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Author Topic: More thoughts (and pictures) on crossovers, phase, and delay  (Read 1323 times)

Frank Koenig

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More thoughts (and pictures) on crossovers, phase, and delay
« on: October 26, 2018, 01:53:56 pm »

The other thread (Lab: Delay mains vs subs) drifted away from the OP's question (mea culpa) and then kind of went off the rails. So I start fresh here.

REQUIREMENTS and COMMON TERMINOLOGY

When developing a crossover we're trying to meet four requirements, at least, in the neighborhood of the crossover frequency:

1. Adequately attenuate the out-of-band signal to each passband to suppress undesirable response, protect (the HF) from excessive out-of-band power, and to maintain the desired polar response, especially in the case on non-collocated pass-bands.

2.  Achieve an approximately flat magnitude response.

3. Achieve an approximately matched phase delay near the crossover frequency, which is essential for 2, above.

4. Achieve an approximately matched group delay near the crossover frequency.

For this discussion I'm assuming a single-input, single-output model of the speaker, which in practice is derived from a statistic (average, for example) of the speaker's response over the angles of interest. There is usually more than one "right" way to do this, although, as with most things, there are many more wrong ways. For example, we often are faced with the option of inverting the polarity of a pass-band while adding a delay of 180 deg at the crossover frequency to one or the other passband. Let's call this the half-cycle-offset (HCO).

The phase delay of a system is minus the phase divided by the frequency.

Dp = -phi(w) / w

(I'm using "w" as a lower-case omega. Pardon my ASCII.)

If the phase is expressed in radians and the frequency in radians / second, then the phase delay has the unit of seconds. For an amplitude-modulated signal, where the envelope varies slowly with respect to the bandwidth of the system, the phase delay represents the delay of the carrier.

The group delay of a system is minus the derivative of the phase with respect to frequency, in other words, minus the slope of the phase.

Dg = -d phi(w) / dw

If the phase is expressed in radians and the frequency in radians / second, then the group delay, too, has the unit of seconds. For an amplitude-modulated signal, where the envelope varies slowly with respect to the bandwidth of the system, the group delay represents the delay of the envelope.*

Filters must be of sufficient strength (slope, cutoff frequency) to achieve requirement 1. Getting the phase curves to cross at the crossover frequency achieves requirement 3.  Getting the phase curves to have equal slope in the vicinity of the crossover frequency achieves requirement 4. Getting the phase curves to overlap (spoon?) in the vicinity of the crossover frequency achieves both 3 and 4. Requirement 2 can usually be satisfied (assuming 3 already is) by the choice of filter behavior in the transition region (Butterworth, Linkwitz-Riley, etc.) and cutoff frequency. Upstream EQ can also aid in achieving 2.

TONE BURST ENVELOPES and GROUP DELAY

I've been using tone-burst (TB) responses in the time domain to verify designs and to try to gain an intuitive understanding of the tradeoffs. Overlapping LF and HF TB responses are somewhat comforting that I'm on the right track, but can still be ambiguous with respect to HCO. Computing and plotting the envelopes of the TBs changes this. I obtain the envelope from the magnitude of the corresponding analytic signal which is obtained from the discrete Hilbert transform of the time series. This is a common technique and is how, for example, energy decay curves are obtained from the impulse response. It works beautifully in this case, with no numerical artifacts that I can see.

Looking at the envelopes this way makes the effect of group delay very clear. In the example below (a 15 in. direct radiator and an HF horn crossed at ~450 Hz) the two cases differ by an HCO. The second case (images 4 -6), where the phase curves overlap, shows almost perfect coincidence of the TB envelopes of the LF and HF, whereas the first case does not. Interestingly, the first case has closer time alignment between the beginnings of the impulse responses, as well as slightly less overall group delay variation over frequency.

The big question, of course, is what is the perceptual significance? I'm going with the, now conventional, wisdom that group delay matching is primary, but I really don't know. In the case of subs, where relative gain adjustments might be made on the fly, there is clear value in phase matching over a range of frequencies so that alignment is preserved when we push up the sub fader.

FORCING PHASE OVERLAP

The availability of FIR filters opens up the possibility of forcing phase overlap over a wide range of frequencies as never before in applications where processing delay is acceptable (not for subs, except for playback only, but easily for main speakers with > ~400 Hz crossovers). To be clear, these would not be "flat-phase", symmetrical impulse response filters but rather non-causal phase equalizers that could be applied to either or both pass-bands to widen the overlap and, perhaps, allow the simultaneous satisfaction of other requirements, such as impulse alignment. I haven't gone there yet but have most of the computational pieces that need to be strung together. What I need to think about is how to constrain the problem sufficiently but not excessively -- there is a dangerous amount of flexibility. As always, your thoughts are welcome.
 
