thesis_sthompson

UNIVERSITY OF CALIFORNIA, SAN DIEGO

Constant Envelope OFDM Phase Modulation

A dissertation submitted in partial satisfaction of the

requirements for the degree

Doctor of Philosophy

in

Electrical Engineering (Communications Theory and Systems)

by

Steve C. Thompson

Committee in charge:

Professor James R. Zeidler, ChairProfessor John G. Proakis, Co-ChairProfessor Robert R. BitmeadProfessor William S. HodgkissProfessor Laurence B. Milstein

2005

Copyright

Steve C. Thompson, 2005

All rights reserved.

The dissertation of Steve C. Thompson is ap-

proved, and it is acceptable in quality and form

for publication on microfilm:

Co-Chair

Chair

University of California, San Diego

2005

iii

“Before PhD,I chopped wood and carried water;

After PhD,I chopped wood and carried water.”

—[Slightly modified] Zen saying

“I wish I could be more moderate in my desires. But I can’t, so there is no rest.”

—John Muir, 1826

“I know this: a man got to do what he got to do. . . ”

—Casy, The Grapes of Wrath, John Steinbeck, 1939

iv

TABLE OF CONTENTS

Signature Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Vita and Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

Abstract of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 An Introduction to OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 ISI-Free Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 A Multicarrier Modulation . . . . . . . . . . . . . . . . . . . . . . 5

1.1.3 Discrete-Time Signal Processing . . . . . . . . . . . . . . . . . . . 8

1.2 Problems with OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Constant Envelope Waveforms . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Constant Envelope OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1 More OFDM Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.1 The Cyclic Prefix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.2 Discrete-Time Model . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.3 Block Modulation with FDE . . . . . . . . . . . . . . . . . . . . . 20

2.1.4 System Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 PAPR Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Power Amplifier Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Effects of Nonlinear Power Amplification . . . . . . . . . . . . . . . . . . . 30

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2.4.1 Spectral Leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.2 Performance Degradation . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.3 System Range and PA Efficiency . . . . . . . . . . . . . . . . . . . 35

2.5 PAPR Mitigation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Constant Envelope OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1 Signal Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Performance of Constant Envelope OFDM in AWGN . . . . . . . . . . . . . . . 58

4.1 The Phase Demodulator Receiver . . . . . . . . . . . . . . . . . . . . . . . 59

4.1.1 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.1.2 Effect of Channel Phase Offset . . . . . . . . . . . . . . . . . . . . 65

4.1.3 Carrier-to-Noise Ratio and Thresholding Effects . . . . . . . . . . 66

4.1.4 FIR Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2 The Optimum Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2.1 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.2 Asymptotic Properties . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3 Phase Demodulator Receiver versus Optimum . . . . . . . . . . . . . . . . 78

4.4 Spectral Efficiency versus Performance . . . . . . . . . . . . . . . . . . . . 80

4.5 CE-OFDM versus OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5 Performance of CE-OFDM in Frequency-Nonselective Fading Channels . . . . . 86

6 Performance of CE-OFDM in Frequency-Selective Channels . . . . . . . . . . . 94

6.1 MMSE versus ZF Equalization . . . . . . . . . . . . . . . . . . . . . . . . 96

6.1.1 Channel Description . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.1.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.1.3 Discussion and Observations . . . . . . . . . . . . . . . . . . . . . 103

6.2 Performance Over Frequency-Selective Fading Channels . . . . . . . . . . 108

6.2.1 Channel Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.2.2 Simulation Procedure and Preliminary Discussion . . . . . . . . . 112

6.2.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

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7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

A Generating Real-Valued OFDM Signals with the Discrete Fourier Transform . . 124

A.1 Signal Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

A.2 Spectral Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

B More on the OFDM Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

C Sample Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

C.1 GNU Octave Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

C.2 Gnuplot Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Production Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

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LIST OF FIGURES

1.1 Representation of a wireless channel with multipath. . . . . . . . . . . . . 2

1.2 A wireless channel in time and frequency. . . . . . . . . . . . . . . . . . . 2

1.3 Intersymbol interference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 OFDM with cyclic prefix (CP). . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Subcarrier and overall spectrum. (N = 16; |I0,k| = 1, for all k) . . . . . . 7

1.6 OFDM converts wideband channel to N narrowband frequency bins. . . . 8

1.7 Frequency offset causes ICI. (εfo = 0.25) . . . . . . . . . . . . . . . . . . . 9

1.8 A typical OFDM signal (N = 16). The PAPR is 9.5 dB. . . . . . . . . . . 10

1.9 Power amplifier transfer function. . . . . . . . . . . . . . . . . . . . . . . . 11

1.10 Comparison of OFDM and CE-OFDM signals. . . . . . . . . . . . . . . . 13

2.1 Sampling instances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Circular convolution with channel and the inverse channel. . . . . . . . . . 21

2.3 Block modulation with cyclic prefix and FDE. . . . . . . . . . . . . . . . . 21

2.4 OFDM is a special case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 OFDM system diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.6 Complementary cumulative distribution functions. (N = 64) . . . . . . . 25

2.7 PAPR CCDF lower bound (2.31) for N = 2k, k = 5, 6, . . . , 10. . . . . . . . 26

2.8 AM/AM (solid) and AM/PM (dash) conversions (SSPA=thick, TWTA=thin)for various backoff ratios K. . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.9 Fractional out-of-band power of OFDM with ideal PA and with TWTAmodel at various input power backoff. (N = 64, IBO in dB) . . . . . . . . 31

2.10 Spectral growth versus IBO. (N = 64) . . . . . . . . . . . . . . . . . . . . 31

2.11 Performance of QPSK/OFDM with nonlinear power amplifier with variousinput power backoff levels. (N = 64) . . . . . . . . . . . . . . . . . . . . . 33

2.12 Performance of M -PSK/OFDM with SSPA. (N = 64) . . . . . . . . . . . 34

2.13 The potential range of system is reduced with input backoff; the range isreduced further from nonlinear amplifier distortion. . . . . . . . . . . . . . 36

2.14 Power amplifier efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.15 Block diagram. The system is evaluated with and without PAPR reduction. 38

2.16 Unclipped OFDM signal (9.25 dB PAPR). The rings have radius Amax

which correspond to various clipping ratios γclip (dB). . . . . . . . . . . . 39

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2.17 PAPR CCDF of clipped OFDM signal for various γclip (dB). [N = 64] . . 40

2.18 PAPR of clipped signal as a function of the clipping ratio. (N = 64) . . . 40

2.19 A comparison of the total degradation curves of clipped and unclippedM -PSK/OFDM systems. (N = 64) . . . . . . . . . . . . . . . . . . . . . . 41

3.1 The CE-OFDM waveform mapping. . . . . . . . . . . . . . . . . . . . . . 43

3.2 Instantaneous signal power. . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Basic concept of CE-OFDM. . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 Phase discontinuities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.5 Continuous phase CE-OFDM signal samples, over L blocks, on the com-plex plane. (2πh = 0.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.6 Estimated fractional out-of-band power. (N = 64) . . . . . . . . . . . . . 52

3.7 Double-sided bandwidth as a function of modulation index. (N = 64) . . 53

3.8 Power density spectrum. (N = 64, 2πh = 0.6) . . . . . . . . . . . . . . . . 54

3.9 Fractional out-of-band power. (N = 64, 2πh = 0.6) . . . . . . . . . . . . . 55

3.10 CE-OFDM versus OFDM. (N = 64) . . . . . . . . . . . . . . . . . . . . . 56

3.11 CE-OFDM versus OFDM with nonlinear PA. (N = 64) . . . . . . . . . . 57

4.1 Phase demodulator receiver. . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2 Bandpass to baseband conversion. . . . . . . . . . . . . . . . . . . . . . . 60

4.3 Discrete-time phase demodulator. . . . . . . . . . . . . . . . . . . . . . . . 62

4.4 Performance with and without phase offsets. System 1 (S1) has phaseoffsets {(θi + φ0) ∈ [0, 2π)}, and System 2 (S2) doesn’t (θi + φ0 = 0).[M = 2, N = 64, J = 8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.5 Threshold effect at low CNR. (M = 8, N = 64, J = 8, 2πh = 0.5) . . . . . 68

4.6 Threshold effect at low CNR, various 2πh. (M = 8, N = 64, J = 8) . . . 68

4.7 Performance for various filter parameters Lfir, fcut/W .(M = 2, N = 64, J = 8, 2πh = 0.5 and Eb/N0 = 10 dB) . . . . . . . . . . 69

4.8 Magnitude response of various Hamming FIR filters. . . . . . . . . . . . . 70

4.9 CE-OFDM performance with and without FIR filter.(M = 2, N = 64, J = 8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.10 The optimum receiver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.11 Correlation functions ρm,n(K). . . . . . . . . . . . . . . . . . . . . . . . . 76

4.12 CE-OFDM optimum receiver performance. (M = 2, N = 8) . . . . . . . . 77

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4.13 All unique ρm,n(K) for M = 2, N = 4 DCT modulation. . . . . . . . . . . 78

4.14 Phase demodulator receiver versus optimum. (N = 64) . . . . . . . . . . . 79

4.15 Noise samples PDF versus Gaussian PDF. (Eb/N0 = 30 dB) . . . . . . . . 80

4.16 Performance ofM -PAM CE-OFDM. (N = 64, †=leftmost curve, ‡=rightmostcurve) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.17 Spectral efficiency versus performance. . . . . . . . . . . . . . . . . . . . . 82

4.18 A comparison of CE-OFDM and conventional OFDM. (M = 2, N = 64) . 85

5.1 Performance of CE-OFDM in flat fading channels. (N = 64) . . . . . . . 88

5.2 A simplified two-region model. (M = 8, N = 64, 2πh = 0.6) . . . . . . . . 90

5.3 A (n+ 1)-region model. (M = 8, N = 64, 2πh = 0.6) . . . . . . . . . . . . 91

5.4 Performance of CE-OFDM in flat fading channels. (Circle=Rayleigh;square=Rice, K = 3 dB; triangle=Rice, K = 10 dB. Solid line=Semi-analytical curve, (5.15); points=simulation. N = 64) . . . . . . . . . . . . 92

5.5 Comparison of semi-analytical technique (5.15) with (5.10) and (5.11).(M = 4, N = 64, 2πh = 1.2) . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.1 CE-OFDM system with frequency-selective channel. . . . . . . . . . . . . 96

6.2 Channel D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.3 Channel A results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.4 Channel B results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.5 Channel C results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.6 Channel D results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.7 Channel E results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.8 Channel F results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.9 Fundamental characteristic functions and quantities [(6.21)–(6.25)] of thefour channel models considered. . . . . . . . . . . . . . . . . . . . . . . . . 113

6.10 Performance results. (Multipath results are labeled with circle and trian-gle points; the Rayleigh, L = 1 result is that of the frequency-nonselectivechannel model. M = 4, N = 64, 2πh = 1.0) . . . . . . . . . . . . . . . . . 115

6.11 Single path versus multipath. (M = 4, N = 64, Channel Cf, MMSE) . . . 119

6.12 CE-OFDM versus QPSK/OFDM. (SSPA model, Channel Cf, N = 64,MMSE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

B.1 “OFDM” search on IEEE Xplore [222]. . . . . . . . . . . . . . . . . . . . . 130

B.2 Papers, filed and piled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

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B.3 Running average of papers read per day. . . . . . . . . . . . . . . . . . . . 132

B.4 Year histogram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

B.5 Projected year histogram? . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

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LIST OF TABLES

6.1 Channel samples of frequency-selective channels. . . . . . . . . . . . . . . 97

6.2 Channel model parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.3 Data symbol contribution per tone for mn(t), n =1, 2, and 3. . . . . . . . 118

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ACKNOWLEDGEMENTS

I want to first thank my advisors, Professors Zeidler and Proakis, for giving me the

chance to do this work, for the encouragement, and for the guidance. I want to thank

Professor Milstein for the many helpful technical conversations and for his many sug-

gestions. Thanks to Professors Bitmead and Hodgkiss for taking the time to participate

as committee members. Also, thanks to Professor Proakis for carefully proofreading the

draft manuscripts of this thesis.

Thanks to UCSD’s Center for Wireless Communications for providing a good en-

vironment for conducting research; thanks to its industrial partners for the financial

support.

Thanks to my wife, Shannon, for the emotional and caloric support. Thanks to

Chaney the cat for waking me up in the morning. Thanks to my friends for fun support.

Thanks to my fellow graduate students in Professor Zeidler’s research group for the

camaraderie. Special thanks to Ahsen Ahmed for helpful collaboration over the past

couple years. Thanks to my family. Also, thanks to Karol Previte for her support early

in my graduate student existence.

Thanks to my teachers: Professors Duman, Masry, Milstein, Pheanis, and Wolf, to

name only a few.

Finally, I would like to thank the countless developers, documentation writers, bug

reporters, and users of the free software I’ve benefited from during the course of my PhD.

The text in this thesis, in part, was originally published in the following papers, of

which I was the primary researcher and author: S. C. Thompson, J. G. Proakis, and

J. R. Zeidler, “Constant Envelope Binary OFDM Phase Modulation,” in Proc. IEEE

Milcom, vol. 1, Boston, Oct. 2003, pp. 621–626; S. C. Thompson, A. U. Ahmed, J.

G. Proakis, and J. R. Zeidler, “Constant Envelope OFDM Phase Modulation: Spectral

Containment, Signal Space Properties and Performance,” in Proc. IEEE Milcom, vol. 2,

Monterey, Oct. 2004, pp. 1129–1135; S. C. Thompson, J. G. Proakis, and J. R. Zeidler,

“Noncoherent Reception of Constant Envelope OFDM in Flat Fading Channels,” in Proc.

IEEE PIMRC, Berlin, Sept. 2005; and S. C. Thompson, J. G. Proakis, and J. R. Zeidler,

“The Effectiveness of Signal Clipping for PAPR Reduction and Total Degradation in

OFDM Systems,” in Proc. IEEE Globecom, St. Louis, Dec. 2005.

xiii

VITA

December 22, 1976 Born, Mesa, Arizona

1997–1998 Associate EngineerInter-Tel, Chandler, Arizona

Summer 1998 Summer InternshipLos Alamos National LaboratoryLos Alamos, New Mexico

1999 BSc in Electrical EngineeringArizona State University, Tempe, Arizona

Summer 2001 Summer InternshipSPAWAR Systems Center, San Diego, California

2001 MSc in Electrical EngineeringUniversity of California at San Diego, La Jolla, California

2001–2005 Research AssistantCenter for Wireless CommunicationsUniversity of California at San Diego, La Jolla, California

Summer 2004 Summer InternshipSPAWAR Systems Center, San Diego, California

2005 PhD in Electrical EngineeringUniversity of California at San Diego, La Jolla, California

PUBLICATIONS

S. C. Thompson, J. G. Proakis, and J. R. Zeidler, “Constant Envelope Binary OFDMPhase Modulation,” in Proc. IEEE Milcom, vol. 1, Boston, Oct. 2003, pp. 621–626.

S. C. Thompson, A. U. Ahmed, J. G. Proakis, and J. R. Zeidler, “Constant EnvelopeOFDM Phase Modulation: Spectral Containment, Signal Space Properties and Perfor-mance,” in Proc. IEEE Milcom, vol. 2, Monterey, Oct. 2004, pp. 1129–1135.

S. C. Thompson, A. U. Ahmed, J. G. Proakis, and J. R. Zeidler, “Constant EnvelopeOFDM Phase Modulation,” submitted to IEEE Transactions on Communications.

S. C. Thompson, J. G. Proakis, and J. R. Zeidler, “Noncoherent Reception of ConstantEnvelope OFDM in Flat Fading Channels,” in Proc. IEEE PIMRC, Berlin, Sept. 2005.

S. C. Thompson, J. G. Proakis, and J. R. Zeidler, “The Effectiveness of Signal Clip-ping for PAPR Reduction and Total Degradation in OFDM Systems,” in Proc. IEEEGlobecom, St. Louis, Dec. 2005.

xiv

S. C. Thompson, J. G. Proakis, and J. R. Zeidler, “The Effectiveness of Signal Clippingfor PAPR Reduction and Total Degradation in OFDM Systems,” in preparation.

S. C. Thompson, A. U. Ahmed, J. G. Proakis, and J. R. Zeidler, M -ary PAM ConstantEnvelope OFDM,” in preparation.

S. C. Thompson, J. G. Proakis, and J. R. Zeidler, “Performance of CE-OFDM inFrequency-Nonselective Fading Channels,” in preparation.

S. C. Thompson, J. G. Proakis, and J. R. Zeidler, “Performance of CE-OFDM inFrequency-Selective Channels,” in preparation.

xv

ABSTRACT OF THE DISSERTATION

Constant Envelope OFDM Phase Modulation

by

Steve C. Thompson

Doctor of Philosophy in Electrical Engineering (Communications Theory and

Systems)

University of California San Diego, 2005

Professor James R. Zeidler, Chair

Professor John G. Proakis, Co-Chair

Orthogonal frequency division multiplexing (OFDM) is a popular modulation technique

for wireless digital communications. It provides a relatively straightforward way to ac-

commodate high data rate links over harsh wireless channels characterized by severe

multipath fading. OFDM has two primary drawbacks, however. The first is a high sen-

sitivity to time variations in the channel caused by Doppler, carrier frequency offsets,

and phase noise. The second, and the focus of this thesis, is that the OFDM waveform

has high amplitude fluctuations, a drawback known as the peak-to-average power ratio

(PAPR) problem. The high PAPR makes OFDM sensitive to nonlinear distortion caused

by the transmitter’s power amplifier (PA). Without sufficient power backoff, the system

suffers from spectral broadening, intermodulation distortion, and, consequently, perfor-

mance degradation. High levels of backoff reduce the efficiency of the PA. For mobile

battery-powered devices this is a particularly detrimental problem due to limited power

resources.

A new PAPR mitigation technique is presented. In constant envelope OFDM (CE-

OFDM), the high PAPR OFDM signal is transformed to a constant envelope 0 dB PAPR

waveform by way of angle modulation. The constant envelope signal can be efficiently

amplified with nonlinear power amplifiers thus achieving greater power efficiency. In

xvi

this thesis, the fundamental aspects of the CE-OFDM modulation are studied, including

the signal spectrum, the signal space, optimum performance, and the performance of

a practical phase demodulator receiver. Performance is evaluated over a wide range of

multipath fading channel models. It is shown that CE-OFDM outperforms conventional

OFDM when taking into account the effects of the power amplifier.

This work was done at UCSD’s Center for Wireless Communication, under the “Mo-

bile OFDM Communications” project (CoRe research grant 00-10071).

xvii

Chapter 1

Introduction

Humans have always found ways to communicate, over space and over time. From

the messenger pigeon to the Pony Express, from the message in a bottle to cave drawings,

smoke signals and beacons, people have used inventive techniques, techniques derived

from their natural environment, to share information. A particularly good natural re-

source for communication is electricity for its speed and ability to be controlled with

devices like capacitors, microprocessors, electronic memory storage and batteries. Com-

munication was profoundly enhanced with Morse’s telegraph (1837), Bell’s telephone

(1876), Edison’s phonograph (1887), and Marconi’s radio (1896). From these early inven-

tions, communications technology has advanced with global telephone networks, satellite

communications, and magnetic storage systems; and with the rise of the internet and

digital computers, digital communications—the transfer of bits (1’s and 0’s) from one

point to another—has become important.

In particular, wireless digital communications is currently under intensive research,

development and deployment to provide high data rate access plus mobility. One chal-

lenge in designing a wireless system is to overcome the effects of the wireless channel,

which is characterized as having multiple transmission paths and as being time vary-

ing [421, 427]. Figure 1.1 illustrates a link with four reflecting paths between points A

and B. These reflections are caused by physical objects in the environment. Due to the

relative mobility between the points and the possibility that the reflecting objects are

mobile, the channel changes with time.

1

2

� �� ���

�� Propagation paths

point B

point A

Figure 1.1: Representation of a wireless channel with multipath.

An example profile of the channel in Figure 1.1 is shown in Figure 1.2(a). Each path

has its own associated delay and power. The first path arrives at the receiver 0.5 µs after

the signal is transmitted; the last path arrives with a 14 µs delay. The Fourier transform

of the profile yields the frequency-domain representation shown in Figure 1.2(b). The

channel is viewed over a 2 MHz range centered at the center frequency fc. Notice that

the channel power fluctuates by 30 dB (a factor of 1000) over the frequency range. The

dispersion in the time domain leads to frequency-selectivity in the frequency domain.

Time (µs)

Path

pow

er

14121086420

1

0.1

0.01

(a) Time domain.

Frequency, f − fc (MHz)

Channel

pow

er(d

B)

10.50−0.5−1

5

0

-5

-10

-15

-20

-25

-30

(b) Frequency domain.

Figure 1.2: A wireless channel in time and frequency.

In general, a digital communication system maps bits to kb-bit data symbols. In a

conventional single carrier system, the symbols are then transmitted serially. The signal

waveform of such a system is

s(t) =∑

i

Iig(t− iTs), (1.1)

where t is the time variable, {Ii} are the data symbols, Ts is the symbol period, and g(t)

is a transmit pulse shape. For time-dispersive channels, such as the 4-path example in

3

Figure 1.2, interference is caused from symbol to symbol. This intersymbol interference

(ISI) is illustrated in Figure 1.3. For simplicity, g(t) is rectangular. The channel is

represented by its time-variant impulse response h(τ, t), where τ is a propagation delay

variable. The received signal is expressed mathematically as [387, p. 97]

r(t) = s(t) ∗ h(τ, t) + n(t)

=

∫ ∞

−∞h(τ, t)s(t− τ)dτ + n(t),

(1.2)

where ∗ represents the linear convolution operator and n(t) is additive noise. The effect of

the time-dispersive channel is shown to smear symbol 1 into symbol 2, therefore creating

intersymbol interference.

Transmitter Channel Receiverr(t)s(t)

.. . .. .

|h(τ, t)|

τ

ISI

symbol 1 symbol 2

s(t)

t

r(t)

t00 2TsTs 2TsTs

Figure 1.3: Intersymbol interference.

The severity of the ISI depends on the symbol period relative to the channel’s max-

imum propagation delay, τmax. Consider transmitting the signal in (1.1) over the 2

MHz channel in Figure 1.2. The signal bandwidth is roughly proportional to the sym-

bol rate 1/Ts Hz. Therefore making s(t) a 2 MHz signal, Ts = (2 × 106)−1 = 0.5 µs.

Since the maximum propagation delay of the channel is τmax = 14 µs, the ISI spans

τmax/Ts = (14 µs) / (0.5 µs) = 28 symbols. (For comparison, the ISI in Figure 1.3 spans

less than one symbol.) Such severe ISI must be corrected at the receiver in order to

provide reliable communication.

The traditional approach to combating intersymbol interference is with time-domain

equalizers [421]. There are many types, ranging in complexity and in effectiveness.

The optimum maximum-likelihood (ML) receiver is the most effective but is typically

impractical due to its high complexity, which grows exponentially with the ISI length.

Linear equalizers are much simpler, having a complexity which grows roughly linearly

with ISI length, but perform much worse than the optimum receiver. Nonlinear decision

feedback equalizers (DFEs) have similar complexity as the linear type and have better

performance.

4

All of these techniques require knowledge of the channel, which is estimated by

transmitting a training sequence which is known at the receiver. Then by comparing

the received signal to what was transmitted, an estimate of h(τ, t) is made. There are

various algorithms available for the estimation process, each having its own complexity,

convergence rate, and stability. The least-mean-square (LMS) algorithm is the most

stable and the least complex, but suffers from a slow convergence rate. The recursive

least-square (RLS or Kalman) algorithm, on the other hand, converges quickly, but has

higher complexity and can be unstable.

For scenarios like the example above with an ISI spanning 28 symbols, conventional

equalization becomes difficult. Training times become long and convergence of the chan-

nel estimator is problematic, especially for time-varying channels. In the example, 2×106

symbols/s are transmitted. Using a QPSK (quadrature phase-shift keying) signal con-

stellation, which maps kb = 2 bits per symbol, the bit rate is 4 Mb/s. Such a bit rate is

desired in current wireless systems, and in many cases demand for many tens of Mb/s

is common.

1.1 An Introduction to OFDM

To meet the demanding data rate requirements, alternative techniques have been

considered. One approach, orthogonal frequency division multiplexing, has become ex-

ceedingly popular. OFDM has been implemented in wireline applications such as digital

subscriber lines (DSL) [95], in wireless broadcast applications such as digital audio and

video broadcasting (DAB and DVB) and in-band on-channel (IBOC) broadcasting [392].

It has been used in wireless local area networks (LANs) under the IEEE 802.11 and the

ETSI HYPERLAN/2 standards [552]. OFDM is being developed for ultra-wideband

(UWB) systems; cellular systems; wireless metropolitan area networks (MANs), under

the IEEE 802.16 (WiMax) standard; and for other wireline systems such as power line

communication (PLC) [119, 160, 264, 604].

1.1.1 ISI-Free Operation

OFDM’s main appeal is that it supports high data rate links without requiring

conventional equalization techniques. Instead of transmitting symbols serially, OFDM

5

sends N symbols as a block. The OFDM block period, TB, is thus N times longer than

the symbol period. Continuing the example above, and choosing N = 300, the block

period is TB = NTs = 300 × 0.5 µs = 150 µs, which is more than 10 times the duration

of the channel’s impulse response. ISI is avoided by inserting a guard interval between

successive blocks during which a cyclic prefix is transmitted. The interval duration, Tg,

is designed such that Tg ≥ τmax so that the channel is absorbed in the guard interval

and the OFDM block is uncorrupted. This is illustrated in the figure below. Selecting a

guard interval Tg = 15 µs for the channel in Figure 1.2 results in a transmission efficiency

ηt = TB/(TB +Tg) = 150/165 ≈ 0.91. Therefore, with a small reduction in efficiency, ISI

is eliminated.

ISI-free block

CP OFDM block

TB

τ

r(t)

t

s(t)

t

|h(τ, t)|

Tg

Figure 1.4: OFDM with cyclic prefix (CP).

1.1.2 A Multicarrier Modulation

The OFDM signal can be expressed as1

s(t) =∑

i

[

N−1∑

k=0

Ii,kej2πfkt

]

g(t− iTB). (1.3)

The pulse shape, g(t), is typically rectangular:

g(t) =

1, 0 ≤ t < TB,

0, otherwise.(1.4)

Notice that the N data symbols {Ii,k}N−1k=0 are transmitted during the ith block. The

set of complex sinusoids {exp (j2πfkt)}N−1k=0 are referred to as subcarriers. The center

1For simplicity, the guard interval is excluded from the signal definition in (1.3). The guard intervaland cyclic prefix is discussed in Chapter 2.

6

frequency of the kth subcarrier is fk = k/TB and the subcarrier spacing, 1/TB Hz, makes

the subcarriers orthogonal over the block interval, expressed mathematically as

1

TB

∫ TB

0

(

ej2πfk1t)∗ (

ej2πfk2t)

dt =1

TB

∫ TB

0ej2π(fk2

−fk1)tdt

=

1, k1 = k2,

0, k1 6= k2,

(1.5)

where (·)∗ represents the complex conjugate operation. The subcarrier orthogonality can

also be viewed in the frequency domain. Consider the 0th OFDM block:

s(t) =

N−1∑

k=0

I0,kej2πfkt, 0 ≤ t < TB. (1.6)

The frequency-domain representation is

S(f) = F {s(t)} (f) = TBe−j2πfTB/2

N−1∑

k=0

I0,k sinc

[(

f − k

TB

)

TB

]

, (1.7)

where F{·}(f) is the Fourier transform and

sinc(x) =

1, x = 0,

sinπxπx , otherwise.

(1.8)

Figure 1.5 plots |S(f)/TB| for N = 16 subcarriers and data symbols with normalized

amplitudes. The individual subcarrier spectra are also plotted. Notice that at the kth

subcarrier frequency, k/TB, the kth subcarrier has a peak and all the other subcarriers

have zero-crossings. Therefore, the subcarriers, while tightly packed (which improves

spectral efficiency), are non-interfering (i.e. orthogonal).

Figure 1.5 also demonstrates that OFDM is a multicarrier modulation, as opposed

to a single carrier modulation like the signal in (1.1). In general, a transmitted bandpass

signal is [421, p. 151]

x(t) = <{

s(t)ej2πfct}

, (1.9)

where fc is the carrier frequency. For single carrier,

xsc(t) =∑

i

|Ii| cos [2πfct+ arg(Ii)] g(t− iTs); (1.10)

while for multicarrier,

xmc(t) =∑

i

{

N−1∑

k=0

|Ii,k| cos[

2π

(

fc +k

TB

)

t+ arg(Ii,k)

]

}

g(t− iTB). (1.11)

7

OverallSubcarrier

Normalized frequency, fTB

Spec

trum

magnitude,|S

(f)/T

B|

181614121086420-2

1.2

1

0.8

0.6

0.4

0.2

0

Figure 1.5: Subcarrier and overall spectrum. (N = 16; |I0,k| = 1, for all k)

For single carrier each symbol occupies the entire signal bandwidth, while for multicarrier

the bandwidth is split into many frequency bands (also referred to as frequency bins).

Notice that the multicarrier signal transmits the N data symbols in parallel over multiple

carriers each centered at (fc + k/TB) Hz, k = 0, 1, . . . , N − 1.

By properly designing the subcarrier spacing, each frequency bin is made frequency-

nonselective. The wideband frequency-selective channel is converted into N contigu-

ous narrowband frequency-nonselective bins. Figure 1.6 shows 18 bins in the range

[−0.9,−0.78] MHz for the N = 300 OFDM system over the channel in Figure 1.2(b).

Notice that the channel gain per bin varies over a 15 dB range. The OFDM modulation

can be optimized for the channel by sending more bits in frequency bins with high gain

and fewer bits in frequency bins with low gain. This technique, known as bit loading,

requires a fairly stable channel, one that can be accurately measured. For this reason,

bit loading is more common in wireline systems and stationary wireless systems than in

wireless systems with high mobility.

Frequency selectivity is the frequency-domain dual of intersymbol interference. Trans-

mitting the single carrier signal over the 2 MHz channel results in a frequency-selective

response. For OFDM, the overall channel is frequency-selective but for each bin the chan-

8

Frequency bins

Frequency, f − fc (MHz)

Channel

pow

er(d

B)

−0.8−0.85−0.9

5

0

-5

-10

-15

-20

-25

-30

Figure 1.6: OFDM converts wideband channel to N narrowband frequency bins.

nel is frequency non-selective and thus ISI is avoided. Therefore, Figure 1.6 illustrates a

frequency-domain interpretation of how OFDM avoids intersymbol interference.

