An exercise in converting a speaker to time-phase coherent

I've been doing some simulation on how to convert a regular speaker to time-phase coherent. The software is Xsim which has enough basic stuffs for basic simulation.

But first let's get some terminology out of the way. The definitions use freq. and phase response so there won't be any ambiguity. I am sure different people might have different terminologies but with respect to what I did, here are three types of first order speaker, from easiest to hardest.

Here are three:
1. First order/No Time-Phase coherence: this speaker will use first electrical order, but there is no time-phase coherence. It will not be able to produce a proper step response. It's more or less conventional with the exception that it uses first order filters.

2. First order/Time-Phase Coherence, BUT NO "Time coincidence"
(which will be explained in #3).
This speaker will be able to produce a proper step response, BUT and an important BUT. It may not be able to produce an excess phase of 0 degree from say 50Hz to 20KHz. This means that the speaker, for example, may have a phase shift of 50 degree or more or could be a full 360 degree at 15KHz, but only 5 deg at 500Hz. That is its excess phase will vary especially at higher frequencies as the tweeter approaching 20KHz. John Atkinson would agree this speaker meets his definition of "Time-Phase Coherence" since it could produce a proper step response. My guess is most speakers that were measured by John Atkinson would fall into this category. I've seen some measurements done on Vandersteens speakers and I was like ... hmmm... I am not quite sure. But the most stringent definition is reserved for #3.

3. First order/Time-Phase Coherence AND Time-coincident: this is the most difficult definition for any speaker to meet. That is it has to be able to produce a proper step response like in #2, BUT it also has to be able to have a 0 degree of excess phase from 1Hz - 20KHz. In reality, no speaker will have absolutely 0 degree, but the variations should be very small. I believe Thiel claims that their speakers excess phase shift is only a few degrees (less than ten). To be honest, I am not sure many speakers in the entire history can meet this definition.

So to summarize, you have three distinct possibilities from easiest to hardest:
1. First order/No Time-Phase coherence
2. First order/Time-Phase coherence BUT NO "Time coincident"
3. First order/Time-Phase coherence AND "Time coincident" (the most stringent)
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This is #1 case: First order / no time-coherent.

This is also the starting point. This speaker will then be converted to time-phase coherent later on.

As you can see from the attached pic below, it is basic a first order, with tweeter inverted, so it's not that too different from any other speaker.


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Case #2: This is case #2. Time-Phase coherent BUT no Time-Coincidence because it has excess phase toward 20KHz. It will produce a proper step-response.

First let's make the tweeter polarity positive. But now we have a dip at the xover point because of the phase mismatch. See the first pic.

One way to fix that is introduce a fix time-delay on the tweeter. For example, if the xover point is at 3KHz, calculate the phase mismatch, then dial in a fix delay on the tweeter so that the phase of the mid and tweeter would match at 3KHz. See the second attached pic. This can be done with DSP for example. But the problem is you end up having an excess phase at high frequency as you can see on the second pic. The consequence of this is the initial spike on the step response.

I think you may be able to see this on Troels latest speaker design. He uses DSP to adjust the delay on the tweeter. But his step response show a rather big spike on the step response.

I suppose this is OK, but we need a speaker that can meet #3 criteria! No excess phase.


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So let's see how we can convert the original xover to one that can meet the #3 criteria!

Let's look at an ideal first order time-phase coherent/time coincident - absolute zero phase from 1Hz to 20KHz.

The simulation attached shows an example. Notice 0 degree phase shift from 1Hz to 20KHz. Also notice how the tweeter phase and woofer phase track each other at 90 degree, but the overall phase of the system is 0 degree.


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Case #3: Let's take a first shot at the original xover and convert it to time-phase coherent/time-coincident.

Notice the excess phase is very close to zero. Yes, ideally the excess phase should be zero, but in real world condition, it's very difficult. Maybe with modify baffle or some baffle tilting or tuning can help minimize the excess phase.

Compare to the non-coherent version #1, #1 has an excess phase of -180 degree.

But xover may be able to be fine-tuned to improve the phase matching at low frequency and high frequency.


