In personal audio systems, sounds should propagate toward the listening point and attenuate beyond the listening point. This study deals with controlling directivity and distance attenuation using loudspeaker arrays. The array signal processing is based on tangent line method (TLM), which can generate acoustic beams following arbitrary convex trajectories. A curvilinear acoustic beam is produced as an envelope of tangent lines, i.e., straight acoustic beams. Specifying the envelope length enables controlling distance attenuation while enhancing directivity. In this study, the TLM for arbitrary circular trajectories is formulated. Optimization algorithm is applied to identify the trajectory maximizing acoustic contrast.

## 1. Introduction

Delivering sounds to a listener and maintaining silence elsewhere is beneficial when listeners and non-listeners share a common space. In such personal audio systems, sounds should propagate toward the listening point and attenuate beyond the listening point. Directivity and distance attenuation must be controlled using loudspeaker arrays. Delay-and-sum (DAS) beamformer is an established technique for controlling directivity.^{1} The basic idea of DAS is to generate a plane wave propagating toward a designated angle by aligning the wavefronts of spherical waves. The mechanism enhances directivity but reduces distance attenuation. The reduced distance attenuation is advantageous when it is desired to deliver sounds as far as possible. However, reduced distance attenuation is a disadvantage in the above personal audio systems. While enhancing the directivity, the distance attenuation beyond the listening point should be enhanced or maintained, at least in comparison with a spherical wave.

Such spatial distributions of the sound pressure level (SPL) can be realized by several methods. Acoustic contrast control (ACC),^{2–9} energy difference control (EDC),^{5,8,10} and pressure matching (PM)^{8,10–14} are well-known techniques for pursuing the designated SPL distributions by optimizing the amplitude and phase of each acoustic source. ACC maximizes the acoustic energy contrast between the listening and non-listening zones. EDC maximizes the acoustic energy difference between the listening and non-listening zones. PM minimizes the error between desired and reproduced pressures at the control points. In principle, these methods have no limitation on the shape of the SPL distribution. Though attentions were not explicitly paid to the distance attenuation in most studies, it was reported in the literatures^{9,13} that the distance attenuation can be enhanced by PM and ACC. In another approach, the near field of a loudspeaker array can be enhanced by maximizing the ratio of reactive acoustic power to active acoustic power.^{15–17} Possible applications of this method are headrest-type and wearable audio devices because the listening point must be very close to the loudspeaker array.

Alternatively, curvilinear acoustic beams can be used for controlling the directivity and distance attenuation. Generation of curvilinear acoustic beams is based on caustic theory and differently formulated by Zhang *et al.*^{18} and Zhao *et al.*^{7,19,20} The methodology is referred to as tangent line method (TLM),^{20} because a curvilinear acoustic beam is produced as the envelope of the tangent lines, i.e., straight acoustic beams. In principle, arbitrary convex trajectories are realizable by TLM.^{18,19} The envelope length is controllable. Therefore, it is possible to control the distance attenuation while enhancing the directivity, as opposed to DAS. In earlier studies by the authors,^{9,21} the TLM for circular trajectories has been applied to deliver sounds to a listening point. As expected, the sounds propagated toward the listening point and were attenuated beyond the listening point. However, $ 0 , R$ was specified as the center of the circular trajectories in the case study, where $R$ is the radius and the acoustic sources are aligned on the $x$-axis. The employed trajectories were not optimized in terms of acoustic contrast between the listening point and other locations. Hence, the acoustic contrast can be improved by using other trajectories.

In this study, the TLM for circular trajectories with arbitrary centers and radii is formulated, and the driving signals of the acoustic sources are analytically derived. Moreover, TLM is applied to deliver sounds to a listening point. The optimal circular trajectory that maximizes the acoustic contrast between the listening point and the far field is identified through an iterative algorithm. Numerical examples are presented to compare the directivity and distance attenuation achieved by the optimal TLM with those achieved by DAS, ACC, and non-optimal TLM. The optimal TLM is also demonstrated experimentally to verify the validity of the theory. As mentioned previously, arbitrary convex trajectories, e.g., parabolic and Bezier curves, are realizable by TLM. There is no rational reason to specify circular trajectories. This study is a preliminary investigation on the sound delivery to a listening point using TLM.

