On the Optimisation of Multi-Degree-of-Freedom Acoustic Impedances of Low-Frequency Electroacoustic Absorbers for Room Modal Equalisation

Low-frequency electroacoustic absorbers have recently been developed as a solution for the modal equalisation. Firstly investigated in waveguides, the technique consists in matching the acoustic impedance at a closed-box loudspeaker diaphragm to the characteristic acoustic impedance of air. Extending the results in a duct to rooms brings up several challenges. Some parameters, such as the position and orientation of absorbers, the total area, as well as the acoustic impedance achieved at the diaphragms may influence the performance, especially in terms of modal decay time reduction. In this paper, the optimal values of a purely resistive acoustic impedance at an absorber diaphragm, whose area varies, are first investigated under normal incidence and grazing incidence in a finite-length waveguide. The optimal acoustic resistance values are then investigated for a given position, orientation, and total area of absorbers in rooms of different size. From these results, the target acoustic impedances with multiple degrees of freedom are defined with a view to assign to the absorber diaphragms. These impedances are then optimised from a global criterion, so that these impedances approach at best the different optimal resistance values found to minimise the modal decay times. Finally, an experimental evaluation of the performance of the electroacoustic absorber in a waveguide is provided.


Introduction
Room modes cause unevendistributions in space and frequencyo ft he sound field and alter the temporal acoustic response, resulting in long decay times [1].This effect is particularly significant in the low-frequencyrange, where the modal density is low, and prevents suitable reproduction and perception of the musical content.Different strategies have been investigated to address this problem.Conventional passive absorbers are mainly used to reduce high-frequencyr eflections, butt heya re too bulky and not efficient enough for the lowfrequencies [2].Optimal room ratios [3,4,5] and optimal placements of one or multiple sound sources [6,7,8,9] were also investigated to reduce the audible effects caused by the resonances.Active corrections have receivedm uch attention in the last decades for the low-frequencyr oom equalisation.Equalising the frequencyresponse at asingle listen-Received8June 2017, accepted 9October 2017.
ing position might cause ad egradation of the frequency responses at other locations.With multiple-input/-output techniques, the equalisation zone can be significantly extended [10,11,12].Several local and global criteria were proposed, such as the minimisation of the potential energy [13], the minimisation of the sound power [14], as well as the minimisation of the modal decay times [15].For rectangular rooms and symmetrical loudspeaker arrangement, the equalisation can also be achievedb ys imulating ap rogressive plane wave,t hanks to secondary loudspeakers located at the opposite wall of the sound sources, with error sensors [16,17], or only with appropriate delay and gain [18].Nevertheless, these control methods may be costly and time consuming, and complicated to implement in rooms of irregular shape, or theym ight require ac alibration as soon as the furniture is moved.
Another approach is the active absorption through the control of acoustic impedances.The concept of electroacoustic absorber represents an alternative solution for the modal equalisation.First developed in waveguides, the technique consists in matching the specifica coustic impedance at aloudspeaker diaphragm to the characteris-tic specificacoustic impedance of air with sensor [19], or without sensor [20].An efficient sound absorption may be achievedoverabroad frequencyrange, resulting in asignificant damping of the first modes.Since it is obviously not realistic to coverall the walls of aroom, these electroacoustic absorbers should have aphysical effective surface area much smaller than the total area of the room.The optimisation of the impedance locations on the walls of an acoustic cavity wasi nvestigated in [21], to minimise the sound levelgenerated by avelocity source.Analysing the distribution of the sound field in rooms with agiven geometry makes it already possible to knowwhere to place the absorbers for maximal performance (preferably in corners for rectangular rooms for example).Depending on the location and area of the absorber relative to the total area of the domain, the target specificacoustic impedance may differ from the characteristic specifica coustic impedance of air.The optimal acoustic impedance of absorbers under grazing incidence in flowd ucts wasi nvestigated from a hybrid passive/active impedance control based on the pressure release behind ar esistive layer [22,23].The target acoustic impedance should be determined from ar epresentative criterion that maximises the performance of the electroacoustic absorber for the modal equalisation at any location.Such aq uantity could be deriveds oa st om inimise the dynamics of the sound pressure levelatdifferent locations as in [11].However, with ar easonable number of frequencyresponses that are dependent on the locations of the source and listener,i tw ill only give an approximate value of the dynamics of the sound pressure level.Since the modal decay times are critical for the perception of low-frequencyp roblems [24], this quantity may also be used as ac riterion as in [15].The main advantages are that the modal decay time is related to the corresponding eigenfrequency, thus the damping coefficient of the mode.Therefore, it is independent of the locations of the source and listener in the room, and it is easily derived from analytical solutions or simulations.The modal decay times can be approximated analytically in rectangular rooms [25] or in rooms of irregular shape [26,27] with different sound absorption coefficients on walls.As aresult, the minimisation of the modal decay times is quite representative of the performance for the modal equalisation in rooms.This global criterion will be used in the following, in order to findthe target acoustic impedances that will be assigned to the electroacoustic absorber diaphragms.
The paper is organised as follows: Section 2investigates the optimal values of apurely resistive acoustic impedance at an absorber located at the end of aduct of finite length (normal incidence), with varying area.Then, the absorber is located along the wall of the duct near one of both terminations (grazing incidence).Section 3i nvestigates the optimal acoustic resistance values in rooms of different size, where the area and orientation of absorbers are fixed.In Section 4, multi-degree-of-freedom (MDOF)target acoustic impedances are defined and optimised, so that these impedances approach at best the different optimal acoustic resistance values found to minimise the modal decay times.Finally,i nS ection 5, an experimental evaluation of the performance of the electroacoustic absorber in awaveguide is provided, by applying the hybrid sensor-/shunt-based impedance control developed in [19].

