Y. Hong, T. Thaj, and E. Viterbo, Delay-Doppler Communications: Principles and Applications, Elsevier, 2022.
This book provides a general overview of the fundamental concepts, the underlying principles, the future research directions, and the potential applications of DD communications. After revisiting OFDM modulation, this book reveals the mathematical and physical relations between different domains for representing channels and waveforms. The key technical aspects of OTFS including detection, channel estimation, integration with MIMO, and multiuser MIMO are also covered.

R. Hadani, S. Rakib, M. Tsatsanis, A. Monk, A. Goldsmith, A. F. Molisch, and R. Calderbank. “Orthogonal Time Frequency Space Modulation," in Proc. IEEE Wireless Communications and Networking Conference, San Francisco, US, March 2017.
This is the first publication in which the name of OTFS was introduced. This paper starts from the DD domain signal representation point of view and compares the performance of OTFS to the classic OFDM modulation. The merits of OTFS, including its conceptual connection with Radar, new coupling relationship between information and channels, robustness against interference, and linear scaling of spectral efficiency with respect to the number of antennas are revealed. The DD domain equalization and precoding are also briefly discussed.

Z. Wei, W. Yuan, S. Li, J. Yuan, G. Bharatula, R. Hadani, and L. Hanzo, “Orthogonal Time-Frequency Space Modulation: A Promising Next-Generation Waveform,” IEEE Wireless Communications, vol. 28, no. 4, pp. 136-144, August 2021.
This magazine article introduces OTFS modulation conceived for communications over high-mobility environments by providing an easy-reading overview of its fundamental concepts, highlighting the challenges and potential solutions as well as exploring new promising areas for future research. Commencing from the channel characteristics of high-mobility communications, the paper presents the basic concepts and properties of the DD domain channels, the DD domain multiplexing, the OTFS transceiver architecture and signal waveform, as well as the OTFS system design criteria. Three fundamental research problems related to the OTFS waveform, including channel estimation, data detection, and channel coding are discussed, and pertinent preliminary results are provided.

L. Xiao, S. Li, Y. Qian, D, Chen, and T. Jiang, “An Overview of OTFS for Internet of Things: Concepts, Benefits and Challenges,” IEEE Internet of Things Journal, vol. 9, no. 10, pp. 7596-7618, May 2022.
This paper provides a comprehensive overview of OTFS for Internet of Things (IoT), including its original concept, system design, and future design guidelines. The basic OTFS transmission model and its matrix form representation are first introduced. Then, the waveform design, prefix selection, signal detection, and performance evaluation are presented. Finally, discussions concerning challenges and future research directions are given.

S. K. Mohammed, “Derivation of OTFS Modulation from First Principles,” IEEE Transactions on Vehicular Technology, vol. 70, no. 8, pp. 7619-7636, August 2021.
This paper derives the OTFS modulation from first principles. The paper rigorously derives an orthonormal basis of approximately time and bandwidth limited signals which are also localized in the DD domain. Furthermore, this paper shows that irrespective of the amount of Doppler shift, the received DD domain basis signals are localized in the DD domain w.r.t the DD resolution. Finally, this paper verifies that the degree of localization of the DD domain basis signals is inversely related to the time duration of the transmit signal.

G. D. Surabhi, R. M. Augustine, and A. Chockalingam, “On the Diversity of Uncoded OTFS Modulation in Doubly-Dispersive Channels,” IEEE Transactions on Wireless Communications, vol. 18, no. 6, pp. 3049-3063, June 2019.
This paper studies the diversity performance of the uncoded OTFS systems. Specifically, this paper shows that the asymptotic diversity order of OTFS modulation is one. However, the potential for a higher order diversity is witnessed before the diversity one regime takes over in the finite signal-to-noise ratio (SNR) regime. Furthermore, a simple phase rotation scheme for OTFS is proposed using transcendental numbers, based on which OTFS transmission is guaranteed to achieve the full diversity.

