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25th January 2013, 11:07 AM  #21 
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Join Date: May 2002
Location: Switzerland

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 bafflestep 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 highshelf would be the weapon of choice. This simple solution now has the disadvantage that we can't take the woofers topend 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 phaseresponse 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 groupdelay 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 welltamed 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 groupdelay 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 lowend 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 midfrequency 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 highshelf and firstorder lowpass chain can be implemented in hardware in two simple ways. Both are using one OPAMP, three resistors and two capacitors. One variant can be used only for Qvalues <= 0.5 and the other one is more flexible. Regards Charles Last edited by phase_accurate; 25th January 2013 at 11:10 AM. 
25th January 2013, 02:53 PM  #22  
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Join Date: Feb 2008

Quote:
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Greg 

25th January 2013, 03:13 PM  #23 
diyAudio Member
Join Date: May 2002
Location: Switzerland

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 
25th January 2013, 03:24 PM  #24  
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Greg 

25th January 2013, 03:30 PM  #25 
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Join Date: May 2002
Location: Switzerland

The socalled 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 lowend support by a larger woofer. Regards Charles 
25th January 2013, 04:01 PM  #26  
diyAudio Member
Join Date: Feb 2008

Quote:
Such a system would seem to be ideal for the Besselderived subtractivedelay crossovers. The subwoofer LPF rollsoff 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 rollsoff at 12db/octave, then with some delay (and perhaps a bit of EQ) that natural rolloff can be incorporated as an approximation of the perfectreconstruction subtractivedelay. Mathematically it's pretty easy to express; implementation might be a bit more difficult. Greg 

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