On the Achievable Rate of Generalized Spatial Modulation Using Multiplexing Under a Gaussian Mixture Model

Spatial modulation (SM) is a new modulation technique where the information to be sent is encoded using one or more symbols and the subspace over which the symbols are transmitted. Unfortunately, a general capacity analysis that encompasses different forms of SM systems has not been developed. In this paper, we consider a general form of SM, where the number of transmitted data streams is allowed to vary. We refer to this form of SM as generalized spatial modulation with multiplexing (GSMM). A Gaussian mixture model (GMM) is shown to accurately model the transmitted spatially modulated signal using a precoding framework. Using this transmit model, a closed-form expression for the achievable rate when operating over Rayleigh fading channels is evaluated, and tight upper and lower bounds for the achievable rate are proposed. The expressions of the achievable rate obtained are flexible enough to accommodate any form of SM—where any subspace can be used for the transmission—by adjusting the precoding set. Simulations are presented to show the tightness of the proposed bounds. The effect of the system dimensions and a comparison with other prominent capacity results are also demonstrated in simulations.