The effect of reverberation on personal audio devices n-Ga lvez,a) Stephen J. Elliott, and Jordan Cheer Marcos F. Simo Institute of Sound and Vibration Research, University of Southampton, Southampton, Hampshire SO17 1BJ, United Kingdom
(Received 18 December 2013; revised 7 March 2014; accepted 11 March 2014) Personal audio refers to the creation of a listening zone within which a person, or a group of people, hears a given sound program, without being annoyed by other sound programs being reproduced in the same space. Generally, these different sound zones are created by arrays of loudspeakers. Although these devices have the capacity to achieve different sound zones in an anechoic environment, they are ultimately used in normal rooms, which are reverberant environments. At high frequencies, reflections from the room surfaces create a diffuse pressure component which is uniform throughout the room volume and thus decreases the directional characteristics of the device. This paper shows how the reverberant performance of an array can be modeled, knowing the anechoic performance of the radiator and the acoustic characteristics of the room. A formulation is presented whose results are compared to practical measurements in reverberant environments. Due to reflections from the room surfaces, pressure variations are introduced in the transfer responses of the array. This aspect is assessed by means of simulations where random noise is added to create uncertainties, and by performing measurements in a real environment. These results show how the robustness of an array is increased when it is designed for use in a reverberant environment. C 2014 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4869681] V PACS number(s): 43.38.Hz, 43.60.Gk, 43.55.Jz, 43.60.Pt [MRB]
I. INTRODUCTION
Reproduction of different sound programs for different users in a common space is becoming widespread. There is also an increasing concern about selective audio reproduction, as different users may not want to listen to an audio program that is not coming from their own device. These objectives can be achieved using “personal audio” devices, which have been discussed by Druyvesteyn and Garas,1 with reference to systems used to create a selective reproduction of the soundfield in a room. Personal audio systems have also been considered for sound reproduction in individual seats using headrest loudspeakers,2 to restrict sound radiation for mobile phones,3,4 to direct the sound coming from a television toward a determined spatial region,5 or to create different sound zones inside a car.6 Conventional acoustic arrays perform poorly at frequencies where the wavelength of radiation is greater than the aperture size of the array,7 unless optimal, or superdirective, beamforming techniques are used. Such techniques have been widely used in the context of sensor arrays,8–10 and by reciprocity have been applied to acoustical source arrays.5,11,12 These techniques do, however, need large gains at frequencies with high spatial correlation, such that the transfer responses from each source to a given point are very similar.13 These large gains are accompanied by greater sensitivity to uncertainties in the acoustic environment.8,14 Although the final application of such personal audio systems is in a normal, everyday room, the behavior of these systems has generally been studied in anechoic conditions, and the effects of reverberation are not well understood. The a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
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performance of a single directional acoustical radiator in a reverberant field was studied by Beranek,15 who derived an expression for the total acoustic field depending on the acoustic characteristics of the room and the radiator. A similar formulation was used to obtain the directional characteristics of a source in a reverberant environment by Druyvesteyn and Garas.1 Measurements of the performance of a line array in reverberant environments have also been presented.16 Other recent work has analyzed the performance of different transfer response models for creating the filters of a superdirective array in a reverberant environment.17 It is clear that the directional characteristics of an acoustical radiator are changed when it is placed in a reverberant room, but it is not so clear what the effect of the reverberation is on the robustness of a superdirective beamformer. The robustness of these radiators is defined in terms of how much the directional response is affected by mismatches in source sensitivity and position. In superdirective antennas, for example, it has been observed how the practical response is far from the theoretical superdirective value if random errors are introduced in the antenna weightings.18 The characteristics of these errors and how to obtain robust beamforming in their response has been studied for microphone arrays, assuming that the errors have a certain probability factor.19 Work on loudspeaker arrays has also studied the effect of errors in source sensitivity and position, and how they affect the ratio of mean square pressures between two spatial zones.14,20 Generally, these acoustical radiators are made robust to changes in their transfer responses by applying regularization,12,14 which also controls the power used to drive the array sources. Recent work has suggested that the space-averaged uncertainties introduced into the transfer responses of an array by reverberation can act analogously to regularization.14
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C 2014 Acoustical Society of America V
This paper investigates the behavior of two personal audio arrays inside reverberant environments. First, superdirective array theory and the creation of least squares filters21 are considered. In order to assess the performance of both devices, the acoustic contrast, as defined by Choi and Kim,22 is used as a metric, which quantifies the mean square pressure ratio between two spatial zones. A formulation is then introduced, which allows the acoustic contrast to be estimated in a reverberant environment, based on the free field directivity of the source and the acoustic characteristics of the environment. Finally, we discuss the behavior of both arrays in a reverberant environment in terms of robustness, and show by means of simulations and measured results, how even without spatial averaging, the reverberation acts as regularization. This constrains the amount of power used and makes the array robust to source mismatches. II. EXPERIMENTAL PROCEDURE
The loudspeaker array is assumed to be placed in a 2D circular control zone, formed by 48 microphones, as shown in Fig. 1. For the experiments presented here, the observation distance, r, is 2 m. A “bright zone,”22 the spatial area where the acoustic pressure is to be maximized, is defined so that it extends from an angle, h, of 0 –15 ; while a “dark zone,” the region in which the acoustic pressure is to be minimized, extends from an angle of 22.5 –352.5 . Two examples of line arrays have been considered, both of them using eight independent sources in the horizontal plane, so that superdirective beamforming is created in this direction. Both of the arrays use phase shift sources,5,23 with a rear port in each cabinet tuned to give a hypercardioid radiation pattern, which minimizes the power input to the reverberant field.24 The first array, 1 8, is shown in the upper part of Fig. 2. The second array, 4 8, which is shown in the lower part of Fig. 2, has four loudspeakers in the vertical direction, which are all driven in phase to obtain natural
FIG. 2. (Color online) View of the two line arrays used in the study; The 1 8 array (top), and the 4 8 array (bottom).
beamforming in the vertical plane. The sources of both arrays are spaced 3.5 cm apart, while the individual sources of the 4 8 array are spaced 5 cm vertically. Both of these arrays have the same theoretical free field performance if measured in a horizontal control geometry. III. SUPERDIRECTIVE BEAMFORMING A. Control and performance metrics definition
The transfer responses between each source of the array and each control microphone can be measured. This allows the formulation of an inverse problem, whose solution will ideally achieve a desired pressure distribution at each frequency. Hence, the formulation presented here assumes radiation at a single frequency. The pressure distribution at all control points can be rearranged into two control vectors, pB and pD of sizes NB 1 and ND 1 respectively. The first of them refers to the pressures of the bright zone, given by pB ¼ ZB q;
(1)
and the second to the pressures of the dark zone, where the mean square pressure is to be minimized, pD ¼ ZD q;
FIG. 1. 2D control geometry used for creating the filters and for measuring the directive performance of both arrays. Black dots represent the dark zone control microphones and open white circles represent the bright zone control microphones. The stars represent the eight sources of the array, and r the observation distance. J. Acoust. Soc. Am., Vol. 135, No. 5, May 2014
(2)
where ZB and ZD are the (NB M) and (ND M) matrices of transfer responses from the volume velocities of each element of the array of M sources, q, to each element in pB and pD. The overall directive performance of the array can be quantified by the acoustic contrast between the mean square pressures in the bright and dark control zones, which is defined as C¼
ND pH ND qH ZH B pB B ZB q ¼ ; H H H NB pD pD NB q ZD ZD q
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(3)
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where H denotes the Hermitian, complex conjugate, transpose and the ratio ND =NB makes the contrast independent of the number of control points in the bright and dark zones, provided they are sufficiently dense. Although the contrast can be directly optimized using the acoustic contrast maximization technique,22 it is only used here as a metric of performance. The normalized array effort is defined as the norm of all the volume velocities in the array, divided by the norm of the volume velocity of a single monopole, which generates the same pressure as that produced by the array in the center of the bright zone, qMON. The normalized array effort can thus be written as AE ¼
qH q jqMON j2
:
(4)
This quantity is also proportional to the amount of electrical power used to drive the array, assuming the electroacoustic interactions between the transducers of the array are negligible. The array effort can be limited to a certain value, preventing damage to the array sources. Array effort and acoustic contrast are dimensionless quantities, whose levels are generally plotted in dB. B. Regularized least squares filters
A widely used superdirective approach is the least squares inverse method,21 which minimizes the error between the generated sound field and a desired sound field. In the work presented here, this method is used for the creation of the filters that drive the array, as it has been shown to provide a good balance between directive performance and audio quality.5 The least squares formulation requires the selection of a target pressure, which provides a desired value of the amplitude and phase of the sound field at each control point. In the least squares formulation both control matrices are rearranged to form a new one, Z, with dimensions N M, of the form ZB ; (5) Z¼ ZD where N ¼ NB þ ND is the total number of control points. The vector of pressures at the N control points due to the array is then defined as p ¼ Zq:
