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Old 25th January 2013, 10:07 AM   #21
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The best compromise regarding overlap etc is IMO a filter with a Q value of 0.5 (aperiodic).
For every filter with a Q equal or less than 0.5 the denominator can be broken up into two first order parts, while a Q value 0f 0.5 is the easist of all cases because both parts are equal.

The higpass will be built by EQing the FR accordingly. We want to go for a Q of 0.5. The pole frequency (-6dB point for a Q value of 0.5) will be one octave below the acoustic crossover frequency.

The lowpass part

(1+sT/0.5)/(1+sT/0.5+s^2T^2)

can be split up (don't know the appropriate term in English) in several ways.

The one already shown as circuit example was a 2nd order lowpass multiplied by the sum of a differentiator and a linear function. This had the advantage of being able to adapt to the woofer's upper rolloff to some degree.
OTOH we usually don't like differentiators that much in practical circuits.

Another way of splitting the lowpass function is the sum of a second order lowpass and a 2nd order bandpass. In hardware this would be easy (although there wouldn't be a big advantage because of the increased amount of components). But the easy DSP crossovers wouldn't like that because of the summing function.


The third way to split the lowpass function would be the product of a first order lowpass and a first order lead filter ("high shelf"):


(1+sT/0.5)/(1+sT/0.5+s^2T^2)=((1+2sT)/(1+sT)) * (1/(1+sT))

Theoretically it should be possible to abuse the woofer's baffle-step function for the lead filter if it should happen in the proper frequency range accidentally. But for most applications the use of a first order lowpass followed by a high-shelf would be the weapon of choice. This simple solution now has the disadvantage that we can't take the
woofers top-end response into consideration that easily with the crossover function alone. But the extended EQ functions of DSP crossovers would allow easy driver EQing. And there is another thing that these crossovers offer and which cannot be implemented easily in an analog way: Delay!!!
The phase-response a the upper end of the woofers respnse is usually the worst contributor to bad signal summing. OTOH a lowpass can be regarded as having a more or less constant group-delay within its passband. So it is advisible to not try to exend the woofers response at the upper end in order to achieve the proper phase response but rather EQ it to behave like a well-tamed lowpass and then delay the FR accordingly. It would even be possible to add another lowpass to the woofer's response as long as it is not too close to the xover frequency. The best situation would be if the woofer's natural response and the additional filter form a Bessel filter (which has constant group-delay within its passband) but that may be asked too much.

So our crossover would look like that:

FR:

[HPF] -> [LT] -> [BSC] - > [delay] -> [misc EQ]

HPF: Highpass filter emulating the woofer's low-end response or even emulationg its equlised low end response if necessary.
LT: Linkwitz transform making the FR behave like a 2nd order highpass having a pole frequency of 0.5 * fcrossover and a Q of 0.5
BSC: Baffle step compensation
delay: Delay equal to the group delay of the woofer's natural (or equalised) lowpass behaviour.
misc EQ: Equaliser stages used to tame the FRs frequency response.

Woofer:

[LPF] -> [HS] -> [misc EQ]

LPF: First order lowpass with a -3dB frequency of 0.5 * fcrossover
HS: 6dB high shelf with a mid-frequency of 0.71 * f crossover
misc EQ: Equaliser stages used to tame the FRs frequency response, eventually including an additional lowpass.

An LT might also be used for the woofer in order to extend its frequency range but take care of the summing with the FRs response.

Hope this will help for playing around a little for those interested.

I will soon post rules of thumb for the asymmetrical 3rd/1st as well and also for the symmetrical 2nd/2nd.

BTW: The high-shelf and first-order lowpass chain can be implemented in hardware in two simple ways. Both are using one OP-AMP, three resistors and two capacitors. One variant can be used only for Q-values <= 0.5 and the other one is more flexible.

Regards

Charles

Last edited by phase_accurate; 25th January 2013 at 10:10 AM.
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Old 25th January 2013, 01:53 PM   #22
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Quote:
Originally Posted by phase_accurate View Post
Here we are dealing with the classic constant-voltage crossovers that don't use any form of subtractive delay simply because of two reasons: They can't be built easily using analog methods (although I know at least one practical example which is a "mass grave" of allpass filters)
True -- they can be created in analog, but not easily. (Ng and Rothenberg used analog delay lines.)
Quote:
or digitally using these DSP crossovers that are configured by the use of a convenient but restricted GUI.
If you can create a Bessel lowpass filter and a delay within your GUI, then you can create the kinds of crossovers described in my paper.
Quote:
Forgot to get into the crossover order discussion. I am totally aware that a second order acoustical rolloff doesn't exactly mean perfect driver protection.
That seems to be the most common complaint about the 2nd order HF rolloff of the Gaussian/Bessel-derived subtractive delay crossovers. I guess my response is that, if it's a problem, then raise the crossover frequency or use a better HF driver.

Greg
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Old 25th January 2013, 02:13 PM   #23
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The "HF drivers" used here are fullrangers and therefore much more robust than domes. So the method shown here should work quite often. It is not accidentally that I posted it under the fullrange forum.

The GUIs of the cheap and convenient DSP crossovers let you define Bessel and delay. So far so good. But they usually only let you define chains of filters without any branching, addin or subtracting. And this is what would be needed for filters you describe.

But they would definitley work nicely for FAST systems, even those that only give a second order response for the highpass section.

Regards

Charles
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Old 25th January 2013, 02:24 PM   #24
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Quote:
Originally Posted by phase_accurate View Post
The "HF drivers" used here are fullrangers and therefore much more robust than domes. So the method shown here should work quite often. It is not accidentally that I posted it under the fullrange forum.
Frankly, I hadn't noticed that this was in the fullrange forum; my mistake. I guess I am a bit mystified as to why crossovers are being discussed for fullrange systems.
Quote:
The GUIs of the cheap and convenient DSP crossovers ... usually only let you define chains of filters without any branching, addin or subtracting.
Ah, that would be a problem.

Greg
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Old 25th January 2013, 02:30 PM   #25
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The so-called FAST systems are increasingly popular amongst DIYers.

FAST = Fullrange And Subwoofer Topology

And there is also some interpretations like "Fullrange Assisted ...." but I can't remember exactly how it goes. But both mean the same: A small fullranger that is getting low-end support by a larger woofer.

Regards

Charles
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Old 25th January 2013, 03:01 PM   #26
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Quote:
Originally Posted by phase_accurate View Post
FAST = Fullrange And Subwoofer Topology
A small fullranger that is getting low-end support by a larger woofer.
Thank you; now I understand.

Such a system would seem to be ideal for the Bessel-derived subtractive-delay crossovers. The subwoofer LPF rolls-off at an extremely steep rate, so the subwoofer "gets out of the way" very quickly and does not intrude upon the fullrange's splendid response from midbass upward. Furthermore, if the fullrange naturally rolls-off at 12db/octave, then with some delay (and perhaps a bit of EQ) that natural rolloff can be incorporated as an approximation of the perfect-reconstruction subtractive-delay. Mathematically it's pretty easy to express; implementation might be a bit more difficult.

Greg
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