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We consider a two--tile reconfigurable intelligent surface (RIS) assistedwireless network with a two-antenna transmitter and receiver over Rayleighfading. We show that the average received signal-to-noise-ratio (SNR) optimaltransmission and combining vectors are given by the left and right singularspaces of the RIS-receiver and transmit-RIS channel matrices, respectively.Moreover, the optimal phases at the two tiles of the RIS are determined by thephases of the elements of the latter spaces. To further study the effect ofphase compensation, we statistically characterize the average SNR of allpossible combinations of transmission and combining directions pertaining tothe latter singular spaces by deriving novel expressions for the outageprobability and throughput of each of those modes. Furthermore, for comparison,we derive the corresponding expressions in the absence of RIS. Our results showan approximate SNR improvement of $2$\,dB due to the phase compensation at theRIS.

Performance Analysis of a Two-Tile Reconfigurable Intelligent Surface Assisted $2\times 2$ MIMO System