OTFS Modulation: Theory and Applications

Grasp the future of wireless communication with this groundbreaking introduction Research and development are already underway on the sixth generation (6G) of wireless communication technology. The new requirements of 6G that arise from challenging new use cases render physical layer waveforms such as CDMA and OFDM inadequate. The OTFS waveform answers these new requirements, and recent research suggests it will play a decisive role in the future of wireless communication. OTFS Modulation: Theory and Applications provides the first ever foundational textbook that introduces this growing, state-of-the-art, field of research from first principles. Beginning with a thorough discussion of the fundamental principles of OTFS, both physical and theoretical, it rigorously situates OTFS modulation in a mathematical framework analogous to more familiar waveforms. The result is a groundbreaking contribution to communication theory and a must-have volume for wireless communication researchers. Readers will find: An expert author team including the inventor of OTFS modulation Detailed discussion of topics including the Zak theory of linear time-varying systems, delay-Doppler communication and radar sensing, machine learning, and many more MATLAB™ code for OTFS transceiver implementation OTFS Modulation: Theory and Applications is ideal for researchers, engineers, graduate and advanced undergraduate students, and standardization professionals working with wireless communication, signal processing, and radar sensing.

Table of Contents:
Preface xv Acknowledgements xvii Acronyms xxi 1 Introduction 1 1.1 Cellular Mobile Evolution 2 1.1.1 Moore’s Law Drives Rate Increase 4 1.2 Multipath Fading Channels 5 1.2.1 Frequency Domain Characterization 6 1.2.2 Time Domain Characterization 7 1.2.3 Impact of High Dopplers 9 1.2.4 Delay-Doppler Domain Characterization 12 1.2.4.1 Urban Multi-lane Scenario − An Example 14 1.3 Communication Waveforms for Doubly Selective Channels 14 1.3.1 OTFS – A Modulation Waveform for Doubly Selective Channels 16 1.3.2 OTFS Realization – Two Approaches 16 1.4 Waveforms for Radar Sensing 17 1.4.1 OTFS for Integrated Sensing and Communication 18 1.5 Organization of the Book 19 2 Delay-Doppler Signaling and OTFS Modulation 23 2.1 Delay-Doppler Domain 28 2.2 Time and Frequency Domain Modulation 29 2.2.1 Channel Interaction of a TD Pulse 30 2.2.2 Channel Interaction of a FD Pulse 32 2.3 Delay-Doppler (DD) Domain Modulation 34 2.3.1 Origin of Quasi-periodicity 37 2.4 Channel Interaction of a DD Domain Pulse 39 2.4.1 The Channel Interaction is Predictable 40 2.4.2 The Channel Interaction is Non-fading 43 2.4.3 The Channel Interaction is Non-stationary 43 2.5 Time- and Bandwidth-Limited DD Domain Carrier Waveforms 43 2.5.1 Example in Fig. 2.17 47 2.5.2 Orthogonality of Pulsones 49 2.5.3 Optimality as Time- and Bandwidth-limited Signals 50 2.5.4 TDM as a Limiting Case 50 2.5.5 FDM as a Limiting Case 50 2.5.6 TD Pulsones Encode Wireless Channel Dynamics 50 2.5.7 The Fourier Transform as a Composition 51 2.6 Zak-OTFS Modulation and I/O Relation 51 2.6.1 Generalized Transceiver Signal Processing 52 2.6.2 Zak-OTFS Modulation 54 2.6.3 Zak-OTFS Receiver 56 2.6.4 Zak-OTFS I/O Relation 57 2.7 Predictability of the Zak-OTFS I/O Relation in the Crystalline Regime 58 2.7.1 Non-predictability of the Zak-OTFS I/O Relation in the Non-crystalline Regime 61 2.7.2 Crystalline Decomposition 62 2.7.3 Identification of Linear Time-Varying Channels 63 2.7.4 Error in Prediction of the Zak-OTFS