Hi Bill
I want to make sure that you understand my suggestion because what you keep pointing to as problems, aren't in my way of thinking.
Take for example you have arrays of data at many angles, but for now, they were taken in a square pattern, not the usual circular one. Since in 2D rectangular the radiation from a rectangle is a Fourier series of aperture velocity distribution V(x,y), you can just reverse this and take the radiation pattern and predict the velocity distribution V(x,y) with the inverse transform. The fact is that V(x,y) need only be a fraction of the data in the full far field data set, because so much of that data is the far field is correlated to itself, i.e. way more data than needed. Its the radiation equivalent of MP3. Typically the data set for V(x,y) would be several hundred times (could even be 1000's in some cases) smaller than the huge array of far field data. And many other features fall out. Like you have an infinite spatial resolution and the same frequency resolution as the far field data. I reduce the frequency data to log spacing and reduce 2048 points down to 200 points in a Zwicker ERB scheme. and there is no sorting trough data set, all of the knowable information is in V(x,y) and it will predict any field point. Limitation here is that you will have a rectangular construction which is not quite as accurate as a circular one, or better yet and elliptical one. But it may also be just fine because in the far field it is hard to tell a circular source from a rectangular one.
And modern processors are highly efficient at Cosine and Sine transforms since in video thousands of these 2D transforms are done per second in an MPEG video.
PS. Symmetry or not falls out directly from the transform or actually isn't even an issue.
PSS in spherical coordinates, the sum of the squares of the modal coefficients is the sound power, no need to integration or storage of the actual sound power, you always have it. I am sure that something like this has to be true in rectangular as well, I just don;t know exactly what it is.
I want to make sure that you understand my suggestion because what you keep pointing to as problems, aren't in my way of thinking.
Take for example you have arrays of data at many angles, but for now, they were taken in a square pattern, not the usual circular one. Since in 2D rectangular the radiation from a rectangle is a Fourier series of aperture velocity distribution V(x,y), you can just reverse this and take the radiation pattern and predict the velocity distribution V(x,y) with the inverse transform. The fact is that V(x,y) need only be a fraction of the data in the full far field data set, because so much of that data is the far field is correlated to itself, i.e. way more data than needed. Its the radiation equivalent of MP3. Typically the data set for V(x,y) would be several hundred times (could even be 1000's in some cases) smaller than the huge array of far field data. And many other features fall out. Like you have an infinite spatial resolution and the same frequency resolution as the far field data. I reduce the frequency data to log spacing and reduce 2048 points down to 200 points in a Zwicker ERB scheme. and there is no sorting trough data set, all of the knowable information is in V(x,y) and it will predict any field point. Limitation here is that you will have a rectangular construction which is not quite as accurate as a circular one, or better yet and elliptical one. But it may also be just fine because in the far field it is hard to tell a circular source from a rectangular one.
And modern processors are highly efficient at Cosine and Sine transforms since in video thousands of these 2D transforms are done per second in an MPEG video.
PS. Symmetry or not falls out directly from the transform or actually isn't even an issue.
PSS in spherical coordinates, the sum of the squares of the modal coefficients is the sound power, no need to integration or storage of the actual sound power, you always have it. I am sure that something like this has to be true in rectangular as well, I just don;t know exactly what it is.
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Hi Earl,
How many points would have to be taken at distance to get a reasonable estimate of the velocity distribution? Most people have been taking data at same distance (1m ideally) and in equal angle steps. In one arbitrary axis for a round driver, or in the horizontal and vertical for others. Would that work for the 2D transform scheme?
Unfortunately it's rare that a diyer goes in less than maybe 5 degree steps (15 degree is probably more usual) because of the difficulty of determining where in 3D space to put the microphone (or rotating the speaker) when taking measurements. Automated turntables aren't usually in the toolset.
What do you mean by 'modal coefficients'? Are these of the velocity distribution?
