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Single Carrier Orthogonal Multiple Access Technique
for Broadband Wireless Communications
DISSERTATION
Submitted in Partial Fulfillment
Of the Requirements for the
Degree of
DOCTOR OF PHILOSOPHY (Electrical Engineering)
at the
POLYTECHNIC UNIVERSITY
by
Hyung G. Myung
January 2007
Approved:
___________________
Department Head
Copy No. _____________ ________________________
Date
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Copyright by
Hyung G. Myung
2007
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Approved by the Guidance Committee:
Major: Electrical Engineering
______________________________
David J. Goodman, Ph.D
Professor of
Electrical and Computer Engineering
______________________________
Peter Voltz, Ph.D
Associate Professor of
Electrical and Computer Engineering
______________________________
Elza Erkip, Ph.D
Associate Professor of
Electrical and Computer Engineering
______________________________
Donald Grieco
Senior Manager of
InterDigital Communications Corporation
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Microfilm or other copies of this dissertation are obtainable from:
UMI Dissertation Publishing
Bell & Howell Information and Learning
300 North Zeeb Road
P.O. Box 1346
Ann Arbor, Michigan 481061346
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Curriculum Vitae
Hyung G. Myung received the B.S. and M.S. degrees in electronics engineering from Seoul
National University, South Korea in 1994 and in 1996, respectively, and the M.S. degree in
applied mathematics from Santa Clara University, California in 2002. He received his Ph.D.
degree from the Electrical and Computer Engineering Department of Polytechnic University,
Brooklyn, NY in January of 2007. From 1996 to 1999, he served in the Republic of Korea Air
Force as a lieutenant officer, and from 1997 to 1999, he was with Department of Electronics
Engineering at Republic of Korea Air Force Academy as an academic instructor. From 2001 to
2003, he was with ArrayComm, San Jose, CA as a software engineer. During the summer of
2005, he was an assistant research staff at Communication & Networking Lab of Samsung
Advanced Institute of Technology. Also from February to August of 2006, he was an intern at
Air Interface Group of InterDigital Communications Corporation, Melville, NY. Since January
of 2007, he is with Qualcomm/Flarion Technologies, Bedminster, NJ as a senior engineer. His
research interests include DSP for communications and wireless communications.
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To
Christ my savior,
Hyun Joo, and Ho
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Acknowledgements
During the past three years working towards my PhD degree, I was very fortunate enough to
come across many great individuals and I am very grateful for it. I would like to give the utmost
gratitude to my thesis advisor, professor David J. Goodman. Not only was he generous enough
to guide my thesis research during his busy schedules, but he was also my role model as a great
engineer and teacher. Through numerous discussions and oneonone meetings, I learned so
much from him and I greatly appreciate all the advice and wisdom, big and small.
I would like to thank the members of the guidance committee, professor Peter Voltz,
professor Elza Erkip, and Donald Grieco, for their time and valuable feedback on my research. I
am honored to have them on the committee. I also wish to express my special appreciation to
Dr. Junsung Lim and Kyungjin Oh with whom I carried out joint research on SCFDMA
resource scheduling. I thank them for the many hours spent together doing research and
encouraging each other. I am also grateful to my parents for their support and encouragement to
pursue the PhD study.
Lastly, I would like to thank my wife and soul mate Hyun Joo who stood behind me rain or
shine. Her loving and caring words were sources of encouragement to me and I am deeply
grateful for them.
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An Abstract
Single Carrier Orthogonal Multiple Access Technique
for Broadband Wireless Communications
by
Hyung G. Myung
Advisor: David J. Goodman
Submitted in Partial Fulfillment of the Requirements
For the Degree of Doctor of Philosophy
January 2007
Broadband wireless mobile communications suffer from multipath frequencyselective fading.
Orthogonal frequency division multiplexing (OFDM) and orthogonal frequency division
multiple access (OFDMA), which are multicarrier communication techniques, have become
widely accepted primarily because of its robustness against frequency selective fading channels.
Despite the many advantages, OFDM and OFDMA suffer a number of drawbacks; high peak
toaverage power ratio (PAPR), a need for an adaptive or coded scheme to overcome spectral
nulls in the channel, and high sensitivity to frequency offset.
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Single carrier frequency division multiple access (SCFDMA) which utilizes single carrier
modulation at the transmitter and frequency domain equalization at the receiver is a technique
that has similar performance and essentially the same overall structure as those of an OFDMA
system. One prominent advantage over OFDMA is that the SCFDMA signal has lower PAPR.
SCFDMA has drawn great attention as an attractive alternative to OFDMA, especially in the
uplink communications where lower PAPR greatly benefits the mobile terminal in terms of
transmit power efficiency and manufacturing cost. SCFDMA is currently a working assumption
for the uplink multiple access scheme in 3rd Generation Partnership Project Long Term
Evolution (3GPP LTE).
In this thesis, we first give a detailed overview of an SCFDMA system. We then analyze
analytically and numerically the peak power characteristics and propose a peak power reduction
method that uses symbol amplitude clipping technique. We show that subcarrier mapping
scheme and pulse shaping are significant factors that affect the peak power characteristics and
that symbol amplitude clipping method is an effective way to reduce the peak power without
compromising the link performance. We investigate multiple input multiple output (MIMO)
spatial multiplexing technique in an SCFDMA system using unitary precoded transmit eigen
beamforming with practical limitations. We also investigate channeldependent scheduling for an
uplink SCFDMA system taking into account the imperfect channel information in the form of
feedback delay. To accommodate both low and high mobility users simultaneously, we propose a
hybrid subcarrier mapping method using orthogonal code spreading on top of SCFDMA and
show that it can have higher capacity gain than that of conventional subcarrier mapping scheme.
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List of Contents
Curriculum Vitae v
Acknowledgements
An Abstract
List of Figures xii
List of Tables xvi
Chapter 1 Introduction 1
1.1. Evolution of Cellular Wireless Communications.................................................................................1
1.2. 3GPP Long Term Evolution ...................................................................................................................2
1.3. Single Carrier FDMA................................................................................................................................5
1.4. Objectives and Contributions..................................................................................................................6
1.5. Organization...............................................................................................................................................8
1.6. Nomenclature...........................................................................................................................................10
Chapter 2 Channel Characteristics and Frequency Multiplexing 13
2.1. Characteristics of Wireless Mobile Communications Channel........................................................13
2.2. Orthogonal Frequency Division Multiplexing (OFDM) ..................................................................17
2.3. Single Carrier with Frequency Domain Equalization (SC/FDE)....................................................19
2.4. Summary and Conclusions.....................................................................................................................22
Chapter 3 Single Carrier FDMA 23
3.1. Overview of SCFDMA System...........................................................................................................25
3.2. Subcarrier Mapping.................................................................................................................................28
3.3. Time Domain Representation of SCFDMA Signals........................................................................31
3.4. SCFDMA and OFDMA.......................................................................................................................36
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3.5. SCFDMA and DSCDMA/FDE........................................................................................................38
3.6. SCFDMA Implementation in 3GPP LTE Uplink............................................................................40
3.7. Summary and Conclusions.....................................................................................................................45
Chapter 4 MIMO SCFDMA 47
4.1. Spatial Diversity and Spatial Multiplexing in MIMO Systems .........................................................48
4.2. MIMO Channel .......................................................................................................................................49
4.3. SCFDMA Transmit EigenBeamforming with Unitary Precoding ...............................................53
4.4. Summary and Conclusions.....................................................................................................................60
Chapter 5 Peak Power Characteristics of an SCFDMA Signal: Analytical Analysis 63
5.1. Upper Bound for IFDMA with Pulse Shaping ..................................................................................64 5.2. Modified Upper Bound for LFDMA and DFDMA..........................................................................70
5.3. Comparison with OFDM.......................................................................................................................71
5.4. Summary and Conclusion......................................................................................................................72
Chapter 6 Peak Power Characteristics of an SCFDMA Signal: Numerical Analysis 74
6.1. PAPR of Single Antenna Transmission Signals..................................................................................76
6.2. PAPR of Multiple Antenna Transmission Signals .............................................................................80
6.3. Peak Power Reduction by Symbol Amplitude Clipping....................................................................83
6.4. Summary and Conclusions.....................................................................................................................88
Chapter 7 ChannelDependent Scheduling of Uplink SCFDMA Systems 90
7.1. ChannelDependent Scheduling in an Uplink SCFDMA System..................................................91
7.2. Impact of Imperfect Channel State Information on CDS...............................................................96
7.3. Hybrid Subcarrier Mapping ................................................................................................................ 106
7.4. Summary and Conclusions.................................................................................................................. 110
Chapter 8 Conclusions 111
Appendix A Derivations of the Upper Bounds in Chapter 5 115
Bibliography 126
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List of Figures
Figure 2.1: Delay profile and frequency response of 3GPP 6tap typical urban (TU6) Rayleigh
fading channel in 5 MHz band. ..................................................................................................... 14
Figure 2.2: Time variation of 3GPP TU6 Rayleigh fading channel in 5 MHz band with 2GHz
carrier frequency............................................................................................................................... 16
Figure 2.3: Transmitter and receiver structures of SC/FDE and OFDM. ..................................... 19
Figure 2.4: Dissimilarities between OFDM and SC/FDE................................................................. 21
Figure 3.1: Transmitter and receiver structure of SCFDMA and OFDMA systems. .................. 24
Figure 3.2: Raisedcosine filter. .............................................................................................................. 27
Figure 3.3: Generation of SCFDMA transmit symbols.................................................................... 28
Figure 3.4: Subcarrier mapping modes; distributed and localized. ................................................... 29
Figure 3.5: An example of different subcarrier mapping schemes for N = 4, Q = 3 and M = 12.
............................................................................................................................................................ 30
Figure 3.6: Subcarrier allocation methods for multiple users (3 users, 12 subcarriers, and 4
subcarriers allocated per user)........................................................................................................ 30
Figure 3.7: Time symbols of different subcarrier mapping schemes. .............................................. 35
Figure 3.8: Amplitude of SCFDMA signals........................................................................................ 35
Figure 3.9: Dissimilarities between OFDMA and SCFDMA........................................................... 37
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Figure 3.10: DSCDMA with FDE. ...................................................................................................... 38
Figure 3.11: Spreading with the roles of data sequence and signature sequence exchanged for
spreading signature of {1, 1, 1, 1} with a data block size of 4................................................. 39
Figure 3.12: Basic subframe structure in the time domain. .............................................................. 40
Figure 3.13: Physical mapping of a block in RF frequency domain ( f c: carrier center frequency).
............................................................................................................................................................ 41
Figure 3.14: Generation of a block. ...................................................................................................... 41
Figure 3.15: FDM and CDM pilots for three simultaneous users with 12 total subcarriers. ........ 45
Figure 4.1: Description of a MIMO channel with N t transmit antennas and N r receive antennas.
............................................................................................................................................................ 50
Figure 4.2: Block diagram of a spatial multiplexing MIMO SCFDMA system. ........................... 53
Figure 4.3: Simpified block diagram of a unitary precoded TxBF SCFDMA MIMO system.... 54
Figure 4.4: Inputoutput characteristics of the quantizers. ................................................................ 57
Figure 4.5: FER performance of a 2x2 SCFDMA unitary precoded TxBF system with feedback
averaging and quantization. ............................................................................................................ 58
Figure 4.6: FER performance................................................................................................................. 59
Figure 4.7: FER performance of a 2x2 SCFDMA unitary precoded TxBF system with feedback
delays of 2, 4, and 6 TTI’s. ............................................................................................................. 61
Figure 5.1: CCDF of instantaneous power for IFDMA with BPSK modulation and different
values of rolloff factor α . ............................................................................................................. 68
Figure 5.2: CCDF of instantaneous power for IFDMA with QPSK modulation and different
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values of rolloff factor α . ............................................................................................................. 69
Figure 5.3: CCDF of instantaneous power for LFDMA with BPSK modulation and different
values of input block size N . .......................................................................................................... 70
Figure 5.4: CCDF of instantaneous power for IFDMA, LFDMA, and OFDM. For IFDMA, we
consider rolloff factor of 0.2. ...................................................................................................... 72
Figure 6.1: A theoretical relationship between PAPR and transmit power efficiency for ideal class
A and B amplifiers. .......................................................................................................................... 75
Figure 6.2: Comparison of CCDF of PAPR for IFDMA, DFDMA, LFDMA, and OFDMA
with total number of subcarriers M = 512, number of input symbols N = 128, IFDMA
spreading factor Q = 4, DFDMA spreading factor Qɶ = 2, and α (rolloff factor) = 0.22.. 78
Figure 6.3: Comparison of CCDF of PAPR for IFDMA and LFDMA with M = 256, N = 64, Q
= 4, Qɶ = 2, and α (rolloff factor) of 0, 0.2, 0.4, 0.6, 0.8, and 1. ............................................ 79
Figure 6.4: Precoding in the frequency domain is convolution and summation in the time
domain.k refers to the subcarrier number.................................................................................... 80
Figure 6.5: CCDF of PAPR for 2x2 unitary precoded TxBF............................................................ 81
Figure 6.6: Impact of quantization and averaging of the precoding matrix on PAPR.................. 82
Figure 6.7: PAPR comparison with other MIMO schemes. .............................................................. 82
Figure 6.8: Three types of amplitude limiter........................................................................................ 85
Figure 6.9: Block diagram of a symbol amplitude clipping method for SCFDMA MIMO
transmission with M t transmit antenna......................................................................................... 86
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Figure 6.10: CCDF of symbol power after clipping. .......................................................................... 86
Figure 6.11: Link level performance for clipping. ............................................................................... 87
Figure 6.12: PSD of the clipped signals................................................................................................ 87
Figure 7.1: Comparison of aggregate throughput with M = 256 system subcarriers, N = 8
subcarriers per user, bandwidth = 5 MHz, and noise power per Hz = 160 dBm................ 94
Figure 7.2: Average user data rate as a function of user distance with M = 256 system subcarriers,
N = 8 subcarriers per user, bandwidth = 5 MHz, and noise power per Hz = 160 dBm..... 95
Figure 7.3: Block diagram of an uplink SCFDMA system with adaptive modulation and CDS
for K users. ....................................................................................................................................... 97
Figure 7.4: System throughput vs. SNR for the 8 classes of QAM................................................ 103
Figure 7.5: Aggregate throughput with CDS and adaptive modulation......................................... 104
Figure 7.6: Aggregate throughput with CDS and constant modulation (16QAM) with mobile
speed of 60 km/h.......................................................................................................................... 104
Figure 7.7: Aggregate throughput with CDS and adaptive modulation with feedback delay of 3
ms and different mobile speeds. .................................................................................................. 105
Figure 7.8: Conventional subcarrier mapping and hybrid subcarrier mapping. ............................ 107
Figure 7.9: Block diagram of an SCCFDMA system...................................................................... 108
Figure 7.10: Comparison between SCFDMA and SCCFDMA in terms of occupied subcarriers
for the same number of users...................................................................................................... 108
Figure 7.11: Aggregate throughputs for hybrid subcarrier mapping method and other
conventional subcarrier mapping methods with CDS and adaptive modulation................. 109
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List of Tables
Table 2.1: Transmission bandwidths of current / future cellular wireless standards..................... 15
Table 3.1: Parameters for uplink SCFDMA transmission scheme in 3GPP LTE. ....................... 42
Table 3.2: Number of RU’s and number of subcarrriers per RU for LB........................................ 43
Table 4.1. Summary of feedback overhead vs. performance loss. .................................................... 60
Table 6.1: 99.9percentile PAPR for IFDMA, DFDMA, LFDMA, and OFDMA ........................ 78
Table 7.1: SNR boundaries for adaptive modulation. ....................................................................... 103
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Chapter 1Chapter 1Chapter 1Chapter 1
IntroductionIntroductionIntroductionIntroduction
1.1. Evolution of Cellular Wireless Communications
During the 1950s and 1960s, researchers at AT&T Bell Laboratories and companies around
the world developed the idea of cellular radiotelephony. The concept of cellular wireless
communications is to break the coverage zone into small cells and reuse the portions of the
available radio spectrum. In 1979, the world’s first cellular system was deployed by Nippon
Telephone and Telegraph (NTT) in Japan and thus began the evolution of cellular wireless
communications [1], [2].
The first generation of cellular wireless communication systems utilized analog
communication techniques and its focus was on accommodating voice traffic. Frequency
modulation (FM) and frequency division multiple access (FDMA) were the basis of the first
generation systems. AMPS (Advanced Mobile Phone System) in US and ETACS (European
Total Access Cellular System) in Europe were among the first generation systems.
The second generation systems saw the advent of digital communication techniques
which greatly improved spectrum efficiency. Also they vastly enhanced the voice quality and
made possible the packet data transmission. The main multiple access schemes are time
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division multiple access (TDMA) and code division multiple access (CDMA). GSM (Global
System for Mobile) which is based on TDMA and IS95 which is based on CDMA are two
most widely accepted second generation systems.
In the mid1980s, the concept for IMT2000 (International Mobile Telecommunications
2000) was born at the ITU (International Telecommunication Union) as the third generation
(3G) system for mobile communications [3]. Key objectives of IMT2000 are to provide
seamless global roaming and to provide seamless delivery of services over a number of media
via higher data rate link. In 2000, a unanimous approval of the technical specifications for 3G
system under the brand IMT2000 was made and UMTS/WCDMA (Universal Mobile
Telecommunications System/Wideband CDMA) and cdma2000 are two prominent standards
under IMT2000 both of which are based on CDMA. IMT2000 provides higher transmission
rates; a minimum speed of 2 Mbps for stationary or walking users and 348 kbps in a moving
vehicle whereas second generation systems only provide speeds ranging from 9.6 kbps to 28.8
kbps. Since the initial standardization, both WCDMA and cdma2000 have evolved into so
called “3.5G”; UMTS through HSD/UPA (High Speed Downlink/Uplink Packet Access) and
cdma2000 through 1xEVDO Rev A (1x Evolution DataOptimized Revision A).
Currently, 3rd Generation Partnership Project Long Term Evolution (3GPP LTE) is
considered as the prominent path to the next generation of cellular system beyond 3G.
1.2. 3GPP Long Term Evolution
3GPP’s work on the evolution of the 3G mobile system started with the Radio Access
Network (RAN) Evolution workshop in November 2004 [4]. Operators, manufacturers, and
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research institutes presented more than 40 contributions with views and proposals on the
evolution of the Universal Terrestrial Radio Access Network (UTRAN) which is the
foundation for UMTS/WCDMA systems. They identified a set of high level requirements at
the workshop; reduced cost per bit, increased service provisioning, flexibility of the use of
existing and new frequency bands, simplified architecture and open interfaces, and allow for
reasonable terminal power consumption. With the conclusions of this workshop and with
broad support from 3GPP members, a feasibility study on the Universal Terrestrial Radio
Access (UTRA) and UTRAN Long Term Evolution started in December 2004. The objective
was to develop a framework for the evolution of the 3GPP radio access technology towards a
highdatarate, lowlatency, and packetoptimized radio access technology. The study focused
on means to support flexible transmission bandwidth of up to 20 MHz, introduction of new
transmission schemes, advanced multiantenna technologies, signaling optimization,
identification of the optimum UTRAN network architecture, and functional split between
RAN network nodes.
The first part of the study resulted in an agreement on the requirements for the Evolved
UTRAN (EUTRAN). Key aspects of the requirements are as follows [5].
• Peak data rate: Instantaneous downlink peak data rate of 100 Mbps within a 20
MHz downlink spectrum allocation (5 bps/Hz) and instantaneous uplink peak
data rate of 50 Mbps (2.5 bps/Hz) within a 20 MHz uplink spectrum allocation.
• Controlplane capacity: At least 200 users per cell should be supported in the
active state for spectrum allocations up to 5 MHz.
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• Userplane latency: Less than 5 ms in an unloaded condition (i.e. single user with
single data stream) for small IP packet.
• Mobility: EUTRAN should be optimized for low mobile speed from 0 to 15
km/h. Higher mobile speeds between 15 and 120 km/h should be supported with
high performance. Mobility across the cellular network shall be maintained at
speeds from 120 to 350 km/h (or even up to 500 km/h depending on the
frequency band).
• Coverage: Throughput, spectrum efficiency, and mobility targets should be met
for 5 km cells and with a slight degradation for 30 km cells. Cells ranging up to
100 km should not be precluded.
• Enhanced multimedia broadcast multicast service (EMBMS).
• Spectrum flexibility: EUTRA shall operate in spectrum allocations of different
sizes including 1.25 MHz, 1.6 MHz, 2.5 MHz, 5 MHz, 10 MHz, 15 MHz, and 20
MHz in both uplink and downlink.
• Architecture and migration: Packetbased single EUTRAN architecture with
provision to support systems supporting realtime and conversational class traffic
and support for an endtoend qualityofservice (QoS).
• Radio resource management: Enhanced support for endtoend QoS, efficient
support for transmission of higher layers, and support of load sharing and policy
management across different radio access technologies.
