

74


0

10

P AP Rpreset = 4 dB

−1

10

WLAN, 16-QAM Orig. PLC ACS, J = 2 ACS, J = 4 ACS, J = 8 Orig. PLC ACS, J = 2 ACS, J = 4 ACS, J = 8 Orig. PLC ACS, J = 2 ACS, J = 4 ACS, J = 8

−2

10

−3

10

−4

10

3

(a)

−2

P AP Rpreset = 5 dB

3.2 3.4 3.6 3.8 4

10

WLAN, 16-QAM Original OFDM Orig. P AP Rpreset = 4 dB ACS, J = 8, P AP Rpreset = 4 dB Orig. P AP Rpreset = 4.5 dB ACS, J = 8, P AP Rpreset = 4.5 dB Orig. P AP Rpreset = 5 dB ACS, J = 8, P AP Rpreset = 5 dB

−3

10

Bit Error Rate


−1

10


−4

10

−5

10

−6

10

−7

10 4.2 4.4 4.6 4.8 5

5.2 5.4 5.6 5.8 6 6.2

P AP R0 (dB)

0

(b)

3

6

9

12

15

18

21

24

27

Eb /N0 (dB)

Figure 6.22. (a) CCDFs of PAPR (b) BER with ACS+PLC on WLAN with 16-QAM

0

10

−1

10

Ac = σ x 104.5/20 Ac = σ x 105/20

4/20

Ac = σ x 10

WLAN, 16-QAM

−2

10

−1

WLAN, 16-QAM Orig. PEC ACS, J = 2 ACS, J = 4 ACS, J = 8 Orig. PEC ACS, J = 2 ACS, J = 4 ACS, J = 8 Orig. PEC ACS, J = 2 ACS, J = 4 ACS, J = 8

−2

10

−3

10

−4

10

3

(a)

3.2 3.4 3.6 3.8 4

−3

10

Bit Error Rate


10

−4

10

Original OFDM −5

Orig. Ac = σx 10 4/20

10

ACS, J = 8, Ac = σx 10 4/20 Orig. Ac = σx 10 4.5/20

−6

10

ACS, J = 8, Ac = σx 10 4.5/20 Orig. Ac = σx 10 5/20

−7

10

ACS, J = 8, Ac = σx 10 5/20 4.2 4.4 4.6 4.8 5

5.2 5.4 5.6 5.8 6 6.2

P AP R0 (dB)

0

(b)

3

6

9

12

15

18

21

24

27

Eb /N0 (dB)

Figure 6.23. (a) CCDFs of PAPR (b) BER with ACS+PEC on WLAN with 16-QAM

0

10

Ac = σ x 105/20 Ac = σ x 10

Ac = σ x 104.5/20

−1

10

Ac = σ x 105.5/20

4.5/20

Ac = σ x 105/20 WLAN, 64-QAM Orig. OFDM Orig. PEC ACS, J = 4 ACS, J = 8 ACS, J = 2 Orig. PEC ACS, J = 2 ACS, J = 4 ACS, J = 8 Orig. PEC ACS, J = 2 ACS, J = 4 ACS, J = 8

−1

−2

10

−3

10

−4

10

(a)

WLAN, 64-QAM Orig. PEC ACS, J = 2 ACS, J = 4 ACS, J = 8 Orig. PEC ACS, J = 2 ACS, J = 4 ACS, J = 8 Orig. PEC ACS, J = 2 ACS, J = 4 ACS, J = 8 3.8

4.1

4.4

Bit Error Rate


10

−2

10

−3

10

−4

4.7

5

5.3

5.6

P AP R0 (dB)

5.9

6.2

6.5

10

(b)

0

3

6

9

Ac = σ x 105.5/20

12

15

18

21

Eb /N0 (dB)

Figure 6.24. (a) CCDFs of PAPR (b) BER with ACS+PEC on WLAN with 64-QAM

75

24

27


0

10

A = σ x 105/20 , c = 0.568

−1

A = σ x 104.5/20 , c = 0.465

10

−1

A = σ x 105/20 , c = 0.568

A = σ x 104.5/20 , c = 0.465

−2

Bit Error Rate


10

WLAN, 64-QAM

10

Orig. ENLC ACS, J = 2 ACS, J = 4 ACS, J = 8 Orig. ENLC ACS, J = 2 ACS, J = 4 ACS, J = 8

−3

10

WLAN, 64-QAM Orig. OFDM Orig. ENLC ACS, J = 2 ACS, J = 4 ACS, J = 8 Orig. ENLC ACS, J = 2 ACS, J = 4 ACS, J = 8

−2

10

−3

10

−4

10

4.1

(a)

4.4

4.7

5

5.3

5.6

5.9

6.2

0

(b)

P AP R0 (dB)

3

6

9

12

15

18

21

24

27

Eb /N0 (dB)

Figure 6.25. (a) CCDFs of PAPR (b) BER with ACS+ENLC on WLAN with 64-QAM

0

10

−1

P AP Rpreset = 5 dB

P AP Rpreset = 4 dB

10

Original OFDM Orig. PLC, P AP Rpreset = 5 dB ACS, J = 8, P AP Rpreset = 5 dB Orig. PLC, P AP Rpreset = 4.5 dB ACS, J = 8, P AP Rpreset = 4.5 dB Orig. PLC, P AP Rpreset = 4 dB ACS, J = 8, P AP Rpreset = 4 dB

16-QAM. For ACS, J = 8 −3

WLAN, Orig. PLC WLAN, ACS+PLC WLAN, Orig. PLC WLAN, ACS+PLC WLAN, Orig. PLC WLAN, ACS+PLC WiMAX, Orig. PLC WiMAX, ACS+PLC WiMAX, Orig. PLC WiMAX, ACS+PLC WiMAX, Orig. PLC WiMAX, ACS+PLC

−2

10

−3

10

2.7

(a)

−2

10

3

3.3

3.6

3.9

Bit Error Rate


P AP Rpreset = 4.5 dB −1

WLAN, 16-QAM

10

10

−4

10

−5

10

−6

10

−7

10 4.2

4.5

4.8

5.1

5.4

5.7

6

P AP R0 (dB)

3

(b)

6

9

12

15

18

21

24

27

Eb /N0 (dB)

Figure 6.26. (a) CCDFs of PAPR (b) BER with ACS+PLC on WLAN and WiMAX with 16-QAM

0

10


Orig. PEC APC, J = 2 APC, J = 4 APC, J = 8 ACS, J = 2 ACS, J = 4 ACS, J = 8 Orig. PLC APC, J = 2 APC, J = 4 APC, J = 8 ACS, J = 2 ACS, J = 4 ACS, J = 8


10

−2

10

−3

10

(a)

4.4

4.6

4.8

5

Bit Error Rate

WiMAX, 64-QAM −1

PEC+APC: Acj = σxj 10 5.5/20 PEC+ACS: Ac = σx 10 5.5/20 5.2

5.4

5.6

PEC: Acj = σxj 10 5.5/20 (APC) Ac = σx 10 5.5/20 (ACS)

−2

10

WiMAX, 64-QAM

−3

10

Orig. OFDM Orig. PEC APC, J = 2 APC, J = 4 APC, J = 8 ACS, J = 8 Orig. PLC APC, J = 2 APC, J = 4 APC, J = 8 ACS, J = 8

−4

10

−5

10

5.8

P AP R0 (dB)

6

6.2

6.4

6.6

(b)


9

12

15

18

21

24

Eb /N0 (dB)

Figure 6.27. (a) CCDFs of PAPR (b) BER with APC+PLC and ACS+PLC on WiMAX with 64-QAM

76

27


6.5.4

Out-of-band Interference (OBI) Levels in APC and ACS Schemes


Figs. 6.28 and 6.29 show plots of power spectral density (PSD) of companded signals.

0

−5

−10

−15

Four coinciding curves: Orig. ENLC, APC+ENLC, J = 2, 4, 8, with Aj = σxj 105/20

WLAN, 16-QAM Five coinciding curves: Orig. PLC, APC, J = 2, 4, 8, ACS, J = 8, with P AP Rpreset =4 dB

Four coinciding curves: Orig. PEC, APC, J = 2, 4, 8, with Acj = σxj 104.5/20

Two coinciding curves: Orig. PLC, ACS, J = 8, with P AP Rpreset =4.5 dB

−20

−25

Original OFDM

−30 −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Normalized Frequency (xπ rad/sample)

Figure 6.28. PSDs of signals transformed by fixed compander and APC and ACS schemes with PLC, PEC and ENLC, applied on WLAN with 16-QAM


10

WiMAX, 64-QAM 5 0 −5

−10 −15

Seven coinciding curves: Orig. PEC, APC+PEC, J = 2, 4, 8, with Acj = σxj 105.5/20 , ACS+PEC, J = 2, 4, 8 with Ac = σx 105.5/20

Seven coinciding curves: Orig. PLC, APC+PLC, J = 2, 4, 8, ACS+PLC, J = 2, 4, 8, with P AP Rpreset = 6 dB

−20 −25

Original OFDM

−30 −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1


Figure 6.29. PSDs of signals transformed by fixed compander and APC and ACS schemes with PLC and PEC, applied on WiMAX with 64-QAM

