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Homework 9

  1. Suppose $ \{X_t, t \in \Rbb\}$ is a random process with power spectral density

    $\displaystyle S_X(\omega) = \frac{1}{(1+\omega^2)^2}
$

    Find the autocorrelation function of $ X_t$ .
  2. Suppose that $ \omega$ is a random variable with p.d.f. $ f_\omega$ and $ \theta$ is a random variable independent of $ \omega$ uniformly distributed in $ (-\pi,\pi)$ . Define a random process by

    $\displaystyle X_t = a \cos(\omega t + \theta), \qquad t \in \Rbb
$

    where $ a$ is a constant. Find the power spectral density of $ \{X_t\}$ .
  3. Suppose events occur randomly in $ T = [0,\infty)$ in the following way:
    1. The numbers of events in nonoverlapping intervals are independent of one another.
    2. $ P($ exactly one event in  $ (t,t+\Delta t)) =
\lambda(t)\Delta t + o(\Delta t)$ , where $ \lambda(t)$ is a continuous nonnegative function on $ [0,\infty)$ .
    3. $ P($ more than one event in an interval of length  $ \Delta t)
= o(\Delta t)$ .
    Define a random process $ \{X_t, t \in T\}$ by $ X_0 = 0$ and $ X_t$ is the number of events occurring in $ (0,t]$ .
    1. For $ t > s \geq 0$ , show that $ (X_t - X_s)$ is a Poisson random variable with parameter $ \int_s^t \lambda(x) dx$ .
    2. Find the mean and autocorrelation functions of $ \{X_t\}$ .

  4. Suppose that $ \{X_t, t \in \Rbb\}$ is a w.s.s., zero-mean, Gaussian random process with autocorrelation function $ R_X(\tau),
\tau \in \Rbb$ and power spectral density $ S_X(\omega), \omega \in
\Rbb$ . Define the random process $ \{Y_t, t \in \Rbb\}$ by $ Y_t =
(X_t)^2, t \in \Rbb$ . Find the mean, autocorrelation, and power spectral density of $ \{Y_t, t \in \Rbb\}$ .
  5. Suppose $ U$ and $ V$ are independent random variables with $ E[U]
= E[V] = 0$ and $ \var(U) = \var(V) = 1$ . Define random processes by

    $\displaystyle X_t = U \cos t + V \sin T \qquad \qquad Y_t = U \sin t + V \cos t,
\qquad t \in \Rbb.
$

    Find the autocorrelation and cross-correlation function s of $ \{X_t, t \in \Rbb\}$ and $ \{Y_t, t \in \Rbb\}$ . Are $ \{X_t\}$ and $ \{Y_t\}$ jointly wide sense stationary? Are they individually wide sense stationary?

Copyright 2008, Todd Moon. Cite/attribute Resource . admin. (2006, June 13). Homework Assignments. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/hw9.html. This work is licensed under a Creative Commons License Creative Commons License