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The estimation and the compensation of particular amplitude distortions, typically affecting the consumer-grade receivers in broadband satellite communications, are considered. The distortion filter is modeled as a linear function of the frequency with unknown slope. We present a novel slope estimator, analytically derived by exploiting the spectral properties of the distorted signal, which results to be unbiased and does not exhibit false locks in the scenarios of practical interest. Since an ideal slope compensator would require an unmanageable filtering, we focus on a low-complexity compensation filter that ensures nearly the same performance. Extensive simulation results prove the effectiveness and the robustness of the proposed solution, which can also work blindly at very low values of the signal-to-noise ratio.