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Old 8th July 2012, 11:56 PM   #41
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Quote:
Originally Posted by gberchin View Post
It's just another design constraint.
Easier said than done. Currently there is no direct, deterministic way to inject such design constraint in the conventional windowed design protocol you are referring above. You will rely on test and trial. You will end up reinventing the wheel, just like you said.
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Old 9th July 2012, 01:18 AM   #42
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Originally Posted by CopperTop View Post
Something I missed totally earlier was the possibility of the high pass being an exact complement of the low pass in the crossover region, so that the ringing of the woofer is 'neutralised' by the opposite ringing of the tweeter (give or take the discrepancies due to off-axis listening). If this is the case (or can be made the case), is the issue of overshoot one of a waste of power and/or needless displacement of the driver?
YES ! You'll be shaking the two membranes for nothing. Even if the two membranes tend to acoustically neutralize, from a mechanical point of view they will generate different inertia forces (different membrane mass), and those forces will be applied on non-coincident baffle points (not a coax speaker). For sure there will be a parasitic vibration conveyed by the speaker cabinet.

Your crossover won't be transparent. Your baffle enclosure will continuously play an extra-sound, correlated to your crossover design. Not what we want in high quality HiFi.

Above, I wrote "tend to acoustically neutralize" because even at the acoustic level, the neutralization process gets somewhat random, like being influenced by the speaker position in the room and other factors that may remain hidden. Try visualizing the fundamental construction difference between a 5 inch woofer membrane, and a dome or ribbon tweeter. When you have sound bouncing back to them, and reflected from them again, kind of standing waves (and this includes the reflected sound from inside the cabinet), it happens in a completely different, potentially random way, depending on your body position and your couch, facing the baffle cabinet.

Now you understand that if you are coming with a FIR exhibiting a long significant preshoot, most specialists will say "no, thank you" even if the woofer preshoot cancels the tweeter preshoot. Lots of specialists consider that any preshoot level above -20dB will destroy the listening pleasure.

Unfortunately, by experience (or By Parceval's theorem), we observe that in the real world, the sharpest the transition band and the slope (of your lowpass filter), the longer the impulse response.

Unfortunately, by experience we know that a linear-phase system has the impulse response extending before the main peak. This means preshoot.

Unfortunately, we know that both time-domain complementary (sum = unity) and frequency-domain complementary (lowpass slope same as highpass slope), only can happen with linear-phase filters.

Consider the following situation. You want to implement a steep lowpass acting like a Butterworth 48dB/octave (this is a 8th-order). You know by advance (Perceval's theorem) that you are going to have a quite long impulse response, with considerable preshoot and ringing. If you design the lowpass as linear-phase, you are obliged to have half the impulse response translating in preshoot, and the other half translating in ringing.
You may then decide to depart from a phase-linear lowpass, and tailor the phase response in such a way that only 10% of the impulse response is preshoot, and 90% ringing. Knowing that using FIRs, you have full independent control over the phase, you can get such particular lowpass exhibiting the frequency behaviour of a Butterworth 48dB/octave, however different through the phase behaviour.
Job done would you say?
Not at all because you still need to deal with the complementary highpass.
You have no degree of freedom, as for being complementary in time-domain, you need to generate the highpass by Dirac less lowpass.
What do you observe?
You observe that the resulting highpass has not the same slope as the lowpass. It has an inferior slope. Your highpass is complementary in time-domain, but not complementary in frequency-domain.
The tweeter doesn't get properly isolated from the bass frequencies.
That's the usual problem when dealing with analog complementary filters. You can view them as infinite lenght FIRs having no preshoot, only ringing. Using LTspice, design a 48db/octave Butterworth. Generate the complementary highpass by subtracting the lowpass signal from the input signal.
Guess the highpass slope.
It is only 6dB/octave!

No need to say, I'm positively impressed by the Philips DSS-930 crossover design, flirting with the preshoot audibility limit (look the woofer target time domain response), providing symmetric (lowpass and highpass) 36dB/octave slopes, and monotonic amplitude curves. All this with relatively short FIRs.

They decided for 36dB/octave lowpass and highpass slopes.
A nice generalization would be to make a redesign featuring 24dB/octave lowpass and highpass slopes. There will be less preshoot.
Another nice generalization would be to make a redesign featuring a 48db/octave lowpass slope, and a 24db/octave highpass slope.
Using such design protocol, is it feasible to end up with a 24db/octave lowpass slope, and a 48db/octave highpass slope?

