I wonder if the input signal (CT) violates Shannon-Nyquist Theorem for a given sampling rate, is there any chance for the overall system not to be LTI although discrete time system is LTI?
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Assuming a system like this:
x(t) -> C/D -> DSP -> D/C -> y(t)
where $x(t)$ and $y(t)$ are sufficiently bandlimited and the C/D and D/C blocks are ideal, then the system behaves as if it were LTI. Note that the system is not truly LTI, since it is not LTI for all inputs.
In particular, if the process of discretizing $x(t)$ produces aliases, then the output $y(t)$ can potentially contain frequencies not present in the input. Since an LTI system cannot create new frequencies, then in conclusion the system is not LTI.