[45] S. Barbarossa and R. Torti, âChirped-OFDM For Transmissions over. Time-Varying Channels ... ICASSP, vol. 4, Salt Lake City, May 2001, pp. 2377â2380.
UNIVERSITY OF CALIFORNIA, SAN DIEGO Constant Envelope OFDM Phase Modulation A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Electrical Engineering (Communications Theory and Systems) by Steve C. Thompson
Committee in charge: Professor Professor Professor Professor Professor
James R. Zeidler, Chair John G. Proakis, Co-Chair Robert R. Bitmead William S. Hodgkiss Laurence B. Milstein
2005
Copyright Steve C. Thompson, 2005 All rights reserved.
The dissertation of Steve C. Thompson is approved, and it is acceptable in quality and form for publication on microfilm:
Co-Chair
Chair
University of California, San Diego 2005
iii
“Before PhD, I chopped wood and carried water; After PhD, I chopped wood and carried water.” —[Slightly modified] Zen saying
“I wish I could be more moderate in my desires. But I can’t, so there is no rest.” —John Muir, 1826
“I know this: a man got to do what he got to do. . . ” —Casy, The Grapes of Wrath, John Steinbeck, 1939
iv
TABLE OF CONTENTS
Signature Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Vita and Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv Abstract of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi 1
2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1
. . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.1
ISI-Free Operation . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.2
A Multicarrier Modulation . . . . . . . . . . . . . . . . . . . . . .
5
1.1.3
Discrete-Time Signal Processing . . . . . . . . . . . . . . . . . . .
8
1.2
Problems with OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.3
Constant Envelope Waveforms . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4
Constant Envelope OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5
Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
An Introduction to OFDM
OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1
More OFDM Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.1
The Cyclic Prefix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.2
Discrete-Time Model . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.3
Block Modulation with FDE . . . . . . . . . . . . . . . . . . . . . 20
2.1.4
System Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2
PAPR Statistics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3
Power Amplifier Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4
Effects of Nonlinear Power Amplification . . . . . . . . . . . . . . . . . . . 30
v
2.5 3
4
2.4.1
Spectral Leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.2
Performance Degradation . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.3
System Range and PA Efficiency . . . . . . . . . . . . . . . . . . . 35
PAPR Mitigation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 37
Constant Envelope OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.1
Signal Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2
Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Performance of Constant Envelope OFDM in AWGN . . . . . . . . . . . . . . . 58 4.1
4.2
The Phase Demodulator Receiver . . . . . . . . . . . . . . . . . . . . . . . 59 4.1.1
Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.1.2
Effect of Channel Phase Offset . . . . . . . . . . . . . . . . . . . . 65
4.1.3
Carrier-to-Noise Ratio and Thresholding Effects . . . . . . . . . . 66
4.1.4
FIR Filter Design
. . . . . . . . . . . . . . . . . . . . . . . . . . . 69
The Optimum Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2.1
Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.2
Asymptotic Properties . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3
Phase Demodulator Receiver versus Optimum . . . . . . . . . . . . . . . . 78
4.4
Spectral Efficiency versus Performance . . . . . . . . . . . . . . . . . . . . 80
4.5
CE-OFDM versus OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5
Performance of CE-OFDM in Frequency-Nonselective Fading Channels . . . . . 86
6
Performance of CE-OFDM in Frequency-Selective Channels . . . . . . . . . . . 94 6.1
6.2
MMSE versus ZF Equalization . . . . . . . . . . . . . . . . . . . . . . . . 96 6.1.1
Channel Description . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.1.2
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.1.3
Discussion and Observations . . . . . . . . . . . . . . . . . . . . . 103
Performance Over Frequency-Selective Fading Channels . . . . . . . . . . 108 6.2.1
Channel Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2.2
Simulation Procedure and Preliminary Discussion
6.2.3
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
vi
. . . . . . . . . 112
7
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
A Generating Real-Valued OFDM Signals with the Discrete Fourier Transform . . 124 A.1 Signal Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 A.2 Spectral Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 B More on the OFDM Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 C Sample Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 C.1 GNU Octave Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 C.2 Gnuplot Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Production Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
vii
LIST OF FIGURES
1.1
Representation of a wireless channel with multipath. . . . . . . . . . . . .
2
1.2
A wireless channel in time and frequency. . . . . . . . . . . . . . . . . . .
2
1.3
Intersymbol interference. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.4
OFDM with cyclic prefix (CP). . . . . . . . . . . . . . . . . . . . . . . . .
5
1.5
Subcarrier and overall spectrum. (N = 16; |I 0,k | = 1, for all k) . . . . . .
7
1.6
OFDM converts wideband channel to N narrowband frequency bins. . . .
8
1.7
Frequency offset causes ICI. (fo = 0.25) . . . . . . . . . . . . . . . . . . .
9
1.8
A typical OFDM signal (N = 16). The PAPR is 9.5 dB. . . . . . . . . . . 10
1.9
Power amplifier transfer function. . . . . . . . . . . . . . . . . . . . . . . . 11
1.10 Comparison of OFDM and CE-OFDM signals. . . . . . . . . . . . . . . . 13 2.1
Sampling instances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2
Circular convolution with channel and the inverse channel. . . . . . . . . . 21
2.3
Block modulation with cyclic prefix and FDE. . . . . . . . . . . . . . . . . 21
2.4
OFDM is a special case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5
OFDM system diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6
Complementary cumulative distribution functions. (N = 64)
2.7
PAPR CCDF lower bound (2.31) for N = 2 k , k = 5, 6, . . . , 10. . . . . . . . 26
2.8
AM/AM (solid) and AM/PM (dash) conversions (SSPA=thick, TWTA=thin) for various backoff ratios K. . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.9
Fractional out-of-band power of OFDM with ideal PA and with TWTA model at various input power backoff. (N = 64, IBO in dB) . . . . . . . . 31
. . . . . . . 25
2.10 Spectral growth versus IBO. (N = 64) . . . . . . . . . . . . . . . . . . . . 31 2.11 Performance of QPSK/OFDM with nonlinear power amplifier with various input power backoff levels. (N = 64) . . . . . . . . . . . . . . . . . . . . . 33 2.12 Performance of M -PSK/OFDM with SSPA. (N = 64) . . . . . . . . . . . 34 2.13 The potential range of system is reduced with input backoff; the range is reduced further from nonlinear amplifier distortion. . . . . . . . . . . . . . 36 2.14 Power amplifier efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.15 Block diagram. The system is evaluated with and without PAPR reduction. 38 2.16 Unclipped OFDM signal (9.25 dB PAPR). The rings have radius A max which correspond to various clipping ratios γ clip (dB). . . . . . . . . . . . 39 viii
2.17 PAPR CCDF of clipped OFDM signal for various γ clip (dB). [N = 64] . . 40 2.18 PAPR of clipped signal as a function of the clipping ratio. (N = 64) . . . 40 2.19 A comparison of the total degradation curves of clipped and unclipped M -PSK/OFDM systems. (N = 64) . . . . . . . . . . . . . . . . . . . . . . 41 3.1
The CE-OFDM waveform mapping. . . . . . . . . . . . . . . . . . . . . . 43
3.2
Instantaneous signal power. . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3
Basic concept of CE-OFDM. . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4
Phase discontinuities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5
Continuous phase CE-OFDM signal samples, over L blocks, on the complex plane. (2πh = 0.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.6
Estimated fractional out-of-band power. (N = 64) . . . . . . . . . . . . . 52
3.7
Double-sided bandwidth as a function of modulation index. (N = 64) . . 53
3.8
Power density spectrum. (N = 64, 2πh = 0.6) . . . . . . . . . . . . . . . . 54
3.9
Fractional out-of-band power. (N = 64, 2πh = 0.6) . . . . . . . . . . . . . 55
3.10 CE-OFDM versus OFDM. (N = 64) . . . . . . . . . . . . . . . . . . . . . 56 3.11 CE-OFDM versus OFDM with nonlinear PA. (N = 64) . . . . . . . . . . 57 4.1
Phase demodulator receiver. . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2
Bandpass to baseband conversion. . . . . . . . . . . . . . . . . . . . . . . 60
4.3
Discrete-time phase demodulator. . . . . . . . . . . . . . . . . . . . . . . . 62
4.4
Performance with and without phase offsets. System 1 (S1) has phase offsets {(θi + φ0 ) ∈ [0, 2π)}, and System 2 (S2) doesn’t (θ i + φ0 = 0). [M = 2, N = 64, J = 8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5
Threshold effect at low CNR. (M = 8, N = 64, J = 8, 2πh = 0.5) . . . . . 68
4.6
Threshold effect at low CNR, various 2πh. (M = 8, N = 64, J = 8) . . . 68
4.7
Performance for various filter parameters L fir , fcut /W . (M = 2, N = 64, J = 8, 2πh = 0.5 and Eb /N0 = 10 dB) . . . . . . . . . . 69
4.8
Magnitude response of various Hamming FIR filters. . . . . . . . . . . . . 70
4.9
CE-OFDM performance with and without FIR filter. (M = 2, N = 64, J = 8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.10 The optimum receiver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.11 Correlation functions ρm,n (K). . . . . . . . . . . . . . . . . . . . . . . . . 76 4.12 CE-OFDM optimum receiver performance. (M = 2, N = 8) . . . . . . . . 77
ix
4.13 All unique ρm,n (K) for M = 2, N = 4 DCT modulation. . . . . . . . . . . 78 4.14 Phase demodulator receiver versus optimum. (N = 64) . . . . . . . . . . . 79 4.15 Noise samples PDF versus Gaussian PDF. (E b /N0 = 30 dB) . . . . . . . . 80 4.16 Performance of M -PAM CE-OFDM. (N = 64, †=leftmost curve, ‡=rightmost curve) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.17 Spectral efficiency versus performance. . . . . . . . . . . . . . . . . . . . . 82 4.18 A comparison of CE-OFDM and conventional OFDM. (M = 2, N = 64) . 85 5.1
Performance of CE-OFDM in flat fading channels. (N = 64)
. . . . . . . 88
5.2
A simplified two-region model. (M = 8, N = 64, 2πh = 0.6) . . . . . . . . 90
5.3
A (n + 1)-region model. (M = 8, N = 64, 2πh = 0.6) . . . . . . . . . . . . 91
5.4
Performance of CE-OFDM in flat fading channels. (Circle=Rayleigh; square=Rice, K = 3 dB; triangle=Rice, K = 10 dB. Solid line=Semianalytical curve, (5.15); points=simulation. N = 64) . . . . . . . . . . . . 92
5.5
Comparison of semi-analytical technique (5.15) with (5.10) and (5.11). (M = 4, N = 64, 2πh = 1.2) . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.1
CE-OFDM system with frequency-selective channel. . . . . . . . . . . . . 96
6.2
Channel D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.3
Channel A results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.4
Channel B results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.5
Channel C results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.6
Channel D results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.7
Channel E results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.8
Channel F results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.9
Fundamental characteristic functions and quantities [(6.21)–(6.25)] of the four channel models considered. . . . . . . . . . . . . . . . . . . . . . . . . 113
6.10 Performance results. (Multipath results are labeled with circle and triangle points; the Rayleigh, L = 1 result is that of the frequency-nonselective channel model. M = 4, N = 64, 2πh = 1.0) . . . . . . . . . . . . . . . . . 115 6.11 Single path versus multipath. (M = 4, N = 64, Channel C f , MMSE) . . . 119 6.12 CE-OFDM versus QPSK/OFDM. (SSPA model, Channel C f , N = 64, MMSE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 B.1 “OFDM” search on IEEE Xplore [222]. . . . . . . . . . . . . . . . . . . . . 130 B.2 Papers, filed and piled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 x
B.3 Running average of papers read per day. . . . . . . . . . . . . . . . . . . . 132 B.4 Year histogram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 B.5 Projected year histogram? . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
xi
LIST OF TABLES
6.1
Channel samples of frequency-selective channels. . . . . . . . . . . . . . . 97
6.2
Channel model parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.3
Data symbol contribution per tone for m n (t), n =1, 2, and 3. . . . . . . . 118
xii
ACKNOWLEDGEMENTS I want to first thank my advisors, Professors Zeidler and Proakis, for giving me the chance to do this work, for the encouragement, and for the guidance. I want to thank Professor Milstein for the many helpful technical conversations and for his many suggestions. Thanks to Professors Bitmead and Hodgkiss for taking the time to participate as committee members. Also, thanks to Professor Proakis for carefully proofreading the draft manuscripts of this thesis. Thanks to UCSD’s Center for Wireless Communications for providing a good environment for conducting research; thanks to its industrial partners for the financial support. Thanks to my wife, Shannon, for the emotional and caloric support. Thanks to Chaney the cat for waking me up in the morning. Thanks to my friends for fun support. Thanks to my fellow graduate students in Professor Zeidler’s research group for the camaraderie. Special thanks to Ahsen Ahmed for helpful collaboration over the past couple years. Thanks to my family. Also, thanks to Karol Previte for her support early in my graduate student existence. Thanks to my teachers: Professors Duman, Masry, Milstein, Pheanis, and Wolf, to name only a few. Finally, I would like to thank the countless developers, documentation writers, bug reporters, and users of the free software I’ve benefited from during the course of my PhD. The text in this thesis, in part, was originally published in the following papers, of which I was the primary researcher and author: S. C. Thompson, J. G. Proakis, and J. R. Zeidler, “Constant Envelope Binary OFDM Phase Modulation,” in Proc. IEEE Milcom, vol. 1, Boston, Oct. 2003, pp. 621–626; S. C. Thompson, A. U. Ahmed, J. G. Proakis, and J. R. Zeidler, “Constant Envelope OFDM Phase Modulation: Spectral Containment, Signal Space Properties and Performance,” in Proc. IEEE Milcom, vol. 2, Monterey, Oct. 2004, pp. 1129–1135; S. C. Thompson, J. G. Proakis, and J. R. Zeidler, “Noncoherent Reception of Constant Envelope OFDM in Flat Fading Channels,” in Proc. IEEE PIMRC, Berlin, Sept. 2005; and S. C. Thompson, J. G. Proakis, and J. R. Zeidler, “The Effectiveness of Signal Clipping for PAPR Reduction and Total Degradation in OFDM Systems,” in Proc. IEEE Globecom, St. Louis, Dec. 2005. xiii
VITA December 22, 1976
Born, Mesa, Arizona
1997–1998
Associate Engineer Inter-Tel, Chandler, Arizona
Summer 1998
Summer Internship Los Alamos National Laboratory Los Alamos, New Mexico
1999
BSc in Electrical Engineering Arizona State University, Tempe, Arizona
Summer 2001
Summer Internship SPAWAR Systems Center, San Diego, California
2001
MSc in Electrical Engineering University of California at San Diego, La Jolla, California
2001–2005
Research Assistant Center for Wireless Communications University of California at San Diego, La Jolla, California
Summer 2004
Summer Internship SPAWAR Systems Center, San Diego, California
2005
PhD in Electrical Engineering University of California at San Diego, La Jolla, California
PUBLICATIONS S. C. Thompson, J. G. Proakis, and J. R. Zeidler, “Constant Envelope Binary OFDM Phase Modulation,” in Proc. IEEE Milcom, vol. 1, Boston, Oct. 2003, pp. 621–626. S. C. Thompson, A. U. Ahmed, J. G. Proakis, and J. R. Zeidler, “Constant Envelope OFDM Phase Modulation: Spectral Containment, Signal Space Properties and Performance,” in Proc. IEEE Milcom, vol. 2, Monterey, Oct. 2004, pp. 1129–1135. S. C. Thompson, A. U. Ahmed, J. G. Proakis, and J. R. Zeidler, “Constant Envelope OFDM Phase Modulation,” submitted to IEEE Transactions on Communications. 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. S. C. Thompson, J. G. Proakis, and J. R. Zeidler, “The Effectiveness of Signal Clipping for PAPR Reduction and Total Degradation in OFDM Systems,” in Proc. IEEE Globecom, St. Louis, Dec. 2005. xiv
S. C. Thompson, J. G. Proakis, and J. R. Zeidler, “The Effectiveness of Signal Clipping for PAPR Reduction and Total Degradation in OFDM Systems,” in preparation. S. C. Thompson, A. U. Ahmed, J. G. Proakis, and J. R. Zeidler, M -ary PAM Constant Envelope OFDM,” in preparation. S. C. Thompson, J. G. Proakis, and J. R. Zeidler, “Performance of CE-OFDM in Frequency-Nonselective Fading Channels,” in preparation. S. C. Thompson, J. G. Proakis, and J. R. Zeidler, “Performance of CE-OFDM in Frequency-Selective Channels,” in preparation.
xv
ABSTRACT OF THE DISSERTATION
Constant Envelope OFDM Phase Modulation by Steve C. Thompson Doctor of Philosophy in Electrical Engineering (Communications Theory and Systems) University of California San Diego, 2005 Professor James R. Zeidler, Chair Professor John G. Proakis, Co-Chair Orthogonal frequency division multiplexing (OFDM) is a popular modulation technique for wireless digital communications. It provides a relatively straightforward way to accommodate high data rate links over harsh wireless channels characterized by severe multipath fading. OFDM has two primary drawbacks, however. The first is a high sensitivity to time variations in the channel caused by Doppler, carrier frequency offsets, and phase noise. The second, and the focus of this thesis, is that the OFDM waveform has high amplitude fluctuations, a drawback known as the peak-to-average power ratio (PAPR) problem. The high PAPR makes OFDM sensitive to nonlinear distortion caused by the transmitter’s power amplifier (PA). Without sufficient power backoff, the system suffers from spectral broadening, intermodulation distortion, and, consequently, performance degradation. High levels of backoff reduce the efficiency of the PA. For mobile battery-powered devices this is a particularly detrimental problem due to limited power resources. A new PAPR mitigation technique is presented. In constant envelope OFDM (CEOFDM), the high PAPR OFDM signal is transformed to a constant envelope 0 dB PAPR waveform by way of angle modulation. The constant envelope signal can be efficiently amplified with nonlinear power amplifiers thus achieving greater power efficiency. In xvi
this thesis, the fundamental aspects of the CE-OFDM modulation are studied, including the signal spectrum, the signal space, optimum performance, and the performance of a practical phase demodulator receiver. Performance is evaluated over a wide range of multipath fading channel models. It is shown that CE-OFDM outperforms conventional OFDM when taking into account the effects of the power amplifier. This work was done at UCSD’s Center for Wireless Communication, under the “Mobile OFDM Communications” project (CoRe research grant 00-10071).
xvii
Chapter 1
Introduction Humans have always found ways to communicate, over space and over time. From the messenger pigeon to the Pony Express, from the message in a bottle to cave drawings, smoke signals and beacons, people have used inventive techniques, techniques derived from their natural environment, to share information. A particularly good natural resource for communication is electricity for its speed and ability to be controlled with devices like capacitors, microprocessors, electronic memory storage and batteries. Communication was 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, satellite communications, and magnetic storage systems; and with the rise of the internet and digital computers, digital communications—the transfer of bits (1’s and 0’s) from one point to another—has become important. In particular, wireless digital communications is currently under intensive research, development and deployment to provide high data rate access plus mobility. One challenge in designing a wireless system is to overcome the effects of the wireless channel, which is characterized as having multiple transmission paths and as being time varying [421, 427]. Figure 1.1 illustrates a link with four reflecting paths between points A and B. These reflections are caused by physical objects in the environment. Due to the relative mobility between the points and the possibility that the reflecting objects are mobile, the channel changes with time.
1
2
point B point A
Propagation paths
Figure 1.1: Representation of a wireless channel with multipath. An example profile of the channel in Figure 1.1 is shown in Figure 1.2(a). Each path has its own associated delay and power. The first path arrives at the receiver 0.5 µs after the signal is transmitted; the last path arrives with a 14 µs delay. The Fourier transform of the profile yields the frequency-domain representation shown in Figure 1.2(b). The channel is viewed over a 2 MHz range centered at the center frequency f c . Notice that the channel power fluctuates by 30 dB (a factor of 1000) over the frequency range. The dispersion in the time domain leads to frequency-selectivity in the frequency domain.
5 Channel power (dB)
Path power
1
0.1
0 -5 -10 -15 -20 -25
0.01 0
2
4
6 8 10 Time (µs)
12
-30−1
14
−0.5 0 0.5 Frequency, f − fc (MHz)
1
(b) Frequency domain.
(a) Time 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 a conventional single carrier system, the symbols are then transmitted serially. The signal waveform of such a system is s(t) =
X i
Ii g(t − iTs ),
(1.1)
where t is the time variable, {Ii } are the data symbols, Ts is the symbol period, and g(t)
is a transmit pulse shape. For time-dispersive channels, such as the 4-path example in
3 Figure 1.2, interference is caused from symbol to symbol. This intersymbol interference (ISI) is illustrated in Figure 1.3. For simplicity, g(t) is rectangular. The channel is represented by its time-variant impulse response h(τ, t), where τ is a propagation delay variable. The received signal is expressed mathematically as [387, p. 97] r(t) = s(t) ∗ h(τ, t) + n(t) Z ∞ h(τ, t)s(t − τ )dτ + n(t), =
(1.2)
−∞
where ∗ represents the linear convolution operator and n(t) is additive noise. The effect of
the time-dispersive channel is shown to smear symbol 1 into symbol 2, therefore creating intersymbol interference. r(t)
s(t) Transmitter
s(t)
Channel
Receiver
|h(τ, t)|
r(t) ISI
. ..
symbol 1 symbol 2 0
Ts
2Ts
... t
τ
0
Ts
2Ts
t
Figure 1.3: Intersymbol interference. The severity of the ISI depends on the symbol period relative to the channel’s maximum propagation delay, τmax . Consider transmitting the signal in (1.1) over the 2 MHz channel in Figure 1.2. The signal bandwidth is roughly proportional to the symbol rate 1/Ts Hz. Therefore making s(t) a 2 MHz signal, T s = (2 × 106 )−1 = 0.5 µs.
Since the maximum propagation delay of the channel is τ max = 14 µs, the ISI spans
τmax /Ts = (14 µs) / (0.5 µs) = 28 symbols. (For comparison, the ISI in Figure 1.3 spans less than one symbol.) Such severe ISI must be corrected at the receiver in order to provide reliable communication. The traditional approach to combating intersymbol interference is with time-domain equalizers [421]. There are many types, ranging in complexity and in effectiveness. The optimum maximum-likelihood (ML) receiver is the most effective but is typically impractical due to its high complexity, which grows exponentially with the ISI length. Linear equalizers are much simpler, having a complexity which grows roughly linearly with ISI length, but perform much worse than the optimum receiver. Nonlinear decision feedback equalizers (DFEs) have similar complexity as the linear type and have better performance.
4 All of these techniques require knowledge of the channel, which is estimated by transmitting a training sequence which is known at the receiver. Then by comparing the received signal to what was transmitted, an estimate of h(τ, t) is made. There are various algorithms available for the estimation process, each having its own complexity, convergence rate, and stability. The least-mean-square (LMS) algorithm is the most stable and the least complex, but suffers from a slow convergence rate. The recursive least-square (RLS or Kalman) algorithm, on the other hand, converges quickly, but has higher complexity and can be unstable. For scenarios like the example above with an ISI spanning 28 symbols, conventional equalization becomes difficult. Training times become long and convergence of the channel estimator is problematic, especially for time-varying channels. In the example, 2×10 6 symbols/s are transmitted. Using a QPSK (quadrature phase-shift keying) signal constellation, which maps kb = 2 bits per symbol, the bit rate is 4 Mb/s. Such a bit rate is desired in current wireless systems, and in many cases demand for many tens of Mb/s is common.