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(9)
where b is a regularization parameter which is used to give a reasonable trade-off between directive performance and electrical power used to drive the array sources.5 The personal audio performance of an array optimized using the least squares method depends strongly on the definition of the vector of target pressures, pT. The acoustic contrast maximization algorithm can be used as a benchmark, to compare with the performance of the least squares solution with different target pressures.5,6 Initial simulations of the behavior of the line array can be performed using point monopole sources, whose transfer impedances are defined as ejkr ; (10) 4pr pffiffiffiffiffiffiffi where j ¼ 1, x represents the angular frequency, q0 ¼ 1.21 kg m3 is the air density, k is the wavenumber and r represents the distance between the monopole and the point where the pressure is being measured. Simulations using an array of eight point monopoles spaced 35 mm apart have been performed with a control zone as shown in Fig. 1 with an observation distance r ¼ 2 m, using the transfer functions from the fourth source to each bright zone control point as target pressure. The results of these simulations, in terms of acoustic contrast and array effort, are shown in Fig. 3, which shows how the array behaves with different levels of regularization. The regularization parameter, b, is selected frequency by frequency so that the normalized array effort does not exceed a certain amount. Theoretically, the directivity of a superdirective array can be made constant at every frequency below the aliasing limit, however, this would need to use a very large amount of power, as observed in Fig. 3, and it is not practical for obvious reasons. A more practically reasonable figure to use for the array effort would be 6 dB, which is the maximum power used throughout the majority of the results presented in this work. Z ¼ jxq0
IV. ESTIMATION OF REVERBERANT DIRECTIVITY
(7)
The least squares solution is the distribution of array source strengths that achieves an acoustic field closest to pT. In order to minimize the mean error, a cost function, J, is defined as J ¼ eH e þ bqH q ¼ ðpT pÞH ðpT pÞ þ bqH q:
q ¼ ½ZH Z þ bI1 ZH pT ;
(6)
An error vector, e, is also introduced, which determines the difference between the physical acoustic field and the desired, target, acoustical field, pT, written as e ¼ pT p:
Substituting Eq. (6) into Eq. (8), differentiating with respect to the real and imaginary parts of q and setting these equal to 0 leads to the following solution in the case of an overdetermined system, such that N > M,
(8)
A. Reverberant directivity formulation
When an acoustical radiator operates inside a reverberant environment, the pressure field it generates will have two components. The first component will be the direct field generated by the source, which is the same as the pressure field created under anechoic conditions. The other component is the reverberant pressure, which is due to all the reflections from the enclosure. In order to estimate the pressure contribution that a source makes to the reverberant field, a knowledge of the acoustic power that it produces is required. If the source is not a point monopole source, the total acoustic n-Ga lvez et al.: Reverberation on personal audio devices Simo
the room walls. The space-average squared reverberant pressure is related to the power radiated by the source, W, by15 hjpR j2 i ¼
4q0 c0 W; R
(13)
with R¼
S^a ; 1 ^a
(14)
where h i denotes spatial averaging, S represents the surface area of the enclosure walls, and ^a is the average absorption coefficient of the walls. This equation allows us to calculate the reverberant pressure that any source produces inside a reverberant environment, once the radiated acoustic power and the absorptive characteristics of the room are known. The radiated power in a diffuse field is assumed to be the same as that radiated into a free space, as originally shown for a monopole.25 Once the reverberant pressure is calculated, this can be combined with the direct pressures from the source, allowing the calculation of the acoustic contrast. The space-average mean square pressures at each control point at a distance r are then written hjpBR ðnb ; rÞji2 ¼ jZB ðnb ; rÞqj2 þ hjpR j2 i;
(15)
hjpDR ðnd ; rÞj2 i ¼ jZD ðnd ; rÞqj2 þ hjpR j2 i;
(16)
and
where nb ¼ 1 : NB and nd ¼ 1 : ND refer to the individual microphones of each of the control zones. FIG. 3. (Color online) Acoustic contrast, C, and array effort, AE, for a line array of eight point monopole sources in the geometry of Fig. 1, for various degrees of regularization.
power radiated can be determined by integrating the intensities over the surface of a sphere.15 Assuming the pressure is measured in the far field, the total radiated acoustic power under free field conditions is given by W¼
r2 q0 c0
ð 2p ð p 0
jp2 ð/; h; rÞjjsinð/Þjd/dh;
(11)
0
where r is the radius of the sphere, c0 ¼ 343 ms1 is the speed of sound in air, p is the acoustic pressure and / and h are the polar and azimuthal angles. If the pressure is only known at a discrete set of control points, however, the integral can be approximated by the summation W¼
2p=D p=D r2 Xh X/ 2 jp ð/n ; hm ; rÞjjsin /n jD/ Dh ; q0 c0 m¼1 n¼1
B. Experimental results
In order to check the validity of the above formulation, its accuracy is compared with measurements of contrast in normal environments. The directive performance of the two arrays described above has been assessed inside an audio listening room26 and inside a classroom. The reverberation times of both rooms against frequency can be observed in Fig. 4, and the dimensions and Schroeder frequencies27 of both rooms, fSCH, are shown in Table I. The audio room has a low reverberation time and a homogeneous distribution of
(12)
where Dh ¼ 2p/NH and D/ ¼ p=NV represent the angle in radians between each horizontal and vertical measurement point, where NH is the number of horizontal measurements and NV is the number of vertical measurements. Under steady-state conditions the power input of a source into a diffuse field is balanced by the absorption of
FIG. 4. (Color online) Reverberation times for the two environments where the reverberant directivities of the array have been studied.
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TABLE I. Geometrical data and Schroeder frequency for the two environments where the reverberant directivities of the array have been studied.
Classroom Audio Room
V (m3)
S (m2)
fSCH (Hz)
86.7 73.4
115.3 112.2
203 111
absorption along its surfaces, while the classroom has a larger reverberation time and a more irregular distribution of absorptive surfaces. The arrays have been placed in the center of the room, at a height of 1 m, and an arc of microphones was placed at 2 m from the array to form a control geometry as shown in Fig. 1. Eyring’s formulation28 has been used to estimate the absorption of the rooms surfaces to allow the mean square pressure in each room to be calculated from the power output of the array, using Eq. (13). The performance that both arrays achieve in the rooms using least squares filters calculated with free field transfer responses has been measured. The performance has also been measured in an anechoic chamber along with the horizontal and vertical transfer responses, which have been sampled at 48 points horizontally and vertically. Values at coordinates different from ðh1 ¼ 0 ; /1 ¼ 0 Þ, have been linearly interpolated, so that each value at /n 6¼ /1 is equal to the values of the horizontal measurement slide, h1, normalized by /n =/1 . This leads to a spherical measurement geometry consisting of a total of 1106 measurement points, which is shown in Fig. 5. By multiplying the transfer responses with the same set of filters as used for the performance measurements, the radiated power of the source can be estimated. The plots of Fig. 6 show the measured acoustic contrast results for the two arrays in the free field and in reverberant conditions, together with the values of the contrast calculated using Eqs. (15) and (16). The array effort needed by the filters in free field conditions has been limited to be lower than 6 dB. For the 1 8 array, a large reduction in performance is observed when it is placed in the reverberant environments, which at 7 kHz is of about 12 dB when placed in the audio room and of about 18 dB when placed in the classroom. The performance of the 4 8 array is less affected by the reverberant field, due to the greater vertical
FIG. 5. (Color online) Measurement geometry used to estimate the power radiated by the two arrays. The closed circles represent the original measured control points and the dots represent the control points which have been obtained by interpolation of the original measured points. The stars represent the eight sources of the array.
directivity of the individual sources, even though its horizontal directivity in the free field is slightly less than the 1 8 array. The acoustic contrast for the 4 8 array in the audio room is about 18 dB at 6 kHz, which is around 5 dB higher than the 1 8 array. In the classroom the 4 8 array obtains an acoustic contrast of about 12 dB at 6 kHz, also around 5 dB higher than the 1 8 array. The predicted reverberant performance of both arrays in the audio room is quite close to the measured reverberant performance. In this environment the results show a mean difference between 500 Hz and 8 kHz of 1.2 dB for the case of the 1 8 array, and of 0.3 dB for the case of the 4 8 array. The predicted reverberant performance in the classroom is, however, not so close to the measured results, with a mean difference of 1.8 dB for the 1 8 array and 1.7 dB for the 4 8 array. Contrary to the audio room, which has an even absorption all along its surfaces, the absorption of the surfaces of the classroom is much more uneven. This contributes to a less diffuse reverberant field, in which measurements of performance at a single point have a greater deviation from the space-averaged result. The directivities of the 4 8 array are shown in Fig. 7 at a number of discrete frequencies, for free field and both
FIG. 6. (Color online) Acoustic contrast and array effort results for the 1 8 array (left hand side plot) and the 4 8 array (right hand side plot). The free field performance is shown by the dashed-dotted lines. The measured reverberant performance is shown by the solid lines and the simulated reverberant performance by the dashed lines, being the audio room results plotted in thin line and the classroom results plotted in thick line. 2658
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FIG. 7. (Color online) Directivities at discrete frequencies for the 4 8 array. The dashed-dotted line represents the free field results, the solid line represents the measured reverberant results and the dashed line represents the simulated reverberant results.
