When to use Laplace or Fourier Transform for Circuit Analysis

Hey Guys,

so I have an electronic system, whose exact structure is unknown. I can however measure what it does to a specific input. This means I have two vectors, which represent current/voltage signals for the input/output. Theoretically I could represent the transfer function of the system in the Laplace-Domain if I knew all the components. Since I do not have an analytic functions for the output, I can only solve this problem numerically. Matlab however does not provide a method for a numerical Laplace transform. That's why I have been using the FFT in order to get the transfer function (Dividing FFT of the output by FFT of input). Is this even legit? Can a transfer function of such a system be represented in the Fourier domain? Or do I lose some information by doing so? Does someone have an alternative solution to this problem? Basically I would like to get an analytical representation of the system's transfer function. Thanks!