  Massive MIMO communication systems, by virtue of utilizing very large number
of antennas, have a potential to yield higher spectral and energy efficiency in
comparison with the conventional MIMO systems. In this paper, we consider
uplink channel estimation in massive MIMO-OFDM systems with frequency selective
channels. With increased number of antennas, the channel estimation problem
becomes very challenging as exceptionally large number of channel parameters
have to be estimated. We propose an efficient distributed linear minimum mean
square error (LMMSE) algorithm that can achieve near optimal channel estimates
at very low complexity by exploiting the strong spatial correlations and
symmetry of large antenna array elements. The proposed method involves solving
a (fixed) reduced dimensional LMMSE problem at each antenna followed by a
repetitive sharing of information through collaboration among neighboring
antenna elements. To further enhance the channel estimates and/or reduce the
number of reserved pilot tones, we propose a data-aided estimation technique
that relies on finding a set of most reliable data carriers. We also analyse
the effect of pilot contamination on the mean square error (MSE) performance of
different channel estimation techniques. Unlike the conventional approaches, we
use stochastic geometry to obtain analytical expression for interference
variance (or power) across OFDM frequency tones and use it to derive the MSE
expressions for different algorithms under both noise and pilot contaminated
regimes. Simulation results validate our analysis and the near optimal MSE
performance of proposed estimation algorithms.
