Affine Frequency Division Multiplexing (AFDM) is a brand new chirp-based multi-carrier (MC) waveform for high mobility communications, with promising advantages over Orthogonal Frequency Division Multiplexing (OFDM) and other MC waveforms. Existing AFDM research focuses on wireless communication at high carrier frequency (CF), which typically considers only Doppler frequency shift (DFS) as a result of mobility, while ignoring the accompanied Doppler time scaling (DTS) on waveform. However, for underwater acoustic (UWA) communication at much lower CF and propagating at speed of sound, the DTS effect could not be ignored and poses significant challenges for channel estimation. This paper analyzes the channel frequency response (CFR) of AFDM under multi-scale multi-lag (MSML) channels, where each propagating path could have different delay and DFS/DTS. Based on the newly derived input-output formula and its characteristics, two new channel estimation methods are proposed, i.e., AFDM with iterative multi-index (AFDM-IMI) estimation under low to moderate DTS, and AFDM with orthogonal matching pursuit (AFDM-OMP) estimation under high DTS. Numerical results confirm the effectiveness of the proposed methods against the original AFDM channel estimation method. Moreover, the resulted AFDM system outperforms OFDM as well as Orthogonal Chirp Division Multiplexing (OCDM) in terms of channel estimation accuracy and bit error rate (BER), which is consistent with our theoretical analysis based on CFR overlap probability (COP), mutual incoherent property (MIP) and channel diversity gain under MSML channels.

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