A review of techniques and challenges in outdoor sound field control

Pierangelo Libianchi 1

d&b audiotechnik GmbH & Co. KG Eugen-Adol ff -Str. 134, 71522 Backnang, Germany

Finn T. Agerkvist 2

DTU Ørsteds Plads, Building 352, 2800 Kgs. Lyngby

Elena Shabalina 3

d&b audiotechnik GmbH & Co. KG Eugen-Adol ff -Str. 134, 71522 Backnang, Germany

ABSTRACT The application of sound field control to outdoor live events at low frequencies is a recent one due to increased concerns regarding noise pollution and stronger regulations. Here is presented an overview of the techniques being recently investigated based on model, data or hybrid approaches. The approaches presented here provide encouraging results but they all deal with the problem at relatively short distances (approximately 100 m). Translating these results to larger distances is going to be a challenge as a new set of problems needs to be addressed. An overview of the most relevant issues encountered in long range applications such as uneven terrain, properties of the medium, ground and obstacles interactions is also presented.

1. INTRODUCTION

The e ff ects that noise has on the health of individuals are well know by now [1]. This has led to increasingly strict law regarding noise emissions. Previous legislation focused on A-weighted measurements to assess such emission e ff ectively leaving out low frequency emissions. Nowadays, updated laws are starting to use C-weighting and more encompassing measures that will make it harder to organize outdoor live events since they can be a nuisance for non participants and can be powerful noise emitters and contain a strong low frequency component which can propagate over large distances with minimal atmospheric attenuation [2]. In the coming years, the proliferation of such regulations might lead to the break up of the large events we have today to smaller ones to reduce their noise footprint. Alternatively, such events could be moved further away from residential areas

1 pierangelo.libianchi@dbaudio.com

2 ftag@dtu.dk

3 elena.shabalina@dbaudio.com

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

but this will require additional infrastructures and be damaging for wildlife with both consequences going against the UN sustainability goals [3]. Sound reinforcement system have been optimized to limit radiation o ff axis to limit noise pollution. This still leaves strong emission on axis where an audience is present. Noise barriers are not the best solution in this case due to limited performances at low frequencies. The use of active noise barriers can improve such performances but, as pointed out in [4], they can introduce reflections in the audience area. Furthermore, many of the events considered here take place on borrowed spaces instead of permanent venues so installing physical barriers is just not an option in many cases. Recently, [5] proposed to use active noise control to weaken such emissions. The system employed is shown in fig. 1 and consists of an additional set of control sources placed behind the audience. These sources are used to generate a secondary field that matches, in specific area, the sound field coming from the stage but with opposite phase to weaken its emissions. In this way it is possible to generate a quiet area usually referred to as dark zone. It is important for this system to not be noticeable by the audience. This can be achieved by either using cardiod control sources or an additional array of sources to limit the spill into the audience area [6]. Another important aspect to consider in this type of application is the non-linear nature of the transducers. It is very important that the filters designed for the control source do not introduce any amplification of the driving signal to avoid distortion from the loudspeakers as it could drastically reduce the performance of the system.

Figure 1: A sketch of a possible sound field control setup.

This field is still in its infancy but di ff erent strategies have been proposed. In this work we will look at these strategies by first looking at the general problem and its formulation. We will see that the general problem can be further divided in two sub-problems. How these two sub-problems are dealt with is the root of the di ff erences between the di ff erent strategies. This will be the focus of the first part of this paper. The second part will present a brief overview of some of the main challenges when distances larger than 100 m are involved. We will look at di ff erent models that deal with outdoor sound propagation, their advantages or disadvantages and finally, describe two of the most important model parameters.

2. THE PRESSURE MATCHING PROBLEM

All the methods presented here have the ultimate purpose of generating a set of filters to apply to a set of control sources. These filters should be designed in way such that the sound field synthesized by the control sources matches the primary field in the dark zone to weaken the noise emissions. The general problem is a pressure matching problem and can be formulated as follows:

min q || p − Hq || 2 2 , (1)

H ∈ C MxL is the secondary path transfer function matrix from L control sources to M microphones, p ∈ C M is the primary sound field in the dark zone and q ∈ C L are the volume velocities of the secondary sources. This problem can be further divided in two sub-problems: how to design the filters and how to obtain p and H . The ways we can deal with these two sub-problems constitute the main di ff erences between the approaches used until now in this type of application. The first sub-problem will be the main focus of the first part of this section while the second part will focus on the second sub-problem.

