References & Further Reading
References
- R. Hadani, S. Rakib, M. Tsatsanis, A. Monk, A. J. Goldsmith, A. F. Molisch, R. Calderbank, Orthogonal Time Frequency Space Modulation, 2017
The continuous-time DD input-output relation is in §II-III. Contains the fundamental statement of the 2D convolution form.
- P. Raviteja, K. T. Phan, Y. Hong, E. Viterbo, Interference Cancellation and Iterative Detection for Orthogonal Time Frequency Space Modulation, 2018
The discrete-form derivation we follow most closely. §III contains the explicit DD channel matrix structure.
- S. K. Mohammed, R. Hadani, A. Chockalingam, G. Caire, OTFS — A Mathematical Foundation for Communication and Radar Sensing in the Delay-Doppler Domain, 2022
Derives the DD input-output relation from the Zak transform directly, with the crystalline-symmetry viewpoint used in §4.
- P. A. Bello, Characterization of Randomly Time-Variant Linear Channels, 1963
The four-function diagram on which this chapter's continuous-time derivation rests.
- P. Raviteja, Y. Hong, E. Viterbo, E. Biglieri, Effective Diversity of OTFS Modulation, 2019
Establishes the diversity-order result that follows from the sparsity of §5. The effective diversity equals the number of distinct $(\ell_i, k_i)$ pairs.
- J. G. Proakis and M. Salehi, Digital Communications, McGraw-Hill, 5th ed., 2007
Ch. 14 covers WSSUS channels and the classical block-fading assumption referenced in §4.
- T. Zemen and C. F. Mecklenbrauker, Time-Variant Channel Estimation Using Discrete Prolate Spheroidal Sequences, 2005
A basis-expansion approach for extending beyond block fading. Relevant to exercise 15 and to Chapter 10 of this book.
- T. Strohmer and S. Beaver, Optimal OFDM Design for Time-Frequency Dispersive Channels, 2003
Early work on pulse shaping for TF-dispersive channels. Provides the mathematical background for the DD-convolution structure.
- D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005
Ch. 2 and 3 cover the classical fading-channel formulation that OTFS refactors into the DD domain.
- L. Liu et al., The COST 2100 MIMO channel model, 2012
Empirical path counts and parameters referenced in the sparsity discussion of §5. Confirms $P \leq 20$ for realistic terrestrial channels.
- 3GPP, 3GPP TR 38.901: Study on channel model for frequencies from 0.5 to 100 GHz, 2023
The standardized channel model for 5G/6G. Section 7 specifies discrete-path counts, delay-Doppler parameters for standardized deployment scenarios.
- S. Yan, L. Sanguinetti, G. Caire, Channel Estimation for OTFS in Doubly-Selective Channels with Off-Grid Paths, 2024
Extends the discrete DD channel model of this chapter to handle off-grid (fractional) paths. Relevant to Chapter 10.
Further Reading
Additional resources for the DD channel model.
Historical: linear time-varying system theory
Kailath, 'Measurements on time-variant communication channels' (IRE Trans. Inf. Theory, 1962); Zadeh, 'Frequency analysis of variable networks' (Proc. IRE, 1950)
The prehistory of what became Bello's 1963 synthesis. Kailath specifically anticipates the delay-Doppler picture.
Advanced: the Heisenberg-Weyl group and OTFS
Folland, *Harmonic Analysis in Phase Space* (1989), Ch. 1
The group-theoretic foundation of DD-plane translations. The Zak transform is a concrete realization of the Heisenberg-Weyl representation.
Implementation: FFT-accelerated DD convolution
Van Loan, *Computational Frameworks for the Fast Fourier Transform* (1992), Ch. 4
Standard reference for efficient Kronecker-FFT implementations relevant to the DD channel matrix of §3.
Measurement: empirical DD channels
COST 2100 Final Report (2012); Rappaport et al., *Wireless Communications: Principles and Practice* (2nd ed., 2002), Ch. 4-5
Empirical validation of the $P \leq 20$ sparsity assumption that underlies this chapter's theorems.
OTFS vs OFDM from a DD perspective
Yuan, Wei, Schober, 'Orthogonal Time Frequency Space — Part I' (IEEE Comm. Mag. 2023)
Part I of the three-part tutorial. Provides a readable re-derivation of the DD-convolution form that we prove rigorously here.