PASSIVE Linear Phase (Quasi Transient Perfect) XO s

This thread is only for the discussion of passive crossovers, active that can be implemented
as passive or to explain some theory is fine but otherwise start another thread about them.

There have been several studies on the audibility of the phase distortions typically found
in multi-way loudspeakers that usually find, in typical reverberant home rooms they are
not audible. Most studies state that the distortions are not audible. One study found
that simple transient sounds such as banging blocks through headphones exhibited a
minor changer in timbre. It's probably been 20 years since I read these and I'll try to find
a list of references - I posted them on the Bass List in the mid 1990s. Here is one:
Lipshitz, Stanly P., Pocock, Mark, and Vanderkooy, John, "On the Audibility of Midrange Phase Distortion in Audio Systems,' J. Audio Eng. Soc., Vol. 30, No, 9, Sept. 1982, pp 580-595.
See also:
Audibility of Phase Distortion

Please start a new thread for further discussion on phase audibility - not here.

I'm interested in passive solutions that provide linear phase simply to understand the methods
and theory.

I've followed the literature closely from about the mid 1970s to 2000, less so since then,
The obvious solution is first order but they have many disadvantages. Then we had filler
driver solutions that I mentioned many years ago in this thread:

"A Novel Approach to Linear Phase Loudspeakers Using Passive Crossover Networks" Erik Baekgaard May 1977

I thought that this was invented by Baekgaard but then found this reference to a 1967
paper: Kido-Yamanaka crossover was first described by Bunkichi Yamanaka of Matsushita Corp. (Panasonic) in 1967.
A Unique Loudspeaker Crossover Design with Waveform Fidelity

Most simple 1st order systems have horrible off axis response.

Another approximate solution was proposed by John Bau and released as a commercial
product, the SPICA TC-50, in 1983. Bau states that he started with a 2nd order Bessel low-pass
and then found that a first order high pass with delay led to an approximate linear phase
result. This site offers a lot of info but not on the theory:
Spica TC-50 Product Information, The Spica Speaker Enthustiast

I have owned these speakers and they have very poor off axis response. This type of
solution would benefit greatly from coincident drivers.

Jeff B. hints at a third order solution with overlap, also as an approximate
solution in this thread at the PE Tech Talk Forum:
Third Order Transient Perfect Passive Crossover -

Techtalk Speaker Building, Audio, Video Discussion Forum

No design equations are provided and I've not seen polar plots for the off axis response.

DDF claims here that he has been promoting this type of solution since the mid 1990s
but he also has not provided design equations:
Third Order Transient Perfect Passive Crossover -

Techtalk Speaker Building, Audio, Video Discussion Forum

DDF makes this comment that might offer a bit more insight:
It sounds like Jeff locked onto a crossover target I know I locked into myself a long time
ago (and as did George Short at North Creek) as the best compromise for most system designs, and one I've tried to socialize over the years:
- sloped baffle
- soft knee thirds with staggered xover frequencies and mini-ripple response through xover

- flat response on axis through xover, but one easily tuned if a slight dip or bump is desired
- high out of band attenuation
- low complexity and parts count
- soft knees reduce tweeter "flare" off axis and provide better driver integration
- sloped baffle allows the power response to more closely mimic the on axis through xover
- low GD variation over frequency
- it sounds bloody good!

DDF suggests that his design is the same as Jeff B's but I read soft knee 3rd order for DDF
and overlapped 3rd order for Jeff B.
Jeff B. confirms the above where he writes:
"Then your crossover and mine are quite different afterall. John was correct when he speculated that I followed a similar path to his QTP 2nd Order filter. What I did doesn't match what you have described above though. "

And also (I don't know who Vance is) post #113:
"You described Vance's spread 3rd order as an example which results in two different Fc values that further separate the lowpass and highpass sections resulting in a more in-phase -6dB crossover point; something like a Linkwitz-Riley response with ripple. Mine on the other hand goes in the opposite direction where the lowpass and highpass are not spread but overlapped with each other, then the transfer functions are modified so that they sum to a flat response. The overlap creates the minimum phase summation."

Neither has provided design equations so I'm not sure about this if anyone can provide
them that would be helpful. I could experiment in the simulator but equations would be better.

John K. Wrote about a QTP second order network here that is on the Wayback machine:
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andy_c writes, with response from John K. in post #142:

Interesting thread. After reading it a bit, I began to wonder if it were really possible to have such a symmetrical crossover network and have it still be TP. Then I did a bit of digging, and was surprised at the results. Sorry if these ideas have been spelled out previously, but I did not see any mention of the actual transfer functions in this thread.

