Non-Orthogonal Multiple Access Schemes for Future Cellular...
Transcript of Non-Orthogonal Multiple Access Schemes for Future Cellular...
Non-Orthogonal Multiple Access Schemes forFuture Cellular Systems
Lei Wen
Submitted for the Degree ofDoctor of Philosophy
from theUniversity of Surrey
5G Innovation CentreInstitute for Communication Systems
Faculty of Engineering and Physical SciencesUniversity of Surrey
Guildford, Surrey GU2 7XH, U.K.
March 2016
c⃝ Lei Wen 2016
Abstract
Non-orthogonal multiple access (NOMA) is an emerging technology for future cellularsystems in order to accommodate more users via non-orthogonal resource allocation,especially when the number of users/devices exceeds the available degrees of freedom,resulting in an overloaded condition. To date, low density signature (LDS) and sparsecode multiple access (SCMA) are two promising NOMA techniques. However, researchin this area is still in its infancy and there are still several open issues in the LD-S/SCMA transceiver design. This thesis aims to address some of these challenges. Thecontributions are summarized as follows.
1:-LDS-OFDM and low density parity check (LDPC) codes both can be representedby a bipartite graph. Inspired by their similar structure, we construct a joint sparsegraph combining the single graphs of LDS-OFDM and LDPC codes, namely joint sparsegraph for OFDM (JSG-OFDM). A joint detection and decoding scenario is proposedusing message passing algorithm (MPA). Design guidelines for the joint sparse graphare derived through extrinsic information transfer (EXIT) chart analysis. Simulationresults show that the JSG-OFDM outperforms existing techniques such as GO-MC-CDMA, LDS-OFDM and turbo structured LDS-OFDM.
2:-Due to the higher power and spectral efficiency, the filter-bank multi-carrier (FBMC)technique is a promising alternative to OFDM. We use a low density graph to modelthe weight matrix of intrinsic interference in the isotropic orthogonal transform algo-rithm (IOTA) filtered FBMC system. In addition, such graph is combined with LDSand LDPC codes to form a joint sparse graph for FBMC-IOTA (JSG-IOTA). Basedon MPA, a joint detection and decoding scheme is designed for JSG-IOTA, and thejoint sparse graph is optimized by EXIT chart analysis. Numerical results show thesuperiority of JSG-IOTA to conventional techniques.
3:-In SCMA, the processes of bit to symbol mapping and LDS spreading are com-bined together. We investigate multi-dimensional SCMA codebooks, and the designrules are derived to maximize the constellation shaping gain. Moreover, we propose toconstruct SCMA codebooks by copy-and-permute operation on protographs and three-dimensional (3D) constellation shaping. Simulation results show that SCMA outper-forms LDS with high-order constellations, and the proposed optimization methods canfurther improve the SCMA performance.
Key words: Non-orthogonal Multiple Access, 5G, Low Density Signature, Sparse CodeMultiple Access, Joint Sparse Graph, Joint Receiver, Joint Detection and Decoding
Email: [email protected]
Acknowledgements
First of all, I would like to indicate that it is my great fortune to have pursued myPh.D. study under the supervision of Dr Pei Xiao and Dr Muhammad Ali Imran. Ihave learned greatly from their strong knowledge, out-of-box thinking, and exceptionalinsight opinions on the technical as well as practical area of my work. Many thanks fortheir continuous support and encouragement. I would also thank Prof. Rahim Tafazollifor his useful directions and feedbacks for the improvement of my research work of myPh.D. A very special thanks goes out to my friends and colleagues in 5GIC, RaziehRazavi, Mohammed AL-Imari, Man Su, Chang He, Yuchao Zhou, Gaojie Chen, JieZhong, Lei Zhang, Yinan Qi, Yue Cao, who made my years in the University of Surreyso enjoyable. Furthermore, I would thank my wife, my son and my parents for theirsupport.
List of Acronyms
2G The Second Generation2D Two-dimensional3D Three-dimensional3G The Third Generation3GPP 3rd Generation Partnership Project Access Interference4G The Forth Generation5G The Fifth GenerationAMC Adaptive Modulation And CodingAPP A Posteriori ProbabilityAWGN Additive White Gaussian NoiseBER Bit Error RateBPSK Binary Phase Shift KeyingBS Base StationCDMA Code Division Multiple AccessCND Check Node DetectorCNDD Check Node Detector-DecoderCP Cyclic PrefixCQI Channel Quality IndicatorCSI Channel State InformationDECT Digital Enhanced Cordless TelecommunicationsDFT Discrete Fourier TransformeNB Enhanced NodeBEGC Equal Gain CombiningEGF Extended Gaussian FunctionEXIT Extrinsic Information TransferFBMC Filter-bank Multi-carrierFEC Forward Error CorrectionFFT Fast Fourier TransformGO-MC-CDMA Group Orthogonal Multi-Carrier Code Division Multiple AccessGSM Global System For Mobile CommunicationsICI Intercarrier InterferenceiDEN Integrated Digital Enhanced NetworkIFFT Inverse Fast Fourier Transformi.i.d independently and identically distributedIND Intrinsic-Interference Nodes DecoderIOTA Isotropic Orthogonal Transform Algorithm
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ISI Intersymbol InterferenceITU International Telecommunication UnionJSG Joint Sparse GraphJSG-OFDM Joint Sparse Graph Orthogonal Frequency Division MultiplexJSG-IOTA Joint Sparse Graph Isotropic Orthogonal Transform AlgorithmLDGM Low Density Generator CheckLDPC Low Density Parity CheckLDS Low Density SignatureLDS-CDMA Low Density Signature Code Division Multiple AccessLDS-OFDM Low Density Signature Orthogonal Frequency Division MultiplexLDWM Low Density Weight MatrixLLR Log-Likelihood RatioLT Luby TransformLTE Long Term EvolutionMAI Multiple Access InterferenceMAP Maximum A PosterioriMC-CDMA Multi-Carrier Code Division Multiple AccessMCS Modulation And Coding SchemeMIMO Multiple-Input Multiple-OutputML Maximum LikelihoodMMSE Minimum Mean-Square ErrorMPA Message Passing AlgorithmMRC Maximum Ratio CombiningMTC Machine Type CommunicationMUD Multiuser DetectionMUI Multiuser InterferenceMUSA Multiuser Shared AccessNOMA Non-orthogonal Multiple AccessOFDM Orthogonal Frequency Division multiplexingOFDMA Orthogonal Frequency Division Multiple AccessOMA Orthogonal Multiple AccessOQPSK Offset Quadrature Phase-shift KeyingPDA Probabilistic Data AssociationPDC Personal Digital CellularPDF Probability Density FunctionPDMA Pattern Division Multiple AccessPIC Parallel Interference CancellationPMF Probability Mass FunctionPND Parity-check Node DecoderPON Passive Optical NetworkPPIC Partial Parallel Interference CancellationP/S Parallel To SerialQAM Quadrature Amplitude ModulationQoS Quality of ServiceQPSK Quadrature Phase Shift KeyingRA Repeat-AccumulateRACH Random-access channel
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RB Resource BlockRRM Radio Resource ManagementSCMA Sparse Code Multiple AccessSIC Successive Interference CancellationSIMO Single-input Multiple-outputSISO Soft-Input Soft-OutputSNR Signal to Noise RatioS/P Serial To ParallelTDMA Time Division Multiple AccessVND Variable Node DetectorVNDD Variable Node Detector-DecoderWBE Welch Bound EqualityWiMAX Worldwide Interoperability For Microwave AccessZF Zero Forcing
List of Symbols and Notations
A Transmit power gaincn The nth chip, also represents chip nodedc,lds Number of symbols that are superimposed at one chipdc,ldwm Number of intrinsic-interference nodes connected to one chip nodedi,ldwm Number of chip nodes connected to one intrinsic-interference nodedp,ldpc Number of variable nodes connected to one parity-check nodedv,ldpc Number of parity-check nodes connected to one variable nodedv,lds Number of chips that are spread by one symbolDCND(x) Degree distribution polynomials of chip nodesDCNDD(x) Degree distribution polynomials of chip nodesDIND(x) Degree distribution polynomials of intrinsic-interference nodesDPND(x) Degree distribution polynomials of parity-check nodesDV NDD(x) Degree distribution polynomials of variable nodesEk Channel gain for the kth userH Low density parity-check matrices for LDPC codesHk Parity-check matrix for the kth userIA,CND&PND Mutual information between CND&PND and a priori LLRIA,CNDD&PND Mutual information between CNDD&PND and a priori LLRIA,IND&V NDD Mutual information between IND&VNDD and a priori LLRIA,V NDD Mutual information between VNDD and a priori LLRIE,CND&PND Mutual information between CND&PND and extrinsic LLRIE,CNDD&PND Mutual information between CNDD&PND and extrinsic LLRIE,IND&V NDD Mutual information between IND&VNDD and extrinsic LLRIE,V NDD Mutual information between VNDD and extrinsic LLRJ Number of parity-check equations of LDPC codesK Number of usersLcn→in,u LLR delivered from chip node cn to intrinsic-interference node in,uLcn→vk,m LLR delivered from chip node cn to variable node vk,mLin,u→cn LLR delivered from intrinsic-interference node in,u to chip node cnLpj→vk,m LLR delivered from parity-check node pj to variable node vk,mLvk,m Final estimation of the variable node vk,mLvk,m→cn LLR delivered from variable node vk,m to chip node cnLvk,m→pj LLR delivered from variable node vk,m to parity-check node pjM Data length of each userN Number of chipspk,j The jth parity-check equation of the kth user
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r[n] Received vectors that spread on the nth chip
rk,m Received spreading sequence for the symbol m of the kth userrnk,m Received signature gain at the nth chip of the variable node vk,mS Low density spreading signaturesSk Spreading matrix for the kth userv Transmitted vectorvk,m The mth symbol of the kth user, also represents variable nodev̂k,m Estimated value of the variable node vk,mv[n] Transmitted vectors containing the symbols spread on the nth chip
y Received signalyn Received signal corresponding to the nth chipz AWGNzn AWGN on the nth chipσ2A Variance of the noiseκn,k,m Normalization coefficientψn Set of symbols interfered on chip cnψn/(k,m) Set of symbols (excluding vk,m) interfered on chip cnεk,m Set of chips spread by vk,mεk,m/n Set of chips (excluding cn) spread by vk,mϕj Set of symbols connected to parity-check node pk,jϕj/(k,m) Set of symbols (excluding vk,m) connected to parity-check node pk,jωk,m Set of parity-check nodes connected to vk,mωk,m/j Set of parity-check nodes (excluding pk,j) connected to vk,mΓ SCMA codebook set
Contents
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivations and Research Objectives . . . . . . . . . . . . . . . . . . . . 3
1.3 Research Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Survey on Multiple Access Techniques and Their Receiver Design 13
2.1 Multiple Access Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Time Division Multiple Access (TDMA) . . . . . . . . . . . . . . 14
2.1.2 Code Division Multiple Access (CDMA) . . . . . . . . . . . . . . 15
2.1.3 Orthogonal Frequency-Division Multiple Access (OFDMA) . . . 16
2.1.4 Multi-carrier Code Division Multiple Access (MC-CDMA) . . . . 18
2.1.5 Low Density Signature (LDS) . . . . . . . . . . . . . . . . . . . . 20
2.1.5.1 LDS-CDMA . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.5.2 LDS-OFDM . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.6 Sparse Code Multiple Access (SCMA) . . . . . . . . . . . . . . . 25
2.1.7 Multiuser Shared Access (MUSA) . . . . . . . . . . . . . . . . . 27
2.1.8 Pattern Division Multiple Access (PDMA) . . . . . . . . . . . . 28
2.2 Multiuser Detection Techniques . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.1 Optimum Multiuser Detector . . . . . . . . . . . . . . . . . . . . 29
2.2.2 Linear Multiuser Detector . . . . . . . . . . . . . . . . . . . . . . 30
2.2.2.1 Minimum Mean Square Error (MMSE) . . . . . . . . . 31
2.2.2.2 Decorrelator . . . . . . . . . . . . . . . . . . . . . . . . 31
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2.2.3 Nonlinear Multiuser Detector . . . . . . . . . . . . . . . . . . . . 32
2.2.3.1 Successive Interference Cancellation (SIC) . . . . . . . 32
2.2.3.2 Parallel Interference Cancelation (PIC) . . . . . . . . . 32
2.2.3.3 Probabilistic Data Association (PDA) . . . . . . . . . . 33
2.2.3.4 Message Passing Algorithm (MPA) . . . . . . . . . . . 34
2.3 Receiver Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Joint Sparse Graph for OFDM (JSG-OFDM) System 40
3.1 JSG-OFDM System Model . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Joint Detection and Decoding for JSG-OFDM . . . . . . . . . . . . . . . 43
3.2.1 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.2 Updating of Chip Nodes and Parity-Check Nodes . . . . . . . . . 45
3.2.3 Updating of Variable Nodes . . . . . . . . . . . . . . . . . . . . . 46
3.2.4 Estimation and Syndrome Computing . . . . . . . . . . . . . . . 46
3.3 EXIT Chart Analysis of JSG-OFDM . . . . . . . . . . . . . . . . . . . . 47
3.3.1 Iterative Structure of the Joint Sparse Graph . . . . . . . . . . . 48
3.3.2 EXIT Chart Analysis Over AWGN Channel . . . . . . . . . . . . 49
3.3.2.1 EXIT Curve for VNDD . . . . . . . . . . . . . . . . . . 49
3.3.2.2 EXIT Curve for CND&PND . . . . . . . . . . . . . . . 51
3.3.2.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.3 EXIT Chart Analysis Over Multipath Fading Channels . . . . . 53
3.4 EXIT Chart Based Design of Joint Sparse Graph . . . . . . . . . . . . . 54
3.4.1 Degree Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4.2 Short Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.5.1 Evaluation Configuration . . . . . . . . . . . . . . . . . . . . . . 60
3.5.2 BER Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.5.3 Convergence Behavior . . . . . . . . . . . . . . . . . . . . . . . . 64
3.5.4 Performance of Different Users . . . . . . . . . . . . . . . . . . . 64
3.5.5 Near-far Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.5.6 Multipath Diversity . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.5.7 Comparison with MMSE Detector . . . . . . . . . . . . . . . . . 67
3.5.8 Detection Complexity Comparison . . . . . . . . . . . . . . . . . 67
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
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4 Joint Sparse Graph for FBMC-IOTA (JSG-IOTA) System 71
4.1 JSG-IOTA System Model . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2 Joint Detection and Decoding . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 EXIT Chart Analysis of JSG-IOTA . . . . . . . . . . . . . . . . . . . . . 83
4.3.1 Iterative Structure of the Joint Sparse Graph . . . . . . . . . . . 83
4.3.2 EXIT Chart Analysis . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3.2.1 EXIT Curve for IND&VNDD . . . . . . . . . . . . . . . 86
4.3.2.2 EXIT Curve for CNDD&PND . . . . . . . . . . . . . . 87
4.3.2.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.4 EXIT Chart Assisted Joint Sparse Graph Design . . . . . . . . . . . . . 90
4.4.1 Degree Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.4.2 Short Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.4.3 Maximum Achievable Throughput . . . . . . . . . . . . . . . . . 93
4.5 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.5.1 BER Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.5.2 Convergence Behavior . . . . . . . . . . . . . . . . . . . . . . . . 97
4.5.3 Performance of Different Users . . . . . . . . . . . . . . . . . . . 98
4.5.4 Dynamic Subcarrier Allocation . . . . . . . . . . . . . . . . . . . 99
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5 Sparse Code Multiple Access (SCMA) 101
5.1 Criteria of SCMA Codebook Design . . . . . . . . . . . . . . . . . . . . 101
5.1.1 SCMA System Model . . . . . . . . . . . . . . . . . . . . . . . . 101
5.1.2 Multi-stage Optimization Approach of SCMA Codebooks . . . . 103
5.1.2.1 Low Density Signature . . . . . . . . . . . . . . . . . . 103
5.1.2.2 Constellation Points . . . . . . . . . . . . . . . . . . . . 104
5.1.2.3 Mother Multi-dimensional Constellation . . . . . . . . . 104
5.1.2.4 Constellation Function Operators . . . . . . . . . . . . 109
5.1.3 Performance Comparison with LDS . . . . . . . . . . . . . . . . 110
5.2 Design of SCMA Codebooks Based on Protographs . . . . . . . . . . . . 111
5.2.1 Extension of SCMA Codewords by Copy Operation . . . . . . . 111
Contents xiii
5.2.2 Construction of SCMA Codebooks by Copy-and-permute Oper-ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2.3 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . 114
5.3 Design of 3D SCMA Codebooks . . . . . . . . . . . . . . . . . . . . . . . 115
5.3.1 Limitations of 2D SCMA Codebooks . . . . . . . . . . . . . . . . 115
5.3.2 Construction of SCMA codebooks with dv,lds of 3 . . . . . . . . . 116
5.3.3 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . 116
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6 Conclusions and Future Works 119
6.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
A Appendix 123
Bibliography 125
List of Figures
Fig. 1.1 Throughput (capacity) region in downlink . . . . . . . . . . . . . . 4
Fig. 1.2 Throughput (capacity) region in uplink . . . . . . . . . . . . . . . . 4
Fig. 2.1 LDS-CDMA system . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Fig. 2.2 Illustration of a LDS spreader . . . . . . . . . . . . . . . . . . . . . 22
Fig. 2.3 LDS iterative structure . . . . . . . . . . . . . . . . . . . . . . . . . 22
Fig. 2.4 LDS-OFDM system . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Fig. 2.5 SCMA system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Fig. 2.6 MUSA system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Fig. 2.7 Receiver structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Fig. 3.1 JSG-OFDM system model . . . . . . . . . . . . . . . . . . . . . . . 42
Fig. 3.2 Iterative structure of the joint detection and decoding in JSG-OFDM 48
Fig. 3.3 Folded view of the joint sparse graph . . . . . . . . . . . . . . . . . 49
Fig. 3.4 EXIT chart over AWGN Channel at Eb/N0 = 9 dB . . . . . . . . . 52
Fig. 3.5 EXIT chart over ITU Pedestrian Channel B at Eb/N0 = 13 dB . . 55
Fig. 3.6 BER versus IA,V NDD for the joint sparse graph . . . . . . . . . . . 56
Fig. 3.7 EXIT chart for different degree distributions . . . . . . . . . . . . . 58
Fig. 3.8 EXIT chart for different schemes . . . . . . . . . . . . . . . . . . . . 59
Fig. 3.9 Performance of 100% loaded systems . . . . . . . . . . . . . . . . . 62
Fig. 3.10 Performance of 150% loaded systems . . . . . . . . . . . . . . . . . 62
Fig. 3.11 Maximum effective throughput of JSG-OFDM . . . . . . . . . . . 63
Fig. 3.12 Performance at different iterations for JSG-OFDM . . . . . . . . . 64
Fig. 3.13 Performance of different users in JSG-OFDM . . . . . . . . . . . . 65
Fig. 3.14 Near-far effect of JSG-OFDM . . . . . . . . . . . . . . . . . . . . . 66
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List of Figures xv
Fig. 3.15 Performance of JSG-OFDM over different multipath channels . . . 67
Fig. 3.16 Performance comparison with MMSE detector . . . . . . . . . . . 68
Fig. 4.1 JSG-IOTA transmitter model . . . . . . . . . . . . . . . . . . . . . 72
Fig. 4.2 JSG-IOTA receiver model . . . . . . . . . . . . . . . . . . . . . . . 73
Fig. 4.3 Iterative structure of the joint sparse graph . . . . . . . . . . . . . . 83
Fig. 4.4 Tree structure of the joint sparse graph . . . . . . . . . . . . . . . . 85
Fig. 4.5 Folded view of the joint sparse graph . . . . . . . . . . . . . . . . . 85
Fig. 4.6 EXIT chart at Eb/N0 = 12 dB . . . . . . . . . . . . . . . . . . . . . 89
Fig. 4.7 EXIT chart for different degree distributions . . . . . . . . . . . . . 92
Fig. 4.8 EXIT chart for different schemes . . . . . . . . . . . . . . . . . . . . 93
Fig. 4.9 Maximum effective throughput of the joint sparse graph . . . . . . 94
Fig. 4.10 Performance of different systems with 200% loading . . . . . . . . 96
Fig. 4.11 Performance of different systems with 300% loading . . . . . . . . 96
Fig. 4.12 Performance on different iterations at Eb/N0 = 12 dB . . . . . . . 97
Fig. 4.13 Performance of different users . . . . . . . . . . . . . . . . . . . . . 98
Fig. 4.14 Performance of different subcarrier allocation schemes . . . . . . . 99
Fig. 5.1 Merging of QAM modulator and LDS spreading in a SCMA encoder 102
Fig. 5.2 16-point SCMA constellation for dv,lds = 2 . . . . . . . . . . . . . . 107
Fig. 5.3 16-point SCMA constellation with 9-projection-point for dv,lds = 2 . 108
Fig. 5.4 Performance of 150% loaded LDS and SCMA . . . . . . . . . . . . 110
Fig. 5.5 Extension of SCMA codewords by copy operation . . . . . . . . . . 112
Fig. 5.6 Construction of SCMA codebooks by copy-and-permute operation . 113
Fig. 5.7 Performance of 200% loaded SCMA by different construction methods114
Fig. 5.8 4-point SCMA constellation for dv,lds of 2 . . . . . . . . . . . . . . . 117
Fig. 5.9 4-points SCMA constellation for dv,lds of 3 . . . . . . . . . . . . . . 117
Fig. 5.10 Performance of 200% loaded SCMA by different dv,lds . . . . . . . 118
List of Tables
TABLE 2.1 Comparisons of multiple access techniques . . . . . . . . . . . . 39
TABLE 3.1 System parameters . . . . . . . . . . . . . . . . . . . . . . . . . 52
TABLE 3.2 Degree distribution . . . . . . . . . . . . . . . . . . . . . . . . . 57
TABLE 3.3 JSG-OFDM scenarios . . . . . . . . . . . . . . . . . . . . . . . . 61
TABLE 3.4 Equivalent number of addition operations for detection . . . . . 69
TABLE 4.1 System parameters . . . . . . . . . . . . . . . . . . . . . . . . . 88
TABLE 4.2 Degree distribution . . . . . . . . . . . . . . . . . . . . . . . . . 91
TABLE 4.3 JSG-IOTA scenarios . . . . . . . . . . . . . . . . . . . . . . . . 93
TABLE 5.1 SCMA codewords for 16-point . . . . . . . . . . . . . . . . . . . 107
TABLE 5.2 SCMA codewords for 16-point with 9-projection-point . . . . . 109
TABLE A.1 Summary of the joint detection and decoding . . . . . . . . . . 124
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Chapter 1
Introduction
1.1 Background
The explosive traffic growth in wireless communications has motivated many research
activities in both academic and industrial communities. Following the large-scale com-
mercialization of the forth generation (4G) networks, the fifth generation (5G) of mo-
bile communications, which is expected to be standardlized in 2017 and commercialized
towards year 2020 and beyond, has become a focal point for global research and de-
velopment [1] [2]. To satisfy requirements of future cellular systems, some enhanced
technologies have been recently proposed for 5G, e.g. massive multiple-input multiple-
output (MIMO), millimeter wave communications and ultra dense network. With re-
source demanding applications such as mobile internet and Internet of Things (IoT),
air interface techniques need to be designed to achieve improved spectrum utilization
and resource management [3]. Generally speaking, a new air interface for future cel-
lular systems should consist of building blocks and configuration mechanisms such as
advanced multiple access schemes, powerful forward error correction coding, adaptive
multi-carrier waveforms and so on [4] [5]. With these blocks and mechanisms, 5G wire-
less networks can offer significant improvements in coverage and user experience, and
are able to accommodate the a wide variety of user services, spectrum bands and traffic
levels.
The goal in the design of cellular systems is to be able to accommodate as much traffic
1
2 Chapter 1. Introduction
as possible (this is called capacity in cellular terminology) in a given bandwidth with
some reliability. Multiple access technologies allow multiple sources communicate with
the network simultaneously. In the history of wireless communications from the first
generation (1G) to 4G, multiple access techniques have been the key to distinguish
different wireless systems. It is well known that frequency division multiple access
(FDMA) for the first generation (1G), time division multiple access (TDMA) mostly
for the second generation (2G), code division multiple access (CDMA) for the third
generation (3G), and orthogonal frequency division multiple access (OFDMA) for 4G
are the primary multiple access techniques. In these conventional schemes, different
users are allocated to orthogonal resources in either the time/frequency/code domain
in order to avoid or alleviate interuser interference, thus they can be classified as or-
thogonal multiple access (OMA) techniques. In current mobile communication systems
such as Long-Term Evolution (LTE) and LTE-Advanced, OMA techniques have been
adopted, e.g. OFDMA. Ideally, no interference exists among multiple users due to the
orthogonal resource allocation in OMA, simple detection techniques can thus be used
to separate different users’ signals. In other words, the users in each cell are allocated
the resources exclusively and there is no inter-user interference, hence, low-complexity
detection approaches can be implemented on the receiver side to retrieve the users’
signals.
The fast growth of mobile internet has propelled more than 1000-fold data traffic in-
crease for the cellular netwoks. Therefore, how to maximize the spectral efficiency
becomes one of the key challenges to handle such explosive data traffic. Moreover, due
to the rapid development of IoT, 5G systems need to support the massive connectiv-
ity of users and/or devices to meet the demand for low latency, low-cost devices, and
diverse service types. Theoretically, it is known that OMA cannot always achieve the
sum-rate capacity of multiuser wireless systems. Apart from that, in OMA schemes,
the maximum number of supported users is limited by the degree of freedom and the
scheduling granularity of orthogonal resources. This issue is more prominent when fair-
ness among the users is considered [6]. For convenience, we define the system loading
in Definition 1.1.
Definition 1.1 (system loading). Consider a multiuser system with K users and N
1.2. Motivations and Research Objectives 3
dimensions, where the dimensions mean any available degrees of freedom including
chips, subcarriers, I/Q channels, antennas and polarisation. The system loading is
described as the ratio of the number of supported users to the number of dimensions
and is denoted as ρ = K/N . The system is respectively called in under-loaded, fully-
loaded and overloaded conditions when ρ < 1, ρ = 1, and ρ > 1.
1.2 Motivations and Research Objectives
Recently, non-orthogonal multiple access (NOMA), including power domain NOMA
and code domain NOMA, has been attracting a lot of attention. The main difference
between these two groups of NOMA is whether utilizing the spreading technique. Dif-
ferent from conventional OMA, the NOMA schemes are highly expected to improve the
spectral efficiency and accommodate much more users via non-orthogonal resource al-
location. Basically, NOMA allows controllable interference by non-orthogonal resource
allocation with a tolerable increase in the receiver complexity. Compared to OMA, the
main advantages of NOMA include the following.
• Improved spectral efficiency: Fig. 1.1 shows the throughput (capacity) compar-
ison of OMA and NOMA in downlink [7–10], where two users in the additive
white Gaussian noise (AWGN) channel are considered as an example without
loss of generality. The h1 and h2 are complex channel coefficients of the two user-
s, the N0,1 and N0,2 are the power spectral density of Gaussian noise of the two
users, the ptotal is the sum of the two users’ transmission power. It can be seen
that the maximum total throughput is achieved when all the transmission power
is allocated to user 1 only, which is achieved by both OMA and NOMA. However,
the throughput region of NOMA is much wider than that of OMA. For example,
if we want R2 to be 0.8 b/s, the achievable R1 for NOMA is approximately 2-
fold higher than that for OMA. This is because the throughput of user 1 with a
high ptotal| h1 |2/N0,1 is bandwidth-limited rather than power-limited and super-
position coding with user 2 allows user 1 to use the full bandwidth while being
allocated only a small amount of transmission power because of power sharing
4 Chapter 1. Introduction
with user 2. Thus, user 1 imparts only a small amount of interference to user 2.
