{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T02:22:12Z","timestamp":1785550932518,"version":"3.56.0"},"reference-count":31,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,11,22]],"date-time":"2022-11-22T00:00:00Z","timestamp":1669075200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Dutch Technology Foundation TTW"},{"name":"Integrated Cooperative Automated Vehicles (i-CAVE)"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In orthogonal time frequency space (OTFS) modulation, information-carrying symbols reside in the delay-Doppler (DD) domain. By operating in the DD domain, an appealing property for communication arises: time-frequency (TF) dispersive channels encountered in high-mobility environments become time-invariant. OTFS outperforms orthogonal frequency division multiplexing (OFDM) in high-mobility scenarios, making it an ideal waveform candidate for 6G. Generally, OTFS is considered a pre- and postprocessing step for OFDM. However, the so-called Zak transform provides the fundamental relation between the DD and time domain. In this work, we propose an OTFS system based on the discrete Zak transform (DZT). To this end, we discuss the DZT and establish the input\u2013output relation for time-frequency (TF) dispersive channels solely by the properties of the DZT. The presented formulation simplifies the derivation and analysis of the input\u2013output relation of the TF dispersive channel in the DD domain. Based on the presented formulation, we show that operating in the DD incurs no loss in capacity.<\/jats:p>","DOI":"10.3390\/e24121704","type":"journal-article","created":{"date-parts":[[2022,11,22]],"date-time":"2022-11-22T08:25:28Z","timestamp":1669105528000},"page":"1704","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["Orthogonal Time Frequency Space Modulation Based on the Discrete Zak Transform"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6520-2469","authenticated-orcid":false,"given":"Franz","family":"Lampel","sequence":"first","affiliation":[{"name":"Information and Communication Theory Lab, Signal Processing Systems Group, Department of Electrical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7162-0325","authenticated-orcid":false,"given":"Hamdi","family":"Joudeh","sequence":"additional","affiliation":[{"name":"Information and Communication Theory Lab, Signal Processing Systems Group, Department of Electrical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2172-3051","authenticated-orcid":false,"given":"Alex","family":"Alvarado","sequence":"additional","affiliation":[{"name":"Information and Communication Theory Lab, Signal Processing Systems Group, Department of Electrical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7316-3471","authenticated-orcid":false,"given":"Frans M. J.","family":"Willems","sequence":"additional","affiliation":[{"name":"Information and Communication Theory Lab, Signal Processing Systems Group, Department of Electrical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Monk, A., Hadani, R., Tsatsanis, M., and Rakib, S. (2016). OTFS\u2014Orthogonal Time Frequency Space. arXiv.","DOI":"10.1109\/WCNC.2017.7925924"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Hadani, R., Rakib, S., Kons, S., Tsatsanis, M., Monk, A., Ibars, C., Delfeld, J., Hebron, Y., Goldsmith, A.J., and Molisch, A.F. (2018). Orthogonal Time Frequency Space Modulation. arXiv.","DOI":"10.1109\/WCNC.2017.7925924"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Hadani, R., Rakib, S., Tsatsanis, M., Monk, A., Goldsmith, A.J., Molisch, A.F., and Calderbank, R. (2017, January 19\u201322). Orthogonal Time Frequency Space Modulation. Proceedings of the 2017 IEEE Wireless Communications and Networking Conference (WCNC), San Francisco, CA, USA.","DOI":"10.1109\/WCNC.2017.7925924"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Hadani, R., Rakib, S., Molisch, A.F., Ibars, C., Monk, A., Tsatsanis, M., Delfeld, J., Goldsmith, A., and Calderbank, R. (2017, January 4\u20139). Orthogonal Time Frequency Space (OTFS) modulation for millimeter-wave communications systems. Proceedings of the 2017 IEEE MTT-S International Microwave Symposium (IMS), Honololu, HI, USA.","DOI":"10.1109\/MWSYM.2017.8058662"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"6501","DOI":"10.1109\/TWC.2018.2860011","article-title":"Interference Cancellation and Iterative Detection for Orthogonal Time Frequency Space Modulation","volume":"17","author":"Raviteja","year":"2018","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"5951","DOI":"10.1109\/TWC.2020.2998583","article-title":"On the Effectiveness of OTFS for Joint Radar Parameter Estimation and Communication","volume":"19","author":"Gaudio","year":"2020","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Murali, K.R., and Chockalingam, A. (2018, January 11\u201316). On OTFS Modulation for High-Doppler Fading Channels. Proceedings of the 2018 Information Theory and Applications Workshop (ITA), San Diego, CA, USA.","DOI":"10.1109\/ITA.2018.8503182"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"7619","DOI":"10.1109\/TVT.2021.3069913","article-title":"Derivation of OTFS Modulation from First Principles","volume":"70","author":"Mohammed","year":"2021","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1109\/MSP.2013.2269702","article-title":"Time-Frequency Foundations of Communications: Concepts and Tools","volume":"30","author":"Matz","year":"2013","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1109\/78.564174","article-title":"Discrete Zak transforms, polyphase transforms, and applications","volume":"45","author":"Hlawatsch","year":"1997","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1775","DOI":"10.1002\/j.1538-7305.1966.tb02435.x","article-title":"Synthesis of band-limited orthogonal signals for multichannel data transmission","volume":"45","author":"Chang","year":"1966","journal-title":"Bell Syst. Tech. J."