First order/Time Phase-Coherent speakers discussions

Hello!
I'm inventor of well-known  (infamous?) Infinite-Slope crossover system currently marketed by Joseph Audio. My patent license agreement with them expired in 2005 and at that time I went into blessed retirement at age 67.  In 2017 I was pulled out of retirement by three audiophiles owning Joseph Audio with the question, can your invention be improved? I said no but they insisted and I gave up to shut them up and returned to my notebooks (thankfully saved and in local tech museum) to study network topology and see if I have any new ideas.
The major problem hinges on simultaneous realization of flat frequency response and uniformgroup delay in three-dimensional acoustic space of the listening room. Actually impossible but at least a good approximation is the best that we can do and I come close in with Joseph Audio's products as is well known.
Mathematically one  can achieve flat frequency response and linear phase (flat group delay) using a single very good 4" driver without crossover in a transmission-line box. This system will have nearly perfect performance over a limited frequency range and with limited  loudness capability. Next, a 2-way first-order crossover is theoretically perfect in the math, but when realized in a speaker system, it is "perfect" only in the so-called "sweet-spot"  where the sonic outputs of the drivers add correctly without acoustic wave interference.
In 2018 at age 80 I attacked the problem again. With study of my notes, I came upon an idea which may work: Combine the idea (1) of so-called "constant-resistant" network theory with my already realized "infinite-slope" theory (2) (based on high-selectivity filters in radio circuits).
I worked up a schematic-diagram of this new crossover idea using circuit-analysis models in a computer to start.  The results looked promising, and at some month's work developed a 2-way crossover model  in virtual cyberspace having both optimum frequency and delay responsesimultaneously!  Now it was time to build a physical crossover and try it in a prototype speaker system and found a quick-easy way to proceed:

I ordered the so-called "Solstice" loudspeaker kit from Parts Express , built kit, and installed my
2-way prototype crossover. At first I did not expect anything unusual, just another pretty-good sounding speaker system. I fired up my test equipment and made frequency, phase tests, and determined that system had good frequency response but surprisingly, flat group delay above the cabinet bass resonances!  A trip to the anechoic chamber at Binghamton University's Tech Center confirmed my measurements.  Now it is time to listen!

Played a CD of John Pizzarelli "Dear Mr. Cole" and the sound hit me so hard I burst into tears!Never have I heard sound like this from a loudspeaker box! The room disappeared and I heard the band!  Switching to my Pearls, I heard a good loudspeaker system. Now time to call my pesky audiophile friends, do listen with everyone astounded! We all hear something magic! I, almost with accident, had hit upon something unexpected!
We repeated test of my prototype in audio showrooms with three listeners against systems in the $30,000+ price range with same results, the little 2-way prototype was clear winner having obvious easily-heard sonic improvement. There was uniform spectral-energy sound throughout the entire listening space, with uniform sound without  "sweet-spot"  with all hearing music coming from a nearly perfect "orchestra" instead of a set of speaker boxes.

I had to develop (successfully) a 3-way crossover so invention could be installed in Joseph Pearls, with the same astonishing results. Stay  tuned everyone!
Patent on invention filed July 2019.

RIMO.

Hello Andy!
I enjoy your posts. Thoughtful and intelligent; we are both music lovers and audiophiles. I need to explain again how I got myself stuck into giant speaker project as described in my very first post ever on any website, being forced into doing this by one of the four (now five!) audiophile friends who pushed me at age 81 (now) to jump into speaker design again.  After retrieval of my notes from museum and study of topology (a favorite of mine from graduate-school network-theory courses)  I tried a merger of "constant-resistance" network theory into my "Infinite-Slope" in my previous patents.  First try was a 2-way series crossover at 2KHz. Installed this into Parts-Express 2-way speaker kit with test measurements in my home lab and also in anechoic chamber of Tech Center at Binghamton university.  Waterfall plats, frequency-response, input impedance, delay response, all looked good, even surprisingly so. Even more so as I examined in detail match at the 2Khz crossover, it was seamless with no evidence of a "join" in either frequency or phase response. It was time to listen to music on this box and as in my earlier post, I burst into tears at first listen (I'm Italian!).  Never in my life did I hear that kind of sound come from a speaker box!  Emphasis needed here: I'm no genius. This result was a pure accident of discovery, nothing more, and certainly not expected. My audiophile friends came over for listen and forced me into building three Parts Express kits to listen to until I could come up with a 3-way crossover for their Joseph Pearls (Successful as mentioned in last post)!  I should mention a funny effect I observe in audio showroom where we were comparing my invention with other speaker systems, as someone wanders in and after a few steps they stop dead suddenly at the sound!  It happens every time one hears my invention for the first time.  In closing I'm actually sorry I hit upon this thing, I nearly gave up and burn my notes and abandon patent, but it's too late; I need to see this thing through. Did a public demo of invention in August before an audience of 300 at a symposium with speakers as subject with incredible results (I hope to see this publicized in Winter quarterly journal of organization which invited me), one speaker manufacturer there asked me to try my invention; gave me data on their drivers. This is a Winter project ongoing.
RIMO

Hello Andy,
Thanks for Interesting recent post. Both you and I find my results with invention crossover a mystery.  Earlier infinite-slope crossovers worked nearly "perfectly" (maybe just OK) inside each other's band-pass, as characteristic to "brick-wall" filters make each other's loads (the drivers in speaker system) operate completely independent from each other. There is no acoustic interference among drivers in infinite-slope systems which eliminates  "sweet-spot"  errors acoustically. Amplitude response is flat all over listening space, dependent only on the polar response of each driver. However phase response is another problem which in my earlier work I either ignored or tried to fix with only slightly improved results. FFT analysis showed group-delay "bumps" at crossovers over a narrow frequency range around each crossover, I left things alone  as an unsolvable problem believing error was inaudible. Many nears ago in Grad School during a course in acoustics I attempted to devise an experiment to determine if we were able to hear "phase" as we hear "amplitude". It became a difficult problem; I could not even devise a methodology to formulate an experiment which would result in useful data. I got a "B" for trying!  Thus I became sure that we were "deaf" to absolute phase, or phase errors in reproduced sound over speakers. I know we hear relative phase in midrange very well with out 2 ears, as God designed us.(to avoid danger, etc.)  However FFT analysis on my new invention reveals clean waterfall plots and seamless amplitude and delay at crossovers!  I assume without proof that accidental combination of constant-resistance with infnite-slope compensates for "bumps" at crossovers. Everyone who listens to invention describes sound as "coherent" or "all-there", etc. Everyone listening for first time immediately states that they have never heard a speaker system sounding like real music as this one! Square waves maybe??? System test reveals good square-waves well over range 150Hz to 2.5Khz, at higher frequencies the 3rd and 5th harmonics become weak and waveform degenerates into sine wave. Square waves not too pretty having ragged edges due to edge diffraction from box.

A weak hypothesis regarding loss of group-delay "bumps" is somehow due to an adjustment of "overlap" at crossover frequency which compensates for delay bumps. Maybe you have an idea in this regard! If so your name can go on patent!

RIMO.

Hello!

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:                  
                    -St
(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
(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.


RIMO