In contrast, OMA has to allocate a significant fraction of bandwidth to user 2
to increase its throughput, and this causes severe degradation in the throughput
of user 1 whose throughput is bandwidth-limited. As for the uplink, Fig. 1.2
shows the throughput (capacity) comparison of OMA and NOMA with two users
in the AWGN channel [7–10]. We can see that the throughput region of NOMA
is also wider than that for OMA. If we want R2 to be 0.8 b/s, the achievable R1
for NOMA is approximately 60% higher than that for OMA. Therefore, in both
downlink and uplink transmissions, NOMA can improve the spectral efficiency
compared to OMA.
Fig. 1.1: Throughput (capacity) region in downlink
Fig. 1.2: Throughput (capacity) region in uplink
• Massive connectivity: In future cellular communications, the number of users or
parallel data streams will inevitably exceeds the available dimensions as the de-
mand for the spectrum is increasing while the bandwidth is fixed. Under such
an overloaded condition, it is impossible to obtain the orthogonality of received
signatures, consequently the performance of the bit error rate (BER), the block
error rate (BLER) and the system throughput are limited by the severe multiuser
interference (MUI). The non-orthogonal resource allocation in NOMA indicates
1.2. Motivations and Research Objectives 5
that the number of supported users or parallel data streams is not strictly limited
by the amount of available dimensions and their scheduling granularity [11, 12].
Therefore, in both downlink and uplink transmissions, NOMA can accommo-
date significantly more users than OMA by using non-orthogonal resource allo-
cation [13–15]. In other words, NOMA can support fully-loaded and overloaded
transmissions. For example, NOMA can achieve a reasonable good performance
when the system loading is 200% [11, 16]. As for the maximum number of non-
orthogonally multiplexed users, it is shown in [10] and [17] that when the system
loading is 200%, the capacity gain of NOMA is significantly better than that of
OMA, i.e., about 60% improvement compared with OMA. However, the further
gain by continually increasing the system loading from 200% to 400% is relatively
small, i.e., approximately 63% improvement compared with OMA. This indicates
that it is sufficient to multiplex non-orthogonally a moderate number of users to
obtain the most from the gain of NOMA.
Although NOMA has above advantages, research in the area, especially in the code
domain NOMA, is still in its infancy and there are several open issues needed to be
figured out.
• Code domain NOMA schemes, including low density signature (LDS), sparse code
multiple access (SCMA), multiuser shared access (MUSA) and pattern division
multiple access (PDMA) [16,18], utilizes a low density signature for the spreading.
However, such low density signature is usually generated without careful design
and optimization. For instance, according to the graph theory, many parameters
such as degree distributions and cycle structures will affect the graph’s perfor-
mance. Nevertheless, these factors has not been studied in detail for the NOMA
schemes, and it is still an open question to design good low density signatures.
• On the receiver side, the message passing algorithm (MPA) is employed on the
low density signature to perform multiuser detection (MUD), and the MUI can
be effectively alleviated. However, the convergence behavior of MPA for NOMA
is not optimal. For example, the iteration number of MPA is relatively high.
6 Chapter 1. Introduction
In addition, although existing receivers in NOMA utilize Turbo-style iterations
between the multiuser detector and the channel decoder to improve the BER
performance, there is still a large gap to the optimum performance. Therefore, it
is necessary to further improve the receiver structure and the performance.
• The existing NOMA schemes are mainly applied in OFDM systems. In future
cellular networks, more advanced waveforms my be utilized. Hence, it is urgent
to research the combination of NOMA techniques and advanced waveforms other
than OFDM.
• The constellation shaping is an interesting topic to the NOMA technique. Al-
though a constellation shaping gain can be obtained in SCMA, further improve-
ment such as multi-dimensional constellations should be studied.
Given the above analysis, this thesis aims to address the listed problems by designing
and optimizing NOMA schemes (in particular, LDS and SCMA techniques). Due to
the fact that the base station in the uplink can afford a relatively high complexity
of MPA, we only focus on uplink transmissions in the thesis. We start with LDS
based NOMA, and propose a joint sparse graph which combines LDS-OFDM and low
density parity-check (LDPC) codes. A joint receiver performing detection and decoding
simultaneously on the joint sparse graph is designed. Different from a Turbo receiver,
the joint receiver has no additional interleaver/deinterleaver between the detector and
the decoder, and no outer-inner iterations. Messages are propagated in a double-door-
double-open/close manner to perform joint detection and decoding. An analytical and
optimization framework based on extrinsic information (EXIT) chart is also proposed
for the joint sparse graph. Such idea is then extended to waveforms other than OFDM,
i.e., filter-bank multi-carrier (FBMC) with isotropic orthogonal transform algorithm
(IOTA) pulse function. In FBMC, there is no need to insert any guard interval [19,
20], and a frequency well-localized pulse shaping leads to higher power and spectral
efficiency compared with OFDM [21–24]. By combining LDS, LDPC and FBMC-
IOTA, it is justified that code domain NOMA can be applied in conjunction with
more advanced waveforms for future mobile networks. Furthermore, to improve the
BER performance of code domain NOMA by exploiting constellation diversity, LDS
1.3. Research Contributions 7
is enhanced to combine multi-dimensional constellation shaping, which becomes the
SCMA scheme. Protograph based codebook and three-dimensional (3D) constellation
are respectively designed for SCMA. By utilizing constellation shaping gain, SCMA
outperforms LDS with high-order modulation schemes.
Objectives of this thesis can be summarized as follows:
• Conduct literature survey on LDS/SCMA schemes for wireless communications;
• Design LDS/SCMA joint receivers for the uplink with affordable complexity;
• Optimize the transreceivers structure of LDS/SCMA schemes.
1.3 Research Contributions
Throughout the course of this Ph.D. study, the following contributions are made:
Design of joint sparse graph for OFDM systems: JSG-OFDM
• The low density signature of LDS-OFDM is represented as a bipartite graph. As
a capacity-approaching code over AWGN channel [25], LDPC code can also be
expressed by a bipartite graph. According to their graph model, we propose to
construct a joint sparse graph which includes the low density signature of LDS-
OFDM and the low density parity-check matrices of LDPC codes. We refer it
as joint sparse graph for OFDM. Unlike any existing sparse graph that is only
used in one specific field such as LDS-OFDM, LDPC code, low density generator
matrix (LDGM) code, repeat-accumulate (RA) code, Luby transform (LT) code
and Raptor code, our proposed JSG-OFDM is based on a novel joint sparse graph
which combines NOMA and FEC techniques. The idea behind the joint sparse
graph is to change the interference pattern being observed by each user, and limit
the amount of interference occured on each chip.
• To the best of our knowledge, there does not exist a multiple access system in
which detection and decoding are performed simultaneously on one sparse graph.
Based on the MPA and the joint sparse graph, we design a joint receiver for the
8 Chapter 1. Introduction
JSG-OFDM. A joint detection and decoding scenario, which performs detection
and decoding at the same time on the entire sparse graph, is presented. There are
significant differences between JSG-OFDM and LDS-OFDM. The LDS-OFDM
receiver is a separate receiver, i.e., detection and decoding are performed and op-
timized individually. The turbo structured LDS-OFDM receiver performs detec-
tion and decoding iteratively in a turbo style. An extra interleaver/de-interleaver
and outer-inner iterations are dominant characters to the turbo receiver. Howev-
er, our proposed receiver of JSG-OFDM is a joint receiver, where detection and
decoding are performed jointly on the entire graph. There is no turbo structure
in the JSG-OFDM, but the detection and decoding information can be freely
exchanged, thus the JSG-OFDM is a novel scheme and different from existing
systems such as LDS-OFDM and turbo structured LDS-OFDM.
• Analysis of a joint sparse graph is different from that of a single sparse graph. We
depict the iterative structure for JSG-OFDM receiver in details, and use the EXIT
chart to analyse the convergence behavior of the joint detection and decoding in
the receiver.
• According to the EXIT chart analysis, two important factors which affect the per-
formance of the joint sparse graph are investigated: degree distributions and short
cycles. As a result, design guidelines for the joint sparse graph are derived. With
offline optimization of the joint sparse graph, JSG-OFDM outperforms similar
multiple access systems. In other words, all the optimizations (for a wide range
of parameters) are carried out in advance at the design stage and the receiver can
use the optimum graph directly.
Design of joint sparse graph for FBMC systems with IOTA function based pulse: JSG-
IOTA
• The intrinsic interference from real and imaginary branches in FBMC-IOTA
transmissions is usually discarded in existing systems. In fact, the intrinsic in-
terference can be estimated by a weight matrix which defines neighboring time-
frequency positions around the signal of interest, but such weight matrix has
1.3. Research Contributions 9
never been studied from graphical view. In this thesis, we creatively regard the
weight matrix as a generator matrix, and the intrinsic interference as parity sym-
bols. In addition, we choose the most significant positions around the signal of
interest to estimate the intrinsic interference, consequently the weight matrix can
be modeled as a low density graph which is referred to as low density weight ma-
trix (LDWM). This is essentially the principle of block coding, and the embedded
weight matrix can be exploited to improve the system performance.
• According to the graphical model, we propose to construct a joint sparse graph
which includes the LDWM of intrinsic interference of FBMC-IOTA, the low den-
sity signature of LDS and the low density parity-check matrix of LDPC code. It
is referred to as joint sparse graph for IOTA, and it is a novel scheme which com-
bines multi-carrier modulation (LDWM), NOMA (LDS) and FEC (LDPC codes)
techniques.
• Based on the MPA and the joint sparse graph, we design a joint detection and
decoding algorithm which exploit the redundancy introduced by the intrinsic in-
terference and LDPC codes. In the JSG-IOTA receiver, multiuser detection,
intrinsic interference decoding and channel decoding are jointly performed on one
entire graph. In light of this finding, we investigate how to effectively utilize the
coding nature of the FBMC-IOTA system. To demonstrate the advantages of
JSG-IOTA, we also develop LDS-IOTA and turbo structured LDS-IOTA, both
of which have never been previously studied in the literature. Similar to OFD-
M systems, there are significant differences between the LDS-IOTA, the turbo
structured LDS-IOTA and the JSG-IOTA. There is no turbo structure in the
JSG-IOTA, but the detection and decoding information can be exchanged, thus
the JSG-IOTA receiver is different from that of existing systems.
• We analyse the iterative structure of the joint receiver of JSG-IOTA in details, and
utilize the EXIT chart to predict the convergence behaviour of the joint sparse
graph. By optimizing the joint sparse graph, JSG-IOTA outperforms similar
systems.
Design of sparse code multiple access: SCMA
10 Chapter 1. Introduction
• The state-of-the-art SCMA codebook sets only cover very short codewords. To
extend the graph size, a copy operation can be performed. However, simple com-
bination of small graphs cannot exhibit advantages of the sparse graph, hence
the performance may not be optimal. In this thesis, we propose to use the pro-
tograph as a basic template of a small SCMA graph, and to construct a larger
SCMA codebook by the copy-and-permute operation. The extra permutation is
designed to obtain interleaving gain and eliminate short cycles. By doing so, the
SCMA system performance can be improved.
• Existing SCMA codebook is strictly designed for two-dimensional (2D) constel-
lation shaping, i.e., dv,lds = 2. For MPA on sparse graphs, the node with only
two edges cannot fully utilize the most of dependable message which comes from
other nodes. To further exploit the shaping gain, we propose to design SCMA
codebooks with 3D constellation shaping, i.e., dv,lds = 3. Simulation results show
that 3D codebooks outperform 2D codebooks.
1.4 Thesis Outline
The remainder of this thesis is organized as follows:
Chapter 2: Survey on Multiple Access Techniques and Their Receiver De-
sign
In this chapter, well-known multiple access techniques are reviewed and discussed. S-
ince MUD is required in multiuser transmissions to detect users’ symbols, different
kinds of MUDs are reviewed and their potential strengths and weaknesses in an over-
loaded condition are investigated. Moreover, for coded systems, three typical receiver
types are introduced.
Chapter 3: Joint Sparse Graph for OFDM (JSG-OFDM) System
In this chapter, a joint sparse graph combining single graphs of LDS-OFDM and LDPC
codes is constructed. Based on the graphical model, a joint detection and decoding
algorithm is presented. The iterative structure of JSG-OFDM receiver is illustrated
and its EXIT chart is researched. In addition, design guidelines for the joint sparse
1.4. Thesis Outline 11
graph are derived through the EXIT chart analysis. By offline optimization of the joint
sparse graph, numerical results show that the JSG-OFDM brings about 1.5 - 1.8 dB
performance improvement at BER of 10−5 over similar well-known systems such as
group-orthogonal multi-carrier code division multiple access (GO-MC-CDMA), LDS-
OFDM, and turbo structured LDS-OFDM.
Chapter 4: Joint Sparse Graph for FBMC-IOTA (JSG-IOTA) System
In this chapter, the idea of the graphical model is extended to FBMC system, and a
single sparse graph is modeled to express the weight matrix which is used to calculate
the intrinsic interference in the IOTA function. Then a joint sparse graph for FBMC-
IOTA system is proposed. Such joint sparse graph combines single graphs of LDWM,
LDS and LDPC codes, which represent multi-carrier modulation, NOMA and FEC
techniques, respectively. By employing MPA, an approach for joint detection and
decoding is presented on the joint sparse graph. The iterative structure of JSG-IOTA
receiver is illustrated, and its EXIT chart is analysed. Moreover, similar to that of JSG-
OFDM, design guidelines for the joint sparse graph of JSG-IOTA are derived through
the EXIT chart analysis. Numerical results show the superiority of JSG-IOTA to
similar systems such as OFDM, FBMC-IOTA, LDS-OFDM, JSG-OFDM, LDS-IOTA
and turbo structured LDS-IOTA.
Chapter 5: Sparse Code Multiple Access (SCMA)
In this chapter, SCMA is investigated and optimized. Different from the joint sparse
graph, SCMA combines LDS and symbol mapping modules. By merging of QAM
modulator and LDS spreading, constellation shaping gain is exploited in the overloaded
transmission. We propose two ways to optimize the SCMA codebooks: copy-and-
permute operation on protographs to form larger codebooks, and 3D constellation
shaping to construct more efficient codebooks. Simulation resutls show that SCMA
outperforms LDS with high-order constellations, and the SCMA performance can be
further improved by the proposed optimization approaches.
Chapter 6: Conclusions and Future Works
This chapter provides a conclusive summary of the insights and findings acquired by
the investigations presented in this thesis. Furthermore, some future work is suggested
12 Chapter 1. Introduction
to address the open issues related to NOMA schemes. The aim is to make NOMA a
viable solution for future cellular networks.
Throughout this thesis, we use the following notations. Variables and constants are
represented in lowercase and uppercase, respectively. Vectors and matrices are denoted
by lowercase and uppercase, respectively, both in bold case. B and C denote the binary
and the complex field, respectively. All vectors are defined as column vectors. The
superscript T represents transpose of a vector or a matrix. Re {·} (or the superscript
R) and Im {·} (or the superscript I) denote the real and imaginary part of a complex
signal, respectively.
1.5 Publications
L. Wen, R. Razavi, M. A. Imran and P. Xiao, Design of joint sparse graph for OFDM
system, IEEE Transactions on Wireless Communications, vol. 14, no. 4, pp. 1823-
1836, November 2014.
L. Wen, P. Xiao, R. Razavi, M. A. Imran and M. Al-Imari, Joint sparse graph for
FBMC-IOTA system, IEEE Transactions on Signal Processing, 2015, submitted and
under review.
L. Wen, P. Xiao, M. A. Imran and R. Razavi, Fast convergence and reduced complexity
receiver design for LDS-OFDM, in IEEE International Symposium on Personal Indoor
and Mobile Radio Communications (PIMRC), pp. 918-912, September, 2014.
L. Wen, P. Xiao, M. A. Imran and R. Tafazolli, joint sparse graph design for multiple
access systems, Patent, No. 1403382.3, February 2014.
L. Wen, R. Razavi, P. Xiao, M. A. Imran and R. Tafazolli, Novel multi-carrier multiple
access scheme JSG-IOTA, Patent, No. 1506036.1, April 2015.
J. Zhong, P. Xiao, R. Tafazolli, L. Wen and G. Chen, Scalable frequency division
multiplexing (SFBM), Patent, No. 1519102.6, October 2015.
Chapter 2
Survey on Multiple Access
Techniques and Their Receiver
Design
In cellular systems, it is necessary to design efficient multiple access schemes that
enable several multiple users to gain access and communicate simultaneously. There
are a number of requirements that multiple access schemes must be able to meet, such
as ability to handle several users without mutual interference and to maximise the
spectrum efficiency. In this chapter various multiple access techniques together with
their receiver design are presented.
2.1 Multiple Access Techniques
Multiple access techniques is central to the way in which the radio technology of the
cellular system functions. In this section, state-of-the-art multiple access techniques
are presented.
13
14 Chapter 2. Survey on Multiple Access Techniques and Their Receiver Design
2.1.1 Time Division Multiple Access (TDMA)
TDMA allows several users to use the same frequency channel by dividing the sig-
nal into different time slots. The users transmit in rapid succession, one after the
other, each using its own time slot. This allows multiple stations to share the same
transmission medium (e.g. radio frequency channel) while using only a part of its
channel capacity [26]. TDMA is applied to 2G cellular systems, i.e., Global System
for Mobile Communications (GSM), and also used in IS-136, Personal Digital Cellular
(PDC), integrated Digital Enhanced Network (iDEN) and Digital Enhanced Cordless
Telecommunications (DECT) standard for portable phones. It is also used extensively
in satellite systems, combat-net radio systems, and Passive Optical Network (PON) for
upstream traffic from premises to the operator [27].
In TDMA systems, the synchronization is achieved by sending timing advance com-
mands from the base station which instructs the mobile phone to transmit earlier and
by how much. This compensates for the propagation delay resulting from the light
speed velocity of radio waves. The mobile phone is not allowed to transmit for its en-
tire time slot, leaving a guard interval at the end of each time slot. As the transmission
moves into the guard period, the mobile network adjusts the timing advance to syn-
chronize the transmission. Initial synchronization of a phone requires even more care.
Before a mobile transmits there is no way to actually know the offset required in the
system. For this reason, an entire time slot has to be dedicated to mobiles attempting
to contact the network; this is known as the random-access channel (RACH) in GSM
system. The mobile attempts to broadcast at the beginning of the time slot, as received
from the network.
A major advantage of TDMA is that the radio part of the mobile only needs to listen
and broadcast for its own time slot. For the rest of the time, the mobile can carry
out measurements on the network, detecting surrounding transmitters on different fre-
quencies. A disadvantage of TDMA is that it creates interference at a frequency which
is directly connected to the time slot length. This is the buzz which can sometimes
be heard if a phone is left next to a radio or speakers. Another disadvantage is that
the “dead time” between time slots limits the potential bandwidth of a TDMA chan-
2.1. Multiple Access Techniques 15
nel. These are implemented in part because of the difficulty in ensuring that different
terminals transmit at exactly the times required. Handsets that are moving will need
to constantly adjust their timings to ensure their transmission is received at precisely
the right time, because as they move further from the base station, their signal will
take longer to arrive. This also means that TDMA systems have very hard limits on
cell sizes in terms of range, though in practice systems the power levels required to
receive and transmit over distances greater than the supported range would be mostly
impractical anyway.
2.1.2 Code Division Multiple Access (CDMA)
As a dominant technique in 3G cellular systems, CDMA allows many users to access a
given frequency allocation [28] [29]. CDMA is based on spectrum spreading, meaning
that a wider radio spectrum is used than the data rate of each of the transferred bit
streams, and several message signals are transferred simultaneously over the same carri-
er frequency, utilizing different spreading codes. Each symbol is spread by a user-unique
high bandwidth pseudo-noise binary sequence, called chips. The wide bandwidth makes
it possible to send with a very low signal-to-noise ratio, meaning that the transmission
power can be reduced to a level below the level of the noise and co-channel interference
(cross talk) from other message signals sharing the same frequency. Different users
are given access to the system by allocating different spreading codes. The scheme has
been likened to being in a room filled with people all speaking different languages. Even
though the noise level is very high, it is still possible to understand someone speaking
in your own language. The receiver mixes down to baseband and then re-multiplies
with the binary pseudo-noise sequence. This effectively removes the pseudo-noise signal
and what remains (of the desired signal) is just the transmitted data. Signals that use
different spreading codes are not decodable, and are discarded in the process. In other
words, the base station uses one code to receive the signal from one mobile, and another
spreading code to receive the signal from a second mobile. Thus in the presence of a
variety of signals it is possible to receive only the required one.
Most of the existing MUD techniques for CDMA systems perform poorly in highly load-
16 Chapter 2. Survey on Multiple Access Techniques and Their Receiver Design
ed conditions [30] [31]. Therefore, the number of users supported in a CDMA cellular
system is typically less than spreading factor, thus the system is said to be under loaded.
References [32] and [33] show the performance comparison of different MUD techniques.
The Branch and Bound technique has been shown to approach the performance of the
optimum MUD technique with typically lower computational complexity by limiting
the search space. However, in an above 100% loaded condition, the detector complexi-
ty increases exponentially with the number of available users, which is intractable for
practical implementation. Actually by using brute-force search, a complexity order
of O(| X |K
)is required, where X is the constellation alphabet and | X | denotes the
cardinality of the set X. On the other hand, when a conventional signature structure is
employed, from the resultant interference pattern, it is easy to see that, the existence
of a strong interferer affects the performance of all users in the system. Therefore more
affordable alternatives that yield a comparable performance to optimum MUD should
be used. Various signature optimization algorithms have been proposed in order to
overcome this high loaded condition problem from the transmit-end [34–37]. The se-
quences which meet the Welch-bound-equality (WBE) [38–40] are able to minimize the
variance of the MUI for symbol-synchronous memoryless channel. In [41] and [42], a
class of overloaded signature is developed for uncoded CDMA systems, where different
inputs give rise to different outputs. Consequently its sequence alphabets are uniquely
detectable. Nevertheless, with above sequences, each user will experience interference
from all the other users’ data symbols, and the system performance is not ideal under
overloaded conditions.
2.1.3 Orthogonal Frequency-Division Multiple Access (OFDMA)
To meet the need for faster and more reliable data services, multi-carrier techniques
have drawn much attention. Multi-carrier systems using overlapping but orthogonal
subcarriers were investigated since the 1960s. However, the use of such systems at the
time was difficult due to the large number of filters and modulators as well as oscilla-
tors required. A major reduction in the required equipment complexity occurred when
the generation of OFDM signals using the discrete fourier transform (DFT) is present-
ed [43]. With the DFT implementation, frequency division is achieved by baseband
2.1. Multiple Access Techniques 17
processing instead of bandpass filtering. As one of the most important multi-carrier
techniques, OFDM is based on dividing the transmitted bitstream into multiple sub-
streams and sending these over different orthogonal subcarriers. In OFDM, the data
rate achieved on each subcarrier is considerably less than the total data rate. In ad-
dition, the bandwidth occupied by each subcarrier is much less than the total system
bandwidth. A cyclic prefix (CP) has to be inserted in OFDM to mitigate the intercarri-
er interference (ICI) and the intersymbol interference (ISI). The number of subcarriers
is selected such that each subcarrier has a bandwidth less than the coherence band-
width of the channel, in order for the subcarriers to experience relatively flat fading.
This allows OFDM to efficiently resist the effect of frequency selective fading, and the
rate and power can be adjusted on each subcarrier individually. A large continuous
block of spectrum is not needed for high rate multi-carrier communications, and sev-
eral contiguous blocks of smaller size can be used instead. This provides flexibility in
spectrum allocation and spectrum management.
OFDMA is an efficient extension of the OFDM technique to a multiuser scenario, and
it has been adopted as a core technique in 4G cellular networks, i.e., the 3rd Generation
Partnership Project Long Term Evolution (3GPP-LTE) [44] [45] and the Worldwide
Interoperability for Microwave Access (WiMAX) [46–48]. In OFDMA, the set of or-
thogonal subcarriers is divided into several mutually exclusive subsets and then each
subset is allocated to transmission of a user signal. Different numbers of subcarriers
can be assigned to different users, to support differentiated Quality of Service (QoS),
i.e. to control the data rate and error probability individually for each user. The sys-
tem spectrum of OFDMA is divided into a number of channels; each channel consists
of a cluster of a number of consecutive orthogonal OFDM subcarriers. As subcarriers
are orthogonal, intra-cell interference is significantly reduced. In OFDMA systems, the
transmission rate of a channel is variable based on the user allocated to this channel
due to the use of Adaptive Modulation and Coding (AMC). Each enhanced NodeB
(eNB) collects the Channel Quality Indicator (CQI) reports which are derived from
the downlink received reference signal quality and fed back from the users. The CQI
is then used to determine the Modulation and Coding Scheme (MCS) for a channel.
Same MCS is used for all subcarriers in a resource block (RB) allocated to a given user,
18 Chapter 2. Survey on Multiple Access Techniques and Their Receiver Design
and different MCS can be allocated to different resource blocks. Since different users
perceive different channel qualities, a bad channel (due to deep fading and narrowband
interference) for one user may still be favorable to other users. Thus, OFDMA exploit-
s the multiuser diversity by avoiding assigning bad channels, which is an important
feature in OFDMA [49] [50].
Compared with single-carrier systems, OFDMA provides significant advantages in terms
of high spectrum efficiency, robustness against multipath fading channels, resistance to
multiuser interference, simplified equalization, and so on. In addition, OFDMA is
considered as highly suitable for broadband wireless networks, due to scalability and
MIMO-friendliness, as well as ability to take advantage of channel frequency selectiv-
ity [51]. As in OFDMA user-data symbols are assigned directly to subchannels, the
frequency domain diversity will not be achievable at modulation symbols level. Thus
it is crucial to incorporate properly designed FEC coding and interleaving schemes to
obtain this diversity at a later stage.
2.1.4 Multi-carrier Code Division Multiple Access (MC-CDMA)
The combination of OFDM and CDMA, known as multi-carrier code division multiple
access (MC-CDMA), has gained attention as a powerful transmission technique [52] [53].
The MC-CDMA concept is based on OFDM signaling with spreading in the frequency
domain, so multiple access is thus achieved by using distinct spreading sequences for
different users similar to more conventional CDMA systems based on time domain
spreading. In other words, MC-CDMA applies spreading in the frequency domain by
mapping a different chip of the spreading sequence to an individual OFDM subcarrier,
thus it is essentially equivalent to performing the inverse fast Fourier transform (IFFT)
operation on a CDMA signal [54]. For high user data rates, the data symbols are first
serial-to-parallel (S/P) converted into substreams, and then each substream is spread in
the frequency domain with a signature sequence. After spreading, the signal is passed
through a subcarrier multiplexer, parallel-to-serial (P/S) converted. A guard interval
with cyclic extensions similar to OFDM is inserted between symbols to counter ISI
caused by multipath fading. MC-CDMA scheme transmits in parallel chips of a spread
2.1. Multiple Access Techniques 19
data symbol on different subcarriers and therefore offers diversity.
Similar to OFDM, the MC-CDMA signal is made up of a series of equal amplitude
subcarriers. Unlike OFDM, where each subcarrier transmits a different symbol, MC-
CDMA transmits the same data symbol over each subcarrier. At the MC-CDMA
receiver a coherent detection method is employed to despread the signal. The received
signal energy can be collected in the frequency domain with moderate complexity even
when the signal bandwidth is large. The received signal, after downconversion and dig-
itization, is first coherently detected with FFT, then multiplied by a gain factor. Equal
gain combining (EGC) and maximum ratio combining (MRC) are standard combin-
ing techniques used in MC-CDMA receivers [55]. The advantage of using combining
techniques is that even though individual branches may not have sufficient SNR, their
combined sum increases the probability of detection by increasing the SNR of a given
signal. In EGC all branches are given equal weight irrespective of signal amplitude, but
the signals from each branch are co-phased to avoid signals arriving at the same time.
In MRC each signal is multiplied by a weight factor depending on the signal strength.
Strong signals are amplified, whereas weak signals are attenuated. Like EGC, MRC
signals are also co-phased to avoid signal cancellations [56].
