© The Author(s), Published by EDP Sciences, 2026

1 Introduction

Several studies in recent years have focused on the active control of the acoustic field scattered by any surface [1–6]. Some aimed to reduce the detectability of objects buried in the ground or underwater, but another potential application is the creation of anechoic or semi-anechoic rooms which are effective at low-frequency. In this context, a demonstration room was built at the Laboratoire de Mécanique et d’Acoustique (LMA) with the goal of creating a hybrid semi-anechoic room. Figure 1 illustrates the concept: a conventional coating will absorb reflections from the walls and ceiling at medium and high frequencies while an array of microphones and off-the-shelf loudspeakers installed near the walls will allow for the estimation and control of the scattered pressure at the lower frequencies, down to the cut-off frequency of the loudspeakers. In practice active control is intended in the 80–200 Hz range in the LMA room.

Figure 1. A hybrid active/passive semi-anechoic room.

The control strategy envisaged for this room is based on accurately estimating the pressure scattered by a surface, which is achieved by filtering total pressure measurements [7, 8]. The necessary filters are derived from the discretization of an integral representation of the scattered pressure inside the volume. They are independent of the noise sources present and can be obtained by solving an inverse problem generated using a source with a known free-field radiation pattern. Global control of the pressure scattered by the single one reflective wall of a semi-anechoic room was thus successfully implemented experimentally at LMA [9]. This was achieved using a set of pressure microphones spread over the reflective wall. Estimation and control at these microphones ensured global reduction of the 1-wall scattered pressure. The case of the 5 absorbing walls of a semi-anechoic room poses one more difficulty: control of the pressure close to these walls does not theoretically involve global control of the pressure inside the room at the Dirichlet eigenfrequencies of the volume delimited by the surface covered by the surface minimization microphones and its image with respect to the reflective wall; indeed the strategy known as Boundary Pressure Control (BPC) on a closed surface has singular frequencies [10]. However previous 2D simulations [11] have suggested how to deal with these singular frequencies. It was also shown that it is possible to estimate the scattered pressure sparsely, using a small number of total pressure measurements using well-chosen microphones on all the reflective surfaces.

This article presents 3D numerical simulations of the LMA room using a modal model and monopoles, which was used to determine the placement of the transducers, as well as a Finite Element (FE) model that will be used later to model sound sources and sound-absorbing materials on the room walls and ceiling more realistically. The article also addresses the sparse estimation of the scattered pressure, with the implementation of real-time control in mind.

The remainder of the paper is organized as follows. Section 2 presents the principles of the proposed active control strategy, formulated using a modal model of acoustic monopoles inside a lightly damped rectangular cavity. A comprehensive simulation in the frequency domain, using this model, of control in the LMA room equipped with 36 wall-mounted sources and 59 microphones is presented in Section 3. The efficient transducer configuration selected using the modal model is used to design the FE model which is presented in Section 4. This model provides verification of the control method, involving more realistic control sources, and offers numerical results of the room frequency response with spatial visualizations of the achieved control effect. Section 5 introduces the sparse estimation framework for reconstructing the scattered pressure using optimal group-lasso and sub-optimally oriented least-squares minimization techniques. The objective of this section is to assess the minimum number of microphones required to accurately estimate the scattered pressure at each minimization point. Section 6 concludes the paper with final remarks and outlines directions for future work.

2 Control strategy and modal room model

2.1 Global control of the scattered pressure

The control strategy developed at LMA and its implementation in several situations have been presented in detail in [7–9, 12–14]. The key idea is that the pressure p _sca scattered by a surface S at point r can be written at every frequency as a surface integral over S of total pressure p _tot using the Green’s function G of the propagation medium without surface S. If the normal vector to surface S points outward from the propagation medium, and r ₀ denotes the location of the integration point, the scattered pressure is given by:

$$\begin{array}{c}\hfill p_{\mathrm{sca}}(\mathbf{r})={\int \int }_S[G(\mathbf{r}|{\mathbf{r}}_0)\frac{\partial p_{\mathrm{tot}}({\mathbf{r}}_0)}{\partial n_0}-p_{\mathrm{tot}}({\mathbf{r}}_0)\frac{\partial G(\mathbf{r}|{\mathbf{r}}_0)}{\partial n_0}]\mathrm{d}{S}_0.\end{array}$$(1)