How many points would have to be taken at distance to get a reasonable estimate of the velocity distribution? Most people have been taking data at same distance (1m ideally) and in equal angle steps. In one arbitrary axis for a round driver, or in the horizontal and vertical for others. Would that work for the 2D transform scheme?
Unfortunately it's rare that a diyer goes in less than maybe 5 degree steps (15 degree is probably more usual) because of the difficulty of determining where in 3D space to put the microphone (or rotating the speaker) when taking measurements. Automated turntables aren't usually in the toolset.
What do you mean by 'modal coefficients'? Are these of the velocity distribution?
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Hi Bill
I guess this stuff is harder to grasp than I had thought.
I have worked out all your questions for the spherical case, but not the rectangular planar case. They would have to be worked out, but the principles would be exactly the same.
Think of your old physics courses and a membrane stretched across a frame. When it vibrates it vibrates in modes and any motion of this diaphragm can be analyzed as a sum of its modes. Now think of your radiating disk as the same thing, except that the boundary conditions are different - the slope of modes must be zero at the edges not the displacement, so this gives a different set of modes (cosines in the case of a rectangular source, Bessel functions for a circular one and Nueman functions for an ellipse.) For each mode there is a unique radiation pattern. Since any motion of the disk can be analyzed as a sum of its modes then likewise any radiation can also be analyzed the same way. If the disk is perfectly rigid then there is only one mode and only one radiation pattern. In that case a single measurement point is all you need to completely define the sound radiation.
But in reality it's going to take several modes for an accurate summation to the measured field at higher frequencies (at low frequencies only one point is all that is ever required for a monopole). Each mode will have a "cut-in" phenomena such that it has no effect below a certain ka value. Knowing the modal cut-in values, one can define how many modes are required to get to 20 kHz (or whatever.) In the spherical case it is somewhere in the 12 - 15 modes. Since this is a matrix problem it takes exactly the same number of measurement points as the number of modes desired. Again, in the spherical case 13 points completely defines the radiation of any axi-symmetric device. But, in the spherical case they do not want to be equally distributed, but want a finer resolution near the axis then away from it so I use 0, 5, 10, 15, 20, 30, 40, 50, 60, 80, 100, 120, 150 and 180. This yields, and this has been proven, the exact same results as one gets taking data every 2 degrees. In the rectangular domain I am not sure what the optimal data locations would be.
Now each mode has a value at each frequency and so to get the power response (again in spherical) one simply sums the squares of the value of each mode at each frequency. Calibrating this to an absolute power is difficult, but if one only want relative power and DI then its since the DI has to go to zero at LFs (assuming a closed box, its a little more complicated for a dipole.)
So 'yes' the "modal coefficients" are both the velocity distribution and the polar response.
A "cool" thing that one can do with this data is to reconstruct the source velocity at any frequency. SO lets say you have a dip in the response that you don't understand, if it has an acoustic source then you will see this in the velocity reconstruction. This is how I discovered the cause of the axial hole in the Abbey - I reconstructed the source velocity at that frequency and found that there was a wave moving across the mouth, i.e. a standing wave across the mouth caused by reflections at the edge. I could actually see it.
In principle, and this field is called Acoustic Holography, one could map out the cone motion from the radiation pattern, but this would require a high precision in the data taking, usually done with a fixed array of microphones.
As to the rotation problem, I just made a wooden stand with two plates on top where one rotates. 5 degree increments on this surface is easy to do. My plates have melamine surfaces and I use grease between them so that rotating a 80 lb speaker on this stand is no problem at all. The mic stays fixed. The stand cost maybe $20. Do one angle then rotate the speaker 90 degrees and do it again and you have pretty much all your data. Of course, in the case of the disk in a plane, you should take the data with this plane in place and that makes for a complicated mess. Another reason why spherical is so useful - no need for an "infinite plane".
Finally, one does not even have to be in the far field to do this (in the spherical case) and Klippel does it very close to the source, but this requires very accurate placement of the mic, hence the $20,000 price tag. I back up to 1-1.5 meters and positioning is not so important. Once you have the source velocities you have all the field data, all angles and all radii, even near field.
