Personal tools
  •  
You are here: Home Electrical and Computer Engineering Stochastic Processes Homework Assignments

Homework Assignments

Document Actions
  • Content View
  • Bookmarks
  • CourseFeed

Homework 10

  1. Suppose $ \{X_t,t \geq 0\}$ is a Wiener process. Define a process $ \{Y_t, t \geq 0\}$ by

    $\displaystyle Y_t = X_{t+D} - X_t
$

    for a fixed positive number $ D$ .
    1. Find the mean and autocorrelation functions of $ \{Y_t\}$ .
    2. Show that $ \{Y_t\}$ is stationary and find its spectrum.

  2. Suppose $ \{X_t\}$ is $ \{Y_t\}$ are zero mean and individually and jointly W.S.S. Show that the mean-square error associated with the noncausal Wiener filter for estimation of $ X_t$ from $ \{Y_t, t
\in \Rbb\}$ is

    $\displaystyle \frac{1}{2\pi}\int{-\infty}^\infty [S_X(\omega) -
\frac{\vert S_{XY}(\omega)\vert^2}{S_Y(\omega)}]\,d\omega.
$

  3. Suppose $ Y_t = S_t + N_t$ for $ t \in \Rbb$ , where $ \{S_t\}$ and $ \{N_t\}$ are zero-mean, W.S.S., and orthogonal. Suppose that we wish to estimate

    $\displaystyle X_t = \int_{-\infty}^\infty k(t-\tau)S_\tau \,d\tau
$

    with an estimate of the form

    $\displaystyle \Xhat_t = \int_{-\infty}^\infty h(t-\tau)Y_\tau\,d\tau,
$

    where $ k$ and $ h$ are impulse responses of linear time-invariant systems. Show that

    $\displaystyle E[(X_t - \Xhat_t)^2] = \frac{1}{2\pi}
\int_{-\infty}^\infty[\vert...
... - H(\omega)\vert^2 S_S(\omega) +
\vert H(\omega)\vert^2 S_N(\omega)]\,d\omega
$

    where $ K$ and $ H$ are the transfer functions of $ k$ and $ h$ , respectively, and $ S_S$ and $ S_N$ are the power spectral densities of $ \{S_t\}$ and $ \{N_t\}$ . (Note the case that $ k(t) =
\delta(t-\lambda)$ for some fixed $ \lambda \in \Rbb$ .
  4. Consider the situation of the previous problem with $ k(t) =
\delta(t-\lambda)$ ,

    $\displaystyle S_S(\omega) = \frac{A^2}{\alpha^2 + \omega^2} \qquad \qquad
S_N(\omega) = \frac{N_0}{2}
$

    1. Find the noncausal Wiener filter for estimating $ X_t$ from $ \{Y_t, t
\in \Rbb\}$ . Find the corresponding mean-square error.
    2. Find the causal Wiener filter for estimating $ X_t$ from $ \{Y_\tau, \tau \leq t\}$ . Consider $ \lambda < 0 $ and $ \lambda
\geq 0$ .

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/hw10.html. This work is licensed under a Creative Commons License Creative Commons License