Post date: Feb 11, 2014 4:26:49 PM
Laplace Transforms exist only for physically realizable signals. For example, exp(t^2) which grows faster than an exponential does not have a Laplace transform defined. And it also cannot be physically generated.
But, if you restrict it from 0 to T, and then after that signal = 0, then Laplace transform will be defined.
Note: even a ramp function is not practically realizable (as it keeps growing with time). However, a Laplace transform is defined for a ramp. In that sense, the restriction is more in the sense of functions growing faster than an exponential.