Similar to CDMA systems, non-orthogonality of received effective signatures in MC-
CDMA causes MUI [57] [58], and implementation of optimum MUDs is not practi-
cal due to their prohibitively high computational complexity. A technique named
group-orthogonal MC-CDMA (GO-MC-CDMA) is proposed to handle the overload-
ed transmission [59] [60]. It partitions the available subcarriers into different groups
and distributes users among these groups, then each group behaves as an independent
MC-CDMA system with a smaller number of users. In [61], an iterative interference
cancellation approach is developed for an overloaded Walsh-Hadamard-spread MC-
CDMA system. However, these techniques fail to achieve good performance under
highly-overloaded conditions.
20 Chapter 2. Survey on Multiple Access Techniques and Their Receiver Design
2.1.5 Low Density Signature (LDS)
Wireless cellular technologies are continuously evolving to meet the increasing demands
for the massive connectivity. For conventional multiple access techniques via code do-
main multiplexing, each user spreads the original data using a given spreading sequence,
where elements of the spreading sequence usually take nonzero values, which are opti-
mized under certain criteria, e.g., good auto- and/or cross-correlation properties. These
spreading sequences are orthogonal to each other to avoid MUI, and naturally have high
density, which means majority chips have nonzero values. The drawback is that, each
user will see the interference coming from all other users at the chip level, and it can-
not easily achieve satisfactory performance under overloaded conditions. To deal with
these problems, a multiple access technique named low density signature has been pro-
posed. Several milestones in this area have been achieved, leading to a flurry of further
research.
2.1.5.1 LDS-CDMA
The LDS concept is first proposed for CDMA systems [62–66]. Fig. 2.1 (a) shows the
block diagram of the LDS-CDMA transreceivers with K users and N chips. We can
see that after FEC encoding and symbol mapping, the data are sent to a specially de-
signed spreader. Instead of optimizing the N -chips sequences, the scheme intentionally
arranges each user to spread its data, vk,m, over very limited chips, cn. More explicitly,
the spreading sequences have small number of nonzero values and the rest chips are
zero valued, hence the resultant signature matrix becomes very sparse. Basic principles
of LDS are i) changing the interference pattern being seen by each user; ii) limiting
the amount of interference occurred on each user.
2.1. Multiple Access Techniques 21
cN
AW
GN
cN
xv
1,1
v1
,2
v1,M
c1
c2
use
r 1
FE
C
enco
der
sym
bo
l
map
per
rad
io
chan
nel
xv
K,1
vK
,M
c1
c2
use
r K
FE
C
enco
der
sym
bo
l
map
per
rad
io
chan
nel
vK
,2
use
r 1
use
r K
FE
C
dec
od
er
FE
C
dec
od
er
SIS
O
iter
ativ
e
det
ecto
r
Fig.2.1:LDS-C
DMA
system
22 Chapter 2. Survey on Multiple Access Techniques and Their Receiver Design
Fig. 2.2 illustrates the LDS principle by using a simple exemplary system with 5 chips
and 10 data symbols, where chip nodes and variable nodes respectively represent chips
and data symbols. It can be seen that each symbol is spread over 2 chips. Each chip is
used by 4 symbols that may belong to different users.
chip nodes
variable nodes x x x x x x x x xx
Fig. 2.2: Illustration of a LDS spreader
To elaborate the LDS structure more clearly, the sets of chip nodes and variable nodes
are associated with chip node detector (CND) and variable node detector (VND), re-
spectively. The iterative structure in the receiver is depicted in Fig. 2.3 and it closely
follows the LDS in Fig. 2.2. CND and VND can be expressed by mathematical func-
tions, and exchange soft messages through the edge interleave. It has been proved that
LDS-CDMA significantly outperforms conventional CDMA systems under overloaded
conditions [64].
CND 1
To FEC decoder
VND
From OFDM demodulator
-
-
Interleaving (edges)
Key:
CND: Chip node detector (corresponding to the chip nodes)
VND: Variable node detector (corresponding to the user nodes)
Fig. 2.3: LDS iterative structure
2.1. Multiple Access Techniques 23
2.1.5.2 LDS-OFDM
As an extension, LDS is applied to OFDM systems [17,67–69]. Reference [70] extends
LDS to a rateless scenario, i.e., the low density signature is a dynamic graph when
data are transmitted. By doing so, the system spectral efficiency is improved, but the
receiver complexity is higher than a fixed rate LDS. Fig. 2.4 (a) shows the block diagram
of the LDS-OFDM transreceiver. It is similar to conventional MC-CDMA systems.
In MC-CDMA, after FEC encoding and symbol mapping, each modulated symbol
is multiplied with a N -chips spreading sequence which is subsequently transmitted
over orthogonal subcarriers in IFFT module. Cyclic prefix (CP) has to be inserted to
eliminate ISI and ICI. The IFFT and CP insertion comprise the OFDM modulator.
Compared with conventional MC-CDMA transmitter, the main difference in Fig. 2.4
(a) is that the spreading signature has low density. Due to the LDS structure, each data
symbol is spread over very limited chips. Each chip is transmitted over an orthogonal
subcarrier, and each subcarrier is only used by a limited number of data symbols
that may belong to different users. Each user, transmitting on given subcarriers, will
experience interference from only a small number of other users’ data symbols. In other
words, the number of users symbols that are superimposed on each chip is much less
than the total number of data symbols. Meanwhile, the number of chips that are spread
by each symbol is much less than the total number of chips.
The philosophy of LDS is that if a fraction of signal of some user is superimposed
by a fraction of signals coming from a relatively small number of interferers, then the
search-space should be moderate, thus detection technique with affordable complexity
can be used to recover the corrupted signal. Moreover, apart from being practical for
implementation, the LDS structure also benefits from having the intrinsic interference
diversity by avoiding strong interferers to corrupt all chips of a user. Therefore, LDS
is an effective technique for fully-loaded and overloaded transmissions. The drawback
of LDS is that its performance degrades with high order constellations.
24 Chapter 2. Survey on Multiple Access Techniques and Their Receiver Design
cN
AW
GN
cN
xv
1,1
v1
,2
v1,M
c1
c2
use
r 1
FE
C
enco
der
sym
bo
l
map
per
OF
DM
mo
du
lato
r
rad
io
chan
nel
xv
K,1
vK
,M
c1
c2
use
r K
FE
C
enco
der
sym
bo
l
map
per
OF
DM
mo
du
lato
r
rad
io
chan
nel
vK
,2
OF
DM
dem
od
ula
tor
use
r 1
use
r K
FE
C
dec
od
er
FE
C
dec
od
er
SIS
O
iter
ativ
e
det
ecto
r
Fig.2.4:LDS-O
FDM
system
2.1. Multiple Access Techniques 25
2.1.6 Sparse Code Multiple Access (SCMA)
The recently proposed SCMA is an enhanced version of LDS-OFDM [71–73]. In LDS-
OFDM, a LDS spreader expands a QAM symbol to a sequence of complex symbols by
using a given low density signature. Hence, a LDS spreader can be seen as a process in
which a number of coded bits are mapped to a sequence of complex symbols. From this
point of view, the QAM mapper block and the LDS spreader can be merged together to
directly map a set of bits to a complex vector so called a SCMA codeword. With this
interpretation, a simple LDS spreading action is generalized to a coding process which
in turn raises a new problem in terms of complex multidimensional codeword design
rather than a relatively simple low density signature design. The SCMA characters can
be summarized as follows:
1) Binary domain data are directly encoded to multidimensional complex domain code-
word selected from a predefined codebook set.
2) Multiple access is achievable by generating multiple codebooks, one for each layer
or user.
3) Codeword of the codebook is sparse such that MPA multiuser detection technique
is applicable to detect the multiplexed codeword with a moderate complexity.
4) Like LDS-OFDM, SCMA can be overloaded such that the number of multiplexed
layers can be more than the spreading factor.
Fig. 2.5 shows a SCMA with 6 users where each user has a predefined codebook. All
codewords in the same codebook contain zeros in the same two dimensions, and the
positions of zeros in different codebooks are distinct to facilitate the collision avoidance
of any two users. For each user, two bits are mapped to a complex codeword. Code-
words for all users are multiplexed over four shared orthogonal resources (e.g., OFDM
subcarriers).
The main difference between LDS-OFDM and SCMA is that a multi-dimensional con-
stellation is designed for SCMA to generate codebooks, which brings the shaping gain
that is not possible for LDS-OFDM [74]. Here, shaping gain is the gain in the average
symbol energy when the shape of a constellation is changed. For the concatenated
26 Chapter 2. Survey on Multiple Access Techniques and Their Receiver Design
Fig. 2.5: SCMA system
approach with high-order constellations, the multi-dimensional constellation can be
optimized to obtain shaping gain, then codebooks are generated based on the multi-
dimensional constellation. The design criteria of multi-dimensional constellations in-
clude [74]: i) minimization of the average energy per constellation point; ii) maxi-
mization of the diversity order; iii) maximization of the minimum product distance;
iv) minimization of the product kissing number for the product distance. The SCMA
codebook design is a complicated problem, since different layers are multiplexed with
different codebooks. As the appropriate design criterion and specific solution to the
multi-dimensional problem are still unknown, a multi-stage approach has been pro-
posed to realize a suboptimal solution. Specifically, a complex constellation which is
called the mother constellation is first optimized to improve the shaping gain, and then
some codebook-specific operations are performed to the mother constellation to gener-
ate the constellation for each codebook. Three typical operations are phase rotation,
complex conjugate, and dimensional shuffling of the constellation. In the generated
constellations after codebook-specific operations, each constellation point is multiplied
with a low density matrix to generate a codeword. In this way, SCMA codebooks can
be obtained.
Other work on SCMA includes the blind detection for uplink grant-free multiple access
[75] and irregular SCMA schemes [76]. Although SCMA performs better than LDS,
the codebooks still need to be improved by the means of graph optimization and the
constellation design.
2.1. Multiple Access Techniques 27
2.1.7 Multiuser Shared Access (MUSA)
MUSA is another NOMA technique via code domain multiplexing, and it is suitable
for overloaded transmissions [77]. Fig. 2.6 shows the MUSA system. Multiple spread-
ing sequences constitute a pool from which each user can randomly pick one of the
sequences. Note that for the same user, different spreading sequences may also be
used for different symbols, which may further improve the performance via interference
averaging. Then all spreading symbols are transmitted over the same time-frequency
resources. The spreading sequences should have low cross-correlation and can be M -
ary. MUSA differs from MC-CDMA in that it is basically synchronous transmission
mechanism when users signals arrive at the base station, while MC-CDMA doesn’t
have this kind of synchronism requirement in the uplink. In addition, MUSA uses non-
binary spreading sequences, while binary spreading sequences are usually considered in
classical MC-CDMA systems.
Fig. 2.6: MUSA system
In downlink MUSA, users are separated into different groups. In each group, different
users’ symbols are mapped to different constellations in a way to ensure Gray mapping
28 Chapter 2. Survey on Multiple Access Techniques and Their Receiver Design
in the combined constellation of superposed signals. The combined constellation is
determined not only by the modulation order of each user, but also by the transmit
power partition among multiplexed users. Orthogonal sequences can be used to spread
the superposed symbols to obtain time or frequency diversity gain. Gray mapping of
the combined constellation reduces the reliance on advanced receivers, therefore less
processing-intensive receivers such as symbol-level SIC can be used.
2.1.8 Pattern Division Multiple Access (PDMA)
PDMA is a NOMA scheme that can be realized in multiple domains [78]. At the
transmitter, PDMA uses non-orthogonal patterns which are designed to maximize the
diversity and minimize the overlaps of multiple users. Then, multiplexing can be real-
ized in code domain, power domain, space domain or their combinations. Multiplexing
in code domain is similar to that in LDS, but the number of subcarriers connected
to the same symbols in the factor graph can be different. At the receiver, MPA is
performed for interference cancellation. In the case of multiplexing in power domain,
power allocation needs to be carefully considered under the total power constraint.
SIC can be also used at the receiver according to SNR difference among multiplexed
users. Multiplexing in space domain, i.e., spatial PDMA, can be combined with the
multiple-antenna technique. The advantage of spatial PDMA compared with multiuser
MIMO is that PDMA doesn’t require joint precoding to realize spatial orthogonality,
which significantly reduces system complexity. In addition, multiple domains can be
combined in PDMA to make full use of various available wireless resources. The system
model of code domain PDMA is similar to that of MUSA, but the spreading matrix in
MUSA is designed with the following principles:
1) The number of groups with different number of nonzero elements in the spreading
sequence should be maximized.
2) The number of the overlapped spreading sequences which have the same number of
nonzero elements should be minimized.
MPA is used at the receiver to separate different users’ signals. The design principle
for spreading matrices in code domain PDMA is to facilitate interference cancellation.
2.2. Multiuser Detection Techniques 29
2.2 Multiuser Detection Techniques
Interference, which originates from channel distortion and from out-of-cell interference,
is unavoidable in wireless systems, even when using orthogonal multiplexing techniques
such as CDMA or OFDMA. Multiuser detection (MUD) deals with demodulation of
the mutually interfering digital streams of information that occur in wireless communi-
cations, and is also investigated for demodulation in low-power inter-chip and intra-chip
communications [79] [80]. Multiuser detection encompasses both receiver technologies
devoted to joint detection of all the interfering signals or to single-user receivers which
are designed to recover only one user but are robustified against multiuser interference
in addition to background noise. By exploiting the structure of the interfering signals,
multiuser detection is an effective way to eliminate the MUI, and can increase spectral
efficiency, receiver sensitivity, and the number of users the system can sustain. Main
performance metrics for MUD techniques include BER performance and near-far effect.
In the following, different multiuser detection techniques are discussed.
2.2.1 Optimum Multiuser Detector
Amaximum likelihood (ML) sequence estimator for a digital pulse-amplitude-modulated
sequence in the presence of finite ISI and white Gaussian noise is studied in [81] and [82].
Such an ML estimator comprises a sampled linear filter and a recursive nonlinear pro-
cessor which utilizes the Viterbi algorithm. The optimal ML receiver for multiuser de-
tection is formulated, e.g., in [83]. It shows an improvement over conventional decoder
by orders of magnitude and achieves the best performance in terms of the probability
of errors. Using the ML detection, the optimal solution can be expressed as
v̂ = arg maxv∈XK
p(y|v) (2.1)
where K, v, y and X represent the user number, the transmitted symbol, the received
signal and the constellation alphabet, respectively.
The ML MUD is optimum in terms of minimizing the block error probability, when the
transmitted data is independently and identically distributed (i.i.d). Therefore, inte-
grating a priori knowledge into the ML function leads to maximum a posteriori (MAP)
30 Chapter 2. Survey on Multiple Access Techniques and Their Receiver Design
detection. MAP detector is implemented by applying two strategies: individually and
jointly optimum detection.
The jointly optimum MAP detection maximizes the joint a posteriori probability mass
function (PMF) of the transmitted symbols v, i.e., p(v|y), while the individually op-
timum MAP detection maximizes the a posteriori PMF, p(vk|y), of the transmitted
symbol for the k-th user by evaluating y. The estimation of vk with individual MAP
detection can be written as
v̂k = argmaxvk∈X
p(vk|y) (2.2)
A posteriori PMF for vk can be found by calculating the marginal of the joint a pos-
teriori PMF, therefore, in order to minimize the symbol error probability, (2.2) can be
written as
v̂k = argmaxvk∈X
∑v ∈
vkXK
p(v|y) (2.3)
Let Pk(vk) be the priori probability of symbol vk, k = 1, ...,K, then according to Bayes’
rule we have
p(v|y) ∝ p(y|v)P (v) (2.4)
where,
P (v) =
K∏k=1
Pk(vk) (2.5)
is the joint a priori PMF of all users’ symbols assuming that they are independent to
each other. Hence, the estimation function can be modified to
v̂k = argmaxvk∈X
∑v ∈
vkXK
p(y|v)K∏l=1
Pl(vl) (2.6)
The optimum MUD’s computational complexity increase exponentially with the user
number K. This high complexity is one of the reasons that optimum MUD is not a
popular choice in practical systems.
2.2.2 Linear Multiuser Detector
Linear detectors, which apply a linear operator or filter to the output of the matched
filter or the received signal, have been well-studied in the literature because of their
2.2. Multiuser Detection Techniques 31
efficiency and simplicity [84]. These detectors have much lower complexity compared
to the optimal detector. The most common linear detectors are the minimum mean
square error (MMSE) detector and the decorrelator.
2.2.2.1 Minimum Mean Square Error (MMSE)
The MMSE detector implements the linear mapping which minimises the mean square
error between the actual data symbols and the soft outputs of the detector, and it
provides good performance with much lower complexity compared to the optimum
detector [85]. Considering that this type of MUD detect the actual data symbols based
on both noise statistics and the received signal powers, better BER performance is
expected compared to the decorrelator detector. However, as the performance of linear
MMSE detector depends on the received power of the interfering users, there is some
performance degradation due to the near-far effect problem [86]. It has been discussed
in [87] that the MAI-plus-noise is approximately Gaussian distribution. This property
is particularly useful in the performance analysis of multiuser communication systems
involving the MMSE detector. The limitations of the linear MMSE detector have been
investigated in [88]. It is reported that the MMSE detector is not able to handle the
highly loaded and overloaded conditions, or in other words, the error dose not vanish
as SNR increases.
2.2.2.2 Decorrelator
The decorrelator inverts the channel matrix leaving the received signal without inter-
ference but by doing so also enhances the noise [89]. The advantage of the decorrelator
is that no knowledge of the receive power is necessary and its performance is indepen-
dent of the power of interfering users so that the near-far problem is avoided. Both
MMSE detector and decorrelator handle near-far situations equally well. Indeed, as
the interference powers tend to infinity, the MMSE detector converges to the decorre-
lator. Like the MMSE detector, the decorrelator fails to perform satisfactorily under
overloaded conditions as the desired as well as the interferers’ signal subspace become
rank-deficient.
32 Chapter 2. Survey on Multiple Access Techniques and Their Receiver Design
2.2.3 Nonlinear Multiuser Detector
Nonlinear detectors can achieve better performance than linear detectors. The popular
nonlinear MUD techniques include the successive interference cancellation (SIC), the
parallel interference cancelation (PIC), the probabilistic data association (PDA), the
message passing algorithm (MPA) and so on.
2.2.3.1 Successive Interference Cancellation (SIC)
SIC is an intuitive approach to joint detection of the received symbols from differ-
ent users in multiuser systems [90–92]. It is applied to a multiuser system that its
transmissions are not orthogonal and all the users share the available time and fre-
quency degrees of freedom. In this scheme the effect of estimated symbol of each user
is removed from the observation used for estimating other users’ symbols. It results
in fewer number of interferers seen by other users if the decision is correct, however,
an incorrect decision will result in doubling the interference [93]. In other words, the
first user is affected by all the interferes, whereas the user in succession experience less
interference as the interference cancellation progresses. Therefore, in order to utilize
this scheme, the user ordering mechanism should be incorporated. The SIC process
should start by estimating the signal that can be detected with highest certainty. All
subsequent detections will subtract the contribution of already estimated symbols from
the received aggregate signal before detecting other signals. This cancels the effect of
detected symbols from all subsequent detections. This process continues iteratively for
all received signals from different users.
2.2.3.2 Parallel Interference Cancelation (PIC)
As discussed earlier, a SIC receiver cancels the MAI user by user in each stage such
that the remaining users see less and less MAI, while a PIC scheme cancels the MAI
for all the users simultaneously at each decision stage [94]. As a result, PIC takes a
multi-stage structure and has shorter processing delay, compared to SIC. Also, there
is no need for user ordering since all users will receive the same treatment in removing
2.2. Multiuser Detection Techniques 33
their respective MAI. Generally, PIC uses hard decision, and in some cases, fails to
guarantee performance improvements as the iterative process goes on, due to erroneous
interference estimates in cancellation. This is because a poor estimation of the MAI at
early stages will not lead to performance improvement at later stages, therefore it is not
wise to completely use that information to estimate the MAI. To solve this problem,
partial parallel interference cancellation (PPIC) is proposed [95] [96]. PPIC alleviates
the performance degradation by weight method and acquires performance superior
to the conventional PIC with no increase in implementation complexity. Therefore,
PPIC alleviates the above-mentioned problem by introducing a weight in each stage
to mitigate the effect of error propagation. In [97], an adaptive least-mean-squares
PIC (LMS-PIC) structure is proposed, where for each stage the weights are obtained
by minimizing the mean-squared error between the received signal and its estimate
through the LMS algorithm. Their results show a substantial performance improvement
over the brute-force PIC and the conventional PPIC schemes. The performance of SIC
and PIC schemes is highly dependent on the first estimate being fed to the interference
cancelation detector, and at a heavy system load, the update of user symbol estimations
becomes unreliable and may degrade the performance significantly.
2.2.3.3 Probabilistic Data Association (PDA)
The PDA filter introduced in [98] is a highly successful approach to target tracking
in the case that measurements are unlabeled and may be spurious. Since then, it has
been applied in many different areas, including digital communications. In the area of
digital communications, the PDA algorithm is a reduced complexity alternative to the
ML detector. Near-optimal results are demonstrated for a PDA-based MUD for CDMA
systems [99]. Since the computational complexity of an optimal MUD is prohibitive,
this suboptimal solution is proposed to provide reliable decisions with relatively low
complexity. However it should be noted that as concluded in [100] PDA performs well
only for low order modulation schemes, i.e. Binary Phase Shift Keying (BPSK) and
Quadrature Phase Shift Keying (QPSK), and not for higher order modulations.
The PDA detector is based on approximating the MAI component by a Gaussian
34 Chapter 2. Survey on Multiple Access Techniques and Their Receiver Design
random variable. Therefore, considering that the thermal noise is also Gaussian random
variable, the total effective noise can be modeled as Gaussian random variable too.
Furthermore, PDA algorithm is applied sequentially and is performed repeatedly in
every iteration stage. Therefore, similar to SIC, this algorithm is also sensitive to the
order of detected users because PDA algorithm works successively from one user to
another. However, this technique uses soft values instead of hard values, the effect
of error propagation is expected to be less severe. In [99], it is suggested that if the
users are appropriately sorted a performance improvement can be achieved. Again, the
PDA’s performance degrades rapidly under overloaded conditions.
2.2.3.4 Message Passing Algorithm (MPA)
The algorithm iteratively computes the distribution of variables in graph-based models
and comes under different names, depending on the context. These names include:
the sum-product algorithm (SPA), the belief propagation algorithm (BPA), and the
message passing algorithm (MPA). The term “message passing” usually refers to all
such iterative algorithms, including SPA, BPA and their approximations [101–105].
MPA is very efficient for belief propagation on sparse graphs [106, 107], thus we give
some definitions about graphical models [108].
Definition 2.1 (sparse graph). A bipartite graph is a graph (nodes connected by edges)
whose nodes may be separated into two types, and edges only connect two nodes of
different types. If the number of edges in a graph is close to the maximal number of
edges (almost fully connect), it is a dense graph. On the contrary, a graph with only a
few edges, is a sparse graph. The distinction between dense and sparse graphs is the
number of edges.
Definition 2.2 (degree distribution). Degree of a node is the number of edges the
node connected to other kinds of nodes, while degree distribution is the probability
distribution of these degrees in a graphical model. If the degrees are constant values,
the graph is a regular graph. Otherwise, it is an irregular graph.
Definition 2.3 (cycle and girth). A cycle in a graph refers to a finite set of connected
edges, the edge starts and ends at the same node, and it satisfies the condition that no
2.2. Multiuser Detection Techniques 35
node (except the initial and final node) appears more than once. Girth of a graph is
the length of the shortest cycle. Apparently, girth should be equal to or greater than
4, and length-4 cycles manifest themselves in the corresponding matrix as four 1’s that
lie on the corners of a sub-matrix.
As for the bipartite graph shown in Fig. 2.2, degrees of chip nodes and variable nodes
are 4 and 2, respectively. Hence it is a regular graph. Bold lines in the figure represent
the shortest cycle of length-6, meaning that the girth of such graph is 6. MPA calculates
the marginal distribution for each node on a sparse graph [109] [110]. The key points
of a clear presentation of MPA are the use of a tensorial representation of the messages
along the edges in the graphical model, and transformations of the graph so that MPA
can be written as equations involving the message updating between different types of
nodes via edges. In a typical run, each node of the graph calculates iteratively from the
previous values of the neighbouring information (two nodes are said to be neighbours
if they are connected by an edge). The information exchanged is the soft value that
represents the reliability of the symbol related to an edge. Degree distributions and
short cycles affect the performance of MPA on sparse graphs significantly [111–113].
On a acyclic (without cycle) graph, the estimated marginal distribution converges to
the true value in a finite number of iterations. Nevertheless, graphs inevitably contain
cycles. It is known that the graph containing length-4 cycles will converge in most cases,
but the probabilities obtained might be incorrect. These graphs may fail to converge,
or oscillate between multiple states over repeated iterations. To analyse and predict
the convergence behavior of MPA on the graphical model, different mathematical tools,
such as the density evolution (DE) [114] [115], the Gaussian approximation (GA) [116]
and the extrinsic information transfer (EXIT) chart [117–122], were developed. In
particular, EXIT chart is a useful tool to analyse the transfer of information between the
soft-input soft-output (SISO) constituents, and it provides an approximate visualization
of the process of belief propagation. In this thesis, we mainly use the EXIT chart to
analyse the MPA on sparse graphs, hence it is necessary to give some definitions of
entropy and mutual information about the EXIT chart [123].
Definition 2.4 (entropy). The entropy H(x) of a continuous random variable x ∈ χ
36 Chapter 2. Survey on Multiple Access Techniques and Their Receiver Design
with probability density function p(x) is defined as
H(x) = −∫χp(x) log p(x)dx (2.7)
Definition 2.5 (conditional entropy). The conditional entropy H(y|x) of a pair of
continuous random variables (x, y) ∈ χ2 is defined as
H(y|x) = −∫χp(x)
∫χp(y|x) log p(y|x)dxdy
= −∫χ
∫χp(x, y) log p(y|x)dxdy
(2.8)
where p(x, y) is the joint probability density function of x and y, p(y|x) is the probability
density function of y conditioned on x.
Definition 2.6 (mutual information). The mutual information I(x, y) between two
continuous random variables x and y is defined as
I(x, y) = −∫χ
∫χp(x, y) log
p(x, y)
p(x)p(y)dxdy
= −∫χ
∫χp(x, y) log
p(x|y)p(x)
dxdy
= −∫χ
∫χp(x, y) log p(x)dxdy −
(−∫χ
∫χp(x, y) log p(x|y)dxdy
)= H(x)−H(x|y)
(2.9)
Intuitively, according to Shannon’s information theory, the entropy gives a measure
of the uncertainty of the random variable, while the mutual information of two ran-
dom variables is a quantity that measures the mutual dependence of the two random
variables and it is the basis of the channel capacity.
MPA has been successfully applied to decoding of sparse graph codes such as LDPC
codes [124–126], low density generator matrix (LDGM) codes [127], repeat-accumulate
(RA) codes [128], Luby transform (LT) codes [129], Raptor codes [130] [131] and so
on. In addition, MPA also has been applied to LDPC coded large-MIMO systems [132]
[133]. For MUD, it is inefficient to apply MPA for detection in conventional multiple
access systems, as their spreading signatures are dense graphs, consequently leading
to prohibitive computational complexity. Unlike conventional systems, LDS-CDMA,
LDS-OFDM and SCMA are based on the sparse graphical model, thus iterative MPA
becomes feasible and can attain satisfactory performance at a relatively low complexity.
2.3. Receiver Structures 37
2.3 Receiver Structures
At the receiver, MUD and FEC decoding are performed to combat MUI and fading
effects as well as the channel noise. As shown in Fig. 2.7, a receiver that performs
detection and decoding i) jointly (optimal) is referred to as Type-A receiver, ii) in-
dependently is referred to as Type-B receiver, and iii) iteratively (between detection
and decoding) is referred to as Type-C receiver [134–136]. The separate detection and
decoding (Type-B receiver) has the lowest complexity, but its performance is subopti-
mal. Turbo equalization that performs detection and decoding in an iterative manner
(Type-C receiver) is known to achieve better performance than the Type-B receiver,
and its computational complexity is several times higher than that of the Type-B re-
ceiver [137] [138]. Although the information is exchanged between the detector and the
decoder in a turbo receiver, it cannot perform the joint detection and decoding as is
the case of the optimal Type-A receiver. In the literature, a Type-A receiver, which
takes into account the knowledge of channels, the FEC decoding, the de-spreading and
the de-interleaving, usually becomes infeasible in practical systems, as it amounts to
essentially trying to fit all possible sequences of transmitted bits to the received data
− a task of prohibitive complexity. Therefore, the goal in this research is to design
a joint receiver which achieve near-optimum performance of the Type-A receiver with
affordable computational costs.