The wide set of options initially identified by the early LTE work was narrowed down in
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December 2005 to a working assumption that the downlink would use Orthogonal Frequency
Division Multiple Access (OFDMA) and the uplink would use Single Carrier Frequency
Division Multiple Access (SCFDMA). Supported downlink data modulation schemes are
QPSK, 16QAM, and 64QAM, and possible uplink data modulation schemes are π/2shifted
BPSK, QPSK, 8PSK and 16QAM. They agreed the use of Multiple Input Multiple Output
(MIMO) scheme with possibly up to four antennas at the mobile side and four antennas at the
base station. Reusing the expertise from the UTRAN, they agreed to the same channel coding
type as UTRAN (turbo codes). They agreed to a transmission time interval (TTI) of 1 ms to
reduce signaling overhead and to improve efficiency.
The study item phase ended in September 2006 and the LTE works are scheduled to
conclude in early 2008 and produce a technical standard. More technical details on 3GPP LTE
are at [6] and [7].
1.3. Single Carrier FDMA
Ever increasing demand for higher data rate is leading to utilization of wider transmission
bandwidth. Broadband wireless mobile communications suffer from multipath frequency
selective fading. For broadband multipath channels, conventional time domain equalizers are
impractical for complexity reason.
Orthogonal frequency division multiplexing (OFDM), which is a multicarrier
communication technique, has become widely accepted primarily because of its robustness
against frequencyselective fading channels which are common in broadband mobile wireless
communications [8]. Orthogonal frequency division multiple access (OFDMA) is a multiple
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access scheme which is an extension of OFDM to accommodate multiple simultaneous users.
OFDM/OFDMA technique is currently adopted in wireless LAN (IEEE 802.11a & 11g),
WiMAX (IEEE 802.16), and 3GPP LTE downlink systems. Despite the benefits of OFDM
and OFDMA, they suffer a number of drawbacks including: high peaktoaverage power ratio
(PAPR), a need for an adaptive or coded scheme to overcome spectral nulls in the channel,
and high sensitivity to frequency offset.
Single carrier frequency division multiple access (SCFDMA) which utilizes single carrier
modulation and frequency domain equalization is a technique that has similar performance
and essentially the same overall complexity as those of OFDMA system [9]. One prominent
advantage over OFDMA is that the SCFDMA signal has lower PAPR because of its inherent
single carrier structure. SCFDMA has drawn great attention as an attractive alternative to
OFDMA, especially in the uplink communications where lower PAPR greatly benefits the
mobile terminal in terms of transmit power efficiency and manufacturing cost. SCFDMA has
two different subcarrier mapping schemes; distributed and localized . In distributed subcarrier
mapping scheme, user’s data occupy a set of distributed subcarriers and we achieve frequency
diversity. In localized subcarrier mapping scheme, user’s data inhabit a set of consecutive
localized subcarriers and we achieve frequencyselective gain through channeldependent
scheduling (CDS). SCFDMA is currently a working assumption for the uplink multiple access
scheme in 3GPP LTE [10].
1.4. Objectives and Contributions
• Introduction to SCFDMA: SCFDMA is a new radio interface technique that is currently
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being adopted in 3GPP LTE uplink. In this thesis, we give a detailed overview of an
SCFDMA system and explain its transmit and receive process. We also illustrate the
similarities to and differences with an OFDMA system.
• Time domain representation of the transmit symbols of an SCFDMA signal: In this thesis, we
derive the time domain representation of the transmit symbols for each subcarrier
mapping scheme.
• Analysis of unitary precoded transmit eigenbeamforming (TxBF) for SCFDMA with limited
feedback: In this thesis, we numerically analyze the performance of a unitary precoded
TxBF SCFDMA system with limited feedback. We show the impacts of feedback
quantization/averaging and feedback delay on the link level performance and also on
the transmit PAPR characteristics.
• Analytical analysis of the peak power of an SCFDMA signal : In this thesis, we derive an
analytical upperbound on the distribution of the instantaneous power of an SC
FDMA signal using Chernoff bound. We also characterize the peak power distribution
for each of the subcarrier mapping scheme. We show analytically that an SCFDMA
signal has indeed lower peak power than an OFDM signal.
• PAPR characteristics of an SCFDMA signal : PAPR is an important measure that affects
the transmit power efficiency. In this thesis, we numerically characterize the PAPR for
different subcarrier mapping considering pulse shaping. We consider both single
antenna and multiple antenna transmissions.
• Peak power reduction by symbol amplitude clipping method : In this thesis, we propose a symbol
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amplitude clipping method to reduce the peak power of the SCFDMA transmit signal.
The proposed method effectively reduces the peak power while hardly affecting the
link level performance.
• Channeldependent resource scheduling with hybrid subcarrier mapping : Channeldependent
scheduling (CDS) results in high capacity gain when channel state information (CSI) is
accurate but the gain decreases when the quality of CSI becomes poor. In this thesis,
we propose a hybrid subcarrier mapping scheme utilizing orthogonal code spreading
for cases where there are both high and low mobility users at the same time. Our
proposed scheme yields higher capacity gain when high mobility users are dominant.
1.5. Organization
The following is the outline of the remainder of the thesis.
Chapter 2 gives a general overview of OFDM and frequency domain equalization. We first
characterize the wireless mobile communications channel. Then, we give an overview of
OFDM which is a popular multicarrier modulation technique, and we explain single carrier
modulation with frequency domain equalization (SC/FDE) and compare it with OFDM.
Chapter 3 introduces SCFDMA. We first give an overview of SCFDMA and explain the
transmission and reception operations in detail. We describe the two flavors of subcarrier
mapping schemes in SCFDMA, distributed and localized, and briefly compare the two, and
we derive the time domain representation of SCFDMA transmit signal for each subcarrier
mapping mode. Then, we give an indepth comparison between SCFDMA and OFDMA and
we also compare SCFDMA with direct sequence spread spectrum code division multiple
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access (DSCDMA) with frequency domain equalization. Lastly, we illustrate in detail the SC
FDMA implementation in the physical layer according to 3GPP LTE uplink and we describe
the reference signal structure of SCFDMA.
Chapter 4 investigates MIMO techniques for an SCFDMA system. We first give a general
overview of MIMO concepts and describe the parallel decomposition of a MIMO channel for
narrowband and wideband transmission. Then, we illustrate the realization of MIMO spatial
multiplexing in SCFDMA. We introduce the SCFDMA TxBF with unitary precoding
technique and numerically analyze the link level performance with practical considerations.
Chapter 5 analyzes the distribution of the instantaneous peak power of an SCFDMA
signal and shows analytically that an SCFDMA signal has statistically lower peak power than
an OFDM signal. Using the Chernoff bound, we first derive an upperbound of the
complementary cumulative distribution function (CCDF) of the instantaneous power for
IFDMA with pulse shaping and show the bounds for BPSK and QPSK with raisedcosine
pulse shaping filter. We also derive a modified upperbound of the CCDF of the
instantaneous power for LFDMA without pulse shaping. Then, we compare the results with
the analytical CCDF of PAPR for OFDM.
Chapter 6 investigates the PAPR characteristics of an SCFDMA signal numerically. We
first characterize the PAPR for single antenna transmission of SCFDMA. We investigate the
PAPR properties for different subcarrier mapping schemes. Then, we analyze the PAPR
characteristics for multiple antenna transmission. Specifically, we numerically analyze the
CCDF of PAPR for a 2x2 unitary precoded TxBF SCFDMA system described in chapter 4.
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Lastly, we propose a symbol amplitude clipping method to reduce peak power and show the
link level performance and frequency domain aspects of the proposed clipping method.
Chapter 7 presents a channeldependent scheduling (CDS) method for an SCFDMA
system in the uplink communications. We first give a general overview of CDS in an uplink
SCFDMA system. We also analyze the capacity for the two subcarrier mapping schemes.
Then, we investigate the impact of imperfect channel state information (CSI) on CDS. We
analyze the data throughput of an SCFDMA system with uncoded adaptive modulation and
CDS when there is a feedback delay. Lastly, we propose a hybrid subcarrier mapping scheme
using direct sequence spreading technique on top of SCFDMA modulation. We show the
throughput improvement of the hybrid subcarrier mapping over the conventional subcarrier
mapping schemes.
Chapter 8 presents a summary of the work and concluding remarks.
1.6. Nomenclature
3GPP 3rd Generation Partnership Project
BER Bit Error Rate
BPSK Binary Phase Shift Keying
CAZAC Constant Amplitude Zero AutoCorrelation
CCDF Complementary Cumulative Distribution Function
CDM Code Division Multiplexing
CDMA Code Division Multiple Access
CDS ChannelDependent Scheduling
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CP Cyclic Prefix
CSI Channel State Information
DFDMA Distributed Frequency Division Multiple Access
DFT Discrete Fourier Transform
FDE Frequency Domain Equalization
FDM Frequency Division Multiplexing
FDMA Frequency Division Multiple Access
FER Frame Error Rate
FFT Fast Fourier Transform
GSM Global System for Mobile
IBI InterBlock Interference
ICI InterCarrier Interference
IDFT Inverse Discrete Fourier Transform
ISI InterSymbol Interference
IFDMA Interleaved Frequency Division Multiple Access
LB Long Block
LFDMA Localized Frequency Division Multiple Access
LMMSE Linear Minimum Mean Square Error
LTE Long Term Evolution
MIMO Multiple Input Multiple Output
MMSE Minimum Mean Square Error
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OFDM Orthogonal Frequency Division Multiplexing
OFDMA Orthogonal Frequency Division Multiple Access
PAPR PeaktoAverage Power Ratio
PSD Power Spectral Density
QAM Quadrature Amplitude Modulation
QPSK Quadrature Phase Shift Keying
RU Resource Unit
SB Short Block
SC Single Carrier
SC/FDE Single Carrier with Frequency Domain Equalization
SCFDMA Single Carrier Frequency Division Multiple Access
SFBC SpaceFrequency Block Coding
SM Spatial Multiplexing
SNR SignaltoNoise Ratio
SVD Singular Value Decomposition
TDM Time Division Multiplexing
TDMA Time Division Multiple Access
TTI Transmission Time Interval
TxBF Transmit EigenBeamforming
WCDMA Wideband Code Division Multiple Access
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Chapter 2Chapter 2Chapter 2Chapter 2
Channel Characteristics and Frequency MuChannel Characteristics and Frequency MuChannel Characteristics and Frequency MuChannel Characteristics and Frequency Multiplexingltiplexingltiplexingltiplexing
In this chapter, we first characterize the wireless mobile communications channel. In section 2.2,
we give an overview of orthogonal frequency division multiplexing (OFDM) which is a popular
multicarrier modulation technique and in section 2.3, we explain single carrier modulation with
frequency domain equalization (SC/FDE) and compare it with OFDM.
2.1. Characteristics of Wireless Mobile Communications Channel
In a wireless mobile communication system, a transmitted signal propagating through the
wireless channel often encounters multiple reflective paths until it reaches the receiver [11]. We
refer to this phenomenon as multipath propagation and it causes fluctuation of the amplitude
and phase of the received signal. We call this fluctuation multipath fading and it can occur either
in large scale or in small scale. Largescale fading represents the average signal power attenuation
or path loss due to motion over large areas. Smallscale fading occurs due to small changes in
position and we also call it as Rayleigh fading since the fading is often statistically characterized
with Rayleigh probability density function (pdf). Rayleigh fading in the propagation channel,
which generates intersymbol interference (ISI) in the time domain, is a major impairment in
wireless communications and it significantly degrades the link performance.
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0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
Time [µsec]
A m p l i t u d e [ l i n e a r ]
3GPP 6Tap Typical Urban (TU6) Channel Delay Profile
0 1 2 3 4 50
0.5
1
1.5
2
2.5
Frequency [MHz]
C h a n n e l G a i n [ l i n e a r ]
Frequency Response of 3GPP TU6 Channel in 5MHz Band
Figure 2.1: Delay profile and frequency response of 3GPP 6tap typical urban (TU6) [12] Rayleigh
fading channel in 5 MHz band.
When characterizing the Rayleigh fading channel, we can categorize it into either flat fading
channel or frequencyselective fading channel. Flat fading occurs when the coherence bandwidth,
which is inversely proportional to channel delay spread, is much larger than the transmission
bandwidth whereas frequencyselective fading happens when the coherence bandwidth is much
smaller than the transmission bandwidth. We show an example of the impulse response and
frequency response of a frequencyselective fading channel in Figure 2.1.
We can also characterize the multipath fading channel in terms of the degree of time
variation of the channel; slow fading and fast fading. The timevarying nature of the channel is
directly related to the movement of the user and the user’s surrounding, and the degree of the
time variation is associated with the Doppler frequency. Doppler frequency f d is given by
d
v f
λ = (2.1)
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where v is the relative speed of the user and λ is the wavelength of the carrier. For a given
Doppler frequency f d , the spacedtime correlation function R( ∆t ) specifies the extent to which
there is correlation between the channel’s response in ∆t time interval and it is given by
( ) ( )0 2 d R t J f t π ∆ = ∆ (2.2)
where J 0( ⋅ ) is the zeroorder Bessel function of the first kind. As the Doppler frequency
increases, the correlation decreases at a given time interval. An example of the time variation of
a fading channel is illustrated in Figure 2.2.
As wireless multimedia applications become more widespread, demand for higher data rate
is leading to utilization of a wider transmission bandwidth. For example, Global System for
Mobile Communications (GSM) system, which is a popular second generation cellular system,
uses transmission bandwidth of only 200 kHz but the next generation cellular standard 3GPP
Long Term Evolution (LTE) envisions bandwidth of up to 20 MHz which is 100 times the
bandwidth of GSM. Table 2.1 illustrates the transmission bandwidth in current and future
cellular wireless standards.
Table 2.1: Transmission bandwidths of current / future cellular wireless standards.
Generation Standard Transmission Bandwidth
GSM 200 kHz2G
IS95 (CDMA) 1.25 MHz
WCDMA 3Gcdma2000 5 MHz
3GPP LTE Up to 20 MHz3.5 ~ 4G
WiMAX (IEEE 802.16) Up to 20 MHz
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0
1
2
3
4
5
0
1
2
3
4
5
0
5
Time [msec]
Mobile speed = 3 km/h (5.6 Hz doppler)
Frequency [MHz]
C h a n n e l G a i n [ l i n e a r ]
(a)
0
1
2
3
4
5
0
1
2
3
4
5
0
5
Time [msec]
Mobile speed = 60 km/h (111 Hz doppler)
Frequency [MHz]
C h a n n e l G a i n
[ l i n e a r ]
(b)
Figure 2.2: Time variation of 3GPP TU6 Rayleigh fading channel in 5 MHz band with 2GHz
carrier frequency; (a) user speed = 3 km/h (Doppler frequency = 5.6 Hz); (b) user speed = 60
km/h (Doppler frequency = 111 Hz).
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With a wider transmission bandwidth, frequency selectivity of the channel becomes more
severe and thus the problem of ISI becomes more serious. In a conventional single carrier
communication system, time domain equalization in the form of tap delay line filtering is
performed to eliminate ISI. However, in case of a wide band channel, the length of the time
domain filter to perform equalization becomes prohibitively large since it linearly increases with
the channel response length.
2.2. Orthogonal Frequency Division Multiplexing (OFDM)
One way to mitigate the frequencyselective fading seen in a wide band channel is to use a
multicarrier technique which subdivides the entire channel into smaller subbands, or subcarriers.
Orthogonal frequency division multiplexing (OFDM) is a multicarrier modulation technique
which uses orthogonal subcarriers to convey information. In the frequency domain, since the
bandwidth of a subcarrier is designed to be smaller than the coherence bandwidth, each
subchannel is seen as a flat fading channel which simplifies the channel equalization process. In
the time domain, by splitting a highrate data stream into a number of lowerrate data stream
that are transmitted in parallel, OFDM resolves the problem of ISI in wide band
communications. More technical details on OFDM are at [8], [13], [14], [15], [16], and [17].
In summary, OFDM has the following advantages:
• For a given channel delay spread, the implementation complexity is much lower than
that of a conventional single carrier system with time domain equalizer.
• Spectral efficiency is high since it uses overlapping orthogonal subcarriers in the
frequency domain.
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• Modulation and demodulation are implemented using inverse discrete Fourier
transform (IDFT) and discrete Fourier transform (DFT), respectively, and fast
Fourier transform (FFT) algorithms can be applied to make the overall system
efficient.
• Capacity can be significantly increased by adapting the data rate per subcarrier
according to the signaltonoise ratio (SNR) of the individual subcarrier.
Because of these advantages, OFDM has been adopted as a modulation of choice by many
wireless communication systems such as wireless LAN (IEEE 802.11a and 11g) and DVBT
(Digital Video BroadcastingTerrestrial).
However, it suffers from the following drawbacks [18], [19]:
• High peaktoaverage power ratio (PAPR): The transmitted signal is a superposition
of all the subcarriers with different carrier frequencies and high amplitude peaks
occur because of the superposition.
• High sensitivity to frequency offset: When there are frequency offsets in the
subcarriers, the orthogonality among the subcarriers breaks and it causes inter
carrier interference (ICI).
• A need for an adaptive or coded scheme to overcome spectral nulls in the channel:
In the presence of a null in the channel, there is no way to recover the data of the
subcarriers that are affected by the null unless we use rate adaptation or a coding
scheme.
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Channel N 
pointIDFT
Equalization N 
pointDFT
SC/FDE
OFDM
DetectRemove
CP{ }n x Add
CP/PS
* CP: Cyclic Prefix, PS: Pulse Shaping
Channel Equalization N 
pointDFT
DetectRemove
CP
N 
pointIDFT
Add
CP/PS
{ }n x
Figure 2.3: Transmitter and receiver structures of SC/FDE and OFDM.
2.3. Single Carrier with Frequency Domain Equalization (SC/FDE)
For broadband multipath channels, conventional time domain equalizers are impractical because
of the complexity (very long channel impulse response in the time domain). Frequency domain
equalization (FDE) is more practical for such channels. Single carrier with frequency domain
equalization (SC/FDE) technique is another way to fight the frequencyselective fading channel.
It delivers performance similar to OFDM with essentially the same overall complexity, even for
long channel delay [18], [19]. Figure 2.3 shows the block diagram of SC/FDE and compares it
with that of OFDM.
In the transmitter of SC/FDE, we add a cyclic prefix (CP), which is a copy of the last part
of the block, to the input data at the beginning of each block in order to prevent interblock
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interference (IBI) and also to make linear convolution of the channel impulse response look like
a circular convolution. It should be noted that circular convolution problem exists for any FDE
since multiplication in the DFTdomain is equivalent to circular convolution in the time domain
[20]. When the data signal propagates through the channel, it linearly convolves with the channel
impulse response. An equalizer basically attempts to invert the channel impulse response and
thus channel filtering and equalization should have the same type of convolution, either linear or
circular convolution. One way to resolve this problem is to add a CP in the transmitter that will
make the channel filtering look like a circular convolution and match the DFTbased FDE.
Another way is not to use CP but perform an “overlap and save” method in the frequency
domain equalizer to emulate the linear convolution [20].
SC/FDE receiver transforms the received signal to the frequency domain by applying DFT
and does the equalization process in the frequency domain. Most of the wellknown time
domain equalization techniques, such as minimum meansquare error (MMSE) equalization,
decision feedback equalization, and turbo equalization, can be applied to the FDE and the
details of the frequency domain implementation of these techniques are found in [21], [22], [23],
[24], [25], and [26]. After the equalization, the signal is brought back to the time domain via
IDFT and detection is performed.
Comparing the two systems in Figure 2.3, it is interesting to find the similarity between the
two. Overall, they both use the same communication component blocks and the only difference
between the two diagrams is the location of the IDFT block. Thus, one can expect the two
systems to have similar link level performance and spectral efficiency.
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Equalizer
Equalizer
Equalizer
Detect
Detect
Detect
Equalizer IDFT DetectSC/FDE
OFDM DFT
DFT
(a)
OFDM symbol
SC/FDE symbols
time (b)
Figure 2.4: Dissimilarities between OFDM and SC/FDE; (a) different detection processes in the
receiver; (b) different modulated symbol durations.
However, there are distinct differences that make the two systems perform differently as
illustrated in Figure 2.4. In the receiver, OFDM performs data detection on a persubcarrier
basis in the frequency domain whereas SC/FDE does it in the time domain after the additional
IDFT operation. Because of this difference, OFDM is more sensitive to a null in the channel
spectrum and it requires channel coding or power/rate control to overcome this deficiency. Also,
the duration of the modulated time symbols are expanded in the case of OFDM with parallel
transmission of the data block during the elongated time period.
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In summary, SC/FDE has advantages over OFDM as follows:
• Low PAPR due to single carrier modulation at the transmitter.
• Robustness to spectral null.
• Lower sensitivity to carrier frequency offset.
• Lower complexity at the transmitter which will benefit the mobile terminal in
cellular uplink communications.
Single carrier FDMA (SCFDMA) is an extension of SC/FDE to accommodate multiuser
access, which will be the subject of the next chapter.
