Technical Notes Volume 2, Number 1

1. Introduction:

This technical note presents issues that should be considered when producing loudspeaker system performance predictions with commercially available software programs such as EASE (Acoustic Design Ahnert, Germany) and CADP2 (JBL). Although EASE is utilized for the discussion and examples in this document, the topics presented are important when using any acoustical prediction program. The most important topics to survey are the frequency and angular resolutions used during the prediction process and the interaction between multiple loudspeaker systems of different types. The methods that are employed to create three-dimensional datasets from two-dimensional datasets are also of concern. The following discussion presents comparative examples of expected prediction errors, describes the conditions producing these inaccuracies, and suggests possible solutions.

The EASE manual suggests that loudspeaker system measurements be made every ten degrees using 1/1 octave bands of pink noise centered at 125, 250, 500, 1000, 2000, 4000 and 8000 Hz. The manual also recommends that measurements should be taken in a reflection-free environment to ensure accurate measurements. To satisfy these requirements, all of the loudspeaker systems in the following examples were measured in an anechoic chamber using a Fast Fourier Transform analyzer. The results were processed to obtain the equivalent of octave-band, pink noise, sound pressure level measurements obtained at one point in space (please refer to the appendix for details on this method). When generating predictions, there are a number of possible methods that may be employed to obtain different types of information regarding loudspeaker systems and their interaction within a given venue. In the context of this document, the primary focus is on the accurate representation of array interaction and prediction. Therefore, direct-field SPL predictions that do not include room effects will be surveyed.

Two different prediction techniques, EASE and a high-resolution complex data method, are utilized for comparisons made in this document. The high-resolution method employs a 1/36^th octave complex frequency resolution (magnitude and phase) and 1-degree angular resolution dataset to predict the octave-band, direct-field sound pressure level (SPL) distribution. This data resolution conforms to the guidelines set forth in AES-5id-1997 "AES information document for room acoustics and sound reinforcement systems" [1]. As described in AES-5id, calculations derived from a dataset consisting of 1/36^th octave 1-degree information will not deviate by more than 1 dB on average from high-resolution measurement data. Therefore, this high-resolution prediction method will be capable of serving as a reference for comparisons made to low-resolution prediction programs.

2. Resolution concerns and array predictions:

The current version of EASE requires a frequency and angular resolution of one point per octave and every ten degrees, respectively. This requirement results in a seven-point frequency response (125 Hz to 8 kHz) at each angle. EASE also assumes horizontal symmetry, which reduces the dataset to a 19x19 matrix (0° to 180° horizontal by -90° to +90° vertical). This dataset is capable of accurately representing the octave-band frequency response and ten-degree polar response of a well-behaved loudspeaker system. Please note that this octave-band SPL is a statistic representing the response of a loudspeaker system when driven with an octave band of pink noise and measured every ten degrees.

The ten-degree polar response only represents single measurement points in space, and does not represent a spatial average (this is the way EASE requires the data). As discussed in AES-5id, an angular resolution of ten-degrees can result in errors of 10 dB on average. Results can be misleading when this low-resolution dataset is used to predict the interaction of multiple loudspeaker systems.

The per-frequency phase response and propagation delay of the loudspeaker system is not included in the dataset. This simplification implies that all frequencies have the same effective acoustical center and time origin, and therefore a perfectly flat system phase response emanating from this point. When a loudspeaker system is entered into EASE, it is located within the room model relative to its effective acoustical center. The only way to include propagation delay is to manually enter this information when performing room predictions. All predictions that include complex data derive phase information from the displacement of sources entered in this manner.

EASE allows the end-user to select various options when producing direct-field SPL predictions. One of these options, called "Interference," is of utmost importance to our discussion. When the calculation of "Interference" is not activated, EASE predicts the SPL distribution produced by the sum of the individual magnitude responses of each loudspeaker, without phase information. When "Interference" has been selected, EASE predicts the SPL distribution produced by the complex summation of sources. This predicted distribution retains linear phase information from the acoustic center that is capable of showing cancellations and additions resulting from the physical distance between acoustic sources. Our empirical tests show that due to the simplifications made to the acoustic center and phase characteristics of a given loudspeaker system, "Interference" should be used at some frequencies and not at others to produce more accurate predictions. Figures 1 through 3 are provided to further demonstrate the use of "Interference" in EASE.

Figure 1 shows the 500 Hz octave-band EASE prediction of a three-speaker array without "Interference;" Figure 2 shows the same prediction considering "Interference." The reader is asked to notice the differences between the two predictions. The prediction including "Interference" shows a narrower coverage pattern, due to the constructive and destructive interaction provided by the additional phase information. Which one of these predictions is more accurate?

Figure 3 shows the 500 Hz octave-band high-resolution prediction of the same three-loudspeaker array. This prediction was obtained using the AES-5id compliant high-resolution complex prediction method, as described in the appendix. In this case, it is evident that the prediction without "Interference" produces the more accurate results at this center frequency.

