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Basic Concepts of Random Processes

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

Joint properties of Two Random Processes

Suppose we have two r.p.s $\{X_t, t\in T\}$ , $\{Y_t, t \in T\}$ .
\begin{definition}
The {\bf cross correlation} function is
\begin{displaymath}R_{XY}(t,s) = E[X_t Y_s].
\end{displaymath}\end{definition}
Properties of the cross correlation function:

  1. $R_{XY}(t,s) = R_{YX}(s,t)$ (symmetry)
  2. $\vert R_{XY}(t,s)\vert \leq \sqrt{R_X(t,t) R_Y(s,s)}$ (Schwartz inequality)
  3. $\vert R_{XY}(t,s)\vert \leq \frac{1}{2}[R_X(t,t) + R_Y(s,s)]$

\begin{definition}
The random processes $X_t$ and $Y_t$ are {\bf orthogonal} if
$R_{XY}(t,s) = 0$ for all $s,t \in T$.
\end{definition}

\begin{definition}
The {\bf cross-covariance} function of $X_t$ and $Y_t$ is
\begin{displaymath}C_{XY}(t,s) = \cov(X_t,Y_s).
\end{displaymath}\end{definition}

\begin{definition}
The random processes $X_t$ and $Y_t$ are {\bf jointly wide...
... $h \in T$. In this case, we write $R_{XY}(t+h,t) = R_{XY}(h)$.
\end{definition}
Properties of $R_{XY}(h)$ for jointly WSS:
  1. $R_{XY}(0) = R_{YX}(0)$
  2. $R_{XY}(h) = R_{YX}(-h)$
  3. If $X$ and $Y$ are individually WSS, then $\vert R_{XY}(h)\vert \leq \sqrt{R_X(0)R_Y(0)}.$ and $\vert R_{XY}(h)\vert \leq
\frac{1}{2}[ R_{X}(0) + R_Y(0)].$

\begin{definition}
If $X_t$ and $Y_t$ are jointly WSS random processes, the {...
...{-i \omega k} R_{XY}(k) & T = \Zbb.
\end{cases}\end{displaymath}\end{definition}
Properties of Spectra:
  1. $S_{XY}(\omega) = S_{XY}^*(\omega).$
  2. If $X_t$ and $Y_t$ are individually and jointly WSS then

    \begin{displaymath}S_{XY}(\omega)\vert^2 \leq S_X(\omega) S_Y(\omega).
\end{displaymath}

  3. $\Real S_{XY}(\omega) = \Real S_{XY}(-\omega)$ and $\Imag(S_{XY}(\omega)) = -\Imag(S_{XY}(\omega)).$
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_10.htm. This work is licensed under a Creative Commons License Creative Commons License