Orthogonal Time Frequency Space Modulation: A Waveform for 6G (River Publishers Series in Communications)

Front Cover
OTFS: Orthogonal Time Frequency Space Modulation
Contents
Preface
Acknowledgements
List of Figures
List of Tables
1 Introduction
	1.1 Background
	1.2 1G - 2G
	1.3 2G - 3G
	1.4 3G - 4G
	1.5 Fifth Generation (5G) Mobile Communication Systems
	1.6 6G
2 A Summary of Waveforms for Wireless Channels
	2.1 Introduction
		2.1.1 Chapter Outline
	2.2 Mathematical Foundation to Time-Frequency Analysis
		2.2.1 Hilbert Space
		2.2.2 Norm on Vector Space
		2.2.3 Linear Operators on Hilbert Space
			2.2.3.1 Functional in Hibert Space
			2.2.3.2 Adjoint Operator
		2.2.4 Orthonormal Basis for Hilbert Space
		2.2.5 Sequence Space l2(N)
		2.2.6 Function Spaces
		2.2.7 Fourier Transform
			2.2.7.1 Operators on L2(R)
		2.2.8 Frames in Hilbert Spaces
			2.2.8.1 Frame Operator
			2.2.8.2 Reisz Basis
			2.2.8.3 Tight Frame
			2.2.8.4 Dual Frame
		2.2.9 Gabor Transform
	2.3 Time–Frequency Foundations
		2.3.1 Time–Frequency Uncertainty Principle
		2.3.2 Short Time Fourier Transform
			2.3.2.1 Properties
		2.3.3 Ambiguity Function
	2.4 Linear Time Varying Channel
		2.4.1 Delay-Doppler Spreading Function (SH(,))
		2.4.2 Time-Varying Transfer Function (LH(t,f))
		2.4.3 Time-Varying Impulse Response (h(t,))
		2.4.4 Linear Time Invariant (LTI) Channel
		2.4.5 Stochastic Description
		2.4.6 Under-Spread Property of Wireless Channel
		2.4.7 Physical Discrete Path Model
			2.4.7.1 Virtual Channel Representation: Sampling in Delay-Doppler Domain
	2.5 Waveform Design in Gabor Setting
		2.5.1 Digital Communication in Gabor System
		2.5.2 Waveform Design of Rectangular Lattice
			2.5.2.1 Ideal Eigenfunction of H
		2.5.3 Approximate Eigen Function for LTV Channel
	2.6 OFDM
		2.6.1 Channel
		2.6.2 Receiver
	2.7 5G Numerology
		2.7.1 Genesis
	2.8 Windowed OFDM
		2.8.1 Transmitter
		2.8.2 Receiver
	2.9 Filtered OFDM
		2.9.1 Transmitter
		2.9.2 Receiver Processing
	2.10 Filter Bank Multi-Carrier
		2.10.1 Cosine Modulated Tone
		2.10.2 Filter Characteristics
		2.10.3 Simplified Filter Characteristics
		2.10.4 MMSE Equalizer for FBMC
	2.11 Universal Filtered Multi-Carrier
		2.11.1 Structure of UFMC Transceiver
		2.11.2 System Model for UFMC
		2.11.3 Output of the Receiver for the UFMC Transceiver Block Diagram
	2.12 Generalized Frequency Division Multiplexing (GFDM)
		2.12.1 Introduction
			2.12.1.1 Chapter Conents
		2.12.2 GFDM System in LTI Channel
			2.12.2.1 Transmitter
			2.12.2.2 Self-interference in GFDM
			2.12.2.3 Receiver
			2.12.2.4 Two Stage Equalizer
			2.12.2.5 One-Stage Equalizer
		2.12.3 GFDM in Gabor System
			2.12.3.1 Discrete Gabor Transform
			2.12.3.2 Critically Sampled Gabor Transform
		2.12.4 Bit Error Rate Computation for MMSE Receiver
