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

TL;DR

This paper analyzes the ambiguity function of AFDM waveform in ISAC signaling, deriving formulas for pulse shaping effects and proposing parameter adjustments to improve weak target detection.

Contribution

It provides a comprehensive AF analysis of AFDM, OFDM, and OCDM waveforms, and introduces a novel AFDM parameter design to mitigate sidelobe depressions.

Findings

AF depends on parameter c1 and data kurtosis, but not c2.

All waveforms show sidelobe depressions affecting weak target detection.

Adjusting c1 in AFDM can mitigate sidelobe effects.

Abstract

This paper investigates the ambiguity function (AF) of the emerging affine frequency division multiplexing (AFDM) waveform for 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 $c_1$ and the kurtosis of random communication data, while being independent of the parameter $c_2$. As a step further, we conduct a comprehensive analysis on the AFs 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 AFs, 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 $c_1$, 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, a closed-form expression of the average squared DPAF for pulse-shaped random AFDM waveform is 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 modulation.