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28th July 2012, 03:12 AM  #121 
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That is correct. A Gaussian follows the equation of the form "exp(x^2)". Us of any power other than 2 is not a Gaussian. That is why I am so careful to label an equation of the form "exp(x^N)" (N not equal to 2) as "Gaussianlike".

28th July 2012, 03:24 AM  #122  
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28th July 2012, 02:29 PM  #123  
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If you want your tweeter to sound ill, to exceded its Xmax at high listening volumes, and overheat, that's exactly what you need. Can you please describe the cases in which an acoustic 2nd order highpass works well, for a tweeter? 

28th July 2012, 04:17 PM  #124 
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I just finished writing iDFT_Lab mini, dealing with linear phase complementary symmetric FIR crossovers.
"linear phase" Both the lowpass and highpass exhibit a linear phase. "complementary" The sum of lowpass and highpass is equal to unity, both in amplitude and phase. "symmetric" When you view the amplitude plots with a log fequency scale, the lowpass slope and shape are the same flavour as the highpass slope and shape. Now about the details. For the crossover to be symmetric, the lowpass attenuation curve must exhibit certain properties. Let us draw the lowpass amplitude using linear Y (from zero to one  this is the transmission factor) and log X (10 Hz, 1 kHz, 10 kHz, etc ...). The X midpoint is thus 1 kHz. Say 1kHz is Fc, the crossover frequency. We know that at Fc, the lowpass amplitude is 0.5. See the attached .jpg sketch 0) Let us draw the (1 kHz, 0.5) point in green, the crossing point 1) Let us draw an arbitrary lowpass amplitude on the log frequency scale spanning from 100 Hz to Fc. 2) Because of the symmetry that we are targeting, the corresponding highpass amplitude from Fc to 10 kHz, also got defined. 3) Because of the lowpass/highpass complementarity, the corresponding lowpass amplitude from Fc to 10 kHz, also got defined (dy1 and dy2 in red). 4) Because of the lowpass/highpass complementarity, the corresponding highpass amplitude from 10 Hz to Fc, also got defined (dy1 and dy2 in blue). Looking at the four curve segments that just got defined, we realize that there is a central symmetry, with the particular (1 kHz, 0.50) point acting as centre. The DC to Fs/2 lowpass amplitude curve serves as input for the iDFT. The iDFT output is the corresponding impulse response in time domain. This way we get the lowpass FIR coefficients. Knowing the lowpass impulse response, we compute the complementary highpass impulse response, the usual way. This way we get the highpass FIR coefficients. A particular straightforward way to get a Nthorder lowpass amplitude curve exhibiting the symmetry requirement is : For i = 0 To N F = FS * (i / N) If F < Fcut Then GAIN(i) = 1  (((F / Fcut) ^ order) / 2.0) If F = Fcut Then GAIN(i) = 0.5 If F > Fcut Then GAIN(i) = (((Fcut / F) ^ order) / 2.0) Next There are many other ways to generate lowpass amplitude curves exhibiting the symmetry requirement. For a given filter slope (or order), one could try softening the lowpass edge close to Fc, in order to see the effect on the time domain preshoot and ringing. Currently I have no idea how to simply (mathematically) soften the lowpass edge close to Fc. And I would hate editing the iDFT input by hand. Any suggestion welcome. Steph Last edited by steph_tsf; 28th July 2012 at 04:22 PM. 
28th July 2012, 07:52 PM  #125  
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EDIT: I missed the "acoustic" in your statement, first time through, so you are correct about the level staying constant below 800 Hz. But by that time the signal has already been attenuated by 25 dB. That's a factor of 1/316 in power. Quote:
Furthermore, and I failed to make this point explicitly before, there are more crossovers than just tweeter crossovers. You said that 2ndorder highpass is inadequate in most situations. Well, when crossing over from a subwoofer to a fullrange speaker, a common situation, 2ndorder highpass is perfectly adequate. When crossing over from a woofer to a midrange, a common situation, 2ndorder highpass is often adequate. When crossing over from a midrange to a robust tweeter, a common situation, 2ndorder highpass is sometimes adequate. I'm just saying that the design must be judged on a casebycase basis, and summarily dismissing any configuration as "inadequate" is illadvised. Last edited by gberchin; 28th July 2012 at 08:08 PM. Reason: Acknowledge "acoustic" 2ndorder. 

28th July 2012, 10:15 PM  #126  
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I took the 800 Hz example for dealing with a quite robust tweeter, as starting point. Yes indeed, such tweeter can cope with a 2nd order acoustic highpass. It will thus survive a LipshitzVanderkooy delay compensated crossover featuring a high order Bessel (pseudo Gaussian) lowpass as kernel. However, nowadays there are interesting miniature tweeters, inexpensive, exhibiting quite high natural highpass cutoff frequencies: Dayton ND28 (1,200 Hz) Dayton ND20 (1,600 Hz) Dayton ND16 (2,200 Hz) Visaton CP13 (3,000 Hz) If you target a 3500 Hz 2nd order acoustic highpass with a Dayton ND16, the ratio between 3,500 Hz and 2,200 Hz is 1.59 so the 2ndorder crossover will attenuate the deep bass by only 8 dB. The Dayton ND16 will exceed the Xmax. On top of this, being so small, it cannot dissipate heat. For such a tweeter, a 3rd order acoustic highpass (or higher) is mandatory. Doing so, the subjective results are very pleasing, considering the price. With digital, and FIRs in particular, it costs the same price, targeting a 2nd order acoustic highpass, or a 6th order acoustic highpass. 

28th July 2012, 10:40 PM  #127  
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29th July 2012, 04:51 AM  #128  
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29th July 2012, 11:34 AM  #129  
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The way I picture it, it is not possible for a mere engineer to identify some arbitrary new practical problem to be solved, and then retrospectively acquire the fundamental skills and insight required to exceed the state of the art 'by the book'. All he can do is to play catchup, and apply the work of others to produce a passable result. Looking at this page Chebyshev polynomials  Wikipedia, the free encyclopedia it's as though I'm reading something written by someone from another planet. But without that level of maths, does that mean I can never truly understand filters (of which a Chebyshev is just one of many) and should absolutely entertain no ideas of ever exceeding the 'state of the art' myself? I suppose I could stop trying to take short cuts and embark on some serious mathematical training. By the time I was 90 I might be able to make the link between Gegenbauer and Jacobi Polynomials myself. I might then think "Right! Now it's time to make a seriously optimal crossover filter". And I might make a filter that optimised some obscure property. But, instead, by embracing ignorance and harnessing the stupendous power of computers, it looks as though a horny handed son of toil really can spend a few evenings bashing out some C and get moderately close to the 'stateoftheart'  in a practical sense. I remember when I first experimented with a neural network, experiencing a giddiness as I realised that I now possessed a tool that would enable me to approach myriad engineering problems in the future, confident that if push came to shove, I could throw computing power at the problem and match or exceed the purely mathematical approach. This crossover filter thing gives me a similar feeling. (By rights, computers should be nowhere near as powerful as they are. It's only the supremely unimportant applications, such as games, that has developed them to the point where, for an application like audio, the power of a common PC is practically limitless. I'm not sure that everyone understands that, yet). 

29th July 2012, 02:00 PM  #130  
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I am weary of this discussion. 

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