2.4. Summary and Conclusions
Broadband mobile wireless channel suffers from severe frequencyselective fading which causes
the variation of received signal strength. OFDM is a multicarrier technique that overcomes the
frequencyselective fading impairment by transmitting data over narrower subbands in parallel.
Despite the many benefits, OFDM has limits including: high peaktoaverage power ratio
(PAPR), high sensitivity to frequency offset, and a need for an adaptive or coded scheme to
overcome spectral nulls in the channel.
Single carrier modulation with frequency domain equalization (SC/FDE) technique is
another way to mitigate the frequencyselective fading. SC/FDE delivers performance similar to
OFDM with essentially the same overall complexity and has advantages including; low PAPR,
robustness to spectral null, lower sensitivity to carrier frequency offset, lower complexity at the
transmitter which will benefit the mobile terminal in cellular uplink communications.
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Chapter 3Chapter 3Chapter 3Chapter 3
Single Carrier FDMASingle Carrier FDMASingle Carrier FDMASingle Carrier FDMA
Single carrier frequency division multiple access (SCFDMA), which utilizes single carrier
modulation and frequency domain equalization, is a technique that has similar performance
and essentially the same overall complexity as those of orthogonal frequency division multiple
access (OFDMA) system. SCFDMA is an extension of single carrier modulation with
frequency domain equalization (SC/FDE) to accommodate multipleuser access. One
prominent advantage over OFDMA is that the SCFDMA signal has lower peaktoaverage
power ratio (PAPR) because of its inherent single carrier structure [9]. SCFDMA has drawn
great attention as an attractive alternative to OFDMA, especially in the uplink communications
where lower PAPR greatly benefits the mobile terminal in terms of power efficiency. It is
currently a working assumption for uplink multiple access scheme in 3GPP Long Term
Evolution (LTE) or Evolved UTRA [6], [7], [10].
In this chapter, we first give an overview of SCFDMA and explain the transmission and
reception operations in detail. In section 3.2, we describe the two flavors of subcarrier
mapping schemes in SCFDMA and briefly compare the two. In section 3.3, we derive the
time domain representations of SCFDMA transmit signal for each subcarrier mapping mode.
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In section 3.4, we give an indepth comparison between SCFDMA and OFDMA. In
section 3.5, we compare SCFDMA with direct sequence spread spectrum code division
multiple access (DSCDMA) with frequency domain equalization and show the similarities
between the two. In section 3.6, we illustrate in detail the SCFDMA implementation in the
physical layer according to 3GPP LTE uplink. In section 3.7, we describe the reference (pilot)
signal structure of SCFDMA.
SubcarrierMapping
Channel
N pointIDFT
SubcarrierDe
mapping/Equalization
M pointDFT
DetectRemove
CP
N pointDFT
M pointIDFT
{ }n x Add CP/ PS
DAC/ RF
RF/ ADC
SCFDMA
OFDMA
* CP: Cyclic Prefix, PS: Pulse Shaping
SubcarrierMapping
Channel
SubcarrierDe
mapping/Equalization
M pointDFT
DetectRemove
CP
M pointIDFT
{ }n x Add CP
/ PSDAC/ RF
RF/ ADC
{ }k X
{ }m xɶ
{ }l X ɶ
Figure 3.1: Transmitter and receiver structure of SCFDMA and OFDMA systems.
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3.1. Overview of SCFDMA System
Figure 3.1 shows a block diagram of an SCFDMA system. SCFDMA can be regarded as
DFTspread OFDMA, where time domain data symbols are transformed to frequency domain
by DFT before going through OFDMA modulation. The orthogonality of the users stems
from the fact that each user occupies different subcarriers in the frequency domain, similar to
the case of OFDMA. Because the overall transmit signal is a single carrier signal, PAPR is
inherently low compared to the case of OFDMA which produces a multicarrier signal.
The transmitter of an SCFDMA system converts a binary input signal to a sequence of
modulated subcarriers. At the input to the transmitter, a baseband modulator transforms the
binary input to a multilevel sequence of complex numbers xn in one of several possible
modulation formats. The transmitter next groups the modulation symbols { xn} into blocks
each containing N symbols. The first step in modulating the SCFDMA subcarriers is to
perform an N point DFT to produce a frequency domain representation X k of the input
symbols. It then maps each of the N DFT outputs to one of the M (> N ) orthogonal
subcarriers that can be transmitted. If N = M / Q and all terminals transmit N symbols per
block, the system can handle Q simultaneous transmissions without cochannel interference. Q
is the bandwidth expansion factor of the symbol sequence. The result of the subcarrier
mapping is the set l X ɶ ( l = 0, 1, 2…, M 1) of complex subcarrier amplitudes, where N of the
amplitudes are nonzero. As in OFDMA, an M point IDFT transforms the subcarrier
amplitudes to a complex time domain signal m xɶ . Each m xɶ then are transmitted sequentially.
The transmitter performs two other signal processing operations prior to transmission. It
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inserts a set of symbols referred to as a cyclic prefix (CP) in order to provide a guard time to
prevent interblock interference (IBI) due to multipath propagation. The transmitter also
performs a linear filtering operation referred to as pulse shaping in order to reduce outof
band signal energy.
In general, CP is a copy of the last part of the block, which is added at the start of each
block for a couple of reasons. First, CP acts as a guard time between successive blocks. If the
length of the CP is longer than the maximum delay spread of the channel, or roughly, the
length of the channel impulse response, then, there is no IBI. Second, since CP is a copy of
the last part of the block, it converts a discrete time linear convolution into a discrete time
circular convolution. Thus transmitted data propagating through the channel can be modeled
as a circular convolution between the channel impulse response and the transmitted data block,
which in the frequency domain is a pointwise multiplication of the DFT frequency samples.
Then, to remove the channel distortion, the DFT of the received signal can simply be divided
by the DFT of the channel impulse response pointwise or a more sophisticated frequency
domain equalization technique can be implemented.
One of the commonly used pulse shaping filter is the raisedcosine filter. The frequency
domain and time domain representations of the filter are as follows.
1,0
2
1 1 1( ) 1 cos ,
2 2 2 2
10 ,
2
T f T
T T P f f f
T T T
f T
α
π α α α
α
α
−≤ ≤
− − + = + − ≤ ≤
+
≥
(3.1)
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( ) ( )2 2 2
sin / cos / ( )
/ 1 4 /
t T t T p t
t T t T
π πα
π α = ⋅
−(3.2)
where T is the symbol period and α is the rolloff factor.
Figure 3.2 shows the raisedcosine filter graphically in the frequency domain and time
domain. Rolloff factor α changes from 0 to 1 and it controls the amount of outofband
radiation; α = 0 generates no outofband radiation and as α increases, the outofband
radiation increases. In the time domain, the pulse has higher side lobes when α is close to 0
and this increases the peak power for the transmitted signal after pulse shaping. We further
investigate the effect of pulse shaping on the peak power characteristics in chapters 5 and 6.
Figure 3.3 details the generation of SCFDMA transmit symbols. There are M subcarriers,
among which N (< M ) subcarriers are occupied by the input data. In the time domain, the
input data symbol has symbol duration of T seconds and the symbol duration is compressed
0
0.2
0.4
0.6
0.8
1
Frequency
P(f)
0.2
0
0.2
0.4
0.6
0.8
p(t)
Time
α = 0α = 0.5
α = 1
α = 0α = 0.5
α = 1
Figure 3.2: Raisedcosine filter.
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to T ɶ = ( N / M )⋅T seconds after going through SCFDMA modulation.
The receiver transforms the received signal into the frequency domain via DFT, demaps
the subcarriers, and then performs frequency domain equalization. Because SCFDMA uses
single carrier modulation, it suffers from intersymbol interference (ISI) and thus equalization
is necessary to combat the ISI. The equalized symbols are transformed back to the time
domain via IDFT, and detection and decoding take place in the time domain.
3.2. Subcarrier Mapping
There are two methods to choose the subcarriers for transmission as shown in Figure 3.4;
distributed subcarrier mapping and localized subcarrier mapping. In the distributed subcarrier
mapping mode, DFT outputs of the input data are allocated over the entire bandwidth with
zeros occupying the unused subcarriers, whereas consecutive subcarriers are occupied by the
DFT outputs of the input data in the localized subcarrier mapping mode. We will refer to the
localized subcarrier mapping mode of SCFDMA as localized FDMA (LFDMA) and
DFT( N point)
IDFT( M point)
SubcarrierMapping{ }n
x{ }k X
{ }m xɶ
{ }l X ɶ
N
T ɶ
N
T ɶ
M N
N T T
M
>
= ⋅ɶ
M
T
: number of data symbols,
,
N M
T T ɶ : symbol durations
Figure 3.3: Generation of SCFDMA transmit symbols. There are M total number of subcarriers,
among which N (< M ) subcarriers are occupied by the input data.
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distributed subcarrier mapping mode of SCFDMA as distributed FDMA (DFDMA). The
case of M = Q⋅ N for the distributed mode with equidistance between occupied subcarriers is
called Interleaved FDMA (IFDMA) [27], [28], [24]. IFDMA is a special case of SCFMDA
and it is very efficient in that the transmitter can modulate the signal strictly in the time
domain without the use of DFT and IDFT.
An example of SCFDMA transmit symbols in the frequency domain for N = 4, Q = 3
and M = 12 is illustrated in Figure 3.5 and a multiuser perspective is shown in Figure 3.6.
From a resource allocation point of view, subcarrier mapping methods are further divided
into static and channeldependent scheduling (CDS) methods. CDS assigns subcarriers to
users according to the channel frequency response of each user. For both scheduling methods,
distributed subcarrier mapping provides frequency diversity because the transmitted signal is
Zeros
0 X
Zeros
1 X
2 X
1 N X −
0 X
1 N X −
1 X
Zeros
Zeros
Distributed Mode Localized Mode
0 X ɶ
1 M X −
ɶ
0 X ɶ
1 M X −
ɶZeros
Figure 3.4: Subcarrier mapping modes; distributed and localized.
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0 0 0 0 0 0 0 0 X 0 X 1 X 2 X 3
frequency
0 0 0 0 0 0 0 0 X 0 X 1 X 2 X 3
{ } :k X X 0 X 1 X 2 X 3
{ } :n x x0 x1 x2 x3
DFT
21
0
, 4 N
j nk N k n
n
X x e N
π −−
=
= = ∑
{ } IFDMAl X ,
~
0 0 00 0 0 0 0 X 0 X 1 X 2 X 3{ } DFDMAl X ,
~
{ } LFDMAl X ,
~
M = Q N
12= 34
M > Q N
12> 24
M = Q N
12= 34
*M : Total number of subcarriers, N : Data block size, Q : Bandwidth spreading factor
Figure 3.5: An example of different subcarrier mapping schemes for N = 4, Q = 3 and M = 12.
subcarriers
Terminal 1
Terminal 2
Terminal 3
subcarriers
Distributed Mode Localized Mode
Figure 3.6: Subcarrier allocation methods for multiple users (3 users, 12 subcarriers, and 4 subcarriers
allocated per user).
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spread over the entire bandwidth. With distributed mapping, CDS incrementally improves
performance. By contrast, CDS is of great benefit with localized subcarrier mapping because
it provides significant multiuser diversity. We will discuss this aspect in more detail in chapter
7.
3.3. Time Domain Representation of SCFDMA Signals
We derive the time domain symbols without pulse shaping for each subcarrier mapping
scheme. In the subsequent derivations, we will follow the notations in Figure 3.3.
3.3.1. Time Domain Symbols of IFDMA
For IFDMA, the frequency samples after subcarrier mapping { }l X ɶ can be described as follows.
/ , (0 1)
0 , otherwise
l Q
l
X l Q k k N X
= ⋅ ≤ ≤ −=
ɶ (3.3)
where 0 1l M ≤ ≤ − and M Q N = ⋅ .
Let m N q n= ⋅ + ( 0 1q Q≤ ≤ − , 0 1n N ≤ ≤ − ). Then,
( )
( )mod
1 1 22
0 0
1 2
0
1 2
0
1 1 1
1 1
1 1
1 1 N
mm M N j k j l N M
m Nq n l k
l k
Nq n N j k
N k
k
n N j k
N k
k
n m
x x X e X e M Q N
X eQ N
X eQ N
x xQ Q
π π
π
π
− −
+= =
+−
=
−
=
= = = ⋅
= ⋅
= ⋅
= =
∑ ∑
∑
∑
ɶɶ ɶ
(3.4)
The resulting time symbols { }m xɶ are simply a repetition of the original input symbols
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{ }n x with a scaling factor of 1/Q in the time domain.
When the subcarrier allocation starts from r
th
subcarrier ( 0 1r Q< ≤ − ), then,
/ , (0 1)
0 , otherwise
l Q r
l
X l Q k r k N X
− = ⋅ + ≤ ≤ −=
ɶ (3.5)
( )
( )mod
1 1 22
0 0
1 2 2
0
1 22
0
2 2
1 1 1
1 1
1 1
1 1
N
m mr m M N j k j l N M M
m Nq n l k
l k
Nq n mr N j k j N M
k
k
n mr N j k j N M
k
k
mr mr j j
M M n m
x x X e X e M Q N
X e eQ N
X e eQ N
e x e xQ Q
π π
π π
π π
π π
− − +
+= =
+−
=
−
=
= = = ⋅
= ⋅
= ⋅ ⋅
= ⋅ = ⋅
∑ ∑
∑
∑
ɶɶ ɶ
(3.6)
Thus, there is an additional phase rotation of 2
mr j
M eπ
when the subcarrier allocation
starts from r th subcarrier instead of subcarrier zero. This phase rotation will also apply to the
other subcarrier mapping schemes as well in the same case.
3.3.2. Time Domain Symbols of LFDMA
For LFDMA, the frequency samples after subcarrier mapping { }l
X ɶ can be described as
follows.
, 0 1
0 , 1
ll
X l N X
N l M
≤ ≤ −=
≤ ≤ −
ɶ (3.7)
Let m Q n q= ⋅ + , where 0 1n N ≤ ≤ − and 0 1q Q≤ ≤ − . Then,
1 1 22
0 0
1 1 1Qn qm M N j l j lQN M
m Qn q l l
l l
x x X e X e M Q N
π π +− −
+= =
= = = ⋅∑ ∑ɶɶ ɶ (3.8)
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/ , (0 1)
0 , otherwise
l Q
l
X l Q k k N X
= ⋅ ≤ ≤ −=
ɶɶ
ɶ (3.11)
where 0 1l M ≤ ≤ − , M Q N = ⋅ , and 1 Q Q≤ <ɶ . Let m Q n q= ⋅ + ( 0 1n N ≤ ≤ − , 0 1q Q≤ ≤ − ).
Then,
1 2
0
1 2
0
1( )
1 1
m M j l
M m Q n q l
l
Qn q N j Qk QN
k
k
x x X e M
X eQ N
π
π
−
⋅ +=
+−
=
= = ⋅
= ⋅
∑
∑ɶ
ɶɶ ɶ
(3.12)
If q = 0, then,
( )
( ) ( )( )
mod
modmod mod
1 12 2
0 0
1 12 2
0 0
1 1 1 1
1 1 1 1
1 1
N
N N N
Qn n N N j Qk j Qk QN N
m Q n k k
k k
Q nQ n N N j k j k N N
k k
k k
Q n Q m
x x X e X eQ N Q N
X e X eQ N Q N
x x
Q Q
π π
π π
− −
⋅= =
⋅⋅− −
= =
⋅
= = ⋅ = ⋅
= ⋅ = ⋅
= ⋅ = ⋅
∑ ∑
∑ ∑
ɶ ɶ
ɶɶ
ɶ ɶ
ɶ ɶ
(3.13)
If q ≠ 0, since1 2
0
p N j k
N k p
p
X x eπ − −
=
= ∑ , (3.12) can be expressed as follows after derivation.
( )
12
0 2
1 11
1
Q N j q pQ
m Q n q Qn p Qq p j
N QN
x x x e
Q N
e
π
π
−
⋅ + − = +
= = − ⋅
−
∑ɶ
ɶ ɶɶ ɶ (3.14)
It is interesting to see that the time domain symbols of DFDMA have the same structure
as those of LFDMA.
Figure 3.7 shows an example of the time symbols for each subcarrier mapping mode
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x0 x1 x2 x3
x0
x1
x2
x3
{ }n x
x0
x1
x2
x3
x0
x1
x2
x3
* * * * * * * * x0
x2
x0
x2
time
* * * * * * * * x0 x1 x2 x3
{ },m IFDMAQ x⋅ ɶ
{ },m DFDMAQ x⋅ ɶ
{ },m LFDMAQ x⋅ ɶ
3
, ,
0
* , : complex weightk m k k m
k
c x c=
= ⋅∑
Figure 3.7: Time symbols of different subcarrier mapping schemes.
10 20 30 40 50 600
0.1
0.2
0.3
0.4
0.5
Symbol
A m p l i t u d e [ l i n e a r ]
IFDMA
LFDMA
DFDMA
Figure 3.8: Amplitude of SCFDMA signals.
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based on Figure 3.5 for M = 12, N = 4, Q = 3, and Qɶ = 2. Figure 3.8 shows the amplitude of
the signal for each subcarrier mapping for M = 64, N = 16, Q = 4, and Qɶ = 3 and we can see
more fluctuation and higher peak for LFDMA and DFDMA.
3.4. SCFDMA and OFDMA
Figure 3.1 includes a block diagram of an OFDMA transmitter. It has much in common with
SCFDMA. The only difference is the presence of the DFT in SCFDMA. For this reason SC
FDMA is sometimes referred to as DFTspread or DFTprecoded OFDMA. Other similarities
between the two include: blockbased data modulation and processing, division of the
transmission bandwidth into narrower subbands, frequency domain channel equalization
process, and the use of CP.
However, there are distinct differences that make the two systems perform differently as
illustrated in Figure 3.9. In terms of data detection at the receiver, OFDMA performs it on a
persubcarrier basis whereas SCFDMA does it after additional IDFT operation. Because of
this difference, OFDMA is more sensitive to a null in the channel spectrum and it requires
channel coding or power/rate control to overcome this deficiency. Also, the duration of the
modulated time symbols are expanded in the case of OFDMA with parallel transmission of
the data block during the elongated time period whereas SCFDMA modulated symbols are
compressed into smaller chips with serial transmission of the data block, much like a direct
sequence code division multiple access (DSCDMA) system.
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Subcarrier
Demapping
Equalizer
Equalizer
Equalizer
SubcarrierDe
mapping
Detect
Detect
Detect
Equalizer IDFT DetectSCFDMA
OFDMA DFT
DFT
(a)
OFDMA symbol
SCFDMA symbols*
Input data symbols
* Bandwidth spreading factor : 4 time
(b)
Figure 3.9: Dissimilarities between OFDMA and SCFDMA; (a) different detection processes in
the receiver; (b) different modulated symbol durations.
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3.5. SCFDMA and DSCDMA/FDE
Direct sequence code division multiple access (DSCDMA) with FDE is a technique that
replaces the rake combiner, commonly used in the conventional DSCDMA, with the
frequency domain equalizer [29]. A rake receiver consists of a bank of correlators, each of
which correlate to a particular multipath component of the desired signal. As the number of
multipaths increase, the frequency selectivity in the channel also increases and the complexity
of the rake combiner increases since more correlators are needed. The use of FDE instead of
rake combing can alleviate the complexity problem in DSCDMA. Block diagram of DS
CDMA/FDE is shown in Figure 3.10.
Spreading Channel M 
pointIDFT
Equalization M 
pointDFT
DetectRemove
CP{ }n x
Add
CP/PS
Despreading
Figure 3.10: DSCDMA with FDE.
The transmitter of DSCDMA/FDE is the same as the conventional DSCDMA except
for the addition of CP. The FDE in the receiver removes the channel distortion from the
received chip symbols to recover ISIfree chip symbols. For small spreading factors, the link
performance of the rake receiver significantly degrades because of the interpath interference
and FDE has a much better performance. For large spreading factors, both have similar
performances. More technical details of the DSCDMA/FDE system can be found in [29].
SCFDMA is similar to DSCDMA/FDE in terms of the following aspects:
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• Both spread narrowband data into broader band.
• They achieve processing gain or spreading gain from spreading.
• They both maintain low PAPR because of the single carrier transmission.
An interesting relationship between orthogonal DSCDMA and IFDMA is that by
exchanging the roles of spreading sequence and data sequence, DSCDMA modulation
becomes IFDMA modulation [30], [31]. An example of this observation is illustrated in Figure
x1 x2 x3
time
1 1 1
x1 x1 x1 x1 x2 x2 x2 x2 x3 x3 x3 x3
1
x0
x1
x2
x3
1 1
1 1 1 1 1 1 1 1 1
x0
x1
x2
x3
x0
x1
x2
x3
x0 x1 x2 x3 x0 x1 x2 x3 x0 x1 x2 x3
Signature Sequence
Data Sequence ××××
××××
Data Sequence
Signature Sequence
time
Conventional
Spreading
Exchanged
Spreading
x0 x0 x0 x0
1 1 1 1
x0
x1
x2
x3
x0 x1 x2 x3
1
x0
Figure 3.11: Spreading with the roles of data sequence and signature sequence exchanged for spreading
signature of {1, 1, 1, 1} with a data block size of 4.