Target acoustic impedance in duct
In this section, we study the effects of the area and orientation of an absorber on the value of the target acoustic impedance, with aviewtominimise the modal decay time of the first longitudinal modes of aclosed-closed duct.In the following, the air is considered as alossless medium of propagation.We denote the characteristic specificacoustic impedance of air by Z c = ρc,where ρ is the density of air, which is equal to 1.20 kg•m −3 at 294 K, and c is the sound speed in air,w hich is equal to 343.86 m•s −1 at 1atm and 59 %ofrelative humidity.

Uniform boundary condition under normal incidence
In the general case, where asound plane wave is directed toward an absorber under normal incidence, the reflection coefficient is expressed as where ω = 2πf is the angular frequency, f is the frequency, and Z s abs is the specificacoustic impedance at the absorber diaphragm, which is equal to the sound pressure overthe velocity.The corresponding sound absorption coefficient is defined as Thus, perfect sound absorption is achieved(α=1) when the specifica coustic impedance Z s abs at the absorber diaphragm is equal to the characteristic specifica coustic impedance of air Z c .
Here, we consider aduct of cross-section area S duct and length L with aperfectly rigid termination at one end and an absorber at the other end.The area S abs of the absorber is assumed equal to the cross-section area of the duct.We denote the normalised acoustic impedance at the absorber diaphragm by ζ = Z s abs /Z c = θ + jχ where θ and χ are the normalised acoustic resistance and normalised acoustic reactance respectively.T he normalised input acoustic impedance computed at the rigid end is expressed as where k is the wave number and η = arctan(−jζ) [ 1].
The complexe igenfrequencies f n correspond to the critical values for which the input impedance tends towards zero, namely: for n ∈ N * .
The modal decay time MT60 n of the n th eigenmode, which is defined as the time needed for as ound pressure leveldecrease of 60 dB during the free response of the individual mode, is related to the corresponding damping coefficient δ n [1].It is expressed as where δ n = 2π Im(f n ).The modal decay time of a givenm ode is displayed in Figure 1, for ad uct of length L = 1.70 md epending on the normalised acoustic resistance and normalised acoustic reactance at the absorber diaphragm.Note that the lower the acoustic reactance at the absorber diaphragm, the shorter the modal decay time.
In the case where the normalised acoustic impedance ζ is equal to 1, the modal decay time tends towards zero: there is no more mode.With the aim of performing the best modal equalisation by minimising the modal decay times, the normalised acoustic impedance at the diaphragm will be assumed purely resistive in the following.