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"628","DOI":"10.1109\/TCOM.1971.1090705","article-title":"Data Transmission by Frequency-Division Multiplexing Using the Discrete Fourier Transform","volume":"19","author":"Weinstein","year":"1971","journal-title":"IEEE Trans. Commun. Technol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1109\/LWC.2017.2776942","article-title":"Low Complexity Modem Structure for OFDM-Based Orthogonal Time Frequency Space Modulation","volume":"7","author":"Farhang","year":"2018","journal-title":"IEEE Wirel. Commun. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Richards, M., Scheer, J., and Holm, W. (2010). Principles of Modern Radar, ASciTech Publishing. [1st ed.].","DOI":"10.1049\/SBRA021E"},{"key":"ref_15","unstructured":"Skollnik, M. (2001). Introduction to Radar Systems, McGraw Hill Education. [3rd ed.]."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Bondre, A.S., and Richmond, C.D. (2022, January 21\u201325). Dual-Use of OTFS Architecture for Pulse Doppler Radar Processing. Proceedings of the 2022 IEEE Radar Conference (RadarConf22), New York, NY, USA.","DOI":"10.1109\/RadarConf2248738.2022.9764285"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Lampel, F., Avarado, A., and Willems, F.M. (2022, January 16\u201320). On OTFS using the Discrete Zak Transform. Proceedings of the 2022 IEEE International Conference on Communications Workshops (ICC Workshops), Seoul, Republic of Korea.","DOI":"10.1109\/ICCWorkshops53468.2022.9814589"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1090\/S0002-9939-1984-0749901-6","article-title":"Radar ambiguity functions, the Heisenberg group, and holomorphic theta series","volume":"92","author":"Schempp","year":"1984","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1385","DOI":"10.1103\/PhysRevLett.19.1385","article-title":"Finite Translation in Solid-State Physics","volume":"19","author":"Zak","year":"1967","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","first-page":"23","article-title":"The Zak Transform: A Signal Transform for Sampled Time-Continuous Signals","volume":"43","author":"Janssen","year":"1988","journal-title":"Philips J. Res."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Vetterli, M., Kova\u010devi\u0107, J., and Goyal, V.K. (2014). Foundations of Signal Processing, Cambridge University Press. [1st ed.].","DOI":"10.1017\/CBO9781139839099"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/S1874-608X(98)80003-4","article-title":"Weyl-Heisenberg Systems and the Finite Zak Transform","volume":"Volume 7","author":"Coifman","year":"1998","journal-title":"Signal and Image Representation in Combined Spaces"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Barry, J., Lee, E., and Messerschmitt, D. (2004). Digital Communication, Springer Science+Business Media. [3rd ed.].","DOI":"10.1007\/978-1-4615-0227-2"},{"key":"ref_24","unstructured":"Hlawatsch, F., and Matz, G. (2011). Wireless Communications Over Rapidly Time-Varying Channels, Academic Press. [1st ed.]."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/0165-1684(90)90087-F","article-title":"Discrete Gabor expansions","volume":"21","author":"Wexler","year":"1990","journal-title":"Signal Process."},{"key":"ref_26","unstructured":"Molisch, A. (2011). Wireless Communications, Wiley. [2nd ed.]."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Goldsmith, A. (2005). Wireless Communications, Cambridge University Press. [1st ed.].","DOI":"10.1017\/CBO9780511841224"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"St\u00fcber, G.L. (2017). Principles of Mobile Communications, Springer. [4th ed.].","DOI":"10.1007\/978-3-319-55615-4"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Han, T.S. (2003). Information-Spectrum Methods in Information Theory, Springer. [1st ed.].","DOI":"10.1007\/978-3-662-12066-8"},{"key":"ref_30","first-page":"323","article-title":"General formulation of Shannon\u2019s main theorem in information theory","volume":"33","author":"Dobrushin","year":"1963","journal-title":"Am. Math. Sot. Trans."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1002\/ett.4460100604","article-title":"Capacity of multi-antenna Gaussian channels","volume":"10","author":"Telatar","year":"1999","journal-title":"Eur. Trans. Telecomm."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/12\/1704\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:24:06Z","timestamp":1760145846000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/12\/1704"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,22]]},"references-count":31,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["e24121704"],"URL":"https:\/\/doi.org\/10.3390\/e24121704","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,22]]}}}