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November 25, 2023, 11:21 PM #1
Using Design of Experiments for Subwoofer Integration to a 2Channel Stereo System
Hi Guys,
I thought I'd a share a robust, realworld experimental, statisticallybased technique called Design of Experiments to optimize setup of a REL sub into my 2channel stereo system.
For those interested, first I'm posting an introduction to Design of Experiments here:
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When I started to look at how to effectively integrate my REL R305 subwoofer with a pair of "mains" speakers using a statistical approach called Design of Experiments to optimize integration of subwoofers for a given room with a set of "mains" speakers.
To start with here is an introduction to Design of Experiments.
Description
Design of experiments (DOE) is a powerful tool that can be used in a variety of experimental situations. DOE allows for multiple input factors to be manipulated determining their effect on a desired output (response). By manipulating multiple inputs at the same time, DOE can identify important interactions that may be missed when experimenting with one factor at a time. All possible combinations can be investigated (full factorial) or only a portion of the possible combinations (fractional factorial). Fractional factorials will not be discussed here.
When to Use DOE
Use DOE when more than one input factor is suspected of influencing an output. For example, it may be desirable to understand the effect of temperature and pressure on the strength of a glue bond.
DOE can also be used to confirm suspected input/output relationships and to develop a predictive equation suitable for performing whatif analysis.
DOE Procedure
Acquire a full understanding of the inputs and outputs being investigated. A process flow diagram or process map can be helpful. Utilize subject matter experts as necessary.
Determine the appropriate measure for the output. A variable measure is preferable. Attribute measures (pass/fail) should be avoided. Ensure the measurement system is stable and repeatable.
Create a design matrix for the factors being investigated. The design matrix will show all possible combinations of high and low levels for each input factor. These high and low levels can be generically coded as +1 and 1. For example, a 2 factor experiment will require 4 experimental runs:
Note: The required number of experimental runs can be calculated using the formula 2n where n is the number of factors.
For each input, determine the extreme but realistic high and low levels you wish to investigate. In some cases the extreme levels may be beyond what is currently in use. The extreme levels selected should be realistic, not absurd. For example:
Enter the factors and levels for the experiment into the design matrix. Perform each experiment and record the results. For example:
Calculate the effect of a factor by averaging the data collected at the low level and subtracting it from the average of the data collected at the high level. For example:
Effect of Temperature on strength:
(51 + 57)/2  (21 + 42)/2 = 22.5 lbs
Effect of Pressure on strength:
(42 + 57)/2  (21 + 51)/2 = 13.5 lbs
The interaction between two factors can be calculated in the same fashion. First, the design matrix must be amended to show the high and low levels of the interaction. The levels are calculated by multiplying the coded levels for the input factors acting in the interaction. For example:
Calculate the effect of the interaction as before.
Effect of the interaction on strength:
(21 + 57)/2  (42 + 51)/2 = 7.5 lbs
The experimental data can be plotted in a 3D Bar Chart.
This simpleminded example above shows that there is an interaction between temperature and pressure in the strength of the glue bond. This is one feature of DOEs that is particularly useful when looking at the effect of a number of factors and their effect on the critical functional response.
Now that that is out of the way as intro, let's look at the specific experiment I had in mind in the next post.
The experiment I was trying to reproduce was the one based on the wellknown "Harman curve" showing that listeners perceived a room as having flat room reponse when the measured frequency actually looked more like this. This is because our hearing is not linear at lower frequencies.
Now, one could debate whether the Harman curve is exactly the right curve on which to base this model, but it was a wellcharacterized "orthogonal reference" as we say in science, and as such, it was the one I decided to use for this experiment. Please keep in mind this was just an experiment I did for fun while recovering from a back operation.
What can I say...this is how scientists like myself are "wired".
Given this, I set out to to perform a "DOE" to see if I could determine the optimal settings of speakers, REL sub settings and other factors, like grilles on or off, speaker toein, port plugs, etc. that would optimize the inroom response.
With that in mind, I set out to see what I could do with my setup to obtain, as close as possible, the Harman response curve via DOE.
System used at the time was: ConradJohnson Premier 17LS Preamp, ConradJohnson LP70S, Dynaudio Contour S3.4 speakers, REL R305, MAudio MobilePre USB mike preamp, Dayton NISTtraceable calibrated measurement microphone with calibration file, and Room EQ Wizard Software. The Room EQ Wizard software generates and simultaneously measures the frequency sweep from 10 Hz to 20,000 Hz.
The desired responses were to maximize 20 and 50 Hz repsonse in dB, and minimize the 150 and 500 Hz responses. These 150 and 500 Hz responses were nodes that I wanted to minimize, if possible.
An example trace as my starting point for reference is shown. The purple trace is where I started with the sub in, the brown trace with the sub out, before optimizing things using the DOE approach. The following is the realtime, INROOM measured response for the right speaker:
The goal here was to maximize the 2050Hz node to range, while smothing the 70, and minimizing the 105, 155 and 500 Hz node peaks, so as to emulate the Harman curve graph.
The factors I used for the DOE were sub gain (as clicks up from zero), sub crossover, likewise clicks up from zero, plug or no plug in the speaker reflex port. So, four responses being measured as the result of 3 factors at two different levels (low, high as in the examples above shown).
Setting up a fullfactorial DOE in JMP (a stastical package), here are the experimental matrix I ran per JMP's output for the experimental design and the ACTUAL measurements for each frequency (as measured in dB by room Eq Wizard).
The next post will be the analysis of the data...