2.1. Sub-problem 1: designing the filters All the algorithms presented here can be described by the general block diagram in fig. 2 In general, the reference signal x ( ω ) is provided directly from the mixing desk and the frequency response of the controller Q ( j ω ) is obtained in di ff erent ways which will be the focus of this section. According to the definition of feedback and feedforward from [7] we can make a first distinction between the families of algorithms analyzed here. In a feed-forward method, an error sensor can be used to monitor the performances of the system but it is not used to directly design or modify the filters. The filters are then static and are obtained o ffl ine from p and H no matter how they have been obtained. The feedback approach instead uses the signal from the error sensor(s) to dynamically adapt the frequency response of the filters. Also in this case estimations of p and H might be needed, depending on the implementation, and potentially any method can be used to obtain them.

+

Figure 2: General scheme used in active noise control. The box encloses the only module used in feedback system that is not present in feedforward approaches.

Feed-forward methods

The problem in eq. 1 is usually an ill-posed problem being under-determined ( M ≫ L ). Most of the methods described here employ a regularization parameter to ensure stability of the solution by reducing the e ff ect of noise and / or rounding errors due to finite accuracy. It also controls the balance between the amplitude of the solution and the amplitude of the error / residual. It can also be used to control the spatial properties of the synthesized sound field. In general, the amount of regularization a ff ects the accuracy but also the robustness. Large regularization can help to keep the amplitude of the solution below a given threshold, control the spatial properties of the secondary field and increase the robustness to uncertainties in p and H due to mismatches or changes in weather conditions, but at the expense of performance. All the feed-forward methods can lead to similar performances in terms of insertion losses. What should drive one to chose a method instead of the other are the practical aspects of each of them and the limits and properties of a given application, as will be discussed in the following. Further details about these methods together with a comparison between them can be found in [8].

– Ridge regression

min q || p − Hq || 2 2 + λ || q || 2 2 (2)

In this case a penalty term is added to the original pressure matching problem from eq.1. The regularization parameter λ controls the trade-o ff between accuracy and amplitude of the solution. Larger regularization parameters lead to solution with a smaller amplitude at the cost of increasing the magnitude of the error. Also know as least square problem with Tikhonov regularization, this is one of the most used method to solve a least square problem [9]. The main advantages of this method are that it is easy to implement and computationally e ffi cient. The main drawback is the need to find the optimal regularization parameter. This might require some trial and error. Automatic selection methods like the l-curve [10], generalized cross validation [11], normalized cumulative periodogram [12] or discrepancy principle [9] can help finding the optimal compromise between amplitude of the residual and of the solution. The solutions found this way might not be viable though. There is no guarantee that the solution resulting from this compromise does not amplify the driving signal which is something to be avoided to not incur in non-linear behavior in the transducer spoiling the results. A strict and explicit amplitude constraint cannot be applied. Furthermore, the radiation pattern resulting from such solution might not be not acceptable and one would have to manually tweak the regularization parameter.

– Convex optimization: min q || p − Hq || 2 2 s.t. f i ( q ) < b i (3)

The original problem may be treated as a convex optimization problem and explicit constraints can be used. The constraints can be used to enforce a limit on the amplitude of the solution. Such constraints can be applied to each coe ffi cient of the solution ( | q | ⪯ 1, the limit here is unity gain) or on the array e ff ort ( || q || 2 2 ≤ 0 . 5, a limit of -6 dB was used in [8]). The first option is more desirable as it guarantees that not a single coe ffi cient will produce amplification. On the other hand, sometimes it is too strict and lead to an empty feasible set. In such cases one case use the more relaxed second option. The constraint can be formulated in di ff erent way and additional constraints can be used. This has to be done with care though to avoid empty feasible sets. This method can provide the most accurate solutions since it searches for them only in the feasible set so that any potential solution will comply with the constraints. The drawback is that it is computationally expensive and implementations to solve these problems are available either as external modules or in commercial software. Furthermore, constraints on the radiation pattern can easily lead to empty feasible sets. This can limit the application of this method when it is important to avoid side lobes.

