I recently built a simulation to study sampling, the effects of aliasing and the effects of anti-aliasing filters on the sampled signal.
For fundamental frequencies above the sample band it's obvious one sees 'imposters' in the sampled signal. Using an antialiasing filter I can eliminate imposters.
But if I rather impose a broadband noise (actually white noise) signal into the sampler then it doesn't make much difference whether the anti-aliasing filter is present or not. The peak to peak noise is the same in either case. Of course the bandwidth of the noise has changed.
But furthermore I would expect the (imposter) aliased broadband noise outside the sample band to be superimposed on the broadband noise that is genuinely passed in the sample band thus 'piling up' with a larger peak to peak level.
Why doesn't this happen?
I should mention that my simulation time step is in the MHz and my system under study in the 1 kHz range. So the system is virtually in a continuous world.