I guess this stuff is harder to grasp than I had thought.
I have worked out all your questions for the spherical case, but not the rectangular planar case. They would have to be worked out, but the principles would be exactly the same.
Think of your old physics courses and a membrane stretched across a frame. When it vibrates it vibrates in modes and any motion of this diaphragm can be analyzed as a sum of its modes. Now think of your radiating disk as the same thing, except that the boundary conditions are different - the slope of modes must be zero at the edges not the displacement, so this gives a different set of modes (cosines in the case of a rectangular source, Bessel functions for a circular one and Nueman functions for an ellipse.) For each mode there is a unique radiation pattern. Since any motion of the disk can be analyzed as a sum of its modes then likewise any radiation can also be analyzed the same way. If the disk is perfectly rigid then there is only one mode and only one radiation pattern. In that case a single measurement point is all you need to completely define the sound radiation.
But in reality it's going to take several modes for an accurate summation to the measured field at higher frequencies (at low frequencies only one point is all that is ever required for a monopole). Each mode will have a "cut-in" phenomena such that it has no effect below a certain ka value. Knowing the modal cut-in values, one can define how many modes are required to get to 20 kHz (or whatever.) In the spherical case it is somewhere in the 12 - 15 modes. Since this is a matrix problem it takes exactly the same number of measurement points as the number of modes desired. Again, in the spherical case 13 points completely defines the radiation of any axi-symmetric device. But, in the spherical case they do not want to be equally distributed, but want a finer resolution near the axis then away from it so I use 0, 5, 10, 15, 20, 30, 40, 50, 60, 80, 100, 120, 150 and 180. This yields, and this has been proven, the exact same results as one gets taking data every 2 degrees. In the rectangular domain I am not sure what the optimal data locations would be.
Now each mode has a value at each frequency and so to get the power response (again in spherical) one simply sums the squares of the value of each mode at each frequency. Calibrating this to an absolute power is difficult, but if one only want relative power and DI then its since the DI has to go to zero at LFs (assuming a closed box, its a little more complicated for a dipole.)
So 'yes' the "modal coefficients" are both the velocity distribution and the polar response.
A "cool" thing that one can do with this data is to reconstruct the source velocity at any frequency. SO lets say you have a dip in the response that you don't understand, if it has an acoustic source then you will see this in the velocity reconstruction. This is how I discovered the cause of the axial hole in the Abbey - I reconstructed the source velocity at that frequency and found that there was a wave moving across the mouth, i.e. a standing wave across the mouth caused by reflections at the edge. I could actually see it.
In principle, and this field is called Acoustic Holography, one could map out the cone motion from the radiation pattern, but this would require a high precision in the data taking, usually done with a fixed array of microphones.
As to the rotation problem, I just made a wooden stand with two plates on top where one rotates. 5 degree increments on this surface is easy to do. My plates have melamine surfaces and I use grease between them so that rotating a 80 lb speaker on this stand is no problem at all. The mic stays fixed. The stand cost maybe $20. Do one angle then rotate the speaker 90 degrees and do it again and you have pretty much all your data. Of course, in the case of the disk in a plane, you should take the data with this plane in place and that makes for a complicated mess. Another reason why spherical is so useful - no need for an "infinite plane".
Finally, one does not even have to be in the far field to do this (in the spherical case) and Klippel does it very close to the source, but this requires very accurate placement of the mic, hence the $20,000 price tag. I back up to 1-1.5 meters and positioning is not so important. Once you have the source velocities you have all the field data, all angles and all radii, even near field.
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The following is not related to deep theories above. Just to speaker engineering in practice.
This is certainly okay for boxed speakers. Few ears ago I recommended this kind of custom recipe with Arta Recorder which supported variable angle step. But lately I've used and recommended constant step of 10 degrees due to practical reasons:
* Common measurement programs such as CLIO 10-11 and ARTA 1.9 do not support variable angle step in auto save / turning table mode.