77


6.5.5

Discussions on PAPR, BER and OBI Performances

PAPR Performance • It can be seen that as J increases, the PAPR of the output signal comes closer to respective preset values in both the APC and ACS schemes. Increase in J implies smaller variance of average symbol power in every subset. • Since the variance of S in WLAN symbols is larger than that in WiMAX, so the amount of PAPR reduction, achieved by effectively reducing this variance, is also greater. • It can be seen that in terms of PAPR, APC and ACS perform almost identically. However, they differ in other signal properties and complexity. • We found that the improvement in PAPR reduction performance is very small in most cases if J is increased beyond 8. Interestingly, with increasing value of J, the CCDF curves approach those obtained for conventional companding on 4-QAM based OFDM. This happens because with increasing J, variance of S within each subset is reduced, so the PAPR reduction performance resembles that of 4-QAM, in which S is a constant. • The schemes can be effectively applied to any configuration of any compander. BER Performance • As J increases, both the PAPR and BER are reduced. Particularly, the noise floor at high SNR is lowered, which indicates that the noise due to clipping is reduced. This is observed in both the APC and ACS schemes. • This joint reduction of PAPR and BER is not possible to achieve with conventional, deterministic companding, in which PAPR can only be reduced at the cost of increased BER. This means that the overall performance of the system is significantly improved. • In conventional companding, the larger variance of PAPR around the ideal value implies that the compander under-performs for some symbols and over-performs

78


for others. This means that in many symbols, conventional companding introduces larger amount of noise than that necessary to achieve the desired PAPR. The proposed schemes enable the compander to adapt to the signal properties. This makes the companding operation more precise. Consequently, the redundant companding/clipping distortion is eradicated to a large extent, which is manifested as the reduction in BER. OBI Performance • Irrespective of J, the OBI due to companding distortion remains same in proposed schemes as in corresponding deterministic companding.

In conventional

companding, reduction in PAPR is achieved at the cost of increased OBI, whereas in the proposed schemes, we see that there is no elevation in the average OBI level with decreasing PAPR. • OBI can be related to the average companding noise energy D = E [(T (x) − x)2 ]. The amount of this noise energy differs from subset to subset, but its overall average remains unaffected. This can be explained by the law of total expectation [47], which states that the overall average of D remains same irrespective of J, i.e.,

E [D] =

J X

E [D | sj−1 < S ≤ sj ] Pr[sj−1 < S ≤ sj ]

(6.23)

j=1

This property was confirmed through simulations by evaluating the noise energies in every subset and comparing their average with the theoretical E[D]. • The constant OBI behavior remains consistent with changing transforms, standards and constellations.

6.5.6

Complexity Analysis of APC and ACS

• Evaluation of S: This is common in both schemes, as shown in Fig. 6.13. If symbol-to-energy mapping for the constellation is pre-calculated as a look-up table (LUT), then Nd floating point additions and 1 multiplication are required. • Classification of input symbol: J − 1 comparisons are required to determine the subset index. This is also common in both schemes. 79


• Parameter calculation in APC: Compander parameters can be either pre- calculated or adjusted (2 or 3 floating point multiplications), as explained in Sub-section 6.4.2. • Scaling in ACS: 4Nd floating point multiplications (for scaling in-phase and quadrature components of data symbols at the transmitter and the receiver) are required. Alternatively, scaled constellations can also be pre-calculated to reduce the implementation complexity.

6.5.7

Bandwidth Efficiency

As explained in Sub-sections 6.4.2 and 6.4.3, dlog2 Je bits of side information are required in both schemes. Simulation results indicate that significant performance improvement is achieved for J = 8, for which only 3 bits are needed. Since the proposed schemes have been implemented on 16 and 64-QAM based OFDM which are already more bandwidthefficient, so the relative trade-off is negligibly small. Advantages of the Proposed Classification Method The overhead on channel throughput is small because of the classification method used in the proposed schemes. If the compander were designed to adjust differently for every single value of S, instead of using only J subsets and hence J adaptations, the required number of bits for WLAN with 16-QAM and 64-QAM would be 7 and 10, respectively. For WiMAX, 9 and 12 bits would have been required for 16 and 64-QAM, respectively. Furthermore, the low complexity implementations suggested in Sub-section 6.5.6 would also not be feasible. It should be noted here that errors in side information may also lead to increase in BER. Thus, smaller amount of side information not only reduces the throughput loss, but the probability of occurrence of such errors is also diminished.

80

Chapter 7

Piecewise Linear Companding Transforms In this chapter, new piecewise linear companding transforms are presented. The design criteria for the proposed transform is developed by investigating the relationships between the compander and decompander’s profile and parameters with the system’s performance metrics. Using analysis along with simulations, the companding parameters are related with the bit error rate (BER), out-of-band interference (OBI), amount of companding noise, computational complexity and average power. Based on a set of criteria developed thereof, design of transforms is formulated with the aim of preserving the signal’s attributes as much as possible for a predetermined amount of PAPR reduction. Simulations are carried out to evaluate and compare the proposed scheme with the existing companding transforms to demonstrate the enhancement in PAPR, BER and OBI performances.

7.1

Related Work and Motivation

In most recent researches, including the transforms presented in [10–12, 15–17, 19, 31– 33, 48, 49], companders have been designed by modifying the amplitude distribution of the OFDM signal. However, amplitude distribution modification is an indirect design methodology. This means that it does not relate the compander and decompander’s profile or parameters with the system’s performance metrics. The performance metrics include implementation complexity, output PAPR, error performance and OBI. Rather these properties change as a consequence of the change in signal’s amplitude distribu81

Chapter 7. Piecewise Linear Companding Transforms

tion. It is noteworthy, however, that the amplitude distribution itself is not a measure of the system’s performance. In contrast, the design approach used in the piecewise linear transform, presented in [13], has been found to provide control over output PAPR, implementation complexity and the amount of companding distortion. But in [13], the transform has been designed by targeting to minimize the amplitude distortion and computational complexity at the transmitter, without considering the effects of decompanding on channel-induced noise and BER. Also, the non-invertible nature of clipping operation and frequency contents of companding noise are not properly understood with reference to the profile of companding function. Recently, an iterative companding transform and filtering (ICTF) scheme [18] has been presented in which linear and piecewise linear transforms have been selected due to their low implementation complexity. In Chapter 5, the design flexibility of piecewise linear transform, presented in [13], was used to optimize its PAPR reduction performance, through adaptive companding. Motivated by the preferred application of piecewise linear transforms in companding schemes presented in recent works, new transforms are designed by formulating design criteria for the optimization of BER and OBI performances of the companded signal.

7.2


The design criteria for optimization of companding transforms is developed by studying the relationships between the system’s performance metrics and compander’s profile and parameters. Using simulations or analysis as required, the relationships between the compander’s profile with error performance, amount of invertible and non-invertible companding distortion, spectral contents of companding noise and implementation complexity are investigated. This is done while ensuring that the pre-determined constraints on average power and output PAPR are also simultaneously satisfied. The effect of decompanding on noise is also taken into consideration so that the compander/decompander pair can be designed to optimize the BER. Based on the observations and discussions, a set of rules or guidelines is developed that are used in deriving the proposed transform. Simulation results show that the proposed functions outperform the existing companding transforms in terms of BER, OBI, implementation complexity and scalability for a certain

82


value of desired PAPR and are also more robust to the changing channel conditions.

7.3

General Design Criteria

Companding is a post-modulation operation on OFDM signal amplitude that inevitably introduces non-linear distortion. Consequently, companded signals suffer from elevated BER and OBI as compared to the original signal. The key challenge in the design of companding transforms is to keep the distortion within tolerable limits. In this section, the design criteria for piecewise linear transforms are developed, that essentially preserve the system’s performance as much as possible while realizing a prescribed constraint on PAPR. The foremost aim is to design the profile and parameters of the compander in such a way that the non-linear distortion is restricted to an amount that is absolutely indispensable to the desired amount of PAPR reduction. The general form of the companding transform, composed of P linear functions, is given below.