measured reverberant and simulated reverberant results in the audio room. In the free field case it is possible to observe how the secondary lobes are small and how the main lobe is pointed toward the listening zone, from 0 to 15 . The measured reverberant results follow the same pattern, but the level of the secondary lobes is much larger, and the nulls in the directional response are not at the same angles as in the free field. This is due to the influence of reflections from the room walls. The predicted reverberant directivities, which represent the space-averaged case, show a reasonable match with that of the measured reverberant directivities, being around halfway between the peaks and nulls of the measured responses. V. EFFECT OF REVERBERATION ON SYSTEM’S ROBUSTNESS A. Sensitivity of personal audio systems to mismatches in acoustic environment
Above the Schroeder frequency, the reverberant field is considered diffuse and is made of a large number of contributions from waves traveling in all directions,27 becoming a random function of the excitation frequency and the spatial position at which this pressure is measured. Considering the direct and reverberant components of the pressure field inside a reverberant volume, an element of the transfer matrices ZB or ZD may thus be written as Z ¼ ZF þ ZR ;
(17)
reverberant pressure component amplitude and phase will vary between the response measured at each control point, and can be considered as a random variable whose mean square value is determined by the properties of the room. The reverberant components of the transfer response can thus be viewed as uncertainties in their nominal values, which are given by the direct field components. The matrix of transfer responses Z can be written to account for these uncertainties as Z ¼ Z0 þ DZ;
(18)
where Z0 is a matrix of nominal, free field values and DZ is a matrix of the uncertain, diffuse field, components. Supposing that the array is positioned in a room and the measured transfer responses are used to obtain a set of filters, then this is analogous to substituting Eq. (18) into Eq. (9), where the optimal set of source strengths is now given by 1 H H H ðZ0 þDZÞpT : q ¼ ZH 0 Z0 þDZ Z0 þZ0 DZþDZ DZ (19) If the transfer responses are measured in the same control geometry but in a series of different places inside a room, and the mean value of the estimation is taken, the spaceaveraged results are obtained. If it is assumed that the uncertainties are uncorrelated with the nominal values, so that hZ0 DZi ¼ 0;
(20)
where ZF is the deterministic, free field component related to the direct field and ZR is a reverberant component. The
where 0 is the null matrix. The space-averaged properties of the uncertainties are defined as
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hDZH DZi ¼ DZ ;
(21)
so that hZH Zi ¼ ZH 0 Z0 þ DZ ;
(22)
the optimal set of source strengths is given in this case by q ¼ ½ZH Z þ DZ 1 ZH pT :
(23)
If the uncertainties in the elements of Z were independent, as they would be well above Schroeder’s frequency in a diffuse field, then DZ is proportional to an identity matrix, which causes the optimal, space-averaged, solution to be regularized even in the absence of any constraint on the effort.14 This formulation provides an interesting limiting result for a radiator inside a room, which suggests that on average the reverberant field acts as a regularization term. B. Simulation results
where ZN is an N M matrix of uncertainties and S is an M M matrix which causes the uncertainty added to one of the transfer impedances to be a weighted sum according to Eq. (25) of the uncertainties added to adjacent elements. According to models of partially correlated noise,30 S is hence defined as 2 3 sincðkd0 Þ sincðkd1 Þ sincðkdM1 Þ 6 sincðkd1 Þ 7 sincðkd0 Þ 6 7 S ¼ wS 6 7; .. 4 5 ⯗ . sincðkdM1 Þ sincðkd0 Þ (27) where d0 ¼ 0 is the distance between a loudspeaker and itself, d1 is the distance between a loudspeaker and its adjacent, and dM1 is the distance between a loudspeaker and that which is M 1 sources apart. The factor wS is used to normalize the partially correlated obtained uncertainties, so that they have the same power as the original noise, and is given by
Instead of calculating the ensemble average of a series of measurements, a simulation using an eight point monopole array in a control geometry as that shown in Fig. 1 is presented. A single complex element of the transfer matrices which is contaminated with noise, ZN, can be defined as ZN ¼ Z0 ð1 þ SNRcÞ;
(24)
where Z0 is the free field value, c is a number from a Gaussian random noise generator and signal-to-noise ratio (SNR) is selected to control the root-mean-square (RMS) level of the added noise. The control matrices which are contaminated with noise are denoted ZBN ; ZDN ; and ZN. The noise can be added to the transfer responses a priori, before, or a posteriori, after, the calculation of the optimal set of source strengths, or at both times. If the noise is added a priori of the calculation of the optimal set of source strengths, simulating the influence of the diffuse field, however, it should be take into account that the correlation between different points in a three dimensional field follows a sinc function of kdn,29 i.e., R¼
sinðkdn Þ ¼ sincðkdn Þ; kdn
(25)
where dn is the distance between two points in such space. The reverberant sound field created by a single source at two different microphones will be correlated according to Eq. (25). By the same principle and using acoustic reciprocity, the correlation between the sound fields generated by two adjacent loudspeakers follows an identical pattern. Since the spacing between the microphones is at least seven times greater than the spacing between the sources, the correlation between the elements of the uncertainty introduced in the impedance matrices, Eq. (21), is dominated by the source spacing. The different correlation between each source can be taken into account by defining the matrix of uncertainties, DZ, as DZ ¼ ZN S; 2660
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FIG. 8. (Color online) Acoustic contrast and array effort for an eight point monopoles array where noise is added to the transfer responses before the calculation of the optimal set of source strengths, a priori, to simulate the effect of reverberation, or after the calculation of the optimal set of source strengths, a posteriori, to simulate mismatches in the transfer responses. n-Ga lvez et al.: Reverberation on personal audio devices Simo
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u M uX wS ¼ t sinc2 ðkdN Þ:
(28)
n¼1
The optimal set of transfer functions can then be obtained using Eq. (19). If the noise is introduced a posteriori of the calculation of the optimal set of source strengths, simulating mismatches in the transfer responses which are considered to be uncorrelated along the whole frequency band, the acoustic contrast is then given by C¼
H ND qH ZBN ZBN q : H NB qH ZDN ZDN q
(29)
Figure 8 shows the effect of noise in the transfer responses of an eight point monopole sources array with a control zone as shown in Fig. 1. The following set of configurations are assessed. (1) A free field case is shown as a reference, where anechoic transfer responses are used to calculate the vector of optimal filters and the acoustic contrast is calculated according to Eq. (3). (2) “A priori” shows the performance when a single set of random variations are introduced in the transfer responses before the calculation of the optimal vector of source strengths, as in Eq. (19), analogously to placing the array in a room, which reduces the acoustic contrast with respect to the free field case. In this case the SNR variable in Eq. (24) is selected to include RMS random variations up to five times the nominal value of the
transfer responses, which is representative of changes in response due to the reverberation. The array effort in this configuration is very similar to that obtained for the “a priori, Avg.” case, but with a larger variance. (3) “A priori, Avg.” shows the space-averaged result, as defined in Eq. (23), where the acoustic contrast is obtained by adding noise before the calculation of the transfer responses, as for the “a priori” case, but with the result averaged over 200 times. The space-averaged solution is quite similar to that obtained in the “a priori” case using a single set of random numbers, but with less variance. Above 800 Hz, the array effort is much reduced due to regularization implied by Eq. (23). (4) “A posteriori” shows the effect of introducing noise in the transfer responses after the calculation of the optimal, free field, set of source strengths. The acoustic contrast is estimated according to Eq. (29), where SNR has been selected to include RMS random variations up to 0.25 times the nominal value of the transfer responses, which correspond to a sensitivity error of about 1 dB. In this situation both the acoustic contrast and array effort are reduced. The reduction in acoustic contrast is larger at frequencies where the array effort is greater than 0 dB. (5) “A priori and a posteriori” shows the space-averaged result of including a different random noise before and after calculating the optimal vector of source strengths. At some frequencies the performance is lower than for the “a posteriori” case, due to the random noise introduced before the calculation of the optimal set of source strengths, which has provided a strong regularization. At other frequencies, however, the performance is larger
FIG. 9. (Color online) Measured results for the free field (left hand side plots) and reverberant (right hand side plots) acoustic contrast and array results for the 1 8 array for different assumed mismatches in the fourth source of the array. “GF” stands for the result where point monopole Green functions have been used as transfer responses to create the array filters. “In situ” stands for results where the array filters have been calculated with transfer responses measured in the reverberant environment. J. Acoust. Soc. Am., Vol. 135, No. 5, May 2014
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with respect to the “a posteriori” case. The a posteriori introduced uncertainties cause the array effort to be slightly greater than that of the “a priori” case above 800 Hz. At this point, two observations can be made from these results; first, that the space-averaged result is very close to that when only one average is used. Second, random errors introduced in the transfer responses before the calculation of the optimal source strengths vector make the solution robust to perturbations present after the calculation of the optimal vector of source strengths. C. Measurements of robustness in a reverberant field
In order to assess the effect of reverberation on the robustness of personal audio devices, the performance of the two practical arrays shown in Fig. 2 has been measured in the audio room defined in Sec. V B, using different set of filters. The different sets of filters are (1) “NO error”: This case refers to filters created with transfer responses measured in free field. (2) “1 dB error”: In this situation the sensitivity of the fourth source is assumed to be 1 dB higher when calculating the vector of optimal filters from the free field responses.