## 2. Theory

This section begins with the formulation of TLM for arbitrary circular trajectories. Next, an iterative optimization algorithm is presented to identify the circular trajectory that maximizes acoustic contrast. In addition, the theories of DAS and ACC, which are examined for comparison, are briefly reviewed.

### 2.1 TLM for arbitrary circular trajectories

### 2.2 Iterative algorithm for optimizing circular trajectories

^{4}

The iterative optimization algorithm was inspired by the literature^{22} in which the source placement was optimized for the ACC. In this study, the arc geometry is optimized for the TLM. The iteration for $ a min \u2264 a < a max , b min \u2264 b \u2264 b max$ is performed, where $ a min$, $ a max$, $ b min$, and $ b max$ are the computing ranges. For each $ a , b$, $R$ is determined using Eq. (1). The delay time required to generate the arc is obtained from Eq. (10), where the right and left half-circles are dealt with individually. The resultant $AC$ is calculated according to Eq. (11) and compared among all cases of $ a , b$. Finally, $ a , b$ achieving the greatest $AC$ is identified.

### 2.3 DAS and ACC

^{1}

^{4,6}

^{6}

^{,}

*via*inverse Fourier transform. The time-domain filters are usually implemented as finite impulse response filter (FIR) filters. Because the number of the taps is limited in practice, control frequencies are discretized and there exist non-control frequencies. The time-domain performance of the ACC is degraded at non-control frequencies.

^{20}

## 3. Numerical examples

In this section, the optimal TLM is compared with the monopole, DAS, ACC, and non-optimal TLM. As an example, the case of $d=0.05\u2009m$, $N=21$, $L= N \u2212 1d=1\u2009m$, and $ R 0=L=1\u2009m$ was investigated. Because $d=0.05\u2009m$ is almost equal to a half wavelength at $3400\u2009Hz$, spatial aliasing is avoidable up to this frequency.^{23} The upper limit of the speech frequency band is $3400\u2009Hz$. Respectively, $3400\u2009Hz$, $1700\u2009Hz$, and $850\u2009Hz$ were examined as high, middle, and low frequencies. Moreover, the listening points corresponding to $ \theta 0=30\xb0,\u200960\xb0,\u2009and\u200990\xb0$ were considered as examples. In this regard, however, only the results of $1700\u2009Hz$ and $60\xb0$ are presented here for avoiding repetition. The results of the other cases are provided as supplementary material.^{25}

In the case of the monopole, a single source at $ L / 2 , 0= 0.5 \u2009 m , 0 \u2009 m$ was driven. The resultant sound pressure distribution is shown in Fig. 2. The sound pressure level is normalized by referring to the listening point and is referred to as the normalized sound pressure level (NSPL). The NSPL at the listening point is $0\u2009dB$. The contour interval of all contour plots in this paper is $10\u2009dB$. The listening point is indicated by a star. In the case of the monopole, the contour plot is identical, regardless of the frequency and $ \theta 0$. The omnidirectionality and distance attenuation of a spherical wave were observed.