– Subspace / Iterative methods: min q || p − ˆHq || 2 2 (4)

There a few methods of this kind and the common thread is to replace H with a lower rank approximation ˆ H to provide regularization. This can be done by applying a decomposition to H such as singular value decomposition, eigenvalue decomposition or principal component analysis. A subset of the vectors in the basis obtained in the previous step will provide the low rank approximation. The vector of pressure p can be projected onto this subspace and be used to compute the solution [13]. This selection process is what provides regularization and allows to select components with desired spatial properties. The drawbacks are the lack of an amplitude constraint and the decomposition and selection process can be time consuming. The first problem can be addressed by using a fixed basis such as discrete cosine transform, discrete Fourier transform, etc. This solution does not fix the lack of an amplitude constraint and the basis chosen is not adapted to the current problem though. Iterative algorithm such

as the conjugate gradient least square can alleviate both of these problems. In this case the regularization, and so the amplitude of the solution and its spatial properties, is provided in discrete steps and controlled by the number of iterations. This allows to include stopping criteria designed specifically for a given application. In this way is possible to control the radiation patter and, to some extent, the amplitude of the solution. An explicit constraint on the amplitude of the solution can be enforced by pairing this method with the active set-type method described in [14].

Feedback methods

In this family of methods the di ff erence between primary and secondary sound fields is captured by the error sensors and used to update the frequency response of the filters. These methods also set to minimize eq. 1 using gradient descent. Taking the derivatives of eq. 1 with respect to the coe ffi cients and setting it to 0, one can obtain the following update rule:

w ( n + 1) = w ( n ) − µ e ( n ) ˆr ( n ) , (5)

where w ∈ R N are the coe ffi cients of the filter of length N , µ is convergence factor or slope rate, e is the error measured at the microphone and ˆr ∈ R N is either the reference signal x ( n ), as in traditional least mean square (LMS), or the reference signal filtered by the secondary path to improve stability as in FxLMS [7]. This implementation is in the time domain and n represent the time step. Furthermore, it is for one control source and one microphone. A multichannel approach is in [15] together with some implementation details to improve performances and reduce computational e ff ort. The downside of this method is that it requires the measurement or simulation of the secondary path transfer functions. It is important to have an accurate estimation of them since the FxLMS algorithm will not converge if the phase di ff erence is larger than 90 ◦ [16]. This can be a limitation when large distances are involved and the system has to work for long periods of time. The atmospheric conditions change over time introducing an estimation error that gets larger with distance as the influence of the medium gets larger as well. These methods can achieve large noise reduction at the position of the microphone. Notably the best performances are achieved in places not accessible by people. The extension of the quiet are can be increase is the system consist of enough error microphone placed close to enough to sample the incoming wavefront. In this way the entire wavefront is weakened and the quiet are can be extended beyond the microphones. The size of this extension depends on the density of the sampling and the complexity of the topology. Another way to introduce losses in areas that are not occupied by the microphones is proposed in [17]. This work uses virtual sensor placing to create quiet areas away from the sensors. The traditional FxLMS is coupled to a model used to predict the transfer function in positions where there are no physical sensors. In this way it is possible to achieve large reductions in position that are now accessible. Finally, [18], introduces a new method that allows reduction in a large area by surrounding in it with control sources and error microphones. This makes this quite area accessible but it requires many microphones and sources. This is usually a limit with traditional FxLMS approaches. In this work, the authors propose an eigenspace and wave-domain adaptive filtering to overcome this issue. The filters are derived from a circular harmonics representation of the reference and the error signals. The filter coe ffi cients, after being computed in this transformed domain, are then transformed back before being applied to the control sources. In this way they minimize the cross-correlation between channels and, once they are decoupled, treat them as separate single adaptive filtering problems.