I/O Relation 64 2.8 Matrix-vector Description of the I/O Relation 67 2.8.1 Zak-OTFS 67 2.8.2 Tdm 68 2.8.3 Fdm 69 2.9 Impact of Fading in the Crystalline Regime 70 2.10 Model-free Operation in the Crystalline Regime 73 2.10.1 Model-dependent Operation 73 2.10.2 Model-free Operation 74 2.11 Summary 77 Appendix 2.A Cascade of Two Doubly Spread Channels and Twisted Convolution 78 2.a.1 Properties of Twisted Convolution 81 Appendix 2.B Channel Action on DD Domain Signal 83 Appendix 2.C Zak Transform 84 Appendix 2.D Derivation of (2.30) and (2.31) 85 Appendix 2.E Proof of Non-overlapping Received Impulses in a-response 86 Appendix 2.F Inverse Zak Transforms 87 2.f.1 Inverse Time-Zak Transform 87 2.f.2 Derivation of (2.38) 87 2.f.3 Inverse Frequency-Zak Transform 88 Appendix 2.G Twisted Convolution Preserves Quasi-periodicity 88 Appendix 2.H Derivation of TD Pulsone Expression in (2.41) 89 Appendix 2.I Derivation of FD Pulsone Expression in (2.44) 90 Appendix 2.J TDM I/O Relation 90 2.j.1 Proof of Theorem 2.5 91 Appendix 2.K FDM I/O Relation 92 2.k.1 Proof of Theorem 2.6 93 Appendix 2.L Discrete DD Domain Signals 94 2.l.1 Impulse Signal in Discrete DD Domain 95 Appendix 2.M Derivation of Zak-OTFS Modulator Architecture 96 Appendix 2.N Derivation of Zak-OTFS Receiver 97 Appendix 2.O Proof of Theorem 2.1 99 3 Approximations of OTFS Modulation 101 3.1 Pulsones as a Basis for TD Signals 109 3.2 Generating Pulsones Using the Heisenberg Transform 110 3.3 Generating Time- and Bandwidth-Limited Pulsones 113 3.3.1 Comparing MC Pulsones with Zak Pulsones 117 3.4 Multicarrier OTFS (MC-OTFS) Modulation 118 3.4.1 Two-step MC-OTFS Modulator 119 3.4.2 Zak Transform-Based MC-OTFS Modulator 122 3.5 MC-OTFS Receiver 123 3.5.1 Two-step MC-OTFS Receiver 124 3.5.2 Zak Transform-Based MC-OTFS Receiver 125 3.6 MC-OTFS I/O Relation 126 3.6.1 MC-OTFS I/O for a Two-Step Transceiver 126 3.6.2 MC-OTFS I/O Relation for Zak Transform-Based Transceiver 129 3.7 Comparing MC-OTFS to Zak-OTFS 130 Appendix 3.A Proof of Theorem 3.1 133 Appendix 3.B Proof of Corollary 3.14 134 Appendix 3.C Proof of Theorem 3.2 134 Appendix 3.D SFT and Inverse SFT 135 Appendix 3.E Proof of Theorem 3.3 138 Appendix 3.F Proof of Theorem 3.4 138 Appendix 3.G Proof of Theorem 3.5 139 Appendix 3.H Impact of Windowing on a Periodic DD Domain Impulse 140 Appendix 3.I Proof of Theorem 3.6 141 Appendix 3.J Proof of Theorem 3.7 142 Appendix 3.K Proof of Theorem 3.9 143 Appendix 3.L Proof of Theorem 3.10 144 Appendix 3.M Proof of Theorem 3.11 145 Appendix 3.N Derivation of Bi-orthogonality Condition 145 Appendix 3.O Proof of Corollary 3.5 146 Appendix 3.P Derivation of I/O Relation for the Two-step MC-OTFS Transceiver 146 Appendix 3.Q Proof of Theorem 3.13 146 Appendix 3.R Proof of Corollary 3.6 147 4 Delay-Doppler Diversity in OTFS 149 4.1 Diversity in SISO OTFS 150 4.1.1 System Model 150 4.1.2 Diversity Analysis 154 4.1.3 Full Diversity with Phase Rotation 157 4.2 Diversity in MIMO-OTFS 161 4.2.1 MIMO-OTFS Diversity Analysis 162 4.2.2 MIMO-OTFS with Phase Rotation 163 4.3 Diversity in Space–Time Coded OTFS 165 4.3.1 STC-OTFS Scheme 166 4.3.1.1 Encoding and Decoding 166 4.3.1.2 Diversity Analysis 167 4.3.1.3 Phase Rotation in STC-OTFS 168 4.4 Diversity in OTFS with Antenna Selection 170 4.4.1 MIMO-OTFS with RAS 170 4.4.2 STC-OTFS with RAS 172 4.4.3 Rank of Multi-antenna OTFS Systems 173 4.4.4 Analysis of Multiantenna OTFS with RAS 173 4.4.5 Rank Deficient Multi-antenna OTFS Systems with RAS 178 5 OTFS Signal Detection 183 5.1 Linear Detection 184 5.1.1 Reduced Complexity