2.4 Summary
In this chapter, we have presented different multiple access techniques, including T-
DMA, CDMA, OFDMA, MC-CDMA, LDS-CDMA, LDS-OFDM, SCMA, MUSA and
PDMA. Advantages and drawbacks of these multiple access techniques are discussed.
Considering that non-orthogonal multiple access systems require MUD, we have pre-
sented several popular MUD techniques. The complexity of the optimum MUD grows
exponentially with the number of users which make it practically prohibitive. Although
linear MUD techniques are among the most popular ones, their performance is not op-
timal. Nonlinear detectors such as SIC, PIC, and PDA cannot achieve satisfactory
38 Chapter 2. Survey on Multiple Access Techniques and Their Receiver Design
(b) Type-B
Detector
Deinterleaver
Decoder
(a) Type-A
Joint receiver
(c) Type-C
Detector
Decoder
Deinterleaver Interleaver
Fig. 2.7: Receiver structures
performance in overloaded conditions. In this regard the application of MPA to sparse
graphical based NOMA schemes such as LDS-CDMA, LDS-OFDM and SCMA can
enable NOMA schemes to obtain good performance under overloaded conditions. TA-
BLE 2.1 summarizes different multiple access techniques and their comparisons. It is
an open issue to design joint receivers for NOMA to achieve near-optimum performance
with affordable computational complexity.
2.4. Summary 39
TABLE
2.1:Compariso
nsofmultiple
access
tech
niques
TDMA
CDMA
OFMDA
MC-C
DMA
LDS-C
DMA
LDS-O
FDM
SCMA
MUSA
PDMA
Multi-carrier
No
No
Yes
Yes
No
Yes
Yes
Yes
Yes
Spreading
No
Yes
No
Yes
Yes
Yes
Yes
Yes
Yes
Non
-binary
chips
No
No
No
No
No
No
No
Yes
No
Con
stellationshap
ing
No
No
No
No
No
No
Yes
No
No
Sparsestructure
No
No
No
No
Yes
Yes
Yes
Yes
Yes
Han
dle
overload
ing
No
No
No
No
Yes
Yes
Yes
Yes
Yes
Receiver
MMSE
Rake
MMSE
MMSE
MPA
MPA
MPA
SIC
MPA
NOMA
No
No
No
No
Yes
Yes
Yes
Yes
Yes
Chapter 3
Joint Sparse Graph for OFDM
(JSG-OFDM) System
LDS-OFDM and LDPC codes are NOMA and FEC techniques, respectively. Both
of them can be expressed by a bipartite graph. In this chapter, we construct a joint
sparse graph combining LDS-OFDM and LDPC codes, namely JSG-OFDM. A joint
detection and decoding is presented. The iterative structure of JSG-OFDM receiver is
illustrated, and its EXIT chart is researched. Furthermore, design guidelines for the
joint sparse graph are derived through the EXIT chart analysis. By offline optimization
of the joint sparse graph, JSG-OFDM outperforms similar well-known systems such as
GO-MC-CDMA, LDS-OFDM and turbo structured LDS-OFDM.
3.1 JSG-OFDM System Model
Consider the uplink communications withK users transmitting to the same base station
where the base station and each user are equipped with a single antenna. The block
diagram of JSG-OFDM system is shown in Fig. 3.1. Let the processing gain to be
N , and each user has a data vector consisting of M data symbols. Let J be the
number of parity-check equations in LDPC code. Perfect channel state information
(CSI) is available at the receiver. We assume that the system is applied in machine type
communication (MTC) network (where MTC devices transmit short-burst messages,
40
3.1. JSG-OFDM System Model 41
such as temperature, meter reading, etc.), thus binary phase shift keying (BPSK) is
used in the transmission.
At the transmitter in Fig. 3.1, the functional blocks are similar to a MC-CDMA
system. In conventional MC-CDMA, after FEC encoding and symbol mapping, data
symbols are multiplied with a spreading signature (a random sequence of chips) and
subsequently OFDM modulation is arranged to modulate the chips onto respective
subcarrier frequencies. The main difference in JSG-OFDM transmitter is that the
spreading signature has low density by the use of zero padding, which means a large
number of chips in the sequence are zeros. Due to the low density signature, each
symbol is only spread over a limited number of chips. Each user’s generated chip
is transmitted over an orthogonal subcarrier, and each subcarrier is only used by a
limited number of symbols that may belong to different users. Each user, transmitting
on given subcarriers, will experience interference from only a small number of other
users’ data symbols. More explicitly, the number of symbols that are superimposed
on each chip is much less than the total number of symbols, dc,lds ≪ (K ×M), where
dc,lds is the number of symbols that are superimposed at one chip. Meanwhile, the
number of chips that are spread by each symbol is much less than the total number of
chips, dv,lds ≪ N , where dv,lds is the number of chips that are spread by one symbol.
In other words, the spreading sequences have a maximum of dv,lds nonzero values and
N − dv,lds zero values, then they are interleaved uniquely for each user such that the
resultant signature matrix becomes very sparse. The interleaving process is designed
so that at each received chip there exists a contribution of, instead of all users, only a
small number of users. Consequently, the interference pattern being seen by each user
is different.
As for the receiver in Fig. 3.1, there are three types of nodes: chip nodes cn(n ∈ [1, N ]),
variable nodes vk,m(k ∈ [1,K],m ∈ [1,M ]) and parity-check nodes pk,j(k ∈ [1,K], j ∈
[1, J ]), representing the nth chip, themth data symbol and the jth parity-check equation
of the kth user, respectively. A single graph, as labelled with LDS in the receiver,
represents the low density signature due to LDS-OFDM [68] [17]. The other single
graphs, as labelled with LDPC in the receiver, represent the low density parity-check
matrices due to LDPC codes [124]. These two types of single graphs belong to the
42 Chapter 3. Joint Sparse Graph for OFDM (JSG-OFDM) System
parity-check nodes variable nodes chip nodes
LDPC LDS
OFDM
demodulator
v1,1
v1,2
vK,1
vK,2
v1,M
vK,M
user 1
user K
p1,J
p1,1
pK,J
pK,1
c1
c2
cN
JSG
AWGN
user 1 LDPC
encoder
symbol
mapper
OFDM
modulator
radio
channel
user K LDPC
encoder
symbol
mapper
OFDM
modulator
radio
channel
spreading
spreading
zero
padding
zero
padding
Block diagram for JSG-OFDM transmitter
Block diagram for JSG-OFDM receiver
Fig. 3.1: JSG-OFDM system model
3.2. Joint Detection and Decoding for JSG-OFDM 43
techniques of NOMA and FEC coding, respectively. In our proposal, variable nodes
are used to connect the other two types of nodes (chip nodes and parity-check nodes)
through low density edges. Therefore, the receiver becomes a joint sparse graph labelled
with JSG in the figure. As such, the LDS structure and the LDPC codes are perfectly
linked together. In the joint sparse graph, users’ signals that are using the same chip
will be superimposed, and the number of symbols that interfere with each other at
one chip is much less than the total number of symbols, so the system can perform
well in overloaded conditions. The joint sparse graph is arranged to process the chips
from the received signals to reconstitute the transmitted data. It is noteworthy that
applying MPA on the joint sparse graph performs not only detection, but also decoding
at the same time. Furthermore, the receiver of JSG-OFDM (Type-A receiver in Fig.
2.7) is different from that of the turbo structured LDS-OFDM (Type-C receiver in
Fig. 2.7) [17], as there is no outer-inner turbo style iteration here. Hence, the JSG-
OFDM is based on a joint sparse graph which combines NOMA and sparse graph coding
techniques. To fit the stream format of practical communications, we can extend the
data length, such that one frame contains multiple other than one OFDM symbol. In
the next section, joint detection and decoding on such graph will be described.
3.2 Joint Detection and Decoding for JSG-OFDM
Based on the system model introduced in Section 3.1, we present the joint detection
and decoding on the joint sparse graph.
Let sk,m = [s1k,m, ..., sNk,m]
T ∈ CN×1 be the spreading sequence for the the mth symbol
of the kth user, and hk,j = [h1k,j , ..., hMk,j ]
T ∈ BM×1 be the coefficients for the jth parity-
check equation of the kth user. The spreading signature and the parity-check matrix
for the kth user are Sk = [sk,1, ..., sk,M ] ∈ CN×M and Hk = [hk,1, ...,hk,J ] ∈ BM×J ,
respectively. As for the multi-user scenario, let S = [S1, ...,SK ] ∈ CN×MK and H =
[H1, ...,HK ] ∈ BM×JK be the spreading signatures and the parity-check matrices of
LDPC codes, respectively. We also define A = diag(a1, ..., aK) ∈ CK×K as the transmit
amplitude of users, and Ek = diag(ek,1, ..., ek,N ) ∈ CN×N as the corresponding channel
gain for the kth user, where ak and ek,n are both scalars. Moreover, ψn = {(k,m) :
44 Chapter 3. Joint Sparse Graph for OFDM (JSG-OFDM) System
snk,m ̸= 0} and εk,m = {n : snk,m ̸= 0} are the set of data symbols (which may belong
to different users) that interfere on chip cn and the set of chips that vk,m is spread on,
respectively; ϕk,j = {(k,m) : hmk,j ̸= 0} and ωk,m = {(k, j) : hmk,j ̸= 0} are the set of
data symbols that connect to parity-check node pk,j and the set of parity-check nodes
that connect to vk,m, respectively.
In JSG-OFDM, each user’s chip will be transmitted over an orthogonal subcarrier.
Therefore, the received spreading sequence for the mth symbol of the kth user can be
represented by rk,m = akEksk,m. In particular, the received signature gain at the nth
chip of the variable node vk,m is rnk,m = akek,nsnk,m. For the uplink MC-CDMA, the
received signal corresponding to the nth chip (subcarrier) is written as
yn =
K∑k=1
M∑m=1
rnk,mvk,m + zn (3.1)
where zn is the AWGN with variance σ2A and mean zero. Considering that in JSG-
OFDM, the signature has a limited number of non-zero positions, we can express the
received signal at the nth chip (subcarrier) as
yn =∑
(k,m)∈ψn
rnk,mvk,m + zn (3.2)
Let Lvk,m→cn and Lvk,m→pk,j be the log-likelihood ratio (LLR) delivered from the vari-
able node vk,m to the chip node cn and the parity-check node pk,j , respectively. The
LLR delivered from the chip node cn and the parity-check node pk,j to the variable node
vk,m are given by Lcn→vk,m and Lpk,j→vk,m , respectively. Lvk,m is the final estimation
of the variable node vk,m. In a typical run, each message will be updated iteratively
from the previous values of the neighboring LLR. The joint detection and decoding can
be presented as messages updating between different types of nodes via edges, which is
explained as follows.
3.2.1 Initialization
Assuming there is no a priori probability available, initial LLRs are set to zeros.
Lvk,m→cn = 0, ∀k, ∀m, ∀n (3.3)
Lvk,m→pk,j = 0, ∀k, ∀m,∀j (3.4)
3.2. Joint Detection and Decoding for JSG-OFDM 45
3.2.2 Updating of Chip Nodes and Parity-Check Nodes
LLRs of the chip nodes and the parity-check nodes are calculated at the same time.
For the chip nodes,
Lcn→vk,m = f(vk,m|yn, Lvk′,m′→cn , (k′,m′) ∈ ψn \ (k,m)) (3.5)
where ψn \ (k,m) is the set of data symbols (excluding vk,m) that interfere on the chip
cn. In order to approximate the maximum a posteriori (MAP) probability detector,
the right hand side of (3.5) represents marginalization function, which is based on (3.2),
and can be written as
f(vk,m|yn, Lvk′,m′→cn , (k′,m′) ∈ ψn \ (k,m))
= log(
N∑n=1
p(yn|v)pn(v|vk,m))
= log(
N∑n=1
p(yn|v)∏
(k′,m′)∈ψn\(k,m)
pn(vk′,m′))
(3.6)
where v is the transmitted vector, the conditional probability density function p(yn|v)
and a priori probability pn(vk′,m′) are given as
p(yn|v) =1√2πσA
exp(− 1
2σ2A∥ yn − rT[n]v[n] ∥2) (3.7)
pn(vk′,m′) = exp(Lvk′,m′→cn) (3.8)
where v[n] and r[n] denote the vector containing the symbols transmitted by every
user that spread its data on the nth chip and their corresponding effective received
signature values, respectively. As can be seen from (3.6), based on the received chip
yn and a priori input information pn(vk′,m′), extrinsic values are calculated for all the
constituent bits involved in (3.2). Substituting (3.7) and (3.8) into (3.6), the message
update becomes
Lcn→vk,m = κn,k,mmaxv[n]
∗(∑
(k′,m′)∈ψn\(k,m)
Lvk′,m′→cn − 1
2σ2A∥ yn − rT[n]v[n] ∥2) (3.9)
where κn,k,m denotes the normalization coefficient and
max∗(a, b) , log(ea + eb) = max(a, b) + log(1 + e−|a−b|) (3.10)
46 Chapter 3. Joint Sparse Graph for OFDM (JSG-OFDM) System
The LLR of the parity-check nodes is updated as
Lpk,j→vk,m = α−1(∑
(k′,m′)∈ϕk,j\(k,m)
α(Lvk′,m′→pk,j )) (3.11)
where ϕk,j \ (k,m) is the set of data symbols (excluding vk,m) that connect to the
parity-check node pk,j , and
α(x) = sign(x)× (− log tan| x |2
) (3.12)
where sign(x) represents the sign of x, and the inverse of α(x) is
α−1(x) = (−1)sign(x) × (− log tan| x |2
) (3.13)
3.2.3 Updating of Variable Nodes
In the single graph case, variable nodes only gather information from one type of nodes
(chip nodes or parity-check nodes) [64] [126]. However, in the joint sparse graph, the
updating of Lvk,m→cn not only receives chip nodes information, but also ultilizes the
information that comes from parity-check nodes.
Lvk,m→cn =∑
n′∈εk,m\n
Lcn′→vk,m +∑
j∈ωk,m
Lpk,j→vk,m (3.14)
where εk,m \ n is the set of chips (excluding cn) that vk,m is spread on.
Similarly, calculation of Lvk,m→pk,j also involves the information from both sides, i.e.
Lvk,m→pk,j =∑
j′∈ωk,m\j
Lpk,j′→vk,m +∑
n∈εk,m
Lcn→vk,m (3.15)
where ωk,m \ j is the set of parity-check nodes (excluding pk,j) that connect to the
variable node vk,m.
3.2.4 Estimation and Syndrome Computing
In the single graph case of LDS-OFDM, a posteriori probability of the transmitted
symbol can only be calculated after a fixed number of iterations, as there is no criterion
to determine whether the iterative message has converged [66]. Fortunately, in the
3.3. EXIT Chart Analysis of JSG-OFDM 47
joint sparse graph, parity-check nodes are available, thus it is possible to terminate the
joint detection and decoding by syndrome computing. A posteriori probability of the
transmitted symbol vk,m is calculated as
Lvk,m =∑
n∈εk,m
Lcn→vk,m +∑
j∈ωk,m
Lpk,j→vk,m (3.16)
The estimated value of the variable node vk,m is obtained by making a hard decision,
v̂k,m = argmaxvk,m
Lvk,m (3.17)
If the syndrome computing for each user equals to zero, or the maximum iteration
number is reached, the process is terminated. Otherwise, the iteration goes on.
The implementation algorithm of the joint detection and decoding is summarized in
TABLE A.1. The key features of the joint detection and decoding include: i) there
is no additional interleaver/de-interleaver between the detector and the decoder; ii)
there is no outer-inner Turbo-fashion iterations; and iii) the messages are propagated
in a double-door-double-open/close manner which means that chip nodes and parity-
check nodes update at the same time while variable update at another time. The extra
interleaver/de-interleaver and outer-inner iterations are dominant characters of a Turbo
receiver, and the double-door-double-open/close updating strategy is different in the
way that a Turbo receiver works. Therefore, the JSG-OFDM joint receiver is different
from a Turbo-type receiver. One of the advantages of the iterative receiver for JSG-
OFDM is its ability to support high loads while maintaining satisfactory performance
and affordable complexity. In the next section, we carry out theoretical analysis of the
joint sparse graph using EXIT chart.
3.3 EXIT Chart Analysis of JSG-OFDM
EXIT chart is a useful tool to analyse the transfer of information between SISO con-
stituents, and it provides an approximate visualization of the process of belief prop-
agation [117–122]. However, EXIT chart has not been applied to any joint sparse
graph. In this section, we shall explain how EXIT charts can be utilized to analyse the
convergence behavior of the joint receiver of JSG-OFDM.
48 Chapter 3. Joint Sparse Graph for OFDM (JSG-OFDM) System
3.3.1 Iterative Structure of the Joint Sparse Graph
Before applying EXIT charts to the JSG-OFDM, we depict the iterative structure of
the joint sparse graph. According to the joint detection and decoding presented in Sec-
tion 3.2, variable nodes calculate the extrinsic messages to chip nodes using a priori
information which they receive from other connected chip nodes and parity-check n-
odes. Meanwhile, based on the received a priori information, variable nodes calculate
the extrinsic messages to parity-check nodes. The same rule applies to the extrinsic
messages that the chip nodes and parity-check nodes send to variable nodes. To e-
valuate the transformation of extrinsic information in the joint sparse graph, the sets
of chip nodes, variable nodes and parity-check nodes are referred to as chip nodes de-
tector (CND), variable node detector-decoder (VNDD) and parity-check node decoder
(PND), respectively. Fig. 3.2 shows the structure of the iterative detector and decoder
in JSG-OFDM. As depicted in the figure, the extrinsic LLR that has been passed on
are considered as a priori information by the other detector or decoder. The edge
interleavers connect the different type of nodes, each of which represents a sparse sig-
nature or matrix. It is worth noting that such iterative structure is more complicated
than any previous single graph which only has the left part labelled as LDS [64] or the
right part labelled as LDPC [126] in this figure.
Chip Node
Detector 1
ld s
ld s
Parity Check Node
Decoder
1
ld p c
ld p c
To receiver
Variable Node
Detector & Decoder
From OFDM demodulator
-
-
-
-
LDS LDPC
Fig. 3.2: Iterative structure of the joint detection and decoding in JSG-OFDM
Additionally, Fig. 3.3 shows a folded view of the joint sparse graph. Based on the joint
detection and decoding, chip nodes and parity check nodes update their messages at the
same time, then variable nodes calculate the LLR delivered to chip nodes and parity
check nodes simultaneously. Therefore, it is reasonable to place chip nodes (rectangles)
and parity-check nodes (triangles) on one side, while to draw variable nodes (circles)
3.3. EXIT Chart Analysis of JSG-OFDM 49
on the other side. The chip nodes are used to be spread on for the variable nodes
through low density edges, which are represented by bold lines. The parity-check
nodes, belonging to different users, are connected to corresponding groups of variable
nodes through sparse edges, which are represented by independent groups of dash lines.
Multiple access, FEC coding and the combination of several single sparse graphs, are
clearly depicted in the figure. This provides a basis for the following analysis.
p1, J
p1, 1
c1
c2
cN
v1, 1
v1, 2
vK, 1
vK, 2
v1, M
vK, M
chip nodes
of user 1
of user K
PK, J
PK, 1
of user 1
variable nodes
of user K
parity check nodes
parity check nodes
variable nodes
Fig. 3.3: Folded view of the joint sparse graph
3.3.2 EXIT Chart Analysis Over AWGN Channel
3.3.2.1 EXIT Curve for VNDD
In this thesis, IA,V NDD refers to the average mutual information between the bits on
the VNDD edges and the a priori LLR, IE,V NDD is the average mutual information
between the bits on the VNDD edges and the extrinsic LLR. In order to compute
an EXIT curve for variable nodes, Lcn→vk,m and Lpk,j→vk,m are modelled as the soft
output of an AWGN channel when the inputs are interleaved bits. Then the mutual
50 Chapter 3. Joint Sparse Graph for OFDM (JSG-OFDM) System
information between the variable node’s extrinsic messages and actual values of symbols
on the edges is calculated. A priori LLR can be calculated by
A = µAx+ zn (3.18)
where zn is an independent Gaussian random variable with variance σ2A and mean zero;
x is the original bits on the graph edge. Furthermore, we have
µA =σ2A2
(3.19)
The mutual information IA,V NDD = I(X;A) can be calculated by [123]
IA,V NDD =1
2
∑x=−1,1
∫ +∞
−∞pA(β|X = x) log2
2pA(β|X = x)
pA(β|X = −1) + pA(β|X = 1)dβ
(3.20)
Since the conditional probability density function pA(β|X = x) depends on LLR of A,
we can write
IA,V NDD(σA) =1−∫ +∞
−∞
exp(−((β − σ2A/2)2/2σ2A))√
2πσAlog2(1 + e−β)dβ (3.21)
For abbreviation we define
B(σ) , IA,V NDD(σA = σ) (3.22)
with
limσ→0
B(σ) = 0 (3.23)
limσ→∞
B(σ) = 1 (3.24)
where σ ≥ 0. Considering (3.14) and (3.15) together with the fact that the sum of
two normally distributed random variables is also normally distributed with the mean
and variance equal to the sum of theirs, the EXIT function of a variable node can be
expressed as
IE,V NDD(IA,V NDD, dv,lds, dv,ldpc) = B(√
(dv,lds + dv,ldpc − 1)(B−1(IA,V NDD))2)
(3.25)
where dv,lds is the effective spreading factor and dv,ldpc is the number of parity-check
nodes connected to one variable node. Therefore, unlike single sparse graph where only
one type of node is considered [64] [124], both LDS and LDPC nodes affect the VNDD
performance.
3.3. EXIT Chart Analysis of JSG-OFDM 51
3.3.2.2 EXIT Curve for CND&PND
Let IA,CND&PND refers to the average mutual information between the bits on the
CND&PND edges and the a priori LLR, IE,CND&PND is the average mutual informa-
tion between the bits on the CND&PND edges and the extrinsic LLR. A chip node
has incoming messages from the connected variable nodes and the OFDM demodula-
tor, whereas a parity-check node only has messages coming from neighbored variable
nodes. The output LLR of chip nodes and parity-check nodes are calculated by (3.9)
and (3.11), respectively. We model Lvk,m→cn and Lvk,m→pk,j as the output of an AWGN
channel that the input is the corresponding transmitted bit, and then calculate the mu-
tual information of the output with regards to the actual value on the edges. Due to the
complexity of the calculation in chip nodes and parity-check nodes, their EXIT curves
are computed by simulations over AWGN channel. The probability density function
for extrinsic information is determined by Monte Carlo simulation with histogram mea-
surements [120], the mutual information between the extrinsic information and the bits
on the joint graph edges, is subsequently calculated.
3.3.2.3 Analysis
System parameters are listed in TABLE 3.1 for the following EXIT chart analysis.
The number of chip nodes is less than that of FFT size, and the unused subcarriers
serve as the guard band. Considering the effective transmitted bits, there are 120 chips
available, where each subcarrier can transmit at least 2 bits due to I/Q channels. As
BPSK is used, each variable node represents a 1-bit symbol. Thus the system loading
is calculated as: number of variable nodes / (number of chip nodes × I/Q channels),
i.e., 240/(120 × 2) = 100%. The number of users or variable nodes is related to the
system loading. The effective spreading factor is set to 3 in order to achieve a good
trade-off between frequency diversity and complexity [17]. Fig. 3.4 shows the EXIT
charts over AWGN channel at Eb/N0 = 9 dB. For comparisons, we also plot curves
of LDS-OFDM system, where there are only two sets of nodes in the single graph of
LDS-OFDM: chip node detector (CND) and variable node detector (VND). According
to Fig. 3.4, we can concluded:
52 Chapter 3. Joint Sparse Graph for OFDM (JSG-OFDM) System
TABLE 3.1: System parameters
Number of users 6
Number of chip nodes 120
Number of variable nodes 240
Number of parity-check nodes 120
FFT size 128
System loading 100%
Number of variable nodes connected to each chip node dc,lds = 6
Effective spreading factor dv,lds = 3
Number of parity-check nodes connected to each variable node dv,ldpc = 3
Number of variable nodes connected to each parity-check node dp,ldpc = 6
Modulation BPSK
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IA,CND&PND
& IE,VNDD
I E,C
ND
&P
ND
& I A
,VN
DD
VND in LDS−OFDMCND in LDS−OFDMVNDD in JSG−OFDMCND&PND in JSG−OFDMJMUDD trajectory in JSG−OFDM
Fig. 3.4: EXIT chart over AWGN Channel at Eb/N0 = 9 dB
3.3. EXIT Chart Analysis of JSG-OFDM 53
1) The curve of VND in LDS-OFDM is higher than that of VNDD in JSG-OFDM. This
is because the edge numbers connecting to variable nodes are different between single
graph of LDS-OFDM and joint graph of JSG-OFDM, which has seen illustrated in Fig.
3.2 and Fig. 3.3. As a result, according to (3.25), these two curves are different.
2) The curve of CND in LDS-OFDM is higher than that of CND&PND in JSG-OFDM
when IA,CND&PND ≤ 0.95. This follows from the fact that the CND could receive
information from both OFDM demodulator and neighbored variable nodes, but the
PND only uses the information from the connected variable nodes (shown in Fig. 3.1).
Hence, the PND pulls down the average extrinsic information of CND&PND in JSG-
OFDM at the first few iterations.
3) The intersection point of VNDD and CND&PND in JSG-OFDM, is higher than that
of VND and CND in LDS-OFDM. This phenomenon implies that the JSG-OFDM has
better ability to eliminate the MUI than LDS-OFDM.
4) The simulation trajectory of the joint detection and decoding in JSG-OFDM is
also plotted in Fig. 3.4, which is marked by the dotted line and labelled by JMUDD
which means joint multiuser detection and decoding. We can see that the iterative
process starts with IA,CND&PND = 0 since no prior information is available for the
CND&PND in the beginning. In the following steps, the output LLR is exchanged
between the two solid curves. The trajectory approximately follows the transfer curves
of the components in JSG-OFDM, which indicates that the EXIT charts analysis is
valid for the joint sparse graph. The minor discrepancy is due to the finite size of edge
interleaver in the joint sparse graph.
3.3.3 EXIT Chart Analysis Over Multipath Fading Channels
As OFDM is used to combat the negative effect of multipath, we analyse the EXIT chart
over multipath fading channels in the sequel. Note that the EXIT chart technique is not
limited to the AWGN channel, it can also be applied to multipath fading channels when
perfect channel state information is available at the receiver [117] [118]. An EXIT chart
assumes that the probability density function of the exchanged messages approaches a
Gaussian-like distribution with increasing number of iterations. Consequently, it can
54 Chapter 3. Joint Sparse Graph for OFDM (JSG-OFDM) System
be applied to multipath fading channels as long as the trajectory follows the curves of
the receiver components.
Similar to the analysis for the AWGN channel, the EXIT function of a variable node
is the same as (3.25), which means the fading does not have any effect on the perfor-
mance of VNDD, however its effect will be on the CND&PND because it calculates
its messages based on the received signal from the fading channel. Due to the com-
plexity of calculation in (3.9) and (3.11), their EXIT curves are drawn by simulations.
Considering that ITU Pedestrian Channel B is adopted in 3GPP and is a typical mul-
tipath fading channel model, Fig. 3.5 illustrates the EXIT charts for LDS-OFDM and
JSG-OFDM over ITU Pedestrian Channel B at Eb/N0 = 13 dB [139]. As shown in
the figure, JSG-OFDM has a higher intersection point than LDS-OFDM. Fig. 3.5 also
shows the simulation trajectory of the joint detection and decoding of JSG-OFDM,
which is marked by the dotted line. Thus the EIXT chart analysis is verified and visu-
alized by the simulated trajectory, and the actual trajectory is well predicted at various
iterations with only a marginal difference. This validates the EXIT chart analysis for
multipath fading channels, and it will be further confirmed in Section 3.5.