(3) “3 dB error”: In this situation the sensitivity of the fourth source is assumed to be 3 dB higher when calculating the vector of optimal filters from the free field responses. (4) “GF”: The filters have been designed using point monopole Green functions to model the transfer responses, as previously performed by Ref. 17. The array effort is limited to be lower than 0 dB, as the point monopole Green functions are quite different from the measured transfer responses, and a large amount of regularization is used to overcome the differences in the response. (5) “In situ”: For this case, the transfer responses have been measured in the room, so that the reverberant component is present in Eq. (19). Measurements of this kind have also been performed in Ref. 31, where the authors suggested that the room reflections could effectively increase the number of degrees of freedom of the array. The performance measured with these sets of filters is shown for the 1 8 array in Fig. 9, and for the 4 8 array in Fig. 10. The contrast for the free field is reduced significantly if a mismatch of 1 dB or 3 dB is assumed when calculating the optimal vector of source strengths, or when this is obtained using point monopole Green functions. It can also be observed how the array effort is reduced when mismatches
FIG. 10. (Color online) Measured results for the free field (left hand side plots) and reverberant (right hand side plots) acoustic contrast and array effort results for the 4 8 array when different mismatches exist in the fourth source of the array. “GF” stands for the result where point monopole Green functions have been used as transfer responses to create the array filters. “In situ” stands for results where the array filters have been calculated with transfer responses measured in the reverberant environment. 2662
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are present in the transfer responses, with the reduction in acoustic contrast and array effort being proportional to the magnitude of the mismatches. These differences are much smaller when the performance of the different filters is measured under reverberant conditions. Similar effects of noise and reverberation on the performance are also observed in the results for the 4 8 array, shown in Fig. 10, where the results obtained in a reverberant field again present a close performance. VI. CONCLUSIONS
This paper has investigated how the performance of a loudspeaker array for a personal audio device is affected when it is located in a reverberant environment. The two main effects of the reverberation are first, the reverberation generates an extra pressure component, which on average, is of uniform level in the reverberant space and reduces the free field performance of the source, and second, the reverberation acts as regularization for the solution of the optimal vector of source strengths. The extra pressure component due to reflections from the room surfaces is added to the radiation of the array in each direction, which diminishes the directional characteristics of the radiator. If a large level difference between two spatial zones is desired inside a room, the line array has to be very directional in a 3D sense. As an example, the reduction in the acoustic contrast produced by a superdirective array using eight phase shift hypercardioid sources is considered inside a listening room. The acoustic contrast inside a reverberant space is reduced much less if each individual source is made more directive in the vertical plane, as in the 4 8 array investigated here. A formulation has also been introduced that allows the performance of an acoustic radiator to be estimated in a reverberant environment. The results of this formulation have been compared with measurements of the two arrays inside a reverberant environment, showing a good agreement. This theory can thus be used at the design stage of a line array to predict its characteristics so that the desired sound control is obtained in a room. The effect that the reverberation imposes on the robustness of a line array has also been considered. It has been shown that small mismatches in the transfer responses lead to a large reduction in the free field directivity, but do not lead to such a large reduction in performance when the directivity is measured in a reverberant environment. Even a single measurement of the response in such a reverberant environment, when used to calculate the filters of the array, can lead to significant regularization of the system and enhanced robustness to uncertainties in the response between the loudspeakers and the microphones. 1
W. F. Druyvesteyn and J. Garas, “Personal sound,” J. Audio Eng. Soc. 45, 685–701 (1997). 2 S. J. Elliott and M. Jones, “An active headrest for personal audio,” J. Acoust. Soc. Am. 119, 2702–2709 (2006).
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S. J. Elliott, J. Cheer, H. Murfet, and K. R. Holland, “Minimally radiating sources for personal audio,” J. Acoust. Soc. Am. 128(4), 1721–1728 (2010). 4 J. Cheer, S. J. Elliott, Y. Kim, and J.-W. Choi, “Practical implementation of personal audio in a mobile device,” J. Audio Eng. Soc. 61, 290–300 (2013). 5 M. F. Sim on Galvez, S. J. Elliott, and J. Cheer, “A superdirective array of phase shift sources,” J. Acoust. Soc. Am. 132, 746–756 (2012). 6 J. Cheer, S. J. Elliott, and M. F. Sim on Galvez, “Design and implementation of a car cabin personal audio system,” J. Audio Eng. Soc. 61, 412–424 (2013). 7 F. Fahy and J. Walker, Advanced Applications in Acoustics, Noise & Vibration (Spon Press, London, 2004), Chap. II.3, pp. 100–113. 8 B. D. V. Veen and K. M. Buckley, “Beamforming: A versatile approach to spatial filtering,” IEEE ASSP Mag. 5, 4–24 (1988). 9 H. Cox, R. M. Zeskind, and T. Kooij, “Practical supergain,” IEEE Trans. Acoust. Speech. Signal Process. 34, 393–398 (1986). 10 J. Bitzer and K. U. Simmer, “Superdirective microphone arrays,” in Microphone Arrays (Springer, Berlin, 2001), Chap. 2, pp. 19–38. 11 E. Mabande and W. Kellermann, “Towards superdirective beamforming with loudspeaker arrays,” in 19th International Congress on Acoustics, Madrid (2007). 12 M. M. Boone, W.-H. Cho, and J.-G. Ih, “Design of a highly directional endfire loudspeaker array,” J. Audio Eng. Soc. 57, 309–325 (2009). 13 E. A. Jorswieck, “Optimal transmission strategies and impact of correlation in multiantenna systems with different types of channel state information,” IEEE Trans. Signal Process. 52, 3440–3453 (2004). 14 S. J. Elliott, J. Cheer, J.-W. Choi, and Y. Kim, “Robustness and regularization of personal audio systems,” IEEE Trans. Audio Speech Lang. Process. 20, 2123–2133 (2012). 15 L. L. Beranek, Acoustics (AIP, New York, 1987), 490 pp. 16 B. E. Anderson, B. Moser, and K. L. Gee, “Loudspeaker line array educational demonstration,” J. Acoust. Soc. Am. 131, 2394–2400 (2012). 17 F. Olivieri, M. Shin, F. M. Fazi, P. A. Nelson, and P. Otto, “Loudspeaker array processing form multi-zone audio reproduction based on analytical and measured eletroacustical transfer functions,” in AES 52nd International Conference (Guildford, UK, 2013). 18 E. N. Gilbert and S. P. Morgan, “Optimum design of directive antenna arrays subject to random variations,” Bell Syst. Tech. J. 34, 637–663 (1955). 19 S. Doclo and M. Moonen, “Superdirective beamforming robust against microphone mismatch,” IEEE Trans. Audio Speech Lang. Process. 15, 617–631 (2007). 20 J.-Y. Park, J.-W. Choi, and Y.-H. Kim, “Acoustic contrast sensitivity to transfer function errors in the design of a personal audio system,” J. Acoust. Soc. Am. 134, EL112–EL118 (2013). 21 O. Kirkeby, P. A. Nelson, H. Hamada, and F. Ordu~ na-Bustamante, “Fast deconvolution of multichannel systems using regularization,” IEEE Trans. Audio Speech Lang. Process. 6(2), 189–194 (1998). 22 J. W. Choi and Y. H. Kim, “Generation of an acoustically bright zone with an illuminated region using multiple sources,” J. Acoust. Soc. Am. 111, 1695–1700 (2002). 23 T. J. Holmes, “The acoustic resistance box—A fresh look at an old principle,” J. Audio Eng. Soc. 34, 981–989 (1985). 24 H. F. Olson, Acoustical Engineering (Van Nostrand, Princeton, NJ, 1957), pp. 295–297. 25 R. H. Lyon, “Statistical analysis of power injection and response in structures and rooms,” J. Acoust. Soc. Am. 45, 545–565 (1969). 26 “ISVR listening room,” URL http://www.isvr.co.uk/faciliti/listroom.htm (Last viewed 16/12/2013). 27 H. Kutruff, Room Acoustics (Spon Press, London, 2009), p. 84. 28 C. F. Eyring, “Reverberation time in ‘dead’ rooms,” J. Acoust. Soc. Am. 1, 217–241 (1930). 29 R. K. Cook, R. V. Waterhouse, R. D. Berendt, S. Edelman, and M. C. Thompson, “Measurement of correlation coefficients in reverberant sound fields,” J. Acoust. Soc. Am. 27, 1072–1077 (1955). 30 W. M. Hartmann and Y. J. Cho, “Generating partially correlated noise a comparison of methods,” J. Acoust. Soc. Am. 130, 292–301 (2011). 31 S. Yon, M. Tanter, and M. Fink, “Sound focusing in rooms: The spatio-temporal inverse filter,” J. Acoust. Soc. Am. 114, 3044–3052 (2003).
