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.

翻译：仿射频率分复用（AFDM）是一种全新的基于啁啾的多载波（MC）波形，适用于高移动性通信，相比正交频分复用（OFDM）及其他MC波形具有显著优势。现有AFDM研究聚焦于高载频（CF）无线通信场景，通常仅考虑移动性引起的多普勒频移（DFS），而忽略伴随的多普勒时间缩放（DTS）对波形的影响。然而，对于以声速传播且载频低得多的水声（UWA）通信而言，DTS效应不可忽略，并对信道估计构成重大挑战。本文分析了AFDM在多尺度多时延（MSML）信道下的信道频率响应（CFR），其中每条传播路径可能具有不同的时延与DFS/DTS。基于新推导的输入输出公式及其特性，提出了两种新的信道估计方法：在低至中等DTS条件下采用AFDM迭代多索引（AFDM-IMI）估计，在高DTS条件下采用AFDM正交匹配追踪（AFDM-OMP）估计。数值结果验证了所提方法相比原始AFDM信道估计方法的有效性。此外，在MSML信道下，基于CFR重叠概率（COP）、互不相干特性（MIP）和信道分集增益的理论分析，所得到的AFDM系统在信道估计精度和误码率（BER）方面均优于OFDM及正交啁啾分复用（OCDM），与理论分析结果一致。