With a suitable microphone mesh, this integral can be accurately approximated by the finite sum:

$$\begin{array}{c}\hfill p_{\mathrm{sca}}(\mathbf{r})\simeq \sum _{k=1}^K{g}_k{p}_{\mathrm{tot}}({\mathbf{r}}_k),\end{array}$$(2)

where the g _k coefficients, resulting from the Green’s function and generalized impedances, do not depend on the acoustic sources inside the volume [13]. They can therefore be identified, for example, by solving an inverse problem generated using a “reference” source, with known radiation in the absence of surface S, which is moved across a targeted measuring volume. If, at point r and angular frequency ω, p _tot designates the vector of pressures for all the positions of the reference source, p _dir the vector of pressures that this source would generate without surface S, and P _K the matrix in which each column is the pressure measured on the K microphones which mesh surface S, then the vector g of coefficients in equation (2) can be obtained by minimizing the least-square index:

$$\begin{array}{c}\hfill J_{\mathbf{g}}(\omega )={\parallel \mathbf{g}{\mathbf{P}}_K-({\mathbf{p}}_{\mathrm{tot}}-{\mathbf{p}}_{\mathrm{dir}})\parallel }_2^2+\lambda {\parallel \mathbf{g}\parallel }_2^2,\end{array}$$(3)

in which penalty coefficient λ weights the 2-norm of vector solution g. Several numerical methods allow the efficient solving of this linear least-square problem. For the simulations presented in this paper, the Matlab function lsqminnorm, giving g for penalization coefficient λ approaching 0, was used. It implements QR factorization of matrix P _K , which is numerically faster than explicit pseudo-inversion with Singular Value Decomposition.

Once g is obtained at all frequencies of interest, the pressure scattered by surface S in the presence of an unknown source can be computed by linear filtering of total pressure measurements at the microphone mesh. Empirical evidence has shown that the pressure scattered by a wall can be accurately estimated in its vicinity, where the ratio of scattered pressure to total pressure is maximum. However, this is a more challenging task further away from a wall because the difference p _tot − p _dir is more sensitive to errors [15].

In the case of a semi-anechoic room, it is therefore possible to estimate accurately the low-frequency scattered pressure in the vicinity of the ceiling and walls. An array of microphones on the walls, and their virtual symmetric counterparts with respect to the floor, form the mesh of a closed surface. Canceling the scattered pressure on this surface implies cancelling it globally in the inner volume, except at the Dirichlet resonance frequencies [10]. With a finite set of minimization points on the surface, several solutions can be envisaged to regularize this global control strategy, known as Boundary Pressure Control (BPC), around these resonance frequencies: some asymmetry and randomness can be added to the microphone locations as in [11]; additional microphones can be inserted inside the initial mesh, which is analogous to the Combined Helmholtz Integral Equation Formulation (CHIEF) method for regularization of Boundary Element Methods [16].

From a practical point of view, installing the control sources over the walls (close to the scattering sources) and the minimization microphones close to the lining, allows to keep the control setup away from the measuring volume where an experiment may then be installed conveniently.

2.2 Modal modelling of the LMA room

Figure 2 shows the 5.34 × 4.22 × 2.77 m³ room constructed at LMA and equipped near the walls and ceilings with 36 off-the-shelf loudspeakers and 59 inexpensive microphones. Absorbing materials will later cover the active device, ensuring the room is semi-anechoic at higher frequencies.

Figure 2. The LMA room with wall loudspeakers and microphones for control of low-frequency reflections. Red ovals have been drawn around a few microphones.