3.4 EXIT Chart Based Design of Joint Sparse Graph
According to graph theory, there are different parameters affecting the performance
of a sparse graph. Two important factors are degree distribution of nodes and short
cycle [108, 111, 112]. In this section, EXIT charts are used for the optimization of the
joint sparse graph.
3.4.1 Degree Distribution
The degree of a node is equal to the number of edges connecting that node to other
nodes in the graph, while degree distribution is the probability distribution of the
degree. For the joint sparse graph, let DCND(x), DPND(x) and DV NDD(x) denote the
degree distribution polynomials of chip nodes, parity-check nodes and variable nodes,
3.4. EXIT Chart Based Design of Joint Sparse Graph 55
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IA,CND&PND
& IE,VNDD
I E,C
ND
&P
ND
& I A
,VN
DD
VND in LDS−OFDMCND in LDS−OFDMVNDD in JSG−OFDMCND&PND in JSG−OFDMJMUDD trajectory in JSG−OFDM
Fig. 3.5: EXIT chart over ITU Pedestrian Channel B at Eb/N0 = 13 dB
respectively. They are defined as
DCND(x) =
dc,lds∑d=1
PCND(d)xd−1 (3.26)
DPND(x) =
dp,ldpc∑d=1
PPND(d)xd−1 (3.27)
DV NDD(x) =
dv,lds+dv,ldpc∑d=1
PV NDD(d)xd−1 (3.28)
where PCND(d), PPND(d) and PV NDD(d) are fractions of edges related to corresponding
nodes. As 3 is suitable for the effective spreading factor, we fix dv,lds = 3. However,
other degree distributions need to be optimized.
In the information theory, the mutual information is a measure of the variables’ mu-
tual dependence [120]. Ideally, in order to the exchange extrinsic information be-
tween the components to a convergence point such that an arbitrarily low BER can be
achieved, the EXIT curves should not intersect before reaching the (IA, IE) = (1, 1)
point [117–119]. This implies that given IA = 1, we have IE = 1 and provided that
56 Chapter 3. Joint Sparse Graph for OFDM (JSG-OFDM) System
this condition is satisfied, a so-called open-convergence tunnel appears in the EXIT
chart. If however, the two curves intersect at a point lower than the (1, 1) point, it
forms a semi-convergence tunnel, ant it will yield a higher BER than the scheme with
an intersection at the (1, 1) point. In order to investigate how the position of the
intersection point in the EXIT chart affects the BER performance, Fig. 3.6 shows the
achievable BER as a function of the average mutual information IA,V NDD for the joint
sparse graph presented in TABLE 3.1 over ITU Pedestrian Channel B. This figure gives
an indication of the minimum required IA,V NDD in order to achieve a target BER. For
example, to achieve the BER of 10−3/10−4/10−5, the point of intersection should be at
least at IA,V NDD = 0.86/0.93/0.96. Hence, to reach the maximum dependence between
detected symbols and their real values, the goal of an iterative system is to approach
the (1, 1) point in the EXIT chart.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
−6
10−5
10−4
10−3
10−2
10−1
100
IA,VNDD
BE
R
Fig. 3.6: BER versus IA,V NDD for the joint sparse graph
The joint sparse graph presented in TABLE 3.1 is a regular graph. Although its
intersection point is higher than that of LDS-OFDM, which is shown in Fig. 3.4
and Fig. 3.5, the average level of CND&PND in JSG-OFDM is much lower than that
of CND in LDS-OFDM. Our approach is based on invoking EXIT chart analysis for
optimizing the shape of the EXIT curve in order to achieve the (1, 1) point while
3.4. EXIT Chart Based Design of Joint Sparse Graph 57
keeping the tunnel open. As can be seen from (3.9), the information coming from
the OFDM demodulator serves as input source of the joint sparse graph, and it is
fed into chip nodes in each iteration of the joint detection and decoding. According
to Section 3.2, chip nodes could receive information from both OFDM demodulator
and neighbouring variable nodes, while parity-check nodes only uses the information
from the connected variable nodes excluding any direct channel knowledge, thus the
PND pulls down the average extrinsic information of CND&PND in JSG-OFDM at
the first few iterations. As mentioned previously, the mutual information determines
the dependence between detected symbols and their exact values, thus the EXIT curve
level has to be considered. If the proportion of the edges related to PND is reduced
properly, the curve of CND&PND can be lifted up, and the adjusted curves of the
EXIT chart enable us to create a near-capacity scheme. As a further benefit, we are
able to shift the EXIT functions close to the (1, 1) point in order to obtain a lower
BER. Based on the above analysis, several schemes of degree distribution are shown in
TABLE 3.2, where Dega is the case of a regular joint sparse graph presented in TABLE
3.1, other schemes are for irregular joint sparse graphs. We can see that compared to
Dega, Degc and Degd slightly decrease the degree of PND, then the polynomials of
VNDD are altered accordingly. On the contrary, Degb increases the density of edges of
PND.
TABLE 3.2: Degree distribution
Scheme DCND(x) DPND(x) DV NDD(x)
Dega x5 x5 x5
Degb 0.15x4 + 0.7x5 + 0.15x6 0.09x5 + 0.91x6 0.5x5 + 0.5x6
Degc 0.05x4 + 0.9x5 + 0.05x6 0.4x3 + 0.6x4 0.7x4 + 0.3x5
Degd 0.03x4 + 0.94x5 + 0.03x6 0.8x3 + 0.2x4 0.9x4 + 0.1x5
Fig. 3.7 shows EXIT charts of different degree distributions over ITU Pedestrian Chan-
nel B at Eb/N0 = 13 dB. ForDegc andDegd, the proportion of the edges related to PND
is reduced, consequently, as seen in Fig. 3.7, the average mutual information between
the bits and the extrinsic values of CND&PND are both increased. More important-
58 Chapter 3. Joint Sparse Graph for OFDM (JSG-OFDM) System
ly, their intersection point becomes higher and closer to the (1, 1) point than that of
Dega, which means better performance can be achieved. By contrast, Degb increases
the density of edges of PND, but its average mutual information and intersection point
are both dropped. Thus the degree distribution is a key factor to the JSG-OFDM
performance. Although the curves of Degc and Degd almost overlap, Degd is a more
suitable choice since it has lower density.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IA,CND&PND
& IE,VNDD
I E,C
ND
&P
ND
& I A
,VN
DD
VNDD of Dega
CND&PND of Dega
VNDD of Degb
CND&PND of Degb
VNDD of Degc
CND&PND of Degc
VNDD of Degd
CND&PND of Degd
Fig. 3.7: EXIT chart for different degree distributions
3.4.2 Short Cycles
Usually, short cycles are only considered and avoided in single graphs such as LDPC
codes and LDS-OFDM. Nevertheless, in the joint sparse graph, length-4 cycles are
easy to be regenerated without careful design. The results presented in Fig. 3.7 do not
consider the impact of cycles, therefore, their girths equal to 4.
Fig. 3.8 shows EXIT charts for different girths of the joint sparse graph, over ITU
Pedestrian Channel B at Eb/N0 = 13 dB. It should be emphasized that the restriction
of the girth is applied to the joint graph rather than any single graph. We choose the
optimal Degd to be the degree distribution. In the case when the girth equals to 6, it is
not difficult to remove length-4 cycles by computer search. When the girth equals to 8,
3.4. EXIT Chart Based Design of Joint Sparse Graph 59
the computer search is time-consuming to eliminate length-4 and length-6 cycles, and
some degrees need to be modified marginally (labelled by Deg′d): DCND(x) is 0.027x4+
0.946x5+0.027x6, DPND(x) is 0.83x3+0.17x4 and DV NDD(x) is 0.915x
4+0.085x5. It
can be seen from this figure that curves of matrices with girth of 4 and 6 are very close
to each other, whereas matrix with girth of 8 outperforms others. This is due to the
same degree distribution adopted by matrices with girth of 4 and 6, while the degree
distribution of matrix with girth of 8 is slightly different. Although the EXIT chart
only depends on the degree distribution, in order to improve system performance, we
suggest to remove cycles of length of 4 and 6 when constructing a joint sparse graph, as
short cycles may lead to failure of message convergence or oscillation between multiple
states over repeat iterations. Ideally, in a cycle-free graph, the belief will converge to
the exact a posteriori probability after a finite number of iterations. Nevertheless,
cycles cannot be avoided, and the propagated information may lead to inaccurate a
posteriori probability. Therefore, degree distribution and short cycle both affect the
JSG-OFDM performance.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IA,CND&PND
& IE,VNDD
I E,C
ND
&P
ND
&I A
,VN
DD
VNDDCND&PND, girth = 4CND&PND, girth = 6CND&PND, girth = 8
Fig. 3.8: EXIT chart for different schemes
60 Chapter 3. Joint Sparse Graph for OFDM (JSG-OFDM) System
3.5 Performance Evaluation
In this section, the JSG-OFDM schemes are simulated and compared with other well-
known systems.
3.5.1 Evaluation Configuration
JSG-OFDM are evaluated and compared with existing well-known multiple access sys-
tems such as GO-MC-CDMA, LDS-OFDM and turbo structured LDS-OFDM. The
simulations are conducted over ITU Pedestrian Channel B, and the system parameters
are listed in TABLE 3.1. For fair comparisons, a half rate quasi-cyclic LDPC code is
adopted by all the systems [125]. For GO-MC-CDMA, Welch-bound-equality (WBE)
is used, and the spreading codes are constructed by algorithms developed in [34]. For
GO-MC-CDMA, the number of subcarriers per group is set to 4 and maximum likeli-
hood detection is employed per group [60]. For LDS-OFDM, the LDS is optimized by
EXIT chart, and an iterative detector is used [68]. For turbo structured LDS-OFDM,
there are six outer-inner turbo iterations between the detector and the decoder, and
the turbo structure is optimized by the EXIT chart in [17]. For JSG-OFDM, based on
the analysis of Section 3.4, three scenarios of the joint sparse graph that have different
degree distribution and girth are given in TABLE 3.3. The maximum iterations are
limited to six for LDS-OFDM and JSG-OFDM. Moreover, in the case when only one
user is active in the uplink transmission, the theoretical single user bound can be ob-
tained by the matched filter which optimally combines the transmitted symbol from all
the paths, similar to a maximum ratio combiner, to maximize the SNR, and thereby
minimizes error probability. All the investigated systems are compared to the optimal
single user bound.
3.5.2 BER Comparison
Fig. 3.9 shows BER results for systems with a load of 100%. As we can see from this
figure, performance of LDS-OFDM is inferior to that of GO-MC-CDMA and turbo
structured LDS-OFDM. Meanwhile, JSG-OFDM outperforms all the other systems.
3.5. Performance Evaluation 61
TABLE 3.3: JSG-OFDM scenarios
Scenario Degree distribution Girth
scenario− 1 Dega 6
scenario− 2 Degb 4
scenario− 3 Deg′d 8
Due to the inherent advantage of the joint sparse graph, JSG-OFDM achieves better
performance than turbo structured LDS-OFDM. With the optimized degree distribu-
tion (Deg′d) and cycle structure (girth of 8), JSG-OFDM scenario − 3 achieves the
best performance. Its performance improvements at BER of 10−5 are: 0.6 dB over
JSG-OFDM scenario − 1, 1.1 dB over JSG-OFDM scenario − 2, 1.5 dB over turbo
structured LDS-OFDM, 1.6 dB over GO-MC-CDMA and 1.8 dB over LDS-OFDM,
respectively. Note that even if JSG-OFDM scenario− 3 is adopted, there is still a gap
to the optimal single user bound. Moreover, for JSG-OFDM scenario − 3, physical
layer framing is adopted and tested, i.e., 10 OFDM symbols constitute a frame. As
its curve outperforms that of JSG-OFDM scenario− 3, the joint sparse graph can be
extended to fit the data length in practical systems.
In addition, the performance for systems with a load of 150% is also evaluated and
shown in Fig. 3.10. It can be seen that the BER results of 150% loading are inferior
to that of 100% loading. For a load of 150%, JSG-OFDM still outperforms other
systems, and the scenario−3 of JSG-OFDM achieves the best performance. Compared
with LDS-OFDM, GO-MC-CDMA and turbo structured LDS-OFDM, the JSG-OFDM
scenario − 3 can obtain about 1.5 - 1.8 dB gain in the medium to high SNR region.
Apparently, as the system loading increases, the gap to the optimal single user bound
becomes wider.
To reveal the theoretical threshold of the joint sparse graph, we analyse its maximum
achievable throughput by EXIT charts. It was proved in [118] and [119] that the
capacity of an iterative system is equal to the area under the EXIT curve of the inner
code, provided that the bit stream input to the receiver has independently distributed
bits and the MAP algorithm is used for the detection or the decoding. For the joint
62 Chapter 3. Joint Sparse Graph for OFDM (JSG-OFDM) System
0 2 4 6 8 10 12 14 16 18 2010
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Eb/N
0(dB)
BE
R
GO−MC−CDMA, separative detection and decoding
LDS−OFDM, separative detection and decoding
Turbo structured LDS−OFDM, turbo iterative detection and decoding
JSG−OFDM scenario−1, joint detection and decoding
JSG−OFDM scenario−2, joint detection and decoding
JSG−OFDM scenario−3, joint detection and decoding
JSG−OFDM scenario−3, 10 OFDM symbols in 1 frame, joint detection and decoding
single user bound
Fig. 3.9: Performance of 100% loaded systems
0 2 4 6 8 10 12 14 16 18 2010
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Eb/N
0(dB)
BE
R
GO−MC−CDMA, separative detection and decoding
LDS−OFDM, separative detection and decoding
Turbo structured LDS−OFDM, turbo iterative detection and decoding
JSG−OFDM scenario−1, joint detection and decoding
JSG−OFDM scenario−2, joint detection and decoding
JSG−OFDM scenario−3, joint detection and decoding
JSG−OFDM scenario−3, 10 OFDM symbols in 1 frame, joint detection and decoding
single user bound
Fig. 3.10: Performance of 150% loaded systems
3.5. Performance Evaluation 63
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 170
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
capa
city
(bi
t/sym
bol)
JSG−OFDM capacity
2.5 dB
1.1 dB
Fig. 3.11: Maximum effective throughput of JSG-OFDM
sparse graph, assuming that the area under the EXIT curve of the CND&PND is
represented byA, then the maximum achievable throughput at a particular Eb/N0 value
is given by A(Eb/N0). In other words, if A is calculated for different Eb/N0 values, the
threshold of the joint sparse graph can be evaluated. Fig. 3.11 quantifies the maximum
effective throughput of the joint sparse graph, where the SNR represent Es/N0, and
the horizontal dotted line represents the throughput of the scheme considered. More
explicitly, 0.5 bit/symbol is the effective throughput of half rate coded and BPSK
modulated system. The circle and the rectangle are respectively located at the SNR
required for 100% and 150% loaded JSG-OFDM scenario − 3 to achieve an identical
throughput at a target BER of 10−5. The SNR values shown next to the circle and the
rectangle indicate the distance to the capacity. As can be seen that the 100% loaded
JSG-OFDM scenario − 3 is capable of operating within 1.1 dB from the capacity
curve. When the system loading is increased to 150%, the gap becomes 2.5 dB. Hence,
as expected, the lower the system loading employed, the closer the joint sparse graph
operates to capacity.
64 Chapter 3. Joint Sparse Graph for OFDM (JSG-OFDM) System
3.5.3 Convergence Behavior
To show the convergence behavior of JSG-OFDM, Fig. 3.12 depicts the performance
at different iterations of 100% loaded JSG-OFDM scenario − 1. As expected, over
AWGN channel at Eb/N0 = 9 dB, the BER stops falling down after 5 iterations, which
is accurately predicted by the joint detection and decoding trajectory presented in
Fig. 3.4. Moreover, over ITU Pedestrian Channel B at Eb/N0 = 13 dB, the receiver
needs 4 iterations to reach the message convergence, which concurs with the trajectory
prediction plotted in Fig. 3.5. Therefore, EXIT chart analysis and convergence behavior
of the joint sparse graph are verified by BER simulations.
1 2 3 4 5 610
−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Number of iterations
BE
R
AWGN at Eb/N
0 = 9 dB
ITU Pedestrian Channel B at Eb/N
0= 13 dB
Fig. 3.12: Performance at different iterations for JSG-OFDM
3.5.4 Performance of Different Users
To gain more insight to the joint sparse graph, we present another result showing the
performance of individual users. Fig. 3.13 illustrates the performance of the worst user
and the best user in 100% loaded JSG-OFDM scenario− 3. It shows that some users
have better performance than the others. To be more precise, we can see that at low
3.5. Performance Evaluation 65
SNR region the performance gap is not as obvious as in the high SNR region. This
phenomenon can be explained by the dominating effect of noise at low SNRs.
0 2 4 6 8 10 12 14 16 18 2010
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Eb/N
0 (dB)
BE
R
Worst userBest userOverall
Fig. 3.13: Performance of different users in JSG-OFDM
3.5.5 Near-far Effect
The near-far problem is a condition in which a receiver captures a strong signal and
thereby makes it impossible for the receiver to detect a weaker signal. The joint sparse
graph does not give equal multiuser efficiency, and it does not result in the same
performance for all the users as indicated by Fig. 3.13, so it is necessary to investigate
the near-far effect of the JSG-OFDM. Fig. 3.14 shows the performance of near-far
resistance for JSG-OFDM scenario− 3 with different loads. The simulation is carried
out for the case when Eb/N0 = 16 dB for the first user, and Eb/N0 of other users is
different. The BER of the first user is plotted against ∆Eb/N0 which represents the
difference in Eb/N0 between the user of interest and the other users. More explicitly,
66 Chapter 3. Joint Sparse Graph for OFDM (JSG-OFDM) System
when the first user’s SNR Eb/N0 = 16 dB, the other users’ SNR equals to ∆Eb/N0 plus
16 dB. It can be seen that unequal received power has a minor effect on the performance
of user of interest under different loadings. It is due to the iterative processing, or in
other words the near-far problem can be alleviated by the joint sparse graph and the
MPA. We can conclude that JSG-OFDM is robust against unequal received powers.
−3 −2 −1 0 1 2 310
−6
10−5
10−4
10−3
10−2
∆Eb/N
0(dB)
BE
R
100% loaded JSG−OFDM150% loaded JSG−OFDM
Fig. 3.14: Near-far effect of JSG-OFDM
3.5.6 Multipath Diversity
To test the multipath diversity of the joint sparse graph, we simulate the JSG-OFDM
scenario− 3 over different multipath channel models, i.e., ITU Pedestrian Channel A,
and the ITU Pedestrian Channel B. Fig. 3.15 shows the comparison results, and it can
be seen that the performance of ITU Pedestrian Channel A is inferior to that of ITU
Pedestrian Channel B, indicating that the multipath diversity can be exploited to the
JSG-OFDM system.
3.5. Performance Evaluation 67
0 2 4 6 8 10 12 14 16 18 2010
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Eb/N
0(dB)
BE
R
JSG−OFDM scenario−3, ITU Pedestrian Channel BJSG−OFDM scenario−3, ITU Pedestrian Channel A
Fig. 3.15: Performance of JSG-OFDM over different multipath channels
3.5.7 Comparison with MMSE Detector
JSG-OFDM is further compared with MC-CDMAwhich is a conventional OMA scheme.
We set the numbers of users and variable nodes to be 12 and 480, respectively, while
keeping the 120 chips and the BPSK modulation, thus the system loading is increased
to 480/(120×2) = 200%. A linear MMSE detector, which is the optimal linear detector
that maximizes SINR, is used in MC-CDMA. Fig. 3.16 shows the comparison result. It
can be seen that, under the overloaded condition, JSG-OFDM outperforms MC-CDMA
significantly. At BER of 3 × 10−3, there is more than 6.5 dB gain by employing JSG-
OFDM. Therefore, the joint sparse graph is a much more effective technique to handle
overloaded transmissions.
3.5.8 Detection Complexity Comparison
As the same LDPC code is applied to all the investigated systems and the decoding
complexity is therefore the same, we focus on the comparisons of detection complexity.
Let X be the constellation alphabet for the transmitted symbol, which is related to the
68 Chapter 3. Joint Sparse Graph for OFDM (JSG-OFDM) System
0 2 4 6 8 10 12 14 16 18 2010
−5
10−4
10−3
10−2
10−1
100
Eb/N
0(dB)
BE
R
MC−CDMAJSG−OFDM
Fig. 3.16: Performance comparison with MMSE detector
modulation order. In GO-MC-CDMA, the detector complexity increases exponential-
ly with the number of symbols in each group, which results in a complexity order of
O(| X |8
). In LDS-OFDM, the detector complexity increases exponentially with the
number of symbols per subcarrier which is dc,lds, thus its complexity order is O(| X |6
).
In turbo structured LDS-OFDM, the detector complexity is several times higher than
that of LDS-OFDM. Apparaently, the complexity of the LDS-OFDM detector is less
than that of GO-MC-CDMA and turbo structured LDS-OFDM. Thus, we compare the
complexity between JSG-OFDM and LDS-OFDM. It is noteworthy that due to the
similar MPA, the detection complexity order of the JSG-OFDM is O(| X |6
)which is
the same as that of LDS-OFDM. However, as the convergence behavior and the inter-
section point of the EXIT chart are different between LDS-OFDM and JSG-OFDM,
the complexity is slightly different, which will be discussed in the sequel.
We express the complexity of detection in terms of equivalent additions. The basic
operations performed by MPA include: i) addition, ii) subtraction, iii) multiplication
by ±1, iv) division by 2, v)comparison and vi) max(x, y). The i) − v) operations
correspond to one equivalent addition, and the vi) operation corresponds to two equiv-
alent additions, since it first uses a compare operation to compare the two input values
3.6. Summary 69
and then stores the result in a register [140]. TABLE 3.4 summarizes the computation-
al requirements for a load of 100%. On one hand, for LDS-OFDM and JSG-OFDM
(including scenario − 1 and scenario − 3), the equivalent addition operations drop
dramatically when Eb/N0 increases, which means channel conditions affect the number
of iterations significantly. This is due to the property of the message passing algorith-
m, more explicitly, as shown in the joint detection and decoding, the iteration can be
terminated promptly if the syndrom equals to zero. Therefore, the number of addi-
tion operations is a function of Eb/N0. On the other hand, the receiver complexity of
JSG-OFDM, including scenario − 1 and scenario − 3, is obviously higher than that
of LDS-OFDM. This is related to the more complicated process on the variable n-
odes in the JSG-OFDM system. However, by careful design of the joint sparse graph,
the detection complexity of JSG-OFDM scenario− 3 is less than that of JSG-OFDM
scenario− 1. This can be attributed to the optimization of degree distribution which
reduces the density of the graph.
TABLE 3.4: Equivalent number of addition operations for detection
System 4 dB 8 dB 12 dB 16 dB 20 dB
LDS-OFDM 249019 179565 112956 85971 63502
JSG-OFDM scenario− 1 396513 286375 180572 136693 101237
JSG-OFDM scenario− 3 391206 282613 175934 132067 96326
3.6 Summary
In this chapter, a JSG-OFDM system has been proposed and analysed. Unlike any
previous single graph, the JSG-OFDM is based on a joint sparse graph which combines
NOMA (LDS-OFDM) and FEC coding (LDPC codes). The system framework of JSG-
OFDM is designed, and the joint detection and decoding, which applies MPA on the
joint sparse graph, is presented. JSG-OFDM is a novel system that jointly performs
detection and decoding on one entire sparse graph (Type-A receiver in Fig. 2.7). In
addition, the iterative structure of JSG-OFDM receiver is illustrated. Its convergence
behavior is analysed by EXIT charts and indicated by the trajectories over different
70 Chapter 3. Joint Sparse Graph for OFDM (JSG-OFDM) System
channels. Furthermore, degree distribution and short cycle of the joint sparse graph
are investigated using EXIT chart. It is shown that these two parameters affect the
performance of the joint detection and decoing significantly. According to the design
guidelines of the joint sparse graph, three scenarios of JSG-OFDM are presented, and
similar well-known multiple access systems are compared. The simulation results show
that the JSG-OFDM scenario−3 (degree distribution of Deg′d and girth of 8) achieves
the best performance. Its performance improvement is mainly due to the offline op-
timization of the joint sparse graph to exploit frequency domain diversity in addition
to avoiding strong interference to corrupt all the subcarriers. In general, by carefully
designing for specific applications, JSG-OFDM can improve the LDS based system’s
performance.
Chapter 4
Joint Sparse Graph for
FBMC-IOTA (JSG-IOTA)
System
As a promising NOMA, LDS has, however, never been applied to FBMC systems. These
motivate us to investigate overloaded FBMC systems by synergistically using techniques
of the joint sparse graph. In this chapter, we use a sparse graph to express the weight
matrix which is used to calculate the intrinsic interference in FBMC systems with IOTA
function based pulse. We further propose a joint sparse graph for the FBMC-IOTA
system, namely JSG-IOTA. The joint sparse graph combines single graphs of LDWM,
LDS and LDPC codes, which represent multi-carrier modulation, NOMA and FEC
coding techniques, respectively. By employing MPA, a joint detection and decoding
algorithm based on the joint sparse graph is presented. The iterative structure of
JSG-IOTA receiver is illustrated, and its EXIT chart is analysed. Furthermore, design
guidelines for the joint sparse graph are derived through the EXIT chart analysis.
Numerical results show the superiority of JSG-IOTA to similar systems such as OFDM,
IOTA, LDS-OFDM, LDS-IOTA and turbo structured LDS-IOTA.
71
72 Chapter 4. Joint Sparse Graph for FBMC-IOTA (JSG-IOTA) System
4.1 JSG-IOTA System Model
Consider an uplink MIMO communications with K users transmitting to the same base
station where each user and the base station are equipped with NT and NR antennas,
respectively. The JSG-IOTA system is schematically shown in Fig. 4.1 and Fig. 4.2.
Let the processing gain be N , and each user has a data vector consisting ofM symbols.