1.1
An Introduction to OFDM
To meet the demanding data rate requirements, alternative techniques have been considered. One approach, orthogonal frequency division multiplexing, has become exceedingly popular. OFDM has been implemented in wireline applications such as digital subscriber lines (DSL) [95], in wireless broadcast applications such as digital audio and video broadcasting (DAB and DVB) and in-band on-channel (IBOC) broadcasting [392]. It has been used in wireless local area networks (LANs) under the IEEE 802.11 and the ETSI HYPERLAN/2 standards [552]. OFDM is being developed for ultra-wideband (UWB) systems; cellular systems; wireless metropolitan area networks (MANs), under the IEEE 802.16 (WiMax) standard; and for other wireline systems such as power line communication (PLC) [119, 160, 264, 604].
1.1.1
ISI-Free Operation
OFDM’s main appeal is that it supports high data rate links without requiring conventional equalization techniques. Instead of transmitting symbols serially, OFDM
5 sends N symbols as a block. The OFDM block period, T B , is thus N times longer than the symbol period. Continuing the example above, and choosing N = 300, the block period is TB = N Ts = 300 × 0.5 µs = 150 µs, which is more than 10 times the duration
of the channel’s impulse response. ISI is avoided by inserting a guard interval between
successive blocks during which a cyclic prefix is transmitted. The interval duration, T g , is designed such that Tg ≥ τmax so that the channel is absorbed in the guard interval
and the OFDM block is uncorrupted. This is illustrated in the figure below. Selecting a
guard interval Tg = 15 µs for the channel in Figure 1.2 results in a transmission efficiency ηt = TB /(TB + Tg ) = 150/165 ≈ 0.91. Therefore, with a small reduction in efficiency, ISI
is eliminated.
s(t) Tg
TB
CP
OFDM block
t
|h(τ, t)|
τ r(t) ISI-free block
t
Figure 1.4: OFDM with cyclic prefix (CP).
1.1.2
A Multicarrier Modulation
The OFDM signal can be expressed as1 # " −1 X NX s(t) = Ii,k ej2πfk t g(t − iTB ). i
(1.3)
k=0
The pulse shape, g(t), is typically rectangular: 1, 0 ≤ t < TB , g(t) = 0, otherwise.
(1.4)
−1 Notice that the N data symbols {Ii,k }N k=0 are transmitted during the ith block. The
−1 set of complex sinusoids {exp (j2πf k t)}N k=0 are referred to as subcarriers. The center 1
For simplicity, the guard interval is excluded from the signal definition in (1.3). The guard interval and cyclic prefix is discussed in Chapter 2.
6 frequency of the kth subcarrier is f k = k/TB and the subcarrier spacing, 1/TB Hz, makes the subcarriers orthogonal over the block interval, expressed mathematically as Z TB Z TB ∗ 1 1 ej2πfk1 t ej2πfk2 t dt = ej2π(fk2 −fk1 )t dt TB 0 TB 0 1, k1 = k2 , = 0, k1 6= k2 ,
(1.5)
where (·)∗ represents the complex conjugate operation. The subcarrier orthogonality can
also be viewed in the frequency domain. Consider the 0th OFDM block: s(t) =
N −1 X
I0,k ej2πfk t ,
k=0
0 ≤ t < TB .
(1.6)
The frequency-domain representation is S(f ) = F {s(t)} (f ) = TB e
−j2πf TB /2
N −1 X
I0,k
k=0
where F{·}(f ) is the Fourier transform and 1, sinc(x) = sin πx , πx
k sinc f − TB , TB
x = 0,
(1.7)
(1.8)
otherwise.
Figure 1.5 plots |S(f )/TB | for N = 16 subcarriers and data symbols with normalized amplitudes. The individual subcarrier spectra are also plotted. Notice that at the kth
subcarrier frequency, k/TB , the kth subcarrier has a peak and all the other subcarriers have zero-crossings. Therefore, the subcarriers, while tightly packed (which improves spectral efficiency), are non-interfering (i.e. orthogonal). Figure 1.5 also demonstrates that OFDM is a multicarrier modulation, as opposed to a single carrier modulation like the signal in (1.1). In general, a transmitted bandpass signal is [421, p. 151] n o x(t) = < s(t)ej2πfc t ,
where fc is the carrier frequency. For single carrier, X xsc (t) = |Ii | cos [2πfc t + arg(Ii )] g(t − iTs );
(1.9)
(1.10)
i
while for multicarrier, ( −1 ) X NX k t + arg(Ii,k ) g(t − iTB ). |Ii,k | cos 2π fc + xmc (t) = TB i
k=0
(1.11)
7
Spectrum magnitude, |S(f )/TB |
1.2
Subcarrier Overall
1
0.8
0.6
0.4
0.2
0
-2
0
2
4
6 8 10 Normalized frequency, f TB
12
14
16
18
Figure 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 multicarrier the bandwidth is split into many frequency bands (also referred to as frequency bins). Notice that the multicarrier signal transmits the N data symbols in parallel over multiple carriers each centered at (fc + k/TB ) Hz, k = 0, 1, . . . , N − 1. By properly designing the subcarrier spacing, each frequency bin is made frequencynonselective. The wideband frequency-selective channel is converted into N contiguous narrowband frequency-nonselective bins. Figure 1.6 shows 18 bins in the range [−0.9, −0.78] MHz for the N = 300 OFDM system over the channel in Figure 1.2(b).
Notice that the channel gain per bin varies over a 15 dB range. The OFDM modulation can be optimized for the channel by sending more bits in frequency bins with high gain and fewer bits in frequency bins with low gain. This technique, known as bit loading, requires a fairly stable channel, one that can be accurately measured. For this reason, bit loading is more common in wireline systems and stationary wireless systems than in wireless systems with high mobility. Frequency selectivity is the frequency-domain dual of intersymbol interference. Transmitting the single carrier signal over the 2 MHz channel results in a frequency-selective response. For OFDM, the overall channel is frequency-selective but for each bin the chan-
8
Channel power (dB)
5 0 -5 -10 -15 -20 Frequency bins
-25 -30 −0.9
−0.85 Frequency, f − fc (MHz)
−0.8
Figure 1.6: OFDM converts wideband channel to N narrowband frequency bins. nel is frequency non-selective and thus ISI is avoided. Therefore, Figure 1.6 illustrates a frequency-domain interpretation of how OFDM avoids intersymbol interference.
1.1.3
Discrete-Time Signal Processing
Thus far, two of OFDM’s primary advantages have been discussed: the elimination of ISI and the ability to optimize the modulation with bit loading. The third appeal of OFDM is that the modulation and demodulation is done in the discrete-time domain with the inverse fast Fourier transform (IFFT) and fast Fourier transform (FFT), respectively. This is seen by sampling s(t) in (1.6) at N equally spaced time instances: y[i] ≡ s(t)|t=iTB /N =
N −1 X
I0,k ej2πki/N ,
k=0
i = 0, 1, . . . N − 1,
(1.12)
which is the inverse discrete Fourier transform (IDFT) of the symbol vector I 0 = [I0,0 , I0,1 , . . . , I0,N −1 ]. Therefore, s(t) is generated at the transmitter with an IDFT folN −1 lowed by a digital-to-analog (D/A) converter. The frequency-domain symbols {I 0,k }k=0
can be expressed as
I0,k =
N −1 1 X y[i]e−j2πkn/N , N i=0
k = 0, 1, . . . N − 1,
(1.13)
which is the discrete Fourier transform (DFT) performed on the time-domain samples. Consequently, the symbols are demodulated at the receiver with an analog-to-digital (A/D) converter followed by a DFT.
9 The IDFT/DFT is performed efficiently with IFFT/FFT algorithms. Doing so is much simpler than performing the modulation/demodulation in the continuous-time domain with N orthogonally tuned oscillators. Moreover, the signal processing can be performed in software, making OFDM suitable for software defined radios (SDRs) [185].
1.2
Problems with OFDM
OFDM has two primary drawbacks. The first is sensitivity to imperfect frequency synchronization which is common for mobile applications. This sensitivity arises from the close subcarrier spacing. Figure 1.5 shows that the subcarriers are properly orthogonal at f = k/TB , k = 0, 1, . . . , N − 1. However, if the frequency synthesizer at the
receiver is misaligned by, say, fo /TB Hz, where −0.5 < fo < 0.5, the subcarriers are
not orthogonal and therefore interfering with one another. This intercarrier interference
(ICI) is illustrated in Figure 1.7: assuming that the receiver is tuned to (k + fo )/TB Hz rather than at the ideal k/TB Hz, the N − 1 neighboring subcarriers interfere with the demodulation of the kth subcarrier. The intercarrier interference causes ISI—and
potentially high irreducible error floors. The second problem with OFDM is that the signal has large amplitude fluctuations
Spectrum magnitude, |S(f )/TB |
caused by the summation of the complex sinusoids. The real and imaginary part of the
1
0.2
0.04 k−1
k + fo k Normalized frequency, f TB
Figure 1.7: Frequency offset causes ICI. ( fo = 0.25)
k+1
10 OFDM signal is < {s(t)} =
N −1 X
< {I0,k } cos (2πkt/TB ) − = {I0,k } sin (2πkt/TB ) ,
(1.14)
= {s(t)} =
N −1 X
< {I0,k } sin (2πkt/TB ) + = {I0,k } cos (2πkt/TB ) ,
(1.15)
and
k=0
k=0
respectively. Figure 1.8(a) shows the real and imaginary parts of an example OFDM signal with N = 16 subcarriers. Also plotted are the individually modulated sinusoids. Notice that each sinusoids has a constant amplitude, but when summing the sinusoids the resulting OFDM signal fluctuates over a large range. The instantaneous signal power, |s(t)|2 = NB , the signal vector is zero-padded. Since
NDFT > Ng , the channel samples are zero-padded: h[i] = 0 for i = N c , . . . , NDFT − 1.
Figure 2.2 shows a block diagram representing the calculation of (2.24). The effect of the channel is simply a DFT followed by a multiplier bank (H[k]), which is then followed by an IDFT. Also shown is the inverse channel which is a DFT followed by a multiplier bank (1/H[k]) followed by an IDFT. Thus the transmit samples s[i] can be reconstructed by passing the receive samples r[i] through the inverse channel.
2.1.3
Block Modulation with FDE
The inverse channel structure in Figure 2.2 corrects the distortion caused by the channel in the frequency domain, and is therefore called a frequency-domain equalizer
21
Channel s[i]
DFT
H[k]
IDFT
r[i]
IDFT
s[i]
Inverse channel r[i]
1 H[k]
DFT
Figure 2.2: Circular convolution with channel and the inverse channel.
Frequency-domain equalizer
Data {Ik }
Modulator
Channel
DFT
Multiplier bank
IDFT
Demodulator
Data {Iˆk }
Figure 2.3: Block modulation with cyclic prefix and FDE.
Frequency-domain equalizer
Data {Ik }
IDFT
Data {Ik }
Channel
DFT
Multiplier bank
IDFT
IDFT
Channel
DFT
Multiplier bank
Figure 2.4: OFDM is a special case.
DFT
Data {Iˆk }
Data {Iˆk }
22 (FDE). Such an equalizer can be used only when the effect of the channel is a circular convolution. This is the case for OFDM, but isn’t unique to OFDM since any modulation can use a cyclic prefix. This observation was first identified by Sari et al. [462] and suggests a more general modulation approach: block modulation with cyclic prefix and frequency-domain equalization. Figure 2.3 shows a simplified block diagram of such a system. (The insertion of the cyclic prefix at the transmitter and removal at the receiver is implied but not included in the diagram for simplicity.) For the special case of OFDM, the modulation is a IDFT and the demodulation is a DFT as shown in Figure 2.4. Notice that the DFT and IDFT cancel each other and the resulting diagram depicts the conventional OFDM system. The multiplier bank at the output of the DFT is often referred to as a one-tap equalizer, one complex multiplication per frequency bin. This operation is required for data symbols that rely on coherent demodulation, such as M -ary phase-shift keying (M -PSK) and M -ary quadrature-amplitude modulation (M -QAM). As Sari et al. pointed out, OFDM doesn’t eliminate the equalization problem (associated with conventional single carrier modulation); rather, OFDM converts the problem to the frequency domain. Since Sari’s original paper, there has been a considerable number of publications focused on the block modulation technique using conventional single carrier modulations [8, 30, 54, 107, 116, 132, 142, 153, 154, 196, 197, 245, 388, 460, 461, 463, 481, 533, 565, 574].
2.1.4
System Diagram
The block diagram in Figure 2.4 conceptually illustrates the OFDM system. Figure 2.5 shows a more detailed description of OFDM’s functional blocks. The encoder adds redundancy to the bit stream for error control. The encoded bits are then mapped to the data symbols I k . In general, the data symbols are complex numbers which result from mapping the bits to points on the complex plane. Next, the symbols are serial-to-parallel (S/P) converted and processed by the IDFT. The cyclic prefix is added and the signal samples, s[i], are passed through the digital-to-analog (D/A) converter to obtain the continuous-time OFDM signal s(t). Finally, the signal is amplified and transmitted.
23 Transmitter
Bits 01101
Encoder
11101
Mapper
Ik
S/P
Add CP
IDFT
s[i]
D/A
P/S
s(t)
Power amplifier
Receiver
r(t)
A/D
Remove CP
r[i]
S/P
DFT
Iˆk
Equalize C[k]
Detector
11001
P/S
Decoder
Bits 01101
Figure 2.5: OFDM system diagram. At the receiver, the inverse operations are performed. First, the received signal, r(t), is sampled to obtain the discrete-time sequence r[i]. The guard interval samples are removed, the DFT is performed and each frequency bin is equalized by a complex multiplication. The estimated data symbols, Iˆk , are processed by the detector which outputs a stream of estimated receive bits, and the decoder attempts to correct any bit errors that may have occurred. As discussed in Section 1.2, one of OFDM’s key drawbacks is the high peak-toaverage power ratio. Nonlinearities in the power amplifier distort the transmitted signal and large input power backoff is required which results in low amplifier efficiency. In the next sections the impact of the PA is studied. But first, the statistical properties of the PAPR are discussed.
24
2.2
PAPR Statistics
The peak-to-average power ratio of the OFDM signal is best viewed statistically. For any given block interval, the PAPR is a random quantity since it depends on the data −1 symbols {Ik }N k=0 . Assuming that they’re selected randomly from a set of M complex
numbers, there are M N unique symbol sequences, and thus M N unique OFDM waveforms per block. Of these waveforms, some have a high PAPR, while others have a relatively low PAPR. Therefore, it is desirable to understand the statistical distribution of this quantity. The OFDM signal is s(t) =
N −1 X
Ik ej2πfk t ,
k=0
0 ≤ t < TB .
(2.27)
The signal during the guard interval is ignored since it has no impact on the PAPR distribution. M -PSK data symbols are assumed, therefore |I k | = 1 for all k. The
average power of s(t) is
Ps =
1 TB
Z
0
TB
|s(t)|2 dt = N.
(2.28)
The peak-to-average power ratio is defined as . PAPRs = max |s(t)|2 Ps . t∈[0,TB )
(2.29)
Notice that the absolute maximum signal power is N 2 , so the PAPR can be as high as N . However, the likelihood that all the subcarriers align in phase is extremely low. For example, as pointed out in [381], a N = 32 subcarrier system having 4-ary data symbols and a block period of TB = 100 µs obtains the theoretical maximum PAPR once every 3.7 million years. Thus it is more meaningful to describe the PAPR statistically rather than in absolute terms. Since the average signal power is a constant, the randomness of the PAPR depends on the randomness of the instantaneous power |s(t)| 2 , and more specifically, the maximum instantaneous power over 0 ≤ t < TB . For large N , the real and imaginary parts of
s(t) are accurately modeled as Gaussian random processes (due to the application of the central limit theorem [394, 421]). Consequently, the instantaneous signal power is chi-squared distributed with two degrees of freedom [421, p. 41], and the complementary
25 cumulative distribution function (CCDF) of the normalized instantaneous signal power is approximated as P
|s(t)|2 >x Ps
≈ e−x .
(2.30)
A lower bound of the peak-to-average power ratio’s CCDF is [515] P (PAPRs > x) ' 1 − (1 − e−x )N ,
(2.31)
where 1 − (1 − e−x )N is an approximation to the CCDF of the PAPR of the sequence {s(t)|t=iTB /N ; i = 0, 1, . . . N −1} [173]. The PAPR of the discrete-time sequence provides
a lower bound to the continuous-time signal since peaks can occur between sampling times. 100
100 Simulation Approximation (2.30)
Simulation Lower bound (2.31)
10−1 CCDF, P (PAPRs > x)
` ´ CCDF, P |s(t)|2 /Ps > x
10−1
10−2
10−3
10−2
10−3
10−4 0
2
4
6 x (dB)
(a) Instantaneous power.
8
10
10−4 4
6
8
10 x (dB)
12
14
(b) Peak-to-average power ratio.
Figure 2.6: Complementary cumulative distribution functions. (N = 64) Figure 2.6(a) compares a simulated instantaneous power CCDF with the approximation in (2.30). This demonstrates the accuracy of the Gaussian approximation to the real and imaginary part of s(t). Figure lower bound in (2.31). 2.6(b) compares PAPR simulation results to the The bound is shown to be within 1 dB of the simulated result for lower values of x. The 0.0001 PAPR is shown to be at around 11.25 dB, and at this
26 level the bound is tight. Notice that essentially all OFDM blocks have a PAPR greater than 6 dB, 10% have a PAPR greater than 8.5 dB, and 0.5% have a PAPR greater than 10 dB. For the results in Figure 2.6, the number of subcarriers is N = 64 and QPSK data symbols (4-ary PSK) are used, that is, I k ∈ {±1, ±j}. While the symbols constellation
has little impact on the PAPR statistics, the number of subcarriers does. Figure 2.7 shows the lower bound (2.31) over a range N = 32 to N = 1024. Notice that the 0.001
PAPR is 1 dB larger for N = 512 than for N = 32. For the N = 64 system, the PAPR is greater than 8 dB for roughly 10% of the time. For the N = 1024 system, however, the PAPR is greater than 8 dB nearly all of the time. 100
k
5
6
7
8
9
10
CCDF, P (PAPRs > x)
10−1
10−2
10−3
10−4 4
5
6
7
8 x (dB)
9
10
11
12
Figure 2.7: PAPR CCDF lower bound (2.31) for N = 2 k , k = 5, 6, . . . , 10.
2.3
Power Amplifier Models
To determine the impact of the PAPR on system performance, power amplifier models must be defined. Two models commonly used in the research literature are the solid-state power amplifier (SSPA) model and the Saleh traveling-wave tube amplifier (TWTA) model [454]. They are described here and then used in Section 2.4 for performance evaluation.
27 In general, modeling nonlinear power amplifiers is complicated (see [233, chap. 5]). A common simplification is to assume that the PA is a memoryless nonlinearity, and therefore has a frequency-nonselective response. For example, if the PA input is sin (t) = A(t) exp[jφ(t)],
(2.32)
sout (t) = G[A(t)] exp j{φ(t) + Φ[A(t)]} ,
(2.33)
the output is
where G(·) and Φ(·) are known as the AM/AM and AM/PM conversions, respectively. The SSPA model is expressed as G(A) = h
g0 A 1 + (A/Asat )
2p
i1/2p , and Φ(A) = 0,
(2.34)
where g0 is the amplifier gain, Asat is the input saturation level, and p controls the AM/AM sharpness of the saturation region. For this model the AM/PM conversion is
assumed to be negligibly small. Though widely known as the Rapp model [426], (2.34) should be credited to the original work by A. J. Cann, published a decade earlier in the IEEE literature [71]. Cann’s formula is obtained with the simple manipulation: G(A) =g0 h =g0 h
=g0 h
A 1 + (A/Asat )2p A 1 + (A/Asat )2p Asat 1 + (Asat /A)2p
i1/2p i1/2p ×
[(Asat /A)2p ]1/2p [(Asat /A)2p ]1/2p
(2.35)
i1/2p ,
which is precisely the nonlinearity presented in Cann’s paper. Saleh’s TWTA model is expressed as [110] G(A) =
αφ A2 g0 A , and Φ(A) = . 1 + β φ A2 1 + (A/Asat )2
(2.36)
Notice that the AM/PM conversion, determined by the constants α φ and βφ , is non-zero. The TWTA model is therefore more nonlinear than the SSPA model.
28 To reduce nonlinear distortion in the amplified OFDM signal, input power backoff (IBO) is required. It is defined as [375] A2sat , Pin
IBO =
(2.37)
where Pin = E{|sin (t)|2 } = E{A2 (t)} is the average power of the input signal. Equiva-
lently, (2.37) can be written as
Pin =
A2sat ; IBO
(2.38)
thus, given Asat and IBO, the input signal power can be scaled accordingly to satisfy (2.38).
Assuming that the PAPR of the input signal is PAPR in , the peak power can be written as Pmax = PAPRin · Pin = where K=
A2 PAPRin 2 Asat = sat , IBO K
IBO PAPRin
(2.39)
(2.40)
is defined as the backoff ratio. Notice that for K > 1 the backoff is greater than the input signal’s PAPR; for K < 1 the backoff is less than the input PAPR. Now, the maximum
value of the input, Amax = max |A(t)|, can be written in terms of the backoff ratio and
the input saturation level:
Amax =
p Asat Pmax = √ . K
(2.41)
Figure 2.8 shows the AM/AM (solid lines) and AM/PM (dashed lines) conversions for the SSPA (thick lines) and TWTA (thin lines) models for various backoff ratios K.
For the SSPA model, p = 2; for the TWTA model, α φ = π/12 and βφ = 1/4. The x-axis is normalized to the maximum input level A max , and the y-axis is normalized to the
maximum output level g0 Asat . For K = −10 dB the IBO is one-tenth the input signal
PAPR, and thus the nonlinearity is severe. One the other hand, for K = 10 dB the
IBO is ten times the input signal PAPR and the PA response is nearly linear. As stated
above, the non-zero AM/PM conversion of the TWTA model makes it more nonlinear than the SSPA model. Insight can be gained by comparing Figure 2.6(b) and Figure 2.8. For example, assuming that the backoff is IBO = 6 dB, the conversions are never as linear as the K = 3 dB curves (the PAPR is a always greater than 3 dB) and are more nonlinear
29
1 Normalized output value, G(A)/g0 Asat
Normalized output value, G(A)/g0 Asat
1
0.5
0
−0.5
0
−0.5
−1 −1
0.5
−1 −0.5 0 0.5 Normalized input value, A/Amax
−1
1
(a) K = 10 dB
1 Normalized output value, G(A)/g0 Asat
Normalized output value, G(A)/g0 Asat
1
(b) K = 3 dB
1
0.5
0
−0.5
0.5
0
−0.5
−1 −1
−0.5 0 0.5 Normalized input value, A/Amax
−1 −0.5 0 0.5 Normalized input value, A/Amax
(c) K = −3 dB
1
−1
−0.5 0 0.5 Normalized input value, A/Amax
1
(d) K = −10 dB
Figure 2.8: AM/AM (solid) and AM/PM (dash) conversions (SSPA=thick, TWTA=thin) for various backoff ratios K.
30 than the K = −3 dB curves for about 5% of the OFDM blocks (the 0.05 PAPR is 9 dB).
Therefore, even with a large IBO of 6 dB, the PA can impose high nonlinear distortion
on the transmitted signal. Also, the degree of distortion for a given OFDM block is random (given a fixed IBO) since the PAPR for a given block is random.
2.4
Effects of Nonlinear Power Amplification
Power amplifier nonlinearities cause spectral leakage and performance degradation to OFDM systems. These undesirable effects can be reduced with increase input backoff. This is an unsatisfactory solution, however, since PA efficiency reduces with IBO. Also, reducing the average transmit power reduces the operational range of the system. In this section these various issues are studied.