Non-uniform boundary condition under normal incidence
If the absorber area S abs is smaller than the cross-section area of the duct S duct and the remaining surface is considered as ahard wall, the boundary condition at this end is non uniform.Near this termination, the field is locally not uniform.When S abs S duct ,t he hypothesis of an almost uniform sound pressure seems reasonable.The particle velocity is equal to that of the absorber on the diaphragm, and equal to zero on the surrounding wall area.By conservation of the volume flow, this termination has an approximate effective normalised acoustic impedance ζ ef f = (S duct /S abs )ζ,which differs from the previous case study by afactor corresponding to the ratio of the absorber area to the duct cross-section area.
If the absorber area is substantially smaller than the duct cross-section area, the approximation does not hold any longer.A nanalytical approach is required to decompose the sound field on the transverse modes, as proposed in [28,29].This semi-analytical approach requires numerical computations to approximate some integrals.It leads to a complexn onlinear problem of size n>1, caused by the contributions of numerous transverse modes, to solvet he sharp boundary between the absorber diaphragm and surrounding hard wall.The longitudinal eigenmodes and corresponding eigenfrequencies of the resonator,w hich are dependent on the absorber area S abs ,c an also be determined with apractical approach, namely using afinite element method, which would be more easily generalised to different geometries.
Fort he case where the absorber area is different from the duct cross-section area (S abs = S duct ), the results are obtained with the help of ac ommercial finite element method software.The duct has as quare cross-section, of width a = 30 cm, and length L = 1.70 m.The absorber is modelled by adisk centred on the duct termination, whose area S abs varies from 3.14 cm 2 to 314 cm 2 (that is ar adius varying from 1cmto10cm).T oidentify the optimal acoustic resistance for every configuration, the normalised acoustic impedance at the absorber diaphragm is assumed purely resistive and constant (ζ = θ).This normalised acoustic resistance θ on the disk varies from 0.1 to 10.Every model is meshed with quadratic elements of minimal size of 0.05 cm and maximal size of 5cm.Fore ach configuration, the eigenmodes are computed, then their corresponding values of MT60 are computed (see Equation 5).
The results for the first three modes are presented in Figure 2. The modal decay times MT 60 are displayed as af unction of the normalised acoustic resistance θ and absorber area S abs .T hese modal decay times are longer for non-uniform boundary conditions relative to the ideal case, where the absorber diaphragm covers the whole termination.Fora bsorbers of sufficient area, there exists an optimal acoustic impedance value for each mode for which the modal decay time is shortened as much as possible.Fore xample, if the absorber covers about 35 %o ft he duct cross-section area (that is S abs = 314 cm 2 ), the target normalised acoustic impedance should be set to about 0.39.Note that this impedance value decreases with the area S abs .F or the first mode, an absorber of area S abs = 150 cm 2 ,whose normalised acoustic resistance θ is equal to 1, has the same performance on this modal decay time as an absorber of S abs = 50 cm 2 with θ = 0.33, or even an absorber of S abs = 25 cm 2 with θ = 0.17.
Thus, if we are able to design absorbers with al ow acoustic resistance value at the diaphragms for agiven set of eigenfrequencies, the absorber area can be lowrelative to the total area of the room walls.The total number of absorbers will essentially depend on howt he design, location, and orientation of these absorbers in the room can interact with the modes in the transverse directions (grazing incidence)and if theycan provide an efficient damping for these modes as well, so as to minimise the modal decay times.

Non-uniform condition under grazing incidence
Under grazing incidence (that is θ 90 • ), the sound absorption coefficient should be close to zero according to Equations ( 1), ( 2).A na nalytic expression for the optimal acoustic impedance of an absorptive surface covering one of the walls of an infinitely long rectangular duct wasf ound in [30].The analytical study givesa no ptimal impedance that only depends on the frequencyand width of the duct.On the other hand, ad uct of finite length inevitably involves reflections at both ends, resulting in other optimal values.Starting from the previous model, we study the case, where the absorber is located along the wall and whose center is at 15 cm from the duct end, so as to stay close to high sound pressure levels for the first modes.Both ends have perfectly rigid terminations; meshing and parameters are the same as in the model presented in Section 2.2.
The results for the first three modes are presented in Figure 3.The modal decay times MT 60 are also displayed depending on the normalised acoustic resistance θ and absorber area S abs .Unexpectedly,the first three modal decay times are actually shortened quite well.Its damping effect is slightly better for the first mode than under normal incidence.It is comparable for the second mode with slightly lower normalised acoustic resistances.But it is lower for the third mode.To understand these results, the sound pressure isosurfaces are displayed in Figure 4for the first mode, in the case where θ = 0.25 and S abs = 314 cm 2 .Note howthe wave fronts are locally deformed by the presence of the absorber and bend toward the absorber surface.The particle velocity is thus locally almost normal to the absorber diaphragm, instead of the expected grazing incidence.Note that this effect wasp reviously reported in experimental results with alined duct with grazing flowat mid frequencies [23].
These results suggest that the absorber orientation has a small influence on the absorption of am ode with ag iven spatial structure, as long as the absorber remains sufficiently close to am aximum of sound pressure levelf or this mode.

