TABLE OF CONTENTS Models for random message signals and noise encountered in communication systems are developed in this chapter. Random signals cannot be explicitly described prior to their occurrence, and noises cannot be described by deterministic functions of time. However, when observed over a long period, a random signal or noise may exhibit certain regularities that can be described in terms of probabilities and statistical averages. Such a model, in the form of a probabilistic description of a collection of functions of times, is called a random process.
Consider a random experiment with outcomes ? and a sample space S. If to every outcome ? ? S we assign a real-valued time function X( t, ?), we create a random (or stochastic) process. A random process X( t, ?) is therefore a function of two parameters, the time t and the outcome ?. For a specific ?, say, ? i, we have a single time function X( t, ? i) = x i( t). This time function is called a sample function. The totality of all sample functions is called an ensemble. For a specific time t j, X( t j, ?) = X j denotes a random variable. For fixed t(= t j) and fixed ?(= ? i), X( t j,