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3.11. We could see that the result of the spreading with exchanged roles is in the form of
IFDMA modulated symbols in Figure 3.7.
One advantage of SCFDMA over DSCDMA/FDE is that channel dependent resource
scheduling is possible to exploit frequency selectivity of the channel.
3.6. SCFDMA Implementation in 3GPP LTE Uplink
In this section, we describe the physical layer implementation of SCFDMA in 3GPP LTE
frequency division duplex (FDD) uplink according to [10].
3.6.1. Modulation and Channel Coding
Available data modulation schemes are π/2offset BPSK, QPSK, 8PSK, and 16QAM. Turbo
code based on 3GPP UTRA Release 6 is used for forward error correcting (FEC) code.
3.6.2. Subframe Structure
In 3GPP LTE, the basic unit of a transmission is a subframe. Figure 3.12 shows the basic
subframe structure in the time domain.
LB #1SB
#1 C P
C P LB #2 C
P LB #3 C P LB #4 C
P LB #5 C P SB
#2 C P LB #6 C
P
1 subframe = 0.5 ms
Figure 3.12: Basic subframe structure in the time domain.
A subframe which has duration of 0.5 ms consists of six long blocks (LB) and two short
blocks (SB). Cyclic prefix (CP) is added in front of each block. Long blocks are used for
control and/or data transmission and short blocks are used for reference (pilot) signals for
coherent demodulation and/or control/data transmission. Both localized and distributed
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subcarrier mapped data use the same subframe structure. The minimum transmission time
interval (TTI) for uplink is equal to the subframe duration, 0.5 ms. It is possible to
concatenate multiple subframes into longer uplink TTI’s and the TTI configuration can be
either semistatic or dynamic in such case. In 3GPP LTE uplink, subcarriers in the guard band
area are intentionally set to zero amplitude to maintain similar subcarrier structure as that of
downlink OFDMA, which is illustrated in Figure 3.13.
Frequency (Subcarrier)
Total Bandwidth ( M subcarriers)
Occupied Subcarriers
f c
Figure 3.13: Physical mapping of a block in RF frequency domain (f c : carrier center frequency).
Figure 3.14 illustrates the generation of a block and Table 3.1 shows the numerology for
different spectrum allocations.
SerialtoParallel
M 
pointIDFT
N 
pointDFT
Zeros
{ }0 1 1, , N
x x x −… ParalleltoSerial
{ }0 1 1, , M x x x −ɶ ɶ ɶ…
SubcarrierMapping
s
u b c a r r i e r
0
M  1
Zeros
Figure 3.14: Generation of a block.
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Table 3.1: Parameters for uplink SCFDMA transmission scheme in 3GPP LTE.
Bandwidth(MHz)
Subframe
duration
(ms)
LB size (µµµµs/# of
occupied subcarriers
/FFT size)
SB size (µµµµs/# of
occupied subcarriers
/FFT size)
CP duration
(µµµµs/# of
subcarriers)
20 0.5 66.67/1200/2048 33.33/600/1024(4.13/127)
or (4.39/135)
15 0.5 66.67/900/1536 33.33/450/768(4.12/95)
or (4.47/103)
10 0.5 66.67/600/1024 33.33/300/512(4.1/63)
or (4.62/71)
5 0.5 66.67/300/512 33.33/150/256(4.04/31)
or (5.08/39)
2.5 0.5 66.67/150/256 33.33/75/128(3.91/15)
or (5.99/23)
1.25 0.5 66.67/75/128 33.33/38/64(3.65/7)
or (7.81/15)
Note that the first CP in the subframe has longer duration to include ramp up and ramp
down time.
For data and control transmission, we can organize the subcarrier resources in the
frequency domain into a number of resource units (RU). In other words, we can assign the
subcarriers in groups or chunks of certain number rather than individually. Each RU consists
of a number of localized or distributed subcarriers within a long block. Table 3.2 gives the
number of RU’s and the number of subcarriers per RU for different bandwidth allocations.
The number of subcarriers per RU may change in the future based on the outcome of the
current interference coordination study. One or more RU’s can be assigned to a user. When
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more than one localized RU’s are assigned to a user, they should be contiguous in frequency
domain and when more than one distributed RU’s are assigned, the subcarriers assigned
should be equally spaced.
Table 3.2: Number of RU’s and number of subcarrriers per RU for LB.
Bandwidth (MHz) 1.25 2.5 5.0 10.0 15.0 20.0
Number of available subcarriers 75 150 300 600 900 1200
Number of available RU’s 3 6 12 24 36 48
Number of subcarriers per RU 25 25 25 25 25 25
3.6.3. Reference (Pilot) Signal Structure
Uplink reference signals are transmitted within the two short blocks as mentioned in 3.6.2.
They are received and used at the base station for uplink channel estimation in coherent
demodulation/detection and for possible uplink channel quality estimation for uplink channel
dependent scheduling. We should note that the bandwidth of each subcarrier for the reference
signal is twice that for the data signal since the short block has half the length of a long block.
We can construct the reference signal using CAZAC (Constant Amplitude Zero Auto
Correlation) sequences such as ZadoffChu polyphase sequences. ZadoffChu CAZAC
sequences are defined as
2
2 , 0,1,2, , 1; for even2
( 1)2 , 0,1,2, , 1; for odd
2
k
r k
j qk k L L L
r k k j qk k L L
L
ea
e
π
π
− + = −
+ − + = −
=
⋯
⋯
(3.15)
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where r is any integer relatively prime to L and q is any integer [32].
The set of ZadoffChu CAZAC sequences has the following properties:
• Constant amplitude.
• Zero circular autocorrelation.
• Flat frequency domain response.
• Circular crosscorrelation between two sequences is low and it has constant
magnitude provided that L is a prime number.
Orthogonality among uplink reference signals can be achieved using the following
methods:
• Frequency division multiplexing (FDM): Each uplink reference signal is
transmitted across a distinct set of subcarriers. This solution achieves signal
orthogonality in the frequency domain.
• Code division multiplexing (CDM): Reference signals are constructed that are
orthogonal in the code domain with the signals transmitted across a common set
of subcarriers. As an example, individual reference signals may be distinguished
by a specific cyclic shift of a single CAZAC sequence.
• Time division multiplexing (TDM): Each user will transmit reference signal in
different short blocks. Orthogonality is achieved in the time domain.
• A combination of the methods above.
Figure 3.15 illustrates an example of FDM and CDM pilots.
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subcarriers
User 1
User 2
User 3
subcarriers
FDM Pilots CDM Pilots
Figure 3.15: FDM and CDM pilots for three simultaneous users with 12 total subcarriers.
3.7. Summary and Conclusions
Single carrier FDMA (SCFDMA) is a new multiple access scheme that is currently being
adopted in 3GPP LTE uplink. SCFDMA utilizes single carrier modulation and frequency
domain equalization, and has similar performance and essentially the same overall complexity
as those of OFDMA system. A salient advantage over OFDMA is that the SCFDMA signal
has lower PAPR because of its inherent single carrier transmission structure. SCFDMA has
drawn great attention as an attractive alternative to OFDMA, especially in the uplink
communications where lower PAPR greatly benefits the mobile terminal in terms of transmit
power efficiency and manufacturing cost.
SCFDMA has two different subcarrier mapping schemes; distributed and localized. In
distributed subcarrier mapping scheme, user’s data occupy a set of distributed subcarriers and
we achieve frequency diversity. In localized subcarrier mapping scheme, user’s data inhabit a
set of consecutive localized subcarriers and we achieve frequencyselective gain through
channeldependent scheduling (CDS). The two flavors of subcarrier mapping schemes give
the system designer a flexibility to adapt to the specific needs and requirements.
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SCFDMA also bears similarities to direct sequence CDMA (DSCDMA) with FDE in
terms of; bandwidth spreading, processing gain from spreading, and low PAPR because of the
single carrier transmission. An advantage of SCFDMA over DSCDMA/FDE is the ability to
utilize channel dependent resource scheduling to exploit frequency selectivity of the channel.
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Chapter 4Chapter 4Chapter 4Chapter 4
MIMO SCMIMO SCMIMO SCMIMO SCFDMAFDMAFDMAFDMA
Multiple input multiple output (MIMO) techniques gathered much attention in recent years as a
way to drastically improve the performance of wireless mobile communications. MIMO
technique utilizes multiple antenna elements on the transmitter and the receiver to improve
communication link quality and/or communication capacity [33]. A MIMO system can provide
two types of gain; spatial diversity gain and spatial multiplexing gain [34]. Spatial diversity improves
the reliability of communication in fading channels and spatial multiplexing increases the
capacity by sending multiple streams of data in parallel through multiple spatial channels.
Transmit eigenbeamforming (TxBF) with unitary precoding is a spatial multiplexing technique
that utilizes the eigenstructure of the channel to generate independent spatial channels.
Practical considerations of TxBF that affect the performance and the overhead of the system
are precoder quantization/averaging and feedback delay.
In this chapter, we first give a general overview of MIMO concepts. In section 4.2, we
describe the parallel decomposition of a MIMO channel for narrowband and wideband
transmission. We also illustrate the realization of MIMO spatial multiplexing in SCFDMA. In
section 4.3, we introduce the SCFDMA TxBF with unitary precoding technique and
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numerically analyze the link level performance with practical considerations.
4.1. Spatial Diversity and Spatial Multiplexing in MIMO Systems
The basic idea behind spatial diversity techniques is to combat channel fading by having multiple
copies of the transmitted signal go through independently fading propagation paths. At the
receiver, multiple independently faded replicas of the transmitted data signal are coherently
combined to achieve a more reliable reception. Spatial diversity gain is the SNR exponent of the
error probability and intuitively, it corresponds to the number of independently faded paths. It is
well known that in a system with N t transmit antennas and N r receive antennas, the maximal
diversity gain is N t ⋅ N r assuming the path gains between the individual transmit and receive
antenna pairs are independently and identically distributed (i.i.d.) Rayleighfaded [35]. Smart
antenna techniques [36], Alamouti transmit diversity scheme [37], and spacetime coding [38] are
some of well known spatial diversity techniques.
If spatial diversity is a means to combat fading, spatial multiplexing is a way to exploit
fading to increase the data throughput. In essence, if the path gains among individual transmit
receive antenna pairs fade independently such that the channel matrix is wellconditioned,
multiple parallel spatial channels can be created and data rate can be increased by transmitting
multiple streams of data through the spatial channels. Spatial multiplexing is especially important
in the highSNR regime where the system is degreeoffreedom limited as opposed to power
limited in the lowSNR regime. A scheme achieves a spatial multiplexing gain of r if the
supported data rate approaches r ⋅log( SNR ) (bits/s/Hz) and we can think of the multiplexing
gaing as the total number of degrees of freedom to communicate. In a system with N t transmit
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antennas and N r receive antennas for high SNR, Foschini [39] and Telatar [40] showed that the
capacity of the channel increases linearly with m = min( N t , N r ) if the channel gains among the
antenna pairs are i.i.d. Rayleighfaded. Thus there is m number of parallel channels or m number
of degrees of freedom. BLAST (Bell Labs SpaceTime Architecture) [39] is a typical spatial
multiplexing technique.
For a given MIMO channel, both diversity and multiplexing gains can be achieved
simultaneously but there is a fundamental tradeoff between the two gains. For example, Zheng
and Tse [41] showed that the optimal diversity gain achievable by any coding scheme of block
length larger than N t + N r − 1 with multiplexing gain m (integer) is precisely ( N t − m)⋅( N r − m)
for i.i.d. Rayleigh slowlyfading channel with N t transmit and N r receive antennas. This implies
that out of the total resource of N t transmit and N r receive antennas, it is as though m transmit
and m receive antennas were used for multiplexing and the remaining N t − m transmit and N r −
m receive antennas provided the diversity. In summary, higher spatial diversity gain comes at the
price of lowering spatial multiplexing gain and vice versa.
4.2. MIMO Channel
First, we consider the narrowband MIMO channel. A pointtopoint narrowband MIMO
channel with N t transmit antennas and N r receive antennas is illustrated in Figure 4.1. We can
represent such a MIMO system by the following discretetime model.
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1 x
2 x
t N x
1 y
2 y
r N y
11h
21h1r N h
r t N N h
11 1
1
t
r r t
N
N N N
h h
H
h h
=
⋯
⋮ ⋱ ⋮
⋯
Figure 4.1: Description of a MIMO channel with N t transmit antennas and N r receive antennas.
11 11 1 1
1
t
r r r t t r
N
N N N N N N
y n x H
h h y x n
y h h x n
y H x n
= ===
= ⋅ +
⇒ = ⋅ +
⋯
⋮ ⋮ ⋱ ⋮ ⋮ ⋮⋯
(4.1)
where x is the N t ×1 transmitted signal vector, y is the N r ×1 received signal vector, n is
the N r ×1 zeromean complex Gaussian noise, and H is the N r × N t complex matrix of channel
gains hij representing the gain from jth
transmit antenna to ith
receive antenna.
By singular value decomposition (SVD) theorem, channel matrix H can be decomposed as
follows.
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H H UDV = (4.2)
whereU
is an N
r × N
r unitary matrix,V
is an N
t × N
t unitary matrix, D
is an N
r × N
t nonnegative
diagonal matrix, and ( ) H i is a Hermitian (conjugate transpose) operation. The diagonal entries
of D are the nonnegative square roots of the eigenvalues of H HH , the columns of U are the
eigenvectors of H HH , and the columns of V are the eigenvectors of H H H .
We can rewrite (4.1) using (4.2) as
H y UDV x n= + (4.3)
Multiplying H U on both sides of (4.3), it becomes
H H H H
I
H H H
U y U U DV x U n
U y DV x U n
=
= +
= +(4.4)
Let H y U y=ɶ , H x V x=ɶ , and H n U n=ɶ , then,
y Dx n= +ɶ ɶ ɶ (4.5)
where nɶ has the same statistical properties as n since U is a unitary matrix. Thus nɶ is an N r
×1 zeromean complex Gaussian noise.
We can see from (4.5) that the MIMO transmission can be decomposed into up to m =
min( N t , N r ) parallel independent transmissions, which is the basis for the spatial multiplexing
gain.
For a wideband MIMO channel, the entire band can be subdivided into subbands, or
subcarriers. Then, the MIMO channel for each subcarrier becomes a narrowband MIMO
channel. Let the entire band be subdivided in to M subcarriers, then for k th subcarrier
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( 0 1k M ≤ ≤ − ),
11, 1 ,1, 1, 1,
, 1, , , ,
t
r r r t t r
k k k k
k N k k k k
N k N k N N k N k N k
Y N X H
k k k k
H H Y X N
Y H H X N
Y H X N
= ===
= ⋅ +
⇒ = ⋅ +
⋯
⋮ ⋮ ⋱ ⋮ ⋮ ⋮
⋯
(4.6)
where k X is the N t ×1 transmitted signal vector, k
Y is the N r ×1 received signal vector, k N is
the N r ×1 zeromean complex Gaussian noise, and H k is the N r × N t complex matrix of channel
gains H ij,k representing the gain from jth transmit antenna to ith receive antenna. Similarly with a
narrowband MIMO channel, H k can be decomposed into H
k k k U D V and (4.6) can be expressed
as
k k k k Y D X N = +ɶ ɶ ɶ (4.7)
where H
k k k Y U Y =ɶ , H
k k k X V X =ɶ , and H
k k k N U N =ɶ .
We can use spatial multiplexing MIMO techniques in SCFDMA by applying the spatial
processing on a subcarrierbysubcarrier basis, somewhat similar to MIMOOFDM [42], in the
frequency domain after DFT. A block diagram of a spatial multiplexing MIMO SCFDMA
system is shown in Figure 4.2.
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SubcarrierMapping
MIMOChannel
N pointIDFT
Subcarrier
Demapping
M point
DFTDetect
RemoveCP
N pointDFT
M pointIDFT
Add CP /PS
DAC
/ RF
RF
/ ADC
S p a t i a l M a p
p i n g
SubcarrierMapping
N pointDFT
M pointIDFT
Add CP /PS
DAC
/ RF
S p a t i a l C o m b i n i n g
/ E q u a l i z a t i o n
N pointIDFT
Subcarrier
Demapping
M point
DFTDetect
RemoveCP
RF
/ ADC
Figure 4.2: Block diagram of a spatial multiplexing MIMO SCFDMA system.
4.3. SCFDMA Transmit EigenBeamforming with Unitary Precoding
Transmit eigenbeamforming (TxBF) with unitary precoding utilizes the eigenstructure of the
channel to achieve spatial multiplexing [43], [44]. Figure 4.3 illustrates a simpified block diagram
of a unitary precoded TxBF SCFDMA MIMO system. We describe below the basic processes
of unitary precoded TxBF system. We assume the channel is timeinvariant and the channel
estimation is perfect, and we follow the notations in (4.6).
First, the MIMO channel matrix H k for subcarrier k is decomposed as follows using SVD
for each subcarrier at the receiver.
H
k k k k H U D V = (4.8)
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Unitary Precoding
MIMO Channel H k
Receiver(LMMSE)
Feedback Processing:
(Averaging& quantization of V k’s)
⊕
k X ˆ
k k k X V X =ɶ
k k H X
k N
k Y
k Z
( )⋅F ˆ ( )k k V F V =
Figure 4.3: Simpified block diagram of a unitary precoded TxBF SCFDMA MIMO system.
We feedback V k to the transmitter and use it as the precoding matrix for transmission. We
may quantize and average the V k ’s at the receiver to reduce feedback overhead. We denote the
quantization and averaging processes as F (⋅) and ( )ˆk k V F V = . The quantized and averaged ˆ
k V ’s
are then sent to the transmitter via the feedback channel. The transmitter precodes the data
signalk
X with ˆk
V as follows.
ˆk k k X V X =ɶ (4.9)
After the transmitted signal propagates through the MIMO channel H k , the received signal
k Y is represented as follows.
k k k k Y H X N = +ɶ (4.10)
At the receiver, we perform linear MMSE (LMMSE) estimation for joint equalization and
MIMO detection. Let k k k Z A Y = where k A is an N t × N r complex matrix. Then, our goal is to
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findk
A such that mean square error betweenk
X andk
Z ( { }2
k k E X Z − ) is minimum. By
applying orthogonality principal, we obtain the following solution for k A .
( )1
, , ,ˆ ˆ ˆ
H H
k XX k k k XX k k NN k A R H H R H R
−
= + (4.11)
where ˆ ˆk k k H H V = , ,
H
XX k k k R X X = , and ,
H
NN k k k R N N = .
There are two practical factors that impact the performance of precoded TxBF; precoder
quanization/averaging and feedback delay. Since the receiver feedbacks the precoding matrix to
the transmitter, the receiver often quantizes and also averages across all subcarriers the
precoding matrix to reduce feedback overhead. We could expect some degradation in
performance because we lose some of precoder information during quantization and averaging
processes. Feedback delay becomes a major factor in the performance of precoded TxBF when
the channel is changing very fast since the precoder is extracted directly from the current
channel matrix.
In the following sections, we give numerical analysis for a 2x2 uplink MIMO system to
quantify the impact of imperfect feedback on the link level performance. We base the simulation
setup on 3GPP LTE uplink and we use the following simulation parameters and assumptions.
• Carrier frequency: 2.0 GHz.
• Transmission bandwidth: 5 MHz.
• Symbol rate: 7.68 million symbols/sec.
• TTI length: 0.5 ms.
• Number of blocks per TTI: 6 long blocks (LB).
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• Number of occupied subcarriers per LB: 128.
• FFT block size: 512.
• Cyclic Prefix (CP) length: 32 symbols.
• Subcarrier mapping: Distributed.
• Pulse shaping: Time domain squaredroot raisedcosine filter (rolloff factor = 0.22)
with 2x oversampling.
• Channel model: 3GPP SCME C quasistatic Rayleigh fading with mobile speed of 3
km/h [45]. We assume the channel is constant during the duration of a block.
• Antenna configurations: 2 transmit and 2 receive antennas
• Data modulation: 16QAM for 1st stream and QPSK for 2nd stream.
• Channel coding: 1/3rate turbo code.
• We use linear MMSE equalizer described in (4.11).
• We assume no feedback error and perfect channel estimation at the receiver.
4.3.1. Impact of Imperfect Feedback: Precoder Quantization/Averaging
We directly quantize the 2x2 precoding matrix V k in the form of Jacobi rotation matrix which is
a unitary matrix [46]. We use the following matrix to quantize V k .
