Estimation of non-Gaussian random variables in Gaussian noise: Properties of the MMSE

Dongning Guo, Shlomo Shamai, Sergio Verdu

Research output: Chapter in Book/Report/Conference proceedingConference contribution

14 Scopus citations

Abstract

This work studies the properties of the minimum mean-square error (MMSE) of estimating an arbitrary random variable contaminated by Gaussian noise based on the observation. The MMSE can be regarded as a function of the signalto-noise ratio (SNR), as well as a functional or transform of the input distribution. This paper shows that the MMSE is analytic in SNR for every random variable. Simple expressions for the derivatives of the MMSE as a function of the SNR are obtained. Since the input-output mutual information can be written as the integral of the MMSE as a function of SNR, the results also lead to higher derivatives of the mutual information. The MMSE and mutual information's convexity in the SNR and concavity in the input distribution are established. 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. Application of the properties of the MMSE to the scalar Gaussian broadcast channel problem is presented.

Original languageEnglish (US)
Title of host publicationProceedings - 2008 IEEE International Symposium on Information Theory, ISIT 2008
Pages1083-1087
Number of pages5
DOIs
StatePublished - Sep 29 2008
Event2008 IEEE International Symposium on Information Theory, ISIT 2008 - Toronto, ON, Canada
Duration: Jul 6 2008Jul 11 2008

Publication series

NameIEEE International Symposium on Information Theory - Proceedings
ISSN (Print)2157-8101

Other

Other2008 IEEE International Symposium on Information Theory, ISIT 2008
CountryCanada
CityToronto, ON
Period7/6/087/11/08

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Information Systems
  • Modeling and Simulation
  • Applied Mathematics

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    Guo, D., Shamai, S., & Verdu, S. (2008). Estimation of non-Gaussian random variables in Gaussian noise: Properties of the MMSE. In Proceedings - 2008 IEEE International Symposium on Information Theory, ISIT 2008 (pp. 1083-1087). [4595154] (IEEE International Symposium on Information Theory - Proceedings). https://doi.org/10.1109/ISIT.2008.4595154