Ambiguity Function Analysis of AFDM Under Pulse-Shaped Random ISAC Signaling

This paper investigates the ambiguity function (AF) of the emerging affine frequency division multiplexing (AFDM) waveform for random integrated sensing and communication (ISAC) signaling under a pulse shaping regime. Specifically, we first derive the closed-form expression of the average squared discrete period AF (DPAF) for AFDM waveform without pulse shaping, revealing that the AF depends on the parameter c1 and the kurtosis of random communication data, while being independent of the parameter c2. As a step further, we conduct a comprehensive analysis on the DPAFs of various waveforms, including AFDM, orthogonal frequency division multiplexing (OFDM) and orthogonal chirp-division multiplexing (OCDM). Our results indicate that all three waveforms exhibit the same number of regular depressions in the sidelobes of their DPAFs, which incurs performance loss for detecting and estimating weak targets. However, the AFDM waveform can flexibly control the positions of depressions by adjusting the parameter c1, which motivates a novel design approach of the AFDM parameters to mitigate the adverse impact of depressions of the strong target on the weak target. Furthermore, the closed-form expressions of the average squared DPAFs for pulse-shaped AFDM, OFDM and OCDM waveforms are derived, which demonstrates that the pulse shaping filter generates the shaped mainlobe along the delay axis and the rapid roll-off sidelobes along the Doppler axis. Numerical results verify the effectiveness of our theoretical analysis and proposed design methodology for the AFDM waveform.