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Author Topic: A look at optimum smoothing in speaker EQ  (Read 967 times)

Frank Koenig

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A look at optimum smoothing in speaker EQ
« on: June 16, 2020, 02:33:30 pm »

In working on tools for creating speaker settings I've encountered the problem of how much smoothing to apply to the inverse of the combined responses when generating the prototype from which the correction filter is synthesized. The availability of easy-to-synthesize FIR filters, or high-order IIR combined with automated, optimized fitting, allows us to have arbitrarily detailed filters, so the question arises of how much is best. This is not a simple question as it is fundamentally a perceptual one, but there are some statistical observations to be made.

The psychoacoustic dogma is that peaks in the frequency response are more perceptible than dips, and that peaks or dips that span less than the critical bandwidth are less relevant. (Why, exactly, it follows that the required smoothness of the frequency response is determined by critical bandwidth I don't understand, and perhaps it's just an empirical observation.) In any case, this has led me, and likely others, to use a frequency-proportional smoothing of about 1/6 octave at mid to high frequencies with perceptually satisfactory results, but that's not the point here.

Here's the  problem. Too much smoothing leaves correctible frequency response errors on the table. In the limiting case smoothing the prototype filter with infinite bandwidth applies no correction at all. As we reduce the amount of smoothing (narrower smoothing bandwidth) the correction gets better. Indeed, if correcting the response at a single angle the unsmoothed correction can be "perfect". But we're never interested in correcting at a point but rather finding a compromise over the intended coverage angle of the speaker (perhaps weighting some angles more heavily, or discarding outliers, as when a reflection off the grille causes a measurable but imperceptible dip in the response over a narrow range of angles). As the amount of smoothing is decreased further there comes a point of diminishing returns as the variation between the responses at different angles dominates the single, electrical correction. So there likely can be too little smoothing, too. A highly detailed filter requires a longer impulse response, at the very least, and might come at some perceptual cost. (I'm unclear on this last point.)

To investigate this point of diminishing returns I generated smoothed, log-magnitudes (inverses of the electrical correction curves) combined* over multiple angles using different amounts of smoothing. I then calculated and plotted the sums across angles of the standard deviations (SDs) across frequency of the differences between the raw measurements and the smoothed curves. I also calculated the SDs of only the positive differences as that might be a more perceptually representative metric (counting the peaks but not the dips).

Below are plots of one set of 18 measurements smoothed at 1/3 octave and 1/24 octave (colorful), the overlaid corrected responses at those smoothings (all black), and the SDs as a function of smoothing. I've tried this on a few different measurement sets and got similar results. The upshot, perhaps unsurprisingly, is that as the smoothing is reduced the variation in response between different angles rapidly swamps our ability to correct those responses with a single, upstream electrical filter. This result is independent of any psychoacoustic criterion. Even if a finer-grained correction were to sound better, the ability to obtain that correction for multiple angles is not there, at least as measured by variance (or SD). Furthermore, good old 1/6 octave is about as good as we need. Even 1/3 octave (looking at you, graphic EQ) is only slightly worse. This is admittedly a simplistic look at a big problem and perhaps just a pedantic route to the obvious. Ideas welcome in any case. Thanks.

--Frank

*The responses were combine by taking the log of mean magnitudes rather than the mean of the log magnitudes. There is a scale factor ("gamma") involved. I've found this to be a perceptually better way of combining measurements. Taking the mean of the log magnitudes tends to over emphasize the dips at high frequencies, that most speakers exhibit, and results in an overly bright correction. This method accounts for the flat-top look of the overlaid corrected responses.







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Mark Wilkinson

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Re: A look at optimum smoothing in speaker EQ
« Reply #1 on: June 16, 2020, 05:34:00 pm »

Hi Frank, super post.
I bet many manufacturers are studying the exact same thing, since near perfect on-axis response is so easy to accomplish.

I've played with this tradeoff for several years now, .....how much correction to apply, ...ever since i discovered FIR, and automated mag and phase corrections.

Maybe you're familiar, FirDesigner's principal way to specify the level of correction is in terms of 1/x octave smoothing per pass band, (or rather per driver with even the ability to band correction strength within the pass band).

The acid test I've come up with (since i can't begin to do the statistical work you do), is how well do 1/x corrections hold up off-axis.
If corrections continue to produce smoothish curves off-axis, it seems to me they are valid. If not, bozo.
By seat of pants, i tend to agree with 1/6th as home base.

If i have a very well behaved driver when looking at on-axis vs off-axis, or nice driver/horn combo etc,
I take 1/x higher..say to 1/12th. I've even seen 1/24th hold up, albeit rarely.
If the off-axis show's a bad letting go vs on-ax, i drop to 1/3....or give up lol.

