  For massive multiple-input multiple-output (MIMO) systems operating in
frequency-division duplex mode, downlink channel state information (CSI)
acquisition will incur large overhead. This overhead is substantially reduced
when sparse channel estimation techniques are employed, owing to the channel
sparsity in the angular domain. When a sparse channel estimation method is
implemented, the measurement matrix, which is related to the pilot matrix, is
essential to the channel estimation performance. Existing sparse channel
estimation schemes widely adopt random measurement matrices, which have been
criticized for their suboptimal reconstruction performance. This paper proposes
novel data-driven solutions to design the measurement matrix. Model-based
autoencoders are customized to optimize the measurement matrix by unfolding the
classical basis pursuit algorithm. The obtained data-driven measurement
matrices are applied to existing sparse reconstruction algorithms, leading to
flexible hybrid data-driven implementations for sparse channel estimation.
Numerical results show that the proposed data-driven measurement matrices can
achieve more accurate reconstructions and use fewer measurements than the
existing random matrices, thereby leading to a higher achievable rate for CSI
acquisition. Moreover, compared with existing pure deep learning-based sparse
reconstruction methods, the proposed hybrid data-driven scheme, which uses the
novel data-driven measurement matrices with conventional sparse reconstruction
algorithms, can achieve higher reconstruction accuracy.
