In DR V7I1 Part 2, we described the new prediction model and compared the modal decomposition and image model predictions with an experimental measurement.

The Optimization Procedure
Numerical optimization techniques are commonly used to find the best designs for a wide variety of engineering problems. In the context of this paper, a computer is used to search for the best room dimensions. The iterative procedure is illustrated in Figure 1. The user inputs the minimum and maximum values for the width, length and height, and the computer finds the best dimensions within these limits. The computer predicts the modal response of the room and rates the quality of the spectra using a single figure of merit (cost parameter). The computer then tries other dimensions trying to find those that have the lowest figure of merit. A completely random search is too time consuming, and so one of the many search algorithms that have been developed for engineering problems was used. In this case, a simplex method search engine was used, which is not the fastest procedure but is robust and does not require knowledge of cost parameter derivatives.

Figure 1. Flow chart describing the optimization process for the Room Sizer™

In developing a single figure of merit it is necessary to consider what would be the best modal response. It is assumed that the flattest modal response corresponds to the ideal. This is done even though a perfectly flat response can never be achieved, as in the sparse modal region there will always be minima and maxima in the frequency response. The cost parameter used is the squared deviations of the modal response from a least squares straight line. If the modal response level of the nth frequency is Lp,n, then the cost parameter is given by e. Where m and c are the slope and intercept of the best fit line and the sum is carried out over n frequencies, fn. This is illustrated in Figure 2. Consequently, this is a least squares minimization criterion, which is commonly used in engineering. The deviation from a best fit line rather than the mean is used because it is assumed that slow variation in the spectrum can be removed by simple equalization, and what is important is to reduce large local variation. Before calculating the figure of merit, some smoothing over a few adjacent frequency bins is used. This is done to reduce the risk of the optimization routine finding a solution that is overly sensitive to the exact room dimensions. Furthermore, in prediction models very exact minima can be found which would never be replicated in real measurements; the smoothing helps mitigate against this.

Figure 2. The quality metric used in the Room Sizer™ is the squared deviations of the modal response from a least squares straight line

Figure 3. Comparison of the optimized results for a test room with the popular 1:1.4: 1.9 ratio suggested by Louden and the worst solution

Results
The optimizer was run for a wide variety of room sizes: 7m to 11m, 4m to 8m and 3m to 5m. Two hundred solutions were gathered. A frequency range of 20-200Hz was chosen, since the greatest improvement comes from avoiding degenerate modes at lower frequencies, where the modes are relatively sparse. In Figure 3, we show the improvement over the well-know ratio suggested by Louden. In the next issue we will compare the optimized solution with several other optimum dimensional ratio suggestions.

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