A Finite Element model of the LMA room was implemented using COMSOL [17]. Measuring Frequency Response Functions in the LMA room allowed for the adjustment of the wall impedances used as input parameters for the FE computation [14]. Alternatively, a simple model of the LMA room, enabling fast simulation of a control configuration, was developed. This model treats all sources as monopoles and expands the acoustic field using the Neumann eigenmodes ψ _n and eigenfrequencies ω _n /2π, while also incorporating the damping ratios ξ _n computed using the FE model. The pressure p _tot produced at pulsation ω and room location r by a monopole at location r ₀ with flow rate q is thus expressed as:

$$\begin{array}{c}\hfill p_{\mathrm{tot}}=\rho c^{2}j\omega q\sum _{n=0}^N\frac{{\mathit{\psi }}_n^t(\mathbf{r}){\mathit{\psi }}_n({\mathbf{r}}_0)}{({\omega }_n^2+2j{\xi }_n{\omega }_n\omega -{\omega }^2){\int }_V{\mathit{\psi }}_n^2\mathrm{d}V},\end{array}$$(4)

where c is the sound speed, ρ is the density of air, V is the volume of the room and superscript ^t denotes matrix transposition.

The direct pressure (which would be observed without ceiling and walls) can be expressed for a monopole source at location r ₀ by adding the contribution of an image source at r ₀′ resulting from the reflection over the rigid ground:

$$\begin{array}{c}\hfill p_{\mathrm{dir}}=\rho j\omega q(\frac{e^{{}_{-j\frac{\omega }{c}{r}_0}}}{4\pi r_0}+\frac{e^{{}_{-j\frac{\omega }{c}{r}_0^{\prime }}}}{4\pi r_0^{\prime }})·\end{array}$$(5)

The pressure scattered by the ceiling and walls p _sca is therefore obtained as:

$$\begin{array}{c}\hfill p_{\mathrm{sca}}=p_{\mathrm{tot}}-p_{\mathrm{dir}}.\end{array}$$(6)

Using c = 340 m/s for the simulations below, the 1012 usual modes of a rectangular cuboid with rigid boundary (see e.g. [18], p. 582) with eigenfrequencies up to 500 Hz have been taken into account in this model. Of these, 82 have an eigenfrequency of less than 200 Hz, which corresponds to the upper frequency initially targeted.

3 Modal simulation of control in the frequency domain

3.1 Description of the active control configuration

The room modal model enables a fast evaluation of the control achieved by a given transducer configuration. It has therefore been used to test many arrangements of controls source and microphones. We now present the modal simulation of a configuration considered as efficient. This simulation implements optimal control computed in the frequency-domain without causality constraints: at every frequency, the control signals supplied to the wall sources are obtained by minimizing the mean-square of the estimated scattered pressures at the microphones. This gives the best possible control in the presence of perfectly predictable acoustic signals and, in practice, control would be less efficient with random signals. The transducer configuration studied here is partly shown in Figure 3. It involves:

Figure 3. An optimized transducer set-up for simulation of control in the LMA room: primary source whose scattered radiation must be reduced ; ceiling control sources ; front side control sources ; rear side microphones °; locations of the reference source for generating the inverse problem leading to the scattered pressure estimator .

• a primary monopole source on the floor at the room center, for which the active system should minimize the pressure scattered by the walls and ceiling in a measurement zone half the size of the room;

• 36 secondary (control) monopole sources spread over the walls;

• 59 microphones at points where total pressure will be measured and the scattered pressure estimated and minimized. This set involves 56 microphones spread over the walls, as seen Figure 2. They are placed about 40 cm from the walls, leaving room for future passive sound-absorbing material. One microphone was added at mid-height in the center of the room. This microphone regularizes the BPC first singular frequency at around 72 Hz. It is not as efficient as it would be if located on the floor but we wanted to keep it outside the measurement volume. Two more microphones were located on the floor outside the measurement zone, opposite the door. They improve control performance, particularly around 200 Hz;

• positions within the volume, one third the size of the room, where the primary sound sources to be measured may be located. A reference monopole at these locations generates the data for the inverse problem related to equation (3) whose solution gives the coefficients g which maps the total pressure signals to a scattered pressure error signal. These 24 source locations differ from the one used for the primary source in the simulations;

• observation points, not shown in Figure 3, that mesh the central measurement volume above the room floor. These points are used to assess the performance of the boundary scattered pressure control over the measurement volume.

