  This work investigates a multi-user, multi-antenna uplink wireless system,
where multiple users transmit signals to a base station. Previous research has
explored the potential for linear growth in spectral efficiency by employing
multiple transmit and receive antennas. This gain depends on the quality of
channel state information and uncorrelated antennas. However, spatial
correlations, arising from closely-spaced antennas, and channel aging effects,
stemming from the difference between the channel at pilot and data time
instances, can substantially counteract these benefits and degrade the
transmission rate, especially in non-stationary environments. To address these
challenges, this work introduces a real-time beamforming framework to
compensate for the spatial correlation effect. A channel estimation scheme is
then developed, leveraging temporal channel correlations and considering mobile
device velocity and antenna spacing. Subsequently, an expression approximating
the average spectral efficiency is obtained, dependent on pilot spacing, pilot
and data powers, and beamforming vectors. By maximizing this expression,
optimal parameters are identified. Numerical results reveal the effectiveness
of the proposed approach compared to prior works. Moreover, optimal pilot
spacing remains unaffected by interference components such as path loss and the
velocity of interference users. The impact of interference components also
diminishes with an increasing number of transmit antennas.
