{"ID":2831088,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.08352","arxiv_id":"2512.08352","title":"On Discrete Ambiguity Functions of Random Communication Waveforms","abstract":"This paper provides a fundamental characterization of the discrete ambiguity functions (AFs) of random communication waveforms under arbitrary orthonormal modulation with random constellation symbols, which serve as a key metric for evaluating the delay-Doppler sensing performance in future ISAC applications. A unified analytical framework is developed for two types of AFs, namely the discrete periodic AF (DP-AF) and the fast-slow time AF (FST-AF), where the latter may be seen as a small-Doppler approximation of the DP-AF. By analyzing the expectation of squared AFs, we derive exact closed-form expressions for both the expected sidelobe level (ESL) and the expected integrated sidelobe level (EISL) under the DP-AF and FST-AF formulations. For the DP-AF, we prove that the normalized EISL is identical for all orthogonal waveforms. To gain structural insights, we introduce a matrix representation based on the finite Weyl-Heisenberg (WH) group, where each delay-Doppler shift corresponds to a WH operator acting on the ISAC signal. This WH-group viewpoint yields sharp geometric constraints on the lowest sidelobes: The minimum ESL can only occur along a one-dimensional cut or over a set of widely dispersed delay-Doppler bins. Consequently, no waveform can attain the minimum ESL over any compact two-dimensional region, leading to a no-optimality (no-go) result under the DP-AF framework. For the FST-AF, the closed-form ESL and EISL expressions reveal a constellation-dependent regime governed by its kurtosis: The OFDM modulation achieves the minimum ESL for sub-Gaussian constellations, whereas the OTFS waveform becomes optimal for super-Gaussian constellations. Finally, four representative waveforms, namely, SC, OFDM, OTFS, and AFDM, are examined under both frameworks, and all theoretical results are verified through numerical examples.","short_abstract":"This paper provides a fundamental characterization of the discrete ambiguity functions (AFs) of random communication waveforms under arbitrary orthonormal modulation with random constellation symbols, which serve as a key metric for evaluating the delay-Doppler sensing performance in future ISAC applications. A unified...","url_abs":"https://arxiv.org/abs/2512.08352","url_pdf":"https://arxiv.org/pdf/2512.08352v2","authors":"[\"Ying Zhang\",\"Fan Liu\",\"Yifeng Xiong\",\"Weijie Yuan\",\"Shuangyang Li\",\"Le Zheng\",\"Tony Xiao Han\",\"Christos Masouros\",\"Shi Jin\"]","published":"2025-12-09T08:20:45Z","proceeding":"cs.IT","tasks":"[\"cs.IT\",\"eess.SP\"]","methods":"[]","has_code":false}