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Subwoofer Arrays — The Search For The Ideal Line Continues

Part 2 of this series provides an evaluation of performance parameters that include tonal consistency in the audience zone, headroom, and the difference between the audience and mix position.

Part 1 of this series, published last month (here), shows that there are significant differences between the curving algorithms for subwoofer arrays in modern sound reinforcement. Now, in Part 2, the resulting sound fields are analyzed in more detail and their performance is quantified.

A broadly equal experience for the listeners is the goal in the design of modern sound reinforcement systems1. The procedure is a mostly iterative and visual process. A comparison between different system concepts is complicated and usually cannot be quantified. In order to analyze the results of the delay algorithms, an objective metric was developed to describe the sound field in terms of:

  • Tonal consistency in the audience zone
  • Headroom of the sound system
  • The difference between the audience and the mix position

To quantify these criteria, three values are used based on the publication of Hill (2018)2. The tonal consistency SV is calculated by the mean spatial variance of the magnitudes across the audience plane for all analysed frequency bins. The smaller the variance the more consistent is the sound field. Since an even sound pressure level (SPL) over the entire audience is not possible due to propagation losses, the magnitude responses in the range of interest (30 – 120 Hz respectively for all examples here) are normalized to better evaluate the coverage consistency and to reduce the influence of propagation losses on the results. The sound engineer at the front of house position is correcting the mix based on their listening experience. A highly varying sound pressure level might lead to an uneven mix across the audience. The difference between audience and FOH position K can be calculated by the absolute deviation between the sound pressure level at the mixing position and the mean sound pressure level across the audience for all frequencies.

To quantify the headroom HR of the sound system, the difference between the sound pressure level at the mix position and an expected target level (115 dB here) is calculated. If the sound system reaches the target level, it is powerful enough HR = 0 dB. For sound pressure levels below the target level, the actual system headroom is analysed. Therefore, the nominal and maximum sound pressure of a subwoofer at 1 m is used (120 dB and 132 dB respectively for all examples here, based on specifications for typical 18-inch subwoofers). If the actual system headroom minus the level difference surpasses 6 dB, the power of the system is insufficient HR = ∞ dB. The HR criteria within performance metric refers to the change in system headroom, where positive values represent a loss of headroom. A smaller value HR indicates more headroom available in the system. All three criteria get linearized, weighted and form a single array performance rating APR. It categorizes the performance in grades for a linear range from 0 (worst) to 1 (best), as shown in Table 1.

Table 1: Grades Array Performance Rating (APR).

To analyze the performance of the different configurations, a standardized test environment is developed (Figure 1). It calculates the sound field based on the wave equation and complex summation at a receiver point under free-field conditions. For the simulations in this paper, all elements are assumed to be omnidirectional sources. This eliminates the influence of product-specific models and allows only the delay approaches to be analysed. The audience zone is a sound field of 50 m by 40 m that should be covered by an array of 12 elements spaced 1.05 m apart, which ensures that no comb-filtering occurs in the frequency range of interest. The FOH position is located in the center at 25 m distance from the array. The physical configuration leads to a nominal opening angle of 60°. On the surface there are receiver points (crosses) at a maximum distance of 1.4 m, which ensures that no errors occur due to spatial aliasing.

Figure 1: Test environment.

Starting from a ZeroConfig, where no delays are applied, the sound fields for SW1 and SW3 are analysed in more detail. The other algorithms tend to develop similar effects based on their curving strategy. A final overview shows the results of the objective performance criteria at the end of this article.

To evaluate the spatial consistency, the magnitude responses at the receiver points are analysed. Figure 2 shows the normalized magnitude responses for the ZeroConfig.

Figure 2: Normalized Magnitude Responses, ZeroConfig.

The width of the envelope around the magnitude responses serves as a graphical quality criterion for the consistency across the audience. The wider this imaginary tube, the higher the variance and the more inhomogeneous the tonal behavior. A high variance is to be expected with this configuration. This can be seen in the value SV of 6.7 dB, which is significantly higher compared to the following configurations. The desired sound pressure level is not achieved. However, the headroom HR of 2.3 dB is within an acceptable range. Overall, this results in an APR of 0.63, grade C. Figure 3 shows the sound pressure level distributions for the ZeroConfig at 40 Hz and 100 Hz.

Figure 3: Mappings, ZeroConfig.

The spectral range around 40 Hz is typical for the tuning of the bass reflex system of the assumed loudspeakers and is also musically important. 100 Hz represents the frequency range in the crossover spectrum to the main system. High variations in this frequency range lead to additional challenges in tuning the overall system. A clear main beam can be seen at both frequencies. At 40 Hz, there is a strong cancellation about 10 m in front of the FOH, which leads to a very narrow coverage. The main beam is also noticeable at 100 Hz. Pronounced side lobes lead to a high variance of the sound pressure inside and high energy components outside the audience area.

SW3, SW4 and SW6 provide the longest delay times in the parameter study. With the parameters of SW3, a significantly lower variance of the sound pressure level at the measurement points can be observed in Figure 4.

Figure 4: Normalized Magnitude Responses, SW3.