T (x) =

   m1 x + c1 ,       m2 x + c2 ,

x ≤ a1 a1 < x ≤ a2

(7.1)

   : :      m x + c , x > a P P P −1

7.3.1

Constant Average Power

In companding transformation, it is desired to preserve the average power of the signal so that signal-to-noise ratio (SNR) remains unchanged [1]. Hence the parameters m1 , m2 , ..., mP , c1 , c2 , ..., cP and a1 , a2 , ..., aP −1 must satisfy the following equation: Z∞

2

x fA (x)dx = −∞ Za2

σx2

Z∞ =

Za1 T (x)fA (x)dx = (m1 x + c1 )2 fA (x)dx+ 2

−∞

0

(m2 x + c2 )2 fA (x)dx + ... +

a1

Z∞

(mP x + cP )2 fA (x)dx

aP −1

83

(7.2)


7.3.2

Hard Clipping for Peak Power Reduction

Companding is a combination of compression and expansion of signal amplitude. Peak power is reduced by compressing larger amplitudes and the power deficiency, caused by compression, is compensated by expanding the smaller amplitudes, so that the average power remains constant. The power deficiency PD due to compression can be expressed as follows: Z

[x2 − T 2 (x)]fA (x)dx

PD =

(7.3)

x>T (x)

For a desired output PAPR, P AP Rdes , and unchanged average power, according to Eq. (7.2), the maximum amplitude am , that is allowed to be present in the companded OFDM signal, can be computed as follows: am = σx 10P AP Rdes /20

(7.4)

If PD is minimized, then smaller power will be required to be compensated by expansion. This means that the companding noise, that is the difference between transformed and original signals, can be kept at minimum if unnecessary compression is avoided, because then any redundancy in the expansion operation will also be eliminated. In order to achieve P AP Rdes , the companding transform should be such that T (x) ≤ am . Hence, Z

2

Z

2

[x − T (x)]fA (x)dx ≥ x>T (x)

[x2 − a2m ]fA (x)dx

(7.5)

x>T (x)

Z


PD ≥

(7.6)

x>T (x)

PD is going to be minimum when the equality condition, in Eq. (7.6), holds. The equality condition will hold when only the samples with amplitude larger than am are compressed. Z PDmin =


x>am

84

(7.7)


T (x) x (a)

PEC

x (b)

am

EC

x (c)

am

T (x) ≥ x T (x) < x

am

T (x) ≥ x T (x) < x

am

am

NLC

T (x)

T (x)

T (x) ≥ x T (x) < x

am

am

ENLC

T (x)

am T (x)

T (x) ≥ x T (x) < x

T (x) ≥ x T (x) < x

x (d)

PLC

x (e)

am

am

Figure 7.1. Compression and expansion profiles of (a) ENLC, (b) PEC, (c) EC, (d) NLC and, (e) PLC

This means that the transform should be such that it only compresses those samples that are greater than am and compression level is also am .

T (x) =

  TE (x), x ≤ am  am ,

(7.8)

x > am

where TE (x) represents the transformation required for expansion. From Eqs. (7.7) and (7.8), we infer that hard clipping, at amplitude am , is the best solution for peak power reduction, as it meets the constraint on PAPR without introducing any redundant companding distortion. In [10–12, 15–17, 19, 33, 50], companding transforms limit the peak power by soft clipping that results in unnecessary compression. This is illustrated in Figs. 7.1(a)-(d). It can be seen that soft clipping compresses samples that are already less than am and compressed samples are reduced to levels lower than am , both of which are unnecessary to attain the desired PAPR. In Fig. 7.1(e), hard clipping is used that does not introduce any redundant compression, since only amplitudes larger than am are clipped and clipping level is also maximum possible, i.e., am . Moreover, since the compressed part of the signal cannot be recovered without noise amplification, as explained in the next section, so minimizing PD also ensures that the non-invertible distortion in the signal will

85


be minimal. It was also found through simulations that the clipping level is related to the increase in the side-lobe level. It was found that since soft clipping introduced larger amplitude companding noise, so the increase in side-lobe level is more than that in case of soft clipping.

7.3.3

Error Performance Optimization

Bussgang Theorem Companding transforms are generally non-linear transforms on the amplitude of OFDM signal. Since OFDM is a complex Gaussian signal, the companded signal can be modeled using the Bussgang theorem [51]. Bussgang Theorem 1 The Bussgang theorem states that the cross-correlation of a Gaussian signal before and after it has passed through a non-linear operation are equal up to a constant. Let X(t) be a zero mean stationary Gaussian random process and Y (t) = g(X(t)), where g(.) is a non-linear amplitude distortion. If RX (τ ) is the auto-correlation function of X(t), then the cross-correlation RXY (τ ) of X(t) and Y (t) is given by RXY (τ ) = CRX (τ )

(7.9)

where C is a constant that depends only on g(.) The Bussgang theorem can be extended for complex Gaussian signals by representing the non-linear distortion as a combination of AM/AM (amplitude distortion that depends upon the input amplitude) and AM/PM (phase distortion that depends upon amplitude of input) non-linearity [52, 53]. According to [53], the extension of Bussgang theorem to complex signal can be written as a sum of attenuated signal replica and uncorrelated distortion. Since OFDM signal is a zero-mean complex Gaussian signal, the relationship between original and companded signal can be expressed as follows. T (x(t)) = αx(t) + nd (t)

(7.10)

where x(t) is the complex Gaussian OFDM signal, T (x(t)) is the result of non-linear companding transformation on x(t), α is the attenuation constant and nd (t) is uncorre86


lated distortion. α can be found as a function of input and distortion function as follows. Multiplying x∗ (t) on both sides of 7.10. x∗ (t)T (x(t)) = αx∗ (t)x(t) + x∗ (t)nd (t)

(7.11)

Taking expectation on both sides. =0, since x(t) and nd (t) are uncorrelated

E [x∗ (t)T (x(t))] = α | {z }

cross-correlation at τ =0

E [x∗ (t)x(t)] | {z }

+

z }| { E [x∗ (t)nd (t)]

(7.12)

auto-correlation at τ =0

RXT (0) = αRX (0)

(7.13)

where RXT denotes cross-correlation between input X(t) α=

RXT (0) RX (0)

(7.14)

Since α is a constant that depends upon the companding transformation T (x(t)), Eq. (7.13) satisfies the Bussgang theorem for τ = 0. Since RXT (0) = E [x∗ (t)T (x(t))] and T (.) only modifies the amplitude of x(t) and although OFDM is not stationary but α has been shown to be time invariant in [53, 54], so RXT (0) = E [|x| |T (x)|] at any time t. R∞ α=

xT (x)fA (x)dx

−∞

R∞

(7.15) x2 f

A (x)dx

−∞

1 α= 2 σx

Z∞ xT (x)fA (x)dx

(7.16)

−∞

It can be seen that α is the correlation co-efficient of x and T (x). Noise Amplification by Decompanding If the companding transform is completely invertible, the signal can be recovered at the receiver without any distortion. However, the decompanding operation may result in noise amplification. Moreover, the companding transform may be only partially recoverable due to the presence of clipping operation.

87


For constant average power, E T 2 (x) = E (αx + nd )2 = E[α2 x2 ] +

2E[αn x] | {z d }

+E[n2d ]

(7.17)

=0, x and nd are uncorrelated

E T 2 (x) = E (αx + nd )2 = E[α2 x2 ] + E[n2d ]

(7.18)

2 σx2 = α2 σx2 + σnd

(7.19)

2 is the power of uncorrelated noise. where σnd

2 σnd = (1 − α2 )σx2

(7.20)

For the noise or distortion to be small, α2 must be close to 1 to keep the average power constant. The decompanding transform can be described as, T −1 (x) =

x − nd α

(7.21)

The decompanding operation in the presence of AWGN results in noise amplification by a factor α. R(x) = T −1 (T (x) + w) =

w x + nd + w − nd =x+ α α

(7.22)

where R(x) is the recovered signal and w represents AWGN. When clipping operation is also involved in the companding transform, then the recovered signal also contains clipping noise in addition to amplified channel noise.   x + w + nc , if x is a clipped sample R(x) =  x + w , if x is an expanded sample α

(7.23)

where nc is clipping noise. The amount of clipping noise is determined by the clipping level am . This noise will be minimum if hard clipping is used for peak power reduction, as explained in Sub-Section 7.3.2. Maximizing Cross-correlation Coefficient In order to achieve optimum error performance for a given PAPR or alternatively a given clipping level, the cross-correlation co-efficient α should be maximized in order to keep 88


the noise amplification w/α as small as possible. Furthermore, since α is correlation coefficient which is the measure of similarity of original and companded signal, imposing the condition of maximum correlation means that the changes in signal will be minimal. This will help preserve the signal as much as possible while reducing PAPR by the required amount. Also, for most of the amplitudes, the slopes m1 , m2 , . . . , mP , in Eq. (7.1), should be either equal to or greater than one, so that the respective inverse transforms at the decompander do not result in noise amplification. According to Eq. (7.22), the expanded portion of signal can be recovered with tolerable amount of noise amplification. However, the compressed part of the signal cannot be recovered. According to Eq. (7.23), at low SNR, wn dominates, which means BER will be close to that of the original signal. At high SNR, cn dominates and the BER approaches a constant value. Since there will always be a minimum amount of noise present in the signal, even at very high SNR, the OFDM systems employing companders have a noise floor or a performance floor, as observed in simulation results presented in Sub-section 7.7.3. The compander/decompander pair must be designed to keep this noise floor as low as possible. In the proposed transform, this is done by employing hard clipping and partial decompanding, explained in Sub-sections 7.3.2 and 7.3.5, respectively.

7.3.4

Out-of-band Interference (OBI) Reduction

Since companding is a non-linear operation on the signal, so it leads to undesirable changes in its spectral contents due to intermodulation noise. This widens the effective bandwidth of the channel, by raising the side-lobe level and increases adjacent channel interference (ACI), due to high frequency contents in the companding noise. The increased side-lobe level and ACI are collectively termed as out-of-band interference (OBI). Experiment In order to study the relationship between the compander’s profile with the amount of OBI, a simple transform with an inflexion point is used:     x, x ≤ a1    T (x) = x + c2 , a1 < x ≤ am − c2      am , x > a m − c2 89

(7.24)


where c2 is the shift introduced in amplitudes greater than the inflexion point a1 . The transforms for various values of a1 are shown in Fig. 7.2(a). It should be noted that each transform satisfies similar constraints on P AP Rdes and average power. Figs. 7.2(b) and 7.2(c) show the PSD plots of the companding noise, D = T (x) − x, and those of the companded signal, respectively. OFDM signal is generated according to Fixed WiMAX specifications, with 4-QAM as modulation scheme. It can be observed that as a1 increases, high frequency contents in the companding noise, shown in Fig. 7.2(b), are also increased. Similar trend is observed in case of OBI in the companded signal, as shown in Fig. 7.2(c).