Denote J as the number of parity-check equations in the LDPC code. We assume that
perfect CSI is available at the receiver.
user 1
LDPC
encoder
symbol
mapper
LDS
spreader
Re{} IFFT
Im{} IFFT
filter
banks
filter
banks
G(n)
G(n-N/2)
j(n+2(u-1)+1)
j(n+2(u-1))
LDPC
encoder
symbol
mapper
LDS
spreader
Re{} IFFT
Im{} IFFT
filter
banks
filter
banks
G(n)
G(n-N/2)
j(n+2(u-1)+1)
j(n+2(u-1))
user K
LDPC
encoder
symbol
mapper
LDS
spreader
Re{} IFFT
Im{} IFFT
filter
banks
filter
banks
G(n)
G(n-N/2)
j(n+2(u-1)+1)
j(n+2(u-1))
LDPC
encoder
symbol
mapper
LDS
spreader
Re{} IFFT
Im{} IFFT
filter
banks
filter
banks
G(n)
G(n-N/2)
j(n+2(u-1)+1)
j(n+2(u-1))
Fig. 4.1: JSG-IOTA transmitter model
4.1. JSG-IOTA System Model 73
T
ime
Fre
qu
ency
u-2
u-1
u
u+
1
u+
2
n-2
0
0
0
0
0
n-1
0
0
0.2
486
0
0
n
0
-0.5
756
1
0.5
756
0
n+
1
0
0
-0.2
486
0
0
n+
2
0
0
0
0
0
R
eal
bra
nch
I
mag
inar
y b
ranch
(
b)
LD
WM
of
the
intr
insi
c in
terf
erence
of
JSG
-IO
TA
T
ime
Fre
qu
ency
u-2
u-1
u
u+
1
u+
2
n-2
0
0
0
0
0
n-1
0
0
0.2
486
0
0
n
0
0.5
756
0
-0.5
756
0
n+
1
0
0
-0.2
486
0
0
n+
2
0
0
0
0
0
intr
insi
c-in
terf
eren
ce n
od
es
FF
T
filt
er
ban
ks
FF
T
filt
er
ban
ks
G(-
n)
G(N
/2-n
)
j(n+
2(u
-1))
j-(n
+2(u
-1))
S/P
real
bra
nch
1st
eq
ual
izer
P/S
real
bra
nch
N
th e
qu
aliz
er
S/P
imag
bra
nch
1st
eq
ual
izer
imag
bra
nch
N
th e
qu
aliz
er
S/P
S/P
P/S
Re{
}
Intr
insi
c
inte
rfer
ence
Im{}
Intr
insi
c
inte
rfer
ence
par
ity-c
hec
k n
od
es
var
iab
le n
od
es
chip
no
des
LD
PC
LD
S
xv 1
,1
v 1,2
xv K
,1
v K,2
v 1,M
v K,M
use
r 1
use
r K
p1,J
p1,1
pK
,J
pK
,1
c 1
c 2
c N
JSG
i 1,1
i 1,2
i T,N
LD
WM
FF
T
filt
er
ban
ks
FF
T
filt
er
ban
ks
G(-
n)
G(N
/2-n
)
j(n+
2(u
-1))
j-(n
+2(u
-1))
(a)
Blo
ck d
iagra
m o
f th
e JS
G-I
OT
A r
ecei
ver
Fig.4.2:JSG-IOTA
receiver
model
74 Chapter 4. Joint Sparse Graph for FBMC-IOTA (JSG-IOTA) System
At the transmitter shown in Fig. 4.1, a modulated data stream is initially generated in a
conventional manner, by multiplying the data symbols with a LDS signature (a random
sequence of chips) and subsequently applying the IOTA modulation to impose the chips
onto respective subcarrier frequencies. The IOTA modulators can be implemented by
separate IFFT blocks followed by a bank of filters for the I and Q components, which
are combined after polyphase filtering. An N/2 rotation of IFFT output is used to
shift the zero frequency subcarrier to middle position. Unlike conventional systems,
the LDS spreader is configured to guarantee that the ratio of dv,lds to N is small
enough (much less than 1), where dv,lds is the number of chips that are spread by one
symbol. Meanwhile, the number of symbols that are superimposed on each chip, dc,lds,
is much less than the total number of symbols. Consequently, each user will experience
interference from only a small number of other users data symbols.
The JSG-IOTA receiver shown in Fig. 4.2 (a) is implemented by filter banks followed
by the FFT block, where the FFT operations are conducted separately for the I and Q
branches. Due to the multiple transmit and receive antennas as well as multiple subcar-
riers, the equalization is performed on per-subcarrier basis, i.e., signals corresponding
to the same subcarrier at different antennas are jointly equalized. More explicitly, the
processing is arranged to perform serial-to-parallel conversion on the subcarrier’s data
received from the real and the imaginary branches of each antenna, and distribute the
subcarrier’s data to corresponding real and imaginary branch equalizers, according to
the subcarrier index. Subsequently, after parallel-to-serial conversion, the real part of
the signal from the I channel is combined with the imaginary part of the signal from the
Q channel, and further passed to the next processing stage. Usually, the residual signal
from the I and the Q channels is regarded as the intrinsic interference and discarded
in the following process. In fact, the intrinsic interference can be determined by mul-
tiplying neighboring signals with a weight matrix which can be determined using the
ambiguity function of the pulse shaping filter [141] [142]. We find that some elements of
the weight matrix are very small and can be safely ignored in order to reduce the com-
putational complexity, consequently a low density weight matrix is formed and shown
in Fig. 4.2 (b). Therefore, the sparse graphical block in Fig. 4.2 (a) can be configured
to include four types of nodes: intrinsic-interference nodes in,u(n ∈ [1, N ], u ∈ [0,Z]),
4.2. Joint Detection and Decoding 75
chip nodes cn(n ∈ [1, N ]), variable nodes vk,m(k ∈ [1,K],m ∈ [1,M ]) and parity-check
nodes pk,j(k ∈ [1,K], j ∈ [1, J ]), representing the intrinsic interference corresponding
to the symbol on the nth subcarrier during the time of index u, the nth chip, the mth
data symbol and the jth parity-check equation of the kth user, respectively. A single
graph, labeled with LDWM in the receiver, represents the low density weight matrix
due to intrinsic interference of IOTA. Another single graph, labeled with LDS in the
receiver, represents the low density signature due to LDS [64]. The other single sparse
graph, as labeled with LDPC in the receiver, represents the low density parity-check
matrix due to LDPC code [124]. These three types of single sparse graphs stem from
the processes of FBMC-IOTA modulation, NOMA and FEC coding, respectively. In
our proposed JSG-IOTA scheme, chip nodes are used to connect intrinsic-interference
nodes and variable nodes through low density edges, meanwhile, variable nodes are
connected to parity-check nodes by low density edges. As such, a joint sparse graph
is modeled and labeled with JSG in the figure, where LDWM, LDS and LDPC codes
are well linked together. In the joint sparse graph, users’ signals that use the same
chip will be superimposed, and the number of symbols that interfere with each other
at one chip is much less than the total number of symbols, thus the system can achieve
good performance in overloaded conditions. The joint sparse graph is arranged to pro-
cess the chips from the received signals to recover the transmitted data. Note that
applying MPA on the joint sparse graph performs not only detection and interference
utilization, but also decoding at the same time. Furthermore, the JSG-IOTA receiver
is different from the turbo structured receiver, as there is no outer-inner turbo style
iteration. Hence, the JSG-IOTA is based on a joint parse graph which combines IOTA
modulation, NOMA and sparse graph coding techniques. In the next section, joint
detection and decoding on such graph will be described.
4.2 Joint Detection and Decoding
According to the JSG-IOTA system model, we present joint detection and decoding on
the joint sparse graph using MPA.
The spreading signature and the parity-check matrix for the kth user are Sk = [sk,1, ..., sk,M ] ∈
76 Chapter 4. Joint Sparse Graph for FBMC-IOTA (JSG-IOTA) System
CN×M and Hk = [hk,1, ...,hk,J ] ∈ BM×J , respectively. The weight matrix for the
nth chip is Wn = [wn,1, ...,wn,U ] ∈ CK×U . As for the multi-user scenario, let S =
[S1, ...,SK ] ∈ CN×MK , H = [H1, ...,HK ] ∈ BM×JK and W = [W1, ...,WN ] ∈ CK×UN
be the spreading signatures, the parity-check matrices of LDPC codes and the weight
matrices of intrinsic interferences, respectively. We also defineAu = diag(a1,u, ..., aK,u) ∈
CK×K as the transmit amplitude of users during the time of index u, and Ek,u =
diag(ek,1,u, ..., ek,N,u) ∈ CN×N as the corresponding channel gain for the kth user dur-
ing the time of index u, where ak,u and ek,n,u are both scalars. Moreover, ψn =
{(k,m) : snk,m ̸= 0} and εk,m = {n : snk,m ̸= 0} are the set of data symbols (which
may belong to different users) that interfere on chip cn and the set of chips that vk,m
is spread on, respectively; γk = {(n, u) : wkn,u ̸= 0} and ηn,u = {k : wkn,u ̸= 0} are
the set of intrinsic interferences that connect to chip ck and the set of chips that con-
nect to intrinsic-interference node in,u, respectively; ϕk,j = {(k,m) : hmk,j ̸= 0} and
ωk,m = {(k, j) : hmk,j ̸= 0} are the set of data symbols that connect to parity-check node
pk,j and the set of parity-check nodes that connect to vk,m, respectively.
Let Lcn→in,u be the log-likelihood ratio (LLR) delivered from the chip node cn to the
intrinsic-interference node in,u, Lin,u→cn be the LLR delivered from in,u to cn. Similarly,
Lvk,m→cn and Lvk,m→pk,j are the LLRs delivered from the variable node vk,m to the
chip node cn and the parity-check node pk,j , respectively. The LLRs delivered from
cn and pk,j to vk,m are given by Lcn→vk,m and Lpk,j→vk,m , respectively. Lvk,m is the
final estimation of vk,m. In JSG-IOTA, each user’s chip will be transmitted over an
orthogonal subcarrier. Let v′k,m,n,u = snk,mvk,m be the signal at nth chip generated by
the LDS spreader during the time of index u, the received spreading sequence for the
data symbol m of the kth user can be represented by rk,m,n,u = ak,uek,n,uv′k,m,n,u. The
received signal in a multi-carrier system (including CP-based OFDM and FBMC-IOTA)
can be written in a general form as
y(t) =K∑k=1
M∑m=1
N∑n=1
+∞∑u=−∞
rk,m,n,ugn,u(t) + z(t) (4.1)
where z(t) and gn,u(t) are the AWGN with variance σ2A and the synthesis basis which is
obtained by the time-frequency shifted version of the prototype function, respectively.
4.2. Joint Detection and Decoding 77
In the OFDM, the synthesis basis can be expressed as [22]
gn,u(t) =
exp(j2π(n− 1)Ft) uT0 − Tcp ≤ t ≤ uT0 + T
0 otherwise
(4.2)
where F = 1/T is subcarrier frequency spacing, Tcp is the length of CP and T0 = T+Tcp
is OFDM symbol duration.
In JSG-IOTA,
gn,u(t) = exp(j((n− 1) + u)π/2)exp(j2π(n− 1)v0t)g(t− uτ0) (4.3)
where g(t) is the well-localized IOTA pulse filter, and v0τ0 = 1/2. The transmitted
signals have symbol duration τ0 and subcarrier spacing v0. One can either set v0 = F ,
τ0 = T/2 or v0 = F/2, τ0 = T [19]. Here, we adopt the former approach, i.e., the
subcarrier spacing is kept the same as in OFDM, but symbol duration is reduced by
half. The received signal is [23] [143]
y(t) =
K∑k=1
M∑m=1
N∑n=1
+∞∑u=−∞
[rRk,m,n,ugn,2u(t) + rIk,m,n,ugn,2u+1(t)] + z(t)
=K∑k=1
M∑m=1
N∑n=1
+∞∑u=−∞
[rRk,m,n,ug(t− 2uτ0) + jrIk,m,n,ug(t− (2u+ 1)τ0)]
× exp(j((n− 1) + 2u)/2)exp(j2π(n− 1)v0t) + z(t)
(4.4)
The demodulated signal can be expressed as
v̂′R
k,m,n,u = Re
{∫y(t)g∗n,2u(t)dt
}(4.5)
v̂′I
k,m,n,u = Im
{∫y(t)g∗n,2u+1(t)dt
}(4.6)
By sampling y(t) at rate 1/Ts during time interval [uT − τ0, uT + τ0], the received
signal can be written as
y(uT + iTs) =
K∑k=1
M∑m=1
N∑n=1
+∞∑l=−∞
[rRk,m,n,lg(uT + iTs − lT ) + jrIk,m,n,lg(uT + iTs − lT − T/2)]
× exp(jπ((n− 1) + 2l)/2)exp(j2π(n− 1)i/N) + z
(4.7)
78 Chapter 4. Joint Sparse Graph for FBMC-IOTA (JSG-IOTA) System
where i = −N/2, ..., N/2 − 1. Denoting yi[u] = y[uN + i] = y(uT + iTs), the above
equation can be reformed as
yi[u] =∑q
g(qT + iTs)
{N∑n=1
rRk,m,n,u−qexp[jπ((n− 1) + 2u− 2q)/2]exp[j2π(n− 1)i/N ]
}
+∑q
g(qT + iTs −T
2)
{N∑n=1
jrIk,m,n,u−qexp[jπ((n− 1) + 2u− 2q)/2]exp[j2π(n− 1)i/N ]
}
+ z
=∑q
{gi[q]D
iN (r
Rk,m,n,u−q) + gi−N/2[q]D
iN (jr
Ik,m,n,u−q)
}+ z
= gi[u]⊗DiN (r
Rk,m,n,u) + gi−N/2[u]⊗Di
N (jrIk,m,n,u) + z
(4.8)
where ⊗ denotes the convolution operation, and
DiN (xk,m,n,u) =
N∑n=1
xk,m,n,uexp(jπ((n− 1) + 2u)/2)exp(j2π(n− 1)i/N) (4.9)
gi[u] = g[uN + i] = g(uT + iTs) (4.10)
The phase correction (j((n−1)+2u) for the I channel and j((n−1)+2u+1) for the Q channel)
before the IFFT operation is due to the first exponential term in (4.9). An N/2 rotation
of IFFT output is needed here to shift the zero frequency subcarrier to middle position.
At the receiver, by sampling the received signal at rate 1/Ts, (4.5) can be reformed as
v̂′R
k,m,n,u = Re
TsN/2−1∑i=−N/2
+∞∑l=−∞
y(lT + iTs)g∗n,2u(lT + iTs)
= Re
Tsexp[−jπ((n− 1) + 2u)/2]
N/2−1∑i=−N/2
+∞∑l=∞
yi[l]gi[l − u]exp[−j2π(n− 1)i/N ]
= Re
Tsexp[−jπ((n− 1) + 2u)/2]
N/2−1∑i=−N/2
yi[u]⊗ gi[−u]exp[−j2π(n− 1)i/N ]
= Re
Tsj((n−1)+2u)
N/2−1∑i=−N/2
yi[u]⊗ gi[−u]exp[−j2π(n− 1)(i+N/2)/N ]
(4.11)
where
gi[−u] = g[−uN + i] = g(−NT + iTs) (4.12)
4.2. Joint Detection and Decoding 79
By repeating the above process on the imaginary branch of (4.6),
v̂′I
k,m,n,u = Im
{Tsj
−((n−1)+2u)N∑i=1
yi−1[u]⊗ gi−1−N/2[−u]exp[−j2π(n− 1)(i− 1)/N ]
}(4.13)
Owing to the real-orthogonality condition on g(t), i.e., Re{gn,u(t)g
∗n0,u0(t)
}= δn,n0δu,u0 ,
(4.5) and (4.6) can be written as
v̂′k,m,n,u =
∫y(t)g∗n,u(t)dt
= ak,uek,n,uv′k,m,n,u +
∑(n′,u′ )̸=(n,u)
ak,u′ek,n′,u′v′k,m,n′,u′
∫gn′,u′(t)g
∗n,u(t)dt︸ ︷︷ ︸
in,u
+ zn,u
(4.14)
Since the prototype function g(t) is chosen to be well localized both in time and fre-
quency, the intrinsic interference in,u in (4.14) only depends on a restricted set of
time-frequency positions (n′, u′) around the signal of interest. Assuming that the chan-
nel remains relatively constant at those positions, the intrinsic interference in,u can be
approximated as
in,u = ak,uek,n,u∑
(n′,u′) ̸=(n,u)
v′k,m,n′,u′
∫gn′,u′(t)g
∗n,u(t)dt︸ ︷︷ ︸
jv′(i)k,m,n,u
= ak,uek,n,ujv′(i)k,m,n,u
(4.15)
Combining (4.14) and (4.15) yields
v̂′k,m,n,u = ak,uek,n,u[v′k,m,n,u + jv′
(i)k,m,n,u] + zn,u (4.16)
In Fig. 4.2 (a), the input to the real and the imaginary branch equalizers at the
JSG-IOTA receiver can be expressed as
yRk,m,n,u = ak,uek,n,u[v′Rk,m,n,u + jv′
(i)k,m,n,u] + zRn,u (4.17)
yIk,m,n,u = ak,uek,n,u[v′(r)k,m,n,u + jv′
Ik,m,n,u] + zIn,u (4.18)
where v′(i)k,m,n,u and v′
(r)k,m,n,u are the intrinsic interference for the real and imaginary
FFT chains, respectively. Given the knowledge of the channel state, the transmitted
80 Chapter 4. Joint Sparse Graph for FBMC-IOTA (JSG-IOTA) System
signal can be recovered by zero forcing (ZF) equalization, that is
v̂′k,m,n,u = Re{yRk,m,n,u/ak,uek,n,u
}+ jIm
{yIk,m,n,u/ak,uek,n,u
}+ (zRn,u/ak,uek,n,u) + (zIn,u/ak,uek,n,u)
= v′Rk,m,n,u + jv′
Ik,m,n,u + z′n,u
(4.19)
where z′n,u denote the combined noise term. An MMSE equalization can be designed
similarly.
For the joint sparse graph, there are two sets of soft information coming from the in-
trinsic interference: the first one is combined by the Q component of the output of the
real branch equalizer and I component of the output of the imaginary branch equal-
izer, i.e., v′(r)k,m,n,u + jv′
(i)k,m,n,u, while the second one is calculated based on neighboring
time-frequency positions around the signal of interest. If the difference between these
two values is small enough, we reach a decision that its correlated information is highly
reliable, and set Pr(v̂′k,m,n,u = v′k,m,n,u) = 1. Otherwise the soft information needs to
be updated. The weight matrix of the intrinsic interference is represented by a sparse
graph to propagate the information, and each message will be calculated iteratively
from the previous values of the neighbouring nodes. In the joint detection and decod-
ing, intrinsic-interference nodes and variable nodes update at the same time. For the
intrinsic-interference nodes in error,
Lin,u→cn = log
∑cn=1
exp(∑
n′∈ηn,u\n
cn′2 Lcn′→in,u − 1
2σ2A∥ v′(r)k,m,n,u + jv′
(i)k,m,n,u − in,u ∥2)
∑cn=0
exp(∑
n′∈ηn,u\n
cn′2 Lcn′→in,u − 1
2σ2A∥ v′(r)k,m,n,u + jv′
(i)k,m,n,u − in,u ∥2)
(4.20)
In the joint sparse graph, the updating of Lvk,m→cn not only receives chip nodes infor-
mation, but also utilizes the information that comes from parity-check nodes.
Lvk,m→cn =∑
n′∈εk,m\n
Lcn′→vk,m +∑
j∈ωk,m
Lpk,j→vk,m (4.21)
where εk,m \n is the set of chips (excluding cn) that vk,m is spread on. When executing
the MPA for joint detection and decoding, the more types of nodes that are connected
to a variable node, the more reliable information that can be utilized when processing
that variable node.
4.2. Joint Detection and Decoding 81
Similarly, Lvk,m→pk,j also involves message from both sides
Lvk,m→pk,j =∑
j′∈ωk,m\j
Lpk,j′→vk,m +∑
n∈εk,m
Lcn→vk,m (4.22)
where ωk,m \ j is the set of parity-check nodes (excluding pk,j) that connect to the
variable node vk,m.
In terms of chip nodes and parity-check nodes, their extrinsic messages are calculated
at the same time. In the single graph, chip nodes only utilize information from variable
nodes [64]. However, in the joint sparse graph, chip nodes gather information from
both intrinsic-interference nodes and variable nodes,
Lcn→in,u =∑
(n′,u′)∈γn\(n,u)
Lin′,u′→cn +∑
(k,m)∈ψn
Lvk,m→cn (4.23)
For the updating of Lcn→vk,m ,
Lcn→vk,m = f(vk,m|v̂′k,m,n,u, Lin,u→cn , Lvk′,m′→cn , (k′,m′) ∈ ψn \ (k,m)) (4.24)
where ψn \ (k,m) is the set of data symbols (excluding vk,m) that interfere on the chip
cn. To approximate the maximum a posteriori (MAP) probability detector, the right
hand side of (4.24) represents marginalization function and can be written as
f(vk,m|v̂′k,m,n,u, Lin,u→cn , Lvk′,m′→cn , (k′,m′) ∈ ψn \ (k,m))
= log(N∑n=1
p(v̂′k,m,n,u|v)pn(v|vk,m))
= log(
N∑n=1
p(v̂′k,m,n,u|v)∏
(k′,m′)∈ψn\(k,m)
pn(vk′,m′))
(4.25)
where v is the transmitted vector, p(v̂′k,m,n,u|v) and pn(vk′,m′) are given as
p(v̂′k,m,n,u|v) =1√2πσA
exp(− 1
2σ2A∥ v̂′k,m,n,u − rT[n]v[n] ∥2) (4.26)
pn(vk′,m′) = exp(Lvk′,m′→cn) (4.27)
where v[n] and r[n] denote the vector containing the symbols transmitted by every
user that spread its data on the nth chip and their corresponding received signature,
82 Chapter 4. Joint Sparse Graph for FBMC-IOTA (JSG-IOTA) System
respectively. Substituting (4.26) and (4.27) into (4.25), the formula becomes
Lcn→vk,m =
κn,k,mmaxv[n]
∗(∑(n,u)
Lin,u→cn +∑
(k′,m′)∈ψn\(k,m)
Lvk′,m′→cn − 1
2σ2A∥ v̂′k,m,n,u − rT[n]v[n] ∥2)
(4.28)
where κn,k,m denotes the normalization coefficient and
max∗(a, b) , log(ea + eb) (4.29)
The LLR of the parity-check nodes is updated as
Lpk,j→vk,m = α−1(∑
(k′,m′)∈ϕk,j\(k,m)
α(Lvk′,m′→pk,j )) (4.30)
where ϕk,j \ (k,m) is the set of data symbols (excluding vk,m) that connect to the
parity-check node pk,j , and
α(x) = sign(x)× (− log tan(| x | /2)) (4.31)
where sign(x) represents the sign of x, and the inverse of α(x) is
α−1(x) = (−1)sign(x) × (− log tan| x |2
) (4.32)
In the joint sparse graph, parity-check nodes are available, thus it is possible to termi-
nate the iteration by syndrome computing. A posteriori probability of the transmitted
symbo vk,m is calculated as
Lvk,m =∑
n∈εk,m
Lcn→vk,m +∑
j∈ωk,m
Lpk,j→vk,m (4.33)
The estimated value of the variable node vk,m is obtained by making a hard decision, i.e.,
v̂k,m = argmaxvk,m
Lvk,m . If the result of syndrome computing for each user equals to zero,
or the maximum iteration number is reached, the process is terminated. Otherwise,
the iteration goes on.
In the next section, we carry out theoretical analysis of the joint sparse graph using
the EXIT chart.
4.3. EXIT Chart Analysis of JSG-IOTA 83
4.3 EXIT Chart Analysis of JSG-IOTA
EXIT chart is used to construct good iteratively-decoded FEC codes. In this section,
we shall illustrate how EXIT charts can be utilized to analyse the convergence behavior
of joint detection and decoding in JSG-IOTA.
4.3.1 Iterative Structure of the Joint Sparse Graph
First, we explain the iterative structure of the joint sparse graph. According to the
algorithm presented in Section 4.2, the updating of chip nodes is based on a priori in-
formation coming from connected intrinsic-interference nodes and variable nodes. The
same rule applies to the updating of variable nodes, where a priori information coming
from chip nodes and parity-check nodes is utilized. As for intrinsic-interference nodes
and parity-check nodes, their extrinsic messages are updated only based on one side
of a priori information, i.e., connected chip nodes and variable nodes, respectively.
For convenience, the sets of intrinsic-interference nodes, chip nodes, variable nodes and
parity-check nodes are hereinafter referred to as intrinsic-interference nodes decoder
(IND), chip nodes detector-decoder (CNDD), variable node detector-decoder (VND-
D) and parity-check node decoder (PND), respectively. Fig. 4.3 shows the iterative
structure of the joint sparse graph, where the extrinsic LLR is considered as a priori
information by the other detector or decoder. The edge interleavers connect different
types of nodes, each of which represents a sparse signature or matrix. Note that such
an iterative structure is more complicated than any previous single sparse graph which
only has the middle part labeled as LDS [64] or the right part labeled as LDPC [126]
in this figure. Moreover, the iterative part of LDWM has never been proposed before.
Chip Node
Detector & Decoder 1
ld s
ld s
Parity Check Node
Decoder 1
ld p c
ld p c
To receiver
Variable Node
Detector & Decoder
From IOTA demodulator
-
-
-
-
LDS LDPC
Intrinsic Interference Node
Decoder 1
ldw m
ldw m
-
-
LDWM
Fig. 4.3: Iterative structure of the joint sparse graph
84 Chapter 4. Joint Sparse Graph for FBMC-IOTA (JSG-IOTA) System
To explain the JSG-IOTA receiver more clearly, we present a tree structure of the joint
sparse graph in Fig. 4.4. As the channel and IOTA demodulated information is fed
into chip node directly, the tree structure is rooted at chip node cn where we name it
level 0 of the tree. According to Fig. 4.2 and Fig. 4.3, each chip node is linked to some
intrinsic-interference nodes and variable nodes via low density edges, we respectively
use long dash lines and bold lines to distinguish these two connections in Fig. 4.4.
Hence, at level 1 of the tree, there are two types of nodes, i.e., intrinsic-interference
nodes (pentagons) and variable nodes (circles), which are connected to cn at level 0 by
different lines. For the intrinsic-interference nodes at level 1, each of them is linked to
other chip nodes (rectangles) at level 2 via long dash lines. For the variable nodes at
level 1, each of them is not only linked to some chip nodes at level 2 via bold lines, but
also connected to some parity-check nodes (triangles) at level 2 via other type of lines,
i.e., short dash lines. Similarly, the chip nodes and the parity-check nodes at level 2 can
generate more intrinsic-interference nodes and variable nodes via corresponding types
of lines, which are drawn at level 3. Note that at level 3, the intrinsic-interference nodes
have only one type of connection to chip nodes at level 2, while the variable nodes have
two types of connection to chip nodes and parity-check nodes at level 2. Obviously,
such a tree structure is novel as it has four types of nodes and three types of edges.
Note that different levels of the tree have specific types of nodes, i.e., the odd level
(level 1 or 3) consists of intrinsic-interference nodes and variable nodes, while the even
level (level 2) consists of chip nodes and parity-check nodes. Chip nodes and variable
nodes are important due to their bridge function on the tree, and messages of different
nodes can be passed to any other types of nodes via the edges.
Additionally, Fig. 4.5 shows a folded view of the joint sparse graph. Based on the joint
detection and decoding presented in Section 4.2 and the tree structure shown in Fig.
4.4, chip nodes and parity-check nodes update their messages at the same time, then
intrinsic interference-nodes and variable nodes calculate their message simultaneously.
Therefore, it is reasonable to place even level nodes, i.e., chip nodes (rectangles) and
parity-check nodes (triangles), on one side, while to draw odd level nodes, i.e., intrinsic-
interference nodes (pentagons) and variable nodes (circles), on the other side. For chip
nodes, they are connected to intrinsic-interference nodes through low density edges
4.3. EXIT Chart Analysis of JSG-IOTA 85
cn
x x
x x x x x x x xlevel 3
level 2
level 1
level 0
Fig. 4.4: Tree structure of the joint sparse graph
p1, J
p1, 1
c1
c2
cN
xv1, 1
v1, 2
xvK, 1
vK, 2
v1, M
vK, M
chip nodes
of user 1
of user K
PK, J
PK, 1
of user 1
variable nodes
of user K
v
parity check nodes
parity check nodes
variable nodes
i1,1
i1,2
iN,T
intrinsic interference nodes
Fig. 4.5: Folded view of the joint sparse graph
86 Chapter 4. Joint Sparse Graph for FBMC-IOTA (JSG-IOTA) System
(represented by long dash lines), and they are also used to be spread on for variable
nodes through sparse edges (represented by bold lines). Since variable nodes belong
to different users, they are not only linked to chip nodes through low density edges,
but also linked to corresponding groups of parity-check nodes through sparse edges
(represented by independent groups of short dash lines). IOTA modulation, NOMA,
FEC coding and the combination of these single sparse graphs, are clearly depicted in
the figure. This provides a basis for the following analysis.
4.3.2 EXIT Chart Analysis
4.3.2.1 EXIT Curve for IND&VNDD
In this thesis, IA,IND&V NDD refers to the average mutual information between the
bits on the IND&VNDD edges and the a priori LLR, IE,IND&V NDD is the average
mutual information between the bits on the IND&VNDD edges and the extrinsic LLR.
