Reconfigurable Intelligent Surfaces With Channel Training: Spectral Efficiency and Optimal Array Dimensions

In reconfigurable intelligent surfaces (RIS), the reflective elements increase the dimensionality of the overall channel model and, correspondingly, the training requirements. This paper analyzes how channel training and spectral efficiency guide the choice of the operational dimensionality of RIS. We derive an inner bound on the training-based capacity as a function of pilot power, data transmission power, and RIS array dimensions. We study the outcomes and implications of data/pilot power optimization in RIS-assisted systems according to the achievable rate metric. Further, this work sheds light on the tradeoff between the superior array gains of larger RIS on the one hand, and the respective training requirements on the other hand, when optimizing the end-to-end capacity. Beyond a certain critical array size, further increase of array size is not beneficial for spectral efficiency due to the training requirement. This critical size is calculated, and its dependence on the signal-to-noise ratio (SNR) and coherence interval of the channel is clarified. The benefits and drawbacks of different training schemes are analyzed in this context, and demonstrated by simulations.