Target acoustic impedances in rooms
From the results of Section 2, the performance of the absorber for the modal equalisation, through the minimisation of the modal decay times, depends more strongly on the area than on the orientation of the absorber in relation to the mode shape.Forag iven absorber area and afi xed orientation, there exists an optimal acoustic resistance that minimises the decay time of each mode.In this section, the optimal acoustic resistance values are investigated, so that the first modal decay times are shortened as much as possible, and the process is repeated for rooms of different size.Forthis case study,the total area of the absorbers in the rooms as well as their orientation are arbitrarily fixed.
The area of the absorber is chosen equal to 151 cm 2 .To get as ignificant absorption area in the rooms, 16 absorbers are used for every configuration resulting in at otal area S abs equal to 2416 cm 2 .M oreover, the results in Section 2haveshown that the absorber seems to be more efficient, when it is under normal incidence rather than under grazing incidence, except for the first axial mode in duct.As the vertical dimension generally contributes less to the first modes in usual rooms, the orientation of the absorber diaphragms normal to the axial modes along the horizontal axes is preferred.Three rooms denoted R1, R2, and R3, whose dimensions are summarised in Table I, are studied.Four boxes, each constituted of four absorbers located on twoadjacent sides, are placed in the four bottom corners of the three studied rooms, as illustrated for the room R1 in Figure 5( the absorbers are in shaded areas).The dimensions of every box is 0.3 m × 0.3 m × 0.62 m, corresponding to aglobal volume of around 40 dm 3 .
Forthis case study,weintend to simulate the behaviour of actual listening rooms that have as ubstantial acoustic treatment at mid and high frequencies.Even though the wall acoustic reactance may shift the eigenfrequencies [1,26], here we are only interested in the damping coefficients of the modes to estimate the modal decay times (see Equation 5).T othis end, anormalised acoustic impedance ζ wall (ω)i si mposed on all the walls, and we assume this impedance as purely resistive to get acorresponding sound absorption coefficient as proposed in [25,27], and thus a givenc orresponding damping coefficient for the estimation of modal decay times.The acoustic resistance is also frequencydependent and is interpolated from three values equal to 78 at 10 Hz, 38 at 100 Hz, and 18 at 200 Hz, corresponding to sound absorption coefficients (under normal incidence) α = 0.05, α = 0.10, and α = 0.20 respectively,a ccording to Equation (2).T he normalised acoustic impedance of the box surfaces (except the absorber diaphragms)i sa rbitrarily equal to 18.The purely resistive normalised acoustic impedance θ at the absorber di-  aphragms varied from 0.04 to 1, and the meshing and computation conditions are the same as in the previous case studie in Sections 2.2 and 2.3.
The results for rooms R1, R2, and R3 are presented in Figures 6a, 6b, and 6c respectively.T he modal decay times MT 60 are displayed as afunction of the normalised acoustic resistance θ between 20 Hz and 120 Hz.As expected, these modal decay times increase with the size of the room.As seen in Section 2, for each mode it corresponds an optimal acoustic resistance for the absorber diaphragms for which the modal decay time is minimal.In this configuration, the optimal normalised acoustic resistance should be below0.1 for the first modes up to 40 Hz.
The ideal frequency-dependent target acoustic resistance to assign to the diaphragms should be optimised, so as to match at best the optimal resistance value of every mode.As illustrated in Figure 6, the absorbers are quite inefficient to makesome of the modal decay times shorter, whatevert he acoustic resistance value.These modes are thus useless for the optimisation of the target acoustic impedances.From these results, the modes, whose modal decay time varies less than at hreshold, are not considered for the optimisation (for normalised acoustic resistances varying between 0.0315 and 1.25).The threshold is arbitrarily chosen ab it shorter than the modal thresholds found in [24], to only keep the modes that are likely to be sufficiently shortened by the absorbers and might be audibly perceptible.With at hreshold here equal to 66 ms, around 75 %ofall the modes are kept in the three rooms below120 Hz.We suggest this threshold should be adapted in function of the dynamics of the modal decay times that are dependent on the wall impedances.Then, for every remaining mode, only the values in an interval equal to [min(MT 60 ), min(MT 60 ) + τ]a re kept.The interval should be arbitrarily chosen (here τ = 10 ms)t o give ar easonable resistance range around the optimum, mainly for visual representation purposes (how the modal decay times vary depending on the acoustic resistance values at the absorber diaphragms).The results of this selection for the three rooms are summarised in Figure 7, where the modal decay times after selection are projected in the plane (f, θ).Note that fewdifferences can be observed between the three rooms.
Although the optimal acoustic resistance increases with the frequencyo ft he mode, we can hypothesise that the profile of this resistance is the same whatevert he room dimensions.This suggests that the target acoustic impedances will be suitable for anyr oom at equivalent wall acoustic impedances.

Optimisation of multi-degree-of-freedom target acoustic impedances
From the results found in Section 3, the target acoustic impedances that will be assigned to the electroacoustic absorber diaphragms are nowd efined, after introducing the acoustic impedance control principle developed in [19].Then, the target impedances are optimised with the aim of performing the best modal equalisation by minimising the modal decay times in rooms.