			2.12.4.1 MMSE Receiver
			2.12.4.2 SINR Computation
			2.12.4.3 Frequency Selective Fading Channel (FSFC)
			2.12.4.4 Additive White Gaussian Noise Channel (AWGN)
			2.12.4.5 BER Computation
			2.12.4.6 FSFC
			2.12.4.7 AWGN Channel
			2.12.4.8 Results
		2.12.5 Performance Comparison
		2.12.6 Issues with GFDM
			2.12.6.1 High PAPR
			2.12.6.2 High Computational Complexity
	2.13 Precoded GFDM System to Combat Inter Carrier Interference: Performance Analysis
		2.13.1 Section Contents
		2.13.2 Precoded GFDM System
			2.13.2.1 Block IDFT Precoded GFDM
			2.13.2.2 Joint Processing
			2.13.2.3 Two-Stage Processing
			2.13.2.4 DFT Precoded GFDM
			2.13.2.5 SVD Precoded GFDM
			2.13.2.6 BER Performance of Precoding Techniques
			2.13.2.7 Computational Complexity
		2.13.3 Results
			2.13.3.1 BER Evaluation of Precoded Techniques
			2.13.3.2 Complexity Computation
			2.13.3.3 PAPR of Precoding Techniques
	2.14 Chapter Summary
3 OTFS Signal Model
	3.1 Introduction
	3.2 OTFS Signal Generation
	3.3 RCP-OTFS as Block OFDM with Time Interleaving
	3.4 Performance in AWGN Channel
		3.4.1 Receiver for AWGN
		3.4.2 Ber Performance in AWGN
	3.5 Performance in Time Varying Wireless Channel
		3.5.1 The Channel
		3.5.2 Linear Receivers
			3.5.2.1 MMSE Equalization
			3.5.2.2 ZF Receiver for TVMC
			3.5.2.3 BER Evaluation of ZF and MMSE Receiver
	3.6 Chapter Summary
4 Receivers Structures for OTFS
	4.1 Belief Propagation Receiver for a Sparse Systems
		4.1.1 Maximum Apposterior Probability (MAP) Decoding
		4.1.2 Factor Graph Description
		4.1.3 Equalization Algorithm
			4.1.3.1 Initiation
			4.1.3.2 Check Node Update
			4.1.3.3 Variable Node Update
			4.1.3.4 Criteria for Variable Node Decision Update
			4.1.3.5 Termination
		4.1.4 Complexity Analysis
		4.1.5 Results
	4.2 Low Complexity LMMSE Receiver for OTFS
		4.2.1 Channel
		4.2.2 Low Complexity LMMSE Receiver Design for OTFS
			4.2.2.1 Structure of =[HH+2d2I]
			4.2.2.2 Low Complexity LU Factorization of
			4.2.2.3 Computation of
			4.2.2.4 LMMSE Receiver for OFDM over TVC
		4.2.3 Result
			4.2.3.1 Computational Complexity
			4.2.3.2 BER Evaluation
	4.3 Iterative Successive Interference Cancellation Receiver
		4.3.1 Introduction
		4.3.2 LDPC Coded LMMSE-SIC Reciever
		4.3.3 Low Complexity Receiver
			4.3.3.1 Complexity Computation
		4.3.4 Performance Presents Cumulative Distribution
	4.4 Chapter Summary
5 Circulant Pulse Shaped OTFS
	5.1 Chapter Outline
	5.2 Circular Pulse Shaped OTFS (CPS-OTFS)
	5.3 Low Complexity Transmitter for CPS-OTFS
	5.4 Circular Dirichlet Pulse Shaped OTFS (CDPS-OTFS)
	5.5 Remarks on Receiver Complexity
		5.5.1 LMMSE Receiver for GFDM and OFDM over TVC
	5.6 Simulation Results
	5.7 Chapter Summary
6 Channel Estimation in OTFS
	6.1 Delay Doppler Channel Estimation