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(Received 18 December 2013; revised 7 March 2014; accepted 11 March 2014) Personal audio refers to the creation of a listening zone within which a person, or a group of people, hears a given sound program, without being annoyed by other sound programs being reproduced in the same space. Generally, these different sound zones are created by arrays of loudspeakers. Although these devices have the capacity to achieve different sound zones in an anechoic environment, they are ultimately used in normal rooms, which are reverberant environments. At high frequencies, reflections from the room surfaces create a diffuse pressure component which is uniform throughout the room volume and thus decreases the directional characteristics of the device. This paper shows how the reverberant performance of an array can be modeled, knowing the anechoic performance of the radiator and the acoustic characteristics of the room. A formulation is presented whose results are compared to practical measurements in reverberant environments. Due to reflections from the room surfaces, pressure variations are introduced in the transfer responses of the array. This aspect is assessed by means of simulations where random noise is added to create uncertainties, and by performing measurements in a real environment. These results show how the robustness of an array is increased when it is designed for use in a reverberant environment. C 2014 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4869681] V PACS number(s): 43.38.Hz, 43.60.Gk, 43.55.Jz, 43.60.Pt [MRB]
I. INTRODUCTION
Reproduction of different sound programs for different users in a common space is becoming widespread. There is also an increasing concern about selective audio reproduction, as different users may not want to listen to an audio program that is not coming from their own device. These objectives can be achieved using “personal audio” devices, which have been discussed by Druyvesteyn and Garas,1 with reference to systems used to create a selective reproduction of the soundfield in a room. Personal audio systems have also been considered for sound reproduction in individual seats using headrest loudspeakers,2 to restrict sound radiation for mobile phones,3,4 to direct the sound coming from a television toward a determined spatial region,5 or to create different sound zones inside a car.6 Conventional acoustic arrays perform poorly at frequencies where the wavelength of radiation is greater than the aperture size of the array,7 unless optimal, or superdirective, beamforming techniques are used. Such techniques have been widely used in the context of sensor arrays,8–10 and by reciprocity have been applied to acoustical source arrays.5,11,12 These techniques do, however, need large gains at frequencies with high spatial correlation, such that the transfer responses from each source to a given point are very similar.13 These large gains are accompanied by greater sensitivity to uncertainties in the acoustic environment.8,14 Although the final application of such personal audio systems is in a normal, everyday room, the behavior of these systems has generally been studied in anechoic conditions, and the effects of reverberation are not well understood. The a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
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performance of a single directional acoustical radiator in a reverberant field was studied by Beranek,15 who derived an expression for the total acoustic field depending on the acoustic characteristics of the room and the radiator. A similar formulation was used to obtain the directional characteristics of a source in a reverberant environment by Druyvesteyn and Garas.1 Measurements of the performance of a line array in reverberant environments have also been presented.16 Other recent work has analyzed the performance of different transfer response models for creating the filters of a superdirective array in a reverberant environment.17 It is clear that the directional characteristics of an acoustical radiator are changed when it is placed in a reverberant room, but it is not so clear what the effect of the reverberation is on the robustness of a superdirective beamformer. The robustness of these radiators is defined in terms of how much the directional response is affected by mismatches in source sensitivity and position. In superdirective antennas, for example, it has been observed how the practical response is far from the theoretical superdirective value if random errors are introduced in the antenna weightings.18 The characteristics of these errors and how to obtain robust beamforming in their response has been studied for microphone arrays, assuming that the errors have a certain probability factor.19 Work on loudspeaker arrays has also studied the effect of errors in source sensitivity and position, and how they affect the ratio of mean square pressures between two spatial zones.14,20 Generally, these acoustical radiators are made robust to changes in their transfer responses by applying regularization,12,14 which also controls the power used to drive the array sources. Recent work has suggested that the space-averaged uncertainties introduced into the transfer responses of an array by reverberation can act analogously to regularization.14
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C 2014 Acoustical Society of America V
This paper investigates the behavior of two personal audio arrays inside reverberant environments. First, superdirective array theory and the creation of least squares filters21 are considered. In order to assess the performance of both devices, the acoustic contrast, as defined by Choi and Kim,22 is used as a metric, which quantifies the mean square pressure ratio between two spatial zones. A formulation is then introduced, which allows the acoustic contrast to be estimated in a reverberant environment, based on the free field directivity of the source and the acoustic characteristics of the environment. Finally, we discuss the behavior of both arrays in a reverberant environment in terms of robustness, and show by means of simulations and measured results, how even without spatial averaging, the reverberation acts as regularization. This constrains the amount of power used and makes the array robust to source mismatches. II. EXPERIMENTAL PROCEDURE
The loudspeaker array is assumed to be placed in a 2D circular control zone, formed by 48 microphones, as shown in Fig. 1. For the experiments presented here, the observation distance, r, is 2 m. A “bright zone,”22 the spatial area where the acoustic pressure is to be maximized, is defined so that it extends from an angle, h, of 0 –15 ; while a “dark zone,” the region in which the acoustic pressure is to be minimized, extends from an angle of 22.5 –352.5 . Two examples of line arrays have been considered, both of them using eight independent sources in the horizontal plane, so that superdirective beamforming is created in this direction. Both of the arrays use phase shift sources,5,23 with a rear port in each cabinet tuned to give a hypercardioid radiation pattern, which minimizes the power input to the reverberant field.24 The first array, 1 8, is shown in the upper part of Fig. 2. The second array, 4 8, which is shown in the lower part of Fig. 2, has four loudspeakers in the vertical direction, which are all driven in phase to obtain natural
FIG. 2. (Color online) View of the two line arrays used in the study; The 1 8 array (top), and the 4 8 array (bottom).
beamforming in the vertical plane. The sources of both arrays are spaced 3.5 cm apart, while the individual sources of the 4 8 array are spaced 5 cm vertically. Both of these arrays have the same theoretical free field performance if measured in a horizontal control geometry. III. SUPERDIRECTIVE BEAMFORMING A. Control and performance metrics definition
The transfer responses between each source of the array and each control microphone can be measured. This allows the formulation of an inverse problem, whose solution will ideally achieve a desired pressure distribution at each frequency. Hence, the formulation presented here assumes radiation at a single frequency. The pressure distribution at all control points can be rearranged into two control vectors, pB and pD of sizes NB 1 and ND 1 respectively. The first of them refers to the pressures of the bright zone, given by pB ¼ ZB q;
(1)
and the second to the pressures of the dark zone, where the mean square pressure is to be minimized, pD ¼ ZD q;
FIG. 1. 2D control geometry used for creating the filters and for measuring the directive performance of both arrays. Black dots represent the dark zone control microphones and open white circles represent the bright zone control microphones. The stars represent the eight sources of the array, and r the observation distance. J. Acoust. Soc. Am., Vol. 135, No. 5, May 2014
(2)
where ZB and ZD are the (NB M) and (ND M) matrices of transfer responses from the volume velocities of each element of the array of M sources, q, to each element in pB and pD. The overall directive performance of the array can be quantified by the acoustic contrast between the mean square pressures in the bright and dark control zones, which is defined as C¼
ND pH ND qH ZH B pB B ZB q ¼ ; H H H NB pD pD NB q ZD ZD q
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(3)
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where H denotes the Hermitian, complex conjugate, transpose and the ratio ND =NB makes the contrast independent of the number of control points in the bright and dark zones, provided they are sufficiently dense. Although the contrast can be directly optimized using the acoustic contrast maximization technique,22 it is only used here as a metric of performance. The normalized array effort is defined as the norm of all the volume velocities in the array, divided by the norm of the volume velocity of a single monopole, which generates the same pressure as that produced by the array in the center of the bright zone, qMON. The normalized array effort can thus be written as AE ¼
qH q jqMON j2
:
(4)
This quantity is also proportional to the amount of electrical power used to drive the array, assuming the electroacoustic interactions between the transducers of the array are negligible. The array effort can be limited to a certain value, preventing damage to the array sources. Array effort and acoustic contrast are dimensionless quantities, whose levels are generally plotted in dB. B. Regularized least squares filters
A widely used superdirective approach is the least squares inverse method,21 which minimizes the error between the generated sound field and a desired sound field. In the work presented here, this method is used for the creation of the filters that drive the array, as it has been shown to provide a good balance between directive performance and audio quality.5 The least squares formulation requires the selection of a target pressure, which provides a desired value of the amplitude and phase of the sound field at each control point. In the least squares formulation both control matrices are rearranged to form a new one, Z, with dimensions N M, of the form ZB ; (5) Z¼ ZD where N ¼ NB þ ND is the total number of control points. The vector of pressures at the N control points due to the array is then defined as p ¼ Zq:
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(9)
where b is a regularization parameter which is used to give a reasonable trade-off between directive performance and electrical power used to drive the array sources.5 The personal audio performance of an array optimized using the least squares method depends strongly on the definition of the vector of target pressures, pT. The acoustic contrast maximization algorithm can be used as a benchmark, to compare with the performance of the least squares solution with different target pressures.5,6 Initial simulations of the behavior of the line array can be performed using point monopole sources, whose transfer impedances are defined as ejkr ; (10) 4pr pffiffiffiffiffiffiffi where j ¼ 1, x represents the angular frequency, q0 ¼ 1.21 kg m3 is the air density, k is the wavenumber and r represents the distance between the monopole and the point where the pressure is being measured. Simulations using an array of eight point monopoles spaced 35 mm apart have been performed with a control zone as shown in Fig. 1 with an observation distance r ¼ 2 m, using the transfer functions from the fourth source to each bright zone control point as target pressure. The results of these simulations, in terms of acoustic contrast and array effort, are shown in Fig. 3, which shows how the array behaves with different levels of regularization. The regularization parameter, b, is selected frequency by frequency so that the normalized array effort does not exceed a certain amount. Theoretically, the directivity of a superdirective array can be made constant at every frequency below the aliasing limit, however, this would need to use a very large amount of power, as observed in Fig. 3, and it is not practical for obvious reasons. A more practically reasonable figure to use for the array effort would be 6 dB, which is the maximum power used throughout the majority of the results presented in this work. Z ¼ jxq0
IV. ESTIMATION OF REVERBERANT DIRECTIVITY
(7)
The least squares solution is the distribution of array source strengths that achieves an acoustic field closest to pT. In order to minimize the mean error, a cost function, J, is defined as J ¼ eH e þ bqH q ¼ ðpT pÞH ðpT pÞ þ bqH q:
q ¼ ½ZH Z þ bI1 ZH pT ;
(6)
An error vector, e, is also introduced, which determines the difference between the physical acoustic field and the desired, target, acoustical field, pT, written as e ¼ pT p:
Substituting Eq. (6) into Eq. (8), differentiating with respect to the real and imaginary parts of q and setting these equal to 0 leads to the following solution in the case of an overdetermined system, such that N > M,
(8)
A. Reverberant directivity formulation
When an acoustical radiator operates inside a reverberant environment, the pressure field it generates will have two components. The first component will be the direct field generated by the source, which is the same as the pressure field created under anechoic conditions. The other component is the reverberant pressure, which is due to all the reflections from the enclosure. In order to estimate the pressure contribution that a source makes to the reverberant field, a knowledge of the acoustic power that it produces is required. If the source is not a point monopole source, the total acoustic n-Ga lvez et al.: Reverberation on personal audio devices Simo
the room walls. The space-average squared reverberant pressure is related to the power radiated by the source, W, by15 hjpR j2 i ¼
4q0 c0 W; R
(13)
with R¼
S^a ; 1 ^a
(14)
where h i denotes spatial averaging, S represents the surface area of the enclosure walls, and ^a is the average absorption coefficient of the walls. This equation allows us to calculate the reverberant pressure that any source produces inside a reverberant environment, once the radiated acoustic power and the absorptive characteristics of the room are known. The radiated power in a diffuse field is assumed to be the same as that radiated into a free space, as originally shown for a monopole.25 Once the reverberant pressure is calculated, this can be combined with the direct pressures from the source, allowing the calculation of the acoustic contrast. The space-average mean square pressures at each control point at a distance r are then written hjpBR ðnb ; rÞji2 ¼ jZB ðnb ; rÞqj2 þ hjpR j2 i;
(15)
hjpDR ðnd ; rÞj2 i ¼ jZD ðnd ; rÞqj2 þ hjpR j2 i;
(16)
and
where nb ¼ 1 : NB and nd ¼ 1 : ND refer to the individual microphones of each of the control zones. FIG. 3. (Color online) Acoustic contrast, C, and array effort, AE, for a line array of eight point monopole sources in the geometry of Fig. 1, for various degrees of regularization.
power radiated can be determined by integrating the intensities over the surface of a sphere.15 Assuming the pressure is measured in the far field, the total radiated acoustic power under free field conditions is given by W¼
r2 q0 c0
ð 2p ð p 0
jp2 ð/; h; rÞjjsinð/Þjd/dh;
(11)
0
where r is the radius of the sphere, c0 ¼ 343 ms1 is the speed of sound in air, p is the acoustic pressure and / and h are the polar and azimuthal angles. If the pressure is only known at a discrete set of control points, however, the integral can be approximated by the summation W¼
2p=D p=D r2 Xh X/ 2 jp ð/n ; hm ; rÞjjsin /n jD/ Dh ; q0 c0 m¼1 n¼1
B. Experimental results
In order to check the validity of the above formulation, its accuracy is compared with measurements of contrast in normal environments. The directive performance of the two arrays described above has been assessed inside an audio listening room26 and inside a classroom. The reverberation times of both rooms against frequency can be observed in Fig. 4, and the dimensions and Schroeder frequencies27 of both rooms, fSCH, are shown in Table I. The audio room has a low reverberation time and a homogeneous distribution of
(12)
where Dh ¼ 2p/NH and D/ ¼ p=NV represent the angle in radians between each horizontal and vertical measurement point, where NH is the number of horizontal measurements and NV is the number of vertical measurements. Under steady-state conditions the power input of a source into a diffuse field is balanced by the absorption of
FIG. 4. (Color online) Reverberation times for the two environments where the reverberant directivities of the array have been studied.
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TABLE I. Geometrical data and Schroeder frequency for the two environments where the reverberant directivities of the array have been studied.
Classroom Audio Room
V (m3)
S (m2)
fSCH (Hz)
86.7 73.4
115.3 112.2
203 111
absorption along its surfaces, while the classroom has a larger reverberation time and a more irregular distribution of absorptive surfaces. The arrays have been placed in the center of the room, at a height of 1 m, and an arc of microphones was placed at 2 m from the array to form a control geometry as shown in Fig. 1. Eyring’s formulation28 has been used to estimate the absorption of the rooms surfaces to allow the mean square pressure in each room to be calculated from the power output of the array, using Eq. (13). The performance that both arrays achieve in the rooms using least squares filters calculated with free field transfer responses has been measured. The performance has also been measured in an anechoic chamber along with the horizontal and vertical transfer responses, which have been sampled at 48 points horizontally and vertically. Values at coordinates different from ðh1 ¼ 0 ; /1 ¼ 0 Þ, have been linearly interpolated, so that each value at /n 6¼ /1 is equal to the values of the horizontal measurement slide, h1, normalized by /n =/1 . This leads to a spherical measurement geometry consisting of a total of 1106 measurement points, which is shown in Fig. 5. By multiplying the transfer responses with the same set of filters as used for the performance measurements, the radiated power of the source can be estimated. The plots of Fig. 6 show the measured acoustic contrast results for the two arrays in the free field and in reverberant conditions, together with the values of the contrast calculated using Eqs. (15) and (16). The array effort needed by the filters in free field conditions has been limited to be lower than 6 dB. For the 1 8 array, a large reduction in performance is observed when it is placed in the reverberant environments, which at 7 kHz is of about 12 dB when placed in the audio room and of about 18 dB when placed in the classroom. The performance of the 4 8 array is less affected by the reverberant field, due to the greater vertical
FIG. 5. (Color online) Measurement geometry used to estimate the power radiated by the two arrays. The closed circles represent the original measured control points and the dots represent the control points which have been obtained by interpolation of the original measured points. The stars represent the eight sources of the array.
directivity of the individual sources, even though its horizontal directivity in the free field is slightly less than the 1 8 array. The acoustic contrast for the 4 8 array in the audio room is about 18 dB at 6 kHz, which is around 5 dB higher than the 1 8 array. In the classroom the 4 8 array obtains an acoustic contrast of about 12 dB at 6 kHz, also around 5 dB higher than the 1 8 array. The predicted reverberant performance of both arrays in the audio room is quite close to the measured reverberant performance. In this environment the results show a mean difference between 500 Hz and 8 kHz of 1.2 dB for the case of the 1 8 array, and of 0.3 dB for the case of the 4 8 array. The predicted reverberant performance in the classroom is, however, not so close to the measured results, with a mean difference of 1.8 dB for the 1 8 array and 1.7 dB for the 4 8 array. Contrary to the audio room, which has an even absorption all along its surfaces, the absorption of the surfaces of the classroom is much more uneven. This contributes to a less diffuse reverberant field, in which measurements of performance at a single point have a greater deviation from the space-averaged result. The directivities of the 4 8 array are shown in Fig. 7 at a number of discrete frequencies, for free field and both
FIG. 6. (Color online) Acoustic contrast and array effort results for the 1 8 array (left hand side plot) and the 4 8 array (right hand side plot). The free field performance is shown by the dashed-dotted lines. The measured reverberant performance is shown by the solid lines and the simulated reverberant performance by the dashed lines, being the audio room results plotted in thin line and the classroom results plotted in thick line. 2658
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FIG. 7. (Color online) Directivities at discrete frequencies for the 4 8 array. The dashed-dotted line represents the free field results, the solid line represents the measured reverberant results and the dashed line represents the simulated reverberant results.