Last edited by steph_tsf; 9th July 2012 at 01:31 AM.
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Old 9th July 2012, 02:00 AM   #43
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Your baffle enclosure will continuously play an extra-sound, correlated to your crossover design.
Then you really need a better enclosure.

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Unfortunately, by experience (or By Parceval's theorem), we observe that in the real world, the sharpest the transition band and the slope (of your lowpass filter), the longer the impulse response.
Parceval's Relation only says that the energy under the frequency response curve equals the energy under the impulse respose curve. It states nothing about steepness of the transition band or duration of the impulse response.

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Unfortunately, we know that both time-domain complementary (sum = unity) and frequency-domain complementary (lowpass slope same as highpass slope), only can happen with linear-phase filters.
Not true. Consider 1st-order complementary pairs; LPF = w0/(s+w0) and HPF = s/(s+w0). Complementary, same slope, definitely not linear phase.

Quote:
You have no degree of freedom, as for being complementary in time-domain, you need to generate the highpass by Dirac less lowpass.
What do you observe?
You observe that the resulting highpass has not the same slope as the lowpass. It has an inferior slope. Your highpass is complementary in time-domain, but not complementary in frequency-domain.
Not necessarily true. Consider creating a LPF by drawing a straight line between a value of 1.0 (0 dB) at 0 Hz and 0.5 (-6 dB) at your chosen cutoff frequency. Extend the line such that it crosses 0.0 (-infinity dB) at twice the cutoff frequency. Now construct its complement. That will be a line extending from 0.0 (-infinity) at 0 Hz through 0.5 (-6 dB) at the cutoff frequency, to 1.0 (0 dB) at twice the cutoff frequency. Above that, it will be a flat line at 1.0 (0 dB) up to infinite frequency.

This is a non-realizable filter (its impulse response will be infinite in duration), but it shows that arbitrary complementary slopes are theoretically possible. And so we can approximate them to any desired precision.
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Old 9th July 2012, 08:57 AM   #44
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Originally Posted by gberchin View Post
Consider 1st-order complementary pairs; LPF = w0/(s+w0) and HPF = s/(s+w0). Complementary, same slope, definitely not linear phase. Consider creating a LPF by drawing a straight line between a value of 1.0 (0 dB) at 0 Hz and 0.5 (-6 dB) at your chosen cutoff frequency. Extend the line such that it crosses 0.0 (-infinity dB) at twice the cutoff frequency. Now construct its complement. That will be a line extending from 0.0 (-infinity) at 0 Hz through 0.5 (-6 dB) at the cutoff frequency, to 1.0 (0 dB) at twice the cutoff frequency. Above that, it will be a flat line at 1.0 (0 dB) up to infinite frequency. This is a non-realizable filter (its impulse response will be infinite in duration), but it shows that arbitrary complementary slopes are theoretically possible. And so we can approximate them to any desired precision.
It gets very interesting, thanks. Say we define such large, symmetrical transition band having a width equal to Fc. What matters, is the woofer to exhibit a steep slope above F (with extinction at 2F), and the tweeter to exhibit the exact same steep slope below F (with extinction at DC). What computational power is needed for such particular filter featuring Fs=48kHz and Fc=3 KHz?
The woofer -6 dB amplitude corridor will be DC to 3 kHz. The tweeter -6 dB amplitude corridor will be 3 kHz to infinite, reaching a close to 0dB response as soon as 6 kHz.
You said the impulse response will exhibit an infinite duration, in theory. Applying windowing, you may come with a practically realizable FIR. Hopefully not a 32768 tap FIR?
But wait a minute, starting from an infinite duration impulse response, eventually shortened using windowing, don't you have the impression that there will be a long, hence fatal preshoot?
Can you significantly reduce the preshoot duration, by shaping the woofer phase response, without ruining the transition band symmetry and the slope symmetry ?
Instead or relying on a FIR, can yo rely on a digital IIR?
Can it be realized in analog?
I'm not critical about such design. I'm only asking questions.