Please post whatever statistical evidence you continue to find.... I'll study to try to keep up  :)



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David Sturzenbecher

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Re: A look at optimum smoothing in speaker EQ
« Reply #2 on: June 16, 2020, 10:31:26 pm »

Frank,
I am guessing that you have read, reread, and read again all the whitepapers and patents that Dave Gunness and Fulcrum Acoustics (and EAW) has put together on the topic.  If not, that is a very good place to start. (I know the "Improving Loudspeaker Transient Response with Digital Signal Processing" is a good one)

https://www.fulcrum-acoustic.com/audio-technology-insights-resources/whitepapers/

It's been years since I have read them in detail, but I remember the jist of it being that you really need to look into what is causing each anomaly and determine if it's correctable with EQ or not. Taking multiple measurements will of course help you determine this on your own.  I don't know if you can broadly give a bandwidth for smoothing as an 1/12th octave issue at one frequency may be correctable, but a 1/12 octave issue at another might not.
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Frank Koenig

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Re: A look at optimum smoothing in speaker EQ
« Reply #3 on: June 17, 2020, 01:55:15 pm »

Mark: Thank you. It looks like we've been going down the same road on this.  I have not used FirDesigner but probably should get my hands on a copy to see what they've done.  The trouble is it would almost certainly lay waste to my years of effort, although I suppose I did learn a little in the process -- no worse than building my own speakers. On the statistics, don't overrate it. What I've done here is dirt simple, and maybe wrong. I did send a summary to a bio-statistician friend hoping she might see some parallels to other known problems. I haven't heard back...

David: I did indeed read whatever of Gunness's stuff that I could get my hands on around 2012-13 when I got started on this. And I remember well the assertion that you need to understand the physics of the aberration you're trying to correct, which philosophically I agree with. But as a consumer, not a designer, of speaker components, I've taken more of a speaker-as-a-black-box view. At the end of the day all I can do is insert an electrical linear filter upstream of whatever speaker I'm stuck with. The specification of that filter, and the evaluation of its performance, are based on linear system modeling of point acoustic measurements, and of course some listening. Even if I know the physical cause of an aberration I only can correct it to the extent that it shows up in the data. When, more often than not,  I don't know its cause I can still correct it on the basis of the data but need to be more clever in interpreting those data.

The big fuzzy part of all this, whether we know the physics or not, is what really matters perceptually and what tradeoffs to make. The are many things I measure that I can't hear. There are things I think I hear that I don't know how to measure. And there are, of course, innumerable tradeoffs.

What I'd like to do -- no promise how far I'll get -- is get better at analyzing the data. This might inform what and how to measure (such as appropriate angular sampling rate), and put an information-theoretic baseline on what can be corrected. No point in making corrections that are, statistically speaking, white noise. The above little exercise is an example of that.

Let's all keep the discussion open. I'd like to think that we're not that far behind the manufacturers.

--Frank
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Chris Grimshaw

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Re: A look at optimum smoothing in speaker EQ
« Reply #4 on: June 17, 2020, 02:58:28 pm »

Frank, I pulled this from the REW Help section, on the subject of smoothing for measurements:

Quote
Variable smoothing applies 1/48 octave below 100 Hz, 1/3 octave above 10 kHz and varies between 1/48 and 1/3 octave from 100 Hz to 10 kHz, reaching 1/6 octave at 1 kHz. Variable smoothing is recommended for responses that are to be equalised. Psychoacoustic smoothing uses 1/3 octave below 100Hz, 1/6 octave above 1 kHz and varies from 1/3 octave to 1/6 octave between 100 Hz and 1 kHz. It also applies more weighting to peaks by using a cubic mean (cube root of the average of the cubed values) to produce a plot that more closely corresponds to the perceived frequency response. ERB smoothing uses a variable smoothing bandwidth that corresponds to the ear's Equivalent Rectangular Bandwidth, which is (107.77f + 24.673) Hz, where f is in kHz. At low frequencies this gives heavy smoothing, about 1 octave at 50Hz, 1/2 octave at 100 Hz, 1/3 octave at 200 Hz then leveling out to approximately 1/6 octave above 1 kHz.

Which sounds like some direction for the questions you're asking. The author is describing the three not-fractions-of-an-octave smoothing options in REW: Variable, Psychoacoustic, and ERB.


FWIW, when it comes to measuring at different angles, I find it's something that varies from box to box. The speaker I worked on most recently was a Faital 10HX230 in a ported box, and I found the off-axis behaviour to be very good - the peaks and dips largely stayed in the same place, but as I moved off-axis there was a smooth decline towards the HF. I did a few sweeps to start off with and "get a feel for it", and went from there. In the end, I think I just averaged the on-axis and 45-degrees off, with 1/6th octave smoothing.

I do tend to eyeball this stuff, though - I'll pick a few measurements that I believe are representative of where the paying-attention-audience is, average them, and EQ the result.