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Case #3: Here's another example using a three way speaker. This is an old speaker I made awhile ago. It's not an ideal candidate to make a time-phase coherent/time coincidence but it serves to for an illustration. The cabinet has some odd diffraction so there quite a bit of ripples on the freq. response.

Drivers are Seas tweeter, 5.5in mid, 8in woofer.


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A few words regarding the step response:

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:

IMG_0693 | Andy VJ | Flickr

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.

Inverted_overall | Andy VJ | Flickr

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

inverted_stepResponse_zoom | Andy VJ | Flickr

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.

Overall_inverted_freqResponse | Andy VJ | Flickr
As a counterpoint to first order/shallow slope filter, here is a short exchange with the designer of "Infinite Slope filter", Richard Modafferi, being used in Joseph Audio. He posted his comments on another public online forum so I don't think it would be a problem "copy and paste" what he said here.

Richard Modafferi:
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.

It seems interesting that you posted with regard to the "Infinite Slope" in a time-phase coherent thread. It's quite a different philosophical design vs. time-phase coherent with respect to phase shift. I would assume Infinite Slope filter would have higher phase shift using very steep roll-off slope filters. vs. a time-phase coherent design that uses first order filter which has the least amount of phase shift. "Infinite Slope" advantage is minimal over-lap in frequency response between different drivers, whereas time-phase coherent is the complete opposite being having a large overlap. I suppose the disadvantage of "Infinite Slope" is the excess in phase shift?

I was wondering if you could share your opinions on "time-phase coherent" as to the extend it may affect on sound quality. Time-Phase Coherent insists that the phase of the system response (the overall response of a speaker) should be as close to 0 deg. phase shift as much as possible from 1Hz to 20KHz. Thiel design has claimed to achieve +/- 10 deg difference.

With "Infinite Slope", I would assume it would violate the criteria of 0 deg. phase shift. Would you share the amount of phase shift a typical "Infinite Slope" speaker. For example, what is the typical phase shift at 17KHz vs. to something like at 270Hz? For a time-phase coherent design, the phase shift should be very close to 0 degree.

Richard Modafferi:
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.
Here is an impression of someone by the username "Prof" from another forum as to his thoughts on the Thiel speakers and Joseph Audio, a polar opposite in design philosophy, one is optimized in the time domain and the other optimized in the frequency domain.

I own both Thiel speakers (had the 3.7s, now own the 2.7s), with their concentric drivers and time/phase coherence, AND I own the Joseph Audio Perspective speakers (Infinite Slope).

The difference I hear between the two designs is that the Thiels have an imaging precision and density none of my other speakers have ever had, including the Joseph speakers. They "disappear" just that much better than most speakers, but without sounding ghostly or insubstantial in the imaging. Tonally they sound very "right" to my ears. And I would but the Thiels as the most coherent multi-driver speakers I've ever owned (or, I think, heard).

The Joseph speakers though seem to offer even lower apparent distortion in the signal, with a sense that a fine layer of "hash" heard in most speakers seems removed, so the timbral quality of instruments seems even more revealed. (They are also very punchy and fun and image/soundstage great!).

As for other time/phase coherent speakers, the old Dunlavy's also impressed me, doing something very similar to my Thiels.

And yet, having also heard the newer Kii Audio Three speakers a couple of times (DSP speakers time/phase coherent), I didn't find they had the same magically believable tone as I hear in either the Thiel or Joseph speakers. I found myself having to "work" to unravel various instrumental timbres in the mix, where with the Thiels and especially with the Joseph speakers, this is effortless. Don't know why.

Anecdotal observations from owning both the Thiel 3.7 and still owning the 2.7s, while also owning the Joseph Audio Perspective (Infinite Slope) speakers:

I’ve mentioned it before but...

The Thiels image with greater image specificity and precision, especially with a sense of sonic density to the instruments and voices. They are a bit more lush sounding from top to bottom. A bit more balanced dynamically/frequency response. They maintain imaging and tonal balance over a wider area than the Joseph Speakers (at least in my set up, and to my memory. Been a couple months since I had the Thiels set up).

The Thiels are also more perfectly coherent sounding. As I’ve said, try as I might I simply can not "hear" the tweeter in the thiels. The treble area just melds seamlessly with the rest.

