Ambiguity Function Analyses of AFDM Under Random ISAC Signaling

This paper analyzes the ambiguity function of the recently proposed affine frequency division multiplexing (AFDM) waveform under random communications information symbols. The derived expression of the discrete periodic ambiguity function of AFDM (DPAF-AFDM) waveform in closed-form reveals that the DPAF-AFDM is only affected by the parameter c<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf>, independent of the parameter c<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf>. Moreover, the derived expression of DPAF-AFDM can be directly used to analyze the ambiguity function for the existing orthogonal frequency division multiplexing (OFDM) and orthogonal chirp-division multiplexing (OCDM). It is shown that the delay-Doppler sidelobe level of DPAFs of AFDM, OFDM and OCDM take two types of values. Although DPAFs of AFDM, OFDM and OCDM have the same number of low sidelobe as well as the identical expectation of integrated sidelobe level (EISL), AFDM waveform has higher degrees of freedom (DoF) to flexibly control the indices of low sidelobes according to sensing scenarios. Numerical results verify the effectiveness of our theoretical analyses.