ˆ
ˆ
ˆ ˆcos sinˆ
ˆ ˆsin cos
k
k
j
k k
k j
k k
eV
e
φ
φ
θ θ
θ θ
− ⋅=
⋅ (4.12)
where for Bamp bits to quantize the amplitude component,
( )ˆ 2 1 ; 1, , 24 2
amp
amp
B
k Bl l
π θ
∈ − =
⋅ ⋯ (4.13)
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0 1 2 3 4 5 6 7 8 9 10 11 1210
4
103
102
101
100
SNR [dB]
F E R
FER, 2x2 TxBF, SCME C 3km/h, 16QAMQPSK 1/3, Ideal ChEst, No Feedback Delay, Jacobi Direct Quant.
No avr., no quant.
No avr., phaseonly(2 bits) quant.No avr., amp.(1 bit)phase(2 bits) quant.
25 SCs avr., phaseonly(2 bits) quant.
25 SCs avr., amp.(1 bit)phase(2 bits) quant.
12 SCs avr., phaseonly(2 bits) quant.
12 SCs avr., amp.(1 bit)phase(2 bits) quant.
Figure 4.5: FER performance of a 2x2 SCFDMA unitary precoded TxBF system with feedback
averaging and quantization.
Figure 4.6 (a) shows the effect of different quantization bit resolutions (no averaging) and
Figure 4.6 (b) shows the effect of different averaging sizes (1bit amplitude and 2bit phase
quantization). As we use more bits and smaller averaging size, we see less performance
degradation.
Table 4.1 summarizes the tradeoff between feedback overhead and performance loss based
on the results in Figure 4.5. Performance loss is the SNR level increase at 1% FER point with
regards to that of no averaging and no quantization result. As we increase the feedback overhead,
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0 1 2 3 4 5 6 7 8 9 10 11 12
104
103
102
101
100
SNR per Antenna [dB]
F E R
FER, 2x2 TxBF, SCME C 3km/h, 16QAMQPSK 1/3, Ideal ChEst, No Feedback Delay, Jacobi Direct Quant., No avr.
No quant.
Phaseonly (2 bits) quant.
Amp. (1 bit)phase (2 bits) quant.
(a)
0 1 2 3 4 5 6 7 8 9 10 11 1210
4
103
102
101
100
SNR per Antenna [dB]
F E R
FER, 2x2 TxBF, SCME C 3km/h, 16QAMQPSK 1/3, Ideal ChEst, No Feedback Delay, Jacobi Direct Quant.
No avr., no quant.
12 SCs avr., quant.
25 SCs avr., quant.
50 SCs avr., quant.
100 SCs avr., quant.
* 1 bit amp. / 2 bit phase quantization
(b)
Figure 4.6: FER performance; (a) with different quantization bit resolutions (no averaging); (b) with
different averaging sizes (1bit amplitude and 2bit phase quantization).
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that is, increase the amount of precoding information, there is less performance loss.
Table 4.1. Summary of feedback overhead vs. performance loss.
Averaging size
(subcarriers per RU)
# of
RU’s
Quantization
resolution (bits)
Feedback
overhead
Performanc
e loss
25 12 Phase: 2, Amp.: none 2 bits × 12 = 24 bits 1.3 dB
25 12 Phase: 2, Amp.: 1 3 bits × 12 = 36 bits 0.6 dB
12 25 Phase: 2, Amp.: none 2 bits × 25 = 50 bits 1 dB
12 25 Phase: 2, Amp.: 1 3 bits × 25 = 75 bits 0.5 dB
4.3.2. Impact of Imperfect Feedback: Feedback Delay
Delay is inevitable in any realistic feedback system. When the channel is changing fast due to the
user’s motion or change of the environment, the precoder may be outdated and it will affect the
receiver performance. Figure 4.7 shows the impact of feedback delay for different user speeds.
We consider feedback delays of 2, 4, and 6 TTI’s with no feedback quantization or averaging.
We can see from Figure 4.7 (a) that at low speed of 3 km/h, there is only a trivial loss of
performance even for the longest delay of 6 TTI’s. But as the user speed becomes faster, we can
see the performance degradation become more severe as illustrated in Figure 4.7 (b). Clearly, in
high mobility and long feedback delay situations, TxBF with unitary precoding does not perform
well at all.
4.4. Summary and Conclusions
MIMO technique utilizes multiple antenna elements on the transmitter and the receiver to
improve communication link quality and/or communication capacity and it has two aspects in
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terms of performance improvement; spatial diversity and spatial multiplexing. Spatial diversity
improves the reliability of communication in fading channels and spatial multiplexing increases
the capacity by sending multiple streams of data in parallel through multiple spatial channels.
Transmit eigenbeamforming (TxBF) with unitary precoding is a spatial multiplexing
technique that utilizes the eigenstructure of the channel to generate independent spatial
channels. Practical considerations of TxBF that affect the performance and the overhead of the
system are precoder quantization/averaging and feedback delay. Quantizing and averaging the
precoding matrix reduces the feedback overhead but the link performance suffers.
In this chapter, we gave an overview of the realization of MIMO spatial multiplexing in SC
FDMA. We specifically analyzed the unitary precoded TxBF technique on an SCFDMA system.
0 2 4 6 8 1010
4
103
102
101
100FER, 2x2 TxBF, SCME C 3km/h, 16QAMQPSK 1/3, Ideal ChEst, No Quant.
F E R
SNR per Antenna [dB]
No delay
2 TTI delay
4 TTI delay
6 TTI delay
0 5 10 15 20
104
103
102
101
100
SNR per Antenna [dB]
F E R
FER, 2x2 TxBF, SCME C 150km/h, 16QAMQPSK 1/3, Ideal ChEst, No Quant.
No delay
2 TTI delay
4 TTI delay
6 TTI delay
(a) (b)
Figure 4.7: FER performance of a 2x2 SCFDMA unitary precoded TxBF system with feedback
delays of 2, 4, and 6 TTI’s; (a) user speed of 3 km/h; (b) user speed of 150 km/h.
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We showed with numerical simulations that there is a tradeoff between the feedback overhead
and the link performance loss. We also showed that feedback delay is another impairment that
degrades the performance especially for the high speed mobile users.
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Chapter 5Chapter 5Chapter 5Chapter 5
PeakPeakPeakPeak Power CharacPower CharacPower CharacPower Characteristics of an SCteristics of an SCteristics of an SCteristics of an SCFDMA Signal:FDMA Signal:FDMA Signal:FDMA Signal:
Analytical AnalysisAnalytical AnalysisAnalytical AnalysisAnalytical Analysis
In this chapter, we analytically characterize the peak power of SCFDMA modulated signals
using the Chernoff bound. We will analyze the peak power characteristics numerically in
chapter 6. In [47], Wulich and Goldfeld showed that the amplitude of a single carrier (SC)
modulated signal does not have a Gaussian distribution and that it is difficult to analytically
derive the exact form of the distribution. As an alternative, they explained a way to derive an
upper bound for the complementary distribution of the instantaneous power using the
Chernoff bound. We follow their derivation and apply it to SCFDMA modulated signals to
characterize the peak power analytically.
The peak power of any signal x(t ) is the maximum of its squared envelope2
( ) x t .
However, for a continuous random process,2
max ( ) x t could be unbounded. Even for
random process with discrete values where
2
max ( ) x t is bounded, it may occur at very low
probability which is not very useful in practice. Rather, the distribution of 2
( ) x t is more
useful and we describe it with the complementary cutoff probability. For a given w, cutoff
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probability is defined as { ( )2
2
( )Pr ( )
x t x t w F w≤ = , where ( )2
( ) x t F w is the cumulative
distribution function (CDF) of 2
( ) x t , and the complementary cutoff probability is
{ ( )2
2
( )Pr ( ) 1
x t x t w F w≥ = − . { }
2Pr ( ) x t w≥ is also referred to as complementary
cumulative distribution function (CCDF).
In this chapter, we first derive an upperbound of the CCDF of the instantaneous power
for interleaved FDMA (IFDMA) with pulse shaping and show the bounds for BPSK and
QPSK with raisedcosine pulse shaping filter. In section 5.2, we show the modified upper
bound of the CCDF of the instantaneous power for localized FDMA (LFDMA) without
pulse shaping. In section 5.3, we compare the results of sections 5.1 and 5.2 with the analytical
CCDF of PAPR for OFDM.
5.1. Upper Bound for IFDMA with Pulse Shaping
In chapter 3 section 3.3.1, we derived the time domain representation of the IFDMA symbols
and showed that it is a repetition of the original input symbols with a scaling factor of 1/ Q.
Since the IFDMA signal has the same structure as the conventional SC modulation signal, we
first derive the upper bound on the CCDF of the instantaneous power of a conventional SC
modulated signal.
We consider a baseband representation of the conventional SC modulated signal
( , ) ( )k
k
x t s s p t kT ∞
=−∞
= −∑ (5.1)
where { }k k s
∞
=−∞are transmitted symbols, 1 0 1[ , , , , ]s s s s−= … … , p(t ) is the pulse shaping
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filter, and T is the symbol duration. sk belongs to a modulation constellation set C of size B,
that is, { }: 0 1k bs C c b B∈ = ≤ ≤ − and it is uniformly distributed. We assume { }k s ’s are
independent of each other.
Let us define a random variable Z for a given t 0 ∈ [0, T ) as follows.
0 0( , ) ( )k k
k k
Z x t s s p t kT a∞ ∞
=−∞ =−∞
= − =∑ ∑≜ (5.2)
where 0( )k k
a s p t kT = − . { }k a ’s are also mutually independent since { }k s ’s are mutually
independent.
Since sk is uniformly distributed over C and from (5.2),
[ ] [ ]0
1Pr ( ) Pr
k b k ba c p t kT s c
B= − = = = (5.3)
where 0 1b B≤ ≤ − .
Our goal is to characterize the following CCDF.
2 2
0Pr ( , ) Pr Pr x t s w Z w Z δ ≥ = ≥ = ≥ (5.4)
where 0w ≥ and wδ = .
Let us assume the modulation constellation set C is real and its constellation points are
symmetric. Then,
[ ] [ ]
[ ]
Pr Pr Pr
2Pr
Z Z Z
Z
δ δ δ
δ
≥ = ≥ + ≤ −
= ≥(5.5)
Using the Chernoff bound, the following inequality holds [48], [49].
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[ ] { }ˆ ˆPr Z Z e E eνδ ν δ −≥ ≤ (5.6)
where ν is a solution of the following equation.
{ } { } 0 Z Z E Ze E eν ν δ − ⋅ = (5.7)
By expanding { } Z E Zeν and { } Z E eν , (5.7) becomes
{ }
{ }
ˆ
ˆ
k
k
a
k
a
k
E a e
E e
ν
ν δ
∞
=−∞
=∑ (5.8)
for ˆν ν = . By solving (5.8), we obtain ν for a given δ . Since (5.8) is not in a closed form, we
evaluate the solution numerically.
We can upperbound the CCDF2
0Pr ( , ) x t s w ≥
as follows from (5.4), (5.5), and
(5.6).
{ }2 ˆˆ
0 ,Pr ( , ) 2 k a
ub SC
k
x t s w e E e Pν νδ ∞−
=−∞
≥ ≤ ∏ ≜ (5.9)
The details of the derivations for (5.8) and (5.9) are in section A.1 of appendix A.
Since it is impossible to consider the infinite span in (5.8) and (5.9), we limit the span by
only considering K max ≤ k ≤ K max. As long as the IFDMA input block size N is larger than
K max, the instantaneous power distribution of the IFDMA signal will have the same upper
bound as the conventional SC signal since the input symbols are mutually independent within
the span of K max ≤ k ≤ K max.
In the next subsections, we derive the upper bounds specifically for BPSK and QPSK
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modulations. We consider the following raisedcosine pulse.
( ) ( )2 2 2
sin / cos / ( ) / 1 4 / t T t T p t
t T t T π πα
π α = ⋅
−(5.10)
where 0 1α ≤ ≤ is the rolloff factor.
5.1.1. Upper Bound for BPSK
For BPSK, we use the following constellation set.
{ }1,1 BPSK C = − (5.11)
Then, (5.8) becomes
0 0
max
0 0max
ˆ ˆ( ) ( )
0 0
ˆ ˆ( ) ( )
1 1( ) ( )
2 21 1
2 2
p t kT p t kT K
p t kT p t kT k K
p t kT e p t kT e
e e
ν ν
ν ν
δ
− − −
− − −=−
− − −=
+∑ (5.12)
After we determine ν from (5.12), the upper bound in (5.9) becomes
( )max max
0 0
max
2
ˆ ˆˆ ( ) ( )( )
,
1
2
K K p t kT p t kT BPSK
ub SC
k K
P e e eν ν νδ − − −−
=−
+
∏≜ (5.13)
and we upperbound the CCDF as
2( ) ( )
0 ,Pr ( , ) BPSK BPSK
ub SC x t s w P ≥ ≤
(5.14)
The details of the derivations for (5.12) and (5.13) are in section A.2 of appendix A.
Figure 5.1 shows ( )
,
BPSK
ub SC P for K max= 8 , T = 1, and t 0 = T /2 = 0.5 along with the
empirical results using Monte Carlo simulation. We considered rolloff factor α of 0, 0.2, 0.4,
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and 0.6. For the Monte Carlo simulation, we generated 1000 random IFDMAmodulated
blocks with total number of symbols M = 256, number of input symbols N = 64, and
spreading factor Q = 4, and produced a histogram. We can see that the upper bound we
derived in (5.13) is valid compared to the empirical results and that the bound is rather tight in
the tail region of the distribution which we are most interested in. Another interesting
observation is the fact that the peak power at a given probability increases as the rolloff
factor of the raisedcosine filter becomes closer to zero.
5.1.2. Modified Upper Bound for QPSK
Figure 5.2 shows the modified upperbound for QPSK, ( )
,
QPSK
ub SC P , for K max= 8 , T = 1, and
0 2 4 6 8
104
103
102
101
100
w [dB]
P r (  Z  2
> w )
CCDF of Instantaneous Power for IFDMA (BPSK)
α = 0.6
α = 0
α = 0.4
α = 0.2
Figure 5.1: CCDF of instantaneous power for IFDMA with BPSK modulation and different values
of rolloff factor α . Dotted lines are empirical results and solid lines are the upper bounds.
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t 0 = T /2 = 0.5 along with the empirical results using Monte Carlo simulation. The derivation
of ( )
,
QPSK
ub SC P is in section A.3 of appendix A. We considered rolloff factor α of 0, 0.2, 0.4,
and 0.6. For the Monte Carlo simulation, we generated 1000 random IFDMAmodulated
blocks with total number of symbols M = 256, number of input symbols N = 64, and
spreading factor Q = 4, and produced a histogram. To make the upper bound tighter, we used
2γ = (see section A.3 of appendix A). We can see that the modified upper bound we
derived is valid compared to the empirical results. As with the case of BPSK in section 5.1.1,
the peak power at a given probability increases as the rolloff factor of the raisedcosine pulse
0 2 4 6 8 10
104
103
102
101
100
w [dB]
P r (  Z  2
> w )
CCDF of Instantaneous Power of IFDMA (QPSK)
α = 0.6
α = 0
α = 0.4
α = 0.2
Figure 5.2: CCDF of instantaneous power for IFDMA with QPSK modulation and different values
of rolloff factor α . Dotted lines are empirical results and solid lines are the upper bounds.
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shaping filter becomes closer to zero.
5.2. Modified Upper Bound for LFDMA and DFDMA
In chapter 3 sections 3.3.2 and 3.3.3, we showed that the time domain symbols of localized
FDMA (LFDMA) and distributed FDMA (DFDMA) signals have the same structure. Thus
the distributions of the instantaneous power are the same for both signals. We do not consider
pulse shaping in the following analysis because of the complexity of the derivation.
Figure 5.3 shows the modified upper bound for BPSK, Pub,LFDMA, for M = 256 and
different values of N and Q, along with the empirical results using Monte Carlo simulation.
The details of the derivation for Pub,LFDMA is in section A.4 of appendix A. For the Monte
0 2 4 6 8 1010
4
103
102
101
100
w [dB]
P r (  Z  2 > w )
CCDF of Instantaneous Power for LFDMA (BPSK)
N = 4
N = 8
N =32
Figure 5.3: CCDF of instantaneous power for LFDMA with BPSK modulation and different values
of input block size N . Dotted lines are empirical results and solid lines are the upper bounds.
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Carlo simulation, we generated 1000 random LFDMAmodulated blocks and produced a
histogram. To make the upper bound tighter, we used 4
π γ = (see section A.4 of appendix
A). We can see that the upper bound we derived is valid compared to the empirical results. We
can also observe that the peak power at a given probability increases as the input block size
block N increases.
5.3. Comparison with OFDM
We can express the CCDF of the PAPR of an OFDM data block for N subcarriers with
Nyquist rate sampling as follows [50], [51].
{ } ( )Pr 1 1 N
wPAPR w e−≥ = − − (5.15)
Let us assume the input symbols have unit power. Then we can use (5.15) and compare it
with the CCDF of the instantaneous power of IFDMA and that of LFDMA.
Figure 5.4 compares the analytical upper bounds of CCDF for IFDMA and LFDMA
with the theoretical CCDF for OFDM with the same block size. For IFDMA, we consider
rolloff factor of 0.2. For LFDMA and OFDM, the number of occupied subcarriers N is 32
and we do not apply pulse shaping filter. We consider the case of BPSK modulation. We can
see that SCFDMA signals for both subcarrier mapping modes indeed have lower peak power
at a given probability than that of OFDM. Comparing the 99.9percentile instantaneous power
( w such that { }2 3Pr 10 Z w−≥ = ), we see that the peak power of IFDMA is 4 dB lower than
that of OFDM and the peak power of LFDMA is 2.2 dB lower than that of OFDM. We can
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also observe that LFDMA signal has higher peak power than IFDMA signal by 1.8 dB at 99.9
percentile probability.
5.4. Summary and Conclusion
In this chapter, we investigated the peak power characteristics for IFDMA and LFDMA
signals analytically. By using Chernoff bound, we found upper bounds for both subcarrier
mapping schemes and we showed results for IFDMA with BPSK and QPSK modulations and
for LFDMA with BPSK modulation. We saw that the peak power of an SCFDMA signal for
both subcarrier mappings is indeed lower than that of OFDM. We also observed that IFDMA
0 2 4 6 8 10 12
104
103
102
101
100
w [dB]
P r (  Z  2
> w )
CCDF of Instantaneous Power
IFDMA
LFDMA
OFDM
Figure 5.4: CCDF of instantaneous power for IFDMA, LFDMA, and OFDM. For IFDMA,
we consider rolloff factor of 0.2. For LFDMA and OFDM, the number of occupied subcarriers N is
32 and we do not apply pulse shaping filter. We consider the case of BPSK modulation.
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has the lowest peak power among the considered subcarrier mapping schemes of SCFDMA.
With regards to the tightness of the upper bounds, the original upper bounds are rather loose
except for the case of IFDMA/BPSK and we had to modify the original upper bounds to
make them tighter. We prefer numerical analysis in order to precisely characterize the peak
power characteristics for SCFDMA signals, which will be the subject of the next chapter.
Other interesting findings are:
• For IFDMA, the rolloff factor of the raisedcosine pulse shaping filter has a
great impact on the distribution of the instantaneous power. As the rolloff factor
decreases to zero, the peak power at a given probability increases.
• For LFDMA, the input block size is the major factor characterizing the CCDF of
peak power.
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Chapter 6Chapter 6Chapter 6Chapter 6
PPPPeak Powereak Powereak Powereak Power CharacteristicsCharacteristicsCharacteristicsCharacteristics of an SCof an SCof an SCof an SCFDMA SignalFDMA SignalFDMA SignalFDMA Signal::::
NumerNumerNumerNumericalicalicalical AnalysisAnalysisAnalysisAnalysis
Peaktoaveragepower ratio (PAPR) is a performance measurement that is indicative of the
power efficiency of the transmitter. In case of an ideal linear power amplifier where we achieve
linear amplification up to the saturation point, we reach the maximum power efficiency when the
amplifier is operating at the saturation point. A positive PAPR in dB means that we need a
power backoff to operate in the linear region of the power amplifier. We can express the
theoretical relationship between PAPR [dB] and transmit power efficiency as follows [52].
20max
10PAPR
η η −
= ⋅ (6.1)
where η is the power efficiency and η max is the maximum power efficiency. For class A power
amplifier, η max is 50% and for class B, 78.5% [53]. Figure 6.1 shows the relationship graphically
and it is evident from the figure that high PAPR degrades the transmit power efficiency
performance.
A salient advantage of SCFDMA over OFDMA is the lower PAPR because of its inherent
single carrier structure [54]. The lower PAPR is greatly beneficial in the uplink communications
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where the mobile terminal is the transmitter. As we showed in section 3.3, time domain samples
of the SCFDMA modulated signals are different depending on the subcarrier mapping scheme
and we can expect different PAPR characteristics for different subcarrier mapping schemes.
