Convolution Of Step function

  • Thread starter MalB
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  • #1
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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.
 

Answers and Replies

  • #2
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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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