Acoustic impedance control principle
Following as trategy similar to that in [22,31], the target acoustic impedances should be chosen to approach at best the optimal acoustic resistance values found to minimise all the first modal decay times, by keeping the reactive part very small relative to the resistive part.In [19], a method waspresented to achieve adesired specificacoustic impedance at an electroacoustic absorber diaphragm, overab road frequencyr ange, through an hybrid sensor-/shunt-based impedance control.The mechanical part of the closed-box loudspeaker is modeled as asimple massspring -damper system in the low-frequencyrange, that is the mass M ms ,the mechanical compliance C mc accounting for the surround suspension, spider,a nd acoustic compliance of the enclosure, and the mechanical resistance R ms , respectively.I fw ed enote the effective piston area by S d and the force factor of the moving-coil transducer by Bl, the equation of motion of the closed-box loudspeaker diaphragm is derivedfrom Newton'ssecond law, which can be written as where P t (ω)i sthe total sound pressure at the diaphragm, Z m (ω) = jωM ms + R ms + 1/(jωC mc )i st he mechanical impedance of the loudspeaker, V (ω)isthe diaphragm velocity,a nd I (ω)i st he electrical current flowing through the voice coil.Using only one microphone in front of the diaphragm and taking into account the loudspeaker model in the transfer function implemented in ac ontroller,i ti s possible to control the diaphragm dynamic response of the current-drivenloudspeaker.Assuming atarget specific acoustic impedance Z st (ω)i sr ealized at the diaphragm, the transfer function from the total sound pressure P t (ω) at the diaphragm to the electrical current I (ω)c an be derivedfrom Equation ( 6) as The specificacoustic impedance at the diaphragm then becomes

Multi-degree-of-freedom target acoustic impedances
If either the mass or the compliance is completely cancelled, the gain of the transfer function H (ω)i nE quation (7),atlow or high frequencies respectively,isinfinite, which is not technically feasible, because of the limitation of the electrical current delivered by the controller.Ageneral target specifica coustic impedance Z st (ω)w as then introduced in [19], which wasexpressed with one (ortwo) reduction factor(s) µ (or µ M and µ C )and atarget specific acoustic resistance R st as With the formulation of target acoustic impedance in Equation ( 9),a lthough it is possible to modify the centre frequency f c of the electroacoustic absorber,o nly one acoustic resistance value can be assigned to the diaphragm.AMDOF target normalised acoustic impedance is thus defined from n one-degree-of-freedom (one-DOF) impedances in parallel as (10) for n ≥ 2, where (11) is equivalent to the one-DOF acoustic impedance in Equation (9).T he terms ν 2k−1 and ν 2k for k = [1,n]a re factors that decrease the effective mass M ms /ν 2k−1 and effective stiffness 1/(ν 2k C mc )r espectively,s oa st oe xtend the sound absorption bandwidth [19].To keep both the natural mass-and compliance behaviour at lowand high frequencies of the MDOF target normalised acoustic impedance equal to those of the one-DOF target normalised acoustic impedance fixed values, the conditions are

Figureo fm erit of electroacoustic absorbers
From the results obtained in Section 2.3 where the absorber is under grazing incidence, the particle velocity is locally almost normal to the electroacoustic absorber diaphragm.Thus, we hypothesise that the sound wavesa re mainly under normal incidence in front of the absorber diaphragms for the optimisation process.To evaluate the absorption performance of the electroacoustic absorber,w e define afi gure of merit from the sound absorption coefficient in Equation ( 2),w ith respect to the normalised acoustic resistance θ under normal incidence, as This way, the absorption capabilities can be determined in relation to the frequencyf or anyn ormalised acoustic resistance.
In addition, abandwidth BW of efficient sound absorption wasdefined in [19], as the frequencyrange overwhich the total sound intensity in front of the diaphragm is less than twice the total sound intensity in the ideal case (α = 1).This criterion corresponds to athreshold value of minimal efficient sound absorption: As this criterion is relevant in aduct, this threshold value is also applied for the figure of merit of electroacoustic absorbers for the room modal equalisation.