T (x)

am

a1 a1 a1 a1 a1

a1

PSD (dB/rad/sample)

−25

a1 a1 a1 a1 a1

−30 −35

= = = = =

0 0.3σx 0.8σx σx 1.2σx

−40 −45 −1

−0.5

0

0 0.3σx 0.8σx σx 1.2σx

a1 a1 a1 Amplitude x (a) PSD (dB/rad/sample)

a1

= = = = =

0.5

1

Normalized frequency (xπ rad/sample) (b)

a1 a1 a1 a1 a1

−10

−20

= = = = =

0 0.3σx 0.8σx σx 1.2σx

−30

−40 −1

−0.5

0

0.5

1

Normalized frequency (xπ rad/sample) (c)

Figure 7.2. Relationship between compander’s profile and OBI. (a) Companding transforms with changing inflexion points, (b) PSDs of companding noise, D = T (x) − x, and, (c) PSDs of companded signals

Observations and Discussions The observations in Fig. 7.2 can be explained by considering the time domain waveform of companding noise. If a1 is large, then it means that fewer samples will be expanded. Since the expanded signals do not occur consecutively in the time domain waveform, so 90


0

a1 = 1.2σx 0

512

0

a1 = σx 0

512

a1 = 0.8σx 0

512

Re[D]

0

a1 = 0.3σx −0.4

0

512

0

a1 = 0 0

512

0

512

1024

0

a1 = σx −0.4

0

512

1024

1024

Samples

0

a1 = 0.8σx −0.4

0

512

1024

Samples

0

a1 = 0.3σx −0.4

0

512

1024

Samples

0.4

Samples

1024

Samples

0.4

Samples

0.4

−0.4

1024

Samples

0.4

a1 = 1.2σx −0.4

0.4

Im[D]

Re[D]

0 −0.4

1024

Samples

0.4

0

0.4

Im[D]

Re[D]

0.4

−0.4

1024

Samples Im[D]

−0.4

Re[D]

Im[D]

0.4

Im[D]

Re[D]

0.4

0

a1 = 0 −0.4

0

512

1024

Samples

Figure 7.3. Waveforms of companding noise, D = T (x) − x, for different values of inflexion point a1

the companding noise, D = T (x)−x, comprises of sharp, impulse-like transitions, that are being added to the original signal. Noise waveforms for different values of a1 are shown in Fig. 7.3. It can be seen that when a1 = 1.2σx , companding noise consists of larger number of sharper transitions and hence more energy at high frequencies, as observed in Fig. 7.2(b). In contrast, for a1 = 0, noise is more uniformly distributed over the time samples and the only sharp transitions are due to the clipping operation, which is performed for only a small fraction of samples per symbol. Also, the amplitudes of noise samples are smaller, which means that the waveform of the companded signal will closely follow that of the original signal and hence their spectral contents can also be expected to resemble more closely. So only a small amount of noise energy leaks into the adjacent channel, as observed in Figs. 7.2(a), (b). Also, it was discussed in Sub-section 7.3.2, that side-lobe levels are lower with hard clipping, because of the minimization of overall companding noise energy. From the observations and discussions, it can be concluded that the OBI will be reduced if the companding noise is well-distributed among all the samples in the signal. 91


So the profiles of the proposed transforms, in Section 7.5, are selected such that they expand all the samples with amplitude smaller than am , instead of only changing the larger amplitudes as done by the previously designed companders [10, 12, 13, 15, 19], including the functions shown in Fig. 7.1.

7.3.5

Partial Decompanding

Since clipping is not invertible, signal can be only partially recovered by the decompander. It should be noted that even in case of soft clipping, compressed part of the signal cannot be recovered without severe noise amplification. In [10, 15–19, 32, 33], enhanced performance has been reported if the decompanding operation is eliminated altogether. However, elimination of the inverse operation ensues a heightened noise floor in the system at high SNR, as observed in simulation results in Sub-section 7.7.3. In [13], clipping, at amplitude level am , is also included in the decompander function. But in this work, it was discovered, through simulations, that using an identity transform for amplitudes greater than am , instead of clipping at the decompander, yields smaller BER. This may be explained by the fact that the compressed amplitudes in the companded signal are actually larger than am in the original signal and the channel-induced noise inadvertently recovers some portions of the clipped peaks. Secondly, clipping at the decompander may also change the frequency contents of the channel-induced white noise and as a result, the amount of the rejected noise, due to the presence of guard band, may be reduced. Also, since clipping is not invertible, so its decompanding counterpart cannot be derived mathematically. Hence, the decompander function has been selected to be as follows:

T −1 (x) =

  T −1 (x), x ≤ am E  x,

(7.25)

x > am

Since TE (x) is going to be designed according to the criteria given in Sub-section 7.3.3, so TE−1 (x) exists for all x ≤ am and leads to minimal or no noise amplification.

7.3.6

Scalability

It can be observed that the design criteria has been formulated in such a way that all the performance measures are dealt with for a certain value of P AP Rdes . This means that the 92


compander can be conveniently configured to achieve any PAPR at the output. Also, the parameters of the compander can be simply scaled if transmission power changes. The parameters, that are the cut-off amplitudes and intercepts, c1 , c2 , . . . , cP , a1 , a2 , . . . , aP −1 in Eq. (7.1), can be obtained for any transmission power by simply multiplying them by σx0 /σx , where σx02 is the new transmission power. The slopes, m1 , m2 , . . . , mP in Eq. (7.1), are not affected by the change in the required average transmission power. This scalability of parameters can facilitate simpler implementation. The proposed transforms are also adequately simple and flexible to be employed in techniques like ICTF [18] and the schemes proposed in this thesis (adaptive companding, APC and ACS).

7.4

Design Methodology

The criteria developed in Section 7.3 will be used in the design of transforms presented in Section 7.5. Various profiles of piecewise linear compander functions will be implemented and evaluated. The general method used in deriving these transforms is summarized below: (i) Following the analysis and discussion in Sub-section 7.3.2, hard clipping will be used for peak power reduction to keep the companding noise as small as possible. The clipping level am will be selected according to P AP Rdes . The relationship is given in Eq. (7.4). (ii) According to the analysis in Sub-section 7.3.3, profile of TE (x) should be selected such that the slopes are greater than or equal to 1. This is not possible to achieve with the condition of invertibility of transform. So, the transforms are designed such that most of the slopes are greater than or equal to 1. The slopes that are less than 1 are still very close to 1 and are applied for smaller fraction of samples. (iii) According to the observations in 7.3.4, profiles of companders are selected such that the companding noise is well distributed over all the samples, so that OBI level is low. (iv) Partial decompanding is used at the receiver, as explained in Sub-section 7.3.5. (v) The transform will be expressed in terms of piecewise linear function. The slopes, intercepts and inflexion points are the compander parameters to optimize. 93


(vi) A set of solutions for the parameters will be calculated by imposing the constant average power constraint, given in Eq. (7.2). (vii) For each element in the obtained solution set, the cross correlation coefficient α, given in Eq. (7.16), will be calculated. The parameters that yield maximum α is selected as the final solution.

7.5 7.5.1

Proposed Companding Transforms Companding Transform PLC-1

Profiles of Compander and Decompander

am

am

T (x)

T −1 (x)

m2

m1 = 1

1/m2

1 a1 Amplitude x (a)

am

m2 a1 + am (1 − m2 ) Amplitude x (b)

am

Figure 7.4. Transform profiles of PLC-1 (a) compander and (b) decompander

The compander profile is shown in Fig. 7.4. The transform is specified by three linear functions with parameters (m1 , m2 , a1 , am ). As shown in the figure, m1 = 1. The companding function and its inverse are expressed as follows:     x − m2 p + p, 0 ≤ x ≤ a1    TP LC−1 (x) = m2 x + am (1 − m2 ), a1 < x ≤ am      am , x > am

94

(7.26)


where am is the clipping level given by Eq. (7.4) and p = am − a1 .     0,       x + m2 p − p, −1 TP LC−1 (x) = x − (1 − m2 )am   ,    m2     x,

0 ≤ x ≤ −m2 p + p −m2 p + p < x ≤ m2 a1 + am (1 − m2 ) (7.27) m2 a1 + am (1 − m2 ) < x ≤ am x > am

Average Power Constraint The constraint on average power is given as:

σx2

Z∞ =

2

Z∞

x fA (x)dx = 0

Zam

TP2 LC−1 (x)fA (x)dx

0

0

Z∞

(m2 x + am (1 − m2 ))2 fA (x)dx +

a1

σx2

Za1 =

Za1 = (x − m2 p + p)2 fA (x)dx+ (7.28)

a2m fA (x)dx

am

(x + p)2 + m22 p2 − 2m2 p(x + p) fA (x)dx+

0

Zam

m22 (x − am )2 + a2m + 2m2 am (x − am ) fA (x)dx +

a1

Z∞

(7.29) a2m fA (x)dx

am

Eq. (7.29) can be written as a quadratic equation with variable m2 . Co-efficients of m22 , m2 and m02 are given by c2 , c1 and c0 as follows. c2 = p2 I0 (0, a1 ) + I2 (a1 , am ) + a2m I0 (a1 , am ) − 2am I1 (a1 , am )