2.4.1
Spectral Leakage
The first problem considered is spectral leakage. By using the Welch method [422, pp. 911–913], the power density spectrum at the output of the power amplifier can be quickly estimated. The result is used to calculate estimated fractional out-of-band power curves, defined as ˆ FOBP(f )=
Rf 0
ˆ s (x)dx Φ , 0.5Pˆs
f > 0,
ˆ s (f ) is the estimated power density spectrum of the signal and Pˆs = where Φ
(2.42) R∞
ˆ
−∞ Φs (f )df
is the signal power. Figure 2.9 shows the curves for an N = 64 subcarrier OFDM signal amplified by the TWTA power amplifier according to (2.36) at various backoff levels.
Also plotted is the FOBP curve for ideal linear amplification. These results show that at least 6 dB backoff is required by the TWTA to avoid spectral broadening. Figure 2.10 shows the 99.5% bandwidth as a function of IBO. The bandwidth of the undistorted OFDM signal is f = 1.07W . For sufficient backoff, the bandwidth of the nonlinearly amplified signal is the same. However, for IBO < 6 dB, the bandwidth is shown to grow roughly linearly with IBO. For IBO = 1 dB, the 99.5% bandwidth is 73% larger than the undistorted signal. Notice that the spectral leakage is roughly the same for the two amplifier models.
31
100
Fractional out-of-band power
OFDM amplified with: TWTA PA ideal PA
10−1
IBO 0
10−2 2 4 6
10−3 0
0.25
0.5 0.75 1 Normalized frequency, f /W
1.25
1.5
Figure 2.9: Fractional out-of-band power of OFDM with ideal PA and with TWTA model at various input power backoff. (N = 64, IBO in dB)
OFDM amplified with: TWTA PA SSPA PA ideal PA
2.0
99.5% bandwidth, f /W
1.8 1.6 1.4 1.2 1.0 0.8 0
2
4 6 Input power backoff, IBO (dB)
8
Figure 2.10: Spectral growth versus IBO. (N = 64)
10
32
2.4.2
Performance Degradation
Next, the performance degradation caused by nonlinear amplification is considered. The OFDM signal is passed through a PA and then it is corrupted by additive white Gaussian noise (AWGN). The received signal is thus, r(t) = sout (t) + n(t),
(2.43)
where sout (t) is the output of the PA from (2.33) and n(t) is a complex-valued Gaussian additive noise signal having a power density spectrum [421, p. 158] N0 , |f | ≤ Bn /2, Φn (f ) = 0, |f | > Bn /2,
(2.44)
where Bn is the bandwidth of the noise signal. The noise spectrum is assumed to be constant over the effective bandwidth of the information bearing signal and is thus called “white”. The transmitted data symbols are estimated by the correlation in (2.5) then passed to the detector which makes the final decision. This decision is based on the maximum-likelihood (ML) criterion assuming a linear PA; that is, the nearest point in the symbol constellation [421, pp. 242–247]. The performance is estimated by way of computer simulation. Following the convention described in Section 2.1.2, the discrete-time signal representation is used and the sampling rate fsa = JN/TB where J ≥ 1 is the oversampling factor. For the AWGN channel, h(τ ) = δ(τ ), and therefore no guard interval is used. The noise samples {n[i]} are Gaussian distributed and assumed independent: σ 2 , i 1 = i 2 , n E {n[i1 ]n[i2 ]} = 0, i1 6= i2 .
(2.45)
The autocorrelation function of n(t) [the inverse Fourier transform of (2.44)] has zerocrossings at τ = 1/Bn . Thus assuming Bn = fs , (2.45) is satisfied and the noise sample variance is σn2 = fsa N0 . Figure 2.11 shows bit error rate (BER) performance as a function of E b /N0 , where Eb =
R TB
|sout (t)|2 dt Number of bits per block 0
(2.46)
33
10−1 Nonlinear PA Ideal PA
Nonlinear PA Ideal PA 10−1
Bit error rate
Bit error rate
10−2
10−3
10−4
10−5 0
10−2
10−3
10−4
2 4 6 8 10 12 14 Signal-to-noise ratio per bit, Eb /N0 (dB)
10−5 0
5 10 15 20 25 30 Signal-to-noise ratio per bit, Eb /N0 (dB)
(a) SSPA model, IBO = 0, 1, 2, 3, 4, 6, 8 dB;
(b) TWTA model, IBO = 0, 1, . . . , 10, 16 dB;
0 = worst, 8 = best.
0 = worst, 16 = best.
Figure 2.11: Performance of QPSK/OFDM with nonlinear power amplifier with various input power backoff levels. (N = 64) is the energy per bit. The quantity E b /N0 is referred to as the signal-to-noise ratio (SNR) per bit, or simply the SNR. QPSK data symbols are used, and the oversampling factor
is J = 4. For the SSPA results in Figure 2.11(a), the IBO ranges from 0 to 8 dB. At the 0.0001 BER level, the IBO = 0 dB case suffers a 3 dB performance loss compared to ideal AWGN performance, which is [421, pp. 268]. BER = Q where Q(x) =
R∞ x
e−y
2 /2
r
Eb 2 N0
!
,
(2.47)
√ dy/ 2π is the Gaussian Q-function. To avoid degradation, 8
dB of backoff is required. The TWTA results in Figure 2.11(b) use IBO ranging from 0 to 16 dB. Notice the irreducible error floors for IBO ≤ 7 dB. To avoid degradation, 16
dB of backoff is required—8 dB more than for the SSPA case. The greater nonlinearity of the TWTA model is evident from the results in this figure.
Figure 2.12 compares performance for higher-order PSK modulations. For M -PSK
34
10 Ideal PA SSPA: IBO = 3 dB IBO = 6 dB
10−1
M = 16
M = 16
Bit error rate
10−2
M =8
10−3
10−4
M = 2, 4
Total degradation (dB)
8
6
M =8
4
Ideal PA SSPA
M = 2, 4 2
Target BER = 0.001 10−5 0
5 10 15 20 25 30 Signal-to-noise ratio per bit, Eb /N0 (dB)
(a) BER performance.
0 0
2 4 6 8 Input power backoff, IBO (dB)
10
(b) Total degradation.
Figure 2.12: Performance of M -PSK/OFDM with SSPA. (N = 64) the data symbols are Ik ∈ {exp(j2πm/M ); m = 0, 1, . . . , M − 1}. The number of bits per data symbols is log 2 M , therefore the bit energy is R TB |sout (t)|2 dt . Eb = 0 N log2 M
(2.48)
(2.49)
Higher-order constellations are used for increased spectral efficiency at the price of BER performance2 . In Figure 2.12(a) BER results for the SSPA model are shown. (The results for M = 2 and M = 4 are very similar so only M = 2 is plotted.) The higherorder modulations are shown to be more sensitive to the PA nonlinearity. For example, the M = 16 result for IBO = 3 dB has an irreducible error floor at 5 × 10 −3 , while the M = 2, 4 result at the same backoff shows only a 1 dB degradation. When increasing
the backoff to IBO = 6 dB, the error floor for M = 16 drops to 2 × 10 −5 and the 0.001 BER is about 2 dB worse than AWGN. Using IBO = 6 dB for M = 8 results in 2 dB less degradation at the 0.001 bit error rate when compared to using IBO = 3 dB. 2
This is the case for linear modulation formats. This isn’t necessarily the case for nonlinear modulation formats as discussed in Section 4.4.
35 A more revealing way to view performance is in terms of total degradation, as shown in Figure 2.12(b). The total degradation is defined as [121] TD(IBO) = SNRPA (IBO) − SNRAWGN + IBO,
[in dB]
(2.50)
where SNRAWGN is the required signal-to-noise ratio per bit to achieve a target bit error rate in AWGN; SNRPA (IBO) is the required SNR when taking into account the distortion caused by the power amplifier at a given backoff. The “optimum” IBO, denote as IBOopt , minimizes the total degradation, that is, TD(IBOopt ) = TDmin =
min
IBO≥0 dB
TD(IBO).
(2.51)
The target BER for the curves in Figure 2.12(b) is 0.001. Clearly the modulation order influences the degradation. The minimum TD for M = 16 is 7.7 dB at IBO opt = 6.5 dB; for M = 8, TDmin = 5 dB at IBOopt = 3 dB. This can be interpreted as follows: M = 8, while having lower spectral efficiency than M = 16 (3 b/s/Hz vs. 4 b/s/Hz), suffers less degradation and can operate with less backoff, resulting in improved range and higher PA efficiency. The M = 2 and M = 4 examples are shown to have the lowest degradation and are thus the more robust against nonlinear distortion.
2.4.3
System Range and PA Efficiency
The total degradation is directly related to the system’s operational range. Consider a transmitter operating at maximum transmit power. The range is represented by the outermost ring in Figure 2.13. Now assume that the system requires a 3 dB backoff: the range is reduced by one-half, as represented by the middle ring. Any degradation caused by the PA further reduces range, as represented by the innermost circle. Thus the actual range of the system is far less than the potential range of the transmitter. The true capability of the power amplifier is greatly underutilized. To quantify the relationship between the PA efficiency and the power backoff, the theoretical efficiency of a Class A power amplifier is used [374]: ηA =
1 1 × 100%, 2 IBO
IBO ≥ 1.
(2.52)
The efficiency is thus inversely proportional to IBO and the maximum efficiency, 50%, occurs at IBO = 1 (0 dB). The efficiency curve, shown in Figure 2.14, can be used
36
Potential range Potential range w/ IBO Actual range
Figure 2.13: The potential range of system is reduced with input backoff; the range is reduced further from nonlinear amplifier distortion. in conjunction with Figures 2.10 and 2.12(b) to gain insight to the various tradeoffs between PA efficiency, spectral containment, and performance/range. For example, the optimum IBO in terms of total degradation for the 8-PSK SSPA example is IBO opt = 3 dB [Figure 2.12(b)]: however, the bandwidth expansion is 42% (Figure 2.10) and the PA efficiency is ηA = 25% (Figure 2.14). The optimum IBO for the 16-PSK example, 6.5 dB, results in no bandwidth expansion but the PA efficiency is reduced to 11%. The M = 2, 4 systems required minimal IBO for the SSPA, thus maximizing efficiency, but the bandwidth expands by 87%. 50
Class-A PA efficiency, ηA (%)
45 40 35 30 25 20 15 10 5 00
1
2
3
4 5 6 7 Input power backoff, IBO (dB)
Figure 2.14: Power amplifier efficiency.
8
9
10
37
2.5
PAPR Mitigation Techniques
There have been many schemes proposed in the research literature aimed at reducing the impact of the PAPR problem. The goal of any scheme is to reduce the minimum total degradation (for increased range) and the IBO opt (for increased PA efficiency). The various schemes can be placed in one the following three categories: 1. transmitter enhancement techniques, 2. receiver enhancement techniques, or 3. signal transformation techniques. Transmitter enhancement techniques include PAPR reduction schemes and PA linearization schemes. The PAPR reduction schemes can be further divided into distortionless and non-distortionless techniques. Distortionless techniques include coding (see [126,439,508] and reference therein), constellation extension [269], tone reservation [169, 268, 512], trellis-shaping [377], and multiple signal representation {aka selected mapping (SLM) or partial transmit sequences (PTS), see [227] and its references}. Non-distortionless
schemes include signal clipping [27, 138, 290, 382], peak cancellation [330], and peak windowing [403]. The PA linearization schemes attempt to predistort the OFDM signal such that the overall response of the predistorter followed by the PA is linear—essentially equalizing the amplifier. In [230], an LMS algorithm is applied for adaptive predistortion; in [395] a neural network learning technique is used. Parametric techniques, which design a predistorter based on a PA model, have been proposed. In [85, 122, 250, 567] nonlinear polynomial models are used, and in [86] a Volterra-based model is suggested. The second category, receiver enhancement techniques, have been suggested in [513], [376] (maximum-likelihood decoding); in [259, 453] (signal reconstruction), and in [87] (interference cancellation). Finally, the third category includes techniques that are based on transforming the OFDM signal prior to the PA, and applying the inverse transform at the receiver prior to demodulation. This category includes constant envelope OFDM (as studied in the second half of this thesis) which uses a phase modulator as the transformer. In [215, 329, 569–571] a companding transform is suggested.
38
Signal Clipping The remainder of this section focuses on the effectiveness of signal clipping, which has been claimed to be the “simplest” and “most effective” PAPR reduction scheme [27, 87, 290, 375, 377, 380, 382, 391]. The impact of “clipping noise”—the intercarrier interference caused by the clipping process—on system performance has been extensively analyzed [39, 124, 382]. However, a common assumption is that the PA is linear [27, 39, 138, 290, 371, 380, 382, 391]. It is argued here that the effectiveness of a PAPR reduction scheme must be measured not only by PAPR reduction, but by the more meaningful measures of TDmin and IBOopt reduction. It is shown that clipping, while an effective PAPR reduction scheme, does not reduce TD min nor does clipping reduce IBOopt for an OFDM system. This result brings into question the usefulness of non-distortionless PAPR reduction techniques in general. The system under consideration is shown in Figure 2.15. When the switch is “on” the PAPR reducing signal clipper is used. When “off” the system is identical to the one studied in Section 2.4.2. Therefore, the earlier unclipped results serve as a performance benchmark in which to compare the clipped results. The channel, as before, has an impulse response h(τ ) = δ(τ ). off
OFDM s(t) modulator PAPR reducing clipper
sin (t)
PA
sout (t)
h(τ )
r(t)
OFDM demodulator
on n(t)
sclip (t)
Figure 2.15: Block diagram. The system is evaluated with and without PAPR reduction. The input to the clipping block is the OFDM signal s(t) from (2.27), the output is the clipped OFDM signal: sclip (t) =
s(t),
Amax ejψ(t) ,
if |s(t)| ≤ Amax ,
(2.53)
if |s(t)| > Amax ,
where ψ(t) = arg[s(t)]. Therefore, the magnitude of the clipped signal does not exceed Amax and the phase of s(t) is preserved. (This has been called “polar clipping” in the literature [276].) The clipping severity is measured by the clipping ratio, defined as [375] Amax γclip = √ . Ps
(2.54)
39
γclip
20
OFDM signal Clip radius
4
2 10
Imaginary axis
0
0
−10
−20
−20
−10
0 Real axis
10
20
Figure 2.16: Unclipped OFDM signal (9.25 dB PAPR). The rings have radius A max which correspond to various clipping ratios γ clip (dB). Figure 2.16 shows a typical OFDM signal on the complex plane. The dark rings have radius Amax which correspond to clipping ratios γ clip = 0, 2, and 4 dB. The PAPR of sclip (t) is
PAPRclip =
max |sclip (t)|2 t∈[0,T ) . R TB 1 2 dt |s (t)| clip TB 0
(2.55)
Clipping’s effectiveness at reducing PAPR is shown in Figure 2.17. For clipping ratio γclip = 5 dB, the peak-to-average power ratio of the clipped signal is PAPR clip ≤ 10
dB; for γclip = 4 dB, PAPRclip ≤ 8 dB, and so forth. The 0.0001 PAPR improvement,
compared to the unclipped signal, is 1.2 dB for γ clip = 5 dB and by 3.2 dB for the γclip = 4 dB. Figure 2.18 shows PAPRclip as a function of the clipping ratio.
The PAPR of
the unclipped signal is 13 dB3 . Notice that for large γclip , sclip (t) is unclipped, there3
This figure is made by generating 2 × 104 consecutive OFDM blocks. The PAPR of the overall block is 13 dB.
40
100
Clipped Unclipped
P (PAPRclip > x)
10−1
10−2
10−3 γclip 10−4 0
2
4
3
4
5
6 x (dB)
8
10
12
Figure 2.17: PAPR CCDF of clipped OFDM signal for various γ clip (dB). [N = 64]
16 14 Peak-to-average power ratio (dB)
PAPRs 12 10 8 PAPRclip
6 4 2
PAPRclip as γclip → 0
0 −2
2 γclip
−4 −8
−6
−4
−2
0 2 4 Clipping ratio, γclip (dB)
6
8
10
Figure 2.18: PAPR of clipped signal as a function of the clipping ratio. (N = 64)
41 fore PAPRclip = PAPRs . As γclip → 0, the peak and average powers converge, thus
PAPRclip → 0 dB. For the region 3 dB < γclip < 6.5 dB, sclip (t) is clipped so the 2 P . However, the clipping is mild so the average power is peak power is A2max = γclip s
2 P /P = γ 2 . approximately the same as s(t); therefore, PAPR clip ≈ γclip s s clip
Clipping is clearly an effective technique at reducing the PAPR. The question is, does the PAPR reduction translate into reduced total degradation? Figure 2.19 compares the total degradation curves of the unclipped system [from Figure 2.12(b)] with the clipped system. Interestingly, the unclipped results are shown to provide a lower bound for the clipped, reduced PAPR, system results. The clipper is shown to increase both the minimum total degradation and the optimum backoff. For example, using the clipping ratio γclip = 3 dB for the M = 8 case increases the TD min by 0.2 dB; using γclip = 2 dB increases TDmin by 1.2 dB. For M = 16, the γclip = 4 dB result is nearly identical to the unclipped result; γclip = 3 dB increases the degradation by 1.2 dB, and the TD curve associated with γclip = 2 dB is beyond the viewing range of the figure. For M = 2, 4 the PAPR reducing clipping yields nearly identical results as the unclipped system.
10
Total degradation (dB)
M = 16 8
6
M =8
4
Ideal PA Unclipped Clipped: γclip = 4 dB 3 dB 2 dB
M = 2, 4 2
0 0
2
4 6 Input power backoff, IBO (dB)
8
10
Figure 2.19: A comparison of the total degradation curves of clipped and unclipped M -PSK/OFDM systems. (N = 64) Thus the effectiveness of a PAPR reduction scheme should be measured not only by its PAPR reducing capabilities but by its effectiveness in reducing total degradation (which increases range) and reducing the optimum IBO (which increases power amplifier
42 efficiency). The distortion caused by non-distortionless schemes can outweigh the benefit of the reduced PAPR. This is clearly shown to be the case for the clipped N = 64 M PSK/OFDM systems studied in this section. The clipping is shown to reduce the 0.0001 PAPR by > 1 dB, but this reduction does not translate into increased PA efficiency. This result brings into question the validity of the claims that clipping is an effective scheme. In fact, the effectiveness of non-distortionless PAPR reduction schemes in general is suspect. For these types of techniques it is important to take into account the effect of the nonlinear power amplifier. The effectiveness of distortionless PAPR reduction techniques are typically studied in terms of PAPR reduction and complexity. It would be interesting to also study these schemes in terms of total degradation. Does a 3 dB reduction in PAPR results in a 3 dB reducing in IBOopt ? What is the resulting minimum total degradation?
Chapter 3
Constant Envelope OFDM Conventional OFDM systems, even with the use of effective PAPR reduction and/or power amplifier linearization techniques, typically require more input power backoff than convention single carrier systems. Therefore, OFDM is considered power inefficient, which is undesirable particularly for battery-powered wireless systems. The technique described in the remainder of the thesis takes a different approach to the PAPR problem. CE-OFDM can be thought of as a mapping of the OFDM signal to the unit circle, as depicted in Figure 3.1. The instantaneous power of the resulting signal is constant. Figure 3.2 compares the instantaneous power of the OFDM signal and the mapped CE-OFDM signal. For the CE-OFDM signal the peak and average powers are the same, thus the PAPR is 0 dB. Signal Unit circle
⇒ OFDM
CE-OFDM
Figure 3.1: The CE-OFDM waveform mapping.
43
44 5 OFDM CE-OFDM
Instantaneous signal power
4
3
2
1
0
0
0.2
0.4 0.6 Normalized time
0.8
1
Figure 3.2: Instantaneous signal power. The mapping is performed with an angle modulator, specifically, a phase modulator. That is, the OFDM signal is used to phase modulate the carrier. This is in contrast to conventional OFDM which amplitude modulates the carrier. To see this, consider the baseband OFDM waveform m(t) =
N XX i
k=1
Ii,k qk (t − iTB )
(3.1)
where {Ii,k } are the data symbols and {qk (t)} are the orthogonal subcarriers. For con-
ventional OFDM the baseband signal is up-converted to bandpass as n o y(t) = < m(t)ej2πfc t
(3.2)
= Am (t) cos [2πfc t + φm (t)] ,
where Am (t) = |m(t)| and φm (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 viewed
as an amplitude single-sideband modulation.) For CE-OFDM, m(t) is passed through a phase modulator prior to up-conversion. The baseband signal is s(t) = ejαm(t) ,
(3.3)
45 where α is a constant. The bandpass signal is n o y(t) = < s(t)ej2πfc t n o = < ejαAm (t) exp[jφm (t)] ej2πfc t n o = < e−αAm (t) sin φm (t) ej[2πfc t+αAm (t) cos φm (t)]
(3.4)
= e−αAm (t) sin φm (t) cos [2πfc t + αAm (t) cos φm (t)] .
For real-valued m(t), y(t) = cos [2πfc 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 Figure 3.3. At the transmitter, the high PAPR OFDM signal is transformed into a low PAPR signal prior to the power amplifier. At the receiver, the inverse transformation is performed prior to demodulation. Transmitter Transform OFDM modulator
m(t)
Phase modulator
s(t)
Power amplifier
To channel
Receiver Inverse From channel
Phase demodulator
OFDM demodulator
transform
Figure 3.3: Basic concept of CE-OFDM. As mentioned in Section 2.5, other approaches based on signal transformation have been suggested. In particular, [215, 329, 569–571] suggest a companding transform. The companded signal has an increased average power and thus a lower peak-to-average power ratio than conventional OFDM. The PAPR is still large relative to single carrier modulation, however. The advantage of the phase modulator transform is that the resulting signal has the lowest achievable peak-to-average power ratio of 0 dB.
46 The idea of transmitting OFDM by way of angle modulation isn’t entirely new. In fact, Harmuth’s 1960 paper suggest transmitting information by orthogonal time functions with “amplitude or frequency modulation, or any other type of modulation suitable for the transmission of continuously varying [waveforms]” [202]. Using existing FM infrastructure for OFDM transmission has been suggested in [76, 77, 575]. These papers don’t consider the PAPR implications, however. Two conference papers, [101] and [506], on the other hand, suggest using a phase modulator prior to the power amplifier for PAPR mitigation—though intriguing, these papers lack a solid theoretical foundation and ignore fundamental signal properties such as the signal’s power density spectrum. The origin of this work, which is independent of the previous references, stems from work done at the US Navy’s spawar Systems Center, (San Diego, CA). Mike Geile, a principle engineer at Nova Engineering, (Cincinnati, OH), which is the contractor of the OFDM component for JTRS (Joint Tactical Radio System), suggested a low PAPR enhancement to OFDM by phase modulation. The motivation is to reduce the 6 dB backoff used in the JTRS radio. Transmitting OFDM with phase modulation raises several fundamental questions. What is the power density spectrum of the modulation? How is the signal space affected? What is the optimum AWGN performance? What is the performance of a phase demodulator receiver (Figure 3.3)? How does the system perform in a frequency-selective fading channel? These questions, and others, are addressed here. First, the CE-OFDM modulation is defined.
3.1
Signal Definition
As indicated by (3.4), CE-OFDM requires a real-valued OFDM message signal, that is, φm (t) = 0. Therefore the data symbols in (3.1) are real-valued: Ii,k ∈ {±1, ±3, . . . , ±(M − 1)}.
(3.6)
This one dimensional constellation is known as pulse-amplitude modulation (PAM). Thus the data symbols are selected from an M -PAM set. The subcarriers {q k (t)} must also
47 be real-valued. Three possibilities are considered: half-wave cosines, cos πkt/TB , 0 ≤ t < TB , qk (t) = 0, otherwise, for k = 1, 2, . . . , N ; half-wave sines, sin πkt/TB , qk (t) = 0,
0 ≤ t < TB ,
(3.7)
(3.8)
otherwise,
for k = 1, 2, . . . , N ; and full-wave cosines and sines, cos 2πkt/TB , 0 ≤ t < TB ; k ≤ qk (t) = sin 2π(k − N/2)t/TB , 0 ≤ t < TB ; k > 0, otherwise.
