Thanks guys, this is exactly as I was hoping for: an intelligent discussion with some useful modeling added in.
Wesayso, your 23 element sims look very close. I agree that practical polar curves are the combination of array related directivity convolved with driver directivity, assuming they are facing the same direction you can just add the driver polar (in dB) or convolve.
As several of you are using VituixCad I can see the usefulness of it. I've begun "messing around" to learn it and hope to be able to do some multisection eXpanding Array (XA) models when the discussion gets around to that. I had done a lot of simulations at Bose with their proprietary array tools (in Matlab) and had a 7 element model with absolutely wonderful polars for multiple Octaves. There is great potential there.
As a first stab I just did a 3 tweeter array, with the goal to achieve moderate directivity with as little lobing as possible. (Such as a THX array I did at Mac and Snell.) I've got fairly constant directivity and just a hint of vertical lobing with the usuall approach of 3 tweeters tightly packed with the outter two rolled off. You have to be careful to gradually roll off the outer two to prevent excess phase shift from messing things up.
Just a work in progress.
Wesayso, your 23 element sims look very close. I agree that practical polar curves are the combination of array related directivity convolved with driver directivity, assuming they are facing the same direction you can just add the driver polar (in dB) or convolve.
As several of you are using VituixCad I can see the usefulness of it. I've begun "messing around" to learn it and hope to be able to do some multisection eXpanding Array (XA) models when the discussion gets around to that. I had done a lot of simulations at Bose with their proprietary array tools (in Matlab) and had a 7 element model with absolutely wonderful polars for multiple Octaves. There is great potential there.
As a first stab I just did a 3 tweeter array, with the goal to achieve moderate directivity with as little lobing as possible. (Such as a THX array I did at Mac and Snell.) I've got fairly constant directivity and just a hint of vertical lobing with the usuall approach of 3 tweeters tightly packed with the outter two rolled off. You have to be careful to gradually roll off the outer two to prevent excess phase shift from messing things up.
Just a work in progress.
Attachments
Hi NC535,
Interesting sims of the sin X/X 23 element. Probably not the best performer as is. In my paper I built one just to verify if the notion of FFT inverse transform held true. That is, if a flat samples array gives a sin X/X transform, can you go the other way and say that a sin X/X weighted array will give steep sided flat topped polars. It did, but the lobing was as bad as ever and the main front lobe shrinks in proportion to rising frequency. What was needed is DSP filtering that shrinks the sin X/X weighting profile in half for each Octave of rise. Not available then but probably doable now.
Interesting sims of the sin X/X 23 element. Probably not the best performer as is. In my paper I built one just to verify if the notion of FFT inverse transform held true. That is, if a flat samples array gives a sin X/X transform, can you go the other way and say that a sin X/X weighted array will give steep sided flat topped polars. It did, but the lobing was as bad as ever and the main front lobe shrinks in proportion to rising frequency. What was needed is DSP filtering that shrinks the sin X/X weighting profile in half for each Octave of rise. Not available then but probably doable now.
Let me put up part 2 of my 1997 paper.
Here I try to explain how the geometry of an array causes lobing and and the angle of lobing and also the relationship between the FFT and predicted polar shapes. We see that the Arcsin function lets you map the FFT of an array weighting sequence into a polar diagram. Typically the FFT aliases around the Nyquist frequency. This is seen as, as you go up in frequency. First, a side lobe emerges at +-90 degrees. For higher frequencies you will see the lobe swing forward and a second and then a third lobe will form.
The whole relationship between the FFT and array polar curves seems to be accepted today but at the time I had not seen anything in print and was nervous about stating it as fact. In this section I also show the FFT of the 5 element rectangular weighted (all 1's) array and the FFT of the Bessel sequence.
The sin X/X, 23 element array, as explained above, was meant to confirm that the inverse transform worked as well, so you could define a desired polar curve and use the inverse FFT to define the weighting coefficients. We took the standard 23 tweeter array and ended up putting resistive padder networks across all the tweeters to get to the calculated strengths. I think there were a few shortcuts we could take with repeated coefficients. Measurements show the expected polar response with fairly steep cuttoffs, but lobing is as bad as any other array, as shown in figure 24.
I ended this section with speculation about a nearfield model possibly giving a more accurate prediction of off axis radiation than the far field model was. It was clear that the very sharp lobes of the long arrays was not being realized. The observation that "response is uniform if you stay within the end point of the array" wasn't being seen in the modeled results.
