Constant Envelope OFDM Phase Modulation - Dr. James R. Zeidler

UNIVERSITY OF CALIFORNIA, SAN DIEGOConstant Envelope OFDM Phase ModulationA dissertation submitted in partial satisfaction of therequirements for the degreeDoctor of PhilosophyinElectrical Engineering (Communications Theory and Systems)bySteve C. ThompsonCommittee in charge:Professor James R. Zeidler, ChairProfessor John G. Proakis, Co-ChairProfessor Robert R. BitmeadProfessor William S. HodgkissProfessor Laurence B. Milstein2005

CopyrightSteve C. Thompson, 2005All rights reserved.

The dissertation of Steve C. Thompson is approved,and it is acceptable in quality and formfor publication on microfilm:Co-ChairChairUniversity of California, San Diego2005iii

“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, 1939iv

TABLE OF CONTENTSSignature Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iiivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xiixiiiVita and Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAbstract of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 An Introduction to OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.1 ISI-Free Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.2 A Multicarrier Modulation . . . . . . . . . . . . . . . . . . . . . . 51.1.3 Discrete-Time Signal Processing . . . . . . . . . . . . . . . . . . . 81.2 Problems with OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Constant Envelope Waveforms . . . . . . . . . . . . . . . . . . . . . . . . 121.4 Constant Envelope OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.5 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1 More OFDM Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.1 The Cyclic Prefix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.2 Discrete-Time Model . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.3 Block Modulation with FDE . . . . . . . . . . . . . . . . . . . . . 202.1.4 System Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 PAPR Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 Power Amplifier Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4 Effects of Nonlinear Power Amplification . . . . . . . . . . . . . . . . . . . 30v

2.4.1 Spectral Leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4.2 Performance Degradation . . . . . . . . . . . . . . . . . . . . . . . 322.4.3 System Range and PA Efficiency . . . . . . . . . . . . . . . . . . . 352.5 PAPR Mitigation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 373 Constant Envelope OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.1 Signal Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 Performance of Constant Envelope OFDM in AWGN . . . . . . . . . . . . . . . 584.1 The Phase Demodulator Receiver . . . . . . . . . . . . . . . . . . . . . . . 594.1.1 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 624.1.2 Effect of Channel Phase Offset . . . . . . . . . . . . . . . . . . . . 654.1.3 Carrier-to-Noise Ratio and Thresholding Effects . . . . . . . . . . 664.1.4 FIR Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 The Optimum Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.2.1 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 724.2.2 Asymptotic Properties . . . . . . . . . . . . . . . . . . . . . . . . . 784.3 Phase Demodulator Receiver versus Optimum . . . . . . . . . . . . . . . . 784.4 Spectral Efficiency versus Performance . . . . . . . . . . . . . . . . . . . . 804.5 CE-OFDM versus OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . 835 Performance of CE-OFDM in Frequency-Nonselective Fading Channels . . . . . 866 Performance of CE-OFDM in Frequency-Selective Channels . . . . . . . . . . . 946.1 MMSE versus ZF Equalization . . . . . . . . . . . . . . . . . . . . . . . . 966.1.1 Channel Description . . . . . . . . . . . . . . . . . . . . . . . . . . 966.1.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.1.3 Discussion and Observations . . . . . . . . . . . . . . . . . . . . . 1036.2 Performance Over Frequency-Selective Fading Channels . . . . . . . . . . 1086.2.1 Channel Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.2.2 Simulation Procedure and Preliminary Discussion . . . . . . . . . 1126.2.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 114vi

7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121A Generating Real-Valued OFDM Signals with the Discrete Fourier Transform . . 124A.1 Signal Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124A.2 Spectral Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126B More on the OFDM Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . 128C Sample Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134C.1 GNU Octave Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134C.2 Gnuplot Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148Production Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194vii

LIST OF FIGURES1.1 Representation of a wireless channel with multipath. . . . . . . . . . . . . 21.2 A wireless channel in time and frequency. . . . . . . . . . . . . . . . . . . 21.3 Intersymbol interference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 OFDM with cyclic prefix (CP). . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Subcarrier and overall spectrum. (N = 16; |I 0,k | = 1, for all k) . . . . . . 71.6 OFDM converts wideband channel to N narrowband frequency bins. . . . 81.7 Frequency offset causes ICI. (ɛ fo = 0.25) . . . . . . . . . . . . . . . . . . . 91.8 A typical OFDM signal (N = 16). The PAPR is 9.5 dB. . . . . . . . . . . 101.9 Power amplifier transfer function. . . . . . . . . . . . . . . . . . . . . . . . 111.10 Comparison of OFDM and CE-OFDM signals. . . . . . . . . . . . . . . . 132.1 Sampling instances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Circular convolution with channel and the inverse channel. . . . . . . . . . 212.3 Block modulation with cyclic prefix and FDE. . . . . . . . . . . . . . . . . 212.4 OFDM is a special case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5 OFDM system diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6 Complementary cumulative distribution functions. (N = 64) . . . . . . . 252.7 PAPR CCDF lower bound (2.31) for N = 2 k , k = 5, 6, . . . , 10. . . . . . . . 262.8 AM/AM (solid) and AM/PM (dash) conversions (SSPA=thick, TWTA=thin)for various backoff ratios K. . . . . . . . . . . . . . . . . . . . . . . . . . . 292.9 Fractional out-of-band power of OFDM with ideal PA and with TWTAmodel at various input power backoff. (N = 64, IBO in dB) . . . . . . . . 312.10 Spectral growth versus IBO. (N = 64) . . . . . . . . . . . . . . . . . . . . 312.11 Performance of QPSK/OFDM with nonlinear power amplifier with variousinput power backoff levels. (N = 64) . . . . . . . . . . . . . . . . . . . . . 332.12 Performance of M-PSK/OFDM with SSPA. (N = 64) . . . . . . . . . . . 342.13 The potential range of system is reduced with input backoff; the range isreduced further from nonlinear amplifier distortion. . . . . . . . . . . . . . 362.14 Power amplifier efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.15 Block diagram. The system is evaluated with and without PAPR reduction. 382.16 Unclipped OFDM signal (9.25 dB PAPR). The rings have radius A maxwhich correspond to various clipping ratios γ clip (dB). . . . . . . . . . . . 39viii

2.17 PAPR CCDF of clipped OFDM signal for various γ clip (dB). [N = 64] . . 402.18 PAPR of clipped signal as a function of the clipping ratio. (N = 64) . . . 402.19 A comparison of the total degradation curves of clipped and unclippedM-PSK/OFDM systems. (N = 64) . . . . . . . . . . . . . . . . . . . . . . 413.1 The CE-OFDM waveform mapping. . . . . . . . . . . . . . . . . . . . . . 433.2 Instantaneous signal power. . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3 Basic concept of CE-OFDM. . . . . . . . . . . . . . . . . . . . . . . . . . 453.4 Phase discontinuities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.5 Continuous phase CE-OFDM signal samples, over L blocks, on the complexplane. (2πh = 0.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.6 Estimated fractional out-of-band power. (N = 64) . . . . . . . . . . . . . 523.7 Double-sided bandwidth as a function of modulation index. (N = 64) . . 533.8 Power density spectrum. (N = 64, 2πh = 0.6) . . . . . . . . . . . . . . . . 543.9 Fractional out-of-band power. (N = 64, 2πh = 0.6) . . . . . . . . . . . . . 553.10 CE-OFDM versus OFDM. (N = 64) . . . . . . . . . . . . . . . . . . . . . 563.11 CE-OFDM versus OFDM with nonlinear PA. (N = 64) . . . . . . . . . . 574.1 Phase demodulator receiver. . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2 Bandpass to baseband conversion. . . . . . . . . . . . . . . . . . . . . . . 604.3 Discrete-time phase demodulator. . . . . . . . . . . . . . . . . . . . . . . . 624.4 Performance with and without phase offsets. System 1 (S1) has phaseoffsets {(θ i + φ 0 ) ∈ [0, 2π)}, and System 2 (S2) doesn’t (θ i + φ 0 = 0).[M = 2, N = 64, J = 8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.5 Threshold effect at low CNR. (M = 8, N = 64, J = 8, 2πh = 0.5) . . . . . 684.6 Threshold effect at low CNR, various 2πh. (M = 8, N = 64, J = 8) . . . 684.7 Performance for various filter parameters L fir , f cut /W .(M = 2, N = 64, J = 8, 2πh = 0.5 and E b /N 0 = 10 dB) . . . . . . . . . . 694.8 Magnitude response of various Hamming FIR filters. . . . . . . . . . . . . 704.9 CE-OFDM performance with and without FIR filter.(M = 2, N = 64, J = 8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.10 The optimum receiver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.11 Correlation functions ρ m,n (K). . . . . . . . . . . . . . . . . . . . . . . . . 764.12 CE-OFDM optimum receiver performance. (M = 2, N = 8) . . . . . . . . 77ix

4.13 All unique ρ m,n (K) for M = 2, N = 4 DCT modulation. . . . . . . . . . . 784.14 Phase demodulator receiver versus optimum. (N = 64) . . . . . . . . . . . 794.15 Noise samples PDF versus Gaussian PDF. (E b /N 0 = 30 dB) . . . . . . . . 804.16 Performance of M-PAM CE-OFDM. (N = 64, †=leftmost curve, ‡=rightmostcurve) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.17 Spectral efficiency versus performance. . . . . . . . . . . . . . . . . . . . . 824.18 A comparison of CE-OFDM and conventional OFDM. (M = 2, N = 64) . 855.1 Performance of CE-OFDM in flat fading channels. (N = 64) . . . . . . . 885.2 A simplified two-region model. (M = 8, N = 64, 2πh = 0.6) . . . . . . . . 905.3 A (n + 1)-region model. (M = 8, N = 64, 2πh = 0.6) . . . . . . . . . . . . 915.4 Performance of CE-OFDM in flat fading channels. (Circle=Rayleigh;square=Rice, K = 3 dB; triangle=Rice, K = 10 dB. Solid line=Semianalyticalcurve, (5.15); points=simulation. N = 64) . . . . . . . . . . . . 925.5 Comparison of semi-analytical technique (5.15) with (5.10) and (5.11).(M = 4, N = 64, 2πh = 1.2) . . . . . . . . . . . . . . . . . . . . . . . . . . 936.1 CE-OFDM system with frequency-selective channel. . . . . . . . . . . . . 966.2 Channel D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.3 Channel A results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.4 Channel B results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.5 Channel C results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.6 Channel D results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.7 Channel E results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.8 Channel F results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.9 Fundamental characteristic functions and quantities [(6.21)–(6.25)] of thefour channel models considered. . . . . . . . . . . . . . . . . . . . . . . . . 1136.10 Performance results. (Multipath results are labeled with circle and trianglepoints; the Rayleigh, L = 1 result is that of the frequency-nonselectivechannel model. M = 4, N = 64, 2πh = 1.0) . . . . . . . . . . . . . . . . . 1156.11 Single path versus multipath. (M = 4, N = 64, Channel C f , MMSE) . . . 1196.12 CE-OFDM versus QPSK/OFDM. (SSPA model, Channel C f , N = 64,MMSE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120B.1 “OFDM” search on IEEE Xplore [222]. . . . . . . . . . . . . . . . . . . . . 130B.2 Papers, filed and piled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131x

B.3 Running average of papers read per day. . . . . . . . . . . . . . . . . . . . 132B.4 Year histogram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133B.5 Projected year histogram? . . . . . . . . . . . . . . . . . . . . . . . . . . . 133xi

LIST OF TABLES6.1 Channel samples of frequency-selective channels. . . . . . . . . . . . . . . 976.2 Channel model parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.3 Data symbol contribution per tone for m n (t), n =1, 2, and 3. . . . . . . . 118xii

ACKNOWLEDGEMENTSI want to first thank my advisors, Professors Zeidler and Proakis, for giving me thechance to do this work, for the encouragement, and for the guidance. I want to thankProfessor Milstein for the many helpful technical conversations and for his many suggestions.Thanks to Professors Bitmead and Hodgkiss for taking the time to participateas committee members. Also, thanks to Professor Proakis for carefully proofreading thedraft manuscripts of this thesis.Thanks to UCSD’s Center for Wireless Communications for providing a good environmentfor conducting research; thanks to its industrial partners for the financialsupport.Thanks to my wife, Shannon, for the emotional and caloric support. Thanks toChaney 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 thecamaraderie. Special thanks to Ahsen Ahmed for helpful collaboration over the pastcouple years. Thanks to my family. Also, thanks to Karol Previte for her support earlyin my graduate student existence.Thanks to my teachers: Professors Duman, Masry, Milstein, Pheanis, and Wolf, toname only a few.Finally, I would like to thank the countless developers, documentation writers, bugreporters, 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, ofwhich I was the primary researcher and author: S. C. Thompson, J. G. Proakis, andJ. R. Zeidler, “Constant Envelope Binary OFDM Phase Modulation,” in Proc. IEEEMilcom, 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: SpectralContainment, 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 inOFDM Systems,” in Proc. IEEE Globecom, St. Louis, Dec. 2005.xiii

VITADecember 22, 1976Born, Mesa, Arizona1997–1998 Associate EngineerInter-Tel, Chandler, ArizonaSummer 1998Summer InternshipLos Alamos National LaboratoryLos Alamos, New Mexico1999 BSc in Electrical EngineeringArizona State University, Tempe, ArizonaSummer 2001Summer InternshipSPAWAR Systems Center, San Diego, California2001 MSc in Electrical EngineeringUniversity of California at San Diego, La Jolla, California2001–2005 Research AssistantCenter for Wireless CommunicationsUniversity of California at San Diego, La Jolla, CaliforniaSummer 2004Summer InternshipSPAWAR Systems Center, San Diego, California2005 PhD in Electrical EngineeringUniversity of California at San Diego, La Jolla, CaliforniaPUBLICATIONSS. 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 Performance,”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 Clippingfor 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 DISSERTATIONConstant Envelope OFDM Phase ModulationbySteve C. ThompsonDoctor of Philosophy in Electrical Engineering (Communications Theory andSystems)University of California San Diego, 2005Professor James R. Zeidler, ChairProfessor John G. Proakis, Co-ChairOrthogonal frequency division multiplexing (OFDM) is a popular modulation techniquefor wireless digital communications. It provides a relatively straightforward way to accommodatehigh data rate links over harsh wireless channels characterized by severemultipath fading. OFDM has two primary drawbacks, however. The first is a high sensitivityto 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 waveformhas high amplitude fluctuations, a drawback known as the peak-to-average power ratio(PAPR) problem. The high PAPR makes OFDM sensitive to nonlinear distortion causedby the transmitter’s power amplifier (PA). Without sufficient power backoff, the systemsuffers from spectral broadening, intermodulation distortion, and, consequently, performancedegradation. High levels of backoff reduce the efficiency of the PA. For mobilebattery-powered devices this is a particularly detrimental problem due to limited powerresources.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 PAPRwaveform by way of angle modulation. The constant envelope signal can be efficientlyamplified with nonlinear power amplifiers thus achieving greater power efficiency. Inxvi

this thesis, the fundamental aspects of the CE-OFDM modulation are studied, includingthe signal spectrum, the signal space, optimum performance, and the performance ofa practical phase demodulator receiver. Performance is evaluated over a wide range ofmultipath fading channel models. It is shown that CE-OFDM outperforms conventionalOFDM when taking into account the effects of the power amplifier.This work was done at UCSD’s Center for Wireless Communication, under the “MobileOFDM Communications” project (CoRe research grant 00-10071).xvii

Chapter 1IntroductionHumans have always found ways to communicate, over space and over time. Fromthe 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 derivedfrom their natural environment, to share information. A particularly good natural resourcefor communication is electricity for its speed and ability to be controlled withdevices like capacitors, microprocessors, electronic memory storage and batteries. Communicationwas profoundly enhanced with Morse’s telegraph (1837), Bell’s telephone(1876), Edison’s phonograph (1887), and Marconi’s radio (1896). From these early inventions,communications technology has advanced with global telephone networks, satellitecommunications, and magnetic storage systems; and with the rise of the internet anddigital computers, digital communications—the transfer of bits (1’s and 0’s) from onepoint 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 challengein 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 varying[421, 427]. Figure 1.1 illustrates a link with four reflecting paths between points Aand B. These reflections are caused by physical objects in the environment. Due to therelative mobility between the points and the possibility that the reflecting objects aremobile, the channel changes with time.1

2point A¢£Propagation pathspoint B¡¡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 pathhas its own associated delay and power. The first path arrives at the receiver 0.5 µs afterthe signal is transmitted; the last path arrives with a 14 µs delay. The Fourier transformof the profile yields the frequency-domain representation shown in Figure 1.2(b). Thechannel is viewed over a 2 MHz range centered at the center frequency f c . Notice thatthe channel power fluctuates by 30 dB (a factor of 1000) over the frequency range. Thedispersion in the time domain leads to frequency-selectivity in the frequency domain.15Path power0.10.010246 8Time (µs)101214Channel power (dB)0-5-10-15-20-25-30−1−0.5 0 0.5Frequency, f − f c (MHz)1(a) Time domain.(b) Frequency domain.Figure 1.2: A wireless channel in time and frequency.In general, a digital communication system maps bits to k b -bit data symbols. In aconventional single carrier system, the symbols are then transmitted serially. The signalwaveform of such a system iss(t) = ∑ iI i g(t − iT s ), (1.1)where t is the time variable, {I i } are the data symbols, T s is the symbol period, and g(t)is a transmit pulse shape. For time-dispersive channels, such as the 4-path example in

3Figure 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 isrepresented by its time-variant impulse response h(τ, t), where τ is a propagation delayvariable. 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 ofthe time-dispersive channel is shown to smear symbol 1 into symbol 2, therefore creatingintersymbol interference.s(t)r(t)Transmitter Channel Receivers(t)... symbol 1 symbol 2 ...0 T s 2T st|h(τ, t)|τ0r(t)ISIT s2T stFigure 1.3: Intersymbol interference.The severity of the ISI depends on the symbol period relative to the channel’s maximumpropagation delay, τ max . Consider transmitting the signal in (1.1) over the 2MHz channel in Figure 1.2. The signal bandwidth is roughly proportional to the symbolrate 1/T s Hz. Therefore making s(t) a 2 MHz signal, T s = (2 × 10 6 ) −1 = 0.5 µs.Since the maximum propagation delay of the channel is τ max = 14 µs, the ISI spansτ max /T s = (14 µs) / (0.5 µs) = 28 symbols. (For comparison, the ISI in Figure 1.3 spansless than one symbol.) Such severe ISI must be corrected at the receiver in order toprovide reliable communication.The traditional approach to combating intersymbol interference is with time-domainequalizers [421]. There are many types, ranging in complexity and in effectiveness.The optimum maximum-likelihood (ML) receiver is the most effective but is typicallyimpractical due to its high complexity, which grows exponentially with the ISI length.Linear equalizers are much simpler, having a complexity which grows roughly linearlywith ISI length, but perform much worse than the optimum receiver. Nonlinear decisionfeedback equalizers (DFEs) have similar complexity as the linear type and have betterperformance.

4All of these techniques require knowledge of the channel, which is estimated bytransmitting a training sequence which is known at the receiver. Then by comparingthe received signal to what was transmitted, an estimate of h(τ, t) is made. There arevarious algorithms available for the estimation process, each having its own complexity,convergence rate, and stability. The least-mean-square (LMS) algorithm is the moststable and the least complex, but suffers from a slow convergence rate. The recursiveleast-square (RLS or Kalman) algorithm, on the other hand, converges quickly, but hashigher complexity and can be unstable.For scenarios like the example above with an ISI spanning 28 symbols, conventionalequalization becomes difficult. Training times become long and convergence of the channelestimator is problematic, especially for time-varying channels. In the example, 2×10 6symbols/s are transmitted. Using a QPSK (quadrature phase-shift keying) signal constellation,which maps k b = 2 bits per symbol, the bit rate is 4 Mb/s. Such a bit rate isdesired in current wireless systems, and in many cases demand for many tens of Mb/sis common.1.1 An Introduction to OFDMTo meet the demanding data rate requirements, alternative techniques have beenconsidered. One approach, orthogonal frequency division multiplexing, has become exceedinglypopular. OFDM has been implemented in wireline applications such as digitalsubscriber lines (DSL) [95], in wireless broadcast applications such as digital audio andvideo 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 theETSI HYPERLAN/2 standards [552]. OFDM is being developed for ultra-wideband(UWB) systems; cellular systems; wireless metropolitan area networks (MANs), underthe IEEE 802.16 (WiMax) standard; and for other wireline systems such as power linecommunication (PLC) [119, 160, 264, 604].1.1.1 ISI-Free OperationOFDM’s main appeal is that it supports high data rate links without requiringconventional equalization techniques. Instead of transmitting symbols serially, OFDM

5sends N symbols as a block. The OFDM block period, T B , is thus N times longer thanthe symbol period. Continuing the example above, and choosing N = 300, the blockperiod is T B = NT s = 300 × 0.5 µs = 150 µs, which is more than 10 times the durationof the channel’s impulse response. ISI is avoided by inserting a guard interval betweensuccessive blocks during which a cyclic prefix is transmitted. The interval duration, T g ,is designed such that T g ≥ τ max so that the channel is absorbed in the guard intervaland the OFDM block is uncorrupted. This is illustrated in the figure below. Selecting aguard interval T g = 15 µs for the channel in Figure 1.2 results in a transmission efficiencyη t = T B /(T B + T g ) = 150/165 ≈ 0.91. Therefore, with a small reduction in efficiency, ISIis eliminated.s(t)T gCP|h(τ, t)|T BOFDM blocktτr(t)ISI-free blocktFigure 1.4: OFDM with cyclic prefix (CP).1.1.2 A Multicarrier ModulationThe OFDM signal can be expressed as 1s(t) = ∑ [ N−1]∑I i,k e j2πf ktg(t − iT B ). (1.3)i k=0The pulse shape, g(t), is typically rectangular:⎧⎪⎨ 1, 0 ≤ t < T B ,g(t) =⎪⎩ 0, otherwise.(1.4)Notice that the N data symbols {I i,k } N−1k=0are transmitted during the ith block. Theset of complex sinusoids {exp (j2πf k t)} N−1k=0are referred to as subcarriers. The center1 For simplicity, the guard interval is excluded from the signal definition in (1.3). The guard intervaland cyclic prefix is discussed in Chapter 2.

6frequency of the kth subcarrier is f k = k/T B and the subcarrier spacing, 1/T B Hz, makesthe subcarriers orthogonal over the block interval, expressed mathematically as∫1 TB (e ) j2πf ∗ ( )k 1te j2πf k 2tdt = 1 ∫ TBe j2π(f k 2−f k1 )t dtT B 0T⎧ B 0⎪⎨ 1, k 1 = k 2 ,=⎪⎩ 0, k 1 ≠ k 2 ,(1.5)where (·) ∗ represents the complex conjugate operation. The subcarrier orthogonality canalso be viewed in the frequency domain. Consider the 0th OFDM block:s(t) =N−1∑k=0I 0,k e j2πf kt , 0 ≤ t < T B . (1.6)The frequency-domain representation isN−1∑[(S(f) = F {s(t)} (f) = T B e −j2πfT B/2I 0,k sinc f − k ) ]T B , (1.7)T Bwhere F{·}(f) is the Fourier transform and⎧⎪⎨ 1, x = 0,sinc(x) =(1.8)⎪⎩ sin πxπx , otherwise.Figure 1.5 plots |S(f)/T B | for N = 16 subcarriers and data symbols with normalizedamplitudes. The individual subcarrier spectra are also plotted. Notice that at the kthsubcarrier frequency, k/T B , the kth subcarrier has a peak and all the other subcarriershave zero-crossings.k=0Therefore, the subcarriers, while tightly packed (which improvesspectral efficiency), are non-interfering (i.e. orthogonal).Figure 1.5 also demonstrates that OFDM is a multicarrier modulation, as opposedto a single carrier modulation like the signal in (1.1). In general, a transmitted bandpasssignal is [421, p. 151]where f c is the carrier frequency. For single carrier,{x(t) = R s(t)e j2πfct} , (1.9)x sc (t) = ∑ i|I i | cos [2πf c t + arg(I i )] g(t − iT s ); (1.10)while for multicarrier,x mc (t) = ∑ { N−1 ∑[ (|I i,k | cos 2π f c + k )t + arg(I i,k )] } g(t − iT B ). (1.11)TiBk=0

71.2SubcarrierOverallSpectrum magnitude, |S(f)/TB|10.80.60.40.20-2024681012141618Normalized frequency, fT BFigure 1.5: Subcarrier and overall spectrum. (N = 16; |I 0,k | = 1, for all k)For single carrier each symbol occupies the entire signal bandwidth, while for multicarrierthe 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 multiplecarriers each centered at (f c + k/T B ) Hz, k = 0, 1, . . . , N − 1.By properly designing the subcarrier spacing, each frequency bin is made frequencynonselective.The wideband frequency-selective channel is converted into N contiguousnarrowband 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 modulationcan be optimized for the channel by sending more bits in frequency bins with high gainand 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 inwireless systems with high mobility.Frequency selectivity is the frequency-domain dual of intersymbol interference. Transmittingthe single carrier signal over the 2 MHz channel results in a frequency-selectiveresponse. For OFDM, the overall channel is frequency-selective but for each bin the chan-

85Channel power (dB)0-5-10-15-20-25-30−0.9−0.85Frequency, f − f c (MHz)Frequency bins−0.8Figure 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 afrequency-domain interpretation of how OFDM avoids intersymbol interference.1.1.3 Discrete-Time Signal ProcessingThus far, two of OFDM’s primary advantages have been discussed: the eliminationof ISI and the ability to optimize the modulation with bit loading. The third appeal ofOFDM is that the modulation and demodulation is done in the discrete-time domain withthe 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=0I 0,k e j2πki/N , i = 0, 1, . . . N − 1, (1.12)which is the inverse discrete Fourier transform (IDFT) of the symbol vector I 0 =[I 0,0 , I 0,1 , . . . , I 0,N−1 ]. Therefore, s(t) is generated at the transmitter with an IDFT followedby a digital-to-analog (D/A) converter. The frequency-domain symbols {I 0,k } N−1k=0can be expressed asI 0,k = 1 NN−1∑i=0y[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.

