Revision as of 11:26, 26 August 2011

Simplify this summation

$ \sum_{n=-\infty}^\infty n \delta [n]  $

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Answer 1

The answer is 0 ?

Instructor's comment: Yes, it's zero. Can you justify your answer? -pm

Answer 2

The delta function is zero everywhere except when n=0 and since we are multiplying the delta by n the answer would thus be 0.

Instructor's comment: Yes, that's the idea. Now can you justify your answer "in math" instead of "in words"? -pm

Answer 3

The answer is zero since impulse function is 0 everywhere except n = 0.

Instructor's comments: Ok, I guess I am going to have to be a bit more specific. I would like to see a way to answer this question as a sequence of small changes to this expression until you get to zero. Something like

$ \begin{align} \sum_{n=-\infty}^\infty n \delta [n] &= \text{ blah } \\ &= \text{ blih} \\ &= 0 \end{align} $

Can you try that? -pm

Answer 4

$ \begin{align} \sum_{n=-\infty}^\infty n \delta [n] &= ...(-2)\delta[-2] + (-1)\delta[-1] + 0\delta[0] + \delta[1] + 2\delta[2] ... \\ &= ...(-2)\cdot 0 + (-1)\cdot 0 + 0\cdot 1 + 0 + 2\cdot 0... \\ &= ...0 + 0 + 0 + 0 + 0... \\ &= 0 \end{align} $

Instructor's comments: It's awesome to see Purdue alums participate in this collective problem solving! As expected from a graduate of our ECE program, this solution does not contain any mistake. Now a challenge: can anybody do it without using any "dot dot dot"? -pm

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