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Convolution Of Step function

  1. May 22, 2012 #1
    I am trying to see if I understand the convolution process correctly.
    This is from the solved example of my prof's notes:

    x[n] = u[n]
    and h[n] = a-nu[-n] for 0<a<1

    As expected the first step was

    y[n] = x[n]* h[n]

    = Ʃ h[k]x[n-k] -∞<k<∞
    = Ʃ a-ku[-k]u[n-k] -∞<k<∞

    From my understanding u[-k] = 0 for all k > 0 so that would give us the upper limit of k to be 0
    And we would also need for n-k > 0 and when we have those conditions we would just then be left with the summation of a^-k since the other 2 would always evaluate to 1 in that interval.

    But for the solution the cases considered were when n <= 0 and when n >0 and then the answers were

    Ʃ ak -n<k<∞ for the first case (n<=0)
    and Ʃak 0<k<∞

    I dont understand how they came to those intervals. Any help would be appreciated.
     
  2. jcsd
  3. May 22, 2012 #2
    I think i have it figured out.
    I will post my understanding and then maybe someone can comment on whether I was right.
    My idea of k<0 was right but only if n > 0 which would make the second term u[n-k] always positive also giving us a SUM a^-k and then we flip the summation around cause we change the sign and end up with the answer noted for n>0

    for n<0 for n-k >= 0 we need -k >= n or k<= -n
    and then we end up with SUM a^-k and then again flip the limits and end up with the answer noted for n<0
     
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