9The IDFT/DFT is performed efficiently with IFFT/FFT algorithms. Doing so ismuch simpler than performing the modulation/demodulation in the continuous-timedomain with N orthogonally tuned oscillators. Moreover, the signal processing can beperformed in software, making OFDM suitable for software defined radios (SDRs) [185].1.2 Problems with OFDMOFDM has two primary drawbacks. The first is sensitivity to imperfect frequencysynchronization which is common for mobile applications. This sensitivity arises fromthe close subcarrier spacing. Figure 1.5 shows that the subcarriers are properly orthogonalat f = k/T B , k = 0, 1, . . . , N − 1. However, if the frequency synthesizer at thereceiver is misaligned by, say, ɛ fo /T B Hz, where −0.5 < ɛ fo < 0.5, the subcarriers arenot 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 + ɛ fo )/T BHz rather than at the ideal k/T B Hz, the N − 1 neighboring subcarriers interfere withthe demodulation of the kth subcarrier. The intercarrier interference causes ISI—andpotentially high irreducible error floors.The second problem with OFDM is that the signal has large amplitude fluctuationscaused by the summation of the complex sinusoids. The real and imaginary part of theSpectrum magnitude, |S(f)/TB|10.20.04k − 1 k k + ɛ fok + 1Normalized frequency, fT BFigure 1.7: Frequency offset causes ICI. (ɛ fo = 0.25)

10OFDM signal isandR {s(t)} =I {s(t)} =N−1∑k=0N−1∑k=0R {I 0,k } cos (2πkt/T B ) − I {I 0,k } sin (2πkt/T B ) , (1.14)R {I 0,k } sin (2πkt/T B ) + I {I 0,k } cos (2πkt/T B ) , (1.15)respectively. Figure 1.8(a) shows the real and imaginary parts of an example OFDMsignal with N = 16 subcarriers. Also plotted are the individually modulated sinusoids.Notice that each sinusoids has a constant amplitude, but when summing the sinusoidsthe resulting OFDM signal fluctuates over a large range. The instantaneous signal power,|s(t)| 2 = R 2 {s(t)} + I 2 {s(t)}, is plotted in Figure 1.8(b). The ratio between the peakpower and the average power is 144/16 = 9 (or in decibels, 10 log 10 9 ≈ 9.5 dB).121601014086SubcarriersR{s(t)}I{s(t)}120|s(t)| 2Peak powerAverage powerSignal amplitude420-2Power magnitude1008060-440-6-82000.20.40.60.81000.20.40.60.81Normalized time, t/T BNormalized time, t/T B(a) Signal amplitude.(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 witha large linear range capable of accommodating the signal.Otherwise, the circuitry

11distorts the waveform nonlinearly, and nonlinear distortion results in a loss of subcarrierorthogonality which degrades performance.One such nonlinear device is the transmitter’s power amplifier (PA) which is responsiblefor the system’s operational range [424]. Ideally the output of the PA is equal tothe input times a gain factor. In reality the PA has a limited linear region, beyond whichit saturates to a maximum output level. Figure 1.9 shows a representative input/outputcurve, known as the AM/AM conversion. In the linear region the curve matches theideal, but as the input power increases the PA saturates. The most efficient operatingpoint is at the PA’s saturation point, but for signals with large PAPR the operatingpoint must shift to the left keeping the amplification linear. The average input poweris reduced and consequently this technique is called input power backoff (IBO). To keepthe peak power of the input signal less than or equal to the saturation input level, theIBO must be at least equal to the PAPR. Thus the required IBO for the OFDM signalin Figure 1.8 is 9.5 dB. At this backoff the efficiency of a Class A power amplifier isless than 6%. Such an efficiency is detrimental to mobile battery-powered devices whichhave limited power resources. Moreover, the operational range of the system is reducedby a factor of nine 2 .Max outputOptimumOutput powerLinear regionActualSaturation regionBackoffInput powerAM/AM curveOperating pointsIdeal AM/AMFigure 1.9: Power amplifier transfer function.2 IBO of 9.5 dB corresponds to 10 9.5/10 ≈ 9 times less signal power transmitted in channel; the(theoretical) efficiency of a Class A amplifier is 0.5/(10 9.5/10 ) ≈ 0.06 [374].

12Nonlinearities in the transmitter also cause the generation of new frequencies inthe transmitted signal. This intermodulation distortion causes interference among thesubcarriers, and a broadening of the overall signal spectrum. The later causes interferencebetween neighboring systems, an effect known as adjacent channel interference.1.3 Constant Envelope WaveformsConstant envelope (CE) waveforms are appealing since the optimum operating pointin Figure 1.9 is attainable. The baseband CE signal representation iss(t) = Ae jφ(t) , (1.16)where A is the signal amplitude and φ(t) is the information bearing phase signal. Theadvantage of the CE waveform is that the instantaneous power is constant: |s(t)| 2 = A 2 .Consequently, the PAPR is 0 dB and the required backoff is 0 dB. The PA can thereforeoperate at the optimum (saturation) point, maximizing average transmit power (goodfor range) and maximizing PA efficiency (good for battery life). Also, since the linearityrequirement is reduced, nonlinear PAs can be used which are generally more efficientand less expensive than linear PAs. For example, the maximum theoretical efficiency ofa linear Class A power amplifier is 50%, while for a nonlinear Class E PA the maximumtheoretical efficiency is 100% [424].Constant envelope signals are thus ideal in terms of the practical considerations of thepower amplifier. The question is how to embed digital information into φ(t) providinggood performance, spectral economy, and high data rates over the wireless channel.Notice that the single carrier signal in (1.1) is constant envelope when |I i | = 1 and g(t)is rectangular. This type of modulation, however, has large spectral sidelobes whichcause adjacent channel interference. In practice, non-rectangular pulse shapes are usedwhich result in a non-CE signal.Continuous phase modulation (CPM) is a class of signaling that has very low sidelobepower while maintaining the constant envelope property [14,421]. CPM uses memory tosmooth φ(t). The memory, however, increases the complexity of the receiver, which is akey disadvantage of CPM. Also CPM systems have difficulty operating over frequencyselectivechannels [118].

131.4 Constant Envelope OFDMConstant envelope OFDM (CE-OFDM) combines OFDM and constant envelope signaling.The high peak-to-average power ratio OFDM signal is transformed into a CEwaveform. The CE-OFDM signal takes the form of (1.16) where the phase signal is anOFDM waveform.signal:φ(t) = R {s OFDM (t)} =For example, the phase signal can be the real part of the OFDMN−1∑k=0where s OFDM (t) is the signal in (1.6).R {I 0,k } cos (2πkt/T B ) − I {I 0,k } sin (2πkt/T B ) , (1.17)Figure 1.10 compares a conventional OFDMbandpass signal with a bandpass CE-OFDM signal. Both are derived from the samebaseband OFDM message signal.✒OFDM bandpassOFDM message❘CE-OFDM bandpassFigure 1.10: Comparison of OFDM and CE-OFDM signals.The motivation for CE-OFDM is to eliminate the PAPR problem of the conventionalOFDM system. Certainly, this is accomplished since the CE-OFDM signal has the 0 dBPAPR 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 isin conventional OFDM? This thesis aims to answering these questions by analyzing thevarious aspects of the CE-OFDM modulation.

141.5 Thesis OverviewIn Chapter 2 the basics of OFDM is further studied. The effect of the nonlinearpower amplification on OFDM is evaluated. In Chapter 3 the CE-OFDM modulationformat is defined and the spectral properties are studied. The performance aspects ofCE-OFDM in the presence of additive noise are analyzed in Chapter 4. Performanceanalysis is extended to frequency-nonselective fading channels in Chapter 5, and multipathfrequency-selective fading channels in Chapter 6.

Chapter 2OFDMIn 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 Section2.1.1, the cyclic prefix is studied. In Section 2.1.2, the processing of the discrete-timesamples is described, and the equivalence of linear channel convolution and circularchannel convolution is explained. In light of this property, OFDM is considered a specialcase of the more general block modulation with cyclic prefix scheme, as discussed inSection 2.1.3. Finally, in Section 2.1.4 the main functional blocks of the OFDM systemare described.The PAPR statistics are analyzed in Section 2.2 and power amplifier models usedto evaluated system performance are described in Section 2.3. Then in Section 2.4 theeffect of nonlinear power amplification on OFDM systems is studied in terms of spectralleakage (Section 2.4.1), performance degradation (Section 2.4.2), and system range andefficiency (Section 2.4.3). Lastly, the various PAPR mitigation techniques found in theresearch literature are categorized in Section 2.5, and a technique called signal clippingis evaluated in terms of its effectiveness to improve system performance.15

162.1 More OFDM Basics2.1.1 The Cyclic PrefixIn Section 1.1.1 it is claimed that the use of the guard interval results in ISI-freeoperation. This is true so long as a cyclic prefix is transmitted during the interval. Thisis demonstrated below and it is shown that ISI results if anything but the cyclic prefixis transmitted.During the OFDM block interval, the waveform iss(t) =N−1∑k=0I k e j2πf kt , 0 ≤ t < T B , (2.1)where {I k } N−1k=0 are the data symbols, {exp(j2πf kt)} N−1k=0are the subcarriers, N is thetotal number of subcarriers, f k = k/T B is the center frequency of the kth subcarrier andT B is the block period. The guard interval is defined during −T g ≤ t < 0, where T g isthe guard period. To transmit a cyclic prefix, the last T g s of the block is transmittedduring the guard interval:s(t) =N−1∑k=0I k e j2πf k(t+T B ) =N−1∑k=0I k e j2πf kt e j2πk =N−1∑k=0I k e j2πf kt , (2.2)−T g ≤ t < 0. Notice that the above simplification is made due to the periodicity of thesignal. Thus the OFDM signal having a guard interval with cyclic prefix is simplyThe received signal iss(t) =N−1∑k=0I k e j2πf kt , −T g ≤ t < T B . (2.3)r(t) = s(t) ∗ h(τ) + n(t)==∫ ∞−∞∫ τmax0h(τ)s(t − τ)dτ + n(t)h(τ)s(t − τ)dτ + n(t),(2.4)where h(τ) is the time-invariant channel impulse response 1 and n(t) is additive noise.The bounds of integration are simplified since the channel is assumed causal [h(τ) = 01 In (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).

17for τ < 0] and to have a maximum propagation delay τ max [h(τ) = 0 for τ > τ max ]. Thereceived signal during the guard interval, which has interference from the previous block(see Figure 1.4), is ignored and r(t) during 0 ≤ t < T B is processed. An estimate of thek 0 th data symbol is made by correlating r(t) with the k 0 th subcarrier:which expands toÎ k0 = 1T B∫ TB0[ ]r(t) e j2πf ∗k 0t dt, (2.5)whereBut sinceÎ k0 = 1T B∫ TB0= 1T B∫ TB0= 1T BN−1r(t)e −j2πf k t 0 dt[ ∫ ]τmax N−1∑h(τ) I k e j2πfk(t−τ) dτ e −j2πf k t 0 dt + N k00k=0∑∫ τmaxI k h(τ)e −j2πfkτ dτk=00N k0 = 1T B∫ TB∫1 TBe j2πt(f k−f k0 ) dt =T B 00∫ TB0e j2πt(f k−f k0 ) dt + N k0 ,(2.6)n(t)e −j2πf k 0 t dt. (2.7)⎧⎪⎨ 1, k = k 0 ,⎪⎩ 0, k ≠ k 0 ,(2.8)(2.6) simplifies toÎ k0 = I k0 H[k 0 ] + N k0 , (2.9)whereH[k 0 ] =∫ τmaxwhich is the Fourier transform of h(τ) evaluated at f = f k0 .This shows that the N received data symbols {Îk} N−1k=0symbols {I k } N−1k=00h(τ)e −j2πf k 0 τ dτ, (2.10)are equal to the transmittedscaled by the complex-valued channel gains {H[k]}N−1k=0. ISI is avoidedsince the kth symbol isn’t impacted by the N − 1 other symbols. Therefore, using theguard interval with cyclic prefix provides ISI-free operation.Now it is shown that by transmitting a signal other than the cyclic prefix during theguard interval causes ISI. Suppose that the transmitted signal is⎧⎪⎨ b(t), −T g ≤ t < 0,s(t) =∑ ⎪⎩ N−1k=0 I ke j2πfkt , 0 ≤ t < T B ,(2.11)

18where b(t) ≠ ∑ N−1k=0 I ke j2πf kt . The estimate of the k 0 th data symbols isÎ k0 = 1T B∫ TB0= 1T B∫ TB0r(t)e −j2πf k 0 t dt[∫ τmax= A k0 + B k0 + N k0 .0]h(τ)s(t − τ)dτ e −j2πf k t 0 dt + N k0(2.12)The bounds of integration are separated into two segments, [0, T g ] and [T g , T B ]:andA k0 = 1T B∫ Tg0B k0 = 1T B∫ TBT g∫ τmax0∫ τmax0h(τ)s(t − τ)e −j2πf k 0 t dτdt, (2.13)h(τ)s(t − τ)e −j2πf k 0 t dτdt. (2.14)A k0 is a non-zero offset term which is a function of b(t). For the second term, t − τ > 0,thusB k0 = 1T B∫ TB= 1T gT BN−1[ ∫ ]τmax N−1∑h(τ) I k e j2πfk(t−τ) dτ e −j2πf k t 0 dt0k=0∑∫ τmaxI k h(τ)e −j2πfkτ dτk=00∫ TBT ge j2πt(f k−f k0 ) dt.(2.15)Due to the integration bounds for t, the orthogonality condition in (2.8) can’t be appliedto (2.15), and this results in ISI. The estimated data symbol is expressed aswhere C 1 = (T B − T g )/T B , and the interference terms isÎ k0 = I k0 H k0 C 1 + N k0 + ICI, (2.16)ICI = A k0 + 1 ∑∫ TBH[k] I k e j2πt(f k−f k0 ) dt. (2.17)T Bk≠kT g 0The interference is denoted as ICI, intercarrier interference, since the subcarriers are nolonger orthogonal and interfere with one another. This phenomenon was described inSection 1.2 in the context of imperfect frequency synchronization. Therefore, ICI canmanifest itself in more than one way, and when it does the data symbols interfere withone another resulting in ISI.In [358], cyclic prefixed OFDM is compared to zero-padded OFDM [b(t) = 0]. Thezero-padding causes ISI, but has the advantage of being able to recover data symbols

19located at channel zeros. This is in contrast with cyclic prefixed OFDM since, as shownin (2.9), a channel zeros at the kth subcarrier, that is, H[k] = 0, results in an estimateddata symbol that consists entirely of noise. The zero-padded system avoids this problemat the cost of increased receiver complexity due to equalization requirements.2.1.2 Discrete-Time ModelIt is convenient to describe OFDM by a discrete-time model. Consider sampling s(t),h(τ) and r(t) at the sampling rate f sa = JN/T B samp/s, where J ≥ 1 is the oversamplingfactor. The sampling instances are shown in the figure below.−T gSignal sampling· · · · · ·0T Bt−N gT sa −T sa T sa (N B − 1)T sa−N gT sa ≥ −T gτChannel samplingτ max0· · ·T sa (N c − 1)T sa(N c − 1)T sa ≤ τ maxFigure 2.1: Sampling instances.The number of guard samples, N g , and channel samples, N c , are defined as⌊ ⌋TgN g ≡ ≤ T g, (2.18)T sa T saand⌊ ⌋τmaxN c ≡ + 1 ≤ τ max+ 1, (2.19)T sa T sawhere T sa = 1/f sa is the sampling period. The number of samples per block is N B = JN;and, by design, N g ≥ N c . The signal samples areand the channel samples ares[i] = s(t)| t=iTsa , i = −N g , . . . , 0, . . . , N B − 1, (2.20)h[i] = h(τ)| τ=iTsa , i = 0, . . . , N c − 1. (2.21)The received samples are expressed by the linear convolution sumr[i] =N∑c−1m=0h[m]s[i − m] + n[i], i = −N g , . . . , 0, . . . , N B − 1, (2.22)

20where {n[i]} are samples of the noise signal n(t). The guard interval samples are ignoredand the samplesare processed.r[i] =N∑c−1m=0h[m]s[i − m] + n[i], i = 0, . . . , N B − 1 (2.23)The linear convolution in (2.23) is equivalent to a circular convolution since, dueto the cyclic prefix, {s[i − m]} is periodic with period N B .The circular convolutioncan 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 asr[i] = IDFT {H[k]S[k]}N DFT ∑−1N DFT= 1k=0H[k]S[k]e j2πik/N DFT, i = 0, . . . , N B − 1,(2.24)where IDFT{·} represents the inverse discrete Fourier transform;andS[k] =H[k] =N DFT∑−1i=0N DFT ∑−1i=0s[i]e −j2πik/N DFT, k = 0, . . . , N DFT − 1 (2.25)h[i]e −j2πik/N DFT, k = 0, . . . , N DFT − 1 (2.26)are the N DFT -point DFTs of the signal and channel samples, respectively. The DFTsize is, in general, N DFT ≥ N B . If N DFT > N B , the signal vector is zero-padded. SinceN DFT > N g , the channel samples are zero-padded: h[i] = 0 for i = N c , . . . , N DFT − 1.Figure 2.2 shows a block diagram representing the calculation of (2.24). The effect ofthe channel is simply a DFT followed by a multiplier bank (H[k]), which is then followedby an IDFT. Also shown is the inverse channel which is a DFT followed by a multiplierbank (1/H[k]) followed by an IDFT. Thus the transmit samples s[i] can be reconstructedby passing the receive samples r[i] through the inverse channel.2.1.3 Block Modulation with FDEThe inverse channel structure in Figure 2.2 corrects the distortion caused by thechannel in the frequency domain, and is therefore called a frequency-domain equalizer

21Channels[i]DFTH[k]IDFTr[i]Inverse channelr[i]DFT1H[k]IDFTs[i]Figure 2.2: Circular convolution with channel and the inverse channel.Frequency-domain equalizerData{I k }ModulatorChannelDFTMultiplierbankIDFTDemodulatorData{Î k }Figure 2.3: Block modulation with cyclic prefix and FDE.Frequency-domain equalizerData{I k }IDFTMultiplierChannel DFTbankIDFTDFTData{Îk}Data{I k }IDFTChannelDFTMultiplierbankData{Îk}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 circularconvolution. This is the case for OFDM, but isn’t unique to OFDM since any modulationcan use a cyclic prefix. This observation was first identified by Sari et al. [462] andsuggests a more general modulation approach: block modulation with cyclic prefix andfrequency-domain equalization. Figure 2.3 shows a simplified block diagram of sucha system. (The insertion of the cyclic prefix at the transmitter and removal at thereceiver is implied but not included in the diagram for simplicity.) For the special caseof OFDM, the modulation is a IDFT and the demodulation is a DFT as shown in Figure2.4. Notice that the DFT and IDFT cancel each other and the resulting diagram depictsthe conventional OFDM system.The multiplier bank at the output of the DFT is often referred to as a one-tapequalizer, one complex multiplication per frequency bin. This operation is required fordata 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 (associatedwith conventional single carrier modulation); rather, OFDM converts the problemto the frequency domain. Since Sari’s original paper, there has been a considerable numberof publications focused on the block modulation technique using conventional singlecarrier 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 DiagramThe block diagram in Figure 2.4 conceptually illustrates the OFDM system. Figure2.5 shows a more detailed description of OFDM’s functional blocks.The encoder adds redundancy to the bit stream for error control. The encoded bitsare then mapped to the data symbols I k . In general, the data symbols are complexnumbers which result from mapping the bits to points on the complex plane. Next, thesymbols are serial-to-parallel (S/P) converted and processed by the IDFT. The cyclicprefix 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 isamplified and transmitted.

23TransmitterBits01101Encoder11101MapperI kS/PIDFTAddCPP/Ss[i]D/As(t)PoweramplifierReceiverr(t)A/DRemoveCPr[i]S/PDFTEqualizeC[k]P/SÎ k 11001Detector Decoder Bits01101Figure 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 samplesare removed, the DFT is performed and each frequency bin is equalized by a complexmultiplication. The estimated data symbols, Î k , are processed by the detector whichoutputs a stream of estimated receive bits, and the decoder attempts to correct any biterrors that may have occurred.As discussed in Section 1.2, one of OFDM’s key drawbacks is the high peak-toaveragepower ratio. Nonlinearities in the power amplifier distort the transmitted signaland large input power backoff is required which results in low amplifier efficiency. In thenext sections the impact of the PA is studied. But first, the statistical properties of thePAPR are discussed.

242.2 PAPR StatisticsThe peak-to-average power ratio of the OFDM signal is best viewed statistically. Forany given block interval, the PAPR is a random quantity since it depends on the datasymbols {I k } N−1k=0. Assuming that they’re selected randomly from a set of M complexnumbers, there are M N unique symbol sequences, and thus M N unique OFDM waveformsper block.Of these waveforms, some have a high PAPR, while others have arelatively low PAPR. Therefore, it is desirable to understand the statistical distributionof this quantity.The OFDM signal iss(t) =N−1∑k=0I k e j2πf kt , 0 ≤ t < T B . (2.27)The signal during the guard interval is ignored since it has no impact on the PAPRdistribution. M-PSK data symbols are assumed, therefore |I k | = 1 for all k. Theaverage power of s(t) isP s = 1T B∫ TBThe peak-to-average power ratio is defined as0|s(t)| 2 dt = N. (2.28)PAPR s =max |s(t)|2/ P s . (2.29)t∈[0,T B )Notice that the absolute maximum signal power is N 2 , so the PAPR can be as high asN. However, the likelihood that all the subcarriers align in phase is extremely low. Forexample, as pointed out in [381], a N = 32 subcarrier system having 4-ary data symbolsand a block period of T B = 100 µs obtains the theoretical maximum PAPR once every3.7 million years. Thus it is more meaningful to describe the PAPR statistically ratherthan in absolute terms.Since the average signal power is a constant, the randomness of the PAPR depends onthe randomness of the instantaneous power |s(t)| 2 , and more specifically, the maximuminstantaneous power over 0 ≤ t < T B . For large N, the real and imaginary parts ofs(t) are accurately modeled as Gaussian random processes (due to the application ofthe central limit theorem [394, 421]). Consequently, the instantaneous signal power ischi-squared distributed with two degrees of freedom [421, p. 41], and the complementary

25cumulative distribution function (CCDF) of the normalized instantaneous signal poweris approximated as( |s(t)|2PP s)> x ≈ e −x . (2.30)A lower bound of the peak-to-average power ratio’s CCDF is [515]P (PAPR s > 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 providesa lower bound to the continuous-time signal since peaks can occur between samplingtimes.10 0SimulationApproximation (2.30)10 0SimulationLower bound (2.31)CCDF, P `|s(t)| 2 /Ps > x´10 −110 −210 −3CCDF, P (PAPRs > x)10 −110 −210 −30210 −4 (a) Instantaneous power.4 6x (dB)8104610 −4 (b) Peak-to-average power ratio.8 10x (dB)1214Figure 2.6: Complementary cumulative distribution functions. (N = 64)Figure 2.6(a) compares a simulated instantaneous power CCDF with the approximationin (2.30). This demonstrates the accuracy of the Gaussian approximation to thereal and imaginary part of s(t). Figure lower bound in (2.31). 2.6(b) compares PAPRsimulation results to the The bound is shown to be within 1 dB of the simulated resultfor lower values of x. The 0.0001 PAPR is shown to be at around 11.25 dB, and at this

26level the bound is tight. Notice that essentially all OFDM blocks have a PAPR greaterthan 6 dB, 10% have a PAPR greater than 8.5 dB, and 0.5% have a PAPR greater than10 dB.For the results in Figure 2.6, the number of subcarriers is N = 64 and QPSK datasymbols (4-ary PSK) are used, that is, I k ∈ {±1, ±j}. While the symbols constellationhas little impact on the PAPR statistics, the number of subcarriers does. Figure 2.7shows the lower bound (2.31) over a range N = 32 to N = 1024. Notice that the 0.001PAPR is 1 dB larger for N = 512 than for N = 32. For the N = 64 system, the PAPRis 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.10 0k 5 6 7 8 9 1010 −1CCDF, P (PAPRs > x)10 −210 −310 −4 45678x (dB)9101112Figure 2.7: PAPR CCDF lower bound (2.31) for N = 2 k , k = 5, 6, . . . , 10.2.3 Power Amplifier ModelsTo determine the impact of the PAPR on system performance, power amplifier modelsmust be defined. Two models commonly used in the research literature are thesolid-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 performanceevaluation.

