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Basic Concepts of Random Processes

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

Gaussian Random Processes


\begin{definition}
The {\bf Gaussian random process} is a r.p. all of whose fin...
...x}\end{displaymath}is a multidimensional Gaussian distribution.
\end{definition}
For G.R.P. the entire distribution is completely determined by the mean

\begin{displaymath}\begin{bmatrix}\mu_{X}(t_1)  \vdots \mu_X(t_n)
\end{bmatrix}\end{displaymath}

and the covariance

\begin{displaymath}\begin{bmatrix}
\cov(X_{t_1},X_{t_1}) & \cov(X_{t_1},X_{t_2})...
...(X_{t_n},X_{t_2})&\cdots & \cov(X_{t_n},X_{t_n})
\end{bmatrix}\end{displaymath}

which are determined by $\mu_X(t)$ and $R_X(t,s)$ . That is, the entire distribution is determined by just the first two moments. It follows, therefore, that a WSS Gaussian process is also strictly stationary.
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_6.htm. This work is licensed under a Creative Commons License Creative Commons License