That is a tricky problem, isn't it? I ran into a similar problem yesterday. Here is what happened. Yesterday, I wanted to learn how to build a boat. I went to the bookshelf and pulled out a book. It turned out to be a philosophy book and, therefore, I was immediately getting furious. How could it be ... "philosophy"? It does not talk about "boat" at all. I started throwing chairs, dishes, ..., and I almost set the whole house on fire. Suddenly, my 4 - year old son came in and said :"Daddy, put down that wrong book and find the right one, ok?" ...Oh, yeah, ok, ...problem solved ...
So, got a good signal, huh? Not too big and staying in a finite range? And want to take the laplace transform? Unfortunately, the books you got only talk about the laplace transforms of δ(t), δ'(t) functions? So .... may I have my baby's words for you:"put down those wrong books and find the right one, ok?"