Wang et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890213]
Published Online 15 July 2014
Broadband pattern synthesis for circular sensor arrays Yong Wang, Yixin Yang,a) Yuanliang Ma, Zhengyao He, and YuKang Liu School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an, 710072, China [email protected], [email protected], [email protected], [email protected], [email protected]
Abstract: The solutions of pattern synthesis are derived for circular sensor arrays based on the criterion of minimizing the mean square error between the desired and synthesized beampatterns. Specifically, the optimal weighting vector, the output beam, and the minimum mean square error are all expressed in closed-form exactly when the desired beampattern is properly formulated. These results provide a more effective and convenient scheme for designing practical frequency-invariant beamformers. Simulations and experimental results demonstrate the performance of the proposed approach. C 2014 Acoustical Society of America V
PACS numbers: 43.60.Fg, 43.30.Wi [DC] Date Received: March 20, 2014 Date Accepted: July 2, 2014
1. Introduction Circular sensor arrays are widely used in many fields and the performance of this type of array has been studied extensively.1–5 Among different aspects of array signal processing, array pattern synthesis has been an active research area. One primary idea of pattern synthesis is to design the weighting vector such that certain errors (e.g., a mean square error) between the desired and synthesized beampatterns are minimized under some given constraints. Approaches for designing the weighting vector have been proposed by various researchers6,7 and most of them can be applied to arrays with arbitrary geometry. However, for circular arrays, these methods are not optimal and the advantages of this array configuration are not fully exploited. By contrast, the concept of phase mode was used in Refs. 1–4 for circular arrays, which provides a convenient method for achieving the desired beampattern. Unfortunately, the obtained beampattern is not accurate owing to the spatial sampling and series truncation limitation of phase mode theory. Actually, the optimal analytical solutions can be achieved for circular arrays by minimizing the mean square error between the desired and synthesized beampatterns. These solutions are derived accurately in this letter and a more effective scheme for designing practical broadband frequency-invariant (FI) beamformers is thus formulated. The performance of this approach is shown in simulations and experimental results. 2. Theoretical solutions Consider the circular sensor array shown in Fig. 1. The radius is a and M omnidirectional sensors are equally spaced on the x-y plane. An incident plane-wave with unit-magnitude impinges from ðh; /Þ. The manifold vector is expressed as Pðh; /Þ ¼ ½p0 ðh; /Þ; p1 ðh; /Þ; …; pM1 ðh; /ÞT with mth element being pm ðh; /Þ ¼ exp½ika sin h cos pffiffiffiffiffiffithe ffi ð/ /m Þ, where rm ¼ a, i ¼ 1, k ¼ 2p=k is the wavenumber, k denotes the wavelength, a)
Author to whom correspondence should be addressed.
J. Acoust. Soc. Am. 136 (2), August 2014
C 2014 Acoustical Society of America EL153 V
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Wang et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890213]
Published Online 15 July 2014
Fig. 1. Coordinates of circular sensor array.