MMSE Detection 185 5.1.1.1 Complexity Reduction Scheme 1 185 5.1.1.2 Complexity Reduction Scheme 2 187 5.2 Approximate MAP-Based Detection 189 5.2.1 Message Passing Detection 189 5.2.2 Variational Bayes-Based Detection 192 5.2.3 Hybrid MAP and PIC Detection 194 5.2.4 MCMC Sampling-Based Detection 197 5.3 Cross-Domain Detection 198 6 Delay-Doppler Channel Estimation 207 6.1 PN Pilot-Based Estimation 208 6.1.1 Solving for (δi,ωi,αi) 209 6.1.2 Performance Results 211 6.2 Exclusive Pilot Approach 212 6.2.1 Impulse-Based Channel Estimation 212 6.2.1.1 Integer DD and Fractional DD 214 6.2.2 DDIPIC-Based Estimation 215 6.2.2.1 Coarse Estimation 217 6.2.2.2 Fine Estimation 217 6.2.2.3 Stopping Criterion 218 6.2.2.4 Refinement of Parameters 218 6.2.2.5 Choice of ϵ 218 6.2.2.6 Complexity 218 6.2.2.7 Performance 219 6.3 Embedded Pilot Approach 221 6.3.1 Embedded Pilot with Guard Symbols 221 6.3.1.1 Threshold-Based Estimation 221 6.3.2 Embedded Pilot Without Guard Symbols 223 6.3.2.1 System Model 223 6.3.2.2 Embedded Frame Structure 224 6.3.2.3 SBL-Based Estimation 224 6.4 Superimposed Pilot Approach 227 6.4.1 System Model 227 6.4.2 SP-NI Scheme 229 6.4.3 SP-I Scheme 230 6.4.4 Performance 231 7 Multiuser OTFS 233 7.1 Resource Elements 234 7.2 OTFS Uplink 236 7.2.1 Orthogonal Multiple Access (OMA) 237 7.2.1.1 Guard Band (GB)-Based OMA 237 7.2.1.2 Interleaved Delay-Doppler OMA 240 7.2.1.3 Interleaved Time-Frequency OMA 245 7.2.1.4 Performance of OTFS-Based OMA Methods with Ideal Pulses 249 7.2.1.5 Performance of OTFS-Based OMA Methods with Practical Rectangular Pulses 251 7.2.2 Non-orthogonal Multiple Access 256 7.2.2.1 Multiuser MIMO-OTFS Uplink 256 7.2.2.2 Massive MIMO-OTFS 263 7.3 OTFS Downlink 265 7.3.1 Downlink Precoding 266 7.3.1.1 Linear Precoding 267 7.3.1.2 Nonlinear Precoding 270 7.3.2 Massive MIMO-Based OTFS Downlink 271 7.3.2.1 MRT-Based Massive MIMO-OTFS Precoding 271 7.4 Multiuser Channel Estimation 276 7.5 Synchronization 277 7.5.1 Frequency Synchronization 277 7.5.2 Timing Synchronization 277 7.5.2.1 Uplink Timing Synchronization in OFDM-Based Systems 278 7.5.2.2 Uplink Timing Synchronization in OTFS-Based Systems 278 7.5.2.3 Downlink Timing Synchronization in OTFS-Based Systems 283 8 Practical Considerations in OTFS Systems 285 8.1 PAPR of MC-OTFS Signals 286 8.1.1 MC-OTFS Transmit Signal 287 8.1.2 Upper Bound on PAPR 288 8.1.3 CCDF of PAPR 289 8.1.4 Results 290 8.1.4.1 Effect of Increasing M and N on the CCDF of PAPR 290 8.1.4.2 Effect of Pulse Shaping on PAPR 291 8.1.4.3 Comparison with GFDM and OFDM 292 8.2 Impact of Phase Noise on OTFS 292 8.2.1 System Model Without Phase Noise 293 8.2.2 System Model with Phase Noise 294 8.2.3 Performance in the Presence of Phase Noise 295 8.3 Impact of IQ Imbalance on OTFS 298 8.3.1 OTFS System Model with IQI 299 8.3.1.1 System Model with Transmitter IQI 299 8.3.1.2 System Model with Receiver IQI 300 8.3.2 Sensitivity of OTFS and OFDM to IQI 300 8.4 Discrete Zak Approach to OTFS 302 8.4.1 DZT-OTFS System Model 302 8.4.2 Vectorized DD Domain I/O Relation 304 8.4.3 Diversity Analysis of DZT-OTFS 305 8.4.4 Performance Results 307 8.4.4.1 Diversity Performance 307 8.4.4.2 BER Performance 309 8.4.4.3 Complexity 310 Appendix 8.A Digital Implementation of Zak-OTFS Modulation 311 8.a.1 Digital Implementation of the Zak-OTFS Receiver 311 8.a.2 Digital Implementation of the Zak-OTFS Transmitter 314 Appendix 8.B Discrete Zak Transform 316 8.b.1 Discrete Delay and Doppler Shifts 318 8.b.2 Discrete Pulsones 319 9 Deep Learning for OTFS Transceiver Design 327 9.1 Learning Framework 