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Audibility of allpass crossover phase distortion

On the Dipole Models page and under FAQ19 I indicated my preference for a low order crossover between woofer and midrange, because of its reduced phase distortion. In the extreme this would argue for a 6 dB/oct 1st order acoustic crossover, which is exceedingly difficult and costly to realize, because it places stringent demands on linearity and frequency response of the drivers used.

I had convinced myself that the phase distortion of a LR4, 24 dB/oct midrange to tweeter crossover is not audible, Ref.17. Those same tests showed that changes in the phase shift at the very low end of the spectrum influenced the character of some test tones. Together with some comparisons between earlier speaker designs, which seemed to favor a model with 12 dB/oct crossover over the same with 24 dB/oct, I concluded that the woofer to midrange crossover should be of low order. In retrospect I think that it might have been an increase of non-linear amplitude distortion in the midrange driver, which gave the impression of better bass.

Following are test results and a setup that you can readily duplicate, if you like to investigate for yourself the audibility of phase distortion of typical crossovers that have allpass behavior. The acoustic lowpass output from the woofer and the acoustic highpass output from the midrange add to a flat amplitude response, but the phase of the summed output changes non-linearly with frequency. This is allpass behavior and it distorts the time domain waveform. 

Ideally the waveform is transmitted undistorted as with a 1st order 100 Hz Butterworth crossover, when lowpass and highpass outputs are added.

The 2nd order Linkwitz-Riley, 3rd order Butterworth and 1st order Butterworth crossovers form a 1st order allpass when the polarity of the midrange is reversed. The waveform is distorted. The high frequency spectral content forms a sharp spike of opposite polarity to the input, followed by the low frequency content with the same polarity as the input.

The 4th order L-R crossover forms a 2nd order allpass when woofer and midrange outputs are added. Again, the waveform is distorted, though differently from the previous case.

The spectrum of the waveform, that is applied to the 100 Hz crossover, is shown below. It has a 50 Hz fundamental and harmonics at odd multiples thereof. Each spectral component is transmitted in the above crossovers with correct amplitude, but their relative phase is changed, particularly between the regions below and above the 100 Hz crossover frequency.

The waveforms of A, B and C above are quite different and it would not be unreasonable to expect that they sound different. Yet, I have not found a signal for which I can hear a difference. This seems to confirm Ohm's acoustic law that we do not hear waveform distortion. At least it seems to apply to the phase distortion generated by typical allpass crossovers.

Invitation to verify test results

The waveforms above were generated with an active circuit that duplicates the phase behavior of the different crossovers, but has perfectly flat (within +/-0.1 dB) frequency response. The circuit is readily constructed using the WM1 printed circuit board. While the board was not intended for this application it can be adapted by adding a few wire connections and using  jumpers instead of some components. Capacitors may be placed where normally resistors would go and resistors connected in parallel to as in the schematic with the WM1 board component designations below. Only one trace needs to be cut. The circuit design is based on the general allpass configuration

Here is what I would like you to do:

• Build the circuit. Test it for flat frequency response and equal gain in all three sections. Trim component values if necessary.
  
• For listening via a loudspeaker insert the circuit ahead of your power amplifier. The speaker preferably has no crossover near the 100 Hz region, or is a 2-way with good bass extension, to reduce the possibility that its own phase distortion masks the contribution from the test circuit, or that it pushes the total phase distortion above the threshold of detection. A full-range Quad ESL might be a good speaker candidate.
  Quality headphones are probably even more accurate transducers, though observations derived from them may nor translate directly to a loudspeaker in a reverberant environment. Insert the circuit ahead of your headphone amplifier. 
  Listen in mono. 
  
• Listen to your selection of test signals or program material and record any audible differences or lack thereof as you switch between the sections. Record all test conditions so that others can duplicate them, if possible. Note the playback level. Phase distortion may increase the crest factor of the waveform (B above) and lead to non-linear distortion in the speaker and/or the ear producing a change in sound character.
  
• E-mail your observations to me. I will give a summary of the various findings on this page.

I would really appreciate your participation in this investigation. It could help to settle one more issue in knowing what is audible. I am not trying to prove that all allpass crossovers sound the same, they do not in practice, and there are good reasons for it. I merely want greater certainty whether phase distortion is a contributor and, I think, so would you. 

Thank you.

PS:
You might be interested in the mathematics and consider pre-processed test signals instead of building the circuit. We are dealing with three allpass transfer functions.

The reference to compare to:
F0(s) = 1

1st order allpass:
F1(s) = (s  - 1) / (s + 1)

2nd order allpass:
F2(s) = (s² - s 2^1/2 + 1) / (s² + s 2^1/2 + 1)

where  s = s + j w  

First you would need to determine the impulse response for F1 and F2, then translate it from 1 radian to 100 Hz,  and convolve it with the test signal time record. This can be done with MATLAB or similar software.

Test signals might be square-waves of different frequencies, or other artificial signals, and speech, and music samples. The results of the convolution are probably best stored on CD-R for comparative listening tests. While this approach makes the analog circuit construction unnecessary, it must be carefully executed not to introduce digital processing artifacts and it limits the easy choice of test material.

Conclusion

The phase distortion of a 100 Hz acoustic crossover with 12 dB/oct or 24 dB/oct is not audible based upon the above tests. 
This is my conclusion. No one has come forth with either confirming or contradictory observations. I must assume that the whole question did not generate much interest, or that readers had already an opinion that was not to be tested. Maybe you just did not have the time to investigate.

At 100 Hz a difference in acoustic path length of 1/8th wavelength, causing 45 degrees of phase shift, corresponds to 43 cm (17 inch). At 150 Hz crossover this decreases to 29 cm (11 inch) and makes the placement of a separate woofer relative to the midrange that much more critical. I like to keep offsets to less than 1/16th of a wavelength which makes it pretty much mandatory at 150 Hz crossover frequency to integrate the woofer with the midrange cabinet. This is not optimal for the woofer/room interaction which is minimized when the woofer is placed near the side walls.
With the woofer separated from the midrange by some angle, a 12 dB/oct crossover seems preferable to me, because the wider frequency overlap gives a more distributed wave launch than a 24 dB/oct crossover, where the transition in output from woofer to midrange cabinet is more abrupt.

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