S. Li, J. Yuan, W. Yuan, Z. Wei, B. Bai, and D. W. K. Ng, “Performance Analysis of Coded OTFS Systems over High-Mobility Channels,” IEEE Transactions on Wireless Communications, vol. 20, no. 9, pp. 7193–7198, July 2021.
This paper investigates the error performance of coded OTFS systems in terms of the diversity gain and coding gain. In particular, the paper shows that the diversity gain of OTFS systems improves with the number of resolvable paths, while the coding gain declines. A rule-of-thumb code design criterion is also provided that is increasing the Euclidean distance between codeword pairs. Numerical results verify the conclusions of the paper and demonstrate the error performance improvement of coded OTFS compared to the coded OFDM.

L. Gaudio, G. Colavolpe, and G. Caire, “OTFS vs. OFDM in the Presence of Sparsity: A Fair Comparison,” to appear in IEEE Transactions on Wireless Communications, vol. 21, no. 6, pp. 4410-4423, June 2022.
This paper studies the pragmatic capacity of both OTFS and OFDM in the presence of sparse channels. The pragmatic capacity is defined as the achievable rate of the channel induced by the signal constellation and the detector soft-output. Specifically, the paper shows that OTFS enjoys a better rate performance compared to the OFDM counterpart over static channels under practical channel estimation and detection approaches. Meanwhile, it is also shown that OTFS achieves a more robust performance than OFDM over high-mobility channels.

P. Bello, “Characterization of Randomly Time-Variant Linear Channels,” IEEE Transactions on Communications Systems, vol. 11, no. 4, pp. 360-393, December 1963.
This paper provides the mathematical fundamentals of randomly time-variant linear channels. Both the wide-sense stationary (WSS) channel and the uncorrelated scattering (US) channel are introduced, which are shown to be time-frequency duals. Then, specific focus is given to the WSSUS channels, whose scatter function can be fully characterized by using time and frequency variables or delay and Doppler variables, which motivates the DD signal representation of OTFS.

G. Matz, H. Bolcskei, and F. Hlawatsch, “Time-Frequency Foundations of Communications: Concepts and Tools,” IEEE Signal Processing Magazine, vol. 30, no. 6, pp. 87-96, November 2013.
This paper provides a comprehensive overview on the fundamentals of wireless channels, including various aspects ranging from the underlying channel physics to the application for communications. This paper also provides the framework of using TF/DD designs to solve problems for communications, such as pulse shaping and capacity calculation.

A. Tusha, S. Doğan-Tusha, F. Yilmaz, S. Althunibat, K. Qaraqe, and H. Arslan, “Performance Analysis of OTFS Under In-Phase and Quadrature Imbalance at Transmitter,” IEEE Transactions on Vehicular Technology, vol. 70, no. 11, pp. 11761-11771, November 2021.
This paper studies the error performance of OTFS over wireless channels in the presences of in-phase and quadrature (IQ) imbalance. In particular, this paper shows that the IQ imbalance could cause a new type of interference named mirror-Doppler interference in the equivalent channel model, which occurs only along the Doppler dimension. Performance analysis is derived in the presence of IQ imbalance, which shows that IQ imbalance could result in a reduced diversity gain for OTFS transmissions.

H. Groll, E. Zöchmann, S. Pratschner, M. Lerch, D. Schützenhöfer, M. Hofer, J. Blumenstein, S. Sangodoyin, T. Zemen, A. Prokeš, A. F. Molisch, and S. Caban, “Sparsity in the Delay-Doppler Domain for Measured 60 GHz Vehicle-to-Infrastructure Communication Channels,” in Proc. IEEE International Conference on Communications Workshops (ICC Workshops), Shanghai, China, May 2019.
This paper reports experimental results for DD domain channel sparsity in 60GHz vehicle-to-infrastructure communication channels in a street crossing scenario in an urban environment. The experimental results verify that the DD channels are indeed sparse under the considered communication scenarios, where OTFS has demonstrated very well performance.

