Asymptotic capacity of a random channel

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Date
2014
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Sutter, David
Lygeros, John
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2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton). - Piscataway, NJ : IEEE, 2014. - pp. 771-778. - ISBN 978-1-4799-8009-3
Abstract
We consider discrete memoryless channels with input and output alphabet size n whose channel transition matrix consists of entries that are independent and identically distributed according to some probability distribution v on (R≥0, B(R≥0)) before being normalized, where v is such that E[X log X) 2 1 <; ∞, μ 1 := E[X] and μ 2 := E[X log X] for a random variable X with distribution v. We prove that in the limit as n → ∞, the capacity of such a channel converges to μ 2 1 - log μ 1 almost surely and in L 2 . We further show that the capacity of these random channels converges to this asymptotic value exponentially in n.
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004 Computer Science
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52nd Annual Allerton Conference on Communication, Control, and Computing, Sep 30, 2014 - Oct 3, 2014, Monticello, IL, USA
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ISO 690SUTTER, Tobias, David SUTTER, John LYGEROS, 2014. Asymptotic capacity of a random channel. 52nd Annual Allerton Conference on Communication, Control, and Computing. Monticello, IL, USA, Sep 30, 2014 - Oct 3, 2014. In: 2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton). Piscataway, NJ:IEEE, pp. 771-778. ISBN 978-1-4799-8009-3. Available under: doi: 10.1109/ALLERTON.2014.7028532
BibTex
@inproceedings{Sutter2014Asymp-55741,
  year={2014},
  doi={10.1109/ALLERTON.2014.7028532},
  title={Asymptotic capacity of a random channel},
  isbn={978-1-4799-8009-3},
  publisher={IEEE},
  address={Piscataway, NJ},
  booktitle={2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton)},
  pages={771--778},
  author={Sutter, Tobias and Sutter, David and Lygeros, John}
}
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    <dcterms:abstract xml:lang="eng">We consider discrete memoryless channels with input and output alphabet size n whose channel transition matrix consists of entries that are independent and identically distributed according to some probability distribution v on (R≥0, B(R≥0)) before being normalized, where v is such that E[X log X)&lt;sup&gt; 2&lt;/sup&gt; 1 &lt;; ∞, μ &lt;sub&gt;1&lt;/sub&gt; := E[X] and μ&lt;sub&gt; 2&lt;/sub&gt; := E[X log X] for a random variable X with distribution v. We prove that in the limit as n → ∞, the capacity of such a channel converges to μ&lt;sub&gt; 2&lt;/sub&gt; /μ&lt;sub&gt; 1&lt;/sub&gt; - log μ&lt;sub&gt; 1&lt;/sub&gt; almost surely and in L&lt;sup&gt; 2&lt;/sup&gt; . We further show that the capacity of these random channels converges to this asymptotic value exponentially in n.</dcterms:abstract>
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