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[Frank Koenig]

There are many ways to cobble together a chain of filters to flatten a speaker’s frequency response. For the common two-way bi-amped system I had settled on an FIR filter for the HF, a group of bi-quad IIR filters for the LF and IIR hi- and low-pass filters for the crossover. The HF FIR could either be minimum-phase (causal) or include an all-pass to flatten the excess group delay, as desired. I was willing to accept the inevitable phase-lag with increasing frequency resulting from the band limitation of the speaker’s pass-bands and the crossover filters. I toyed with non-causal filters for the crossovers to reduce the overall variation in excess group delay but it gets pretty messy pretty fast. Simply using “zero-phase” (symmetrical impulse response) crossover filters fixes only part of the problem as the phase shift from the electro-acoustic part remains. 

I’m now trying what I believe is a simplification that does what I want with pretty good visibility and control. In short, it’s to flatten the magnitude of the whole system using minimum-phase filters without regard for excess group delay and then applying an FIR all-pass to the whole system to fix that. Here are more detailed steps:

--Take pseudo-anechoic measurement pairs of the HF and LF at a number of angles.

--Combine the HF and LF frequency responses into a single HF/LF pair. (All the information is contained in the log-magnitude and excess group delay. I find this to be the most useful representation.)

--Generate a minimum-phase FIR filter to flatten the HF magnitude.

--Generate a set of bi-quads to flatten the LF magnitude.

--Model the crossover, including the above flattening filters, and choose minimum-phase hi- and low-pass filters and delays that meet driver limitations and provide a large phase overlap at the crossover. (At this point we have perfectly usable settings that flatten the magnitude. You can do a gig with this.)

--Take the full-range result of the model and determine its excess group-delay.

--Generate an FIR all-pass that flattens this excess group delay from below the crossover frequency up to the upper limit of the HF. This fixes the phase lag in the crossover region as well as any excess group delay in the HF.

The DSP I’m using (Powersoft) only allows one FIR in each channel’s chain so it is necessary to convolve the HF minimum-phase coefficients with the all-pass coefficients and to place this filter in both pass bands. When it’s all done the HF and LF EQ chains are identical except for the crossover filters. There is some “unnecessary” computing going on but who cares? An FIR requires the same amount of computation no matter what the coefficients.

The choice of how low to go with the overall all-pass depends on the length of FIR filter available and the tolerable system latency. As you get low on coefficients the all-pass starts to deviate from its specification at the low end.

For what it’s worth the current implementation I’m working with is a PM60-like homemade speaker using a B&C DCX464 HF driver on an 18sound XT1464 horn and two B&C 12NDL76 LF drivers on the plywood folded horn. I’m using B&C’s passive crossover for the coaxial HF driver. The active crossover is at 650 Hz consisting of a 3rd order Butterworth on the HF and a 2nd order Butterworth on the LF. This makes for a lovely alignment of phase and group delay around the crossover frequency. The LF has eight bi-quads (some "out-of-band") and the HF has a minimum-phase FIR. The overall all-pass goes down to 200 Hz resulting in essentially flat phase from there on up. The processing delay is 3 ms and the filter is limited to 384 taps as this is the most the old K-series Powersofts will do.

Having gone through this exercise I’ll add that I’m not convinced that flattening the excess group delay in an otherwise well-behaved system has any audible benefit when used stand-alone. When different boxes need to play together this flexibilty is likely useful. Maybe Paul Klipsch was half-right when he said we cannot hear phase.

--Frank

[Brian Jojade]

Having gone through this exercise I’ll add that I’m not convinced that flattening the excess group delay in an otherwise well-behaved system has any audible benefit when used stand-alone. When different boxes need to play together this flexibilty is likely useful. Maybe Paul Klipsch was half-right when he said we cannot hear phase.

--Frank

The ideal system looks good on paper and sounds good.  However, it's very possible to have a system sound good, but on paper look like a complete mess.  The reality is, you will never get it perfect, so at some point, you hit the laws of diminishing returns.

On the same note, you also can have a system that looks great on paper, but just doesn't sound right.  The engineer then tweaks things to how he likes it to sound anyway, so really, just getting close is all that really matters.

Perfect is impossible, and the closer you get to perfection, there is exponentially more effort needed to inch ever so closer.

[Russell Ault]

{...} Having gone through this exercise I’ll add that I’m not convinced that flattening the excess group delay in an otherwise well-behaved system has any audible benefit when used stand-alone. {...} Maybe Paul Klipsch was half-right when he said we cannot hear phase. {...}

I was lucky enough to take Merlijn van Veen's advanced audio workshop when he was still offering it, and it included two phase-related demonstrations. The first was basically an ABX test of how a second-order all-pass filter at 1kHz sounds where he would play the same chunk of a song twice, once with the all-pass filter inserted and once without, and then ask us to tell him which was which (we couldn't). For the second, he used FIR filtering (with a lot of taps) to take the phase response of a small full-range Meyer PA, which was already phase-flat above ~300 Hz, and lower that "flatness" down to something like 30 Hz; the difference between before and after was truly astonishing (I seem to recall that I sat bolt upright in my seat as soon as the test track started playing).

Taken together, these two experiences suggest to me that we can "hear phase", but only at lower frequencies. Phase is just time, and (as Haas demonstrated) humans tend to have difficulty perceiving small timing offsets. Of course the lower the frequency, the longer (in milliseconds) the group delay for a given amount of phase offset, so it makes to me that second-order all-pass filter at 1kHz wouldn't be audible (after all, one full cycle of phase offset at 1 kHz is only 1 ms), but the same filter at 50 Hz (i.e. 20ms) might very well be.

-Russ

[Frank Koenig]

For the second, he used FIR filtering (with a lot of taps) to take the phase response of a small full-range Meyer PA, which was already phase-flat above ~300 Hz, and lower that "flatness" down to something like 30 Hz; the difference between before and after was truly astonishing (I seem to recall that I sat bolt upright in my seat as soon as the test track started playing).

Thanks Russell. This is very interesting and consistent with some experiments I did a few years back. There I brutally flattened the phase down to maybe 100 Hz -- can’t remember exactly. I wasn’t sure if I perceived any difference on music playback but I could hear a clear difference on 100 Hz square and triangle waves.

I guess the sad result is that the frequency range where any of this appears to matter is also where we can’t do anything about it in a live environment because of latency. Perhaps in really large-scale systems where any extra 10 or 20 ms can be worked in by hanging the speakers that much farther forward it could be done. For living-room playback however -- bring it on.

Merlijn’s demo does help lay to rest the concern that filters with a very long non-causal part might cause audible artifacts. I must confess that while I trust the math I still have a bit of a hard time wrapping my head around it being OK to start making sound many ms before the main event 😊

I’ve rejiggered the processing some since my last post but I won’t bore you all with the details until I’ve gained some experience.

--Frank

[Russell Ault]

{...} I guess the sad result is that the frequency range where any of this appears to matter is also where we can’t do anything about it in a live environment because of latency. {...}

After the initial shock of the second demonstration had worn off a bit I also seem to recall feeling a subtle but profound sense of loss, as if I were experiencing something that I desperately wanted but could never truly have (and, given that this all ties back to the limitations inherent in the "one way only" nature of how we perceive time, I've found that it's rather easy to spiral from "phase correction below 100 Hz for live reinforcement applications" into "full-blow existential crisis" if I'm not careful).

-Russ

[Peter Kowalczyk]

I've found that it's rather easy to spiral from "phase correction below 100 Hz for live reinforcement applications" into "full-blow existential crisis" if I'm not careful).

-Russ

hey man, calm down, its just rock and roll....

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