27In general, modeling nonlinear power amplifiers is complicated (see [233, chap. 5]).A common simplification is to assume that the PA is a memoryless nonlinearity, andtherefore has a frequency-nonselective response. For example, if the PA input iss in (t) = A(t) exp[jφ(t)], (2.32)the output iss out (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 asG(A) =g 0 A[1 + (A/A sat ) 2p] , and Φ(A) = 0, (2.34)1/2pwhere g 0 is the amplifier gain, A sat is the input saturation level, and p controls theAM/AM sharpness of the saturation region. For this model the AM/PM conversion isassumed to be negligibly small.Though widely known as the Rapp model [426], (2.34) should be credited to theoriginal work by A. J. Cann, published a decade earlier in the IEEE literature [71].Cann’s formula is obtained with the simple manipulation:AG(A) =g 0 [1 + (A/A sat ) 2p] 1/2pA=g 0 [1 + (A/A sat ) 2p] 1/2p × [(A sat/A) 2p ] 1/2p[(A sat /A) 2p ] 1/2p(2.35)A sat=g 0 [1 + (A sat /A) 2p] 1/2p ,which is precisely the nonlinearity presented in Cann’s paper.Saleh’s TWTA model is expressed as [110]G(A) =g 0 A1 + (A/A sat ) 2 , and Φ(A) = α φA 21 + β φ A 2 . (2.36)Notice that the AM/PM conversion, determined by the constants α φ and β φ , is non-zero.The TWTA model is therefore more nonlinear than the SSPA model.

28To reduce nonlinear distortion in the amplified OFDM signal, input power backoff(IBO) is required. It is defined as [375]IBO = A2 satP in, (2.37)where P in = E{|s in (t)| 2 } = E{A 2 (t)} is the average power of the input signal. Equivalently,(2.37) can be written asP in = A2 satIBO ; (2.38)thus, given A sat and IBO, the input signal power can be scaled accordingly to satisfy(2.38).Assuming that the PAPR of the input signal is PAPR in , the peak power can bewritten aswhereP max = PAPR in · P in = PAPR inIBOA2 sat = A2 satK , (2.39)K =IBOPAPR in(2.40)is defined as the backoff ratio. Notice that for K > 1 the backoff is greater than the inputsignal’s PAPR; for K < 1 the backoff is less than the input PAPR. Now, the maximumvalue of the input, A max = max |A(t)|, can be written in terms of the backoff ratio andthe input saturation level:A max = √ P max = A sat√K. (2.41)Figure 2.8 shows the AM/AM (solid lines) and AM/PM (dashed lines) conversionsfor the SSPA (thick lines) and TWTA (thin lines) models for various backoff ratios K.For the SSPA model, p = 2; for the TWTA model, α φ = π/12 and β φ = 1/4. The x-axisis normalized to the maximum input level A max , and the y-axis is normalized to themaximum output level g 0 A sat . For K = −10 dB the IBO is one-tenth the input signalPAPR, and thus the nonlinearity is severe. One the other hand, for K = 10 dB theIBO is ten times the input signal PAPR and the PA response is nearly linear. As statedabove, the non-zero AM/PM conversion of the TWTA model makes it more nonlinearthan 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 theK = 3 dB curves (the PAPR is a always greater than 3 dB) and are more nonlinear

2911Normalized output value, G(A)/g0Asat0.50−0.5Normalized output value, G(A)/g0Asat0.50−0.5−1−1−0.5 0 0.5Normalized input value, A/A max1−1−1−0.5 0 0.5Normalized input value, A/A max1(a) K = 10 dB(b) K = 3 dB11Normalized output value, G(A)/g0Asat0.50−0.5Normalized output value, G(A)/g0Asat0.50−0.5−1−1−0.5 0 0.5Normalized input value, A/A max1−1−1−0.5 0 0.5Normalized input value, A/A max1(c) K = −3 dB(d) K = −10 dBFigure 2.8: AM/AM (solid) and AM/PM (dash) conversions (SSPA=thick,TWTA=thin) for various backoff ratios K.

30than 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 distortionon the transmitted signal. Also, the degree of distortion for a given OFDM block israndom (given a fixed IBO) since the PAPR for a given block is random.2.4 Effects of Nonlinear Power AmplificationPower amplifier nonlinearities cause spectral leakage and performance degradationto 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. Inthis section these various issues are studied.2.4.1 Spectral LeakageThe 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 quicklyestimated. The result is used to calculate estimated fractional out-of-band power curves,defined as∫ fFOBP(f) ˆ0=ˆΦ s (x)dx0.5 ˆP, f > 0, (2.42)swhere ˆΦ s (f) is the estimated power density spectrum of the signal and ˆP s = ∫ ∞−∞ ˆΦ s (f)dfis the signal power. Figure 2.9 shows the curves for an N = 64 subcarrier OFDM signalamplified 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 thatat 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 theundistorted OFDM signal is f = 1.07W . For sufficient backoff, the bandwidth of thenonlinearly amplified signal is the same. However, for IBO < 6 dB, the bandwidth isshown 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 samefor the two amplifier models.

3110 0OFDM amplified with: TWTA PAideal PAFractional out-of-band power10 −110 −2IBO024610 −300.250.50.751Normalized frequency, f/WFigure 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)1.251.52.01.8OFDM amplified with: TWTA PASSPA PAideal PA99.5% bandwidth, f/W1.61.41.21.00.80246Input power backoff, IBO (dB)810Figure 2.10: Spectral growth versus IBO. (N = 64)

322.4.2 Performance DegradationNext, the performance degradation caused by nonlinear amplification is considered.The OFDM signal is passed through a PA and then it is corrupted by additive whiteGaussian noise (AWGN). The received signal is thus,r(t) = s out (t) + n(t), (2.43)where s out (t) is the output of the PA from (2.33) and n(t) is a complex-valued Gaussianadditive noise signal having a power density spectrum [421, p. 158]⎧⎪⎨ N 0 , |f| ≤ B n /2,Φ n (f) =(2.44)⎪⎩ 0, |f| > B n /2,where B n is the bandwidth of the noise signal. The noise spectrum is assumed to beconstant 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) thenpassed to the detector which makes the final decision. This decision is based on themaximum-likelihood (ML) criterion assuming a linear PA; that is, the nearest point inthe symbol constellation [421, pp. 242–247].The performance is estimated by way of computer simulation. Following the conventiondescribed in Section 2.1.2, the discrete-time signal representation is used and thesampling rate f sa = JN/T B where J ≥ 1 is the oversampling factor. For the AWGNchannel, h(τ) = δ(τ), and therefore no guard interval is used. The noise samples {n[i]}are Gaussian distributed and assumed independent:⎧⎪⎨ σn, 2 i 1 = i 2 ,E {n[i 1 ]n[i 2 ]} =(2.45)⎪⎩ 0, i 1 ≠ i 2 .The autocorrelation function of n(t) [the inverse Fourier transform of (2.44)] has zerocrossingsat τ = 1/B n . Thus assuming B n = f s , (2.45) is satisfied and the noise samplevariance is σn 2 = f sa N 0 .Figure 2.11 shows bit error rate (BER) performance as a function of E b /N 0 , whereE b =∫ TB0|s out (t)| 2 dtNumber of bits per block(2.46)

3310 −1Nonlinear PAIdeal PA10 −1Nonlinear PAIdeal PA10 −2Bit error rate10 −3Bit error rate10 −210 −310 −410 −410 −5 02 4 6 8 10 12Signal-to-noise ratio per bit, E b /N 0 (dB)1410 −5 05 10 15 20 25Signal-to-noise ratio per bit, E b /N 0 (dB)30(a) SSPA model, IBO = 0, 1, 2, 3, 4, 6, 8 dB;0 = worst, 8 = best.(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 E b /N 0 is referred to as the signal-to-noise ratio (SNR)per bit, or simply the SNR. QPSK data symbols are used, and the oversampling factoris J = 4. For the SSPA results in Figure 2.11(a), the IBO ranges from 0 to 8 dB. Atthe 0.0001 BER level, the IBO = 0 dB case suffers a 3 dB performance loss compared toideal AWGN performance, which is [421, pp. 268].(√ )BER = Q 2 E b, (2.47)N 0where Q(x) = ∫ ∞x e−y2 /2 dy/ √ 2π is the Gaussian Q-function. To avoid degradation, 8dB of backoff is required. The TWTA results in Figure 2.11(b) use IBO ranging from 0to 16 dB. Notice the irreducible error floors for IBO ≤ 7 dB. To avoid degradation, 16dB of backoff is required—8 dB more than for the SSPA case. The greater nonlinearityof the TWTA model is evident from the results in this figure.Figure 2.12 compares performance for higher-order PSK modulations. For M-PSK

341010 −1Ideal PASSPA: IBO = 3 dBIBO = 6 dB8M = 16Bit error rate10 −210 −3M = 8M = 16Total degradation (dB)64M = 2, 4M = 8Ideal PASSPA10 −4M = 2, 42Target BER = 0.001010 −5 (a) BER performance.5 10 15 20 25Signal-to-noise ratio per bit, E b /N 0 (dB)30002 4 6 8Input power backoff, IBO (dB)10(b) Total degradation.Figure 2.12: Performance of M-PSK/OFDM with SSPA. (N = 64)the data symbols areI k ∈ {exp(j2πm/M); m = 0, 1, . . . , M − 1}. (2.48)The number of bits per data symbols is log 2 M, therefore the bit energy isE b =∫ TB0|s out (t)| 2 dtN log 2 M . (2.49)Higher-order constellations are used for increased spectral efficiency at the price of BERperformance 2 . In Figure 2.12(a) BER results for the SSPA model are shown. (Theresults for M = 2 and M = 4 are very similar so only M = 2 is plotted.) The higherordermodulations 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 theM = 2, 4 result at the same backoff shows only a 1 dB degradation. When increasingthe backoff to IBO = 6 dB, the error floor for M = 16 drops to 2 × 10 −5 and the 0.001BER is about 2 dB worse than AWGN. Using IBO = 6 dB for M = 8 results in 2 dBless degradation at the 0.001 bit error rate when compared to using IBO = 3 dB.2 This is the case for linear modulation formats. This isn’t necessarily the case for nonlinear modulationformats as discussed in Section 4.4.

35A more revealing way to view performance is in terms of total degradation, as shownin Figure 2.12(b). The total degradation is defined as [121]TD(IBO) = SNR PA (IBO) − SNR AWGN + IBO, [in dB] (2.50)where SNR AWGN is the required signal-to-noise ratio per bit to achieve a target biterror rate in AWGN; SNR PA (IBO) is the required SNR when taking into account thedistortion caused by the power amplifier at a given backoff. The “optimum” IBO, denoteas IBO opt , minimizes the total degradation, that is,TD(IBO opt ) = TD min =min TD(IBO). (2.51)IBO≥0 dBThe target BER for the curves in Figure 2.12(b) is 0.001. Clearly the modulation orderinfluences the degradation. The minimum TD for M = 16 is 7.7 dB at IBO opt = 6.5dB; for M = 8, TD min = 5 dB at IBO opt = 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 rangeand higher PA efficiency. The M = 2 and M = 4 examples are shown to have the lowestdegradation and are thus the more robust against nonlinear distortion.2.4.3 System Range and PA EfficiencyThe total degradation is directly related to the system’s operational range. Considera transmitter operating at maximum transmit power. The range is represented by theoutermost 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 degradationcaused by the PA further reduces range, as represented by the innermost circle. Thusthe 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, thetheoretical efficiency of a Class A power amplifier is used [374]:η A = 1 1× 100%, IBO ≥ 1. (2.52)2 IBOThe 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

36Potential rangePotential range w/ IBOActual rangeFigure 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 tradeoffsbetween PA efficiency, spectral containment, and performance/range. For example, theoptimum IBO in terms of total degradation for the 8-PSK SSPA example is IBO opt = 3dB [Figure 2.12(b)]: however, the bandwidth expansion is 42% (Figure 2.10) and thePA efficiency is η A = 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%. TheM = 2, 4 systems required minimal IBO for the SSPA, thus maximizing efficiency, butthe bandwidth expands by 87%.5045Class-A PA efficiency, ηA (%)403530252015105001234 5 6Input power backoff, IBO (dB)78910Figure 2.14: Power amplifier efficiency.

372.5 PAPR Mitigation TechniquesThere have been many schemes proposed in the research literature aimed at reducingthe impact of the PAPR problem. The goal of any scheme is to reduce the minimumtotal degradation (for increased range) and the IBO opt (for increased PA efficiency). Thevarious schemes can be placed in one the following three categories:1. transmitter enhancement techniques,2. receiver enhancement techniques, or3. signal transformation techniques.Transmitter enhancement techniques include PAPR reduction schemes and PA linearizationschemes. The PAPR reduction schemes can be further divided into distortionless andnon-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-distortionlessschemes include signal clipping [27,138,290,382], peak cancellation [330], and peak windowing[403].The PA linearization schemes attempt to predistort the OFDM signal such that theoverall response of the predistorter followed by the PA is linear—essentially equalizingthe amplifier. In [230], an LMS algorithm is applied for adaptive predistortion; in [395]a neural network learning technique is used. Parametric techniques, which design apredistorter based on a PA model, have been proposed. In [85, 122, 250, 567] nonlinearpolynomial 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 basedon transforming the OFDM signal prior to the PA, and applying the inverse transform atthe receiver prior to demodulation. This category includes constant envelope OFDM (asstudied 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.

38Signal ClippingThe remainder of this section focuses on the effectiveness of signal clipping, which hasbeen 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 interferencecaused 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 reductionscheme must be measured not only by PAPR reduction, but by the more meaningfulmeasures of TD min and IBO opt reduction. It is shown that clipping, while an effectivePAPR reduction scheme, does not reduce TD min nor does clipping reduce IBO opt foran OFDM system. This result brings into question the usefulness of non-distortionlessPAPR 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 onestudied in Section 2.4.2. Therefore, the earlier unclipped results serve as a performancebenchmark in which to compare the clipped results. The channel, as before, has animpulse response h(τ) = δ(τ).OFDMmodulators(t)PAPRreducingclipperoffs in (t)ons clip (t)PAs out(t)h(τ)n(t)r(t)OFDMdemodulatorFigure 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 isthe clipped OFDM signal:⎧⎪⎨ s(t), if |s(t)| ≤ A max ,s clip (t) =⎪⎩ A max e jψ(t) , if |s(t)| > A max ,(2.53)where ψ(t) = arg[s(t)]. Therefore, the magnitude of the clipped signal does not exceedA max and the phase of s(t) is preserved. (This has been called “polar clipping” in theliterature [276].) The clipping severity is measured by the clipping ratio, defined as [375]γ clip = A max√Ps. (2.54)

3920γ clip 4OFDM signalClip radius1020Imaginary axis0−10−20−20−100Real axis1020Figure 2.16: Unclipped OFDM signal (9.25 dB PAPR). The rings have radius A maxwhich correspond to various clipping ratios γ clip (dB).Figure 2.16 shows a typical OFDM signal on the complex plane. The dark rings haveradius A max which correspond to clipping ratios γ clip = 0, 2, and 4 dB.The PAPR of s clip (t) isPAPR clip =max |s clip(t)| 2t∈[0,T )1T B∫ TB0|s clip (t)| 2 dt . (2.55)Clipping’s effectiveness at reducing PAPR is shown in Figure 2.17. For clipping ratioγ clip = 5 dB, the peak-to-average power ratio of the clipped signal is PAPR clip ≤ 10dB; for γ clip = 4 dB, PAPR clip ≤ 8 dB, and so forth. The 0.0001 PAPR improvement,compared to the unclipped signal, is 1.2 dB for γ clip = 5 dB and by 3.2 dB for theγ clip = 4 dB.Figure 2.18 shows PAPR clip as a function of the clipping ratio. The PAPR ofthe unclipped signal is 13 dB 3 . Notice that for large γ clip , s clip (t) is unclipped, there-3 This figure is made by generating 2 × 10 4 consecutive OFDM blocks. The PAPR of the overall blockis 13 dB.

4010 010 −1ClippedUnclippedP (PAPRclip > x)10 −210 −3γ clip 3 4 510 −4 0246x (dB)81012Figure 2.17: PAPR CCDF of clipped OFDM signal for various γ clip (dB). [N = 64]1614Peak-to-average power ratio (dB)121086420PAPR sPAPR clipPAPR clip as γ clip → 0−2−4γ 2 clip−8−6−4−2 0 2Clipping ratio, γ clip (dB)46810Figure 2.18: PAPR of clipped signal as a function of the clipping ratio. (N = 64)

41fore PAPR clip = PAPR s . As γ clip → 0, the peak and average powers converge, thusPAPR clip → 0 dB. For the region 3 dB < γ clip < 6.5 dB, s clip (t) is clipped so thepeak power is A 2 max = γclip 2 P s. However, the clipping is mild so the average power isapproximately the same as s(t); therefore, PAPR clip ≈ γclip 2 P s/P s = γclip 2 .Clipping is clearly an effective technique at reducing the PAPR. The question is, doesthe PAPR reduction translate into reduced total degradation? Figure 2.19 compares thetotal degradation curves of the unclipped system [from Figure 2.12(b)] with the clippedsystem. Interestingly, the unclipped results are shown to provide a lower bound forthe clipped, reduced PAPR, system results. The clipper is shown to increase both theminimum total degradation and the optimum backoff. For example, using the clippingratio γ clip = 3 dB for the M = 8 case increases the TD min by 0.2 dB; using γ clip = 2 dBincreases TD min by 1.2 dB. For M = 16, the γ clip = 4 dB result is nearly identical to theunclipped result; γ clip = 3 dB increases the degradation by 1.2 dB, and the TD curveassociated with γ clip = 2 dB is beyond the viewing range of the figure. For M = 2, 4 thePAPR reducing clipping yields nearly identical results as the unclipped system.10Total degradation (dB)8642M = 2, 4M = 8M = 16Ideal PAUnclippedClipped: γ clip = 4 dB3 dB2 dB00246Input power backoff, IBO (dB)Figure 2.19: A comparison of the total degradation curves of clipped and unclippedM-PSK/OFDM systems. (N = 64)810Thus the effectiveness of a PAPR reduction scheme should be measured not onlyby its PAPR reducing capabilities but by its effectiveness in reducing total degradation(which increases range) and reducing the optimum IBO (which increases power amplifier

42efficiency). The distortion caused by non-distortionless schemes can outweigh the benefitof 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 0.0001PAPR 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 effectivescheme. In fact, the effectiveness of non-distortionless PAPR reduction schemes ingeneral is suspect. For these types of techniques it is important to take into account theeffect of the nonlinear power amplifier.The effectiveness of distortionless PAPR reduction techniques are typically studiedin terms of PAPR reduction and complexity. It would be interesting to also study theseschemes in terms of total degradation. Does a 3 dB reduction in PAPR results in a 3dB reducing in IBO opt ? What is the resulting minimum total degradation?

Chapter 3Constant Envelope OFDMConventional OFDM systems, even with the use of effective PAPR reduction and/orpower amplifier linearization techniques, typically require more input power backoff thanconvention 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 tothe PAPR problem. CE-OFDM can be thought of as a mapping of the OFDM signal tothe unit circle, as depicted in Figure 3.1. The instantaneous power of the resulting signalis constant. Figure 3.2 compares the instantaneous power of the OFDM signal and themapped CE-OFDM signal. For the CE-OFDM signal the peak and average powers arethe same, thus the PAPR is 0 dB.SignalUnit circle⇒OFDMCE-OFDMFigure 3.1: The CE-OFDM waveform mapping.43

445OFDMCE-OFDM4Instantaneous signal power321000.20.40.6Normalized time0.81Figure 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 toconventional OFDM which amplitude modulates the carrier. To see this, consider thebaseband OFDM waveformm(t) = ∑ iN∑I i,k q k (t − iT B ) (3.1)k=1where {I i,k } are the data symbols and {q k (t)} are the orthogonal subcarriers. For conventionalOFDM the baseband signal is up-converted to bandpass asy(t) = R{m(t)e j2πfct}= A m (t) cos [2πf c t + φ m (t)] ,(3.2)where A m (t) = |m(t)| and φ m (t) = arg[m(t)]. For real-valued m(t), φ m (t) = 0 and y(t)is simply an amplitude modulated signal. (For complex-valued m(t), y(t) can be viewedas an amplitude single-sideband modulation.) For CE-OFDM, m(t) is passed through aphase modulator prior to up-conversion. The baseband signal iss(t) = e jαm(t) , (3.3)

45where α is a constant. The bandpass signal is{s(t)e j2πfct}y(t) = R= R{e jαAm(t) exp[jφm(t)] e j2πfct}= R{e −αAm(t) sin φm(t) j[2πfct+αAm(t) coseφm(t)]}(3.4)= e −αAm(t) sin φm(t) cos [2πf c t + αA m (t) cos φ m (t)] .For real-valued m(t),y(t) = cos [2πf c t + 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 Figure3.3. At the transmitter, the high PAPR OFDM signal is transformed into a lowPAPR signal prior to the power amplifier. At the receiver, the inverse transformation isperformed prior to demodulation.TransmitterTransformOFDMmodulatorm(t)Phasemodulators(t)PoweramplifierTochannelFromchannelReceiverInversePhasedemodulatorOFDMdemodulatortransformFigure 3.3: Basic concept of CE-OFDM.As mentioned in Section 2.5, other approaches based on signal transformation havebeen suggested. In particular, [215, 329, 569–571] suggest a companding transform. Thecompanded signal has an increased average power and thus a lower peak-to-averagepower ratio than conventional OFDM. The PAPR is still large relative to single carriermodulation, however. The advantage of the phase modulator transform is that theresulting signal has the lowest achievable peak-to-average power ratio of 0 dB.

46The idea of transmitting OFDM by way of angle modulation isn’t entirely new.In fact, Harmuth’s 1960 paper suggest transmitting information by orthogonal timefunctions with “amplitude or frequency modulation, or any other type of modulationsuitable for the transmission of continuously varying [waveforms]” [202]. Using existingFM infrastructure for OFDM transmission has been suggested in [76, 77, 575]. Thesepapers 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 amplifierfor PAPR mitigation—though intriguing, these papers lack a solid theoretical foundationand 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 fromwork 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 ofthe OFDM component for JTRS (Joint Tactical Radio System), suggested a low PAPRenhancement to OFDM by phase modulation. The motivation is to reduce the 6 dBbackoff 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 affected?What is the optimum AWGN performance? What is the performance of a phasedemodulator receiver (Figure 3.3)? How does the system perform in a frequency-selectivefading channel? These questions, and others, are addressed here. First, the CE-OFDMmodulation is defined.3.1 Signal DefinitionAs indicated by (3.4), CE-OFDM requires a real-valued OFDM message signal, thatis, φ m (t) = 0. Therefore the data symbols in (3.1) are real-valued:I i,k ∈ {±1, ±3, . . . , ±(M − 1)}. (3.6)This one dimensional constellation is known as pulse-amplitude modulation (PAM). Thusthe data symbols are selected from an M-PAM set. The subcarriers {q k (t)} must also

47be real-valued. Three possibilities are considered: half-wave cosines,⎧⎪⎨ cos πkt/T B , 0 ≤ t < T B ,q k (t) =⎪⎩ 0, otherwise,for k = 1, 2, . . . , N; half-wave sines,⎧⎪⎨ sin πkt/T B , 0 ≤ t < T B ,q k (t) =⎪⎩ 0, otherwise,for k = 1, 2, . . . , N; and full-wave cosines and sines,⎧cos 2πkt/T B , 0 ≤ t < T B ; k ≤⎪⎨N 2 ,q k (t) = sin 2π(k − N/2)t/T B , 0 ≤ t < T B ; k > N 2 ,⎪⎩ 0, otherwise.For each case, the subcarrier orthogonality condition holds:⎧∫ (i+1)TB⎪⎨ E q , k 1 = k 2 ,q k1 (t − iT B )q k2 (t − iT B )dt =iT B⎪⎩ 0, k 1 ≠ k 2 ,(3.7)(3.8)(3.9)(3.10)where E q = T B /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 partof a discrete Fourier transform (DFT), or equivalently by taking a 2N-point DFT of aconjugate symmetric data vector (see Appendix A.)The baseband CE-OFDM signal iss(t) = Ae jφ(t) , (3.11)where A is the signal amplitude. The phase signal during the ith block is written asφ(t) = θ i + 2πhC NN ∑k=1I i,k q k (t − iT B ), iT B ≤ t < (i + 1)T B , (3.12)where h is referred to as the modulation index, and θ i is a memory term (to be describedbelow). The normalizing constant, C N , is set to√2C N ≡Nσ 2 I, (3.13)

48where σ 2 Iis the data symbol variance:σ 2 I = E { |I i,k | 2} = 1 MM∑(2l − 1 − M) 2l=1= M 2 − 1,3(3.14)assuming equally likely signal points, that is, P (I i,k = l) = 1/M, l = ±1, ±3, . . . , ±(M −1), for all i and k. Consequently, the phase signal variance is{ ∫ }σφ 2 = E 1 (i+1)TB[φ(t) − θ i ] 2 dtT B iT B∫= (2πh)2 2 (i+1)TB ∑N N∑T B NσI2 E {I k1 I k2 } q k1 (t − iT B )q k2 (t − iT B )dtiT B= (2πh)2T B2Nσ 2 IN∑k=1∫ TB0k 1 =1 k 2 =1σ 2 I q2 k (t)dt = (2πh)2 ,which is only a function of the modulation index. The signal energy is(3.15)and the bit energy isE s =E b =∫ (i+1)TBiT B|s(t)| 2 dt = A 2 T B , (3.16)E sN log 2 M =A2 T BN log 2 M . (3.17)The term θ i is a memory component designed to make the modulation phase-continuous.At the ith signaling interval boundary, the phase discontinuity isc i = φ(iT B − ɛ) − φ(iT B + ɛ), ɛ → 0. (3.18)Since q k (t) = 0 for t /∈ [0, T B ), it follows thatφ(iT B − ɛ) = Kandφ(iT + ɛ) = KN∑I i−1,k A e (k), (3.19)k=1N∑I i,k A b (k), (3.20)where K ≡ 2πhC N , A b (k) = q k (0) and A e (k) = q k (T B − ɛ), ɛ → 0. Therefore,N∑c i = θ i−1 − θ i + K [I i−1,k A e (k) − I i,k A b (k)] . (3.21)k=1k=1