*"Signal Analysis", Athanansios Papoulis, McGraw Hill, 1977, pp. 123

--Frank














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Mark Wilkinson

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Re: More thoughts (and pictures) on crossovers, phase, and delay
« Reply #1 on: October 27, 2018, 06:08:40 pm »


Frank, all you said made clear sense, thx.
May I ask what  software you're using?  And hardware, if going beyond sims?
Also very curious about the rationale for using tone bursts, instead of the more usual swept sine, MLS, or dual channel ?

Looking at your graphs, I noticed the two phase plots show different x-over freq's.
Did the corresponding envelops come from those?  Or I guess I'm wondering can you get the envelops straight from the tone bursts?

Using REW or Smaart to do look at filtered impulse response, I know we can peer in at 1/3 octave.
 
But in just thinking about 'what if' we were able to look at an impulse gradation as fine as a single frequency...
Well, if we were using complementary crossovers measuring at x-over freq....wouldn't  the 'single freq envelop' be identical for both sides?
IOW, shouldn't the two waves completely coincide...where both initial rise and peaks are identical? For that single x-over freq?

I like to make a perfect impulse either in software or thru dual FFT, and then look at all the initial rise times and peaks in the 1/3 filtered impulses.
It keeps reinforcing the idea that aligning initial rise = phase alignment.
But then again. a perfect impulse implies perfect flat phase.....
IIR slopes get a whole lot more slippery, huh ?  :)


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Frank Koenig

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Re: More thoughts (and pictures) on crossovers, phase, and delay
« Reply #2 on: October 28, 2018, 05:19:47 pm »

Frank, all you said made clear sense, thx.

Mark, nicest thing anyone's said to me in a long time. Thanks :)

Quote
May I ask what software you're using?  And hardware, if going beyond sims?

R language under Windows. ARTA for impulse response acquisition using M-Audio "Fast Track Pro", Earthworks M30. I import the IRs as CSVs into R.

https://www.r-project.org/

Quote
Also very curious about the rationale for using tone bursts, instead of the more usual swept sine, MLS, or dual channel ?

I use tone bursts as a way of visualizing system behavior as they provide localization in time AND frequency and resemble musical sounds. Furthermore, as they are in the time domain they are not very abstract and therefor easy to comprehend. As fascinating and useful as the frequency domain is, it is rather abstract and hard for many to grok. Look at the blowback I got for mentioning negative frequency on this forum. Good I didn't get into Laplace transforms or that there is only one infinity for complex numbers

The IRs are captured by ARTA using MLS derived pseudo-noise. I've used swept sines too, which get a rise out of the neighbors who think a flying saucer is landing.

Quote
Looking at your graphs, I noticed the two phase plots show different x-over freq's.
Did the corresponding envelops come from those?  Or I guess I'm wondering can you get the envelops straight from the tone bursts?

I define the crossover frequency as the frequency where the magnitudes are equal. By this definition they happen to be the same for both cases.

I do get the envelopes from the tone bursts using the analytic signal representation, as I attempted to describe. I can go into greater detail if anyone wants.

Quote
But in just thinking about 'what if' we were able to look at an impulse gradation as fine as a single frequency...
Well, if we were using complementary crossovers measuring at x-over freq....wouldn't  the 'single freq envelop' be identical for both sides?
IOW, shouldn't the two waves completely coincide...where both initial rise and peaks are identical? For that single x-over freq?

Kinda lost me. Not sure what you mean by a single frequency as resolution in one domain must be traded for resolution in the other in accordance with the Uncertainty Relationship. A true single frequency is, of course, a sine wave of doubly infinite duration.

Quote
It keeps reinforcing the idea that aligning initial rise = phase alignment.

After thinking about it some more I believe that aligning impulse responses is a red herring. Consider the thought experiment of two finite-order systems that have no initial delay. Say a "straight wire" h(t) = delta(t) with spectrum  Ha(s) = 1, and a first-order all-pass with spectrum Hb(s) = (s+1)/(s-1). We want the phase delay to be equal at some (crossover) frequency, say w = 1. Phi(Ha(j1)) = 0, indeed  Phi(Ha(jw)) = 0 for all w. Phi(Hb(j1)) = pi / 4. To get Ha and Hb to have equal phase delay at w = 1 we need to delay Ha by pi / 4. The point is when designing a crossover we are interested in the summation of the passbands in the vicinity (typically +/- one octave) of the crossover frequency. The impulse response contains the complete linear behavior of the system, including much that is irrelevant to the crossover. In general, a good crossover alignment does not coincide with impulse alignment.