1.1.3 Discrete-Time Signal Processing

Thus far, two of OFDM’s primary advantages have been discussed: the elimination

of ISI and the ability to optimize the modulation with bit loading. The third appeal of

OFDM is that the modulation and demodulation is done in the discrete-time domain with

the inverse fast Fourier transform (IFFT) and fast Fourier transform (FFT), respectively.

This is seen by sampling s(t) in (1.6) at N equally spaced time instances:

y[i] ≡ s(t)|t=iTB/N =

N−1∑

k=0

I0,kej2πki/N , i = 0, 1, . . . N − 1, (1.12)

which is the inverse discrete Fourier transform (IDFT) of the symbol vector I0 =

[I0,0, I0,1, . . . , I0,N−1]. Therefore, s(t) is generated at the transmitter with an IDFT fol-

lowed by a digital-to-analog (D/A) converter. The frequency-domain symbols {I0,k}N−1k=0

can be expressed as

I0,k =1

N

N−1∑

i=0

y[i]e−j2πkn/N , k = 0, 1, . . . N − 1, (1.13)

which is the discrete Fourier transform (DFT) performed on the time-domain samples.

Consequently, the symbols are demodulated at the receiver with an analog-to-digital

(A/D) converter followed by a DFT.

9

The IDFT/DFT is performed efficiently with IFFT/FFT algorithms. Doing so is

much simpler than performing the modulation/demodulation in the continuous-time

domain with N orthogonally tuned oscillators. Moreover, the signal processing can be

performed in software, making OFDM suitable for software defined radios (SDRs) [185].

1.2 Problems with OFDM

OFDM has two primary drawbacks. The first is sensitivity to imperfect frequency

synchronization which is common for mobile applications. This sensitivity arises from

the close subcarrier spacing. Figure 1.5 shows that the subcarriers are properly orthog-

onal at f = k/TB, k = 0, 1, . . . , N − 1. However, if the frequency synthesizer at the

receiver is misaligned by, say, εfo/TB Hz, where −0.5 < εfo < 0.5, the subcarriers are

not orthogonal and therefore interfering with one another. This intercarrier interference

(ICI) is illustrated in Figure 1.7: assuming that the receiver is tuned to (k + εfo)/TB

Hz rather than at the ideal k/TB Hz, the N − 1 neighboring subcarriers interfere with

the demodulation of the kth subcarrier. The intercarrier interference causes ISI—and

potentially high irreducible error floors.

The second problem with OFDM is that the signal has large amplitude fluctuations

caused by the summation of the complex sinusoids. The real and imaginary part of the

Normalized frequency, fTB

Spec

trum

magnitude,|S

(f)/T

B|

k − 1 k k + 1k + εfo

1

0.2

0.04

Figure 1.7: Frequency offset causes ICI. (εfo = 0.25)

10

OFDM signal is

<{s(t)} =

N−1∑

k=0

<{I0,k} cos (2πkt/TB) −={I0,k} sin (2πkt/TB) , (1.14)

and

={s(t)} =N−1∑

k=0

<{I0,k} sin (2πkt/TB) + ={I0,k} cos (2πkt/TB) , (1.15)

respectively. Figure 1.8(a) shows the real and imaginary parts of an example OFDM

signal with N = 16 subcarriers. Also plotted are the individually modulated sinusoids.

Notice that each sinusoids has a constant amplitude, but when summing the sinusoids

the resulting OFDM signal fluctuates over a large range. The instantaneous signal power,

|s(t)|2 = <2{s(t)} + =2{s(t)}, is plotted in Figure 1.8(b). The ratio between the peak

power and the average power is 144/16 = 9 (or in decibels, 10 log10 9 ≈ 9.5 dB).

={s(t)}<{s(t)}

Subcarriers

Normalized time, t/TB

Sig

nalam

plitu

de

10.80.60.40.20

12

10

8

6

4

2

0

-2

-4

-6

-8

(a) Signal amplitude.

Average powerPeak power

|s(t)|2

Normalized time, t/TB

Pow

erm

agnitude

10.80.60.40.20

160

140

120

100

80

60

40

20

0

(b) Signal power.

Figure 1.8: A typical OFDM signal (N = 16). The PAPR is 9.5 dB.

OFDM’s high peak-to-average power ratio (PAPR) requires system components with

a large linear range capable of accommodating the signal. Otherwise, the circuitry

11

distorts the waveform nonlinearly, and nonlinear distortion results in a loss of subcarrier

orthogonality which degrades performance.

One such nonlinear device is the transmitter’s power amplifier (PA) which is respon-

sible for the system’s operational range [424]. Ideally the output of the PA is equal to

the input times a gain factor. In reality the PA has a limited linear region, beyond which

it saturates to a maximum output level. Figure 1.9 shows a representative input/output

curve, known as the AM/AM conversion. In the linear region the curve matches the

ideal, but as the input power increases the PA saturates. The most efficient operating

point is at the PA’s saturation point, but for signals with large PAPR the operating

point must shift to the left keeping the amplification linear. The average input power

is reduced and consequently this technique is called input power backoff (IBO). To keep

the peak power of the input signal less than or equal to the saturation input level, the

IBO must be at least equal to the PAPR. Thus the required IBO for the OFDM signal

in Figure 1.8 is 9.5 dB. At this backoff the efficiency of a Class A power amplifier is

less than 6%. Such an efficiency is detrimental to mobile battery-powered devices which

have limited power resources. Moreover, the operational range of the system is reduced

by a factor of nine2.

Ideal AM/AMOperating points

AM/AM curve

Saturation regionLinear region

Backoff

Actual

Optimum

Max output

Input power

Outp

ut

pow

er

Figure 1.9: Power amplifier transfer function.

2IBO of 9.5 dB corresponds to 109.5/10 ≈ 9 times less signal power transmitted in channel; the(theoretical) efficiency of a Class A amplifier is 0.5/(109.5/10) ≈ 0.06 [374].

12

Nonlinearities in the transmitter also cause the generation of new frequencies in

the transmitted signal. This intermodulation distortion causes interference among the

subcarriers, and a broadening of the overall signal spectrum. The later causes interference

between neighboring systems, an effect known as adjacent channel interference.

1.3 Constant Envelope Waveforms

Constant envelope (CE) waveforms are appealing since the optimum operating point

in Figure 1.9 is attainable. The baseband CE signal representation is

s(t) = Aejφ(t), (1.16)

where A is the signal amplitude and φ(t) is the information bearing phase signal. The

advantage of the CE waveform is that the instantaneous power is constant: |s(t)|2 = A2.

Consequently, the PAPR is 0 dB and the required backoff is 0 dB. The PA can therefore

operate at the optimum (saturation) point, maximizing average transmit power (good

for range) and maximizing PA efficiency (good for battery life). Also, since the linearity

requirement is reduced, nonlinear PAs can be used which are generally more efficient

and less expensive than linear PAs. For example, the maximum theoretical efficiency of

a linear Class A power amplifier is 50%, while for a nonlinear Class E PA the maximum

theoretical efficiency is 100% [424].

Constant envelope signals are thus ideal in terms of the practical considerations of the

power amplifier. The question is how to embed digital information into φ(t) providing

good performance, spectral economy, and high data rates over the wireless channel.

Notice that the single carrier signal in (1.1) is constant envelope when |Ii| = 1 and g(t)

is rectangular. This type of modulation, however, has large spectral sidelobes which

cause adjacent channel interference. In practice, non-rectangular pulse shapes are used

which result in a non-CE signal.

Continuous phase modulation (CPM) is a class of signaling that has very low sidelobe

power while maintaining the constant envelope property [14,421]. CPM uses memory to

smooth φ(t). The memory, however, increases the complexity of the receiver, which is a

key disadvantage of CPM. Also CPM systems have difficulty operating over frequency-

selective channels [118].

13

1.4 Constant Envelope OFDM

Constant envelope OFDM (CE-OFDM) combines OFDM and constant envelope sig-

naling. The high peak-to-average power ratio OFDM signal is transformed into a CE

waveform. The CE-OFDM signal takes the form of (1.16) where the phase signal is an

OFDM waveform. For example, the phase signal can be the real part of the OFDM

signal:

φ(t) = <{sOFDM(t)} =

N−1∑

k=0

<{I0,k} cos (2πkt/TB) −={I0,k} sin (2πkt/TB) , (1.17)

where sOFDM(t) is the signal in (1.6). Figure 1.10 compares a conventional OFDM

bandpass signal with a bandpass CE-OFDM signal. Both are derived from the same

baseband OFDM message signal.

CE-OFDM bandpass

OFDM bandpass

R

�

OFDM message

Figure 1.10: Comparison of OFDM and CE-OFDM signals.

The motivation for CE-OFDM is to eliminate the PAPR problem of the conventional

OFDM system. Certainly, this is accomplished since the CE-OFDM signal has the 0 dB

PAPR property. The question is: at what cost? What is the performance of CE-

OFDM? What is its bandwidth? Can the guard interval be used in CE-OFDM as it is

in conventional OFDM? This thesis aims to answering these questions by analyzing the

various aspects of the CE-OFDM modulation.

14

1.5 Thesis Overview

In Chapter 2 the basics of OFDM is further studied. The effect of the nonlinear

power amplification on OFDM is evaluated. In Chapter 3 the CE-OFDM modulation

format is defined and the spectral properties are studied. The performance aspects of

CE-OFDM in the presence of additive noise are analyzed in Chapter 4. Performance

analysis is extended to frequency-nonselective fading channels in Chapter 5, and multi-

path frequency-selective fading channels in Chapter 6.

Chapter 2

OFDM

In Sections 1.1 and 1.2 the basic properties of OFDM are identified. In this chapter,

OFDM is studied in more detail. Section 2.1 covers key properties of OFDM. In Section

2.1.1, the cyclic prefix is studied. In Section 2.1.2, the processing of the discrete-time

samples is described, and the equivalence of linear channel convolution and circular

channel convolution is explained. In light of this property, OFDM is considered a special

case of the more general block modulation with cyclic prefix scheme, as discussed in

Section 2.1.3. Finally, in Section 2.1.4 the main functional blocks of the OFDM system

are described.

The PAPR statistics are analyzed in Section 2.2 and power amplifier models used

to evaluated system performance are described in Section 2.3. Then in Section 2.4 the

effect of nonlinear power amplification on OFDM systems is studied in terms of spectral

leakage (Section 2.4.1), performance degradation (Section 2.4.2), and system range and

efficiency (Section 2.4.3). Lastly, the various PAPR mitigation techniques found in the

research literature are categorized in Section 2.5, and a technique called signal clipping

is evaluated in terms of its effectiveness to improve system performance.

15

16

2.1 More OFDM Basics

2.1.1 The Cyclic Prefix

In Section 1.1.1 it is claimed that the use of the guard interval results in ISI-free

operation. This is true so long as a cyclic prefix is transmitted during the interval. This

is demonstrated below and it is shown that ISI results if anything but the cyclic prefix

is transmitted.

During the OFDM block interval, the waveform is

s(t) =

N−1∑

k=0

Ikej2πfkt, 0 ≤ t < TB, (2.1)

where {Ik}N−1k=0 are the data symbols, {exp(j2πfkt)}N−1

k=0 are the subcarriers, N is the

total number of subcarriers, fk = k/TB is the center frequency of the kth subcarrier and

TB is the block period. The guard interval is defined during −Tg ≤ t < 0, where Tg is

the guard period. To transmit a cyclic prefix, the last Tg s of the block is transmitted

during the guard interval:

s(t) =

N−1∑

k=0

Ikej2πfk(t+TB) =

N−1∑

k=0

Ikej2πfktej2πk =

N−1∑

k=0

Ikej2πfkt, (2.2)

−Tg ≤ t < 0. Notice that the above simplification is made due to the periodicity of the

signal. Thus the OFDM signal having a guard interval with cyclic prefix is simply

s(t) =

N−1∑

k=0

Ikej2πfkt, −Tg ≤ t < TB. (2.3)

The received signal is

r(t) = s(t) ∗ h(τ) + n(t)

=

∫ ∞

−∞h(τ)s(t− τ)dτ + n(t)

=

∫ τmax

0h(τ)s(t− τ)dτ + n(t),

(2.4)

where h(τ) is the time-invariant channel impulse response1 and n(t) is additive noise.

The bounds of integration are simplified since the channel is assumed causal [h(τ) = 0

1In (1.2), the received signal is expressed in terms of the time-variant channel impulse response h(τ, t).If the channel is assumed to be time invariant, the impulse response is referred to as simply h(τ ).

17

for τ < 0] and to have a maximum propagation delay τmax [h(τ) = 0 for τ > τmax]. The

received signal during the guard interval, which has interference from the previous block

(see Figure 1.4), is ignored and r(t) during 0 ≤ t < TB is processed. An estimate of the

k0th data symbol is made by correlating r(t) with the k0th subcarrier:

Ik0 =1

TB

∫ TB

0r(t)

[

ej2πfk0t]∗dt, (2.5)

which expands to

Ik0 =1

TB

∫ TB

0r(t)e−j2πfk0

tdt

=1

TB

∫ TB

0

[

∫ τmax

0h(τ)

N−1∑

k=0

Ikej2πfk(t−τ)dτ

]

e−j2πfk0tdt+Nk0

=1

TB

N−1∑

k=0

Ik

∫ τmax

0h(τ)e−j2πfkτdτ

∫ TB

0ej2πt(fk−fk0

)dt+Nk0 ,

(2.6)

where

Nk0 =1

TB

∫ TB

0n(t)e−j2πfk0

tdt. (2.7)

But since

1

TB

∫ TB

0ej2πt(fk−fk0

)dt =

1, k = k0,

0, k 6= k0,(2.8)

(2.6) simplifies to

Ik0 = Ik0H[k0] +Nk0 , (2.9)

where

H[k0] =

∫ τmax

0h(τ)e−j2πfk0

τdτ, (2.10)

which is the Fourier transform of h(τ) evaluated at f = fk0 .

This shows that the N received data symbols {Ik}N−1k=0 are equal to the transmitted

symbols {Ik}N−1k=0 scaled by the complex-valued channel gains {H[k]}N−1

k=0 . ISI is avoided

since the kth symbol isn’t impacted by the N − 1 other symbols. Therefore, using the

guard interval with cyclic prefix provides ISI-free operation.

Now it is shown that by transmitting a signal other than the cyclic prefix during the

guard interval causes ISI. Suppose that the transmitted signal is

s(t) =

b(t), −Tg ≤ t < 0,

∑N−1k=0 Ike

j2πfkt, 0 ≤ t < TB,(2.11)

18

where b(t) 6= ∑N−1k=0 Ike

j2πfkt. The estimate of the k0th data symbols is

Ik0 =1

TB

∫ TB

0r(t)e−j2πfk0

tdt

=1

TB

∫ TB

0

[∫ τmax

0h(τ)s(t− τ)dτ

]

e−j2πfk0tdt+Nk0

= Ak0 +Bk0 +Nk0 .

(2.12)

The bounds of integration are separated into two segments, [0, Tg] and [Tg, TB]:

Ak0 =1

TB

∫ Tg

0

∫ τmax

0h(τ)s(t − τ)e−j2πfk0

tdτdt, (2.13)

and

Bk0 =1

TB

∫ TB

Tg

∫ τmax

0h(τ)s(t− τ)e−j2πfk0

tdτdt. (2.14)

Ak0 is a non-zero offset term which is a function of b(t). For the second term, t− τ > 0,

thus

Bk0 =1

TB

∫ TB

Tg

[

∫ τmax

0h(τ)

N−1∑

k=0

Ikej2πfk(t−τ)dτ

]

e−j2πfk0tdt

=1

TB

N−1∑

k=0

Ik

∫ τmax

0h(τ)e−j2πfkτdτ

∫ TB

Tg

ej2πt(fk−fk0)dt.

(2.15)

Due to the integration bounds for t, the orthogonality condition in (2.8) can’t be applied

to (2.15), and this results in ISI. The estimated data symbol is expressed as

Ik0 = Ik0Hk0C1 +Nk0 + ICI, (2.16)

where C1 = (TB − Tg)/TB, and the interference terms is

ICI = Ak0 +1

TB

∑

k 6=k0

H[k]

∫ TB

Tg

Ikej2πt(fk−fk0

)dt. (2.17)

The interference is denoted as ICI, intercarrier interference, since the subcarriers are no

longer orthogonal and interfere with one another. This phenomenon was described in

Section 1.2 in the context of imperfect frequency synchronization. Therefore, ICI can

manifest itself in more than one way, and when it does the data symbols interfere with

one another resulting in ISI.

In [358], cyclic prefixed OFDM is compared to zero-padded OFDM [b(t) = 0]. The

zero-padding causes ISI, but has the advantage of being able to recover data symbols

19

located at channel zeros. This is in contrast with cyclic prefixed OFDM since, as shown

in (2.9), a channel zeros at the kth subcarrier, that is, H[k] = 0, results in an estimated

data symbol that consists entirely of noise. The zero-padded system avoids this problem

at the cost of increased receiver complexity due to equalization requirements.

2.1.2 Discrete-Time Model

It is convenient to describe OFDM by a discrete-time model. Consider sampling s(t),

h(τ) and r(t) at the sampling rate fsa = JN/TB samp/s, where J ≥ 1 is the oversampling

factor. The sampling instances are shown in the figure below.

t

TB

· · · · · ·0

Signal sampling

−NgTsa −Tsa Tsa (NB − 1)Tsa

τ

τmax

· · ·0

Channel sampling

Tsa (Nc − 1)Tsa

(Nc − 1)Tsa ≤ τmax

−NgTsa ≥ −Tg

−Tg

Figure 2.1: Sampling instances.

The number of guard samples, Ng, and channel samples, Nc, are defined as

Ng ≡⌊

Tg

Tsa

⌋

≤ Tg

Tsa, (2.18)

and

Nc ≡⌊

τmax

Tsa

⌋

+ 1 ≤ τmax

Tsa+ 1, (2.19)

where Tsa = 1/fsa is the sampling period. The number of samples per block is NB = JN ;

and, by design, Ng ≥ Nc. The signal samples are

s[i] = s(t)|t=iTsa , i = −Ng, . . . , 0, . . . , NB − 1, (2.20)

and the channel samples are

h[i] = h(τ)|τ=iTsa , i = 0, . . . , Nc − 1. (2.21)

The received samples are expressed by the linear convolution sum

r[i] =

Nc−1∑

m=0

h[m]s[i−m] + n[i], i = −Ng, . . . , 0, . . . , NB − 1, (2.22)

20

where {n[i]} are samples of the noise signal n(t). The guard interval samples are ignored

and the samples

r[i] =

Nc−1∑

m=0

h[m]s[i−m] + n[i], i = 0, . . . , NB − 1 (2.23)

are processed.

The linear convolution in (2.23) is equivalent to a circular convolution since, due

to the cyclic prefix, {s[i − m]} is periodic with period NB. The circular convolution

can be performed by taking the IDFT of the product of two DFTs [422, pp. 415–420].

Therefore, ignoring the noise samples, (2.23) can be expressed as

r[i] = IDFT {H[k]S[k]}

=1

NDFT

NDFT−1∑

k=0

H[k]S[k]ej2πik/NDFT , i = 0, . . . , NB − 1,(2.24)

where IDFT{·} represents the inverse discrete Fourier transform;

S[k] =

NDFT−1∑

i=0

s[i]e−j2πik/NDFT , k = 0, . . . , NDFT − 1 (2.25)

and

H[k] =

NDFT−1∑

i=0

h[i]e−j2πik/NDFT , k = 0, . . . , NDFT − 1 (2.26)

are the NDFT-point DFTs of the signal and channel samples, respectively. The DFT

size is, in general, NDFT ≥ NB. If NDFT > NB, the signal vector is zero-padded. Since

NDFT > Ng, the channel samples are zero-padded: h[i] = 0 for i = Nc, . . . , NDFT − 1.

Figure 2.2 shows a block diagram representing the calculation of (2.24). The effect of

the channel is simply a DFT followed by a multiplier bank (H[k]), which is then followed

by an IDFT. Also shown is the inverse channel which is a DFT followed by a multiplier

bank (1/H[k]) followed by an IDFT. Thus the transmit samples s[i] can be reconstructed

by passing the receive samples r[i] through the inverse channel.

2.1.3 Block Modulation with FDE

The inverse channel structure in Figure 2.2 corrects the distortion caused by the

channel in the frequency domain, and is therefore called a frequency-domain equalizer

21

DFT IDFT

DFT IDFT

Inverse channel

1

H[k]

Channel

H[k]

r[i]

s[i] r[i]

s[i]

Figure 2.2: Circular convolution with channel and the inverse channel.

DFT IDFTMultiplier

bank

Frequency-domain equalizer

{Ik}

Data Data

{Ik}Modulator DemodulatorChannel

Figure 2.3: Block modulation with cyclic prefix and FDE.

Channel DFT IDFTMultiplier

bank

Frequency-domain equalizer

DFTIDFTData

{Ik}

Data{Ik}

Channel DFTMultiplier

bankIDFTData

{Ik}

Data

{Ik}

Figure 2.4: OFDM is a special case.

22

(FDE). Such an equalizer can be used only when the effect of the channel is a circular

convolution. This is the case for OFDM, but isn’t unique to OFDM since any modulation

can use a cyclic prefix. This observation was first identified by Sari et al. [462] and

suggests a more general modulation approach: block modulation with cyclic prefix and

frequency-domain equalization. Figure 2.3 shows a simplified block diagram of such

a system. (The insertion of the cyclic prefix at the transmitter and removal at the

receiver is implied but not included in the diagram for simplicity.) For the special case

of OFDM, the modulation is a IDFT and the demodulation is a DFT as shown in Figure

2.4. Notice that the DFT and IDFT cancel each other and the resulting diagram depicts

the conventional OFDM system.

The multiplier bank at the output of the DFT is often referred to as a one-tap

equalizer, one complex multiplication per frequency bin. This operation is required for

data symbols that rely on coherent demodulation, such as M -ary phase-shift keying

(M -PSK) and M -ary quadrature-amplitude modulation (M -QAM).

As Sari et al. pointed out, OFDM doesn’t eliminate the equalization problem (asso-

ciated with conventional single carrier modulation); rather, OFDM converts the problem

to the frequency domain. Since Sari’s original paper, there has been a considerable num-

ber of publications focused on the block modulation technique using conventional single

carrier modulations [8, 30, 54, 107, 116, 132, 142, 153, 154, 196, 197, 245, 388, 460, 461, 463,

481, 533, 565, 574].

2.1.4 System Diagram

The block diagram in Figure 2.4 conceptually illustrates the OFDM system. Figure

2.5 shows a more detailed description of OFDM’s functional blocks.

The encoder adds redundancy to the bit stream for error control. The encoded bits

are then mapped to the data symbols Ik. In general, the data symbols are complex

numbers which result from mapping the bits to points on the complex plane. Next, the

symbols are serial-to-parallel (S/P) converted and processed by the IDFT. The cyclic

prefix is added and the signal samples, s[i], are passed through the digital-to-analog

(D/A) converter to obtain the continuous-time OFDM signal s(t). Finally, the signal is

amplified and transmitted.

23

Receiver

A/DRemove

CP S/P DFTEqualizeC[k] P/S

Detector Decoder Bits01101

r(t) r[i]

Ik 11001

Transmitter

S/P IDFTAddCP P/SEncoderBits

01101Mapper

PoweramplifierD/A

11101

s(t)s[i]

Ik

Figure 2.5: OFDM system diagram.

At the receiver, the inverse operations are performed. First, the received signal,

r(t), is sampled to obtain the discrete-time sequence r[i]. The guard interval samples

are removed, the DFT is performed and each frequency bin is equalized by a complex

multiplication. The estimated data symbols, Ik, are processed by the detector which

outputs a stream of estimated receive bits, and the decoder attempts to correct any bit

errors that may have occurred.

As discussed in Section 1.2, one of OFDM’s key drawbacks is the high peak-to-

average power ratio. Nonlinearities in the power amplifier distort the transmitted signal

and large input power backoff is required which results in low amplifier efficiency. In the

next sections the impact of the PA is studied. But first, the statistical properties of the

PAPR are discussed.

24

2.2 PAPR Statistics

The peak-to-average power ratio of the OFDM signal is best viewed statistically. For

any given block interval, the PAPR is a random quantity since it depends on the data

symbols {Ik}N−1k=0 . Assuming that they’re selected randomly from a set of M complex

numbers, there are MN unique symbol sequences, and thus MN unique OFDM wave-

forms per block. Of these waveforms, some have a high PAPR, while others have a

relatively low PAPR. Therefore, it is desirable to understand the statistical distribution

of this quantity.

The OFDM signal is

s(t) =

N−1∑

k=0

Ikej2πfkt, 0 ≤ t < TB. (2.27)

The signal during the guard interval is ignored since it has no impact on the PAPR

distribution. M -PSK data symbols are assumed, therefore |Ik| = 1 for all k. The

average power of s(t) is

Ps =1

TB

∫ TB

0|s(t)|2dt = N. (2.28)

The peak-to-average power ratio is defined as

PAPRs = maxt∈[0,TB)

|s(t)|2/

Ps. (2.29)

Notice that the absolute maximum signal power is N 2, so the PAPR can be as high as

N . However, the likelihood that all the subcarriers align in phase is extremely low. For

example, as pointed out in [381], a N = 32 subcarrier system having 4-ary data symbols

and a block period of TB = 100 µs obtains the theoretical maximum PAPR once every

3.7 million years. Thus it is more meaningful to describe the PAPR statistically rather

than in absolute terms.

Since the average signal power is a constant, the randomness of the PAPR depends on

the randomness of the instantaneous power |s(t)|2, and more specifically, the maximum

instantaneous power over 0 ≤ t < TB. For large N , the real and imaginary parts of

s(t) are accurately modeled as Gaussian random processes (due to the application of

the central limit theorem [394, 421]). Consequently, the instantaneous signal power is

chi-squared distributed with two degrees of freedom [421, p. 41], and the complementary

25

cumulative distribution function (CCDF) of the normalized instantaneous signal power

is approximated as

P

( |s(t)|2Ps

> x

)

≈ e−x. (2.30)

A lower bound of the peak-to-average power ratio’s CCDF is [515]

P (PAPRs > x) ' 1 − (1 − e−x)N , (2.31)

where 1 − (1 − e−x)N is an approximation to the CCDF of the PAPR of the sequence

{s(t)|t=iTB/N ; i = 0, 1, . . . N−1} [173]. The PAPR of the discrete-time sequence provides

a lower bound to the continuous-time signal since peaks can occur between sampling

times.

Approximation (2.30)Simulation

x (dB)

CC

DF,P

`

|s(t

)|2/P

s>x

´

1086420

100

10−1

10−2

10−3

10−4

(a) Instantaneous power.

Lower bound (2.31)Simulation

x (dB)

CC

DF,P

(PA

PR

s>x)

141210864

100

10−1

10−2

10−3

10−4

(b) Peak-to-average power ratio.

Figure 2.6: Complementary cumulative distribution functions. (N = 64)

Figure 2.6(a) compares a simulated instantaneous power CCDF with the approxi-

mation in (2.30). This demonstrates the accuracy of the Gaussian approximation to the

real and imaginary part of s(t). Figure lower bound in (2.31). 2.6(b) compares PAPR

simulation results to the The bound is shown to be within 1 dB of the simulated result

for lower values of x. The 0.0001 PAPR is shown to be at around 11.25 dB, and at this

26

level the bound is tight. Notice that essentially all OFDM blocks have a PAPR greater

than 6 dB, 10% have a PAPR greater than 8.5 dB, and 0.5% have a PAPR greater than

10 dB.

For the results in Figure 2.6, the number of subcarriers is N = 64 and QPSK data

symbols (4-ary PSK) are used, that is, Ik ∈ {±1,±j}. While the symbols constellation

has little impact on the PAPR statistics, the number of subcarriers does. Figure 2.7

shows the lower bound (2.31) over a range N = 32 to N = 1024. Notice that the 0.001

PAPR is 1 dB larger for N = 512 than for N = 32. For the N = 64 system, the PAPR

is greater than 8 dB for roughly 10% of the time. For the N = 1024 system, however,

the PAPR is greater than 8 dB nearly all of the time.

k 1098765

x (dB)

CC

DF,P

(PA

PR

s>x)

121110987654

100

10−1

10−2

10−3

10−4

Figure 2.7: PAPR CCDF lower bound (2.31) for N = 2k, k = 5, 6, . . . , 10.

2.3 Power Amplifier Models

To determine the impact of the PAPR on system performance, power amplifier mod-

els must be defined. Two models commonly used in the research literature are the

solid-state power amplifier (SSPA) model and the Saleh traveling-wave tube amplifier

(TWTA) model [454]. They are described here and then used in Section 2.4 for perfor-

mance evaluation.

27

In general, modeling nonlinear power amplifiers is complicated (see [233, chap. 5]).

A common simplification is to assume that the PA is a memoryless nonlinearity, and

therefore has a frequency-nonselective response. For example, if the PA input is

sin(t) = A(t) exp[jφ(t)], (2.32)

the output is

sout(t) = G[A(t)] exp[

j{φ(t) + Φ[A(t)]}]

, (2.33)

where G(·) and Φ(·) are known as the AM/AM and AM/PM conversions, respectively.

The SSPA model is expressed as

G(A) =g0A

[

1 + (A/Asat)2p]1/2p

, and Φ(A) = 0, (2.34)

where g0 is the amplifier gain, Asat is the input saturation level, and p controls the

AM/AM sharpness of the saturation region. For this model the AM/PM conversion is

assumed to be negligibly small.

Though widely known as the Rapp model [426], (2.34) should be credited to the

original work by A. J. Cann, published a decade earlier in the IEEE literature [71].

Cann’s formula is obtained with the simple manipulation:

G(A) =g0A

[

1 + (A/Asat)2p]1/2p

=g0A

[

1 + (A/Asat)2p]1/2p

× [(Asat/A)2p]1/2p

[(Asat/A)2p]1/2p

=g0Asat

[

1 + (Asat/A)2p]1/2p

,

(2.35)

which is precisely the nonlinearity presented in Cann’s paper.