Weighting function of optimal acoustic resistances
To takeinto account the results found in Section 3for the optimisation of the MDOF target acoustic impedance expressed in Equation ( 10),aprofile θ p (f )isestimated from ap olynomial of second order to fit at best all the normalised acoustic resistances corresponding to the minimal modal decay times illustrated in Figure 7, between 16 Hz and 100 Hz.Above 100 Hz, the profile θ p (f )i sc hosen constant and equal to the normalised acoustic resistance value at 100 Hz.Although the simulations were calculated for only three rooms in Section 3, the optimal acoustic resistances should followthis profile for anyroom.The discrete optimal acoustic resistances illustrated in Figure 6 are then turned into aweighting function, which is defined through aGaussian-based function as where a 1 and a 2 are coefficients of the frequency f and b 1 and b 2 are the constant terms of twol inear functions respectively,a nd g is an overall coefficient of the profile θ p (f ).This way, the maximal magnitude and standard deviation of the Gaussian-based function vary according to both linear functions.The parameter values are chosen to give more importance to the first modes between 20 Hz and 40 Hz, and give less importance to the modes higher than 80 Hz.The coefficient g is also chosen below1 ,s o that the figure of merit, expressed in Equation ( 13),covers Table II.Parameters used for the weighting function of optimal acoustic resistances.at best the minimal values of the optimal acoustic resistance of all the modes depicted in Figure 7.The parameter values of the weighting function W opt (θ, f)a re summarised in Table II.Figure 8illustrates the frequency-normalised acoustic resistance map of the weighting function of optimal normalised acoustic resistances.The red dotted line depicts the profile θ p (f )multiplied by the coefficient g.

Optimisation strategy
To maximise the performance of the electroacoustic absorbers for the room modal equalisation, the normalised acoustic resistances at the diaphragms should be as close as possible of the profile θ p (f ).As it is not possible to assign purely resistive impedances at the loudspeaker diaphragms (see Section 4.2), the acoustic reactance should be taken into account in the optimisation process through the figure of merit defined in Section 4.4, so as to evaluate the global effect of the impedance on the performance of absorbers to damp the modes.Ap arametric optimisation is proposed through an objective method using the simplexsearch method developed in [32].The objective function is defined according to the figure of merit α(f, θ)and weighting function W opt (f, θ)expressed in Equations ( 13) and ( 14) respectively,as By maximising this objective function, expressed in Hz, the figure of merit is maximised depending on the weight-ing function, overf requencya nd normalised acoustic resistance ranges as large as possible, that is α(f, θ) ≥ α th , so as to approach at best the optimal acoustic resistance values of the first room modes.The optimisation is limited to finding the factors ν 2k−1 and ν 2k and target normalised acoustic resistances θ t k for k = [1,n], depending on the fixed values of factors ν M 1 and ν C 1 expressed in Equations ( 12) and ( 12) respectively.

Performance analysis
The performance of the electroacoustic absorbers for the room modal equalisation is estimated considering the Peerless SDS-P830657 loudspeaker mounted in aclosedbox of volume V b = 10 dm 3 .T he equivalent mechanical compliance is C mc = 242.35µm•N −1 and the moving mass is M ms = 14.67 g.Ad iminution of 84 %o ft he effective mass of the loudspeaker is imposed, that is ν M 1 = 6.25, so as to improve the absorption performance [19], and thus minimise as manym odal decay times as possible.Also, the centre frequency f c ,w hich is equal to 84.4 Hz when the electroacoustic absorber is in open circuit, should be lowered through the factor ν C 1 ,s oa st og et closer to the first eigenfrequencies.If we want to ensure aproper functioning of the electroacoustic absorbers up to ac ertain sound pressure leveli nag iven room, the maximal value of electrical current delivered by the controller is limited by technical specifications.As the magnitude of the transfer function H (ω)i nE quation ( 7) is directly dependent on the factors ν M 1 and ν C 1 ,t he factor ν C 1 is chosen equal to 25.00, which results in dividing the centre frequency f c by two.Table III summarizes the parameters for the different cases under study.T he case C0 corresponds to the basic configuration of the electroacoustic absorber in open circuit.Cases C1, C2, and C3 correspond to the one-DOF,two-DOF,and three-DOF target normalised acoustic impedances respectively.
Figure 9i llustrates the real and imaginary parts of the normalised acoustic impedance at the electroacoustic absorber diaphragm computed in cases C1, C2, and C3, relative to the basic configuration (case C0).The red dotted line depicts the profile θ p (f ).Thanks to the one-DOF impedances in parallel, the imaginary part of the target normalised acoustic impedances in cases C2 and C3 is closer to zero overaw ider frequencyb and than that in case C1.The target normalised acoustic resistance is constant in case C1 and is equal to 0.126, it varies between 0.075 and 0.200 in case C2, and between 0.052 and 0.202 in case C3.The higher the number of degrees of freedom of the target normalised acoustic impedance, the larger the variations of the normalised acoustic resistance, and the greater the deviation from the profile θ p (f )for the modes at higher frequencies.
To compare the performance for the room modal equalisation between the four cases, the figure of merit α(θ, f) defined in Equation ( 13) is computed in every case.The set of points of the plane (f, θ)s uch as α(f, θ) = α th is displayed in every case in Figure 10, as well as the   modal decay times after selection illustrated in Figure 7 (inbeige solid lines).The red dotted line depicts the profile θ p (f ).Thanks to the chosen values for the terms ν 1 and ν M 1 making the center frequency f c decrease, the areas defined by this set of points in cases C1, C2, and C3 are larger than that in case C0.Although these areas include the optimal normalised acoustic resistances for the first modes in the three cases, the upper bound along the frequencya xis is higher in case C2 than that in case C1, and even more in case C3.The best possible performance of the electroacoustic absorbers for room modal equalisation is thus expected in case C3, where asignificant effect on modal decay times is expected at frequencies as lowas 25 Hz.Note that the areas include the majority of the minimal values of the optimal normalised acoustic resistances of the first modes, thanks to the coefficient g expressed in Equation ( 14).