P. Raviteja, Y. Hong, E. Viterbo, and E. Biglieri, “Practical Pulse-Shaping Waveforms for Reduced-Cyclic-Prefix OTFS,” IEEE Transactions on Vehicular Technology, vol. 68, no. 1, pp. 957-961, January 2019.
This paper analyses the input–output relation of OTFS system for arbitrary pulse-shaping waveforms using a block-circulant matrix decomposition. The out-of-band radiation and BER performance with different waveforms are compared.

Z. Wei, W. Yuan, S. Li, J. Yuan, and D. W. K. Ng, “Transmitter and Receiver Window Designs for Orthogonal Time Frequency Space Modulation,” IEEE Transactions on Communications, vol. 69, no. 4, pp. 2207–2223, April 2021.
This paper investigates the impact of window functions at transmitter and receiver for OTFS modulation. The TX window can be interpreted as a power allocation in the TF domain, while employing a RX window causes a colored noise. When channel state information (CSI) is available at both the TX and RX, an optimal TX window design that minimizes the detection mean squared error (MSE) can be interpreted as a mercury/water-filling power allocation scheme. When CSI is not available at the TX, a proper window design, such as Dolph-Chebyshev (DC) window, in the TF domain can enhance the channel sparsity and thus improves the channel estimation performance even with a smaller amount of guard space overhead.

B. C. Pandey, S. K. Mohammed, P. Raviteja, Y. Hong, and E. Viterbo, “Low Complexity Precoding and Detection in Multi-user Massive MIMO OTFS Downlink,” IEEE Transactions on Vehicular Technology, vol. 70, no. 5, pp. 4389–4405, May 2021.
This paper applies OTFS modulation to downlink multi-user massive MIMO systems and proposes a novel OTFS based multi-user precoder at the base station (BS) and a corresponding low complexity detector (LCD) at the user terminals (UTs). Owing to the channel hardening effect, the proposed scheme allows for separate demodulation of each DD domain information symbol at the UT. The proposed scheme can achieve a much higher spectral efficiency and a much better error rate performance compared to OFDM based multi-user massive MIMO systems.

R. Bomfin, M. Chafii, A. Nimr, and G. Fettweis, “A Robust Baseband Transceiver Design for Doubly-Dispersive Channels,” IEEE Transactions on Wireless Communications, vol. 20, no. 8, pp. 4781-4796, August 2021.
In this paper, waveform design is considered along with channel estimation based on unique-word (UW), and cyclic-prefix (CP)-free transmission. It is shown that for highly time varying channels, it is advantageous to suppress the CP so that the channel estimation is improved. A framework is developed for choosing the waveform and frame parameters, such as number of subcarriers and subblocks. For the CP-free transmission, it is shown that the waveforms which spread symbols energy in the time-domain per sub-block are the most resilient ones due to equal interference levels of the symbols. As an example of such a waveform is the block-multiplexing (BM) with orthogonal chirp division multiplexing (OCDM). All these findings are supported by thorough mathematical analysis and extensive numerical simulations.

W. Li and J. Preisig, “Estimation of Rapidly Time-Varying Sparse Channels,” IEEE Journal of Oceanic Engineering, vol. 32, no. 4, pp. 927-939, October 2007.
This paper considers the DD domain-based communications in oceanic acoustic environments. Although this paper does not use the specific name OTFS, it is nonetheless a pioneering work in the area of OTFS that utilizes the DD-spread function for time-varying channel representation. The proposed approach can capture both the acoustic channel structure and its dynamics without explicit dynamic channel modelling. Experiments were also conducted to verify the effectiveness of the proposed algorithm.

P. Raviteja, K. T. Phan, and Y. Hong, “Embedded Pilot-aided Channel Estimation for OTFS in Delay-Doppler Channels,” IEEE Transactions on Vehicular Technology, vol. 68, no. 5, pp. 4906–4917, May 2019.
This paper proposes a pilot-aided DD domain channel estimation scheme, where a single pilot impulse with guard symbol around it is inserted in the DD domain. A threshold-based method is employed to estimate the channel with both integer and fractional Doppler shifts. Compared to the case of perfect channel state information, only a marginal performance loss occurs by the proposed channel estimation scheme. The extension of the proposed scheme to MIMO and multi-user cases is also discussed.

