  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.
