When a random process is applied to a linear system, the output is also a random process. Even if the probability distribution of the input signal is known, it is, in general, difficult to determine the probability distribution of the output signal. It is, however, possible to determine some major characteristics of the output random process. The focus of this chapter is on the analysis and processing of random signals in linear systems.
12.1 Stochastic Continuity, Differentiation, and Integration
In this section, the continuous‐time continuous‐valued random processes are exclusively discussed. In system analysis, as a sample function of a random process is a deterministic signal, the system output is a sample function of another random process. However, in order to determine the output random process when the input is a random process, probabilistic methods, to address continuity, differentiation, and integration of an ensemble of sample functions as a whole, are required. Our focus is on the mean‐square sense, as it provides simple tractability along with useful applications in the study of linear systems with random inputs.
12.1.1 Mean‐Square Continuity
A random process X(t) is said to be continuous in the mean‐square or mean‐square continuous, if we have
If all sample functions of a random process are continuous at ...