Fundamentals of Applied Probability and Random Processes, 2nd Edition

10.8 Discrete-Time Random Processes

All the discussion thus far assumes that we are dealing with continuous-time random processes. In this section we extend the discussion to discrete-time random processes {X[n], n = 0, 1, 2, …}, which are also called random sequences. A discrete-time random process can be obtained by sampling a continuous-time random process. Thus, if the sampling interval is T_S, then for such a case we have that

$\mathit{X}\left[\mathit{n}\right]=\mathit{X}\left(\mathit{n}{\mathit{T}}_{\mathit{S}}\right)\mathit{n}=0,\pm 1,\pm 2,\dots$

We provide a summary of the key results as follows.

10.8.1 Mean, Autocorrelation Function and Autocovariance Function

The mean of X[n] is given by

${\mathit{\mu }}_{\mathit{X}}\left[\mathit{n}\right]=\mathit{E}\left[\mathit{X}\left[\mathit{n}\right]\right]$

The autocorrelation function is given ...