Channel Estimation for Active Reconfigurable Intelligent Surfaces Under Amplifier Compression

Active reconfigurable intelligent surfaces amplify the reflected signal using amplifiers integrated into their elements. The signal model adopted throughout the literature treats each amplifier as a constant gain applied jointly to the incident signal and to the thermal noise of the element. Its experimental validation was performed at a single, very low incident power, far below the level at which an amplifier compresses. This article examines channel estimation when the amplifier is driven near saturation. Adopting an established nonlinear surface model, we show that the equivalent gain depends on the pilot power, so the closed-form training results of the linear model no longer hold: the conventional estimator becomes biased, the published training rules lose their optimality, and the estimation error exhibits a floor. We split the amplifier distortion into a static per-element component and a fluctuating component, and derive closed-form expressions for the error variance, the estimator bias, and the resulting floors, together with a Gaussian-equivalent variance bound. The expressions agree with Monte Carlo simulation to within 0.25 dB over twelve configurations. A numerical Bayes-MMSE benchmark that uses the exact Rapp response is at most 1.3 dB better than the Bussgang-based linear estimator, so the linearization costs little. Three consequences follow: the conventional estimator degrades once the pilot power passes a threshold; additional receive antennas, surface elements and training slots do not remove the floor in the simulated regime, because its dominant component is static, and this holds for rank-one, Rician and Rayleigh backward channels alike; and the published rule for the training amplification points the wrong way unless the receiver noise is high or the amplifier headroom is large. We then propose an online gain-tracking scheme that identifies the operating back-off from the ratio of two estimates taken at different pilot powers, needing no baseband processing at the surface, and we quantify its sensitivity to finite samples, channel aging, parameter drift and shape mismatch. Finally, we characterize robustness across amplifier models, element spread, and system scale and identify the region in which compensation helps.