Saleh’s TWTA model is expressed as [110]

G(A) =g0A

1 + (A/Asat)2 , and Φ(A) =

αφA2

1 + βφA2. (2.36)

Notice that the AM/PM conversion, determined by the constants αφ and βφ, is non-zero.

The TWTA model is therefore more nonlinear than the SSPA model.

28

To reduce nonlinear distortion in the amplified OFDM signal, input power backoff

(IBO) is required. It is defined as [375]

IBO =A2

sat

Pin, (2.37)

where Pin = E{|sin(t)|2} = E{A2(t)} is the average power of the input signal. Equiva-

lently, (2.37) can be written as

Pin =A2

sat

IBO; (2.38)

thus, given Asat and IBO, the input signal power can be scaled accordingly to satisfy

(2.38).

Assuming that the PAPR of the input signal is PAPRin, the peak power can be

written as

Pmax = PAPRin · Pin =PAPRin

IBOA2

sat =A2

sat

K , (2.39)

where

K =IBO

PAPRin(2.40)

is defined as the backoff ratio. Notice that for K > 1 the backoff is greater than the input

signal’s PAPR; for K < 1 the backoff is less than the input PAPR. Now, the maximum

value of the input, Amax = max |A(t)|, can be written in terms of the backoff ratio and

the input saturation level:

Amax =√

Pmax =Asat√K. (2.41)

Figure 2.8 shows the AM/AM (solid lines) and AM/PM (dashed lines) conversions

for the SSPA (thick lines) and TWTA (thin lines) models for various backoff ratios K.

For the SSPA model, p = 2; for the TWTA model, αφ = π/12 and βφ = 1/4. The x-axis

is normalized to the maximum input level Amax, and the y-axis is normalized to the

maximum output level g0Asat. For K = −10 dB the IBO is one-tenth the input signal

PAPR, and thus the nonlinearity is severe. One the other hand, for K = 10 dB the

IBO is ten times the input signal PAPR and the PA response is nearly linear. As stated

above, the non-zero AM/PM conversion of the TWTA model makes it more nonlinear

than the SSPA model.

Insight can be gained by comparing Figure 2.6(b) and Figure 2.8. For example,

assuming that the backoff is IBO = 6 dB, the conversions are never as linear as the

K = 3 dB curves (the PAPR is a always greater than 3 dB) and are more nonlinear

29

Normalized input value, A/Amax

Norm

alize

doutp

ut

valu

e,G

(A)/g0A

sat

10.50−0.5−1

1

0.5

0

−0.5

−1

(a) K = 10 dB

Normalized input value, A/AmaxN

orm

alize

doutp

ut

valu

e,G

(A)/g0A

sat

10.50−0.5−1

1

0.5

0

−0.5

−1

(b) K = 3 dB

Normalized input value, A/Amax

Norm

alize

doutp

ut

valu

e,G

(A)/g0A

sat

10.50−0.5−1

1

0.5

0

−0.5

−1

(c) K = −3 dB

Normalized input value, A/Amax

Norm

alize

doutp

ut

valu

e,G

(A)/g0A

sat

10.50−0.5−1

1

0.5

0

−0.5

−1

(d) K = −10 dB

Figure 2.8: AM/AM (solid) and AM/PM (dash) conversions (SSPA=thick,TWTA=thin) for various backoff ratios K.

30

than the K = −3 dB curves for about 5% of the OFDM blocks (the 0.05 PAPR is 9 dB).

Therefore, even with a large IBO of 6 dB, the PA can impose high nonlinear distortion

on the transmitted signal. Also, the degree of distortion for a given OFDM block is

random (given a fixed IBO) since the PAPR for a given block is random.

2.4 Effects of Nonlinear Power Amplification

Power amplifier nonlinearities cause spectral leakage and performance degradation

to OFDM systems. These undesirable effects can be reduced with increase input backoff.

This is an unsatisfactory solution, however, since PA efficiency reduces with IBO. Also,

reducing the average transmit power reduces the operational range of the system. In

this section these various issues are studied.

2.4.1 Spectral Leakage

The first problem considered is spectral leakage. By using the Welch method [422, pp.

911–913], the power density spectrum at the output of the power amplifier can be quickly

estimated. The result is used to calculate estimated fractional out-of-band power curves,

defined as

ˆFOBP(f) =

∫ f0 Φs(x)dx

0.5Ps, f > 0, (2.42)

where Φs(f) is the estimated power density spectrum of the signal and Ps =∫∞−∞ Φs(f)df

is the signal power. Figure 2.9 shows the curves for an N = 64 subcarrier OFDM signal

amplified by the TWTA power amplifier according to (2.36) at various backoff levels.

Also plotted is the FOBP curve for ideal linear amplification. These results show that

at least 6 dB backoff is required by the TWTA to avoid spectral broadening.

Figure 2.10 shows the 99.5% bandwidth as a function of IBO. The bandwidth of the

undistorted OFDM signal is f = 1.07W . For sufficient backoff, the bandwidth of the

nonlinearly amplified signal is the same. However, for IBO < 6 dB, the bandwidth is

shown to grow roughly linearly with IBO. For IBO = 1 dB, the 99.5% bandwidth is 73%

larger than the undistorted signal. Notice that the spectral leakage is roughly the same

for the two amplifier models.

31

ideal PAOFDM amplified with: TWTA PA

IBO

0

2

4

6

Normalized frequency, f/W

Fra

ctio

nalout-

of-band

pow

er

1.51.2510.750.50.250

100

10−1

10−2

10−3

Figure 2.9: Fractional out-of-band power of OFDM with ideal PA and with TWTAmodel at various input power backoff. (N = 64, IBO in dB)

ideal PASSPA PA

OFDM amplified with: TWTA PA

Input power backoff, IBO (dB)

99.5

%bandw

idth

,f/W

1086420

2.0

1.0

1.2

1.4

1.6

1.8

0.8

Figure 2.10: Spectral growth versus IBO. (N = 64)

32

2.4.2 Performance Degradation

Next, the performance degradation caused by nonlinear amplification is considered.

The OFDM signal is passed through a PA and then it is corrupted by additive white

Gaussian noise (AWGN). The received signal is thus,

r(t) = sout(t) + n(t), (2.43)

where sout(t) is the output of the PA from (2.33) and n(t) is a complex-valued Gaussian

additive noise signal having a power density spectrum [421, p. 158]

Φn(f) =

N0, |f | ≤ Bn/2,

0, |f | > Bn/2,(2.44)

where Bn is the bandwidth of the noise signal. The noise spectrum is assumed to be

constant over the effective bandwidth of the information bearing signal and is thus called

“white”. The transmitted data symbols are estimated by the correlation in (2.5) then

passed to the detector which makes the final decision. This decision is based on the

maximum-likelihood (ML) criterion assuming a linear PA; that is, the nearest point in

the symbol constellation [421, pp. 242–247].

The performance is estimated by way of computer simulation. Following the conven-

tion described in Section 2.1.2, the discrete-time signal representation is used and the

sampling rate fsa = JN/TB where J ≥ 1 is the oversampling factor. For the AWGN

channel, h(τ) = δ(τ), and therefore no guard interval is used. The noise samples {n[i]}are Gaussian distributed and assumed independent:

E {n[i1]n[i2]} =

σ2n, i1 = i2,

0, i1 6= i2.(2.45)

The autocorrelation function of n(t) [the inverse Fourier transform of (2.44)] has zero-

crossings at τ = 1/Bn. Thus assuming Bn = fs, (2.45) is satisfied and the noise sample

variance is σ2n = fsaN0.

Figure 2.11 shows bit error rate (BER) performance as a function of Eb/N0, where

Eb =

∫ TB

0 |sout(t)|2dtNumber of bits per block

(2.46)

33

Ideal PANonlinear PA

Signal-to-noise ratio per bit, Eb/N0 (dB)

Bit

erro

rra

te

14121086420

10−1

10−2

10−3

10−4

10−5

(a) SSPA model, IBO = 0, 1, 2, 3, 4, 6, 8 dB;

0 = worst, 8 = best.

Ideal PANonlinear PA

Signal-to-noise ratio per bit, Eb/N0 (dB)

Bit

erro

rra

te

302520151050

10−1

10−2

10−3

10−4

10−5

(b) TWTA model, IBO = 0, 1, . . . , 10, 16 dB;

0 = worst, 16 = best.

Figure 2.11: Performance of QPSK/OFDM with nonlinear power amplifier with variousinput power backoff levels. (N = 64)

is the energy per bit. The quantity Eb/N0 is referred to as the signal-to-noise ratio (SNR)

per bit, or simply the SNR. QPSK data symbols are used, and the oversampling factor

is J = 4. For the SSPA results in Figure 2.11(a), the IBO ranges from 0 to 8 dB. At

the 0.0001 BER level, the IBO = 0 dB case suffers a 3 dB performance loss compared to

ideal AWGN performance, which is [421, pp. 268].

BER = Q

(

√

2Eb

N0

)

, (2.47)

where Q(x) =∫∞x e−y

2/2dy/√

2π is the Gaussian Q-function. To avoid degradation, 8

dB of backoff is required. The TWTA results in Figure 2.11(b) use IBO ranging from 0

to 16 dB. Notice the irreducible error floors for IBO ≤ 7 dB. To avoid degradation, 16

dB of backoff is required—8 dB more than for the SSPA case. The greater nonlinearity

of the TWTA model is evident from the results in this figure.

Figure 2.12 compares performance for higher-order PSK modulations. For M -PSK

34

IBO = 6 dBSSPA: IBO = 3 dB

Ideal PA

M = 2, 4

M = 8

M = 16

Signal-to-noise ratio per bit, Eb/N0 (dB)

Bit

erro

rra

te

302520151050

10−1

10−2

10−3

10−4

10−5

(a) BER performance.

SSPAIdeal PA

Target BER = 0.001

M = 2, 4

M = 8

M = 16

Input power backoff, IBO (dB)

Tota

ldeg

radation

(dB

)

1086420

10

8

6

4

2

0

(b) Total degradation.

Figure 2.12: Performance of M -PSK/OFDM with SSPA. (N = 64)

the data symbols are

Ik ∈ {exp(j2πm/M); m = 0, 1, . . . ,M − 1}. (2.48)

The number of bits per data symbols is log2M , therefore the bit energy is

Eb =

∫ TB

0 |sout(t)|2dtN log2M

. (2.49)

Higher-order constellations are used for increased spectral efficiency at the price of BER

performance2. In Figure 2.12(a) BER results for the SSPA model are shown. (The

results for M = 2 and M = 4 are very similar so only M = 2 is plotted.) The higher-

order modulations are shown to be more sensitive to the PA nonlinearity. For example,

the M = 16 result for IBO = 3 dB has an irreducible error floor at 5 × 10−3, while the

M = 2, 4 result at the same backoff shows only a 1 dB degradation. When increasing

the backoff to IBO = 6 dB, the error floor for M = 16 drops to 2 × 10−5 and the 0.001

BER is about 2 dB worse than AWGN. Using IBO = 6 dB for M = 8 results in 2 dB

less degradation at the 0.001 bit error rate when compared to using IBO = 3 dB.

2This is the case for linear modulation formats. This isn’t necessarily the case for nonlinear modulationformats as discussed in Section 4.4.

35

A more revealing way to view performance is in terms of total degradation, as shown

in Figure 2.12(b). The total degradation is defined as [121]

TD(IBO) = SNRPA(IBO) − SNRAWGN + IBO, [in dB] (2.50)

where SNRAWGN is the required signal-to-noise ratio per bit to achieve a target bit

error rate in AWGN; SNRPA(IBO) is the required SNR when taking into account the

distortion caused by the power amplifier at a given backoff. The “optimum” IBO, denote

as IBOopt, minimizes the total degradation, that is,

TD(IBOopt) = TDmin = minIBO≥0 dB

TD(IBO). (2.51)

The target BER for the curves in Figure 2.12(b) is 0.001. Clearly the modulation order

influences the degradation. The minimum TD for M = 16 is 7.7 dB at IBOopt = 6.5

dB; for M = 8, TDmin = 5 dB at IBOopt = 3 dB. This can be interpreted as follows:

M = 8, while having lower spectral efficiency than M = 16 (3 b/s/Hz vs. 4 b/s/Hz),

suffers less degradation and can operate with less backoff, resulting in improved range

and higher PA efficiency. The M = 2 and M = 4 examples are shown to have the lowest

degradation and are thus the more robust against nonlinear distortion.

2.4.3 System Range and PA Efficiency

The total degradation is directly related to the system’s operational range. Consider

a transmitter operating at maximum transmit power. The range is represented by the

outermost ring in Figure 2.13. Now assume that the system requires a 3 dB backoff:

the range is reduced by one-half, as represented by the middle ring. Any degradation

caused by the PA further reduces range, as represented by the innermost circle. Thus

the actual range of the system is far less than the potential range of the transmitter.

The true capability of the power amplifier is greatly underutilized.

To quantify the relationship between the PA efficiency and the power backoff, the

theoretical efficiency of a Class A power amplifier is used [374]:

ηA =1

2

1

IBO× 100%, IBO ≥ 1. (2.52)

The efficiency is thus inversely proportional to IBO and the maximum efficiency, 50%,

occurs at IBO = 1 (0 dB). The efficiency curve, shown in Figure 2.14, can be used

36

Potential rangePotential range w/ IBOActual range

Figure 2.13: The potential range of system is reduced with input backoff; the range isreduced further from nonlinear amplifier distortion.

in conjunction with Figures 2.10 and 2.12(b) to gain insight to the various tradeoffs

between PA efficiency, spectral containment, and performance/range. For example, the

optimum IBO in terms of total degradation for the 8-PSK SSPA example is IBOopt = 3

dB [Figure 2.12(b)]: however, the bandwidth expansion is 42% (Figure 2.10) and the

PA efficiency is ηA = 25% (Figure 2.14). The optimum IBO for the 16-PSK example,

6.5 dB, results in no bandwidth expansion but the PA efficiency is reduced to 11%. The

M = 2, 4 systems required minimal IBO for the SSPA, thus maximizing efficiency, but

the bandwidth expands by 87%.

Input power backoff, IBO (dB)

Cla

ss-A

PA

effici

ency

,ηA

(%)

109876543210

50

45

40

35

30

25

20

15

10

5

0

Figure 2.14: Power amplifier efficiency.

37

2.5 PAPR Mitigation Techniques

There have been many schemes proposed in the research literature aimed at reducing

the impact of the PAPR problem. The goal of any scheme is to reduce the minimum

total degradation (for increased range) and the IBOopt (for increased PA efficiency). The

various schemes can be placed in one the following three categories:

1. transmitter enhancement techniques,

2. receiver enhancement techniques, or

3. signal transformation techniques.

Transmitter enhancement techniques include PAPR reduction schemes and PA lineariza-

tion schemes. The PAPR reduction schemes can be further divided into distortionless and

non-distortionless techniques. Distortionless techniques include coding (see [126,439,508]

and reference therein), constellation extension [269], tone reservation [169, 268, 512],

trellis-shaping [377], and multiple signal representation {aka selected mapping (SLM)

or partial transmit sequences (PTS), see [227] and its references}. Non-distortionless

schemes include signal clipping [27,138,290,382], peak cancellation [330], and peak win-

dowing [403].

The PA linearization schemes attempt to predistort the OFDM signal such that the

overall response of the predistorter followed by the PA is linear—essentially equalizing

the amplifier. In [230], an LMS algorithm is applied for adaptive predistortion; in [395]

a neural network learning technique is used. Parametric techniques, which design a

predistorter based on a PA model, have been proposed. In [85, 122, 250, 567] nonlinear

polynomial models are used, and in [86] a Volterra-based model is suggested.

The second category, receiver enhancement techniques, have been suggested in [513],

[376] (maximum-likelihood decoding); in [259, 453] (signal reconstruction), and in [87]

(interference cancellation). Finally, the third category includes techniques that are based

on transforming the OFDM signal prior to the PA, and applying the inverse transform at

the receiver prior to demodulation. This category includes constant envelope OFDM (as

studied in the second half of this thesis) which uses a phase modulator as the transformer.

In [215, 329, 569–571] a companding transform is suggested.

38

Signal Clipping

The remainder of this section focuses on the effectiveness of signal clipping, which has

been claimed to be the “simplest” and “most effective” PAPR reduction scheme [27,87,

290,375,377,380,382,391]. The impact of “clipping noise”—the intercarrier interference

caused by the clipping process—on system performance has been extensively analyzed

[39, 124, 382]. However, a common assumption is that the PA is linear [27, 39, 138,

290, 371, 380, 382, 391]. It is argued here that the effectiveness of a PAPR reduction

scheme must be measured not only by PAPR reduction, but by the more meaningful

measures of TDmin and IBOopt reduction. It is shown that clipping, while an effective

PAPR reduction scheme, does not reduce TDmin nor does clipping reduce IBOopt for

an OFDM system. This result brings into question the usefulness of non-distortionless

PAPR reduction techniques in general.

The system under consideration is shown in Figure 2.15. When the switch is “on”

the PAPR reducing signal clipper is used. When “off” the system is identical to the one

studied in Section 2.4.2. Therefore, the earlier unclipped results serve as a performance

benchmark in which to compare the clipped results. The channel, as before, has an

impulse response h(τ) = δ(τ).

PAPRreducingclipper

PA h(τ)OFDM

modulatorOFDM

demodulatorsout(t)s(t)

n(t)

r(t)

sclip(t)

sin(t)off

on

Figure 2.15: Block diagram. The system is evaluated with and without PAPR reduction.

The input to the clipping block is the OFDM signal s(t) from (2.27), the output is

the clipped OFDM signal:

sclip(t) =

s(t), if |s(t)| ≤ Amax,

Amaxejψ(t), if |s(t)| > Amax,

(2.53)

where ψ(t) = arg[s(t)]. Therefore, the magnitude of the clipped signal does not exceed

Amax and the phase of s(t) is preserved. (This has been called “polar clipping” in the

literature [276].) The clipping severity is measured by the clipping ratio, defined as [375]

γclip =Amax√Ps

. (2.54)

39

Clip radiusOFDM signalγclip 4

2

0

Real axis

Imagin

ary

axis

20100−10−20

20

10

0

−10

−20

Figure 2.16: Unclipped OFDM signal (9.25 dB PAPR). The rings have radius Amax

which correspond to various clipping ratios γclip (dB).

Figure 2.16 shows a typical OFDM signal on the complex plane. The dark rings have

radius Amax which correspond to clipping ratios γclip = 0, 2, and 4 dB.

The PAPR of sclip(t) is

PAPRclip =

maxt∈[0,T )

|sclip(t)|2

1TB

∫ TB

0 |sclip(t)|2dt. (2.55)

Clipping’s effectiveness at reducing PAPR is shown in Figure 2.17. For clipping ratio

γclip = 5 dB, the peak-to-average power ratio of the clipped signal is PAPRclip ≤ 10

dB; for γclip = 4 dB, PAPRclip ≤ 8 dB, and so forth. The 0.0001 PAPR improvement,

compared to the unclipped signal, is 1.2 dB for γclip = 5 dB and by 3.2 dB for the

γclip = 4 dB.

Figure 2.18 shows PAPRclip as a function of the clipping ratio. The PAPR of

the unclipped signal is 13 dB3. Notice that for large γclip, sclip(t) is unclipped, there-

3This figure is made by generating 2×104 consecutive OFDM blocks. The PAPR of the overall blockis 13 dB.

40

UnclippedClipped

γclip 3 4 5

x (dB)

P(P

AP

Rclip>x)

121086420

100

10−1

10−2

10−3

10−4

Figure 2.17: PAPR CCDF of clipped OFDM signal for various γclip (dB). [N = 64]

PAPRclip

PAPRclip as γclip → 0

γ2clip

PAPRs

Clipping ratio, γclip (dB)

Pea

k-t

o-a

ver

age

pow

erra

tio

(dB

)

1086420−2−4−6−8

16

14

12

10

8

6

4

2

0

−2

−4

Figure 2.18: PAPR of clipped signal as a function of the clipping ratio. (N = 64)

41

fore PAPRclip = PAPRs. As γclip → 0, the peak and average powers converge, thus

PAPRclip → 0 dB. For the region 3 dB < γclip < 6.5 dB, sclip(t) is clipped so the

peak power is A2max = γ2

clipPs. However, the clipping is mild so the average power is

approximately the same as s(t); therefore, PAPRclip ≈ γ2clipPs/Ps = γ2

clip.

Clipping is clearly an effective technique at reducing the PAPR. The question is, does

the PAPR reduction translate into reduced total degradation? Figure 2.19 compares the

total degradation curves of the unclipped system [from Figure 2.12(b)] with the clipped

system. Interestingly, the unclipped results are shown to provide a lower bound for

the clipped, reduced PAPR, system results. The clipper is shown to increase both the

minimum total degradation and the optimum backoff. For example, using the clipping

ratio γclip = 3 dB for the M = 8 case increases the TDmin by 0.2 dB; using γclip = 2 dB

increases TDmin by 1.2 dB. For M = 16, the γclip = 4 dB result is nearly identical to the

unclipped result; γclip = 3 dB increases the degradation by 1.2 dB, and the TD curve

associated with γclip = 2 dB is beyond the viewing range of the figure. For M = 2, 4 the

PAPR reducing clipping yields nearly identical results as the unclipped system.

2 dB3 dB

Clipped: γclip = 4 dBUnclippedIdeal PA

M = 2, 4

M = 8

M = 16

Input power backoff, IBO (dB)

Tota

ldeg

radation

(dB

)

1086420

10

8

6

4

2

0

Figure 2.19: A comparison of the total degradation curves of clipped and unclippedM -PSK/OFDM systems. (N = 64)

Thus the effectiveness of a PAPR reduction scheme should be measured not only

by its PAPR reducing capabilities but by its effectiveness in reducing total degradation

(which increases range) and reducing the optimum IBO (which increases power amplifier

42

efficiency). The distortion caused by non-distortionless schemes can outweigh the benefit

of the reduced PAPR. This is clearly shown to be the case for the clipped N = 64 M -

PSK/OFDM systems studied in this section. The clipping is shown to reduce the 0.0001

PAPR by > 1 dB, but this reduction does not translate into increased PA efficiency.

This result brings into question the validity of the claims that clipping is an effec-

tive scheme. In fact, the effectiveness of non-distortionless PAPR reduction schemes in

general is suspect. For these types of techniques it is important to take into account the

effect of the nonlinear power amplifier.

The effectiveness of distortionless PAPR reduction techniques are typically studied

in terms of PAPR reduction and complexity. It would be interesting to also study these

schemes in terms of total degradation. Does a 3 dB reduction in PAPR results in a 3

dB reducing in IBOopt? What is the resulting minimum total degradation?

Chapter 3

Constant Envelope OFDM

Conventional OFDM systems, even with the use of effective PAPR reduction and/or

power amplifier linearization techniques, typically require more input power backoff than

convention single carrier systems. Therefore, OFDM is considered power inefficient,

which is undesirable particularly for battery-powered wireless systems.

The technique described in the remainder of the thesis takes a different approach to

the PAPR problem. CE-OFDM can be thought of as a mapping of the OFDM signal to

the unit circle, as depicted in Figure 3.1. The instantaneous power of the resulting signal

is constant. Figure 3.2 compares the instantaneous power of the OFDM signal and the

mapped CE-OFDM signal. For the CE-OFDM signal the peak and average powers are

the same, thus the PAPR is 0 dB.

Unit circleSignal

CE-OFDMOFDM

⇒

Figure 3.1: The CE-OFDM waveform mapping.

43

44

CE-OFDMOFDM

Normalized time

Inst

anta

neo

us

signalpow

er

10.80.60.40.20

5

4

3

2

1

0

Figure 3.2: Instantaneous signal power.

The mapping is performed with an angle modulator, specifically, a phase modulator.

That is, the OFDM signal is used to phase modulate the carrier. This is in contrast to

conventional OFDM which amplitude modulates the carrier. To see this, consider the

baseband OFDM waveform

m(t) =∑

i

N∑

k=1

Ii,kqk(t− iTB) (3.1)

where {Ii,k} are the data symbols and {qk(t)} are the orthogonal subcarriers. For con-

ventional OFDM the baseband signal is up-converted to bandpass as

y(t) = <{

m(t)ej2πfct}

= Am(t) cos [2πfct+ φm(t)] ,(3.2)

where Am(t) = |m(t)| and φm(t) = arg[m(t)]. For real-valued m(t), φm(t) = 0 and y(t)

is simply an amplitude modulated signal. (For complex-valued m(t), y(t) can be viewed

as an amplitude single-sideband modulation.) For CE-OFDM, m(t) is passed through a

phase modulator prior to up-conversion. The baseband signal is

s(t) = ejαm(t), (3.3)

45

where α is a constant. The bandpass signal is

y(t) = <{

s(t)ej2πfct}

= <{

ejαAm(t) exp[jφm(t)]ej2πfct}

= <{

e−αAm(t) sinφm(t)ej[2πfct+αAm(t) cos φm(t)]}

= e−αAm(t) sinφm(t) cos [2πfct+ αAm(t) cosφm(t)] .

(3.4)

For real-valued m(t),

y(t) = cos [2πfct+ αm(t)] . (3.5)

Therefore y(t) is a phase modulated signal.

CE-OFDM can also be thought of as a transformation technique, as shown in Fig-

ure 3.3. At the transmitter, the high PAPR OFDM signal is transformed into a low

PAPR signal prior to the power amplifier. At the receiver, the inverse transformation is

performed prior to demodulation.

OFDMmodulator

Poweramplifier

Tochannel

Phasemodulator

Fromchannel

OFDMdemodulator

Phasedemodulator

Receiver

Transmitter

m(t) s(t)

Inverse

transform

Transform

Figure 3.3: Basic concept of CE-OFDM.

As mentioned in Section 2.5, other approaches based on signal transformation have

been suggested. In particular, [215, 329, 569–571] suggest a companding transform. The

companded signal has an increased average power and thus a lower peak-to-average

power ratio than conventional OFDM. The PAPR is still large relative to single carrier

modulation, however. The advantage of the phase modulator transform is that the

resulting signal has the lowest achievable peak-to-average power ratio of 0 dB.

46

The idea of transmitting OFDM by way of angle modulation isn’t entirely new.

In fact, Harmuth’s 1960 paper suggest transmitting information by orthogonal time

functions with “amplitude or frequency modulation, or any other type of modulation

suitable for the transmission of continuously varying [waveforms]” [202]. Using existing

FM infrastructure for OFDM transmission has been suggested in [76, 77, 575]. These

papers don’t consider the PAPR implications, however. Two conference papers, [101]

and [506], on the other hand, suggest using a phase modulator prior to the power amplifier

for PAPR mitigation—though intriguing, these papers lack a solid theoretical foundation

and ignore fundamental signal properties such as the signal’s power density spectrum.

The origin of this work, which is independent of the previous references, stems from

work done at the US Navy’s spawar Systems Center, (San Diego, CA). Mike Geile,

a principle engineer at Nova Engineering, (Cincinnati, OH), which is the contractor of

the OFDM component for JTRS (Joint Tactical Radio System), suggested a low PAPR

enhancement to OFDM by phase modulation. The motivation is to reduce the 6 dB

backoff used in the JTRS radio.

Transmitting OFDM with phase modulation raises several fundamental questions.

What is the power density spectrum of the modulation? How is the signal space af-

fected? What is the optimum AWGN performance? What is the performance of a phase

demodulator receiver (Figure 3.3)? How does the system perform in a frequency-selective

fading channel? These questions, and others, are addressed here. First, the CE-OFDM

modulation is defined.

3.1 Signal Definition

As indicated by (3.4), CE-OFDM requires a real-valued OFDM message signal, that

is, φm(t) = 0. Therefore the data symbols in (3.1) are real-valued:

Ii,k ∈ {±1,±3, . . . ,±(M − 1)}. (3.6)

This one dimensional constellation is known as pulse-amplitude modulation (PAM). Thus

the data symbols are selected from an M -PAM set. The subcarriers {qk(t)} must also

47

be real-valued. Three possibilities are considered: half-wave cosines,

qk(t) =

cos πkt/TB, 0 ≤ t < TB,

0, otherwise,(3.7)

for k = 1, 2, . . . , N ; half-wave sines,

qk(t) =

sinπkt/TB, 0 ≤ t < TB,

0, otherwise,(3.8)

for k = 1, 2, . . . , N ; and full-wave cosines and sines,

qk(t) =

cos 2πkt/TB, 0 ≤ t < TB; k ≤ N2 ,

sin 2π(k −N/2)t/TB, 0 ≤ t < TB; k > N2 ,

0, otherwise.

(3.9)

For each case, the subcarrier orthogonality condition holds:

∫ (i+1)TB

iTB

qk1(t− iTB)qk2(t− iTB)dt =

Eq, k1 = k2,

0, k1 6= k2,(3.10)

where Eq = TB/2.

In terms of implementation, (3.7) can be computed with a discrete cosine transform

(DCT); (3.8) with a discrete sine transform (DST); and (3.9) by taking the real part

of a discrete Fourier transform (DFT), or equivalently by taking a 2N -point DFT of a

conjugate symmetric data vector (see Appendix A.)

The baseband CE-OFDM signal is

s(t) = Aejφ(t), (3.11)

where A is the signal amplitude. The phase signal during the ith block is written as

φ(t) = θi + 2πhCN

N∑

k=1

Ii,kqk(t− iTB), iTB ≤ t < (i+ 1)TB, (3.12)

where h is referred to as the modulation index, and θi is a memory term (to be described

below). The normalizing constant, CN , is set to

CN ≡√

2

Nσ2I

, (3.13)

48

where σ2I is the data symbol variance:

σ2I = E

{

|Ii,k|2}

=1

M

M∑

l=1

(2l − 1 −M)2

=M2 − 1

3,

(3.14)

assuming equally likely signal points, that is, P (Ii,k = l) = 1/M , l = ±1,±3, . . . ,±(M −1), for all i and k. Consequently, the phase signal variance is

σ2φ = E

{

1

TB

∫ (i+1)TB

iTB

[φ(t) − θi]2 dt

}

=(2πh)2

TB

2

Nσ2I

∫ (i+1)TB

iTB

N∑

k1=1

N∑

k2=1

E {Ik1Ik2} qk1(t− iTB)qk2(t− iTB)dt

=(2πh)2

TB

2

Nσ2I

N∑

k=1

∫ TB

0σ2Iq

2k(t)dt = (2πh)2,

(3.15)

which is only a function of the modulation index. The signal energy is

Es =

∫ (i+1)TB

iTB

|s(t)|2dt = A2TB, (3.16)

and the bit energy is

Eb =Es

N log2M=

A2TB

N log2M. (3.17)

The term θi is a memory component designed to make the modulation phase-continuous.

