I am looking at the H(f) which is the frequency response of an LTI system. What kind of relationship should I expect between the real and imaginary part of this frequency response? Here is an example of the real and imaginary part from experimental measurement. Is there any way to prove if the system is LTI or not?
It is only because the system is LTI that you can talk about a frequency response. If the system were not LTI, the notion of "frequency response" would be either meaningless or - if defined appropriately - it would be a function of frequency and time (as is the case for linear but time-varying systems).
The sole fact that a system is LTI does not imply any relation between the real and imaginary parts of the frequency response. Such a relation exists if the LTI system is causal. In that case, the real and imaginary parts of the frequency response are related by the Hilbert transform.