328 9.1.1 Classification of Machine Learning Programs 328 9.1.1.1 Classification Based on Training Dataset 328 9.1.1.2 Classification Based on Nature of Objective 329 9.1.2 Traditional Program and Machine Learning Program 329 9.1.3 Deep Learning Using Neural Networks 330 9.1.3.1 Fully Connected Neural Networks 330 9.1.3.2 Recurrent Neural Networks 333 9.1.3.3 Convolutional Neural Networks 333 9.1.4 Common Terminologies in the DL Framework 333 9.2 OTFS Signal Detection Using DNNs 335 9.2.1 Full-DNN Detection Approach 335 9.2.2 Symbol-DNN Detection Approach 336 9.2.2.1 Training and Testing 337 9.2.3 Performance Results 337 9.2.3.1 Static Channel with i.i.d. Gaussian Noise 338 9.2.3.2 Static Channel with Non-Gaussian Noise 339 9.2.3.3 Doppler Channel with Non-Gaussian Noise 342 9.2.3.4 Correlated Gaussian Noise in MIMO-OTFS 342 9.3 IQI Estimation/Compensation Using DNNs 344 9.3.1 DNN-Based OTFS Transceiver 344 9.3.1.1 Tx IQI Compensation 344 9.3.1.2 Rx IQI Compensation 345 9.3.1.3 Channel Training and Detection 346 9.3.2 Performance Results 347 9.3.2.1 Tx IQI Compensation 348 9.3.2.2 Rx IQI Compensation 349 9.3.2.3 Channel Training and Detection 350 9.3.2.4 Combined Performance of All DNNs 351 9.4 DD Channel Estimation Using DNNs 353 9.4.1 Estimation Using Embedded Pilots 355 9.4.1.1 Architecture 356 9.4.1.2 Training Methodology 356 9.4.1.3 Inference from the Network 357 9.4.1.4 Performance Results 358 9.4.2 Estimation Using Interleaved Pilots 360 9.4.2.1 Architecture 362 9.4.2.2 Training Methodology 363 9.4.2.3 Estimation of Delay and Doppler Indices 363 9.4.2.4 Complexity 363 9.4.2.5 Performance Results 364 9.4.3 Estimation Using Superimposed Pilots 365 9.4.3.1 Iterative Scheme 366 9.4.3.2 Performance Results 366 10 OTFS for Radar Sensing 371 10.1 Radar Scene 373 10.2 The Radar Ambiguity Function 374 10.2.1 Radar Resolution 376 10.2.2 Ambiguity Functions for TD and FD Pulses 379 10.3 A Good Radar Waveform: Quasi-localization in TD and FD Using Modulated Pulse Trains 381 10.4 Zak Theoretical Framework for Design of Radar Waveforms 383 10.4.1 Ambiguity Function of the Zak-OTFS Pulsone for Sinc Pulse 388 10.5 Summary 389 Appendix 10.A Proof of (10.7) 390 Appendix 10.B Proof of (10.9) 391 Appendix 10.C Proof of Theorem 10.1 392 Appendix 10.D Proof of (10.21) 394 Appendix 10.E Effect of Time and Bandwidth Limitation on the DD Representation of a Waveform 394 Appendix 10.F Peaks of the Ambiguity Function 395 Appendix 10.G Proof of Theorem 10.2 396 Appendix 10.H Proof of Theorem 10.3 397 Appendix 10.I Proof of Theorem 10.4 398 Appendix 10.J Proof of (10.43) and (10.44) 400 Appendix 10.K Proof of Theorem 10.5 400 References 403 Index 415

About the Author :
Saif Khan Mohammed, PhD, is a Professor in the Department of Electrical Engineering at IIT Delhi, India. He worked extensively in industry before transitioning to academia, and his research interests include wireless communication, signal processing, OTFS, and related subjects. Ronny Hadani, PhD, is an Associate Professor in the Department of Mathematics, University of Texas, Austin, USA. He is the co-founder of Cohere Technologies and co-inventor of Cohere’s OTFS modulation scheme. His research focuses on representation theory, harmonic analysis, and signal processing. Ananthanarayanan Chockalingam, PhD, is a Professor in the Department of Electrical and Communication Engineering, Indian Institute of Science, Bangalore, India. He has made pioneering contributions in the area of large MIMO systems. He has vast industry experience in wireless communications.