Single-user MIMO system, Painleve transcendents, and double scaling
Chen, Hongmei; Chen, Min; Blower, Gordon; Chen, Yang
AbstractIn this paper, we study a particular Painleve V (denoted P-V) that arises from multiinput- multi-output wireless communication systems. Such P-V appears through its intimate relation with the Hankel determinant that describes the moment generating function (MGF) of the Shannon capacity. This originates through the multiplication of the Laguerre weight or the gamma density x(alpha)e(-x), x > 0, for alpha > -1 by (1 + x/t)(lambda) with t > 0 a scaling parameter. Here the lambda parameter "generates" the Shannon capacity; see Chen, Y. and McKay, M. R. [IEEE Trans. Inf. Theory 58, 4594-4634 (2012)]. It was found that the MGF has an integral representation as a functional of y(t) and y'(t), where y(t) satisfies the "classical form" of P-V. In this paper, we consider the situation where n, the number of transmit antennas, (or the size of the random matrix), tends to infinity and the signal-to-noise ratio, P, tends to infinity such that s = 4n(2)/P is finite. Under such double scaling, the MGF, effectively an infinite determinant, has an integral representation in terms of a " lesser" P-III. We also consider the situations where alpha = k + 1/2, k is an element of N, and alpha is an element of {0, 1, 2, ...}, lambda is an element of {1, 2, ...}, linking the relevant quantity to a solution of the two-dimensional sine-Gordon equation in radial coordinates and a certain discrete Painleve-II. From the large n asymptotic of the orthogonal polynomials, which appears naturally, we obtain the double scaled MGF for small and large s, together with the constant term in the large s expansion. With the aid of these, we derive a number of cumulants and find that the capacity distribution function is non-Gaussian. Published by AIP Publishing.
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Indexed BySCI
WOS Research AreaPhysics
WOS SubjectPhysics, Mathematical
WOS IDWOS:000418997300026
The Source to ArticleWOS
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Cited Times [WOS]:2   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
CollectionUniversity of Macau
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Chen, Hongmei,Chen, Min,Blower, Gordon,et al. Single-user MIMO system, Painleve transcendents, and double scaling[J]. JOURNAL OF MATHEMATICAL PHYSICS,2017,58(12).
APA Chen, Hongmei,Chen, Min,Blower, Gordon,&Chen, Yang.(2017).Single-user MIMO system, Painleve transcendents, and double scaling.JOURNAL OF MATHEMATICAL PHYSICS,58(12).
MLA Chen, Hongmei,et al."Single-user MIMO system, Painleve transcendents, and double scaling".JOURNAL OF MATHEMATICAL PHYSICS 58.12(2017).
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