(7.30a)

c1 = −2pI1 (0, a1 ) − 2p2 I0 (0, a1 ) + 2am I1 (a1 , am ) − 2a2m I0 (a1 , am )

(7.30b)

c0 = I2 (0, a1 ) + p2 I0 (0, a1 ) + 2pI1 (0, a1 ) + a2m I0 (a1 , ∞) − σx2

(7.30c)

The integrals required in the above equations are given as follows. Zxf I0 (xi , xf ) =

Zxf fA (x)dx =

xi

xi

2 2 2 xf x x 2x exp − 2 dx = − exp − 2 + exp − i2 (7.31) 2 σx σx σx σx

95

Chapter 7. Piecewise Linear Companding Transforms Zxf

Zxf

2 2 xf 2x x I1 (xi , xf ) = xfA (x)dx = x 2 exp − 2 dx = −xf exp − 2 σx σx σx xi xi 2 √ √ xi πσx xf πσx xi + xi exp − 2 + erf − erf σx 2 σx 2 σx

(7.32)

2 Rz exp (−t2 ) dt. where erf(z) = √ π0 Zxf

Zxf

2 2 xf x 2 I2 (xi , xf ) = x fA (x)dx = exp − 2 dx = −xf exp − 2 σx σx xi xi 2 2 2 xf xi x 2 2 + xi exp − 2 + σx − exp − 2 + exp − i2 σx σx σx 2

2x x2 2 σx

(7.33)

Correlation Coefficient Correlation coefficient for TP LC−1 (x) is given by: 1 α= 2 σx

Z∞ xTP LC−1 (x)fA (x)dx

(7.34a)

0

Zam Za1 1 1 α = 2 (x − m2 p + p)xfA (x)dx + 2 x (m2 x + am (1 − m2 )) fA (x)dx σx σx a1

0

+ α=

1 σx2

Z∞ xam fA (x)dx

(7.34b)

am

1 1 1 1 I2 (0, a1 ) + 2 (p − m2 p)I1 (0, a1 ) + 2 m2 I2 (a1 , am ) + 2 am (1 − m2 )I1 (a1 , am ) 2 σx σx σx σx

+ am I1 (am , ∞)

(7.34c)

Optimization of Compander’s Parameters The solution for (a1 , m2 ) is obtained using Algorithm 5. The algorithm first calculates all solutions (a1 , m2 ) that satisfy the average power constraint and then selects the solution with maximum correlation coefficient.

96

Chapter 7. Piecewise Linear Companding Transforms Algorithm 5 Parameters of PLC-1 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19:

Input am , σx . Set a1array = 0 :inc: am , where inc is a small increment. Initialize solution set S={}. for each element a1 in a1array do Find p = am − a1 . Find co-efficients c2 , c1 and c0 using Eq. (7.30). Solve c2 m22 + c1 m2 + c0 = 0 for m2 to get two solutions (m21 , m22 ). Set m2 = m21 or m2 = m22 such that 0 < m2 < 1. Add (a1 , m2 ) to S. end for Initialize αmax =0. for each element of S do Find α using (7.34). if α is greater than αmax then Accept current element of S as the final solution. Set αmax = α. end if end for Output accepted solution for m2 and a1 .

7.5.2

Companding Transform PLC-2

Profiles of Compander and Decompander The compander and decompander profiles are shown in Fig. 7.5. The companding func-

am

am

T (x)

T −1 (x)

m2 1/m2

m1 1/m1 a1 Amplitude x (a)

a1 m1 Amplitude x (b)

am

am

Figure 7.5. Profiles of PLC-2 (a) compander and (b) decompander

tion and its inverse specified using three parameters (m1 , m2 , a1 , am ). The transform functions are expressed as follows:     m1 x, 0 ≤ x ≤ a1    TP LC−2 (x) = m2 x + am (1 − m2 ), a1 < x ≤ am      am , x > am 97

(7.35)


where am is the clipping level given by Eq. (7.4).  x   , 0 ≤ x ≤ m1 a1     m1 TP−1LC−2 (x) = x − am (1 − m2 ) , m1 a1 < x ≤ am  m2     x, x > am

(7.36)

m1 and m2 are related as follows: a1 m1 =m2 a1 + (1 − m2 )am m1 =

(7.37a)

m2 (a1 − am ) + am a1

(7.37b)

m1 =pm2 + q a1 − am am where p = ,q= a1 a1

(7.37c) (7.37d)

m1 and m2 satisfy following constraints, m1 > 1 and m2 < 1

(7.38)

Average Power Constraint The constraint on average power is given as follows:

σx2 =

Z∞

x2 fA (x)dx =

0

Za1 =

Z∞

TP2 LC−2 (x)fA (x)dx

0

m21 x2 fA (x)dx +

Zam

(m2 x + am (1 − m2 ))2 fA (x)dx +

a1

0

Z∞

(7.39) a2m fA (x)dx

am

Za1 Zam Z∞ σx2 = (pm2 + q)2 x2 fA (x)dx + (m2 x + am (1 − m2 ))2 fA (x)dx + a2m fA (x)dx 0

a1

am

(7.40) Za1 Zam 2 2 2 2 2 σx = (p m2 + q + 2pqm2 )x fA (x)dx + m22 x2 fA (x)dx+ 0

Zam a1

a1

a2m 1 + m22 − 2m2 fA (x)dx +

Zam

2m2 am − 2m22 am xfA (x)dx +

a1

Z∞

a2m fA (x)dx

am

(7.41) 98

Chapter 7. Piecewise Linear Companding Transforms Eq. (7.41) is a quadratic equation with variable m2 . Co-efficients of m22 , m2 and m02 are given by c2 , c1 and c0 as follows. c2 = p2 I2 (0, a1 ) + I2 (a1 , am ) + a2m I0 (a1 , am ) − 2am I1 (a1 , am )

(7.42a)

c1 = 2pqI2 (0, a1 ) − 2a2m I0 (a1 , am ) + 2am I1 (a1 , am )

(7.42b)

c0 = q 2 I2 (0, a1 ) + a2m I0 (a1 , am ) + a2m I0 (am , ∞) − σx2

(7.42c)

where I2 , I1 and I0 are same as given in (7.33), (7.32) and (7.31) respectively. Correlation Coefficient The correlation coefficient α for TP LC−2 (x) is given as follows: 1 α= 2 σx 1 α= 2 σx


(7.43a)

0

Za1 0

1 xm1 xfA (x)dx + 2 σx

Zam a1

1 x (m2 x + am (1 − m2 )) fA (x)dx + 2 σx

Z∞ xam fA (x)dx am

(7.43b) α=

1 1 1 m1 I2 (0, a1 ) + 2 m2 I2 (a1 , am ) + 2 am (1 − m2 )I1 (a1 , am ) + am I1 (am , ∞) (7.43c) 2 σx σx σx

Optimization of Compander’s Parameters The solution for (a1 , m1 , m2 ) is obtained by maximizing the correlation coefficient while keeping average signal power unchanged, using Algorithm 6.

7.5.3

Companding Transform PLC-3

Profiles of Compander and Decompander The compander profile is as given in Fig. 7.6. The transform is composed of four linear functions, specified by parameters (m1 , m2 , m3 , a1 , a2 , am ), such that m2 = 1, as indicated

99


Algorithm 6 Parameters of TP LC−2 (x) Input am , σx . Set a1array = 0 :inc: am , where inc is a small increment. Initialize solution set S={}. for each element a1 in a1array do Find p and q using Eq. (7.37). Find co-efficients c2 , c1 and c0 using Eq. (7.42). Solve c2 m22 + c1 m2 + c0 = 0 for m2 to get two solutions (m21 , m22 ). Set m2 = m21 or m2 = m22 such that m2 < 1. Find m1 using Eq. (7.37). if m1 and m2 satisfy constraints in Eq. (7.38) then Add (a1 , m1 , m2 ) to S. end if end for Initialize αmax =0. for each element of S do Find α using Eq. (7.43). if α is greater than αmax then Accept current element of S as the final solution. Set αmax = α. end if end for Output accepted solution for m1 , m2 and a1 .

am

am

m3 m2 = 1

T (x)

1/m3

T −1 (x)

1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22:

1 m1 1/m1 a1

a2 Amplitude x (a)

am

a1 m1 Amplitude x (b)

am

Figure 7.6. Profiles of PLC-3(a) compander and (b) decompander

in Fig. 7.6. The companding function and its inverse are expressed as follows:

TP LC−3 (x) =

   m1 x,       x + a1 (m1 − 1),

0 ≤ x ≤ a1 a1 < x ≤ a2

   m3 x + am (1 − m3 ), a2 < x ≤ am      a , x > am m

100

(7.44)


where am is the clipping level given by Eq. (7.4).  x   ,     m1    x − a1 (m1 − 1), −1 TP LC−3 (x) = x − am (1 − m3 )    ,   m 3     x,

