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Stephen Scharf
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Using Design of Experiments for integration of Subs into Main speaker systems (For JA in particular)

As some, or most of you know, I'm a scientist by profession. As JA also has an engineering background, I thought I would post my experiences with using Design of Experiments as a way to assess integration of subwoofers into a "mains" system.

In recently acquiring a REL subwoofer, and with the data acquisition system that my friend Tim helped (greatly..thanks, Tim) to set me up with, it naturally occured to me that one could use a formal system of experimentation called Design of Experiments (henceforth referred to here as DOE) to formally and reproducibly optimize the integration of the sub with the mains to provide the ideal room response.

Design of Experiments is different than the classic scientific "change one factor, leave everything else the same" (One Factor At A Time or OFAT) approach that scientists and engineers were taught in school and are still primarily using to this day (much to their detriment in many cases). While the basic concepts of DOE have been around since the 1700s, it was originally formalized by the statistician R.A. Fisher. The difference between DOE and the classic OFAT approach is that you can change [i]multiple factors[/i] at a time, set the the factors different "levels" (effectively, high or low), and run a series of a defined pattern of experiments (even in random order), measure the response (the effect you are interested in) and the DOE will tell you which factors are most important to get the response you want, and more importantly, what, if any INTERACTIONS there are between factors.

The ability to examine interactions between factors is one of the most important aspects of DOE, because if there are interactions involved between complex set factors and their resultant responses, you will never be able to deconvolute them without DOE, or if you can, it will not usually be without pure chance or luck, or without spending a lot more time, money, and effort in trying to figure them out, and often, you will still not figure them out in terms of which of the factors have the biggest effect from a statistically valid point of view.

Given this, I set out to to perform a set of experiment to see if I could determine the optimal settings of speakers, REL sub settings and other factors, like grilles on or off, speaker toe-in, port plugs, etc. that would optimize the in-room response.

My next post will be a brief introduction to DOE, and setting up the precepts of the experiments.

Stephen Scharf
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DOE continued...

Design of Experiments (DOE) Tutorial from the American Society of Quality

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 what-if 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

..........................Input A Level.......Input B Level
Experiment #1.........-1......................-1
Experiment #2........-1......................+1
Experiment #3.......+1.......................-1
Experiment #4 ......+1......................+1

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:

Factors................Input -1 Level.......Input +1 Level
Temperature........100 degrees.........200 degrees
Pressure..............50 psi.................100 psi

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:

..........................Input A Level............Input B Level....Interaction
Experiment #1.....-1.............................-1.................+1
Experiment #2.....-1............................+1.................-1
Experiment #3.....+1............................-1................. -1
Experiment #4.....+1...........................+1.................+1

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.
http://photos.imageevent.com/puma_cat/fujif31andf20photos/doe-tutorial1.gif

The effect of each factor can be plotted in a Pareto Chart.

http://photos.imageevent.com/puma_cat/fujif31andf20photos/doe-tutorial2.gif

The negative effect of the interaction is most easily seen when the pressure is set to 50 psi and Temperature is set to 100 degrees. Keeping the temperature at 200 degrees will avoid the negative effect of the interaction and help ensure a strong glue bond.
_________________

This simple-minded 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.

Buddha
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I look forward to the full

I look forward to the full story!

Of great interest will be what your ears tell you and comparing that to what your calculations predict.

Stephen Scharf
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DOE's continued...

Buddha,
I've been meaning to do that, but I haven't figured out how to embed graphics into forum posts yet....and my Design of Experiments results, which are written up, are graphics-rich.

Perhaps Stephen Mejias can provide some tips...

RGibran
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I wonder if this works? Test...
absolutepitch
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DOE and more

The classic DOE uses various combinations of factors and test the extremes so that the landscape of responses is mapped from which you get the optimal result. The example shown with two levels used four experiments. With many levels (or factors) the experiments increase very quickly. A seven factor test will require 128 experiments for a full DOE.

I have used the above methods for few factors. For seven factors, I have used the Taguchi Method where only eight experiments would be needed. What is sacrificed is some of the interactions, and other things I have not taken the time to fully understand. But it works! You may not get the best "optimum", but can get pretty close to it with a lot less work. Then a confirmation test is done to show that it works. It also shows which factors are most influential. Then you can run a smaller full DOE on those to get to the best solution.

One can measure the result and correlate it with listening evaluations. Looking forward to the results and your impressions.

WillWeber
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Interesting, but caution...

This approach assumes several things about the system under test, assumptions that may not be realistic:

  1. Linearity of cause and effect, and of the interactions - this condition can only be approximately realized in an acoustic environment for small changes> Example: Standing waves and interferences of multiple speakers are not only non-linear, they are not even monotonic (phase dependent and cyclical).
  2. Single variable dependence - not likely for all variables> Example: changing only the placement of subwoofer interacts with various room modes and with the other speakers. That takes very highly dimensional matrices to solve, and the requirements for linearity and precision (#3 below) are all the more stringent. It also takes many more experiemnts than 2^n.
  3. Repeatable and sensitive measurement of change effects - not so easy> are we to use our ears as this measurement device? Even with measuring instruments the sound quality is multi-parametric and difficult to interpret even if one could get complete data set sets.

If only life were so simple. Voicing the hifi setup takes great patience, and one cannot hope to find the ultimate optimum setup in any given room. DOE may be of limited help and should be exercised over quite small variations. And then still, how to evaluate and quantify the result of each experiment?

Just keep listening, use knowledge of acoustics as best as we can understand, and enjoy the process. It may go on for quite awhile.

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