Experimental setup
To validate the results found in Section 4, the performance of the electroacoustic absorber wase xperimentally evaluated in each case by measuring the frequencyr esponse of the normalised acoustic impedance at the diaphragm in aw aveguide of length L = 1.97 ma nd internal diameter = 150 mm, according to the ISO 10534-2 standard [33].The measurement procedure wast he same as the one described in [19].The duct wasc losed by electrodynamic loudspeakers in closed boxes of volume V b = 10 dm 3 .The sound source delivered ab and-limited pink noise of bandwidth [2 Hz -2kHz].Three 1/2" PCB3 78B02 microphones were wall-mounted at positions x 1 = 1.02 m, x 2 = 1.51 m, and x 3 = 1.62 mf rom the sound source, measuring the sound pressures p 1 = p(x 1 ,t), p 2 = p(x 2 ,t), and p 3 = p(x 3 ,t).The frequencyresponses H 13 = p 3 /p 1 and H 23 = p 3 /p 2 were processed through aB rüel and Kjaer Pulse multichannel analyser (type 3160).With this setup, the electroacoustic absorber performance is evaluated for plane wavesu nder normal incidence overaf requencyrange 44-1340 Hz.To focus the evaluation on the frequencyrange of interest, the results are displayed up to 500 Hz.
To asssign the different MDOF target acoustic impedances at the diaphragm of an electroacoustic absorber,weused the hybrid sensor-/shunt-based impedance control with the same electroacoustic absorber as the one used in [19], whose loudspeaker model parameters are presented in Section 4.2.Only the electronic part of the impedance control wasd i ff erent.The sound pressure in front of the diaphragm used as input signal for the control wasm easured thanks to a1 /4" electret microphone with its preamplification circuit.The transfer function H (ω) in Equation ( 7) wasi mplemented onto an Analog Devices digital signal processor (Sharc processor), with a measured time delay equal to 18.1 µs.The signals were converted thanks to Analog Devices analog-to-digital and digital-to-analog converters.The voltage controlled current source wasa no perational amplifier-based improved Howland current pump circuit [34].

Acoustic impedance measurement
Cases C0, C1, C2 and C3, whose parameters are summarised in Table III, were used for the measurements.The measured frequencyr esponse of the normalised acoustic impedance in every case is presented in Figure 11.Thanks to the control, both the acoustic resistance and reactance at the diaphragm are modified to reach as close as possible the target acoustic impedances.In cases C1, C2 and C3, even though as light shift in frequencyi sv isible, the magnitude is kept belowone fifth, and the phase is closer to zero overab roader frequencyr ange than in case C0 (when the control is switched off).The slight differences can be attributed to imperfections in the lumped element model and to the frequencyr esponse of the microphone, which wasnot taken into account in the control law.
The figure of merit α(θ, f)e xpressed in Equation ( 13) is computed from the measured frequencyresponses of the normalised acoustic impedance in every case.The set of points of the plane (f, θ)s uch as α(f, θ) = α th is displayed in every case in Figure 12, as well as the modal decay times after selection illustrated in Figure 7(in beige solid lines).
The red dotted line depicts the profile θ p (f ).The measurements are satisfactorily consistent with the corresponding computations illustrated in Figure 10.With three degrees of freedom (case C3), the large area defined by the set of points of the plane (f, θ)covers the whole values of optimal acoustic resistances found in Section 3, except for the first three modes.Nevertheless, with ahigher value of factor ν C 1 ,and if the technical specifications makeitpossible, the first modes could be covered by this area as well.Thus we may expect good performance of electroacoustic absorbers for the equalisation of the first modes in actual small rooms.

