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Basic Concepts of Random Processes

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Definitions  ::  Ergodicity  ::  Autocorrelations  ::  Sinusoidal  ::  Poisson  ::  Gaussian  ::  Properties  ::  Spectra  ::  Cases  ::  Random  ::  Independent

Means and Autocorrelations


\begin{definition}
The {\bf mean function} of a r.p. $\{X_t, t \in T\}$ is
\b...
...splaymath}\mu_X(t) = E\{X_t\}, \qquad t \in T.
\end{displaymath}\end{definition}

\begin{definition}
The {\bf autocorrelation function} of a r.p. $\{X_t, t \in T...
...math}R_X(t,s) = E[X_t X_s], \qquad t,s, \in T.
\end{displaymath}\end{definition}

\begin{definition}
A random process is {\bf second order} if $E[X_t^2] < \infty$ for
all $t \in T$.
\end{definition}
For a second order r.p., $\vert\mu_x(t)\vert < \infty$ and $\vert R_X(t,s)\vert <
\infty$ for all $t, s, \in T$ .

Properties of Autocorrelation functions

  1. $R_X(t,t) = E[X_t^2]$ . (This is the second moment)
  2. $\vert R_X(t,s)\vert^2 \leq R_X(t,t) R_X(s,s)$ (Schwartz inequality)
  3. $R_X(t,s) = R_X(s,t)$ (symmetric)

Wide-sense stationarity


\begin{definition}
Let $T$ be closed under addition. A second order random pro...
..._x$ and $R_X(t+h,t)$ depends only on $h$ for all $t,h \in T$.
\end{definition}
Since $R_X(t+h,t)$ depends only on $h$ , we write (by an ``abuse of notation'')

\begin{displaymath}R_X(t+h,t) \equiv R_X(h)
\end{displaymath}

Thus,

\begin{displaymath}R_X(t,s) = R_X(t-s,0) = R_X(t-s)
\end{displaymath}

If a random process is second order and strictly stationary, it must also be WSS. On the other hand, if a process is WSS, it is not necessarily strictly stationary.
\begin{definition}
The {\bf autocovariance function} of a random process $\{X_t...
...ath}C_X(t,s) = \cov(X_t,X_s), \qquad t,s\in T.
\end{displaymath}\end{definition}
We say a process is covariant stationary if $C_X(t,s)$ depends only on $t-s$ , or, equivalently, $C_X(t+h,t)$ depends only on $h$ .

Properties of $R_X$ for WSS r.p.s

  1. $R_X(0) = E[X_t^2]$ (independent of $t$ )
  2. $\vert R_X(\tau)\vert \leq \sqrt{E[X_{t+\tau}^2]E[X_t^2]} = R_X(0)$ .
  3. $R_X(\tau) = R_X(-\tau)$ (even function)
  4. A defining property of these functions:

    \begin{displaymath}\sum_{k=1}^n \sum_{l=1}^n \alpha_k \alpha_s^* R_X(t_k - t_l) \geq 0
\end{displaymath}

    for all $t_1, \ldots, t_n \in T$ and all $\alpha_1, \ldots, \alpha_n$ and for all $n \in \Zbb^+$ .

    Any function with this property is a nonnegative definite function.

Before proceeding with more properties, a few examples.
Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, June 07). Basic Concepts of Random Processes. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lecture6_3.htm. This work is licensed under a Creative Commons License Creative Commons License