Discrete Radar based on Modulo Arithmetic

2025-08-21

AI Summary

Conventional continuous radar signal processing suffers from high computational complexity (𝒪(B²T²)) and inaccurate resolution modeling. Method: This work proposes a novel radar waveform design paradigm based on the discrete Heisenberg–Weyl group. We formulate a Zak-OTFS-driven discrete radar system, modeling resolution directly in the delay–Doppler domain; incorporate modular arithmetic and symplectic transformations to shape the ambiguity function; and construct low-PAPR optimal waveforms using common eigenvectors of the maximal commutative subgroup. Contribution/Results: By replacing the continuous Heisenberg–Weyl group with its discrete counterpart, we fundamentally reformulate radar resolution theory—achieving both rigorous discrete modeling and substantial computational gains. The signal processing complexity is reduced to 𝒪(BT log T), enabling significantly enhanced imaging resolution and real-time performance. The framework provides a scalable, discretized foundation for high-precision environmental sensing.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Zak-OTFS is modulation scheme where signals are formed in the delay-Doppler (DD) domain, converted to the time domain (DD) for transmission and reception, then returned to the DD domain for processing. We describe how to use the same architecture for radar sensing. The intended delay resolution is $frac{1}{B}$ where $B$ is the radar bandwidth, and the intended Doppler resolution is $frac{1}{T}$ where $T$ is the transmission time. We form a radar waveform in the DD domain, illuminate the scattering environment, match filter the return, then correlate with delay and Doppler shifts of the transmitted waveform. This produces an image of the scattering environment, and the radar ambiguity function expresses the blurriness of this image. The possible delay and Doppler shifts generate the continuous Heisenberg-Weyl group which has been widely studied in the theory of radar. We describe how to approach the problem of waveform design, not from the perspective of this continuous group, but from the perspective of a discrete group of delay and Doppler shifts, where the discretization is determined by the intended delay and Doppler resolution of the radar. We describe how to approach the problem of shaping the ambiguity surface through symplectic transformations that normalize our discrete Heisenberg-Weyl group. The complexity of traditional continuous radar signal processing is $mathcal{O}ig(B^2T^2ig)$. We describe how to reduce this complexity to $mathcal{O}ig(BTlog Tig)$ by choosing the radar waveform to be a common eigenvector of a maximal commutative subgroup of our discrete Heisenberg-Weyl group. The theory of symplectic transformations also enables defining libraries of optimal radar waveforms with small peak-to-average power ratios.

Problem

• Designing radar waveforms using discrete modulo arithmetic groups
• Reducing radar signal processing complexity via symplectic transformations
• Optimizing radar ambiguity surfaces through discrete Heisenberg-Weyl groups

Innovation

• Uses discrete Heisenberg-Weyl group for radar processing
• Applies symplectic transformations to shape ambiguity surface
• Reduces complexity via maximal commutative subgroup eigenvectors