2.2. Sub-problem 2: Obtaining the two sound fields A second sub-problem consists in obtaining the vector of pressures p and the matrix of transfer functions H necessary to compute the filters. This can be done in di ff erent ways. Each way has its own benefits and drawbacks as we will see in this section. However, some general consideration can be made regarding the spacing and positioning of control sources and sensors. The spacing between control sources and sensors and the spacing between sensors a ff ect the condition number of the transfer function matrix which in turn a ff ects the regularization that has to be applied and the set of solutions that can be obtained. In [19] is shown that the condition number of H is at its smallest when the spacing between sensors and between sensors and control sources is equal to the spacing between the control sources. This configuration might not be ideal for this application though. First, a spacing between sources of approximately 2 m would be fine enough to provide control and limited spatial aliasing up to around 120 Hz while the resolution of the measurement grid should be finer than that to properly sample the sound field. Second, the solutions tend to not generalize well when the sensors are placed so close to the sources. There are multiple reasons for it: the secondary wavefront is more curved than the primary one, the rate of decay of the amplitude of the two fields are also very di ff erent and near-field e ff ect are recorded that do not propagate to the far field.

Model based approach

The primary field and secondary path transfer functions can be obtained from simulation using a suitable propagation model. This method is the least demanding in terms of practical e ff ort since it does not require any measurement. In terms of computation time it is less clear since it vastly depends on the complexity of the model used. The accuracy of potential solutions depends on how well the model matches reality. This approach has been proposed in [8] using the complex directivity point source model (CDPS, [20]) to generate H and p . At the moment there is no experimental data available for this approach. This is a simple model that only considers far-field and free-field conditions and a static and homogeneous medium. From simulations, we can expect large insertion losses when these conditions are met. The actual sensitivity of the solution to these uncertainties depends on the algorithm chosen to derive the filters and the amount of regularization. More advanced models should be used in case of reflections or propagation over distances larger than 100 m, where the influence of the atmosphere cannot be neglected. This simple model can help extend the quiet area beyond the dark zone when there are reflection. In general, solutions obtained from transfer function measured in the dark zone do not generalize well beyond it when reflections are involved. The interference pattern sampled is specific to that microphone location and as we move away from it, the interference patter will change. When CDPS is used, one only corrects for the direct field which will keep propagating beyond the dark zone and its properties will not change as quickly as they would if reflections were to be included.

Data based approach

This method has been used in multiple works like [6,21] is quite straightforward since it consists of measuring both p and H . This is the most time consuming of approaches presented here and at the same time potentially the most accurate for a limited amount of time. Both p and H change with time as the weather does [16]. The performance of the solution will then depend on how close the weather conditions match the ones encountered when p and H were measured. The sensitivity to the mismatch will depend instead on the algorithm chosen to derive the filters and the amount of regularization. Some work has been done in [22] to adapt the measurements to changes in temperature and wind and the model used can be e ff ective when the distances involved are not larger than approximately 100 m. From this distance, the e ff ects produced by changes in the properties of the atmosphere requires more complex modelling than what proposed.

[21] presents results from di ff erent measurement sessions using this approach and Ridge regression to obtain the filters. The insertion losses reach a peak of 12-14 dB and at least 8 dB between 30 Hz and 120 Hz in the case where reflections were limited (Refshaleøen, Copenhagen [6,21]). In more complex environments (Tivoli, Copenhagen and Kappa FuturFestival, Turin [21]) the insertion losses dropped to an average of 5 dB with peaks of 8 dB. In these last two cases, the sound fields where much more complex due to reflections. In this case, also the e ff ect of changes in the weather is magnified. In free-field, only the direct field is a ff ected by these changes. When there are reflections, also the reflections will be a ff ected and in di ff erent ways depending on their direction and the direction of the wind dramatically changing the interference pattern that was measured under di ff erent weather conditions.

Hybrid approach

In [23], a dataset of sparsely measured transfer functions is used to fit a propagation model based on a spherical harmonics expansion. Such model is then used to infer the transfer functions at positions that were not measured; drastically reducing the burden in the previous approach. This method first assumes sources with an identical axi-symmetric radiation pattern. In this model, the transfer function between a loudspeaker and a receiver at r = ( r , θ, ϕ ) in spherical coordinates can be expressed in terms of a spherical harmonics expansion:

