Acknowledgements

This book is the outcome of our individual and collaborative research on OTFS. It would not have been possible without the active collaboration, support, help, and encouragement we received from many. Our students and research collaborators from both academia and industry were pivotal for sustaining our research interest on OTFS. Our sincere thanks to all of them.Our deep appreciation and special thanks are due to Prof. Robert Calderbank, Duke University, USA, for his inspiring research collaboration on Zak-OTFS that led to the two seminal papers on Zak-OTFS published in IEEE BITS the Information Theory Magazine.We had the opportunity to deliver talks and tutorials on OTFS on several occasions in various forums that served as ideal platforms for interaction with the research community. These interactions have contributed to our deeper understanding and appreciation of OTFS. We thank the hosts and organizers of these events.

We thank Wiley and IEEE for accepting our proposal to write this book. It was a pleasure to work with the Wiley team throughout this book writing project. We thank Michelle Dunckley, Vishal Paduchuru, Elisha Benjamin, and Kavipriya Ramachandran for their help and guidance at various stages of the project that helped us to keep the project on track. Our special thanks are due to Mustaq Ahamed Noorullah and Sundaramoorthy Balasubramani for patiently working with us in the copy editing and proof reading stages of the production process.We appreciate their meticulous efforts.

A brief note on the cover design of the book is due here.We express our sincere thanks to Dr. Shailesh Rao M. for his help in getting the photograph of the crystalline rock formation in St. Mary’s island, Udupi, Karnataka state, India, used in the cover design. A noticeable structural similarity between the crystalline rocks in the island and the 3D-rendering of the received delay-Doppler power profile in Zak-OTFS in the crystalline regime (see Fig. 2.37) has been an inspiration for us to use the crystalline rock image in the cover design. Of course, the waveform (the pulsone) responsible for the delay-Doppler crystallization phenomenon has also found its place along with the Arabian sea waves in the cover design.

– Saif Khan Mohammed, Ronny Hadani, and Ananthanarayanan Chockalingam

I thank my students for being a part of my research on OTFS and also for constantly pushing me to write a book on OTFS – Venkatesh Khammammetti, Imran Ali Khan, Muhammad Ubadah, Rahul Kumar Jaiswal, Danish Nisar, Jinu Jayachandran, Alok Kumar Sinha, Saurabh Prakash, and Brijesh Chander Pandey. I would like to thank Prof. Emanuele Viterbo and Prof. Yi Hong at Monash University, Australia, for collaborating in the area of multiuser OTFS precoding and synchronization.

I would like to thank my PhD supervisor Prof. A. Chockalingam, IISc, Bangalore (coauthor of this book), for bringing out the best inme in terms of research and fromwhom I have inherited the art of perseverance in research. It is Prof. Chockalingam who introduced me to OTFS in 2017.

This book could not have been a reality without the support from Prof. Kishan and Pramila Gupta, Chair Professor position at IIT Delhi. I would specially thank Dr. Surendra Prasad, Honorary Professor, IIT Delhi, for being a mentor and a source of inspiration at the Department of Electrical Engineering, IIT Delhi. I would also like to thank Dr. M. Balakrishnan, Honorary Professor, IIT Delhi, who was my BTech project supervisor when I was an undergraduate student at IIT Delhi and under whom I learnt the basics of digital hardware and microprocessor design. I thankDr. M. Balakrishnan for introducing me to the world of scientific research and innovation, which is why I chose to be an academic.

I cannot forget to thank my parents for their constant and unconditional support and encouragement and without whom I would not be what I am,my wife and daughter for sacrificing family time so that I could devote more time to book writing and research. I specially thank my wife Shaba for being an unconditional source of support for the last 20 years and also for encouraging me to pursue a career in academics, a decision which I will always be happy about. I also thankmy parents in-law who have always helped me whenever I needed them.

Lastly, I thank Almighty God for enlightening me and my coauthors, which has enabled us to write this book.

