  On the heels of orthogonal time frequency space (OTFS) modulation, the
recently discovered affine frequency division multiplexing (AFDM) is a
promising waveform for the sixth-generation wireless network. In this paper, we
study the widely-used embedded pilot-aided (EPA) channel estimation in
multiple-input multiple-output AFDM (MIMO-AFDM) system with fractional Doppler
shifts. We first formulate the vectorized input-output relationship of
MIMO-AFDM, and theoretically prove that MIMO-AFDM can achieve full diversity in
doubly selective channels. Then we illustrate the implementation of EPA channel
estimation in MIMO-AFDM and unveil that serious inter-Doppler interference
(IDoI) occurs if we try to estimate the channel gain, delay shift, and Doppler
shift of each propagation path. To address this issue, the diagonal
reconstructability of AFDM subchannel matrix is studied and a low-complexity
embedded pilot-aided diagonal reconstruction (EPA-DR) channel estimation scheme
is proposed. The EPA-DR scheme calculates the AFDM effective channel matrix
directly without estimating the three channel parameters, eliminating the
severe IDoI inherently. Since the effective channel matrix is necessary for
MIMO-AFDM receive processing, we believe this is an important step to bring
AFDM towards practical communication systems. Finally, we investigate the
orthogonal resource allocation of affine frequency division multiple access
(AFDMA) system. Simulation results validate the effectiveness of the proposed
EPA-DR scheme.
