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.

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