On Discrete Ambiguity Functions of Random Communication Waveforms
Ying Zhang, Fan Liu, Yifeng Xiong, Weijie Yuan, Shuangyang Li, Le Zheng, Tony Xiao Han, Christos Masouros, Shi Jin
TL;DR
This work analyzes the ambiguity properties of random communication waveforms for joint sensing and communication (ISAC) by developing two frameworks: the discrete periodic ambiguity function (DP-AF) and the fast-slow-time ambiguity function (FST-AF). It derives closed-form expressions for the expected sidelobe levels (ESL) and the expected integrated sidelobe levels (EISL), revealing that DP-AF exhibits an invariant normalized EISL across constellations and faces a no-go restriction preventing a compact 2D low-ambiguity region. In contrast, the FST-AF regime shows constellation-dependent optimality, with OFDM optimal for sub-Gaussian and OTFS optimal for super-Gaussian constellations, while both minimize ESL at every delay-Doppler bin. The results are demonstrated across SC, OFDM, OTFS, and AFDM and validated numerically, offering concrete design guidance for ISAC waveform selection under random data modulation.
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.