Impulse Matching: Finding the Unit Impulse Response

[Corneo]

Impulse Matching

In regards to finding the unit impulse response of a system. We assume that x(t) = \delta (t) and that the intials conditions at t=0^_ are all zero. The impulse response h(t) therefore must consists of the systems's modes for when t \geq 0^+. But why is it that h(t) = A \delta(t) + \text{modes} for t \geq 0?

 

[phantom_photon]

I guess the simplest explanation would be that if the domain of the response is extended to include t = 0 (or 0-) it will have an additional component: the system's response to the impulse while the impulse is being applied, which is of course a scalar multiple of the Dirac delta function.
Since the delta function is non-zero for t = 0- and 0 but not for 0+, the response for t >= 0+ will not involve the delta function at all, just it's residual effect (the system's characteristic oscillations). You will generally only be concerned with the response for t > 0+.
That's what I think it is, but I might be wrong. I appologise for the lack of mathematical rigour in this post.