M X

ˆ h ( r , a ) =

m = 0 a m h (2) m ( kr )P m (cos( θ )) , (6)

where a = [ a 0 , a 1 , . . . , a m ] T ∈ C M + 1 are the complex coe ffi cients, h (2) m are the spherical Hankel functions of the second kind, P m are the Legendre polynomials of order m , M + 1 is the number of modes included and k = 2 π f / c . In dry air, the speed of sound c = √ γ RT where γ = 1 . 401, R = 287 J / kg K is the universal gas constant and T is the temperature in K. Since it is assumed that the sources have the same radiation patter, also the coe ffi cients a m will be the same for all sources so that they can be modelled with parameters ( a , T ). T is modelled here as a frequency dependent parameter that includes the e ff ect of both temperature and wind. This model is then fitted to a set of measured transfer functions h ∈ C LN S between the L loudspeakers and a subset N S of the M sensors. The model to solve, after including the measurement noise n ∈C KN S , is:

h = ˆ h ( a , T ) + n , (7)

where the dependency of the wavenumber on the temperature has been included. The variables a , T and n are treated as stochastic variables following either a proper complex normal distribution or a normal distribution. In this way is possible to write the likelyhood distribution, i.e. the distribution of the measured data conditioned on the unknowns and using Bayes theorem, the posterior distribution is then written in terms of the likelyhood and the priors:

π ( a , T , σ n | h ) ∝ π ( h | a , T , σ n ) π ( a ) π ( T ) π ( σ n ) , (8)

where σ n is the standard deviation of the noise and is an hyperparameter following a normal distribution. One can obtain the parameters a and T by finding the maximum a posteriori (MAP) of the previous expression which is the set of unknowns that maximizes the posterior distribution. The model can then used to compute the transfer function to positions where there is no physical sensor. In this work, a least square solution is found using Tikhonov regularization but other methods could have been used. This method provides the best compromise between the previous two. Much of the accuracy that one has by measuring the transfer function is retained but the cost of the measurement itself is

diminished by having to measure only at limited locations K instead of the entire quiet zone with enough microphones M to properly sample the sound field. [23] shown how this method achieves spatially averaged insertion losses with peaks beyond 20 dB and at least 10 dB broadband. It noted though, how this method works well in free field conditions but it might struggle with more complex topologies with many reflections. In addition, like the other methods, the performance of the system is strongly related to the atmospheric conditions and are a ff ected even by moderate changes. The larger the distance between the control zone and the control sources, the more sensitive is the system to these changes.

3. LONG RANGE CHALLENGES

For propagation over more than 100 m one has to take into account the e ff ects of the ground, the atmosphere and obstacles. These challenges can be reduced to the calculation of p and H . The model approach could be used for this purpose. There are few models developed for outdoor sound propagation able to handle a moving inhomogeneous medium, ground reflections and, in some cases, reflections from obstacles. These models together with some of their main parameters will be the focus of this section. The measurement approach might be too demanding for any real practical application in this case. The amount of microphone depends on the extension of the dark zone and considering the distances involved there are high chances to have signal distribution problems. Even without this problem this approach is not expected to perform well for long amounts of time. As it was said before, the transfer functions will change over time and the error introduced by these changes gets larger with distance. Correcting the transfer functions might not be easy here since e ff ects such as obstacles, ground reflections and refraction must be considered. A hybrid approach might provide a way to use measured data to reduce uncertainty in the model parameters but has not been developed as of yet for this application.

3.1. Outdoor propagation models There are few models suited for outdoor sound propagation. A first distinction can be made into frequency and time domain models. In general, time domain models such as Finite-Di ff erence Time- Domain (FDTD) [24] tend to be more precise allowing to faithfully represent a specific environment. The main drawback is that they are computationally expensive and usually require a long time to provide a solution. The computation time is a factor that narrowly restrict the models that can be used. Average atmospheric properties can be considered constant only in small time windows. Any model that requires any time close to this window would make the solution useless since the atmospheric conditions would have already changed. Frequency domain approaches tends to be less accurate than their time domain counterparts for the simple fact that obstacles cannot be included and admit limited irregularities of the terrain. On the other hand, they usually provide more e ffi cient implementations. Many of the frequency domain models such as Fast Field Programming (FFP) [25], Crank-Nicholson Parabolic Equation (CNPE) [26], Green’s Function Parabolic Equation (GFPE) [27] use the e ff ective wavenumber approximation. The e ff ect of moving medium is include by using the e ff ective speed of sound c e f f = c ( T ) + v · n , where the c ( T ) is the temperature dependant speed of sound, v is the wind velocity vector and n is the normal to the wavefront. This is approximation produces phase distortion and if more than one source is involved even amplitude distortion due to their interaction [24]. Ray tracing with caustic di ff raction field [27] is inherently less accurate at low frequencies even without using the e ff ective wavenumber approximation. Even though the time domain models tend to be more accurate, one has to take into account also the uncertainty in the parameters. Regardless of the model, the ground impedance and the sound speed profile (and indirectly the temperature and wind profiles) are needed and their accuracy will