At the ith signaling interval boundary, the phase discontinuity is

ci = φ(iTB − ε) − φ(iTB + ε), ε→ 0. (3.18)

Since qk(t) = 0 for t /∈ [0, TB), it follows that

φ(iTB − ε) = K

N∑

k=1

Ii−1,kAe(k), (3.19)

and

φ(iT + ε) = K

N∑

k=1

Ii,kAb(k), (3.20)

where K ≡ 2πhCN , Ab(k) = qk(0) and Ae(k) = qk(TB − ε), ε→ 0. Therefore,

ci = θi−1 − θi +K

N∑

k=1

[Ii−1,kAe(k) − Ii,kAb(k)] . (3.21)

49

To guarantee continuous phase, that is, ci = 0, the memory term is set to

θi ≡ θi−1 +K

N∑

k=1

[Ii,kAb(k) − Ii−1,kAe(k)] . (3.22)

Notice that θi depends on θi−1; the OFDM signal at the beginning of the ith block,∑N

k=1 Ii,kAb(k); and the OFDM signal at the end of the (i−1)th block,∑N

k=1 Ii−1,kAe(k).

The recursive relationship can be written as

θi = K

∞∑

l=0

N∑

k=1

[Ii−l,kAb(k) − Ii−1−l,kAe(k)] . (3.23)

Thus, the memory term is a function of all data symbols during and prior to the ith

block.

Figure 3.4 plots the phase discontinuities {ci} at the boundary times t = iTB, i =

0, 1, . . . , 49. In Figure 3.4(a), ci is plotted for memoryless modulation, that is, θi = 0,

for all i; therefore, ci = K∑N

k=1 [Ii−1,kAe(k) − Ii,kAb(k)]. Figure 3.4(b) shows that the

phase discontinuities are eliminated with the use of memory as defined in (3.22).

Normalized time, t/TB

Phase

dis

continuity,c i

50403020100

1.5

1

0.5

0

−0.5

−1

−1.5

(a) Without memory.

Normalized time, t/TB

Phase

dis

continuity,c i

50403020100

1.5

1

0.5

0

−0.5

−1

−1.5

(b) With memory.

Figure 3.4: Phase discontinuities.

The benefit of continuous phase CE-OFDM is a more compact signal spectrum. This

property is studied further in Section 3.2. A second consequence of the memory terms

is the entire unit circle is used for the CE-OFDM phase modulation. This is illustrated

in Figure 3.5 which plots continuous phase CE-OFDM signal samples on the complex

50

(b) L = 100

Starting point

Unit circle

(a) L = 1

Figure 3.5: Continuous phase CE-OFDM signal samples, over L blocks, on the complexplane. (2πh = 0.7)

plane. The modulation index is 2πh = 0.7. Figure 3.5(a) shows signal samples over

L = 1 block, where the phase signal occupies about one-half the unit circle. Viewing

samples over L = 100 blocks, Figure 3.5(b) shows that the phase signal occupies the

entire unit circle.

3.2 Spectrum

CE-OFDM is a complicated nonlinear modulation and a general closed-form expres-

sion for the power density spectrum is not available. The approach taken in [34], [421, pp.

207–217] to calculate the power spectrum of conventional CPM signals can be applied

to CE-OFDM. The Fourier transform of the average autocorrelation function results in

a two-dimensional definite integral. The problem is there are N sinusoidal phase pulses

in CE-OFDM, versus a single phase pulse as in CPM. This makes the integrand very

jagged for all but trivial values of N , and numerical integration algorithms (for example,

those in [328,419]) fail to converge in a timely manner. Insight can be gained by taking

this approach, however. It can be shown that memoryless modulation (θi = 0) results

in spectral lines at the frequencies fk = k/TB, k = 0,±1,±3, . . . [17]. Using memory as

defined by (3.22) eliminates these lines.

Since the Fourier transform approach isn’t computationally feasible, other techniques

are required to understand the CE-OFDM spectrum. The simplest is with the Taylor

51

expansion ex =∑∞

n=0 xn/n!. The CE-OFDM signal, with θi = 0, can be written as

s(t) = Aejσφm(t)

= A

∞∑

n=0

[

(jσφ)n

n!

]

mn(t),(3.24)

where

m(t) = CN∑

i

N∑

k=1

Ii,kqk(t− iTB) (3.25)

is the normalized OFDM message signal. The effective double-sided bandwidth, defined

as the twice the highest frequency subcarrier, of m(t) is

W = 2 × N

2TB=

N

TB. (3.26)

The bandwidth of s(t) is at least W : in (3.24), the n = 0 term contains no information

and thus has zero bandwidth; the n = 1 term is information bearing and has bandwidth

W ; the n = 2 term has a bandwidth 2W ; and so on. Thus, due to the n = 1 term, the

bandwidth of s(t) is at least W , and depending on the modulation index the effective

bandwidth can be greater than W .

The power density spectrum, Φs(f), can be easily estimated by the Welch method

of periodogram averaging [526]. The result, Φs(f) ≈ Φs(f), is used to calculate the

fractional out-of-band power,

FOBP(f) =

∫ f0 Φs(x)dx

0.5Ps≈∫ f0 Φs(x)dx

0.5Ps= ˆFOBP(f), (3.27)

where Ps =∫∞−∞Φs(f)df = Es/TB = A2 is the signal power. Figure 3.6 shows estimated

fractional out-of-band power curves for N = 64 and various 2πh. Due to the normalizing

constant CN these curves are valid for any M . The dashed lines represent the RMS

bandwidth,

Brms = σφW = 2πhN/TB. (3.28)

The RMS (root-mean-square) bandwidth is obtained by borrowing a result from analog

angle modulation [423, pp. 340–343] [437], which assumes a Gaussian message signal;

for large N , the OFDM waveform is well modeled as such (see Section 2.2). The results

in Figure 3.6 shows that Brms accounts for at least 90% of the signal power.

As defined in (3.28), the RMS bandwidth can be less than W , but, as shown by the

Taylor expansion in (3.24), the CE-OFDM bandwidth is at least W . A more suitable

52

Brms

ˆFOBP(f)

2πh

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Normalized frequency, f/W

Fra

ctio

nalout-

of-band

pow

er

21.510.50

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

Figure 3.6: Estimated fractional out-of-band power. (N = 64)

bandwidth is thus

Bs = max(2πh, 1)W. (3.29)

Figure 3.7 plots Bs versus 2πh, and compares it with the 90–99% bandwidths as de-

termined by the Welch method. Notice that (3.29) is an accurate 90–92% bandwidth

measure for 2πh ≥ 1.0. For small modulation index, Bs is a conservative bandwidth.

With 2πh = 0.4, for example, (3.29) accounts for 99.8% of the signal power (from Figure

3.6).

Figure 3.8 compares spectral estimates for CE-OFDM signals with the three sub-

carrier modulations from (3.7), (3.8) and (3.9). The modulation index is 2πh = 0.6.

Memoryless, non-continuous phase CE-OFDM is compared to continuous phase CE-

OFDM (the continuous phase examples are prefixed with “CP”). The estimates are also

53

99%95%92%

Welch: 90%Bs

Modulation index, 2πh

Norm

alize

ddouble

-sid

edbandw

idth

,B/W

21.81.61.41.210.80.60.40.2

3

2.8

2.6

2.4

2.2

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

Figure 3.7: Double-sided bandwidth as a function of modulation index. (N = 64)

compared to the Abramson spectrum [1]:

ΦAb(f) = A2∞∑

n=0

anUn(f), (3.30)

where

an =e−σ

2φσ2n

φ

n!, (3.31)

and

Un(f) =

δ(f), n = 0,

Φm(f), n = 1,

Φm(f)n∗ Φm(f), n > 1.

(3.32)

The weighting factors {an} are Poisson distributed, and∑∞

n=0 an = 1;n∗ denotes the

n-fold convolution, for example x(t)3∗ x(t) = x(t) ∗ x(t) ∗ x(t); and Φm(f) is the power

54

ΦAb(f)terms from (3.30)

Welch estimate

n = 1

n = 4

n = 3

n = 2

DCTDFT

DSTCP-DFT

CP-DCT

Normalized frequency, f/W

Pow

ersp

ectr

um

(dB

)

3210−1−2−3

0

−10

−20

−30

−40

−50

−60

−70

−80

Figure 3.8: Power density spectrum. (N = 64, 2πh = 0.6)

density spectrum of the message signal m(t) according to (3.25):

Φm(f) =TB

2N

N∑

k=1

sinc2

[(

f − k

2TB

)

TB

]

+ sinc2

[(

f +k

2TB

)

TB

]

, (3.33)

where

sinc(x) =

1, x = 0,

sinπxπx , otherwise.

(3.34)

The functions {Un(f)} have the property:∫∞−∞ Un(f)df = 1, for all n [1]. Therefore

the nth term in (3.30) has an an × 100% contribution to the overall spectrum. For

example, the carrier component, represented as δ(f), has a fractional contribution of

e−σ2φ ; Φm(f)

2∗ Φm(f) has a fractional contribution (e−σ2φσ4

φ)/2; and so on. Notice that

for 2πh = 0.2, the carrier component accounts for e−0.22×100 ≈ 96% of the signal power.

(This explains why the 90–92% curves at 2πh = 0.2 in Figure 3.7 are equal zero.)

55

Figure 3.8 plots the n = 1, 2, 3, 4 terms in (3.30), and the resulting sum

ΦAb(f) = A24∑

n=0

anUn(f) ≈ ΦAb(f). (3.35)

The Abramson spectrum is shown to match all estimates over the range |f/W | ≤ 1. For

|f/W | > 1, the spectral height depends on the overall smoothness of the phase signal. For

example, DST has a continuous phase [with or without memory since Ab(k) = Ae(k) = 0,

for all k] and has a lower out-of-band power than memoryless DFT, which isn’t phase-

continuous. Memoryless DFT results in a slightly smoother phase than memoryless DCT

since one-half of the subcarriers have zero-crossings at the signal boundaries [Ab(k) =

Ae(k) = 0, for k = N/2 + 1, . . . , N , and Ab(k) = Ae(k) = 1, otherwise] while DCT

doesn’t [Ab(k) = Ae(k) = 1, for all k]. The smoothest phase results from CP-DCT

which, unlike DST and CP-DFT, has a first derivative equal to zero at the boundary

times t = iTB. Consequently, the CP-DCT is the most spectrally contained.

Figure 3.9 shows estimated fractional out-of-band power curves that correspond to

the signals in Figure 3.8. For reference, conventional OFDM is also plotted. Notice that

the 99% spectral containment at f/W = 0.5 is the same for each signal. The continu-

ous phase CE-OFDM signals are the most spectrally contained and are shown to have

better than 99.99% containment at f/W = 1.25. Over the range 0.5 ≤ f/W ≤ 0.8, This

Brms

OFDMCE-OFDM

DFTDCT

DST

CP-DFT

CP-DCT

Normalized frequency, f/W

Fra

ctio

nalout-

of-band

pow

er

32.521.510.50

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10−8

Figure 3.9: Fractional out-of-band power. (N = 64, 2πh = 0.6)

56

figure shows that the CE-OFDM spectrum has more out-of-band power than conven-

tional OFDM. Since the modulation index controls the CE-OFDM spectral containment,

smaller h can be used if a tighter spectrum is required. The tradeoff is that smaller h

results in worse performance, as will be discussed in the next chapter. Therefore, the

system designer can trade performance for spectral containment, and visa versa.

Figure 3.10 compares CE-OFDM, with CP-DFT modulation over a large range of

modulation index, to conventional OFDM. For 2πh ≤ 0.4 the fractional out-of-band

power of CE-OFDM is always better than OFDM; otherwise CE-OFDM has more out-

of-band power for at least some frequencies f/W > 0.5. The 2πh = 2.0 example has a

broad spectrum, greater than OFDM over all frequencies. Notice that the shape of the

spectrum appears Gaussian shaped. This is due to the fact that for a large modulation

index, the higher-order terms in (3.32) dominate. They are Gaussian shaped due to the

multiple convolutions of (3.33). The shape of “wideband FM” signals is well covered in

the classical works of [1, 341, 437, 472].

OFDMCE-OFDM

2πh

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Normalized frequency, f/W

Fra

ctio

nalout-

of-band

pow

er

21.510.50

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

Figure 3.10: CE-OFDM versus OFDM. (N = 64)

57

Finally, Figure 3.11 compares CE-OFDM and OFDM with nonlinear power ampli-

fication. The OFDM curves (from Figure 2.9) require > 6 dB backoff to avoid spectral

broadening. The CE-OFDM signals have a bandwidth that depends only on the modu-

lation index and are not effected by the PA nonlinearity.

CE-OFDMOFDM, Ideal

OFDM, TWTA

2πh

IBO (dB)

64

02

0.70.60.50.4

Normalized frequency, f/W

Fra

ctio

nalout-

of-band

pow

er

21.510.50

100

10−1

10−2

10−3

10−4

10−5

Figure 3.11: CE-OFDM versus OFDM with nonlinear PA. (N = 64)

Chapter 4

Performance of Constant

Envelope OFDM in AWGN

In this chapter the basic performance properties of CE-OFDM are studied. The

baseband signal, represented by (3.11) and (3.12), is up-converted and transmitted as

the bandpass signal

sbp(t) = <{

s(t)ej2πfct}

= A cos [2πfct+ φ(t)] , (4.1)

where fc is the carrier frequency. The received signal is

rbp(t) = sbp(t) + nw(t), (4.2)

where nw(t) denotes a sample function of the additive white Gaussian noise (AWGN)

process with power density spectrum Φnw(f) = N0/2 W/Hz. The primary focus of the

chapter is to analyze the phase demodulator receiver, depicted by the block diagram

below. An expression for the bit error rate (BER) is derived by making certain high

carrier-to-noise ratio (CNR) approximations. The analytical result is then compared

against computer simulation and it is shown to be accurate for BER < 0.01. It is also

Bandpassfilter

Phasedemodulator

OFDMdemodulator

Todetector

rbp(t)

Figure 4.1: Phase demodulator receiver.

58

59

demonstrated that with the use of a phase unwrapper, the receiver is insensitive to phase

offsets caused by the channel and/or by the memory terms {θi}.

The phase demodulator receiver is a practical implementation of the CE-OFDM

receiver and is therefore of practical interest. However, it isn’t necessarily optimum,

since the optimum receiver is a bank of MN matched filters [421, p. 244], one for each

potentially transmitted signal. In Section 4.2 a performance bound and approximation

for the optimum receiver is derived; and then in Section 4.3, the performance of the

phase demodulator receiver is compared to the optimum result. It is shown that under

certain conditions the phase demodulator receiver has near-optimum performance.

In Section 4.4 CE-OFDM’s spectral efficiency versus performance is compared to

channel capacity. Finally, the chapter is concluded in Section 4.5 with a comparison

between CE-OFDM and conventional OFDM in terms of power amplifier efficiency, total

degradation, and spectral containment.

4.1 The Phase Demodulator Receiver

The phase demodulator receiver essentially consists of a phase demodulator followed

by a conventional OFDM demodulator. Figure 4.2 shows the model used in this analy-

sis. The received signal is first passed through a front-end bandpass filter, centered at

the carrier frequency fc, which limits the bandwidth of the additive noise. Then the

bandpass signal is down-converted to r(t), sampled, and processed in the discrete-time

domain. The conversion from rbp(t) to r(t) is described first1, making use of the following

trigonometric identities:

sin(x) sin(y) =cos(x− y) − cos(x+ y)

2, (4.3)

sin(x) cos(y) =sin(x+ y) + sin(x− y)

2, (4.4)

cos(x) cos(y) =cos(x+ y) + cos(x− y)

2, (4.5)

cos(x) sin(y) =sin(x+ y) − sin(x− y)

2. (4.6)

1This is the standard model used for representing received baseband signals, and more discussion ofthe model can be found in [421, sec. 4.1], [624, sec. 5.5], among other places.

60

Lowpassfilter

Phasedemodulator

OFDMdemodulator

Lowpassfilter

Bandpassfilter

rbp(t)

−2 sin(2πfct)

2 cos(2πfct)

j

r[i]r(t)

t = iTsa

u(t)

Figure 4.2: Bandpass to baseband conversion.

The output of the bandpass filter is

u(t) = sbp(t) + nbp(t), (4.7)

where

nbp(t) = nc(t) cos(2πfct) − ns(t) sin(2πfct) (4.8)

is the result of passing nw(t) through the bandpass filter. The terms nc(t) and ns(t)

are referred to as the in-phase and quadrature components of the narrowband noise,

respectively, and have the power density spectrum

Φnc(f) = Φns(f) =

N0, |f | ≤ Bbpf/2,

0, |f | > Bbpf/2,(4.9)

where Bbpf is the bandwidth of the bandpass filter. Note that Bbpf is assumed to be

sufficiently large so sbp(t) is passed through the front-end filter with negligible distortion

[421, pp. 157–158]. Writing sbp(t) in the form

sbp(t) = sc(t) cos(2πfct) − ss(t) sin(2πfct), (4.10)

where sc(t) = A cos[φ(t)] and ss(t) = A sin[φ(t)], the filter output can then be written as

u(t) = [sc(t) + nc(t)] cos(2πfct) − [ss(t) + ns(t)] sin(2πfct). (4.11)

61

The output of the top (in-phase) branch of the down-converter is2

rc(t) = LP {u(t) × 2 cos(2πfct)}

= LP{[sc(t) + nc(t)] + [sc(t) + nc(t)] cos(4πfct)

− [ss(t) + ns(t)] sin(4πfct)}

= sc(t) + nc(t),

(4.12)

where LP{·} denotes the lowpass component of its argument (i.e., double-frequency terms

are rejected) [624, p. 364]. Likewise, the output of the bottom (quadrature) branch is

rs(t) = LP {u(t) ×−2 sin(2πfct)}

= LP{−[sc(t) + nc(t)] sin(4πfct) + [ss(t) + ns(t)]

− [ss(t) + ns(t)] cos(4πfct)}

= ss(t) + ns(t).

(4.13)

The two are combined to obtain

r(t) = s(t) + n(t), (4.14)

where s(t) is the lowpass equivalent CE-OFDM signal from (3.11), and

n(t) = nc(t) + jns(t) (4.15)

is the lowpass equivalent representation of the bandpass white noise, nbp(t) [421, p. 158].

The power density spectrum of n(t) is [421, p. 158]

Φn(f) =

N0, |f | ≤ Bn/2,

0, |f | > Bn/2,(4.16)

where Bn = Bbpf is the noise bandwidth. The corresponding autocorrelation of n(t)

is [421, p. 158]

φn(τ) = N0sinπBnτ

πτ. (4.17)

The continuous-time receive signal is then sampled at the rate fsa = 1/Tsa samp/s

to obtain the discrete-time signal3

r[i] = s[i] + n[i], i = 0, 1, . . . , (4.18)

2Here, ideal phase coherence and frequency synchronization is assumed. In Section 4.1.2 the effect ofchannel phase offsets is considered.

3Perfect timing synchronization is assumed.

62

FIRfilter

Phaseunwrapper

To OFDMdemodulator

r[i] arg(·)

Phase demodulator

Figure 4.3: Discrete-time phase demodulator.

where s[i] = s(t)|t=iTsa and n[i] = n(t)|t=iTsa . As discussed in Section 2.4.2, the noise

samples {n[i]} are assumed independent:

E {n[i1]n[i2]} =

σ2n, i1 = i2,

0, i1 6= i2;(4.19)

and therefore the sampling rate is fsa = Bn, and σ2n = fsaN0.

The discrete-time phase demodulator studied in this thesis is shown in Figure 4.3.

The finite impulse response (FIR) filter is optional, but has been found effective at

improving performance; arg(·) simply calculates the arctangent of its argument; and the

phase unwrapper is used to minimize the effect of phase ambiguities. As will be shown,

the unwrapper makes the receiver insensitive to phase offsets caused by the channel

and/or by the memory terms.

The output of the phase demodulator is processed by the OFDM demodulator which

consists of the N correlators, one corresponding to each subcarrier. This correlator bank

is implemented in practice with the fast Fourier transform.

4.1.1 Performance Analysis

In this section a bit error rate approximation is derived for the phase demodulator

receiver. Although the receiver operates in the discrete-time domain, it is convenient to

analyze it in the continuous-time domain. The angle of the received signal is

arg[r(t)] = θi + 2πhCN

N∑

k=1

Ii,kqk(t− iTB) + ξ(t), (4.20)

iTB ≤ t < (i+ 1)TB, where

ξ(t) = arctan

[

N(t) sin [Θ(t) − φ(t)]

A+N(t) cos [Θ(t) − φ(t)]

]

(4.21)

is the corrupting noise [624, p. 416]. The terms N(t) and Θ(t) in (4.21) are the envelope

and phase of n(t).

63

The kth correlator in the OFDM demodulator computes

1

TB

∫ (i+1)TB

iTB

arg[r(t)]qk(t− iTB)dt = Si,k +Ni,k + Ψi,k. (4.22)

The signal term is

Si,k =1

TB

∫ (i+1)TB

iTB

[φ(t) − θi]qk(t− iTB)dt

=2πhCNTB

∫ (i+1)TB

iTB

N∑

n=1

Ii,nqn(t− iTB)qk(t− iTB)dt

=2πhCNTB

Ii,kEq = 2πh

√

1

2Nσ2I

Ii,k.

(4.23)

The noise term is

Ni,k =1

TB

∫ (i+1)TB

iTB

ξ(t)qk(t− iTB)dt. (4.24)

For example, with DST subcarrier modulation (3.8),

Ni,k =1

TB

∫ (i+1)TB

iTB

ξ(t) sin [πk(t− iTB)/TB] dt, (4.25)

which can be viewed as a Fourier coefficient of ξ(t) at f = k/2TB Hz. As TB → ∞,

the variance of the coefficient is proportional to the power density spectrum function

evaluated at f = k/2TB [442, pp. 41–43]. It is well known that, given a high CNR, the

noise at the output of a phase demodulator has a power density spectrum [423, p. 410]

Φξ(f) ≈ N0

A2, |f | ≤W/2, (4.26)

where, from (3.26), W = N/TB is the effective bandwidth of φ(t). Moreover, for high

CNR, ξ(t) is well modeled as a sample function of a zero mean Gaussian process. There-

fore, Ni,k is approximated as a zero mean Gaussian random variable with variance [442,

pp. 41–43]

var{Ni,k} ≈ 1

2TBΦξ(f)|f=k/2TB

≈ 1

2TB

N0

A2. (4.27)

This result is the same for DCT and DFT subcarrier modulation.

The third term in (4.22), Ψi,k, is expressed as

Ψi,k =1

TB

∫ (i+1)TB

iTB

θiqk(t− iTB)dt. (4.28)

Since∫ TB

0qk(t)dt = 0, k = 1, 2, . . . , N, (4.29)

64

for DCT and DFT modulations [(3.7), (3.9)], Ψi,k = 0 and therefore has no effect on sys-

tem performance. This highlights an important observation: DST subcarrier modulation

(3.8) is inferior to DCT and DFT since Ψi,k = 0 isn’t guaranteed.

The symbol error rate is computed by determining the probability of error for each

signal point in the M -PAM constellation. For the M − 2 inner points, the probability of

error is

Pinner = P (|Ni,k| > d) = 2P (Ni,k > d), (4.30)

where

d = 2πh

√

1

2Nσ2I

. (4.31)

[Notice that (4.30) is not averaged over i nor k since var{Ni,k}, as approximated by (4.27),

is a constant.] Due to the Gaussian approximation applied to the random variable Ni,k,

Pinner ≈ 2

∫ ∞

d

1√

2πN0/(2A2TB)exp

(

−x2/[

2N0/(2A2TB)

])

dx

= 2

∫ ∞

d[N0/(2A2TB)]−0.5

1√2π

exp(

−x2/2)

dx

= 2Q

(

2πh

√

A2TB

N0Nσ2I

)

= 2Q

(

2πh

√

6 log2M

M2 − 1

Eb

N0

)

.

(4.32)

For the two outer points, the probability of error is

Pouter = P (Ni,k > d) =1

2Pinner. (4.33)

Therefore, the overall symbol error rate is

SER =M − 2

MPinner +

2

MPouter

≈ 2

(

M − 1

M

)

Q

(

2πh

√

6 log2M

M2 − 1

Eb

N0

)

.(4.34)

Notice that for 2πh = 1, (4.34) is equivalent to the SER for conventional M -PAM [483,

pp. 194–195]. For high SNR, the only significant symbol errors are those that occur in

adjacent signal levels, in which case the bit error rate is approximated as [483, p. 195]

BER ≈ SER

log2M≈ 2

(

M − 1

M log2M

)

Q

(

2πh

√

6 log2M

M2 − 1

Eb

N0

)

. (4.35)

65

4.1.2 Effect of Channel Phase Offset

Suppose the channel imposes a phase offset of φ0. The received signal is then

r(t) = s(t)ejφ0 + n(t). (4.36)

The angle of r(t) is

arg[r(t)] = θi + 2πhCN

N∑

k=1

Ii,kqk(t− iTB) + φ0 + ξ(t), (4.37)

iTB ≤ t < (i+ 1)TB. Which is identical to (4.20) with the addition of the channel offset

term. The kth correlator is the same as (4.22), except the third term is

Ψi,k =1

TB

∫ (i+1)TB

iTB

[θi + φ0]qk(t− iTB)dt = 0. (4.38)

Therefore, the phase offset due to the channel has no impact on performance, and the

analytical approximation in (4.35) is applicable.

Figure 4.4 compares the performance of N = 64, M = 2 CE-OFDM with phase offset

{(θi + φ0) ∈ [0, 2π)}, and without (θi + φ0 = 0). The former is referred to as System

1 (S1), the later as System 2 (S2). The system is computer simulated with a sampling

rate fsa = JN/TB, where J = 8 is the oversampling factor4. For Eb/N0 ≥ 10 dB and

2πh ≤ 0.5, S1 and S2 are shown to have identical performance. For these cases the

analytical approximation (4.35) closely matches the simulation results for BER < 0.01.

With the 2πh = 0.7 example, S1 is shown to have a 1 dB performance loss compared

to S2. In this case, the analytical approximation is shown to be overly optimistic. This

demonstrates a limitation of the phase demodulator receiver: for a large modulation

index and low signal-to-noise ratio, the phase demodulator has difficulty demodulating

the noisy samples. The performance of S1 is slightly worse than S2 since the output

of the phase demodulator, the arg(·) block in Figure 4.3, has more phase jumps since

the received phase crosses the π boundary more frequently. Proper phase unwrapping

is therefore required. However, phase unwrapping a noisy signal is a difficult problem

and the unwrapper makes mistakes. As a result the performance degrades slightly. For

a smaller modulation index, the unwrapper works perfectly and the performance of S1

isn’t degraded.

4Also, the FIR filter (see Figure 4.3) has length Lfir = 11 and normalized cutoff frequency fcut/W =0.2. See Section 4.1.4 for more on the filter design.

66

Approx (4.35)System 2System 1

0.10.20.30.50.72πh

Signal-to-noise ratio per bit, Eb/N0 (dB)

Bit

erro

rra

te

302520151050

100

10−1

10−2

10−3

10−4

10−5

Figure 4.4: Performance with and without phase offsets. System 1 (S1) has phase offsets{(θi + φ0) ∈ [0, 2π)}, and System 2 (S2) doesn’t (θi + φ0 = 0). [M = 2, N = 64, J = 8]

4.1.3 Carrier-to-Noise Ratio and Thresholding Effects

The high-CNR approximation made in (4.26), which leads to the BER approximation

(4.35), is a standard technique for analyzing phase demodulator receivers [423, 624]. A

well-known characteristic of such receivers is: at low CNR, below a threshold value, the

approximation is invalid and system performance degrades drastically. In this section, the

CNR is defined and the threshold effect for CE-OFDM is observed by way of computer

simulation.

The CNR at the output of the analog front end, r(t), is

CNR =A2

Pn, (4.39)

where A2 is the carrier power, and

Pn =

∫ ∞

−∞Φn(f)df = BnN0 (4.40)

is the noise power. From (3.17), the carrier power can be written in the form

A2 =EbN log2M

TB; (4.41)

67

thus

CNR =(Eb/N0)N log2M

TBBn. (4.42)

Since the noise samples are assumed independent [see (4.19)],

Bn = fsa = JN/TB, (4.43)

and (4.42) reduces to

CNR =(Eb/N0) log2M

J. (4.44)

Therefore, the carrier-to-noise ratio is proportional to Eb/N0 and M , and inversely pro-

portional to the oversampling factor.

A commonly accepted threshold CNR for analog FM systems is 10 dB [472, pp.

120–138], [501, pp. 87–91]. This threshold level is studied in the following two figures.

In Figure 4.5, simulation results for an M = 8, N = 64, J = 8, 2πh = 0.5 system are

compared to (4.35). In subfigures (a) and (b) the system is below and above the 10

dB threshold, respectively. Clearly, above CNR = 10 dB, the system is observed to be

above threshold, with simulation results closely matching the analytical approximation.

Below 10 dB, the performance begins to deviate from (4.35); and for CNR < 5 dB, the

performance quickly degrades to a bit error rate of 1/2. Figure 4.6 shows results for more

values of 2πh. For each case, 10 dB can be considered an appropriate threshold level.

There is, however, a transition region—that is, a region where the system is useless,

with a BER of 1/2, to where the system is above threshold. This transition region is

difficult to study analytically. Gaining more insight into this issue is a subject for future

investigation.

68

Approx (4.35)Simulation

Carrier-to-noise ratio (dB)

Bit

erro

rra

te

1086420−2

10−1

(a) Below 10 dB threshold.

Approx (4.35)Simulation

Carrier-to-noise ratio (dB)

Bit

erro

rra

te

16151413121110

10−2

10−3

(b) Above 10 dB threshold.

Figure 4.5: Threshold effect at low CNR. (M = 8, N = 64, J = 8, 2πh = 0.5)

Approx (4.35)Simulation

2πh

0.8

0.6

0.4

0.2

Carrier-to-noise ratio (dB)

Bit

erro

rra

te

1086420−2

10−1

10−2

(a) Below 10 dB threshold.

