  Reconfigurable intelligent surface (RIS) is a promising technology that
enables the customization of electromagnetic propagation environments in
next-generation wireless networks. In this paper, we investigate the optimal
pilot power allocation during the channel estimation stage to improve the
ergodic channel gain of RIS-assisted systems under practical imperfect channel
state information (CSI). Specifically, we commence by deriving an explicit
closed-form expression of the ergodic channel gain of a multi-RIS-aided
communication system that takes into account channel estimation errors. Then,
we formulate the pilot power allocation problem to maximize the ergodic channel
gain under imperfect CSI, subject to the average pilot power constraint. Then,
the method of Lagrange multipliers is invoked to obtain the optimal pilot power
allocation solution, which indicates that allocating more power to the pilots
for estimating the weak reflection channels is capable of effectively improving
the ergodic channel gain under imperfect CSI. Finally, extensive simulation
results corroborate our theoretical analysis.
