Estimation in gaussian noise: Properties of the minimum mean-square error

This work studies the minimum mean-square error (MMSE) of estimating an arbitrary random variable from an observation contaminated by Gaussian noise. The MMSE can be regarded as a function of the signal-to-noise ratio (SNR), as well as a functional of the distribution of the random variable. The MMSE is shown to be an analytic function of the SNR, and simple expressions for its first three derivatives are obtained. This paper also shows that the MMSE is convex in the SNR and concave in the input distribution. Moreover, it is shown that there can be only one SNR for which the MMSE of a Gaussian random variable and that of a non-Gaussian random variable coincide. These properties lead to simple proofs of the facts that Gaussian inputs achieve both the secrecy capacity of scalar Gaussian wiretap channels and the capacity of scalar Gaussian broadcast channels