My hunch, having heard various Thiel speakers over the years, is that this is not *entirely* due to the time/phase coherence. This is because, at least in my memory, I didn’t find the old Thiel 3.6s quite as perfectly coherent in the treble - a little bit more of a treble shine poking out with those. The CS6s that I had were smoother, but still had a tiny hollowness in the upper midrange that could take away some of the body of instruments and slightly separate highs from the rest. Subtle, but there. (I think we discussed that back then Thiel was still working out the problem of a bit of interference that could happen between the tweeter and mid - can’t remember if that was due to concentric driver design, or due to challenges in first order crossover. Now that I remember the Meadowlark speakers I had (time/phase coherent) actually had this problem to a much greater degree, I believe it’s a challenge in the time/phase coherent design).

All that seems pretty much solved in the 3.7/2.7 design from what I can tell.

The Joseph speakers have, as I mentioned before and to my ears, a lower level of hash (reduced driver interference?) as their main sonic virtue, which makes the sound more relaxed and un-mechanical sounding. A rare purity of tone. They seem to have a bit finer resolution, and bring out more timbral nuances, differences, in the mixes.They can sound surprisingly huge for their small size. Though I don’t think that’s anything to do with time/phase coherence vs lack of. More driver choice/voicing etc. The Thiels sound a bit more focused and dense in the bass. The Josephs are a bit more "juicy" and punchy, with a bit more "heft" lower down from the Thiels. A bit of added warmth perhaps. But it makes for exciting punchy drums and bass tracks. I can constantly "feel" the bass from the Josephs, where the Thiels would tend to produce the bass happening more holographically "in front of me" behind the speakers.

The Joseph speakers are very coherent - that is one of the characteristics noted in review after review. So it seems their crossover design works to minimize driver interference. Still, it’s only having lived with the Thiels that shows up the Josephs as being slightly less coherent, both from bass to mids and mids to highs. There is a teeny bit more of the high frequencies, the tweeter, "riding on top of the sound" vs the Thiels. But again, the Josephs are more coherent than the majority of speakers I demoed. But the gorgeousness of the upper frequencies are entrancing. Last night I was listening to everything from soundtracks, to rock, to jazz, and the sense of openness, airiness, the aliveness and vividness of tone was like a sonic rainbow. Really pleasing.

Finally, to throw in one more wrench: I was also listening to my little Spendor S3/5s last night, comparing with the Joseph speakers.The Spendors aren’t of course time/phase coherent, but MAN are they coherent! They also sound virtually perfect in coherence. In fact, with vocals, they are THE most coherent sound I’ve heard, even beating the Thiels. But if I’m to ponder why, it could be that the Thiels are super coherent but more revealing, so the artifice of recording technics, which will exaggerate sibilance or color voices, will be more on display.Where the Spendors have a canny balancing act of an under-damped cabinet, and a voicing that likely does a bit of BBCing, which hides a bit of the problem frequencies that tend to show up on voices. So even on sibilant recordings, for instance, the frictives on vocals "sets back" naturally in to voices instead of sounding detached. And they have a richness and roundess that recreates the organic quality of voices. They still astonish me.
Personally, I am a believer in first order design since I think there is a unique and musical sound signature of first order that is not there in higher order designs. Also taken from the same forum above, here is my "instinct" (for the lack of a better word) as to the differences:

Richard Modafferi, Infinite Slope designer, did claim that the group delay within the overlap region is very minimum therefore, at least within this region, the transition should not be a problem in the time domain. But as you said, there is still a problem of phase rotation in which the signal at 20KHz will arrive at 360 degree ahead of the signal at say 300Hz. But the other problem with steep slope is the rapid change in phase which will be addressed below.

