Regarding a Linkwitz Transform... From a practical standpoint, a 2nd order shelf filter can almost always achieve the goal of an LT, although sometimes it may need a little help from a parametric EQ.
The starting point for an LT will usually be a sealed box with a Qtc between 0.6 and 1.2 and an F3 (Fb) which is higher than desired. This system can be “transformed” into having a lower F3, a flat 2-pi response, and a Qtc between 0.6 and 0.8 using just a 2nd order shelf filter, and as I said, a small PEQ. This is not as mathematically elegant as Dr. S. Linkwitz's original method, but it is easy to understand and easy to implement with analog or the commonly available DSP IIR filters.
j.
The starting point for an LT will usually be a sealed box with a Qtc between 0.6 and 1.2 and an F3 (Fb) which is higher than desired. This system can be “transformed” into having a lower F3, a flat 2-pi response, and a Qtc between 0.6 and 0.8 using just a 2nd order shelf filter, and as I said, a small PEQ. This is not as mathematically elegant as Dr. S. Linkwitz's original method, but it is easy to understand and easy to implement with analog or the commonly available DSP IIR filters.
j.
I think we have to put this a bit better into context.
First of all, yes you can combine basically all kinds of filters to get the same result.
The difference these days, that with a DSP you have almost an infinite amount of filter blocks from a practical point of view.
Back in the day, with active analog circuits, this was a VERY different story.
So doing things on a smart way also had a very practical purpose.
That being said, I think there is still a purpose of optimizing DSP filters.
In general it just makes the entire workflow easier and more clear, instead of having a whole bunch of (random) EQ's and shelving filters doing the same thing.
You very quickly lose oversight what the hell is going on.
At higher sample rate smaller DSP IC's like the ADAU1701/1401 have only a very limited amount of filter blocks available.
So with smarter design, and more compact bi-quads that can do specific things, you free up space.
Which can makes the difference in having to use a bigger and more expensive DPS IC.
This is maybe not very interesting for most hobbyist, but for developers and system integrators this could change the play field of options.
Especially when there is a budget constraint (which btw is 90% of most cases)
All that being said, I think there is also still a very good use case for active analog filters as well.
Especially from a price point and performance ratio.
But most people seem to be a bit spoiled and lazy unfortunately, lol
In the end I agree with you that practically speaking it might not be so amazing to have.
But my practical counter argument would be, it's nice to have the option in case you ever need it!
I find it interesting, because my first thought was the same as yours; that if it can be predicted in CAD it must be minimum phase.
But then I thought of the example of comb filtering where the amplitude at nulls could mathematically be zero and thus phase would be undefined. I suppose LEM can also be used to model non-linear aspects of a system.
Is it fair to say the frequency region below a ports operational range is likely to be significantly non minimum phase since it will basically just make noise?
But then I thought of the example of comb filtering where the amplitude at nulls could mathematically be zero and thus phase would be undefined. I suppose LEM can also be used to model non-linear aspects of a system.
Is it fair to say the frequency region below a ports operational range is likely to be significantly non minimum phase since it will basically just make noise?
Coincidently I counted how many bi-quad filter I could run on an ADAU1701 at 44.1khz a few days ago! It was 100 bi-quads, limited by memory, while CPU was about 80% utilised. At 96khz I guess it would be 50 biquads. The ADAU1452 handled over 3000 biquads at 44.1khz.
Yes, because a ADAU1701 has 1024 instructions @ 48kHz
A little less when we include some overhead.
A second order double precision filter block, costs 10 memory blocks.
So yeah that leaves us around roughly 100 bi-quads
For 96kHz that's half.
A 100 taps FIR blocks will take about 103 memory instruction with one additional instruction for each additional tap.
Many years ago I found a great excel sheet to calculate the MIPS usage.
See if I can find it again.
The ADAU145x and newer ADAU146x series are a LOT more powerful.
They can do 20k FIR taps no problem.
Not that will be practically very usable, since it will give you a whopping 40ms delay!
Anyway, one might think that 50 bi-quads is plenty for just a speaker crossover.
But this goes down very fast in a multi-way system, even more so with additional dynamic filters (compressors etc) as well as some additional preset choices etc.
It makes zero sense, since they actually use a ADAU145x (I forgot which type exactly)
This is even very clearly stated in the datasheet of the DSP IC.
So I have no idea why they build in an artificial limit.
Btw, I found the excel sheet again;
https://ez.analog.com/cfs-file/__ke...5F00_Sigma_2D00_100_5F00_MIPS_2D00_Usage.xlsx
source:
https://ez.analog.com/dsp/sigmadsp/f/discussions/65644/adau1701-instruction-usage-table
https://ez.analog.com/dsp/sigmadsp/...ctions-can-be-executed-per-sample-in-sigmadsp
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no, a ported speaker response is a stable 4th order high-pass filter. it is minimum phase because it’s inverse is causal and stable too. This is the same as saying that both its poles and zeros are in the left half plane. You are right that transmission zeros (eg from comb filters) is a borderline since the zeros are on the imaginary axis and a small perturbation can move them to the left plane
Quoting from my own document (somewhat out of context; the quoted section is about adding a highpass filter to a vented box loudspeaker to limit excursion, but it applies to filter design in general):
"This approach is an improvement over the traditional method of dealing with excessive ventedbox cone excursion, which is to add an 'ad hoc' highpass filter in the woofer signal path. Design of this extra highpass filter is often by trial-and-error, so the frequency and phase responses – and thus the group delay and transient responses – of the resulting combination of filter and woofer are not likely to be well-behaved. The formal Linkwitz Transform design technique allows the guesswork and compromises of empirical filter design to be replaced by analytical methods, yielding well-understood and predictable frequency and phase responses."
