Capacity and reliability function for small peak signal constraints

The capacity and the reliability function as the peak constraint tends to zero are considered for a discrete-time memoryless channel with peak constrained inputs. Prelov and van der Meulen showed that under mild conditions the ratio of the capacity to the squared peak constraint converges to one-half the maximum eigenvalue of the Fisher information matrix and if the Fisher information matrix is nonzero, the asymptotically optimal input distribution is symmetric antipodal signaling. Under similar conditions, it is shown in the first part of the paper that the reliability function has the same asymptotic shape as the reliability function for the power-constrained infinite bandwidth white Gaussian noise channel. The second part of the paper deals with Rayleigh-fading channels. For such channels, the Fisher information matrix is zero, indicating the difficulty of transmission over such channels with small peak constrained signals. Asymptotics for the Rayleigh channel are derived and applied to obtain the asymptotics of the capacity of the Marzetta and Hochwald fading channel model for small peak constraints, and to obtain a result of the type of Médard and Gallager for wide-band fading channels.