# How to construct a LTI system with a specific impulse response function, say $h(t) = \delta '(t)$?

I want to ask the question on a generic basis, but using $$\h(t) = \delta '(t)\$$ as my example.

I have Mathematical knowledge on the LTI system, but no hands on experience. In real life, how can we construct a LTI system with specific impulse response function? Say for $$\h(t) = \delta '(t)\$$, I know it is a differentiator, so I would search for "differentiator" (e.g. this one) regarding how to construct it.

How about the other form of $$\h(t)\$$? Is there a generic way to construct LTI system for a specific family of $$\h(t)\$$ (say the family of linear combination of $$\sinc(t)\$$, $$\\delta (t)\$$, $$\\delta '(t)\$$ and other common functions)? Or I need to construct it in an adhoc way for every different $$\h(t)\$$?

Many thanks!

First, you can construct possible systems. That means that getting an impulse response of $$\h\left(t\right) = \delta'\left(t\right)\$$ is right out, because it's non-causal. For that matter, a system that truly has impulse response $$\h\left(t\right) = \delta\left(t\right)\$$ is also out, because anything real responds in a finite amount of time.
In general, if you can start with your desired $$\h(t)\$$ and break it into a weighted sum of impulse responses that you know how to realize in circuitry, then you can generally move forward. Things of the form $$\h(t) = t^n e^{at}\$$ are particularly easy, because (as long as you allow $$\a\$$ to be complex) that's the form that can be realized by an appropriate lumped-element circuit.
You can, in theory, implement approximations of arbitrary $$\h(t)\$$ by making tapped delay lines and summing their outputs -- but that's kind of an expensive wacky old radar technology, not something that's of much use today (unless it's in really expensive, fast radars).