There’s the power-DAC with the digital input going all the way to the analogue conversion at the amp’s output stage, but other than that an active speaker with digital input just refers to the use of a DSP unit which can then receive a digital or analogue signal from the DAC. With an analogue input to the DSP from the DAC an A/D to D/A conversion is needed, but from all I’ve heard this extra conversion stage is of no consequence to the sound. To all of you who believe to the contrary here I might as well suspect it’s simply something it says "on paper."
Active per definition means filtration prior to amplification on signal level (as opposed to passive ditto on the output side of the amp); not necessarily that it’s a bundled package. So, as a separate component solution and per the above my own active-via-DSP setup fits the description.
My DSP unit is a Xilica XP-3060. It has no digital input, so an A/D to D/A conversion is taking place before sending the analogue output signal to three amps, one covering ~600Hz on up (Belles SA30), another ~85 to 600Hz (Lab.Gruppen FP6400) and the last one from ~85Hz on down (Crown Audio K2). Main speakers are Electro-Voice TS9040D LX pro cinema speakers, and the two subs are 15"-loaded tapped horns (20 cf. volume per cab, tuned @23Hz). Filter slopes are L-R 36dB/octave throughout, and a 36dB/octave BW high-pass on the subs @20Hz.
To me going active this way has been revelatory. I can’t relate to the negative and not least reductive remarks by poster @teo_audio above on the use of DSP in a home setting, but there are many ways to configure such a system, so if you want to make general assertions here be specific with regard to the contexts in which you’ve made your observations, or else it has no bearing to me.
As implemented in my setup I find the results to be highly satisfactory. Btw, to those who don’t believe pro cinema speakers can set a sonically high bar in a home setting, well.. have a go at it before any judgement is made (the DH1A compression driver used in my EV speakers is one of the best I’ve heard - a sledgehammer in velvet gloves). Active (via DSP) to my ears generally sounds more resolved, more transiently clean and overall less inhibited vs. a passively configured ditto (my experience, in addition to the one I've drawn from my own setup, is based in particular on passive speakers that have been converted to active config.); more immediate, yet smoother and easier on the ear.
The best systems I’ve heard have been actively configured, pretty much like my own setup. Just my $0.02.