  Stacked intelligent metasurfaces (SIM) is a revolutionary technology, which
can outperform its single-layer counterparts by performing advanced signal
processing relying on wave propagation. In this work, we exploit SIM to enable
transmit precoding and receiver combining in holographic multiple-input
multiple-output (HMIMO) communications, and we study the achievable rate by
formulating a joint optimization problem of the SIM phase shifts at both sides
of the transceiver and the covariance matrix of the transmitted signal.
Notably, we propose its solution by means of an iterative optimization
algorithm that relies on the projected gradient method, and accounts for all
optimization parameters simultaneously. We also obtain the step size
guaranteeing the convergence of the proposed algorithm. Simulation results
provide fundamental insights such the performance improvements compared to the
single-RIS counterpart and conventional MIMO system. Remarkably, the proposed
algorithm results in the same achievable rate as the alternating optimization
(AO) benchmark but with a less number of iterations.
