When it comes to first-order coherent and non-coherent, there has been some misundertanding with regard to the step response. For non coherent speakers, invariably, the polarity of the different drivers will be different, that is the tweeter may be inverted polarity and the woofer will be positive polarity. Therefore in the step response, the tweeter will "APPEAR" (notice capital letter for emphasis) to go negative with respect to the woofer. But that misses one subtle but very important point that will lead to a crucial insight to the phase of coherent vs. non coherent design.
Most people came familiar with step response measurements by way of Stereophile John Atkinson measurements, but his measurements only show the overall step response without the measurements of each single driver. Once each single response is shown, there is further truth to be gained.
I will use one of my project to illustrate the misunderstanding. See pic of speakers here:
https://www.flickr.com/photos/185616271@N05/49106259616/in/album-72157711891600612/
Below is the xover design. Notice the tweeter is inverted polarity and the woofer is positive. Also notice the step response on the lower right corner. Also notice the phase of the speaker response on the upper right corner of the plot. That is the phase starts at about 0 degree, and gradually go to -180 degree. Also notice each individual component freq. response and the cross over point.
https://www.flickr.com/photos/185616271@N05/49106444352/in/album-72157711891600612/
At first glance, the tweeter seems to go completely opposite of the woofer (the initial dip of the step response). The time-coherent proponent will say – wait a minute, that’s all wrong since how can the treble be completely opposite of the woofer? But that is not quite right. The tweeter ONLY opposite of the woofer at very high frequencies, but at low frequencies, below the xover point, the tweeter and the woofer actually go up together.
I here have a zoom in pic of the step response. You can see the tweeter and woofer go up together after the initial high frequencies. It’s subtle but it’s there and it’s very important.
Green – system
Red – tweeter
Yellow - woofer
https://www.flickr.com/photos/185616271@N05/49106444417/in/album-72157711891600612/
What does this mean? If you look at the frequency response below, you see that the phase of the system is only at -180 at 20KHz, not the entire plot. But below the xover point, the tweeter and woofer are more or less “in-phase”. For example, at 2KHz the phase is only at -60 degree, at 3KHz the phase at -80 degree. As I said before, since our perception is not very sensitive at high frequencies, most of us may not “hear the difference”. Most of the musical contents occurr below the xover point which is about 3KHz in this case and that’s where most of the phase shift has not occurred.
Also the phase change is not at a constant -180 degree everywhere but only at exactly at 20KHz. For example, the phase change at 20KHz is only a few degree vs. 19KHz and likewise 19KHz is only a few degree vs. 18KHz … and so on. It’s a gradual shift in phase so our hearing may be able to adjust to it.
But on the other hand, if this -180 degree occurs at 500Hz, trust me, you will definitely HEAR it! I think high order can get away with it because the phase shift occurs at high freq. where our hearing may not be so sensitive.
https://www.flickr.com/photos/185616271@N05/49106492271/in/album-72157711891600612/
So how does an ideal time-phase coherent design look like. Below is a design using ideal perfect driver. Notice how the phase of both the tweeter and woofer tracking each other at 90 degree apart. But the overall phase of the system (the green curve) is at exactly zero.
https://www.flickr.com/photos/185616271@N05/49106060213/in/album-72157711891600612/
But real world system does not look that clean and the phase is not going to be exactly at 0 degree. I’ve modified the xover so the tweeter and woofer are all positive polarity, and the phase of the speaker stays more or less within 0 degree (the green curve). Notice the step response where the tweeter and woofer go up together.
https://www.flickr.com/photos/185616271@N05/49106502197/in/album-72157711891600612/