* Rational speaker designer uses measurement software which supports auto save / turning table mode - especially for multi-way speakers. Easy and fast procedure is one of the best motivators to measure adequate data which is mandatory to get accurate result ~ good sound at once.
* Open baffle constructions with cone drivers are very unsymmetrical. Power response and DI approximation requires about the same amount of measurements from rear than front.
This is the reason why also VituixCAD Diffraction tool exports with constant steps of 10 degrees: 19 or 36 responses per plane depending on user selections (negative angles, vertical plane). "10 deg rule" makes easier to combine measurements and simulations in many ways in order to create also directivity information down to 20 Hz with quasi anechoic measurements at home.
In some cases 5 deg and 15 deg could be valuable. User can measure and simulate and process those directions manually.
This is certainly okay for boxed speakers. Few ears ago I recommended this kind of custom recipe with Arta Recorder which supported variable angle step. But lately I've used and recommended constant step of 10 degrees due to practical reasons:
* Common measurement programs such as CLIO 10-11 and ARTA 1.9 do not support variable angle step in auto save / turning table mode.
* Rational speaker designer uses measurement software which supports auto save / turning table mode - especially for multi-way speakers. Easy and fast procedure is one of the best motivators to measure adequate data which is mandatory to get accurate result ~ good sound at once.
* Open baffle constructions with cone drivers are very unsymmetrical. Power response and DI approximation requires about the same amount of measurements from rear than front.
This is the reason why also VituixCAD Diffraction tool exports with constant steps of 10 degrees: 19 or 36 responses per plane depending on user selections (negative angles, vertical plane). "10 deg rule" makes easier to combine measurements and simulations in many ways in order to create also directivity information down to 20 Hz with quasi anechoic measurements at home.
In some cases 5 deg and 15 deg could be valuable. User can measure and simulate and process those directions manually.
I am not sure that you understand that my choice of angles is not arbitrary. In spherical coordinates the angular functions are Legendre function of order n defined in x, but, and this is the main point, x=cos(theta). If I want to fit the angular functions to the minimum set of data points then they need to be equally spaced in x. But this then means that they will not be equally spaced in theta. What I chose was as close as I could get to the exact values of theta for equal spacing in x.
^I am able to understand that, but my point is that common speaker designer needs just easy, fast and reasonably accurate method, and measurement and simulation tools which support that design method for any construction/source type. All this is already available for everyone without special angle recipes, complex theories or proprietary tools limited for some special task. But I'm not trying to prevent this discussion because brain exercise is claimed to be healthy.
But you see that my method achieves this. With 13 points of my method I get the same accuracy as you will with 45 points of your method. If you want less accuracy then my method could be brought down to 6 points, yielding the same accuracy as yours at 5 degree increments, or 3 or 4 points for 10 degree accuracy. Its all about ease and accuracy, but to me, its accuracy that is king, because simple inadequate measurements mean inadequate acoustic performance. The flaws get covered up by poor accuracy of the data.
PS. Once the code is written, my technique takes no longer to calculate than any other, its just way more accurate and simpler, not more complicated. To the end user all the math is transparent - only the coder needs to understand it.
PSS. You also have to remember that I did all this some twenty years ago when the software packages available would do none of it. I want highly accurate polar maps with power response and DI. I needed it simple because I only had a living room to work in. Sure the math was tough, but hardly anything that I had not done before. The coding took some time, but that's been done for > ten years.
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^Then you just have to make your king method and tool available for everyone and make sure that measurement systems support method in auto save mode without manual or extra measurements and your sw imports data directly without manual/extra processing and give all simulation services and analyses what the best tools today. After that we can uninstalled all other tools as useless crap.
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^Unfortunate reality is that most of the diy speakers are designed by a single frequency response per driver. Method and tool and some measurement systems I was talking about support 1...74 directions per driver. At least possibilities are decades better than de facto standard in diy. It's also tested how many directions with constant angle step is needed to get very close to final power response and DI. It's much less than 45 with boxed speaker. In practice at home, accuracy is defined by amount of measured planes, accuracy of measurements and auxiliary simulations and combining/merging methods of data. Not how clever the (graphical) interpolation of polarmap is in a single plane. This may not apply to you, but it's fact for many or most of the others.