49To guarantee continuous phase, that is, c i = 0, the memory term is set toθ i ≡ θ i−1 + KN∑[I i,k A b (k) − I i−1,k A e (k)] . (3.22)k=1Notice that θ i depends on θ i−1 ; the OFDM signal at the beginning of the ith block,∑ Nk=1 I i,kA b (k); and the OFDM signal at the end of the (i−1)th block, ∑ Nk=1 I i−1,kA e (k).The recursive relationship can be written asθ i = K∞∑l=0 k=1N∑[I i−l,k A b (k) − I i−1−l,k A e (k)] . (3.23)Thus, the memory term is a function of all data symbols during and prior to the ithblock.Figure 3.4 plots the phase discontinuities {c i } at the boundary times t = iT B , i =0, 1, . . . , 49. In Figure 3.4(a), c i is plotted for memoryless modulation, that is, θ i = 0,for all i; therefore, c i = K ∑ Nk=1 [I i−1,kA e (k) − I i,k A b (k)]. Figure 3.4(b) shows that thephase discontinuities are eliminated with the use of memory as defined in (3.22).1.51.5Phase discontinuity, ci10.50−0.5−1Phase discontinuity, ci10.50−0.5−1−1.501020304050−1.501020304050Normalized time, t/T BNormalized time, t/T B(a) Without memory.(b) With memory.Figure 3.4: Phase discontinuities.The benefit of continuous phase CE-OFDM is a more compact signal spectrum. Thisproperty is studied further in Section 3.2. A second consequence of the memory termsis the entire unit circle is used for the CE-OFDM phase modulation. This is illustratedin Figure 3.5 which plots continuous phase CE-OFDM signal samples on the complex

50Unit circleStarting point(a) L = 1(b) L = 100Figure 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 overL = 1 block, where the phase signal occupies about one-half the unit circle. Viewingsamples over L = 100 blocks, Figure 3.5(b) shows that the phase signal occupies theentire unit circle.3.2 SpectrumCE-OFDM is a complicated nonlinear modulation and a general closed-form expressionfor 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 appliedto CE-OFDM. The Fourier transform of the average autocorrelation function results ina two-dimensional definite integral. The problem is there are N sinusoidal phase pulsesin CE-OFDM, versus a single phase pulse as in CPM. This makes the integrand veryjagged 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 takingthis approach, however. It can be shown that memoryless modulation (θ i = 0) resultsin spectral lines at the frequencies f k = k/T B , k = 0, ±1, ±3, . . . [17]. Using memory asdefined by (3.22) eliminates these lines.Since the Fourier transform approach isn’t computationally feasible, other techniquesare required to understand the CE-OFDM spectrum. The simplest is with the Taylor

51expansion e x = ∑ ∞n=0 xn /n!. The CE-OFDM signal, with θ i = 0, can be written aswheres(t) = Ae jσ φm(t)∞∑[ (jσφ ) n ]= Am n (t),n!n=0m(t) = C N∑i(3.24)N∑I i,k q k (t − iT B ) (3.25)k=1is the normalized OFDM message signal. The effective double-sided bandwidth, definedas the twice the highest frequency subcarrier, of m(t) isW = 2 × N2T B= N T B. (3.26)The bandwidth of s(t) is at least W : in (3.24), the n = 0 term contains no informationand thus has zero bandwidth; the n = 1 term is information bearing and has bandwidthW ; the n = 2 term has a bandwidth 2W ; and so on. Thus, due to the n = 1 term, thebandwidth of s(t) is at least W , and depending on the modulation index the effectivebandwidth can be greater than W .The power density spectrum, Φ s (f), can be easily estimated by the Welch methodof periodogram averaging [526].fractional out-of-band power,FOBP(f) =The result, ˆΦs (f) ≈ Φ s (f), is used to calculate the∫ f0 Φ ∫ fs(x)dx ˆΦ 0 s (x)dx≈= FOBP(f),0.5P s 0.5P ˆ(3.27)swhere P s = ∫ ∞−∞ Φ s(f)df = E s /T B = A 2 is the signal power. Figure 3.6 shows estimatedfractional out-of-band power curves for N = 64 and various 2πh. Due to the normalizingconstant C Nbandwidth,these curves are valid for any M. The dashed lines represent the RMSB rms = σ φ W = 2πhN/T B . (3.28)The RMS (root-mean-square) bandwidth is obtained by borrowing a result from analogangle 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 resultsin Figure 3.6 shows that B rms 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 theTaylor expansion in (3.24), the CE-OFDM bandwidth is at least W . A more suitable

5210 0FOBP(f) ˆB rms10 −1Fractional out-of-band power10 −210 −310 −410 −510 −62πh2.01.81.61.41.21.00.80.60.40.200.510 −7 Figure 3.6: Estimated fractional out-of-band power. (N = 64)1Normalized frequency, f/W1.52bandwidth is thusB s = max(2πh, 1)W. (3.29)Figure 3.7 plots B s versus 2πh, and compares it with the 90–99% bandwidths as determinedby the Welch method. Notice that (3.29) is an accurate 90–92% bandwidthmeasure for 2πh ≥ 1.0. For small modulation index, B s is a conservative bandwidth.With 2πh = 0.4, for example, (3.29) accounts for 99.8% of the signal power (from Figure3.6).Figure 3.8 compares spectral estimates for CE-OFDM signals with the three subcarriermodulations 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

5332.82.62.4B sWelch: 90%92%95%99%Normalized double-sided bandwidth, B/W2.221.81.61.41.210.80.60.40.200.20.40.60.81 1.2Modulation index, 2πh1.41.61.82Figure 3.7: Double-sided bandwidth as a function of modulation index. (N = 64)compared to the Abramson spectrum [1]:Φ Ab (f) = A 2∞ ∑n=0a n U n (f), (3.30)whereandφσa n = e−σ2 φ2n, (3.31)n!⎧δ(f), n = 0,⎪⎨U n (f) = Φ m (f), n = 1,⎪⎩ Φ m (f) ∗ n Φ m (f), n > 1.(3.32)The weighting factors {a n } are Poisson distributed, and ∑ ∞n=0 a n = 1; n ∗ denotes then-fold convolution, for example x(t) 3 ∗ x(t) = x(t) ∗ x(t) ∗ x(t); and Φ m (f) is the power

540Welch estimateterms from (3.30)ˆΦ Ab (f)−10n = 2−20n = 3Power spectrum (dB)−30−40−50n = 4n = 1DCTDFTDSTCP-DFT−60−70CP-DCT−80−3−2−101Normalized frequency, f/W23Figure 3.8: Power density spectrum. (N = 64, 2πh = 0.6)density spectrum of the message signal m(t) according to (3.25):Φ m (f) = T B2NN∑k=1sinc 2 [(f −k ) ] [(T B + sinc 2 f +2T Bk ) ]T B , (3.33)2T Bwhere⎧⎪⎨ 1, x = 0,sinc(x) =(3.34)⎪⎩ sin πxπx , otherwise. ∫ ∞The functions {U n (f)} have the property:−∞ U n(f)df = 1, for all n [1]. Thereforethe nth term in (3.30) has an a n × 100% contribution to the overall spectrum.example, the carrier component, represented as δ(f), has a fractional contribution ofe −σ2 φ; Φ m (f) ∗ 2 Φ m (f) has a fractional contribution (e −σ2 φσφ 4 )/2; and so on. Notice thatfor 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.)For

55Figure 3.8 plots the n = 1, 2, 3, 4 terms in (3.30), and the resulting sum4∑ˆΦ Ab (f) = A 2 a n U n (f) ≈ Φ Ab (f). (3.35)n=0The 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. Forexample, DST has a continuous phase [with or without memory since A b (k) = A e (k) = 0,for all k] and has a lower out-of-band power than memoryless DFT, which isn’t phasecontinuous.Memoryless DFT results in a slightly smoother phase than memoryless DCTsince one-half of the subcarriers have zero-crossings at the signal boundaries [A b (k) =A e (k) = 0, for k = N/2 + 1, . . . , N, and A b (k) = A e (k) = 1, otherwise] while DCTdoesn’t [A b (k) = A e (k) = 1, for all k]. The smoothest phase results from CP-DCTwhich, unlike DST and CP-DFT, has a first derivative equal to zero at the boundarytimes t = iT B . Consequently, the CP-DCT is the most spectrally contained.Figure 3.9 shows estimated fractional out-of-band power curves that correspond tothe signals in Figure 3.8. For reference, conventional OFDM is also plotted. Notice thatthe 99% spectral containment at f/W = 0.5 is the same for each signal. The continuousphase CE-OFDM signals are the most spectrally contained and are shown to havebetter than 99.99% containment at f/W = 1.25. Over the range 0.5 ≤ f/W ≤ 0.8, This10 010 −1CE-OFDMOFDMB rmsFractional out-of-band power10 −210 −310 −410 −510 −6DCTDFTDSTCP-DFT10 −7CP-DCT00.510 −8 Figure 3.9: Fractional out-of-band power. (N = 64, 2πh = 0.6)11.52Normalized frequency, f/W2.53

56figure shows that the CE-OFDM spectrum has more out-of-band power than conventionalOFDM. 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 hresults in worse performance, as will be discussed in the next chapter. Therefore, thesystem designer can trade performance for spectral containment, and visa versa.Figure 3.10 compares CE-OFDM, with CP-DFT modulation over a large range ofmodulation index, to conventional OFDM. For 2πh ≤ 0.4 the fractional out-of-bandpower of CE-OFDM is always better than OFDM; otherwise CE-OFDM has more outof-bandpower for at least some frequencies f/W > 0.5. The 2πh = 2.0 example has abroad spectrum, greater than OFDM over all frequencies. Notice that the shape of thespectrum appears Gaussian shaped. This is due to the fact that for a large modulationindex, the higher-order terms in (3.32) dominate. They are Gaussian shaped due to themultiple convolutions of (3.33). The shape of “wideband FM” signals is well covered inthe classical works of [1, 341, 437, 472].10 0CE-OFDMOFDM10 −1Fractional out-of-band power10 −210 −310 −410 −510 −62πh2.01.81.61.41.21.00.80.60.40.200.510 −7 Figure 3.10: CE-OFDM versus OFDM. (N = 64)1Normalized frequency, f/W1.52

57Finally, Figure 3.11 compares CE-OFDM and OFDM with nonlinear power amplification.The OFDM curves (from Figure 2.9) require > 6 dB backoff to avoid spectralbroadening. The CE-OFDM signals have a bandwidth that depends only on the modulationindex and are not effected by the PA nonlinearity.10 0OFDM, TWTAOFDM, IdealCE-OFDMFractional out-of-band power10 −110 −210 −3IBO (dB)024610 −42πh0.40.50.60.710 −5 00.51Normalized frequency, f/W1.52Figure 3.11: CE-OFDM versus OFDM with nonlinear PA. (N = 64)

Chapter 4Performance of ConstantEnvelope OFDM in AWGNIn this chapter the basic performance properties of CE-OFDM are studied.baseband signal, represented by (3.11) and (3.12), is up-converted and transmitted asthe bandpass signalThe{s bp (t) = R s(t)e j2πfct} = A cos [2πf c t + φ(t)] , (4.1)where f c is the carrier frequency. The received signal isr bp (t) = s bp (t) + n w (t), (4.2)where n w (t) denotes a sample function of the additive white Gaussian noise (AWGN)process with power density spectrum Φ nw (f) = N 0 /2 W/Hz. The primary focus of thechapter is to analyze the phase demodulator receiver, depicted by the block diagrambelow. An expression for the bit error rate (BER) is derived by making certain highcarrier-to-noise ratio (CNR) approximations.The analytical result is then comparedagainst computer simulation and it is shown to be accurate for BER < 0.01. It is alsor bp (t)BandpassfilterPhasedemodulatorOFDMdemodulatorTodetectorFigure 4.1: Phase demodulator receiver.58

59demonstrated that with the use of a phase unwrapper, the receiver is insensitive to phaseoffsets caused by the channel and/or by the memory terms {θ i }.The phase demodulator receiver is a practical implementation of the CE-OFDMreceiver and is therefore of practical interest. However, it isn’t necessarily optimum,since the optimum receiver is a bank of M N matched filters [421, p. 244], one for eachpotentially transmitted signal. In Section 4.2 a performance bound and approximationfor the optimum receiver is derived; and then in Section 4.3, the performance of thephase demodulator receiver is compared to the optimum result. It is shown that undercertain conditions the phase demodulator receiver has near-optimum performance.In Section 4.4 CE-OFDM’s spectral efficiency versus performance is compared tochannel capacity. Finally, the chapter is concluded in Section 4.5 with a comparisonbetween CE-OFDM and conventional OFDM in terms of power amplifier efficiency, totaldegradation, and spectral containment.4.1 The Phase Demodulator ReceiverThe phase demodulator receiver essentially consists of a phase demodulator followedby a conventional OFDM demodulator. Figure 4.2 shows the model used in this analysis.The received signal is first passed through a front-end bandpass filter, centered atthe carrier frequency f c , which limits the bandwidth of the additive noise. Then thebandpass signal is down-converted to r(t), sampled, and processed in the discrete-timedomain. The conversion from r bp (t) to r(t) is described first 1 , making use of the followingtrigonometric identities:sin(x) sin(y) =sin(x) cos(y) =cos(x) cos(y) =cos(x) sin(y) =cos(x − y) − cos(x + y),2(4.3)sin(x + y) + sin(x − y),2(4.4)cos(x + y) + cos(x − y),2(4.5)sin(x + y) − sin(x − y).2(4.6)1 This 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.

60Lowpassfilterr bp (t)Bandpassfilteru(t)2 cos(2πf ct)−2 sin(2πf ct)jr(t)r[i]t = iT saPhasedemodulatorOFDMdemodulatorLowpassfilterFigure 4.2: Bandpass to baseband conversion.The output of the bandpass filter isu(t) = s bp (t) + n bp (t), (4.7)wheren bp (t) = n c (t) cos(2πf c t) − n s (t) sin(2πf c t) (4.8)is the result of passing n w (t) through the bandpass filter. The terms n c (t) and n s (t)are referred to as the in-phase and quadrature components of the narrowband noise,respectively, and have the power density spectrum⎧⎪⎨ N 0 , |f| ≤ B bpf /2,Φ nc (f) = Φ ns (f) =(4.9)⎪⎩ 0, |f| > B bpf /2,where B bpf is the bandwidth of the bandpass filter. Note that B bpf is assumed to besufficiently large so s bp (t) is passed through the front-end filter with negligible distortion[421, pp. 157–158]. Writing s bp (t) in the forms bp (t) = s c (t) cos(2πf c t) − s s (t) sin(2πf c t), (4.10)where s c (t) = A cos[φ(t)] and s s (t) = A sin[φ(t)], the filter output can then be written asu(t) = [s c (t) + n c (t)] cos(2πf c t) − [s s (t) + n s (t)] sin(2πf c t). (4.11)

61The output of the top (in-phase) branch of the down-converter is 2r c (t) = LP {u(t) × 2 cos(2πf c t)}= LP{[s c (t) + n c (t)] + [s c (t) + n c (t)] cos(4πf c t)− [s s (t) + n s (t)] sin(4πf c t)}(4.12)= s c (t) + n c (t),where LP{·} denotes the lowpass component of its argument (i.e., double-frequency termsare rejected) [624, p. 364]. Likewise, the output of the bottom (quadrature) branch isr s (t) = LP {u(t) × −2 sin(2πf c t)}= LP{−[s c (t) + n c (t)] sin(4πf c t) + [s s (t) + n s (t)]− [s s (t) + n s (t)] cos(4πf c t)}(4.13)= s s (t) + n s (t).The two are combined to obtainr(t) = s(t) + n(t), (4.14)where s(t) is the lowpass equivalent CE-OFDM signal from (3.11), andn(t) = n c (t) + jn s (t) (4.15)is the lowpass equivalent representation of the bandpass white noise, n bp (t) [421, p. 158].The power density spectrum of n(t) is [421, p. 158]⎧⎪⎨ N 0 , |f| ≤ B n /2,Φ n (f) =⎪⎩ 0, |f| > B n /2,(4.16)where B n = B bpf is the noise bandwidth. The corresponding autocorrelation of n(t)is [421, p. 158]sin πB n τφ n (τ) = N 0 . (4.17)πτThe continuous-time receive signal is then sampled at the rate f sa = 1/T sa samp/sto obtain the discrete-time signal 3r[i] = s[i] + n[i], i = 0, 1, . . . , (4.18)2 Here, ideal phase coherence and frequency synchronization is assumed. In Section 4.1.2 the effect ofchannel phase offsets is considered.3 Perfect timing synchronization is assumed.

62Phase demodulatorr[i]FIRfilterarg(·)PhaseunwrapperTo OFDMdemodulatorFigure 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 noisesamples {n[i]} are assumed independent:⎧⎪⎨ σn 2E {n[i 1 ]n[i 2 ]} =, i 1 = i 2 ,(4.19)⎪⎩ 0, i 1 ≠ i 2 ;and therefore the sampling rate is f sa = B n , and σn 2 = f saN 0 .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 atimproving performance; arg(·) simply calculates the arctangent of its argument; and thephase 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 channeland/or by the memory terms.The output of the phase demodulator is processed by the OFDM demodulator whichconsists of the N correlators, one corresponding to each subcarrier. This correlator bankis implemented in practice with the fast Fourier transform.4.1.1 Performance AnalysisIn this section a bit error rate approximation is derived for the phase demodulatorreceiver. Although the receiver operates in the discrete-time domain, it is convenient toanalyze it in the continuous-time domain. The angle of the received signal isarg[r(t)] = θ i + 2πhC NiT B ≤ t < (i + 1)T B , where[]N(t) sin [Θ(t) − φ(t)]ξ(t) = arctanA + N(t) cos [Θ(t) − φ(t)]N ∑k=1I i,k q k (t − iT B ) + ξ(t), (4.20)(4.21)is the corrupting noise [624, p. 416]. The terms N(t) and Θ(t) in (4.21) are the envelopeand phase of n(t).

63The kth correlator in the OFDM demodulator computesThe signal term is1T B∫ (i+1)TBiT Barg[r(t)]q k (t − iT B )dt = S i,k + N i,k + Ψ i,k . (4.22)S i,k = 1 ∫ (i+1)TB[φ(t) − θ i ]q k (t − iT B )dtT B= 2πhC NT B= 2πhC NT BiT B∫ (i+1)TBiT BN ∑n=1√1I i,k E q = 2πh2NσI2 I i,k .I i,n q n (t − iT B )q k (t − iT B )dt(4.23)The noise term isN i,k = 1T B∫ (i+1)TBiT Bξ(t)q k (t − iT B )dt. (4.24)For example, with DST subcarrier modulation (3.8),N i,k = 1T B∫ (i+1)TBiT Bξ(t) sin [πk(t − iT B )/T B ] dt, (4.25)which can be viewed as a Fourier coefficient of ξ(t) at f = k/2T B Hz. As T B → ∞,the variance of the coefficient is proportional to the power density spectrum functionevaluated at f = k/2T B [442, pp. 41–43]. It is well known that, given a high CNR, thenoise at the output of a phase demodulator has a power density spectrum [423, p. 410]Φ ξ (f) ≈ N 0, |f| ≤ W/2, (4.26)A2 where, from (3.26), W = N/T B is the effective bandwidth of φ(t). Moreover, for highCNR, ξ(t) is well modeled as a sample function of a zero mean Gaussian process. Therefore,N i,k is approximated as a zero mean Gaussian random variable with variance [442,pp. 41–43]var{N i,k } ≈ 12T BΦ ξ (f)| f=k/2TB ≈ 12T BN 0A 2 . (4.27)This result is the same for DCT and DFT subcarrier modulation.The third term in (4.22), Ψ i,k , is expressed asSinceΨ i,k = 1T B∫ (i+1)TBiT Bθ i q k (t − iT B )dt. (4.28)∫ TB0q k (t)dt = 0, k = 1, 2, . . . , N, (4.29)

64for DCT and DFT modulations [(3.7), (3.9)], Ψ i,k = 0 and therefore has no effect on systemperformance. This highlights an important observation: DST subcarrier modulation(3.8) is inferior to DCT and DFT since Ψ i,k = 0 isn’t guaranteed.The symbol error rate is computed by determining the probability of error for eachsignal point in the M-PAM constellation. For the M − 2 inner points, the probability oferror isP inner = P (|N i,k | > d) = 2P (N i,k > d), (4.30)where√1d = 2πh2NσI2 . (4.31)[Notice that (4.30) is not averaged over i nor k since var{N i,k }, as approximated by (4.27),is a constant.] Due to the Gaussian approximation applied to the random variable N i,k ,∫ ∞1P inner ≈ 2 √2πN0 /(2A 2 T B ) exp ( −x 2 / [ 2N 0 /(2A 2 T B ) ]) dx= 2d∫ ∞d[N 0 /(2A 2 T B )] −0.5 1√2πexp ( −x 2 /2 ) dx( √ ) ( √ )A= 2Q 2πh2 T B6 log2 M E bN 0 NσI2 = 2Q 2πhM 2 .− 1 N 0(4.32)For the two outer points, the probability of error isTherefore, the overall symbol error rate isP outer = P (N i,k > d) = 1 2 P inner. (4.33)SER = M − 2MP inner + 2 M P outer( ) ( √ )M − 16 log2 M E b≈ 2 Q 2πhMM 2 .− 1 N 0(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 inadjacent signal levels, in which case the bit error rate is approximated as [483, p. 195]BER ≈SER ( ) ( √ )M − 16log 2 M ≈ 2 log2 M E bQ 2πhM log 2 MM 2 . (4.35)− 1 N 0

654.1.2 Effect of Channel Phase OffsetSuppose the channel imposes a phase offset of φ 0 . The received signal is thenThe angle of r(t) isarg[r(t)] = θ i + 2πhC Nr(t) = s(t)e jφ 0+ n(t). (4.36)N ∑k=1I i,k q k (t − iT B ) + φ 0 + ξ(t), (4.37)iT B ≤ t < (i + 1)T B . Which is identical to (4.20) with the addition of the channel offsetterm. The kth correlator is the same as (4.22), except the third term isΨ i,k = 1T B∫ (i+1)TBiT B[θ i + φ 0 ]q k (t − iT B )dt = 0. (4.38)Therefore, the phase offset due to the channel has no impact on performance, and theanalytical approximation in (4.35) is applicable.Figure 4.4 compares the performance of N = 64, M = 2 CE-OFDM with phase offset{(θ i + φ 0 ) ∈ [0, 2π)}, and without (θ i + φ 0 = 0). The former is referred to as System1 (S1), the later as System 2 (S2). The system is computer simulated with a samplingrate f sa = JN/T B , where J = 8 is the oversampling factor 4 . For E b /N 0 ≥ 10 dB and2πh ≤ 0.5, S1 and S2 are shown to have identical performance.For these cases theanalytical 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 comparedto S2. In this case, the analytical approximation is shown to be overly optimistic. Thisdemonstrates a limitation of the phase demodulator receiver: for a large modulationindex and low signal-to-noise ratio, the phase demodulator has difficulty demodulatingthe noisy samples. The performance of S1 is slightly worse than S2 since the outputof the phase demodulator, the arg(·) block in Figure 4.3, has more phase jumps sincethe received phase crosses the π boundary more frequently. Proper phase unwrappingis therefore required. However, phase unwrapping a noisy signal is a difficult problemand the unwrapper makes mistakes. As a result the performance degrades slightly. Fora smaller modulation index, the unwrapper works perfectly and the performance of S1isn’t degraded.4 Also, the FIR filter (see Figure 4.3) has length L fir = 11 and normalized cutoff frequency f cut/W =0.2. See Section 4.1.4 for more on the filter design.

6610 0System 1System 2Approx (4.35)10 −1Bit error rate10 −210 −32πh0.70.50.30.20.110 −410 −5 05 10 15 20 25Signal-to-noise ratio per bit, E b /N 0 (dB)Figure 4.4: Performance with and without phase offsets. System 1 (S1) has phase offsets{(θ i + φ 0 ) ∈ [0, 2π)}, and System 2 (S2) doesn’t (θ i + φ 0 = 0). [M = 2, N = 64, J = 8]304.1.3 Carrier-to-Noise Ratio and Thresholding EffectsThe 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]. Awell-known characteristic of such receivers is: at low CNR, below a threshold value, theapproximation is invalid and system performance degrades drastically. In this section, theCNR is defined and the threshold effect for CE-OFDM is observed by way of computersimulation.The CNR at the output of the analog front end, r(t), iswhere A 2 is the carrier power, andP n =CNR = A2P n, (4.39)∫ ∞−∞Φ n (f)df = B n N 0 (4.40)is the noise power. From (3.17), the carrier power can be written in the formA 2 = E bN log 2 MT B; (4.41)

67thusCNR = (E b/N 0 )N log 2 MT B B n. (4.42)Since the noise samples are assumed independent [see (4.19)],and (4.42) reduces toB n = f sa = JN/T B , (4.43)CNR = (E b/N 0 ) log 2 M. (4.44)JTherefore, the carrier-to-noise ratio is proportional to E b /N 0 and M, and inversely proportionalto 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 arecompared to (4.35). In subfigures (a) and (b) the system is below and above the 10dB threshold, respectively. Clearly, above CNR = 10 dB, the system is observed to beabove 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, theperformance quickly degrades to a bit error rate of 1/2. Figure 4.6 shows results for morevalues 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 isdifficult to study analytically. Gaining more insight into this issue is a subject for futureinvestigation.

