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Homework Seven

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    1. Chapter 3, Chapter 4.
  • Homework:
    1. (10 pts) Consider the following basis functions:
      $\displaystyle x_1(t) = u(t)-u(t-1) $

      $\displaystyle x_2(t) = u(t-1)-u(t-2) $

      $\displaystyle x_3(t) = u(t-2)-u(t-3) $
      $\displaystyle x_4(t) = u(t-3)-u(t-4) $

      1. Plot the basis functions and satisfy yourself that they are orthonormal.
      2. For the functions
        $\displaystyle f_1(t) = 2u(t) - 3 u(t-1) + u(t-4) $

        $\displaystyle f_2(t) = -2u(t) + 3u(t-1) - u(t-3) $

        $\displaystyle f_3(t) = u(t)-2u(t-1)+2u(t-2)-2u(t-3)+u(t-4) $

        $\displaystyle f_4(t) = u(t) - 3u(t-1)+4u(u-3) - 2u(t-4) $

        defined over the interval $ 0 \leq t \leq 4$ :
        1. Plot the functions.
        2. Express each function $ f_i(t)$ as a linear combination of the basis functions $ x_i(t)$ .
    2. (10 pts) Problem 3.4-7(a)
    3. (5 pts) Problem 3.4-9(a,c,e)
    4. Problem 3.4-11. (There is a mistake in the book. There is nothing to do in this problem, just read it carefully. You should understand that there are such things as Walsh functions.)
    5. (5 pts) Problem 3.4-12(a)
    6. (10 pts) Problem 3.5-4
    7. (5 pts) Problem 3.5-6
    8. (10 pts) A signal $ f_0(t) = \sin t$ passes through a full-wave rectifier, producing the signal $ f(t) = \vert\sin(t)\vert$ . This, in turn, passes through a RC lowpass filter with $ C=1$ F and $ R=1 \Omega$ .
      1. Determine a Fourier series representation for $ f(t)$ .
      2. Determine the Fourier series for the output of the filter, $ y(t)$ .
      3. Determine an expression for the rms power in the ripple. How does this compare to the total power in $ y(t)$ ?
    9. Problem 4.1-1.
Copyright 2008, by the Contributing Authors. Cite/attribute Resource. admin. (2006, June 14). Homework Seven. Retrieved November 23, 2009, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Signals_and_Systems/hw7.html. This work is licensed under a Creative Commons License. Creative Commons License
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