In this chapter, we first characterize the PAPR for single antenna transmission of SC
FDMA. We investigate the PAPR properties for different subcarrier mapping schemes. In
section 6.2, we analyze the PAPR characteristics for multiple antenna transmission. Specifically,
we numerically analyze the CCDF of PAPR for 2x2 unitary precoded TxBF SCFDMA system
described in chapter 4. In section 6.3, we propose a symbol amplitude clipping method to reduce
peak power. We show the link level performance and frequency domain aspects of the proposed
0 2 4 6 8 100
10
20
30
40
50
60
70
80
90
100
PAPR [dB]
E f f i c i e n c y [ % ] Class B
Class A
Figure 6.1: A theoretical relationship between PAPR and transmit power efficiency for ideal class A
and B amplifiers.
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clipping method.
6.1. PAPR of Single Antenna Transmission Signals
In this section, we analyze the PAPR of the SCFDMA signal for each subcarrier mapping mode.
We follow the notations in Figure 3.2 of chapter 3.
Let { }: 0,1, , 1n x n N = −⋯ be data symbols to be modulated. Then,
{ }: 0,1, , 1k
X k N = −⋯ are frequency domain samples after DFT of { : 0,1, , 1n
x n N = −⋯ ,
{ }: 0,1, , 1l
X l M = −ɶ ⋯ are frequency domain samples after subcarrier mapping, and
{ }: 0,1, , 1m
x m M = −ɶ ⋯ are time symbols after IDFT of { }: 0,1, , 1l
X l M = −ɶ ⋯ . We
represent the complex passband transmit signal of SCFDMA x(t ) for a block of data as
1
0
( ) ( )c
M j t
m
m
x t e x p t mT ω
−
=
= −∑ ɶɶ (6.2)
where cω is the carrier frequency of the system, p(t ) is the baseband pulse, and T ɶ is the
symbol duration of the transmitted symbol m xɶ . We consider raisedcosine (RC) pulse and
squaredroot raisedcosine (RRC) pulse which are widely used pulse shapes in wireless
communications.
We define the PAPR as follows for transmit signal x(t ).
2
0
2
0
max ( )peak power of ( )PAPR
1average power of ( )( )
t MT
MT
x t x t
x t x t dt
MT
≤ ≤= =
∫
ɶ
ɶ
ɶ
(6.3)
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Without pulse shaping, that is, using rectangular pulse shaping, symbol rate sampling will
give the same PAPR as the continuous case since SCFDMA signal is modulated over a single
carrier. Thus, we express the PAPR without pulse shaping with symbol rate sampling as follows.
2
0,1, , 1
12
0
maxPAPR
1
mm M
M
m
m
x
x M
= −
−
=
=
∑
⋯ɶ
ɶ(6.4)
Using Monte Carlo simulation, we calculate the CCDF (Complementary Cumulative
Distribution Function) of PAPR, which is the probability that PAPR is higher than a certain
PAPR value PAPR0 ( Pr{PAPR>PAPR0} ). We evaluate and compare the CCDFs of PAPR for
IFDMA, DFDMA, LFDMA, and OFDMA. The following are the simulation setup and
assumptions.
• We generate 104 uniformly random data blocks to acquire the CCDF of PAPR.
• We consider QPSK and 16QAM symbol constellations.
• We truncate the baseband pulse p(t ) from 6 T ɶ to 6 T ɶ time period and we
oversample it by 8 times and we use transmission bandwidth of 5 MHz.
• We consider consecutive chunks for LFDMA.
• We do not apply any pulse shaping in the case OFDMA.
Figure 6.2 contains the plots of CCDF of PAPR for IFDMA, DFDMA, LFDMA, and
OFDMA for total number of subcarriers M = 512, number of input symbols N = 128, IFDMA
spreading factor Q = 4, and DFDMA spreading factor 2Q =ɶ . We compare the PAPR value that
is exceeded with the probability less than 0.1% ( Pr{PAPR>PAPR0} = 103 ), or 99.9percentile
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PAPR. We denote 99.9percentile PAPR as PAPR99.9%. Table 6.1 summarizes the 99.9percentile
PAPR for each subcarrier mapping.
Table 6.1: 99.9percentile PAPR for IFDMA, DFDMA, LFDMA, and OFDMA
Modulation Pulse shaping IFDMA DFDMA LFDMA OFDMA
None 0 dB 7.7 dB 7.7 dB 11.1 dB
RC 6.2 dB 7.7 dB 8.0 dB N/AQPSK
RRC 5.3 dB 7.8 dB 8.7 dB N/A
None 3.2 dB 8.7 dB 8.7 dB 11.1 dB
RC 7.8 dB 8.7 dB 9.0 dB N/A16QAM
RRC 7.2 dB 8.7 dB 9.5 dB N/A
* RC: raisedcosine pulse shaping, RRC: squaredroot raisedcosine pulse shaping
0 2 4 6 8 10 1210
4
103
102
101
100
P r ( P A P R > P A P R
0 )
PAPR0
[dB]
CCDF of PAPR: QPSK, Rolloff = 0.22, Nfft
= 512, Noccupied
= 128
Dotted lines: no PS
Dashed lines: RRC PS
Solid lines: RC PS
IFDMA
DFDMA
LFDMA
OFDMA
0 2 4 6 8 10 1210
4
103
102
101
100
P r ( P A P R > P A P R 0
)
PAPR0
[dB]
CCDF of PAPR: 16QAM, Rolloff = 0.22, Nfft
= 512, Noccupied
= 128
Dotted lines: no PS
Dashed lines: RRC PS
Solid lines: RC PS
IFDMA
DFDMA
LFDMA
OFDMA
(a) (b)
Figure 6.2: Comparison of CCDF of PAPR for IFDMA, DFDMA, LFDMA, and OFDMA
with total number of subcarriers M = 512, number of input symbols N = 128, IFDMA spreading
factor Q = 4 , DFDMA spreading factor Qɶ = 2, and α (rolloff factor) = 0.22; (a) QPSK; (b)
16QAM.
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We can see that all the cases for SCFDMA have indeed lower PAPR than that of OFDMA.
Also, IFDMA has the lowest PAPR, and DFDMA and LFDMA have very similar levels of
PAPR.
Figure 6.3 shows the impact of the rolloff factor α on the PAPR when using raisedcosine
pulse shaping for M = 256, N = 64, and Q = 4. We can see that this impact is more obvious in
the case of IFDMA. As the rolloff factor decreases from 1 to 0, PAPR increases significantly
for IFDMA. This observation is consistent with the results of the instantaneous peak power for
IFDMA with pulse shaping in section 5.1 of chapter 5. This implies that there is a tradeoff
between PAPR performance and outofband radiation since outofband radiation increases
with increasing rolloff factor.
0 2 4 6 8 1010
4
103
102
101
100
P r ( P A P R > P A P R 0
)
PAPR0
[dB]
CCDF of PAPR: QPSK, Nfft
= 256, Noccupied
= 64
Solid lines: without pulse shaping
Dotted lines: with pulse shaping
IFDMA LFDMA
α=0.4α=0.6
α=0.2
α=0
α=0.8
α=1
0 2 4 6 8 1010
4
103
102
101
100
P r ( P A P R > P A P R
0 )
PAPR0
[dB]
CCDF of PAPR: 16QAM, Nfft
= 256, Noccupied
= 64
Solid lines: without pulse shaping
Dotted lines: with pulse shaping
IFDMALFDMA
α=0.4α=0.6α=0.2
α=0
α=0.8
α=1
(a) (b)
Figure 6.3: Comparison of CCDF of PAPR for IFDMA and LFDMA with M = 256, N = 64,
Q = 4, and α (rolloff factor) of 0, 0.2, 0.4, 0.6, 0.8, and 1; (a) QPSK; (b) 16QAM.
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6.2. PAPR of Multiple Antenna Transmission Signals
In this section, we analyze the PAPR characteristics of multiple antenna transmission for SC
FDMA system. Specifically, we investigate unitary precoded transmit eigenbeamforming (TxBF)
for 2 transmit and 2 receiver antennas which we described in section 4.3 of chapter 4. Since it is
difficult to derive the PAPR analytically, we resort to numerical analysis using Monte Carlo
simulation to investigate the PAPR properties of our MIMO SCFDMA system. We use the
parameters in section 4.3 of chapter 4 for the simulations in this section.
As described in section 4.3 of chapter 4, we apply unitary precoding in the frequency
domain after DFT. Precoding in the frequency domain is convolution (filtering) and summation
in the time domain as shown in Figure 6.4. Thus we expect to see an increase in the peak power
because of the filtering and summation.
⋅+⋅
⋅+⋅=
⋅
=
⋅=
=
=
k k k k
k k k k
k
k
k k
k k
k k k
k
k
k
k k
k k
k
SV SV
SV SV
S
S
V V
V V
SV X
S
SS
V V
V V V
,2,22,1,21
,2,12,1,11
,2
,1
,22,21
,12,11
,2
,1
,22,21
,12,11,
{ }{ }
,0 ,1 , 1
,0 ,1 , 1
11 1 12 2
21 1 22 2
, , ,
, , ,
{ } , { }
* *
* *
ij ij ij ij M
l l l l M
ij ij l l
V V V V
S S S S
v IDFT V s IDFT S
v s v s x
v s v s
−
−
=
=
= =
+ =
+
⋯⋯
Frequency domain Time domain
Figure 6.4: Precoding in the frequency domain is convolution and summation in the time domain.k
refers to the subcarrier number.
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4 6 8 10 1210
4
103
102
101
100
PAPR0
[dB]
P r ( P A P R > P A P R
0 )
CCDF of PAPR for Unitary Precoded TxBF
2nd
antenna
(QPSK)
1st
antenna(16QAM)
No precoding
Only summation
2nd
stream zero
Precoded
Figure 6.5: CCDF of PAPR for 2x2 unitary precoded TxBF.
Figure 6.5 shows the CCDF of the PAPR for 2x2 unitary precoded TxBF. To verify that the
filtering aspect of the precoding is the dominant factor in increasing the PAPR, we also show
results for the case when there is only filtering by intentionally setting the 2nd antenna signal to
zero, that is, 20s = (the graph labeled “2nd stream zero”) and for the case when there is only
summation by intentionally replacing the convolution with a scalar multiplication of the first
element of ijv (the graph labeled “Only summation”). With precoding, we can see that there is
an increase of 1.6 dB for the 1st antenna and 2.6 dB for the 2nd antenna in the 99.9percentile
PAPR. We can also observe that the filtering aspect is the main contributor to the increase of
PAPR for precoding.
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4 6 8 10 1210
5
104
103
102
101
100
PAPR0
[dB]
P r ( P A P R > P A P R
0 )
No precoding
(QPSK)
No precoding
(16QAM)
TxBF
(both avr. & quant.)
TxBF
(only quant.)
TxBF
( only avr.)
TxBF
(no avr. & no quant.)
Figure 6.6: Impact of quantization and averaging of the precoding matrix on PAPR.
4 6 8 10 12
104
103
102
101
100
PAPR0
[dB]
P r ( P A P R > P A P R
0 )
SM
SFBC (QPSK)
SFBC (16QAM)
TxBF
(avr. & quant.)
TxBF
(no avr. & no quant.)
Figure 6.7: PAPR comparison with other MIMO schemes.
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Figure 6.6 illustrates the impact of quantization and averaging of the precoding matrix on
the PAPR characteristics. We use direct quantization and averaging process described in section
4.3.1 of chapter 4. We average the channel H over 25 continuous subcarriers and perform direct
quantization of the precoder matrix V using 3 bits (1 bit for amplitude and 2 bits for phase
information).
Without averaging of channel and quantization of precoding matrix, the MIMO TxBF
signal has 1.6 ~2.6 dB higher PAPR99.9% with respect to single antenna transmission. When we
apply both averaging and quantization, PAPR99.9% decreases by 0.7 dB. This is due to the fact
that the averaging and quantization of the precoding matrix smoothes the time filtering, thus
reduces the peak power.
Figure 6.7 compares the CCDF of the PAPR for unitary precoded TxBF with those of
different MIMO schemes. We consider spacefrequency block coding (SFBC) and nonprecoded
spatial multiplexing (SM) for comparison. Since the spatial multiplexing that we consider does
not have any precoding or spatial processing at the transmitter, it has the same PAPR as the case
of single antenna transmission. Without quantization/averaging of precoding matrix, TxBF has
higher PAPR99.9% than that of SFBC by 0.5 ~ 1 dB. However, TxBF and SFBC have similar
PAPR when we have quantization/averaging of the precoding matrix.
6.3. Peak Power Reduction by Symbol Amplitude Clipping
One way to reduce PAPR is to limit or clip the peak power of the transmitted symbols [55].
Depending on the smoothness of the limiter, we can define three types of limiter; hard , soft , and
smooth limiter [56]. The inputout relationship of each type of amplitude limiter for a complex
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symbol is as follows. Note that we leave the phase of the symbol as is and only modify the
amplitude part.
• Hard limiter:
( ) maxm j x
m hard m y g x A e= = ∡ (6.5)
• Soft limiter :
( )max
max max
,
,m
m m
m soft m j x
m
x x A y g x
A e x A
≤= =
>∡ (6.6)
• Smooth limiter:
( ) ( )max maxerf / m j x
m smooth m m y g x A x A e= = ∡ (6.7)
where xm is the input symbol to the limiter, m x∡ is the phase component of xm, ym is the
output symbol of the limiter, Amax > 0, and ( )2
0
2erf
xt x e dt
π
−=
∫ . Figure 6.8 graphically
illustrates the input amplitudeoutput amplitude relationship of each limiter. In the subsequent
analysis, we use the soft limiter for clipping.
The problems associated with clipping are inband signal distortion and generation of out
ofband signal. Because SCFDMA modulation spreads the information data across all the
modulated symbols, inband signal distortion is mitigated when an SCFDMA symbol is clipped.
To analyze the impact of symbol amplitude clipping on the link level performance, we apply
clipping to the unitary precoded TxBF MIMO system that we described in chapter 4 and show
numerical simulation results. We use the parameters and assumptions in section 4.3 of chapter 4
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for the simulations in this section.
Figure 6.9 shows the block diagram of a symbol amplitude clipping method for spatial
multiplexing SCFDMA MIMO transmission for M t transmit antenna. In our current analysis,
we apply the clipping after baseband pulse shaping. We can also consider other sophisticated
clipping methods, such as iterative clipping.
Figure 6.10 shows the CCDF of symbol power with clipping at various levels. With 7 dB
max clipping less than 1% of the symbols are clipped. Note that even with as much as 3 dBmax
PAPR clipping, only about 10% of the modulated symbols are clipped.
Figure 6.11 shows the uncoded bit error rate (BER) and coded frame error rate (FER)
performances when we apply symbol amplitude clipping. We can observe that the performance
degradation due to clipping is almost none for 7 dBmax clipping. The performance degrades
Amax
Amax
m y
m x
Hard limiter
Soft limiter
Smooth limiter
Figure 6.8: Three types of amplitude limiter.
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S p a t i a l P r o c e s s i
n g
Channel
DFT IDFT Amplitude
Clipping
Add
CP/PS
SucarrierMapping
DAC/RF},,{
1,0,0 − N n x x ⋯
IDFT Amplitude
Clipping
Add
CP/PS
SucarrierMapping
DAC/RF
DFT1, 1, 1
{ , , }t t M n M N x x− − −⋯
Figure 6.9: Block diagram of a symbol amplitude clipping method for SCFDMA MIMO
transmission with M t transmit antenna.
0 2 4 6 8 10
104
103
102
101
100
w [dB]
P r (  y m
 2 > w )
CCDF of Symbol Power
3 dBmax clipping
5 dBmax clipping
7 dBmax clipping
No clipping
Figure 6.10: CCDF of symbol power after clipping. ym represents the baseband symbol. Solid line
represents CCDF for antenna 1 and dashed line represents CCDF for antenna 2.
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0 5 10 15 20 25 3010
6
104
102
100
SNR per Antenna [dB]
R a w
B E R
No clipping
7 dBmax clipping
5 dBmax clipping
3 dBmax clipping
0 2 4 6 8 1010
4
103
102
101
100
SNR per Antenna [dB]
F E R
No clipping
7 dBmax clipping
5 dBmax clipping
3 dBmax clipping
(a) (b)
Figure 6.11: Link level performance for clipping; (a) uncoded BER; (b) coded FER.
4 3 2 1 0 1 2 3 460
50
40
30
20
10
0
Normalized Frequency Band
P o w e r [ d B ]
3 dBmax
clipping
5 dBmax
clipping
7 dBmax
clipping
No clipping
Figure 6.12: PSD of the clipped signals.
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slightly more when we apply 5 or 3 dBmax clipping. For 3 dBmax clipping, an error floor starts
to appear in high SNR region for uncoded BER. But when we incorporate forward error
correction coding, it mitigates this effect. Thus, we can use a modest amount of amplitude
clipping to reduce the PAPR for unitary precoded TxBF with SCFDMA without compromising
the link level performance.
Clipping, consequently, will generate both inband and outofband frequency components.
Figure 6.12 shows the power spectral density (PSD) of the clipped signals. For PSD calculation,
we use Hanning window with 1/4 of window overlapping. For 7 dBmax clipping, the spectrum
is almost the same as that of the original signal. More pronounced outofband components
arise when we use 5 or 3 dBmax clipping. Thus, we should control the amount of clipping
depending on the outofband radiation requirements.
6.4. Summary and Conclusions
In this chapter, we analyzed the PAPR of SCFDMA signals for single and multiple antenna
transmissions using numerical simulations. We showed that SCFDMA signals indeed have lower
PAPR compared to OFDMA signals. Also, we have shown that DFDMA and LFDMA incur
higher PAPR compared to IFDMA but compared to OFDMA, it is still lower, though not
significantly. Another noticeable fact is that pulse shaping increases PAPR, thus degrades the
power efficiency and that the rolloff factor in the case of raisedcosine pulse shaping has a
significant impact on PAPR of IFDMA. A pulse shaping filter should be designed carefully in
order to reduce the PAPR without degrading the system performance.
For multiple antenna transmission, we observed that unitary precoding TxBF scheme
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increases the PAPR because of its filtering aspect. It was interesting to find out that averaging
and quantization of the precoder matrix actually reduces the PAPR.
In order to reduce the peak power in a SCFDMA system, we proposed a symbol amplitude
clipping method and applied it to the 2x2 unitary precoded TxBF system. Numerical simulation
results showed that moderate clipping does not affect the link level performance. But outof
band radiation is a concern when we use clipping.
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Chapter 7Chapter 7Chapter 7Chapter 7
ChannelChannelChannelChannelDependent Scheduling of Uplink SCDependent Scheduling of Uplink SCDependent Scheduling of Uplink SCDependent Scheduling of Uplink SCFDMAFDMAFDMAFDMA
SystemsSystemsSystemsSystems
As we illustrated in section 2.1 of chapter 2, wide band wireless channels experience time and
frequencyselective fading because of user’s mobility and multipath propagation. In wide
band multiuser uplink communications, the channel gain of each user is different for
different time and frequency subcarrier when the channels are uncorrelated among users.
Time and frequency resources which are in deep fading for one user may be in excellent
conditions for other users. The resource scheduler in the base station can assign the time
frequency resources to a favorable user which will increase the total system throughput [57],
[58], [59]. We term the class of this adaptive resource scheduling method as channel
dependent scheduling (CDS) which can greatly increase the spectral efficiency (bits/Hz).
Another way to exploit the time and frequencyselectivity in the channel to its advantage
is the adaptive modulation and coding (AMC) scheme along with subband or multicarrier
communications. AMC dynamically adapts the modulation constellation and the channel
coding rate depending on the channel condition and thus improves the energy efficiency and
the data rate [60], [61]. By applying AMC on subbands or subcarriers, we can dramatically
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improve the link quality and increase the data rate for time and frequencyselective fading
channel.
In this chapter, we first give a general overview of CDS in an uplink SCFDMA system.
We also analyze the capacity for the distributed (IFDMA) and localized subcarrier mapping
schemes. In section 7.2, we investigate the impact of imperfect channel state information
(CSI) on CDS. We analyze the data throughput of an SCFDMA system with uncoded
adaptive modulation and CDS when there is a feedback delay. In section 7.3, we propose a
hybrid subcarrier mapping scheme using direct sequence spreading technique on top of SC
FDMA modulation. We show the throughput improvement of the hybrid subcarrier mapping
over the conventional subcarrier mapping schemes.
7.1. ChannelDependent Scheduling in an Uplink SCFDMA System
In this section, we investigate channeldependent resource scheduling for an SCFDMA
system in uplink communications. Specifically, we analyze the capacity gains from CDS for the
two different flavors of subcarrier mapping of SCFDMA. For distributed subcarrier mapping,
we consider IFDMA. A key question of CDS is how we should allocate time and frequency
resources fairly among users while achieving multiuser diversity and frequency selective
diversity. To do so, we introduce utilitybased scheduling where utility is an economic concept
representing level of satisfaction [62], [63], [64], [65], [66]. The choice of a utility measure
influences the tradeoff between overall efficiency and fairness among users. In our studies, we
consider two different utility functions: aggregate user throughput for maximizing system
capacity and aggregate logarithmic user throughput for maximizing proportional fairness. The
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objective is to find an optimum resource assignment for all users in order to maximize the sum
of user utility at each transmission time interval (TTI). If the user throughput is regarded as
the utility function, the resource allocation maximizes ratesum capacity ignoring fairness
among users. Therefore, only the users near the base station who have the best channel
conditions occupy most of the resources. On the other hand, setting the logarithmic user data
rate as the utility function provides proportional fairness.