measured reverberant and simulated reverberant results in the audio room. In the free field case it is possible to observe how the secondary lobes are small and how the main lobe is pointed toward the listening zone, from 0 to 15 . The measured reverberant results follow the same pattern, but the level of the secondary lobes is much larger, and the nulls in the directional response are not at the same angles as in the free field. This is due to the influence of reflections from the room walls. The predicted reverberant directivities, which represent the space-averaged case, show a reasonable match with that of the measured reverberant directivities, being around halfway between the peaks and nulls of the measured responses. V. EFFECT OF REVERBERATION ON SYSTEM’S ROBUSTNESS A. Sensitivity of personal audio systems to mismatches in acoustic environment
Above the Schroeder frequency, the reverberant field is considered diffuse and is made of a large number of contributions from waves traveling in all directions,27 becoming a random function of the excitation frequency and the spatial position at which this pressure is measured. Considering the direct and reverberant components of the pressure field inside a reverberant volume, an element of the transfer matrices ZB or ZD may thus be written as Z ¼ ZF þ ZR ;
(17)
reverberant pressure component amplitude and phase will vary between the response measured at each control point, and can be considered as a random variable whose mean square value is determined by the properties of the room. The reverberant components of the transfer response can thus be viewed as uncertainties in their nominal values, which are given by the direct field components. The matrix of transfer responses Z can be written to account for these uncertainties as Z ¼ Z0 þ DZ;
(18)
where Z0 is a matrix of nominal, free field values and DZ is a matrix of the uncertain, diffuse field, components. Supposing that the array is positioned in a room and the measured transfer responses are used to obtain a set of filters, then this is analogous to substituting Eq. (18) into Eq. (9), where the optimal set of source strengths is now given by 1 H H H ðZ0 þDZÞpT : q ¼ ZH 0 Z0 þDZ Z0 þZ0 DZþDZ DZ (19) If the transfer responses are measured in the same control geometry but in a series of different places inside a room, and the mean value of the estimation is taken, the spaceaveraged results are obtained. If it is assumed that the uncertainties are uncorrelated with the nominal values, so that hZ0 DZi ¼ 0;
(20)
where ZF is the deterministic, free field component related to the direct field and ZR is a reverberant component. The
where 0 is the null matrix. The space-averaged properties of the uncertainties are defined as
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hDZH DZi ¼ DZ ;
(21)
so that hZH Zi ¼ ZH 0 Z0 þ DZ ;
(22)
the optimal set of source strengths is given in this case by q ¼ ½ZH Z þ DZ 1 ZH pT :
(23)
If the uncertainties in the elements of Z were independent, as they would be well above Schroeder’s frequency in a diffuse field, then DZ is proportional to an identity matrix, which causes the optimal, space-averaged, solution to be regularized even in the absence of any constraint on the effort.14 This formulation provides an interesting limiting result for a radiator inside a room, which suggests that on average the reverberant field acts as a regularization term. B. Simulation results
where ZN is an N M matrix of uncertainties and S is an M M matrix which causes the uncertainty added to one of the transfer impedances to be a weighted sum according to Eq. (25) of the uncertainties added to adjacent elements. According to models of partially correlated noise,30 S is hence defined as 2 3 sincðkd0 Þ sincðkd1 Þ sincðkdM1 Þ 6 sincðkd1 Þ 7 sincðkd0 Þ 6 7 S ¼ wS 6 7; .. 4 5 ⯗ . sincðkdM1 Þ sincðkd0 Þ (27) where d0 ¼ 0 is the distance between a loudspeaker and itself, d1 is the distance between a loudspeaker and its adjacent, and dM1 is the distance between a loudspeaker and that which is M 1 sources apart. The factor wS is used to normalize the partially correlated obtained uncertainties, so that they have the same power as the original noise, and is given by
Instead of calculating the ensemble average of a series of measurements, a simulation using an eight point monopole array in a control geometry as that shown in Fig. 1 is presented. A single complex element of the transfer matrices which is contaminated with noise, ZN, can be defined as ZN ¼ Z0 ð1 þ SNRcÞ;
(24)
where Z0 is the free field value, c is a number from a Gaussian random noise generator and signal-to-noise ratio (SNR) is selected to control the root-mean-square (RMS) level of the added noise. The control matrices which are contaminated with noise are denoted ZBN ; ZDN ; and ZN. The noise can be added to the transfer responses a priori, before, or a posteriori, after, the calculation of the optimal set of source strengths, or at both times. If the noise is added a priori of the calculation of the optimal set of source strengths, simulating the influence of the diffuse field, however, it should be take into account that the correlation between different points in a three dimensional field follows a sinc function of kdn,29 i.e., R¼
sinðkdn Þ ¼ sincðkdn Þ; kdn
(25)
where dn is the distance between two points in such space. The reverberant sound field created by a single source at two different microphones will be correlated according to Eq. (25). By the same principle and using acoustic reciprocity, the correlation between the sound fields generated by two adjacent loudspeakers follows an identical pattern. Since the spacing between the microphones is at least seven times greater than the spacing between the sources, the correlation between the elements of the uncertainty introduced in the impedance matrices, Eq. (21), is dominated by the source spacing. The different correlation between each source can be taken into account by defining the matrix of uncertainties, DZ, as DZ ¼ ZN S; 2660
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(26)
FIG. 8. (Color online) Acoustic contrast and array effort for an eight point monopoles array where noise is added to the transfer responses before the calculation of the optimal set of source strengths, a priori, to simulate the effect of reverberation, or after the calculation of the optimal set of source strengths, a posteriori, to simulate mismatches in the transfer responses. n-Ga lvez et al.: Reverberation on personal audio devices Simo
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u M uX wS ¼ t sinc2 ðkdN Þ:
(28)
n¼1
The optimal set of transfer functions can then be obtained using Eq. (19). If the noise is introduced a posteriori of the calculation of the optimal set of source strengths, simulating mismatches in the transfer responses which are considered to be uncorrelated along the whole frequency band, the acoustic contrast is then given by C¼
H ND qH ZBN ZBN q : H NB qH ZDN ZDN q
(29)
Figure 8 shows the effect of noise in the transfer responses of an eight point monopole sources array with a control zone as shown in Fig. 1. The following set of configurations are assessed. (1) A free field case is shown as a reference, where anechoic transfer responses are used to calculate the vector of optimal filters and the acoustic contrast is calculated according to Eq. (3). (2) “A priori” shows the performance when a single set of random variations are introduced in the transfer responses before the calculation of the optimal vector of source strengths, as in Eq. (19), analogously to placing the array in a room, which reduces the acoustic contrast with respect to the free field case. In this case the SNR variable in Eq. (24) is selected to include RMS random variations up to five times the nominal value of the
transfer responses, which is representative of changes in response due to the reverberation. The array effort in this configuration is very similar to that obtained for the “a priori, Avg.” case, but with a larger variance. (3) “A priori, Avg.” shows the space-averaged result, as defined in Eq. (23), where the acoustic contrast is obtained by adding noise before the calculation of the transfer responses, as for the “a priori” case, but with the result averaged over 200 times. The space-averaged solution is quite similar to that obtained in the “a priori” case using a single set of random numbers, but with less variance. Above 800 Hz, the array effort is much reduced due to regularization implied by Eq. (23). (4) “A posteriori” shows the effect of introducing noise in the transfer responses after the calculation of the optimal, free field, set of source strengths. The acoustic contrast is estimated according to Eq. (29), where SNR has been selected to include RMS random variations up to 0.25 times the nominal value of the transfer responses, which correspond to a sensitivity error of about 1 dB. In this situation both the acoustic contrast and array effort are reduced. The reduction in acoustic contrast is larger at frequencies where the array effort is greater than 0 dB. (5) “A priori and a posteriori” shows the space-averaged result of including a different random noise before and after calculating the optimal vector of source strengths. At some frequencies the performance is lower than for the “a posteriori” case, due to the random noise introduced before the calculation of the optimal set of source strengths, which has provided a strong regularization. At other frequencies, however, the performance is larger
FIG. 9. (Color online) Measured results for the free field (left hand side plots) and reverberant (right hand side plots) acoustic contrast and array results for the 1 8 array for different assumed mismatches in the fourth source of the array. “GF” stands for the result where point monopole Green functions have been used as transfer responses to create the array filters. “In situ” stands for results where the array filters have been calculated with transfer responses measured in the reverberant environment. J. Acoust. Soc. Am., Vol. 135, No. 5, May 2014
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with respect to the “a posteriori” case. The a posteriori introduced uncertainties cause the array effort to be slightly greater than that of the “a priori” case above 800 Hz. At this point, two observations can be made from these results; first, that the space-averaged result is very close to that when only one average is used. Second, random errors introduced in the transfer responses before the calculation of the optimal source strengths vector make the solution robust to perturbations present after the calculation of the optimal vector of source strengths. C. Measurements of robustness in a reverberant field
In order to assess the effect of reverberation on the robustness of personal audio devices, the performance of the two practical arrays shown in Fig. 2 has been measured in the audio room defined in Sec. V B, using different set of filters. The different sets of filters are (1) “NO error”: This case refers to filters created with transfer responses measured in free field. (2) “1 dB error”: In this situation the sensitivity of the fourth source is assumed to be 1 dB higher when calculating the vector of optimal filters from the free field responses.