Approx (4.35)Simulation

2πh0.8 0.6 0.4 0.2

Carrier-to-noise ratio (dB)

Bit

erro

rra

te

2422201816141210

10−1

10−2

10−3

(b) Above 10 dB threshold.

Figure 4.6: Threshold effect at low CNR, various 2πh. (M = 8, N = 64, J = 8)

69

4.1.4 FIR Filter Design

The FIR filter preceding the phase demodulator (see Figure 4.3) can improve per-

formance. Figure 4.7 shows BER simulation results of an M = 2, N = 64, J = 8,

2πh = 0.5 system. The SNR is held constant at Eb/N0 = 10 dB. The filter, designed

using the window technique described in [422, pp. 623–630], has a length 3 ≤ Lfir ≤ 101

and a normalized cutoff frequency 0 < fcut/W ≤ 1. Hamming windows are used5. The

performance without a filter is shown to be BER = 0.05, while the analytical approxi-

mation (4.35) is BER = 0.012. For fcut/W ≥ 0.4 all the filtered results are shown to be

better than the unfiltered result. The filters with Lfir > 5 and fcut/W > 0.5 are shown

to have roughly the same performance. The higher-order filters, which have a narrower

transition bands, require fcut/W > 0.5 to yield good performance. This is explained

by noting that the (single-sided) signal bandwidth is at least W/2 Hz. Therefore, the

higher-order filters with fcut/W < 0.5 distort the signal. Notice that the Lfir = 11 filter

has equally good performance so long as fcut/W ≥ 0.1. This is due to the wide transition

band of the lower-order filter.

10161312111975

Lfir = 3

No filter

Approx (4.35)

Normalized cutoff frequency, fcut/W

Bit

erro

rra

te

10.80.60.40.20

10−1

10−2

Figure 4.7: Performance for various filter parameters Lfir, fcut/W .(M = 2, N = 64, J = 8, 2πh = 0.5 and Eb/N0 = 10 dB)

5It has been observed that the window type has negligible impact on performance.

70

Lfir, fcut/W

31, 0.1

101, 0.7 9, 0.1

3, 0.1

9, 0.7

Normalized frequency, f/W

Magnitude

resp

onse

(dB

)

32.521.510.50

0

−20

−40

−60

−80

−100

Figure 4.8: Magnitude response of various Hamming FIR filters.

The figure above shows the magnitude response of the various Hamming FIR filters.

The filters with relatively flat response over |f/W | ≤ 0.5 result in good performance.

The Lfir = 31, fcut/W = 0.1 example is shown to not have this property, and, as shown

in Figure 4.7, has worse BER performance than the other filters.

Figure 4.9 compares the performance of binary (M = 2) CE-OFDM with and without

the FIR filter. The Lfir = 11, fcut/W = 0.2 filter is used. These results show that the

filter becomes important for larger modulation index: for 2πh = 0.1 the filtered and

unfiltered results are the same; for 2πh = 0.3 the filtered performance is a fraction of

a dB better than the unfiltered; for 2πh = 0.7 there is a 2 dB improvement in the

range 10−3 < BER < 10−5. Notice the error floor developing below 10−5. This is a

consequence of imperfect phase demodulation. The filter lowers the error floor resulting

in a 9 dB improvement at BER = 10−6.

71

Approx (4.35)With FIR filter

Without FIR filter

0.10.30.72πh

Signal-to-noise ratio per bit, Eb/N0 (dB)

Bit

erro

rra

te

302520151050

100

10−1

10−2

10−3

10−4

10−5

10−6

Figure 4.9: CE-OFDM performance with and without FIR filter.(M = 2, N = 64, J = 8)

4.2 The Optimum Receiver

As mentioned in the introduction to this chapter, the phase demodulator receiver is

a practical implementation, but not necessarily optimum. In this section, the optimum,

yet impractical, CE-OFDM receiver is studied. Results obtained here are used in the

following section to compare the phase demodulator receiver to optimum performance.

During each block one of MN CE-OFDM signals is transmitted. Consider the mth

bandpass signal

sm(t) = A cos

[

2πfct+ θ0 +K

N∑

k=1

I(m)k qk(t)

]

, 0 ≤ t < TB, (4.45)

where K = 2πhCN . The set of all possible signals, {sm(t)}MN

m=1, is determined by the

set of all possible data symbol vectors {I(m) = [I(m)1 , I

(m)2 , . . . , I

(m)N ]}MN

m=1. The optimum

72

receiver, as shown in Figure 4.10, correlates the received signal, rbp(t) = sm(t) + nw(t),

with each potentially transmitted signal. The detector then selects the largest result [421,

pp. 242–247].

Sampleat t = TB

R TB

0(·)dt

sMN (t)

Outputdecision

Selectthe

largest

s1(t)

R TB

0(·)dt

R TB

0 (·)dt

s2(t)

......

Receivedsignal rbp(t)

Figure 4.10: The optimum receiver.

4.2.1 Performance Analysis

It is desired to obtain an analytical expression for the bit error probability6, P (bit error).

However, there are two other probabilities to consider:

P (signal error) and P (symbol error) .

The first is the probability that the output of the optimum receiver is in error—that is,

the receiver selects a different signal than the one transmitted. The second is the data

symbol error probability. Determining exact expressions for the above probabilities is

intractable for large N . However, upperbounds and approximations can be derived in a

straightforward way, as described below.

6The bit error probability is used interchangeably with the bit error rate. Likewise for the symbolerror probability and symbol error rate.

73

An upperbound for P (signal error) is [373]:

P (signal error) ≤ 1√2π

∫ ∞

−∞

[

1 − [1 −Q(y)]MN−1

]

×

exp

−1

2

y −√

2Es(1 − λ)

N0

2

dy.

(4.46)

The above expression is the probability of detection error for MN signals with equal

correlation −1 ≤ λ ≤ 1. Therefore, it provides an upperbound given that

λ = ρmax = maxm,n;m6=n

ρm,n, (4.47)

where ρm,n is the normalized correlation between sm(t) and sn(t):

ρm,n =1

Es

∫ TB

0sm(t)sn(t)dt. (4.48)

An approximation for P (signal error) is [421, p. 288]

P (signal error) ≈ Kd2minQ

√

d2min

2N0

, (4.49)

where Kd2minis the number of neighboring signal points having the minimum squared

Euclidean distance

d2min = min

m,n;m6=nd2m,n, (4.50)

where

d2m,n =

∫ TB

0[sm(t) − sn(t)]

2dt (4.51)

is the squared Euclidean distance between sm(t) and sn(t). This quantity is related to

the signal correlation as

d2m,n = 2Es(1 − ρm,n), (4.52)

thus

d2min = 2Es(1 − ρmax). (4.53)

Therefore to obtain the performance bound (4.46) and the approximation (4.49) the

signal correlation properties must be studied, and in particular ρmax must be determined.

The normalized correlation between the mth and nth signal, as a function of the phase

74

constant K = 2πhCN , is

ρm,n(K) =1

Es

∫ TB

0sm(t)sn(t)dt

=A2

Es

∫ TB

0cos

[

2πfct+ θ0 +K

N∑

k=1

I(m)k qk(t)

]

×

cos

[

2πfct+ θ0 +K

N∑

k=1

I(n)k qk(t)

]

dt

=A2

2Es

∫ TB

0cos

[

2KN∑

k=1

∆m,n(k)qk(t)

]

dt,

(4.54)

where ∆m,n(k) = 0.5[I(m)k −I(n)

k ]. The double frequency term is ignored since fc � 1/TB

is assumed. Notice that for k where ∆m,n(k) = 0, the data symbols are the same, and

these indices don’t contribute to the correlation. Therefore

ρm,n(K) =A2

2Es

∫ TB

0cos

[

2K

D∑

d=1

∆m,n(kd)qk(t)

]

dt, (4.55)

where {kd}Dd=1 are the indices where the data symbols differ, that is, ∆m,n(kd) 6= 0, and

D is the total number of differences. Writing (4.55) in exponential form yields

ρm,n(K) =A2

2Es

∫ TB

0<{

exp

[

j2K

D∑

d=1

∆m,n(kd)qk(t)

]}

dt

=A2

2Es

∫ TB

0<{

D∏

d=1

exp [j2K∆m,n(kd)qk(t)]

}

dt.

(4.56)

To proceed, the DCT modulation (3.7) is assumed. Making use of the Jacobi-Anger

expansion [580],

eja cos b =∞∑

i=−∞

Ji(a)eji(b+π/2), (4.57)

where Ji(a) is the ith-order Bessel function of the first kind, (4.56) is written as

ρm,n(K) =A2

2Es

∫ TB

0<[

∞∑

i1=−∞

· · ·∞∑

iD=−∞

Ji1 [2K∆m,n(k1)] × · · · × JiD [2K∆m,n(kD)]ejσ(i)

]

dt

=A2

2Es

∫ TB

0

∞∑

i1=−∞

· · ·∞∑

iD=−∞

Ji1 [2K∆m,n(k1)]×

· · · × JiD [2K∆m,n(kD)] cos[ω(i) + ψ(i)]dt,

(4.58)

75

where σ(i) = ω(i) + ψ(i), ω(i) ≡ πtTB

∑Dd=1 idkd and ψ(i) ≡ π

2

∑Dd=1 id. Index values that

result in ω(i) 6= 0 have no contribution, so (4.58) simplifies to

ρi,j(K) =∑

i

D∏

d=1

Ji′i,d [2K∆m,n(kd)] cos[ψ(i′i)], (4.59)

where i′i ≡ [i′i,1, . . . , i′i,D], i = 1, 2, . . ., represent the vectors whereby ω(i′i) = 0. This

result is the same for DST modulation except ψ(i′i) = 0. For DFT modulation, (4.59) is

slightly different since both sinusoids and cosinusoids are used as subcarriers.

For D = 1,

ρm,n(K) = J0[2K∆m,n(k1)]. (4.60)

Therefore the correlation is simply the 0th-order Bessel function. Figure 4.11(a) plots

(4.60) for |∆m,n(k1) = 1|. Also plotted is the envelope of the 0th-order Bessel func-

tion [580, p. 121]. Note that ρm,n(K) doesn’t depend on the subcarrier frequency

fk1 = k1/TB, k1 ∈ {1, 2, . . . , N}, just on the magnitude of the difference |∆m,n(k1)| ∈{1, 2, . . . , (M − 1)}.

For CE-OFDM signals of interest,

ρmax = J0(2K). (4.61)

Figure 4.11(b) plots all unique ρm,n(K) for M = 2, N = 8 DCT subcarrier modulation.

Notice that the largest correlation function is associated with D = 1. For any given

signal, there are N other signals with D = 1: therefore, Kd2min= N , and from (4.49),

the probability of signal error is approximated as

P (signal error) ≈ Kd2minQ

√

d2min

2N0

= NQ(

√

Es[1 − ρmax]/N0

)

≈ NQ(

√

Es[1 − J0(2K)]/N0

)

.

(4.62)

A minimum distance signal error results in one data symbols error. Therefore, the symbol

error probability is approximated as

P (symbol error) ≈ P (signal error)

N≈ Q

(

√

Es[1 − J0(2K)]/N0

)

. (4.63)

For M = 2, one symbol error corresponds to one bit error. For M > 2, a symbol error

can result in 1 to log2M bit errors. Assuming each outcome is equally likely, a symbol

76

p

1/πK

K

ρm

,n(K

)

543210

1

0.5

0

−0.5

(a) D = 1.

J0(2K)

K

ρm

,n(K

)

0.50.40.30.20.10

1

0.8

0.6

0.4

(b) All unique ρm,n(K) for M = 2, N = 8 DCT modulation.

Figure 4.11: Correlation functions ρm,n(K).

77

error results in 1log2 M

∑log2Mi i = 0.5(log2M + 1) bit errors. Thus

P (bit error) ≈ 0.5(log2M + 1)

log2MP (symbol error)

≈ 0.5(log2M + 1)

log2MQ(

√

Es[1 − J0(2K)]/N0

)

.

(4.64)

The bit error probability is bounded by noting that P (bit error) ≤ P (signal error),

and using (4.46) with λ = ρmax = J0(2K):

P (bit error) ≤ 1√2π

∫ ∞

−∞

[

1 − [1 −Q(y)]MN−1

]

×

exp

−1

2

y −√

2Es[1 − J0(2K)]

N0

2

dy.

(4.65)

Figure 4.12 shows simulation results of the optimum receiver for M = 2 and N = 8.

The number of correlators at the receiver is therefore 28 = 256. Two values of modulation

index are plotted: 2πh = 0.3 and 2πh = 0.7 which corresponds to K = 0.15 and

K = 0.35. The upperbound (4.65) is shown to be within 3 dB of the simulated results

for high SNR. The analytical approximation (4.64) is shown to be very accurate.

SimulationBound (4.65)

Approx (4.64)

2πh 0.30.7

Signal-to-noise ratio per bit, Eb/N0 (dB)

Bit

erro

rra

te

211815129630

100

10−1

10−2

10−3

10−4

10−5

10−6

Figure 4.12: CE-OFDM optimum receiver performance. (M = 2, N = 8)

78

4.2.2 Asymptotic Properties

In Figure (4.13) each correlation function is plotted for M = 2, N = 4 DCT modu-

lation. The functions are shown to be bounded by

ρm,n(K) ≤ ρmax(K) ≤√

1

πK, (4.66)

the envelope of the 0th-order Bessel function. Therefore,

d2m,n(K) ≥ d2

min(K) ≥ 2Es(

1 −√

1

πK

)

. (4.67)

Notice that as K → ∞ the CE-OFDM signals become orthogonal. The phase modulator

thus drastically alters the signal space. Prior to the phase demodulator, the OFDM

signal space is described by 2N dimensions (2 per subcarrier). At the output of the

phase modulator, the space is transformed into a MN -dimensional space (due to the

linear independence of the signal set [421, p. 164]); and as the modulation index becomes

very large, a MN -dimensional orthogonal space. However, from (3.29), the bandwidth

tends to infinity as 2πh→ ∞.

p

1/πK

K

ρm

,n(K

)

543210

1

0.5

0

−0.5

Figure 4.13: All unique ρm,n(K) for M = 2, N = 4 DCT modulation.

4.3 Phase Demodulator Receiver versus Optimum

Figure 4.14 shows simulation results for the phase demodulator receiver with N = 64

and for various modulation index values 2πh and modulation order M . The simulation

79

SimulationApprox (4.64)Approx (4.35)

M , 2πh

2, 0.3 4, 0.2

8, 1.2 16, 0.8 16, 0.2

Signal-to-noise ratio per bit, Eb/N0 (dB)

Bit

erro

rra

te

403530252015105

100

10−1

10−2

10−3

10−4

10−5

Figure 4.14: Phase demodulator receiver versus optimum. (N = 64)

results are compared to the analytical approximation (4.35) and the optimum receiver

approximation (4.64). All curves are shown to be essentially identical for BER < 0.01.

This implies that the phase demodulator receiver is nearly optimum. For this to be

true, the phase demodulator must perfectly invert the phase modulation done at the

transmitter, and the noise at the output of the phase demodulator must be “white” and

Gaussian. That is, the OFDM demodulator is optimum given that the input, φ(t)+ξ(t),

is comprised of the transmitted message signal plus an AWGN corrupting signal. As

shown by (4.26), ξ(t) is approximately “white”. The probability density function of ξ(t)

samples is represented by the well-known form [421, p. 268]

pξ(x) =

∫ ∞

0

y

2πσ2n

exp

[

−y2 +A2 − 2yA cos x

2σ2n

]

dy, (4.68)

where σ2n = BnN0 is the power of the noise signal n(t). Figure 4.15 compares (4.68) to

the Gaussian probability density function. The SNR per bit is Eb/N0 = 30 dB. This

shows that ξ(t) is well approximated as Gaussian, and near optimum performance of the

phase demodulator receiver is expected.

80

Gaussianpξ(x)

x

Pro

bability

den

sity

funct

ion,p(x

)

1.510.50−0.5−1−1.5

100

10−5

10−10

10−15

10−20

Figure 4.15: Noise samples PDF versus Gaussian PDF. (Eb/N0 = 30 dB)

4.4 Spectral Efficiency versus Performance

In the previous sections, it is shown that the performance of CE-OFDM is deter-

mined by the modulation index, which, as shown in Section 3.2, also controls the signal

bandwidth. In this section, the spectral efficiency (b/s/Hz) versus performance (Eb/N0

to achieve a target bit error rate) is plotted for a variety of CE-OFDM signals. The

results are compared to channel capacity.

It is first demonstrated that CE-OFDM with modulation index 2πh > 1 can out-

perform the underlying M -PAM subcarrier modulation. Figure 4.16 shows simulation

results7 for M = 2, 4, 8 and 16. The bit error rate is plotted against the SNR per bit on

the bottom x-axis and the carrier-to-noise ratio on the top x-axis. The viewable range

is such that CNR ≥ 5 dB. Notice that for M ≥ 4 and 2πh > 1, CE-OFDM outperforms

M -PAM. This is predicted by (4.35), since for 2πh = 1.0, the expression is equal to the

performance of M -PAM, and for 2πh > 1.0, it is better than M -PAM. For CE-OFDM

to operate in the region 2πh > 1, the carrier-to-noise ratio must be above threshold.

7The oversampling factor is J = 8 for M = 2, 4 and 8, and J = 16 for M = 16. The FIR filter haslength Lfir = 11 and a normalized cutoff frequency 0.2 cycles per sample for M = 2, 4 and 16, and 0.3cycles per sample for M = 8.

81

(4.35)Simulation

2πh ∈ {0.5†, 0.4, . . . , 0.1‡}

Carrier-to-noise ratio (dB)

Signal-to-noise ratio per bit, Eb/N0 (dB)

Bit

erro

rra

te

5 10 15 20

3028262422201816

10−1

10−2

10−3

10−4

10−5

(a) M = 2.

4-PAM(4.35)

Simulation

2πh ∈ {1.0†, 0.9, . . . , 0.1‡}

Carrier-to-noise ratio (dB)

Signal-to-noise ratio per bit, Eb/N0 (dB)B

iter

ror

rate

5 10 15 20 25

3530252015

10−1

10−2

10−3

10−4

10−5

(b) M = 4.

8-PAM(4.35)

Simulation

2πh ∈ {1.5†, 1.2, 1.0, 0.9, . . . , 0.1‡}

Carrier-to-noise ratio (dB)

Signal-to-noise ratio per bit, Eb/N0 (dB)

Bit

erro

rra

te

5 10 15 20 25 30 35

40353025201510

10−1

10−2

10−3

10−4

10−5

(c) M = 8.

16-PAM(4.35)

Simulation

2πh ∈ {2.0†, 1.5, 1.2, 1.1, . . . , 0.1‡}

Carrier-to-noise ratio (dB)

Signal-to-noise ratio per bit, Eb/N0 (dB)

Bit

erro

rra

te

5 10 15 20 25 30 35 40

45403530252015

10−1

10−2

10−3

10−4

10−5

(d) M = 16.

Figure 4.16: Performance of M -PAM CE-OFDM. (N = 64, †=leftmost curve,‡=rightmost curve)

82

To plot the spectral efficiency versus performance, the data rate must be defined,

which for uncoded CE-OFDM is

R =N log2M

TBb/s. (4.69)

Using (3.29) as the effective signal bandwidth, the spectral efficiency is

R/B =R

Bs=

log2M

max(2πh, 1)b/s/Hz. (4.70)

Figure 4.17 shows result for M = 2, 4, 8 and 16. The target bit error rate is 0.0001. For

reference the channel capacity is also plotted, which is expressed as [421, p. 387]

C = B log2

(

1 +C

B

Eb

N0

)

, (4.71)

or equivalently,Eb

N0=

2C/B − 1

C/B. (4.72)

CapacityM = 16M = 8M = 4M = 2

M = 16: 2πh = 2.0, 1.8, . . . , 0.6

M = 8: 2πh = 1.4, 1.2, . . . , 0.4

M = 4: 2πh = 1.0, 0.8, . . . , 0.2

M = 2: 2πh = 0.5, 0.4, 0.3, 0.2

Performance: Eb/N0 (dB) to achieve 0.0001 bit error rate

Spec

traleffi

cien

cy(b

/s/

Hz)

-1.6 0 5 10 15 20 25

1

2

3

4

5

6

7

10

0.5

Figure 4.17: Spectral efficiency versus performance.

There are two main observations to be made. First, for a fixed modulation index,

CE-OFDM has improved spectral efficiency with increase modulation orderM at the cost

of performance degradation. For example consider 2πh = 0.4. The spectral efficiency

83

is 1, 2 and 3 b/s/Hz for M = 2, 4 and 8, respectively. However, M = 4 requires 4 dB

more power than M = 2, and M = 8 requires nearly 5 dB more power than M = 4.

This type of spectral efficiency/performance tradeoff is the same for conventional linear

modulations such as M -PAM, M -PSK and M -QAM [421, p. 282].

The second observation is that CE-OFDM can have both improvements in spectral

efficiency and in performance. Compare M = 2, 2πh = 0.5 with M = 4, 2πh = 1.0,

for example. The spectral efficiency doubles in the later case while also having a 2 dB

performance gain. Conventional CPM systems also have the property of increase spectral

efficiency and performance [14]. However, with CPM the receiver complexity increases

drastically with M (due to phase trellis decoding), which isn’t the case for CE-OFDM.

4.5 CE-OFDM versus OFDM

The total degradation, as defined in Section 2.4.2, is

TD(IBO) = SNRPA(IBO) − SNRAWGN + IBO, [in dB]

where SNRAWGN is the required signal-to-noise ratio required to achieve a target bit error

rate, SNRPA(IBO) is the required SNR when taking into account the nonlinear power

amplifier at a given backoff. Applying the PA model from Section 2.3 to CE-OFDM, the

input signal is

sin(t) = A exp[jφ(t)], (4.73)

and the output is

sout(t) = G(A) exp(

j[φ(t) + Φ(A)])

. (4.74)

The instantaneous nonlinearity results in a constant amplitude and a constant phase

shift. Therefore the PA has no impact on the CE-OFDM performance and no backoff is

needed. The total degradation for CE-OFDM is defined as

TD = SNRPM − SNRsub, (4.75)

where SNRsub is the required SNR for the underlying subcarrier modulation and SNRPM

is the required SNR for the phase modulated CE-OFDM system. By this definition,

the total degradation can be negative since, as observed in Figure 4.16, CE-OFDM can

outperform the underlying subcarrier modulation at the price of lower spectral efficiency.

84

Figure 4.18 compares CE-OFDM with conventional OFDM in terms of PA efficiency,

total degradation and spectral containment. Binary modulation is used in both systems.

The target BER is 10−5 and the number of subcarriers is N = 64. Both the SSPA and

TWTA models are considered. The lowest TD for the TWTA system is 10.5 dB at 8 dB

backoff, which corresponds to an 8% efficiency as shown in Figure 4.18(a). At this backoff

level, the 99.5% bandwidth occupancy is roughly the same as undistorted ideal OFDM

as shown in Figure 4.18(c). For the SSPA model, the lowest TD is 3.8 dB at IBO = 1 dB.

In this case, the PA efficiency is improved to 40% but the bandwidth requirement is 73%

more than ideal OFDM. Since CE-OFDM has a constant envelope, the PA can operate

at IBO = 0 dB thus maximizing amplifier efficiency. The total degradation is 5 dB for

2πh = 0.6 and the corresponding bandwidth requirement is 26% more than ideal OFDM.

For 2πh = 0.4, the total degradation is 8 dB but the bandwidth reduces to f/W = 0.98

which is 8% less than ideal OFDM. This shows that the modulation index for CE-OFDM

can be chosen accordingly to balance performance and bandwidth. Also, since the PA

imposes no additional distortion on the CE-OFDM signal, the resulting spectrum can be

well contained with no power backoff and at the same time have optimal PA efficiency.

85

Input power backoff, IBO (dB)

Cla

ss-A

PA

effici

ency

,ηA

(%)

109876543210

50

45

40

35

30

25

20

15

10

5

0

(a) PA efficiency.

0.60.5

CE-OFDM: 2πh = 0.4OFDM, ideal

OFDM, SSPAOFDM, TWTA

Input power backoff, IBO (dB)

Tota

ldeg

radet

ion

(dB

)

10864200

2

4

6

8

10

12

14

16

(b) Total degradation for target BER 10−5.

0.60.5

CE-OFDM: 2πh = 0.4OFDM, ideal

OFDM, SSPAOFDM, TWTA

Input power backoff, IBO (dB)

99.5

%bandw

idth

,f/W

1086420

1

1.2

1.4

1.6

1.8

2

0.8

(c) Spectral containment.

Figure 4.18: A comparison of CE-OFDM and conventional OFDM. (M = 2, N = 64)

Chapter 5

Performance of CE-OFDM in

Frequency-Nonselective Fading

Channels

In this chapter, performance analysis of the phase demodulator receiver is extended

to fading channels. The lowpass equivalent representation of the received signal is

r(t) = αejφ0s(t) + n(t) (5.1)

where s(t) is the CE-OFDM signal according to (3.11), α and φ0 is the channel amplitude

and phase, respectively, and n(t) is the complex Gaussian noise term represented in

(4.15). The received signal can be written as r(t) =∫∞−∞ h(τ)s(t− τ)dτ +n(t) [see (1.2),

(2.4)], where the channel impulse response is h(τ) = αejφ0δ(τ). In the frequency domain,

the channel is H(f) = F{h(τ)}(f) = αejφ0 , and is thus constant at all frequencies—that

is, the channel is frequency nonselective.

In the previous chapter only the simple case of α = 1 (i.e. no fading) was considered.

In this chapter the channel amplitude is treated as a random quantity. Such a channel

model, since it’s frequency nonselective, is commonly referred to as flat fading. The

signal-to-noise ratio per bit for a given α is

γ = α2 Eb

N0, (5.2)

86

87

and the average SNR per bit is [421, p. 817]

γ = E{γ} = E{

α2} Eb

N0. (5.3)

It is desired to calculate the bit error rate at a given γ, denoted here as BER(γ). This

quantity depends on the statistical distribution of γ. For channels with a line-of-sight

(LOS) component, the probability density function of γ is [483, p. 102]

pγ(x) =(1 +KR)e−KR

γexp

[

−(1 +KR)x

γ

]

I0

[

2

√

KR(1 +KR)x

γ

]

, x ≥ 0, (5.4)

where I0(·) is the 0th-order modified Bessel function of the first kind, and

KR =ρ2

2σ20

(5.5)

is the Rice factor: ρ2 and 2σ20 represent the power of the LOS and scatter component,

respectively [401, p. 40]. For channels without a line-of-sight, ρ → 0 and γ is Rayleigh

distributed [483, p. 101]:

pγ(x) =1

γexp

(

−xγ

)

, x ≥ 0. (5.6)

To obtain BER(γ), the conditional BER is averaged over the distribution of γ [421, p.

817]:

BER(γ) =

∫ ∞

0BER(x)pγ(x)dx. (5.7)

In Section 4.1.1 it is shown that

BER(x) ≈ c1Q(

c2√x)

, (5.8)

where c1 = 2(M − 1)/(M log2M) and c2 = 2πh√

6 log2M/(M2 − 1), so long as the

system is above threshold. For the moment, assume

BER(x) = c1Q(c2√x), for all x ≥ 0. (5.9)

If this were true, the bit error rate for the Ricean channel, described by (5.4), is [483, p.

102]

BERRice(γ) =c1π

∫ π/2

0

(1 +KR) sin2 θ

(1 +KR) sin2 θ + c22γ/2×

exp

[

− KRc22γ/2

(1 +KR) sin2 θ + c22γ/2

]

dθ,

(5.10)

88

and for the Rayleigh channel, as described by (5.6), [483, p. 101]

BERRay(γ) =c12

(

1 −√

c22γ/2

1 + c22γ/2

)

. (5.11)

However, as discussed in Section 4.1.3, the bit error rate of CE-OFDM, as a result of

the threshold effect, isn’t simply expressed by the Q-function for all values of SNR.

Consequently (5.10) and (5.11) are not generally accurate.

Figure 5.1(a) compares simulation results1 to (5.10) for an M = 8, N = 64 system in

the Ricean channel withKR = 10 dB. For 2πh = 0.6 the simulation result closely matches

(5.10) for γ > 15 dB. For lower values of γ, (5.10) is overly optimistic since the system is

more likely to experience channel fades which take the system below threshold—in which

case the bit error rate isn’t accurately represented by the Q-function, that is, (5.9) is

false. For the 2πh = 1.8 example, (5.10) is overly optimistic by at least 3 dB for all

values of γ. This is due to the inaccuracy of the Q-function for large modulation index

cases (see Figure 4.4, for example).

Approx (5.10)Simulation

2πh 1.8 0.6

Average signal-to-noise ratio per bit, γ (dB)

Bit

erro

rra

te

302520151050

100

10−1

10−2

10−3

10−4

10−5

(a) M = 8, Ricean KR = 10 dB.

Approx (5.11)Simulation

2πh 1.2 0.4

Average signal-to-noise ratio per bit, γ (dB)

Bit

erro

rra

te

50403020100

100

10−1

10−2

10−3

10−4

10−5

(b) M = 4, Rayleigh.

Figure 5.1: Performance of CE-OFDM in flat fading channels. (N = 64)

1Unless otherwise stated, the simulation parameters—J , Lfir, normalized cutoff frequency, and soforth—are the same as those used for the result shown in Figure 4.16 (see the footnote in on page 80).

89

Figure 5.1(b) further illustrates the inaccuracy of assuming (5.9). An M = 4, N = 64

system is simulated in the Rayleigh channel. For the low modulation index case of

2πh = 0.4, (5.11) is somewhat accurate. However, for the large modulation index case

of 2πh = 1.2, (5.11) is shown to be off by 5–7 dB.

A Semi-Analytical Approach

The problem with (5.10) and (5.11) is that the conditional bit error rate, BER(x),

is not accurately described by the Q-function at low SNR and/or for large modulation

index. For a limited range of 2πh (for example, the values shown in Figure 4.16) the

following observation can be made: above a certain SNR, say x0, the conditional bit

error rate closely matches the Q-function, that is, (5.8) holds. Therefore (5.7) can be

approximated as

BER(γ) =

∫ x0

0BER(x)pγ(x)dx +

∫ ∞

x0

BER(x)pγ(x)dx

≈∫ x0

0BER(x)pγ(x)dx +

∫ ∞

x0

c1Q(c2√x)pγ(x)dx.

