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Basic Concepts of Random Processes

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

A Sinusoidal Process

Let $T = \Rbb$ . Assume $A$ and $\theta$ are independent r.v.s, with $E[A^2] < \infty$ and $\theta \sim \Uc(-\pi,\pi)$ . Define $\{X_t, t\in T\}$ by

\begin{displaymath}X_t = A \sin(\omega_0 t + \theta)
\end{displaymath}

where $\omega_0$ is a known constant.

A typical realization is a sinusoid.

This is an example of a deterministic random process, that is, a random process determined by random parameters.

\begin{displaymath}\begin{aligned}
\mu_X(t) &= E[A \sin(\omega_0 t + \theta)] = ...
...{-\pi}^\pi \sin(\omega_0 t + \theta)
d\theta = 0.
\end{aligned}\end{displaymath}

We observe that this process is ergodic in the mean -- a time average is equal to the ensemble average.


\begin{displaymath}\begin{aligned}
R_X(t,s) &= E[X_t X_s] = E[A^2 \sin(\omega_0...
...ta \\
&= E[A^2] \frac{1}{2} \cos(\omega_0(t-s)).
\end{aligned}\end{displaymath}

We observe that $R_X(t,s)$ depends only on the time difference $t-s$ . Hence, the r.p. is WSS. Let $\tau = t-s$ . We can write

\begin{displaymath}R_X(\tau) = E[A^2] \frac{\cos(\omega_0 \tau)}{2}.
\end{displaymath}

Checking the properties, observe that we have a (local) maximum at $\tau = 0$ , and that the function is symmetric.
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_4.htm. This work is licensed under a Creative Commons License Creative Commons License