November 25, 2023, 11:22 PM #2
Re: Using Design of Experiments for Subwoofer Integration to a 2Channel Stereo System
Here is the result of analyzing the 20 Hz response of the DOE (warning: large image)
We can see from the actual by predicted plot that the R^2 and R^2adjusted are strongly correlated at 0.97 and 0.94, respectively, indicating that our results (i.e., the "residuals") fit our model really well. This indicates that the DOE "statistical model" fits the observed, realworld data really well. Also the leverage plots (the factors affecting the response (20Hz output), show that Sub gain has a large effect, and quite possibly that the interaction of sub gain and sub crossover may have an effect, as the pvalue is just barely above p>.05, which tends to suggest there may be a significant effect if we gather more data or relax our confidence level from 95% to 90%. In, fact, if we relax our alpha from 5% (the chance we are willing to accept that we are wrong) to 10%, then the Sub crossover point becomes significant. In addition, our analysis of variance with probability of F >0.0016, suggests that that this result occurred from our null hypothesis, that the model does not predict sub 20 Hz performance is very low.
The interaction profile plots also suggest that there is a likely interaction between sub XO setting and sub gain, and the plot lines are not perfectly parallel, but appear to intersect.
Looking at the Prediction Profiler from JMP, comparing Sub gain with Sub XO, we can see that the maximum 20 Hz response is obtained by the Sub gain at 12 clicks up, and sub set at zero clicks, and we could expect a level of 75.1875 dB.
JMP is also cool because it will show you a 3D image of two factors at one time as they affect the desired response in a 3D cube, in this case, the 20 Hz response. As you can see, the upper left corner of the rectangle shows, which is max 20 Hz response is at a sub crossover of zero, and a gain setting of 12.
These analyses show how you can utitlize DOE to predict with good confidence the response from setting the factors (crossover point and gain) to give the desired response at 20 Hz.
I used this approach then to measure the BEFORE and AFTER responses, in this graph, BEFORE is BLUE and AFTER is GREEN. Note how much smoother the overall frequency response is for the green trace.