(5.12)

Determining x0 for a givenM and 2πh, and dealing with∫ x0

0 BER(x)pγ(x)dx in (5.12) are

the problems that remain to obtain an accurate approximation of BER(γ). As observed

in Section 4.1.3 [see Figure 4.6(a)], at low SNR the bit error rate is roughly 1/2. Assume

for the moment that BER(x) = 1/2 for x ≤ x0; then

BER(γ) ≈ 1

2

∫ x0

0pγ(x)dx+

∫ ∞

x0

c1Q(c2√x)pγ(x)dx. (5.13)

This simplified model, referred to as a two-region model since the conditional BER is split

into two regions, is illustrated in Figure 5.2: below x0 the BER is 1/2, otherwise the BER

is equal to the Q-function. Also shown is the observed simulation result. Notice that

the two-region model doesn’t account for the transition region in which BER(x) ≈ 1/2

to where BER(x) ≈ c1Q(c2√x). [For more examples of the transition region, see Figure

4.6.] Consequently, (5.13) is not generally accurate, and a more elaborate approach is

required which accounts for the transition region.

90

Q-function (4.35)Observed (simulation)

Two-region model

Transition region

Signal-to-noise ratio per bit, x (dB)

Conditio

nalbit

erro

rra

te,B

ER

(x)

x0

1

0.01

0.5

0.1

Figure 5.2: A simplified two-region model. (M = 8, N = 64, 2πh = 0.6)

This is done by splitting the SNR region 0 ≤ x ≤ x0 into n sub-regions:∫ x0

0BER(x)pγ(x)dx =

∫ γ1

γ0

BER(x)pγ(x)dx+

∫ γ2

γ1

BER(x)pγ(x)dx+ . . .+

∫ γn

γn−1

BER(x)pγ(x)dx,

(5.14)

where γi > γi−1, i = 1, 2, . . . , n, γ0 = 0 and γn = x0. Due to the analytical difficulty

of describing BER(x) over 0 ≤ x ≤ x0, computer simulation is used. The system is

simulated at SNR values γi, i = 1, 2, . . . , n−1, to get the result BERi, i = 1, 2, . . . , n−1.

It is assumed that BER(x) ≈ BERi for γi ≤ x ≤ γi+1 to obtain the approximation

BER(γ) ≈n−1∑

i=0

∫ γi+1

γi

BERipγ(x)dx+

∫ ∞

γn

c1Q(c2√x)pγ(x)dx. (5.15)

For SNR in the range 0 ≤ x ≤ γ1 the bit error rate is assumed to be BER0 = 1/2. Figure

5.3 illustrates the n+ 1 regions of (5.15). Notice that for n = 1, (5.15) is equivalent to

(5.13). In other words, (5.15), a (n+1)-region model, is a generalization of the two-region

model (5.13).

CE-OFDM systems are simulated in Rayleigh and Ricean (KR = 3 dB and KR = 10

dB) channels. The values of modulation index are as follows: for M = 2, 2πh ≤ 0.6;

91

Q-function (4.35)Observed (simulation)(n+ 1)-region model

← γ0 = −∞

...

. . .

BER2

BER0 = 1/2

Signal-to-noise ratio per bit, x (dB)

Conditio

nalbit

erro

rra

te,B

ER

(x)

γ1 γ2 γ3 γ4 γn−2 γn−1 γn

1

0.01

BER1

BER3

BER4

BERn−2

BERn−1

Figure 5.3: A (n+ 1)-region model. (M = 8, N = 64, 2πh = 0.6)

for M = 4, 2πh ≤ 1.2; for M = 8, 2πh ≤ 1.8; and for M = 16, 2πh ≤ 2.4. The

results are shown in Figure 5.4: the circles represent Rayleigh results; the squares and

triangles represent the Ricean results for KR = 3 dB and KR = 10 dB, respectively.

The solid lines are the results of the semi-analytical approach, (5.15). The transition

region is sampled every 0.5 dB, that is, γi+1 − γi = 0.5 dB, i = 1, 2, . . . , n − 1; the

starting point is γ1 = −5 dB. Therefore γi = 0.5(i − 1) − 5 dB, i = 1, 2, . . . , n. The

sampling continues until BERn < 0.01. For SNR x ≥ γn the conditional bit error rate

is approximated with the Q-function (5.8). This criteria used for γn is based on the

observation that, for the modulation index values under consideration, the Q-function is

accurate for BER < 0.01. As shown in the figure, this semi-analytical approach yields

curves for BER(γ) that closely match simulation.

Figure 5.5 shows the improvement of (5.15) over (5.10) and (5.11). The semi-

analytical approach closely matches the simulation results, even at low SNR, while (5.10)

and (5.11) are overly optimistic by several dB.

The advantage of the technique described in this section is it gives an accurate

result in a small fraction of the time required for direct simulation. For example, the

92

(a) M = 2, 2πh = 0.2

Average SNR per bit, γ (dB)

Bit

erro

rra

te

50403020100

100

10−1

10−2

10−3

10−4

10−5

(b) M = 2, 2πh = 0.6

Average SNR per bit, γ (dB)

Bit

erro

rra

te

50403020100

100

10−1

10−2

10−3

10−4

10−5

(c) M = 4, 2πh = 0.4

Average SNR per bit, γ (dB)

Bit

erro

rra

te

50403020100

100

10−1

10−2

10−3

10−4

10−5

(d) M = 4, 2πh = 1.2

Average SNR per bit, γ (dB)B

iter

ror

rate

50403020100

100

10−1

10−2

10−3

10−4

10−5

(e) M = 8, 2πh = 0.6

Average SNR per bit, γ (dB)

Bit

erro

rra

te

50403020100

100

10−1

10−2

10−3

10−4

10−5

(f) M = 8, 2πh = 1.8

Average SNR per bit, γ (dB)

Bit

erro

rra

te

50403020100

100

10−1

10−2

10−3

10−4

10−5

(g) M = 16, 2πh = 0.8

Average SNR per bit, γ (dB)

Bit

erro

rra

te

50403020100

100

10−1

10−2

10−3

10−4

10−5

(h) M = 16, 2πh = 2.4

Average SNR per bit, γ (dB)

Bit

erro

rra

te

50403020100

100

10−1

10−2

10−3

10−4

10−5

Figure 5.4: Performance of CE-OFDM in flat fading channels. (Circle=Rayleigh;square=Rice, K = 3 dB; triangle=Rice, K = 10 dB. Solid line=Semi-analytical curve,(5.15); points=simulation. N = 64)

93

Semi-analytical technique (5.15)Ricean (KR = 10 dB) approximation (5.10)

Ricean (KR = 10 dB) simulationRicean (KR = 3 dB) approximation (5.10)

Ricean (KR = 3 dB) simulationRayleigh approximation (5.11)

Rayleigh simulation

Average signal-to-noise ratio per bit, γ (dB)

Bit

erro

rra

te

50403020100

100

10−1

10−2

10−3

10−4

10−5

Figure 5.5: Comparison of semi-analytical technique (5.15) with (5.10) and (5.11).(M = 4, N = 64, 2πh = 1.2)

simulated Rayleigh result in Figure 5.5 requires about 6 hours of computer time (on a

workstation with 1 gigabytes of memory and a single 3 gigahertz microprocessor). The

semi-analytical result, on the other hand, requires less than 7 s (to obtain {BERi}, and

perform numerical integration): a speed improvement of 4 orders of magnitude.

The disadvantage, however, is that this technique doesn’t yield a closed-form expres-

sion. As of the time of this writing, such a solution, that is general and accurate, doesn’t

seem possible.

Chapter 6

Performance of CE-OFDM in

Frequency-Selective Channels

In this chapter the performance of CE-OFDM in frequency-selective channels is stud-

ied. The channel is time dispersive having an impulse response h(τ) that can be non-zero

over 0 ≤ τ ≤ τmax, where τmax is the channel’s maximum propagation delay. The received

signal is

r(t) =

∫ ∞

−∞h(τ)s(t− τ)dτ + n(t)

=

∫ τmax

0h(τ)s(t− τ)dτ + n(t),

(6.1)

where s(t) is the CE-OFDM signal according to (3.11) and n(t) is the complex Gaussian

noise term represented by (4.15). The lower bound of integration in (6.1) is due to the

law of causality [401, p. 245]: h(τ) = 0 for τ < 0. The upperbound is τmax since, by

definition of the maximum propagation delay, h(τ) = 0 for τ > τmax.

CE-OFDM has the same block structure as conventional OFDM, with a block period,

TB, designed to be much longer than τmax. A guard interval of duration Tg ≥ τmax is

inserted between successive CE-OFDM blocks to avoid interblock interference. At the

receiver, r(t) is sampled at the rate fsa = 1/Tsa samp/s, the guard time samples are

discarded and the block time samples are processed. Using the discrete-time model

outlined in Section 2.1.2, the processed samples are

rp[i] = r[i] =

Nc−1∑

m=0

h[m]s[i −m] + n[i], i = 0, . . . , NB − 1. (6.2)

94

95

Note that the discarded samples are {r[i]}−1i=−Ng

. Transmitting a cyclic prefix during

the guard interval makes the linear convolution with the channel equivalent to circular

convolution. Thus

rp[i] =1

NDFT

NDFT−1∑

k=0

H[k]S[k]ej2πik/NDFT , i = 0, . . . , NB − 1, (6.3)

where {H[k]} is the DFT of {h[i]} and {S[k]} is the DFT of {s[i]}. The effect of the

channel can be reversed with the frequency-domain equalizer: a DFT followed by a

multiplier bank, followed by an IDFT. The FDE output is

s[i] =1

NDFT

NDFT−1∑

k=0

Rp[k]C[k]ej2πik/NDFT , i = 0, . . . , NB − 1, (6.4)

where {Rp[k]} is the DFT of the processed samples and {C[k]} are the equalizer correc-

tion terms, which are computed as [463]

C[k] =1

H[k](6.5)

for the zero-forcing (ZF) criterion, and

C[k] =H∗[k]

|H[k]|2 + (Eb/N0)−1(6.6)

for the minimum mean-square error (MMSE) criterion.

Ignoring noise (n[i] = 0), the output of the frequency-domain equalizer using (6.5) is

s[i] =1

NDFT

NDFT−1∑

k=0

H[k]S[k]C[k]ej2πik/NDFT

=1

NDFT

NDFT−1∑

k=0

H[k]S[k]1

H[k]ej2πik/NDFT

=1

NDFT

NDFT−1∑

k=0

S[k]ej2πik/NDFT

= s[i], i = 0, . . . , NB − 1.

(6.7)

Therefore, the ZF frequency-domain equalizer perfectly reverses the effect of the channel.

When noise can’t be ignored, the ZF suffers from noise enhancement. For example, a

fade of −30 dB results in a correction term with gain +30 dB, which corrects the channel

but amplifies the noise by a factor of 1000. The MMSE criterion (6.6) takes into account

96

the signal-to-noise ratio, making an optimum trade between channel inversion and noise

enhancement. Notice that the MMSE and ZF are equivalent at high SNR:

limEb/N0→∞

C[k]|MMSE =H∗[k]

|H[k]|2 =1

H[k]= C[k]|ZF. (6.8)

The system under consideration is shown in Figure 6.1. System performance is

estimated by way of computer simulation. The samples {h[i]}, {s[i]} and {n[i]} are

generated then used to calculate the received samples (6.2) which are then processed by

the FDE and the demodulator.

Removeh(τ)

s(t)

n(t)

r(t)FDE

r[i] rp[i]CE-OFDM CE-OFDM

CPModulator Demodulator

Figure 6.1: CE-OFDM system with frequency-selective channel.

The study is separated into two parts. In Section 6.1, the performance of the MMSE

and ZF equalizers are compared over various frequency-selective channels. In Section 6.2,

performance is evaluated for frequency-selective fading channels, in which case {h[i]} is

described statistically. In both sections an N = 64 CE-OFDM system is considered,

with a block period of TB = 128 µs. The subcarrier spacing is 1/TB = 7812.5 Hz and

the mainlobe bandwidth is W = N/TB = 500 kHz. The guard period is Tg = 10 µs,

resulting in a transmission efficiency ηt = 128/138 ≈ 0.93. The simulation uses an

oversampling factor J = 8; therefore the sampling rate is fsa = JN/TB = 4 Msamp/s,

and the sampling period is Tsa = 1/fsa = 0.25 µs.

6.1 MMSE versus ZF Equalization

In this section, the performance of CE-OFDM using the MMSE and ZF frequency-

domain equalizers is compared over six frequency-selective channels.

6.1.1 Channel Description

The channel samples {h[i]}, over the corresponding guard interval [0, 10µs], are

shown in Table 6.1. For Channels A–C the maximum propagation delay is τmax = 0.75

97

Table 6.1: Channel samples of frequency-selective channels.

Delay (µs) Channel A Channel B Channel C Channel D Channel E Channel Fi τi = iTsa h[i] h[i] h[i] h[i] h[i] h[i]

0 0.00 0.59e+j3.04 0.93e−j1.11 0.71e−j0.77 0.14e+j1.99 0.56e−j0.40 0.62e+j0.67

1 0.25 0.80e−j2.22 0.30e−j2.90 0.70e+j2.00 0.47e−j1.01 0.24e+j0.98 0.47e−j0.95

2 0.50 0 0 0 0.61e+j0.26 0.51e−j0.06 0.33e+j2.58

3 0.75 0.10e−j0.37 0.20e+j2.97 0.07e+j0.98 0.42e−j0.01 0.21e−j2.12 0.22e+j0.10

4 1.00 – – – 0.23e+j1.09 0.24e+j1.14 0.25e−j1.92

5 1.25 – – – 0.10e+j1.00 0.11e+j1.64 0.16e−j0.20

6 1.50 – – – 0.18e+j1.82 0.25e−j1.28 0.14e−j2.30

7 1.75 – – – 0.13e+j2.36 0.12e−j0.93 0.21e−j1.14

8 2.00 – – – 0.13e−j0.60 0.27e+j1.82 0.13e+j0.34

9 2.25 – – – 0.12e+j1.00 0.12e+j1.49 0.16e−j2.43

10 2.50 – – – 0.08e−j2.30 0.15e+j0.15 0.17e+j0.36

11 2.75 – – – 0.09e−j1.91 0.19e+j0.23 0.08e−j0.93

12 3.00 – – – 0.13e+j2.99 0.05e+j2.57 0.06e−j1.08

13 3.25 – – – 0.04e−j1.97 0.07e−j0.17 0.05e+j0.13

14 3.50 – – – 0.08e+j1.05 0.04e+j3.00 0.02e+j3.11

15 3.75 – – – 0.08e+j1.01 0.05e−j1.20 0.07e−j2.81

16 4.00 – – – 0.05e+j1.42 0.09e+j0.54 0.05e−j2.87

17 4.25 – – – 0.06e−j0.18 0.12e−j0.10 0.04e−j1.39

18 4.50 – – – 0.09e+j0.56 0.03e+j0.05 0.01e−j0.89

19 4.75 – – – 0.05e+j0.72 0.03e+j0.96 0.02e−j2.00

20 5.00 – – – 0.01e+j3.13 0.02e−j0.33 0.02e+j2.22

21 5.25 – – – 0.05e+j1.11 0.03e−j1.53 0.03e+j0.92

22 5.50 – – – 0.01e+j2.42 0.01e+j0.29 0.01e−j1.56

23 5.75 – – – 0.02e−j1.92 0.02e+j2.58 0.02e+j0.55

24 6.00 – – – 0.02e−j1.20 0.01e−j1.33 0.01e+j2.83

25 6.25 – – – 0.03e+j2.07 0.02e−j1.96 0.02e+j0.48

26 6.50 – – – 0.01e+j0.17 0.02e+j2.29 0.01e+j2.68

27 6.75 – – – 0.01e−j0.93 0.03e+j2.86 0.01e+j2.03

28 7.00 – – – 0.01e+j2.93 0.01e−j0.14 0.01e−j1.76

29 7.25 – – – 0.02e−j2.91 0.01e−j0.36 0.01e−j2.42

30 7.50 – – – 0.01e−j0.76 0.02e+j1.98 0.01e+j1.11

31 7.75 – – – 0.01e−j1.88 0.01e−j2.38 0.01e+j0.01

32 8.00 – – – 0.01e−j2.96 0.01e+j0.19 0.01e+j0.40

33 8.25 – – – 0.01e−j0.89 0.02e+j2.18 0.01e+j1.69

34 8.50 – – – 0.01e−j1.54 0.01e−j2.41 0.01e−j0.49

35 8.75 – – – 0.01e−j3.01 0.01e−j3.11 0.01e+j2.67

36 9.00 – – – – – –37 9.25 – – – – – –38 9.50 – – – – – –39 9.75 – – – – – –40 10.0 – – – – – –

98

µs, which results in Nc = bτmax/Tsac + 1 = b0.75/0.25c + 1 = 4 samples [see (2.19)].

For Channels D–F, τmax = 8.75 µs, thus Nc = b8.75/0.25c + 1 = 36. The channels are

normalized such thatNc−1∑

i=0

|h[i]|2 = 1. (6.9)

Channels A–C are single realizations of an approximation to the maritime channel

model in [350]. Channels D–F are single realizations of a stochastic model which has an

exponential delay power density spectrum1.

Figure 6.2 shows Channel D in the time and frequency domains. In subfigure (a),

|h[i]|2, that is, the power of the time samples, is plotted. In subfigure (b), |H(f ′)|2 is

plotted, where [422, p. 256]

H(f ′) =

Nc−1∑

i=0

h[i]e−j2πf′i, (6.10)

is the Fourier transform of h[i]. The x-axis is scaled as [422, p. 24]

f = f ′fsa Hz, (6.11)

where f ′ is the normalized frequency variable having units cycles/samp [422, p. 16].

Notice that over the signal’s mainlobe frequency range, −250 kHz ≤ f ≤ 250 kHz, the

channel is frequency selective. The magnitude response fluctuates over a 8.5 dB range,

−2.5 dB ≤ |H(f ′)|2 ≤ 6 dB.

The Fourier transform (6.10) is related to the discrete Fourier transform,

H[k] =

Nc−1∑

i=0

h[i]e−j2πik/NDFT , k = 0, . . . , NDFT − 1, (6.12)

as

H[k] = H(f ′k), k = 0, 1, . . . , NDFT − 1, (6.13)

where the discrete set of frequencies {f ′k} are defined as

f ′k ≡

kNDFT

, k = 0, 1, . . . , NDFT2 ,

kNDFT

− 1, k = NDFT2 + 1, . . . , NDFT − 1.

(6.14)

1Stochastic models are discussed in the next section.

99

Propagation delay, iTsa (µs)

|h[i]|2

9876543210

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

(a) Time domain.

20 dB10 dB

Equalizer response, MMSE: Eb/N0 = 0 dBEqualizer response, ZF

Channel D response, |H(f ′)|2

Frequency, f = f ′fsa (kHz)

Magnitude

resp

onse

(dB

)

2001000−100−200

10

5

0

−5

(b) Frequency domain.

Figure 6.2: Channel D.

100

Using a DFT size NDFT = JN = NB and noting (6.11), the frequency samples {H[k]}correspond to the frequencies

fk = f ′kfsa = f ′kJN

TB=

kTB, k = 0, 1, . . . , NDFT

2 ,

kTB

− fsa, k = NDFT2 + 1, . . . , . . . , NDFT − 1.

(6.15)

Included in Figure 6.2(b) is the response of the MMSE and ZF equalizers. The ZF

response, (6.5), is simply the inverse of the channel. The MMSE response, (6.6), is shown

for Eb/N0 = 0, 10, and 20 dB. Notice that at high SNR the MMSE approaches the ZF

equalizer, which is to be expected from (6.8). For this particular channel the MMSE and

ZF are shown to be equivalent for Eb/N0 ≥ 20 dB.

6.1.2 Simulation Results

The N = 64 CE-OFDM system is simulated over Channels A–F. The modulation

order is M = 2, and different values of the modulation index, h, are selected. Due to

the channel normalization (6.9), the simulation results are compared against the simple

AWGN channel. The results are shown in Figures 6.3–6.8. For each case, |h[i]|2 is plotted

in subfigure (a); the channel and equalizer frequency-domain responses are plotted in

subfigure (b); and the bit error rate performance results are shown in subfigure (c).

The results for Channel A are shown in Figure 6.3. Of the six test channels, Channel

A is the most mild in terms of its frequency-domain response. The magnitude response

|H(f ′)|2 spans a 3 dB region in a nearly linearly manner. The equalizers are shown to

effectively correct the channel: the BER curves in Figure 6.3(c) are nearly indistinguish-

able from the simple AWGN curves. Results are plotted for 2πh = 0.1, 0.3 and 0.6. For

the 2πh = 0.6 example at the lower SNR values Eb/N0 < 10 dB, the ZF result is shown

to be slightly worse than the MMSE result; for higher values of SNR the performance of

the two equalizers becomes nearly identical. This is to be expected since, as illustrated

in Figure 6.3(b), their frequency response become the same at high Eb/N0.

Results for 2πh = 0.1, 0.2, 0.4 and 0.6 over Channel B are shown in Figure 6.4.

The frequency response of this channel is more severely varying than Channel A. Over

the signal bandwidth, |H(f ′)|2 spans a 6 dB range. As with the previous example, the

MMSE is shown to slightly outperform the ZF at low SNR (i.e., the 2πh = 0.6 example

for Eb/N0 < 10 dB), but the two equalizers have essentially the same performance at the

101

Propagation delay, iTsa (µs)

|h[i]|2

9876543210

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

(a) Time domain.

20 dB10 dB

MMSE: Eb/N0 = 0 dBZF

Channel A

Frequency, f = f ′fsa (kHz)

Magnitude

resp

onse

(dB

)

2001000−100−200

2

0

−2

−4

−6

(b) Frequency domain.

AWGN approx (4.35)AWGN sim

MMSEZF

0.10.30.62πh

Signal-to-noise ratio per bit, Eb/N0 (dB)

Bit

erro

rra

te

30252015105

10−1

10−2

10−3

10−4

(c) Performance for MMSE and ZF compared to AWGN and (4.35).

Figure 6.3: Channel A results.

102

Propagation delay, iTsa (µs)

|h[i]|2

9876543210

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

(a) Time domain.

30 dB20 dB10 dB

MMSE: Eb/N0 = 0 dBZF

Channel B

Frequency, f = f ′fsa (kHz)

Magnitude

resp

onse

(dB

)

2001000−100−200

4

2

0

−2

−4

−6

(b) Frequency domain.

AWGN approx (4.35)AWGN sim

MMSEZF

0.10.20.40.62πh

Signal-to-noise ratio per bit, Eb/N0 (dB)

Bit

erro

rra

te

30252015105

10−1

10−2

10−3

10−4

(c) Performance for MMSE and ZF compared to AWGN and (4.35).

Figure 6.4: Channel B results.

103

higher SNR values. For BER ≤ 0.001 the degradation caused by the frequency selective

channel, when compared to the simple AWGN result, is slightly less than 1 dB.

Channel C has the most frequency-selective response of the three maritime channel

realizations. As shown in Figure 6.5(b), the magnitude response varies over a 20 dB

range. It is also shown that very high SNR is required for the MMSE response to

approach the ZF response. Over the frequency range −250 kHz ≤ f ′fsa ≤ −200 kHz,

for example, the two are equivalent only for Eb/N0 > 35 dB. This equivalence is also

demonstrated in Figure 6.5(c): for the 2πh = 0.1 example, the ZF performance gradually

approaches the MMSE performance at these high SNR values. Clearly, the large amount

of frequency selectivity of this channel results in a large performance degradation when

compared to the AWGN results. At the bit error rate 0.001, the degradation is 10 dB

for the 2πh = 0.1 case. The improvement of the MMSE is pronounced for 2πh = 0.5.

At the bit error rate 0.001, the MMSE outperforms the ZF by 7 dB, and is only 2 dB

worse than the performance over the simple AWGN channel.

Figures 6.6–6.8 show the results for Channels D–F. As stated earlier, the three chan-

nels are three different realizations of a stochastic model with an exponential delay power

density spectrum. The degree that the each channel varies over the signal bandwidth

progresses from Channel D to Channel F. Channel F, having a 50 dB attenuation at

185 kHz, is the most harsh of the test channels. The results in Figure 6.8(c) show the

dramatic performance degradation as a consequence of the severe frequency selectivity.

An 18 dB loss, compared to the AWGN performance, is experienced for the 2πh = 0.6,

MMSE example at the bit error rate 0.001; the ZF case degrades more than 20 dB further.

A 40 dB loss is suffered for the 2πh = 0.1 and 0.3 cases. These results show that fre-

quency selective channels having deep fades in the signal bandwidth impact performance

greatly.

6.1.3 Discussion and Observations

At this point, several observations can be made. First, the performance of the

equalized CE-OFDM systems studied depends on the amount of frequency selectivity

over the signal bandwidth. For channels with a relatively mild frequency response—

Channels A, B and D, for example—the performance degradation is minor. The noise

enhancement that results from equalizing channels with severe frequency responses—

104

Propagation delay, iTsa (µs)

|h[i]|2

9876543210

0.6

0.5

0.4

0.3

0.2

0.1

0

(a) Time domain.

35 dB30 dB20 dB10 dB

MMSE: Eb/N0 = 0 dBZF

Channel C

Frequency, f = f ′fsa (kHz)

Magnitude

resp

onse

(dB

)

2001000−100−200

20

10

0

−10

−20

(b) Frequency domain.

AWGN approx (4.35)AWGN sim

MMSEZF

0.10.52πh

Signal-to-noise ratio per bit, Eb/N0 (dB)

Bit

erro

rra

te

403530252015105

10−1

10−2

10−3

10−4

(c) Performance for MMSE and ZF compared to AWGN and (4.35).

Figure 6.5: Channel C results.

105

Propagation delay, iTsa (µs)

|h[i]|2

9876543210

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

(a) Time domain.

20 dB10 dB

MMSE: Eb/N0 = 0 dBZF

Channel D

Frequency, f = f ′fsa (kHz)

Magnitude

resp

onse

(dB

)

2001000−100−200

6

4

2

0

−2

−4

−6

−8

(b) Frequency domain.

AWGN approx (4.35)AWGN sim

MMSEZF

0.10.20.62πh

Signal-to-noise ratio per bit, Eb/N0 (dB)

Bit

erro

rra

te

3530252015105

10−1

10−2

10−3

10−4

(c) Performance for MMSE and ZF compared to AWGN and (4.35).

Figure 6.6: Channel D results.

106

Propagation delay, iTsa (µs)

|h[i]|2

9876543210

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

(a) Time domain.

30 dB20 dB10 dB

MMSE: Eb/N0 = 0 dBZF

Channel E

Frequency, f = f ′fsa (kHz)

Magnitude

resp

onse

(dB

)

2001000−100−200

10

5

0

−5

−10

(b) Frequency domain.

AWGN approx (4.35)AWGN sim

MMSEZF

0.10.62πh

Signal-to-noise ratio per bit, Eb/N0 (dB)

Bit

erro

rra

te

403530252015105

10−1

10−2

10−3

10−4

(c) Performance for MMSE and ZF compared to AWGN and (4.35).

Figure 6.7: Channel E results.

107

Propagation delay, iTsa (µs)

|h[i]|2

9876543210

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

(a) Time domain.

30 dB20 dB10 dB

MMSE: Eb/N0 = 0 dBZF

Channel F

Frequency, f = f ′fsa (kHz)

Magnitude

resp

onse

(dB

)

2001000−100−200

40

20

0

−20

−40

(b) Frequency domain.

AWGN approx (4.35)AWGN sim

MMSEZF

0.10.30.6

0.10.30.6 2πh

Signal-to-noise ratio per bit, Eb/N0 (dB)

Bit

erro

rra

te

706560555045403530252015105

10−1

10−2

10−3

10−4

(c) Performance for MMSE and ZF compared to AWGN and (4.35).

Figure 6.8: Channel F results.

108

Channels C, E and F—degrades performance dramatically. Second, the complexity of

the frequency-domain equalizers is determined by the DFT size, not by the number of

non-zero channel terms h[i]. This is in contrast to conventional time-domain equalizers

which have a complexity that depends on the number of paths in the multipath channel.

Last, the MMSE equalizer is more complicated than the ZF equalizer since the SNR

per bit, Eb/N0, must be estimated at the receiver. The results of this study show that

this added complexity doesn’t always translate into improved performance. That is, the

ZF performance is the same as the MMSE performance for many cases—the 2πh ≤ 0.4

cases in Channel B, for example. In other cases, the MMSE performs much better, and

thus estimating Eb/N0 pays substantial dividends—the 2πh = 0.5 case for Channel C

illustrates this point.

As demonstrated in the following section, the MMSE equalizer offers significant im-

provement over the ZF equalizer when averaging performance over many channel real-

izations of a stochastic channel model.

6.2 Performance Over Frequency-Selective Fading Chan-

nels

In contrast to the test channels used in the previous section, which were deterministic

as defined in Table 6.1, the channels used in this section are described statistically.

The mathematical foundation for stochastic time-variant linear channels was pioneered

by Bello [50]; more recently Patzold’s text, Mobile Fading Channels [401], provides a

excellent treatment of the topic, with a focus on the various aspects of simulation. In the

study here, the widely used assumption of WSSUS (wide-sense stationary uncorrelated

scattering) is applied. Also, it is assumed that the channel is composed of discrete paths,

each having an associated gain and discrete propagation delay. This assumption is based

on the Parsons and Bajwa ellipse model for describing multipath channel geometry [401,

p. 244]. The channel’s impulse response is

h(τ) =L−1∑

l=0

alδ(τ − τl), (6.16)

109

where al is the complex channel gain and τl is discrete propagation delay of the lth path;

the total number of paths is represented by L. The propagation delay differences are

∆τl = τl − τl−1 ≡ Tsa, l = 1, 2, . . . ,L − 1. (6.17)

That is, they are set equal to the sampling period of the simulation [401, p. 269]. The

delay of the 0th path is defined as τ0 ≡ 0, thus

τl = lTsa, l = 0, 1, . . . ,L− 1. (6.18)

For each simulation trial, the set of path gains {al}L−1l=0 are generated randomly. Each

gain is complex valued, has a zero mean and a variance

σ2al

= E{

|al|2}

, l = 0, 1, . . . ,L− 1. (6.19)

Both the real and imaginary parts of the path gains are Gaussian distributed [401, p.

