Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

6.2 CHARACTERIZATIONS OF A RANDOM PROCESS

A random process can be defined in a manner similar to that used for a random variable. Assume there exists a sample space containing all possible outcomes of duration of the random process. We denote an outcome by and use this notation for a random process for two reasons: (i) to differentiate it from used previously for the outcomes of an abstract sample space and (ii) to avoid confusion when we consider sinusoidal signals and Fourier transforms based on radian frequency ω. Each outcome of is a deterministic function of time and is called a realization of the random process. We assume there is an underlying probability space that defines the random process. For outcome , represents the realization. The ensemble of realizations, which together ...