Proceedings of the Institute of Acoustics 

 

Loudspeaker simulation vs. measurement: Validating matrix array electro-acoustic models

 

Mert Aslantürk, Holoplot GmbH, Berlin, Germany
Evert W. Start, Holoplot GmbH, Berlin, Germany

 

1 INTRODUCTION

 

This paper introduces compact objective methods to quantify the “goodness of fit” between measurement and simulation of matrix array loudspeakers¹ and investigates the scalability of a simulation model for large matrix arrays.

 

An accurate electro-acoustic model provides a fundamental value to loudspeaker manufacturers and system designers. The simulations using these models² can play a crucial role in product and system design, optimization³, and commissioning efforts. Matrix arrays demand increased complexity in their modeling and simulation process.

 

The first challenge is a lightweight loudspeaker model creation that can take the multi-way, multi layered design with densely populated drivers accurately and efficiently into account for fast and reliable computations. The driver directivity and sensitivity data in the model must be valid for not only a single audio module, but it needs to stay valid when hundreds of modules with thousands of drivers are in joint use in a bigger array.

 

The second challenge is caused by the achievable new array geometries with matrix arrays. Larger matrix arrays have significantly different acoustic boundary conditions compared to small arrays. Therefore, modeling of matrix arrays of varying sizes requires modifications in the complex directivity point source simulation (CDPS) model to take the baffle size and edge diffraction effects into account. In conventional CDPS model it is not possible to change the boundary condition of a loudspeaker (I.e., created larger baffle due to the neighboring loudspeaker in an array) after a model is created. For subwoofers, BEM can improve the modeling accuracy but is computationally intensive⁹. Therefore, another method is needed. Using a proprietary Edge Diffraction Model, the effect of array size on the sensitivity and the complex directivity balloons of the drivers can be calculated efficiently. The details of this model are beyond the scope of this paper.

 

Several comparison reports between measurement and simulation can be found in the literature^4,5,6,7,8 for DSP-enabled line arrays. This paper explores the efficiency of the simulation model for remarkably more complex 2D loudspeaker arrays, namely matrix arrays. Matrix arrays of varying sizes (as big as 3 m tall and 3.2 m wide with 1840 individually amplified DSP channels) are measured and simulated using large circular microphone layouts on the horizontal plane. The deviation between the simulation results and measurements is objectively quantified using the methods introduced in this paper.

 

2 MEASUREMENT SETUP

 

2.1 Loudspeaker Arrays

 

Multiple arrays of gradually increasing size are measured to capture the 'baffle effect’ of Matrix Arrays. A rigging frame was used to build and scale the different arrays. The frame was populated using two types of HOLOPLOT X1 audio modules¹⁰. The MD96 audio module has 78 HF and 18 LF drivers and operates within the frequency bandwidth of 100 Hz to 18kHz. The MD80-S is a sub-woofer-integrated version of the MD96. It contains 64 HF, 16 LF, and one subwoofer driver, and it operates within the frequency bandwidth of 30 Hz to 18kHz.

 

To effectively measure the baffle effect and low-frequency beamforming control, the MD80-S Audio Modules are placed in a checkerboard configuration as well as spatially apart from each other to achieve sufficient low-frequency control. The resulting 1×1, 2×2, 3×3, and 5×4 array configurations are shown in the figures below.

 

The bottom edge of the array is rigged at a 5.06 m distance from the floor to maintain an equal distance between reflecting surfaces. The module placement within the rigging frame is defined in a way to reduce the amount of module relocation during assembly.

 

Figure 1: Positions of 1×1 (Sub-woofer integrated audio module), 2×2, 3×3, and 5×4 matrix arrays deployed in the rigging frame.

 

2.2 Beam Configurations

 

2.2.1 Nomenclature

 

Three beam configurations with various directivity characteristics were deployed and measured with each matrix array.

 

The beamforming configuration nomenclature is specified as ArraySize_Vaaa_Hbbb

ArraySize: Number of audio modules in the matrix array (Rows x Columns)

aaa: Vertical opening angle in degrees.

bbb: Horizontal opening angle in degrees.

 

Measured beam configurations are:

1. V008_H010
2. V030_H060
3. V060_H120

 

2.3 Microphones

 

A circular microphone array was used to capture the horizontal directivity pattern of the measured configurations. Seventy-two microphones were placed at a 10 m distance from the array’s center at an angular resolution of 5º. The circular microphone positions in the venue are shown in the figure below.

