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Basic Concepts of Random Processes

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

Properties of Spectra

  1. Symmetry:

    \begin{displaymath}S_X(\omega) =
\begin{cases}
\int_{-\infty}^\infty \cos(\ome...
...k=-\infty}^\infty \cos(k \omega) R_X(k) & T = \Zbb.
\end{cases}\end{displaymath}

  2. $S_X(\omega) = S_X(-\omega)$ . $S_X(\omega) = S_X^*(\omega)$ .
  3. Inverse:

    \begin{displaymath}\begin{aligned}
R_X(\tau) &= \frac{1}{2\pi} \int_{-\infty}^\i...
...ty}^\infty \cos(\omega
\tau)S_X(\omega) d\omega.
\end{aligned}\end{displaymath}


    \begin{displaymath}\begin{aligned}
R_X(\tau) &= \frac{1}{2\pi} \int_{-\pi}^\pi e...
...y}^\infty \cos(\omega \tau)
S_X(\omega) d\omega.
\end{aligned}\end{displaymath}

  4. If $T = \Rbb$ ,

    \begin{displaymath}R_X(0) = E[X_t^2] = \frac{1}{2\pi} \int_{-\infty}^\infty
S_X(\omega) d\omega.
\end{displaymath}

    if $T = \Zbb$ :,

    \begin{displaymath}R_X(0) = \frac{1}{2\pi} \int_{-\pi}^\pi S_X(\omega) d\omega.
\end{displaymath}

  5. $S_X(\omega) \geq 0$ for all $\omega$ . (This follows from the non-negative definiteness of $R_X$ .

    Any symmetric non-negative definite function having a finite integral is a legitimate spectral density.

    Observe that being nnd and finite integral is analogous to a probability density, so that makes $R_X$ analogous to a characteristic function.

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_8.htm. This work is licensed under a Creative Commons License Creative Commons License