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Simple C-T Signal Energy

  1. Oct 8, 2009 #1
    1. The problem statement, all variables and given/known data
    Given two constants, A and B, what is the energy of the following signal?

    [tex]f(t) = A*rect(t) + B*rect(t-0.5)[/tex]


    2. Relevant equations
    [tex]E_f = \int_{-\infty}^{\infty} |f(t)|^2[/tex]



    3. The attempt at a solution
    [tex]E_f = \int_{-\infty}^{\infty} [A*rect(t) + B*rect(t-0.5)]^2 dt[/tex]
    [tex]= \int_{-\infty}^{\infty} [A^2*rect^2(t) + 2AB*rect(t)rect(t-0.5) + B^2rect^2(t-0.5)] dt[/tex]
    [tex]= A^2\int_{-\infty}^{\infty} rect^2(t) dt + 2AB\int_{-\infty}^{\infty} rect(t)rect(t-0.5) dt + B^2\int_{-\infty}^{\infty} rect^2(t-0.5) dt[/tex]
    [tex]= A^2 + 2AB + B^2[/tex]
    [tex]= (A + B)^2[/tex]

    This seems wrong to me somehow; I guess my real question is does [tex]\int_{-\infty}^{\infty} rect^2(\frac{t}{\tau}) = \tau[/tex]?
     
  2. jcsd
  3. Oct 8, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    What do you mean by "rect(x)"?
     
  4. Oct 8, 2009 #3
    [tex]rect(t) = \left\{
    \begin{array}{11}
    0 & \mbox{if } |t| > \frac{1}{2} \\
    \frac{1}{2} & \mbox{if } |t| = \frac{1}{2} \\
    1 & \mbox{if } |t| < \frac{1}{2}
    \end{array}
    \right.[/tex]
    http://en.wikipedia.org/wiki/Rectangular_function
     
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