On the Gallager Error Exponent Function and the ρ-Mutual Information of Additive Noise Channels

Abstract

We provide the computationally efficient general explicit formula for the error exponent function of an integer order ρ and its normalized version, the ρ-mutual information, which is applicable for additive noise channels with discrete inputs. In particular, we consider the additive white Gaussian and exponential noise channels with various modulation formats. We show that the error exponent function and the ρ-mutual information follow the same qualitative behavior as the Shannon mutual information but, in contrast to the Shannon case, can be expressed in a closed-form that does not impose estimation. We further apply the results to the analysis of the Shannon-Gallager-Berlekamp limiting rate for the sphere packing bound on the minimum error probability, which is the limit of the ρ-mutual information when ρ tends to infinity.