In the case of DAS, 21 sources located within $0\u2009m\u2264x\u22641\u2009m$ were driven according to Eq. (13). The resulting NSPL distribution in the case of $1700\u2009Hz$ and $60\xb0$ is shown in Fig. 2. Directionality was achieved. Moreover, distance attenuation in the DAS was less than that in the monopole. This is because the distance attenuation of a plane wave, which is mimicked by DAS, is less than that of a spherical wave. Owing to the poor distance attenuation, loud sounds were undesirably delivered beyond the listening point. The results of other frequencies and angles were in line with the above tendency (see supplementary material [SuppFig1.tiff]). In addition, the acoustic beams became sharper as the frequency increased. It is well known that the width of an acoustic beam produced by DAS decreases with the ratio of the array length to the acoustic wavelength.^{24}

In the case of ACC, the 21 sources were driven according to the eigenvector with the largest eigenvalue of $ R + \lambda 2 I \u2212 1 z Hz$. The regularization parameter was primarily set as $ \lambda 2=0$ to maximize $AC$ without considering $AE$. The numerical results of these indices achieved by the ACC and other methods are listed in Table 1 for the case of $1700\u2009Hz$ and $60\xb0$. $AE$ in the monopole is $0\u2009dB$, because it is the reference. $AE$ in the DAS was less than that in monopole. $AE$ in the ACC was less than that in the monopole, even with $ \lambda 2=0$. The NSPL distribution produced by the ACC is shown in Fig. 2. Directionality in the ACC was higher than that in the DAS. Distance attenuation in the ACC was steeper than that in the monopole. In other words, the sounds were successfully delivered toward the listening point and attenuated beyond the listening point. The results of other frequencies and angles were in line with this tendency (see supplementary material [SuppTable1.tiff and SuppFig2.tiff]). However, the difference between the ACC and DAS decreased as the frequency decreased in terms of $AC$, $AE$, and the NSPL distributions. At $850\u2009Hz$, the $AE$ in the ACC with $ \lambda 2=0$ was greater than that in the monopole. Therefore, non-zero values were applied to $ \lambda 2$ at this frequency. The values were determined on a trial and error basis so that $AE$ in the ACC could almost be equal to that in the single-frequency optimal TLM, which is discussed later. The non-zero $ \lambda 2$ significantly improved $AE$ while scarcely degrading $AC$. $AE$ in the ACC with non-zero $ \lambda 2$ was less than that in the monopole.

Method . | N
. | λ^{2}
. | a (m)
. | b (m)
. | R (m)
. | AC (dB)
. | AE (dB)
. |
---|---|---|---|---|---|---|---|

Monopole | 1 | NA | NA | NA | NA | 18.1 | 0.0 |

DAS | 21 | NA | NA | NA | NA | 28.4 | −8.4 |

ACC | 21 | 0 | NA | NA | NA | 33.1 | −13.1 |

Non-optimal TLM | 21 | NA | 0.000 | 1.010 | 1.010 | 27.4 | −8.2 |

Single-frequency optimal TLM | 21 | NA | 1.150 | 0.800 | 0.164 | 33.0 | −13.1 |

Frequency-averaged optimal TLM | 21 | NA | 1.050 | 0.850 | 0.053 | 33.0 | −13.1 |

Method . | N
. | λ^{2}
. | a (m)
. | b (m)
. | R (m)
. | AC (dB)
. | AE (dB)
. |
---|---|---|---|---|---|---|---|

Monopole | 1 | NA | NA | NA | NA | 18.1 | 0.0 |

DAS | 21 | NA | NA | NA | NA | 28.4 | −8.4 |

ACC | 21 | 0 | NA | NA | NA | 33.1 | −13.1 |

Non-optimal TLM | 21 | NA | 0.000 | 1.010 | 1.010 | 27.4 | −8.2 |

Single-frequency optimal TLM | 21 | NA | 1.150 | 0.800 | 0.164 | 33.0 | −13.1 |

Frequency-averaged optimal TLM | 21 | NA | 1.050 | 0.850 | 0.053 | 33.0 | −13.1 |