For the "symmetrical" assumption, I took that to mean "high-pass is derived from low-pass via a low-pass to high-pass transformation". To simplify that, I started playing around with normalized transfer functions having a nominal crossover frequency of 1 rad/sec. Then the low-pass to high-pass transformation consists of replacing s with 1/s in the low-pass transfer function to obtain the high-pass. This also means the poles of the high-pass will be the reciprocals of the poles of the low-pass.

In order to be TP, the poles of the high-pass must be the same as those of the low-pass. Can this be made to work? I looked at this denominator polynomial D(s) of the low-pass. Suppose it looks like this:

D(s) = (s + 1)(s^2 + as + 1)

For the pole at s = -1, it is its own reciprocal, so that works. If the quadratic has two real poles, they must have a product equal to 1, so they are reciprocals of each other. So when you take the reciprocals of these poles, they just change places. For the complex pole case of the quadratic, the poles must have a magnitude of 1, so the reciprocal of the pole is just its complex conjugate. Taking the reciprocals of both complex poles interchanges the poles with their conjugates, so you end up where you started. Looks like it can work!

Let the numerator polynomial N(s) of the low-pass be:

N(s) = 1 + bs

If you take b = 1 + a, you get a TP crossover. Let's take a = 1 as an example. The low-pass transfer function HL(s) is this:

HL(s) = (1 + 2s) / (s^3 + 2s^2 + 2s + 1)

To get the high-pass transfer function HH(s), replace s with 1/s in HL(s) and multiply the numerator and denominator of the result by s^3. This gives:

HH(s) = (s^3 + 2s^2) / (s^3 + 2s^2 + 2s + 1)

You can see that HL(s) + HH(s) = 1. This is 12 dB/oct, not 18 dB/oct though. It seems like you'd need more poles to get 18 dB/oct, as the numerator polynomial of the low-pass can't just be a constant.

Response from John K:
Yes, if D(s) is the denominator and N(s) is the numerator, then if D(s) = N(s) you can break the numerator down into parts and have a TP crossover of any order, or a multiway TP crossover. Example, D(1 + s + s^2)

LP = 1/D(s), BP = s/D(s), HP = s^2/D(s) which is a form of the filler driver TP 3-way system. But the problem is, such transfer functions generally have very strange response with peaking in the passband or around the crossover frequency. The inter driver phase differences often exceeds 90 degrees which implies that there must be cancellation to achieve flat response. For example, of the inter-driver phase difference is 120 degrees at the crossover point both the HP and LP response of a 2-ways will be 0dB at the crossover point and you can have as much a 6dB peaking off axis, depending of wave length and driver spacing. At low frequency this isn't so much of a problem, but it does waste power.

When I was working on my ICTA speaker years ago I took this approach for the woofer to mid crossover. Somewhere I have a spread sheet that allow computing these types or crossovers' responses.

Further response by andy_c:
Yes, but splitting N(s) arbitrarily does not yield symmetric lowpass and highpass in general. You can force symmetry by requiring that HH(s) = HL(1/s): that is, by requiring that HH(s) be derived from HL(s) using the lowpass-to-highpass transformation. But this constraint can mess up the TP property, because the poles of HH(s) are the reciprocals of the poles of HL(s). This imposes constraints on D(s), requiring that the set of poles of HL(s) be invariant under the lowpass-to-highpass transformation. That's what I did in my original post. For example, starting with a lowpass having

D(s) = (s + 2)(s^2 + s + 1)

and performing the LP-to-HP transformation can never satisfy both the TP and symmetry constraints simultaneously, because the poles of HH(s) will be different from the poles of HL(s) when 1/s is substituted for s in HL(s). But

D(s) = (s + 1)(s^2 + s + 1)

will work.
John K writes in post #188:
I just did what I did and stopped where I stopped. These are all basically subtractive filter pairs. After reading Small's paper where he was subtracting the LP function, 1/P(s) form 1 to get the HP, which always comes out as asymptotic 1st order, I realized that if P(s) was of high order (3rd, 4th , 5th ...) then you could also form the LP as (1+s)/P(s) or (1 + s + s^2)/P(s) which reduces the asymptotic roll off of the LP section and increases that of the HP.

Q greater than 1 is still MP.

DDF responds in post #193:
For my 4th yr EE project, I started with Small's paper as well. You can also subtract the HP from the LP to generate the LP as the asymptotic first order and therefore maintain full degree of tweeter protection.

However, any function using this approach showed terrible off axis behaviour so I rejected it.

I found far better results using Lipz/Vander's approach of HP(s) + LP(s) = A(s) where A(s) is an all pass function. As you now, LR xovers use squared Butterworths for the A(s), HP(s) and LP(s) targets. They are the only functions with achieve full roll off as they are the only functions which result in both HP and LP being all pole.