Now by introducing processing delay (and FIR filters) we might be able to get phase delay, group delay, and impulse response all to line up (approximately), but I'm not convinced of the subjective value of doing so. There are many perceptual factors that are way above my current pay grade, and to some, perhaps a large, extent a matter of application and taste.
 
Quote
IIR slopes get a whole lot more slippery, huh ?

Yes, and pretty much necessary for subs, at least, in a live situation.

--Frank
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Mark Wilkinson

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Re: More thoughts (and pictures) on crossovers, phase, and delay
« Reply #3 on: October 29, 2018, 06:02:41 pm »



After thinking about it some more I believe that aligning impulse responses is a red herring. Consider the thought experiment of two finite-order systems that have no initial delay. Say a "straight wire" h(t) = delta(t) with spectrum  Ha(s) = 1, and a first-order all-pass with spectrum Hb(s) = (s+1)/(s-1). We want the phase delay to be equal at some (crossover) frequency, say w = 1. Phi(Ha(j1)) = 0, indeed  Phi(Ha(jw)) = 0 for all w. Phi(Hb(j1)) = pi / 4. To get Ha and Hb to have equal phase delay at w = 1 we need to delay Ha by pi / 4. The point is when designing a crossover we are interested in the summation of the passbands in the vicinity (typically +/- one octave) of the crossover frequency. The impulse response contains the complete linear behavior of the system, including much that is irrelevant to the crossover. In general, a good crossover alignment does not coincide with impulse alignment.

Now by introducing processing delay (and FIR filters) we might be able to get phase delay, group delay, and impulse response all to line up (approximately), but I'm not convinced of the subjective value of doing so. There are many perceptual factors that are way above my current pay grade, and to some, perhaps a large, extent a matter of application and taste.
 
Yes, and pretty much necessary for subs, at least, in a live situation.

--Frank

Frank, thanks a lot for sharing all the info I asked about. 
I got a good laugh about sine sweeps scaring neighbors with visions of flying saucers.
I only risk 2-3 high power sweeps on any given day since my test setup points the speaker out over a large lake, which naturally caries sound. for miles. Warning-siren syndrome for sure !!  So I only use outdoor sine sweeps sparingly for harmonic distortion.


The 'what if' that I conveyed poorly in my prior post, was wondering exactly what  high resolution impulse response filtering (say 1/48th oct) centered on x-over frequency,  would look like for both sides of a complementary crossover (say LR24).   I think the high res filtered responses would be practically identical.

Sorry I couldn't keep up with your thought experiment.  Unfortunately, I have no formal classwork in electronics, signal theory, etc.

Before continuing, I guess I should stop and say all my comments about phase alignment, and initial rise alignment have been with drivers out of the picture. 
Simplify, step at a time, my motto :)  Besides drivers aren't so confusing, I just use minimum phase in and out of band to get them as well behaved in their pass-bands and through summation region as much as possible.  (Applying phase-only compensation to the roll offs is a little tricky, but still easier than IIR x-overs.)

Anyway, I do believe aligning impulse responses is equivalent to aligning phase.....so IMO a good impulse is essential to a good crossover.

I think a good IRR impulse will reflect the best possible alignment given the group delay/phase of the x-overs in play.
IOW, a true minimum phase alignment, with any excess (or deficit) phase removed.
The deviations from the ideal minimum phase alignment can be seen as mis-alignment of initial rise times, looking between bandwidth filtered impulse responses.
REW is great at showing this, IME.

My standard speaker tuning process (now including drivers) has become pretty simple really.
Step 1:  driver by driver minimum phase adjustment as previously mentioned, into the FIR file
Step 2: driver by driver phase-only adjustment via FIR if needed
Step 3: apply linear phase crossovers via FIR to all bands, with all the taps available
Step 4: tie all bands together with polarity, delays, and levels
Good as gets soundwise. ROCKS!!!!  Standard playback-only setup.

Step 5: substitute IIR crossovers between sub and main, or blend in a combination of IIR and linear-phase xovers, to get latency down to live acceptably.
This is where looking at initial rise time really helps I think.

