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Old 28th July 2012, 02:12 AM   #121
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Originally Posted by steph_tsf View Post
Today I have no time to scan the entire thread. I vaguely remember that Berchin wrote that one must remain cautious about the wording. I vaguely remember Berchin wrote that "Gaussian" means nothing else than applying a Gauss curve as FIR ...
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 "Gaussian-like".
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Old 28th July 2012, 02:24 AM   #122
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Originally Posted by CopperTop View Post
I have to admit that I have yet to come to terms with the motivation behind all these filter types! For example, the Bessel which you mention appears to be notable because of its constant group delay across the pass band (thanks Wikipedia!). Presumably, though, this is not a factor in a linear phase implementation..?
The Bessel is an IIR approximation of the Gaussian. You have the luxury of huge amounts of processing power at your disposal in your FFT-based implementation. That is not true in all cases, and an 8th-order digital IIR filter requires far less processing than, for example, a 256th-order (or even higher, at low cutoff frequencies) FIR.

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Yes, it's the motivation behind the filter 'brands' that I'm not sure of. As far as I can tell, it is possible to 'dial in' a desired frequency response, and suck-it-and-see what you get in terms of the impulse response ringing. No real maths involved. If that doesn't meet your various criteria, you can then massage the frequency response iteratively until it does. And give up if it doesn't.
That is certaily a viable design procedure, but it is akin to shooting in the dark. The various filter "flavors" are each optimal in some sense, so if what they optimize happens to be important to you, then you can go right to the answer without guessing.

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But in short, if we're designing a digital crossover filter, why do we need the maths and the filter 'brands'?
Because it's nice to actually understand the fundamentals behind what we are doing, rather than simply iterating until we (hopefully) find something that works.
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Old 28th July 2012, 01:29 PM   #123
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Originally Posted by gberchin View Post
Steph, you keep making that statement without justification. The adequacy of a 2nd order highpass is entirely dependent upon the circumstances in which it is used. There are certainly situations in which a 2nd order highpass is insufficient, but to say that this is true in most cases is simply an overstatement. There are many, many cases in which a 2nd order highpass works very well indeed. And, as I mentioned in an earlier post, if it does not then perhaps one is either crossing over at too low a frequency, or using a tweeter inappropriate for the job.
Consider a crossover demanding the tweeter to exhibit an acoustic 2nd-order highpass with F-3db = 3400 Hz. Consider the unfiltered tweeter Bode plot, showing a natural highpass with F-3dB = 800 Hz. The highpass crossover will thus only provide some attenuation between 800 Hz and 3400 Hz. All frequencies below 800 Hz will reach the tweeter, mostly unattenuated. This is what your so-called 2nd-order highpass crossover delivers.
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?
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Old 28th July 2012, 03:17 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 Nth-order 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
Attached Images
File Type: jpg symmetric complementary crossover.jpg (45.1 KB, 78 views)
File Type: jpg iDFT Lab mini.jpg (180.4 KB, 76 views)
Attached Files
File Type: zip iDFT_Lab.zip (218.9 KB, 6 views)
File Type: zip iDFT_Lab.mini.zip (217.8 KB, 6 views)

Last edited by steph_tsf; 28th July 2012 at 03:22 PM.
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Old 28th July 2012, 06:52 PM   #125
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Originally Posted by steph_tsf View Post
Consider a crossover demanding the tweeter to exhibit an acoustic 2nd-order highpass with F-3db = 3400 Hz. Consider the unfiltered tweeter Bode plot, showing a natural highpass with F-3dB = 800 Hz. The highpass crossover will thus only provide some attenuation between 800 Hz and 3400 Hz.
A 2nd-order highpass will provide 25 dB attenuation between 3400 Hz and 800 Hz.

Quote:
All frequencies below 800 Hz will reach the tweeter, mostly unattenuated. This is what your so-called 2nd-order highpass crossover delivers.
At 800 Hz they will be attenuated by 25 dB.

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.

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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?
THIEL uses 1st order crossovers.

Furthermore, and I failed to make this point explicitly before, there are more crossovers than just tweeter crossovers. You said that 2nd-order highpass is inadequate in most situations. Well, when crossing over from a subwoofer to a full-range speaker, a common situation, 2nd-order highpass is perfectly adequate. When crossing over from a woofer to a midrange, a common situation, 2nd-order highpass is often adequate. When crossing over from a midrange to a robust tweeter, a common situation, 2nd-order highpass is sometimes adequate. I'm just saying that the design must be judged on a case-by-case basis, and summarily dismissing any configuration as "inadequate" is ill-advised.

