I understand that if you have a system that is linear and time invariant, that you can perform a Fourier transform on it. But that doesn't mean you need to Fourier transform it. Or does it? Is a linear, time invariant system equivalent to or in some way implies a Fourier transform? Or is the Fourier transform just an optional procedure for arbitrary purposes? What information is derived from the Fourier transform that might reproduce the original signal? And if you can reproduce the original signal, does that mean that the Fourier transform is in some way equivalent to the original signal? Or are there systems that are closed under Fourier transformation, whose functions are transforms of themselves?