68Bit error rateBit error rate10 −2SimulationApprox (4.35)SimulationApprox (4.35)−210 −1 (a) Below 10 dB threshold.0 2 4 6 8Carrier-to-noise ratio (dB)101010 −3 (b) Above 10 dB threshold.11 12 13 14 15Carrier-to-noise ratio (dB)16Figure 4.5: Threshold effect at low CNR. (M = 8, N = 64, J = 8, 2πh = 0.5)2πh0.210 −1SimulationApprox (4.35)Bit error rate10 −10.40.6Bit error rate10 −2SimulationApprox (4.35)0.80.8 0.6 0.4 2πh 0.2−210 −2 (a) Below 10 dB threshold.0 2 4 6 8Carrier-to-noise ratio (dB)101010 −3 (b) Above 10 dB threshold.12 14 16 18 20 22Carrier-to-noise ratio (dB)24Figure 4.6: Threshold effect at low CNR, various 2πh. (M = 8, N = 64, J = 8)

694.1.4 FIR Filter DesignThe FIR filter preceding the phase demodulator (see Figure 4.3) can improve performance.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 E b /N 0 = 10 dB. The filter, designedusing the window technique described in [422, pp. 623–630], has a length 3 ≤ L fir ≤ 101and a normalized cutoff frequency 0 < f cut /W ≤ 1. Hamming windows are used 5 . Theperformance without a filter is shown to be BER = 0.05, while the analytical approximation(4.35) is BER = 0.012. For f cut /W ≥ 0.4 all the filtered results are shown to bebetter than the unfiltered result. The filters with L fir > 5 and f cut /W > 0.5 are shownto have roughly the same performance. The higher-order filters, which have a narrowertransition bands, require f cut /W > 0.5 to yield good performance. This is explainedby noting that the (single-sided) signal bandwidth is at least W/2 Hz. Therefore, thehigher-order filters with f cut /W < 0.5 distort the signal. Notice that the L fir = 11 filterhas equally good performance so long as f cut /W ≥ 0.1. This is due to the wide transitionband of the lower-order filter.L fir = 310 −111 975211016131Bit error rateNo filterApprox (4.35)10 −2 00.20.40.6Normalized cutoff frequency, f cut/WFigure 4.7: Performance for various filter parameters L fir , f cut /W .(M = 2, N = 64, J = 8, 2πh = 0.5 and E b /N 0 = 10 dB)0.815 It has been observed that the window type has negligible impact on performance.

700L fir , f cut/W−209, 0.7Magnitude response (dB)−40−6031, 0.13, 0.1101, 0.7 9, 0.1−80−10000.511.52Normalized frequency, f/W2.53Figure 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 L fir = 31, f cut /W = 0.1 example is shown to not have this property, and, as shownin 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 withoutthe FIR filter. The L fir = 11, f cut /W = 0.2 filter is used. These results show that thefilter becomes important for larger modulation index: for 2πh = 0.1 the filtered andunfiltered results are the same; for 2πh = 0.3 the filtered performance is a fraction ofa dB better than the unfiltered; for 2πh = 0.7 there is a 2 dB improvement in therange 10 −3 < BER < 10 −5 . Notice the error floor developing below 10 −5 . This is aconsequence of imperfect phase demodulation. The filter lowers the error floor resultingin a 9 dB improvement at BER = 10 −6 .

7110 0Without FIR filterWith FIR filterApprox (4.35)10 −110 −2Bit error rate10 −32πh0.70.30.110 −410 −510 −6 05101520Signal-to-noise ratio per bit, E b /N 0 (dB)Figure 4.9: CE-OFDM performance with and without FIR filter.(M = 2, N = 64, J = 8)25304.2 The Optimum ReceiverAs mentioned in the introduction to this chapter, the phase demodulator receiver isa practical implementation, but not necessarily optimum. In this section, the optimum,yet impractical, CE-OFDM receiver is studied. Results obtained here are used in thefollowing section to compare the phase demodulator receiver to optimum performance.During each block one of M N CE-OFDM signals is transmitted. Consider the mthbandpass signals m (t) = A cos[2πf c t + θ 0 + KN∑k=1I (m)kq k (t)], 0 ≤ t < T B , (4.45)where K = 2πhC N . The set of all possible signals, {s m (t)} M Nm=1 , is determined by theset of all possible data symbol vectors {I (m) = [I (m)1 , I (m)2 , . . . , I (m) ]}M Nm=1 . The optimumN

72receiver, as shown in Figure 4.10, correlates the received signal, r bp (t) = s m (t) + n w (t),with each potentially transmitted signal. The detector then selects the largest result [421,pp. 242–247].R TB0 (·)dts 1 (t)R TB0 (·)dtReceivedsignal r bp (t)s 2 (t)..SelectthelargestOutputdecisionR TB0 (·)dts M N (t)Sampleat t = T BFigure 4.10: The optimum receiver.4.2.1 Performance AnalysisIt is desired to obtain an analytical expression for the bit error probability 6 , 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 datasymbol error probability. Determining exact expressions for the above probabilities isintractable for large N. However, upperbounds and approximations can be derived in astraightforward way, as described below.6 The bit error probability is used interchangeably with the bit error rate. Likewise for the symbolerror probability and symbol error rate.

73An upperbound for P (signal error) is [373]:P (signal error) ≤ 1 √2π∫ ∞−∞[1 − [1 − Q(y)] M N −1 ] ×⎧ ⎡ √ ⎤2 ⎨⎫exp⎩ −1 2E s (1 − λ)⎬(4.46)⎣y −⎦2N 0 ⎭ dy.The above expression is the probability of detection error for M Ncorrelation −1 ≤ λ ≤ 1. Therefore, it provides an upperbound given thatλ = ρ max =signals with equalmax ρ m,n, (4.47)m,n; m≠nwhere ρ m,n is the normalized correlation between s m (t) and s n (t):ρ m,n = 1 ∫ TBs m (t)s n (t)dt. (4.48)E s0An approximation for P (signal error) is [421, p. 288]⎛√⎞P (signal error) ≈ K d 2 Q ⎝d 2 min ⎠ , (4.49)min 2N 0where K d 2 is the number of neighboring signal points having the minimum squaredminEuclidean distancewhered 2 m,n =d 2 min =∫ TB0minm,n; m≠n d2 m,n, (4.50)[s m (t) − s n (t)] 2 dt (4.51)is the squared Euclidean distance between s m (t) and s n (t). This quantity is related tothe signal correlation asd 2 m,n = 2E s (1 − ρ m,n ), (4.52)thusd 2 min = 2E s(1 − ρ max ). (4.53)Therefore to obtain the performance bound (4.46) and the approximation (4.49) thesignal correlation properties must be studied, and in particular ρ max must be determined.The normalized correlation between the mth and nth signal, as a function of the phase

74constant K = 2πhC N , isρ m,n (K) = 1 ∫ TBs m (t)s n (t)dtE s 0∫ [= A2 TBcos 2πf c t + θ 0 + KE s 0[= A22E s∫ TB0cos[cos2KN∑k=12πf c t + θ 0 + KI (m)kq k (t)N∑k=1]N∑∆ m,n (k)q k (t) dt,k=1]×I (n)kq k(t)]dt(4.54)where ∆ m,n (k) = 0.5[I (m)k−I (n)k]. The double frequency term is ignored since f c ≫ 1/T Bis assumed. Notice that for k where ∆ m,n (k) = 0, the data symbols are the same, andthese indices don’t contribute to the correlation. Therefore∫ []ρ m,n (K) = A2 TBD∑cos 2K ∆ m,n (k d )q k (t) dt, (4.55)2E s0where {k d } D d=1 are the indices where the data symbols differ, that is, ∆ m,n(k d ) ≠ 0, andD is the total number of differences. Writing (4.55) in exponential form yields∫ { []}ρ m,n (K) = A2 TBD∑R exp j2K ∆ m,n (k d )q k (t) dt2E s0= A22E s∫ TB0d=1d=1{ D}∏R exp [j2K∆ m,n (k d )q k (t)] dt.d=1To proceed, the DCT modulation (3.7) is assumed.expansion [580],e ja cos b =∞∑i=−∞(4.56)Making use of the Jacobi-AngerJ i (a)e ji(b+π/2) , (4.57)where J i (a) is the ith-order Bessel function of the first kind, (4.56) is written as∫ [ρ m,n (K) = A2 TB ∑ ∞ ∞∑R · · ·2E s0i 1 =−∞i D =−∞J i1 [2K∆ m,n (k 1 )] × · · · × J iD [2K∆ m,n (k D )]e jσ(i) ]dt= A22E s∫ TB0∞∑i 1 =−∞· · ·∞∑i D =−∞J i1 [2K∆ m,n (k 1 )]×· · · × J iD [2K∆ m,n (k D )] cos[ω(i) + ψ(i)]dt,(4.58)

75where σ(i) = ω(i) + ψ(i), ω(i) ≡ πtT B∑ Dd=1 i dk d and ψ(i) ≡ π 2∑ Dd=1 i d. Index values thatresult in ω(i) ≠ 0 have no contribution, so (4.58) simplifies toρ i,j (K) = ∑ iD∏J i ′i,d[2K∆ m,n (k d )] cos[ψ(i ′ i)], (4.59)d=1where i ′ i ≡ [i′ i,1 , . . . , i′ i,D ], i = 1, 2, . . ., represent the vectors whereby ω(i′ i ) = 0. Thisresult is the same for DST modulation except ψ(i ′ i ) = 0. For DFT modulation, (4.59) isslightly different since both sinusoids and cosinusoids are used as subcarriers.For D = 1,ρ m,n (K) = J 0 [2K∆ m,n (k 1 )]. (4.60)Therefore the correlation is simply the 0th-order Bessel function. Figure 4.11(a) plots(4.60) for |∆ m,n (k 1 ) = 1|. Also plotted is the envelope of the 0th-order Bessel function[580, p. 121]. Note that ρ m,n (K) doesn’t depend on the subcarrier frequencyf k1= k 1 /T B , k 1 ∈ {1, 2, . . . , N}, just on the magnitude of the difference |∆ m,n (k 1 )| ∈{1, 2, . . . , (M − 1)}.For CE-OFDM signals of interest,ρ max = J 0 (2K). (4.61)Figure 4.11(b) plots all unique ρ m,n (K) for M = 2, N = 8 DCT subcarrier modulation.Notice that the largest correlation function is associated with D = 1. For any givensignal, there are N other signals with D = 1: therefore, K d 2min= N, and from (4.49),the probability of signal error is approximated as⎛√⎞P (signal error) ≈ K d 2 Q ⎝d 2 min ⎠min 2N 0(√ )= NQ Es [1 − ρ max ]/N 0(√ )≈ NQ Es [1 − J 0 (2K)]/N 0 .(4.62)A minimum distance signal error results in one data symbols error. Therefore, the symbolerror probability is approximated asP (symbol error) ≈P (signal error)N(√ )≈ Q Es [1 − J 0 (2K)]/N 0 . (4.63)For M = 2, one symbol error corresponds to one bit error. For M > 2, a symbol errorcan result in 1 to log 2 M bit errors. Assuming each outcome is equally likely, a symbol

7610.5p1/πKρm,n(K)0−0.5012K345(a) D = 1.1J 0 (2K)0.8ρm,n(K)0.60.400.10.2K0.30.40.5(b) All unique ρ m,n(K) for M = 2, N = 8 DCT modulation.Figure 4.11: Correlation functions ρ m,n (K).

771∑ log2 Merror results inlog 2 M ii = 0.5(log 2 M + 1) bit errors. ThusP (bit error) ≈ 0.5(log 2 M + 1)P (symbol error)log 2 M≈ 0.5(log 2 M + 1)Qlog 2 M(√Es [1 − J 0 (2K)]/N 0).(4.64)The bit error probability is bounded by noting that P (bit error) ≤ P (signal error),and using (4.46) with λ = ρ max = J 0 (2K):P (bit error) ≤ 1 √2π∫ ∞−∞[1 − [1 − Q(y)] M N −1 ] ×⎧ ⎡ √⎤2 ⎨⎫exp⎩ −1 2E s [1 − J 0 (2K)]⎬ (4.65)⎣y −⎦2N 0 ⎭ dy.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 2 8 = 256. Two values of modulationindex are plotted: 2πh = 0.3 and 2πh = 0.7 which corresponds to K = 0.15 andK = 0.35. The upperbound (4.65) is shown to be within 3 dB of the simulated resultsfor high SNR. The analytical approximation (4.64) is shown to be very accurate.10 010 −110 −2Bit error rate10 −310 −4Approx (4.64)Bound (4.65)Simulation0.72πh 0.310 −510 −6 036 9 12 15Signal-to-noise ratio per bit, E b /N 0 (dB)1821Figure 4.12: CE-OFDM optimum receiver performance. (M = 2, N = 8)

784.2.2 Asymptotic PropertiesIn Figure (4.13) each correlation function is plotted for M = 2, N = 4 DCT modulation.The functions are shown to be bounded byρ m,n (K) ≤ ρ max (K) ≤√1πK , (4.66)the envelope of the 0th-order Bessel function. Therefore,√ )1d 2 m,n(K) ≥ d 2 min(K) ≥ 2E s(1 − . (4.67)πKNotice that as K → ∞ the CE-OFDM signals become orthogonal. The phase modulatorthus drastically alters the signal space.Prior to the phase demodulator, the OFDMsignal space is described by 2N dimensions (2 per subcarrier). At the output of thephase modulator, the space is transformed into a M N -dimensional space (due to thelinear independence of the signal set [421, p. 164]); and as the modulation index becomesvery large, a M N -dimensional orthogonal space. However, from (3.29), the bandwidthtends to infinity as 2πh → ∞.10.5ρm,n(K)p1/πK0−0.5012K345Figure 4.13: All unique ρ m,n (K) for M = 2, N = 4 DCT modulation.4.3 Phase Demodulator Receiver versus OptimumFigure 4.14 shows simulation results for the phase demodulator receiver with N = 64and for various modulation index values 2πh and modulation order M. The simulation

7910 0Approx (4.35)Approx (4.64)Simulation10 −1Bit error rate10 −210 −3M, 2πh8, 1.2 16, 0.8 16, 0.210 −42, 0.3 4, 0.210 −5 51015 20 25 30Signal-to-noise ratio per bit, E b /N 0 (dB)3540Figure 4.14: Phase demodulator receiver versus optimum. (N = 64)results are compared to the analytical approximation (4.35) and the optimum receiverapproximation (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 betrue, the phase demodulator must perfectly invert the phase modulation done at thetransmitter, and the noise at the output of the phase demodulator must be “white” andGaussian. 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. Asshown 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]∫ ∞[yp ξ (x) =0 2πσn2 exp − y2 + A 2 ]− 2yA cos x2σn2 dy, (4.68)where σ 2 n = B nN 0 is the power of the noise signal n(t). Figure 4.15 compares (4.68) tothe Gaussian probability density function. The SNR per bit is E b /N 0 = 30 dB. Thisshows that ξ(t) is well approximated as Gaussian, and near optimum performance of thephase demodulator receiver is expected.

8010 0p ξ (x)GaussianProbability density function, p(x)10 −510 −1010 −1510 −20 −1.5−1−0.50x0.511.5Figure 4.15: Noise samples PDF versus Gaussian PDF. (E b /N 0 = 30 dB)4.4 Spectral Efficiency versus PerformanceIn the previous sections, it is shown that the performance of CE-OFDM is determinedby the modulation index, which, as shown in Section 3.2, also controls the signalbandwidth. In this section, the spectral efficiency (b/s/Hz) versus performance (E b /N 0to achieve a target bit error rate) is plotted for a variety of CE-OFDM signals. Theresults are compared to channel capacity.It is first demonstrated that CE-OFDM with modulation index 2πh > 1 can outperformthe underlying M-PAM subcarrier modulation. Figure 4.16 shows simulationresults 7 for M = 2, 4, 8 and 16. The bit error rate is plotted against the SNR per bit onthe bottom x-axis and the carrier-to-noise ratio on the top x-axis. The viewable rangeis such that CNR ≥ 5 dB. Notice that for M ≥ 4 and 2πh > 1, CE-OFDM outperformsM-PAM. This is predicted by (4.35), since for 2πh = 1.0, the expression is equal to theperformance of M-PAM, and for 2πh > 1.0, it is better than M-PAM. For CE-OFDMto operate in the region 2πh > 1, the carrier-to-noise ratio must be above threshold.7 The oversampling factor is J = 8 for M = 2, 4 and 8, and J = 16 for M = 16. The FIR filter haslength L fir = 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.

81Carrier-to-noise ratio (dB)5 10 15 2010 −1Simulation(4.35)Carrier-to-noise ratio (dB)5 10 15 20 2510 −1Simulation(4.35)4-PAM10 −210 −2Bit error rate10 −3Bit error rate10 −310 −42πh ∈ {0.5 † , 0.4, . . . , 0.1 ‡ }10 −42πh ∈ {1.0 † , 0.9, . . . , 0.1 ‡ }10 −5 16 18 20 22 24 26 28Signal-to-noise ratio per bit, E b /N 0 (dB)(a) M = 2.3010 −5 15 20 25 30Signal-to-noise ratio per bit, E b /N 0 (dB)(b) M = 4.35Carrier-to-noise ratio (dB)5 10 15 20 25 30 3510 −110 −2Simulation(4.35)8-PAMCarrier-to-noise ratio (dB)5 10 15 20 25 30 35 4010 −110 −2Simulation(4.35)16-PAMBit error rate10 −3Bit error rate10 −310 −42πh ∈ {1.5 † , 1.2, 1.0, 0.9, . . . , 0.1 ‡ }10 −42πh ∈ {2.0 † , 1.5, 1.2, 1.1, . . . , 0.1 ‡ }10 −5 10 15 20 25 30 35Signal-to-noise ratio per bit, E b /N 0 (dB)(c) M = 8.4010 −5 15 20 25 30 35 40 45Signal-to-noise ratio per bit, E b /N 0 (dB)(d) M = 16.Figure 4.16: Performance of M-PAM CE-OFDM. (N = 64, †=leftmost curve,‡=rightmost curve)

82To plot the spectral efficiency versus performance, the data rate must be defined,which for uncoded CE-OFDM isR = N log 2 MT Bb/s. (4.69)Using (3.29) as the effective signal bandwidth, the spectral efficiency isR/B = R B s=log 2 Mmax(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. Forreference the channel capacity is also plotted, which is expressed as [421, p. 387](C = B log 2 1 + C )E b, (4.71)B N 0or equivalently,E bN 0= 2C/B − 1C/B . (4.72)10765M = 2M = 4M = 8M = 16CapacityM = 16: 2πh = 2.0, 1.8, . . . , 0.6M = 8: 2πh = 1.4, 1.2, . . . , 0.4Spectral efficiency (b/s/Hz)432M = 4: 2πh = 1.0, 0.8, . . . , 0.21M = 2: 2πh = 0.5, 0.4, 0.3, 0.20.5-1.6 0 5 10 15 20 25Performance: E b /N 0 (dB) to achieve 0.0001 bit error rateFigure 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 order M at the costof performance degradation. For example consider 2πh = 0.4. The spectral efficiency

83is 1, 2 and 3 b/s/Hz for M = 2, 4 and 8, respectively. However, M = 4 requires 4 dBmore 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 linearmodulations such as M-PAM, M-PSK and M-QAM [421, p. 282].The second observation is that CE-OFDM can have both improvements in spectralefficiency 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 dBperformance gain. Conventional CPM systems also have the property of increase spectralefficiency and performance [14]. However, with CPM the receiver complexity increasesdrastically with M (due to phase trellis decoding), which isn’t the case for CE-OFDM.4.5 CE-OFDM versus OFDMThe total degradation, as defined in Section 2.4.2, isTD(IBO) = SNR PA (IBO) − SNR AWGN + IBO,[in dB]where SNR AWGN is the required signal-to-noise ratio required to achieve a target bit errorrate, SNR PA (IBO) is the required SNR when taking into account the nonlinear poweramplifier at a given backoff. Applying the PA model from Section 2.3 to CE-OFDM, theinput signal iss in (t) = A exp[jφ(t)], (4.73)and the output iss out (t) = G(A) exp ( j[φ(t) + Φ(A)] ) . (4.74)The instantaneous nonlinearity results in a constant amplitude and a constant phaseshift. Therefore the PA has no impact on the CE-OFDM performance and no backoff isneeded. The total degradation for CE-OFDM is defined asTD = SNR PM − SNR sub , (4.75)where SNR sub is the required SNR for the underlying subcarrier modulation and SNR PMis 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 canoutperform the underlying subcarrier modulation at the price of lower spectral efficiency.

84Figure 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 andTWTA models are considered. The lowest TD for the TWTA system is 10.5 dB at 8 dBbackoff, which corresponds to an 8% efficiency as shown in Figure 4.18(a). At this backofflevel, the 99.5% bandwidth occupancy is roughly the same as undistorted ideal OFDMas 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 operateat IBO = 0 dB thus maximizing amplifier efficiency. The total degradation is 5 dB for2πh = 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 = 0.98which is 8% less than ideal OFDM. This shows that the modulation index for CE-OFDMcan be chosen accordingly to balance performance and bandwidth. Also, since the PAimposes no additional distortion on the CE-OFDM signal, the resulting spectrum can bewell contained with no power backoff and at the same time have optimal PA efficiency.

8550Class-A PA efficiency, ηA (%)45403530252015105001234 5 6Input power backoff, IBO (dB)78910(a) PA efficiency.Total degradetion (dB)161412108642OFDM, TWTAOFDM, SSPAOFDM, idealCE-OFDM: 2πh = 0.40.50.600246Input power backoff, IBO (dB)810(b) Total degradation for target BER 10 −5 .99.5% bandwidth, f/W21.81.61.41.21OFDM, TWTAOFDM, SSPAOFDM, idealCE-OFDM: 2πh = 0.40.50.60.80246Input power backoff, IBO (dB)810(c) Spectral containment.Figure 4.18: A comparison of CE-OFDM and conventional OFDM. (M = 2, N = 64)

Chapter 5Performance of CE-OFDM inFrequency-Nonselective FadingChannelsIn this chapter, performance analysis of the phase demodulator receiver is extendedto fading channels. The lowpass equivalent representation of the received signal isr(t) = αe jφ 0s(t) + n(t) (5.1)where s(t) is the CE-OFDM signal according to (3.11), α and φ 0 is the channel amplitudeand 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) = αe jφ 0δ(τ). In the frequency domain,the channel is H(f) = F{h(τ)}(f) = αe jφ 0, and is thus constant at all frequencies—thatis, 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 channelmodel, since it’s frequency nonselective, is commonly referred to as flat fading.signal-to-noise ratio per bit for a given α isTheγ = α 2 E bN 0, (5.2)86

87and the average SNR per bit is [421, p. 817]¯γ = E{γ} = E { α 2} E bN 0. (5.3)It is desired to calculate the bit error rate at a given ¯γ, denoted here as BER(¯γ). Thisquantity 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]p γ (x) = (1 + K R)e −K [Rexp − (1 + K ] [ √]R)x K R (1 + K R )xI 0 2, x ≥ 0, (5.4)¯γ¯γ¯γwhere I 0 (·) is the 0th-order modified Bessel function of the first kind, andK R = ρ22σ 2 0(5.5)is the Rice factor: ρ 2 and 2σ 2 0represent the power of the LOS and scatter component,respectively [401, p. 40]. For channels without a line-of-sight, ρ → 0 and γ is Rayleighdistributed [483, p. 101]:p γ (x) = 1¯γ (exp − x¯γ), x ≥ 0. (5.6)To obtain BER(¯γ), the conditional BER is averaged over the distribution of γ [421, p.817]:BER(¯γ) =In Section 4.1.1 it is shown that∫ ∞0BER(x)p γ (x)dx. (5.7)BER(x) ≈ c 1 Q ( c 2√ x), (5.8)where c 1 = 2(M − 1)/(M log 2 M) and c 2 = 2πh √ 6 log 2 M/(M 2 − 1), so long as thesystem is above threshold. For the moment, assumeBER(x) = c 1 Q(c 2√ 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]BER Rice (¯γ) = c 1π∫ π/20(1 + K R ) sin 2 θ(1 + K R ) sin 2 θ + c 2[2¯γ/2×K R c 2exp −2¯γ/2 ](1 + K R ) sin 2 θ + c 2 2¯γ/2 dθ,(5.10)

88and for the Rayleigh channel, as described by (5.6), [483, p. 101]( √BER Ray (¯γ) = c 121 −)c 2 2¯γ/21 + c 2 2¯γ/2 . (5.11)However, as discussed in Section 4.1.3, the bit error rate of CE-OFDM, as a result ofthe 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 results 1 to (5.10) for an M = 8, N = 64 system inthe Ricean channel with K R = 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 ismore likely to experience channel fades which take the system below threshold—in whichcase the bit error rate isn’t accurately represented by the Q-function, that is, (5.9) isfalse. For the 2πh = 1.8 example, (5.10) is overly optimistic by at least 3 dB for allvalues of ¯γ. This is due to the inaccuracy of the Q-function for large modulation indexcases (see Figure 4.4, for example).10 0SimulationApprox (5.10)10 0SimulationApprox (5.11)10 −110 −1Bit error rate10 −210 −32πh 1.8 0.6Bit error rate10 −210 −32πh 1.2 0.410 −410 −410 −5 0 5 10 15 20 25 30Average signal-to-noise ratio per bit, ¯γ (dB)(a) M = 8, Ricean K R = 10 dB.10 −5 0 10 20 30 40 50Average signal-to-noise ratio per bit, ¯γ (dB)(b) M = 4, Rayleigh.Figure 5.1: Performance of CE-OFDM in flat fading channels. (N = 64)1 Unless otherwise stated, the simulation parameters—J, L fir , 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).

