  In this study, we explore the performance of a reconfigurable reflecting
surface (RIS)-assisted transmit spatial modulation (SM) system for downlink
transmission, wherein the deployment of RIS serves the purpose of blind area
coverage within the channel. At the receiving end, we present three detectors,
i.e., maximum likelihood (ML) detector, two-stage ML detection, and greedy
detector to recover the transmitted signal. By utilizing the ML detector, we
initially derive the conditional pair error probability expression for the
proposed scheme. Subsequently, we leverage the central limit theorem (CLT) to
obtain the probability density function of the combined channel. Following
this, the Gaussian-Chebyshev quadrature method is applied to derive a
closed-form expression for the unconditional pair error probability and
establish the union tight upper bound for the average bit error probability
(ABEP). Furthermore, we derive a closed-form expression for the ergodic
capacity of the proposed RIS-SM scheme. Monte Carlo simulations are conducted
not only to assess the complexity and reliability of the three detection
algorithms but also to validate the results obtained through theoretical
derivation results.
