We analyze the transmission of a noisy signal by systems which are linear for small inputs and saturate at large inputs. Large information-carrying signals are thus distorted in their transmission. We demonstrate conditions where noise addition to such large input signals can reduce the distortion they undergo in the transmission. This is established for both periodic and aperiodic information-carrying signals. This effect is made possible because the noise acts as a random bias, shifting the operating zone of the nonlinearity, on average, into a region more favorable to the signal transmission. These results constitute a new instance of the nonlinear phenomenon of stochastic resonance where addition of noise may reveal beneficial to the signal.