Table 1: Pros and cons of di ff erent outdoor propagation models. Method Pros Cons

Obstacles cannot be included

Properties of the medium and of the

ground cannot change with distance

FFP Fast

Only flat terrain

E ff ective wavenumber assumption

No vertical wind component

CNPE Possible to have an irregular terrain Obstacles cannot be included

Possible to have an irregular terrain

Limited shapes possible for the terrain

E ff ective wavenumber assumption

E ffi cient 3D implementation

GFPE

No vertical wind component

Ray model including

Possible to have irregular terrain Reduced low frequency accuracy

caustic di ff raction field

FDTD Possible to include obstacle

Computationally expensive

Possible to include irregular terrain

a ff ect the final solution. Some more details on these two parameters will be provided in the following sections. In the case of time domain model, all the impedances, from the ground to the obstacles, have to be formulated in the time domain. It seems clear that even though the model approach is viable for this application, further e ff ort should be placed in developing a propagation model suited for the task. The main advantages of the individual methods are resumed in Table 1 and a comparison of the performances of some of them can be found in [28].

3.2. Ground impedance In a downward refracting atmosphere, the sound waves will hit the ground and be reflected multiple times before reaching a sensor or the dark zone. The interactions between, direct, reflected and refracted waves is crucial to accurately model the sound field. It is necessary though, to have a good estimate of the ground impedance. There is a vast literature about modelling the ground impedance and many models have been proposed with di ff erent degrees of complexity and accuracy. Some models tend to work better with certain types of terrain. Due to such extensive literature and the fact that a thorough descriptions of such models is beyond the scope of this work, we refer the reader to [29] for an overview, description and comparison of di ff erent methods.

3.3. Sound speed profile Direct measurement of the sound speed profile can be demanding in terms of equipment and cost. There are many models in the literature for the potential temperature and the wind speed [30–32]. A first selection of such model can be done depending on which state the atmospheric boundary layer (ABL) is in: stable, unstable or neutral. Depending on the state the expressions used in these models can di ff er. The state also a ff ect the height of the ABL which in turn can potentially a ff ect the size of the domain to be modelled. In [33] is shown how in most cases the atmosphere is in a neutral or quasi- neutral state. Furthermore, in [34] they distinguish between a truly neutral and conventionally neutral boundary layer. The first has no heat flux both at the bottom and at the top of the ABL while the second has zero heat flux only at the bottom. The second one is most common of the two. In this type

of conditions, di ff erent models have been proposed and some of them are compared in [35]. It can be seen how the models agree close to the surface, where the Monin-Obukov similarity theory holds. As we move further from the surface the models tend to diverge introducing large uncertainties. The sensitivity of the propagation models to this uncertainty is under investigation at the moment. The models that seems more accurate use quantities related to the top of the ABL which are hard to measure thus can introduce additional uncertainty.

4. CONCLUSIONS

Even though outdoor active noise control is a new field, the results so far for relatively short distances (less than 100 m) have been promising. A variety of approaches are available with di ff erent degrees of complexity, requirements and characteristics. All the methods and their combinations presented here have the potential to deliver satisfactory performances in terms of insertion losses. No combination is generally better than others. One should pick the right combination depending on the characteristics and limitations of the application. Di ff erent methods are able to achieve broadband insertion loss of at least 10 dB in relatively simple topologies. When the environments get more complex and reflections are involved, the results are not as good. More work is required for this type of scenarios by either using more advanced models, adapt the measured transfer functions or used a hybrid method with a model able to include reflections. Larger distances represent a challenge mostly because there is no one outdoor sound model perfectly suited to model the various physical phenomena a ff ecting the sound propagation over a large range and when they do, the computational time required is not acceptable for this kind of application. The bottlenecks are the accuracy and computation time, since no model excel in both aspects. Further research should focus on this topic. The development of an hybrid approach for long distances is also of interest since it could allow to reduce the uncertainty in model parameters through measured data.

ACKNOWLEDGEMENTS

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