Sensing Fast and Distant Targets in Overspread Channels with Zak-OTFS and Modular Arithmetic

Delay-Doppler (DD) modulations, such as Zak-transform-based orthogonal time frequency space (Zak-OTFS) modulation, represent the channel as a sparse set of delay and Doppler shifts, which makes them well suited for radar sensing. Sensing with a single DD frame, however, is unambiguous only when the channel spread factor, i.e., the product of the delay and Doppler spreads, is below one. Long-range mobility scenarios, including drones, airplanes, low Earth orbit (LEO) satellites, and underwater acoustic links, typically violate this crystallization condition, so that a single frame aliases the target delays, Doppler shifts, or both. Existing methods add a time-frequency pilot to a conventional DD frame and resolve the aliasing in only one dimension. We propose a frame design that combines multiple DD bases with different delay and Doppler periods, which together cover an overspread channel, on the same bandwidth and frame size. For two frames, we show that modular arithmetic on the wrapped peak positions uniquely recovers each target by pairing the peaks through their common projection and their gains, and we derive the condition under which this pairing is ambiguous. We implement the design with two superimposed Zak-OTFS frames and propose two receivers: dual-frame sensing (DS), which performs the joint detection directly, and DS with cross-pilot cancellation (DS-XC), which additionally cancels the cross-pilot interference, re-estimates the gains by joint least squares, and removes ghost targets. Both receivers keep a detection rate close to one as the spread factor grows, and DS-XC also achieves the lowest false-alarm rate and channel normalized mean squared error (NMSE). In a 100 km drone scenario, DS-XC detects 98% of the targets, compared with 81% for the best baseline, with a complexity that grows near-linearly with the time-bandwidth product for a fixed number of targets.