Here I try to explain how the geometry of an array causes lobing and and the angle of lobing and also the relationship between the FFT and predicted polar shapes. We see that the Arcsin function lets you map the FFT of an array weighting sequence into a polar diagram. Typically the FFT aliases around the Nyquist frequency. This is seen as, as you go up in frequency. First, a side lobe emerges at +-90 degrees. For higher frequencies you will see the lobe swing forward and a second and then a third lobe will form.
The whole relationship between the FFT and array polar curves seems to be accepted today but at the time I had not seen anything in print and was nervous about stating it as fact. In this section I also show the FFT of the 5 element rectangular weighted (all 1's) array and the FFT of the Bessel sequence.
The sin X/X, 23 element array, as explained above, was meant to confirm that the inverse transform worked as well, so you could define a desired polar curve and use the inverse FFT to define the weighting coefficients. We took the standard 23 tweeter array and ended up putting resistive padder networks across all the tweeters to get to the calculated strengths. I think there were a few shortcuts we could take with repeated coefficients. Measurements show the expected polar response with fairly steep cuttoffs, but lobing is as bad as any other array, as shown in figure 24.
I ended this section with speculation about a nearfield model possibly giving a more accurate prediction of off axis radiation than the far field model was. It was clear that the very sharp lobes of the long arrays was not being realized. The observation that "response is uniform if you stay within the end point of the array" wasn't being seen in the modeled results.
Directivity of the SinX/X array was awesomely well controlled right up to the point where it wasn't. Its easy to see why your successors at McIntosh have gravitated to what looks like an array of 3/4" tweeters in their latest array.
Fine grained DSP and weighting is indeed do-able able now with 8++ channel DACs and amplifiers at reasonable cost and a PC with JRiver for the DSP. I would think a large company like McIntosh or Bose would have the resources to do an integrated electronics and software package for this purpose as well as custom drivers. With so many drivers, an amp for a pair of drivers or even a small group is perfect for a chip amp.
There are no practicality limits simulation.
I assembled the amps, MOTU DAC, and fanless PC, and learned JRiver as well as MiniDSP. I did lots of simulations; but enjoyed my straight line unweighted arrays at my primary LP so much I never got around to rewiring them before moving to a place where line arrays didn't "fit".
I'm looking forward to your treatment of expanding arrays. I never got them quite right in simulation .... That seems to be an easier way of overcoming the HF compromises of a line array while retaining good vertical directivity.
Fine grained DSP and weighting is indeed do-able able now with 8++ channel DACs and amplifiers at reasonable cost and a PC with JRiver for the DSP. I would think a large company like McIntosh or Bose would have the resources to do an integrated electronics and software package for this purpose as well as custom drivers. With so many drivers, an amp for a pair of drivers or even a small group is perfect for a chip amp.
There are no practicality limits simulation.
I assembled the amps, MOTU DAC, and fanless PC, and learned JRiver as well as MiniDSP. I did lots of simulations; but enjoyed my straight line unweighted arrays at my primary LP so much I never got around to rewiring them before moving to a place where line arrays didn't "fit". I'm looking forward to your treatment of expanding arrays. I never got them quite right in simulation .... That seems to be an easier way of overcoming the HF compromises of a line array while retaining good vertical directivity.
The above restriction calls for listening at a specific height precisely on the array's vertical axis?
Are there any other disadvantages? I mean eg. due to strongly coloured floor and ceiling reflections?
Those are killer disadvantages, a sign that a different approach is needed. Look at what @wesayso has done to get significant smoothing of the HF while retaining wide vertical directivity. The pure sinx/x weights proved a theoretical point. For actual listening you need to compromise the smoothing/reduction of combing or tuning of the directivity to keep the vertical window wide enough.
Ok, I see. Then what vertical window is wide enough? What should we aim for?
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Looking forward to seeing this part progress. One small Vituix tip that should be useful in this context, if you right click on the impedance graph you can change it to a vertical directivity graph and then display both directivity planes in the one image.
That's incredible Neal! Wow.
A couple of years back I did a project inspired by the Donald North Audio speakers, where it's basically an MTM with a driver on the Y axis AND the X axis.
The results were very very good. I kinda lost interest because I like to tinker more than I like to finish stuff, but adding elements seemed to help a lot.
https://www.diyaudio.com/community/...st-controlled-directivity-loudspeaker.347025/
Isn't that the (or a) million dollar question? I can name another similar question: what directivity pattern is ideal for stereo listening?
