Impulse Response as Figured from Inverse Transform

Abstract

The impulse response is probably the most important response of a system! It is obtained by applying an impulse (delta) function input and observing the output(s). The impulse itself is a pulse of almost zero width and almost infinite height, subject to the requirement of unity area. The impulse has a very unique and interesting Laplace transform—that of 1! Hence the spectrum of the impulse is very simple and attractive to use. The impulse response itself, on the other hand, also has an interesting transform and it is nothing other than the transfer function H(s) itself. To repeat the Laplace transform of the impulse response is the transfer function H(s). So if we know the transfer function of the system we can figure the impulse response (in time) by simply taking the inverse transform of H(s)! We demonstrate this process on a handful of RLC circuits where we first derive the input/output transfer function (impedance one for example), then do inverse Laplace transform and arrive at the impulse response. Other than deriving the impulse response of those circuits we also shed some light into the meaning of the response, and examine it under various conditions. Another advantage of the impulse response is that once it is known we can figure the response due to another input by simply convolving that input with the impulse response h(t).