 

2.4 Venue

 

The chosen venue for the measurement is Messe Leipzig Halle Eins. The venue dimensions are 144×134 meters with a perimeter ceiling height of 16 meters in the middle area.

 

Figure 2: Venue dimensions, microphone placement, and device under test (DUT) position.

 

 

Figure 3: Picture of 5×4 array deployed in the rigging frame.

 

2.5 Post-Processing

 

A logarithmic sweep was used to capture the impulse responses. The sweep duration was about 1 second. Measurement levels were calibrated using a pistonphone calibrator. The recorded signal level was compensated for the maximum input voltage level of loudspeakers at the post-processing stage. This stage is necessary because the simulation results were obtained considering the maximum input voltage level.

 

The simulated impulse responses consider only direct sound in anechoic conditions, i.e., no room reflections are included. To accomplish a direct sound comparison between simulations and measurements in the venue, time windowing is required to remove reflections as much as possible.

 

 

Figure 4: The shape of the applied time window (black line) on an on-axis measurement.

 

The chosen 15ms window length excludes the first arriving strong reflection from the impulse responses while still leaving enough frequency resolution (~70Hz) to analyze the data at 125 Hz octave band.

 

3 GOODNESS OF FIT

 

Various quantitative measures are defined for the deviation between two data sets (e.g., measured and simulated). The data is assumed to be a function of frequency f_k and position r_l (or angle in the case of polar data).

 

The measured and the simulated transfer functions are defined as H_meas(f_k , r_l) and H_sim(f_k, r_l), respectively. It’s assumed that 1/1-octave frequency smoothing has been applied already.

 

3.1 Spatial Standard Deviation

 

The Relative Standard Deviation (RSD) per frequency band k is defined as:

Note that the definition of the RSD is based on Sound Pressure (in Pa) and not on Sound Pressure Level (in dB). This is argued as follows. SPL is a logarithm of the ratio between the actual Sound Pressure and a fixed reference pressure (20 𝜇Pa). Subtracting decibel values may lead to unexpected and undesirable results. In this specific case, calculating the standard deviation based on the SPL would lead to unduly weighting of the deviation at far off-axis positions. For example, having a simulated off-axis value of -70dB (rel. to the on-axis) and a measured value of -40dB (possibly affected by noise) would lead to an exceptionally large error of 30dB, although both sound pressures are extremely low and have a negligible contribution to the overall sound power of the array. The Spatial Standard Deviation includes both the level offset and the variability between the data sets.

 

The Spatial Standard Deviation (in dB) per frequency band k is given by

 

The Total (i.e., broadband) Relative Standard Deviation (RSD_tot) is defined as:

 

 

The Total Standard Deviation (in dB) is given by

 

3.2 Spatial Deviation

 

To calculate the mean signed deviation (i.e., level offset) between two data sets, the Relative Deviation (RD) per frequency band k is defined as:

 

 

The Spatial Deviation (in dB) per frequency band k is given by

 

The Total (i.e., broadband) Relative Deviation (RD_tot) is defined as:

 

 

Total Deviation (in dB) is given by:

 

 

4 RESULTS

 

Polar diagrams show the horizontal directivity pattern of each configuration. The displayed data is normalized with the maximum SPL value across the microphone positions per frequency band separately for simulated and measured data to enable convenient visual comparison. As a result of this normalization, 0 dB most often can be seen at the on-beam position. Polar plots were generated in 1-octave resolution.

 

 

Figure 5: Polar plots of 1x1_V008_H010

 

 

Figure 6: Polar plots of 1x1_V030_H060

 

 

Figure 7: Polar plots of 1x1_V060_H120

 

 

Figure 8: Polar plots of 2x2_V008_H010

 

 

Figure 9: Polar plots of 2x2_V030_H060

 

 

Figure 10: Polar plots of 2x2_V060_H120

 

 

Figure 11: Polar plots of 3x3_V008_H010

 

 

Figure 12: Polar plots of 3x3_V030_H060

 

 

Figure 13: Polar plots of 3x3_V060_H120

 

 

Figure 14: Polar plots of 5x4_V008_H010

 

 

Figure 15: Polar plots of 5x4_V030_H060

 

 

Figure 16: Polar plots of 5x4_V060_H120

 

A good fit between the predicted and the measured data can be seen in the polar diagrams above, both at the front and the back of the arrays. The SSD and the SD for various array sizes, averaged across different beam settings, are shown in Figure 17 and 18.