I can only answer what I was after for my personal goals, I wanted to achieve the best results at the seated listening height,
checking if those results hold up along a 3 seat couch to cover the whole family. Next up I wanted acceptable results when
standing up. Meaning, the seated listening height should be in the middle of the vertical "bundle" while the standing results
may still show a somewhat rough top end, but remain balanced after smoothing. Another goal I wanted to achieve was to
diminish the effects of floor and ceiling reflections as much as possible. Meaning I wanted to have less energy going towards
the floor and/or ceiling.
A comparison of the vertical directivity of the original unshaded array vs the frequency shaded array (25 drivers, 85,5mm CTC):
The effects of the filters I applied in a single graph (after removing overall EQ):
The drivers are divided in groups of 5 drivers, one group has no filters (moving straight just above -36), all other groups are filtered, with the first
group around the 5 unfiltered drivers divided in 3 below, and two on top. That's why you see another small division at ~3 KHz, as I've put notch
filters on the third driver to get more symmetry or balance in my results. The lowest line is the power the top 5 drivers get. They do most of their
work on the bottom end. You'll see the difference in the unshaded array center (0 - position shifting) vs the shaded array.
The filters used in one array, including two notch filters. Not included in this picture are some tweaks on the top end capacitor values I've
changed after changing my drivers from the Vifa/Peerless TC9 FD18-08 towards the Scan Speak 10F 8414 G10.
To answer the original question; I opted for a ~30 degree vertical bundle to cover the positions I was interested in.
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As to what vertical window is wide enough, that is a very fundamental question. As soon as you get into longish line arrays you have the capability of very high directivity, at least in the far field. I don't think that was ever the objective. At Mac we used to have people stand on a chair and you could get your head just above the top of the tweeter array. As you stood upwards the highs would drop like a stone. Great parlor trick but is that what you want?
As I started getting good performance with the Snell arrays I gradually came to believe that the right goal was the smoothest possible vertical response (minimum vertical frequency response variation) and avoidance of nulls in the listening window. A modest amount of directivity increase was a good thing and would reduce the floor and ceiling bounce, but no need to go crazy in that regard.
With any of the symmetrical arrays (such as Snell XA) you can think of it as first doing a single sided multiway, such as a TMW. When you add the other side (making it WMTMW) you would expect a 6 dB rise in pressure while you had added 3 dB in power. This means an overall d.i. increase of 3 dB, a moderate but useful amount that would reduce the reverberant field by 3dB.
I have never felt the need to push for more than that and feel that achieving vertical resonse smoothness is the goal. If you can achieve a totally lobe free response then you will have a multiway system with undetectable crossover points. That is a very neat result.
As I started getting good performance with the Snell arrays I gradually came to believe that the right goal was the smoothest possible vertical response (minimum vertical frequency response variation) and avoidance of nulls in the listening window. A modest amount of directivity increase was a good thing and would reduce the floor and ceiling bounce, but no need to go crazy in that regard.
With any of the symmetrical arrays (such as Snell XA) you can think of it as first doing a single sided multiway, such as a TMW. When you add the other side (making it WMTMW) you would expect a 6 dB rise in pressure while you had added 3 dB in power. This means an overall d.i. increase of 3 dB, a moderate but useful amount that would reduce the reverberant field by 3dB.
I have never felt the need to push for more than that and feel that achieving vertical resonse smoothness is the goal. If you can achieve a totally lobe free response then you will have a multiway system with undetectable crossover points. That is a very neat result.
That's a slightly different point of view than what I have had, when going for arrays. My goal from the start was to diminish the floor and ceiling bounce.
Having multiple drivers in a long array helps in that regard. As it achieves very smooth vertical response, with minimum frequency response variation
when measured inside a room with floor and ceiling reflections present.
Look at the orange in-room predictions between a single driver vs array... That's my specific goal of using arrays.
They are quite unique in that behavior. Their biggest enemy left are parallel planes. Take care of that and they can be
loud and clear.
Having multiple drivers in a long array helps in that regard. As it achieves very smooth vertical response, with minimum frequency response variation
when measured inside a room with floor and ceiling reflections present.
Look at the orange in-room predictions between a single driver vs array... That's my specific goal of using arrays.
They are quite unique in that behavior. Their biggest enemy left are parallel planes. Take care of that and they can be
loud and clear.
Undetectable crossover points would be down to the wavelength spacing at crossover. We see from our sims that if you surround, say, a tweeter by 2 small mids, if they are all runing at a frequency where they are one wavelength apart then there will be a 90 degree side lobe and a sharp null at a more forward angle. If we could crossover well below that (say when the elements are 0.33 wavelengths apart, then there is no need for a lobe.