From what I can gather so far, the high order suffers the following main two things:

1. As pointed above, phase rotation where high frequencies will arrive earlier than low frequencies and how much it depends on how the filter is designed. It could be 270 or 360 or 180 degree or somewhere in between . At the same time, I am not sure how sensitive our hearing is to this type of phase shift. Our hearing is not sensitive to the sound delay if this delay is within a few mlli second. For example, at 17KHz, 360 degree would equate to 0.05 milli second. Could our hearing tell of this small delay? I mean with diffraction from the baffle, I would assume sound delay from baffle reflection could be more than 0.05 from various high frequency components. On the other hand, at 500Hz, 360 degree would be 2 mill second which is somewhat close to our hearing threshold. My conclusion is as at higher and higher frequencies, this type of phase shift may not something our hearing is sensitive to. Thankfully, most xover even using higher order filter, do not have this type of phase shift issue at low frequencies. And considering most xover crosses at 3KHz below which more or less covers most of the musical contents. So regardless of xover, most of this type of phase shift will only occur at above 3KHz where our hearing may not be so sensitive.

I have designed speakers that is first order time-coherent (no phase shift) and first order NON time-phase coherent. The common denominator here is first order - but one is time coherent but the other in NOT time coherent. The non coherent version has 180 degree phase shift at 20KHz. BUT I had a hard time telling the difference in the sound quality or the intrinsic type of sound between the two xover types. Although I do prefer the time-coherent version, but the frequency response and oxver point between the mid and tweeter are all different between these two version so I don't if most of the differences come from the time-coherence or because of different types of voicing. So where do I think the differences come from? I think most of the differences that we have talked about will be because of #2 below.

2. All drivers have non-linear distortion which means the drivers will produce more extra frequencies than the frequencies putting in at the input. But why would this affect high order more than first order? I think because of of the rapid phase shift of steep filter slope, which in turn produces more over-shoot or more high frequencies energy vs. lower order filter with shallow phase shift. This extra high frequencies energy when applied to the non-linear distortion of the drivers, will produce more extra high frequencies contents that was not part of the original source. This extra high freq. will affect perception and it is consistent with what I and other (Prof) have observed that the treble of high order speaker seem to be riding on to of the music as if it is not part of the music but our mind will include it as part of the re-construction process. What I said above also implies that first order filter will suffer from this as well since it will inevitably have some overshoot (from parasitic such as drivers coils ...) but not to the same degree. This will also implies that if quality drivers are used which have low non-linear distortion, this problem will be minimized even with using high order filter.

#2 also explains why solid state components cannot produce treble as well as tubes (I won't try to go into much details), due to the nature of solid state physics, the electrons movement within the medium will experience high order filtering affect from parasitic, impurities and such. With tubes on the other hands, electrons only have to move in the vacuum from the anode to the cathode without having to go through any other medium therefore won't be affected by any filtering.

And of course solid state amplifier does not have any xover that can explain the differences.
I was playing around with a three-way mocked-up. Basically I use a setup with what I already have. The tweeter is ScanSpeak AirCir, Seas Nextel 5.5in. as mid, and Scanspeak Illuminator 5.5 in as woofer.

Measurement was done at 1Meter between the tweeter and the mid which is about 33in. high. There are quite a bit of ripples on the freq. response probably due to room reflection which also shown in the step response being taking a bit longer time to settle at the low frequencies.

The xover seems a bit complicated, but that's typical of first order, time-phase coherent, time-coincident speakers. The 50uf cap on the tweeter should only affect the lower freq. so it probably will be OK. Actually I don't need it there for freq. response, but I put it there to protect the tweeter.

IMG_0697 | Andy VJ | Flickr

ThreeWayExp | Andy VJ | Flickr
I believe that one of the differences in sound signature of first-order filter is that high order filter, there is a faster change in phase, which results in excess overshoot and undershoot in time domain. And the reasons why were explained in my previous posts.

I here compare different impulse response of different filters. An ideal first-order time-phase coherent has the "best" impulse response. The 3rd order electrical filter has the worst. There are a total of six different filters here:

1. First order ideal
Impulse_ideal | Andy VJ | Flickr

2. Third order electrical: shows the most undershoot, overshoot:
Impulse_3rd_ele_order | Andy VJ | Flickr

3. A Three-Way, First Order, No Time-Phase Coherent
Impulse_3way_noCo | Andy VJ | Flickr

4. Three-Way Time-Phase/Time-Coincident:
Impulse_3way_Coherent_1 | Andy VJ | Flickr

5. Another Three-Way Time-Phase/Time-Coincident:
Impulse_3way_Coherent_2 | Andy VJ | Flickr

6. A Two-Way Time-Phase/Time-Coincident:
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