0 ≤ x ≤ m 1 a1 m1 a1 ≤ x ≤ m3 a2 + (1 − m3 )am

(7.45)

m3 a2 + (1 − m3 )am < x ≤ am x > am

m1 and m3 are related as follows. a2 + a1 (m1 − 1) =m3 a2 + (1 − m3 )am

(7.46a)

a2 + a1 m1 − a1 =m3 a2 + am − m3 am

(7.46b)

m3 =p + qm1 a1 a2 − a1 − am ,q= where p = a2 − am a2 − am

(7.46c) (7.46d)

m1 and m3 satisfy following constraints, m1 > 1 and m3 < 1

(7.47)

Re-writing the transform TP LC−3 (x)

TP LC−3 (x) =

   m1 x,       x + a1 (m1 − 1),

0 ≤ x ≤ a1 a 1 < x ≤ a2

(7.48)

   (qx − qam )m1 + (px + am − pam ), a2 < x ≤ am      a , x > am m Let r = am − pam

TP LC−3 (x) =

   m1 x,       x + a1 (m1 − 1),

(7.49) 0 ≤ x ≤ a1 a1 < x ≤ a2

   (qx − qam )m1 + (px + r), a2 < x ≤ am      a , x > am m

101

(7.50)


Average Power Constraint The constraint on average power is given as:

σx2 =

Z∞

x2 fA (x)dx =

0

Z∞

TP2 LC−3 (x)fA (x)dx +

0

Za2

Za1

m21 x2 fA (x)dx+

0

Zam

2

(x + a1 (m1 − 1)) fA (x)dx + a1

Z∞

2

((qx − qam )m1 + (px + r)) fA (x)dx + a2

a2m fA (x)dx

am

(7.51) σx2 =

Za1

m21 x2 fA (x)dx +

0

Zam

Za2

x2 + a21 (m21 + 1 − 2m1 ) + 2xa1 (m1 − 1) fA (x)dx+

a1 2

(qx − qam )

m21

2

Z∞

+ (px + r) + 2(px + r)(qx − qam )m1 fA (x)dx +

a2m fA (x)dx

am

a2

(7.52) Eq. (7.52) is a quadratic equation with variable m1 . Co-efficients of m21 , m1 and m01 are given by c2 , c1 and c0 as follows. c2 = I2 (0, a1 ) + a21 I0 (a1 , a2 ) + q 2 I2 (a2 , am ) + q 2 a2m I0 (a2 , am ) − 2q 2 a2m I1 (a2 , am ) (7.53a) c1 = −2a21 I0 (a1 , a2 ) + 2a1 I1 (a1 , a2 ) + 2pqI2 (a2 , am ) + 2(qr − pqam )I1 (a2 , am )− qram I0 (a2 , am )

(7.53b)

c0 = I2 (a1 , a2 ) + a21 I0 (a1 , a2 ) − 2a1 I1 (a1 , a2 ) + p2 I2 (a2 , am ) + r2 I0 (a2 , am )+ 2prI1 (a2 , am ) + a2m I0 (am , ∞) − σx2

(7.53c)

where I2 , I1 and I0 are same as given in (7.33), (7.32) and (7.31) respectively.

102


Correlation Coefficient The correlation coefficient α for TP LC−3 (x) is given as: 1 α= 2 σx 1 α= 2 σx 1 σx2

Zam a2


(7.54a)

0

Za1 0

1 xm1 xfA (x)dx + 2 σx

Za2 x (x + a1 (m1 − 1)) fA (x)dx+ a1

1 x (m3 x + (1 − m3 )am ) fA (x)dx + 2 σx

Z∞ xam fA (x)dx

(7.54b)

am

1 1 1 1 α = 2 m1 I2 (0, a1 ) + 2 I2 (a1 , a2 ) + 2 a1 (m1 − 1)I1 (a1 , a2 ) + 2 m3 I2 (a2 , am )+ σx σx σx σx 1 1 (1 − m3 )am I1 (a2 , am ) + 2 am I1 (am , ∞) (7.54c) 2 σx σx Optimization of Compander’s Parameters The solution for (a1 , a2 , m1 , m3 ) is obtained using Algorithm 7. Algorithm 7 Parameters of PLC-3 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25:

Input am , σx . Set a1array = 0 :inc: am , where inc is a small increment. Initialize solution set S={}. for each element a1 in a1array do Set a2array = a1 +inc:inc: am , where inc is a small increment. for each element a2 in a2array do Find p, q and r using Eqs. (7.46) and (7.49). Find co-efficients c2 , c1 and c0 using Eq. (7.53). Solve c2 m21 + c1 m1 + c0 = 0 for m1 to get two solutions (m11 , m12 ). Set m1 = m11 or m1 = m12 such that m1 > 1. Find m3 using Eq. (7.46). if m1 and m3 satisfy constraints in Eq. (7.47) then Add (a1 , a2 , m1 , m3 ) to S. end if end for end for Initialize αmax =0. for each element of S do Find α using Eq. (7.54). if α is greater than αmax then Accept current element of S as the final solution. Set αmax = α. end if end for Output accepted solution for m1 , m3 , a1 and a2 .

103


7.6

Performance Evaluation of PLC-1, PLC-2 and PLC-3

In this section, the proposed companders PLC-1, PLC-2 and PLC-3 are compared in terms of PAPR, BER and OBI.

7.6.1

Simulation Setup

OFDM signal is simulated according to the physical layer specifications in IEEE 802.16d used in Fixed Worldwide Interoperability for Microwave Access (WiMAX). Total number of sub-carriers N in an OFDM symbol is 256, including 192 data bearing carriers, 8 pilot carriers and 56 null carriers (guard band and DC). Oversampling factor L is 4. 4- and 16-QAM are used as modulation schemes.

7.6.2

Simulation Results 0

10

Original OFDM PLC-1, P AP Rdes = PLC-1, P AP Rdes = PLC-2, P AP Rdes = PLC-2, P AP Rdes = PLC-3, P AP Rdes = PLC-3, P AP Rdes =

−1


10

5 4 5 4 5 4

dB dB dB dB dB dB

Fixed WiMAX 4-QAM

−2

10

−3

10

−4

10

3

4

5

6

7

8

9

10

11

12

P AP R0 (dB)

Figure 7.7. CCDFs of PAPR with PLC-1, PLC-2 and PLC-3 applied on WiMAX with 4-QAM

104


−2

10

−3

Bit Error Rate

10

Fixed WiMAX, 4-QAM, over AWGN channel

−4

10


−5

10

−6

10

4

5

5 4 5 4 5 4

dB dB dB dB dB dB

6

7

8

9

10

11

Eb /N0 (dB)

Figure 7.8. BER over AWGN channel with PLC-1, PLC-2 and PLC-3 applied on WiMAX with 4-QAM −1

10


−2

Bit Error Rate

10

−3

10

5 4 5 4 5 4

dB dB dB dB dB dB


−4

10

−5

10

−6

10

0

3

6

9

12

15

18

21

24

27

30

Eb /N0 (dB)

Figure 7.9. BER over AWGN channel with PLC-1, PLC-2 and PLC-3 applied on WiMAX with 16-QAM

7.6.3

Observations and Discussions

• PAPR reduction performance is same is all three companders. The degradation in performance with 16-QAM can be mitigated by using the schemes in Chapter 6. • BER performance of PLC-3 is the best among the three companders. 105



−5


−10 −15 −20

5 4 5 4 5 4

dB dB dB dB dB dB

Fixed WiMAX, 4-QAM

−25 −30 −35 −40 −45 −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1


Figure 7.10. PSDs of signal transformed by PLC-1, PLC-2 and PLC-3 applied on WiMAX with 4-QAM

• OBI is lowest in case of PLC-3 Hence, in the next section, the performance of PLC-3 is compared with those in previous works.

7.7

Performance Comparison with Previous Works

In this section, simulation results are presented for the comparison of proposed companding transform PLC-3 with the companders in existing researches. In Section 7.6, it was shown that among the three proposed transforms, the best performance is obtained with PLC-3. So in this section, performance of PLC-3 is compared with previous works. The transforms selected for comparison include efficient non-linear compander (ENLC) in [15], piecewise exponential compander (PEC) in [12], exponential compander (EC) in [32], non-linear compander (NLC) in [19] and piecewise linear compander (PLC) in [13]. Their profiles are shown in Fig. 7.1. The performance evaluation and comparison is done in terms of PAPR reduction performance, error performance, OBI and implementation complexity.

106

Chapter 7. Piecewise Linear Companding Transforms Table 7.1. Parameters for SUI-1 channel simulation Tap 1

Tap 2

Tap 3

Units

0

0.4

0.9

µs

Power

0

-15

-20

dB

K-Factor

20

0

0

Delay

Table 7.2. Parameters for SUI-5 channel simulation Tap 1

7.7.1

Tap 2

Tap 3

Units

Delay

0

4

10

µs

Power

0

-5

-10

dB

K-Factor

0

0

0

Simulation Setup

OFDM signal is simulated according to the physical layer specifications in IEEE 802.16d used in Fixed Worldwide Interoperability for Microwave Access (WiMAX). Total number of sub-carriers N in an OFDM symbol is 256, including 192 data bearing carriers, 8 pilot carriers and 56 null carriers (guard band and DC). Oversampling factor L is 4. 4- and 16-QAM are used as modulation schemes. Perfect synchronization, ideal channel estimation and zero carrier frequency offset are assumed at the receiver. Stanford university interim (SUI) models given in IEEE 802.16 standard are adopted as multipath channels. The parameters used in simulation of the SUI channels are given in Tables 7.1 and 7.2. Performance with HPA is evaluated using the solid space power amplifier (SSPA) model.