However, A(s) can be set to a low group delay function that can also arrive at HP and LP functions that have inherently pretty good out of band rejection (ie it takes the derived function longer to go asymptoic to first) with the inherent better behaved GD of the chosen A(s) and better tweeter protection than the Small approach. A side benefit was better off axis behaviour for a typical woofer/tweeter layout.

I never published the xover but realize that I should have.
DDF writes in post #269:

Back in the 80s, Lip and Van found that the LR crossover is just one specific solution to the L(s) + H(s) = A(s) equation. It happened to be the one that provided the best lobing (with the drivers vc aligned) and highest immediate roll off rates (by using squared Butterworths). Back then, lobing was an area of relatively intense study in the AES.

Some time earlier, Small had derived L(s) + H(s) = 1 solutions. Big problem with these as I discovered in my early sims in the 80s was that they led to terrible off axis response (Small never looked at this aspect) or your choice of bad tweeter power handling or poor out of band suppression of the woofer/mid.

What I did in my 4th year EE project back in the 80s was derive functions for A(s) that had far better group delay characteristics than LRs but with still decent out of band suppression. I added into the analysis:
- Leach's technique of accounting for tweeter phase on the over all response
- an e^(-as) term to account for the difference in bulk delay between tweeter and woofer at the observation point
- magnitude and phase (ie time) effect due to driver high frequency roll offs
- resonance of drivers, to model their impacts on response and delay

Remember this was the 80s, no one had measurement gear (or powerful enough PCs, I did this on the University's mainframe in Fortran, printed out the graphs using an ASCII plotter) so these additions were necessary to arrive at real world solutions.

This technique of using the right A(s) led to a better trade off IMO between out of band rejection, off axis response (ie "power" response), power handling (ie distortion) and group delay. Note on the Leach technique: back then there were few (if any!) tweeters that could be crossed low (the D25 and the pricy Scans became easily available after this) so its easy to dismiss Leach's idea now, but back then it did have some limited justification.

I spent the next few years researching the subjective tolerance to group delay (which I've freely shared here and elsewhere many times) by studying all the audition journals: JAES, JASA, JASJ, etc etc and eventually decided for myself the attainment of low GD wasn't worth the other trade offs, when implemented passively. I then eventually (mid 90s) locked onto the soft-knee third + tilted baffle approach and felt this was the best compromise in power response, on axis, GD and power handling in a passive crossover. The tilted baffle also allows a better off axis "power" response for a nominally flat in the mids on axis target. One way to understand this is to contemplate that a woofer and tweeter on axis are by physics beaming their highest amplitude in their high frequencies. All off axes will see driver high frequency roll off. By tilting the drivers relative to the observation point, the raw driver response on axis is closer (if not equal) to the average of its average response over all axes. Dick Olsher did something similar with one of his designs, but to the extreme. Even more extreme are "omni" designs, but this is something in between.

I'm qualifying this by saying that this was the best solution I acheived in a passive crossover. Active crossovers or DSP provide significantly more flexibility and lead to different optimizations (per the discussion's diversion to Sound Easy).
Back in the 80s, Lip and Van found that the LR crossover is just one specific solution to the L(s) + H(s) = A(s) equation. It happened to be the one that provided the best lobing (with the drivers vc aligned) and highest immediate roll off rates (by using squared Butterworths).

The advantage of LR (i.e. squared Butterworth) is the fact, that they are symmetrical and that they can be done either actively or passively. If the in-phase x-over is done with other transfer functions than Butterworth the orders get asymmetrical. One can still choose which branch is the steeper one. Lobing will not be as good as LR but still better than others. Group-delay performance is of course depending on the chosen Q value. And they would not be that easy to implement passively because there is some subtraction to be done.

I will try to undig a document that I once made for these.


A passive 1st order transient perfect XO is one of the main reasons my 10F/RS225 FAST speaker design sounds so good. Many people have built it, and they all agree it’s one of the best speakers that they have heard. I have made many speakers over the years, but this one remains the one I still have as my main system that I go back to over and over again. Once you hear a transient perfect speaker, you know what it should sound like and it’s hard to go back to inverted phase LR24 or even BW12 crossovers.

One of the biggest improvements is the realism of percussive sounds which are reproduced with a speaker that can produce a right triangle step response or play square waves excitation. The other benefit of transient perfect speakers and crossovers is the improved stereo soundstage and imaging because phase is flat over main bandwidth.

There is also the B&O Hole Filler XO that achieves the same thing in a 3-way. And if you have DSP, the Harsch XO is also closely related relative to the step response that can be achieved. I have studied these crossovers also.

