{
  "id": "0910.0575",
  "version": "v6",
  "published": "2009-10-03T22:25:09.000Z",
  "updated": "2013-07-13T03:02:31.000Z",
  "title": "A Note on Functional Averages over Gaussian Ensembles",
  "authors": [
    "Gabriel H. Tucci",
    "Maria V. Vega"
  ],
  "comment": "Published in Journal of Probability and Statistics, Vol. 2013, Article ID 941058",
  "categories": [
    "math.PR",
    "cs.IT",
    "math.IT",
    "math.OA"
  ],
  "abstract": "In this work we find a new formula for matrix averages over the Gaussian ensemble. Let ${\\bf H}$ be an $n\\times n$ Gaussian random matrix with complex, independent, and identically distributed entries of zero mean and unit variance. Given an $n\\times n$ positive definite matrix ${\\bf A}$, and a continuous function $f:\\R^{+}\\to\\R$ such that $\\int_{0}^{\\infty}{e^{-\\alpha t}|f(t)|^2\\,dt}<\\infty$ for every $\\alpha>0$, we find a new formula for the expectation $\\E[\\mathrm{Tr}(f({\\bf HAH^{*}}))]$. Taking $f(x)=\\log(1+x)$ gives another formula for the capacity of the MIMO communication channel, and taking $f(x)=(1+x)^{-1}$ gives the MMSE achieved by a linear receiver.",
  "revisions": [
    {
      "version": "v6",
      "updated": "2013-07-13T03:02:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "15B52"
    ],
    "keywords": [
      "gaussian ensemble",
      "functional averages",
      "gaussian random matrix",
      "mimo communication channel",
      "zero mean"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0910.0575T"
    }
  }
}