This paper investigates channel estimation for linear time-varying (LTV) wireless channels under double sparsity, i.e., sparsity in both the delay and Doppler domains. An on-grid approximation is first considered, enabling rigorous hierarchical-sparsity modeling and compressed sensing-based channel estimation. Guaranteed recovery conditions are provided for affine frequency division multiplexing (AFDM), orthogonal frequency division multiplexing (OFDM) and single-carrier modulation (SCM), highlighting the superiority of AFDM in terms of doubly sparse channel estimation. To address arbitrary Doppler shifts, a relaxed version of the on-grid model is introduced by making use of multiple elementary Expansion Models (BEM) each based on Discrete Prolate Spheroidal Sequences (DPSS). Next, theoretical guarantees are provided for the precision of this off-grid model before further extending it to tackle channel prediction by exploiting the inherent DPSS extrapolation capability. Finally, numerical results are provided to both validate the proposed off-grid model for channel estimation and prediction purposes under the double sparsity assumption and to compare the corresponding mean squared error (MSE) and the overhead performance when the different wireless waveforms are used.

翻译：本文研究在双稀疏性（即延迟域和多普勒域均具有稀疏性）条件下的线性时变无线信道估计问题。首先考虑一种网格近似方法，实现了严格的层次稀疏建模与基于压缩感知的信道估计。针对仿射频分复用、正交频分复用及单载波调制波形，给出了保证恢复性能的理论条件，并凸显了仿射频分复用在双稀疏信道估计方面的优越性。为处理任意多普勒频移，通过采用多个基于离散长球面序列的基本扩展模型，提出了网格近似模型的松弛版本。随后，在进一步利用离散长球面序列固有外推能力扩展该模型以解决信道预测问题之前，先给出了该非网格模型精度的理论保证。最后，通过数值结果验证了所提非网格模型在双稀疏假设下进行信道估计与预测的有效性，并比较了不同无线波形对应的均方误差与开销性能。