Absolutely wonderful read! Glad this thread was bumped recently back from 2005 or I would never have found it. I am an electrical engineer myself with significant signal processing background, and I truly appreciate the clarity of Roy’s and Karls explanations here. They are absolutely correct. At the same time, it is easy to see how without formal study of electrical signal mathematics it might be difficult to follow or even counterintuitive. Nowhere is this as obvious as the discussions relating to Phase coherency vs Time coherency (i.e. simultaneity). Time coherency is clearly the stricter condition as Roy points out. I also really liked Karls explanation of the sum of the high and low frequency drivers output after first order crossover as the sum of two phasors separated by 90 degrees at all frequencies. I think I would have further reinforced the fact that it is *perfectly fine* to add these "out of phase" signals because the result is always a constant phase at the average of the two individual phases, which is actually ideal. Without some background in signal processing it might be easy to think these two drivers were fighting each other and causing distortion. Karls post really helped clear this up.
Finally, it was good to see it pointed out correctly that the spatial lobing problem is a multiple emitter issue, and not a crossover issue. Steep crossovers merely narrow the region over which multiple emitters "exist" for that frequency. Thanks for the great read.
I hope there is more discussion on this. I have a pair of speakers with a first order crossover now myself (Dynaudio Special 40), and they really do have a very special sound. Yes, I hear the lobing, but yes, I also hear the outstanding clarity which I assume is the phase and/or time coherency of the drivers in the sweet spot. Still trying to decide what I think of them overall, but so far very impressed. Btw, there are a lot of other LC components on the crossover boards (presumably used for impedance correction of the drivers?). Would like to learn more.