Common Pitfalls in the Application of Stationary Process Theory to Time-Sampled and Modulated Signals

The common practice of applying the theory of stationary stochastic processes to a cyclostationary process by introducing random phase(s) into the probabilistic model in order to stationarize the process can lead to erroneous results, such as incorrect formulas for power spectral density. This is illustrated by showing that commonly used formulas for signals that have undergone frequency conversion or time sampling can be incorrect. The source of error is shown to be inappropriate phase-randomization procedures. The correct procedure is described, and corrected formulas are given. The problem is further illustrated by showing that commonly used resolution and reliability (mean and variance) formulas for spectrum analyzers must be corrected for cyclostationary signals. It is explained that all corrections to formulas reflect the effects of spectral correlation. These effects are inappropriately averaged out by inappropriate phase-randomization procedures. It is further explained that these inappropriate procedures destroy the important property of ergodicity of the probabilistic model.