For each case, the subcarrier orthogonality condition holds: Z (i+1)TB Eq , qk1 (t − iTB )qk2 (t − iTB )dt = iTB 0,
N 2, N 2,
k1 = k2 ,
(3.9)
(3.10)
k1 6= k2 ,
where Eq = TB /2.
In terms of implementation, (3.7) can be computed with a discrete cosine transform (DCT); (3.8) with a discrete sine transform (DST); and (3.9) by taking the real part of a discrete Fourier transform (DFT), or equivalently by taking a 2N -point DFT of a conjugate symmetric data vector (see Appendix A.) The baseband CE-OFDM signal is s(t) = Aejφ(t) ,
(3.11)
where A is the signal amplitude. The phase signal during the ith block is written as φ(t) = θi + 2πhCN
N X k=1
Ii,k qk (t − iTB ),
iTB ≤ t < (i + 1)TB ,
(3.12)
where h is referred to as the modulation index, and θ i is a memory term (to be described below). The normalizing constant, C N , is set to s 2 , CN ≡ N σI2
(3.13)
48 where σI2 is the data symbol variance: σI2
= E |Ii,k |
2
M 1 X = (2l − 1 − M )2 M l=1
=
(3.14)
M2
−1 , 3
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 ) ( Z (i+1)TB 1 [φ(t) − θi ]2 dt σφ2 = E TB iTB Z (i+1)TB X N N X (2πh)2 2 E {Ik1 Ik2 } qk1 (t − iTB )qk2 (t − iTB )dt = TB N σI2 iTB
(3.15)
k1 =1 k2 =1
=
N Z (2πh)2 2 X TB 2 2 σI qk (t)dt = (2πh)2 , TB N σI2 0 k=1
which is only a function of the modulation index. The signal energy is Z (i+1)TB |s(t)|2 dt = A2 TB , Es =
(3.16)
iTB
and the bit energy is Eb =
Es A2 TB = . N log2 M N log 2 M
(3.17)
The term θi is a memory component designed to make the modulation phase-continuous. At the ith signaling interval boundary, the phase discontinuity is ci = φ(iTB − ) − φ(iTB + ),
→ 0.
(3.18)
Since qk (t) = 0 for t ∈ / [0, TB ), it follows that φ(iTB − ) = K and φ(iT + ) = K
N X
Ii−1,k Ae (k),
(3.19)
k=1 N X
Ii,k Ab (k),
(3.20)
k=1
where K ≡ 2πhCN , Ab (k) = qk (0) and Ae (k) = qk (TB − ), → 0. Therefore, ci = θi−1 − θi + K
N X k=1
[Ii−1,k Ae (k) − Ii,k Ab (k)] .
(3.21)
49 To guarantee continuous phase, that is, c i = 0, the memory term is set to θi ≡ θi−1 + K
N X
k=1
[Ii,k Ab (k) − Ii−1,k Ae (k)] .
(3.22)
Notice that θi depends on θi−1 ; the OFDM signal at the beginning of the ith block, PN PN k=1 Ii,k Ab (k); and the OFDM signal at the end of the (i−1)th block, k=1 Ii−1,k Ae (k).
The recursive relationship can be written as θi = K
N ∞ X X l=0 k=1
[Ii−l,k Ab (k) − Ii−1−l,k Ae (k)] .
(3.23)
Thus, the memory term is a function of all data symbols during and prior to the ith block. Figure 3.4 plots the phase discontinuities {c i } at the boundary times t = iTB , i =
0, 1, . . . , 49. In Figure 3.4(a), ci is plotted for memoryless modulation, that is, θ i = 0, P for all i; therefore, ci = K N k=1 [Ii−1,k Ae (k) − Ii,k Ab (k)]. Figure 3.4(b) shows that the 1.5
1.5
1
1 Phase discontinuity, ci
Phase discontinuity, ci
phase discontinuities are eliminated with the use of memory as defined in (3.22).
0.5 0
−0.5
0
−0.5
−1
−1.5 0
0.5
10
20 30 40 Normalized time, t/TB
(a) Without memory.
50
−1
−1.5 0
10
20 30 40 Normalized time, t/TB
50
(b) With memory.
Figure 3.4: Phase discontinuities. The benefit of continuous phase CE-OFDM is a more compact signal spectrum. This property is studied further in Section 3.2. A second consequence of the memory terms is the entire unit circle is used for the CE-OFDM phase modulation. This is illustrated in Figure 3.5 which plots continuous phase CE-OFDM signal samples on the complex
50
Unit circle
Starting point
(a) L = 1
(b) L = 100
Figure 3.5: Continuous phase CE-OFDM signal samples, over L blocks, on the complex plane. (2πh = 0.7) plane. The modulation index is 2πh = 0.7. Figure 3.5(a) shows signal samples over L = 1 block, where the phase signal occupies about one-half the unit circle. Viewing samples over L = 100 blocks, Figure 3.5(b) shows that the phase signal occupies the entire unit circle.
3.2
Spectrum
CE-OFDM is a complicated nonlinear modulation and a general closed-form expression for the power density spectrum is not available. The approach taken in [34], [421, pp. 207–217] to calculate the power spectrum of conventional CPM signals can be applied to CE-OFDM. The Fourier transform of the average autocorrelation function results in a two-dimensional definite integral. The problem is there are N sinusoidal phase pulses in CE-OFDM, versus a single phase pulse as in CPM. This makes the integrand very jagged for all but trivial values of N , and numerical integration algorithms (for example, those in [328, 419]) fail to converge in a timely manner. Insight can be gained by taking this approach, however. It can be shown that memoryless modulation (θ i = 0) results in spectral lines at the frequencies f k = k/TB , k = 0, ±1, ±3, . . . [17]. Using memory as defined by (3.22) eliminates these lines.
Since the Fourier transform approach isn’t computationally feasible, other techniques are required to understand the CE-OFDM spectrum. The simplest is with the Taylor
51 expansion ex =
P∞
n=0 x
n /n!.
The CE-OFDM signal, with θi = 0, can be written as s(t) = Aejσφ m(t) ∞ X (jσφ )n mn (t), =A n! n=0
where m(t) = CN
N XX i
k=1
Ii,k qk (t − iTB )
(3.24)
(3.25)
is the normalized OFDM message signal. The effective double-sided bandwidth, defined as the twice the highest frequency subcarrier, of m(t) is W =2×
N N = . 2TB TB
(3.26)
The bandwidth of s(t) is at least W : in (3.24), the n = 0 term contains no information and thus has zero bandwidth; the n = 1 term is information bearing and has bandwidth W ; the n = 2 term has a bandwidth 2W ; and so on. Thus, due to the n = 1 term, the bandwidth of s(t) is at least W , and depending on the modulation index the effective bandwidth can be greater than W . The power density spectrum, Φs (f ), can be easily estimated by the Welch method ˆ s (f ) ≈ Φs (f ), is used to calculate the of periodogram averaging [526]. The result, Φ
fractional out-of-band power,
FOBP(f ) = where Ps =
R∞
−∞ Φs (f )df
Rf 0
Φs (x)dx ≈ 0.5Ps
Rf 0
ˆ s (x)dx Φ ˆ = FOBP(f ), 0.5Ps
(3.27)
= Es /TB = A2 is the signal power. Figure 3.6 shows estimated
fractional out-of-band power curves for N = 64 and various 2πh. Due to the normalizing
constant CN these curves are valid for any M . The dashed lines represent the RMS bandwidth, Brms = σφ W = 2πhN/TB .
(3.28)
The RMS (root-mean-square) bandwidth is obtained by borrowing a result from analog angle modulation [423, pp. 340–343] [437], which assumes a Gaussian message signal; for large N , the OFDM waveform is well modeled as such (see Section 2.2). The results in Figure 3.6 shows that Brms accounts for at least 90% of the signal power. As defined in (3.28), the RMS bandwidth can be less than W , but, as shown by the Taylor expansion in (3.24), the CE-OFDM bandwidth is at least W . A more suitable
52 100
ˆ FOBP(f ) Brms
10−1
2πh
Fractional out-of-band power
10−2
2.0 1.8 1.6
10−3
1.4 1.2
10−4
1.0 0.8
10−5
0.6 0.4 10−6 0.2 10−7 0
0.5
1 Normalized frequency, f /W
1.5
2
Figure 3.6: Estimated fractional out-of-band power. (N = 64) bandwidth is thus Bs = max(2πh, 1)W.
(3.29)
Figure 3.7 plots Bs versus 2πh, and compares it with the 90–99% bandwidths as determined by the Welch method. Notice that (3.29) is an accurate 90–92% bandwidth measure 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 Figure
3.6). Figure 3.8 compares spectral estimates for CE-OFDM signals with the three subcarrier modulations from (3.7), (3.8) and (3.9). The modulation index is 2πh = 0.6. Memoryless, non-continuous phase CE-OFDM is compared to continuous phase CEOFDM (the continuous phase examples are prefixed with “CP”). The estimates are also
53 3 2.8
Bs Welch: 90% 92% 95% 99%
2.6
Normalized double-sided bandwidth, B/W
2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.2
0.4
0.6
0.8
1 1.2 1.4 Modulation index, 2πh
1.6
1.8
2
Figure 3.7: Double-sided bandwidth as a function of modulation index. (N = 64) compared to the Abramson spectrum [1]: ΦAb (f ) = A
2
∞ X
an Un (f ),
(3.30)
n=0
where
2
an = and
e−σφ σφ2n n!
,
δ(f ), Un (f ) = Φm (f ), Φm (f ) ∗n Φm (f ),
(3.31)
n = 0,
n > 1.
The weighting factors {an } are Poisson distributed, and 3
(3.32)
n = 1,
P∞
n=0 an
n
= 1; ∗ denotes the
n-fold convolution, for example x(t) ∗ x(t) = x(t) ∗ x(t) ∗ x(t); and Φm (f ) is the power
54
Welch estimate terms from (3.30) ˆ Ab (f ) Φ
0
−10 n=2
−20
Power spectrum (dB)
n=3
−30
n=1 n=4
DCT
DFT
−40
−50
DST CP-DFT
−60
CP-DCT
−70
−80−3
−2
−1 0 1 Normalized frequency, f /W
2
3
Figure 3.8: Power density spectrum. (N = 64, 2πh = 0.6) density spectrum of the message signal m(t) according to (3.25): N TB X k k 2 2 Φm (f ) = f+ f− TB + sinc TB , sinc 2N 2TB 2TB
(3.33)
1,
(3.34)
k=1
where sinc(x) =
x = 0,
sin πx , πx
The functions {Un (f )} have the property:
otherwise.
R∞
−∞ Un (f )df
= 1, for all n [1]. Therefore
the nth term in (3.30) has an an × 100% contribution to the overall spectrum. For example, the carrier component, represented as δ(f ), has a fractional contribution of 2
2
2
e−σφ ; Φm (f ) ∗ Φm (f ) has a fractional contribution (e −σφ σφ4 )/2; and so on. Notice that 2
for 2πh = 0.2, the carrier component accounts for e −0.2 ×100 ≈ 96% of the signal power.
(This explains why the 90–92% curves at 2πh = 0.2 in Figure 3.7 are equal zero.)
55 Figure 3.8 plots the n = 1, 2, 3, 4 terms in (3.30), and the resulting sum ˆ Ab (f ) = A2 Φ
4 X
n=0
an Un (f ) ≈ ΦAb (f ).
(3.35)
The Abramson spectrum is shown to match all estimates over the range |f /W | ≤ 1. For |f /W | > 1, the spectral height depends on the overall smoothness of the phase signal. For
example, DST has a continuous phase [with or without memory since A b (k) = Ae (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 DCT since one-half of the subcarriers have zero-crossings at the signal boundaries [A b (k) = Ae (k) = 0, for k = N/2 + 1, . . . , N , and Ab (k) = Ae (k) = 1, otherwise] while DCT doesn’t [Ab (k) = Ae (k) = 1, for all k]. The smoothest phase results from CP-DCT which, unlike DST and CP-DFT, has a first derivative equal to zero at the boundary times t = iTB . Consequently, the CP-DCT is the most spectrally contained. Figure 3.9 shows estimated fractional out-of-band power curves that correspond to the signals in Figure 3.8. For reference, conventional OFDM is also plotted. Notice that the 99% spectral containment at f /W = 0.5 is the same for each signal. The continuous phase CE-OFDM signals are the most spectrally contained and are shown to have better than 99.99% containment at f /W = 1.25. Over the range 0.5 ≤ f /W ≤ 0.8, This 100 CE-OFDM OFDM Brms
Fractional out-of-band power
10−1 10−2 10−3 10−4
DCT
DFT
10−5 DST
10−6
CP-DFT
10−7 CP-DCT
10−8 0
0.5
1
1.5 2 Normalized frequency, f /W
2.5
Figure 3.9: Fractional out-of-band power. (N = 64, 2πh = 0.6)
3
56 figure shows that the CE-OFDM spectrum has more out-of-band power than conventional OFDM. Since the modulation index controls the CE-OFDM spectral containment, smaller h can be used if a tighter spectrum is required. The tradeoff is that smaller h results in worse performance, as will be discussed in the next chapter. Therefore, the system designer can trade performance for spectral containment, and visa versa. Figure 3.10 compares CE-OFDM, with CP-DFT modulation over a large range of modulation index, to conventional OFDM. For 2πh ≤ 0.4 the fractional out-of-band power of CE-OFDM is always better than OFDM; otherwise CE-OFDM has more out-
of-band power for at least some frequencies f /W > 0.5. The 2πh = 2.0 example has a broad spectrum, greater than OFDM over all frequencies. Notice that the shape of the spectrum appears Gaussian shaped. This is due to the fact that for a large modulation index, the higher-order terms in (3.32) dominate. They are Gaussian shaped due to the multiple convolutions of (3.33). The shape of “wideband FM” signals is well covered in the classical works of [1, 341, 437, 472]. 100 CE-OFDM OFDM 10−1 2πh Fractional out-of-band power
10−2
2.0 1.8 1.6
10−3
1.4 1.2
10−4
1.0 0.8
10−5
0.6 0.4 10−6 0.2 10−7 0
0.5
1 Normalized frequency, f /W
1.5
Figure 3.10: CE-OFDM versus OFDM. (N = 64)
2
57 Finally, 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 spectral broadening. The CE-OFDM signals have a bandwidth that depends only on the modulation index and are not effected by the PA nonlinearity. 100 OFDM, TWTA OFDM, Ideal CE-OFDM
Fractional out-of-band power
10−1 IBO (dB) 0 2 4 6
10−2
10−3
10−4
10−5 0
2πh
0.5
0.4
0.5
0.6
1 Normalized frequency, f /W
0.7
1.5
Figure 3.11: CE-OFDM versus OFDM with nonlinear PA. (N = 64)
2
Chapter 4
Performance of Constant Envelope OFDM in AWGN In this chapter the basic performance properties of CE-OFDM are studied. The baseband signal, represented by (3.11) and (3.12), is up-converted and transmitted as the bandpass signal n o sbp (t) = < s(t)ej2πfc t = A cos [2πfc t + φ(t)] ,
(4.1)
where fc is the carrier frequency. The received signal is rbp (t) = sbp (t) + nw (t),
(4.2)
where nw (t) denotes a sample function of the additive white Gaussian noise (AWGN) process with power density spectrum Φ nw (f ) = N0 /2 W/Hz. The primary focus of the chapter is to analyze the phase demodulator receiver, depicted by the block diagram below. An expression for the bit error rate (BER) is derived by making certain high carrier-to-noise ratio (CNR) approximations. The analytical result is then compared against computer simulation and it is shown to be accurate for BER < 0.01. It is also
rbp (t)
Bandpass filter
Phase demodulator
OFDM demodulator
Figure 4.1: Phase demodulator receiver.
58
To detector
59 demonstrated that with the use of a phase unwrapper, the receiver is insensitive to phase offsets caused by the channel and/or by the memory terms {θ i }. The phase demodulator receiver is a practical implementation of the CE-OFDM receiver and is therefore of practical interest. However, it isn’t necessarily optimum, since the optimum receiver is a bank of M N matched filters [421, p. 244], one for each potentially transmitted signal. In Section 4.2 a performance bound and approximation for the optimum receiver is derived; and then in Section 4.3, the performance of the phase demodulator receiver is compared to the optimum result. It is shown that under certain conditions the phase demodulator receiver has near-optimum performance. In Section 4.4 CE-OFDM’s spectral efficiency versus performance is compared to channel capacity. Finally, the chapter is concluded in Section 4.5 with a comparison between CE-OFDM and conventional OFDM in terms of power amplifier efficiency, total degradation, and spectral containment.
4.1
The Phase Demodulator Receiver
The phase demodulator receiver essentially consists of a phase demodulator followed by a conventional OFDM demodulator. Figure 4.2 shows the model used in this analysis. The received signal is first passed through a front-end bandpass filter, centered at the carrier frequency fc , which limits the bandwidth of the additive noise. Then the bandpass signal is down-converted to r(t), sampled, and processed in the discrete-time domain. The conversion from rbp (t) to r(t) is described first1 , making use of the following trigonometric identities: cos(x − y) − cos(x + y) , 2 sin(x + y) + sin(x − y) sin(x) cos(y) = , 2 cos(x + y) + cos(x − y) cos(x) cos(y) = , 2 sin(x + y) − sin(x − y) cos(x) sin(y) = . 2 sin(x) sin(y) =
1
(4.3) (4.4) (4.5) (4.6)
This is the standard model used for representing received baseband signals, and more discussion of the model can be found in [421, sec. 4.1], [624, sec. 5.5], among other places.
60 Lowpass filter 2 cos(2πfc t) rbp (t)
Bandpass u(t) filter
r(t)
r[i]
t = iTsa
Phase demodulator
OFDM demodulator
−2 sin(2πfc t) j Lowpass filter
Figure 4.2: Bandpass to baseband conversion. The output of the bandpass filter is u(t) = sbp (t) + nbp (t),
(4.7)
nbp (t) = nc (t) cos(2πfc t) − ns (t) sin(2πfc t)
(4.8)
where
is the result of passing nw (t) through the bandpass filter. The terms n c (t) and ns (t) are referred to as the in-phase and quadrature components of the narrowband noise, respectively, and have the power density spectrum N0 , |f | ≤ Bbpf /2, Φnc (f ) = Φns (f ) = 0, |f | > Bbpf /2,
(4.9)
where Bbpf is the bandwidth of the bandpass filter. Note that B bpf is assumed to be
sufficiently large so sbp (t) is passed through the front-end filter with negligible distortion [421, pp. 157–158]. Writing sbp (t) in the form sbp (t) = sc (t) cos(2πfc t) − ss (t) sin(2πfc t),
(4.10)
where sc (t) = A cos[φ(t)] and ss (t) = A sin[φ(t)], the filter output can then be written as u(t) = [sc (t) + nc (t)] cos(2πfc t) − [ss (t) + ns (t)] sin(2πfc t).
(4.11)
61 The output of the top (in-phase) branch of the down-converter is 2 rc (t) = LP {u(t) × 2 cos(2πfc t)} = LP{[sc (t) + nc (t)] + [sc (t) + nc (t)] cos(4πfc t)
(4.12)
− [ss (t) + ns (t)] sin(4πfc t)} = sc (t) + nc (t), where LP{·} denotes the lowpass component of its argument (i.e., double-frequency terms are rejected) [624, p. 364]. Likewise, the output of the bottom (quadrature) branch is rs (t) = LP {u(t) × −2 sin(2πfc t)} = LP{−[sc (t) + nc (t)] sin(4πfc t) + [ss (t) + ns (t)]
(4.13)
− [ss (t) + ns (t)] cos(4πfc t)} = ss (t) + ns (t). The two are combined to obtain r(t) = s(t) + n(t),
(4.14)
where s(t) is the lowpass equivalent CE-OFDM signal from (3.11), and n(t) = nc (t) + jns (t)
(4.15)
is the lowpass equivalent representation of the bandpass white noise, n bp (t) [421, p. 158]. The power density spectrum of n(t) is [421, p. 158] N0 , |f | ≤ Bn /2, Φn (f ) = 0, |f | > Bn /2,
(4.16)
where Bn = Bbpf is the noise bandwidth. The corresponding autocorrelation of n(t)
is [421, p. 158] φn (τ ) = N0
sin πBn τ . πτ
(4.17)
The continuous-time receive signal is then sampled at the rate f sa = 1/Tsa samp/s to obtain the discrete-time signal 3 r[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 of channel phase offsets is considered. 3 Perfect timing synchronization is assumed.
62 Phase demodulator r[i]
FIR filter
Phase unwrapper
arg(·)
To OFDM demodulator
Figure 4.3: Discrete-time phase demodulator. where s[i] = s(t)|t=iTsa and n[i] = n(t)|t=iTsa . As discussed in Section 2.4.2, the noise samples {n[i]} are assumed independent: E {n[i1 ]n[i2 ]} =
σ 2 , n
0,
i1 = i2 ,
(4.19)
i1 6= i2 ;
and therefore the sampling rate is f sa = Bn , and σn2 = fsa N0 . The discrete-time phase demodulator studied in this thesis is shown in Figure 4.3. The finite impulse response (FIR) filter is optional, but has been found effective at improving performance; arg(·) simply calculates the arctangent of its argument; and the phase unwrapper is used to minimize the effect of phase ambiguities. As will be shown, the unwrapper makes the receiver insensitive to phase offsets caused by the channel and/or by the memory terms. The output of the phase demodulator is processed by the OFDM demodulator which consists of the N correlators, one corresponding to each subcarrier. This correlator bank is implemented in practice with the fast Fourier transform.
4.1.1
Performance Analysis
In this section a bit error rate approximation is derived for the phase demodulator receiver. Although the receiver operates in the discrete-time domain, it is convenient to analyze it in the continuous-time domain. The angle of the received signal is arg[r(t)] = θi + 2πhCN
N X k=1
iTB ≤ t < (i + 1)TB , where
Ii,k qk (t − iTB ) + ξ(t),
N (t) sin [Θ(t) − φ(t)] ξ(t) = arctan A + N (t) cos [Θ(t) − φ(t)]
(4.20)
(4.21)
is the corrupting noise [624, p. 416]. The terms N (t) and Θ(t) in (4.21) are the envelope and phase of n(t).
63 The kth correlator in the OFDM demodulator computes Z (i+1)TB 1 arg[r(t)]qk (t − iTB )dt = Si,k + Ni,k + Ψi,k . TB iTB
(4.22)
The signal term is Si,k
1 = TB
Z
(i+1)TB
[φ(t) − θi ]qk (t − iTB )dt
iTB
2πhCN = TB
Z
N (i+1)TB X
iTB
n=1
2πhCN Ii,k Eq = 2πh = TB The noise term is Ni,k =
1 TB
Z
Ii,n qn (t − iTB )qk (t − iTB )dt
s
(4.23)
1 Ii,k . 2N σI2
(i+1)TB iTB
ξ(t)qk (t − iTB )dt.
