If you have a positive unidirectional pulse, which seems to be what the Tact actually uses, what do you get from a speaker through a microphone. It cannot be an identical positive unidirectional pulse! The speaker/microphone system is a bandpass system which has to, at least, be adding at best about one additional "cycle." It cannot reproduce DC, so a signal with DC offset has to be bent somewhat around that limitation.
The response you could get could start with a brief negative leading edge cycle, the positive cycle of the large body of the pulse itself, and a trailing negative restoration cycle. So 3 half cycles from 1, and I think that's about the best that can be done from any speaker without DC capability (the best actually might be more like two half cycles, depending on duration). Assuming the pulse is long enough to have significant low frequencies, like 1-10ms, so perfect reproduction of the pulse would otherwise require DC capability. (The Tact seems to use two pulses in measuring actually, for higher and lower frequencies, and have separate correction algorithms for each.)
In fact, that is the kind of thing I see if I measure bandpass curves electronically from my crossover, not even going through speakers. In fact, doing this led me to believe it is not a good idea to use very high order Linkwitz Riley LR48 in the extreme treble. You get at least an extra cycle of ringing from that, in the electronic signal itself, and Butterworth 24 gives about the cleanest impulse short of single pole. Strangely, at lower frequencies used for my subwoofer crossover, the high order LR48 high pass impulse electronic response looked more OK, or at least just lays on top of the panel response less objectionably. Note that you can't entirely judge a crossover by looking just at its high or low pass section in isolation; used together the combination should approximate some kind of ideal. But the ideal ideal is most often an "all-pass" response which shifts phase, only acoustic 6dB/octave crossovers can do better. Nevertheless, in practice the ideal will not be achieved, so it's best if each drive signal stays as simple as possible.
After much work (!), that's also what I can get from my very complicated system. Actually the impulse looks like about 3 full cycles, with the subsequent two being greatly reduced in level, plus the usual digital aliasing stuff around the edges. My truly great achievement was getting a combined system response, including sub and tweeter playing at uncompromised levels, that has an impulse response that looks barely different than the Acoustat alone playing by itself. I never expected that, I only expected that the acoutstat-by-itself would look a bit simpler, because I hadn't thought about the issues very much.
And another enduring question is what is the correct polarity? I believe the correct polarity is not that of the leading transient, which may be always out-of-polarity in the ideal bandpass case, but that of the larger square wave what follows. BTW, I inverted the tweeters to give them a puzzle-fitting response. Just by themselves, the acoustats seem to have out-of-polarity leading edge and half cycle followed by full positive cycle, somehow I needed to invert the tweeters to make it fit precisely.* (Strangely, the tweeters actually have a kind of quasi-DC response... More about that discovery later...)
I wonder if some would claim the correct polarity has the leading edge in polarity, so I could be wrong. Perhaps it depends on the number of octaves in the bandpass, and what the bandpass functions are. And perhaps also it becomes fundamentally a subjective question in the case of a system with sufficient group delay (as probably most are). Do you want the bass in polarity or the treble in polarity (assuming you have to make the choice)? If the leading edge defines the treble, but a larger partial cycle follows in opposite but correct polarity, that indicates the treble is out-of-polarity but the midrange--if-not-the-bass--is in polarity.
Another interesting (and now very important) issue is the fundamental panel resonance. I believe there is some fundamental panel resonance in the region of 55 Hz. That resonance is how the speaker maintains frequency response to 40 Hz in spite of being a narrow dipole, I think. Well, it has that resonance when manufactured, but age probably causes membrane to get loose, hence less controlled, which may mean higher resonance harmonics, and that is what seems to be happening, I get buzzing around 110 Hz (on one recording anyway). Well, I need to fix the speaker, the Acoustats use HS65 which is actually 6.5 mil heat shrink. The fix is to use hair dryer to shrink the membrane back to tightness. It's supposed to last about forever if you keep doing that. Other than that, the panels cannot be repaired, you can scavenge old units or make new ones using the same principles (people have claimed to do that and say they never want to go back, and one beauty of the acoustat design is that there is nothing in it that couldn't be done in a garage with readily available and cheap materials).
In the meantime, and this is what I did before and now discovered I must do, I raise the crossover point to 121 Hz and the problem disappears at reasonable levels. I thought I could get away with lowering the crossover point (and with brand new panels, I ought to be able to do so) but apparently not for now.
But anyway, my original question relates to this because we have to consider the panel not as a DC tracking system but a fundamentally resonant system, with fundamental low frequency "drumhead" resonance and high frequency "breakup" resonance. These affect how impulse is going to look, even in the absense of crossovers, reflections, diffractions, etc.
Speaking of which, I should dig out the old impulse picture from an ESL63, which I thought looked pretty good.