**Optimizing Procedure**

The computer optimizing process finds the best position for the listener and loudspeakers by an iterative process that is illustrated in fig. 1. The short- and long-term spectra are predicted by the image source method. From these spectra, a cost parameter (described below) is derived that characterizes the quality of the sound produced. Then new listener and loudspeaker positions are repeatedly tried until a minimum in the cost parameter is found indicating that the best positions are found. The movement of the listener and loudspeaker positions is carried out using a search engine following a standard minimization procedure1.

**Performance Index**

It is assumed that the best position within the room is represented by the position where the short-term and long-term spectra have the flattest frequency response. This then motivates the production of a performance index that measures how much the true frequency spectrum deviates from a flat response. Previous work by the authors2,3 has shown that a standard deviation function is a good measure for characterizing how even the pressure scattered from diffusers are. Consequently, a standard deviation was also used here and evaluated typically from 20 to 300 Hz. Some smoothing over a few adjacent frequencies is carried out to simulate the effect of spatial averaging, which would naturally happen in actual listening rooms. Otherwise, there is a risk that the optimization routine will find a solution that is overly sensitive to the exact solution position. To form the single performance index required, the indices for each of the two spectra are added. More or less importance can be given to the Speaker-boundary Interference Response or Modal Response by the use of a weighting parameter. Typically the two spectra are equally weighted, but this is a program option.

**The Optimization Procedure**

The optimization procedure used is a standard simplex routine. The simplex has a series of nodes, which are different points in the error space (representing different listener and loudspeaker positions). These nodes move around the space until either (i) the difference in the error parameter between the worst and best nodes is less than some tolerance variable (e.g., 10�6) or (ii) a maximum number of iterations is exceeded (set at 500). Most of the time the program stops because of (i). The simplex routine has the advantage of being a robust system, which does not require first derivatives for calculations. Unfortunately, derivatives of the spectra with respect to the listener and loudspeaker positions are not immediately available from a numerical method such as the image source model. The penalty for using a function-only optimization routine is that the number of iterations used to find a solution is longer, and so the procedure takes longer.

The user of the program inputs various parameters to define the optimization procedure. These can be input via standard Windows dialog boxes, making the program user friendly. The optimization routines cannot be used completely without some interpretation of the results from the user. For example, there is a tendency for the routines to want to place the source and receivers on the room boundaries, as this will minimize the interference effects. Obviously, it is not always possible to build the loudspeakers into the walls, and so this may not be a useful solution. Furthermore, there is a risk that the optimization routines will place the loudspeakers close enough to the surfaces to reduce the interference within the frequency band selected (say 20�300 Hz), ignoring the fact that there may be audible interference effects just outside this frequency range. Also, the best solution found for the bass response will not always be optimized for stereo imaging, physical listener, and loudspeaker placement and other factors.

Therefore, the user is given the opportunity to limit the search range for the listener and loudspeakers. The limits of the listener and independent loudspeakers are defined in terms of rectangular volumes, determined by the minimum and maximum coordinates. The program allows the user to find the most appropriate solution within these imposed limits. The listener and loudspeakers can vary within the rectangular volume limits for an optimum practical solution. These limit constraints are applied to the simplex routine by brute force. For example, if the simplex routine asks for a prediction for a point outside the listener�s rectangle, the program forces the co-ordinates of the point onto the nearest edge of the listener�s constraint boundaries.

The loudspeakers can all be treated as independently varying. However, in most listening situations, certain loudspeaker positions are determined by others. For example, in a simple stereo pair, both loudspeakers are related by mirror symmetry about the plane passing through the center of the room. As the number of loudspeakers increase in the 5.1 home theater and multichannel music surround formats, we can make the program more efficient by taking advantage of positional relationships between the speakers. It is usual to search the room and find several minima. The use of geometric constraints increases the chance of finding the global minimum.

To accomplish this, a system of independent and dependent loudspeakers is adopted with each dependent loudspeaker position being determined by an independent loudspeaker. For a stereo pair example, the left front loudspeaker can be considered the independent loudspeaker and the right front loudspeaker can be defined as the dependent loudspeaker with its position determined by a simple mirror image of the independent loudspeaker�s position about the center line of the room. The program allows mirror symmetry operations with respect to the x, y, z planes passing through the center of the room or with respect to the x, y, z planes passing through the listener�s position. Mirror symmetry about planes passing through the variable listener position allows constraints to be imposed on the rear surround speakers. For example, in a 5.1 multichannel music format with five matching speakers equi-spaced from the listener, we can set up constraint relationships with one independent speaker (left front) and four dependent speakers (center, right front, left surround, and right surround).

To include the lessons we have learned about good stereo imaging, the program makes use of a stereo constraint. The stereo constraint refers to the normal angular constraints between a stereo pair and the listener that are applied to give a good stereo image. In the program, this constraint is applied by ensuring that the ratio of the distances between the listener to the center point of the speaker plane, and the distance between the stereo pair, is within a specified range. The default is between 0.88 (equilateral triangle) and 1.33. Applying such a nonlinear constraint to the simplex routine can only be achieved by brute force. If a position, which violates the constraint, is required, the simplex routine moves both the listener and loudspeaker positions to the nearest points in the room that comply with the constraint. The simplex routine can accommodate such abuses, but there is a risk that this will slow the procedures finding of the optimum position. When optimizing independent loudspeakers, like subwoofers for example, this constraint need not be applied.

Next time we�ll take a look at THX home theaters, multichannel music, and subwoofers, as well as comparisons between direct radiator and dipole surrounds.