Coulumb Fluid, Painleve Transcendents, and the Information Theory of MIMO Systems
Coulumb Fluid, Painleve Transcendents, and the Information Theory of MIMO Systems
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DOI：
10.1109/tit.2012.2195154
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发表时间：
2012-07-01
影响因子：
2.5
通讯作者：
McKay, Matthew R.
中科院分区：
文献类型：
--
作者：
Chen, Yang;McKay, Matthew R.
本文计算了多输入多输出（MIMO）天线无线通信系统应用中出现的两个重要信息论量：多天线高斯信道的互信息分布和有限长度信道码可达到的误差概率的Gallager随机编码上界。我们证明了支撑这两个量的数学问题是由经典权函数的变形版本生成的某些汉克尔行列式的计算。对于单用户MIMO系统，它是一个变形的拉盖尔权值；对于多用户MIMO系统，它是一个变形的雅可比权值。我们采用两种不同的方法来表征这些汉克尔行列式。首先，我们利用相应的单正交多项式的阶梯算子给出了用Painleve微分方程表示的Hankel行列式的精确表征。这是单用户MIMO场景的painlev和多用户场景的painlevi。然后，我们引入库仑流体线性统计方法来推导MIMO互信息分布和误差概率的封闭近似，尽管在大矩阵维度上是有效的，但即使在矩阵维度很小的情况下也显示出准确的结果。专注于单用户互信息分布，我们在库仑流体线性统计近似的帮助下使用精确的painlevel V表示，以更深入地了解天线数量和信噪比（SNR）的缩放行为。除其他外，这些结果使我们能够研究随着天线数量增加而分布的渐近高斯性，并研究当信噪比增加时这种近似何时以及为什么会失效。在Painleve的基础上，我们还导出了以封闭形式显式计算任意数目的渐近均值和方差的校正项的递归公式，以及任意数目的高阶累积量的封闭形式渐近表达式。利用这些累积量，我们提出了新的互信息分布的封闭近似，该近似不仅在总体上，而且在停机概率感兴趣的尾部区域都非常准确。
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 Painleve differential equations. This turns out to be a Painleve V for the single-user MIMO scenario and a Painleve 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 Painleve 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 Painleve, 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.