3.2 Unconstrained optimal control

Let G be the matrix in which each row g relates the total pressure at all the microphones to the scattered pressure at one microphone, as introduced in Section 2.1. If ${\mathbf{p}}_{\mathrm{tot}}^{(1)}$ denotes the total pressure from a primary source at all the microphones, $\mathbf{G}{\mathbf{p}}_{\mathrm{tot}}^{(1)}$ is an estimate of the primary scattered pressure at all the microphones. If H denotes the transfer matrix from the control source inputs u to the microphone outputs, optimal active control can then be obtained at one frequency by minimizing the sum of the primary scattered pressure and the secondary total pressure:

$$\begin{array}{c}\hfill J_{\mathbf{u}}={\parallel \mathbf{G}{\mathbf{p}}_{\mathrm{tot}}^{(1)}+\mathbf{H}\mathbf{u}\parallel }_2^2+\lambda {\parallel \mathbf{u}\parallel }_2^2.\end{array}$$(7)

In practice the Matlab function lsqminnorm was used to compute the optimal control signals u, with penalization coefficient λ approaching zero, as for the computation of vectors g.

Note that calculating the secondary source signals independently at each frequency does not take into account the causality constraint. Therefore the results below could be approached in real-time only for control of pure tone noises or by using feedforward control with a perfectly correlated and anticipating reference signal [19]. In the case of a primary noise source which is driven by an electrical signal, such as a loudspeaker, this signal can be made as anticipating as required by introducing delay in the primary source input. In other cases, the selection of an appropriate reference signal for feedforward control will be an additional challenge. The optimal control theorical results given below may therefore be difficult to mimic in real-time in the case of a complex wideband source (e.g. such as jet noise), as usual in feedforward noise control.

3.3 Control results

Figure 4 shows the spatial average, over the 910 points in the measurement area, of the modulus of the total pressure divided by the direct pressure from the primary source. This ratio measures the deviation of the total field in the measurement area from the field that would be produced by the primary source in a half-free space. It has similarities with the indicator used in ISO standard 26101 which requires a ±2.5 dB maximum deviation at low-frequency for the certification of a semi-anechoic room. The figure compares the results obtained without control, with control of the scattered pressure at the observation microphones inside the measurement volume, and with control of the scattered pressure at the surrounding minimization microphones either directly provided by the model or estimated using matrix G that maps the total pressures to the scattered pressures. The horizontal dashed lines in Figure 4 indicate a ±2.5 dB deviation from perfect semi-anechoic conditions. The vertical dot lines in Figure 4 give an indication of the first singular frequencies of the BPC problem in the room. These have been computed as the eigenfrequencies of a 4.54 × 4.42 × 4.74 m³ rectangular cuboid with zero-pressure boundary, which is the volume enclosed by a mesh of microphones placed 40 cm from the walls and ceiling and its mirror image with respect to the floor. Since not all wall microphones are located exactly 40 cm from the walls of the LMA room, these singular frequencies are approximate.

Figure 4. Spatial average, over the room measurement zone, of the RMS of total pressure divided by direct pressure.

It can be observed that:

• without control, the field deviation is governed by room resonances up to 200 Hz, and the total pressure deviates significantly from the direct pressure. Beyond 220 Hz, the higher modal density and the estimated damping rate (around 2%) lead to a total pressure closer to the direct pressure;

• when the control sources are operated to minimize the mean square of the scattered pressure at the 910 locations within the measurement volume, the field deviation is nearly 0 dB up to 175 Hz and then fluctuates within the range of ±0.5 dB thereafter. Wall sources would therefore be capable of very effectively reducing the scattered pressure in the measurement area if it could be directly sensed;

• when minimizing the mean square of the exact scattered pressure over the 59 microphones, or its estimation using equation (2), the field deviation in the measurement area is less than ±2.5 dB below 200 Hz. Therefore control appears to be effective and global at low frequencies. Furthermore, control of the estimated scattered pressure is very similar to control of the exact scattered pressure: the linear estimator derived from the inverse problem using the reference monopole therefore seems adequate.

Figure 5 displays the RMS average of the secondary source flow rates divided by the primary source flow rate. The required output level for the secondary sources appears to be approximately 15 dB lower than that of the primary source, which means that real control sources should be carefully designed to deal with high-level low-frequency primary sources.

Figure 5. Average of the secondary monopole flow rates divided by the primary monopole flow rate.