7.7.2

PAPR Reduction Performance

Fig. 7.11 shows empirical curves of CCDF of PAPR for original and companded signals. PLC, PEC and ENLC are configured to specific values of output PAPR. The proposed compander PLC-3 is also configured to the same values of PAPR, so that the BER and PSDs of the companded signals can be fairly compared in the next sub-sections. NLC and EC cannot be tuned to any desired PAPR, so their most commonly used configurations are simulated for comparison. It can be seen that the proposed compander can be conveniently configured to attain any PAPR reduction performance at the output.

107

Chapter 7. Piecewise Linear Companding Transforms 0

10

Original OFDM PLC, P AP Rpreset = 4 dB PLC, P AP Rpreset = 4.5 dB PLC, P AP Rpreset = 5 dB NLC PEC, Ac = σx 10 4/20


−1

10

ENLC, c = 0.465, A = σx 10 4.5/20 EC, d = 1 PLC-3, P AP Rdes = 5 dB PLC-3, P AP Rdes = 4.5 dB PLC-3, P AP Rdes = 4 dB

−2

10

−3

10

Fixed WiMAX, 4-QAM −4

10

4

5

6

7

8

9

10

11

12

P AP R0 (dB)

Figure 7.11. PAPR reduction performances of PLC, NLC, PEC, ENLC, EC and PLC-3

7.7.3

Error Performance

BER over AWGN Channel Figs. 7.12 and 7.13 show error performance evaluation and comparison of the proposed compander/decompander pair over AWGN channel for 4-QAM and 16-QAM modulation, respectively. ENLC, PEC, NLC and EC are simulated without decompanding at the receiver, as proposed in [12, 15, 19, 32], whereas in case of PLC both the compander and decompander are simulated [13]. It can be seen that the proposed compander can efficiently trade between PAPR and BER, i.e., the error performance gracefully degrades with decreasing P AP Rdes . The proposed transform also outperforms the existing companding schemes. For same amount of PAPR reduction, the proposed transform performs better than PLC, PEC and ENLC. It also performs better than EC and NLC as it yields smaller BER, at average PAPR of 4.5 dB, as compared to EC, at average PAPR of 4.76 dB. Similarly, the proposed PLC-3 scheme gives smaller BER at average PAPR of 4 dB as compared to NLC at average PAPR of 4.25 dB. It can also be observed from Fig. 7.13, that by using partial decompanding and hard clipping in the proposed transform, noise floor of the system, observable at high SNR, is considerably lowered as compared to ENLC, NLC, PEC, EC and PLC.

108


−2


10

−3

Bit Error Rate

10


−4

10


−5

10

−6

10

4

5

6

7

8

9

10

11

Eb /N0 (dB)

Figure 7.12. BER over AWGN channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 4-QAM Original OFDM PLC, P AP Rpreset = 4 dB PLC, P AP Rpreset = 4.5 dB PLC, P AP Rpreset = 5 dB NLC PEC, Ac = σx 10 4/20

−1

10

−2

Bit Error Rate

10


−3

10

−4

10


−5

10

−6

10

0

3

6

9

12

15

18

21

24

27

30

Eb /N0 (dB)

Figure 7.13. BER over AWGN channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 16-QAM

BER over SUI-1 Channel Figs. 7.14 and 7.15 show error performance evaluation and comparison over multi-path fading channels, for which the Stanford University Interim (SUI) models are adopted. It was found that generally, the noise amplification due to decompanding has more severe effects in the presence of fading channels and channel estimation, which may be attributed

109

Chapter 7. Piecewise Linear Companding Transforms −1

10


Bit Error Rate

−2

10


−3

10

Fixed WiMAX, 4-QAM over SUI-1 channel −4

10

0

2

4

6

8

10

12

14

16

18

Eb /N0 (dB)

Figure 7.14. BER over SUI-1 channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 4-QAM

−1

10

Fixed WiMAX, 16-QAM, over SUI-1 channel

−2

Bit Error Rate

10


−3

10

−4


10

−5

10

0

2

4

6

8

10

12

14

16

18

20

22

24

Eb /N0 (dB)


to amplitude scaling of the signal introduced by scattering. In case of SUI-1 (Rician) channel, the proposed compander/decompander pair has been found to outperform all the existing transforms under consideration. It should also be noted that the relative degradation observed for PLC in fading channels, as compared to the AWGN channel, is due to the fact that its decompander introduces considerable noise amplification at the receiver, which is further amplified by channel estimation. The proposed compander/decompander pair performs significantly

110


better than PLC, because we have carefully designed both the compander and decompander, according to the criteria in Sub-sections 7.3.3 and 7.3.5. It also performs better than ENLC, NLC, EC and PEC that are simulated without decompanding. BER over SUI-5 Channel The results for BER evaluation over SUI-5 channel are shown in Figs. 7.16 and 7.17. In this case, the effects of decompanding are even more severe and it was found that without decompanding, all the six companders under consideration yield approximately equal BER, which is similar to the BER of the original OFDM signal. It can be observed that the performance of the proposed scheme (with decompanding), at very high Eb /N0 , slightly degrades as compared to PEC, ENLC, NLC and EC (without decompanding). But it also performs significantly better than PLC (with decompanding) for a wide range of Eb /N0 , which again demonstrates that the performance of the decompander in the proposed scheme has been found to be more enhanced as compared to PLC, since it results in a much smaller amount of noise amplification, even in the presence of channel estimation.

Fixed WiMAX, 4-QAM, over SUI-5 channel Original OFDM PLC, P AP Rpreset = 4 dB PLC, P AP Rpreset = 4.5 dB PLC, P AP Rpreset = 5 dB NLC PEC, Ac = σx 10 4/20

Bit Error Rate

−2

10


−3

10

4

6

8

10

12

14

16

18

20

22

24

26

Eb /N0 (dB)


111

28


−1

10

Bit Error Rate

Fixed WiMAX, 16-QAM, over SUI-5 channel Original OFDM PLC, P AP Rpreset = 4 dB PLC, P AP Rpreset = 4.5 dB PLC, P AP Rpreset = 5 dB NLC PEC, Ac = σx 10 4/20

−2

10


−3

10

4

8

12

16

20

24

28

Eb /N0 (dB)

Figure 7.17. BER over SUI-5 channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 16-QAM Table 7.3. Summary of error performance comparison BER (x10−5 ) Schemes

Mean PAPR (dB)

AWGN

SUI-1 (Rician)

SUI-5 (Rayleigh)

Eb /N0 = 10 dB, 4-QAM

Eb /N0 = 15 dB, 16-QAM

Eb /N0 = 8 dB, 4-QAM

Eb /N0 = 16 dB, 16-QAM

Eb /N0 = 20 dB, 4-QAM

Eb /N0 = 20 dB, 16-QAM

PLC

4

3.200

38.49

310.2

164.7

237.7

1284

PLC

4.5

1.700

10.28

285.6

67.29

255.5

971.2

PLC

5

1.225

2.538

276.1

32.33

240.9

753.7

NLC

4.25

2.313

25.18

291

49.49

200.7

458.6

PEC

4

2.750

45.74

310.2

69.46

207.1

458.6

ENLC

4.5

1.563

12.27

278.7

35.22

206.3

439

EC

4.76

1.800

17.13

292

39.51

195.9

446.7

PLC-3

5

0.800

1.375

266.5

19.22

210.6

431

PLC-3

4.5

1.112

4.844

282.3

27.39

214.9

484.9

PLC-3

4

2.075

18.18

300.7

51.52

224.2

601

Summary of Error Performance When considering the overall performance, shown in Figs. 7.12–7.17, it can be concluded that the proposed transform performs significantly better than the existing transforms and is also more robust to the changing channel conditions for a wide range of SNR. The error performance evaluation and comparison is summarized in Table 7.3.

