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Random matrix models, double-time Painlevé equations, and wireless relaying

### Abstract

This paper gives an in-depth study of a multiple-antenna wireless communication scenario in which a weak signal received at an intermediate relay station is amplified and then forwarded to the final destination. The key quantity determining system performance is the statistical properties of the signal-to-noise ratio (SNR) γ at the destination. Under certain assumptions on the encoding structure, recent work has characterized the SNR distribution through its moment generating function, in terms of a certain Hankel determinant generated via a deformed Laguerre weight. Here, we employ two different methods to describe the Hankel determinant. First, we make use of ladder operators satisfied by orthogonal polynomials to give an exact characterization in terms of a “double-time” Painlevé differential equation, which reduces to Painlevé V under certain limits. Second, we employ Dyson's Coulomb fluid method to derive a closed form approximation for the Hankel determinant. The two characterizations are used to derive closed-form expressions for the cumulants of γ, and to compute performance quantities of engineering interest.

© 2013 AIP Publishing LLC

Received 22 December 2012
Accepted 15 May 2013
Published online 06 June 2013

Acknowledgments:
N. S. Haq is supported by an EPSRC grant. M. R. McKay is supported by the Hong Kong Research Grants Council under Grant No. 616911.

Article outline:

I. INTRODUCTION
A. Amplify and forward wireless relay model
B. Wireless communication performance measures
C. Statistical characterization of the SNR γ
D. Alternative characterization of the SNR γ
II. PAINLEVÉ CHARACTERIZATION VIA THE LADDER OPERATOR FRAMEWORK
III. COULOMB FLUID METHOD FOR LARGE *n* ANALYSIS
A. Preliminaries of the Coulomb fluid method
B. Coulomb fluid calculations for the SNR moment generating function
C. SER performance measure analysis based on Coulomb fluid
D. Coulomb fluid analysis of large *n* cumulants of SNR
1. Case 1: β = 0
2. Case 2: β > 0
IV. POWER SERIES EXPANSION FOR *H* _{ n }(*s*, *v*)
A. Analysis of κ_{1}
B. Analysis of κ_{2}
C. Beyond κ_{1} and κ_{2}
D. Comparison of cumulants obtained from ODEs with those obtained from determinant representation
V. LARGE *n* CORRECTIONS OF CUMULANTS OBTAINED FROM COULOMB FLUID
A. Large *n* expansion of κ_{1}
B. Large *n* expansion of κ_{2}
C. Beyond κ_{1} and κ_{2}
VI. ASYMPTOTIC PERFORMANCE ANALYSIS BASED ON COULOMB FLUID
A. The case of β = 0
1. High SNR analysis of the symbol error rate
2. High SNR analysis of the probability density function of γ
B. The case of β ≠ 0 (with *N* _{ R } < *N* _{ D })
1. High SNR analysis of the symbol error rate
2. High SNR analysis of the probability density function of γ
VII. CHARACTERIZING *A* _{0} THROUGH PAINLEVÉ *V*
VIII. CONCLUSION

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