4 FE-based simulation of the active control method

4.1 Implementation of the FE model

A three-dimensional FE model of the LMA active semi-anechoic room was developed using COMSOL Multiphysics 6.3 for the purpose of verifying the control results. The computational domain is represented by a rectangular cavity with a rigid floor, while all other surfaces are assigned frequency-dependent impedance boundary conditions [17] based on data estimated in the physical room. A total of 36 secondary loudspeakers are installed at 10 cm from the wall and ceiling, as illustrated in Figure 6. Each speaker is modeled as a rigid circular piston subject to a normal acceleration boundary condition, mounted on a rigid enclosure. Because of the enclosure, which is one of the elements of the installation that the FE model will help optimize, the control sources are not located exactly at the ceiling and on the walls, unlike the control monopoles in the modal model. The primary source is also not located exactly at the floor in the FE model.

Figure 6. Model geometry for calculating the total pressure field in the room (cross-sectional view).

The complex amplitude of the loudspeaker imposed acceleration is defined as the control variable, determined from equation (7) at each angular frequency ω. The process necessitates the preliminary determination of the transfer functions between the successively activated secondary sources and the microphones. This methodology facilitates the consistent evaluation of the active semi-anechoic room across a range of frequencies, thereby enabling direct comparison between the uncontrolled and controlled states.

Simulations are performed within the frequency range 20 Hz to 250 Hz, employing a frequency-domain parametric sweep with a step size of 0.1 Hz. A quadratic finite-element discretization is employed, with the mesh density being selected to guarantee a minimum of six elements per wavelength at the highest computed frequency. The primary source is modeled as a monopole with a flow rate of 10⁻⁴ m³/s, positioned at the center of the room at a height of 0.2 m.

4.2 Computed results in the measurement volume

Figure 7 shows the room frequency response at an observation microphone located at the coordinates (3 m, 2 m, 0.4 m) within the controlled volume, with the control system off and on. In the uncontrolled room, distinct standing wave behavior can be observed, characterized by marked modal peaks. The total pressure field is mainly the result of the pressure field scattered on the walls. When the control is switched on, the modal peaks can no longer be distinguished, and the room frequency response at the observation point is approaching that calculated for a source radiating in a semi-free field.

Figure 7. Computed frequency response at a selected observation microphone in the measurement volume, with the control system off and on. Also shown are the direct pressure in a free half-space, p dir, and the scattered pressure p sca obtained using equation (6).

4.3 Pressure field visualisation with Boundary Pressure Control off/on

The spatial visualization of the pressure field at selected resonance frequencies further illustrates the control effect. Figures 8–10 illustrate the total pressure distribution at the selected room modes (200), (201) and (421). In the uncontrolled room, the spatial mode shapes are consistent with the expected low-order eigenfunctions of a rectangular cavity. The presence of loudspeakers and walls with very low damping merely modifies this behavior. When the control is on, the pressure nodes and anti-nodes areas that are characteristic of rigid-wall standing waves are mitigated, resulting in a more spatially uniform field of lower amplitude. The cross-sectional view of Figures 8–10 allows to distinguish the spherical spreading (1/R decrease of pressure) within the central (measuring) volume, while this behavior is distorted for locations closer to the walls. Therefore, activating the control system has a significant impact on the room modal features. The amplitude of the modal peaks in the low-frequency range has decreased, which indicates that the secondary sources are effectively reducing the pressure field scattered by the walls. Even in this lightly damped scenario, the combination of passive wall impedance and optimized active control can approximate semi-anechoic conditions over a broad low-frequency range. However, as Figure 10 shows, the effect of active control diminishes as the mode order increases. Consequently, the pressure field no longer resembles that of a source that is purely radiating in a semi-free field. This effect is expected to be mitigated by combining the active control system with a layer of sound-absorbing materials.

Figure 8. Visualizations of the pressure field amplitude for room mode (200) at 64.4 Hz, with the control system switched off (left) and switched on (right).

Figure 9. Visualizations of the pressure field amplitude for room mode (201) at 90.0 Hz, with the control system switched off (left) and switched on (right).

Figure 10. Visualizations of the pressure field amplitude for room mode (421) at 164.7 Hz, without (left) or with control (right).

