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Basic Concepts of Random Processes

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

Cases when $R_X(\tau)$ does not have a transform


\begin{example}
Recall the random sinusoid had
\begin{displaymath}R_X(\tau) = \...
...as opposed to $\delta$ functions), there is no Fourier
transform.
\end{example}
We examine this and other cases that are WSS but do not have a Fourier transform (in the conventional sense).

  1. Suppose $T = \Rbb$ . Then $R_X(\tau)$ is continuous and is the autocorrelation function of a WSS r.p. if and only if there is a c.d.f. $G_X$ satisfying $G_X(b) = 1-G_X(b)$ such that

    \begin{displaymath}R_X(\tau)/R_X(0) = \int_{-\infty}^\infty e^{i \omega \tau}
dG_X(\omega).
\end{displaymath}

    This transform is called the Fourier-Stieltjes transform.
  2. Suppose $T = \Zbb$ . Then $R_X(k)$ is the autocorrelation function of a WSS r.p. if and only if there exists a $G_X$ satisfying $G_X(b) = 1-G_X(b)$ such that

    \begin{displaymath}R_X(k)/R_K(0) = \int_{-\pi}^\pi e^{i \omega k} dG_X(\omega)
\end{displaymath}

Thus the a.c.f. acts like a characteristic function, but also has symmetry.

If $S_X(\omega)$ exists, then

\begin{displaymath}G_X(\omega) = \int_{-\infty}^\infty S_X(\zeta)d\zeta/(2\pi R_X(0)).
\end{displaymath}

In this case, $S_X(\omega)/2\pi R_X(0)$ is, in fact, a p.d.f.

More generally, $2\pi R_X(0)G_X$ is a spectral distribution of $\{X_t\}$ .
\begin{example}
Random sinusoid.
\begin{displaymath}R_X(\tau)/R_X(0) = \cos(\o...
...mega_0) + u(\omega-\omega_0)]
\end{displaymath}(right-continuous).
\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_9.htm. This work is licensed under a Creative Commons License Creative Commons License