On Discrete Ambiguity Functions of Random Communication Waveforms
🤖 AI Summary
This work addresses the delay-Doppler sensing performance evaluation of stochastic communication waveforms in integrated sensing and communication (ISAC) systems, systematically characterizing the fundamental statistical properties of their discrete ambiguity functions (AFs). A unified analytical framework is proposed, jointly modeling periodic and fast-slow-time AFs; finite Weyl–Heisenberg group matrix representations are introduced to reveal geometric constraints on sidelobe distributions, enabling the first rigorous proof of the intrinsic “non-simultaneous optimality” limitation in the two-dimensional delay-Doppler domain. Closed-form expressions for the expected sidelobe level (ESL) and expected integrated sidelobe level (EISL) are derived, showing that the normalized EISL is a constant for orthogonal waveforms. Crucially, modulation constellation kurtosis is identified as the decisive metric governing the relative optimality of sub-Gaussian (e.g., OFDM) versus super-Gaussian (e.g., OTFS) waveforms. Numerical experiments validate the theoretical performance bounds across SC, OFDM, OTFS, and AFDM waveforms.
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
Characterizes discrete ambiguity functions of random communication waveforms for ISAC sensing
Derives closed-form expressions for expected sidelobe levels in DP-AF and FST-AF frameworks
Identifies optimal waveforms based on constellation kurtosis for minimizing sidelobes
Innovation
Methods, ideas, or system contributions that make the work stand out.
Unified analytical framework for discrete periodic and fast-slow time ambiguity functions
Matrix representation using finite Weyl-Heisenberg group for structural insights
Closed-form expressions linking sidelobe levels to constellation kurtosis for optimization
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