In order to compute an EXIT curve for intrinsic nodes and variable nodes, Lcn→in,u ,
Lcn→vk,m and Lpk,j→vk,m are modelled as Gaussian-like distributions. Then the mutual
information between the variable node’s extrinsic messages and actual values of symbols
on the edges is calculated. A priori LLR can be calculated by
A = µAx+ zn (4.34)
where zn is an independent Gaussian random variable with variance σ2A and mean
zero; x is the original bits on the graph edge; µA = σ2A/2. The mutual information
IA,IND&V NDD = I(X;A) can be calculated by
IA,IND&V NDD =1
2
∑x=−1,1
∫ +∞
−∞pA(β|X = x) log2
2pA(β|X = x)
pA(β|X = −1) + pA(β|X = 1)dβ
(4.35)
Since the conditional probability density function pA(β|X = x) depends on LLR of A,
we can write
IA,IND&V NDD(σA) =1−∫ +∞
−∞
exp(−((β − σ2A/2)2/2σ2A))√
2πσAlog2(1 + e−β)dβ (4.36)
4.3. EXIT Chart Analysis of JSG-IOTA 87
For abbreviation we define
B(σ) , IA,IND&V NDD(σA = σ) (4.37)
with
limσ→0
B(σ) = 0 (4.38)
limσ→∞
B(σ) = 1 (4.39)
where σ ≥ 0. Considering (4.20), (4.21) and (4.22) together with the fact that the sum
of several normally distributed random variables is also normally distributed with the
mean and variance equal to the sum of theirs, the EXIT function of IND&VNDD can
be expressed as
IE,IND&V NDD(IA,IND&V NDD, di,ldwm, dv,lds, dv,ldpc)
= B(√
(di,ldwm + dv,lds + dv,ldpc − 1)(B−1(IA,IND&V NDD))2)(4.40)
where di,ldwm is the degree of intrinsic-interference node in the sparse graph of LDWM,
dv,lds and dv,ldpc are the degrees of variable node in sparse graphs of LDS and LDPC
codes, respectively. Unlike the single sparse graph where only one type of node is
considered [64] [126], both LDWM, LDS and LDPC nodes affect the IND&VNDD
performance.
4.3.2.2 EXIT Curve for CNDD&PND
Let IA,CNDD&PND refers to the average mutual information between the bits on the
CNDD&PND edges and the a priori LLR, IE,CNDD&PND is the average mutual in-
formation between the bits on the CNDD&PND edges and the extrinsic LLR. A chip
node has incoming messages from the connected intrinsic-interference nodes and vari-
able nodes in addition to the FBMC-IOTA demodulator, whereas a parity-check node
only has messages coming from neighbored variable nodes. The output LLR of chip
nodes and parity-check nodes are calculated by (4.23), (4.28) and (4.30). We model
Lin,u→cn , Lvk,m→cn and Lvk,m→pk,j as the output of multipath fading channels, and then
calculate the mutual information of the output with regards to the actual value on the
edges. Considering the complexity of the calculation in chip nodes and parity-check
nodes, their EXIT curves are computed by simulations over multipath fading channels.
88 Chapter 4. Joint Sparse Graph for FBMC-IOTA (JSG-IOTA) System
4.3.2.3 Analysis
System parameters are listed in TABLE 4.1 for the following EXIT chart analysis. Fig.
4.6 illustrates the EXIT charts over SUI-3 channel at Eb/N0 = 12 dB, and it can be
concluded:
TABLE 4.1: System parameters
Number of users 4
Number of chip nodes 64
Number of variable nodes 128
Number of intrinsic-interference nodes 128
Number of parity-check nodes 64
FFT size 64
Multiuser SIMO 1× 4
System loading 200%
Chips linked to each variable node dv,lds = 2
Variable nodes linked to each chip dc,lds = 4
Chips linked to each intrinsic-interference node di,ldwm = 4
Intrinsic-interference nodes linked to each chip dc,ldwm = 4
Parity-check nodes linked to each variable node dv,ldpc = 3
Variable nodes linked to each parity-check node dp,ldpc = 6
Modulation OQPSK
1) The curve of VND in LDS-IOTA is higher than that of IND&VNDD in JSG-IOTA.
This is because the edge number connecting to VND in LDS-IOTA is different from
that connecting to IND&VNDD in JSG-IOTA. As a result, according to (4.40), these
two curves are different.
2) The curve of CND in LDS-IOTA is higher than that of CNDD&PND in JSG-IOTA
when IA,CNDD&PND ≤ 0.97. This follows from the fact that the CNDD could receive
information from the IOTA demodulator and neighbored variable nodes as well as
intrinsic-interference nodes, but the PND only uses the information from the connected
variable nodes (shown in Fig. 4.3). Hence, the PND pulls down the average extrinsic
information of CNDD&PND at the first few iterations.
4.3. EXIT Chart Analysis of JSG-IOTA 89
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IA,CNDD&PND
& IE,IND&VNDD
I E,C
ND
D&
PN
D &
IA
,IN
D&
VN
DD
VND in LDS−IOTACND in LDS−IOTAtrajectory in LDS−IOTAIND&VNDD in JSG−IOTACNDD&PND in JSG−IOTAtrajectory in JSG−IOTA
Fig. 4.6: EXIT chart at Eb/N0 = 12 dB
3) The intersection point of IND&VNDD and CNDD&PND in JSG-IOTA, is higher
than that of VND and CND in LDS-IOTA. This phenomenon implies that the JSG-
IOTA is more capable of eliminating the MUI than LDS-IOTA.
4) The detection and decoding trajectories in LDS-IOTA and JSG-IOTA are plotted in
Fig. 4.6, and are marked by blue solid lines and red dotted lines, respectively. For each
trajectory, we can see that the iterative process starts with IA,CNDD&PND = 0 since
no prior information is available to the CND or CNDD&PND in the beginning. In the
following steps, the output LLR is exchanged between the two corresponding curves.
The trajectories closely follow the transfer curves of the components in LDS-IOTA and
JSG-IOTA, which indicates that the EXIT charts analysis is valid for the joint sparse
graph.
5) For LDS-IOTA, the trajectory shows that at least seven iterations are necessary to
reach the intersection point. Whereas for JSG-IOTA, the trajectory indicates that only
five iterations are needed to reach the intersection point. This is attributed to the more
efficient message passing and the more reliable information utilized by chip nodes and
90 Chapter 4. Joint Sparse Graph for FBMC-IOTA (JSG-IOTA) System
variable nodes in JSG-IOTA than that in LDS-IOTA. Therefore, the JSG-IOTA can
reach its intersection point faster than the LDS-IOTA.
4.4 EXIT Chart Assisted Joint Sparse Graph Design
In this section, we discuss how to use EXIT charts to optimize the joint sparse graph.
4.4.1 Degree Distribution
let DIND(x), DCNDD(x), DV NDD(x), and DPND(x) denote the degree distribution
polynomials of intrinsic-interference nodes, chip nodes, variable nodes and parity-check
nodes, respectively. They are defined as
DIND(x) =
di,ldwm∑d=1
PIND(d)xd−1 (4.41)
DCNDD(x) =
dc,ldwm+dc,lds∑d=1
PCNDD(d)xd−1 (4.42)
DV NDD(x) =
dv,lds+dv,ldpc∑d=1
PV NDD(d)xd−1 (4.43)
DPND(x) =
dp,ldpc∑d=1
PPND(d)xd−1 (4.44)
where PIND(d), PCNDD(d), PVNDD(d) and PPND(d) are the fractions of all edges con-
nected to corresponding nodes. According to Fig. 4.2 (b), each intrinsic-interference
node is determined by four neighboring time-frequency positions around the signal of
interest, hence we fix di,ldwm = 4. However, other degree distributions can be optimized
by EXIT chart analysis.
The joint sparse graph in TABLE 4.1 is a regular graph without any optimization.
Although its intersection point is higher than that of LDS-IOTA, which is shown in
Fig. 4.6, the average level of CNDD&PND in JSG-IOTA is much lower than that of
CND in LDS-IOTA. Similar to JSG-OFMD, our approach here is based on invoking
EXIT chart analysis for optimizing the shape of the EXIT curve in order to achieve
4.4. EXIT Chart Assisted Joint Sparse Graph Design 91
the (1, 1) convergence point. According to Section 4.3, the information coming from
the IOTA demodulator serves as input source of the joint sparse graph, and it is fed
into chip nodes in each iteration of the joint detection and decoding as well as intrinsic
interference utilization. As mentioned previously, the mutual information determines
the dependence between detected symbols and their exact values, thus the EXIT curve
level has to be considered. If the density of the edges in PND is reduced, the EXIT curve
of CNDD&PND should be lifted up and a higher intersection point can be achieved.
Therefore, several schemes of degree distribution are developed in TABLE 4.2, where
Dega is the case of a regular joint sparse graph presented in TABLE 4.1, other schemes
are developed for irregular joint sparse graphs. We can see that compared to Dega,
Degc and Degd slightly decrease the degree of PND, then the polynomials of VNDD
and CNDD are altered accordingly. On the contrary, Degb increases the density of
edges of PND.
TABLE 4.2: Degree distribution
Scheme DIND(x) DCNDD(x) DVNDD(x) DPND(x)
Dega x3 x7 x4 x5
Degb x3 0.1x6 + 0.8x7 + 0.1x8 0.575x4 + 0.425x5 0.15x5 + 0.85x6
Degc x3 0.075x6 + 0.85x7 + 0.075x8 0.315x3 + 0.685x4 0.63x4 + 0.37x5
Degd x3 0.045x6 + 0.91x7 + 0.045x8 0.375x3 + 0.625x4 0.75x4 + 0.25x5
Fig. 4.7 shows EXIT charts for different degree distribution over SUI-3 channel at
Eb/N0 = 12 dB. In TABLE 4.2, Degc and Degd slightly decrease the weight of the
PND, consequently in Fig. 4.7, the average mutual information between the bits and
the extrinsic values of CNDD&PND are both increased. More importantly, their in-
tersection points become higher, which means better performance can be achieved. By
contrast, Degb increases the density of edges of PND, but its average mutual informa-
tion and intersection point are both dropped. Therefore, a lower density of PND in
general results in a scheme that is more robust to interference on the joint sparse graph.
It is noteworthy that although the PND weight of Degd is less than that of Degc, but
the intersection point of Degd is lower than that of Degc, indicating Degc outperforms
92 Chapter 4. Joint Sparse Graph for FBMC-IOTA (JSG-IOTA) System
Degd. This is related to the function of PND, i.e., performing parity-check for error cor-
rection. As explained above, it is necessary to design the degree distributions carefully
and strike a balance between the optimization processing of the joint sparse graph and
the original function of the parity-check node. This ensures efficient operations of belief
propagation executed at the receiver of JSG-IOTA. For the joint sparse graph, Degc is
a preferable choice since it has the highest intersection point among these schemes.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IA,CNDD&PND
& IE,IND&VNDD
I E,C
ND
D&
PN
D &
IA
,IN
D&
VN
DD
IND&VNDD of Dega
CNDD&PND of Dega
IND&VNDD of Degb
CNDD&PND of Degb
IND&VNDD of Degc
CNDD&PND of Degc
IND&VNDD of Degd
CNDD&PND of Degd
Fig. 4.7: EXIT chart for different degree distributions
4.4.2 Short Cycle
Fig. 4.8 shows EXIT charts for different girths of the joint sparse graph over SUI-3
channel at Eb/N0 = 12 dB. We choose the optimal Degc to be the degree distribution.
When the girth equals to 8, some degrees need to be modified marginally (labelled by
Deg′c): DIND(x) is x3, DCNDD(x) is 0.07x
6 + 0.86x7 + 0.07x8, DV NDD(x) is 0.31x3 +
0.69x4 and DPND(x) is 0.62x4 +0.38x5. It can be seen from this figure that similar to
JSG-OFDM, the curves of matrices with girth of 4 and 6 are very close to each other,
whereas matrix with girth of 8 outperforms others. Therefore, degree distribution and
short cycle both affect the system performance.
4.4. EXIT Chart Assisted Joint Sparse Graph Design 93
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
IA,CNDD&PND
& IE,IND&VNDD
I E,C
ND
D&
PN
D &
IA
,IN
D&
VN
DD
IND&VNDDCNDD&PND, girth = 4, Deg
c
CNDD&PND, girth = 6, Degc
CNDD&PND, girth = 8, Deg’c
Fig. 4.8: EXIT chart for different schemes
4.4.3 Maximum Achievable Throughput
TABLE 4.3: JSG-IOTA scenarios
Scenario Degree distribution Girth
scenario− 1 Dega 6
scenario− 2 Degb 4
scenario− 3 Deg′c 8
To reveal the theoretical threshold of the joint sparse graph, we analyse its maximum
achievable throughput by EXIT charts. Similar to JSG-OFDM, assuming that the area
under the EXIT curve of the CNDD&PND is represented by A, then the maximum
achievable throughput at a particular Eb/N0 value is given by A(Eb/N0). For com-
parisons, three scenarios of the JSG-IOTA that have different degree distributions and
girths are given in TABLE 4.3, and their maximum throughputs of the joint sparse
graph are quantified in Fig. 4.9, where the SNR represent Es/N0. It can be seen
that, as expected, JSG-IOTA scenario− 3 owns the highest capacity in the scenarios.
94 Chapter 4. Joint Sparse Graph for FBMC-IOTA (JSG-IOTA) System
Compared to the maximum achievable throughput of scenario−3, there are respective
degradations for that of scenario− 1 and scenario− 2. Thus scenario− 3 is the best
option for JSG-IOTA, which will be further verified by performance simulations.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 170
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
SNR (dB)
Cap
acity
(bi
ts/s
ymbo
l)
JSG−IOTAscenario−1JSG−IOTAscenario−2JSG−IOTAscenario−3
Fig. 4.9: Maximum effective throughput of the joint sparse graph
4.5 Performance Evaluation
In this section, JSG-IOTA is evaluated and compared with systems such as OFDM,
IOTA, LDS-OFDM, LDS-IOTA and turbo structured LDS-IOTA, where OFDM and
IOTA represent conventional systems without LDS structure. The performance is eval-
uated using Monte Carlo simulations over SUI-3 channel, and the simulation parameters
are listed in TABLE 4.1. For fair comparisons, a half rate quasi-cyclic LDPC code is
adopted by all the systems [125]. For OFDM-based systems (including OFDM and
LDS-OFDM), we use QPSK modulation. For IOTA-based systems (including IOTA,
LDS-IOTA, turbo structured LDS-IOTA and JSG-IOTA), we use offset quadrature
phase-shift keying (OQPSK) modulation, and the number of filter taps for each sub-
carrier is 5. For LDS-based systems (including LDS-OFDM and LDS-IOTA), the LDS
4.5. Performance Evaluation 95
is optimized by EXIT charts, and iterative detectors are used [68]. For turbo struc-
tured LDS-IOTA, there are eight outer-inner turbo iteration between the detector and
the decoder, and the turbo structure is optimized by EXIT charts [17]. The maximum
iterations are limited to eight for LDS-OFDM, LDS-IOTA and JSG-IOTA.
4.5.1 BER Comparison
Fig. 4.10 shows BER results for systems with a load of 200%. As we can see that, LDS-
OFDM and LDS-IOTA significantly outperform OFDM and IOTA, respectively. This
is due to the LDS structure which can effectively eliminate the MUI and exploit the
frequency diversity under overloaded conditions. Meanwhile, IOTA and LDS-IOTA
respectively yield slight gain over OFDM and LDS-OFDM. Hence, compared with
OFDM systems, IOTA-based systems achieve improved power efficiency (better BER
performance) and spectral efficiency due to the elimination of CP. Among all the IOTA-
based systems, the performance of LDS-IOTA is inferior to that of turbo structured
LDS-IOTA and JSG-IOTA. Moreover, each of the three JSG-IOTA scenarios achieves
superior performance over other systems. With the degree distribution (Deg′c) and the
cycle structure (girth of 8), JSG-IOTA scenario− 3 achieves the best performance. Its
performance improvements at BER of 10−6 are: 0.3 dB over JSG-IOTA scenario− 1,
0.7 dB over JSG-IOTA scenario− 2, 1.3 dB over turbo structured LDS-IOTA, 1.5 dB
over LDS-IOTA, and 1.9 dB over LDS-OFDM, respectively. Furthermore, to investigate
how the IOTA intrinsic interference utilization affects the system performance, we also
simulate the JSG-IOTA scenario − 3 excluding the low density weight matrix of the
intrinsic interference, which is referred to as JSG-IOTA scenario−3 without LDWM in
the figure. It can be seen that, compared with JSG-IOTA scenario− 3, there is a loss
of about 0.5 dB when JSG-IOTA scenario− 3 without LDWM is adopted. Therefore,
the intrinsic interference utilization of the joint sparse graph can profit extra gain to
the system.
In addition, the performance of 300% loaded systems is simulated and shown in Fig.
4.11. Similarly to the 200% loading case, JSG-IOTA scenario−3 still achieves the best
performance. Compared with LDS-OFDM, LDS-IOTA and turbo structured LDS-
96 Chapter 4. Joint Sparse Graph for FBMC-IOTA (JSG-IOTA) System
6 8 10 12 14 16 18 20
10−6
10−5
10−4
Eb/N
0(dB)
BE
R
OFDMIOTALDS−OFDMLDS−IOTATurbo structured LDS−IOTAJSG−IOTAscenario−1JSG−IOTAscenario−2JSG−IOTAscenario−3JSG−IOTAscenario−3 without LDWM
Fig. 4.10: Performance of different systems with 200% loading
6 8 10 12 14 16 18 2010
−6
10−5
10−4
Eb/N
0(dB)
BE
R
OFDMIOTALDS−OFDMLDS−IOTATurbo structured LDS−IOTAJSG−IOTAscenario−1JSG−IOTAscenario−2JSG−IOTAscenario−3JSG−IOTAscenario−3 without LDWM
Fig. 4.11: Performance of different systems with 300% loading
4.5. Performance Evaluation 97
IOTA, JSG-IOTA scenario− 3 attains approximately 1.3 - 1.9 dB gain in the medium
to high SNR region.
4.5.2 Convergence Behavior
The convergence behavior of 200% loaded LDS-IOTA and JSG-IOTA scenario− 1 at
Eb/N0 = 12 dB is shown in Fig. 4.12, indicating the performance of the two systems
at different iterations. At the first iteration, the performance of LDS-IOTA is almost
identical to that of JSG-IOTA. However, as the iterative process proceeds, BER of JSG-
IOTA drops much faster than that of LDS-IOTA, indicating a better convergence rate
can be achieved by employing the joint sparse graph. After 5 iterations, the performance
of JSG-IOTA becomes saturated, whereas LDS-IOTA needs 7 iterations to reach its
lowest BER. Such results concur with the trajectory prediction of the EXIT chart
plotted in Fig. 4.6. Therefore, the EXIT chart analysis and the convergence behavior
are verified by the BER simulation results.
1 2 3 4 5 6 7 810
−6
10−5
10−4
10−3
10−2
10−1
100
Number of iterations
BE
R
LDS−IOTAJSG−IOTA
Fig. 4.12: Performance on different iterations at Eb/N0 = 12 dB
98 Chapter 4. Joint Sparse Graph for FBMC-IOTA (JSG-IOTA) System
4.5.3 Performance of Different Users
In Fig. 4.13, we show the performance of the best user and the worst user in LDS-IOTA
and JSG-IOTA scenario− 3 for the system loading of 200%. It indicates that for both
two schemes, some users have poorer performance than the others, and in the low SNR
region the performance degradation is not as obvious as in the high SNR region. This
phenomenon can be explained by observing the signal constellation at each chip and
the dominating effect of noise at SNRs. Therefore, it can be seen that the joint sparse
graph does not give equal multiuser efficiency or in other words it does not result in the
same performance for all the users. But the performance gap between the best user and
the worst user in JSG-IOTA is slightly smaller than that of LDS-IOTA, which means a
fairer and a more uniform user experience can be achieved by utilizing the joint sparse
graph.
6 8 10 12 14 16 18 20
10−6
10−5
10−4
Eb/N
0(dB)
BE
R
Worst user in LDS−IOTABest user in LDS−IOTAOverall in LDS−IOTAWorst user in JSG−IOTABest user in JSG−IOTAOverall in JSG−IOTA
Fig. 4.13: Performance of different users
4.5. Performance Evaluation 99
4.5.4 Dynamic Subcarrier Allocation
In JSG-IOTA system, the generated chips at the output of the LDS spreader are
mapped to the subcarriers of the employed IOTA signal. In the simulations we have
carried out so far, we assume that a static subcarrier allocation is employed, i.e., at
transmitters, individual subcarriers are allocated to users for transmission without the
knowledge of CSI. However, to maximize the sum-rate with fairness consideration for
JSG-IOTA, it is necessary to investigate dynamic subcarrier allocations. In other word-
s, transmitters periodically estimate the uplink channel states of all subcarriers for each
user. Based on the obtained CSI, transmitters assign the subcarriers and power to each
user through a reliable signaling channel. Fig. 4.14 shows the effect of the subcarrier
allocation in 200% loaded JSG-IOTA scenario−3, where the dynamic scheme allocates
subcarriers to the user who has the largest channel gain [69]. As revealed by the figure,
the dynamic subcarrier allocation improves the performance significantly. Compared
to the static subcarrier allocation, there is about 2 - 3 dB gain. It should be noted
that for the dynamic subcarrier allocation, CSI is required at both transmitters and
receivers, while for the static subcarrier allocation, CSI is only required at receivers.
6 8 10 12 14 16 18 2010
−7
10−6
10−5
10−4
Eb/N
0(dB)
BE
R
Static subcarrier allocationDynamic subcarrier allocation
Fig. 4.14: Performance of different subcarrier allocation schemes
100 Chapter 4. Joint Sparse Graph for FBMC-IOTA (JSG-IOTA) System
4.6 Summary
In this chapter, a joint sparse graph combing pulse shaping property (LDWM of FBMC-
IOTA), NOMA (LDS) and FEC (LDPC codes) was proposed. Differing from any single
graph, in the JSG-IOTA receiver, multiuser detection, intrinsic interference utilization
and channel decoding are jointly performed on one entire sparse graph. Hence it is a
Type-A receiver shown in Fig. 2.7. EXIT charts are utilized to analyse the JSG-IOTA
receiver in details, consequently the design guidelines and the threshold of the joint
sparse graph were obtained. Simulations show that the JSG-IOTA scenario−3 (degree
distribution of Deg′c and girth of 8) achieves the best performance. Its performance
improvement is mainly due to the intrinsic interference utilization and the capability
of the joint sparse graph.
Chapter 5
Sparse Code Multiple Access
(SCMA)
Although LDS performs well under overloaded conditions, its performance is not ideal
with high-order constellations. Based on the LDS technique, SCMA is proposed by
introducing constellation shaping gain [71]. It is a codebook-based NOMA technique
via code domain multiplexing. In this chapter, present novel codebook design for SCMA
systems. The design criteria of SCMA codebooks are discussed, and its performance
comparison with LDS is evaluated. Considering long codewords’ construction and high-
degree distribution, two approaches are proposed to optimize the SCMA performance,
i.e., copy-and-permute operation on protographs and 3D codebook design.
5.1 Criteria of SCMA Codebook Design
The SCMA system model and the codebook design criteria are presented in this section.
5.1.1 SCMA System Model
In SCMA, according to predefined codebook sets, bit streams are directly mapped to
sparse codewords, and the whole process can be interpreted as a coding procedure from
the binary domain to a multidimensional complex domain as shown in Fig. 5.1.
101
102 Chapter 5. Sparse Code Multiple Access (SCMA)
LDS
Spreader
FEC
Encoder QAM
Mapper
SCMA Encoder
Fig. 5.1: Merging of QAM modulator and LDS spreading in a SCMA encoder
Multidimensional lattice constellation design is a challenging problem which has been
studied in different aspects of communications. The SCMA codebook design is even
more complicated as multiple layers are multiplexed with different codebooks. A SCMA
encoder is defined as v = f(b), where b and v respectively represent the bit sequence
and the SCMA codeword vector. The N dimensional complex codeword v is a sparse
vector with dv,lds ≪ N non-zero entries. Let c denote an dv,lds dimensional complex
constellation point such that: c = g(b). A SCMA encoder can be redefined as f , sg
where s simply maps the dv,lds dimensions of a constellation point to an N dimensional
SCMA codeword. In fact, s is the low density spreading in LDS. The resulting code-
book contains of M codewords each consisting of N complex values from which only
maximum dv,lds are non-zero specified by s. The difference between LDS and SCMA
is that LDS only consider spreading, i.e., f , s, while SCMA considers both spreading
and constellation such that f , sg.
A SCMA encoder contains K separate layers or users each defined by
Γk = fk(Sk, gk(b);M,dv,lds, N), k ∈ [1,K] (5.1)
where S = {S1, ...,SK} and Γ = {Γ1, ...,ΓK} represent the low density signature
and the SCMA codebook set. The constellation function gk generates the constel-
lation set. The matrix Sk maps the constellation point to SCMA codeword to for-
m the codeword set vk. In summary, a SCMA code can be represented by Γ =
f([Sk]Kk=1, [gk]
Kk=1;K,M, dv,lds, N). SCMA codewords are multiplexed overN shared or-
thogonal resources, e.g. OFDM subcarriers. The received signal after the synchronous
layer multiplexing can be expressed as
y =
K∑k=1
EkSkgk(bk) + z (5.2)
5.1. Criteria of SCMA Codebook Design 103
where Ek and z are the corresponding channel gain for the kth user and the AWGN,
respectively.
The SCMA detection is the same to that of the LDS detection [68]. The algorithm
iteratively updates the belief associated to the edges in the factor graph by passing the
extrinsic information of constellation points between resource nodes and layer nodes.
Note that resource nodes and layer nodes in SCMA respectively correspond to chip
nodes and variable nodes in LDS. At each resource node of SCMA, the received signal
is combined with the extrinsic information passed by the resource nodes through the
rest of the edges connected to that resource node. At each layer node, the a priori
information of the transmitted layers (usually assumed to be uniformly distributed)
is combined with the extrinsic information passed by the resource nodes through the
rest of edges connected to that layer node. Finally, after a few iterations, the detector
converges to reliable soft information of the transmitted layers.
5.1.2 Multi-stage Optimization Approach of SCMA Codebooks
The codebook design problem of SCMA can be defined as
Γ = argmaxS,g
f([Sk]Kk=1, [gk]
Kk=1;K,M, dv,lds, N) (5.3)
Reference [74] points out that the solution of this multi-dimensional problem is still
unknown, a multi-stage optimization approach is the suboptimal solution for the issue.
We summarize the multi-stage optimization procedures according to [74] as follows.
5.1.2.1 Low Density Signature
As described before, the mapping matrix is actually the low density signature of the
LDS scheme, and it is used to determine the number of layers interfering at each
resource node which in turn defines the complexity of the MPA detection. The sparser
the codewords structure, the lower complexity is the MPA detection. Optimization of
the mapping matrix is the same as the LDS scheme, which has been discussed and
analyzed in Chapter 3.
104 Chapter 5. Sparse Code Multiple Access (SCMA)
5.1.2.2 Constellation Points
Having the mapping matrix S+, the optimization problem of an SCMA is reduced to
g+ = argmaxgf([S+, g;K,M, dv,lds, N) (5.4)
The problem is to define K different dv,lds dimensional constellations each containing
M points. To simplify the optimization problem, the constellation points of the layers
are modeled based on a mother constellation and the layer-specific operators, i.e. gk ,
(∆k)g, k ∈ [1,K], where ∆k denotes a constellation operator. According to the model,
the SCMA code optimization problem turns into
g+, [∆+k ]Kk=1 = arg max
g,[∆k]Kk=1
f([S+, g = [(∆k)g]Kk=1;K,M, dv,lds, N) (5.5)
As a suboptimal approach to the above problem, the mother constellation and the
operators are determined separately.
5.1.2.3 Mother Multi-dimensional Constellation
The target is to design a multi-dimensional lattice constellation. A compact multi-
dimensional constellation can be designed by minimizing the average alphabet energy
for a given minimum Euclidian distance between the constellation points. A unitary
operation (lattice rotation) can be applied directly on the lattice constellation without
sacrificing the Euclidian distance of the constellation. The lattice rotation can be op-
timized to improve the product distance of the constellation and induce dependency
among the lattice dimensions, and reduce the number of projections per lattice di-
mensions to reduce the detection complexity. Also, the dimensional power properties
of the constellation can be changed by lattice rotation, for better convergence of the
MPA receiver by taking advantage of near-far effect of the overlaid codeword. Shaping
gain of multi-dimensional constellations is the major source of the performance gain of
SCMA over LDS. After optimizing the constellation set, the corresponding constella-
tion function is defined to setup the mapping rule between the binary words and the
5.1. Criteria of SCMA Codebook Design 105
constellation alphabet points. Following the Gray mapping rule, the binary words of
any two closet constellation points can have a Hamming distance of 1.