 

Figure 17: The spatial standard deviation values for all measured configurations. “Overall” values are arithmetic averages of calculated deviation values across all beam settings per matrix array configuration

 

The figure above shows that the calculated overall total deviation values stay consistent across the different matrix array configurations; in other words, the used electroacoustic models are scalable. The mean value of the overall standard deviation for all array sizes is 1.6 dB.

 

Slightly higher spatial standard deviation values are found in 1×1 array configurations. The potential contributors to this deviation are the mechanical tolerances in the measurement setup, the rigging frame, and the non-populated sections surrounding the 1×1 array as seen in Figure 1. The rigging frame used to fly the multiple array size combinations is not included in the simulation model. Any noise originating from the frame due to edge diffraction, rattling, resonances, or loose parts is negatively associated with additional deviation.

 

Overall spatial deviation values further ensure the consistency of simulated results across the configurations, as shown in the figure below.

 

 

Figure 18: Spatial deviation values of all measured configurations

 

The used simulation algorithms implement a diffraction model, a proprietary feature that predicts the effect of the matrix array size (so-called baffle effect) on the array’s radiation pattern. The importance of using the diffraction model with matrix arrays can be seen when a simulation runs without this model. Significant deviations can be seen in the figure below at the frequencies where the diffraction model interacts, 125 Hz, 250 Hz, and 500 Hz bands, especially behind the 5×4 array. This improvement in the lower octave bands when using the diffraction model proves the benefits of its implementation, mostly in two-dimensional arrays. No changes are expected, nor seen in the octave bands where the diffraction model is no longer relevant, i.e., frequencies from 2kHz.

 

Figure 19: Polar plot of 5x4_V008_H010. The diffraction model is de-activated in the simulation

 

5 CONCLUSIONS

 

In this paper, the prediction accuracy and the scalability of the modified CDPS simulation model are validated for matrix arrays. Arrays in four different sizes (the biggest one as big as 3 m tall, 3.2 m wide with 1840 individually amplified DSP channels) are configured with three distinctly different beam characteristics and measured horizontally in a 5° resolution microphone array in a sufficiently large environment.

 

The introduced metrics quantify the goodness of fit between simulations and the measurements. These metrics offer a compact way of checking the deviation across big data sets. They are instrumental in quantifying level offset and the variability between the data sets.

 

The benefits of using the proprietary diffraction model are visible when this one is disabled, and quantitative assessment is done between simulations and measurements. A significant improvement in predictions is achieved when the diffraction model is active, especially behind the matrix arrays.

 

The mean overall standard deviation for all measured configurations is 1.6 dB. This result concludes that the modified CDPS model accurately predicts the performance of matrix arrays regardless of the array size and beam configurations.

 

6 REFERENCES

 

1. E.W. Start, “Loudspeaker matrix arrays: Challenging the way we create and control the sound”, (Reproduced sound) Vol 44, Part 23 (2022).
2. Meyer, D. G., “Computer simulation of loudspeaker directivity., J. Audio Eng., Vol 32(5), 294-315. (1984).
3. G.W.J. van Beuningen & E.W. Start, “Optimizing directivity properties of DSP-controlled loudspeaker arrays”, (Reproduced sound), Vol 22 part 6,17–37. (2000).
4. Feistel, S., Thompson, A. and Ahnert, W., Methods, and limitations of line source simulation, in ‘Audio Engineering Society Convention 125’. (2008).
5. Feistel, S. and Ahnert, W., The effect of sample variation among cabinets of a line array on simulation accuracy, in ‘Audio Engineering Society Convention 127’. (2009).
6. S. Feistel, W. Ahnert, C. Hughes, and B. Olson, "Simulating the Directivity Behaviour of Loudspeakers with Crossover Filters" in ‘Audio Engineering Society Convention 127’. (2007).
7. E.W. Start & G.W.J van Beuningen, “Design and application of DDS-controlled cardioid loudspeaker arrays”, Proceedings of the Insititute of Acoustics (Reproduced sound), Vol 25(8)- (2003).
8. E.W. Start & G.W.J van Beuningen, “Analysis of DDS-controlled loudspeaker arrays by near field acoustic holography”, (Reproduced sound), Vol23(8), (2001).
9. E.W. Start & G.W.J van Beuningen, “Optimisation of DDS controlled loudspeaker arrays using a hybrid PSM-BEM model”, (Reproduced sound) Vol 27(5), (2005).
10. www.holoplot.com