This turns out to be a challenge for the top most crossover, but not a big deal for lower crossover points for 3 and 4 way arrays.
This turns out to be a challenge for the top most crossover, but not a big deal for lower crossover points for 3 and 4 way arrays.
I just changed my avatar that had a picture of the waveguide...
The primary goal was the most compact cluster for the central 3 elements. It turned out that front mounting the mids and rear mounting the tweeter helped the height dimension. The recess flare for the tweeter followed what I knew of constant directivity flares (at least approximately). The horizontal flare angle was made quite wide (say, maybe, 120 degrees). Giving it all a pinch in the vertical direction (and therefor conforming to the midrange perimeter) meant that the vertical angle was fairly narrow perhaps 60 degrees (these numbers all from memory). This all contributes to the general desired for wide horizontal and reasonably narrow vertical. At the same time any sort of waveguide on the tweeter leads to "low frequency" gain which goes directly into power handling and hence the ability to crossover a little lower.
A lot of nice synergies here.
P.S. we had an NC machinist a few miles away that did the XA Reference plate out of a solid plate of aluminum of abut 15 mm thick. He did a beautiful job and the raw piece required no filling.
P.P.S. the XA 55, 75 and 90 used a cast piece and some small mids custom made for us by parent company Boston Acoustics.
The primary goal was the most compact cluster for the central 3 elements. It turned out that front mounting the mids and rear mounting the tweeter helped the height dimension. The recess flare for the tweeter followed what I knew of constant directivity flares (at least approximately). The horizontal flare angle was made quite wide (say, maybe, 120 degrees). Giving it all a pinch in the vertical direction (and therefor conforming to the midrange perimeter) meant that the vertical angle was fairly narrow perhaps 60 degrees (these numbers all from memory). This all contributes to the general desired for wide horizontal and reasonably narrow vertical. At the same time any sort of waveguide on the tweeter leads to "low frequency" gain which goes directly into power handling and hence the ability to crossover a little lower.
A lot of nice synergies here.
P.S. we had an NC machinist a few miles away that did the XA Reference plate out of a solid plate of aluminum of abut 15 mm thick. He did a beautiful job and the raw piece required no filling.
P.P.S. the XA 55, 75 and 90 used a cast piece and some small mids custom made for us by parent company Boston Acoustics.
I definitely want to see more about your expanding arrays but going back to line arrays for a moment, here are vertical polar map and vertical line chart up to 45 degrees off axis for a 75xLineArray where the top and bottom 25 drivers model floor and ceiling image drivers. Here we see the wide vertical directivity we experience with floor to ceiling arrays in our homes
Attachments
Hi NC535,
At Mac I did a little bit of array plus floor and ceiling bounce modeling. The model had to include a gap between the central (real) array and the floor and ceiling reflections as the array didn't full stretch to the boundaries. I recall a slight improvement but not really a big change. If your long array gives high directivity (drops like a stone above and below the endpoints) then I would think the difference would be nominal. I think that this, as a general case, is different than, say, a small tower speaker on the floor. Such a system has little overall directivity and the bounces are of a point source nature. The array is far from a point source and has very significant directivity.
I do believe that achieving vertical directivity and reducing floor and ceiling bounce is "a good thing". We are very good at ignoring lateral reflections but we know that vertical reflections are not so easy to ignore and end up as comb filter type response variations. At the same Floyd Toole is whispering in my ear: "all things in moderation" suggesting that supression of floor bounces to some degree gets you to a threshold of audibility beyond which there is no perceived improvement.
Does the border of the "red zone" in your sims equal the end points of array length? The constant directivity is impressive.
At Mac I did a little bit of array plus floor and ceiling bounce modeling. The model had to include a gap between the central (real) array and the floor and ceiling reflections as the array didn't full stretch to the boundaries. I recall a slight improvement but not really a big change. If your long array gives high directivity (drops like a stone above and below the endpoints) then I would think the difference would be nominal. I think that this, as a general case, is different than, say, a small tower speaker on the floor. Such a system has little overall directivity and the bounces are of a point source nature. The array is far from a point source and has very significant directivity.
I do believe that achieving vertical directivity and reducing floor and ceiling bounce is "a good thing". We are very good at ignoring lateral reflections but we know that vertical reflections are not so easy to ignore and end up as comb filter type response variations. At the same Floyd Toole is whispering in my ear: "all things in moderation" suggesting that supression of floor bounces to some degree gets you to a threshold of audibility beyond which there is no perceived improvement.
Does the border of the "red zone" in your sims equal the end points of array length? The constant directivity is impressive.