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For example, the intuitive explanation for this convolution integral is as follows:

Any input can be represented by using time shifted and weighted impulses in succession, and the output at any time t is the superposition of individual impulse responses obtained from each impulse.

This makes sense, but when I try plot it on paper, something looks wrong.

For input x(t)=1...(x>=0) and impulse response h(t)=e

^{-t}, output y(t)= 1-e

^{-t}

Input:

Impulse reaponse:

Output:

Here, output y(0)=0, but impulse response h(0)=1. What makes y(0)=0 then?

Also, if I pick any instant t=a, the superposition of all the exponentially decaying impulse reaponses at t=a does not seem to give me the correct value of y(a). Where is my mistake? I feel I am missing something very fundamental here.

Thanks!