(3) “3 dB error”: In this situation the sensitivity of the fourth source is assumed to be 3 dB higher when calculating the vector of optimal filters from the free field responses. (4) “GF”: The filters have been designed using point monopole Green functions to model the transfer responses, as previously performed by Ref. 17. The array effort is limited to be lower than 0 dB, as the point monopole Green functions are quite different from the measured transfer responses, and a large amount of regularization is used to overcome the differences in the response. (5) “In situ”: For this case, the transfer responses have been measured in the room, so that the reverberant component is present in Eq. (19). Measurements of this kind have also been performed in Ref. 31, where the authors suggested that the room reflections could effectively increase the number of degrees of freedom of the array. The performance measured with these sets of filters is shown for the 1 8 array in Fig. 9, and for the 4 8 array in Fig. 10. The contrast for the free field is reduced significantly if a mismatch of 1 dB or 3 dB is assumed when calculating the optimal vector of source strengths, or when this is obtained using point monopole Green functions. It can also be observed how the array effort is reduced when mismatches
FIG. 10. (Color online) Measured results for the free field (left hand side plots) and reverberant (right hand side plots) acoustic contrast and array effort results for the 4 8 array when different mismatches exist in the fourth source of the array. “GF” stands for the result where point monopole Green functions have been used as transfer responses to create the array filters. “In situ” stands for results where the array filters have been calculated with transfer responses measured in the reverberant environment. 2662
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are present in the transfer responses, with the reduction in acoustic contrast and array effort being proportional to the magnitude of the mismatches. These differences are much smaller when the performance of the different filters is measured under reverberant conditions. Similar effects of noise and reverberation on the performance are also observed in the results for the 4 8 array, shown in Fig. 10, where the results obtained in a reverberant field again present a close performance. VI. CONCLUSIONS
This paper has investigated how the performance of a loudspeaker array for a personal audio device is affected when it is located in a reverberant environment. The two main effects of the reverberation are first, the reverberation generates an extra pressure component, which on average, is of uniform level in the reverberant space and reduces the free field performance of the source, and second, the reverberation acts as regularization for the solution of the optimal vector of source strengths. The extra pressure component due to reflections from the room surfaces is added to the radiation of the array in each direction, which diminishes the directional characteristics of the radiator. If a large level difference between two spatial zones is desired inside a room, the line array has to be very directional in a 3D sense. As an example, the reduction in the acoustic contrast produced by a superdirective array using eight phase shift hypercardioid sources is considered inside a listening room. The acoustic contrast inside a reverberant space is reduced much less if each individual source is made more directive in the vertical plane, as in the 4 8 array investigated here. A formulation has also been introduced that allows the performance of an acoustic radiator to be estimated in a reverberant environment. The results of this formulation have been compared with measurements of the two arrays inside a reverberant environment, showing a good agreement. This theory can thus be used at the design stage of a line array to predict its characteristics so that the desired sound control is obtained in a room. The effect that the reverberation imposes on the robustness of a line array has also been considered. It has been shown that small mismatches in the transfer responses lead to a large reduction in the free field directivity, but do not lead to such a large reduction in performance when the directivity is measured in a reverberant environment. Even a single measurement of the response in such a reverberant environment, when used to calculate the filters of the array, can lead to significant regularization of the system and enhanced robustness to uncertainties in the response between the loudspeakers and the microphones. 1
W. F. Druyvesteyn and J. Garas, “Personal sound,” J. Audio Eng. Soc. 45, 685–701 (1997). 2 S. J. Elliott and M. Jones, “An active headrest for personal audio,” J. Acoust. Soc. Am. 119, 2702–2709 (2006).
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S. J. Elliott, J. Cheer, H. Murfet, and K. R. Holland, “Minimally radiating sources for personal audio,” J. Acoust. Soc. Am. 128(4), 1721–1728 (2010). 4 J. Cheer, S. J. Elliott, Y. Kim, and J.-W. Choi, “Practical implementation of personal audio in a mobile device,” J. Audio Eng. Soc. 61, 290–300 (2013). 5 M. F. Sim on Galvez, S. J. Elliott, and J. Cheer, “A superdirective array of phase shift sources,” J. Acoust. Soc. Am. 132, 746–756 (2012). 6 J. Cheer, S. J. Elliott, and M. F. Sim on Galvez, “Design and implementation of a car cabin personal audio system,” J. Audio Eng. Soc. 61, 412–424 (2013). 7 F. Fahy and J. Walker, Advanced Applications in Acoustics, Noise & Vibration (Spon Press, London, 2004), Chap. II.3, pp. 100–113. 8 B. D. V. Veen and K. M. Buckley, “Beamforming: A versatile approach to spatial filtering,” IEEE ASSP Mag. 5, 4–24 (1988). 9 H. Cox, R. M. Zeskind, and T. Kooij, “Practical supergain,” IEEE Trans. Acoust. Speech. Signal Process. 34, 393–398 (1986). 10 J. Bitzer and K. U. Simmer, “Superdirective microphone arrays,” in Microphone Arrays (Springer, Berlin, 2001), Chap. 2, pp. 19–38. 11 E. Mabande and W. Kellermann, “Towards superdirective beamforming with loudspeaker arrays,” in 19th International Congress on Acoustics, Madrid (2007). 12 M. M. Boone, W.-H. Cho, and J.-G. Ih, “Design of a highly directional endfire loudspeaker array,” J. Audio Eng. Soc. 57, 309–325 (2009). 13 E. A. Jorswieck, “Optimal transmission strategies and impact of correlation in multiantenna systems with different types of channel state information,” IEEE Trans. Signal Process. 52, 3440–3453 (2004). 14 S. J. Elliott, J. Cheer, J.-W. Choi, and Y. Kim, “Robustness and regularization of personal audio systems,” IEEE Trans. Audio Speech Lang. Process. 20, 2123–2133 (2012). 15 L. L. Beranek, Acoustics (AIP, New York, 1987), 490 pp. 16 B. E. Anderson, B. Moser, and K. L. Gee, “Loudspeaker line array educational demonstration,” J. Acoust. Soc. Am. 131, 2394–2400 (2012). 17 F. Olivieri, M. Shin, F. M. Fazi, P. A. Nelson, and P. Otto, “Loudspeaker array processing form multi-zone audio reproduction based on analytical and measured eletroacustical transfer functions,” in AES 52nd International Conference (Guildford, UK, 2013). 18 E. N. Gilbert and S. P. Morgan, “Optimum design of directive antenna arrays subject to random variations,” Bell Syst. Tech. J. 34, 637–663 (1955). 19 S. Doclo and M. Moonen, “Superdirective beamforming robust against microphone mismatch,” IEEE Trans. Audio Speech Lang. Process. 15, 617–631 (2007). 20 J.-Y. Park, J.-W. Choi, and Y.-H. Kim, “Acoustic contrast sensitivity to transfer function errors in the design of a personal audio system,” J. Acoust. Soc. Am. 134, EL112–EL118 (2013). 21 O. Kirkeby, P. A. Nelson, H. Hamada, and F. Ordu~ na-Bustamante, “Fast deconvolution of multichannel systems using regularization,” IEEE Trans. Audio Speech Lang. Process. 6(2), 189–194 (1998). 22 J. W. Choi and Y. H. Kim, “Generation of an acoustically bright zone with an illuminated region using multiple sources,” J. Acoust. Soc. Am. 111, 1695–1700 (2002). 23 T. J. Holmes, “The acoustic resistance box—A fresh look at an old principle,” J. Audio Eng. Soc. 34, 981–989 (1985). 24 H. F. Olson, Acoustical Engineering (Van Nostrand, Princeton, NJ, 1957), pp. 295–297. 25 R. H. Lyon, “Statistical analysis of power injection and response in structures and rooms,” J. Acoust. Soc. Am. 45, 545–565 (1969). 26 “ISVR listening room,” URL http://www.isvr.co.uk/faciliti/listroom.htm (Last viewed 16/12/2013). 27 H. Kutruff, Room Acoustics (Spon Press, London, 2009), p. 84. 28 C. F. Eyring, “Reverberation time in ‘dead’ rooms,” J. Acoust. Soc. Am. 1, 217–241 (1930). 29 R. K. Cook, R. V. Waterhouse, R. D. Berendt, S. Edelman, and M. C. Thompson, “Measurement of correlation coefficients in reverberant sound fields,” J. Acoust. Soc. Am. 27, 1072–1077 (1955). 30 W. M. Hartmann and Y. J. Cho, “Generating partially correlated noise a comparison of methods,” J. Acoust. Soc. Am. 130, 292–301 (2011). 31 S. Yon, M. Tanter, and M. Fink, “Sound focusing in rooms: The spatio-temporal inverse filter,” J. Acoust. Soc. Am. 114, 3044–3052 (2003).
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