I've found a material explaining the sample and hold operation pretty well, but there's one thing I can't understand.
Next, we would like to take this ideally sampled signal and hold the values between samples. As we have a train of impulses with the correct areas, we need a “block” that takes an impulse with area A, transforms it into a rectangular pulse of height A that starts at the time at which the delta function is input to the block, and persists for exactly Ts seconds. A little bit of thought shows that what we need is a linear, time-invariant (LTI) filter whose impulse response, h(t), is 1 between t = 0 and t = Ts and is zero elsewhere.
How do we know this is the impulse response that will give us a pulse of HEIGHT A, not area A? Could anyone explain this "a little bit of thought" part? Thanks in advance.