1) Design metrics and rotated constellations
Large minimum Euclidean distance of a multi-dimensional constellation ensures a good
performance of the SCMA system with a small number of layers where there are no
collisions between the layers over a resource node. Once the number of layers grows, two
or more layers may collide over a resource node. Under this condition, it is important
to induce dependency among the nonzero elements of codewords to be able to recover
colliding codewords from the other resource nodes. In addition, power imbalance across
the dimensions of codewords introduces near-far effect among colliding layers. It helps
MPA detector to operate more effectively to remove interferences among layers.
Having a constellation with a desirable Euclidian distance profile, a unitary rotation
can be applied to the base constellation in order to control dimensional dependency and
power variation of the constellation while maintaining the Euclidian distance profile.
Similar to the code design for communications over fading channels, a unitary rotation
can be applied to maximize the minimum product distance of the constellation. There-
fore, the design objective encapsulates both the sum distance and the product distance
between the points in the mother constellation.
2) Rotated lattice constellations
In general, the base constellation can be any multi-dimensional constellation with a
maximized minimum Euclidean distance. At low rates, constellation can be designed
by heuristic optimization, but at higher rates a structured construction way is required.
Lattice constellation is a structural approach of the base constellation design. As a
special case of lattice constellations, we can consider the base constellation to be formed
by orthogonal QAMs on different complex planes. It is equivalent to a constellation
from the lattice. Gray labeling is an advantage of this type of lattice constellations.
3) Shuffling multi-dimensional constellations in real and imaginary axes
If a complex constellation is built such that its real part is independent of its imaginary
part, it can help reduce the decoding complexity while maintaining dependency among
106 Chapter 5. Sparse Code Multiple Access (SCMA)
the complex dimensions of the resulted multi-dimensional constellation. Using this
technique, the complexity order of MPA reduces from Mdc,lds to Mdc,lds/2 which results
in significant complexity saving especially for large constellation sizes.
A shuffling is to construct a dv,lds dimensional complex mother constellation from Carte-
sian product of two dv,lds dimensional real constellations, where each of them is con-
structed by the same method described in the previous section. One of these two dv,lds
dimensional real constellations corresponds to the real part of the points of the complex
mother constellation and the other one corresponds to the imaginary part.
Fig. 5.2 shows an example of the shuffling to construct a 16-point SCMA mother
constellation with two nonzero positions (dv,lds = 2). The optimum rotation angle
that maximizes the minimum product distance is tan−1((1 +√5)/2) [74]. First, two
QPSK constellations are rotated using the optimum angle. Each point of the rotated
constellations can be queued according to the axis value: X1(11 01 10 00), X2(10 11
00 01), Y1(11 01 10 00), Y2(10 11 00 01). Subsequently, a shuffling is performed to
separate real and imaginary parts. One of these two real constellations corresponds
to the real part of the points of the complex mother constellation and the other one
corresponds to the imaginary part.
To explain the SCMA principle more clearly, we show a 150% loaded SCMA as an
example. There are 6 users and 4 subcarriers, each user is spread on 2 subcarriers,
thus N = 4, K = 6 and dv,lds = 2. We use the 16-point constellation presented in
Fig. 5.2 as the constellation scheme. The LDS matrix is shown in (5.6), where a0 = 1,
a1 = exp(j2π27 ), a2 = exp(j4π27 ) [63]. Note that each row and each column of the LDS
matrix in (5.6) represent a subcarrier and a user, respectively. TABLE 5.1 illustrates
the generated SCMA codewords which is a combination of the LDS matrix in (5.6) and
the constellation in Fig. 5.2.
a0 a1 a2 0 0 0
a1 0 0 a0 a2 0
0 a2 0 a1 0 a0
0 0 a0 0 a1 a2
(5.6)
5.1. Criteria of SCMA Codebook Design 107
16 Points Constellation for dv,lds = 2
a
b
Fig. 5.2: 16-point SCMA constellation for dv,lds = 2
TABLE 5.1: SCMA codewords for 16-point
Information bits SCMA codeword
User 1 0000 [(a+ ja)a0, (b+ jb)a1, 0, 0]
User 2 0101 [(−b− jb)a1, 0, (a+ ja)a2, 0]
User 3 0011 [(a− ja)a2, 0, 0, (b− jb)a0]
User 4 0101 [0, (−b− jb)a0, (a+ ja)a1, 0]
User 5 1100 [0, (−a+ ja)a2, 0, (−b+ jb)a1]
User 6 0111 [0, 0, (−b− ja)a0, (a− jb)a2]
108 Chapter 5. Sparse Code Multiple Access (SCMA)
4) Rotation to Minimize Number of Projection Points
For the sake of simplicity of the MPA detection, it is more desirable to use mother
constellations that have a smaller number of projections per resource node (or complex
dimension). Let m denote the number of projections per complex dimensions of an
M point constellation. It is obvious that m < M . As m decreases, the complexity
of the corresponding MPA detector is also reduced by mdc,lds . During the process of
mother constellation design, the rotation matrix can be set in a way so as to lead to
the lower number of projected points. It makes the minimum product distance equal
to zero and degrades the performance of the SCMA system. Consequently, there is
a trade-off between the performance requirement and the complexity in this case. As
an example, Fig. 5.3 shows a solution with 9-projections per complex dimension of a
16-point constellation [74].
16 Points Constellation with 9 projection points for dv,lds = 2
a
a
Fig. 5.3: 16-point SCMA constellation with 9-projection-point for dv,lds = 2
Similarly, we show the generated SCMA codewords for 16-points with 9-projection-
point in TABLE 5.2. Such SCMA codebook is a combination of the LDS matrix in
(5.6) and the constellation in Fig. 5.3.
5.1. Criteria of SCMA Codebook Design 109
TABLE 5.2: SCMA codewords for 16-point with 9-projection-point
Information bits SCMA codeword
User 1 0000 [(a+ ja)a0, (0 + j0)a1, 0, 0]
User 2 0101 [(0 + j0)a1, 0, (a+ ja)a2, 0]
User 3 0011 [(a− ja)a2, 0, 0, (0 + j0)a0]
User 4 0101 [0, (0 + j0)a0, (a+ ja)a1, 0]
User 5 1100 [0, (−a+ ja)a2, 0, (0 + j0)a1]
User 6 0111 [0, 0, (0− ja)a0, (a+ j0)a2]
5.1.2.4 Constellation Function Operators
By having a solution for the mother constellation, the original SCMA optimization
problem is further reduced to
[∆+k ]Kk=1 = arg max
[∆k]Kk=1
f([S+, g = [(∆k)g+]Kk=1;K,M, dv,lds, N) (5.7)
Here, we limit the operators to those with unitary representation over real domain
which guarantees that the Euclidian distances between different codewords are not al-
tered. Three typical operators are complex conjugate, phase rotation, and dimensional
permutation of the lattice constellation. The codebooks of different SCMA layers are
constructed based on the mother constellation g and a layer-specific operator ∆k for
layer k. The task of the MPA detector is to separate the interfering symbols in an
iterative fashion. As a basic rule, interfering symbols at a resource node are more
easily separated if their power level is more diverse. Intuitively, the strongest sym-
bol is first detected and then it helps the rest to be detected by removing the next
strongest symbols, consecutively. Based on this reasoning, the mother constellation
must have a diverse average power level over the constellation dimensions. This target
can be achieved by appropriate rotation of the lattice constellation as discussed in the
previous section. Assuming the dimensional power level of the mother constellation is
diverse enough, the permutation operators of the SCMA codebooks must be selected
in a way to capture as much power diversity as possible over the interfering layers. The
110 Chapter 5. Sparse Code Multiple Access (SCMA)
power variation over the layers can be optimized following the approach described in
the following: the permutation of each codebook set is designed to avoid interfering the
same dimensions of a mother constellation over a resource node.
5.1.3 Performance Comparison with LDS
In the following, the performance of SCMA is evaluated and the performance gain of
SCMA over LDS is quantified by a simple exemplary case. SCMA codewords or LDS
modulated signatures are carried over OFDM subcarriers in SUI-3 channel [144]. The
SCMA and LDS follow the sparse graph with basic parameters: N = 4, K = 6 and
dv,lds = 2. The users number is 6 and the subcarrier number is 4, hence the system
loading is 150%. Four and sixteen point constellations are considered in the simulations.
According to the results shown in Fig. 5.4, for both constellations, SCMA outperforms
LDS in terms of BER, confirming that SCMA provides shaping gains over LDS due to
the multi-dimensional codebooks. In addition, the 16-point SCMA with 9-projection-
point, which is shown in Fig. 5.3, is also evaluated. It can be seen that, although the
9-projection-point scheme reduces the detection complexity, its BER is much higher
than the 16-point SCMA scheme.
6 8 10 12 14 16 18 20 22 24 26 28 3010
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Eb/N
0 (dB)
BE
R
LDS 4−pointSCMA 4−pointLDS 16−pointSCMA 16−pointSCMA 16−point with 9−projection−point
Fig. 5.4: Performance of 150% loaded LDS and SCMA
5.2. Design of SCMA Codebooks Based on Protographs 111
5.2 Design of SCMA Codebooks Based on Protographs
Existing SCMA codebooks, e.g., those presented in [71–75], are strictly designed for
very short codewords, i.e., N = 4 and K = 6, which means a 4-point FFT is adopted
in the OFDM modulation. In practical multi-carrier systems, the subcarrier number is
definitely larger than 4. For example, 128, 256, 512 or even larger FFT is used in prac-
tical OFDM systems. To increase the subcarrier (resource node) number, reference [74]
points out that several short SCMA codewords can be linked together to form a long
codeword. However, simple combination of short SCMA codewords ignores some im-
portant characters of sparse graph and cannot achieve ideal transmission performance.
In this section, we design the SCMA codebooks using protographs.
5.2.1 Extension of SCMA Codewords by Copy Operation
For convenience, we define the existing short SCMA codewords as a template called a
protograph. Note that the protograph was proposed for LDPC codes’ design [145] [146],
but it has never been applied to any NOMA technique. A protograph can be any Tanner
graph, typically one with a relatively small number of nodes. As a simple example, we
consider a protograph shown in Fig. 5.5 (a). This small Tanner graph consists of 2
resource nodes and 3 layer nodes, connected by 5 edges, and can be recognized as a
protograph of a (N = 2, K = 3) SCMA codeword. According to [74], to increase
the subcarrier number, a larger graph can be obtained by a copy operation, which is
illustrated in Fig. 5.5 (b). In this figure, the protograph has been copied P times,
where the P is determined by the FFT size requirement. Here the same-type vertices
of the P copies are in close proximity, but the overall graph consists of P disconnected
subgraphs.
The advantage of the extension method shown in Fig. 5.5 is the simplicity. On the
transmitter side, protographs only need to be copied to form a larger Tanner graph,
without any other operation. On the receiver side, as each copy of the protograph is
exactly the same to each other, a low complexity protograph decoder can be utilized
for all protographs. Nevertheless, there are some drawbacks of this method:
112 Chapter 5. Sparse Code Multiple Access (SCMA)
resource nodes
layer nodes
edges
(a) Protograph (b) Copy p times
Fig. 5.5: Extension of SCMA codewords by copy operation
1) Each protograph in Fig. 5.5 (b) is independent subgraph, consequently when iterative
detection is performed, each node cannot receive soft message from other protographs
or far-distant nodes. In graph theory, one of the main advantages of the sparse graph
is that far-distant nodes’ message can be propagated to correct errors.
2) Separate protographs cannot obtain interleaving gain. According to the LDPC
theory, intrinsic interleaver of the sparse graph is an important factor to achieve a
satisfactory performance.
3) The large graph generated by protographs is not optimized. For example, there is a
length-4 cycle in the protograph in Fig. 5.5 (a). These short cycles still exist in Fig.
5.5 (b) as no extra processing is performed. It has been proved in Chapter 3 and 4 that
short cycle will degrade the performance of an iterative detector.
Therefore, although the extension is straightforward, the generated SCMA codeword
may not be optimal.
5.2.2 Construction of SCMA Codebooks by Copy-and-permute Op-
eration
To solve above problems, we propose an improved approach to construct SCMA code-
books based on protographs. In our proposal, the protograph still serves as a blueprint
for constructing large SCMA codewords of arbitrary size, but the large Tanner graph
is optimized. This operation consists of first making several copies of the protograph,
and then permuting the edges among the disconnected protographs. The main steps
are illustrated in Fig. 5.6 and introduced as follows.
5.2. Design of SCMA Codebooks Based on Protographs 113
(b) Copy p times (c) Permute the edges (a) Protograph
resource nodes
layer nodes
edges
Fig. 5.6: Construction of SCMA codebooks by copy-and-permute operation
1) Choose a protograph with a relatively low number of nodes. In Fig. 5.6 (a), we take
the same protograph shown in Fig. 5.5 (a) as an example.
2) Copy the protograph P times. In Fig. 5.6 (b), the protograph has been copied
P times, which is similar to that of Fig. 5.5 (b). After the replication, we obtain a
temporary graph, where each set has a size P times larger than the corresponding sets
in the protograph.
3) Permute the edges of the nodes in the P replicas of the protograph to obtain the
final graph. The permutations are performed in a way to maximize the girth of the
large graph. In Fig. 5.6 (c), the edges of the P copies of the protograph have been per-
muted among the P copies of the corresponding layer nodes and resource nodes. More
explicitly, we set a target girth in advance, and calculate the girth of the joint graph
in a typical run. If the girth is less than the target, we permute the edge connections
of two protographs which have at least P/2 protographs’s distance. The requirement
for the minimum distance is to make sure that the message in the iterative detection
comes from the node as far as possible, which will lead to a better convergence. The
permutation operation continues repeatedly until the target girth is achieved. After
the permutation, the P subgraphs are interconnected and shown in Fig. 5.6 (c), con-
sequently a new (N = 2P , K = 3P ) SCMA codeword is derived.
In general, we can apply the copy-and-permute operation to any protograph to derive
larger graphs of various sizes. When the protograph is replicated P times, each proto-
graph edge becomes a bundle of P edges, connecting P layer nodes to P resource nodes.
The copies of the protograph are interconnected by permuting these layer-to-resource
pairings within each bundle. The derived graph is the graph of a SCMA codeword P
times as large as the SCMA codeword corresponding to the protograph, with the same
114 Chapter 5. Sparse Code Multiple Access (SCMA)
overloadings and degree distributions.
As can be seen from Fig. 5.6, the key of our proposal is the permutation operation.
Similar to the optimization of the joint sparse graph, the design criteria of the permu-
taion operation is to increase the girth of the graph. By doing so, separate protographs
become correlated, meaning that the message from far-distant nodes can be exchanged.
In addition, the large sized interleaving is exploited in the graph structure. More im-
portantly, short cycles can be eliminated during the optimization processing.
5.2.3 Performance Evaluation
6 8 10 12 14 16 18 20 22 24 26 28 3010
−6
10−5
10−4
10−3
10−2
10−1
100
Eb/N
0 (dB)
BE
R
SCMA 4−point by copy operationSCMA 4−point by copy−and−permute operationSCMA 8−point by copy operationSCMA 8−point by copy−and−permute operation
Fig. 5.7: Performance of 200% loaded SCMA by different construction methods
To compare the above two construction approaches, Fig. 5.7 shows BER results for
SCMA over SUI-3 channel. A (N = 8, K = 16) protograph with 8 resource nodes and
16 layer nodes is employed. We assume that the FFT size is 128, thus 16 protographs
(P = 16) are copied or copied-and-permuted to generate the SCMA codebooks. The
system overloading is 200%. Four and eight point constellations are considered in the
simulation. It can be seen that the copy-and-permute operation outperforms the copy
5.3. Design of 3D SCMA Codebooks 115
operation. For both constellations, there is about 1 dB gain in the medium to high
SNR region. According to the analysis presented in Section 5.2.2, such gain comes from
interleaving and graph optimization by the copy-and-permute operation. Therefore, the
graph design is very important to the construction of SCMA codebooks.
5.3 Design of 3D SCMA Codebooks
State-of-the-art SCMA codebooks are strictly designed for very limited effective spread-
ing factor, i.e., dv,lds = 2, which means a 2D rotation is considered to the lattice con-
stellation [71–75]. To the best of our knowledge, there does not exist a SCMA codebook
with dv,lds larger than 2. In this section, we aim to design 3D SCMA codebooks.
5.3.1 Limitations of 2D SCMA Codebooks
In Chapter 3, it has been proved analytically and numerically that the performance of
LDS is mainly determined by the degree distributions of different nodes. SCMA is built
upon LDS, thus the analysis presented in Chapter 3 is also valid for SCMA. The advan-
tage of SCMA codebooks with dv,lds of 2 is the low complexity, as the computational
complexity of MPA is directly related to the density of a graphical model. The smaller
dv,lds is, the lower complexity can be achieved. In LDPC codes, by optimizing the
degree distribution, i.e., by varying the column weights and row weights of the parity
check matrix, the performance can be improved. Similar to the LDPC decoding, in SC-
MA, the layer nodes with high-degree can obtain more information from their adjacent
resource nodes so that it tends to converged more quickly than the low-degree layer
nodes. These layer nodes offer more extrinsic information in initial iterations which can
be fed back to enhance the detection performance. More explicitly, the more reliable
information, provided by the high-degree nodes, can be used to improve the detection
performance even in poor channel conditions. In terms of LLR propagation, the LLRs
for the high-degree nodes converge quickly to large values, whereas the low-degree n-
odes converge slowly to smaller LLRs. The more reliable LLR information provided by
the high-degree nodes, can be used to provide more reliable LLR for computing soft
116 Chapter 5. Sparse Code Multiple Access (SCMA)
symbol average value. The enhanced soft symbols can be used to improve the detection
performance in the following iterations. Hence, dv,lds of 2 represents the weakest link
in the SCMA codebook and its message may converge slowly than that high-degree
nodes.
5.3.2 Construction of SCMA codebooks with dv,lds of 3
Due to the fact that dv,lds of 2 is not the optimum degree distribution, we develop 3D
SCMA codebooks, i.e., dv,lds = 3, to bridge this gap. In this case, more than 2 rotated
constellations are needed to generate the mother constellation.
Fig. 5.8 shows an example of the shuffling to construct a 4-point SCMA constellation
with dv,lds of 2, which is similar to that of Fig. 5.2. To extend the dv,lds to 3, it is
necessary to consider the BPSK in 3D constellations, which is shown in Fig. 5.9. First
of all, two BPSK points in 3D constellations, which are marked by dotted circles, are
respectively rotated using the same angle to different directions. The optimum rotation
angle is tan−1((1+√5)/2) [74]. Each point of the rotated constellation is then marked
by a solid cycle and can be queued according to the axis value: X1(0 1), X2(1 0),
X3(1 0), Y1(0 1), Y2(1 0), Y3(0 1). Subsequently, a shuffling is performed to separate
these axes. This idea is a straightforward extension to that of 2D constellation, where
three complex mother constellations are generated. By doing so, a 4-point SCMA
constellation with dv,lds of 3 is constructed.
5.3.3 Performance Evaluation
The performance of 200% loaded SCMA with 4-point constellation and different dv,lds
is simulated over SUI-3 channel and shown in Fig. 5.10. The FFT size is 128. For
dv,lds of 2, the optimized SCMA codebooks, which is generated by copy-and-permute
operation in section 5.2.2, is employed for the comparison. For dv,lds of 3, three different
constellations corresponding to the three edges of each layer node, are generated by the
approach shown in Fig. 5.9. According to Fig. 5.10, by increasing dv,lds and rigorous
constellation design, the SCMA performance can be further improved. At BER of
10−5, 4 dB gain can be achieved by dv,lds = 3 compared to dv,lds = 2. It should be
5.3. Design of 3D SCMA Codebooks 117
X2
X1
Y2
Y1
Rotated BPSK 1 Rotated BPSK 2
0 0
1 1
Y1
X1
01 11
00 10
Y2
X2
10 00
11 01
4 Points Constellation points for dv,lds = 2
Fig. 5.8: 4-point SCMA constellation for dv,lds of 2
X3
X2
X1
0
1
Rotated BPSK 1
Y3
Y2
Y1
0
1
Rotated BPSK 2
Y1
X1
01 11
00 10
Y2
X2
10 00
11 01
Y3
X3
11 01
10 00
4 Points Constellation points for dv,lds = 3
Fig. 5.9: 4-points SCMA constellation for dv,lds of 3
118 Chapter 5. Sparse Code Multiple Access (SCMA)
noted that according to the graph theory and the analysis of degree distribution, the
SCMA performance will not be improved infinitely when dv,lds increases. Also larger
dv,lds means higher computational complexity. Therefore, in practical system design, it
is necessary to strike a balance between the performance requirement and the receiver
complexity.
6 8 10 12 14 16 18 20 22 24 26 28 3010
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Eb/N
0 (dB)
BE
R
SCMA 4−point for dv,lds
= 2
SCMA 4−point for dv,lds
= 3
Fig. 5.10: Performance of 200% loaded SCMA by different dv,lds
5.4 Summary
As an enhanced version of LDS, SCMA combines the techniques of multi-dimensional
constellation shaping and the low density spreading. It has become a promising can-
didate for overloaded transmissions with high-order constellations in 5G networks. In
this chapter, the SCMA system model and codebook design criteria are studied. Based
on this investigation, we optimize the SCMA structure by copy-and-permute operation
on protographs, and design SCMA codebooks with dv,lds of 3. Simulation results show
that the proposed approaches improve the SCMA performance.
Chapter 6
Conclusions and Future Works
This chapter provides a summary of this thesis’ contributions, and discusses how the
proposed algorithms in this thesis contributed to the field of NOMA.The perspective
regarding future works is further discussed.
6.1 Summary of Contributions
In Chapter 3, inspired by the similar structures of LDS-OFDM and LDPC codes, a joint
sparse graph for OFDM systems is proposed. Variable nodes act as bridge functions
to connect chip nodes and parity-check nodes, then the low density signature of LDS-
OFDM and the low density parity-check matrix of LDPC codes are naturally linked to
form a joint sparse graph. Such joint sparse graph is different from any technique of
single graphs, as it combines NOMA (LDS-OFDM) and FEC techniques (LDPC codes).
On the receiver side, a joint receiver performing simultaneous detection and decoding
is designed. The joint detection and decoding is different from turbo-style iterations.
In addition, to examine the behavior of the joint sparse graph, EXIT charts are utilized
to analyze the convergence property of MPA. Based on EXIT charts analysis, design
principles of the joint sparse graph are studied, illustrating that degree distributions
and short cycles determine the system’s performance. Simulations show the superiority
of JSG-OFDM over GO-MC-CDMA, LDS-OFDM and turbo structured LDS-OFDM.
119
120 Chapter 6. Conclusions and Future Works
In Chapter 4, the idea of the joint sparse graph is extended to the FBMC-IOTA wave-
form. Considering that there is intrinsic interference of the real and imaginary branches
of FBMC transmissions, we propose to model such intrinsic interference by a low densi-
ty weight matrix. Then a joint sparse graph containing LDWM, LDS and LDPC codes
is designed. The key insight is that all of these techniques are based on low density
structure, and MPA is the common algorithm for the receiver. Therefore, a joint re-
ceiver for JSG-IOTA to jointy detect and decode becomes possible. Such joint sparse
graph combines multi-carrier modulation (LDWM), multiple access (LDS) and channel
coding (LDPC codes) techniques, and the iterative structure as well as convergence
behavior is analyzed by EXIT charts. Similar to JSG-OFDM, design guidelines for the
JSG-IOTA are provided.
In Chapter 5, SCMA is studied, and its codebook design’s criteria are investigated.
Different from LDS, the processes of bit to symbol mapping and LDS spreading are
merged in the SCMA encoder. As a result, compared with LDS, SCMA with multi-
dimensional constellation codebooks can attain the extra constellation shaping gain. To
construct longer SCMA codewords, a copy-and-permute operation on the protograph
is applied to the SCMA scheme. Moreover, 3D constellation with dv,lds of 3 is proposed
for the SCMA codebook design. We show that SCMA outperforms LDS with high-
order constellations, and the SCMA performance can be further improved by optimized
designs.
In general, NOMA can overcome some problems of OMA based techniques via non-
orthogonal resource allocation, and is an advantageous technique in future wireless
communications. We have developed code domain NOMA techniques for uplink in this
thesis. We start with LDS based NOMA. In order to design a joint receiver to achieve a
better BER performance, LDS is researched and optimized in three directions: i) LDS
+ LDPC codes; ii) LDS + advanced waveforms; and iii) LDS + multi-dimensional
constellation shaping. By careful design of the LDS graph structure, code domain
NOMA can be improved and optimized. Our proposed NOMA schemes can be applied
to MTC systems. Compared with OMA techniques, code domain NOMA increases
the supported user number and spectral efficiency, while the disadvantages include a
relative higher receiver complexity and the uncertainty of randomly constructed sparse
6.2. Future Works 121
graphs. More explicitly, not all of the low density signatures perform well under fully-
loaded and overloaded conditions. Even with identical parameters such as dv,lds and
dc,lds, randomly generated low density signatures by computer searching may behave
significantly different from each other. It is hard for a single factor to determine and
improve the LDS performance, e.g. the degree distribution, to meet the performance
requirements in future wireless communications. Therefore, it is important to construct
good low density signatures based on overall considerations for practical systems.
6.2 Future Works
Recently, the research activities of LDS and SCMA have attracted increasing attention
due to their ability to support overloaded transmissions. With the contributions in
mind, the following three topics for future investigations are highlighted:
1. Although code domain NOMA including LDS and SCMA have been analyzed and
optimized in the thesis, the codebook design and optimization are still open issues.
Different from techniques of sparse graph coding, LDS and SCMA are multiple
access schemes for the multiuser scenario, and the channel condition of different
users is random and dynamic. Thus the research in this area is more challenging
than conventional sparse graphs. To date, there does not exist a systematic
approach for the system design. It is thus of vital importance to establish an
unified framework for the design and optimization of NOMA schemes.
2. Due to the use of MPA for the detection of LDS and SCMA, the receiver com-
plexity is a big challenge in practical applications, especially when the size of the
sparse graph is large or the constellation order is high. It is necessary to continue
the research on receiver design, by finding approaches other than MPA to reduce
the receiver complexity while maintaining satisfactory performance.
3. SCMA blind detection should be further developed to include decoding of user’s
data with no complete knowledge of SCMA codebook sets. This is of particular
interest to the vehiclar communications, where the users’ number is dynamic, and
122 Chapter 6. Conclusions and Future Works
the roadside access point is random. This necessitates the blind detection of LDS
sparse graph or SCMA codebooks, the MPA cannot be performed otherwise.
4. Although current code domain NOMA mainly focuses on uplink transmissions, it
is possible to design downlink LDS/SCMA schemes by designing low complexity
nonlinear precoding techniques, which is an interesting topic for future research.
The output of this thesis and future research contribute to the state-of-the-art on
mobile communications, and can be used by industry to design, optimize and investigate
performance of future communication systems.
124 Appendix A. Appendix
TABLE A.1: Summary of the joint detection and decoding
INIT If no priors available, set Lvk,m→cn = 0, Lvk,m→pk,j = 0, ∀k, ∀m, ∀n, ∀j
For Iteration = 1 : Maximum iteration number
For n = 1 : N
For ∀k,m ∈ ψn
Update Lcn→vk,m using (3.9)
End
End
For j = 1 : J
For ∀k,m ∈ ϕj
Update Lpk,j→vk,m using (3.11)
End
End
For k = 1 : K
For m = 1 : M
For n ∈ εk,m, j ∈ ωk,m
Update Lvk,m→cn using (3.14)
Update Lvk,m→pk,j using (3.15)
Update Lvk,m using (3.16)
End
Hard Decision according to (3.17)
End
End
If syndrome computing for each user equals to zero, Quit the iterations
End
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