Introduction to the Laplace Transform

Given a continuous time LTI system with impulse response , recall that the response (if it's well defined) to the input signal is , where is the transfer function of the system. Now let's calculate the response to the input signal in terms of the impulse response:

From this, we see that the integral above in the square brackets must be the transfer function:

The right hand side of the above equation is known as the Laplace Transform of the signal . Thus, the transfer function of an LTI system is the Laplace transform of the system's impulse response. The region of convergence of the transfer function is thus the set of complex numbers for which the Laplace transform of the impulse response converges.