Figures 4 through 6 present the same examples as Figures 1 through 3, now at a center frequency of 2 kHz. Figure 4 shows the EASE prediction of the three-loudspeaker array without "Interference", Figure 5 shows the same EASE prediction with "Interference", and Figure 6 shows the high-resolution prediction method. Comparing Figure 6 to Figures 4 and 5, it is evident that the prediction with "Interference" produces the most accurate results at this center frequency in the near field. However, the prediction that does not include "Interference" is most accurate in the far field response of the array.

As this first set of Figures shows, "Interference" plays a key role in the accuracy of an EASE prediction. "Interference" should be carefully selected on a per-frequency and distance basis in order to obtain accurate results. With atypical sources, however, the best way to obtain accurate results is by making use of increased frequency and angular resolutions.

3. Dissimilar sound sources:

In the previous discussion, an array of three identical loudspeakers was used to discuss prediction accuracy. EASE produced comparable results only because all of the loudspeakers in the prediction were the same. EASE cannot correctly characterize the interaction of multiple loudspeakers having different magnitude and phase characteristics.

Figures 7 and 8 illustrate the problems encountered with dissimilar sources. Two different loudspeakers were arrayed and predicted with EASE and with the high-resolution method. The high-resolution method uses 1/36^th octave frequency response data that contains both the magnitude and phase responses of each acoustic source on a per-frequency basis. Thus, the high-resolution method takes both physical offsets and internal source phase responses into account. As the two figures depict, the two prediction methods do not provide similar results. The high-resolution method shows the SPL distribution that can be measured and verified with a tool as simple as an SPL meter. In this case, EASE does not predict an SPL distribution that would match measurements, since it assumes a perfectly linear phase response.

Figures 7 and 8 reveal an important limitation in the effective use of low-resolution prediction programs. Predictions based upon two or more dissimilar sources may be unreliable, since there is no phase information contained in the loudspeaker system datasets. The internal source phase responses must be taken into account to accurately predict the total system response. It is suggested that only identical loudspeakers be used when producing predictions in low-resolution prediction programs that do not include loudspeaker system phase information.

4. Interpolation:

Another concern with using EASE is the ability to create a three-dimensional dataset from only the horizontal and vertical polar responses of a given loudspeaker system. In order to do this, EASE utilizes an elliptical interpolation scheme. For well-behaved loudspeaker systems that exhibit a true acoustical center across their operating bandwidth¹, this method is a reasonable approximation.
¹The UPA-1P was used in this example. Its exceptional acoustic center properties are the subject of a pending patent.

Figure 9 shows the amount of error that may be expected when using this interpolation scheme with a well-behaved loudspeaker system. A sample loudspeaker system was measured on a 40-degree diagonal off the vertical axis. At 500 Hz, the maximum error from a direct measurement compared to the interpolated data is 3.2 dB. At 2 kHz, the maximum error increased to 9.5 dB. Both of these maximum error values occur behind the loudspeaker system. In the front hemisphere of the loudspeaker system, which is most important in the prediction process, maximum errors are reduced to 1.5 dB at 500 Hz and 2 dB at 2.1 kHz.

This interpolation error is an important error to characterize, since it is one of the primary sources of error in all of the measured loudspeaker system catalogs available for EASE. This error places a limitation on the expected accuracy of all EASE-based predictions. It is suggested that this source of error be kept in mind when assessing the accuracy of predictions.

5. Conclusion - Low-resolution loudspeaker system data

Despite our reservations based upon these issues, Meyer Sound has supplied low-resolution data for most of the Self-Powered Concert and UltraSeries loudspeaker systems. The primary reason for supplying this information is to allow experienced consultants and contractors to present predictions of Meyer Sound Loudspeakers in comparison to other manufacturers' products. Meyer Sound does not advocate the use of low-resolution prediction methods without a thorough understanding of their limitations and an in-depth knowledge of electroacoustics and room acoustics. The examples and figures discussed do not fully encapsulate all of the issues regarding error, but are intended to be a reference for establishing the accuracy of array prediction results.

All of Meyer Sound's self-powered loudspeaker systems are phase corrected, which means that they will work well within the simplified phase capabilities of low-resolution prediction programs. Care should be taken when using a non-phase corrected product in low-resolution prediction programs, since the differences in acoustic center on a per-frequency basis will produce inaccurate array predictions.

The loudspeaker systems that we have included for prediction in low-resolution prediction programs are acoustically well behaved. Meyer Sound has gone to great lengths to make these low-resolution datasets as accurate as possible. Our own measurements of other loudspeaker system manufacturers' products show that the low-resolution datasets released by these manufacturers are not representative of their products' performance. Care should be taken when using loudspeaker system data from discrepant sources.