For example, with DST subcarrier modulation (3.8), Z (i+1)TB 1 ξ(t) sin [πk(t − iTB )/TB ] dt, Ni,k = TB iTB
(4.24)
(4.25)
which can be viewed as a Fourier coefficient of ξ(t) at f = k/2T B Hz. As TB → ∞, the variance of the coefficient is proportional to the power density spectrum function
evaluated at f = k/2TB [442, pp. 41–43]. It is well known that, given a high CNR, the noise at the output of a phase demodulator has a power density spectrum [423, p. 410] Φξ (f ) ≈
N0 , A2
|f | ≤ W/2,
(4.26)
where, from (3.26), W = N/TB is the effective bandwidth of φ(t). Moreover, for high CNR, ξ(t) is well modeled as a sample function of a zero mean Gaussian process. Therefore, Ni,k is approximated as a zero mean Gaussian random variable with variance [442, pp. 41–43] 1 1 N0 Φξ (f )|f =k/2TB ≈ . 2TB 2TB A2 This result is the same for DCT and DFT subcarrier modulation. var{Ni,k } ≈
The third term in (4.22), Ψi,k , is expressed as Z (i+1)TB 1 Ψi,k = θi qk (t − iTB )dt. TB iTB Since Z
0
(4.27)
(4.28)
TB
qk (t)dt = 0,
k = 1, 2, . . . , N,
(4.29)
64 for DCT and DFT modulations [(3.7), (3.9)], Ψ i,k = 0 and therefore has no effect on system performance. 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 each signal point in the M -PAM constellation. For the M − 2 inner points, the probability of
error is
Pinner = P (|Ni,k | > d) = 2P (Ni,k > d),
(4.30)
s
(4.31)
where d = 2πh
1 . 2N σI2
[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 , Z ∞ 1 p Pinner ≈ 2 exp −x2 / 2N0 /(2A2 TB ) dx 2πN0 /(2A2 TB ) d Z ∞ 1 √ exp −x2 /2 dx =2 (4.32) 2π d[N0 /(2A2 TB )]−0.5 s ! ! r A2 TB 6 log 2 M Eb . = 2Q 2πh = 2Q 2πh M 2 − 1 N0 N0 N σI2
For the two outer points, the probability of error is 1 Pouter = P (Ni,k > d) = Pinner . 2
(4.33)
Therefore, the overall symbol error rate is M −2 2 Pinner + Pouter M M ! r 6 log 2 M Eb M −1 Q 2πh . ≈2 M M 2 − 1 N0
SER =
(4.34)
Notice that for 2πh = 1, (4.34) is equivalent to the SER for conventional M -PAM [483, pp. 194–195]. For high SNR, the only significant symbol errors are those that occur in adjacent signal levels, in which case the bit error rate is approximated as [483, p. 195] ! r SER 6 log 2 M Eb M −1 BER ≈ Q 2πh . (4.35) ≈2 log2 M M log2 M M 2 − 1 N0
65
4.1.2
Effect of Channel Phase Offset
Suppose the channel imposes a phase offset of φ 0 . The received signal is then r(t) = s(t)ejφ0 + n(t).
(4.36)
The angle of r(t) is arg[r(t)] = θi + 2πhCN
N X k=1
Ii,k qk (t − iTB ) + φ0 + ξ(t),
(4.37)
iTB ≤ t < (i + 1)TB . Which is identical to (4.20) with the addition of the channel offset term. The kth correlator is the same as (4.22), except the third term is Ψi,k
1 = TB
Z
(i+1)TB iTB
[θi + φ0 ]qk (t − iTB )dt = 0.
(4.38)
Therefore, the phase offset due to the channel has no impact on performance, and the analytical approximation in (4.35) is applicable. Figure 4.4 compares the performance of N = 64, M = 2 CE-OFDM with phase offset {(θi + φ0 ) ∈ [0, 2π)}, and without (θi + φ0 = 0). The former is referred to as System 1 (S1), the later as System 2 (S2). The system is computer simulated with a sampling
rate fsa = JN/TB , where J = 8 is the oversampling factor 4 . For Eb /N0 ≥ 10 dB and 2πh ≤ 0.5, S1 and S2 are shown to have identical performance. For these cases the analytical approximation (4.35) closely matches the simulation results for BER < 0.01.
With the 2πh = 0.7 example, S1 is shown to have a 1 dB performance loss compared to S2. In this case, the analytical approximation is shown to be overly optimistic. This demonstrates a limitation of the phase demodulator receiver: for a large modulation index and low signal-to-noise ratio, the phase demodulator has difficulty demodulating the noisy samples. The performance of S1 is slightly worse than S2 since the output of the phase demodulator, the arg(·) block in Figure 4.3, has more phase jumps since the received phase crosses the π boundary more frequently. Proper phase unwrapping is therefore required. However, phase unwrapping a noisy signal is a difficult problem and the unwrapper makes mistakes. As a result the performance degrades slightly. For a smaller modulation index, the unwrapper works perfectly and the performance of S1 isn’t degraded. 4
Also, the FIR filter (see Figure 4.3) has length Lfir = 11 and normalized cutoff frequency fcut /W = 0.2. See Section 4.1.4 for more on the filter design.
66 100 System 1 System 2 Approx (4.35)
Bit error rate
10−1
10−2
10−3
0.7
2πh
0.5
0.3
0.2
0.1
10−4
10−5 0
5
10 15 20 25 Signal-to-noise ratio per bit, Eb /N0 (dB)
30
Figure 4.4: Performance with and without phase offsets. System 1 (S1) has phase offsets {(θi + φ0 ) ∈ [0, 2π)}, and System 2 (S2) doesn’t (θ i + φ0 = 0). [M = 2, N = 64, J = 8]
4.1.3
Carrier-to-Noise Ratio and Thresholding Effects
The high-CNR approximation made in (4.26), which leads to the BER approximation (4.35), is a standard technique for analyzing phase demodulator receivers [423, 624]. A well-known characteristic of such receivers is: at low CNR, below a threshold value, the approximation is invalid and system performance degrades drastically. In this section, the CNR is defined and the threshold effect for CE-OFDM is observed by way of computer simulation. The CNR at the output of the analog front end, r(t), is CNR = where A2 is the carrier power, and Pn =
Z
A2 , Pn
(4.39)
∞
Φn (f )df = Bn N0
(4.40)
−∞
is the noise power. From (3.17), the carrier power can be written in the form A2 =
Eb N log 2 M ; TB
(4.41)
67 thus CNR =
(Eb /N0 )N log2 M . TB Bn
(4.42)
Since the noise samples are assumed independent [see (4.19)], Bn = fsa = JN/TB ,
(4.43)
and (4.42) reduces to CNR =
(Eb /N0 ) log 2 M . J
(4.44)
Therefore, the carrier-to-noise ratio is proportional to E b /N0 and M , and inversely proportional to the oversampling factor.
A commonly accepted threshold CNR for analog FM systems is 10 dB [472, pp. 120–138], [501, pp. 87–91]. This threshold level is studied in the following two figures. In Figure 4.5, simulation results for an M = 8, N = 64, J = 8, 2πh = 0.5 system are compared to (4.35). In subfigures (a) and (b) the system is below and above the 10 dB threshold, respectively. Clearly, above CNR = 10 dB, the system is observed to be above threshold, with simulation results closely matching the analytical approximation. Below 10 dB, the performance begins to deviate from (4.35); and for CNR < 5 dB, the performance quickly degrades to a bit error rate of 1/2. Figure 4.6 shows results for more values of 2πh. For each case, 10 dB can be considered an appropriate threshold level. There is, however, a transition region—that is, a region where the system is useless, with a BER of 1/2, to where the system is above threshold. This transition region is difficult to study analytically. Gaining more insight into this issue is a subject for future investigation.
Bit error rate
Bit error rate
68
10−2
Simulation Approx (4.35)
10−1 Simulation Approx (4.35)
−2
0 2 4 6 8 Carrier-to-noise ratio (dB)
10−310
10
11 12 13 14 15 Carrier-to-noise ratio (dB)
16
(b) Above 10 dB threshold.
(a) Below 10 dB threshold.
Figure 4.5: Threshold effect at low CNR. (M = 8, N = 64, J = 8, 2πh = 0.5)
2πh
Simulation Approx (4.35)
10−1
0.4
10−1
Bit error rate
Bit error rate
0.2
0.6
10−2
Simulation Approx (4.35)
0.8
0.6
0.4
2πh
0.2
0.8
10−2 −2
0 2 4 6 8 Carrier-to-noise ratio (dB)
(a) Below 10 dB threshold.
10
10−310
12 14 16 18 20 22 Carrier-to-noise ratio (dB)
24
(b) Above 10 dB threshold.
Figure 4.6: Threshold effect at low CNR, various 2πh. (M = 8, N = 64, J = 8)
69
4.1.4
FIR Filter Design
The FIR filter preceding the phase demodulator (see Figure 4.3) can improve 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 /N0 = 10 dB. The filter, designed
using the window technique described in [422, pp. 623–630], has a length 3 ≤ L fir ≤ 101
and a normalized cutoff frequency 0 < f cut /W ≤ 1. Hamming windows are used5 . The
performance without a filter is shown to be BER = 0.05, while the analytical approxi-
mation (4.35) is BER = 0.012. For fcut /W ≥ 0.4 all the filtered results are shown to be
better than the unfiltered result. The filters with L fir > 5 and fcut /W > 0.5 are shown
to have roughly the same performance. The higher-order filters, which have a narrower transition bands, require fcut /W > 0.5 to yield good performance. This is explained by noting that the (single-sided) signal bandwidth is at least W/2 Hz. Therefore, the higher-order filters with fcut /W < 0.5 distort the signal. Notice that the L fir = 11 filter has equally good performance so long as f cut /W ≥ 0.1. This is due to the wide transition
band of the lower-order filter.
Lfir = 3 5 7 9 11 21 31 61 101
Bit error rate
10−1
No filter
Approx (4.35) 10−2 0
0.2
0.4 0.6 Normalized cutoff frequency, fcut /W
0.8
Figure 4.7: Performance for various filter parameters L fir , fcut /W . (M = 2, N = 64, J = 8, 2πh = 0.5 and Eb /N0 = 10 dB) 5
It has been observed that the window type has negligible impact on performance.
1
70
0 Lfir , fcut /W
Magnitude response (dB)
−20
9, 0.7
3, 0.1 −40 101, 0.7
9, 0.1
−60 31, 0.1
−80
−100 0
0.5
1
1.5 2 Normalized frequency, f /W
2.5
3
Figure 4.8: Magnitude response of various Hamming FIR filters. The figure above shows the magnitude response of the various Hamming FIR filters. The filters with relatively flat response over |f /W | ≤ 0.5 result in good performance.
The Lfir = 31, fcut /W = 0.1 example is shown to not have this property, and, as shown in Figure 4.7, has worse BER performance than the other filters. Figure 4.9 compares the performance of binary (M = 2) CE-OFDM with and without the FIR filter. The Lfir = 11, fcut /W = 0.2 filter is used. These results show that the filter becomes important for larger modulation index: for 2πh = 0.1 the filtered and unfiltered results are the same; for 2πh = 0.3 the filtered performance is a fraction of a dB better than the unfiltered; for 2πh = 0.7 there is a 2 dB improvement in the range 10−3 < BER < 10−5 . Notice the error floor developing below 10 −5 . This is a consequence of imperfect phase demodulation. The filter lowers the error floor resulting in a 9 dB improvement at BER = 10−6 .
71 100 Without FIR filter With FIR filter Approx (4.35) 10−1
Bit error rate
10−2
10−3
0.7
2πh
0.3
0.1
10−4
10−5
10−6 0
5
10 15 20 Signal-to-noise ratio per bit, Eb /N0 (dB)
25
30
Figure 4.9: CE-OFDM performance with and without FIR filter. (M = 2, N = 64, J = 8)
4.2
The Optimum Receiver
As mentioned in the introduction to this chapter, the phase demodulator receiver is a practical implementation, but not necessarily optimum. In this section, the optimum, yet impractical, CE-OFDM receiver is studied. Results obtained here are used in the following section to compare the phase demodulator receiver to optimum performance. During each block one of M N CE-OFDM signals is transmitted. Consider the mth bandpass signal "
sm (t) = A cos 2πfc t + θ0 + K
N X
(m) Ik qk (t)
k=1
#
,
0 ≤ t < TB ,
(4.45)
N
where K = 2πhCN . The set of all possible signals, {s m (t)}M m=1 , is determined by the (m)
set of all possible data symbol vectors {I (m) = [I1
(m)
, I2
(m)
N
, . . . , IN ]}M m=1 . The optimum
72 receiver, as shown in Figure 4.10, correlates the received signal, r bp (t) = sm (t) + nw (t), with each potentially transmitted signal. The detector then selects the largest result [421, pp. 242–247]. R TB 0
(·)dt
R TB
(·)dt
s1 (t) 0
s2 (t)
Select the largest
Received signal rbp (t) .. .
.. .
R TB 0
sM N (t)
Output decision
(·)dt Sample at t = TB
Figure 4.10: The optimum receiver.
4.2.1
Performance Analysis
It 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 data symbol error probability. Determining exact expressions for the above probabilities is intractable for large N . However, upperbounds and approximations can be derived in a straightforward way, as described below. 6
The bit error probability is used interchangeably with the bit error rate. Likewise for the symbol error probability and symbol error rate.
73 An upperbound for P (signal error) is [373]: Z ∞h i 1 M N −1 √ P (signal error) ≤ × 1 − [1 − Q(y)] 2π −∞ 2 s 1 2E (1 − λ) s dy. exp − y − 2 N0
(4.46)
The above expression is the probability of detection error for M N signals with equal correlation −1 ≤ λ ≤ 1. Therefore, it provides an upperbound given that λ = ρmax =
max ρm,n ,
(4.47)
m,n; m6=n
where ρm,n is the normalized correlation between s m (t) and sn (t): ρm,n =
1 Es
Z
TB
sm (t)sn (t)dt.
An approximation for P (signal error) is [421, p. 288] s P (signal error) ≈ Kd2 Q min
where Kd2
min
(4.48)
0
d2min 2N0
,
(4.49)
is the number of neighboring signal points having the minimum squared
Euclidean distance d2min = where d2m,n =
Z
0
min d2m,n ,
m,n; m6=n
TB
[sm (t) − sn (t)]2 dt
(4.50)
(4.51)
is the squared Euclidean distance between s m (t) and sn (t). This quantity is related to the signal correlation as d2m,n = 2Es (1 − ρm,n ),
(4.52)
d2min = 2Es (1 − ρmax ).
(4.53)
thus
Therefore to obtain the performance bound (4.46) and the approximation (4.49) the signal 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
74 constant K = 2πhCN , is ρm,n (K) =
1 Es
A2 = Es
TB
Z
0 TB
Z
sm (t)sn (t)dt "
cos 2πfc t + θ0 + K
0
N X
(m) Ik qk (t)
k=1
"
cos 2πfc t + θ0 + K =
A2 2Es
(m)
where ∆m,n (k) = 0.5[Ik
Z
"
TB
cos 2K 0
N X
#
×
(n) Ik qk (t)
k=1
N X
#
#
(4.54) dt
∆m,n (k)qk (t) dt,
k=1
(n)
− Ik ]. The double frequency term is ignored since f c 1/TB
is assumed. Notice that for k where ∆ m,n (k) = 0, the data symbols are the same, and these indices don’t contribute to the correlation. Therefore " # Z D X A 2 TB cos 2K ρm,n (K) = ∆m,n (kd )qk (t) dt, 2Es 0
(4.55)
d=1
where {kd }D d=1 are the indices where the data symbols differ, that is, ∆ m,n (kd ) 6= 0, and
D is the total number of differences. Writing (4.55) in exponential form yields #) ( " Z D X A 2 TB ∆m,n (kd )qk (t) dt < exp j2K ρm,n (K) = 2Es 0 d=1 (D ) Z Y A 2 TB < exp [j2K∆m,n (kd )qk (t)] dt. = 2Es 0
(4.56)
d=1
To proceed, the DCT modulation (3.7) is assumed. Making use of the Jacobi-Anger expansion [580], e
ja cos b
=
∞ X
Ji (a)eji(b+π/2) ,
(4.57)
i=−∞
where Ji (a) is the ith-order Bessel function of the first kind, (4.56) is written as " ∞ Z ∞ X X A 2 TB < ··· ρm,n (K) = 2Es 0 iD =−∞ i1 =−∞ # Ji1 [2K∆m,n (k1 )] × · · · × JiD [2K∆m,n (kD )]ejσ(i) dt =
A2 2Es
Z
TB 0
∞ X
i1 =−∞
···
∞ X
Ji1 [2K∆m,n (k1 )]×
iD =−∞
· · · × JiD [2K∆m,n (kD )] cos[ω(i) + ψ(i)]dt,
(4.58)
75 where σ(i) = ω(i) + ψ(i), ω(i) ≡
PD
πt TB
d=1 id kd
and ψ(i) ≡
π 2
PD
d=1 id .
Index values that
result in ω(i) 6= 0 have no contribution, so (4.58) simplifies to ρi,j (K) =
D XY i
Ji0i,d [2K∆m,n (kd )] cos[ψ(i0i )],
(4.59)
d=1
where i0i ≡ [i0i,1 , . . . , i0i,D ], i = 1, 2, . . ., represent the vectors whereby ω(i 0i ) = 0. This result is the same for DST modulation except ψ(i 0i ) = 0. For DFT modulation, (4.59) is slightly different since both sinusoids and cosinusoids are used as subcarriers. For D = 1, ρm,n (K) = J0 [2K∆m,n (k1 )].
(4.60)
Therefore the correlation is simply the 0th-order Bessel function. Figure 4.11(a) plots (4.60) for |∆m,n (k1 ) = 1|. Also plotted is the envelope of the 0th-order Bessel func-
tion [580, p. 121]. Note that ρm,n (K) doesn’t depend on the subcarrier frequency fk1 = k1 /TB , k1 ∈ {1, 2, . . . , N }, just on the magnitude of the difference |∆ m,n (k1 )| ∈ {1, 2, . . . , (M − 1)}.
For CE-OFDM signals of interest, ρmax = J0 (2K).
(4.61)
Figure 4.11(b) plots all unique ρm,n (K) for M = 2, N = 8 DCT subcarrier modulation. Notice that the largest correlation function is associated with D = 1. For any given signal, there are N other signals with D = 1: therefore, K d2
min
the probability of signal error is approximated as s P (signal error) ≈ Kd2 Q min
= N , and from (4.49),
d2min 2N0
p Es [1 − ρmax ]/N0 p ≈ NQ Es [1 − J0 (2K)]/N0 . = NQ
(4.62)
A minimum distance signal error results in one data symbols error. Therefore, the symbol error probability is approximated as P (symbol error) ≈
p P (signal error) Es [1 − J0 (2K)]/N0 . ≈Q N
(4.63)
For M = 2, one symbol error corresponds to one bit error. For M > 2, a symbol error can result in 1 to log 2 M bit errors. Assuming each outcome is equally likely, a symbol
76
1
0.5 ρm,n (K)
p 1/πK
0
−0.5 0
1
2
3
4
5
K
(a) D = 1. 1
J0 (2K)
ρm,n (K)
0.8
0.6
0.40
0.1
0.2
0.3
0.4
K
(b) All unique ρm,n (K) for M = 2, N = 8 DCT modulation.
Figure 4.11: Correlation functions ρ m,n (K).
0.5
77 error results in
1 log2 M
Plog2 M i
i = 0.5(log 2 M + 1) bit errors. Thus
0.5(log 2 M + 1) P (symbol error) log2 M 0.5(log 2 M + 1) p ≈ Q Es [1 − J0 (2K)]/N0 . log2 M
P (bit error) ≈
(4.64)
The bit error probability is bounded by noting that P (bit error) ≤ P (signal error),
and using (4.46) with λ = ρmax = J0 (2K): Z ∞h i N 1 P (bit error) ≤ √ 1 − [1 − Q(y)]M −1 × 2π −∞ 2 s 1 2Es [1 − J0 (2K)] dy. exp − y − 2 N0
(4.65)
Figure 4.12 shows simulation results of the optimum receiver for M = 2 and N = 8.
The number of correlators at the receiver is therefore 2 8 = 256. Two values of modulation index are plotted: 2πh = 0.3 and 2πh = 0.7 which corresponds to K = 0.15 and K = 0.35. The upperbound (4.65) is shown to be within 3 dB of the simulated results for high SNR. The analytical approximation (4.64) is shown to be very accurate. 100
10−1
Bit error rate
10−2
10−3 Approx (4.64) Bound (4.65) Simulation
10−4
0.7
0.3
2πh
10−5
10−6 0
3
6
9 12 15 Signal-to-noise ratio per bit, Eb /N0 (dB)
18
21
Figure 4.12: CE-OFDM optimum receiver performance. (M = 2, N = 8)
78
4.2.2
Asymptotic Properties
In Figure (4.13) each correlation function is plotted for M = 2, N = 4 DCT modulation. The functions are shown to be bounded by r
1 , πK the envelope of the 0th-order Bessel function. Therefore, r ρm,n (K) ≤ ρmax (K) ≤
d2m,n (K) ≥ d2min (K) ≥ 2Es
1−
1 πK
(4.66)
!
.
(4.67)
Notice that as K → ∞ the CE-OFDM signals become orthogonal. The phase modulator thus drastically alters the signal space. Prior to the phase demodulator, the OFDM
signal space is described by 2N dimensions (2 per subcarrier). At the output of the phase modulator, the space is transformed into a M N -dimensional space (due to the linear independence of the signal set [421, p. 164]); and as the modulation index becomes very large, a M N -dimensional orthogonal space. However, from (3.29), the bandwidth tends to infinity as 2πh → ∞. 1
ρm,n (K)
0.5 p 1/πK
0
−0.5 0
1
2
3
4
5
K
Figure 4.13: All unique ρm,n (K) for M = 2, N = 4 DCT modulation.
4.3
Phase Demodulator Receiver versus Optimum
Figure 4.14 shows simulation results for the phase demodulator receiver with N = 64 and for various modulation index values 2πh and modulation order M . The simulation
79 100 Approx (4.35) Approx (4.64) Simulation
Bit error rate
10−1
10−2
10−3
8, 1.2
M , 2πh
2, 0.3
10−4
10−5 5
16, 0.8
10
16, 0.2
4, 0.2
15 20 25 30 Signal-to-noise ratio per bit, Eb /N0 (dB)
35
40
Figure 4.14: Phase demodulator receiver versus optimum. (N = 64) results are compared to the analytical approximation (4.35) and the optimum receiver approximation (4.64). All curves are shown to be essentially identical for BER < 0.01. This implies that the phase demodulator receiver is nearly optimum. For this to be true, the phase demodulator must perfectly invert the phase modulation done at the transmitter, and the noise at the output of the phase demodulator must be “white” and Gaussian. That is, the OFDM demodulator is optimum given that the input, φ(t) + ξ(t), is comprised of the transmitted message signal plus an AWGN corrupting signal. As shown by (4.26), ξ(t) is approximately “white”. The probability density function of ξ(t) samples is represented by the well-known form [421, p. 268] 2 Z ∞ y y + A2 − 2yA cos x pξ (x) = exp − dy, 2πσn2 2σn2 0
(4.68)
where σn2 = Bn N0 is the power of the noise signal n(t). Figure 4.15 compares (4.68) to the Gaussian probability density function. The SNR per bit is E b /N0 = 30 dB. This shows that ξ(t) is well approximated as Gaussian, and near optimum performance of the phase demodulator receiver is expected.
