I am looking at the description of LTI systems in the time domain.


Intuitively, I'd have guessed it would be the composition of the input function and some "system function".
$$ y(t) = f(x(t)) = (f\circ x)(t)$$
Where $x(t)$ is the input, $y(t)$ output and $f(x)$ a "system function".


Why is it not that way? Could such a "system function" be found for, say, an R-C-Circuit?


The actual output function y(t), is defined as
$$ y(t) = (h \* x)(t) $$
Where $h(t)$ is the response to a dirac impulse.
This is hard to grasp for me. Why is it so? I have looked at various explanations, drawings of rectangles becoming infinitely narrow, which I sort of understood, but it is still "hard to grasp"! I am looking for a simple explanation in one or two sentences here.


<http://en.wikipedia.org/wiki/LTI_system_theory>



Why is the output of an LTI system the convolution of the input funtion and the impulse response?
Well, it's just as simple as this: the output at any moment reflects the effect of the input at just that moment, plus the lingering effect after one second of the input from one second before, plus the lingering effect after two seconds of the input from two seconds before, plus the lingering effect after three seconds of the input from three seconds ago, etc. [And half a second and 2.8 seconds and so on as well...]. That is, the sum (or, rather, integral) over all t of "The input from t seconds ago" \* "The amount of lingering effect a unit of input contributes after t seconds". That is, precisely a convolution of the input signal and the impulse response.