		6.1.1 Pilot Structure
		6.1.2 Delay-Doppler Channel Estimation
		6.1.3 Channel Equalization
		6.1.4 Performance of Channel Estimation
		6.1.5 VSB OFDM Overview
			6.1.5.1 Transmitter
			6.1.5.2 Receiver
		6.1.6 Pilot Power in OTFS and VSB-OFDM
		6.1.7 Results
	6.2 Time Domain Channel and Equalization
		6.2.1 System Model
			6.2.1.1 Transmitter
		6.2.2 Effects of Residual Synchronization Errors
			6.2.2.1 Integer Delay and Integer Doppler Values
			6.2.2.2 Integer Delay and Fractional Doppler Values
		6.2.3 Equivalent Channel Matrix for OTFS Including Synchronization Errors
			6.2.3.1 OTFS Channel Matrices
		6.2.4 Estimation of Equivalent Channel Matrix
			6.2.4.1 Pilot Structure in Delay-Doppler Domain
			6.2.4.2 Channel Estimation
			6.2.4.3 Time Domain Interpretation of the Channel Estimation
		6.2.5 LMMSE Equalization
			6.2.5.1 Structure of q =[qq+2d2I]
			6.2.5.2 Computation of
			6.2.5.3 Computation Complexity
		6.2.6 LDPC Coded LMMSE-SIC Reciever
		6.2.7 Unified Framework for Orthogonal Multicarrier Systems
		6.2.8 Results
			6.2.8.1 Block Error Rate (BLER) Performance
	6.3 Conclusions
		6.3.1 Proof of Theorem 1
		6.3.2 Proof of Theorem 2
		6.3.3 PROOF: Delay-Doppler Input-Output Relation
7 Nonorthogonal Multiple Access with OTFS
	7.1 OTFS Signal Model
	7.2 Delay-Doppler Power-Domain NOMA-OTFS
		7.2.1 De-Do PD-NOMA-OTFS Downlink
			7.2.1.1 Transmit Signal Model
			7.2.1.2 Receiver Processing, SINR and SE Analysis
		7.2.2 De-Do PD-NOMA-OTFS Uplink
			7.2.2.1 Transmit Signal Model
			7.2.2.2 Receiver Processing, SINR and SE Analysis
	7.3 Power Allocation Schemes Among Download NOMA-OTFS Users
		7.3.1 Fixed Power Allocation (FPA)
		7.3.2 Fractional Transmit Power Allocation (FTPA)
			7.3.2.1 Average SNR Based FTPA
			7.3.2.2 Channel Norm Based FTPA
		7.3.3 Power Allocation for Weighed Sum Rate Maximization (WSRM)
			7.3.3.1 Average SNR Based WSRM
			7.3.3.2 Instantaneous Channel Information Based WSRM
	7.4 Link Level Performance Analysis of NOMA-OTFS Systems
		7.4.1 Downlink MMSE SIC Receiver with LDPC Coding
			7.4.1.1 Processing at First User
			7.4.1.2 Processing at Second User
		7.4.2 Uplink MMSE SIC Receiver with LDPC Coding
	7.5 Simulation Results and Discussion
		7.5.1 System Level Spectral Efficiency Results
			7.5.1.1 Comparison between NOMA/OMA-OTFS
			7.5.1.2 Comparison between OTFS and OFDM Performances
			7.5.1.3 Comparison of Various NOMA Power Allocation Schemes
			7.5.1.4 Extracting NOMA Gain in OTFS with User Channel Heterogeneity
		7.5.2 Link Level Performance of NOMA-OTFS
			7.5.2.1 Performance of NOMA-OTFS in Downlink
			7.5.2.2 Performance of NOMA-OTFS in Uplink
	7.6 Conclusion
A OTFS Channel Matrix (Ideal)
References
Index
About the Authors
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