They are characterized by stronger deviations at 40 Hz and 110 Hz. Around 70 Hz, the deviation is lowest. From a performance point of view, the system is not sufficiently dimensioned for the intended situation (HR = ∞ dB). In addition, the difference between the sound pressure level at the FOH and the average sound pressure level in the audience area increases. In total, the individual values lead to an unchanged APR of 0.63 compared to the ZeroConfig. Here the influence of the performance criteria becomes clear. Despite improved uniformity, the optimization of the array does not meet the requirements of HR and K, which is directly reflected in the value of APR. The spatial sound pressure level distribution in Figure 5 shows a better fit to the audience zone at 40 Hz.

Figure 5: Mappings, SW3.

The main lobe is much wider and covers almost the entire area with a sound pressure level reduction of 6 dB compared to the FOH. The longer delay times lead to a splitting of the main maximum at a frequency of 100 Hz. In this case, the sound pressure level at FOH is lower than in the adjacent areas. This behaviour can lead to wrong decisions by the sound engineer, resulting in a very bass-heavy and undefined sound in the audience. Secondary maxima are significantly lower with the parameters of SW3. However, more sound energy extends outside the edges of the audience zone, which can have a critical impact on noise emissions for local residents.

When optimizing the array using SW1, the normalized magnitude responses have the smallest deviations (Figure 6).

Figure 6: Normalized Magnitude Responses, SW1.

The values show also a very good consistency of the sound field both within the audience area and between the audience and the FOH. A closer look reveals increasing deviations in the higher frequency range above 100 Hz. In terms of performance, the system is in the acceptable range, even though the target level is not reached. Nevertheless, SW1 reproduces the highest sound pressure level at the FOH position of all optimization algorithms. The good results for the individual criteria achieve an APR of grade A. A pronounced main beam can be seen for both frequency ranges in Figure 7.

Figure 7: Mappings, SW1.

There are no local minima. Almost the entire audience zone next to and behind the FOH position is in a range of maximum +/-6 dB relative level. The main maximum follows the straight edges of the audience area. However, the width is minimally underestimated with the nominal opening angle of 60°.


All delay algorithms result in a more homogeneous sound pressure level distribution in the audience. Due to the longer delay times of SW3 (representative for the other algorithms), the sound pressure level in the main radiation direction is reduced, which is why the system can’t fulfil the requirements regarding the headroom criterion. Furthermore, these algorithms divide the main lobe for higher frequencies. That means that lower sound pressure levels can be expected in the centre of the audience zone, where the FOH might be located. In general, the curving algorithm of SW1 achieved the best results in the evaluation using the APR. This is also evident from investigations on other system configurations and array arrangements. The shorter delay times (see Part 1 of this series), which increase slowly to the outer elements, lead to a pronounced main lobe with a high efficiency in the main radiation direction. Furthermore, the algorithm of SW1 achieves a high consistency of the sound field in the whole frequency range. The sound field is characterized by defined boundaries along the side lines of the rectangular audience zone.

Table 2 summarizes the results for all curving algorithms. The results agree with the findings described earlier. The consistency across the audience can be improved with all approaches. The algorithms with a high increase in delay values suffer from insufficient system headroom. Only SW1 and SW5 provide an acceptable sound pressure level at the FOH. This results in a grade C for algorithms SW3, SW4 and SW6. SW2 and SW5 achieve a grade B. Only the curving algorithm of SW1 achieves an APR of grade A.

Table 2: Results Array Performance Rating.

This analysis shows that with an optimized beamforming algorithm for the subwoofer configuration, consistency in the audience and sound emission to the outside can be improved. The variations between the curving approaches result in significant differences in the sound field reproduction that should be considered when designing a system.

Further research shows also that a subwoofer arc configuration in front of the stage is most effective in reducing the sound pressure level at the sides compared to a left-right or flown subwoofer configuration3. However, the noise exposure for the nearest audience members is significantly higher, which can lead to hearing damage4. In addition, it can be shown that level consistency can be further improved by flying the subwoofers, both in terms of efficiency and alignment with the main system5. With this in mind, Part 3 will focus on the verification of the results and compare measurements with the simulation for the example system presented.

  1. Audio Engineering Society Technical Committee on Acoustics and Sound Reinforcement (AES TC-ASR). (2020). Understanding And Managing Sound Exposure And Noise Pollution At Outdoor Events (AESTD1007.1.20-05) (Technical Document).
  2. Hill, A. J. (2018, 23.-26. May). Live Sound Subwoofer System Performance Quantification (Convention Paper). Audio Engineering Society Convention 144, Milan.
  3. Mouterde, T. & Corteel, E. (2021, October). On the comparison of flown and ground-stacked subwoofer configurations regarding noise pollution (Convention Paper). Audio Engineering Society Convention 151, Las Vegas and Online.
  4. AES TC-ASR (2020)
  5. Corteel, E. et al. (2018, 17.-20. October). On the efficiency of flown vs. ground-stacked subwoofer configurations (Convention Paper). Audio Engineering Society Convention 145, New York.
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