November 25, 2023, 11:23 PM #3
Re: Using Design of Experiments for Subwoofer Integration to a 2Channel Stereo System
One of the really powerful things about DOEs, is you can build a good model, it can be use to show the "response" of the measurement(s) under study.
Now, most folks are just taught in school about fairly simple approaches to characterizing a response, e.g. using a linear regression, where the output is plotted as straight line based on the levels of the input factors.
An example are the plots from the Sub's DOE showing the influence of Sub Crossover (XO) and Gain on the model. These plots are simple linear regression plots. Basically, you turn Knob "X" and you get response "Y" changing in a linear proportion to the change in X.
But...DOEs are much more powerful than that, and can shown the nature of....interactions. For example, take the Time*Temp interaction when doing something as simple as...baking cookies. It's not Time and it's not Temp that produces a properly baked cookie, it's the INTERACTION of Time*Temp that produces a properlybaked coookie. This is important to understand because University typically does not teach or demonstrate the influences of INTERACTIONS to students when teaching science or engineering. Instead they teach OFAT: One Factor At a Time. Unfortunatly, often times the real world simply does not work in as simplistic a manner as OFAT.
Here's an example of what DOEs can also do: this is a response surface plot from the Sub DOE above depicting the INTERACTION between sub GAIN and sub CROSSOVER at the 155 Hz room mode. As is obvious, the fuctional response is not a simple staight line e.g. in a linear regression. It's 3dimensional shape that has curvature and complexity to it.
And..if you rotate the response surface cube around, you see more of the actual shape of the 3dimensional response curve of the 155 Hz room mode.
The operative point here being is to really fully characterize and thus, understand, what's going on your room with your components and speakers interacting with the room itself (remember, Danny Richie will tell you that the room itself is one of the most important factors), tools such as DOEs provide INSIGHT into the settings you'll need to obtain the best, realworld response.
Don'tcha just love...real SCIENCE?

November 25, 2023, 11:31 PM #4
Re: Using Design of Experiments for Subwoofer Integration to a 2Channel Stereo System
The nice thing about using DOEs is, it's takes the guesswork out of optimizing a system. You literally obtain Fstatistics and pvalues that specify how accurately the DOE model "explains" the entirety of the measurement data for your system under experimentation. In statistics, we generally set our confidence level at 95% confidence, which means the alpharisk (which the chance you're willing to reject the null hypothesis when it is in fact true, i.e., accept the risk of the model "being wrong") is set to 0.05 (5%).
What this means, basically, is that any DOE model with a pvalue < 0.05 is considered to be statistically significant with respect to having 95% confidence or greater. In the case of this DOE model for integrating the REL sub, our pvalue is 0.0016, which means we have 99.84% confidence that this model explains the actual data observed. And...that's pretty darn good statistical confidence.
DOEs are incredibly powerful tools and you can use it for many audio applications. For example, you could use it to optimize speaker placement, or cartridge alignment and setup.

November 25, 2023, 11:45 PM #5
Re: Using Design of Experiments for Subwoofer Integration to a 2Channel Stereo System
I applaud the significant amount of effort you put into this!
"We are all billionaires because of the huge piles sweet dough rolling in from our participation in the ongoing campaign to suck your collective brains through a stirrerstraw."  PTA Editorial Staff

November 26, 2023, 01:21 AM #6
Re: Using Design of Experiments for Subwoofer Integration to a 2Channel Stereo System
Ha! Thanks, Michael. I did the DOE on this some years back while at home over the holidays recovering from a back surgery. I spent 10 years teaching this kinda stuff to PhDlevel scientists and engineers while working as a Design for Six Sigma Master Black Belt in Biotech. 😺
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AudioShark forum is a leading forum site for High End Audio Discussion, Stereo System Discussion, Home Theater System Discussion, Best Home Stereo System Discussion, Home Theater Installation Discussion etc.
The AudioShark forum was created for sharing the passion of highend Audio. We have Audiophiles from all over the world participating and sharing their knowledge. From novice to experts, you will find a friendly environment for discussing about High End Audio, Stereo System, Home Theater System, Home Stereo System, Home Theater Installation, Amplifiers, Speakers, Subwoofers, Integrated System, Acoustic treatments & Digital Room Corrections and many more.
At AudioShark, we also have incorporated an exciting Marketplace where members can peruse terrific buys on used gear, as well as meet dealers and discuss the purchase of new gear.
We are as crazy about this hobby as you are! So come on in and join us! Audioshark.org the Friendliest Audio Forum!
Industry Participation Disclosure : The owner and administrator of Audioshark is the owner of Suncoast Audio LLC in Sarasota Florida. Suncoast Audio has a full brick and mortar presence in Sarasota with several great show rooms with many world class brands. More information can be found at http://www.suncoastaudio.com
Audioshark is a community of like minded individuals. Audioshark welcomes participation from all manufacturers and owners of all brands and products. It is our belief that online forums provide a community of like minded audiophiles and music lovers to encourage the growth of this wonderful hobby.
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