and the superscript T indicates the transpose. /m ¼ mb is the azimuth angle of the mth sensor and b ¼ 2p=M. The beampattern is Bðh; /Þ ¼ wH ðh0 ; /0 ÞPðh; /Þ;
(1)
where w is the weighting vector, ðh0 ; /0 Þ is the preset steering direction, and the superscript H indicates the Hermitian transpose. In this letter, only the horizontal beampatterns are desired and there is h ¼ p=2. The pattern synthesis method is to find the least mean square error approximation to the desired beampattern Bd ð/Þ. The mean square error is defined as ð 1 2p F ð/ÞjBd ð/Þ Bð/Þj2 d/; (2) d ¼ hF ð/Þ½Bd ð/Þ Bð/Þ; Bd ð/Þ Bð/Þi ¼ 2p 0 where F ð/Þ is a positive real weighting function, assigning more or less importance to certain angles, and F ð/Þ ¼ 1 will be assumed Ð 2p in the following parts of this letter. The expression h;i is given by hx; yi ¼ ð2pÞ1 0 xð/Þy ð/Þd/; where the superscript asterisk indicates complex conjugation. The mean square error can be written as a quadratic function d ¼ wH qw wH q qH w þ n;
(3)
where q ¼ hPð/Þ; Pð/Þi ¼
1 2p
ð 2p
Pð/Þ PH ð/Þd/;
(4)
0
q ¼ hPð/Þ; Bd ð/Þi;
(5)
n ¼ hBd ð/Þ; Bd ð/Þi:
(6)
The optimal weighting vector minimizing the mean square error can thus be derived as w ¼ q1 q:
(7)
It is noted that the matrix q is equivalent to the noise covariance matrix in a cylindrically isotropic noise field and its elements are qmm0 ¼ qs ¼ J0 ðkDrs Þ;
EL154 J. Acoust. Soc. Am. 136 (2), August 2014
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Wang et al.: Broadband pattern synthesis
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Wang et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890213]
Published Online 15 July 2014
where J0 ðÞ is the 0th-order cylindrical Bessel function, Drs ¼ 2a sinðsb=2Þ is the spacing between the mth and m0 th sensors, and s ¼ jm m0 j. It is easily obtained that qs ¼ qMs which means q is a circulant matrix and its eigenvector and eigenvalue are ismb vm ¼ M 1=2 ½1 eimb …eiðM1Þmb T , km ¼ RM1 , respectively.8 In addition, the eigens¼0 qs e vectors corresponding to different eigenvalues are mutually orthogonal. Equation (7) can be rewritten as " # M1 X 1 w¼ vm vH (9) m hPð/Þ; Bd ð/Þi; k m m¼0 where the inverse matrix of q is applied. Substitution of Eq. (9) into Eq. (1) yields the modified beampattern which is Bð/Þ ¼ xH Eð/Þ ¼
M1 X
xm Em ð/Þ;
(10)
m¼0
where Eð/Þ ¼ ½E0 ð/Þ; E1 ð/Þ; …; EM1 ð/ÞT is the modified manifold vector and its elements are 1=2 Em ð/Þ ¼ vH m Pð/Þ ¼ M
M1 X
eismb ps ð/Þ
(11)
s¼0
and x ¼ ½x0 ; x1 ; …; xM1 T is the modified weighting vector with elements xm ¼ hEm ð/Þ; Bd ð/Þi=km :
(12)
Combining Eqs. (1) and (10) gives w ¼ Vx; where V ¼ ½v0 ; v1 ; …; vM1 . This equation shows that the classical weighting vector w can be easily obtained from the modified weighting vector x. From Eq. (11), it can be inferred that 0 ðm 6¼ m0 Þ; H H hEm ð/Þ; Em0 ð/Þi ¼ vm qvm0 ¼ km0 vm vm0 ¼ (13) km ðm ¼ m0 Þ: Substitution of Eq. (7) into Eq. (3) yields the minimum mean square error which is d ¼ n qH q1 q ¼ n
M1 X
jhEm ð/Þ; Bd ð/Þij2 =km :
(14)
m¼0
It is noteworthy that hEm ð/Þ; Bd ð/Þi ¼ vH m q; which means it is not necessary to calculate hEm ð/Þ; Bd ð/Þi one by one and the computation complexity is only determined by q. 3. Broadband FI beamforming In this section, it is found that the solutions of broadband FI pattern synthesis can be all expressed in closed-form when the desired beampattern is formulated in an appropriate form. Since the purpose is to achieve a FI beampattern over a range of frequencies, the desired beampattern can be obtained using an M-element uniform circular array with a reference radius ar at a single reference frequency. The general desired beampattern is formulated as9 Bd ðkr ar ; /Þ ¼
M 1 X
xm ðkr ar ÞEm ðkr ar ; /Þ;
(15)
m¼0