Chris
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Frank Koenig

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Re: A look at optimum smoothing in speaker EQ
« Reply #5 on: June 17, 2020, 04:09:31 pm »

Variable smoothing applies 1/48 octave below 100 Hz, 1/3 octave above 10 kHz and varies between 1/48 and 1/3 octave from 100 Hz to 10 kHz, reaching 1/6 octave at 1 kHz. Variable smoothing is recommended for responses that are to be equalised. Psychoacoustic smoothing uses 1/3 octave below 100Hz, 1/6 octave above 1 kHz and varies from 1/3 octave to 1/6 octave between 100 Hz and 1 kHz. It also applies more weighting to peaks by using a cubic mean (cube root of the average of the cubed values) to produce a plot that more closely corresponds to the perceived frequency response. ERB smoothing uses a variable smoothing bandwidth that corresponds to the ear's Equivalent Rectangular Bandwidth, which is (107.77f + 24.673) Hz, where f is in kHz. At low frequencies this gives heavy smoothing, about 1 octave at 50Hz, 1/2 octave at 100 Hz, 1/3 octave at 200 Hz then leveling out to approximately 1/6 octave above 1 kHz.

I dunno ??? Hate to say it but that sounds like a bunch of studiophool mumbo-jumbo to me, although I appreciate your posting it.  First off, their (unsupported) recommendations are all over the map. 1/48 octave or 1/3 octave below 100Hz, which is it? Furthermore, for 1/48 octave smoothing to have any effect at 100 Hz you'd need a data window of almost 700 ms so that the inherent smoothing of the window would not dominate. The second and third recommendations are at least close to our notion of critical  bandwidth in humans, which has some support -- 1/6 octave above a few 100 Hz sounds familiar.

Be that as it may, the idea of smoothing that varies with frequency is worth investigating. It seems to me that in order to go about this rationally we need to know two lower limits as a function of frequency. One is based on the information content of the data, which depends on the specific speaker and how it's measured, and the other, which is fixed but less well defined, is based our knowledge of perception.

--Frank
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Chris Grimshaw

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Re: A look at optimum smoothing in speaker EQ
« Reply #6 on: June 17, 2020, 07:08:46 pm »

Yep, there's certainly a range of recommendations in that text. Of the three, the "Psychoacoustic" one makes the most sense on an intuitive level. 1/48 below 100Hz wouldn't be useful, and neither would be 1-octave smoothing - I'm certain I can hear more resolution at LF than that.

All that said, I did a little digging around and couldn't find much science behind what was written in the REW help file, so it's worth taking with some salt.


I'm now wondering if smoothing/weighting should be tied in with the equal loudness contours. If there was a dip at 19kHz, would we bother to EQ it back in? I don't think I would. If the dip was at 3kHz, though, I'd be working hard to figure out what was going on and how to fix it (EQ or otherwise).

Chris
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Timo Beckman

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Re: A look at optimum smoothing in speaker EQ
« Reply #7 on: July 07, 2020, 10:12:54 am »

Did you check the new FIR Designer beta?

You can load an on-axi trac and correct the respons the way you see fit and simultaneously see the effect of those on traces taken at other mic positions...

https://timobeckmangeluid.wordpress.com/2020/07/07/fir-designer-m-beta-supplementary-responses-window/
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Frank Koenig

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Re: A look at optimum smoothing in speaker EQ
« Reply #8 on: July 07, 2020, 06:44:37 pm »

You can load an on-axi trac and correct the respons the way you see fit and simultaneously see the effect of those on traces taken at other mic positions...
https://timobeckmangeluid.wordpress.com/2020/07/07/fir-designer-m-beta-supplementary-responses-window/

Interesting. This is exactly what I've been working on in my code. The conclusion I'm coming to is that for a speaker with pretty good polar uniformity correcting onax (or an average over a narrow range of angles near onax) at high resolution can really clean things up. I've been playing with data from a B&C 8CXN51 8" coax and fixing onax fixes +/- 20 deg very well. The FIR filter was synthesized from a 1/12 octave smoothed frequency response. (Before and after onax TFE plots below, and the offax corrected by onax look almost as good.) When I look at a large, loud PA speaker, that will remain unnamed for the moment, while the polar uniformity from a tonality standpoint isn't bad, the fine wiggles are so uncorrelated from one angle to the next that I see no point in going beyond 1/6 octave. The corrected TFE plot for any angle, other than the one on which a high-resolution correction is based, looks almost as bad as no correction at all.

I've been playing with statistical clustering, using the kmeans() function in R, in an attempt to cluster measurements taken at different angles to choose the subset to use for "fine" correction. Dunno if this will go anywhere.

I need to get back to my old outdoor measurement spot on the Stanford campus and get more data to work with. Trouble is they've got a small tent city of COVID testing stations there now and I don't want to get anywhere near it, unless I'm getting tested.

--Frank

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Timo Beckman

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Re: A look at optimum smoothing in speaker EQ
« Reply #9 on: July 08, 2020, 07:56:08 am »

I'm just a mis-user of software so i've got no idea what the statement below means ;-)
The new supplementary response option is something i've been waiting for (had no time to find a work around before the beta came so now there's no need to).

"I've been playing with statistical clustering, using the kmeans() function in R, in an attempt to cluster measurements taken at different angles to choose the subset to use for "fine" correction. Dunno if this will go anywhere."

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Re: A look at optimum smoothing in speaker EQ
« Reply #9 on: July 08, 2020, 07:56:08 am »


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