89Figure 5.1(b) further illustrates the inaccuracy of assuming (5.9). An M = 4, N = 64system is simulated in the Rayleigh channel. For the low modulation index case of2πh = 0.4, (5.11) is somewhat accurate. However, for the large modulation index caseof 2πh = 1.2, (5.11) is shown to be off by 5–7 dB.A Semi-Analytical ApproachThe 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 modulationindex. For a limited range of 2πh (for example, the values shown in Figure 4.16) thefollowing observation can be made: above a certain SNR, say x 0 , the conditional biterror rate closely matches the Q-function, that is, (5.8) holds. Therefore (5.7) can beapproximated asBER(¯γ) =≈∫ x00∫ x00BER(x)p γ (x)dx +BER(x)p γ (x)dx +∫ ∞x∫ 0∞x 0BER(x)p γ (x)dxc 1 Q(c 2√ x)pγ (x)dx.(5.12)Determining x 0 for a given M and 2πh, and dealing with ∫ x 00BER(x)p γ (x)dx in (5.12) arethe problems that remain to obtain an accurate approximation of BER(¯γ). As observedin Section 4.1.3 [see Figure 4.6(a)], at low SNR the bit error rate is roughly 1/2. Assumefor the moment that BER(x) = 1/2 for x ≤ x 0 ; thenBER(¯γ) ≈ 1 2∫ x00p γ (x)dx +∫ ∞x 0c 1 Q(c 2√ x)pγ (x)dx. (5.13)This simplified model, referred to as a two-region model since the conditional BER is splitinto two regions, is illustrated in Figure 5.2: below x 0 the BER is 1/2, otherwise the BERis equal to the Q-function. Also shown is the observed simulation result. Notice thatthe two-region model doesn’t account for the transition region in which BER(x) ≈ 1/2to where BER(x) ≈ c 1 Q(c 2√ x). [For more examples of the transition region, see Figure4.6.] Consequently, (5.13) is not generally accurate, and a more elaborate approach isrequired which accounts for the transition region.

901Transition regionConditional bit error rate, BER(x)0.50.1Two-region modelObserved (simulation)Q-function (4.35)0.01x 0Signal-to-noise ratio per bit, x (dB)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 ≤ x 0 into n sub-regions:∫ x00BER(x)p γ (x)dx =∫ γ1γ∫0γ2BER(x)p γ (x)dx+(5.14)BER(x)p γ (x)dx + . . . + BER(x)p γ (x)dx,γ 1 γ n−1where γ i > γ i−1 , i = 1, 2, . . . , n, γ 0 = 0 and γ n = x 0 . Due to the analytical difficultyof describing BER(x) over 0 ≤ x ≤ x 0 , computer simulation is used. The system issimulated at SNR values γ i , i = 1, 2, . . . , n−1, to get the result BER i , i = 1, 2, . . . , n−1.It is assumed that BER(x) ≈ BER i for γ i ≤ x ≤ γ i+1 to obtain the approximationn−1∑BER(¯γ) ≈i=0∫ γi+1∫ ∞∫ γn√BER i p γ (x)dx + c 1 Q(c 2 x)pγ (x)dx. (5.15)γ i γ nFor SNR in the range 0 ≤ x ≤ γ 1 the bit error rate is assumed to be BER 0 = 1/2. Figure5.3 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-regionmodel (5.13).CE-OFDM systems are simulated in Rayleigh and Ricean (K R = 3 dB and K R = 10dB) channels. The values of modulation index are as follows: for M = 2, 2πh ≤ 0.6;

911BER 0 = 1/2Conditional bit error rate, BER(x)BER 1BER 3BER 4.BER n−2BER n−10.01(n + 1)-region modelObserved (simulation)Q-function (4.35)BER 2← γ 0 = −∞γ 1 γ 2 γ 3 γ 4 . . . γ n−2 γ n−1 γ nSignal-to-noise ratio per bit, x (dB)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. Theresults are shown in Figure 5.4: the circles represent Rayleigh results; the squares andtriangles represent the Ricean results for K R = 3 dB and K R = 10 dB, respectively.The solid lines are the results of the semi-analytical approach, (5.15). The transitionregion is sampled every 0.5 dB, that is, γ i+1 − γ i = 0.5 dB, i = 1, 2, . . . , n − 1; thestarting point is γ 1 = −5 dB. Therefore γ i = 0.5(i − 1) − 5 dB, i = 1, 2, . . . , n. Thesampling continues until BER n < 0.01. For SNR x ≥ γ n the conditional bit error rateis approximated with the Q-function (5.8). This criteria used for γ n is based on theobservation that, for the modulation index values under consideration, the Q-function isaccurate for BER < 0.01. As shown in the figure, this semi-analytical approach yieldscurves for BER(¯γ) that closely match simulation.Figure 5.5 shows the improvement of (5.15) over (5.10) and (5.11). The semianalyticalapproach 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 accurateresult in a small fraction of the time required for direct simulation. For example, the

92Bit error rate(a) M = 2, 2πh = 0.210 −5 0 10 20 30 40 5010 −5 0 10 20 30 4010 010 0 (b) M = 2, 2πh = 0.610 −110 −110 −210 −210 −310 −310 −410 −4Average SNR per bit, ¯γ (dB)Average SNR per bit, ¯γ (dB)Bit error rate50Bit error rate(c) M = 4, 2πh = 0.410 −5 0 10 20 30 40 5010 −5 0 10 20 30 4010 010 0 (d) M = 4, 2πh = 1.210 −110 −110 −210 −210 −310 −310 −410 −4Average SNR per bit, ¯γ (dB)Average SNR per bit, ¯γ (dB)Bit error rate50Bit error rate(e) M = 8, 2πh = 0.610 −5 0 10 20 30 40 5010 −5 0 10 20 30 4010 010 0 (f) M = 8, 2πh = 1.810 −110 −110 −210 −210 −310 −310 −410 −4Average SNR per bit, ¯γ (dB)Average SNR per bit, ¯γ (dB)Bit error rate50Bit error rate(g) M = 16, 2πh = 0.810 −5 0 10 20 30 40 5010 −5 0 10 20 30 4010 010 0 (h) M = 16, 2πh = 2.410 −110 −110 −210 −210 −310 −310 −410 −4Average SNR per bit, ¯γ (dB)Average SNR per bit, ¯γ (dB)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)Bit error rate50

9310 010 −1Rayleigh simulationRayleigh approximation (5.11)Ricean (K R = 3 dB) simulationRicean (K R = 3 dB) approximation (5.10)Ricean (K R = 10 dB) simulationRicean (K R = 10 dB) approximation (5.10)Semi-analytical technique (5.15)Bit error rate10 −210 −310 −410 −5 0102030Average signal-to-noise ratio per bit, ¯γ (dB)Figure 5.5: Comparison of semi-analytical technique (5.15) with (5.10) and (5.11).(M = 4, N = 64, 2πh = 1.2)4050simulated Rayleigh result in Figure 5.5 requires about 6 hours of computer time (on aworkstation with 1 gigabytes of memory and a single 3 gigahertz microprocessor). Thesemi-analytical result, on the other hand, requires less than 7 s (to obtain {BER i }, andperform numerical integration): a speed improvement of 4 orders of magnitude.The disadvantage, however, is that this technique doesn’t yield a closed-form expression.As of the time of this writing, such a solution, that is general and accurate, doesn’tseem possible.

Chapter 6Performance of CE-OFDM inFrequency-Selective ChannelsIn this chapter the performance of CE-OFDM in frequency-selective channels is studied.The channel is time dispersive having an impulse response h(τ) that can be non-zeroover 0 ≤ τ ≤ τ max , where τ max is the channel’s maximum propagation delay. The receivedsignal isr(t) ==∫ ∞−∞∫ τmax0h(τ)s(t − τ)dτ + n(t)h(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 Gaussiannoise term represented by (4.15). The lower bound of integration in (6.1) is due to thelaw of causality [401, p. 245]: h(τ) = 0 for τ < 0. The upperbound is τ max since, bydefinition of the maximum propagation delay, h(τ) = 0 for τ > τ max .CE-OFDM has the same block structure as conventional OFDM, with a block period,T B , designed to be much longer than τ max . A guard interval of duration T g ≥ τ max isinserted between successive CE-OFDM blocks to avoid interblock interference. At thereceiver, r(t) is sampled at the rate f sa = 1/T sa samp/s, the guard time samples arediscarded and the block time samples are processed.outlined in Section 2.1.2, the processed samples arer p [i] = r[i] =N∑c−1m=0Using the discrete-time modelh[m]s[i − m] + n[i], i = 0, . . . , N B − 1. (6.2)94

95Note that the discarded samples are {r[i]} −1i=−N g. Transmitting a cyclic prefix duringthe guard interval makes the linear convolution with the channel equivalent to circularconvolution. Thusr p [i] = 1N DFT ∑−1N DFTk=0H[k]S[k]e j2πik/N DFT, i = 0, . . . , N B − 1, (6.3)where {H[k]} is the DFT of {h[i]} and {S[k]} is the DFT of {s[i]}. The effect of thechannel can be reversed with the frequency-domain equalizer: a DFT followed by amultiplier bank, followed by an IDFT. The FDE output isŝ[i] = 1N DFT ∑−1N DFTk=0R p [k]C[k]e j2πik/N DFT, i = 0, . . . , N B − 1, (6.4)where {R p [k]} is the DFT of the processed samples and {C[k]} are the equalizer correctionterms, which are computed as [463]C[k] = 1H[k](6.5)for the zero-forcing (ZF) criterion, andC[k] =for the minimum mean-square error (MMSE) criterion.H ∗ [k]|H[k]| 2 + (E b /N 0 ) −1 (6.6)Ignoring noise (n[i] = 0), the output of the frequency-domain equalizer using (6.5) isŝ[i] = 1N DFT ∑−1H[k]S[k]C[k]e j2πik/N DFTN DFTk=0N DFT ∑−11H[k]S[k]N DFT H[k] ej2πik/N DFTk=0N DFT∑−1S[k]e j2πik/N DFTN DFTk=0= 1= 1= s[i], i = 0, . . . , N B − 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, afade of −30 dB results in a correction term with gain +30 dB, which corrects the channelbut amplifies the noise by a factor of 1000. The MMSE criterion (6.6) takes into account

96the signal-to-noise ratio, making an optimum trade between channel inversion and noiseenhancement. Notice that the MMSE and ZF are equivalent at high SNR:lim C[k]| MMSE = H∗ [k]E b /N 0 →∞ |H[k]| 2 = 1H[k] = C[k]| ZF. (6.8)The system under consideration is shown in Figure 6.1.System performance isestimated by way of computer simulation.The samples {h[i]}, {s[i]} and {n[i]} aregenerated then used to calculate the received samples (6.2) which are then processed bythe FDE and the demodulator.CE-OFDMModulators(t)h(τ)r(t)r[i]RemoveCPr p[i]FDECE-OFDMDemodulatorn(t)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 MMSEand 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]} isdescribed statistically.In both sections an N = 64 CE-OFDM system is considered,with a block period of T B = 128 µs. The subcarrier spacing is 1/T B = 7812.5 Hz andthe mainlobe bandwidth is W = N/T B = 500 kHz. The guard period is T g = 10 µs,resulting in a transmission efficiency η t = 128/138 ≈ 0.93. The simulation uses anoversampling factor J = 8; therefore the sampling rate is f sa = JN/T B = 4 Msamp/s,and the sampling period is T sa = 1/f sa = 0.25 µs.6.1 MMSE versus ZF EqualizationIn this section, the performance of CE-OFDM using the MMSE and ZF frequencydomainequalizers is compared over six frequency-selective channels.6.1.1 Channel DescriptionThe channel samples {h[i]}, over the corresponding guard interval [0, 10 µs], areshown in Table 6.1. For Channels A–C the maximum propagation delay is τ max = 0.75

97Table 6.1: Channel samples of frequency-selective channels.Delay (µs) Channel A Channel B Channel C Channel D Channel E Channel Fi τ i = iT sa 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 +j0.671 0.25 0.80e −j2.22 0.30e −j2.90 0.70e +j2.00 0.47e −j1.01 0.24e +j0.98 0.47e −j0.952 0.50 0 0 0 0.61e +j0.26 0.51e −j0.06 0.33e +j2.583 0.75 0.10e −j0.37 0.20e +j2.97 0.07e +j0.98 0.42e −j0.01 0.21e −j2.12 0.22e +j0.104 1.00 – – – 0.23e +j1.09 0.24e +j1.14 0.25e −j1.925 1.25 – – – 0.10e +j1.00 0.11e +j1.64 0.16e −j0.206 1.50 – – – 0.18e +j1.82 0.25e −j1.28 0.14e −j2.307 1.75 – – – 0.13e +j2.36 0.12e −j0.93 0.21e −j1.148 2.00 – – – 0.13e −j0.60 0.27e +j1.82 0.13e +j0.349 2.25 – – – 0.12e +j1.00 0.12e +j1.49 0.16e −j2.4310 2.50 – – – 0.08e −j2.30 0.15e +j0.15 0.17e +j0.3611 2.75 – – – 0.09e −j1.91 0.19e +j0.23 0.08e −j0.9312 3.00 – – – 0.13e +j2.99 0.05e +j2.57 0.06e −j1.0813 3.25 – – – 0.04e −j1.97 0.07e −j0.17 0.05e +j0.1314 3.50 – – – 0.08e +j1.05 0.04e +j3.00 0.02e +j3.1115 3.75 – – – 0.08e +j1.01 0.05e −j1.20 0.07e −j2.8116 4.00 – – – 0.05e +j1.42 0.09e +j0.54 0.05e −j2.8717 4.25 – – – 0.06e −j0.18 0.12e −j0.10 0.04e −j1.3918 4.50 – – – 0.09e +j0.56 0.03e +j0.05 0.01e −j0.8919 4.75 – – – 0.05e +j0.72 0.03e +j0.96 0.02e −j2.0020 5.00 – – – 0.01e +j3.13 0.02e −j0.33 0.02e +j2.2221 5.25 – – – 0.05e +j1.11 0.03e −j1.53 0.03e +j0.9222 5.50 – – – 0.01e +j2.42 0.01e +j0.29 0.01e −j1.5623 5.75 – – – 0.02e −j1.92 0.02e +j2.58 0.02e +j0.5524 6.00 – – – 0.02e −j1.20 0.01e −j1.33 0.01e +j2.8325 6.25 – – – 0.03e +j2.07 0.02e −j1.96 0.02e +j0.4826 6.50 – – – 0.01e +j0.17 0.02e +j2.29 0.01e +j2.6827 6.75 – – – 0.01e −j0.93 0.03e +j2.86 0.01e +j2.0328 7.00 – – – 0.01e +j2.93 0.01e −j0.14 0.01e −j1.7629 7.25 – – – 0.02e −j2.91 0.01e −j0.36 0.01e −j2.4230 7.50 – – – 0.01e −j0.76 0.02e +j1.98 0.01e +j1.1131 7.75 – – – 0.01e −j1.88 0.01e −j2.38 0.01e +j0.0132 8.00 – – – 0.01e −j2.96 0.01e +j0.19 0.01e +j0.4033 8.25 – – – 0.01e −j0.89 0.02e +j2.18 0.01e +j1.6934 8.50 – – – 0.01e −j1.54 0.01e −j2.41 0.01e −j0.4935 8.75 – – – 0.01e −j3.01 0.01e −j3.11 0.01e +j2.6736 9.00 – – – – – –37 9.25 – – – – – –38 9.50 – – – – – –39 9.75 – – – – – –40 10.0 – – – – – –

98µs, which results in N c = ⌊τ max /T sa ⌋ + 1 = ⌊0.75/0.25⌋ + 1 = 4 samples [see (2.19)].For Channels D–F, τ max = 8.75 µs, thus N c = ⌊8.75/0.25⌋ + 1 = 36. The channels arenormalized such thatN∑c−1i=0|h[i]| 2 = 1. (6.9)Channels A–C are single realizations of an approximation to the maritime channelmodel in [350]. Channels D–F are single realizations of a stochastic model which has anexponential delay power density spectrum 1 .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 isplotted, where [422, p. 256]H(f ′ ) =N∑c−1i=0h[i]e −j2πf ′i , (6.10)is the Fourier transform of h[i]. The x-axis is scaled as [422, p. 24]f = f ′ f sa 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, thechannel 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,asH[k] =N∑c−1i=0h[i]e −j2πik/N DFT, k = 0, . . . , N DFT − 1, (6.12)H[k] = H(f ′ k ), k = 0, 1, . . . , N DFT − 1, (6.13)where the discrete set of frequencies {fk ′ } are defined asf ′ k ≡ ⎧⎪⎨⎪ ⎩kN DFT,kN DFT− 1,k = 0, 1, . . . , N DFT2,k = N DFT2+ 1, . . . , N DFT − 1.(6.14)1 Stochastic models are discussed in the next section.

990.40.350.30.25|h[i]| 20.20.150.10.0500123 4 5 6Propagation delay, iT sa (µs)789(a) Time domain.Magnitude response (dB)1050Channel D response, |H(f ′ )| 2Equalizer response, ZFEqualizer response, MMSE: E b /N 0 = 0 dB10 dB20 dB−5−200−1000Frequency, f = f ′ f sa (kHz)100200(b) Frequency domain.Figure 6.2: Channel D.

100Using a DFT size N DFT = JN = N B and noting (6.11), the frequency samples {H[k]}correspond to the frequencies⎧f k = f k ′ f sa = f k′ JN⎪⎨ kT= B,T B ⎪⎩ kT B− f sa ,k = 0, 1, . . . , N DFT2,k = N DFT2+ 1, . . . , . . . , N DFT − 1.(6.15)Included in Figure 6.2(b) is the response of the MMSE and ZF equalizers. The ZFresponse, (6.5), is simply the inverse of the channel. The MMSE response, (6.6), is shownfor E b /N 0 = 0, 10, and 20 dB. Notice that at high SNR the MMSE approaches the ZFequalizer, which is to be expected from (6.8). For this particular channel the MMSE andZF are shown to be equivalent for E b /N 0 ≥ 20 dB.6.1.2 Simulation ResultsThe N = 64 CE-OFDM system is simulated over Channels A–F. The modulationorder is M = 2, and different values of the modulation index, h, are selected. Due tothe channel normalization (6.9), the simulation results are compared against the simpleAWGN channel. The results are shown in Figures 6.3–6.8. For each case, |h[i]| 2 is plottedin subfigure (a); the channel and equalizer frequency-domain responses are plotted insubfigure (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, ChannelA 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 toeffectively correct the channel: the BER curves in Figure 6.3(c) are nearly indistinguishablefrom the simple AWGN curves. Results are plotted for 2πh = 0.1, 0.3 and 0.6. Forthe 2πh = 0.6 example at the lower SNR values E b /N 0 < 10 dB, the ZF result is shownto be slightly worse than the MMSE result; for higher values of SNR the performance ofthe two equalizers becomes nearly identical. This is to be expected since, as illustratedin Figure 6.3(b), their frequency response become the same at high E b /N 0 .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. Overthe signal bandwidth, |H(f ′ )| 2 spans a 6 dB range. As with the previous example, theMMSE is shown to slightly outperform the ZF at low SNR (i.e., the 2πh = 0.6 examplefor E b /N 0 < 10 dB), but the two equalizers have essentially the same performance at the

1010.70.62|h[i]| 20.50.40.30.20.1Magnitude response (dB)0−2−4−6ChannelMMSE: E b /N 0 = 0 dBZFA10 dB20 dB0012 3 4 5 6 7Propagation delay, iT sa (µs)89−200−100 0 100Frequency, f = f ′ f sa (kHz)200(a) Time domain.(b) Frequency domain.10 −1MMSEZFAWGN simAWGN approx (4.35)10 −2Bit error rate10 −32πh0.60.30.151010 −4 (c) Performance for MMSE and ZF compared to AWGN and (4.35).152025Signal-to-noise ratio per bit, E b /N 0 (dB)30Figure 6.3: Channel A results.

1020.90.84|h[i]| 20.70.60.50.40.30.20.1Magnitude response (dB)20−2−4−6ChannelMMSE: E b /N 0 = 0 dBZFB10 dB20 dB30 dB0012 3 4 5 6 7Propagation delay, iT sa (µs)89−200−100 0 100Frequency, f = f ′ f sa (kHz)200(a) Time domain.(b) Frequency domain.10 −1MMSEZFAWGN simAWGN approx (4.35)10 −2Bit error rate10 −32πh0.60.40.20.151010 −4 (c) Performance for MMSE and ZF compared to AWGN and (4.35).152025Signal-to-noise ratio per bit, E b /N 0 (dB)30Figure 6.4: Channel B results.

103higher SNR values. For BER ≤ 0.001 the degradation caused by the frequency selectivechannel, 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 channelrealizations. As shown in Figure 6.5(b), the magnitude response varies over a 20 dBrange. It is also shown that very high SNR is required for the MMSE response toapproach the ZF response. Over the frequency range −250 kHz ≤ f ′ f sa ≤ −200 kHz,for example, the two are equivalent only for E b /N 0 > 35 dB. This equivalence is alsodemonstrated in Figure 6.5(c): for the 2πh = 0.1 example, the ZF performance graduallyapproaches the MMSE performance at these high SNR values. Clearly, the large amountof frequency selectivity of this channel results in a large performance degradation whencompared to the AWGN results. At the bit error rate 0.001, the degradation is 10 dBfor 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 dBworse 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 channelsare three different realizations of a stochastic model with an exponential delay powerdensity spectrum. The degree that the each channel varies over the signal bandwidthprogresses from Channel D to Channel F. Channel F, having a 50 dB attenuation at185 kHz, is the most harsh of the test channels. The results in Figure 6.8(c) show thedramatic 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 frequencyselective channels having deep fades in the signal bandwidth impact performancegreatly.6.1.3 Discussion and ObservationsAt this point, several observations can be made. First, the performance of theequalized CE-OFDM systems studied depends on the amount of frequency selectivityover the signal bandwidth. For channels with a relatively mild frequency response—Channels A, B and D, for example—the performance degradation is minor. The noiseenhancement that results from equalizing channels with severe frequency responses—

104|h[i]| 20.60.50.40.30.2Magnitude response (dB)20100−10ChannelMMSE: E b /N 0 = 0 dBZFC10 dB20 dB30 dB35 dB0.1−200012 3 4 5 6 7Propagation delay, iT sa (µs)89−200−100 0 100Frequency, f = f ′ f sa (kHz)200(a) Time domain.(b) Frequency domain.10 −1MMSEZFAWGN simAWGN approx (4.35)10 −2Bit error rate10 −32πh0.50.151010 −4 (c) Performance for MMSE and ZF compared to AWGN and (4.35).15 20 25 30Signal-to-noise ratio per bit, E b /N 0 (dB)3540Figure 6.5: Channel C results.

1050.460.354|h[i]| 20.30.250.20.150.1Magnitude response (dB)20−2−4ChannelMMSE: E b /N 0 = 0 dBZFD10 dB20 dB0.05−60012 3 4 5 6 7Propagation delay, iT sa (µs)89−8−200−100 0 100Frequency, f = f ′ f sa (kHz)200(a) Time domain.(b) Frequency domain.10 −1MMSEZFAWGN simAWGN approx (4.35)10 −2Bit error rate10 −32πh0.60.20.151010 −4 (c) Performance for MMSE and ZF compared to AWGN and (4.35).152025Signal-to-noise ratio per bit, E b /N 0 (dB)3035Figure 6.6: Channel D results.

106|h[i]| 20.350.30.250.20.150.1Magnitude response (dB)1050−5ChannelMMSE: E b /N 0 = 0 dBZFE10 dB20 dB30 dB0.050012 3 4 5 6 7Propagation delay, iT sa (µs)89−10−200−100 0 100Frequency, f = f ′ f sa (kHz)200(a) Time domain.(b) Frequency domain.10 −1MMSEZFAWGN simAWGN approx (4.35)10 −2Bit error rate10 −32πh0.60.151010 −4 (c) Performance for MMSE and ZF compared to AWGN and (4.35).15 20 25 30Signal-to-noise ratio per bit, E b /N 0 (dB)3540Figure 6.7: Channel E results.

1070.4|h[i]| 20.350.30.250.20.150.1Magnitude response (dB)40200−20ChannelMMSE: E b /N 0 = 0 dBZFF10 dB20 dB30 dB0.05−400012 3 4 5 6 7Propagation delay, iT sa (µs)89−200−100 0 100Frequency, f = f ′ f sa (kHz)200(a) Time domain.(b) Frequency domain.10 −1MMSEZFAWGN simAWGN approx (4.35)10 −2Bit error rate10 −30.60.30.10.6 0.3 0.12πh510152010 −4 (c) Performance for MMSE and ZF compared to AWGN and (4.35).25 30 35 40 45 50 55Signal-to-noise ratio per bit, E b /N 0 (dB)606570Figure 6.8: Channel F results.

