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30th July 2012, 12:49 AM  #1 
diyAudio Member
Join Date: Mar 2008

IIRbased crossover: no thanks!
Hello,
Did you know that a LinkwitzRiley XO can deliver a perfectly reconstructed phase? From the attached .jpg it gets obvious that we are going to live a few more decades with LinkwitzRiley crossovers, this time using FIRs. The 4thorder LR (two cascaded 2ndorder Butterworth) exhibits a fair low amount of preshoot and ringing. The phase and the amplitude are perfectly reconstructed, when adding the lowpass and the highpass. This is something an analog LinkwitzRiley XO cannot do. This is something a digital IIRbased LinkwitzRiley XO cannot do. I went on www.linkwitzlab.com Over there I have not found such FIRbased LinkwitzRiley. Can somebody explain this to me? Where is it located? Such FIRbased LinkwitzRiley is easy to obtain. You decide how many taps your FIR will be (101 taps in our case). You decide about the sampling frequency (48 kHz) and the 3dB lowpass frequency (3400 Hz). You apply the module function of the Butterworth lowpass, in function of the frequency. You cascade two such module functions. Code is here: If Me.Lowpass_List.SelectedIndex = 7 Then 'Butterworth Me.F_Label.Text = "F3dB(Hz)" Me.Order_List.Enabled = True Me.Double_Box.Enabled = True If Me.Order_List.SelectedIndex = 1 Then n0 = 1 d0 = 1 d2 = 0 d4 = 1 d6 = 0 d8 = 0 d10 = 0 d12 = 0 End If For i = 0 To N F = FS * (i / N) Fr = F / Fcut GAIN(i) = n0 / ((d0 + d2 * Fr ^ 2 + d4 * Fr ^ 4 + d6 * Fr ^ 6 + d8 * Fr ^ 8 + d10 * Fr ^ 10 + d12 * Fr ^ 12)) ^ 0.5 If Me.Double_Box.Checked = True Then GAIN(i) = GAIN(i) * GAIN(i) PHASE(i) = 2 * Math.PI * (MID * i / N) 'take FIR delay in account for the phase Next End If The above code builds the iDFT GAIN array and iDFT PHASE array. Watch out the phase, it got defined as a pure delay equal to half the FIR delay. You then execute the iDFT. As result you obtain the timedomain impulse response, that you regard as the required lowpass FIR coefficients. The complementary highpass gets computed, basing on the lowpass FIR coefficients. The highpass FIR is the lowpass FIR with negative sign, with one notable exception which is the FIR midpoint highpass, where one must obey the rule: highpass FIR midpoint = (1.00  lowpass FIR midpoint) This way, the reconstruction (lowpass + highpass) will equal a Dirac occurring on the FIR midpoint. Meaning that the reconstruction is perfect in amplitude, and in phase. Add some windowing function if you want, and you are done. IIRbased crossover: no thanks! Any comments welcome. Last edited by steph_tsf; 30th July 2012 at 01:04 AM. 
30th July 2012, 09:30 AM  #2 
diyAudio Member
Join Date: Dec 2009
Location: Md

Very nice. Lots of thought. Now for the monkey wrench. Drivers. I can't remember the last time I had a mathematically symmetrical crossover in amplitude, let alone phase. We need to sum the results of the acoustic output of drivers at close to where they are very nonlinear. All a big bag of tradeoffs. Active makes it easier, as you are not dealing with variable impedance of the load, the amp has to do that. Still, offset crossover points and different slopes are what I usually wind up with. ( DCX on bench).
So, how to use your analysis to include the asymmetry from the drivers? I believe, though have not played with it, Bodizio UE tries to do this. No idea if it is as comprehensive as you explain in your approach. 
30th July 2012, 10:35 AM  #3 
diyAudio Member
Join Date: Dec 2004
Location: Israel

Inverse of driver's impulse response multiplied by impulse response of this filter = perfect slope for the xover...
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30th July 2012, 11:21 AM  #4 
diyAudio Member
Join Date: Jun 2009

where are the phase responses ??

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