We consider a generalization of the Laplace transform of Poisson shot noise defined as an integral transform with respect to a matrix exponential. We denote this integral transform as the {\em matrix Laplace transform} given its similarity to the Laplace-Stieltjes transform. We establish that the matrix Laplace transform is in general a natural matrix function extension of the typical scalar Laplace transform, and that the matrix Laplace transform of Poisson shot noise admits an expression that is analogous to the expression implied by Campbell's theorem for the Laplace functional of a Poisson point process. We demonstrate the utility of our generalization of Campbell's theorem in two important applications: the characterization of a Poisson shot noise process and the derivation of the complementary cumulative distribution function (CCDF) of signal to interference and noise (SINR) models with phase-type distributed fading powers. In the former application, we demonstrate how the higher order moments of a linear combination of samples of a Poisson shot noise process may be obtained directly from the elements of its matrix Laplace transform. We further show how arbitrarily tight approximations and bounds on the CCDF of this object may be obtained from the summation of the first row of its matrix Laplace transform. For the latter application, we show how the CCDF of SINR models with phase-type distributed fading powers may be obtained in terms of an expectation of the matrix Laplace transform of the interference and noise, analogous to the canonical case of SINR models with Rayleigh fading.

翻译：本文考虑泊松散粒噪声拉普拉斯变换的一种推广形式，该形式被定义为关于矩阵指数的积分变换。鉴于其与拉普拉斯-斯蒂尔杰斯变换的相似性，我们将此积分变换称为{\em 矩阵拉普拉斯变换}。我们证明矩阵拉普拉斯变换在一般情况下是典型标量拉普拉斯变换的自然矩阵函数扩展，并且泊松散粒噪声的矩阵拉普拉斯变换所满足的表达式，与坎贝尔定理所隐含的泊松点过程拉普拉斯泛函表达式具有类比关系。我们在两个重要应用中展示了坎贝尔定理这一推广形式的实用性：泊松散粒噪声过程的特征刻画，以及具有相位型分布衰落功率的信干噪比（SINR）模型互补累积分布函数（CCDF）的推导。在前一应用中，我们展示了如何直接从泊松散粒噪声过程样本线性组合的矩阵拉普拉斯变换元素中获取其高阶矩。我们进一步说明如何通过该对象矩阵拉普拉斯变换首行元素之和，获得其CCDF的任意紧逼近与边界。对于后一应用，我们展示了具有相位型分布衰落功率的SINR模型CCDF，如何通过干扰与噪声的矩阵拉普拉斯变换的期望值表示，这与瑞利衰落条件下SINR模型的典型情形具有类比性。