In the case of the non-optimal TLM, the right half-circles, whose center was $ 0 , R$, were adopted. These were chosen as examples because such circles were investigated in an earlier work.^{9} The arc to be generated for $60\xb0$ is shown in Fig. 3. $R$ of the arc is listed in Table 1. The 21 sources were driven according to Eq. (10). The resulting NSPL distribution is shown in Fig. 2. The arc-shaped contour plot indicates that prescribed arc (Fig. 3) was generated. The directionality of the non-optimal TLM was higher than that of the monopole. The distance attenuation in the non-optimal TLM was steeper than that in DAS. Nevertheless, ACC outperformed the non-optimal TLM in terms of directivity and distance attenuation. The results of other frequencies and angles were in line with the above trend (see supplementary material [SuppTable1.tiff, SuppTLM3a.tiff, and SuppTLM3b.tiff]). Moreover, the widths of the arc-shaped acoustic beams decreased with increasing frequency. This is reasonable because an arc consists of tangent lines produced by the principles of DAS. Moreover, the TLM is based on the caustic theory, which is a high-frequency approximation.^{18,19} The TLM is also regarded as the stationary phase method, which works well, especially at high frequencies.^{20} These explanations are in line with the observed relationship between the beam width and frequency. Since the TLM works better at higher frequencies, the applicable scope is not limited to the speech frequency band.

In the case of the optimal TLM, the iteration for $ a min \u2264 a < a max , b min \u2264 b \u2264 b max$ was performed, where $ a min= L / 2\u22122.5\u2009m$, $ a max= L / 2+2.5\u2009m$, $ b min=\u22121\u2009m$, and $ b max=2\u2009m$. The intervals for $a$ and $b$ were $0.05\u2009m$. The optimal arc for $1700\u2009Hz$ and $60\xb0$ is shown in Fig. 3. The values of $ a , b$ and $R$ are listed in Table 1. The optimal arcs are dependent on the frequency because the optimizations were performed for each frequency. The 21 sources were driven according to Eq. (10). The resulting NSPL distribution is shown in Fig. 2. The directivity and distance attenuation were significantly improved compared to the non-optimal TLM and were comparable to those of the ACC. According to Table 1, the single-frequency optimal TLM was outperformed at some extent by ACC in terms of $AC$ and $AE$, because ACC optimizes these indices. The results of other frequencies and angles were in line with the above trend (see supplementary material [SuppTable1.tiff, SuppFig4a.tiff, and SuppFig4b.tiff]). As mentioned previously, the difference between the DAS and ACC decreased with decreasing frequency in terms of $AC$, $AE$, and the NSPL distributions. In summary, the performances of the DAS, ACC, and single-frequency optimal TLM became similar as the frequency decreased. The ACC and single-frequency optimal TLM gained advantages over the DAS especially at the middle and high frequencies. As mentioned above, the principle of the TLM explains better performances at higher frequencies. Moreover, the optimal arcs for each angle depend on the frequency. For cases where $R$ was too small to graphically show the arc, an enlarged view is presented in the figure (see supplementary material [SuppFig4b.tiff]). In the case of $90\xb0$, $R$ was almost zero. Such extremely small arcs can be interpreted as concentrating all tangent lines, that is, all straight acoustic beams, on the listening point. However, the relationship between the optimized circular trajectories and wavelength is not clear and will be investigated in the future work. The TLM can be realized simply by delay-time control, if—and only if—the arcs to be generated are specified, regardless of the frequency. The advantage of simplicity cannot be achieved in the single-frequency optimal TLM. Therefore, the frequency-averaged optimal TLM is considered.

