@phusis - my experiments have been very simple. No special filtering or DSP. Mainly with desktop or small monitors. I just unplug the speaker without the amp from the speaker with the amp. This is a speaker-level connection. Then I plug the speaker without amp to my main amp. Often, it sounds better. I make no claim that this happens with more costly speakers. But it makes me think that ’affordable’ active speakers will often sound better with a better amp. The amp really matters. Of course this is the case with passive speakers too. And I think that the quality of the amp is more important than if it is placed in a speaker cabinet or outside it.
My experiments can be ’shot down’ since in a sense they are grossly unfair. The amps I’ve used for comparing are much more costly than the speakers. You cant get the sound from the Atma-sphere MA-1 or the Krell FPB600 from inside a compact active speaker. Not that I know of. So my only point, in describing the experiments, is to draw attention to the quality of the amp in the active speakers - I think this is often overlooked.
An argument for active speakers is that the amp and speakers can be more closely matched and tuned to each other. Yet I have not been gripped by this, with the low cost active speakers I have tried. Instead, the big amps just made the speaker sound better. Interestingly, this main effect was the same even with two quite different amps (tube, solid state). My guess is that ’matching’ in affordable active speakers is only approximate, "good enough", so and so many watts drive them to required volume. The amp and the matching are hopefully much better with mid to top level active speakers - I have not tried.