References & Further Reading

References

  1. R. Hadani, S. Rakib, M. Tsatsanis, A. Monk, A. J. Goldsmith, A. F. Molisch, R. Calderbank, Orthogonal Time Frequency Space Modulation, 2017

    Section III defines the ISFFT and SFFT as we use them here. Essential primary reference.

  2. P. Raviteja, K. T. Phan, Y. Hong, E. Viterbo, Interference Cancellation and Iterative Detection for Orthogonal Time Frequency Space Modulation, 2018

    The standard detection-theory reference. Section III carefully derives the ISFFT/SFFT signs used by most of the OTFS literature today.

  3. G. B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, 1989

    Rigorous mathematical treatment of the symplectic Fourier transform, the Heisenberg-Weyl group, and their role in harmonic analysis. Chapter 1 is the core reference.

  4. K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, 2001

    Chapter 12 covers the symplectic Fourier transform in the context of time-frequency analysis. Complements Folland with a more signal-processing-friendly style.

  5. S. K. Mohammed, R. Hadani, A. Chockalingam, G. Caire, OTFS — A Mathematical Foundation for Communication and Radar Sensing in the Delay-Doppler Domain, 2022

    The Zak-OTFS formulation, which avoids the ISFFT/OFDM two-step construction entirely and works directly with the discrete Zak transform. Highly recommended for a second pass on OTFS mathematics.

  6. G. D. Surabhi, M. K. Ramachandran, A. Chockalingam, Peak-to-Average Power Ratio of OTFS Modulation, 2019

    Quantifies OTFS PAPR and compares with OFDM. Establishes that OTFS PAPR is approximately the same as OFDM for Gaussian-like data.

  7. W. Yuan, R. Schober, G. Caire, Orthogonal Time Frequency Space (OTFS) Modulation — Part III: ISAC and Potential Applications, 2024

    Part III of a three-part tutorial. Parts I and II (Yuan et al., 2023) cover the basic modulation and detection. Part III's ISAC perspective anchors the sensing chapters later in this book.

  8. M. K. Ramachandran, A. Chockalingam, MIMO-OTFS in High-Doppler Fading Channels: Signal Detection and Channel Estimation, 2018

    Extends OTFS to MIMO and derives the corresponding SFFT/ISFFT variants for multi-antenna systems. Relevant for Chapter 16 of this book.

  9. G. D. Surabhi, R. Madhava Augustine, A. Chockalingam, On the Diversity of Uncoded OTFS Modulation in Doubly-Dispersive Channels, 2019

    Proves that OTFS achieves full DD diversity. Used in Chapter 9 of this book. The SFT's spreading property is central to their proof.

  10. W. Zhu, M. Zhao, S. Li, Z. Wei, Fractional Doppler Effects in OTFS Modulation: An Equivalent Channel Representation, 2022

    Relevant for Chapter 10: treats fractional Doppler as SFT sidelobe leakage. Useful cross-reference.

Further Reading

For readers who want to dig deeper into the mathematical structure of the symplectic Fourier transform or see its application in other domains.

  • Symplectic geometry and phase-space physics

    Folland, *Harmonic Analysis in Phase Space* (1989); de Gosson, *Symplectic Methods in Harmonic Analysis* (2011)

    De Gosson extends Folland's framework to the setting relevant for quantum mechanics and signal processing, with explicit discussion of the metaplectic representation — the natural home of linear canonical transforms.

  • Efficient 2D FFT and OFDM implementations

    Oppenheim & Schafer, *Discrete-Time Signal Processing* (3rd ed., 2010), Ch. 10; Lin & Phoong, 'OFDM transmitters: Analog representation and DFT-based implementation' (IEEE Trans. SP, 2003)

    For readers implementing OTFS on real hardware: the 2D FFT structure and its overlap with OFDM silicon are developed in detail.

  • Zak-OTFS vs Hadani-Rakib OTFS

    Mohammed, Hadani, Chockalingam, Caire (2022) — sections V–VII

    Clarifies when the two formulations give the same result and when they differ (at the pulse-shaping level). A mathematical companion to Chapter 6.

  • Linear canonical transforms and OTFS generalizations

    Mohammed et al., 'Linear Canonical Transform OTFS' (IEEE TWC 2024)

    Shows how LCT-OTFS generalizes the SFT to other symplectic rotations — an extension particularly relevant for channels with structured Doppler (not just shift).

  • Rigorous proof of the ISFFT/SFFT unitarity

    Gröchenig (2001), Ch. 12, Theorem 12.1.1

    A self-contained proof using the general theory of tensor-product bases on $\mathbb{C}^{MN}$.

ExercisesCh 4 →