5 Sparse estimation of the scattered pressure

The previous simulations were based on an estimation of the scattered pressure at 59 minimization microphones through linear filtering of the total pressure at all these microphones. In practice, this estimation step would dramatically increase the amount of real-time computation required for adaptive control using the usual Filtered-Reference Least Mean Square algorithm (FxLMS). The use of a Filtered-Error LMS [19] instead would reduce the computational burden because the estimation of the scattered pressure can be combined off-line with the usual filtering of the error signals [8], but this approach introduces delay in the control and it requires longer Finite Impulse Response filters. An alternative is to filter a reduced number of pressure signals for the estimation of each scattered pressure minimization signal.

A previous 2D study [11] showed that the scattered pressure at each minimization microphone could be accurately estimated from a small number of total pressure measurements, provided they were taken on all the walls and ceiling of the room and not just in the vicinity of the estimation point. In 3D, picking up a subset of microphones from geometric considerations is not as easy as in 2D and a systematic way of selecting microphones for sparse estimation is desirable. Two optimization processes have been investigated for application in the LMA room. Because of the required computation time, only the modal model has been used for the microphone selections discussed below.

5.1 Lasso and group-lasso optimization

Deep Learning studies have given rise to numerical algorithms designed to select optimal subsets in large sets of variables available for linear prediction. For example lasso optimization, introduced originally in [20] with LASSO as the acronym for Least Absolute Shrinkage and Selection Operator, minimizes the index:

$$\begin{array}{c}\hfill J_{\mathbf{x}}=\frac{1}{2}{\parallel \mathbf{A}x-\mathbf{b}\parallel }_2^2+\lambda {\parallel \mathbf{x}\parallel }_1.\end{array}$$(8)

Using a 1-norm for the penalization term in this equation tends to nullify the components of solution vector x which do not contribute most to the minimization, and the weight λ effectively controls the number of non-zero components of the solution vector.

Lasso optimization, which deals with real numbers in equation (8), cannot be directly applied to the selection of pressure signals in the LMA room because the vector g which minimizes equation (3) index at angular frequency ω is complex. Furthermore, for practical implementation, the pressure signals selected for the estimation of one scattered pressure signal must be the same at all frequencies. With complex data one may consider the index with real numbers analog to equation (8):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill J_{\mathbf{x}}& =\frac{1}{2}\parallel \left(\begin{array}{cc}\mathfrak{R}(\mathbf{A})& -\mathfrak{I}(\mathbf{A})\\ \mathfrak{I}(\mathbf{A})& \mathfrak{R}(\mathbf{A})\end{array}\right)\left(\begin{array}{c}\mathfrak{R}(\mathbf{x})\\ \mathfrak{I}(\mathbf{x})\end{array}\right)\hfill \\ \hfill & -{\left(\begin{array}{c}\mathfrak{R}(\mathbf{b})\\ \mathfrak{I}(\mathbf{b})\end{array}\right)\parallel }_2^2+\lambda \sum _{k=1}^K\left|{\mathbf{x}}_k\right|,\hfill \end{array}\end{array}$$(9)

where the penalization term is the sum of the modulus of each complex component x _k of vector x. This penalization term is therefore equal to the 1-norm of a vector whose components are the 2-norm of the vector whose real and imaginary parts of x _k constitute the two components. Optimization with such a penalization term is referred to as a group-lasso problem, for the solution of which efficient numerical methods have also been developed [21]. Lasso and related methods have been used for several years for source selections in acoustics, see e.g. [22].

A comparison of equations (8), (3) and (9) shows that the single-frequency minimization index for estimating the scattered pressure at a given point can be written as a group-lasso problem by using A = P _K ^t , b = (p _tot − p _dir) ^t and x = g ^t .

The computation of vector g which minimizes a multi-frequency combination of indexes such as in equation (3) can also be formulated as a group-lasso problem, although much more data must be processed than in the one-frequency case. However, because the complexity of the acoustic field in a room increases with frequency, a set of pressure sensors which enable estimation of the scattered pressure at a high frequency will most likely be adequate to estimate it at lower frequency. A single high-frequency group-lasso optimization may therefore be adequate for the selection of a subset or pressure signals to estimate one scattered pressure signal at all the lower frequencies.

