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I'm having trouble finding this sum

  1. Aug 7, 2011 #1
    1. The problem statement, all variables and given/known data

    find

    [tex]\sum_{-N1}^{+N1}e^{-j\omega n}[/tex]

    2. Relevant equations



    3. The attempt at a solution

    Let [tex]\lambda = e^{-j\omega} [/tex]

    we have

    [tex]\sum_{-N1}^{+N1}\lambda ^{n} = \sum_{-N1}^{-1}\lambda ^{n} + \sum_{0}^{+N1} \lambda ^{n}[/tex]

    for the first i have

    [tex]S = \lambda ^{-N1} + \lambda ^{-N1+1} + \lambda ^{-N1+2} + ... + \lambda ^{-2} + \lambda ^{-1}[/tex]

    [tex]-\lambda S = -\lambda ^{-N1+1} - \lambda ^{-N1+2} - \lambda ^{-N1+3} - ... - \lambda ^{-1} - \lambda ^{0} [/tex]

    hence

    [tex] S = \frac{\lambda ^{-N1} - 1}{1-\lambda }[/tex]

    for the second i have

    [tex] S2 = \lambda^{0} + \lambda^{1} + ... + \lambda^{N1-1} + \lambda^{N1}[/tex]
    [tex] -\lambda S2 = -\lambda^{1} - \lambda^{2} - ... - \lambda^{N1} - \lambda^{N1+1}[/tex]

    hence

    [tex] S2 = \frac{1 - \lambda^{N1+1}}{1-\lambda}[/tex]

    so the sum is

    [tex]\frac{1-\lambda^{N1+1} + \lambda^{-N1} - 1}{1-\lambda} = \frac{\lambda^{-N1} - \lambda^{N1+1}}{1-\lambda} = \frac{e^{j \omega N1} - e^{-j \omega N1}e^{-j\omega}}{1-e^{-j\omega}}[/tex]

    but the book says [tex]\frac{sin\omega(N1 + \frac{1}{2})}{sin(\frac{\omega}{2})}[/tex]
     
  2. jcsd
  3. Aug 7, 2011 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Multiply numerator and denominator by e^(j*w/2). Now remember e^(jx)-e^(-jx)=2jsin(x).
     
  4. Aug 7, 2011 #3
    thanks a lot :)
     
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