DAFT-s-AFDM Enabled ISAC Systems: Ambiguity Function Analysis and Waveform Design

Discrete affine Fourier transform spread affine frequency division multiplexing (DAFT-s-AFDM) is a promising waveform for integrated sensing and communication (ISAC) due to its low peak-to-average power ratio, robustness to Doppler shifts, and reduced multiuser interference in the uplink transmission. This paper presents a comprehensive ambiguity function (AF) analysis of DAFT-s-AFDM and derives the closed-form expression for the AF magnitude expectation. Several key insights into the impact of DAFT-s-AFDM parameters on ISAC performance are revealed, thus providing concrete guidance for the subsequent waveform design. Building on these insights, a novel probabilistic constellation shaping (PCS) framework is proposed for ISAC waveform enhancement, where the communication throughput and the sensing AF characteristics are jointly optimized by addressing a multi-objective problem. An efficient algorithm based on a closed-form bit error rate expression is developed to obtain the Pareto-optimal solutions. Extensive simulations validate the theoretical results and that the proposed PCS-enhanced DAFT-s-AFDM can significantly outperform the classical counterparts, achieving a superior and highly controllable tradeoff between the dual-functional performances.