Let me repeat that I'm not against different and smarter methods. Programmers just need public and well documented description with code snippets or working implementation with some language, and designers need compatible measurement system. Simple praising of one theoretical/non-existing method over the existing does not help anybody.
Let me repeat that I'm not against different and smarter methods. Programmers just need public and well documented description with code snippets or working implementation with some language, and designers need compatible measurement system. Simple praising of one theoretical/non-existing method over the existing does not help anybody.
Earl and Bill,
Thank you for this thread!
This has been one of the most exciting reads in my years or frequenting this site because over the years, I've thought about how my own speaker measurement setup could be improved. But since Klippel released their Near Field Scanner, I've been working on a thought experiment on how to DIY something with similar capabilities; this has led me to investigating Earl's methods, as well as reading Weinreich's paper among other things.
I believe that, owing to the DIY CNC community, a piece of hardware that can accurately place a microphone can be built for a reasonable amount of money. The software is open source, the necessary motors, controllers, slides, bearings, and whatnot aren't terribly expensive.
What isn't readily available is a way to take the measured data and turn it into something useful.
I apologize as as this is an off topic thought, but if there's any interest in pursuing it (perhaps even as a software set and matching hardware designs for the DIY community), I would love to continue this in a dedicated thread, or private communication.
Thank you for this thread!
This has been one of the most exciting reads in my years or frequenting this site because over the years, I've thought about how my own speaker measurement setup could be improved. But since Klippel released their Near Field Scanner, I've been working on a thought experiment on how to DIY something with similar capabilities; this has led me to investigating Earl's methods, as well as reading Weinreich's paper among other things.
I believe that, owing to the DIY CNC community, a piece of hardware that can accurately place a microphone can be built for a reasonable amount of money. The software is open source, the necessary motors, controllers, slides, bearings, and whatnot aren't terribly expensive.
What isn't readily available is a way to take the measured data and turn it into something useful.
I apologize as as this is an off topic thought, but if there's any interest in pursuing it (perhaps even as a software set and matching hardware designs for the DIY community), I would love to continue this in a dedicated thread, or private communication.
Hi Bill,
Here are a few measurements from my line arrays. I can provide more, if you need them.
http://www.diyaudio.com/forums/mult...line-array-using-vifa-tc9-13.html#post4584857
Great work so far!
Here are a few measurements from my line arrays. I can provide more, if you need them.
http://www.diyaudio.com/forums/mult...line-array-using-vifa-tc9-13.html#post4584857
Great work so far!
I'll help either way. I keep all this to myself for many years because it was a competitive advantage, but now (that I have retired) I'd just like to see it get developed for the audio community.
^Especially if advertised number of required measurements (13) relies on unidirectional radiator in symmetric baffle in one plane, or different source types require different angle recipe, some knowledge and planning before off-axis responses are captured 
But we will see possible flaws and narrow-mindedness in claims as soon as first working implementation is published.
But we will see possible flaws and narrow-mindedness in claims as soon as first working implementation is published.
hello Earl,
Just as aslepekis remarked, this is one of the most valuable threads I' ve encountered in a long time. Let my try to recap& please correct me where I misunderstand:
By measuring just 13 of axis points in a hemisphere ( your measuring cradle, so to speak) , you are able to "map" the entire behaviour of an axi-symmetric driver.
You mention 0, 5, 10, 15, 20, 30, 40, 50, 60, 80, 100, 120, 150 and 180 degrees. Is that the outcome of the Legendre spacing you mention?
Furthermore, would your technique, which seems a sort of inverse FEM, in principle allow to indicate where exactly on the cone surface the break-ups take place?
