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Basic Concepts of Random Processes

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

More properties and definitions

Recall that a second-order process is WSS if its mean function is constant and its autocorrelation function depends only on the difference of its arguments.


\begin{definition}
Suppose $T = \Rbb$. The {\bf power spectral density} (or {\b...
...ega k} R_X(k),}
\end{displaymath}assuming the transform exists.
\end{definition}
A sufficient condition for the existence of $S_X$ is that $\int_{-\infty}^\infty \vert R_X(\tau)\vert d\tau < \infty$ or $\sum_{k=-\infty}^\infty \vert R_X(k)\vert < \infty$ .


\begin{example}
Suppose $R_X(\tau) = \sigma^2 e^{-\beta\vert\tau\vert}$. Then
\...
...e-band (e.g., nearly
white) noise which has been lowpass filtered.
\end{example}

\begin{example}
The Ideal Low Pass Process. Suppose
\begin{displaymath}S_X(\ome...
...$, etc. If the
signal were Gaussian, it would also be independent.
\end{example}

\begin{example}
Let $R_X(k) = \sigma^2 r^{\vert k\vert}$ for $\vert r\vert < 1...
... \frac{\sigma^2(1-r^2)}{1-2 r \cos \omega + r^2}.
\end{displaymath}\end{example}

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_7.htm. This work is licensed under a Creative Commons License Creative Commons License