In considering the optimization problem of CDS for multicarrier multiple access, it is
theoretically possible to assume that the scheduler can assign subcarriers individually.
Allocating individual subcarriers is, however, a prohibitively complex combinatorial
optimization problem in systems with a large number of subcarriers and user terminals.
Moreover, assigning subcarriers individually would introduce unacceptable control signaling
overhead. In practice, we allocate the resource in chunks, which are disjoint sets of subcarriers.
We refer to this chunk as a resource unit (RU). As a practical matter for SCFDMA, chunk
based transmission is desirable since the input data symbols are grouped into a block for DFT
operation before subcarrier mapping. We will consider only chunkbased scheduling in the
remainder of this section. With regards to chunk structure, consecutive subcarriers comprise a
chunk for LFDMA and distributed subcarriers with equidistance for IFDMA.
Even with subcarriers assigned in chunks, optimum scheduling is extremely complex for
two reasons: 1) The objective function is complicated, consisting of nonlinear and discrete
constraints dependent on the combined channel gains of the assigned subcarriers; and 2) there
is a total transmit power constraint for each user. Furthermore, the optimum solution entails
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combinatorial comparisons with high complexity. Instead of directly solving the optimization
problem, we can use a suboptimal chunk allocation scheme for both IFDMA and LFDMA to
obtain most of the benefits of CDS. For LFDMA, we can apply a greedy chunk selection
method where each chunk is assigned to the user who can maximize the marginal utility when
occupying the specific chunk. For IFDMA, we can achieve the benefit of multiuser diversity
by selecting users in order of the estimated marginal utility based on the average channel
condition over the entire set of subcarriers. The users with higher channel gains may occupy a
larger number of chunks than users with lower channel gains. The details of the chunk
selection method are in [64] and [65].
Our throughput measure in this study is the sum of the upper bound on user
throughputs given by Shannon’s formula, C = BW ⋅log(1+SNR) where BW is the effective
bandwidth depending on the number of occupied subcarriers and SNR is signaltonoise ratio
of a block.
Figures 7.1 and 7.2 are the results of computer simulations of SCFDMA with 256
subcarriers spread over a 5 MHz band. They compare the effects of channel dependent
subcarrier allocation (S) with static (roundrobin) scheduling (R) for LFDMA and IFDMA. In
all of the examples, the scheduling took place with chunks containing 8 subcarriers and we
assume perfect knowledge of the channel state information.
Figure 7.1 shows two effects of applying different utility functions in the scheduling
algorithm. In the two graphs in Figure 7.1, the utility functions are the sum of user
throughputs and the sum of the logarithm of user throughputs, respectively. Each graph
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shows the aggregate throughput as a function of the number of simultaneous transmissions.
Figure 7.2 shows the expected user throughput at each distance from the base station for the
same utility functions. The simulation results in the figures use the following abbreviations: R
LFDMA (static round robin scheduling of LFDMA), SLFDMA (CDS of LFDMA), R
IFDMA (static round robin scheduling of IFDMA), and SIFDMA (CDS of IFDMA).
Figure 7.1 shows that for throughput maximization (utility = bit rate), the advantage of
channel dependent scheduling over round robin scheduling increases as the number of users
increases. This is because the scheduler selects the closer users who can transmit at higher data
rate. If there are more users, the possibility of locating users at closer distance to the base
station increases. As a result, the CDS achieves significant improvements for both IFDMA
4 8 16 32 64 1285
10
15
20
25
30
35
40
45
Number of users
A g g r e g a t e t h r o u g h p u t [ M b p s ]
RLFDMA
SLFDMA
RIFDMASIFDMA
4 8 16 32 64 1285
10
15
20
25
30
35
40
45
Number of users
A g g r e g a t e t h r o u g h p u t [ M b p s ]
RLFDMA
SLFDMA
RIFDMASIFDMA
(a) (b)
Figure 7.1: Comparison of aggregate throughput with M = 256 system subcarriers, N = 8 subcarriers
per user, bandwidth = 5 MHz, and noise power per Hz = 160 dBm; (a) utility function: sum of
user throughputs; (b) utility function: sum of the logarithm of user throughputs.
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and LFDMA. In the case of logarithmic rate utility, the CDS gain stops increasing beyond
approximately 32 users. With 32 users, maximizing logarithmic rate utility can increase system
capacity by a factor of 1.8 for LFDMA and 1.26 for IFDMA relative to static scheduling.
Figure 7.2 shows that the CDS scheme based on the logarithm of user throughput as a
utility function provides proportional fairness whose gains are shared among all users, whereas
the CDS gains are concentrated to the users near the base station when the user throughput is
considered as the utility function.
For static subcarrier assignment (round robin scheduling), a system with users each
transmitting at a moderate data rate is better off with IFDMA while LFDMA works better in
a system with a few high date rate users. Due to the advantages of lower outage probability
0.2 0.4 0.6 0.8 10
1
2
3
4
5
User distance from target BS [km]
A v e r a g e u s e r d a t a r a t e [ M b p s ]
RLFDMA
SLFDMA
RIFDMASIFDMA
0.2 0.4 0.6 0.8 10
1
2
3
4
5
User distance from target BS [Km]
A v e r a g e u s e r d a t a r a t e [ M b p s
]
RLFDMA
SLFDMA
RIFDMASIFDMA
(a) (b)
Figure 7.2: Average user data rate as a function of user distance with M = 256 system subcarriers, N
= 8 subcarriers per user, bandwidth = 5 MHz, and noise power per Hz = 160 dBm; (a) utility
function: sum of user throughputs; (b) utility function: sum of the logarithm of user throughputs.
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and lower PAPR, IFDMA is an attractive approach to static subcarrier scheduling. For CDS,
LFDMA has the potential to achieve considerably higher data rate. The results show that CDS
increases system throughput by up to 80% relative to static scheduling for LFDMA but the
increase is only 26% for IFDMA. We can exploit the scheduling gains in LFDMA to reduce
power consumption and PAPR by using power control to establish a power margin instead of
increasing system capacity.
7.2. Impact of Imperfect Channel State Information on CDS
In section 7.1, we assumed perfect knowledge of the channel state information (CSI). In a
practical wireless system, CSI is often not known at all or only part of the information is
available due to limited feedback mechanism and channel estimation error. Also, the CSI may
become outdated because of the feedback delay. In a centralized resource management
scheme, the base station makes all the decision regarding the uplink transmission parameters
based on the uplink channel quality and feeds back the transmission parameter information to
the mobile terminal. A delay in the feedback mechanism is inevitable in a practical feedback
system and the feedback delay has a major impact on the system performance especially when
the channel is fading rapidly.
In this section, we investigate the impact of imperfect CSI on CDS in uplink
communications by considering outdated CSI due to feedback delay. We assume the channel
estimation is perfect and also assume there is no error during the feedback signaling. Since
there is no general closed form for the capacity equation for outdated CSI, we resort to
numerical simulations in the context of an SCFDMA system with uncoded adaptive
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modulation to measure the data throughput.
Figure 7.3 describes the system under investigation. We consider K number of users
orthogonally and independently sending data to the base station and we assume that the
channel responses are independent of each other. The resource scheduler in the base station
performs the following two tasks upon acquiring the channel information for all users. First,
the scheduler searches for an optimal set of users that maximizes the total capacity. Next, for
each user, it decides the modulation constellation based on the SNR of the chunk allocated to
the user. The scheduler, then, sends the chunk allocation and modulation information back to
Channel K
Resource Scheduler
Channel 2
S u b c a r r i e r
M a p p i n g
Channel 1IDFTCP /PS
DFT
C o n s t e l l a t i o n
M a p p i n g
SCFDMA Receiver
User 1
User 2
User K
Data flow
Control signal flow
Mobile terminals Base station
Figure 7.3: Block diagram of an uplink SCFDMA system with adaptive modulation and CDS for
K users.
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each user via the feedback control signaling channel. When there is a delay in the feedback
channel, it creates two sources of performance degradation. First, the chunk allocation
becomes outdated by the time the users transmit and it no longer is the optimal allocation.
Second, the modulation constellation for each user no longer matches the channel condition at
the time of transmission. Because of these two factors, we can expect significant decrease in
the total capacity for timevarying channels.
In our analysis, we introduce a time delay between the time instance of the channel
estimation ( t 1 ) and the actual time of data transmission from the mobile terminals ( t 2 ). We
consider a quasistatic multipath fading channel in which channel response is constant over a
TTI but varies between each TTI depending on a correlation parameter. For a Rayleigh fading
channel, we represent the correlation coefficient for time delay ∆t = t 2  t 1 as
0 0( ) (2 / ) (2 ) D
t J v t J f t ρ π λ π ∆ = ∆ = ∆ (7.1)
where 0( ) J i is the zeroorder Bessel function, f D = v/ λ is the maximum Doppler frequency, v
is the velocity of the mobile terminal, and λ is the wavelength of the carrier signal [11].
We model the multipath fading channel at t 1 with an Rpath discrete time channel impulse
response, 1( ) ( )t h n , as follows.
( )1 1
1( ) ( )
0( )
Rt t
r rel r r
r s
h n A P w n T
τ τ δ
−
=
= ⋅ ⋅ ⋅ − ∑ (7.2)
where { 1( )t
r w }’s are zero mean complex Gaussian i.i.d. noise process, τ r is the propagation
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delay of path r , T s is the symbol duration, A is a normalization parameter such that the average
power of the channel equals to 1, Prel(τ r ) is the relative power of the delay profile model for
path r , and δ (n) is the discrete time Diracdelta function. Note that {τ r }’s are integermultiples
of T s.
Using (7.1), we can generate the multipath fading channel at t 2 , 2( ) ( )t h n , as follows.
( )2 2
1( ) ( )
0
( ) ( ) R
t t r rel r r
r s
h n A P w nT
τ τ δ
−
=
= ⋅ ⋅ ⋅ −∑ (7.3)
and
( ) ( )2 1( ) ( ) 21t t
r r r w t w t n ρ ρ = ∆ ⋅ + − ∆ ⋅ (7.4)
where nr is a zero mean complex Gaussian noise process.
We will use (7.2) and (7.3) to analyze the effect of the outdated channels on our system.
In [64], we derived the SNR of user k , γ k , as follows based on [67] for an MMSE equalizer
when a set of subcarriers I sub,k is assigned to the user.
,
1
,
,
11
1
1sub k
k i k
i I i k N
γ γ
γ
−
∈
= −
+ ∑
(7.5)
where N is the number of subcarriers allocated to the user and ,i k γ is the SNR of subcarrier
i for user k . For equal power allocation scheme, we can express,i k γ as
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( ) ,
, 2
/ k i k
i k
i
P N H γ
σ
⋅= (7.6)
where Pk is the total transmit power of user k , H i,k is the channel gain of subcarrier i for user
k , and 2
iσ is the noise power of subcarrier i.
In our simulations, we apply the effect of the time delay during feedback in the following
manner. We first calculate the received SNRs for the subcarrier allocations and select the
modulation constellations based on the channels at timet 1 and then calculate the received
SNRs for the throughput calculations based on the channels at time t 2.
For the numerical simulations, we consider the following setup and assumptions.
• Carrier frequency: 2 GHz.
• Transmission bandwidth: 5 MHz.
• Duration of a TTI: 0.5 ms.
• Blocks per TTI: 7. We assume there is no CP in a block for simplicity of the
simulation.
• The size of the subcarrier chunk is the same among all users and each user
occupies only one subcarrier chunk.
• The number of users is K , the total number of subcarriers M , the size of a chunk
N , the number of chunks Q, and M = Q⋅ N .
• Channel: 3GPP TU6 i.i.d. quasistatic Rayleigh fading channel [12]. We assume
that the channel is constant during the duration of a TTI. We also assume that all
the users have the same path loss to the base station.
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• Modulation format: Quadrature amplitude modulation (QAM). We consider S
QAM where S = 2
B
and B is the number of bits per symbol. We have 8 classes of
QAM where B ∈ {1, 2, … , 8}. B = 1 corresponds to BPSK and B = 2 to QPSK.
• We assume equal transmit power and equal receive noise power for all users.
• We use aggregate user throughput as the utility function of CDS.
We define the system throughput as the number of information bits per second received
without error. Let user k be assigned Sk QAM where 2 k B
k S = and Bk is the number of bits
per symbol. Then we express the system throughput for user k as follows [68], [69].
( )7
,k k k k
TTI
N BTP PSR S
T γ
⋅ ⋅= ⋅ (7.7)
where T TTI is the duration of a TTI, γ k is the received SNR for user k , and ( )PSR i is the
packet success rate (PSR) which we define as the probability of receiving an information
packet correctly. The aggregate throughput TP total is the sum of the K individual throughput
measures in (7.7).
For an uncoded packet with a size of L bits, PSR becomes
( ) ( ){ }, 1 ,L
ePSR S P Sγ γ = − (7.8)
where ( ),eP S γ is the BER for a given constellation size S and SNR γ .
For an additive white Gaussian noise (AWGN) channel with SQAM modulation and
ideal coherent detection, we can upperbound ( ),eP S γ as follows [70].
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( )1.5
( 1), 0.2 S
eP S e
γ
γ
−
−≤ (7.9)
Using this upper bound, PSR in (7.8) becomes
( )1.5
( 1), 1 0.2
L
SPSR S e
γ
γ
−
−
= −
(7.10)
and the system throughput for user k becomes
1.5
( 1)7
1 0.2
k
k
L
Sk k
TTI
N B
TP eT
γ −
− ⋅ ⋅
= ⋅ − (7.11)
Then, the total system throughput for K number of users is
1
K
total k
k
TP TP=
= ∑ (7.12)
Figure 7.4 shows the system throughput vs. SNR for the 8 classes of QAM. We use L =
100 bits per packet and N
= 32 subcarriers per chunk. Based on Figure 7.4, we set the SNR
boundaries for the adaptive modulation. Table 7.1 describes the SNR boundaries for our
analysis.
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Table 7.1: SNR boundaries for adaptive modulation.
SNR (dB) Bits per symbol ( B) Modulation
> 8.7 1 BPSK
8.7 ~ 13.0 2 QPSK
13.0 ~ 16.7 3 8QAM
16.7 ~ 20.0 4 16QAM
20.0 ~ 23.3 5 32QAM
23.3 ~ 26.6 6 64QAM
26.6 ~ 29.6 7 128QAM
29.6 < 8 256QAM
0 5 10 15 20 25 30 35 400
0.5
1
1.5
2
2.5
3
3.5
4
T h r o u g h p u t [ M b p s ]
SNR [dB]
B = 1
B = 2
B = 3
B = 4
B = 5
B = 6
B = 7
B = 8
Figure 7.4: System throughput vs. SNR for the 8 classes of QAM. B is the number of bits per
symbol.
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0 1 2 3 4 52
4
6
8
10
12
14
16
18
Feedback delay [ms]
A g g r e g a t e t h r o u g h p u t [ M b
p s ]
mobile speed = 3 km/h (fD
= 5.6 Hz)
LFDMA: Static
LFDMA: CDS
IFDMA: Static
IFDMA: CDS
0 1 2 3 4 5
2
4
6
8
10
12
14
16
18
Feedback delay [ms]
A g g r e g a t e t h r o u g h p u t [ M b
p s ]
mobile speed = 60 km/h (fD
= 111 Hz)
LFDMA: Static
LFDMA: CDS
IFDMA: StaticIFDMA: CDS
(a) (b)
Figure 7.5: Aggregate throughput with CDS and adaptive modulation; (a) mobile speed = 3 km/h
(Doppler = 5.6 Hz); (b) mobile speed = 60 km/h (Doppler = 111 Hz).
0 1 2 3 4 50
2
4
6
8
10
12
14
16
18
Feedback delay [ms]
A g g r e g a t e t h r o u g h p u t [ M b p s ]
mobile speed = 60 km/h (fD
= 111 Hz)
LFDMA: Static
LFDMA: CDS
IFDMA: Static
IFDMA: CDS
Figure 7.6: Aggregate throughput with CDS and constant modulation (16QAM) with mobile
speed of 60 km/h.
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Figures 7.5 and 7.6 show the results of the Monte Carlo simulations of the investigated
system. Total number of subcarriers is M = 256, the number of chunks is Q = 16, the number
of subcarriers per chunk is N = 16, and total number of users is K = 64. We perform 2000
independent iterations of the simulation.
Figure 7.5 shows the aggregate throughputs with CDS and adaptive modulation for
different mobile speeds. For low mobile speed, we can see that the feedback delay does not
affect the system throughput. However, at high mobile speed, the throughput degrades as the
feedback delay increases and the degradation is most significant for LFDMA with CDS. This
implies that LFDMA is very sensitive to the quality of CSI when we apply CDS for fast fading
0 20 (37) 40 (74) 60 (111) 80 (148)2
4
6
8
10
12
14
16
18
Mobile speed [km/h] (Doppler [Hz])
A g g r e g a t e t h r o u g h p u t [ M b p s ]
Feedback delay = 3 ms
LFDMA: Static
LFDMA: CDS
IFDMA: Static
IFDMA: CDS
Figure 7.7: Aggregate throughput with CDS and adaptive modulation with feedback delay of 3 ms
and different mobile speeds.
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channel. Another interesting observation is that the static roundrobin scheduling also suffers
from throughput decrease. This is due to the fact that even though there is no chunk
allocation mismatch, outdated CSI causes the adaptive modulation to be mismatched with the
current channel condition.
Figure 7.6 shows the aggregate throughput with CDS for mobile speed of 60 km/h and
constant modulation (16QAM) instead of adaptive modulation. It further verifies the
observation in Figure 7.5 (b) for static scheduling that the cause of the throughput decrease is
the adaptive modulation process. We do not see any throughput decrease for the static
scheduling when we use constant modulation in the same channel condition.
Figure 7.7 shows the aggregate throughput with CDS and adaptive modulation with
feedback delay of 3 ms and different mobile speeds. We can see the performance impact of
the mobility of the user on each type of scheduling method. Similarly with Figure 7.5 (b),
LFDMA with CDS suffers the most throughput decrease when the mobile speed is high.
7.3. Hybrid Subcarrier Mapping
As we can see from Figure 7.7, localized subcarrier mapping yields much higher throughput
for low speed users when we use CDS but the throughput of LFDMA with CDS becomes
lower when the users are moving at high speed. In this situation, static roundrobin scheduling
using distributed subcarrier mapping is advantageous since it requires no overhead and no
computation. Also, IFDMA has an additional advantage in PAPR. Thus, localized subcarrier
mapping with CDS is more advantageous for low mobility users and IFDMA with static
scheduling is better for users with high mobility. When there are low mobility users and high
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mobility users at the same, accommodating both types of subcarrier mapping can lead to
higher data capacity.
Conventional SCFDMA cannot efficiently accommodate both distributed and localized
mapping schemes since subcarriers must not overlap to maintain orthogonality among users.
Using orthogonal direct sequence spread spectrum technique prior to SCFDMA modulation,
both mapping can coexist with overlapping subcarriers as illustrated in Figure 7.8.
subcarriers
Terminal 1
Terminal 2
Terminal 3
subcarriers
Conventional Hybrid
Spreading Code 1
Spreading Code 2
Figure 7.8: Conventional subcarrier mapping and hybrid subcarrier mapping.
We refer to this multiple access scheme as single carrier codefrequency division multiple
access (SCCFDMA). Figure 7.9 shows the block diagram of an SCCFDMA system and
Figure 7.10 compares SCFDMA and SCCFDMA in terms of occupied subcarriers for the
same number of users.
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SubcarrierMapping
Channel
N 
pointIDFT
Subcarrier De
mapping/Equalization
M 
pointDFT
Detect
RemoveCP
N 
point
DFT
M 
point
IDFT
Add
CP/
PS
SpreadingDe
spreadingSCFDMA Modulation SCFDMA Demodulation
Figure 7.9: Block diagram of an SCCFDMA system.
subcarriers
User 1
User 2
User 3
subcarriers
SCFDMA
SCCFDMA
User 4
User 1
User 2
User 3
User 4
Code 1
Code 2
Q (# of users) = 4 Q (# of users) = Qcode×Q frequency= 2×2 = 4
Figure 7.10: Comparison between SCFDMA and SCCFDMA in terms of occupied subcarriers
for the same number of users.
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0
2
4
6
8
1012
14
16
25/75 50/50 75/25
Low/high mobility users percentile [%/%]
A g g r e g a t e t h r o u g h p u t
[ M b p s ] IFDMA: static
Hybrid
LFDMA: CDS
Figure 7.11: Aggregate throughputs for hybrid subcarrier mapping method and other conventional
subcarrier mapping methods with CDS and adaptive modulation.