Kind regards,
Eelco
Just as aslepekis remarked, this is one of the most valuable threads I' ve encountered in a long time. Let my try to recap& please correct me where I misunderstand:
By measuring just 13 of axis points in a hemisphere ( your measuring cradle, so to speak) , you are able to "map" the entire behaviour of an axi-symmetric driver.
You mention 0, 5, 10, 15, 20, 30, 40, 50, 60, 80, 100, 120, 150 and 180 degrees. Is that the outcome of the Legendre spacing you mention?
Furthermore, would your technique, which seems a sort of inverse FEM, in principle allow to indicate where exactly on the cone surface the break-ups take place?
Kind regards,
Eelco
Hi Boden
It's good to see people who are interested in advancing the way they do things to more modern approaches.
If I measure in only one plane I can map only in that plane. If the device is axi-symmetric then a 1/2 plane is sufficient. If I had two drivers and want to measure the vertical interaction, I would need a few points in the vertical plane to do that.
More like an inverse BEM than FEM. The answer to this question gets more complicated because of resolution. The reconstruction resolution from far field data is a 1/2 lambda, just like it is for Radar. But this can be improved by going into the near field and including what are called evanescent waves, waves that decay exponentially from the source and do not propagate to the far field and as such are not in far field data. This is exactly the Near-field Acoustic Holography problem and is the subject of an entire book by my colleage at PSU Earl Williams called Fourier Acoustics.
I would do this problem this way - I once considered building such a system, but never did. Put an axi-symmetric device in a large baffle (size is TBD, but big enough that the edge diffraction does not mess up the data. Then measure near-field on a hemisphere and use a Bessel transform to calculate the velocity pattern in the aperture of the source. This would yield a highly accurate and detailed velocity distribution in the aperture and one could use other techniques to map back to the cone profile. With software this could be done by a single mic and a DIY with a hand rotated stand. But this is not really the level of detail that would interest a DIY.
It's good to see people who are interested in advancing the way they do things to more modern approaches.
If I measure in only one plane I can map only in that plane. If the device is axi-symmetric then a 1/2 plane is sufficient. If I had two drivers and want to measure the vertical interaction, I would need a few points in the vertical plane to do that.
Yes, that is correct, the Legendre functions run between x = -1 and 1 where x = cos(theta) so the mapping from theta to x is not linear.
More like an inverse BEM than FEM. The answer to this question gets more complicated because of resolution. The reconstruction resolution from far field data is a 1/2 lambda, just like it is for Radar. But this can be improved by going into the near field and including what are called evanescent waves, waves that decay exponentially from the source and do not propagate to the far field and as such are not in far field data. This is exactly the Near-field Acoustic Holography problem and is the subject of an entire book by my colleage at PSU Earl Williams called Fourier Acoustics.
I would do this problem this way - I once considered building such a system, but never did. Put an axi-symmetric device in a large baffle (size is TBD, but big enough that the edge diffraction does not mess up the data. Then measure near-field on a hemisphere and use a Bessel transform to calculate the velocity pattern in the aperture of the source. This would yield a highly accurate and detailed velocity distribution in the aperture and one could use other techniques to map back to the cone profile. With software this could be done by a single mic and a DIY with a hand rotated stand. But this is not really the level of detail that would interest a DIY.
Thank you very much! I certainly would like to see something like this available to to audio community too, after all, with the right tools amazing things can be made. And being able to "see" what a speaker is doing is imperative to making a better speaker.
I sent you a PM with some questions, but I'm going to start a new thread on this topic so Bill's doesn't go too far off topic...
For those interested, there was another thread that contained a great comparison of how well the method could reconstruct fine polar details from the coarse set of angles Gedlee uses.
Theoretical source data set:
Measurement tehcnology-Post#58
Reconstruction Results:
Measurement tehcnology-Post#67
Resolution Comparison:
Measurement tehcnology-Post#87
Theoretical source data set:
Measurement tehcnology-Post#58
Reconstruction Results:
Measurement tehcnology-Post#67
Resolution Comparison:
Measurement tehcnology-Post#87
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