First order/Time Phase-Coherent speakers discussions

"The game is done! I’ve won! I’ve won!"

I would like to use this thread to talk about this subject which I find rather fascinating and somewhat difficult to get my hands on. I went through a course in electromagnetism in college and I have to say this is even more confusing and you won’t find the answer in calculus, physics, Einstein relativity be damned it’s not in there either and definitely not in quantum physics. Listening to the "experts" from Vandersteens and Stereophile but ultimately it all came down to a missing link sort of argument ... something like this:
"Since if a speaker can produce a step response correctly, therefore it is time-phase coherent, and therefore it must be "good".

It’s like saying humans come from chimps since they share 90% genetic content with us, but we can’t find any missing links or evidence. FYI, we share a lot of gene with the corn plants as well. Another argument I’ve heard from John Atkinson that lacks any supporting evidence and he said that if everything else being equal, time-phase coherence tends to produce a more coherent and superior soundstage, but to the best of my knowledge, nobody has been able to produce some semblance of evidence since there is no way to compare apples to apples. Speaker "A" may have better soundstage simply because it’s a BETTER design, and the claim "time-phase coherent" is just a red herring. There’s no way one can say the "goodness" from "time-phase coherence" because you can’t compare apples to apples. Ultimately it’s a subjective quantification.

I’ve been doing some simulation and I will post some of my findings with graphs, plots, actual simulation runs so that we are discussing on subjective personal opinions. Some of my findings actually shows that intentionally making time-phase may result in inferior phase problem and NOT better! (will be discussed more in detail).

Having said all that, I am actually in favor of first order/time-phase coherent if POSSIBLE. I am not in favor of time-phase coherence just for the sake of it. It’s just that there are a lot of mis-information out there that hopefully this will clear those out. Well hopefully ...

Here my preliminary outline:

1. My "subjective" impression of what is "musicality" and how it’s related to first order filters.
2. Interpretation of step-response. I’ve read a lot of online writing with regard to the interpretations but I think a lot of them are wrong. A proper interpretation is presented with graphs and simulations.
3. A simulation of an 1st order and higher order filters with ideal drivers and why time-phase coherence is only possible with 1st order filter. This part will use ideal drivers. The next part will use real world drivers.
4. A simulation with actual drivers and how to design a 1st order/time phase coherent speaker. Discuss pros and cons. And why time-phase coherence may actually have phase issues.
5. Discuss real world examples of time-phase coherence with Thiel’s and Vandersteens speakers (and why I suspect they may not ultimately be time-phase coherent in the strictest sense).
6. I’ll think of something real to say here ... :-)
Contrast audio loudspeakers.
Own designed parts. Only from naturall materials.
Only single one capacitor in crossover-network! Frequency response of tweeter 10 KHz!

If to say about 3-band  speakers, situation more interesting. WITHOUTT FILTEREDDD BASS DRIVER.... 


Audiophile friend urged me to jump into this discussion with some of my ideas.  Here goes my take on phase-coherent speakers and first-order crossovers:
A perfect transition from an electrical signal to acoustic sound is defined by the simple so-called all-pass transfer function defined by:                  
(1) f(s) = Ke          (note: this dumb website does not allow math symbols, sorry)

For perfect fidelity in acoustic space this equation defines it, where S is the complex frequency variable S = p+jw  and K,T are real positive constants and =2.718......
One can realize a loudspeaker crossover in Laplace-Transform form by expanding (1) in an infinite series and taking only the first term in (2) to yield (3), as:
(2)   F(s) = e        = 1/(1 + St + (St)    + etc.......)
Taking just the first term yields

(3)  f1(S) =  1/ (1 + St)
Laplace transform of (2) represents circuit consisting of a coil of T Henries in series with a 1-ohmresistor. (You need to imagine circuit because website cannot draw schematic symbols!)
Replacing variable S with its inverse 1/S in (3) yields St/(1 + St) or, a single capacitor and resistor in series:

(4)  f2(S) = St/(1 + St)
Equation (3) is transform of a woofer crossover in circuit, equation (4) is transform of a tweeter in
a crossover circuit. Now add (3) and (4) to obtain:
(5)  f1(S)   +  f2(S) = 1/(1 + St) + St/(1 + St) = 1.
Sum adds to a constant, the number one! The crossover is perfect, a constant. No variables are present. Speaker will be flat in amplitude and linear in phase!!!! Crossover uses one coil and one capacitor!!!!  Simple!!!!

Here I show why people are in love with first-order crossovers.  Realized carefully, a first-order crossover system will work reasonably well and satisfy almost anyone with decent sound.