W. Shen, L. Dai, J. An, P. Fan, and R. W. Heath, “Channel Estimation for Orthogonal Time Frequency Space (OTFS) Massive MIMO,” IEEE Transactions on Signal Processing, vol. 67, no. 16, pp. 4204–4217, July 2019.
This paper transforms the time-variant massive MIMO channels into the delay-Doppler-angle three-dimensional (3D) channel in OTFS massive MIMO systems and reveals the structured sparsity of the 3D channel. Then, this paper proposes a structured orthogonal matching pursuit algorithm-based channel estimation scheme for massive MIMO-OTFS systems by exploiting the normal sparsity along the delay dimension, block sparsity along the Doppler dimension, and burst sparsity along the angle dimension. A superior channel estimation performance can be achieved by the proposed scheme.

F. Liu, Z. Yuan, Q. Guo, Z. Wang, and P. Sun, “Message Passing based Structured Sparse Signal Recovery for Estimation of OTFS Channels with Fractional Doppler Shifts,” IEEE Transactions on Wireless Communications, vol. 20, no. 12, pp. 7773-7785, December 2021.
This paper proposes a parametric DD domain channel estimation model, which is further formulated as a structured signal recovery problem. With a factor graph representation of the problem, a message passing algorithm is developed to estimate the channel gains and fractional Doppler shifts jointly. The Cramer-Rao lower bound (CRLB) for the estimation is derived to verify the performance of the proposed algorithm. This paper shows that the proposed algorithm significantly outperforms the existing algorithms, and it is able to work with multiple pilot symbols to achieve significant PAPR reduction.

H. B. Mishra, P. Singh, A. K. Prasad, and R. Budhiraja, “OTFS Channel Estimation and Data Detection Designs with Superimposed Pilots,” IEEE Transactions on Wireless Communications, vol. 21, no. 4, pp. 2258-2274, April 2022.
This paper considers a superimposed pilot-based channel estimation and data detection framework for OTFS systems. Both pilot and data symbols are superimposed in all the delay-Doppler (DD) bins in an OTFS frame. The superimposed pilot (SP) idea is attractive because it avoids throughout loss due to pilots at the cost of estimation/detection complexity. The paper proposes two SP-aided channel estimation and data detection designs: SP-noniterative (SP-NI) and SP-iterative (SP-I). The SP-NI design exploits OTFS channel sparsity by performing minimum mean square error (MMSE) channel estimation in the DD domain. It then detects data using a computationally efficient message passing algorithm, which again exploits the DD domain channel sparsity. Good performance of the proposed framework is reported.

S. Li, W. Yuan, Z. Wei, and J. Yuan, “Cross Domain Iterative Detection for Orthogonal Time Frequency Space Modulation,” IEEE Transactions on Wireless Communications, vol. 21, no. 4, pp. 2227–2242, April 2022.
This paper introduces a cross domain iterative detection for OTFS modulation, which is suitable for both the integer and fractional Doppler cases. The key innovation of this detection is that basic estimation/detection approaches are applied to both the time domain and DD domain and the extrinsic information from two domains are iteratively updated according to the unitary transformation. Error performance is analyzed based on the derived state evolution, which shows that this detection can approach the performance of the optimal MAP detection.

P. Raviteja, K. T. Phan, Y. Hong, and E. Viterbo, “Interference Cancellation and Iterative Detection for Orthogonal Time Frequency Space Modulation,” IEEE Transactions on Wireless Communications, vol. 17, no. 10, pp. 6501–6515, October 2018.
This paper presents the massage passing algorithm for OTFS modulation. After a careful study on the input-output relationship in different domains, concise DD domain system models are presented with both the rectangular and ideal shaping pulses. Then, the message passing algorithm is introduced, which applies the Gaussian approximation to the intersymbol interference (ISI) in order to reduce the detection complexity. Extensive simulations are presented, which verify the effectiveness of the proposed detection for OTFS systems.