108Channels C, E and F—degrades performance dramatically. Second, the complexity ofthe frequency-domain equalizers is determined by the DFT size, not by the number ofnon-zero channel terms h[i]. This is in contrast to conventional time-domain equalizerswhich 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 SNRper bit, E b /N 0 , must be estimated at the receiver. The results of this study show thatthis added complexity doesn’t always translate into improved performance. That is, theZF performance is the same as the MMSE performance for many cases—the 2πh ≤ 0.4cases in Channel B, for example. In other cases, the MMSE performs much better, andthus estimating E b /N 0 pays substantial dividends—the 2πh = 0.5 case for Channel Cillustrates this point.As demonstrated in the following section, the MMSE equalizer offers significant improvementover the ZF equalizer when averaging performance over many channel realizationsof a stochastic channel model.6.2 Performance Over Frequency-Selective Fading ChannelsIn contrast to the test channels used in the previous section, which were deterministicas defined in Table 6.1, the channels used in this section are described statistically.The mathematical foundation for stochastic time-variant linear channels was pioneeredby Bello [50]; more recently Pätzold’s text, Mobile Fading Channels [401], provides aexcellent treatment of the topic, with a focus on the various aspects of simulation. In thestudy here, the widely used assumption of WSSUS (wide-sense stationary uncorrelatedscattering) 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 basedon the Parsons and Bajwa ellipse model for describing multipath channel geometry [401,p. 244]. The channel’s impulse response isL−1∑h(τ) = a l δ(τ − τ l ), (6.16)l=0

109where a l is the complex channel gain and τ l is discrete propagation delay of the lth path;the total number of paths is represented by L. The propagation delay differences are∆τ l = τ l − τ l−1 ≡ T sa , l = 1, 2, . . . , L − 1. (6.17)That is, they are set equal to the sampling period of the simulation [401, p. 269]. Thedelay of the 0th path is defined as τ 0 ≡ 0, thusτ l = lT sa , l = 0, 1, . . . , L − 1. (6.18)For each simulation trial, the set of path gains {a l } L−1l=0gain is complex valued, has a zero mean and a varianceare generated randomly. Eachσ 2 a l= E { |a l | 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 |a l | 2 is Rayleigh distributed. Also, the channels are normalizedsuch thatL−1∑σa 2 l= 1. (6.20)l=0As outlined in Pätzold’s text (pp. 276–279) the parameters σ 2 a l, τ l and L determinethe fundamental characteristic functions and quantities of the channel models, such as thedelay power spectral density and the delay spread 2 . The relevant formulas are expressedbelow.• Delay power spectral density:L−1∑S ττ (τ) = σa 2 lδ(τ − τ l ). (6.21)l=0• Average delay:L−1∑B ττ (1) = σa 2 lτ l . (6.22)l=02 The 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 Pätzold’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 ∑ ((σ al τ l ) 2 −l=0B (1)ττ) 2(6.23)• Frequency correlation function:L−1∑r ττ (v ′ ) = σa 2 le −j2πv′ τ l(6.24)l=0The variable v ′ is referred to as the frequency separation variable [401, p. 278].• Coherence bandwidth: The coherence bandwidth is the smallest positive valueB C which fulfils |r ττ (B C )| = 0.5|r ττ (0)|; which, due to (6.20) and (6.24), is equivalentto∣ ∣ ∣∣∣∣ L−1∑∣∣∣∣σa 2 le −j2πB Cτ l− 1 = 0. (6.25)2l=0Notice that B C is the 3 dB bandwidth of r ττ (v ′ ).6.2.1 Channel ModelsCE-OFDM is simulated over four frequency-selective fading channel models. Table6.2 defines the parameters {σa 2 l} and {τ l }. Channel A f and B f are similar to the maritimechannel models in [350] 3 . Both have a secondary path with a 5 µs propagation delay.Channel A f has a weak secondary path (one-tenth, i.e., −10 dB, the power of the primarypath); Channel B f has a stronger secondary path (one-half, i.e., −3 dB, the power of theprimary path).Channel C f has an exponential delay power spectral density:σ 2 a l ,C = ⎧⎪⎨⎪ ⎩C Cf e −τ l/2µs ,0 ≤ τ l ≤ 8.75 µs,0, otherwise,(6.26)where/ 35∑C Cf = 1 exp(−τ l /2e-6) = 0.1188 . . . (6.27)l=0is the normalizing constant used to guarantee (6.20). Note that the maximum propagationdelay is 8.75 µs.3 To avoid notational ambiguities, the channel model labels in this section have the subscript “f”(“fading”).

111Table 6.2: Channel model parameters.Path no. Delay (µs) Channel A f Channel B f Channel C f Channel D fl τ l = lT sa σa 2 l ,A σa 2 l ,B σa 2 l ,C σa 2 l ,D0 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

112The last model, Channel D f , has a uniform delay power density spectrum:σ 2 a l ,D = ⎧⎪⎨⎪ ⎩C Df ,0 ≤ τ l ≤ 8.75 µs,0, otherwise,(6.28)where the normalizing constant isC Df = 1/36. (6.29)In Figure 6.9 the delay power density spectrum (6.21) and the frequency correlationfunction (6.24) are plotted for each of the four models. The corresponding averagedelay (6.22), delay spread (6.23) and coherence bandwidth (6.25) for each model islabeled. Notice that Channel D f has the smallest coherence bandwidth, B C = 67 kHz.For Channel A f the coherence bandwidth isn’t finite since, as shown in subfigure (b),|r ττ (v ′ )| > −3 dB for all frequency separation values 4 .6.2.2 Simulation Procedure and Preliminary DiscussionThe average performance of various CE-OFDM systems is evaluated over the fourstochastic channel models. This is done by randomly generating {a l }—which, as statedabove, are complex-valued quantities, drawn from the Gaussian distribution, with zeromean and variance {σa 2 l}—computing the received samples (6.2), then processing thesamples with the frequency-domain equalizer and the CE-OFDM demodulator. At eachaverage E b /N 0 considered, the simulation runs for at least 20,000 bit errors, or until100,000,000 bits are transmitted, whichever happens first. This corresponds to manythousands of channel realizations 5 . Some channel realizations result in very poor performance(for example, see Figure 6.8), while others result in a bit error rates not muchworse than that of the simple AWGN channel. This performance difference is attributedto the severity of the channel’s frequency response, as observed with the several examplesin 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, thechannel may be fading such that the gain is less than unity, resulting in degraded performance.The likelihood of a deep channel fade depends on the number of independent4 For Channel A f , min |r ττ(v ′ )| = min ˛˛ 10+ 111 11 exp(−j2πv′ 5 µs)˛˛ =95 Example simulation code can be found in Appendix C.11 > 1 2≈ −3 dB.

11310 log 10 [Sττ (τ)]0−5−10−15−20−25(a) Delay power spectral density, Channel A fB ττ (1) = 0.45 µsB ττ (2) = 1.44 µs10 log 10 [rττ (v ′ )](b) Frequency correlation function, Channel A f0−3B C → ∞−6−9−12−30012 3 4 5 6 7Propagation delay, τ (µs)8910−15−450 −300 −150 0 150 300 450Frequency separation, v ′ (kHz)10 log 10 [Sττ (τ)]0−5−10−15−20−25(c) Delay power spectral density, Channel B fB ττ (1) = 1.67 µsB ττ (2) = 2.36 µs10 log 10 [rττ (v ′ )](d) Frequency correlation function, Channel B f0−3B C−6−9−12−30012 3 4 5 6 7Propagation delay, τ (µs)8910−15−450 −300 −150 0 74 150 300 450Frequency separation, v ′ (kHz)10 log 10 [Sττ (τ)]0−5−10−15−20−25(e) Delay power spectral density, Channel C fB ττ (1) = 1.78 µsB ττ (2) = 1.75 µs10 log 10 [rττ (v ′ )](f) Frequency correlation function, Channel C f0−3B C−6−9−12−30012 3 4 5 6 7Propagation delay, τ (µs)8910−15−450 −300 −150 0 140 300 450Frequency separation, v ′ (kHz)10 log 10 [Sττ (τ)]0−5−10−15−20−25(g) Delay power spectral density, Channel D fB ττ (1) = 4.38 µsB ττ (2) = 2.60 µs10 log 10 [rττ (v ′ )](h) Frequency correlation function, Channel D f0−3B C−6−9−12−30012 3 4 5 6 7 8Propagation delay, τ (µs)910−15−450 −300 −150 0 67 150 300 450Frequency separation, v ′ (kHz)Figure 6.9: Fundamental characteristic functions and quantities [(6.21)–(6.25)] of thefour channel models considered.

114propagation paths [the WSSUS assumption makes each path in (6.16) independent]. It isunlikely that multiple paths fade simultaneously. For this reason, channels characterizedby 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 D f can be said to have the most multipath diversity: the gain of a given realizationdepends on 36 independent paths, each having, on average, an equal contribution.Channel A f can be said to have the least amount of multipath diversity: over 90% of thechannel gain depends on a single path. Channel B f has more multipath diversity thanChannel A f 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 alsoon the way in which the power is distributed over the paths, as determined by {σa 2 l}. [Itis worth noting that the frequency-nonselective channel models considered in Chapter 5have L = 1 path of which 100% of the channel gain depends (σ 2 a 1= 1), and thus thesechannels have no multipath diversity.] In the results that follow, the impact of multipathdiversity—and its frequency-domain dual frequency diversity—on CE-OFDM systems isstudied.6.2.3 Simulation ResultsThe simulation results of this study are presented over three figures: Figure 6.10compares the performance of a CE-OFDM system, with fixed modulation order M andmodulation index h, over the four channel models; Figure 6.11 compares the performanceof a CE-OFDM system with fixed M but varying h over Channel C f ; and Figure 6.12compares the performance of constant envelope and conventional OFDM systems, in thepresence of power amplifier nonlinearities, over Channel C f . For each case, the numberof subcarriers is N = 64.In Figure 6.10, performance results of an M = 4, N = 64, 2πh = 1.0 CE-OFDMsystem are plotted. The simulation results over the multipath channel models A f –D f arelabeled with circles and triangles; the MMSE equalized results have solid lines connectingthe points, while the ZF equalized results use dashed lines. For reference, the performanceof the system over the simple AWGN channel is plotted (with dash-dot lines) along withthe performance over the Rayleigh frequency-nonselective fading channel (representedby the thick solid line). These results show the significant performance improvement

11510 −1MMSE: Channel A fB fC fD fZF: Channel A fB fC fD fRayleigh, L = 1AWGNAWGN approx (4.35)Bit error rate10 −210 −310 −4 51015 20 25 30 35Average signal-to-noise ratio per bit, E b /N 0 (dB)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)40that 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 D f . These results also show the impact ofmultipath diversity. Consider the MMSE results. For E b /N 0 > 15 dB, the performanceover Channels A f –D f is better than the performance over the frequency-nonselectiveRayleigh (L = 1 path) channel. For BER ≤ 0.001, the performance over the multipathchannels is at least 5 dB better than the performance over the single path channel.Notice that Channel D f , which has the most multipath diversity, results in a betterperformance that all the other channels. The performance over Channel B f , which hasmore multipath diversity than Channel A f , is in fact better than the performance overChannel A f . These results indicate that the CE-OFDM receiver exploits the multipathdiversity of the channel.The fact that constant envelope OFDM exploits multipath diversity is an interestingresult since conventional OFDM doesn’t. This was shown in Section 2.1.1; specifically,

116by (2.9).So long as the duration of the guard interval is greater than or equal tothe channel’s maximum propagation delay, that is, T g ≥ τ max , and a cyclic prefix istransmitted during the guard interval, the performance of OFDM in a time-dispersivechannel is equivalent to flat fading performance. In other words, the multipath fadingperformance is the same as single path fading performance. In the context of Section2.1.1, 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 thechannel isn’t leveraged 6 .To understand why CE-OFDM has improved performance over multipath fadingchannels (compared to single path fading channels) while OFDM doesn’t, it is best toview the problem in the frequency domain. The frequency domain dual to multipathdiversity is frequency diversity.It can be said that OFDM lacks frequency diversityas well. As identified in Section 1.1.2, the wideband frequency-selective fading channelis converted into N contiguous frequency-nonselective fading channels. Therefore anyfrequency diversity inherent to the channel—that is, over the signal bandwidth the frequencyresponse of the channel varies, which can be taken advantage of by the receiverto 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 thechannel since the phase modulator, in effect, spreads the data symbol energy in thefrequency domain. This can be seen by viewing the CE-OFDM waveform by the Taylorseries expansion [see Section 3.2, (3.24)]:[s(t) = A1 + jσ φ m(t) − σ2 φ2 m2 (t) − j σ3 φ6 m3 (t) + . . .], (6.30)0 ≤ t < T B , where A is the signal amplitude, σ 2 φ = (2πh)2 is the phase signal variance,and m(t) = C N∑ Nk=1 I kq k (t), 0 ≤ t < T B , C N= √ 6/N(M 2 − 1), is the normalizedOFDM message signal. The higher-order terms m n (t), n ≥ 2, results in a frequencyspreading of the data symbols. This property is best demonstrated by way of a simpleexample.Example 6.2.1Consider a CE-OFDM waveform with an OFDM message signal composed of N = 2 orthogonal6 Note 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.

117cosine subcarriers modulated with binary data symbols (M = 2):2∑m(t) = I k cos 2πkt/T B , 0 ≤ t < T B , (6.31)k=1where I k ∈ {±1}, k = 1, 2. Assume that the modulation index, h, is such that the higher-orderterms m 2 (t) and m 3 (t) contribute to the make up of s(t) according to (6.30). It is desired towrite m 2 (t) and m 3 (t) in terms of I 1 , I 2 and {cos 2πkt/T B }. This task requires some algebra,but is simply done. For notational simplicity, let’s definec k ≡ cos 2πkt/T B . (6.32)Thus, (6.31) is written asm(t) = I 1 c 1 + I 2 c 2 . (6.33)The second-order term is calculated asm 2 (t) = (I 1 c 1 + I 2 c 2 )(I 1 c 1 + I 2 c 2 )= ( 0.5I 2 1 + 0.5I 2 2)c0 + (I 1 I 2 ) c 1 + ( 0.5I 2 1)c2 + (I 1 I 2 ) c 3 + ( 0.5I 2 2)c4 ,(6.34)and the third-order term asm 3 (t) = [ ( 0.5I 2 1 + 0.5I 2 2)c0 + (I 1 I 2 ) c 1 + ( 0.5I 2 1)c2+ (I 1 I 2 ) c 3 + ( 0.5I 2 2)c4](I1 c 1 + I 2 c 2 )= ( 0.75I 2 1 I 2)c0 + ( 0.75I 3 1 + 1.5I 1I 2 2)c1 + ( 1.25I 2 1 I 2 + 0.5I 3 2)c2(6.35)+ ( 0.25I1 3 + 0.75I 1 I2 2 )c3 + ( 0.75I1I 2 )2 c4+ ( 0.75I 1 I2 2 )c5 + ( 0.25I2) 3 c6 .The expansions above are represented in Table 6.3. The data symbol contribution at eachtone cos 2πkt/T B , k = 0, 1, . . . , 6, for m(t), m 2 (t) and m 3 (t) is shown. Referring to the tonesas frequency bins, it can be said that for m(t) the two data symbols are simply contained inthe k = 1 and k = 2 frequency bins. For the second-order term, m 2 (t), the data symbols mixacross the k = 0, 1, 2, 3, and 4 frequency bins. For m 3 (t), the data symbols mix across thek = 0, 1, . . . , 6 frequency bins.The simple example above shows how the data symbols spread across multiple frequencybins. In general, it can be said that the N data symbols that constitute theconstant envelope OFDM signal are not simply confined to N frequency bins—as is thecase with conventional OFDM. The phase modulator mixes and spreads—albeit in anonlinear and exceedingly complicated manner—the data symbols in frequency, whichgives the CE-OFDM system the potential to exploit the frequency diversity in the channel.This isn’t necessarily the case, however. For small values of modulation index,

118Table 6.3: Data symbol contribution per tone for m n (t), n =1, 2, and 3.kth tone, cos 2πkt/T B0 1 2 3 4 5 6m(t) – I 1 I 2 – – – –m 2 (t)0.5I 2 1 ,0.5I 2 2m 3 (t) 0.75I 2 1 I 2I 1 I 2 0.5I 2 1 I 1 I 2 0.5I 2 2 – –0.75I 3 1 ,1.5I 1 I 2 21.25I 2 1 I 2,0.5I 3 20.25I 3 1 ,0.75I 1 I 2 20.75I 2 1 I 2 0.75I 1 I 2 2 0.25I 3 2where only the first two terms in (6.30) contribute, that is,s(t) ≈ A [1 + jσ φ m(t)] , (6.36)the CE-OFDM signal doesn’t have the frequency spreading given by the higher-orderterms. In this case, the CE-OFDM signal is essentially equivalent to a conventionalOFDM signal, jσ φ m(t), (plus a relatively large DC term, A) and therefore doesn’t havethe ability to exploit the frequency diversity of the channel. Simply put, CE-OFDMhas frequency diversity when the modulation index is large and doesn’t have frequencydiversity 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 Rayleighflat fading channel and over the multipath fading model Channel C f . To demonstrate thatCE-OFDM with a small modulation index lacks frequency diversity, results for 2πh = 0.1are shown. Notice that the single path and multipath performance is essentially thesame. By contrast, for the large modulation index example 2πh = 1.1, the multipathperformance is significantly better than the single path performance. For example, atthe bit error rate 0.001 the multipath performance is over 10 dB better than the singlepath performance.In the final figure, Figure 6.12, the performance of constant envelope OFDM iscompared to conventional OFDM in the presence of power amplifier nonlinearities. TheSSPA model (see Section 2.3) is used at various input backoff levels. The x-axis isadjusted to account for the negative impact of input power backoff. The systems aresimulated over Channel C f . For the OFDM system, QPSK data symbols are used. Threedifferent CE-OFDM systems are tested: M = 4, 2πh = 0.9; M = 8, 2πh = 2.0; andM = 16, 2πh = 3.0. The advantage of the CE-OFDM systems is twofold. First, theCE-OFDM systems operate with IBO = 0 dB. Second, the CE-OFDM systems exploit

11910 0MultipathSingle path10 −1Bit error rate10 −210 −32πh1.10.110 −4 51015 20 25 30 35 40Average signal-to-noise ratio per bit, E b /N 0 (dB)4550Figure 6.11: Single path versus multipath. (M = 4, N = 64, Channel C f , MMSE)the frequency diversity inherent to the channel.At the bit error rate 0.001 the CE-OFDM systems outperform the OFDM systemby at least 10 dB. At this bit error rate, the OFDM system has essentially the sameperformance with backoff levels of 6 and 10 dB; therefore, IBO = 6 dB is preferred sincethe performance is the same but the power efficiency is higher (see Figure 2.14). Evenso, the 6 dB backoff required by the OFDM system is still far less desirable as the 0 dBbackoff used by the CE-OFDM system. Notice that the OFDM system with IBO = 0dB 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 atlow SNR due to the threshold effect (as studied in Section 4.1.3). Over the region 0 dB≤ E b /N 0 ≤ 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 largemodulation index values (2πh = 2.0 and 2πh = 3.0 respectively) which results in spectralbroadening. Roughly speaking, the spectral efficiency of the QPSK/OFDM system is 2b/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 and1.3 b/s/Hz, respectively.Making a direct comparison between CE-OFDM and conventional OFDM is difficult

12010 −1Bit error rate10 −210 −3OFDM: IBO = 0 dB3 dB6 dB10 dBCE-OFDM: M = 4, 2πh = 0.9M = 8, 2πh = 2.0M = 16, 2πh = 3.010 −4 05101520 25E b /N 0 + IBO (dB)Figure 6.12: CE-OFDM versus QPSK/OFDM. (SSPA model, Channel C f , N = 64,MMSE)303540due to the various parameters involved (M, 2πh, IBO, etc.), and due to the fact thatsystem requirements vary from system to system. For example, if power amplifier efficiencyis the most important requirement, then the input power backoff of 0 dB shouldbe chosen. At this backoff level, the OFDM system has a very high irreducible errorfloor due to the power amplifier distortion, while the CE-OFDM system is relativelyunaffected. Alternatively, if operation at low SNR is important, then CE-OFDM maynot be well suited due to the threshold effect.The results in this chapter show that CE-OFDM can perform quite well in multipathfading channels—so long as the channel information (i.e., {H[k]}) is known at the receiverand so long as the added complexity of the frequency-domain equalizer (i.e., two extraFFTs) 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 couldprovide for interesting results.

Chapter 7ConclusionsIn this thesis the peak-to-average power ratio problem associated with orthogonalfrequency division multiplexing is evaluated. The PAPR statistics are studied and theeffect of power amplifier nonlinearities as a function of power backoff is evaluated bycomputer simulation. It is shown that the amount of backoff required to reduce spectralgrowth and performance degradation is significant: 6–10 dB depending on the subcarriermodulation used. Large backoff is an unsatisfactory solution for battery-powered systemssince PA efficiency is low.A signal transformation method for solving the PAPR problem is presented andanalyzed. The high PAPR OFDM signal is transformed to a 0 dB PAPR constant envelopewaveform. At the receiver, the inverse transform is performed prior to the OFDMdemodulator. For the CE-OFDM technique described, phase modulation is used. Theeffect of the phase modulator on the transmitted signal’s spectrum is studied. It is shownthat the modulation index controls the spectral containment. The modulation index alsocontrols the system performance. The optimum receiver is analyzed and a performancebound and approximation is derived. For a large modulation index, the CE-OFDM signalsbecome less correlated which improves detection performance. The approximationof the optimum receiver closely matches simulation results. It also closely matches aderived bit error rate approximation for a practical phase demodulator receiver. For asmall modulation index and high signal-to-noise ratio, the phase demodulator receiver isnearly optimum. For a larger modulation index the phase demodulator receiver becomessub-optimum due to the limitations of the phase demodulator and phase unwrapper.121

122This problem can be suppressed with the use of a properly designed finite impulse responselowpass filter which precedes the phase demodulator.The simulation results of the CE-OFDM performance curves use an oversamplingfactor of J = 8. Future work includes experimenting with lower sampling rates forreduced receiver complexity. The performance of the phase demodulator is a crucialelement to the overall CE-OFDM performance. Therefore, further research is neededto evaluate more advanced phase demodulation techniques such as digital phase-lockedloops.Phase modulation is used exclusively in this work. It would be interesting to evaluateCE-OFDM frequency modulation systems and compare them to the results in this thesis.In terms of performance over frequency-selective fading channels, the frequencydomainequalizer requires knowledge of the channel. Many conventional OFDM systems(those that don’t use differentially encoded modulations) also require channel state information.Thus techniques for channel estimation in OFDM has been extensively researched[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 complexitysingular value decomposition (SVD) approaches, to CE-OFDM is a subject for futureinvestigation. The impact of imperfect channel state information on the performance ofthe frequency-domain equalizer is of interest.CE-OFDM might be used as a stand-alone modulation technique or as a supplementto an existing OFDM system. For example, a conventional OFDM system is designed forsevere multipath channels. However, at times the channel might be relatively benign sothe OFDM systems is an overkill and, due to power backoff, inefficient. An adaptive radiomight sense times where power efficient CE-OFDM, which requires minimal backoff, ismore applicable. Such a system can adaptively switch between conventional and constantenvelope modes.For systems, such as power-limited satellite communications, where a constant envelopeis very desirable, if not required, CE-OFDM might be a viable alternative toconvention continuous phase modulation systems which are complex due to phase trellisdecoding and sensitive to multipath. CE-OFDM is relatively robust in multipath fadingchannels with the use of the frequency-domain equalizer. Depending on the channelcondition, equalization might not be required, therefore reducing receiver complexity.

123For example, a channel characterized by a two-path model with a weak secondary path,CE-OFDM might provide acceptable performance without equalization. CPM systemsin 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 Researchunder an STTR (small business technology transfer) initiative with UCSD beingthe university partner. The goal of the prototype is to offer a second low-power modefor the existing JTRS (Joint Tactical Radio System) wideband component which usesOFDM. Research challenges that remain include evaluating CE-OFDM with many subcarriers(in this thesis, only 64 subcarriers are used), considering different equalizationtechniques, developing synchronization schemes and studying the impact of channel codingand the effects of time-varying channels.Additional future work includes comparing CE-OFDM with other block modulationtechnique in terms of PAPR, spectral efficiency, power amplifier efficiency, performanceand complexity. There has been an increasing amount of attention given to conventionalsingle carrier modulation with the addition of a cyclic prefix which allows for frequencydomainequalization [107, 154, 460, 463, 574]. However, most single carrier modulationshave a non-constant envelope due to pulse shaping and multilevel QAM symbol constellations.A study is needed to compare these modulation techniques to CE-OFDMtaking into account the effects of the PA at various backoff levels. Also, using CPM witha cyclic prefix is an interesting idea. Comparing the complexity and spectral efficiency ofsuch a technique with CE-OFDM would be interesting. Such research will help provideinsight into good designs for future wireless digital communication systems that requirepower efficiency and high data rates.