A number of our current production loudspeaker systems are not included in this initial release of data. This is primarily due to the fact that the angular and frequency resolutions are too coarse. For example, the Sound Beam^TM (SB-1) is so high in Q that the angular resolution of low-resolution prediction programs cannot represent the loudspeakers' response at all. As low-resolution prediction programs continue to advance in terms of resolution and prediction accuracy, we will reevaluate these issues and include those loudspeaker systems capable of being represented.

APPENDIX

This appendix describes, in detail, the high-resolution complex prediction method used in this document as a comparative reference with EASE predictions. The details of the data acquisition and measurement system utilized are also included for further clarification:

Measurement / Acquisition System:

Placed inside the anechoic chamber, the automated loudspeaker positioner rotates a loudspeaker with a center of rotation exactly four meters away from the measurement microphone location. The acquisition computer controls the movement of the positioner and obtains sixteen five-millisecond time records, sampled at 50 kHz, at each measurement location. For these examples, measurements were obtained at each degree and a random pink noise excitation was used to stimulate the loudspeaker under test. Each five-millisecond dataset is recursively convolved using the McClellan-Parks algorithm for optimal FIR filters to produce eight successive bands of data with progressively longer time records, ending with a 640 millisecond time record for the lowest band. Each of the eight resulting datasets is windowed using the appropriate length Hanning function, and 256-point FFT's are performed. Each of these datasets is then used to provide a specific portion of the frequency spectrum. This algorithm is performed on both the source and measurement channels to produce a single complex transfer function. The sixteen resulting complex transfer functions are then vector averaged to produce the final representative complex transfer function. This piecewise linear approximation to a constant-Q transform affords greater than 1/36^th octave resolution from 80 to 20000 Hz. The resulting 360x464 complex matrix is then post-processed to produce polar plots, pressure responses, visualizations and lower resolution representations.

Loudspeaker data for EASE was derived by integrating these high-resolution frequency responses on a per 1/1 octave basis, in order to obtain an estimate of single octave SPL using equal energy per octave weighted noise (pink noise). The mathematical constructs of this technique are detailed as follows:

The complex acoustic pressure will be represented by

Equation 1

Equation 2

Estimated Total Sound Pressure =

Equation 3

Equation 6

Estimated Total Sound Pressure =

Equation 7

Equation 7 provides an estimate of sound pressure if white noise was used to drive the speaker. The integration must be weighted by 1/f to obtain the sound pressure that corresponds to driving the speaker with a pink noise source (equal energy per octave):

Equation 8

Estimated Total Sound Pressure from Pink Noise =

Equation 9

This algorithm is utilized to produce the high-resolution prediction of octave-band sound pressure levels shown in this document.

In order to confirm the validity of this approximation scheme, a series of tests were performed. Figure 10 shows the results of one of these tests. Again, measurements were performed in an anechoic chamber. A B&K 2660 Investigator (SPL meter) was used along with a B&K 1405 Noise Generator (set to pink noise) to produce the measured octave-band SPL plots. The algorithm described previously in this appendix was used to produce the calculated octave-band SPL plots.

For this example, both of these measurement methods were used to produce on-axis and 30-degree below vertical axis comparisons, as depicted in figure 10. Across the entire spectrum, at both angles and regardless of relative level, the discrepancy between the measured and calculate octave-band sound pressure levels is shown to be less than 2 dB. This calculated method has been used to produce all of the low-resolution EASE datasets.

REFERENCES / FURTHER READING:

1. F. Seidel and H. Staffeldt. AES Information Document for Room Acoustics and Sound Reinforcement Systems - Loudspeaker Modeling and Measurement - Frequency and Angular Resolution for Measuring, Presenting, and Predicting Loudspeaker Polar Data (AES-5id-1997). Journal of the Audio Engineering Society, 46(3):195-216, 1998.

2. Pierce. Acoustics: An Introduction to its physical principles and Applications. American Institute of Physics for the Acoustical Society of America, Woodbury, NY, 1991.

3. Leo L. Beranek. Acoustics. American Institute of Physics for the Acoustical Society of America, New York, NY, 1986.

4. ANSI. Specification for Octave-Band and Fractional Octave-Band Analog and Digital Filters. American National Standards Institute ANSI, Acoustical Society of America, 1986. ANSI S1.11- 1986, ASA 65-1986.

5. ANSI. Acoustical Terminology. American National Standards Institute ANSI, Acoustical Society of America, 1994. ANSI S1.1-1994, ASA 111-1994.

6. John Meyer. Precision Transfer Function Measurements Using Program Material as the Excitation Signal. Proceedings of the AES 11th International Conference, Audio Test and Measurement, 1992, Audio Engineering Society.

7. William Press, et al. Numerical Recipies - The Art of Scientific Computing. Cambridge University Press, Cambridge, 1988.