While there exists a great body of research on reflection optimization of a reconfigurable intelligent surface (RIS), these optimizations assume independent phase shift and amplitude for the RIS reflection coefficients. Moreover, the choice of phase is typically assumed to be continuous over the full range. In practice, the phase shift and the amplitude are coupled, and the phase choices are limited to a discrete set. In our work, we consider a practical RIS model with coupled phase shift and amplitude and limited phase choices. For the coupled RIS model, given a configuration set (which is a discrete set of coupled reflection coefficient choices that an RIS element can take), we develop an efficient method for capacity maximization by finding the optimal reflection coefficients of the RIS elements. Our method has a complexity linear with the number of RIS elements and the number of discrete reflection coefficient choices. We also develop a method that optimally selects the configuration set of the system.

We address the reflection optimization problem for a reconfigurable intelligent surface (RIS), where the RIS elements feature a set of non-uniformly spaced discrete phase shifts. This is motivated by the actual behavior of practical RIS elements, where it is shown that a uniform phase shift assumption is not realistic. A problem is formulated to find the optimal reflection amplitudes and reflection phase shifts of the RIS elements such that the channel capacity of the target user is maximized. We first prove that in the optimal configuration, each RIS element is either turned off or operates at maximum amplitude. We then develop a method that finds the optimal reflection amplitudes and phases with complexity linear with the number of RIS elements. An interesting insight into the reflection optimization problem is also provided.