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.

翻译：本文针对未来ISAC应用中作为时延-多普勒感知性能关键评估指标的随机通信波形，建立了任意正交调制下随机星座符号离散模糊函数（AF）的基础理论表征。研究构建了适用于两类AF的统一分析框架：离散周期AF（DP-AF）与快慢时间AF（FST-AF），其中后者可视为DP-AF的小多普勒近似。通过分析AF平方的期望，我们推导出DP-AF与FST-AF框架下期望旁瓣电平（ESL）与期望积分旁瓣电平（EISL）的精确闭式表达式。对于DP-AF，我们证明所有正交波形的归一化EISL均相同。为获得结构洞察，我们引入基于有限Weyl-Heisenberg（WH）群的矩阵表示，其中每个时延-多普勒偏移对应作用于ISAC信号的WH算子。该WH群视角揭示了最低旁瓣的严格几何约束：最小ESL仅能沿一维截线或在广泛分散的时延-多普勒单元集合上实现。因此，任何波形均无法在紧致二维区域内达到最小ESL，从而在DP-AF框架下导出不存在全域最优解（no-go）的结论。对于FST-AF，闭式ESL与EISL表达式揭示了由峰度主导的星座依赖机制：对于亚高斯星座，OFDM调制可实现最小ESL；而对于超高斯星座，OTFS波形则达到最优。最后，我们在两种框架下检验了四种典型波形（SC、OFDM、OTFS与AFDM），并通过数值算例验证了所有理论结果。