5.2 Sparse estimation using group-lasso

For the simulations leading to the results given in this section, optimizations using the group-lasso method were performed at 150 Hz, a frequency below which 41 modes resonate. The control in the LMA room is intended for frequencies of up to 200 Hz, but it appeared that microphone subsets derived at 200 Hz led to poor sparse estimation of the scattered sound pressure at 200 Hz and below. On the contrary, subsets derived at 150 Hz led to good estimation up to 150 Hz and estimation good enough for control from 150 to 200 Hz.

Because the maximum number of microphones selected by group-lasso is automatically limited by the number of reference source positions used to generate the minimization index, 6 × 5 × 3 = 90 locations were also used here, instead of the 24 locations shown in Figure 3. In this way, monitoring parameter λ in equation (9) allows to derive estimation sets with possibly more than 24 microphones.

Figure 11 displays the exact scattered pressure at one randomly chosen microphone and its estimates using the filters resulting from group-lasso optimization on data generated with the reference source when the weight factor λ in equation (9) varies by several orders of magnitude. The weight factor effectively monitors the number of total pressure measurements which are taken for the estimation. Using more than 9 microphones provide decent estimates of the scattered pressure up to 220 Hz. Using only 4 microphones leads to a significantly worse estimate. For this simulation the primary source was a monopole with a 10⁻⁴ m³/s flow-rate.

Figure 11. Exact scattered pressure at microphone #3 and its estimates using group-lasso with various weight factors.

Global control in the room requires the derivation of microphone sets for estimating the scattered pressure at each of the minimization points. Unfortunately, the number of microphones selected at 150 Hz with the same group-lasso weight factor varies significantly depending on the minimization point considered. As a result, to calculate a control with a number of microphones that varies little between points, we need to adjust the value of λ for each minimization point. Figure 12 shows the ISO 26101-like control result when parameter λ has been adjusted “by hand” at each of the 59 minimization points, so that the number of microphones selected for estimation varies little from point to point, around 15 microphones. The vertical dotted lines again correspond to the critical frequencies of the wall control problem, see [11]. In almost the whole frequency range, using about 15 signals for estimation at each minimization location has performances similar to the ones obtained using all 59 signals. This figure confirms that, for global control up to 220 Hz, the scattered pressure can be estimated using a reduced number of microphones. However the group-lasso optimization does not allow this number to be monitored straightforwardly for all minimization points simultaneously. In practice, achieving good convergence of the group-lasso solver required careful attention, which prompted the search for alternative methods of selecting sensor subsets.

Figure 12. Spatial average, over the room measurement zone, of total pressure divided by direct pressure with control of the group-lasso estimation of the scattered pressure on the microphone array.

Since the subsets of microphones used for sparse estimation of the diffuse pressure vary from one minimization microphone to another, it is possible that, in the end, all microphones will be needed to estimate the whole set of error signals. However, the goal of sparse estimation is not to reduce the number of microphones; these are all wired inside the LMA room and could be used for real-time computation. The goal is to reduce the number of pressure signals that need to be filtered for the estimation. Using different subsets of microphones increases the complexity of a real-time calculation program, but not the amount of computation required.

5.3 Estimation using oriented Least Square

The minimization of equation (3) least square index leads to a solution vector g of which the component modulus indicate the weights put on the microphone signals for the scattered pressure estimation. A subset with a given number of microphones can therefore be selected at one frequency by picking up the microphones with the largest weights. Unlike the group-lasso approach, this microphone selection is not optimal because it does not result from an index minimization. However the microphones that contribute most to estimation using all the microphones are good candidates for estimation with a reduced number of microphones. A second least square optimization using only these selected microphones then gives the optimal non-null components of vector g. In contrast to group-lasso, the number of microphones required can be easily assigned to each minimization point without the need for iterative computation.

Since the subset of microphones that contributes most to minimizing the error index when all microphones are used is not necessarily the one that, with the same number of microphones, would actually minimize that index, this method results in a suboptimal subset rather than the truly optimal one. Figure 13 displays the simulated control results when the scattered pressured at the minimization points is estimated from only the 5 microphones contributing most to the least square estimation with 59 microphones at 150 Hz. Control with 5-microphone estimation for each minimization signal has comparable performance to control with 59-microphone estimation.

Figure 13. Spatial average, over the room measurement zone, of total pressure divided by direct pressure with control of the oriented least square estimation of the scattered pressure on the microphone array.