G. D. Surabhi and A. Chockalingam, “Low-Complexity Linear Equalization for OTFS Modulation,” IEEE Communications Letters, vol. 24, no. 2, pp. 330-334, February 2020.
This paper investigates the low-complexity realizations of both (ZF) zero-forcing and minimum mean square error (MMSE) detections for OTFS systems. After studying the properties of the DD domain effective channel matrix, simplified implementations for both ZF and MMSE are derived based on the eigenvalue decomposition. Compared to the conventional implementation of ZF and MMSE, the proposed approach only requires a logarithmically increased complexity without admitting any performance degradation.

W. Yuan, Z. Wei, J. Yuan, and D. W. K. Ng, “A Simple Variational Bayes Detector for Orthogonal Time Frequency Space (OTFS) Modulation,” IEEE Transactions on Vehicular Technology, vol. 69, no. 7, pp. 7976–7980, July 2020.
This paper considers a more general variational Bayes framework for designing detector for OTFS modulation. This paper approximates the joint distribution of OTFS data symbols by a much simpler one in the sense of minimum relative entropy. Through variational calculus, the marginal distribution of a single OTFS symbol is derived. The benefits of the proposed variational Bayes detector include low-complexity implementation and convergence-guaranteed property.

Z. Zhou, L. Liu, J. Xu, and R. Calderbank, “Learning to Equalize OTFS,” to appear in IEEE Transactions on Wireless Communications, 2022.
This paper considers a neural network-based supervised learning framework for OTFS equalization. Utilizing reservoir computing, a special recurrent neural network, the resulting one-shot online learning is sufficiently flexible to cope with channel variations among different OTFS frames. The main benefit is that the proposed scheme does not rely on the knowledge of explicit channel state information. Moreover, a better tradeoff between the processing complexity and the equalization performance is achieved for OTFS modulation compared to its neural network-based counterparts for OFDM.

P. Singh, A. Gupta, H. B. Mishra, and R. Budhiraja, “Low-Complexity ZF/MMSE MIMO-OTFS Receivers for High-Speed Vehicular Communication,” IEEE Open Journal of the Communications Society, vol. 3, pp. 209-227, 2022.
In this paper, low-complexity ZF and MMSE detectors are considered for OTFS-MIMO systems. Expressions are derived for the signal-to-interference-and-noise-ratio (SINR) for the imperfect CSI scenario. Low-complexity non-linear extensions to these detectors is also considered and are shown to achieve better bit error rate performance than message passing-based OTFS detection.

V. Khammammetti and S. K. Mohammed, “OTFS-Based Multiple-Access in High Doppler and Delay Spread Wireless Channels,” IEEE Wireless Communications Letters, vol. 8, no. 2, pp. 528-531, April 2019.
This paper proposes a novel uplink multiple-access method for multi-user OTFS systems. The proposed method achieves multi-user interference free communication without the need for guard bands. Guard band-based multiple-access methods use guard bands to reduce interference between information transmitted by different users in the DD domain and therefore suffer from loss in spectral efficiency as no information is transmitted in the guard bands. Simulations confirm the better performance of the proposed method when compared to guard band-based methods. In the proposed method, the DD domain resource elements allocated to each user are spaced apart at regular intervals in both the delay and the Doppler domain. The corresponding TF domain signal can be restricted to a subset of all available time-frequency resource elements. Allocation of non-overlapping resource elements in the DD and the TF domains ensures minimal multi-user interference.

Z. Ding, R. Schober, P. Fan, and H. Vincent Poor, “OTFS-NOMA: An Efficient Approach for Exploiting Heterogenous User Mobility Profiles,” IEEE Transactions on Communications, vol. 67, no. 11, pp. 7950-7965, November 2019.
Compared to orthogonal multiple-access scheme where users are allocated distinct non-overlapping resources, this paper considers a non-orthogonal multiple-access (NOMA) method where a high mobility user uses OTFS modulation for communication whereas low-mobility users use OFDM on the same physical resource. Successive cancellation decoding is used to decode the information of all users at the base station. Simulation results show that OTFS-NOMA can achieve enhanced spectral efficiency in scenarios with heterogeneous user mobility profiles.

