  This paper investigates double RIS-assisted MIMO communication systems over
Rician fading channels with finite scatterers, spatial correlation, and the
existence of a double-scattering link between the transceiver. First, the
statistical information is driven in closed form for the aggregated channels,
unveiling various influences of the system and environment on the average
channel power gains. Next, we study two active and passive beamforming designs
corresponding to two objectives. The first problem maximizes channel capacity
by jointly optimizing the active precoding and combining matrices at the
transceivers and passive beamforming at the double RISs subject to the
transmitting power constraint. In order to tackle the inherently non-convex
issue, we propose an efficient alternating optimization algorithm (AO) based on
the alternating direction method of multipliers (ADMM). The second problem
enhances communication reliability by jointly training the encoder and decoder
at the transceivers and the phase shifters at the RISs. Each neural network
representing a system entity in an end-to-end learning framework is proposed to
minimize the symbol error rate of the detected symbols by controlling the
transceiver and the RISs phase shifts. Numerical results verify our analysis
and demonstrate the superior improvements of phase shift designs to boost
system performance.