Here is a comparison of the frequency and time domain responses of various vented-box alignments, and a similar closed box.
https://community.klipsch.com/index...ubs-vs-1-15-sub/&tab=comments#comment-2555558
In addition to the oscillation after the end of the input signal, also note that all of the alignments, including the closed box, respond very similarly to the first two cycles of the input signal. So the "transient attack" is not lost in a vented system.
Full disclosure: I am "Edgar" on that forum.
https://community.klipsch.com/index...ubs-vs-1-15-sub/&tab=comments#comment-2555558
In addition to the oscillation after the end of the input signal, also note that all of the alignments, including the closed box, respond very similarly to the first two cycles of the input signal. So the "transient attack" is not lost in a vented system.
Full disclosure: I am "Edgar" on that forum.
I was gonna talk about this part via a DM, but I guess it's interesting to have a open discussion about this.
Because this statement isn't actually fully true.
Combining and summing of different filters and systems works on exactly the same way.
So in other words, we can perfectly mathematically calculate what HP filter we need to get a certain target curve, since we know the Q-factor of each system as well as the slope.
Practically speaking, one just has to just follow up some of those tables and fill in the parameters to get the exact same overal system response.
Or just use a solving script to get to a certain target curve.
VituixCAD can do this for example.
In fact, it has a function where it will just resolve all parameters of a bi-quad as function of a certain target curve.
Absolutely NOT to sound harsh or bad, but basically doing all the math solving for you.
(within certain limits obviously)
But I 100% agree that it's a lot less elegant.
Yes! We cannot lose sight of the fact that Thiele/Small parameters and transfer functions are just models. And that drivers are never, never perfect. So formal design techniques applied to loudspeaker and crossover design are just the first step in a long process.
@gberchin
The most important thing I forgot to add in my previous post, is that those "solving scripts" can be practically handy sometimes.
What they don't do, is providing mathematical and scientific proof that certain concepts work.
Worst of all, you very quickly loose any grasp of what is going on.
So it becomes incredibly hard to see if things still make sense, because; garbage in = garbage out.
The reason why your paper is so very incredible, because it closes off a (little) chapter!
Personally I see these things and concepts in conjunction with each other.
We have the physics and math to know concepts work, we have certain solving scripts to fix it quickly for us
My only feedback on your paper would be to briefly talk about practical problems in reality.
Like you just said, how speakers actually behave in the real world can be quite different.
Which will mess up some of the variables an parameters, especially for vented systems (since it's an higher order system)
The most important thing I forgot to add in my previous post, is that those "solving scripts" can be practically handy sometimes.
What they don't do, is providing mathematical and scientific proof that certain concepts work.
Worst of all, you very quickly loose any grasp of what is going on.
So it becomes incredibly hard to see if things still make sense, because; garbage in = garbage out.
The reason why your paper is so very incredible, because it closes off a (little) chapter!
Personally I see these things and concepts in conjunction with each other.
We have the physics and math to know concepts work, we have certain solving scripts to fix it quickly for us
My only feedback on your paper would be to briefly talk about practical problems in reality.
Like you just said, how speakers actually behave in the real world can be quite different.
Which will mess up some of the variables an parameters, especially for vented systems (since it's an higher order system)
I guess I'd comment that the practical problems are universal; they have always existed and they always will. In spite of that, the analytical models (T/S parameters, transfer functions, etc.), while not perfect, are a really good first step. They are far better than the old-world method of building a box, measuring and listening to it, adjusting it as best you can, deciding that you don't like it, building another box, etc., and trying to formulate "rules of thumb" from the results. (I have some books and magazines from my father's era that describe exactly that.)
I know, you know and some other people know that as well.
Fact is that some people take things very literally and don't (want to) understand how that sits within practical context.
Even people with backgrounds who should know better.
Is this entirely true in practice? My thinking is that an octave (just a figure picked from the air) below a ports tuning frequency, air displacement inside the port will grow very large with minimal desirable sound output. Essentially, the signal to noise ratio will become unfavourable between desired output and unwanted noise caused by turbulence, vortex shedding and other mechanical vibrations. Those aspects, I believe, would introduce a proportion of non minimum-phase behaviour, correct?
Then again, turbulence is the only noise source among those I mentioned that reliably introduces low frequency noise.
These cases you're describing are all a kind of form of compression.
Which is completely non-linear to begin with.
So God knows what happens to the phase response.
Welcome to the deep rabbit hole of non-linear thermodynamics.
It is fair to say a port will just make noise at any frequency.
But obviously it is a good platform to learn control theory.
!Sheesh, you think wasps are bad, This little monster loved to make port entry at warp speed when she still fit.
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Exactly - I feel that transient decay does not get the attention deserved when considering LF fidelity. I believe that the rise time is less relevant than several cycles of ringing after a transient signal ceases, when the resonant enclosure unloads the energy its been input.