After studying the loudspeaker crossover issues in the 1977-1983 time frame, I came to the conclusion that the Linkwitz-Riley was by far the best kind of crossover for most drivers. Generally speaking the 24-dB per octave (LR24) would be the best crossover choice. Linkwitz concluded that the group delay (a compromise) introduced in the overall response of the crossover (by design, the drivers are always in phase with each other through the crossover region) was not audible and of negligible importance.
Only a few crossover designs, like the 6db per octave acoustic crossover used in Thiel, Vandersteen, and some others, can achieve total lack of group delay in the summed response. And that requires multiple other design compromises.
When the Behringer DCX 2496 digital crossover came out, we were treated to LR48 achieved in hirez digital. What could be better? That's what i have adopted uncritically since 2005; I now use the Behringer in both living room and bed room systems.
Now, I have figured out how to examine the impulse and frequency response of the crossover network by itself, and I am not so sure LR48 is the best choice. For augentation purposes, where there is no perfect cancellation of phase artifacts, Butterworth 24db per octoave (BU24) looks like the choice that gives the best compromise between steep cutoff and lack of visible time dispersion.
LR24 might be a better choice if you had a perfect acoustic LR24, something very hard to achieve in practice, but given lack of perfection, the best bet is probably to minimize time dispersion in each crossover member with one that provides less dispersion
In a quasi-augmentation mode, or as a solo highpass network, the LR24 increases time dispersion AND reduces sharpness of cutoff compared to BU24. Im not sure of the advantages of the Bessel, it may have he steepest cutoff, but it is marred by 20dB passband irregularities. It might be OK for supertweeter where, say, above 20K you don't care about passband irregularities.
I think it's possible that in the low frequency crossover (the highpass on the acoustats is currently set to 104Hz) LR48 works OK, and is extremely beneficial in reducing panel flap at high volume levels.
The visible difference between BU24 and LR48 is pretty small at 104Hz, the LR48 does have a bit of initial out-of=polarity undershoot, and somewhat more overshoot on the trailing edge followed by slow recovery. Whereas the BU24 almost looks untouched, the perfect Tact impulse (that is, as perfect as the Tact gets) almost. But it just doesn't phase me much, at least on screen.
But what has been driving me apoplectic about this is the supertweeter highpass. It turns out that the highpass signal from LR48 at 15.5 or 20 kHz has 3-4 cycles of ringing at 20kHz. That's all there is, it doesn't look like an impulse at all, just ringing. Superimposed on a perfect impulse, it smears it out considerably. My time domain purist friends should be laughing at me now.
That's exactly what I've been seeing in the system impuse time response. And it bugged me so much I refused to print it yesterday. Now I've figured it out, at least partly. Even if the supertweeter is reproducing the signal that it receives perfectly, that impulse response looks like 4 cycles of ringing at 20kHz because that is the signal that the crossover is providing.
Now in the context of a perfect LR48 crossover, with perfect high and low frequency drivers crossing over, the majority of the phase anomalies might well cancel out (I am not entirely sure of this...) and it wouldn't look so bad. Maybe.
But in the context of the kind of slap dash (if infinitely pondered) systems which are the only kind I can put together, not being able to hire a team of engineers, it's looking to me like simpler is better, probably BU24, the time domain smearing is cut in half or less, in fact it doesn't look like ringing anymore, it looks like a double pulse, which probably adds nicely with the low frequency system pulse.