| This thesis applies the theory of random processes (stochastic differential equations) and random processes with underlying structure (switched stochastic differential equations) to problems in electrical and biomedical engineering, specifically in the areas of propagation (satellite and maritime), cellular ion channel response, and performance of a digital mobile communication system in non-Gaussian noise. Within each topic, the use of random processes is fundamental to obtaining a solution that can be used for abstraction in other simulations, such as discrete event-based system simulation. The mathematical tools of Fokker-Planck and generally the Kramers-Moyal expansion are used to develop analytical solutions that are checked against known theoretical results and numerical simulation. Each application requires a different approach: e.g., the maritime antenna is a purely analytical solution, rain fade modeling in a satellite channel is a combination of data mining from published data sources along with a random process with structure framework. The chapter on ion channels incorporate multidimensional switching in a statistical mechanical environment and incorporates elements of path integration. Finally, the phase noise simulation solves the Kramers-Moyal expansion by solving for differential intensity coefficients in the wrapped phase space and then applies semi-analytical techniques to solve for BPSK performance in a compound noise environment. Introductory chapters and an appendix are provided for additional information outside of the primary application chapters themselves. |