Conclusion
In this paper,the effect of electroacoustic absorbers on the minimisation of the modal decay times wasi nvestigated.Fort he simulation, the absorbers were modelled by flat disks, where apurely resistive acoustic impedance wasassigned to each absorber.T wo numerical studies in ad showed that the absorber causes as ignificant decrease of the modal decay times, whether its diaphragm is oriented normal to the propagation dimension (normal incidence)  or parallel to it (grazing incidence).Fore very mode, the optimal acoustic resistance is dependent on the mode and absorber area: the larger the area, the lower the modal decay times.The effect of the room dimensions has also been studied from the simulation of three rectangular rooms of different dimensions.Agiven layout of apossible actual implementation of electroacoustic absorbers, with afixedtotal area, wasproposed.It wasshown that the optimal acoustic resistances for which aminimal modal decay time is obtained, has the same frequency-dependent profile, whatevert he room dimensions for ag iven wall impedance value.As the reactive part of the acoustic impedances at the absorber diaphragms is dominant at very lowand high frequencies, further work should include the effect of the acoustic reactances at the diaphragms in the modelling.
Then, taking into account the results of the simulations with the givenlayout of absorbers in the bottom corners of the studied rooms, multi-degree-of-freedom target acoustic impedances at the diaphragms were optimised to maximise the performance for the modal equalisation.The acoustic impedances approach at best the optimal resistance value found to minimise each modal decay time, by keeping the reactive part very small relative to the resistive part.
Finally,the performance of the electroacoustic absorber wase xperimentally evaluated in aw aveguide, by measuring the acoustic impedance for each optimised MDOF target acoustic impedance.The best performance is obtained for the three-DOF acoustic impedance, providing the best fit to the optimal profile of acoustic resistances.Although higher DOF target acoustic impedances could also be achieved, the performance might be limited by technological specifications, such as the maximal value of electrical current delivered by the controller.
The performance of the electroacoustic absorbers for the modal equalisation wasalso evaluated in actual listening rooms, whose results can be found in [35], confirming the efficiencyofthe concept to shorten the modal decay times.The method presented in this paper could also be used for other applications requiring specificv alues of acoustic impedance overagiven frequencyrange.

Figure 1 .
Figure 1.(Colour online)N ormalised acoustic resistance -normalised acoustic reactance map of the modal decay time of a givenmode in the 1D ideal case of length L = 1.7m.

Figure 2 .
Figure 2. Normalised acoustic resistance -a bsorber area maps of the decay times of the (a) mode 1, (b) mode 2, and (c) mode 3 computed for an absorber under normal incidence in aduct.

Figure 3 .
Figure 3. (Colour online)Normalised acoustic resistance -absorber area maps of the decay times of the (a) mode 1, (b)mode 2, and (c) mode 3computed for an absorber under grazing incidence in aduct.

Figure 4 .
Figure 4. (Colour online)S ound pressure isosurfaces for the mode 1i nt he duct with the absorber under grazing incidence (rear view),whose normalised acoustic resitance is θ = 0.25 and area is S abs = 314 cm 2 .

Figure 5 .
Figure 5. Geometry of the finite element model of the room S with 16 absorbers (inblue)located in the bottom corners.

Figure 6 .
Figure 6.(Colour online)Modal decay times between 20 and 120 Hz depending on the normalised acoustic resistance computed for 16 absorbers located in bottom corners of the rooms (a) R1, (b) R2, and (c) R3.

FrequencyFigure 7 .
Figure 7. Projection in the plane (f, θ)ofmodal decay times after selection computed for 16 absorbers located in bottom corners of the rooms R1, R2, and R3.

Figure 8 .
Figure 8. (Colour online)Frequency-normalised acoustic resistance map of the weighting function for the optimisation of the target normalised acoustic resistance.The red dotted line depicts the profile θ p (f )multiplied by the coefficient g.

Figure 9 .
Figure 9. Real and imaginary parts of the target normalised acoustic impedance at the electroacoustic absorber diaphragm, computed for the one-, two-and, three-DOF target impedances optimised from the objective function of area overthreshold, relative to the basic configuration.The red dotted line depicts the profile θ p (f ).

Figure 10 .
Figure 10.Set of points of the plane (f, θ)such as α(f, θ) = α th computed in cases C0, C1, C2, and C3.The beige solid lines depict the projection in the plane (f, θ)ofmodal decay times after selection computed for 16 absorbers located in bottom corners of the rooms R1, R2, and R3.The red dotted line depicts the profile θ p (f ).

Figure 12 .
Figure 12.Set of points of the plane (f, θ)such as α(f, θ) = α th computed in cases C0, C1, C2, and C3 from the corresponding measured acoustic impedances.The beige solid lines depict the projection in the plane (f, θ)o fm odal decay times after selection computed for 16 absorbers located in bottom corners of the rooms R1, R2, and R3.The red dotted line depicts the profile θ p (f ).

Table I .
Dimensions of the rooms.

Table III .
Parameter values of the optimised one-, two-, and three-DOF target normalised acoustic impedances (cases C1, C2, and C3) relative to the basic configuration of the open circuit electroacoustic absorber (case C0).