Mutual Information and Minimum Mean-square Error in Gaussian Channels
Abstract
This paper deals with arbitrarily distributed finite-power input signals observed through an additive Gaussian noise channel. It shows a new formula that connects the input-output mutual information and the minimum mean-square error (MMSE) achievable by optimal estimation of the input given the output. That is, the derivative of the mutual information (nats) with respect to the signal-to-noise ratio (SNR) is equal to half the MMSE, regardless of the input statistics. This relationship holds for both scalar and vector signals, as well as for discrete-time and continuous-time noncausal MMSE estimation. This fundamental information-theoretic result has an unexpected consequence in continuous-time nonlinear estimation: For any input signal with finite power, the causal filtering MMSE achieved at SNR is equal to the average value of the noncausal smoothing MMSE achieved with a channel whose signal-to-noise ratio is chosen uniformly distributed between 0 and SNR.
Cite
@article{arxiv.cs/0412108,
title = {Mutual Information and Minimum Mean-square Error in Gaussian Channels},
author = {Dongning Guo and Shlomo Shamai and Sergio Verdu},
journal= {arXiv preprint arXiv:cs/0412108},
year = {2007}
}