112


Output Amplitude

Asat

1

0.8

0.6

0.4

p=2 p=4

0.2

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Input Amplitude

Figure 7.18. Input amplitude versus output amplitude characteristic curve of SSPA

Output Power (dB)

0

−5

−10

−15

−20 −20

p=2 p=4 −15

−10

−5

0

5

10

Input Power (dB)

Figure 7.19. Input power versus output power characteristic curve of SSPA

7.7.4

Performance with HPA

In order to evaluate the performance of companded signals passing through HPA, solid state power amplifier (SSPA) model is used. The input-output characteristics of the SSPA are given as follows [13]: |y(t)| = "

|x(t)| 2p #1/2p |x(t)| 1+ Asat

113

(7.55)


Input versus output characteristics of SSPA are shown in Figs. 7.18 and 7.19 for different values of p. The parameter p determines the extent of non-linearity of the power amplifier near the saturation region. Smaller value of p indicates higher degree of non-linearity. Asat is the saturation level of the amplifier. The efficiency of the amplifier depends on its input back-off (IBO) which must be greater than or equal to the PAPR of the signal [1]. IBO is defined as follows: IBO (dB) = Psat (dB) − Pavg (dB)

(7.56)

Using the model in Eq. (7.55), BER of companded signals passing through SSPA was evaluated over a range of values of p and IBO. Fixed WiMAX, 4-QAM, with SSPA (p = 2, IBO=PAPR+0.5 dB) over AWGN channel

−2

Bit Error Rate

10


−3

10

−4

10


−5

10

2

3

4

5

6

7

8

9

10

11

Eb /N0 (dB)

Figure 7.20. BER with SSPA (p = 2) over AWGN channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 4-QAM

Following conclusions can be drawn from this performance evaluation: • For highly non-linear HPA (small p and small IBO), the effect of HPA’s non-linearity has more dominant effect as compared to companding distortion. As a result, all companders at same average PAPR perform almost identically. • For large p and IBO, the proposed transform surpasses others in performance. • In the presence of HPA pre-distortion or linearization [6], companded signals will pass through HPA with minimal distortion at IBO≈PAPR. In this case, the proposed transform will surpass the others in performance. 114


Fixed WiMAX, 4-QAM, with SSPA (p = 4, IBO=PAPR+0.5 dB) over AWGN channel

−2

Bit Error Rate

10


−3

10


−4

10

2

3

4

5

6

7

8

9

10

11

Eb /N0 (dB)

Figure 7.21. BER with SSPA (p = 4) over AWGN channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 4-QAM −1

10


−2

Bit Error Rate

10


−3

10


−4

10

3

6

9

12

15

18

21

24

Eb /N0 (dB)

Figure 7.22. BER with SSPA (p = 2, IBO=0.5 dB) over AWGN channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 16-QAM

7.7.5

Out-of-band Interference (OBI) Levels

Figs. 7.26 and 7.27 show PSDs of original and companded signals. It can be seen that the proposed PLC-3 transform performs better in terms of side-lobe level and ACI. For same amount of PAPR reduction, the PLC-3 transform results in smaller OBI than ENLC, PEC and PLC. At average PAPR of 4 dB, the ACI using PLC-3 transform is found to be lower than that with PLC at PAPR of 5 dB, PEC with PAPR of 4 dB, NLC with 115

Chapter 7. Piecewise Linear Companding Transforms −1

10


−2

Bit Error Rate

10


−3

10

−4

10


−5

10

−6

10

3

6

9

12

15

18

21

24

Eb /N0 (dB)

Figure 7.23. BER with SSPA (p = 4, IBO=0.5 dB) over AWGN channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 16-QAM −1

10

Fixed WiMAX, 16-QAM, with SSPA (p = 2, IBO=PAPR+1 dB) over AWGN channel

−2

Bit Error Rate

10


−3

10

−4

10


−5

10

3

6

9

12

15

18

21

24

Eb /N0 (dB)

Figure 7.24. BER with SSPA (p = 2, IBO=1 dB) over AWGN channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 16-QAM

PAPR of 4.25 dB and EC with PAPR 4.76 dB. At average PAPR of 4.5 dB, ACI with the proposed transform is lower than that in case of ENLC with same PAPR. Side-lobe level, in case of the proposed PLC-3 transform, is also reduced as compared to other transforms for same amount of PAPR reduction. This means that it gives smaller PAPR at lower OBI levels as compared to all the existing transforms under consideration. This is due to the fact that the proposed transform distributes the companding noise among 116


−1

10

Fixed WiMAX, 16-QAM, with SSPA (p = 4, IBO=PAPR+1 dB) over AWGN channel

−2

Bit Error Rate

10


−3

10

−4

10


−5

10

−6

10

3

6

9

12

15

18

21

24

Eb /N0 (dB)

Figure 7.25. BER with SSPA (p = 4, IBO=1 dB) over AWGN channel for PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 16-QAM


−5


−10 −15


−20 −25 −30

Fixed WiMAX, 4-QAM

−35 −40 −45 −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1


Figure 7.26. PSDs of signals transformed by PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 4-QAM

all the samples, thereby limiting the high frequency contents in the companded signal, as discussed in Sub-section 7.3.4. In contrast, the existing companders only transform higher amplitudes that introduces higher frequency companding noise. Performance is summarized in Table 7.4.

117




0 −5 −10


−15 −20

Fixed WiMAX 16-QAM

−25 −30 −35 −40 −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1


Figure 7.27. PSDs of signals transformed by PLC, NLC, PEC, ENLC, EC and PLC-3 applied on WiMAX with 16-QAM Table 7.4. Summary of OBI performance comparison PSD (dB/rad/sample) Schemes Mean PAPR (dB) Normalized Normalized frequency= 0.4π frequency= 0.8π

7.7.6

PLC

4

-31.59

-40.91

PLC

4.5

-33.20

-42.41

PLC

5

-34.75

-43.55

NLC

4.25

-31.83

-43.43

PEC

4

-30.75

-41.36

ENLC

4.5

-32.52

-43.99

EC

4.76

-29.79

-38.80

PLC-3

5

-34.95

-44.37

PLC-3

4.5

-33.50

-43.99

PLC-3

4

-32.01

-43.66


Piecewise linear transforms are the least complex functions among the companding transforms since they, at most, require only one floating point multiplication and one addition per sample. Computational complexity, as given in Table 7.5, has been evaluated and compared in terms of MATLAB floating point operations or flops [12, 13]. Since amplitude calculation is common in all cases, so only the arithmetic operations in the compander functions are considered. For fair comparison, all the configurable transforms under con118

Chapter 7. Piecewise Linear Companding Transforms Table 7.5. Comparison of implementation complexity Flops/sample Schemes

PLC, P AP Rpreset = 4.5 dB

Transmitter

Receiver

1

1

Number of Companded Samples (xNL) A Rc fA (dx)dx = Ai

Total flops (xNL) 1.192

0.596 R∞ NLC

8

7

√ σx / 6

fA (dx)dx = 12.7 0.85

PEC, Ac = σx 104.5/20 dB

11

9

R∞ Ai

ENLC, A = σx 104.5/20 , c = 0.465 dB

35

PLC-3, P AP Rdes = 4.5 dB

1

45

fA (dx)dx = 0.66 R∞

fA (dx)dx =

cσx

13.12 64.48

0.806 1

aRm

fA (dx)dx = 0.94

1.88

0

sideration, namely ENLC, PLC, PEC and the proposed transform, are set to same output PAPR, i.e., 4.5 dB. In cases where decompander is not used, the computations at the receiver can be neglected. Also, it should be noted that although the number of companded samples in the proposed scheme is larger, but for most of the amplitudes in TP LC−1 and TP LC−3 , only one floating point addition is required, as compared to a large number of multiplications and other complex operations required in other transforms.

119

Chapter 8

Conclusion and Future Work 8.1

Conclusions

In this thesis, the effects of OFDM system parameters on the performance of companding transforms were explored. It was found that companding schemes do not perform well for systems with relatively smaller number of sub-carriers and higher order QAM modulation. In order to mitigate this problem, adaptive companding, adjustable-parameter companding (APC) and adaptive constellation scaling (ACS) were proposed. Adaptive companding is a general scheme that can be designed for any OFDM specifications. APC and ACS are specifically for higher order QAM, like 16 and 64-QAM. Detailed performance evaluation of the schemes were done by considering different constellations, number of sub-carriers and companding transforms. It was found that in case of 4-QAM, PAPR reduction performance can be improved without increasing the BER and OBI. In case of 16 and 64-QAM, the proposed schemes have the ability to jointly improve PAPR and BER, while keeping OBI unchanged. This is not possible to achieve in conventional companding in which PAPR can only be improved by compromising BER and OBI. Performance analysis also shows that the increase in complexity and data rate loss in the proposed modifications is very small. Therefore, in terms of overall performance of the system, the proposed schemes perform significantly better than conventional companding. In the second part of the thesis, design optimization of piecewise linear companding transforms was considered. Experiment/analysis based studies were conducted to understand the relationship between compander design and system’s performance metrics. Three new transforms were presented that were designed with the aim of collectively

120

Chapter 8. Conclusion and Future Work

optimizing BER, OBI and computational complexity. Simulation results show that the new transforms outperformed the ones in previous works from all of the above-mentioned aspects.

8.2

Future Work

Further improvement in error performance can be achieved by integrating reconstruction algorithms with the decompander for regrowing clipped signals. Decompander’s performance in case of fading channels was found to degrade. So more possibilities of refinements in the decompanding operation in the presence of channel estimation can be explored. Compander design can also be related with some other performance metrics used to measure the effects of non-linearities in OFDM systems. The performance metrics include Error Vector Magnitude (EVM), Spectral Mask and Raw Cubic Metric (RCM). Companding operation in the presence of transmit filter and/or band-limited channel can be explored and corresponding modifications in the decompander can be designed. Stochastic modeling of OFDM systems used in mobile internet access, like Mobile WiMAX, can be done and companding transforms can be designed. In these systems, probability distribution of signal’s amplitude and phase changes due to the addition of large number of BPSK modulated pilot carriers. It will also be interesting to consider integrating companding transforms with the HPA linearization algorithm. This can be expected to reduce the overall complexity of the system.

121

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