Finding the Probability Distribution Functions of $S$ -Parameters and Their Monte Carlo Simulation

This paper presents the probability distribution function (PDF) of the ratio of two random waves. This result is used to obtain the PDF of S-parameters random errors in magnitude (in decibels) and phase, which are the quantities that most engineers work with. These results are further used on the development of a Monte Carlo simulation method in order to predict the variability of frequency-domain measurements. Experiments are performed to identify and characterize frequency-domain random errors, such as instrument noise, connector repeatability, and calibration variations, in measurement systems. By comparing with real measurement data, it is shown that random-error effects can be accurately estimated by the PDF's obtained and the Monte Carlo technique.