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Signal Windowing Help

  1. Nov 6, 2011 #1
    1. The problem statement, all variables and given/known data

    [itex]p[n]=x[n]w[n][/itex], where [itex]w[n][/itex] is the rectangular window.



    [itex]x[n]=\sum_{k=-∞}^{∞} δ[n-k][/itex]

    [itex]w[n]= 1,[/itex] for [itex]-M≤n≤M;[/itex]
    [itex] = 0[/itex] otherwise​



    1. What is [itex]X(e^{jw})[/itex]?
    2. What is [itex]P(e^{jw})[/itex] when...
    M=1?
    M=10?​

    [itex]j[/itex] is the imaginary number. (it's the same as [itex]i[/itex].)

    2. Relevant equations
    N/A


    3. The attempt at a solution

    [itex]X(e^{jw})=2\pi\sum_{k=-∞}^{∞}δ[n-2\pi k][/itex]

    I found [itex]X(e^{jw}[/itex] fairly easily, but I can't find [itex]P(e^{jw})[/itex] for either case of M. The answers from the book are below. Can someone tell me how I solve for this? I don't see why [itex]X(e^{jw})[/itex] is so different from [itex]P(e^{jw})[/itex]

    [itex]P(e^{jw})=e^{jw}+1+e^{-jw}=1+2cos(\omega)[/itex] when M=1
    [itex]P(e^{jw})=\frac{sin(21 \omega/2}{\omega/2}[/itex] when M=1
     
  2. jcsd
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