### 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 language | English (US) |
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Title of host publication | Proceedings - 2008 IEEE International Symposium on Information Theory, ISIT 2008 |

Pages | 1083-1087 |

Number of pages | 5 |

DOIs | |

State | Published - Sep 29 2008 |

Event | 2008 IEEE International Symposium on Information Theory, ISIT 2008 - Toronto, ON, Canada Duration: Jul 6 2008 → Jul 11 2008 |

### Publication series

Name | IEEE International Symposium on Information Theory - Proceedings |
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ISSN (Print) | 2157-8101 |

### Other

Other | 2008 IEEE International Symposium on Information Theory, ISIT 2008 |
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Country | Canada |

City | Toronto, ON |

Period | 7/6/08 → 7/11/08 |

### All Science Journal Classification (ASJC) codes

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

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## Cite this

*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