80
pξ (x) Gaussian
Probability density function, p(x)
100
10−5
10−10
10−15
10−20 −1.5
−1
−0.5
0 x
0.5
1
1.5
Figure 4.15: Noise samples PDF versus Gaussian PDF. (E b /N0 = 30 dB)
4.4
Spectral Efficiency versus Performance
In the previous sections, it is shown that the performance of CE-OFDM is determined by the modulation index, which, as shown in Section 3.2, also controls the signal bandwidth. In this section, the spectral efficiency (b/s/Hz) versus performance (E b /N0 to achieve a target bit error rate) is plotted for a variety of CE-OFDM signals. The results are compared to channel capacity. It is first demonstrated that CE-OFDM with modulation index 2πh > 1 can outperform the underlying M -PAM subcarrier modulation. Figure 4.16 shows simulation results7 for M = 2, 4, 8 and 16. The bit error rate is plotted against the SNR per bit on the bottom x-axis and the carrier-to-noise ratio on the top x-axis. The viewable range is such that CNR ≥ 5 dB. Notice that for M ≥ 4 and 2πh > 1, CE-OFDM outperforms M -PAM. This is predicted by (4.35), since for 2πh = 1.0, the expression is equal to the
performance of M -PAM, and for 2πh > 1.0, it is better than M -PAM. For CE-OFDM to operate in the region 2πh > 1, the carrier-to-noise ratio must be above threshold. 7
The oversampling factor is J = 8 for M = 2, 4 and 8, and J = 16 for M = 16. The FIR filter has length Lfir = 11 and a normalized cutoff frequency 0.2 cycles per sample for M = 2, 4 and 16, and 0.3 cycles per sample for M = 8.
81
Carrier-to-noise ratio (dB) 10 15
5 10−1
20
Carrier-to-noise ratio (dB) 10 15 20 25
5 10−1
Simulation (4.35)
Simulation (4.35) 4-PAM 10−2 Bit error rate
Bit error rate
10−2
10−3
2πh ∈ {0.5† , 0.4, . . . , 0.1‡ }
10−4
10−5
10−3
2πh ∈ {1.0† , 0.9, . . . , 0.1‡ }
10−4
16 18 20 22 24 26 28 30 Signal-to-noise ratio per bit, Eb /N0 (dB)
10−5
15 20 25 30 35 Signal-to-noise ratio per bit, Eb /N0 (dB)
(a) M = 2.
5 10−1
10
(b) M = 4.
Carrier-to-noise ratio (dB) 15 20 25 30
35
5 10−1
10
Carrier-to-noise ratio (dB) 15 20 25 30 35
Simulation (4.35) 8-PAM
Simulation (4.35) 16-PAM 10−2 Bit error rate
Bit error rate
10−2
10−3
10−4
10−5 10
40
2πh ∈ {1.5† , 1.2, 1.0, 0.9, . . . , 0.1‡ }
15 20 25 30 35 40 Signal-to-noise ratio per bit, Eb /N0 (dB)
(c) M = 8.
10−3
10−4
10−5
2πh ∈ {2.0† , 1.5, 1.2, 1.1, . . . , 0.1‡ }
15 20 25 30 35 40 45 Signal-to-noise ratio per bit, Eb /N0 (dB)
(d) M = 16.
Figure 4.16: Performance of M -PAM CE-OFDM. (N = 64, †=leftmost curve, ‡=rightmost curve)
82 To plot the spectral efficiency versus performance, the data rate must be defined, which for uncoded CE-OFDM is R=
N log 2 M b/s. TB
(4.69)
Using (3.29) as the effective signal bandwidth, the spectral efficiency is R/B =
log2 M R = b/s/Hz. Bs max(2πh, 1)
(4.70)
Figure 4.17 shows result for M = 2, 4, 8 and 16. The target bit error rate is 0.0001. For reference the channel capacity is also plotted, which is expressed as [421, p. 387] C Eb , (4.71) C = B log 2 1 + B N0 or equivalently, Eb 2C/B − 1 = . N0 C/B
(4.72)
10
7 6
M =2 M =4 M =8 M = 16 Capacity
M = 16: 2πh = 2.0, 1.8, . . . , 0.6 M = 8: 2πh = 1.4, 1.2, . . . , 0.4
Spectral efficiency (b/s/Hz)
5 4 3
2 M = 4: 2πh = 1.0, 0.8, . . . , 0.2
1 M = 2: 2πh = 0.5, 0.4, 0.3, 0.2
0.5-1.6
0
5 10 15 20 Performance: Eb /N0 (dB) to achieve 0.0001 bit error rate
25
Figure 4.17: Spectral efficiency versus performance. There are two main observations to be made. First, for a fixed modulation index, CE-OFDM has improved spectral efficiency with increase modulation order M at the cost of performance degradation. For example consider 2πh = 0.4. The spectral efficiency
83 is 1, 2 and 3 b/s/Hz for M = 2, 4 and 8, respectively. However, M = 4 requires 4 dB more power than M = 2, and M = 8 requires nearly 5 dB more power than M = 4. This type of spectral efficiency/performance tradeoff is the same for conventional linear modulations such as M -PAM, M -PSK and M -QAM [421, p. 282]. The second observation is that CE-OFDM can have both improvements in spectral efficiency and in performance. Compare M = 2, 2πh = 0.5 with M = 4, 2πh = 1.0, for example. The spectral efficiency doubles in the later case while also having a 2 dB performance gain. Conventional CPM systems also have the property of increase spectral efficiency and performance [14]. However, with CPM the receiver complexity increases drastically with M (due to phase trellis decoding), which isn’t the case for CE-OFDM.
4.5
CE-OFDM versus OFDM
The total degradation, as defined in Section 2.4.2, is TD(IBO) = SNRPA (IBO) − SNRAWGN + IBO,
[in dB]
where SNRAWGN is the required signal-to-noise ratio required to achieve a target bit error rate, SNRPA (IBO) is the required SNR when taking into account the nonlinear power amplifier at a given backoff. Applying the PA model from Section 2.3 to CE-OFDM, the input signal is sin (t) = A exp[jφ(t)],
(4.73)
sout (t) = G(A) exp j[φ(t) + Φ(A)] .
(4.74)
and the output is
The instantaneous nonlinearity results in a constant amplitude and a constant phase shift. Therefore the PA has no impact on the CE-OFDM performance and no backoff is needed. The total degradation for CE-OFDM is defined as TD = SNRPM − SNRsub ,
(4.75)
where SNRsub is the required SNR for the underlying subcarrier modulation and SNR PM is the required SNR for the phase modulated CE-OFDM system. By this definition, the total degradation can be negative since, as observed in Figure 4.16, CE-OFDM can outperform the underlying subcarrier modulation at the price of lower spectral efficiency.
84 Figure 4.18 compares CE-OFDM with conventional OFDM in terms of PA efficiency, total degradation and spectral containment. Binary modulation is used in both systems. The target BER is 10−5 and the number of subcarriers is N = 64. Both the SSPA and TWTA models are considered. The lowest TD for the TWTA system is 10.5 dB at 8 dB backoff, which corresponds to an 8% efficiency as shown in Figure 4.18(a). At this backoff level, the 99.5% bandwidth occupancy is roughly the same as undistorted ideal OFDM as shown in Figure 4.18(c). For the SSPA model, the lowest TD is 3.8 dB at IBO = 1 dB. In this case, the PA efficiency is improved to 40% but the bandwidth requirement is 73% more than ideal OFDM. Since CE-OFDM has a constant envelope, the PA can operate at IBO = 0 dB thus maximizing amplifier efficiency. The total degradation is 5 dB for 2πh = 0.6 and the corresponding bandwidth requirement is 26% more than ideal OFDM. For 2πh = 0.4, the total degradation is 8 dB but the bandwidth reduces to f /W = 0.98 which is 8% less than ideal OFDM. This shows that the modulation index for CE-OFDM can be chosen accordingly to balance performance and bandwidth. Also, since the PA imposes no additional distortion on the CE-OFDM signal, the resulting spectrum can be well contained with no power backoff and at the same time have optimal PA efficiency.
85
Class-A PA efficiency, ηA (%)
50 45 40 35 30 25 20 15 10 5 00
1
2
3
4 5 6 7 Input power backoff, IBO (dB)
8
9
10
(a) PA efficiency.
Total degradetion (dB)
16 14 12 10
OFDM, TWTA OFDM, SSPA OFDM, ideal CE-OFDM: 2πh = 0.4 0.5 0.6
8 6 4 2 00
2
4 6 Input power backoff, IBO (dB)
8
10
(b) Total degradation for target BER 10−5 .
99.5% bandwidth, f /W
2
OFDM, TWTA OFDM, SSPA OFDM, ideal CE-OFDM: 2πh = 0.4 0.5 0.6
1.8 1.6 1.4 1.2 1 0.8 0
2
4 6 Input power backoff, IBO (dB)
8
10
(c) Spectral containment.
Figure 4.18: A comparison of CE-OFDM and conventional OFDM. (M = 2, N = 64)
Chapter 5
Performance of CE-OFDM in Frequency-Nonselective Fading Channels In this chapter, performance analysis of the phase demodulator receiver is extended to fading channels. The lowpass equivalent representation of the received signal is r(t) = αejφ0 s(t) + n(t)
(5.1)
where s(t) is the CE-OFDM signal according to (3.11), α and φ 0 is the channel amplitude and phase, respectively, and n(t) is the complex Gaussian noise term represented in R∞ (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 jφ0 δ(τ ). In the frequency domain, the channel is H(f ) = F{h(τ )}(f ) = αe jφ0 , and is thus constant at all frequencies—that
is, the channel is frequency nonselective.
In the previous chapter only the simple case of α = 1 (i.e. no fading) was considered. In this chapter the channel amplitude is treated as a random quantity. Such a channel model, since it’s frequency nonselective, is commonly referred to as flat fading. The signal-to-noise ratio per bit for a given α is γ = α2
86
Eb , N0
(5.2)
87 and the average SNR per bit is [421, p. 817] Eb γ¯ = E{γ} = E α2 . N0
(5.3)
It is desired to calculate the bit error rate at a given γ¯ , denoted here as BER(¯ γ ). This quantity depends on the statistical distribution of γ. For channels with a line-of-sight (LOS) component, the probability density function of γ is [483, p. 102] # " s −K R KR (1 + KR )x (1 + KR )x (1 + KR )e exp − I0 2 , pγ (x) = γ¯ γ¯ γ¯
x ≥ 0,
(5.4)
where I0 (·) is the 0th-order modified Bessel function of the first kind, and KR =
ρ2 2σ02
(5.5)
is the Rice factor: ρ2 and 2σ02 represent the power of the LOS and scatter component, respectively [401, p. 40]. For channels without a line-of-sight, ρ → 0 and γ is Rayleigh
distributed [483, p. 101]:
pγ (x) =
1 x exp − , γ¯ γ¯
x ≥ 0.
(5.6)
To obtain BER(¯ γ ), the conditional BER is averaged over the distribution of γ [421, p. 817]: BER(¯ γ) =
Z
∞
BER(x)pγ (x)dx.
(5.7)
0
In Section 4.1.1 it is shown that √ BER(x) ≈ c1 Q c2 x ,
where c1 = 2(M − 1)/(M log 2 M ) and c2 = 2πh
p
system is above threshold. For the moment, assume √ BER(x) = c1 Q(c2 x),
(5.8)
6 log 2 M/(M 2 − 1), so long as the
for all x ≥ 0.
(5.9)
If this were true, the bit error rate for the Ricean channel, described by (5.4), is [483, p. 102] BERRice (¯ γ) =
c1 π
Z
π/2 0
(1 + KR ) sin2 θ × (1 + KR ) sin2 θ + c22 γ¯ /2 KR c22 γ¯ /2 dθ, exp − (1 + KR ) sin2 θ + c22 γ¯ /2
(5.10)
88 and for the Rayleigh channel, as described by (5.6), [483, p. 101] s ! c1 c22 γ¯ /2 BERRay (¯ γ) = 1− . 2 1 + c22 γ¯ /2
(5.11)
However, as discussed in Section 4.1.3, the bit error rate of CE-OFDM, as a result of the threshold effect, isn’t simply expressed by the Q-function for all values of SNR. Consequently (5.10) and (5.11) are not generally accurate. Figure 5.1(a) compares simulation results 1 to (5.10) for an M = 8, N = 64 system in the Ricean channel with KR = 10 dB. For 2πh = 0.6 the simulation result closely matches (5.10) for γ¯ > 15 dB. For lower values of γ¯ , (5.10) is overly optimistic since the system is more likely to experience channel fades which take the system below threshold—in which case the bit error rate isn’t accurately represented by the Q-function, that is, (5.9) is false. For the 2πh = 1.8 example, (5.10) is overly optimistic by at least 3 dB for all values of γ¯ . This is due to the inaccuracy of the Q-function for large modulation index cases (see Figure 4.4, for example). 100
100 Simulation Approx (5.10)
Simulation Approx (5.11) 10−1
10−2 2πh
1.8
0.6
10−3
Bit error rate
Bit error rate
10−1
10−4
10−5 0
10−2 2πh
1.2
0.4
10−3
10−4
5 10 15 20 25 30 Average signal-to-noise ratio per bit, γ ¯ (dB)
(a) M = 8, Ricean KR = 10 dB.
10−5 0
10 20 30 40 50 Average 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, Lfir , normalized cutoff frequency, and so forth—are the same as those used for the result shown in Figure 4.16 (see the footnote in on page 80).
89 Figure 5.1(b) further illustrates the inaccuracy of assuming (5.9). An M = 4, N = 64 system is simulated in the Rayleigh channel. For the low modulation index case of 2πh = 0.4, (5.11) is somewhat accurate. However, for the large modulation index case of 2πh = 1.2, (5.11) is shown to be off by 5–7 dB.
A Semi-Analytical Approach The problem with (5.10) and (5.11) is that the conditional bit error rate, BER(x), is not accurately described by the Q-function at low SNR and/or for large modulation index. For a limited range of 2πh (for example, the values shown in Figure 4.16) the following observation can be made: above a certain SNR, say x 0 , the conditional bit error rate closely matches the Q-function, that is, (5.8) holds. Therefore (5.7) can be approximated as BER(¯ γ) =
Z
x0
BER(x)pγ (x)dx +
0
≈
Z
0
x0
BER(x)pγ (x)dx +
Z
∞
BER(x)pγ (x)dx
Zx0∞
√
(5.12)
c1 Q(c2 x)pγ (x)dx.
x0
Determining x0 for a given M and 2πh, and dealing with
R x0 0
BER(x)pγ (x)dx in (5.12) are
the problems that remain to obtain an accurate approximation of BER(¯ γ ). As observed in Section 4.1.3 [see Figure 4.6(a)], at low SNR the bit error rate is roughly 1/2. Assume for the moment that BER(x) = 1/2 for x ≤ x 0 ; then Z Z ∞ √ 1 x0 c1 Q(c2 x)pγ (x)dx. pγ (x)dx + BER(¯ γ) ≈ 2 0 x0
(5.13)
This simplified model, referred to as a two-region model since the conditional BER is split into two regions, is illustrated in Figure 5.2: below x 0 the BER is 1/2, otherwise the BER is equal to the Q-function. Also shown is the observed simulation result. Notice that the two-region model doesn’t account for the transition region in which BER(x) ≈ 1/2 √ to where BER(x) ≈ c1 Q(c2 x). [For more examples of the transition region, see Figure
4.6.] Consequently, (5.13) is not generally accurate, and a more elaborate approach is required which accounts for the transition region.
90
Transition region 1
Conditional bit error rate, BER(x)
0.5
0.1
Two-region model Observed (simulation) Q-function (4.35)
0.01
x0 Signal-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: Z x0 Z γ1 BER(x)pγ (x)dx+ BER(x)pγ (x)dx = 0 γ0 Z γ2 Z γn BER(x)pγ (x)dx + . . . + BER(x)pγ (x)dx, γ1
(5.14)
γn−1
where γi > γi−1 , i = 1, 2, . . . , n, γ0 = 0 and γn = x0 . Due to the analytical difficulty of describing BER(x) over 0 ≤ x ≤ x0 , computer simulation is used. The system is
simulated at SNR values γi , i = 1, 2, . . . , n − 1, to get the result BER i , i = 1, 2, . . . , n − 1. It is assumed that BER(x) ≈ BERi for γi ≤ x ≤ γi+1 to obtain the approximation Z ∞ n−1 X Z γi+1 √ c1 Q(c2 x)pγ (x)dx. BERi pγ (x)dx + BER(¯ γ) ≈ (5.15) i=0
γi
γn
For SNR in the range 0 ≤ x ≤ γ1 the bit error rate is assumed to be BER 0 = 1/2. Figure 5.3 illustrates the n + 1 regions of (5.15). Notice that for n = 1, (5.15) is equivalent to
(5.13). In other words, (5.15), a (n+1)-region model, is a generalization of the two-region model (5.13). CE-OFDM systems are simulated in Rayleigh and Ricean (K R = 3 dB and KR = 10 dB) channels. The values of modulation index are as follows: for M = 2, 2πh ≤ 0.6;
91 1 BER0 = 1/2 BER1 BER3 Conditional bit error rate, BER(x)
BER2 BER4 .. . BERn−2
BERn−1
(n + 1)-region model Observed (simulation) Q-function (4.35)
0.01
← γ0 = −∞
γ1
... γn−2 γ2 γ3 γ4 Signal-to-noise ratio per bit, x (dB)
γn−1
γn
Figure 5.3: A (n + 1)-region model. (M = 8, N = 64, 2πh = 0.6) for M = 4, 2πh ≤ 1.2; for M = 8, 2πh ≤ 1.8; and for M = 16, 2πh ≤ 2.4. The results are shown in Figure 5.4: the circles represent Rayleigh results; the squares and
triangles represent the Ricean results for K R = 3 dB and KR = 10 dB, respectively. The solid lines are the results of the semi-analytical approach, (5.15). The transition region is sampled every 0.5 dB, that is, γ i+1 − γi = 0.5 dB, i = 1, 2, . . . , n − 1; the
starting point is γ1 = −5 dB. Therefore γi = 0.5(i − 1) − 5 dB, i = 1, 2, . . . , n. The
sampling continues until BERn < 0.01. For SNR x ≥ γn the conditional bit error rate
is approximated with the Q-function (5.8). This criteria used for γ n is based on the observation that, for the modulation index values under consideration, the Q-function is accurate for BER < 0.01. As shown in the figure, this semi-analytical approach yields curves for BER(¯ γ ) that closely match simulation. Figure 5.5 shows the improvement of (5.15) over (5.10) and (5.11).
The semi-
analytical approach closely matches the simulation results, even at low SNR, while (5.10) and (5.11) are overly optimistic by several dB. The advantage of the technique described in this section is it gives an accurate result in a small fraction of the time required for direct simulation. For example, the
92
10−3 10−4 10 20 30 40 Average SNR per bit, γ¯ (dB)
(c) M = 4, 2πh = 0.4 Bit error rate
10−2 10−3 10−4 10−5 0
100
10 20 30 40 Average SNR per bit, γ¯ (dB)
(e) M = 8, 2πh = 0.6
10−3 10−4
100
(g) M = 16, 2πh = 0.8
10−3 10−4 10 20 30 40 Average SNR per bit, γ¯ (dB)
50
50
(d) M = 4, 2πh = 1.2
10−2 10−3 10−4 10 20 30 40 Average SNR per bit, γ¯ (dB)
50
(f) M = 8, 2πh = 1.8
10−1 10−2 10−3 10−4
100
10−2
10 20 30 40 Average SNR per bit, γ¯ (dB)
10−1
10−5 0
50
10−1
10−5 0
10−4
100
10−2
10 20 30 40 Average SNR per bit, γ¯ (dB)
10−3
10−5 0
50
10−1
10−5 0
10−2
100
10−1
(b) M = 2, 2πh = 0.6
10−1
10−5 0
50
Bit error rate
Bit error rate
Bit error rate
10−2
100
Bit error rate
100
10−1
10−5 0
Bit error rate
(a) M = 2, 2πh = 0.2
Bit error rate
Bit error rate
100
10 20 30 40 Average SNR per bit, γ¯ (dB)
50
(h) M = 16, 2πh = 2.4
10−1 10−2 10−3 10−4 10−5 0
10 20 30 40 Average SNR per bit, γ¯ (dB)
50
Figure 5.4: Performance of CE-OFDM in flat fading channels. (Circle=Rayleigh; square=Rice, K = 3 dB; triangle=Rice, K = 10 dB. Solid line=Semi-analytical curve, (5.15); points=simulation. N = 64)
93 100 Rayleigh simulation Rayleigh approximation (5.11) Ricean (KR = 3 dB) simulation Ricean (KR = 3 dB) approximation (5.10) Ricean (KR = 10 dB) simulation Ricean (KR = 10 dB) approximation (5.10) Semi-analytical technique (5.15)
Bit error rate
10−1
10−2
10−3
10−4
10−5 0
10
20 30 Average signal-to-noise ratio per bit, γ¯ (dB)
40
50
Figure 5.5: Comparison of semi-analytical technique (5.15) with (5.10) and (5.11). (M = 4, N = 64, 2πh = 1.2) simulated Rayleigh result in Figure 5.5 requires about 6 hours of computer time (on a workstation with 1 gigabytes of memory and a single 3 gigahertz microprocessor). The semi-analytical result, on the other hand, requires less than 7 s (to obtain {BER i }, and perform numerical integration): a speed improvement of 4 orders of magnitude.
The disadvantage, however, is that this technique doesn’t yield a closed-form expression. As of the time of this writing, such a solution, that is general and accurate, doesn’t seem possible.
Chapter 6
Performance of CE-OFDM in Frequency-Selective Channels In this chapter the performance of CE-OFDM in frequency-selective channels is studied. The channel is time dispersive having an impulse response h(τ ) that can be non-zero over 0 ≤ τ ≤ τmax , where τmax is the channel’s maximum propagation delay. The received signal is
r(t) = =
∞
Z
h(τ )s(t − τ )dτ + n(t)
Z−∞ τmax 0
(6.1)
h(τ )s(t − τ )dτ + n(t),
where s(t) is the CE-OFDM signal according to (3.11) and n(t) is the complex Gaussian noise term represented by (4.15). The lower bound of integration in (6.1) is due to the law of causality [401, p. 245]: h(τ ) = 0 for τ < 0. The upperbound is τ max since, by definition 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, TB , designed to be much longer than τmax . A guard interval of duration Tg ≥ τmax is inserted between successive CE-OFDM blocks to avoid interblock interference. At the
receiver, r(t) is sampled at the rate f sa = 1/Tsa samp/s, the guard time samples are discarded and the block time samples are processed. Using the discrete-time model outlined in Section 2.1.2, the processed samples are rp [i] = r[i] =
NX c −1 m=0
h[m]s[i − m] + n[i], 94
i = 0, . . . , NB − 1.
(6.2)
95 Note that the discarded samples are {r[i]} −1 i=−Ng . Transmitting a cyclic prefix during the guard interval makes the linear convolution with the channel equivalent to circular
convolution. Thus rp [i] =
1 NDFT
NDFT X−1
H[k]S[k]ej2πik/NDFT ,
k=0
i = 0, . . . , NB − 1,
(6.3)
where {H[k]} is the DFT of {h[i]} and {S[k]} is the DFT of {s[i]}. The effect of the channel can be reversed with the frequency-domain equalizer: a DFT followed by a
multiplier bank, followed by an IDFT. The FDE output is sˆ[i] =
1 NDFT
NDFT X−1
Rp [k]C[k]ej2πik/NDFT ,
k=0
i = 0, . . . , NB − 1,
(6.4)
where {Rp [k]} is the DFT of the processed samples and {C[k]} are the equalizer correc-
tion terms, which are computed as [463]
1 H[k]
(6.5)
H ∗ [k] |H[k]|2 + (Eb /N0 )−1
(6.6)
C[k] = for the zero-forcing (ZF) criterion, and C[k] =
for the minimum mean-square error (MMSE) criterion. Ignoring noise (n[i] = 0), the output of the frequency-domain equalizer using (6.5) is sˆ[i] =
=
=
1 NDFT 1 NDFT 1 NDFT
= s[i],
NDFT X−1 k=0 NDFT X−1 k=0 NDFT X−1
H[k]S[k]C[k]ej2πik/NDFT
H[k]S[k]
1 j2πik/NDFT e H[k]
(6.7)
S[k]ej2πik/NDFT
k=0
i = 0, . . . , NB − 1.
