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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 $< 0 $" align="middle" border="0" height="34" width="50" /> and $ \lambda \geq 0$.

Copyright 2008, Todd Moon. Cite/attribute Resource. admin. (2006, June 13). Homework Assignments. Retrieved November 23, 2009, 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
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