Last edited by steph_tsf; 9th July 2012 at 08:59 AM.
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Old 9th July 2012, 09:07 AM   #45
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Originally Posted by gberchin View Post
Consider 1st-order complementary pairs; LPF = w0/(s+w0) and HPF = s/(s+w0). Complementary, same slope, definitely not linear phase.
You dropped the context. I wrote this in the context of filters exhibiting no relative phase shifts and a perfect reconstruction. I thus never considered the 1st-order complementary pair to be a candidate.
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Old 9th July 2012, 09:24 AM   #46
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Originally Posted by gberchin View Post
Then you really need a better enclosure.
Efficient engineering is about preventing, not about curing. All things remaining equal, you'll get a better, less costly product if you don't allow the transducers to move "for nothing". In first instance, you want to avoid preshoot, as it degrades the listening pleasure. After this, you shall remain very cautious about ringing because even if it tends to be masked by the main response, it adds wasted energy transmitted to the speaker cabinet. After programming long FIRs, after executing them at great effort using powerful DSPs or x86 computers, do you really want to end up with your speaker cabinet continuously playing the crossover smears? Trying to cure the defects, you'll end up with a speaker cabinet made in unobtainium, or weighting hundreds of kilos. By the way, as explained above, your speaker will remain quite random, when exposed to standing waves (room coupling) and reflexions (your body and your couch, just in front). If planes were designed this way, none would flight.
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Old 9th July 2012, 12:13 PM   #47
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What would your advice be for me and my setup? Basically I can run any filters I like, so where should I start?

I'm thinking along the lines of simply calculating the desired frequency amplitude response of the high pass and low pass using 'a formula', and then windowing the resulting impulse response. Presumably real only, as phase will be zero. Steph, should I make the crossover frequency a simple factor of the sample rate, as I think you suggested earlier?

Can you suggest some good formulae to start with? Something along these lines I have lifted from a paper somewhere?:
Quote:

|LP| = 1/√((1-fn)2 + fn/Q2) (1)

where f is frequency normalized by the crossover frequency, f = ω/ωc. This expression yields an amplitude response with an nth order roll off and a response with magnitude
20 Log(Q) at f = 1.0. For n equal to 2 the amplitude given be Eq(1) is consistent with that of any standard 2nd order filter, Butterworth, Bessel or Linkwitz/Riley, for example. For n greater than 2 the response is only consistent with symmetric higher order filters. These include Butterworth filters (n even or odd, Q = 0.707) and Linkwitz/Riley filter (n even, Q = 0.5), but not higher order Bessel filters. The corresponding HP filter amplitude is
____________
|HP| = fn /√((1-fn)2 + fn/Q2)
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Old 9th July 2012, 02:17 PM   #48
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Originally Posted by steph_tsf View Post
Efficient engineering is about preventing, not about curing.
I would argue that efficient engineering is about effective compromise, but that is really not the matter at hand.

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After this, you shall remain very cautious about ringing because even if it tends to be masked by the main response, it adds wasted energy transmitted to the speaker cabinet.
No; according to Parseval's Relation, for a given frequency response magnitude, the corresponding energy in the time domain is exactly the same, no matter how it's "moved around" by the phase response. This means that the energy in the impulse response of the linear-phase version of a filter is exactly the same as the energy in the impulse response of the minimum-phase version of the same filter.
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Old 9th July 2012, 02:29 PM   #49
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Originally Posted by steph_tsf View Post
But wait a minute, starting from an infinite duration impulse response, eventually shortened using windowing, don't you have the impression that there will be a long, hence fatal preshoot?
Yes. That's why I'm an advocate of using Gaussian kernels (or higher-order, "Gaussian-like" kernels). I'm just pointing out that there are other options, if you are willing to tolerate overshoot and/or ringing.

Quote:
Can you significantly reduce the preshoot duration, by shaping the woofer phase response, without ruining the transition band symmetry and the slope symmetry ?
I'm not sure. You're straying from theoretical concepts to issues of design. As you are the designer, it is up to you to make the necessary compromises.

Quote:
Instead or relying on a FIR, can yo rely on a digital IIR?
Can it be realized in analog?
I'm not critical about such design. I'm only asking questions.
Again, though there is a theoretical aspect to your questions, they are more in the realm of design. As the designer, it is up to you to determine the optimum implementation.
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Old 9th July 2012, 02:42 PM   #50
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Quote:
Originally Posted by gberchin View Post
I would argue that efficient engineering is about effective compromise, but that is really not the matter at hand.
Its an interesting aside - I'd argue that compromise is inimicable to engineering design. Real engineering is about optimisation, not compromise. The difference being that optimisation in the design space occurs in the quantum realm where synergies occur; compromise is in the classical (zero-sum) world.
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