I give myself 15ms latency to work with for live...(nobody has ever mentioned latency issues with horn-subs properly aligned for live. 15ms isn't much above that, and I've had no problems at gigs yet using 15ms.) Still Rocks!!!

Again, sorry I can't respond to your thought experiment. Need more school for sure  :-[
I have some pretty good proofs, or what i think are proofs, of initial rise = phase alignment if you or anyone wants to go into it.  Don't want to bore ....

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Frank Koenig

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Re: More thoughts (and pictures) on crossovers, phase, and delay
« Reply #4 on: October 30, 2018, 03:32:54 pm »

My tuning/crossover procedure, while evolving, is not far off from yours. In three-way systems I've worked on so far I've used a passive crossover mid-high and bi-amped low-mid.

After developing a passive mid-high crossover, suffering the difficulties of optimizing passive L-C-R networks in the face of wild and potentially inconsistent driver impedance variations (a subject for another time), I apply an FIR filter to the mid-high band that covers all the sins. This includes some combination of minimum-phase magnitude (and hence phase) compensation, group delay compensation, and low-mid crossover high-pass. (I derive these separately, convolve, and optimally window at the end.) When working with a "good" speaker I've often left it at just minimum-phase compensation, as I've seen little evidence of minor, smooth, overall group delay variation being perceptible on music. (I might hear a difference on 100 Hz triangle waves.) Having said that, I'm aware of the importance of phase and delay matching when trying to get disparate systems to play together. I guess my philosophy is that since I don't really know what I'm doing, first do no harm -- use a light touch. In some cases (foldback) I use minimum-phase only to minimize overall latency.

On the low to mid-high crossover I have the full flexibility of bi-amping, mainly the ability to use pure delay, and freedom from driver impedance weirdness. My goals remain as stated in the original post.

I played some more with the notion of impulse alignment, which is the term that I'm applying to delaying one or the other passband to get the impulse responses to start at the same time. While intuitively compelling, and perhaps of perceptual value, I don't see it as the primary goal, and clearly not sufficient to ensure a good crossover.

Here's an idealized example I ran. It generated lots of graphs that I'm too lazy to post now but will put up on Dropbox with a link if anyone cares. Forget about real speakers for a moment. Consider a mildly crummy crossover that consists of high- and low-pass 3rd order Butterworth filters with identical cutoff frequencies. One of them has inverted polarity. This crossover, by the way, is a bit of a classic as the sum is an all-pass, which is compelling. The trouble is that the phase is 90 deg out at the crossover, which will cause odd polar behavior for non-collocated* drivers. In the absence of delay the impulse responses, of course, align. Adding a quarter cycle of delay to the HF causes the phase to match, the group delay to match fairly well, and the magnitude to get a mild bump. I would argue that this is likely a better compromise, if only because the polar response will be more uniform across the crossover, even though the impulse responses do not align.

This is an ideal example -- real world speakers usually throw much worse at us. There likely exist constraints under which impulse alignment is sufficient for phase and group delay alignment at the crossover, and if I were a better and more ambitious mathematician I could lay out what they are. Mark may be satisfying these constraints in a high-latency, playback-only situation. But I maintain that in the common case when these constraints fail to be satisfied, that good phase and delay behavior in the crossover region is primary.

Thank you, Mark, for engaging me on this. I'm happy to put up source code or pictures. I know all the real speaker designers are rolling their eyes at this, no doubt tedious, discussion. But please realize, with so much at the boundary between theory and application shrouded in commercial secrecy, that us amateurs need to reinvent stuff for ourselves. And nothing helps my understanding so much as trying to explain it, however failingly, to others.

*In this context I favor "collocated" over "coincident", reserving the latter for when time rather than space is involved.

--Frank
« Last Edit: October 30, 2018, 03:36:03 pm by Frank Koenig »
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Nathan Riddle

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Re: More thoughts (and pictures) on crossovers, phase, and delay
« Reply #5 on: October 30, 2018, 03:44:24 pm »

Frank, Mark,

I just want to say I really enjoy listening (reading?) to you two.

I understand about half of what you say, but I really have to think when I read what you write and that's a good thing!

Thanks for teaching me something new every day (time, season, period?).
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Frank Koenig

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Re: More thoughts (and pictures) on crossovers, phase, and delay
« Reply #6 on: October 30, 2018, 09:49:31 pm »

Frank, Mark,

I just want to say I really enjoy listening (reading?) to you two.

I understand about half of what you say, but I really have to think when I read what you write and that's a good thing!

Thanks for teaching me something new every day (time, season, period?).

