chapter  3
45 Pages

Random variables, sequences, and stochastic processes

WithAlexander D. Poularikas, Zayed M. Ramadan

This chapter discusses Random variables, sequences, and stochastic processes for adaptive filtering primer. To obtain a formal definition of a discrete-time stochastic process, an experiment is considered with a finite or infinite number of unpredictable outcomes from a sample space, each one occurring with a probability. Next, by some rule a deterministic sequence is assigned to elements of the sample space. The sample space, the probabilities of each outcome, and the sequences constitute a discrete-time stochastic process or random sequence. A realization is one member of a set called the ensemble of all possible results from the repetition of an experiment. The process that produces an ensemble of realizations and whose statistical characteristics do not change with time is called stationary. For a WSS process, the correlation function asymptotically goes to zero and, therefore, one can find its spectrum using the discrete-time Fourier transform.