## 4. Experiment

In this section, experimental results are presented to validate the theory of frequency-averaged optimal TLM. As shown in Fig. 4(a), $N=16$ loudspeakers were used in the experiment. The gap between the loudspeakers was $d=0.05\u2009m$. The array length was $L= N \u2212 1d=0.75\u2009m$. The case of $ R 0=L=0.75\u2009m$ and $ \theta 0=60\xb0$ was observed. The iterative optimization algorithm was performed, where $ a min= L / 2\u22122.5\u2009m$, $ a max= L / 2+2.5\u2009m$, $ b min=\u22121\u2009m$, and $ b max=2\u2009m$. The intervals for $a$ and $b$ were $0.05\u2009m$. $AC$ was averaged between $300$ and $3400\u2009Hz$ at an interval of $10\u2009Hz$. The frequency-averaged optimal arc is shown in Fig. 4(b). The center and radius were $ a , b= 0.775 \u2009 m , 0.650 \u2009 m$ and $R=0.025\u2009m$, respectively. The 16 loudspeakers were driven by a digital signal processor, according to Eq. (10), where the sampling rate was $44.1\u2009kHz$. The source signal was white noise. The sound pressure radiating from the loudspeaker array was measured in an anechoic room at Kogakuin University using a sound level meter, with a sample rate of $48\u2009kHz$. The measurement points were positioned on two half-circles. Their centers were equally $ L / 2 , 0$, and their radii were $ R 0=0.75\u2009m$ and $2 R 0=1.5\u2009m$. The interval between the neighboring measurement points was $15\xb0$. The 1/3 octave band analysis was performed on the measured data. The bands centered at $3150$, $1600$, and $800\u2009Hz$, and the overall from $315$ to $3150\u2009Hz$ were evaluated. In this regard, however, only the result at $1600\u2009Hz$ is presented here for avoiding repetition. The results of the other frequencies are provided with supplementary material.

The measured NSPL distribution at $1600\u2009Hz$ is shown in Fig. 4(c). The blue and red circles in the figure depict the measured values at $ R 0=0.75\u2009m$ and $2 R 0=1.5\u2009m$, respectively. The simulated results for $1600\u2009Hz$ is also plotted for comparison. The blue and red lines depict the simulated values for $ R 0=0.75\u2009m$ and $2 R 0=1.5\u2009m$, respectively. The single-frequency analysis was conducted in the simulation. The measured and simulated results agreed with each other. The directivity toward $60\xb0$ was successfully realized. The achieved distance attenuation was equal to that of a spherical wave or more. Theoretically, the distance attenuation in the spherical wave is $6.0\u2009dB$ when the distance is doubled from $ R 0$ to $2 R 0$. On the other hand, the measured distance attenuation in the TLM was $7.0\u2009dB$ at $1600\u2009Hz$. The results of other frequencies were in line with this trend (see supplementary material [SuppFig6.tiff]). As expected, the gained directivity and distance attenuation decreased with decreasing frequency.

## 5. Conclusions

In this study, the TLM for circular trajectories with arbitrary center and radius has been formulated, where point acoustic sources are assumed to be aligned. The delay time for each source was derived analytically. Arc-shaped acoustic beams with arbitrary centers and radii can be generated by driving a linear loudspeaker array, according to the delay times. Delivering sounds to a listening point using the TLM has also been discussed. Arc-shaped acoustic beams were used to deliver sounds toward the listening point while attenuating sounds beyond the listening point. The center and radius of the arcs were optimized to maximize the acoustic contrast between the listening point and the far field. The optimization was performed on a frequency-averaged basis by using an iterative algorithm. In the presented numerical examples, the directivity and distance attenuation achieved by the frequency-averaged optimal TLM were comparable to those achieved by the ACC. The advantage of the proposed method over the ACC is the simplicity in implementation. Multiple channels of FIR filters are required for the ACC but not for the frequency-averaged optimal TLM. This is because the TLM is a delay-time control, like the traditional DAS. Determining a number of the FIR taps and evaluating performance degradations at discretized non-control frequencies are no concern of the TLM. The frequency-averaged optimal TLM was also demonstrated in an experiment. The measured data corresponded to the simulated results.

## Acknowledgments

This study is related to collaborative research with CASIO COMPUTER CO., Ltd. Moreover, Kogakuin University applied for a patent related to this study.

## REFERENCES

*Acoustical Engineering*

*Acoustic Systems and Digital Processing for Them*(The Institute of Electronics, Information and Communication Engineers, 1995) (in Japanese).