Nathan, thanks for the interest. It's OK to ask questions. No guarantee that I can answer them or that I'll be correct, but I'll try.

I put together some plots from the last example. I'll get them put up here or on Dropbox in due course.

--Frank
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Mark Wilkinson

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Re: More thoughts (and pictures) on crossovers, phase, and delay
« Reply #7 on: October 31, 2018, 07:59:43 am »

Many thanks to you Frank, for your kind words.
And I really appreciate you taking the time to explain your procedures at a level I can follow....
which BTW, everything in your last post made clear sense to me....
and I'm in complete agreement with you (other than our assessments of the value of impulse alignment  :))

Looking forward to your BW3 experiment. 
Occasionally, I get called by guys to lend a hand in time-aligning their subs for a gig,
and of course, sometimes there is never enough setup time to even measure flight time.
I just slap in BW3's with the idea of being half right either way, eh? 

And yes, I'm sure the real speaker guys read this stuff and shake their heads, but hey, we gotta learn somehow, huh.
 
and Nathan, thx to you too...hope all is going well with testing/setting up all the rigs you have going...
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Frank Koenig

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Re: More thoughts (and pictures) on crossovers, phase, and delay
« Reply #8 on: October 31, 2018, 11:43:30 am »

It appears that I've filled my allotment of attachments on the forum and can't put up any more. This would be fine except I haven't figured out how to remove the old ones. Anyway, here's a link to three panels each of which shows TB response, phase, TB envelope, and impulse response for the ideal 3rd order Butterworths. The first is with 1/4 cycle HF delay at the crossover. This is what I would use given the choice between the three. The second is with 1/4 cycle LF delay. What a disaster! The third is with no delay and therefor aligned impulse responses. -F
https://www.dropbox.com/sh/zsgta3fw3juwuv2/AAAQfKwvyjNa8ue_e-t5xNrWa/Audio?dl=0&subfolder_nav_tracking=1
« Last Edit: October 31, 2018, 11:45:32 am by Frank Koenig »
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Mark Wilkinson

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Re: More thoughts (and pictures) on crossovers, phase, and delay
« Reply #9 on: October 31, 2018, 01:35:36 pm »

It appears that I've filled my allotment of attachments on the forum and can't put up any more. This would be fine except I haven't figured out how to remove the old ones. Anyway, here's a link to three panels each of which shows TB response, phase, TB envelope, and impulse response for the ideal 3rd order Butterworths. The first is with 1/4 cycle HF delay at the crossover. This is what I would use given the choice between the three. The second is with 1/4 cycle LF delay. What a disaster! The third is with no delay and therefor aligned impulse responses. -F
https://www.dropbox.com/sh/zsgta3fw3juwuv2/AAAQfKwvyjNa8ue_e-t5xNrWa/Audio?dl=0&subfolder_nav_tracking=1

Hi Frank, I vote for #3....looks great to me other than the low-pass inversion.
Tried it out, removing the inversion.
First plot below is 100Hz BW3 HP and LP, aligned via initial rise, both with +polarity.

Second is same thing, but with crossover freqs spread out to get rid of the small mag hump, and to put x-over freq right on 100Hz
I simply spread each 10Hz from center....so,  90Hz LP, and 110Hz HP.


edit:  I also got an error: uploader full message, when trying to post...I guess it's a forum prob, or we're being blackballed lol
Anyway, pls do try TB method getting rid of the inversion, with initial rise alignment...it will be interesting to see if TB method gives anything different than the pretty good looking REW sine sweep
I'll try again later to post plots.

double edit:  pls ignore any previous comments about preferring same polarities....I now see that costs an extra 180 degrees of wrap vs opposite polarities

« Last Edit: October 31, 2018, 04:48:08 pm by Mark Wilkinson »
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Luke Geis

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Re: More thoughts (and pictures) on crossovers, phase, and delay
« Reply #10 on: October 31, 2018, 03:36:33 pm »

It has been my understanding that the crossover corner frequency is where the bulk of the phase issue arise. We oversimplify the filters effects by simply saying that a 2 pole or 4 pole filter has 180* and 360* of phase shift respectively. We tend to forget phase lead and phase lag around the corner frequency and that the phase of the upper passband starts at +45* per pole and settles to 0* of phase while the lower passband starts at 0* of phase shift and shifts to its final relative phase. This means that there is a fair number of frequencies in which the phase of two different drivers are never going to properly align no matter what we do unless we make a frequency dependent phase inverting filter.