M. Li, S. Zhang, F. Gao, P. Fan, and O. A. Dobre, “A New Path Division Multiple Access for the Massive MIMO-OTFS Networks,” IEEE Journal on Selected Areas in Communications, vol. 39, no. 4, pp. 903-918, April 2021.
In this paper, a novel OTFS based multi-user massive MIMO system has been proposed where users are allocated non-overlapping resource elements in the three-dimensional (3D) angle-delay-Doppler (ADD) domain unlike previous resource allocation schemes done in the two-dimensional DD and TF domain. Resource allocation in the 3D ADD allows for more separation between users when compared to 2D allocation and therefore low-complexity MRC in the 3D ADD domain achieves good performance.

L. Gaudio, M. Kobayashi, G. Caire, and G. Colavolpe, “On the Effectiveness of OTFS for Joint Radar Parameter Estimation and Communication,” IEEE Transactions on Wireless Communications, vol. 19, no. 9, pp. 5951–5965, September 2020.
This paper is the first study in the literature to analyze the OTFS-based sensing and communication performances. Maximum likelihood estimator and Cramér-Rao lower bound for range and velocity estimation are derived. For the communication receiver, the canonical sum-product algorithm rules relying on the factor graph framework is adopted. The results show that for the dual-functional sensing and communications use cases, OTFS signaling can provide as accurate parameter estimation as the dedicated radar waveforms, e.g., FMCW. Moreover, OTFS performs better than OFDM in terms of channel capacity and overhead cost.

W. Yuan, Z. Wei, S. Li, J. Yuan, and D. W. K. Ng, “Integrated Sensing and Communication-assisted Orthogonal Time Frequency Space Transmission for Vehicular Networks,” IEEE Journal of Selected Topics in Signal Processing, vol. 15, no. 6, pp. 1515–1528, December 2021.
This paper presents a framework of OTFS-based integrated sensing and communication (ISAC) system in vehicular networks. By adopting the slot time-variation of OTFS channel, the delay and Doppler shifts obtained from the sensing echoes can be used for inferring the communications channel, which is also characterized by delays and Doppler shifts. Consequently, a new data frame structure is designed which can make full use of the delay/Doppler grids and the overhead for channel estimation is reduced. An iterative receiver is proposed to recursively update the channel information and to detect data symbols. Simulation results show that the sensing information can enhance the performance of OTFS communications. This paper serves as a good reference for the researches in unifying communication and sensing in the same delay Doppler domain.

K. Wu, J. A. Zhang, X. Huang, and Y. J. Guo, “OTFS-based Joint Communication and Sensing for Future Industrial IoT,” to appear in IEEE Internet of Things Journal, 2022.
This paper is the first work considering OTFS-based ISAC system in industrial Internet-of-Things (IIoT). Using a series of waveform pre-processing, the impact of communication data symbols is removed in the time-frequency domain. A high-accuracy and off-grid method is then proposed for estimating range and velocity, which experiences a very low complexity dominated by two-dimensional DFT. Further SINR analysis is derived for the proposed algorithm. Extensive simulations have validated the near-optimal sensing performance and high energy efficiency of OTFS in IIoT applications.

S. Li, W. Yuan, C. Liu, Z. Wei, J. Yuan, B. Bai, and D. W. K. Ng, “A Novel ISAC Transmission Framework based on Spatially-spread Orthogonal Time Frequency Space Modulation,” IEEE Journal on Selected Areas in Communications, vol. 40, no. 6, pp. 1854-1872, June 2022.
In this paper, a spatial spread OTFS (SS-OTFS)-based ISAC framework is developed. With spatial spreading/de-spreading enabled discretized angular domain, insightful effective models for communications and radar sensing can be derived. In particular, three use cases, i.e., beam tracking, power allocation, and angle estimation are studied with a special structure of sensing matrix. The most important conclusion in this paper is that the power allocation should be designed leaning towards sensing in practical SS-OTFS-based ISAC system.