Figure 7.11 compares the aggregate throughput between hybrid subcarrier mapping
method and conventional subcarrier mapping methods when we apply CDS and adaptive
modulation. In hybrid subcarrier mapping method, we apply LFDMA with CDS to low
mobility users and IFDMA with static scheduling to high mobility users. Total number of
subcarriers is M = 256, the number of chunks is Q = 16, the number of subcarriers per chunk
is N = 16, the total number of users is K = 64, and the feedback delay is 3 ms. Low mobility
users have velocity of 3 km/h and high mobility users have velocity of 60 km/h. In the figure,
the label ‘X/Y’ means that Xpercentile of the users have low mobility and Ypercentile of the
users have high mobility. Clearly, when high mobility users are the majority of the entire users
(the graphs labeled ‘25/75’ and ‘50/50’), hybrid subcarrier mapping scheme yields higher data
throughput than the conventional subcarrier mapping schemes.
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7.4. Summary and Conclusions
In this chapter, we investigated channeldependent scheduling (CDS) for an uplink SCFDMA
system. Specifically, we analyzed the capacity gains from CDS for the two different flavors of
subcarrier mapping of SCFDMA. We also looked into the impact of imperfect channel state
information (CSI) on CDS by considering the feedback delay.
We showed that localized subcarrier mapping yields highest aggregate data throughput
when we use CDS. However, we also showed that LFDMA is very sensitive to the quality of
CSI and the capacity gain quickly decreases when the channel changes very fast. For high
mobility users, IFDMA with static roundrobin scheduling is more suitable.
To accommodate both low and high mobility users simultaneously, we proposed a hybrid
subcarrier mapping method using orthogonal code spreading on top of SCFDMA. We
showed that the hybrid subcarrier mapping scheme has capacity gains over conventional
subcarrier mapping schemes when there are both low and high mobility users at the same time.
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Chapter 8Chapter 8Chapter 8Chapter 8
ConclusionsConclusionsConclusionsConclusions
Single carrier FDMA (SCFDMA) is a new multiple access scheme that is currently being
adopted in uplink of 3GPP LTE. SCFDMA utilizes single carrier modulation and frequency
domain equalization, and has similar performance and essentially the same overall complexity as
those of OFDMA system. A salient advantage over OFDMA is that the SCFDMA signal has
lower PAPR because of its inherent single carrier transmission structure. SCFDMA has drawn
great attention as an attractive alternative to OFDMA, especially in the uplink communications
where lower PAPR greatly benefits the mobile terminal in terms of transmit power efficiency
and manufacturing cost.
In this thesis, we gave a detailed overview of an SCFDMA system and compared it with
the OFDMA system and also with the direct sequence CDMA (DSCDMA) with frequency
domain equalization (FDE) system. We also explained the different subcarrier mapping schemes
in SCFDMA; distributed and localized. The two flavors of subcarrier mapping schemes give the
system designer a flexibility to adapt to the specific needs and requirements.
Along with the introduction of SCFDMA, we investigated the followings.
• Realization of MIMO spatial multiplexing in SCFDMA with practical limitations : We
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specifically analyzed the unitary precoded transmit eigenbeamforming (TxBF)
technique on an SCFDMA system. We showed with numerical simulations that there is
a tradeoff between the feedback overhead and the link performance loss. We also
showed that feedback delay is another impairment that degrades the performance
especially for the high speed mobile users.
• Analysis of the peak power characteristics of SCFDMA signals : By using Chernoff bound and
using Monte Carlo simulations, we characterized the peak power of an SCFDMA
signal both analytically and numerically. We saw that the peak power of an SCFDMA
signal for any subcarrier mapping is indeed lower than that of OFDM and observed
that IFDMA has the lowest peak power among the considered subcarrier mapping
schemes of SCFDMA. For IFDMA, the rolloff factor of the raisedcosine pulse
shaping filter has a great impact on the distribution of the instantaneous power. For
LFDMA, the input block size is the major factor characterizing the distribution of peak
power. For multiple antenna transmission, we observed that unitary precoded TxBF
scheme increases the peak power because of its filtering aspect. Interestingly, averaging
and quantization of the precoder matrix actually reduces the peak power.
• Peak power reduction by symbol amplitude clipping : In order to reduce the peak power in an
SCFDMA system, we proposed a symbol amplitude clipping method and applied it to
the 2x2 unitary precoded TxBF system. Numerical simulation results showed that
moderate clipping does not affect the link level performance. But outofband radiation
is a concern when we use clipping.
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• Channeldependent resource scheduling for SCFDMA: We investigated channeldependent
scheduling (CDS) for an uplink SCFDMA system. Specifically, we analyzed the capacity
gains from CDS for the two different flavors of subcarrier mapping of SCFDMA. We
also looked into the impact of imperfect channel state information (CSI) on CDS by
considering the feedback delay. We showed that localized subcarrier mapping yields
highest aggregate data throughput when we use CDS. However, we also showed that
LFDMA is very sensitive to the quality of CSI and the capacity gain quickly decreases
when the channel changes very fast. For high mobility users, IFDMA with static round
robin scheduling is more suitable since it does not require any computation and the
PAPR is low. To accommodate both low and high mobility users simultaneously, we
proposed a hybrid subcarrier mapping method using orthogonal code spreading on top
of SCFDMA. We showed that the hybrid subcarrier mapping scheme has capacity
gains over conventional subcarrier mapping schemes when there are both low and high
mobility users at the same time.
For future work, we list the following items.
• Impact of carrier frequency offset on the system performance: In this thesis, we did not
consider carrier frequency offset which is a common impairment at the receiver. Its
impact on the link level performance and capacity is an important issue for a practical
implementation.
• Peak power characteristics of SCFDMA pilot reference signals: We can apply the same
numerical analysis to the pilot signals proposed by 3GPP LTE and characterize the peak
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power characteristics.
• More sophisticated clipping methods, such as iterative clipping, for peak power
reduction: We showed that clipping is a very effective way to reduce peak power without
compromising the link performance. Yet, outofband radiation is a problem for
clipping. In order to mitigate the outofband radiation associated with clipping, we can
apply clipping before and after the pulse shaping and iterate the process until we reach a
satisfactory level of performance.
• Impact of channel estimation error on unitary precoded TxBF MIMO SCFDMA
system: In our analysis, we assumed that the channel estimation is without error. In real
systems, channel estimation error is unavoidable and this will have a significant impact
on our unitary precoded TxBF MIMO SCFDMA system.
• Impact of channel estimation error on CDS considering error correction coding:
Channel estimation error adds to the performance degradation in adaptive modulation
scheme and channeldependent resource scheduling scheme. We can analyze its impact
on capacity using our adaptive modulation/CDS system.
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Appendix AAppendix AAppendix AAppendix A
DerivationsDerivationsDerivationsDerivations of theof theof theof the Upper BoundUpper BoundUpper BoundUpper Boundssss in Chapter 5in Chapter 5in Chapter 5in Chapter 5
A.1 Derivation of (5.8) and (5.9)
We can expand { } Z E Zeν as follows.
{ }
{ } { }
exp l
k l
k l
a Z
k l k
k l k l
a a
k
k ll k
a a
k
k ll k
E Ze E a a E a e
E a e e
E a e E e
ν ν
ν ν
ν ν
ν ∞∞ ∞ ∞
=−∞ =−∞ =−∞ =−∞
∞∞
=−∞ =−∞≠
∞∞
=−∞ =−∞≠
= =
=
=
∑ ∑ ∑ ∏
∑ ∏
∑ ∏
(A.1)
We can also expand { } Z E eν similarly as follows.
{ }
{ }
exp l
l
a Z
l
l l
a
l
E e E a E e
E e
ν ν
ν
ν ∞∞
=−∞ =−∞
∞
=−∞
= =
=
∑ ∏
∏
(A.2)
By applying (A.1) and (A.2) to (5.7) for ˆν ν = , we get the following.
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{ } { } { }
{ } { } { }
ˆ ˆ ˆ
ˆ ˆ ˆ
0k l l
k l l
a a a
k
k l ll k
a a a
k
k l ll k
E a e E e E e
E a e E e E e
ν ν ν
ν ν ν
δ
δ
∞ ∞∞
=−∞ =−∞ =−∞≠
∞ ∞∞
=−∞ =−∞ =−∞≠
− ⋅ =
= ⋅
∑ ∏ ∏
∑ ∏ ∏(A.3)
By dividing each side with { }ˆ la
l
E eν
∞
=−∞∏ , we obtain
{ }
{ }
ˆ
ˆ
k
k
a
k
ak
E a e
E e
ν
ν δ
∞
=−∞
=∑ (A.4)
Using (A.2), (5.6) becomes
{ } { }ˆˆPr k a
k
Z e E eν νδ δ
∞−
=−∞
≥ ≤ ∏ (A.5)
A.2 Derivations of (5.12) and (5.13)
From (5.11),
{ }0 0( ), ( )
k BPSK
k
s C
a p t kT p t kT
∈
∈ − − −(A.6)
Also,
{ } { }
{ } { }0 0
11 1
2
1( ) ( )
2
k k
k k
P s P s
P a p t kT P a p t kT
= − = = =
= − − = = − =
(A.7)
Using (A.6) and (A.7), (5.8) becomes
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0 0
max
0 0max
ˆ ˆ( ) ( )
0 0
ˆ ˆ( ) ( )
1 1( ) ( )
2 2
1 12 2
p t kT p t kT K
p t kT p t kT k K
p t kT e p t kT e
e e
ν ν
ν ν
δ
− − −
− − −=−
− − −=
+∑ (A.8)
After we determine ν from (A.8), the upper bound in (5.9) becomes
{ }
( )
( )
max
max
max
0 0
max
max max
0 0
max
max m
0 0
max
ˆˆ( )
,
ˆ ˆˆ ( ) ( )
2 1ˆ ˆˆ ( ) ( )
2
ˆ ˆˆ ( ) ( )
2
1 12
2 2
12
2
1
2
k
K a BPSK
ub SC
k K
K p t kT p t kT
k K
K K p t kT p t kT
k K
K K p t kT p t kT
k K
P e E e
e e e
e e e
e e e
ν νδ
ν ν νδ
ν ν νδ
ν ν νδ
−
=−
− − −−
=−
+− − −−
=−
− − −−
=−
= +
= +
= +
∏
∏
∏
≜
ax
∏
(A.9)
and we upperbound the CCDF as
2( ) ( )
0 ,Pr ( , ) BPSK BPSK
ub SC x t s w P ≥ ≤
(A.10)
A.3 Modified Upper Bound Derivation for QPSK
For complex modulation constellations, we can express (5.4) as follows from the probability
regions in Figure A.1.
2 2
0
2 2
Pr ( , ) Pr
Pr Pr
2 2
Pr Pr
re im
re im
x t s w Z w
w w Z Z
Z Z δ δ
≥ = ≥
≤ ≥ + ≥
′ ′ = ≥ + ≥
(A.11)
where
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{ }Re Z
{ }Im Z
ww−
w−
w 2 Z w=
2w
2w
2w−
2w−
{ }Re2
w Z = −
{ }Im2
w Z =
{ }Im2
w Z = −
{ }Re2
w Z =
Figure A.1: Probablity regions for complex modulation constellations.
{ } { } { }max max
max max
0Re Re Re ( )K K
re k k
k K k K
Z Z a s p t kT =− =−
= = = −∑ ∑ (A.12)
,
{ } { } { }max max
max max
0Im Im Im ( )
K K
im k k
k K k K
Z Z a s p t kT =− =−
= = = −∑ ∑ (A.13)
, and2
wδ ′ = . Since (A.11) is not a tightlybounded inequality, we can instead use
wδ γ ′ = to produce a tighter bound by adjusting 1
2γ > empirically.
Similarly with (5.5),
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[ ]
[ ]
Pr 2Pr
Pr 2Pr
re re
im im
Z Z
Z Z
δ δ
δ δ
′ ′ ≥ = ≥
′ ′ ≥ = ≥
(A.14)
We consider a QPSK constellation sk such that { } { }1 1
Re , Im ,2 2
k k s s
∈ −
and
{ }Re k s and { }Im k s are independent and uniformly distributed. Then,
[ ] [ ]Pr Prre im Z Z δ δ ′ ′≥ = ≥ (A.15)
Thus (A.11) becomes
[ ] [ ]{ }
[ ]
2
0Pr ( , ) Pr Pr
2 Pr Pr
4Pr
re im
re im
re
x t s w Z Z
Z Z
Z
δ δ
δ δ
δ
′ ′ ≥ ≤ ≥ + ≥
′ ′= ≥ + ≥
′= ≥
(A.16)
From (A.12),
{ }{ }
{ }
max
max
ˆ Reˆ
Prk
K a
re
k K Z e E e
ν νδ
δ ′−
=−′≥ ≤ ∏ (A.17)
where ν is the solution of the following equation.
{ } { }{ }{ }{ }
max
max
ˆ Re
ˆ Re
Re k
k
aK
k
ak K
E a e
E e
ν
ν δ
=−
′=∑ (A.18)
Then, we can upperbound the CCDF2
0Pr ( , ) x t s w ≥
as follows.
{ }{ }max
max
2 ˆ Reˆ ( )
0 ,Pr ( , ) 4 k
K a QPSK
ub SC
k K
x t s w e E e Pν νδ ′−
=−
≥ ≤ ∏ ≜ (A.19)
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where wδ γ ′ = .
Similarly with the case of BPSK in section 5.1.1, we can express ( )
,
QPSK
ub SC P as follows.
max max0 0
max
ˆ ˆ2 1( ) ( )
ˆ( ) 2 2,
1
2
K K p t kT p t kT BPSK
ub SC
k K
P e e e
ν ν
νδ
−− − −
−
=−
= +
∏ (A.20)
A.4 Modified Upper Bound for LFDMA and DFDMA
In this section, we derive the upper bound of the CCDF for LFDMA symbols which will also
apply to the case of DFDMA.
In chapter 3 section 3.3.2, we derived the time domain symbols after LFDMA modulation
without pulse shaping as follows. We follow the symbol notations in section 3.3 of chapter 3.
( )mod
12
( )20
1, 0
1 11 , 0
1
N m
q N m Q n q j pQ
n p q j p N Q N
x qQ
x x x
e qQ N
e
π
π
−⋅ +
− + =⋅
=
= =
⋅ − ⋅ ≠
− ∑
ɶ ɶ (A.21)
where m Q n q= ⋅ + , 0 1n N ≤ ≤ − , and 0 1q Q≤ ≤ − .
We can expressmQ x⋅ ɶ as follows when 0q ≠ .
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12
( )20
( )
1
( ) ( )0
( 1)
11
1
1
( 2 )sin
q N j pQ
m n p q j p
N Q N
n p q j
q q q N Q N N j j j pQ Q Q
n p q n p q j j p
N Q N N Q N
N q Qn j
QN
xQ x e
N
e
e xe e e
N e e
qe j
Q
π
π
π
π π π
π π
π
π
−
−+ =
⋅
−− +
⋅ −−
− −− + +=
⋅ ⋅
− −
⋅ = − ⋅
−
= − ⋅
−
= ⋅ − ⋅
∑
∑
ɶ
( ),
1
0
( 1) 1
0
ˆ
( , , )
( 1) 1
0
1
( )( 2 )sin
sin
( )sin
ˆ( , , )
p
LFDMAn q
p j
N N p
p
N q Qn p N j jQN N
p
p
x
c n q p
N q Qn N jQN
p
p
Z
e x
N Qn q p j
QN N
q
Qe e x
Qn q p N
QN N
e c n q p x
π
π π
π
π π
π
π π
−
=
− − −
=
− − −
=
+− ⋅ −
= ⋅
+ ⋅ −
= ⋅ ⋅
∑
∑
∑
≜
≜
≜
(A.22)
where( )
( , , ) sin sinq Qn q p
c n q p N Q QN N
π π π +
= ⋅ −
, ( )ˆ exp p p x j p N xπ = ⋅ , and
1( )
, 0ˆ( , , )
N LFDMA
n q p p Z c n q p x
−
== ⋅∑ .
Our objective now is to characterize the following CCDF of an LFDMA signal.
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{ } { }{ }
{ } { }{ }
( ),
( ),
ˆ Reˆ( )
,
ˆ Imˆ( )
,
Pr Re
Pr Im
LFDMAre n qre
LFDMAim n qim
Z LFDMA
n q
Z LFDMA
n q
Z e E e
Z e E e
ν ν δ
ν ν δ
δ
δ
′−
′−
′≥ ≤
′≥ ≤
(A.26)
where ˆreν and imν are the solutions of the following equations, respectively.
{ } { }{ } { }{ }{ } { }{ } { }{ }
( ) ( ), ,
( ) ( ), ,
Re Re( )
,
Im Im( )
,
Re 0
Im 0
LFDMA LFDMAn q n q
LFDMA LFDMAn q n q
Z Z LFDMA
n q
Z Z LFDMA
n q
E Z e E e
E Z e E e
ν ν
ν ν
δ
δ
′− ⋅ =
′− ⋅ =
(A.27)
From (A.22)
{ } { }
{ } { }
{ } { }
1( )
,
0
1
0
( , , )
1
0
1( )
,
0
ˆRe ( , , ) Re
( , , ) cos Re sin Im
( , , )
ˆIm ( , , ) Im
( , , ) sin Re
re
N LFDMA
n q p
p
N
p p
p
d n q p
N
re
p
N LFDMA
n q p
p
Z c n q p x
p pc n q p x x
N N
d n q p
Z c n q p x
pc n q p
N
π π
π
−
=
−
=
−
=
−
=
= ⋅
= ⋅ −
=
= ⋅
= ⋅
∑
∑
∑
∑
≜
{ } { }1
0
( , , )
1
0
cos Im
( , , )
im
N
p p
p
d n q p
N
im
p
p x x
N
d n q p
π −
=
−
=
+
=
∑
∑
≜
(A.28)
where ( ) { } ( ) { }( , , ) ( , , ) cos Re sin Imre p pd n q p c n q p p N x p N xπ π
= ⋅ − and
( ) { } ( ) { }( , , ) ( , , ) sin Re cos Imim p pd n q p c n q p p N x p N xπ π = ⋅ + .
Using (A.28),
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{ } { }{ }( )
,Re( )
,
1 1
0 0
11( , , )
0 0
11( , , ) ( , , )
0 0
Re
( , , ) exp ( , , )
( , , )
( , , )
( , , )
LFDMAn q
re
re re
Z LFDMA
n q
N N
re re
p l
N N d n q l
re
p l
N N d n q p d n q l
re
p ll p
re
E Z e
E d n q p d n q l
E d n q p e
E d n q p e e
E d n q p
ν
ν
ν ν
ν − −
= =
−−
= =
−−
= =≠
=
=
=
=
∑ ∑
∑ ∏
∑ ∏
{ } { }
{ } { }{ }
{ } { }
( ),
11( , , ) ( , , )
0 0
Im( )
,
11( , , ) ( , , )
0 0
Im
( , , )
re re
LFDMAn q
im im
N N d n q p d n q l
p ll p
Z LFDMA
n q
N N d n q p d n q l
im
p ll p
e E e
E Z e
E d n q p e E e
ν ν
ν
ν ν
−−
= =≠
−−
= =≠
=
∑ ∏
∑ ∏ (A.29)
and
{ }{ }{ }
{ }{ } { }
( )
,
( ),
11
Re ( , , )
0 0
1( , , )
0
1Im ( , , )
0
exp ( , , ) LFDMA
n q re
re
LFDMAn q im
N N
Z d n q lre
l l
N d n q l
l
N Z d n q l
l
E e E d n q l E e
E e
E e E e
ν ν
ν
ν ν
ν −−
= =
−
=
−
=
= =
=
=
∑ ∏
∏
∏
(A.30)
From (A.29) and (A.30), (A.27) becomes
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{ }
{ }
{ }
{ }
ˆ ( , , )1
ˆ ( , , )0
ˆ ( , , )1
ˆ ( , , )0
( , , )
( , , )
re re
re re
im im
im im
d n q p N
re
d n q p p
d n q p N
im
d n q p p
E d n q p e
E e
E d n q p e
E e
ν
ν
ν
ν
δ
δ
−
=
−
=
′=
′=
∑
∑(A.31)
and we can upperbound the CCDF of an LFDMA signal as follows.
{ }
{ }
{ }
{ }{ } { }
( ) ( ), ,
12
0
11ˆ ˆRe Imˆ ˆ
0 1
1 1 11ˆ ˆ ˆ ˆ( , , ) ( , , )
0 1 0 0
,
1Pr
12
12
LFDMA LFDMAre n q im n qre im
re re re im im im
M
m
m
Q N Z Z
n q
Q N N N d n q l d n q l
n q l l
ub LFDMA
Q x w M
e E e e E e
M
e E e e E e M
P
ν ν ν δ ν δ
ν δ ν ν δ ν
−
=
−−′ ′− −
= =
− − −−′ ′− −
= = = =
⋅ ≥
≤ +
≤ +
∑
∑∑
∑∑ ∏ ∏
ɶ
≜
(A.32)
where ˆreν and imν are the solutions of (A.31) and wδ γ ′ = .
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