Last edited by gberchin; 28th July 2012 at 07:08 PM. Reason: Acknowledge "acoustic" 2nd-order.
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Old 28th July 2012, 09:15 PM   #126
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Quote:
Originally Posted by gberchin View Post
EDIT: I missed the "acoustic" in your statement, first time through, so you are correct about the level staying constant below 800 Hz.
Yes indeed, it is the acoustic transfer function that we are interested in.

Quote:
Originally Posted by gberchin View Post
But by that time the signal has already been attenuated by 25 dB. That's a factor of 1/316 in power.
Agree !

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 Lipshitz-Vanderkooy 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 2nd-order 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.
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Old 28th July 2012, 09:40 PM   #127
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Originally Posted by steph_tsf View Post
Yes indeed, such tweeter can cope with a 2nd order acoustic highpass.
It will thus survive a Lipshitz-Vanderkooy delay compensated crossover featuring a high order Bessel (pseudo Gaussian) lowpass as kernel.
Once again, a Bessel-derived matched-delay crossover is either an Ng/Rothenberg crossover or it is a Berchin crossover, but in no way is it a Lipshitz-Vanderkooy crossover. Either credit the originators of the matched-delay subtractive configuration, or credit the originator of the optimal-transient Gaussian/Bessel configuration, but do not credit people who originated neither the structure nor the optimal configuration.

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However, nowadays there are interesting miniature tweeters, inexpensive, exhibiting quite high natural highpass cutoff frequencies:
Granted. But not everybody is using complementary crossovers for only those drivers. There is a whole world of other drivers and configurations that you simply ignore in your analysis. A 2nd-order highpass crossover may not be suitable for the drivers that you want to use, but there are other drivers that other people want to use, for which 2nd-order crossovers are quite suitable.

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With digital, and FIRs in particular, it costs the same price, targeting a 2nd order acoustic highpass, or a 6th order acoustic highpass.
Not strictly true. The longer the impulse response, the larger the FIR filter that implements it, and the higher the computational burden. Furthermore, as I mentioned earlier, when the cutoff frequency is very low, the cost of a FIR implementation may be completely prohibitive. And with an IIR implementation, a complementary highpass characteristic beyond approximately 3rd-order may be impossible.
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Old 29th July 2012, 03:51 AM   #128
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Originally Posted by gberchin View Post
Once again, a Bessel-derived matched-delay crossover is either an Ng/Rothenberg crossover or it is a Berchin crossover, but in no way is it a Lipshitz-Vanderkooy crossover. Either credit the originators of the matched-delay subtractive configuration, or credit the originator of the optimal-transient Gaussian/Bessel configuration, but do not credit people who originated neither the structure nor the optimal configuration.
Between the early eighties and the late nineties, there must have been dozens of people having tweaked the Lipshitz-Vanderkooy, putting a Bessel as lowpass kernel, concluding that the poor highpass slope that's resulting transforms it into something barely usable. You are talking about "optimizing". Where is the optimization ? Once you use a Bessel as lowpass kernel in a LV-delay compensated crossover, the only choices you have are the Bessel order, and the Bessel -3dB cutoff frequency. The only "optimization" is to adjust the delay line for matching the Bessel group delay. Right? Sorry if I missed something important. I'm only asking a question. I need some more input. Any reply much appreciated.
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Old 29th July 2012, 10:34 AM   #129
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Originally Posted by gberchin View Post
...Because it's nice to actually understand the fundamentals behind what we are doing, rather than simply iterating until we (hopefully) find something that works.
Hard to disagree, of course. But I think "understanding" means different things to different people. Presumably the purist approach is to create a filter merely by manipulating equations, without looking at anything so grubby as a frequency response or impulse response. Sadly, I realise that having tried to picture how an impulse response acts a filter by plotting points and displaying a values in a computer program marks me out as a mathematical loser straight away!

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 catch-up, 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 'state-of-the-art' - 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).
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Old 29th July 2012, 01:00 PM   #130
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Originally Posted by steph_tsf View Post
Between the early eighties and the late nineties, there must have been dozens of people having tweaked the Lipshitz-Vanderkooy, putting a Bessel as lowpass kernel ...
That is an empty argument. Archimedes had knowledge of The Calculus, but credit is given to Newton and Leibniz for formalizing it.

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... concluding that the poor highpass slope that's resulting transforms it into something barely usable.
Barely usable? In the past few messages I have given numerous examples of how useful it can be. Yet your narrow view of a world consisting of nothing but delicate tweeters has not changed.

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You are talking about "optimizing". Where is the optimization ?
It provides optimal transient response for a given bandwidth. Please read the paper.

I am weary of this discussion.
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