On the construction of a random process with given power spectrum and probability density function

Often random processes with a given probability density function (pdf) and a given power spectrum are desired. Usually a discrete-time linear system is used to shape the power spectrum of an i.i.d. excitation process in the desired manner but there does not exist a solution to simultaneously adjust the output pdf. In this paper a method is introduced to adjust the pdf at the output of a given linear system to almost any desired shape by computing the required pdf of the excitation process, which can easily be shaped by a memoryless nonlinear system without any impact on the power spectrum. The method is based on Hermite polynomials to obtain an approximation of the pdf, cumulants to describe the effect of the linear system on the pdf, and a combinatorial algorithm to complete the Hermitian coefficients from cumulants, and vice versa. A numerical example gives evidence of the practical applicability.