T. Thaj and E. Viterbo, “OTFS Modem SDR Implementation and Experimental Study of Receiver Impairment Effects,” in Proc. IEEE International Conference on Communications Workshops (ICC Workshops), Shanghai, China, May 2019.
A software defined radio (SDR) Design and Implementation of OTFS modem is presented in this paper. The effects of DC offset and carrier frequency offset on the receiver performance from real time experiments conducted on the implemented OTFS modem in a real indoor wireless channel. Performance of OTFS modulation is compared to that of OFDM modulation using the same hardware setup and environment, which shows the superior performance of OTFS in doubly selective channels. The testbed is useful to study and develop a variety of OTFS receiver algorithms.

T. Blazek and D. Radovic, “Performance Evaluation of OTFS over Measured V2V Channels at 60 GHz,” in Proc. IEEE MTT-S International Conference on Microwaves for Intelligent Mobility (ICMIM), Linz, Austria, November 2020.
This paper presents an analysis of the Orthogonal Time Frequency Space (OTFS) modulation scheme when applied to realistic vehicular channel situations. Measured millimeter wave vehicular channels and typical physical layer settings are used for performance analysis. The experimental results show that the trade-off between channel conditions that are easy to equalize and channel conditions that allow OFTS to exploit the two-dimensional diversity.

Y. Ma, G. Ma, N. Wang, Z. Zhong, and B. Ai, “OTFS-TSMA for Massive Internet of Things in High-Speed Railway,” IEEE Transactions on Wireless Communications, vol. 21, no. 1, pp. 519-531, January 2022.
This paper jointly designs OTFS and TSMA, and propose OTFS-TSMA for high-speed railway massive internet of things. The principle of OTFS-TSMA transceiver is described, where OTFS and TSMA are improved respectively. Two-dimension cyclic shift of DD domain elements in OTFS is transformed into cyclic shift of Doppler elements, segments, symbols and chips by the proposed resource allocation and interleaving schemes. This paper also summarizes several potential researches of OTFS-TSMA related to high-speed railway.

W. Kozek and A. F. Molisch, “Non-orthogonal Pulse Shapes for Multicarrier Communications in Doubly Dispersive Channels,” IEEE Journal on Selected Areas in Communications, vol. 16, no. 8, pp. 1579-1589, October 1998.
This paper introduces a fundamental analysis of time-frequency modulation using a set of pulses generated by time and frequency-shifted versions of a prototype pulse, which is known as Weyl–Heisenberg (WH) system. The goal is to design optimal synthesis and analysis prototype pulses that minimize inter-symbol and intercarrier interference, assuming a priori knowledge on the channel DD spread function. The paper reviews the conditions of the size of the time-frequency lattice TF and its relation to pulse localization and recovery. Accordingly, it is shown that the optimal solution follows incomplete Riesz bases under-critical grid (TF sufficiently larger than 1). Such bases are well-localized in the frequency domain and enable more robustness in doubly-dispersive channel. The work is extended by several appendixes that provide detailed mathematical analysis as well as an introduction to the digital implementation of the corresponding modulation system.

N. Michailow, M. Matthe, I. S. Gaspar, A. N. Caldevilla, L. C. Mendes, A. Festag, and G. Fettweis, “Generalized Frequency Division Multiplexing for 5th Generation Cellular Networks,” IEEE Transactions on Communications, vol. 62, no. 9, pp. 3045-3061, September 2014.
This paper presents a comprehensive overview on Generalized Frequency Division Multiplexing (GFDM). GFDM is a flexible multicarrier modulation designed by circular time-frequency shifting of a prototype pulse. It includes multiple degrees of freedom, allowing different waveforms to be implemented in a single framework, which can be reconfigured to fit different use case requirements. This work covers different aspects, including a consistent mathematical representation of the GFDM modulator and the derivation of different types of demodulators. Moreover, analytical analyses of symbol error performance in different channel models as well as the bit error rate performance for coded GFDM transmission with linear and iterative receivers are performed. The paper also introduces space-time coding techniques for MIMO-GFDM. In addition, concrete GFDM settings are suggested for different type of services.  Finally, a prototype implementation on hardware platforms is presented.

