  The doubly selective (DS) channel estimation in the large-scale
multiple-input multiple-output (MIMO) systems is a challenging problem due to
the large number of the channel coefficients to be estimated, which requires
unaffordable and prohibitive pilot overhead. In this paper, firstly we conduct
the analysis about the common sparsity of the basis expansion model (BEM)
coefficients among all the BEM orders and all the transmit-receive antenna
pairs. Then a novel pilot pattern is proposed, which inserts the guard pilots
to deal with the inter carrier interference (ICI) under the superimposed pilot
pattern. Moreover, by exploiting the common sparsity of the BEM coefficients
among different BEM orders and different antennas, we propose a block
distributed compressive sensing (BDCS) based DS channel estimator for the
large-scale MIMO systems. Its structured sparsity leads to the reduction of the
pilot overhead under the premise of guaranteeing the accuracy of the
estimation. Furthermore, taking consideration of the block structure, a pilot
design algorithm referred to as block discrete stochastic optimization (BDSO)
is proposed. It optimizes the pilot positions by reducing the coherence among
different blocks of the measurement matrix. Besides, a linear smoothing method
is extended to large-scale MIMO systems to improve the accuracy of the
estimation. Simulation results verify the performance gains of our proposed
estimator and the pilot design algorithm compared with the existing schemes.