In the case of a 24db per octave filter ( a 4th order filter ), we have 360* of phase shift occurring between the hi and lo band. The upper band would start at 180* and settle to 0* while the low band would start at 0* and shift 180* out of phase where it would eventually settle at. The 360* of phase shift only actually exists around the crossover point, but as we get lower and higher in frequency relative to the crossover point the phase shift is actually only 180*.

This is where all-pass filters help. A first-order allpass filter provides 180* of phase shift regardless of frequency. This means that when used on the upper passband that we could have the lows and hi's in proper phase outside of the crossover region ( both now lagging by 180* ), but it doesn't address the phase shift within the crossover region. Keep in mind that the upper passband would now start at 360* and then settle to 180*. Within the crossover region, there will always be some amount of phase mismatch, but at least the relative phase beyond the crossover region would be in proper phase. This doesn't really address the issues of frequency choice ( for the crossovers ) and the physical placement and resultant time-based issues. 

We simplify the group delay utilizing the IR and or FFT information and then simply align the phase slopes till they best match. Honestly, this is probably the best way. If we were to widen the window and average less for a more realistic outcome, we would probably find that we can never truly align the subs and the mains at all frequencies that pass between them.

Lead-Lag Compensators do exist but are not implemented in audio as of yet. It may be the next step in the correction of crossover networks. I think with this form of technology we can get better results. The problem isn't so much the resultant phase difference as much as the altering phase directions within a crossover network.
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Mark Wilkinson

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Re: More thoughts (and pictures) on crossovers, phase, and delay
« Reply #11 on: October 31, 2018, 03:51:54 pm »

Luke, what about complementary crossovers...e.g. linkwitz-riley...that share the same phase trace throughout summation region?

 
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Luke Geis

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Re: More thoughts (and pictures) on crossovers, phase, and delay
« Reply #12 on: October 31, 2018, 06:45:25 pm »

LR crossovers do solve the phase issue through the crossover region, yes, but they still exhibit the total phase shift from the two passbands.

An LR filter is simply a cascaded BW filter. An LR filter is a 2x summation of two BW filters. So two 6db BW filters cascaded will add up to a 12db LR filter. It will have an equal phase shift at all frequencies through the crossover region, but it will still exhibit the total phase shift of the pole count. In the case of a 12db LR filter that would be a 180* phase shift. You still have to play god in other words.

The biggest issue is still the physical difference in time and space of the drivers in interest. Electrically we can add them up and alter things to be more in line, but the acoustic outcome is what matters. The big question is how do we measure it and what is the best way to make it align? As we know if we move any number of feet up or down, left or right, we will change the time-space point of incident and thus the phase and acoustic summation result. What yields the best outcome?
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Mark Wilkinson

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Re: More thoughts (and pictures) on crossovers, phase, and delay
« Reply #13 on: October 31, 2018, 07:15:29 pm »

LR crossovers do solve the phase issue through the crossover region, yes, but they still exhibit the total phase shift from the two passbands.

An LR filter is simply a cascaded BW filter. An LR filter is a 2x summation of two BW filters. So two 6db BW filters cascaded will add up to a 12db LR filter. It will have an equal phase shift at all frequencies through the crossover region, but it will still exhibit the total phase shift of the pole count. In the case of a 12db LR filter that would be a 180* phase shift. You still have to play god in other words.

The biggest issue is still the physical difference in time and space of the drivers in interest. Electrically we can add them up and alter things to be more in line, but the acoustic outcome is what matters. The big question is how do we measure it and what is the best way to make it align? As we know if we move any number of feet up or down, left or right, we will change the time-space point of incident and thus the phase and acoustic summation result. What yields the best outcome?

Yep on the butterworths...I've tried to attach a chart on how the orders act, for any following along...(if upload is working yet)

And sure, acoustic summation is all that finally counts...and measured to what position?  But those are issues I think beyond what's being discussed...

So far, we've kinda just been focusing on the relationship between phase and impulse response....

Nope..uploader still full when tried to post...
Here's the chart without good formatting..still should make sense i hope

      Bu crossovers   
         
              deg    dB gain
order pol     @ xover   @ xover
1   equal   90   0
1   opposed   270   0
2   equal   180   -total
2   opposed   0   3
3   equal   270   0
3   opposed   90   0
4   equal   0   3
4   opposed   180   -total
5   equal   270   0
5   opposed   90   0
6   equal   90   -total
6   opposed   180   3
7   equal   90   0
7   opposed   270   0
8   equal   0   3
8   opposed   180   -total

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