X. Ouyang and J. Zhao, “Orthogonal Chirp Division Multiplexing,” IEEE Transactions on Communications, vol. 64, no. 9, pp. 3946-3957, September 2016.
This paper presents a comprehensive formulation of the Orthogonal Chirp Division Multiplexing (OCDM) waveform described using the discrete Fresnel transform (DFnT), which can be seen as the DFT-based OTFS waveform with additional quadratic phases. The main benefit of OCDM is that the pulses spread the data symbols over the time and frequency domains. The paper proposes efficient digital signal processing for the modulation and equalization. The performance of OCDM is evaluated using different types of linear and iterative receivers, and it is shown that the OCDM system can efficiently exploit multipath diversity. Therefore, it outperforms the standard OFDM and it is more resilient to the interference caused by insufficient cyclic prefix guard interval.

A. Nimr, M. Chafii, M. Matthe, and G. Fettweis, “Extended GFDM Framework: OTFS and GFDM Comparison,” in Proc., IEEE Global Communications Conference (GLOBECOM), Abu Dhabi, UAE, December 2018.
In this paper, the conventional Generalized Frequency Division Multiplexing (GFDM) is revised and represented in a time-frequency block matrix structure and reinterpreted as a four-step process. Based on this, a unified architecture in the time and frequency domains is presented and additional degrees of freedom are introduced without any costs to extend the flexibility of GFDM. Using the extended GFDM, the relationship between GFDM and OTFS is represented as a simple permutation of the time-frequency block. Moreover, the paper introduces the potential of the extended GFDM as a unified implementation framework for many waveforms, with the possibility of replacing the involved DFT matrices by other unitary transforms such as Walsh-Hadamard transform. The paper provides analytical formulas for the symbol error rate and demonstrates the effects of symbol spreading in a time-variant channel. 

R. Bomfin, A. Nimr, M. Chafii, and G. Fettweis, “A Robust and Low-Complexity Walsh-Hadamard Modulation for Doubly-Dispersive Channels,” IEEE Communications Letters, vol. 25, no. 3, pp. 897-901, March 2021.
This letter presents a generic linear modulation framework, where the waveform is defined by a matrix with a specific structure. In particular, the paper focuses on the design of modulation that enables robust wireless communications over doubly-dispersive channels by spreading the data symbols in the time and frequency domains. The structure of the modulation matrix to be examined is similar to the OTFS matrix. However, in contrast to OTFS and Walsh-Hadamard transform (WHT) spreading with full matrices, this work introduces sparse spreading that can be controller based on the channel selectivity. Aiming at low-complexity implementation, sparse WHT (SWHT) precoding is proposed, which has similar performance as DFT spreading but requires less computational resources. Furthermore, it is shown that properly chosen sparse spreading is sufficient to obtain the same benefits as full spreading. These results are confirmed and demonstrated through simulations with iterative receivers.

K. P. Arunkumar and C. R. Murthy, “Orthogonal Delay Scale Space Modulation: A New Technique for Wideband Time-Varying Channels,” to appear in IEEE Transactions on Signal Processing, 2022.
This paper extends the concepts of OTFS to the orthogonal delay scale space (ODSS) that is suitable for communications over wideband time-varying channels, such as UWA communications. The ODSS takes advantage of a two-dimensional transformation from the Fourier-Mellin domain to the delay-scale in ODSS, which improves the bit-error-rate. The error performances of ODSS, OTFS, and OFDM over wideband time-varying channels are also numerically compared under low complexity receivers.