Unified Framework and Comparative Analysis of Delay-Doppler Communication Systems

Delay-Doppler (DD) system translates a time-varying channel into quasi-static time invariant channel by modulating the information in DD domain. This serves as a core principle behind orthogonal time frequency space (OTFS) and orthogonal delay Doppler multiplexing (ODDM), the two prominent DD systems. A comparison of inverse symplectic finite Fourier transform (ISFFT-OTFS) and Zak-OTFS is already presented in the DD literature. But such a comparison between Zak-OTFS and ODDM remained as a research gap. In this paper, we proposed a unified framework for modeling and analyzing of any DD system. Specifically, we investigate five representative schemes: (ISFFT-OTFS), continuous Zak transform-based OTFS (CZT-OTFS), discrete Zak transform-based OTFS (DZT-OTFS), continuous ODDM (C-ODDM) and discrete ODDM (D-ODDM). A generalized approach is introduced to construct time-domain basis functions from sub-pulse trains for any DD system. Using this, we formulated two such structures namely the constant phase pulsone (CPP) and varying phase pulsone (VPP). Later we mathematically showed that CPP and VPP captures the Zak processing techniques and DD domain multicarrier processing respectively. Using the definitions of CPP and VPP we showed that D-ODDM and DZT-OTFS are equivalent and phase sampled C-ODDM is CZT-OTFS. We also concluded that as the number of delay bins <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> increases, the performance gap between C-ODDM and CZT-OTFS decreases. Then we analyzed the interference of CPP and VPP by examining the fraction of minimally interfered symbols as a function of normalized Doppler shift. Results indicate that VPP experiences significantly lower interference than CPP. We also establish structural relationships among DD systems by comparing their time-domain basis functions and DD representations. Moreover, we evaluate the diversity gain of each system variant under a doubly dispersive channel. Among all DD schemes, C-ODDM exhibits superior diversity and error performance due to effective decoupling of delay and Doppler. These observations are validated through both analytical derivations and numerical simulations.