– Saif Khan Mohammed

My OTFS journey began when I first met Shlomo Rakib in 2008 during one of my lectures on the canonical basis of eigen vectors of the finite Fourier transform (FFT). We immediately resonated on the personal level, however, only by the end of 2009, we started to work on a project together. The early discussions were conducted on Skype, after working hours, before miraculously converging to an intuitive idea of a communication method based on delay-Doppler shifts. From that point on, over a period of three years, we established the basic principles of OTFS, built a radio implementation, demonstrated its superior performance to a third party and built a company around it called Cohere Technologies. Shlomo has always been a source of knowledge and inspiration for me, and I am grateful for having the opportunity of learning and working side by side with him until today.

I would like to acknowledge the imminent contribution of Clayton Ambrose and Norm Rayes during these early years of development. Clayton, single handedly put together a comprehensive demonstration platform that provided deep insights about the interaction between the OTFS waveform and the wireless channel – confirming our theoretical hypothesis. Jointly with Clayton we performed extensive over-the-air testing under various mobility conditions both indoor and outdoor, at times employing unorthodox methods like driving a car with one hand and holding an antenna with another or holding a transmitter in front of a rotating fan. Norm joined our small team shortly after Clayton and put together the first OTFS radio and conducted a successful over-the-air demonstration to Sprint (telecom company) in their Kansas campus. This facilitated the first substantial Venture capital investment in Cohere. Norm always impressed me as a “renaissance man” – an orchestra of hardware and software engineers distilled in one man!

Although OTFS basic transceiver was in place as early as 2013, it took another several years to reveal its underlying mathematical structure. During this second period of development, I had the pleasure of working closely, as part of a due diligence process, with John Campbell from Telstra and Giovanni Vannucci from Bell labs. Their insights helped tremendously in explaining the value of OTFS and clarifying its conceptual structure.

Another important stimulator was Cohere’s attempt to promote orthogonal time frequency space (OTFS) into the emerging 3GPP 5G standard. Since orthogonal frequency division multiplexing (OFDM) was the waveform of choice at that time, our focus revolved around a multi-carrier variant of the waveform, referred to in this book as MC-OTFS. I would like to thank Anton Monk, the director of the Cohere standards team and my coauthor of the OTFS white paper, who led the audacious effort of promoting OTFS into 3GPP. Anton put together a team of experts and established strategic relations with telecom partners and carriers. The standards team wrote multiple high-quality technical contributions to 3GPP which laid the basis for all future developments. I would like to acknowledge the contribution of Shachar Kons, Yoav Hevron, Michael Tsatsanis, Anthony Ekpenyong, Cristian Ibars, and Paul Harris. Their deep knowledge of theory and vast practical experience combined with a healthy dose of skepticism helped mature OTFS into a complete architecture with comprehensive performance evaluation.

The cherry on top was our conference paper that was published in 2017 and constituted the first peer reviewed publication that exposed OTFS to the academic community. I would like to thank Prof. Andrea Goldsmith, who served as chair of Cohere Technical Advisory Board (TAB), Prof. AndreasMolisch and Prof. Robert Calderbank, both members of Cohere TAB, for putting the time and effort in organizing the writing of this pioneering paper.

In my mind, there were always two parallel derivations of OTFS. One was using the Zak transform giving rise to what we refer to in this book as Zak-OTFS and the other was using the symplectic finite Fourier transform (SFFT), giving rise to MC-OTFS. Until around 2017, I believed that the two derivations are mathematically equivalent and it is a matter of convenience which one to use. However, as often happens in mathematics, beliefs are proved to be false. Eventually, I came to realize that the two derivations are not equivalent and, moreover, got convinced that the Zak-theoretic derivation is the “correct” one while the SFFT derivation is merely an approximation. This realization marks the third period of OTFS development. Most of my work in Cohere on this topic at an early stage was conducted in close collaboration with my former PhD student Jim Delfeld who had numerous key insights and helped develop comprehensive simulations. A later stage collaboration which continues until today is conducted with Shachar Kons who made fundamental contributions to the development and implementation of a complete transceiver structure.

In parallel to the research activity, there was a continuous effort by Cohere to develop a viable product based on OTFS technology. This parallel path turns out to reinforce the research, revealing gaps and inspiring new ideas and directions. The process of building a product is...