Therefore, the ZF frequency-domain equalizer perfectly reverses the effect of the channel. When noise can’t be ignored, the ZF suffers from noise enhancement. For example, a fade of −30 dB results in a correction term with gain +30 dB, which corrects the channel
but amplifies the noise by a factor of 1000. The MMSE criterion (6.6) takes into account
96 the signal-to-noise ratio, making an optimum trade between channel inversion and noise enhancement. Notice that the MMSE and ZF are equivalent at high SNR: lim
Eb /N0 →∞
C[k]|MMSE =
H ∗ [k] 1 = = C[k]|ZF . 2 |H[k]| H[k]
(6.8)
The system under consideration is shown in Figure 6.1. System performance is estimated by way of computer simulation. The samples {h[i]}, {s[i]} and {n[i]} are
generated then used to calculate the received samples (6.2) which are then processed by the FDE and the demodulator. CE-OFDM Modulator
s(t)
r(t) h(τ )
r[i]
Remove CP
rp [i] FDE
CE-OFDM Demodulator
n(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 MMSE and ZF equalizers are compared over various frequency-selective channels. In Section 6.2, performance is evaluated for frequency-selective fading channels, in which case {h[i]} is
described statistically. In both sections an N = 64 CE-OFDM system is considered, with a block period of TB = 128 µs. The subcarrier spacing is 1/T B = 7812.5 Hz and the mainlobe bandwidth is W = N/TB = 500 kHz. The guard period is Tg = 10 µs, resulting in a transmission efficiency η t = 128/138 ≈ 0.93. The simulation uses an
oversampling factor J = 8; therefore the sampling rate is f sa = JN/TB = 4 Msamp/s, and the sampling period is Tsa = 1/fsa = 0.25 µs.
6.1
MMSE versus ZF Equalization
In this section, the performance of CE-OFDM using the MMSE and ZF frequencydomain equalizers is compared over six frequency-selective channels.
6.1.1
Channel Description
The channel samples {h[i]}, over the corresponding guard interval [0, 10 µs], are
shown in Table 6.1. For Channels A–C the maximum propagation delay is τ max = 0.75
97
Table 6.1: Channel samples of frequency-selective channels. i
Delay (µs) τi = iTsa
Channel A h[i]
Channel B h[i]
Channel C h[i]
Channel D h[i]
Channel E h[i]
Channel F h[i]
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.0
0.59e+j3.04 0.80e−j2.22 0 0.10e−j0.37 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –
0.93e−j1.11 0.30e−j2.90 0 0.20e+j2.97 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –
0.71e−j0.77 0.70e+j2.00 0 0.07e+j0.98 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –
0.14e+j1.99 0.47e−j1.01 0.61e+j0.26 0.42e−j0.01 0.23e+j1.09 0.10e+j1.00 0.18e+j1.82 0.13e+j2.36 0.13e−j0.60 0.12e+j1.00 0.08e−j2.30 0.09e−j1.91 0.13e+j2.99 0.04e−j1.97 0.08e+j1.05 0.08e+j1.01 0.05e+j1.42 0.06e−j0.18 0.09e+j0.56 0.05e+j0.72 0.01e+j3.13 0.05e+j1.11 0.01e+j2.42 0.02e−j1.92 0.02e−j1.20 0.03e+j2.07 0.01e+j0.17 0.01e−j0.93 0.01e+j2.93 0.02e−j2.91 0.01e−j0.76 0.01e−j1.88 0.01e−j2.96 0.01e−j0.89 0.01e−j1.54 0.01e−j3.01 – – – – –
0.56e−j0.40 0.24e+j0.98 0.51e−j0.06 0.21e−j2.12 0.24e+j1.14 0.11e+j1.64 0.25e−j1.28 0.12e−j0.93 0.27e+j1.82 0.12e+j1.49 0.15e+j0.15 0.19e+j0.23 0.05e+j2.57 0.07e−j0.17 0.04e+j3.00 0.05e−j1.20 0.09e+j0.54 0.12e−j0.10 0.03e+j0.05 0.03e+j0.96 0.02e−j0.33 0.03e−j1.53 0.01e+j0.29 0.02e+j2.58 0.01e−j1.33 0.02e−j1.96 0.02e+j2.29 0.03e+j2.86 0.01e−j0.14 0.01e−j0.36 0.02e+j1.98 0.01e−j2.38 0.01e+j0.19 0.02e+j2.18 0.01e−j2.41 0.01e−j3.11 – – – – –
0.62e+j0.67 0.47e−j0.95 0.33e+j2.58 0.22e+j0.10 0.25e−j1.92 0.16e−j0.20 0.14e−j2.30 0.21e−j1.14 0.13e+j0.34 0.16e−j2.43 0.17e+j0.36 0.08e−j0.93 0.06e−j1.08 0.05e+j0.13 0.02e+j3.11 0.07e−j2.81 0.05e−j2.87 0.04e−j1.39 0.01e−j0.89 0.02e−j2.00 0.02e+j2.22 0.03e+j0.92 0.01e−j1.56 0.02e+j0.55 0.01e+j2.83 0.02e+j0.48 0.01e+j2.68 0.01e+j2.03 0.01e−j1.76 0.01e−j2.42 0.01e+j1.11 0.01e+j0.01 0.01e+j0.40 0.01e+j1.69 0.01e−j0.49 0.01e+j2.67 – – – – –
98 µs, which results in Nc = bτmax /Tsa c + 1 = b0.75/0.25c + 1 = 4 samples [see (2.19)].
For Channels D–F, τmax = 8.75 µs, thus Nc = b8.75/0.25c + 1 = 36. The channels are normalized such that
NX c −1 i=0
|h[i]|2 = 1.
(6.9)
Channels A–C are single realizations of an approximation to the maritime channel model in [350]. Channels D–F are single realizations of a stochastic model which has an exponential delay power density 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 0 )|2 is plotted, where [422, p. 256]
0
H(f ) =
NX c −1
0
h[i]e−j2πf i ,
(6.10)
i=0
is the Fourier transform of h[i]. The x-axis is scaled as [422, p. 24] f = f 0 fsa
Hz,
(6.11)
where f 0 is the normalized frequency variable having units cycles/samp [422, p. 16]. Notice that over the signal’s mainlobe frequency range, −250 kHz ≤ f ≤ 250 kHz, the channel is frequency selective. The magnitude response fluctuates over a 8.5 dB range,
−2.5 dB ≤ |H(f 0 )|2 ≤ 6 dB. The Fourier transform (6.10) is related to the discrete Fourier transform, H[k] =
NX c −1
h[i]e−j2πik/NDFT ,
i=0
k = 0, . . . , NDFT − 1,
(6.12)
as H[k] = H(fk0 ),
k = 0, 1, . . . , NDFT − 1,
where the discrete set of frequencies {f k0 } are defined as k , k = 0, 1, . . . , NDFT NDFT 2 , 0 fk ≡ k − 1, k = NDFT + 1, . . . , NDFT − 1. NDFT 2 1
Stochastic models are discussed in the next section.
(6.13)
(6.14)
99
0.4 0.35 0.3
|h[i]|2
0.25 0.2 0.15 0.1 0.05 0 0
1
2
3 4 5 6 Propagation delay, iTsa (µs)
7
8
(a) Time domain.
Channel D response, |H(f 0 )|2 Equalizer response, ZF Equalizer response, MMSE: Eb /N0 = 0 dB 10 dB 20 dB
Magnitude response (dB)
10
5
0
−5
−200
−100
0 Frequency, f = f 0 fsa (kHz)
(b) Frequency domain.
Figure 6.2: Channel D.
100
200
9
100 Using a DFT size NDFT = JN = NB and noting (6.11), the frequency samples {H[k]} correspond to the frequencies k, TB 0 0 JN = fk = fk fsa = fk TB k − fsa , TB
k = 0, 1, . . . , NDFT 2 , k=
NDFT 2
(6.15)
+ 1, . . . , . . . , NDFT − 1.
Included in Figure 6.2(b) is the response of the MMSE and ZF equalizers. The ZF
response, (6.5), is simply the inverse of the channel. The MMSE response, (6.6), is shown for Eb /N0 = 0, 10, and 20 dB. Notice that at high SNR the MMSE approaches the ZF equalizer, which is to be expected from (6.8). For this particular channel the MMSE and ZF are shown to be equivalent for Eb /N0 ≥ 20 dB.
6.1.2
Simulation Results
The N = 64 CE-OFDM system is simulated over Channels A–F. The modulation order is M = 2, and different values of the modulation index, h, are selected. Due to the channel normalization (6.9), the simulation results are compared against the simple AWGN channel. The results are shown in Figures 6.3–6.8. For each case, |h[i]| 2 is plotted in subfigure (a); the channel and equalizer frequency-domain responses are plotted in subfigure (b); and the bit error rate performance results are shown in subfigure (c). The results for Channel A are shown in Figure 6.3. Of the six test channels, Channel A is the most mild in terms of its frequency-domain response. The magnitude response |H(f 0 )|2 spans a 3 dB region in a nearly linearly manner. The equalizers are shown to
effectively correct the channel: the BER curves in Figure 6.3(c) are nearly indistinguish-
able from the simple AWGN curves. Results are plotted for 2πh = 0.1, 0.3 and 0.6. For the 2πh = 0.6 example at the lower SNR values E b /N0 < 10 dB, the ZF result is shown to be slightly worse than the MMSE result; for higher values of SNR the performance of
the two equalizers becomes nearly identical. This is to be expected since, as illustrated in Figure 6.3(b), their frequency response become the same at high E b /N0 . Results for 2πh = 0.1, 0.2, 0.4 and 0.6 over Channel B are shown in Figure 6.4. The frequency response of this channel is more severely varying than Channel A. Over the signal bandwidth, |H(f 0 )|2 spans a 6 dB range. As with the previous example, the MMSE is shown to slightly outperform the ZF at low SNR (i.e., the 2πh = 0.6 example
for Eb /N0 < 10 dB), but the two equalizers have essentially the same performance at the
101
0.7 2 Magnitude response (dB)
0.6
0.4 0.3 0.2 0.1 0
0
−2 Channel A ZF MMSE: Eb /N0 = 0 dB 10 dB 20 dB
−4
−6 0
1
2 3 4 5 6 7 Propagation delay, iTsa (µs)
8
9
−200
−100 0 100 Frequency, f = f 0 fsa (kHz)
(b) Frequency domain.
(a) Time domain. 10−1
ZF MMSE AWGN sim AWGN approx (4.35)
10−2 Bit error rate
|h[i]|2
0.5
10−3
10−4 5
0.6
2πh
10
0.3
0.1
15 20 25 Signal-to-noise ratio per bit, Eb /N0 (dB)
(c) Performance for MMSE and ZF compared to AWGN and (4.35).
Figure 6.3: Channel A results.
30
200
102
0.9 4
0.8 Magnitude response (dB)
0.7
0.5 0.4 0.3 0.2 0.1 0
0
1
2 3 4 5 6 7 Propagation delay, iTsa (µs)
8
9
0 −2
Channel B ZF MMSE: Eb /N0 = 0 dB −4 10 dB 20 dB 30 dB −6 −200
−100 0 100 Frequency, f = f 0 fsa (kHz)
(b) Frequency domain.
(a) Time domain. 10−1
ZF MMSE AWGN sim AWGN approx (4.35)
10−2 Bit error rate
|h[i]|2
0.6
2
10−3
10−4 5
0.6
2πh
10
0.4
0.2
0.1
15 20 25 Signal-to-noise ratio per bit, Eb /N0 (dB)
(c) Performance for MMSE and ZF compared to AWGN and (4.35).
Figure 6.4: Channel B results.
30
200
103 higher SNR values. For BER ≤ 0.001 the degradation caused by the frequency selective
channel, when compared to the simple AWGN result, is slightly less than 1 dB.
Channel C has the most frequency-selective response of the three maritime channel realizations. As shown in Figure 6.5(b), the magnitude response varies over a 20 dB range. It is also shown that very high SNR is required for the MMSE response to approach the ZF response. Over the frequency range −250 kHz ≤ f 0 fsa ≤ −200 kHz, for example, the two are equivalent only for E b /N0 > 35 dB. This equivalence is also demonstrated in Figure 6.5(c): for the 2πh = 0.1 example, the ZF performance gradually approaches the MMSE performance at these high SNR values. Clearly, the large amount of frequency selectivity of this channel results in a large performance degradation when compared to the AWGN results. At the bit error rate 0.001, the degradation is 10 dB for the 2πh = 0.1 case. The improvement of the MMSE is pronounced for 2πh = 0.5. At the bit error rate 0.001, the MMSE outperforms the ZF by 7 dB, and is only 2 dB worse than the performance over the simple AWGN channel. Figures 6.6–6.8 show the results for Channels D–F. As stated earlier, the three channels are three different realizations of a stochastic model with an exponential delay power density spectrum. The degree that the each channel varies over the signal bandwidth progresses from Channel D to Channel F. Channel F, having a 50 dB attenuation at 185 kHz, is the most harsh of the test channels. The results in Figure 6.8(c) show the dramatic performance degradation as a consequence of the severe frequency selectivity. An 18 dB loss, compared to the AWGN performance, is experienced for the 2πh = 0.6, MMSE example at the bit error rate 0.001; the ZF case degrades more than 20 dB further. A 40 dB loss is suffered for the 2πh = 0.1 and 0.3 cases. These results show that frequency selective channels having deep fades in the signal bandwidth impact performance greatly.
6.1.3
Discussion and Observations
At this point, several observations can be made. First, the performance of the equalized CE-OFDM systems studied depends on the amount of frequency selectivity over the signal bandwidth. For channels with a relatively mild frequency response— Channels A, B and D, for example—the performance degradation is minor. The noise enhancement that results from equalizing channels with severe frequency responses—
104
0.6
Magnitude response (dB)
0.5
0.3 0.2
10
0
−10
0.1
−20 0
1
2 3 4 5 6 7 Propagation delay, iTsa (µs)
8
9
−200
−100 0 100 Frequency, f = f 0 fsa (kHz)
(b) Frequency domain.
(a) Time domain. 10−1
ZF MMSE AWGN sim AWGN approx (4.35)
10−2 Bit error rate
|h[i]|2
0.4
0
Channel C ZF MMSE: Eb /N0 = 0 dB 10 dB 20 dB 30 dB 35 dB
20
10−3
10−4 5
0.5
2πh
10
0.1
15 20 25 30 Signal-to-noise ratio per bit, Eb /N0 (dB)
35
(c) Performance for MMSE and ZF compared to AWGN and (4.35).
Figure 6.5: Channel C results.
40
200
105
6
0.35
4 Magnitude response (dB)
0.4
0.3
0.2 0.15 0.1
−2 −4 −6
0.05 0
Channel D ZF MMSE: Eb /N0 = 0 dB 10 dB 20 dB
0
0
1
2 3 4 5 6 7 Propagation delay, iTsa (µs)
8
9
−8
−200
−100 0 100 Frequency, f = f 0 fsa (kHz)
200
(b) Frequency domain.
(a) Time domain. 10−1
ZF MMSE AWGN sim AWGN approx (4.35)
10−2 Bit error rate
|h[i]|2
0.25
2
10−3
10−4 5
0.6
2πh
10
0.2
15 20 25 Signal-to-noise ratio per bit, Eb /N0 (dB)
0.1
30
(c) Performance for MMSE and ZF compared to AWGN and (4.35).
Figure 6.6: Channel D results.
35
106
10
0.35
Channel E ZF dB dB dB dB
Magnitude response (dB)
0.3
0.2 0.15 0.1
5
0
−5
0.05 −10 0
0
1
2 3 4 5 6 7 Propagation delay, iTsa (µs)
(a) Time domain.
8
9
−200
−100 0 100 Frequency, f = f 0 fsa (kHz)
(b) Frequency domain.
10−1 ZF MMSE AWGN sim AWGN approx (4.35)
10−2 Bit error rate
|h[i]|2
0.25
MMSE: Eb /N0 = 0 10 20 30
10−3
10−4 5
0.6
2πh
10
0.1
15 20 25 30 Signal-to-noise ratio per bit, Eb /N0 (dB)
35
(c) Performance for MMSE and ZF compared to AWGN and (4.35).
Figure 6.7: Channel E results.
40
200
107
0.4 40 Magnitude response (dB)
0.35 0.3
0.2 0.15
20
0
−20
0.1 0.05 0
−40 0
1
2 3 4 5 6 7 Propagation delay, iTsa (µs)
8
−200
9
−100 0 100 Frequency, f = f 0 fsa (kHz)
(b) Frequency domain.
(a) Time domain. 10−1
ZF MMSE AWGN sim AWGN approx (4.35)
10−2 Bit error rate
|h[i]|2
0.25
Channel F ZF MMSE: Eb /N0 = 0 dB 10 dB 20 dB 30 dB
10−3
0.6 0.6
10−4 5
10
0.3
15
20
0.1
0.3
0.1
2πh
25 30 35 40 45 50 55 Signal-to-noise ratio per bit, Eb /N0 (dB)
60
65
(c) Performance for MMSE and ZF compared to AWGN and (4.35).
Figure 6.8: Channel F results.
70
200
108 Channels C, E and F—degrades performance dramatically. Second, the complexity of the frequency-domain equalizers is determined by the DFT size, not by the number of non-zero channel terms h[i]. This is in contrast to conventional time-domain equalizers which have a complexity that depends on the number of paths in the multipath channel. Last, the MMSE equalizer is more complicated than the ZF equalizer since the SNR per bit, Eb /N0 , must be estimated at the receiver. The results of this study show that
this added complexity doesn’t always translate into improved performance. That is, the ZF performance is the same as the MMSE performance for many cases—the 2πh ≤ 0.4
cases in Channel B, for example. In other cases, the MMSE performs much better, and
thus estimating Eb /N0 pays substantial dividends—the 2πh = 0.5 case for Channel C illustrates this point.
As demonstrated in the following section, the MMSE equalizer offers significant improvement over the ZF equalizer when averaging performance over many channel realizations of a stochastic channel model.
6.2
Performance Over Frequency-Selective Fading Channels
In contrast to the test channels used in the previous section, which were deterministic as defined in Table 6.1, the channels used in this section are described statistically. The mathematical foundation for stochastic time-variant linear channels was pioneered by Bello [50]; more recently P¨atzold’s text, Mobile Fading Channels [401], provides a excellent treatment of the topic, with a focus on the various aspects of simulation. In the study here, the widely used assumption of WSSUS (wide-sense stationary uncorrelated scattering) is applied. Also, it is assumed that the channel is composed of discrete paths, each having an associated gain and discrete propagation delay. This assumption is based on the Parsons and Bajwa ellipse model for describing multipath channel geometry [401, p. 244]. The channel’s impulse response is h(τ ) =
L−1 X l=0
al δ(τ − τl ),
(6.16)
109 where al is the complex channel gain and τl is discrete propagation delay of the lth path; the total number of paths is represented by L. The propagation delay differences are ∆τl = τl − τl−1 ≡ Tsa ,
l = 1, 2, . . . , L − 1.
(6.17)
That is, they are set equal to the sampling period of the simulation [401, p. 269]. The delay of the 0th path is defined as τ0 ≡ 0, thus τl = lTsa ,
l = 0, 1, . . . , L − 1.
(6.18)
For each simulation trial, the set of path gains {a l }L−1 l=0 are generated randomly. Each
gain is complex valued, has a zero mean and a variance σa2l = E |al |2 ,
l = 0, 1, . . . , L − 1.
(6.19)
Both the real and imaginary parts of the path gains are Gaussian distributed [401, p. 267]; thus the envelope |al |2 is Rayleigh distributed. Also, the channels are normalized
such that
L−1 X
σa2l = 1.
(6.20)
l=0
As outlined in P¨atzold’s text (pp. 276–279) the parameters σ a2l , τl and L determine
the fundamental characteristic functions and quantities of the channel models, such as the
delay power spectral density and the delay spread 2 . The relevant formulas are expressed below. • Delay power spectral density: Sτ τ (τ ) =
L−1 X l=0
• Average delay: Bτ(1) τ =
σa2l δ(τ − τl ).
L−1 X
σa2l τl .
(6.21)
(6.22)
l=0
2
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 power spectral density” is used here. In P¨ atzold’s text, a clear distinction is made between stochastic channel models, which provide the theoretical and mathematical foundations, and “deterministic” channel models which are generated in software or hardware for simulation purposes. For the sake of simplicity, this distinction isn’t stressed here (which results in a slightly different notation for the expressed formulas in his text). Also, since only time-invariant channels are considered in this thesis, the Doppler power spectral density, time correlation function and coherence time (see [401, pp. 277–279]) are not discussed.
110 • Delay spread: Bτ(2) τ
v uL−1 uX (1) 2 (σal τl )2 − Bτ τ =t
(6.23)
l=0
• Frequency correlation function: 0
rτ τ (v ) =
L−1 X
0
σa2l e−j2πv τl
(6.24)
l=0
The variable v 0 is referred to as the frequency separation variable [401, p. 278]. • Coherence bandwidth: The coherence bandwidth is the smallest positive value
BC which fulfils |rτ τ (BC )| = 0.5|rτ τ (0)|; which, due to (6.20) and (6.24), is equiv-
alent to
L−1 1 X 2 −j2πBC τl σ e − = 0. al 2
(6.25)
l=0
Notice that BC is the 3 dB bandwidth of rτ τ (v 0 ).
6.2.1
Channel Models
CE-OFDM is simulated over four frequency-selective fading channel models. Table 6.2 defines the parameters {σa2l } and {τl }. Channel Af and Bf are similar to the maritime channel models in [350]3 . Both have a secondary path with a 5 µs propagation delay.
Channel Af has a weak secondary path (one-tenth, i.e., −10 dB, the power of the primary
path); Channel Bf has a stronger secondary path (one-half, i.e., −3 dB, the power of the
primary path).
Channel Cf has an exponential delay power spectral density: CC e−τl /2µs , 0 ≤ τl ≤ 8.75 µs, f 2 σal ,C = 0, otherwise,
(6.26)
exp(−τl /2e-6) = 0.1188 . . .
(6.27)
where
C Cf = 1
X 35 l=0
is the normalizing constant used to guarantee (6.20). Note that the maximum propagation delay is 8.75 µs. 3
To avoid notational ambiguities, the channel model labels in this section have the subscript “f” (“fading”).
111
Table 6.2: Channel model parameters. Path no. l
Delay (µs) τl = lTsa
Channel Af σa2l ,A
Channel Bf σa2l ,B
Channel Cf σa2l ,C
Channel Df σa2l ,D
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.0
10/11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1/11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2/3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1/3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1.18e-1 1.04e-1 9.25e-2 8.16e-2 7.20e-2 6.36e-2 5.61e-2 4.95e-2 4.37e-2 3.85e-2 3.40e-2 3.00e-2 2.65e-2 2.33e-2 2.06e-2 1.82e-2 1.60e-2 1.41e-2 1.25e-2 1.10e-2 9.75e-3 8.60e-3 7.59e-3 6.70e-3 5.91e-3 5.22e-3 4.60e-3 4.06e-3 3.58e-3 3.16e-3 2.79e-3 2.46e-3 2.17e-3 1.92e-3 1.69e-3 1.49e-3 0 0 0 0 0
1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 0 0 0 0 0
112 The last model, Channel Df , has a uniform delay power density spectrum: CD , 0 ≤ τl ≤ 8.75 µs, f 2 σal ,D = 0, otherwise,
(6.28)
where the normalizing constant is
CDf = 1/36.