10F/8424 & RS225-8 FAST / WAW Ref Monitor

Here is the XO design - besides good smooth wide bandwidth drivers as starting points (10F/8424 and RS225-8) a key element is that the tweeter and woofer are of the same polarity:

Modeled response:

Here is the XO implementation:

Here is the measured frequency and phase response - Notice the phase varies by only +/-22.5deg from 100Hz to 12kHz:

Here is the step response:

Here is the speaker:
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Yes, it was either put tweeter in a waveguide or place on top in a separate chamber with a deep 3in stepped baffle to get Enough setback, or place on bottom so that it approximates a 3in delay at rough listening position with ear at woofer axis. Putting on bottom achieves this for a surprisingly wide sweet spot. The XO is 1st order electro-acoustic as modeled by Xsim based on measured raw response and impedance sweep. Electrically it might be higher order with extra filters.
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I am not an advocate of "filler drivers" as a solution to realise linear phase crossovers. Nevertheless there is a set of solutions that I am not aware have been published explicitly elsewhere, one of which usefully exploits an LR4 alignment as its basis. In this particular crossover, the low-pass and high-pass sections retain their fourth-order characteristics, but the "filler" still has limited first-order roll-offs. In the general set, as with any like arrangement, improving the filler section's characteristics requires peaks in the low- and high-pass sections that cause more problems than they solve. But to the solution at hand...

The second-order Butterworth polynomial is defined as:

B2(s) = 1 + sqrt(2).s + s^2

The fourth-order Linkwitz-Riley polynomial is obtained (for good reason) from square of B2(s) where:

LR4(s) = B2(s)^2 = 1 + 2.sqrt(2).s + 4.s^2 + 2.sqrt(2).s^3 + s^4

From which we derive the low-pass and high-pass filters as:

L(s) = 1/LR4(s) and H(s) = s^4/LR4(s)

with the well-known result that produces a uniform magnitude response and a non-linear, all-pass phase response A2(s) where:

L(s) + H(s) = B2(-s)/B2(s) = A2(s)

and where usefully in attempting to arrange a uniform power output:

arg{L(s)} = arg{H(s)}

But we might also notice that via a bit of mathematical manipulation, we can rewrite the expression for LR4(s) as:

LR4(s) = 1 + 2.sqrt(2).s x [1 + sqrt(2).s + s^2] + s^4 = 1 + 2.sqrt(2).s.B2(s) + s^4

Which is to say that, if we set the filler response as a band-pass filter, F(s), where:

F(s) = 2.sqrt(2).s.B2(s)/LR4(s) = 2.sqrt(2).s/B2(s)

then the summed response changes to:

L(s) + F(s) + H(s) = [B2(-s) + 2.sqrt(2).s]/B2(s) = B2(s)/B2(s) = 1

The summed output therefore has both uniform magnitude and linear phase. Whether this is a desirable solution is another question: As I said at the start, I am not an advocate of such approaches to implementing "transient perfect", linear phase crossovers, but this particular approach at least gives the experimenter the chance to switch instantly between a "transient perfect" filler driver arrangement and a conventional, optimal LR4 crossover so they can find out for themselves :)

And if this has all been stated previously, my apologies, but I would be grateful if someone could credit the originator appropriately.
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john k...

2004-08-10 2:50 am
Actually, this has been done to death. The reality is that ALL filler driver crossovers are just manipulations of subtractive crossovers.

Given ANY LP and ANY HP, (they don't have to be of the same order or family) a filler can be found to give flat amplitude and phase as

1 - (LP + HP) = FL.

Also note that there is nothing magical about the filler. It's just a complication of the simple 2-way subtractive filter,

1 - LP = HP' or

1 - HP = LP'

Now, since

1 - HP - LP = FL you also have

1- LP =( HP + FL) = HP'


1 - HP = (LP + FL) = LP'

The question is not what can or can't be done mathematically, but rather the characteristics of FL, HP' and LP', depending on how it is viewed. For the LR4 with filler, the problem is that the filler has a peak amplitude of +6dB at the crossover point. This gives rise to horrible off axis response. As we all recognize, the basic LR4 has symmetric polar response with nulls above and below the on axis position. With the filler, the HP and LP sill still sum to a nulls at those off axis positions leaving only the filler's response. Thus where the was an off axis null, there will now be and off axis peak of +6dB.

So the reference would be back to Small's paper on subtractive crossovers.
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Actually, this has been done to death.

Oddly I have never seen it in any of the literature, at least not explicitly so.

...This gives rise to horrible off axis response

That is my conclusion too, but it does allow instant switching back to LR4 to hear the differences by simply muting the filler driver for people to easily hear for themselves :)
It is quite easy to use the approximation of a subtractive crossover with a widerange driver combined with a well-behaved woofer (even better in MTM). The lower the order the easier. "2nd order" HP with "1st order" LP is the easiest because it gives the least overlap and the lowest bump in the "derived" path.
For everything else the use of other crossover topologies is advised.


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