Ambiguity Function Analysis of AFDM Signals for Integrated Sensing and Communications

Affine frequency division multiplexing (AFDM) is a promising chirp-based waveform with high flexibility and resilience, making it well-suited for next-generation wireless networks, particularly in high-mobility scenarios. In this paper, we investigate the ambiguity functions (AFs) of AFDM signals, which fundamentally characterize their range and velocity estimation capabilities in both monostatic and bistatic settings. Specifically, we first derive the auto-ambiguity function (AAF) of an AFDM chirp subcarrier, revealing its "spike-like" local property and "periodic-like" global property along the rotated delay and Doppler dimensions. This structure naturally forms a parallelogram for each localized pulse of the AAF of the AFDM chirp subcarrier, enabling unambiguous target sensing. Then, we study the cross-ambiguity function (CAF) between two different AFDM chirp subcarriers, which exhibits the same local and global properties as the AAF but with an additional shift along the Doppler dimension. We then extend our analysis to the AF of various typical AFDM frames, considering both deterministic pilot and random data symbols. In particular, we demonstrate that inserting guard symbols in AFDM facilitates interference-free sensing. Simulation results validate our theoretical findings, highlighting AFDM's strong potential for ISAC applications.