267]; thus the envelope |al|2 is Rayleigh distributed. Also, the channels are normalized

such thatL−1∑

l=0

σ2al

= 1. (6.20)

As outlined in Patzold’s text (pp. 276–279) the parameters σ2al

, τl and L determine

the fundamental characteristic functions and quantities of the channel models, such as the

delay power spectral density and the delay spread2. The relevant formulas are expressed

below.

• Delay power spectral density:

Sττ (τ) =

L−1∑

l=0

σ2alδ(τ − τl). (6.21)

• Average delay:

B(1)ττ =

L−1∑

l=0

σ2alτl. (6.22)

2The phrase “delay power spectral density” is also commonly referred to as “power delay profile”(PDP) or “multipath intensity profile” (MIP). For the sake of being consistent with [401], “delay powerspectral density” is used here. In Patzold’s text, a clear distinction is made between stochastic channelmodels, which provide the theoretical and mathematical foundations, and “deterministic” channel modelswhich are generated in software or hardware for simulation purposes. For the sake of simplicity, thisdistinction isn’t stressed here (which results in a slightly different notation for the expressed formulasin his text). Also, since only time-invariant channels are considered in this thesis, the Doppler powerspectral density, time correlation function and coherence time (see [401, pp. 277–279]) are not discussed.

110

• Delay spread:

B(2)ττ =

√

√

√

√

L−1∑

l=0

(σalτl)2 −

(

B(1)ττ

)2(6.23)

• Frequency correlation function:

rττ (v′) =

L−1∑

l=0

σ2ale−j2πv

′τl (6.24)

The variable v′ is referred to as the frequency separation variable [401, p. 278].

• Coherence bandwidth: The coherence bandwidth is the smallest positive value

BC which fulfils |rττ (BC)| = 0.5|rττ (0)|; which, due to (6.20) and (6.24), is equiv-

alent to∣

∣

∣

∣

∣

L−1∑

l=0

σ2ale−j2πBCτl

∣

∣

∣

∣

∣

− 1

2= 0. (6.25)

Notice that BC is the 3 dB bandwidth of rττ (v′).

6.2.1 Channel Models

CE-OFDM is simulated over four frequency-selective fading channel models. Table

6.2 defines the parameters {σ2al} and {τl}. Channel Af and Bf are similar to the maritime

channel models in [350]3. Both have a secondary path with a 5 µs propagation delay.

Channel Af has a weak secondary path (one-tenth, i.e., −10 dB, the power of the primary

path); Channel Bf has a stronger secondary path (one-half, i.e., −3 dB, the power of the

primary path).

Channel Cf has an exponential delay power spectral density:

σ2al,C

=

CCfe−τl/2µs, 0 ≤ τl ≤ 8.75µs,

0, otherwise,(6.26)

where

CCf= 1

/ 35∑

l=0

exp(−τl/2e-6) = 0.1188 . . . (6.27)

is the normalizing constant used to guarantee (6.20). Note that the maximum propaga-

tion delay is 8.75 µs.

3To avoid notational ambiguities, the channel model labels in this section have the subscript “f”(“fading”).

111

Table 6.2: Channel model parameters.

Path no. Delay (µs) Channel Af Channel Bf Channel Cf Channel Df

l τl = lTsa σ2al,A

σ2al,B

σ2al,C

σ2al,D

0 0.00 10/11 2/3 1.18e-1 1/361 0.25 0 0 1.04e-1 1/362 0.50 0 0 9.25e-2 1/363 0.75 0 0 8.16e-2 1/364 1.00 0 0 7.20e-2 1/365 1.25 0 0 6.36e-2 1/366 1.50 0 0 5.61e-2 1/367 1.75 0 0 4.95e-2 1/368 2.00 0 0 4.37e-2 1/369 2.25 0 0 3.85e-2 1/3610 2.50 0 0 3.40e-2 1/3611 2.75 0 0 3.00e-2 1/3612 3.00 0 0 2.65e-2 1/3613 3.25 0 0 2.33e-2 1/3614 3.50 0 0 2.06e-2 1/3615 3.75 0 0 1.82e-2 1/3616 4.00 0 0 1.60e-2 1/3617 4.25 0 0 1.41e-2 1/3618 4.50 0 0 1.25e-2 1/3619 4.75 0 0 1.10e-2 1/3620 5.00 1/11 1/3 9.75e-3 1/3621 5.25 0 0 8.60e-3 1/3622 5.50 0 0 7.59e-3 1/3623 5.75 0 0 6.70e-3 1/3624 6.00 0 0 5.91e-3 1/3625 6.25 0 0 5.22e-3 1/3626 6.50 0 0 4.60e-3 1/3627 6.75 0 0 4.06e-3 1/3628 7.00 0 0 3.58e-3 1/3629 7.25 0 0 3.16e-3 1/3630 7.50 0 0 2.79e-3 1/3631 7.75 0 0 2.46e-3 1/3632 8.00 0 0 2.17e-3 1/3633 8.25 0 0 1.92e-3 1/3634 8.50 0 0 1.69e-3 1/3635 8.75 0 0 1.49e-3 1/3636 9.00 0 0 0 037 9.25 0 0 0 038 9.50 0 0 0 039 9.75 0 0 0 040 10.0 0 0 0 0

112

The last model, Channel Df, has a uniform delay power density spectrum:

σ2al,D

=

CDf, 0 ≤ τl ≤ 8.75µs,

0, otherwise,(6.28)

where the normalizing constant is

CDf= 1/36. (6.29)

In Figure 6.9 the delay power density spectrum (6.21) and the frequency correlation

function (6.24) are plotted for each of the four models. The corresponding average

delay (6.22), delay spread (6.23) and coherence bandwidth (6.25) for each model is

labeled. Notice that Channel Df has the smallest coherence bandwidth, BC = 67 kHz.

For Channel Af the coherence bandwidth isn’t finite since, as shown in subfigure (b),

|rττ (v′)| > −3 dB for all frequency separation values4.

6.2.2 Simulation Procedure and Preliminary Discussion

The average performance of various CE-OFDM systems is evaluated over the four

stochastic channel models. This is done by randomly generating {al}—which, as stated

above, are complex-valued quantities, drawn from the Gaussian distribution, with zero

mean and variance {σ2al}—computing the received samples (6.2), then processing the

samples with the frequency-domain equalizer and the CE-OFDM demodulator. At each

average Eb/N0 considered, the simulation runs for at least 20,000 bit errors, or until

100,000,000 bits are transmitted, whichever happens first. This corresponds to many

thousands of channel realizations5. Some channel realizations result in very poor per-

formance (for example, see Figure 6.8), while others result in a bit error rates not much

worse than that of the simple AWGN channel. This performance difference is attributed

to the severity of the channel’s frequency response, as observed with the several examples

in Section 6.1.

The performance also depends on the gain of the channel realization. Due to (6.20)

the channel gain, on average, is normalized to unity; however, for a given trial, the

channel may be fading such that the gain is less than unity, resulting in degraded per-

formance. The likelihood of a deep channel fade depends on the number of independent

4For Channel Af, min |rττ (v′)| = min˛

˛

1011

+ 111

exp(−j2πv′5 µs)˛

˛ = 911

> 12≈ −3 dB.

5Example simulation code can be found in Appendix C.

113

B(2)ττ = 1.44 µs

B(1)ττ = 0.45 µs

(a) Delay power spectral density, Channel Af

Propagation delay, τ (µs)

10

log10[S

ττ(τ

)]

109876543210

0

−5

−10

−15

−20

−25

−30

BC →∞

(b) Frequency correlation function, Channel Af

Frequency separation, v′ (kHz)

10lo

g10[r

ττ(v

′)]

0−150−300−450 300150 450

0

−3

−6

−9

−12

−15

B(2)ττ = 2.36 µs

B(1)ττ = 1.67 µs

(c) Delay power spectral density, Channel Bf

Propagation delay, τ (µs)

10

log10[S

ττ(τ

)]

109876543210

0

−5

−10

−15

−20

−25

−30

BC

(d) Frequency correlation function, Channel Bf

Frequency separation, v′ (kHz)

10lo

g10[r

ττ(v

′)]

740−150−300−450 300150 450

0

−3

−6

−9

−12

−15

B(2)ττ = 1.75 µs

B(1)ττ = 1.78 µs

(e) Delay power spectral density, Channel Cf

Propagation delay, τ (µs)

10

log10[S

ττ(τ

)]

109876543210

0

−5

−10

−15

−20

−25

−30

BC

(f) Frequency correlation function, Channel Cf

Frequency separation, v′ (kHz)

10lo

g10[r

ττ(v

′)]

1400−150−300−450 300 450

0

−3

−6

−9

−12

−15

B(2)ττ = 2.60 µs

B(1)ττ = 4.38 µs

(g) Delay power spectral density, Channel Df

Propagation delay, τ (µs)

10

log10[S

ττ(τ

)]

109876543210

0

−5

−10

−15

−20

−25

−30

BC

(h) Frequency correlation function, Channel Df

Frequency separation, v′ (kHz)

10lo

g10[r

ττ(v

′)]

670−150−300−450 300150 450

0

−3

−6

−9

−12

−15

Figure 6.9: Fundamental characteristic functions and quantities [(6.21)–(6.25)] of thefour channel models considered.

114

propagation paths [the WSSUS assumption makes each path in (6.16) independent]. It is

unlikely that multiple paths fade simultaneously. For this reason, channels characterized

by multiple propagation paths possess a type of diversity known at multipath diversity—

which can be exploited by the receiver. Of the four models considered in this study,

Channel Df can be said to have the most multipath diversity: the gain of a given realiza-

tion depends on 36 independent paths, each having, on average, an equal contribution.

Channel Af can be said to have the least amount of multipath diversity: over 90% of the

channel gain depends on a single path. Channel Bf has more multipath diversity than

Channel Af since the gain is distributed more equally between the two paths. That is,

the multipath diversity depends not only on the number of independent paths but also

on the way in which the power is distributed over the paths, as determined by {σ2al}. [It

is worth noting that the frequency-nonselective channel models considered in Chapter 5

have L = 1 path of which 100% of the channel gain depends (σ2a1 = 1), and thus these

channels have no multipath diversity.] In the results that follow, the impact of multipath

diversity—and its frequency-domain dual frequency diversity—on CE-OFDM systems is

studied.

6.2.3 Simulation Results

The simulation results of this study are presented over three figures: Figure 6.10

compares the performance of a CE-OFDM system, with fixed modulation order M and

modulation index h, over the four channel models; Figure 6.11 compares the performance

of a CE-OFDM system with fixed M but varying h over Channel Cf; and Figure 6.12

compares the performance of constant envelope and conventional OFDM systems, in the

presence of power amplifier nonlinearities, over Channel Cf. For each case, the number

of subcarriers is N = 64.

In Figure 6.10, performance results of an M = 4, N = 64, 2πh = 1.0 CE-OFDM

system are plotted. The simulation results over the multipath channel models Af–Df are

labeled with circles and triangles; the MMSE equalized results have solid lines connecting

the points, while the ZF equalized results use dashed lines. For reference, the performance

of the system over the simple AWGN channel is plotted (with dash-dot lines) along with

the performance over the Rayleigh frequency-nonselective fading channel (represented

by the thick solid line). These results show the significant performance improvement

115

AWGN approx (4.35)AWGN

Rayleigh, L = 1Df

Cf

Bf

ZF: Channel Af

Df

Cf

Bf

MMSE: Channel Af

Average signal-to-noise ratio per bit, Eb/N0 (dB)

Bit

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10−2

10−3

10−4

Figure 6.10: Performance results. (Multipath results are labeled with circle and trianglepoints; the Rayleigh, L = 1 result is that of the frequency-nonselective channel model.M = 4, N = 64, 2πh = 1.0)

that is to be had by using the MMSE equalizer. At the bit error rate 0.001, for example,

MMSE outperforms ZF by 10 dB for Channel Df. These results also show the impact of

multipath diversity. Consider the MMSE results. For Eb/N0 > 15 dB, the performance

over Channels Af–Df is better than the performance over the frequency-nonselective

Rayleigh (L = 1 path) channel. For BER ≤ 0.001, the performance over the multipath

channels is at least 5 dB better than the performance over the single path channel.

Notice that Channel Df, which has the most multipath diversity, results in a better

performance that all the other channels. The performance over Channel Bf, which has

more multipath diversity than Channel Af, is in fact better than the performance over

Channel Af. These results indicate that the CE-OFDM receiver exploits the multipath

diversity of the channel.

The fact that constant envelope OFDM exploits multipath diversity is an interesting

result since conventional OFDM doesn’t. This was shown in Section 2.1.1; specifically,

116

by (2.9). So long as the duration of the guard interval is greater than or equal to

the channel’s maximum propagation delay, that is, Tg ≥ τmax, and a cyclic prefix is

transmitted during the guard interval, the performance of OFDM in a time-dispersive

channel is equivalent to flat fading performance. In other words, the multipath fading

performance is the same as single path fading performance. In the context of Section

2.1.1, this property was considered beneficial since ISI is avoided. In the context here,

however, this property is considered a weakness since the multipath diversity of the

channel isn’t leveraged6.

To understand why CE-OFDM has improved performance over multipath fading

channels (compared to single path fading channels) while OFDM doesn’t, it is best to

view the problem in the frequency domain. The frequency domain dual to multipath

diversity is frequency diversity. It can be said that OFDM lacks frequency diversity

as well. As identified in Section 1.1.2, the wideband frequency-selective fading channel

is converted into N contiguous frequency-nonselective fading channels. Therefore any

frequency diversity inherent to the channel—that is, over the signal bandwidth the fre-

quency response of the channel varies, which can be taken advantage of by the receiver

to obtain performance better than flat fading—is not exploited by the OFDM receiver.

CE-OFDM, in contrast, has the ability to exploit the frequency diversity of the

channel since the phase modulator, in effect, spreads the data symbol energy in the

frequency domain. This can be seen by viewing the CE-OFDM waveform by the Taylor

series expansion [see Section 3.2, (3.24)]:

s(t) = A

[

1 + jσφm(t) −σ2φ

2m2(t) − j

σ3φ

6m3(t) + . . .

]

, (6.30)

0 ≤ t < TB, where A is the signal amplitude, σ2φ = (2πh)2 is the phase signal variance,

and m(t) = CN∑N

k=1 Ikqk(t), 0 ≤ t < TB, CN =√

6/N(M2 − 1), is the normalized

OFDM message signal. The higher-order terms mn(t), n ≥ 2, results in a frequency

spreading of the data symbols. This property is best demonstrated by way of a simple

example.

Example 6.2.1

Consider a CE-OFDM waveform with an OFDM message signal composed of N = 2 orthogonal

6Note that OFDM systems typically employ channel coding and frequency-domain interleaving, whichoffers diversity. However, since this thesis only deals with uncoded systems, these topics are beyond itsscope—and are topics for further research.

117

cosine subcarriers modulated with binary data symbols (M = 2):

m(t) =

2∑

k=1

Ik cos 2πkt/TB, 0 ≤ t < TB, (6.31)

where Ik ∈ {±1}, k = 1, 2. Assume that the modulation index, h, is such that the higher-order

terms m2(t) and m3(t) contribute to the make up of s(t) according to (6.30). It is desired to

write m2(t) and m3(t) in terms of I1, I2 and {cos 2πkt/TB}. This task requires some algebra,

but is simply done. For notational simplicity, let’s define

ck ≡ cos 2πkt/TB. (6.32)

Thus, (6.31) is written as

m(t) = I1c1 + I2c2. (6.33)

The second-order term is calculated as

m2(t) = (I1c1 + I2c2)(I1c1 + I2c2)

=(

0.5I21 + 0.5I2

2

)

c0 + (I1I2) c1 +(

0.5I21

)

c2 + (I1I2) c3 +(

0.5I22

)

c4,(6.34)

and the third-order term as

m3(t) =[ (

0.5I21 + 0.5I2

2

)

c0 + (I1I2) c1 +(

0.5I21

)

c2

+ (I1I2) c3 +(

0.5I22

)

c4]

(I1c1 + I2c2)

=(

0.75I21I2)

c0 +(

0.75I31 + 1.5I1I

22

)

c1 +(

1.25I21I2 + 0.5I3

2

)

c2

+(

0.25I31 + 0.75I1I

22

)

c3 +(

0.75I21I2)

c4

+(

0.75I1I22

)

c5 +(

0.25I32

)

c6.

(6.35)

The expansions above are represented in Table 6.3. The data symbol contribution at each

tone cos 2πkt/TB, k = 0, 1, . . . , 6, for m(t), m2(t) and m3(t) is shown. Referring to the tones

as frequency bins, it can be said that for m(t) the two data symbols are simply contained in

the k = 1 and k = 2 frequency bins. For the second-order term, m2(t), the data symbols mix

across the k = 0, 1, 2, 3, and 4 frequency bins. For m3(t), the data symbols mix across the

k = 0, 1, . . . , 6 frequency bins.

The simple example above shows how the data symbols spread across multiple fre-

quency bins. In general, it can be said that the N data symbols that constitute the

constant envelope OFDM signal are not simply confined to N frequency bins—as is the

case with conventional OFDM. The phase modulator mixes and spreads—albeit in a

nonlinear and exceedingly complicated manner—the data symbols in frequency, which

gives the CE-OFDM system the potential to exploit the frequency diversity in the chan-

nel. This isn’t necessarily the case, however. For small values of modulation index,

118

Table 6.3: Data symbol contribution per tone for mn(t), n =1, 2, and 3.

kth tone, cos 2πkt/TB

0 1 2 3 4 5 6

m(t) – I1 I2 – – – –

m2(t)0.5I2

1 ,0.5I2

2

I1I2 0.5I21 I1I2 0.5I2

2 – –

m3(t) 0.75I21I2

0.75I31 ,

1.5I1I22

1.25I21I2,

0.5I32

0.25I31 ,

0.75I1I22

0.75I21I2 0.75I1I

22 0.25I3

2

where only the first two terms in (6.30) contribute, that is,

s(t) ≈ A [1 + jσφm(t)] , (6.36)

the CE-OFDM signal doesn’t have the frequency spreading given by the higher-order

terms. In this case, the CE-OFDM signal is essentially equivalent to a conventional

OFDM signal, jσφm(t), (plus a relatively large DC term, A) and therefore doesn’t have

the ability to exploit the frequency diversity of the channel. Simply put, CE-OFDM

has frequency diversity when the modulation index is large and doesn’t have frequency

diversity when the modulation index is small.

This property is demonstrated in Figure 6.11. Simulation results of an M = 4, N =

64 CE-OFDM system are shown. The system is simulated over the single path Rayleigh

flat fading channel and over the multipath fading model Channel Cf. To demonstrate that

CE-OFDM with a small modulation index lacks frequency diversity, results for 2πh = 0.1

are shown. Notice that the single path and multipath performance is essentially the

same. By contrast, for the large modulation index example 2πh = 1.1, the multipath

performance is significantly better than the single path performance. For example, at

the bit error rate 0.001 the multipath performance is over 10 dB better than the single

path performance.

In the final figure, Figure 6.12, the performance of constant envelope OFDM is

compared to conventional OFDM in the presence of power amplifier nonlinearities. The

SSPA model (see Section 2.3) is used at various input backoff levels. The x-axis is

adjusted to account for the negative impact of input power backoff. The systems are

simulated over Channel Cf. For the OFDM system, QPSK data symbols are used. Three

different CE-OFDM systems are tested: M = 4, 2πh = 0.9; M = 8, 2πh = 2.0; and

M = 16, 2πh = 3.0. The advantage of the CE-OFDM systems is twofold. First, the

CE-OFDM systems operate with IBO = 0 dB. Second, the CE-OFDM systems exploit

119

Single pathMultipath

0.11.12πh

Average signal-to-noise ratio per bit, Eb/N0 (dB)

Bit

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100

10−1

10−2

10−3

10−4

Figure 6.11: Single path versus multipath. (M = 4, N = 64, Channel Cf, MMSE)

the frequency diversity inherent to the channel.

At the bit error rate 0.001 the CE-OFDM systems outperform the OFDM system

by at least 10 dB. At this bit error rate, the OFDM system has essentially the same

performance with backoff levels of 6 and 10 dB; therefore, IBO = 6 dB is preferred since

the performance is the same but the power efficiency is higher (see Figure 2.14). Even

so, the 6 dB backoff required by the OFDM system is still far less desirable as the 0 dB

backoff used by the CE-OFDM system. Notice that the OFDM system with IBO = 0

dB results in an irreducible error floor just below the bit error rate 0.1.

The results in Figure 6.12 also highlight the poor performance of CE-OFDM at

low SNR due to the threshold effect (as studied in Section 4.1.3). Over the region 0 dB

≤ Eb/N0 ≤ 10 dB, the OFDM system performs better than the CE-OFDM system. Also,

it should be noted that the M = 8 and M = 16 CE-OFDM systems shown have large

modulation index values (2πh = 2.0 and 2πh = 3.0 respectively) which results in spectral

broadening. Roughly speaking, the spectral efficiency of the QPSK/OFDM system is 2

b/s/Hz, which, according to (4.70), is about the same as the M = 4, 2πh = 0.9 CE-

OFDM system. The M = 8 and M = 16 systems have spectral efficiencies of 1.5 and

1.3 b/s/Hz, respectively.

Making a direct comparison between CE-OFDM and conventional OFDM is difficult

120

M = 16, 2πh = 3.0M = 8, 2πh = 2.0

CE-OFDM: M = 4, 2πh = 0.910 dB6 dB3 dB

OFDM: IBO = 0 dB

Eb/N0 + IBO (dB)

Bit

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10−2

10−3

10−4

Figure 6.12: CE-OFDM versus QPSK/OFDM. (SSPA model, Channel Cf, N = 64,MMSE)

due to the various parameters involved (M , 2πh, IBO, etc.), and due to the fact that

system requirements vary from system to system. For example, if power amplifier effi-

ciency is the most important requirement, then the input power backoff of 0 dB should

be chosen. At this backoff level, the OFDM system has a very high irreducible error

floor due to the power amplifier distortion, while the CE-OFDM system is relatively

unaffected. Alternatively, if operation at low SNR is important, then CE-OFDM may

not be well suited due to the threshold effect.

The results in this chapter show that CE-OFDM can perform quite well in multipath

fading channels—so long as the channel information (i.e., {H[k]}) is known at the receiver

and so long as the added complexity of the frequency-domain equalizer (i.e., two extra

FFTs) is acceptable. Further work is needed to study the effects of channel coding,

time-varying channels, phase noise, and so forth. Also, a thorough study comparing CE-

OFDM, OFDM and single carrier frequency-domain equalizer (SC-FDE) systems could

provide for interesting results.

Chapter 7

Conclusions

In this thesis the peak-to-average power ratio problem associated with orthogonal

frequency division multiplexing is evaluated. The PAPR statistics are studied and the

effect of power amplifier nonlinearities as a function of power backoff is evaluated by

computer simulation. It is shown that the amount of backoff required to reduce spectral

growth and performance degradation is significant: 6–10 dB depending on the subcarrier

modulation used. Large backoff is an unsatisfactory solution for battery-powered systems

since PA efficiency is low.

A signal transformation method for solving the PAPR problem is presented and

analyzed. The high PAPR OFDM signal is transformed to a 0 dB PAPR constant enve-

lope waveform. At the receiver, the inverse transform is performed prior to the OFDM

demodulator. For the CE-OFDM technique described, phase modulation is used. The

effect of the phase modulator on the transmitted signal’s spectrum is studied. It is shown

that the modulation index controls the spectral containment. The modulation index also

controls the system performance. The optimum receiver is analyzed and a performance

bound and approximation is derived. For a large modulation index, the CE-OFDM sig-

nals become less correlated which improves detection performance. The approximation

of the optimum receiver closely matches simulation results. It also closely matches a

derived bit error rate approximation for a practical phase demodulator receiver. For a

small modulation index and high signal-to-noise ratio, the phase demodulator receiver is

nearly optimum. For a larger modulation index the phase demodulator receiver becomes

sub-optimum due to the limitations of the phase demodulator and phase unwrapper.

121

122

This problem can be suppressed with the use of a properly designed finite impulse re-

sponse lowpass filter which precedes the phase demodulator.

The simulation results of the CE-OFDM performance curves use an oversampling

factor of J = 8. Future work includes experimenting with lower sampling rates for

reduced receiver complexity. The performance of the phase demodulator is a crucial

element to the overall CE-OFDM performance. Therefore, further research is needed

to evaluate more advanced phase demodulation techniques such as digital phase-locked

loops.

Phase modulation is used exclusively in this work. It would be interesting to evaluate

CE-OFDM frequency modulation systems and compare them to the results in this thesis.

In terms of performance over frequency-selective fading channels, the frequency-

domain equalizer requires knowledge of the channel. Many conventional OFDM systems

(those that don’t use differentially encoded modulations) also require channel state in-

formation. Thus techniques for channel estimation in OFDM has been extensively re-

searched [105, 144, 204, 213, 251, 257, 259, 273, 469, 547]. Applying the known techniques,

such as linear minimum mean-squared error (LMMSE) estimation and reduced complex-

ity singular value decomposition (SVD) approaches, to CE-OFDM is a subject for future

investigation. The impact of imperfect channel state information on the performance of

the frequency-domain equalizer is of interest.

CE-OFDM might be used as a stand-alone modulation technique or as a supplement

to an existing OFDM system. For example, a conventional OFDM system is designed for

severe multipath channels. However, at times the channel might be relatively benign so

the OFDM systems is an overkill and, due to power backoff, inefficient. An adaptive radio

might sense times where power efficient CE-OFDM, which requires minimal backoff, is

more applicable. Such a system can adaptively switch between conventional and constant

envelope modes.

For systems, such as power-limited satellite communications, where a constant en-

velope is very desirable, if not required, CE-OFDM might be a viable alternative to

convention continuous phase modulation systems which are complex due to phase trellis

decoding and sensitive to multipath. CE-OFDM is relatively robust in multipath fading

channels with the use of the frequency-domain equalizer. Depending on the channel

condition, equalization might not be required, therefore reducing receiver complexity.

123

For example, a channel characterized by a two-path model with a weak secondary path,

CE-OFDM might provide acceptable performance without equalization. CPM systems

in the other hand require high quality coherent channels.

In the near term a CE-OFDM prototype is being developed by Nova Engineering

(Cincinnati, OH). This work is being funded by the United States Office of Naval Re-

search under an STTR (small business technology transfer) initiative with UCSD being

the university partner. The goal of the prototype is to offer a second low-power mode

for the existing JTRS (Joint Tactical Radio System) wideband component which uses

OFDM. Research challenges that remain include evaluating CE-OFDM with many sub-

carriers (in this thesis, only 64 subcarriers are used), considering different equalization

techniques, developing synchronization schemes and studying the impact of channel cod-

ing and the effects of time-varying channels.

Additional future work includes comparing CE-OFDM with other block modulation

technique in terms of PAPR, spectral efficiency, power amplifier efficiency, performance

and complexity. There has been an increasing amount of attention given to conventional

single carrier modulation with the addition of a cyclic prefix which allows for frequency-

domain equalization [107, 154, 460, 463, 574]. However, most single carrier modulations

have a non-constant envelope due to pulse shaping and multilevel QAM symbol con-

stellations. A study is needed to compare these modulation techniques to CE-OFDM

taking into account the effects of the PA at various backoff levels. Also, using CPM with

a cyclic prefix is an interesting idea. Comparing the complexity and spectral efficiency of

such a technique with CE-OFDM would be interesting. Such research will help provide

insight into good designs for future wireless digital communication systems that require

power efficiency and high data rates.

Appendix A

Generating Real-Valued OFDM

Signals with the Discrete Fourier

Transform

For some applications, a real-valued OFDM signal is required. This can be done by

taking a DFT of a conjugate symmetric vector. The spectral efficiency of the real-valued

OFDM signal is the same as the spectral efficiency of the complex-valued OFDM signal.

A.1 Signal Description

The baseband OFDM signal is typically written as

x(t) =N−1∑

k=0

Xkej2πkt/TB , 0 ≤ t < TB, (A.1)

where N is the number of subcarriers, {Xk}N−1k=0 are the data symbols and TB is the

block period. Sampling x(t) at N equally spaced intervals over 0 ≤ t < TB yields the

sequence,

x[i] = x(t)|t=iTB/N =N−1∑

k=0

Xkej2πki/N , i = 0, 1, . . . , N − 1, (A.2)

which is the inverse discrete Fourier transform (IDFT) of the vector

X = [X0, X1, . . . , XN−1]. (A.3)

124

125

The sequence is complex-valued in general. However it can be made real-valued by

making X conjugate symmetric:

XN/2+k = X∗N/2−k (A.4)

and

X0 = XN/2 = 0. (A.5)

The IDFT is then

x[i] =

N−1∑

k=1

Xkej2πki/N

=

N/2−1∑

k=1

XN/2−kej2π(N/2−k)i/N +XN/2+ke

j2π(N/2+k)i/N

=

N/2−1∑

k=1

XN/2−kej2π(N/2−k)i/N +X∗N/2−ke

j2π(N/2+k)i/N ,

(A.6)

i = 0, 1, . . . , N − 1. But since

ej2π(N/2+k)i/N = ej2π(N/2+k)i/Ne−j2πNi/N

= ej2π(−N/2+k)i/N

= e−j2π(N/2−k)i/N ,

(A.7)

(A.6) can be written as

x[i] =

N/2−1∑

k=1

XN/2−kej2π(N/2−k)i/N +X∗N/2−ke

−j2π(N/2−k)i/N , (A.8)

i = 0, 1, . . . , N − 1. Using the identity A+A∗ = 2<{A},

x[i] = 2<

N/2−1∑

k=1

XN/2−kej2π(N/2−k)i/N

= 2<

N/2−1∑

k=1

Xkej2πki/N

, i = 0, 1, . . . , N − 1.

(A.9)

And since <{AB} = <{A}<{B} − ={A}={B},

x[i] = 2

N/2−1∑

k=1

<{Xk} cos(2πki/N) −={Xk} sin(2πki/N), (A.10)

126

i = 0, 1, . . . , N − 1. Thus, x[i] is real. Passing the sequence through a D/A converter

yields the continuous-time real-valued OFDM signal:

x(t) = 2

N/2−1∑

k=1

<{Xk} cos(2πkt/TB) −={Xk} sin(2πkt/TB). (A.11)

Now, suppose the data symbols are derived from a M 2-QAM (quadrature-amplitude

modulation) constellation; that is,

Xk = <{Xk} + j={Xk}, (A.12)

where

<{Xk},={Xk} ∈ {±1,±3,±(M − 1)}, for all k. (A.13)

In other words, the real and imaginary components are derived from M -PAM (pulse-

amplitude modulation) constellations. Therefore, processing M 2-QAM data with the

IDFT, (A.11) is a real-valued M -PAM OFDM signal.

