Hi Guys, Back in 2010, I started to look at subwoofer integration 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 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: 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 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. The experiment I was trying to reproduce was the one based on the graph from research from Harnan 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: 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. With that in mind, I set out to see what I could do with my set up to emulate that via DOE. System used at the time was: Conrad-Johnson Premier 17LS Preamp, Conrad-Johnson LP70S, Dynaudio Contour S3.4 speakers, REL R-305, M-Audio MobilePre USB mike preamp, Dayton 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 real-time, IN-ROOM measured response for the right speaker: The goal here was to maximize the 20-50Hz node to range, while smothing the 70, and minimizing the 105, 155 and 500 Hz node peaks, so as to emulate the JBL 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 full-factorial 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...

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^2 adjusted are strongly correlated at 0.97 and 0.94, respectively, indicating that our results fit our model pretty 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 p-value 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 3-D image of two factors at one time as they affect the desired response in a 3-D 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 to determine where to set the subwoofer controls & speaker port plug for all the frequencies I wanted to optimize, then measured the BEFORE and AFTER responses, in this graph, BEFORE is BLUE and AFTER is GREEN. Note the similarity of the overall frequency response for the green trace to the ideal trace above from Harman labs. I then confirmed these settings by actual listening tests, and was very happy with the results. I am still using these settings today.