Sample and hold - LTI filter

Question

I've found a material explaining the sample and hold operation pretty well, but there's one thing I can't understand.

http://www.springer.com/cda/content/document/cda_downloaddocument/9781848001183-c1.pdf?SGWID=0-0-45-484012-p173780368

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.

Answer 1

An impulse of "area" $A$ means $A\delta(t)$. The nomenclature arises because $\displaystyle \int_{-\infty}^\infty A\delta(t) \,\mathrm dt = A,$ that is, the "area under the curve" is $A$.

Since the system is linear and time-invariant, and the response to input $\delta(t)$ is a pulse of height $1$ and duration from $0$ to $T_s$, the response to $A\delta(t)$ is a pulse of height $A$ and duration from $0$ to $T_s$. Its "area" is $AT_s$, that is, $A$ multiplied by the area $T_s$ of the response to the unit impulse.

Answer 2

Some little bits of thought:

• the system behaves constantly at any time, turning an impulse into a rectangle => it is time invariant. One can also say that the system needs to transform an impulse into a rectangle but does not now when it will arrive;
• linear, because if two impulses are superimposed then the heights will be added, not mixed or masked in any other way;
• the time range between t and Ts should be obvious;
• the desired output is the height of the impulse (which carries the information), but this value needs to be kept during a certain amount of time Ts to be able to be observed. Thus, what the sample-and-hold should output is a rectangle height A and width Ts. The sample part needs to detect the amplitude A of the signal, while the hold part need to maintain it during Ts seconds.