Appendix AGenerating Real-Valued OFDMSignals with the Discrete FourierTransformFor some applications, a real-valued OFDM signal is required. This can be done bytaking a DFT of a conjugate symmetric vector. The spectral efficiency of the real-valuedOFDM signal is the same as the spectral efficiency of the complex-valued OFDM signal.A.1 Signal DescriptionThe baseband OFDM signal is typically written asx(t) =N−1∑k=0X k e j2πkt/T B, 0 ≤ t < T B , (A.1)where N is the number of subcarriers, {X k } N−1k=0 are the data symbols and T B is theblock period. Sampling x(t) at N equally spaced intervals over 0 ≤ t < T B yields thesequence,x[i] = x(t)| t=iTB /N =N−1∑k=0X k e j2πki/N , i = 0, 1, . . . , N − 1, (A.2)which is the inverse discrete Fourier transform (IDFT) of the vectorX = [X 0 , X 1 , . . . , X N−1 ].(A.3)124

125The sequence is complex-valued in general.making X conjugate symmetric:However it can be made real-valued byX N/2+k = X ∗ N/2−k(A.4)andThe IDFT is thenX 0 = X N/2 = 0.(A.5)x[i] ===N−1∑k=1N/2−1∑k=1N/2−1∑k=1X k e j2πki/NX N/2−k e j2π(N/2−k)i/N + X N/2+k e j2π(N/2+k)i/NX N/2−k e j2π(N/2−k)i/N + X ∗ N/2−k ej2π(N/2+k)i/N ,(A.6)i = 0, 1, . . . , N − 1. But sincee j2π(N/2+k)i/N = e j2π(N/2+k)i/N e −j2πNi/N= e j2π(−N/2+k)i/N(A.7)= e −j2π(N/2−k)i/N ,(A.6) can be written asx[i] =N/2−1∑k=1X N/2−k e j2π(N/2−k)i/N + X ∗ N/2−k e−j2π(N/2−k)i/N ,(A.8)i = 0, 1, . . . , N − 1. Using the identity A + A ∗ = 2R{A},⎧⎫⎨N/2−1∑⎬x[i] = 2R X⎩ N/2−k e j2π(N/2−k)i/N ⎭k=1⎧⎫ (A.9)⎨N/2−1∑⎬= 2R X⎩ k e j2πki/N ⎭ , i = 0, 1, . . . , N − 1.k=1And since R{AB} = R{A}R{B} − I{A}I{B},N/2−1∑x[i] = 2 R{X k } cos(2πki/N) − I{X k } sin(2πki/N),k=1(A.10)

126i = 0, 1, . . . , N − 1. Thus, x[i] is real. Passing the sequence through a D/A converteryields the continuous-time real-valued OFDM signal:N/2−1∑x(t) = 2 R{X k } cos(2πkt/T B ) − I{X k } sin(2πkt/T B ).k=1(A.11)Now, suppose the data symbols are derived from a M 2 -QAM (quadrature-amplitudemodulation) constellation; that is,X k = R{X k } + jI{X k },(A.12)whereR{X k }, I{X k } ∈ {±1, ±3, ±(M − 1)}, for all k. (A.13)In other words, the real and imaginary components are derived from M-PAM (pulseamplitudemodulation) constellations. Therefore, processing M 2 -QAM data with theIDFT, (A.11) is a real-valued M-PAM OFDM signal.A.2 Spectral EfficiencyComplex-valued baseband signals are transmitted as bandpass signals, centered ata carrier frequency f c Hz. This is the case for the complex-valued signal in (A.1). Thetransmitted signal is represented ass 1 (t) = R{x(t)e j2πfct} .(A.14)In the frequency domain, x(t) is shifted to the right by f c Hz, and the subcarriers arecentered at f c , f c + 1/T B , f c + 2/T B , . . . , f c + (N − 1)/T B Hz. The effective bandwidth ofthe signal is therefore N/T B Hz. Each data symbol represents log 2 M bits (i.e., they areassumed to be selected from a M-ary constellation), therefore the spectral efficiency isS 1 =Bits per second (b/s)Bandwidth (Hz)= N log 2 M/T BN/T B= log 2 M b/s/Hz. (A.15)The real-valued OFDM signal in (A.11) has the same spectral efficiency as thecomplex-valued signal, so long as it is transmitted at baseband.Transmitting thesignal as-is, R{X k }, k = 1, 2, . . . , (N/2) − 1, modulate cosine subcarriers centered at1/T B , 2/T B , . . . , [(N/2) − 1]/T B Hz; and likewise, I{X k }, k = 1, 2, . . . , (N/2) − 1, modulatesine subcarriers at the same frequencies. The effective bandwidth of the signal is

127(N/2)/T B Hz 1 , and since the real and imaginary parts of X k represent 0.5 log 2 M bits,the spectral efficiency of the real-valued OFDM signal isS 2 =Bits per second (b/s)Bandwidth (Hz)= 2 × 0.5(N/2) log 2 M/T B(N/2)/T B= log 2 M 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 complexvaluedsignal if the real-valued signal is translated up to a carrier frequency. This is due tothe fact that the cosine and sine subcarriers in (A.11) have a double sideband spectrum:that is, cos(2πkt/T B ) [or sin(2πkt/T B )] has a spectral components at ±k/T B Hz. [Thisisn’t the case for the complex-valued signal, which has complex sinusoids: exp(j2πkt/T B )has a spectral component only at k/T B Hz and is thus considered single sideband.] Thecarrier frequency is typically much larger than the signal bandwidth, so the frequencytranslation brings all the negative frequencies to the positive side: −(N/2)/T B + f c ≫ 0.Consequently, the passband transmission of (A.11) results in a signal with double thebandwidth and 1/2 the spectral efficiency.1 Only the positive frequencies, f ≥ 0, count.

Appendix BMore on the OFDM LiteratureThe first OFDM-like radio to be found in the research literature is the Kineplex systempresented 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 phasemodulated. 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 identifythe Doppler sensitivity of such a radio was by P. A. Bello [51]. Significant theoreticalcontributions 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 OFDMmodulation [579]. This observation was made six years after Cooley and Tukey publisheddetails of the fast Fourier transform; these developments were significant since all modernOFDM systems are based on the FFT.A decade passed with little mention of OFDM in the literature. Then, in the early80’s researchers from IBM’s Watson Research Center suggests OFDM for a wireline DSLtypeapplication [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 OFDM128

129to mobile systems.In the late 80’s and early 90’s OFDM received wide interest for the applicationsof DSL and for wireless digital broadcasting. Kalet and Zervos compare OFDM tosingle carrier with decision feedback equalization [248, 614]. The acceptance of OFDMinto 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 electronicsapplication. In terms of digital broadcasting, OFDM has been accepted for the EuropeanDAB and DVB standards [162, 477, 552]. In the US, OFDM is being used for IBOCbroadcasting [221, 392].OFDM is being applied to indoor wireless local area networks under the IEEE 802.11and the ETSI HYPERLAN/2 standards [552]. And as mentioned in Chapter 1, OFDMis being developed for ultra-wideband systems; cellular systems; wireless metropolitanarea networks; and for power line communication [119, 160, 264, 604].Active OFDM research continues. The major focus in the OFDM literature includesOFDM’s sensitivity to Doppler, phase noise, carrier frequency offsets, and nonlinearities.Channel estimation and synchronization techniques are of interest, along with techniquesto address the PAPR problem.Literature Survey StatisticsThe OFDM literature is immense, so a detail discussion of it here would be overlyambitious. The bibliography of this thesis does provide a somewhat current snapshot ofthe OFDM literature. Also included in the bibliography are papers dealing with generaldigital communications, continuous phase modulation, FM analog communications,power amplifiers, computer simulation techniques, and other miscellaneous papers thathave, 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 aredisplayed below.

130First, to get an idea of the size of the literature, Figure B.1 shows the result ofsearching 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 whenincluding papers presented at IEEE conferences.1400Journal papersJournal plus conference papers4500Journal papersJournal plus conference papers120040003500100030008002500Papers600Papers200015004001000200500019881992199620002004019881992199620002004(a) Papers each year.(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-specificpapers, there are many interesting and fundamental papers dealing with the generalarea of digital communications and information theory. Being familiar with the relevantliterature, which may include several thousands of papers published over many decades,is the goal, however long-term it may be.

131600550500450Papers400350300250200150Oct 2004Jan 2005FiledApr 2005Jul 2005PiledOct 2005Figure B.2: Papers, filed and piled.This figure shows the number of filed and the number of piled papers as a functionof 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 twoparagraphs. A piled paper is in queue waiting to be filed. As the figure shows, the pileis in good health. In late Spring 2005, a concerted effort was made to “kill the pile”. Itbriefly dipped below 150 papers, but the literature is too large—and the battle continues.

1328Papers read per day (log scale)421Oct 2004Jan 2005Apr 2005Jul 2005Oct 2005Running averageDaily pointsFigure 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 showsa histogram of the filed papers’ publication year. One unknown is the true papers-ofinterestcount. A simple model might be: 20 papers per year from 1920–1960; 50 papersper year from 1960–1980; and 100 papers per year from 1980 to present. A histogram ofthis projected goal in relation to the current progress is shown in Figure B.5. Accordingto the model, 4300 papers are of interest, of which roughly 3700 have yet to be filed. Say350 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 papersto be from the past. It would therefore take 3700/250 = 14.8 years to “kill the pile”.

133706050Papers4030201019301940195019601970198019902000Figure B.4: Year histogram.100Desired?80Papers604020Current192019401960198020002020Figure B.5: Projected year histogram?

Appendix CSample CodeThe simulations were performed using GNU Octave [188] and the figures were generatedwith Gnuplot [189]. In this appendix sample code is provided.C.1 GNU Octave CodeBelow is GNU Octave code used to obtain the results for the Channel C f , MMSEcurve in Figure 6.10. The code can easily be adapted to obtain other results, as outlinedbelow.% 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 sqrtshortrun=0; % equals 0 or 1if shortrun % (use for speed/testing)Trans_max=1e5;% max bits sent per SNRTrans_min=2e4;% min bits sent per SNRError_min=2e1;% min errors per SNRelse % long run (use for accuracy/final result)Trans_max=100e6;% max bits sent per SNRTrans_min=1e6;% min bits sent per SNRError_min=2e5;% min errors per SNRendtargetBER=1e-5;SNRmax=50;% target BER% max SNR (dB)134

135io=1;A=1;M=4;modh=1.0/(2*pi);N=64;TB=128e-6;J=8;Fsa=J*N/TB;Tsa=1/Fsa;Tg=10e-6;TF=Tg+TB;Ng=Tg*Fsa;NB=TB*Fsa;NF=TF*Fsa;ip=[Ng:NF-1]+io;Ndft=512;taumax=9e-6;Nc=taumax*Fsa;Nr=Nc+NF-1;L=8;% index offset% signal amplitude% modulation order% modulation index% number of subcarriers% block time% oversampling factor% sampling rate% sampling period% guard time% frame time% samples per guard interval% samples per symbol% samples per frame% processing indices% DFT size (for equalizer)% maximum delay spread of channel (sec)% number of channel taps% number of received samples% blocks/channel realization (vectorize)%% Bit and symbol mappings (depends on modulation order)if M==2SymMap=[-1;1];% data symbol mappingBitMap=[0; 1];% bit mappingendif M==4SymMap=[-3;-1;1;3]; % data symbol mappingBitMap=[...% bit mapping0 0; 0 1; 1 1; 1 0];endif M==8SymMap=[-7:2:7]’;% data symbol mappingBitMap=[...% bit mapping0 0 0; 0 0 1; 0 1 1; 0 1 0; 1 1 0; 1 1 1; 1 0 1; 1 0 0];endif M==16SymMap=(-15:2:15)’; % data symbol mappingBitMap=[...% bit mapping0 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];endvarI=sum(SymMap.^2)/M;% variance of data

136CN=sqrt(2/(N*varI));% normalizing constant%% Subcarrier matrixt=0:Tsa:(TB-Tsa);% time vectorW=zeros(NB,N);% initialize unitary matrixfor k=1:N/2% W is a set of orth. sines and cosinesW(:,k)=cos(2*pi*k*t/TB)’;endfor k=(N/2+1):NW(:,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 detailsMf=11;% filter lengthn1=0:(Mf-1);% filter sample indexd=(Mf-1)/2;% delayn2=(d+1):(d+NB);% desired, delayed indicesfc=0.2;% normalized cutoff frequency (cyc/samp)wc=2*pi*fc;% normalized cutoff frequency (rad/samp)h1=zeros(1,Mf);% initializefor i=1:Mf% compute coefficientsif n1(i)==((Mf-1)/2)h1(i)=wc/pi;elseh1(i)=sin(wc*(n1(i)-(Mf-1)/2))/(pi*(n1(i)-(Mf-1)/2));endendw1=0.54-0.46*cos(2*pi*n1/(Mf-1)); % Hamming windowhf=h1.*w1;% windowed filter coefficients%% Channel delay power spectral density (exponential)t=[0:Nc-1]’*Tsa;% time vectorp=1/tauRms*exp(-t/2e-6); % delay PDS% ------- Simulation -----------------------------------------------BER=0;% initialize BER vectorEbN0_dB=0;% initialize SNR vectordx=2.5;% SNR step sizeiSNR=1;% SNR countergo=1;% initialize loopwhile go% run until max SNR conditionError_num=0; Trans_num=0; % initializewhile Trans_num

137%% Generate L blocksin=ceil(M*rand(N,L));I=SymMap(in);m=CN*W*I;theta0=2*pi*rand(1,L)-pi;phi=zeros(NF,L);for i=1:Lphi(:,i)=[2*pi*modh*m(NB-Ng+1:NB,i)+theta0(i);...2*pi*modh*m(:,i)+theta0(i)];ends=A*exp(j*phi);% random symbol index% data symbols% OFDM message signal% memory terms (assume uniform)% initialize CE-OFDM phase signal% cyclic prefix% CE-OFDM signal%% Determine noise powerEs=sum(sum(abs(s).^2))*Tsa; % signal energyEb=Es/(L*N*log2(M)); % bit energyEbN0=10^(EbN0_dB(iSNR)/10); % SNRN0=Eb./EbN0;% noise spectral height%% Channeltmp=sqrt(1/2)*(randn(Nc,1)+j*randn(Nc,1)); % Gaussian vectorCh=sqrt(p/sum(p)).*tmp; % channel (normalize average power)%% Received signal plus noise (to be processed by FDE)rp=zeros(NB,L);% initializefor i=1:Ltmp1=(conv(Ch,s(:,i))).’; % received samplestmp1=tmp1(ip);% discard cyclic prefixtmp2=sqrt(1/2)*(randn(NB,1)+j*randn(NB,1)); % complex Gaussiannoise=sqrt(N0*Fsa)*tmp2; % Gaussian noiserp(:,i)=tmp1+noise; % received samples plus noiseend%% Frequency-domain equalizerH=fft(Ch,Ndft);% channel gainsC=conj(H)./(abs(H).^2+EbN0^(-1)); % correction term (MMSE)X=fft(rp,Ndft);% to frequency domainhatS=X.*(C*ones(1,L)); % equalizex=ifft(hatS,Ndft); % to time domain%% Filter signalhats=zeros(NB,L);for i=1:Ltmp=(conv(hf,x(:,i))).’;hats(:,i)=tmp(n2);% initialize% filtered signal% filtered signal, desired indices

138end%% Demodulate and detecthatphi=unwrap(angle(hats)); % phase demodulateIhat=W’*hatphi/((2*pi*modh*CN)*NB*1/2); % matched-filter outputinHat=min(round((Ihat+(M-1))/2)+io,M); % index estimate, (=1)Errors=sum(sum(BitMap(in,:)~=BitMap(inHat,:))); % bit errorsError_num=Error_num+Errors; % cumulative bit errorsTrans_num=Trans_num+L*N*log2(M); % cumulative bits%% Display (optional)if rem(Trans_num,10*L*N*log2(M))==0 % print-frequencyclcprintf([’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)endend % end this SNRBER(iSNR)=Error_num/Trans_num; % bit error rate for current SNR%% Test for max SNR conditionif BER(iSNR)=SNRmaxgo=0;else % keep goingiSNR=iSNR+1;EbN0_dB(iSNR)=EbN0_dB(iSNR-1)+dx;endend % end simulation%% Plotsemilogy(EbN0_dB,BER)%% Savetmp=[EbN0_dB’ BER’];save -ascii data tmpTo 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 toC=1./H;% correction term (ZF)The other fading channels are generated by changing the code that defines the channel.For Channel A f :

139%% Channel delay power spectral density (two-path)tau=[0 5e-6];% path delayspower_dB=[0 -10];% path power (dB)power=10.^(power_dB/10); % path powerfor n=1:length(tau)i=tau(n)*Fs;% path indexp(i+io,1)=power(n); % delay PSDendp=[p; zeros(Nc-length(p),1)]; % zero-padFor Channel B f :%% Channel delay power spectral density (two-path)tau=[0 5e-6];% path delayspower_dB=[0 -3];% path power (dB)power=10.^(power_dB/10); % path powerfor n=1:length(tau)i=tau(n)*Fs;% path indexp(i+io,1)=power(n); % delay PSDendp=[p; zeros(Nc-length(p),1)]; % zero-padFor Channel D f :%% Channel delay power spectral density (uniform)tau=[0:Nc-1]’*Ts;% discrete propagation delaysp=ones(size(t));% delay PSDAdditionally, the above template can be used for conventional OFDM with someminor alterations.C.2 Gnuplot CodeThe majority of the figures in this thesis were generated with Gnuplot. Below issample 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 8set format "$%g$"set logscale y 10set format y "$10^{%T}$"

140set ticscale 0.5set border 31 linewidth 0.5set gridset size 1.0,1.4set key width -23.5 height 1 box lw 0.1 41.4,1.5e-1set output "p_ber"# Define line styles.set style line 1 lt 1 lw 1 pt 9 ps 1.0set style line 11 lt 3 lw 1 pt 9 ps 1.0set style line 2 lt 1 lw 1 pt 6 ps 1.0set style line 22 lt 3 lw 1 pt 6 ps 1.0set style line 3 lt 1 lw 1 pt 7 ps 1.0set style line 33 lt 3 lw 1 pt 7 ps 1.0set style line 4 lt 1 lw 1 pt 8 ps 1.0set style line 44 lt 3 lw 1 pt 8 ps 1.0set style line 5 lt 1 lw 3set style line 6 lt 5 lw 3set 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

AbbreviationsA/D analog-to-digital converterAM/AM amplitude/amplitude conversion of power amplifierAM/PM amplitude/phase conversion of power amplifierAWGN additive white Gaussian noisebbitBER bit error rateCCDF complementary cumulative distribution functionCE constant envelopeCE-OFDM constant envelope OFDMCNR carrier-to-noise ratioCP cyclic prefixCPM continuous phase modulationdB decibels, 10 log 10 (·)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 System141

142kHz 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-squaressecondS/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)

SymbolsSet Theory∈ is an element of/∈ is not an element of[·] closed interval[·) open interval{x n } N n=1 set of elements x 1 , x 2 , . . . , x NOperators and Miscellaneous Symbolsarg(·) argumentcos(·) cosineDFT{·} discrete Fourier transforme 2.71828182845905. . .e (·) exponential functionexp(·) exponential functionE{·} expected valueF{·}(f) Fourier transformI 0 (·) 0th-order modified Bessel function of the first kindIDFT{·} inverse discrete Fourier transformI{·} imaginary√partj−1J i (·) ith-order Bessel function of the first kindmax maximummin minimumlim limitln(·) natural loglog x (·) log base xLP{·} lowpass componentP (·) probabilityQ(·) Gaussian Q-functionR{·} real partsin(·) sinesinc(·) sinc functionvar{·} variance143

144x(t) x as a function of tx[i] discrete-time samples of x at the ith indexδ(·) delta functionπ 3.14159265358979. . .∞ infinity∫ b∫ a (·)dx definite integral(·)dx indefinite integral∏ Nn=1multiple product∑ Nn=1multiple sumn! factorialx → a x approaches ax ∗ y x convolved with y| · | absolute value(·) ∗ complex conjugate⌈·⌉ ceiling function⌊·⌋ floor function= equal≡ equal by definition≠ 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 thanPower AmplifierA maxA satg 0G(·)pα φ , β φη AKΦ(·)maximum input levelinput saturation levelgainAM/AM conversionssharpness parameter for the SSPA modelAM/PM parameters for the TWTA modelefficiency of Class-A power amplifierbackoff ratioAM/PM conversions

145Channel2σ02a lB CB ττ(1)B ττ(2)Ch(τ, t)h(τ)h[i]H[k]K RLr ττ (v ′ )S ττ (τ)v ′∆τ lρσ 2 a lττ lτ maxscatter component power of frequency-nonselective channelcomplex-valued gain of the lth pathcoherence bandwidthaverage delaydelay spreadchannel capacitytime-variant channel impulse responsetime-invariant channel impulse responsesamples of the channel impulse responsediscrete Fourier transform of h[i]Rice factornumber of discrete pathsfrequency correlation functiondelay power spectral densityfrequency separation variablepropagation delay difference between τ l and τ l−1 , that is, ∆τ l = τ l − τ l−1line-of-sight component power of frequency-nonselective channelaverage power of the lth pathcontinuous propagation delaydiscrete propagation delay of the lth pathmaximum propagation delaySignalAA b (k)A e (k)A maxB bpfB nB rmsB sC[k]C Nd 2 m,nd 2 m,n (K)d 2 minDE bE b /N 0E qE xfsignal amplitudethe value of the kth subcarrier at the beginning of the block intervalthe value of the kth subcarrier at the end of the block intervalclip levelbandwidth of bandpass filternoise bandwidthroot-mean-square bandwidtheffective bandwidth of CE-OFDM signalfrequency-domain equalizer termsnormalizing constantsquared Euclidean distance between mth and nth signalsquared Euclidean distance between mth and nth signal as a function of thephase constantminimum squared Euclidean distancetotal number of data symbol differencesenergy per bitsignal-to-noise ratio per bitsubcarrier energyenergy of signal xfrequency variable (cycles/s)

146f ′f cf saFOBP(f)FOBP(f) ˆg(t)hIÎJk bKK d 2minL firm(t)Mn(t)n[i]n bp (t)n c (t)n s (t)n w (t)NN 0 /2N cN BN gp γ (x)p ξ (x)P xPAPR xq k (t)r(t)r bp (t)RR/Bs(t)s[i]s bp (t)s c (t)s s (t)S(f)S[k]tnormalized frequency variable (cycles/samp)carrier (or center) frequency (cycles/s)sampling rate (samp/s)fractional out-of-band powerestimated fractional out-of-band powerpulse shapemodulation indexdata symbolestimated data symboloversampling factorbits per symbolphase signal constant, K = 2πhC Nnumber of neighboring signal points having minimum squared Euclideandistance d 2 minfilter lengthmessage signalmodulation order of data symbol constellationlowpass complex-valued zero mean additive Gaussian noisesamples of n(t)bandpass representation of n(t) [bandpass Gaussian noise]in-phase component of n bp (t)quadrature component of n bp (t)white Gaussian noisenumber of subcarriersspectral height of additive white Gaussian noisenumber of channel samplesnumber of block samplesnumber of guard samplesprobability density function of signal-to-noise ratio per bitprobability density function of ξ(t) samplesaverage power of signal xthe peak-to-average power ratio of signal xkth subcarrierlowpass equivalent representation of received signalbandpass representation of r(t)rate, b/sspectral efficiency, b/s/Hzlowpass equivalent representation of transmitted signalsamples of s(t)bandpass representation of s(t)in-phase component of s bp (t)quadrature component of s bp (t)frequency domain representation of s(t)discrete Fourier transform of s[i]time variable

147T B block periodT g guard periodT s symbol periodT sa sampling periodW effective bandwidth of OFDM signal, W = N/T B∆ m,n (k) data symbol difference between mth and nth signal at the kth subcarrierγ signal-to-noise ratio per bit (used interchangeably with E b /N 0 )¯γ average signal-to-noise ratio per bitγ clip clipping ratioɛ fo normalized carrier frequency offsetη t transmission efficiencyθ i memory term during ith CE-OFDM block intervalξ(t) noise at the output of phase demodulatorσI2 data symbol varianceσn2 variance of noise samples, n[i]σφ2 phase signal varianceρ m,n correlation between mth and nth signalρ m,n (K) correlation between mth and nth signal as a function of the phase constantρ max maximum correlation among signalsφ(t) phase signalφ n (t) noise autocorrelation functionΦ Ab (f) Abramson spectrumˆΦ Ab (f) estimated Abramson spectrumΦ x (f) power density spectrum of signal x.ˆΦ x (f) estimated power density spectrum of signal x.

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Production NotesThis thesis was typeset using the L A TEX document preparation system [348]. Thebibliography was managed using BibTeX (with help from bibtool). All numerical work,including the computer simulations, was done with GNU Octave [188]. The block diagramswere drawn using Xfig [597] and all of the other figures were generated with Gnuplot[189] (using the pslatex driver). The source files were backed up and synchronizedamong multiple computers using rsync. The LATEX output was converted to PostScriptusing dvips; the PostScript was converted to PDF (portable document format) usingGhostscript. The size of the PDF output is 1.9 megabytes.The work was done at UCSD on a Dell Precision 370 workstation running the DebianGNU/Linux operating system [131]. The X11 window system provided the graphicaluser interface; the window manager used was IceWM. Typically the work was conductedacross several rxvt terminal emulators—arranged across multiple workspaces—runningbash. All work was done using the Vim (Vi improved) text editor [523]. PDF outputwas viewed using xpdf. The work was also done at various locations throughout the SanDiego 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 workstationwhich has a 3 gigahertz microprocessor and 1 gigabyte of memory.194