(6.29)
In Figure 6.9 the delay power density spectrum (6.21) and the frequency correlation function (6.24) are plotted for each of the four models. The corresponding average delay (6.22), delay spread (6.23) and coherence bandwidth (6.25) for each model is labeled. Notice that Channel Df has the smallest coherence bandwidth, B C = 67 kHz. For Channel Af the coherence bandwidth isn’t finite since, as shown in subfigure (b), |rτ τ (v 0 )| > −3 dB for all frequency separation values 4 .
6.2.2
Simulation Procedure and Preliminary Discussion
The average performance of various CE-OFDM systems is evaluated over the four stochastic channel models. This is done by randomly generating {a l }—which, as stated
above, are complex-valued quantities, drawn from the Gaussian distribution, with zero
mean and variance {σa2l }—computing the received samples (6.2), then processing the samples with the frequency-domain equalizer and the CE-OFDM demodulator. At each
average Eb /N0 considered, the simulation runs for at least 20,000 bit errors, or until 100,000,000 bits are transmitted, whichever happens first. This corresponds to many
thousands of channel realizations5 . Some channel realizations result in very poor performance (for example, see Figure 6.8), while others result in a bit error rates not much worse than that of the simple AWGN channel. This performance difference is attributed to the severity of the channel’s frequency response, as observed with the several examples in Section 6.1. The performance also depends on the gain of the channel realization. Due to (6.20) the channel gain, on average, is normalized to unity; however, for a given trial, the channel may be fading such that the gain is less than unity, resulting in degraded performance. The likelihood of a deep channel fade depends on the number of independent 4
˛ ˛ 10 1 + 11 exp(−j2πv 0 5 µs)˛ = For Channel Af , min |rτ τ (v 0 )| = min ˛ 11 5 Example simulation code can be found in Appendix C.
9 11
>
1 2
≈ −3 dB.
113
(a) Delay power spectral density, Channel Af
(b) Frequency correlation function, Channel Af 0
(1)
Bτ τ = 0.45 µs
−5
10 log 10 [rτ τ (v 0 )]
10 log10 [Sτ τ (τ )]
0 (2)
Bτ τ = 1.44 µs
−10 −15 −20
0
1
2 3 4 5 6 7 8 Propagation delay, τ (µs)
9
10
−9
−15 −450
(c) Delay power spectral density, Channel Bf
10 log 10 [rτ τ (v 0 )]
10 log10 [Sτ τ (τ )]
(1)
(2) Bτ τ
−10
= 2.36 µs
−15 −20
−3 −6 −9
0
1
2 3 4 5 6 7 8 Propagation delay, τ (µs)
9
10
−15 −450
(e) Delay power spectral density, Channel Cf
450
(f) Frequency correlation function, Channel Cf
(1)
Bτ τ = 1.78 µs
−5
(2) Bτ τ
−10
10 log 10 [rτ τ (v 0 )]
10 log10 [Sτ τ (τ )]
−300 −150 0 74 150 300 Frequency separation, v 0 (kHz)
0
0 = 1.75 µs
−15 −20
−3
BC
−6 −9
−12
−25 0
1
2 3 4 5 6 7 8 Propagation delay, τ (µs)
9
10
−15 −450
(g) Delay power spectral density, Channel Df
−300 −150 0 140 300 Frequency separation, v 0 (kHz)
450
(h) Frequency correlation function, Channel Df 0
0 (1) Bτ τ (2) Bτ τ
−5
−10
= 4.38 µs
10 log 10 [rτ τ (v 0 )]
10 log10 [Sτ τ (τ )]
BC
−12
−25
= 2.60 µs
−15 −20
−3
BC
−6 −9
−12
−25 −30
450
(d) Frequency correlation function, Channel Bf
Bτ τ = 1.67 µs
−5
−30
−300 −150 0 150 300 Frequency separation, v 0 (kHz)
0
0
−30
BC → ∞
−6
−12
−25 −30
−3
0
1
2 3 4 5 6 7 8 Propagation delay, τ (µs)
9
10
−15 −450
−300 −150 0 67 150 300 Frequency separation, v 0 (kHz)
450
Figure 6.9: Fundamental characteristic functions and quantities [(6.21)–(6.25)] of the four channel models considered.
114 propagation paths [the WSSUS assumption makes each path in (6.16) independent]. It is unlikely that multiple paths fade simultaneously. For this reason, channels characterized by multiple propagation paths possess a type of diversity known at multipath diversity— which can be exploited by the receiver. Of the four models considered in this study, Channel Df can be said to have the most multipath diversity: the gain of a given realization depends on 36 independent paths, each having, on average, an equal contribution. Channel Af can be said to have the least amount of multipath diversity: over 90% of the channel gain depends on a single path. Channel B f has more multipath diversity than Channel Af since the gain is distributed more equally between the two paths. That is, the multipath diversity depends not only on the number of independent paths but also on the way in which the power is distributed over the paths, as determined by {σ a2l }. [It is worth noting that the frequency-nonselective channel models considered in Chapter 5
have L = 1 path of which 100% of the channel gain depends (σ a21 = 1), and thus these
channels have no multipath diversity.] In the results that follow, the impact of multipath
diversity—and its frequency-domain dual frequency diversity—on CE-OFDM systems is studied.
6.2.3
Simulation Results
The simulation results of this study are presented over three figures: Figure 6.10 compares the performance of a CE-OFDM system, with fixed modulation order M and modulation index h, over the four channel models; Figure 6.11 compares the performance of a CE-OFDM system with fixed M but varying h over Channel C f ; and Figure 6.12 compares the performance of constant envelope and conventional OFDM systems, in the presence of power amplifier nonlinearities, over Channel C f . For each case, the number of subcarriers is N = 64. In Figure 6.10, performance results of an M = 4, N = 64, 2πh = 1.0 CE-OFDM system are plotted. The simulation results over the multipath channel models A f –Df are labeled with circles and triangles; the MMSE equalized results have solid lines connecting the points, while the ZF equalized results use dashed lines. For reference, the performance of the system over the simple AWGN channel is plotted (with dash-dot lines) along with the performance over the Rayleigh frequency-nonselective fading channel (represented by the thick solid line). These results show the significant performance improvement
115
MMSE: Channel Af Bf Cf Df ZF: Channel Af Bf Cf Df Rayleigh, L = 1 AWGN AWGN approx (4.35)
10−1
Bit error rate
10−2
10−3
10−4 5
10
15 20 25 30 35 Average signal-to-noise ratio per bit, Eb /N0 (dB)
40
Figure 6.10: Performance results. (Multipath results are labeled with circle and triangle points; the Rayleigh, L = 1 result is that of the frequency-nonselective channel model. M = 4, N = 64, 2πh = 1.0) that is to be had by using the MMSE equalizer. At the bit error rate 0.001, for example, MMSE outperforms ZF by 10 dB for Channel D f . These results also show the impact of multipath diversity. Consider the MMSE results. For E b /N0 > 15 dB, the performance
over Channels Af –Df is better than the performance over the frequency-nonselective
Rayleigh (L = 1 path) channel. For BER ≤ 0.001, the performance over the multipath channels is at least 5 dB better than the performance over the single path channel.
Notice that Channel Df , which has the most multipath diversity, results in a better performance that all the other channels. The performance over Channel B f , which has more multipath diversity than Channel A f , is in fact better than the performance over Channel Af . These results indicate that the CE-OFDM receiver exploits the multipath diversity of the channel. The fact that constant envelope OFDM exploits multipath diversity is an interesting result since conventional OFDM doesn’t. This was shown in Section 2.1.1; specifically,
116 by (2.9). So long as the duration of the guard interval is greater than or equal to the channel’s maximum propagation delay, that is, T g ≥ τmax , and a cyclic prefix is transmitted during the guard interval, the performance of OFDM in a time-dispersive
channel is equivalent to flat fading performance. In other words, the multipath fading performance is the same as single path fading performance. In the context of Section 2.1.1, this property was considered beneficial since ISI is avoided. In the context here, however, this property is considered a weakness since the multipath diversity of the channel isn’t leveraged6 . To understand why CE-OFDM has improved performance over multipath fading channels (compared to single path fading channels) while OFDM doesn’t, it is best to view the problem in the frequency domain. The frequency domain dual to multipath diversity is frequency diversity. It can be said that OFDM lacks frequency diversity as well. As identified in Section 1.1.2, the wideband frequency-selective fading channel is converted into N contiguous frequency-nonselective fading channels. Therefore any frequency diversity inherent to the channel—that is, over the signal bandwidth the frequency response of the channel varies, which can be taken advantage of by the receiver to obtain performance better than flat fading—is not exploited by the OFDM receiver. CE-OFDM, in contrast, has the ability to exploit the frequency diversity of the channel since the phase modulator, in effect, spreads the data symbol energy in the frequency domain. This can be seen by viewing the CE-OFDM waveform by the Taylor series expansion [see Section 3.2, (3.24)]: "
s(t) = A 1 + jσφ m(t) −
σφ2 2
2
m (t) − j
σφ3 6
3
#
m (t) + . . . ,
(6.30)
0 ≤ t < TB , where A is the signal amplitude, σφ2 = (2πh)2 is the phase signal variance, p P and m(t) = CN N 6/N (M 2 − 1), is the normalized k=1 Ik qk (t), 0 ≤ t < TB , CN =
OFDM message signal. The higher-order terms m n (t), n ≥ 2, results in a frequency spreading of the data symbols. This property is best demonstrated by way of a simple example. Example 6.2.1 Consider a CE-OFDM waveform with an OFDM message signal composed of N = 2 orthogonal 6 Note that OFDM systems typically employ channel coding and frequency-domain interleaving, which offers diversity. However, since this thesis only deals with uncoded systems, these topics are beyond its scope—and are topics for further research.
117 cosine subcarriers modulated with binary data symbols (M = 2): m(t) =
2 X
Ik cos 2πkt/TB ,
k=1
0 ≤ t < TB ,
(6.31)
where Ik ∈ {±1}, k = 1, 2. Assume that the modulation index, h, is such that the higher-order terms m2 (t) and m3 (t) contribute to the make up of s(t) according to (6.30). It is desired to
write m2 (t) and m3 (t) in terms of I1 , I2 and {cos 2πkt/TB }. This task requires some algebra,
but is simply done. For notational simplicity, let’s define
ck ≡ cos 2πkt/TB .
(6.32)
m(t) = I1 c1 + I2 c2 .
(6.33)
Thus, (6.31) is written as
The second-order term is calculated as m2 (t) = (I1 c1 + I2 c2 )(I1 c1 + I2 c2 ) = 0.5I12 + 0.5I22 c0 + (I1 I2 ) c1 + 0.5I12 c2 + (I1 I2 ) c3 + 0.5I22 c4 ,
and the third-order term as m3 (t) = 0.5I12 + 0.5I22 c0 + (I1 I2 ) c1 + 0.5I12 c2 + (I1 I2 ) c3 + 0.5I22 c4 (I1 c1 + I2 c2 ) = 0.75I12 I2 c0 + 0.75I13 + 1.5I1 I22 c1 + 1.25I12I2 + 0.5I23 c2 + 0.25I13 + 0.75I1 I22 c3 + 0.75I12 I2 c4 + 0.75I1 I22 c5 + 0.25I23 c6 .
(6.34)
(6.35)
The expansions above are represented in Table 6.3. The data symbol contribution at each
tone cos 2πkt/TB , k = 0, 1, . . . , 6, for m(t), m2 (t) and m3 (t) is shown. Referring to the tones as frequency bins, it can be said that for m(t) the two data symbols are simply contained in the k = 1 and k = 2 frequency bins. For the second-order term, m2 (t), the data symbols mix across the k = 0, 1, 2, 3, and 4 frequency bins. For m3 (t), the data symbols mix across the k = 0, 1, . . . , 6 frequency bins.
The simple example above shows how the data symbols spread across multiple frequency bins. In general, it can be said that the N data symbols that constitute the constant envelope OFDM signal are not simply confined to N frequency bins—as is the case with conventional OFDM. The phase modulator mixes and spreads—albeit in a nonlinear and exceedingly complicated manner—the data symbols in frequency, which gives the CE-OFDM system the potential to exploit the frequency diversity in the channel. This isn’t necessarily the case, however. For small values of modulation index,
118 Table 6.3: Data symbol contribution per tone for m n (t), n =1, 2, and 3.
m(t) m2 (t) m3 (t)
0 – 0.5I12 , 0.5I22 0.75I12 I2
1 I1
kth tone, cos 2πkt/TB 2 3 4 I2 – –
5 –
6 –
I1 I2
0.5I12
I1 I2
0.5I22
–
–
0.75I13 , 1.5I1 I22
1.25I12 I2 , 0.5I23
0.25I13 , 0.75I1 I22
0.75I12 I2
0.75I1 I22
0.25I23
where only the first two terms in (6.30) contribute, that is, s(t) ≈ A [1 + jσφ m(t)] ,
(6.36)
the CE-OFDM signal doesn’t have the frequency spreading given by the higher-order terms. In this case, the CE-OFDM signal is essentially equivalent to a conventional OFDM signal, jσφ m(t), (plus a relatively large DC term, A) and therefore doesn’t have the ability to exploit the frequency diversity of the channel. Simply put, CE-OFDM has frequency diversity when the modulation index is large and doesn’t have frequency diversity when the modulation index is small. This property is demonstrated in Figure 6.11. Simulation results of an M = 4, N = 64 CE-OFDM system are shown. The system is simulated over the single path Rayleigh flat fading channel and over the multipath fading model Channel C f . To demonstrate that CE-OFDM with a small modulation index lacks frequency diversity, results for 2πh = 0.1 are shown. Notice that the single path and multipath performance is essentially the same. By contrast, for the large modulation index example 2πh = 1.1, the multipath performance is significantly better than the single path performance. For example, at the bit error rate 0.001 the multipath performance is over 10 dB better than the single path performance. In the final figure, Figure 6.12, the performance of constant envelope OFDM is compared to conventional OFDM in the presence of power amplifier nonlinearities. The SSPA model (see Section 2.3) is used at various input backoff levels. The x-axis is adjusted to account for the negative impact of input power backoff. The systems are simulated over Channel Cf . For the OFDM system, QPSK data symbols are used. Three different CE-OFDM systems are tested: M = 4, 2πh = 0.9; M = 8, 2πh = 2.0; and M = 16, 2πh = 3.0. The advantage of the CE-OFDM systems is twofold. First, the CE-OFDM systems operate with IBO = 0 dB. Second, the CE-OFDM systems exploit
119 100 Multipath Single path
Bit error rate
10−1
10−2
10−3
10−4 5
2πh
10
1.1
15 20 25 30 35 40 Average signal-to-noise ratio per bit, Eb /N0 (dB)
0.1
45
50
Figure 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 system by at least 10 dB. At this bit error rate, the OFDM system has essentially the same performance with backoff levels of 6 and 10 dB; therefore, IBO = 6 dB is preferred since the performance is the same but the power efficiency is higher (see Figure 2.14). Even so, the 6 dB backoff required by the OFDM system is still far less desirable as the 0 dB backoff used by the CE-OFDM system. Notice that the OFDM system with IBO = 0 dB results in an irreducible error floor just below the bit error rate 0.1. The results in Figure 6.12 also highlight the poor performance of CE-OFDM at low SNR due to the threshold effect (as studied in Section 4.1.3). Over the region 0 dB ≤ Eb /N0 ≤ 10 dB, the OFDM system performs better than the CE-OFDM system. Also,
it should be noted that the M = 8 and M = 16 CE-OFDM systems shown have large modulation index values (2πh = 2.0 and 2πh = 3.0 respectively) which results in spectral
broadening. Roughly speaking, the spectral efficiency of the QPSK/OFDM system is 2 b/s/Hz, which, according to (4.70), is about the same as the M = 4, 2πh = 0.9 CEOFDM system. The M = 8 and M = 16 systems have spectral efficiencies of 1.5 and 1.3 b/s/Hz, respectively. Making a direct comparison between CE-OFDM and conventional OFDM is difficult
120
Bit error rate
10−1
10−2
10−3
10−4 0
OFDM: IBO = 0 dB 3 dB 6 dB 10 dB CE-OFDM: M = 4, 2πh = 0.9 M = 8, 2πh = 2.0 M = 16, 2πh = 3.0 5
10
15
20 25 Eb /N0 + IBO (dB)
30
35
40
Figure 6.12: CE-OFDM versus QPSK/OFDM. (SSPA model, Channel C f , N = 64, MMSE) due to the various parameters involved (M , 2πh, IBO, etc.), and due to the fact that system requirements vary from system to system. For example, if power amplifier efficiency is the most important requirement, then the input power backoff of 0 dB should be chosen. At this backoff level, the OFDM system has a very high irreducible error floor due to the power amplifier distortion, while the CE-OFDM system is relatively unaffected. Alternatively, if operation at low SNR is important, then CE-OFDM may not be well suited due to the threshold effect. The results in this chapter show that CE-OFDM can perform quite well in multipath fading channels—so long as the channel information (i.e., {H[k]}) is known at the receiver
and so long as the added complexity of the frequency-domain equalizer (i.e., two extra FFTs) is acceptable. Further work is needed to study the effects of channel coding, time-varying channels, phase noise, and so forth. Also, a thorough study comparing CEOFDM, OFDM and single carrier frequency-domain equalizer (SC-FDE) systems could provide for interesting results.
Chapter 7
Conclusions In this thesis the peak-to-average power ratio problem associated with orthogonal frequency division multiplexing is evaluated. The PAPR statistics are studied and the effect of power amplifier nonlinearities as a function of power backoff is evaluated by computer simulation. It is shown that the amount of backoff required to reduce spectral growth and performance degradation is significant: 6–10 dB depending on the subcarrier modulation used. Large backoff is an unsatisfactory solution for battery-powered systems since PA efficiency is low. A signal transformation method for solving the PAPR problem is presented and analyzed. The high PAPR OFDM signal is transformed to a 0 dB PAPR constant envelope waveform. At the receiver, the inverse transform is performed prior to the OFDM demodulator. For the CE-OFDM technique described, phase modulation is used. The effect of the phase modulator on the transmitted signal’s spectrum is studied. It is shown that the modulation index controls the spectral containment. The modulation index also controls the system performance. The optimum receiver is analyzed and a performance bound and approximation is derived. For a large modulation index, the CE-OFDM signals become less correlated which improves detection performance. The approximation of the optimum receiver closely matches simulation results. It also closely matches a derived bit error rate approximation for a practical phase demodulator receiver. For a small modulation index and high signal-to-noise ratio, the phase demodulator receiver is nearly optimum. For a larger modulation index the phase demodulator receiver becomes sub-optimum due to the limitations of the phase demodulator and phase unwrapper.
121
122 This problem can be suppressed with the use of a properly designed finite impulse response lowpass filter which precedes the phase demodulator. The simulation results of the CE-OFDM performance curves use an oversampling factor of J = 8. Future work includes experimenting with lower sampling rates for reduced receiver complexity. The performance of the phase demodulator is a crucial element to the overall CE-OFDM performance. Therefore, further research is needed to evaluate more advanced phase demodulation techniques such as digital phase-locked loops. Phase modulation is used exclusively in this work. It would be interesting to evaluate CE-OFDM frequency modulation systems and compare them to the results in this thesis. In terms of performance over frequency-selective fading channels, the frequencydomain equalizer 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 complexity singular value decomposition (SVD) approaches, to CE-OFDM is a subject for future investigation. The impact of imperfect channel state information on the performance of the frequency-domain equalizer is of interest. CE-OFDM might be used as a stand-alone modulation technique or as a supplement to an existing OFDM system. For example, a conventional OFDM system is designed for severe multipath channels. However, at times the channel might be relatively benign so the OFDM systems is an overkill and, due to power backoff, inefficient. An adaptive radio might sense times where power efficient CE-OFDM, which requires minimal backoff, is more applicable. Such a system can adaptively switch between conventional and constant envelope modes. For systems, such as power-limited satellite communications, where a constant envelope is very desirable, if not required, CE-OFDM might be a viable alternative to convention continuous phase modulation systems which are complex due to phase trellis decoding and sensitive to multipath. CE-OFDM is relatively robust in multipath fading channels with the use of the frequency-domain equalizer. Depending on the channel condition, equalization might not be required, therefore reducing receiver complexity.
123 For example, a channel characterized by a two-path model with a weak secondary path, CE-OFDM might provide acceptable performance without equalization. CPM systems in the other hand require high quality coherent channels. In the near term a CE-OFDM prototype is being developed by Nova Engineering (Cincinnati, OH). This work is being funded by the United States Office of Naval Research under an STTR (small business technology transfer) initiative with UCSD being the university partner. The goal of the prototype is to offer a second low-power mode for the existing JTRS (Joint Tactical Radio System) wideband component which uses OFDM. Research challenges that remain include evaluating CE-OFDM with many subcarriers (in this thesis, only 64 subcarriers are used), considering different equalization techniques, developing synchronization schemes and studying the impact of channel coding and the effects of time-varying channels. Additional future work includes comparing CE-OFDM with other block modulation technique in terms of PAPR, spectral efficiency, power amplifier efficiency, performance and complexity. There has been an increasing amount of attention given to conventional single carrier modulation with the addition of a cyclic prefix which allows for frequencydomain equalization [107, 154, 460, 463, 574]. However, most single carrier modulations have a non-constant envelope due to pulse shaping and multilevel QAM symbol constellations. A study is needed to compare these modulation techniques to CE-OFDM taking into account the effects of the PA at various backoff levels. Also, using CPM with a cyclic prefix is an interesting idea. Comparing the complexity and spectral efficiency of such a technique with CE-OFDM would be interesting. Such research will help provide insight into good designs for future wireless digital communication systems that require power efficiency and high data rates.
Appendix A
Generating Real-Valued OFDM Signals with the Discrete Fourier Transform For some applications, a real-valued OFDM signal is required. This can be done by taking a DFT of a conjugate symmetric vector. The spectral efficiency of the real-valued OFDM signal is the same as the spectral efficiency of the complex-valued OFDM signal.
A.1
Signal Description
The baseband OFDM signal is typically written as x(t) =
N −1 X
Xk ej2πkt/TB ,
k=0
0 ≤ t < TB ,
(A.1)
−1 where N is the number of subcarriers, {X k }N k=0 are the data symbols and TB is the
block period. Sampling x(t) at N equally spaced intervals over 0 ≤ t < T B yields the
sequence,
x[i] = x(t)|t=iTB /N =
N −1 X
Xk ej2πki/N ,
k=0
i = 0, 1, . . . , N − 1,
(A.2)
which is the inverse discrete Fourier transform (IDFT) of the vector X = [X0 , X1 , . . . , XN −1 ]. 124
(A.3)
125 The sequence is complex-valued in general. However it can be made real-valued by making X conjugate symmetric: ∗ XN/2+k = XN/2−k
(A.4)
X0 = XN/2 = 0.
(A.5)
and
The IDFT is then x[i] =
N −1 X
Xk ej2πki/N
k=1
N/2−1
=
X
XN/2−k ej2π(N/2−k)i/N + XN/2+k ej2π(N/2+k)i/N
X
∗ ej2π(N/2+k)i/N , XN/2−k ej2π(N/2−k)i/N + XN/2−k
(A.6)
k=1
N/2−1
=
k=1
i = 0, 1, . . . , N − 1. But since ej2π(N/2+k)i/N = ej2π(N/2+k)i/N e−j2πN i/N = ej2π(−N/2+k)i/N
(A.7)
= e−j2π(N/2−k)i/N , (A.6) can be written as N/2−1
x[i] =
X
∗ XN/2−k ej2π(N/2−k)i/N + XN/2−k e−j2π(N/2−k)i/N ,
(A.8)
k=1
i = 0, 1, . . . , N − 1. Using the identity A + A ∗ = 2