A.2 Spectral Efficiency

Complex-valued baseband signals are transmitted as bandpass signals, centered at

a carrier frequency fc Hz. This is the case for the complex-valued signal in (A.1). The

transmitted signal is represented as

s1(t) = <{

x(t)ej2πfct}

. (A.14)

In the frequency domain, x(t) is shifted to the right by fc Hz, and the subcarriers are

centered at fc, fc +1/TB, fc +2/TB, . . . , fc +(N − 1)/TB Hz. The effective bandwidth of

the signal is therefore N/TB Hz. Each data symbol represents log2M bits (i.e., they are

assumed to be selected from a M -ary constellation), therefore the spectral efficiency is

S1 =Bits per second (b/s)

Bandwidth (Hz)=N log2M/TB

N/TB= log2M b/s/Hz. (A.15)

The real-valued OFDM signal in (A.11) has the same spectral efficiency as the

complex-valued signal, so long as it is transmitted at baseband. Transmitting the

signal as-is, <{Xk}, k = 1, 2, . . . , (N/2) − 1, modulate cosine subcarriers centered at

1/TB, 2/TB, . . . , [(N/2) − 1]/TB Hz; and likewise, ={Xk}, k = 1, 2, . . . , (N/2) − 1, mod-

ulate sine subcarriers at the same frequencies. The effective bandwidth of the signal is

127

(N/2)/TB Hz1, and since the real and imaginary parts of Xk represent 0.5 log2M bits,

the spectral efficiency of the real-valued OFDM signal is

S2 =Bits per second (b/s)

Bandwidth (Hz)=

2 × 0.5(N/2) log2M/TB

(N/2)/TB= log2M b/s/Hz. (A.16)

Therefore the spectral efficiency is the same as for the complex case.

However, the spectral efficiency of the real-valued signal is 1/2 that of the complex-

valued signal if the real-valued signal is translated up to a carrier frequency. This is due to

the fact that the cosine and sine subcarriers in (A.11) have a double sideband spectrum:

that is, cos(2πkt/TB) [or sin(2πkt/TB)] has a spectral components at ±k/TB Hz. [This

isn’t the case for the complex-valued signal, which has complex sinusoids: exp(j2πkt/TB)

has a spectral component only at k/TB Hz and is thus considered single sideband.] The

carrier frequency is typically much larger than the signal bandwidth, so the frequency

translation brings all the negative frequencies to the positive side: −(N/2)/TB +fc � 0.

Consequently, the passband transmission of (A.11) results in a signal with double the

bandwidth and 1/2 the spectral efficiency.

1Only the positive frequencies, f ≥ 0, count.

Appendix B

More on the OFDM Literature

The first OFDM-like radio to be found in the research literature is the Kineplex sys-

tem presented by Mosier et. al in 1958 [354]. Developed at the Collins Radio Company,

Burbank, CA, the radio used 20 tones separated by 110 Hz, each differentially phase

modulated. This paper caused some interest and some controversy as indicated by E.

D. Sunde’s (Bell Laboratories) comments found at the end of the journal paper.

In his 1960 paper [202], H. F. Harmuth, a researcher at General Dynamics, Rochester,

NY, suggested multiplexing orthogonal waveforms. Then, in 1967 M. S. Zimmerman et.

al described a 34 subcarrier military radio named Kathryn. The first paper to identify

the Doppler sensitivity of such a radio was by P. A. Bello [51]. Significant theoretical

contributions were made by B. R. Saltzberg and R. W. Chang of Bell Laboratories

[83, 84, 455]. In 1970 Chang was issued US patent 3,488,445 on OFDM [82].

Weinstein and Ebert, in 1971, where to first to suggest using a DFT for OFDM

modulation [579]. This observation was made six years after Cooley and Tukey published

details of the fast Fourier transform; these developments were significant since all modern

OFDM systems are based on the FFT.

A decade passed with little mention of OFDM in the literature. Then, in the early

80’s researchers from IBM’s Watson Research Center suggests OFDM for a wireline DSL-

type application [408]. They were the first to suggest bit loading. Around this time,

Japanese researcher suggest OFDM for wireless communications [207–209] (also see [6]).

L. J. Cimini’s 1985 paper [102] generated interest when he suggested applying OFDM

128

129

to mobile systems.

In the late 80’s and early 90’s OFDM received wide interest for the applications

of DSL and for wireless digital broadcasting. Kalet and Zervos compare OFDM to

single carrier with decision feedback equalization [248, 614]. The acceptance of OFDM

into xDSL standards was lead primarily by Stanford University’s J. M. Cioffi et al.

[9, 61, 95–97, 105, 446]. Now, OFDM is widely deployed for this consumer electronics

application. In terms of digital broadcasting, OFDM has been accepted for the European

DAB and DVB standards [162, 477, 552]. In the US, OFDM is being used for IBOC

broadcasting [221, 392].

OFDM is being applied to indoor wireless local area networks under the IEEE 802.11

and the ETSI HYPERLAN/2 standards [552]. And as mentioned in Chapter 1, OFDM

is being developed for ultra-wideband systems; cellular systems; wireless metropolitan

area networks; and for power line communication [119, 160, 264, 604].

Active OFDM research continues. The major focus in the OFDM literature includes

OFDM’s sensitivity to Doppler, phase noise, carrier frequency offsets, and nonlinearities.

Channel estimation and synchronization techniques are of interest, along with techniques

to address the PAPR problem.

Literature Survey Statistics

The OFDM literature is immense, so a detail discussion of it here would be overly

ambitious. The bibliography of this thesis does provide a somewhat current snapshot of

the OFDM literature. Also included in the bibliography are papers dealing with gen-

eral digital communications, continuous phase modulation, FM analog communications,

power amplifiers, computer simulation techniques, and other miscellaneous papers that

have, in some way, contributed to this work.

Conducting a 100% thorough literature review in this field, over the course of a PhD,

is a formidable, if impossible, task. Some statistics of the current author’s attempt are

displayed below.

130

First, to get an idea of the size of the literature, Figure B.1 shows the result of

searching for “OFDM” in the IEEE online literature database. As of the year 2004,

there are over 800 OFDM-specific IEEE journal papers and over 4300 papers when

including papers presented at IEEE conferences.

Journal plus conference papersJournal papers

Paper

s

20042000199619921988

1400

1200

1000

800

600

400

200

0

(a) Papers each year.

Journal plus conference papersJournal papers

Paper

s

20042000199619921988

4500

4000

3500

3000

2500

2000

1500

1000

500

0

(b) Cumulative paper count.

Figure B.1: “OFDM” search on IEEE Xplore [222].

So, there are many papers to read and to learn from. Besides the OFDM-specific

papers, there are many interesting and fundamental papers dealing with the general

area of digital communications and information theory. Being familiar with the relevant

literature, which may include several thousands of papers published over many decades,

is the goal, however long-term it may be.

131

PiledFiled

Paper

s

Oct 2005Jul 2005Apr 2005Jan 2005Oct 2004

600

550

500

450

400

350

300

250

200

150

Figure B.2: Papers, filed and piled.

This figure shows the number of filed and the number of piled papers as a function

of time, spanning my final year as a PhD student. A filed paper has been printed out,

read, added to a citation list (using BibTeX), and briefly summarized in one or two

paragraphs. A piled paper is in queue waiting to be filed. As the figure shows, the pile

is in good health. In late Spring 2005, a concerted effort was made to “kill the pile”. It

briefly dipped below 150 papers, but the literature is too large—and the battle continues.

132

Daily pointsRunning average

Paper

sre

ad

per

day

(log

scale

)

Oct 2005Jul 2005Apr 2005Jan 2005Oct 2004

8

4

2

1

Figure B.3: Running average of papers read per day.

Figure B.3 shows the running average of papers read per day, and Figure B.4 shows

a histogram of the filed papers’ publication year. One unknown is the true papers-of-

interest count. A simple model might be: 20 papers per year from 1920–1960; 50 papers

per year from 1960–1980; and 100 papers per year from 1980 to present. A histogram of

this projected goal in relation to the current progress is shown in Figure B.5. According

to the model, 4300 papers are of interest, of which roughly 3700 have yet to be filed. Say

350 papers are read per year (which, according to Figure B.3, isn’t entirely unreasonable).

Of these 350 papers, assume that 100 are current-year, leaving the remaining 250 papers

to be from the past. It would therefore take 3700/250 = 14.8 years to “kill the pile”.

133

Paper

s

20001990198019701960195019401930

70

60

50

40

30

20

10

Figure B.4: Year histogram.

Desired?

Current

Paper

s

202020001980196019401920

100

80

60

40

20

Figure B.5: Projected year histogram?

Appendix C

Sample Code

The simulations were performed using GNU Octave [188] and the figures were gen-

erated with Gnuplot [189]. In this appendix sample code is provided.

C.1 GNU Octave Code

Below is GNU Octave code used to obtain the results for the Channel Cf, MMSE

curve in Figure 6.10. The code can easily be adapted to obtain other results, as outlined

below.

% GNU Octave code for M=4, N=64, 2pih=1, Channel Cf result.

% Written by: Steve Thompson

% ------- Simulation parameters ------------------------------------

% for a good time, max min sqrt

shortrun=0; % equals 0 or 1

if shortrun % (use for speed/testing)

Trans_max=1e5; % max bits sent per SNR

Trans_min=2e4; % min bits sent per SNR

Error_min=2e1; % min errors per SNR

else % long run (use for accuracy/final result)

Trans_max=100e6; % max bits sent per SNR

Trans_min=1e6; % min bits sent per SNR

Error_min=2e5; % min errors per SNR

end

targetBER=1e-5; % target BER

SNRmax=50; % max SNR (dB)

134

135

io=1; % index offset

A=1; % signal amplitude

M=4; % modulation order

modh=1.0/(2*pi); % modulation index

N=64; % number of subcarriers

TB=128e-6; % block time

J=8; % oversampling factor

Fsa=J*N/TB; % sampling rate

Tsa=1/Fsa; % sampling period

Tg=10e-6; % guard time

TF=Tg+TB; % frame time

Ng=Tg*Fsa; % samples per guard interval

NB=TB*Fsa; % samples per symbol

NF=TF*Fsa; % samples per frame

ip=[Ng:NF-1]+io; % processing indices

Ndft=512; % DFT size (for equalizer)

taumax=9e-6; % maximum delay spread of channel (sec)

Nc=taumax*Fsa; % number of channel taps

Nr=Nc+NF-1; % number of received samples

L=8; % blocks/channel realization (vectorize)

%% Bit and symbol mappings (depends on modulation order)

if M==2

SymMap=[-1;1]; % data symbol mapping

BitMap=[0; 1]; % bit mapping

end

if M==4

SymMap=[-3;-1;1;3]; % data symbol mapping

BitMap=[... % bit mapping

0 0; 0 1; 1 1; 1 0];

end

if M==8

SymMap=[-7:2:7]’; % data symbol mapping

BitMap=[... % bit mapping

0 0 0; 0 0 1; 0 1 1; 0 1 0; 1 1 0; 1 1 1; 1 0 1; 1 0 0];

end

if M==16

SymMap=(-15:2:15)’; % data symbol mapping

BitMap=[... % bit mapping

0 0 0 0; 0 0 0 1; 0 0 1 1; 0 0 1 0; 0 1 1 0; 0 1 1 1; ...

0 1 0 1; 0 1 0 0; 1 1 0 0; 1 1 0 1; 1 1 1 1; 1 1 1 0; ...

1 0 1 0; 1 0 1 1; 1 0 0 1; 1 0 0 0];

end

varI=sum(SymMap.^2)/M; % variance of data

136

CN=sqrt(2/(N*varI)); % normalizing constant

%% Subcarrier matrix

t=0:Tsa:(TB-Tsa); % time vector

W=zeros(NB,N); % initialize unitary matrix

for k=1:N/2 % W is a set of orth. sines and cosines

W(:,k)=cos(2*pi*k*t/TB)’;

end

for k=(N/2+1):N

W(:,k)=sin(2*pi*(k-N/2)*t/TB)’;

end

%% Design FIR filter: improves performance of phase demodulator

%% See Proakis’s DSP text for design details

Mf=11; % filter length

n1=0:(Mf-1); % filter sample index

d=(Mf-1)/2; % delay

n2=(d+1):(d+NB); % desired, delayed indices

fc=0.2; % normalized cutoff frequency (cyc/samp)

wc=2*pi*fc; % normalized cutoff frequency (rad/samp)

h1=zeros(1,Mf); % initialize

for i=1:Mf % compute coefficients

if n1(i)==((Mf-1)/2)

h1(i)=wc/pi;

else

h1(i)=sin(wc*(n1(i)-(Mf-1)/2))/(pi*(n1(i)-(Mf-1)/2));

end

end

w1=0.54-0.46*cos(2*pi*n1/(Mf-1)); % Hamming window

hf=h1.*w1; % windowed filter coefficients

%% Channel delay power spectral density (exponential)

t=[0:Nc-1]’*Tsa; % time vector

p=1/tauRms*exp(-t/2e-6); % delay PDS

% ------- Simulation -----------------------------------------------

BER=0; % initialize BER vector

EbN0_dB=0; % initialize SNR vector

dx=2.5; % SNR step size

iSNR=1; % SNR counter

go=1; % initialize loop

while go % run until max SNR condition

Error_num=0; Trans_num=0; % initialize

while Trans_num<=Trans_min | ...

(Error_num<=Error_min & Trans_num<=Trans_max)

137

%% Generate L blocks

in=ceil(M*rand(N,L)); % random symbol index

I=SymMap(in); % data symbols

m=CN*W*I; % OFDM message signal

theta0=2*pi*rand(1,L)-pi; % memory terms (assume uniform)

phi=zeros(NF,L); % initialize CE-OFDM phase signal

for i=1:L % cyclic prefix

phi(:,i)=[2*pi*modh*m(NB-Ng+1:NB,i)+theta0(i);...

2*pi*modh*m(:,i)+theta0(i)];

end

s=A*exp(j*phi); % CE-OFDM signal

%% Determine noise power

Es=sum(sum(abs(s).^2))*Tsa; % signal energy

Eb=Es/(L*N*log2(M)); % bit energy

EbN0=10^(EbN0_dB(iSNR)/10); % SNR

N0=Eb./EbN0; % noise spectral height

%% Channel

tmp=sqrt(1/2)*(randn(Nc,1)+j*randn(Nc,1)); % Gaussian vector

Ch=sqrt(p/sum(p)).*tmp; % channel (normalize average power)

%% Received signal plus noise (to be processed by FDE)

rp=zeros(NB,L); % initialize

for i=1:L

tmp1=(conv(Ch,s(:,i))).’; % received samples

tmp1=tmp1(ip); % discard cyclic prefix

tmp2=sqrt(1/2)*(randn(NB,1)+j*randn(NB,1)); % complex Gaussian

noise=sqrt(N0*Fsa)*tmp2; % Gaussian noise

rp(:,i)=tmp1+noise; % received samples plus noise

end

%% Frequency-domain equalizer

H=fft(Ch,Ndft); % channel gains

C=conj(H)./(abs(H).^2+EbN0^(-1)); % correction term (MMSE)

X=fft(rp,Ndft); % to frequency domain

hatS=X.*(C*ones(1,L)); % equalize

x=ifft(hatS,Ndft); % to time domain

%% Filter signal

hats=zeros(NB,L); % initialize

for i=1:L

tmp=(conv(hf,x(:,i))).’; % filtered signal

hats(:,i)=tmp(n2); % filtered signal, desired indices

138

end

%% Demodulate and detect

hatphi=unwrap(angle(hats)); % phase demodulate

Ihat=W’*hatphi/((2*pi*modh*CN)*NB*1/2); % matched-filter output

inHat=min(round((Ihat+(M-1))/2)+io,M); % index estimate, (<=M)

inHat=max(inHat,1); % (>=1)

Errors=sum(sum(BitMap(in,:)~=BitMap(inHat,:))); % bit errors

Error_num=Error_num+Errors; % cumulative bit errors

Trans_num=Trans_num+L*N*log2(M); % cumulative bits

%% Display (optional)

if rem(Trans_num,10*L*N*log2(M))==0 % print-frequency

clc

printf([’MMSE, fading ChC, EQ, M=%d, 2pih=%1.1f, J=%d, ’...

’fc=%1.1f, EbN0=%2.1f, Trans_num=%d, ’...

’Error_num=%d, BER=%1.1e’], M, 2*pi*modh, J, fc,...

EbN0_dB(end), Trans_num, Error_num, Error_num/Trans_num)

end

end % end this SNR

BER(iSNR)=Error_num/Trans_num; % bit error rate for current SNR

%% Test for max SNR condition

if BER(iSNR)<targetBER | EbN0_dB(iSNR)>=SNRmax

go=0;

else % keep going

iSNR=iSNR+1;

EbN0_dB(iSNR)=EbN0_dB(iSNR-1)+dx;

end

end % end simulation

%% Plot

semilogy(EbN0_dB,BER)

%% Save

tmp=[EbN0_dB’ BER’];

save -ascii data tmp

To get other results, the above code is used with different values of M , 2πh, equalizersettings, and/or channel definitions. The ZF equalizer is simulated by changing theequalizer to

C=1./H; % correction term (ZF)

The other fading channels are generated by changing the code that defines the channel.For Channel Af:

139

%% Channel delay power spectral density (two-path)

tau=[0 5e-6]; % path delays

power_dB=[0 -10]; % path power (dB)

power=10.^(power_dB/10); % path power

for n=1:length(tau)

i=tau(n)*Fs; % path index

p(i+io,1)=power(n); % delay PSD

end

p=[p; zeros(Nc-length(p),1)]; % zero-pad

For Channel Bf:

%% Channel delay power spectral density (two-path)

tau=[0 5e-6]; % path delays

power_dB=[0 -3]; % path power (dB)

power=10.^(power_dB/10); % path power

for n=1:length(tau)

i=tau(n)*Fs; % path index

p(i+io,1)=power(n); % delay PSD

end

p=[p; zeros(Nc-length(p),1)]; % zero-pad

For Channel Df:

%% Channel delay power spectral density (uniform)

tau=[0:Nc-1]’*Ts; % discrete propagation delays

p=ones(size(t)); % delay PSD

Additionally, the above template can be used for conventional OFDM with some

minor alterations.

C.2 Gnuplot Code

The majority of the figures in this thesis were generated with Gnuplot. Below is

sample code which generates Figure 6.10.

# Tell Gnuplot what kind of plot to generate and give it

# some parameters.

set term pslatex monochrome dashed rotate 8

set format "$%g$"

set logscale y 10

set format y "$10^{%T}$"

140

set ticscale 0.5

set border 31 linewidth 0.5

set grid

set size 1.0,1.4

set key width -23.5 height 1 box lw 0.1 41.4,1.5e-1

set output "p_ber"

# Define line styles.

set style line 1 lt 1 lw 1 pt 9 ps 1.0

set style line 11 lt 3 lw 1 pt 9 ps 1.0

set style line 2 lt 1 lw 1 pt 6 ps 1.0

set style line 22 lt 3 lw 1 pt 6 ps 1.0

set style line 3 lt 1 lw 1 pt 7 ps 1.0

set style line 33 lt 3 lw 1 pt 7 ps 1.0

set style line 4 lt 1 lw 1 pt 8 ps 1.0

set style line 44 lt 3 lw 1 pt 8 ps 1.0

set style line 5 lt 1 lw 3

set style line 6 lt 5 lw 3

set style line 7 lt 5 lw 1

# Define labels.

set xlabel ’[t]{Average signal-to-noise ratio per bit,\

$\mathcal{E}_\text{b}/N_0$ (dB)}’

set ylabel ’Bit error rate’

# Now, plot. (The data files are in a make-believe

# directory called ‘results’

plot [5:44][1e-4:2e-1]\

"results/MMSE/ChA" t ’MMSE: Channel A’ w lp ls 1,\

"results/MMSE/ChB" t ’B’ w lp ls 2,\

"results/MMSE/ChC" t ’C’ w lp ls 3,\

"results/MMSE/ChD" t ’D’ w lp ls 4,\

"results/ZF/ChA/" t ’ZF: Channel A’ w lp ls 11,\

"results/ZF/ChB/" t ’B’ w lp ls 22,\

"results/ZF/ChC/" t ’C’ w lp ls 33,\

"results/ZF/ChD/" t ’D’ w lp ls 44,\

"results/flat" t ’Rayleigh, $\mathcal{L}=1$’ w l ls 5,\

"results/AWGN" t ’AWGN’ w l ls 6,\

"results/approx" t ’AWGN approx \eqref{eqn:approx}’ w l ls 7

Abbreviations

A/D analog-to-digital converterAM/AM amplitude/amplitude conversion of power amplifierAM/PM amplitude/phase conversion of power amplifierAWGN additive white Gaussian noiseb bitBER bit error rateCCDF complementary cumulative distribution functionCE constant envelopeCE-OFDM constant envelope OFDMCNR carrier-to-noise ratioCP cyclic prefixCPM continuous phase modulationdB decibels, 10 log10(·)D/A digital-to-analog converterDAB digital audio broadcastingDC direct currentDFE decision feedback equalizerDFT discrete Fourier transformDSL digital subscriber lineDVB digital video broadcastingETSI European Telecommunications Standards InstituteFDE frequency-domain equalizerFFT fast Fourier transformFIR finite impulse responseFOBP fractional out-of-band powerHz Hertz (1 cycle/s)IBO input power backoffIBOC in-band on-channelICI intercarrier interferenceIDFT inverse discrete Fourier transformIEEE Institute of Electrical and Electronic EngineersIFFT inverse fast Fourier transformISI intersymbol interferenceJTRS Joint Tactical Radio System

141

142

kHz kilohertz (1 thousand cycles/s)LAN local area networkLMMSE linear minimum mean-squared errorLMS least-mean-squareLOS line-of-signalM -PSK M -ary phase-shift keyingM -PAM M -ary pulse-amplitude modulationM -QAM M -ary quadrature-amplitude modulationMAN metropolitan area networkMb/s megabits per second (1 million b/s)Msamp megasample (1 million samples)MHz megahertz (1 million cycles/s)ML maximum-likelihoodOFDM orthogonal frequency division multiplexingP/S parallel-to-serial conversionPA power amplifierPAM pulse-amplitude modulationPAPR peak-to-average power ratioPLC power line communicationPSK phase-shift keyingQAM quadrature-amplitude modulationQPSK quadrature phase-shift keyingRLS recursive least-squareRMS root-mean-squares secondS/P serial-to-parallel conversionsamp sampleSC-FDE single carrier frequency-domain equalizerSER symbol error rateSDR software defined radioSNR signal-to-noise ratioSSPA solid-state power amplifierSTTR small business technology transferSVD singular value decompositionTWTA traveling-wave tube amplifierUWB ultra-widebandW WattsWSSUS wide-sense stationary uncorrelated scatteringµs microsecond (1/1,000,000 s)

Symbols

Set Theory

∈ is an element of/∈ is not an element of[·] closed interval[·) open interval{xn}Nn=1 set of elements x1, x2, . . . , xN

Operators and Miscellaneous Symbols

arg(·) argumentcos(·) cosineDFT{·} discrete Fourier transforme 2.71828182845905. . .

e(·) exponential functionexp(·) exponential functionE{·} expected valueF{·}(f) Fourier transformI0(·) 0th-order modified Bessel function of the first kindIDFT{·} inverse discrete Fourier transform={·} imaginary partj

√−1

Ji(·) ith-order Bessel function of the first kindmax maximummin minimumlim limitln(·) natural loglogx(·) log base xLP{·} lowpass componentP (·) probabilityQ(·) Gaussian Q-function<{·} real partsin(·) sinesinc(·) sinc functionvar{·} variance

143

144

x(t) x as a function of tx[i] discrete-time samples of x at the ith indexδ(·) delta functionπ 3.14159265358979. . .∞ infinity∫ ba (·)dx definite integral∫

(·)dx indefinite integral∏Nn=1 multiple product

∑Nn=1 multiple sum

n! factorialx→ a x approaches ax ∗ y x convolved with y| · | absolute value(·)∗ complex conjugated·e ceiling functionb·c floor function= equal≡ equal by definition6= not equal≈ approximately equal≤ less than or equal to≥ greater than or equal to< strictly less than> strictly greater than� much less than� much greater than

Power Amplifier

Amax maximum input levelAsat input saturation levelg0 gainG(·) AM/AM conversionsp sharpness parameter for the SSPA modelαφ, βφ AM/PM parameters for the TWTA modelηA efficiency of Class-A power amplifierK backoff ratioΦ(·) AM/PM conversions

145

Channel2σ2

0 scatter component power of frequency-nonselective channelal complex-valued gain of the lth pathBC coherence bandwidth

B(1)ττ average delay

B(2)ττ delay spread

C channel capacityh(τ, t) time-variant channel impulse responseh(τ) time-invariant channel impulse responseh[i] samples of the channel impulse responseH[k] discrete Fourier transform of h[i]KR Rice factorL number of discrete pathsrττ (v

′) frequency correlation functionSττ (τ) delay power spectral densityv′ frequency separation variable∆τl propagation delay difference between τl and τl−1, that is, ∆τl = τl − τl−1

ρ line-of-sight component power of frequency-nonselective channelσ2al

average power of the lth pathτ continuous propagation delayτl discrete propagation delay of the lth pathτmax maximum propagation delay

Signal

A signal amplitudeAb(k) the value of the kth subcarrier at the beginning of the block intervalAe(k) the value of the kth subcarrier at the end of the block intervalAmax clip levelBbpf bandwidth of bandpass filterBn noise bandwidthBrms root-mean-square bandwidthBs effective bandwidth of CE-OFDM signalC[k] frequency-domain equalizer termsCN normalizing constantd2m,n squared Euclidean distance between mth and nth signal

d2m,n(K) squared Euclidean distance between mth and nth signal as a function of the

phase constantd2min minimum squared Euclidean distanceD total number of data symbol differencesEb energy per bitEb/N0 signal-to-noise ratio per bitEq subcarrier energyEx energy of signal xf frequency variable (cycles/s)

146

f ′ normalized frequency variable (cycles/samp)fc carrier (or center) frequency (cycles/s)fsa sampling rate (samp/s)FOBP(f) fractional out-of-band power

ˆFOBP(f) estimated fractional out-of-band powerg(t) pulse shapeh modulation indexI data symbol

I estimated data symbolJ oversampling factorkb bits per symbolK phase signal constant, K = 2πhCNKd2min

number of neighboring signal points having minimum squared Euclideandistance d2

min

Lfir filter lengthm(t) message signalM modulation order of data symbol constellationn(t) lowpass complex-valued zero mean additive Gaussian noisen[i] samples of n(t)nbp(t) bandpass representation of n(t) [bandpass Gaussian noise]nc(t) in-phase component of nbp(t)ns(t) quadrature component of nbp(t)nw(t) white Gaussian noiseN number of subcarriersN0/2 spectral height of additive white Gaussian noiseNc number of channel samplesNB number of block samplesNg number of guard samplespγ(x) probability density function of signal-to-noise ratio per bitpξ(x) probability density function of ξ(t) samplesPx average power of signal xPAPRx the peak-to-average power ratio of signal xqk(t) kth subcarrierr(t) lowpass equivalent representation of received signalrbp(t) bandpass representation of r(t)R rate, b/sR/B spectral efficiency, b/s/Hzs(t) lowpass equivalent representation of transmitted signals[i] samples of s(t)sbp(t) bandpass representation of s(t)sc(t) in-phase component of sbp(t)ss(t) quadrature component of sbp(t)S(f) frequency domain representation of s(t)S[k] discrete Fourier transform of s[i]t time variable

147

TB block periodTg guard periodTs symbol periodTsa sampling periodW effective bandwidth of OFDM signal, W = N/TB

∆m,n(k) data symbol difference between mth and nth signal at the kth subcarrierγ signal-to-noise ratio per bit (used interchangeably with Eb/N0)γ average signal-to-noise ratio per bitγclip clipping ratioεfo normalized carrier frequency offsetηt transmission efficiencyθi memory term during ith CE-OFDM block intervalξ(t) noise at the output of phase demodulatorσ2I data symbol varianceσ2n variance of noise samples, n[i]σ2φ phase signal variance

ρm,n correlation between mth and nth signalρm,n(K) correlation between mth and nth signal as a function of the phase constantρmax maximum correlation among signalsφ(t) phase signalφn(t) noise autocorrelation functionΦAb(f) Abramson spectrum

ΦAb(f) estimated Abramson spectrumΦx(f) power density spectrum of signal x.

Φx(f) estimated power density spectrum of signal x.

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[13] F. Amoroso, “Pulse and Spectrum Manipulation in the Minimum (Frequency)Shift Keying (MSK) Format,” IEEE Trans. Commun., vol. com-24, no. 3, pp.381–384, Mar. 1976.

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Production Notes

This thesis was typeset using the LATEX document preparation system [348]. The

bibliography was managed using BibTeX (with help from bibtool). All numerical work,

including the computer simulations, was done with GNU Octave [188]. The block dia-

grams were drawn using Xfig [597] and all of the other figures were generated with Gnu-

plot [189] (using the pslatex driver). The source files were backed up and synchronized

among multiple computers using rsync. The LATEX output was converted to PostScript

using dvips; the PostScript was converted to PDF (portable document format) using

Ghostscript. The size of the PDF output is 1.9 megabytes.

The work was done at UCSD on a Dell Precision 370 workstation running the Debian

GNU/Linux operating system [131]. The X11 window system provided the graphical

user interface; the window manager used was IceWM. Typically the work was conducted

across several rxvt terminal emulators—arranged across multiple workspaces—running

bash. All work was done using the Vim (Vi improved) text editor [523]. PDF output

was viewed using xpdf. The work was also done at various locations throughout the San

Diego area on a Dell Inspiron 4000 laptop computer running the same software.

This thesis were printed on a Hewlett Packard LaserJet 1300n printer.

In terms of compilation time, this thesis takes roughly 10 s to compile on the work-

station which has a 3 gigahertz microprocessor and 1 gigabyte of memory.

194