References & Further Reading
References
- J. Zak, Finite Translations in Solid-State Physics, 1967
The original Zak transform paper. Two pages, solid-state-physics motivated, but the definition and quasi-periodicity are stated clearly.
- A. J. E. M. Janssen, The Zak Transform: a Signal Transform for Sampled Time-Continuous Signals, 1988
Janssen's paper bringing the Zak transform into signal processing. Establishes the connection to Gabor analysis and proves the unitarity. Most accessible treatment for engineers.
- K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, 2001
The definitive mathematical reference for time-frequency analysis, including the Zak transform and Gabor frames. Chapters 8 and 12 are essential.
- H. Bölcskei and F. Hlawatsch, Discrete Zak transforms, polyphase transforms, and applications, 1997
The authoritative reference for the discrete Zak transform. Establishes the unitary property and the link to polyphase decomposition.
- R. Hadani, S. Rakib, M. Tsatsanis, A. Monk, A. J. Goldsmith, A. F. Molisch, R. Calderbank, Orthogonal Time Frequency Space Modulation, 2017
The Zak-OTFS construction is in Section II-B. Brief but rigorous.
- S. K. Mohammed, R. Hadani, A. Chockalingam, G. Caire, OTFS — A Mathematical Foundation for Communication and Radar Sensing in the Delay-Doppler Domain, 2022
A comprehensive tutorial on Zak-OTFS that unifies the DD-plane perspective for both communications and sensing. Highly recommended for a second pass.
- C. Heil and D. F. Walnut, Fundamental Papers in Wavelet Theory, Princeton University Press, 2011
Includes an English translation of Zak's 1967 paper with annotations, useful for historical context.
- D. Gabor, Theory of Communication, 1946
Gabor's foundational paper proposing the Gabor expansion — the original 'signal on a TF lattice' framework that OTFS now operationalizes.
- R. Balian, Un principe d'incertitude fort en théorie du signal ou en mécanique quantique, 1981
The Balian–Low theorem in its original (French) formulation. Shows that no smooth prototype pulse can achieve simultaneous time-frequency localization at critical Gabor density.
- F. Low, Complete Sets of Wave Packets, 1985
Low's independent proof of the same theorem, published four years later. Both are readable short papers.
- G. B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, 1989
Mathematical reference for the Heisenberg-Weyl group and its role in time-frequency analysis. Chapter 1 contains the Zak transform treated as a projective representation.
Further Reading
For readers who want a deeper mathematical or historical perspective on the Zak transform.
Historical development of the Zak transform in physics
Zak, 'Magnetic translation group' (Phys. Rev. 1964) and 'Dynamics of electrons in solids in external fields' (Phys. Rev. 1968)
The papers where Zak developed the transform for his own physics problem. Useful for seeing the problem the transform was invented to solve.
Rigorous Gabor analysis and frame theory
Daubechies, *Ten Lectures on Wavelets* (1992); Casazza & Kutyniok (eds.), *Finite Frames* (2013)
Daubechies provides a gentle introduction; Casazza–Kutyniok covers the finite-dimensional case closer to the DZT used in OTFS.
Discrete Zak transform in communications
Mohammed & Caire, 'DZT-OTFS: A Natural Zak-Transform-Based Design' (2023)
A direct application of the DZT to OTFS transceiver design, building on Bölcskei & Hlawatsch (1997).
Pulse shaping for OTFS
Raviteja, Phan, Hong (2019), 'Practical pulse-shaping waveforms for reduced-cyclic-prefix OTFS'; OTFS Chapter 20 in this book
How to choose a realistic (non-rectangular) prototype pulse that still has good Zak-domain properties.
Quantum mechanics and OTFS: the deeper connection
Folland (1989) Ch. 1; Grossmann, 'Parity operator and quantization of $\delta$-functions' (Commun. Math. Phys. 1976)
The Heisenberg-Weyl structure of the DD plane is the same structure that underlies quantization in quantum mechanics. For readers with physics background, the analogy illuminates the uniqueness of the DD domain.