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Coulumb fluid, painlevé transcendents, and the information theory of MIMO systems
Chen Y.1,2; McKay M.R.3
Source PublicationIEEE Transactions on Information Theory

In this paper, we compute two important information-theoretic quantities which arise in the application of multiple-input multiple-output (MIMO) antenna wireless communication systems: the distribution of the mutual information of multiantenna Gaussian channels, and the Gallager random coding upper bound on the error probability achievable by finite-length channel codes. We show that the mathematical problem underpinning both quantities is the computation of certain Hankel determinants generated by deformed versions of classical weight functions. For single-user MIMO systems, it is a deformed Laguerre weight; for multiuser MIMO systems, it is a deformed Jacobi weight. We apply two different methods to characterize each of these Hankel determinants. First, we employ the ladder operators of the corresponding monic orthogonal polynomials to give an exact characterization of the Hankel determinants in terms of Painlevé differential equations. This turns out to be a Painlevé V for the single-user MIMO scenario and a Painlevé VI for the multiuser scenario. We then introduce Coulomb fluid linear statistics methods to derive closed-form approximations for the MIMO mutual information distribution and the error probability which, although formally valid for large matrix dimensions, are shown to give accurate results even when the matrix dimensions are small. Focusing on the single-user mutual information distribution, we then employ the exact Painlevé V representation with the help of the Coulomb fluid linear statistics approximation to yield deeper insights into the scaling behavior in terms of the number of antennas and signal-to-noise ratio (SNR). Among other things, these results allow us to study the asymptotic Gaussianity of the distribution as the number of antennas increase, and to investigate when and why such approximations break down as the SNR increases. Based on the Painlevé, we also derive recursive formulas for explicitly computing in closed form any desired number of correction terms to the asymptotic mean and variance, as well as closed-form asymptotic expressions for any desired number of higher order cumulants. Using these cumulants, we propose new closed-form approximations to the mutual information distribution which are shown to be very accurate, not only in the bulk but also in the tail region of interest for the outage probability. 

KeywordChannel Capacity Multiple-input Multiple-output (Mimo) Systems Orthogonal Polynomials Random Matrix Theory
URLView the original
Indexed BySCI
WOS Research AreaComputer Science ; Engineering
WOS SubjectComputer Science, Information Systems ; Engineering, Electrical & Electronic
WOS IDWOS:000305575000032
The Source to ArticleScopus
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Cited Times [WOS]:51   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
Corresponding AuthorChen Y.
Affiliation1.Univ London Imperial Coll Sci Technol & Med, Dept Math, London W2 1NY, England
2.Univ Macau, Dept Math, Taipa, Peoples R China
3.Hong Kong Univ Sci & Technol, Dept Elect & Comp Engn, Kowloon, Hong Kong, Peoples R China
First Author AffilicationUniversity of Macau
Corresponding Author AffilicationUniversity of Macau
Recommended Citation
GB/T 7714
Chen Y.,McKay M.R.. Coulumb fluid, painlevé transcendents, and the information theory of MIMO systems[J]. IEEE Transactions on Information Theory,2012,58(7):4594.
APA Chen Y.,&McKay M.R..(2012).Coulumb fluid, painlevé transcendents, and the information theory of MIMO systems.IEEE Transactions on Information Theory,58(7),4594.
MLA Chen Y.,et al."Coulumb fluid, painlevé transcendents, and the information theory of MIMO systems".IEEE Transactions on Information Theory 58.7(2012):4594.
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