The optimization methods above were tested with many numbers of microphones. As Figure 11 shows, group-lasso may lead to poor estimation of the scattered pressure when less than a dozen microphones are selected. On the contrary the selection based on least-square leads to a good estimation as soon as more than 5 microphones are used. Figures 12 and 13 were intended to show control results with comparable estimation quality with both selections, hence the respective numbers of microphones selected for the figures.

6 Conclusion

The present article presented the numerical simulation of the semi-anechoic room currently under development at LMA. One objective of the simulations was firstly to show that the sets of microphones and secondary sources in the room allowed for accurate estimation and efficient active control of the pressure field scattered by the walls and ceiling. A second objective was to test methods to select subsets of microphones for sparse estimation of the scattered pressure in order to reduce the number of pressure signals to be filtered for the real-time computation of error signals. A three-dimensional analytical model with realistic modal damping ratios was used for the room simulation and the evaluation of two sparse estimation methods for reconstructing the wall-scattered pressure field. The numerical results obtained using group-lasso and least squares regularization techniques show that pressure signals are enough to estimate and control the wall-scattered pressure in the room at each of the 59 minimization microphones. In the end data computed using 36 monopole point sources and 59 microphones positioned near the walls and ceiling indicate that control should allow to approach semi-anechoic conditions below 200 Hz. This result was also confirmed by FE simulations, which were carried out with baffled 36 pistons and 59 microphones using about the same locations throughout the room as in the analytical model.

At the time of writing, the experimental estimation of frequency response functions between all the sources and the microphones is being conducted in the LMA room. This will allow to simulate active control using experimental data as opposed to the room modal model. A series of offline control experiments are to be conducted in the actual room. These will include replay experiments, using primary and control sources, which will be monitored with pre-computed signals. Meanwhile, FE simulation should help to optimize the control sources and their coupling with the passive lining of the room, for a variety of primary sources. The last step will be to implement real-time adaptive control of the scattered field inside the room; this is a longer-term objective.

Funding

This work received support from the french government under the France 2030 investment plan, as part of the Initiative d’Excellence d’Aix-Marseille Université – AMIDEX AMX-23-REC-COFIRE-CG-08

Conflicts of interest

Authors declared no conflict of interests.

Data availability statement

Data are available on request from the authors.

References

All Figures

	Figure 1. A hybrid active/passive semi-anechoic room.
In the text

	Figure 2. The LMA room with wall loudspeakers and microphones for control of low-frequency reflections. Red ovals have been drawn around a few microphones.
In the text

	Figure 3. An optimized transducer set-up for simulation of control in the LMA room: primary source whose scattered radiation must be reduced ; ceiling control sources ; front side control sources ; rear side microphones °; locations of the reference source for generating the inverse problem leading to the scattered pressure estimator .
In the text

	Figure 4. Spatial average, over the room measurement zone, of the RMS of total pressure divided by direct pressure.
In the text

	Figure 5. Average of the secondary monopole flow rates divided by the primary monopole flow rate.
In the text

	Figure 6. Model geometry for calculating the total pressure field in the room (cross-sectional view).
In the text

	Figure 7. Computed frequency response at a selected observation microphone in the measurement volume, with the control system off and on. Also shown are the direct pressure in a free half-space, p dir, and the scattered pressure p sca obtained using equation (6).
In the text

	Figure 8. Visualizations of the pressure field amplitude for room mode (200) at 64.4 Hz, with the control system switched off (left) and switched on (right).
In the text

	Figure 9. Visualizations of the pressure field amplitude for room mode (201) at 90.0 Hz, with the control system switched off (left) and switched on (right).
In the text

	Figure 10. Visualizations of the pressure field amplitude for room mode (421) at 164.7 Hz, without (left) or with control (right).
In the text

	Figure 11. Exact scattered pressure at microphone #3 and its estimates using group-lasso with various weight factors.
In the text

	Figure 12. Spatial average, over the room measurement zone, of total pressure divided by direct pressure with control of the group-lasso estimation of the scattered pressure on the microphone array.
In the text

	Figure 13. Spatial average, over the room measurement zone, of total pressure divided by direct pressure with control of the oriented least square estimation of the scattered pressure on the microphone array.
In the text