where kr denotes the reference wavenumber, and this expression is similar to Eq. (10). The elements of the weighting vector x have the property xMm ¼ ð1Þm xm when the
J. Acoust. Soc. Am. 136 (2), August 2014
Wang et al.: Broadband pattern synthesis EL155
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Wang et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890213]
Published Online 15 July 2014
element number is even and they can be calculated according to the desired requirements.9 Substitution of Eq. (15) into Eq. (12) gives xm ðkaÞ ¼
M1 X
xm0 ðkr ar ÞhEm ðka; /Þ; Em0 ðkr ar ; /Þi=km ¼
m0 ¼0
M1 X
vm0 Þ=km ; xm0 ðkr ar ÞðvH mq
m0 ¼0
(16) where q ¼
1 2p
ð 2p
Pðka; /ÞPH ðkr ar ; /Þd/:
(17)
0
And its elements can be derived as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ¼ J0 ð ðkr ar Þ2 þ ðkaÞ2 2kr ar ka cosðsbÞÞ: q
(18)
Ms and q is also a circulant matrix. Therefore, Eq. (16) can s ¼ q It is obvious that q be simplified to be xm ðkaÞ ¼
k m xm ðkr ar Þ km
(19) PM1
s eismb . by employing Eq. (13). In Eq. (19), k m is the eigenvalue of q and k m ¼ s¼0 q It is seen that each entry of the actual weighting vector at another frequency is the corresponding entry of the desired weighting vector multiplying by a coefficient, and the final synthesized beampattern is rewritten as Bðka; /Þ ¼
M 1 X km k m¼0 m
xm ðkr ar ÞEm ðka; /Þ:
(20)
The coefficient k m =km actually offsets the difference between Em ðkr ar ; /Þ and Em ðka; /Þ for achieving the FI beampatterns, cf. Eqs. (15) and (20), and the desired and synthesized beampatterns will be equal when ka ¼ kr ar . The classical weighting vector can be obtained using w ¼ Vx and the FI beampatterns can also be synthesized by employing Eq. (1). It is evident that these final synthesized FI beampatterns are optimal in terms of their minimum mean square errors with the desired beampattern, and there are no approximations involved during these derivations. P 2^ Substitution of Eq. (15) into Eq. (6) produces n ¼ M1 m¼0 jxm ðkr ar Þj k m ; where ^ k m is the eigenvalue of the circulant matrix q at the reference frequency, and the elements of q should always be kept similar to Eq. (8) despite how the weighting vector of the desired beampattern is obtained. The minimum mean square error in Eq. (14) will be correspondingly changed to d¼
M1 X
jxm ðkr ar Þj2 ð^k m jk m j2 =km Þ:
(21)
m¼0
It is observed that the mean square error is in closed-form and it is only dependent on the desired weighting vector and the eigenvalues of the related circulant matrixes. Additionally, the method described in this letter can be easily implemented in time-domain and the coefficients of finite impulse response filters can be obtained using the method presented in Ref. 10. 4. Simulation computations and experimental results The desired beampattern is obtained using a 24-element uniform circular array with kr ar ¼ 6 and the desired values of x0 x12 are listed in Table 1. Note that other
EL156 J. Acoust. Soc. Am. 136 (2), August 2014
Wang et al.: Broadband pattern synthesis
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Wang et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890213]
Published Online 15 July 2014
Table 1. Entries of the weighting vector for desired beampattern. x0 0.0554 x7 0.0700i
x1 0.0675i
x2 0.0734
x3 0.0258i
x4 0.0434
x5 0.0433i
x8 0.0576
x9 0.0298i
x10 0.0106
x11 0.0029i
x12 0.0015
x6 0.0578
entries x13 x23 can be calculated by employing the relation xMm ¼ ð1Þm xm .9 As shown in Fig. 2(a), all the sidelobe levels of the desired beampattern are kept under 15 dB. The uniform circular array of interest also consists of 24 elements. In Fig. 2(b), the minimum mean square error curve computed using Eq. (21) has a dip at ka ¼ 6, which means the synthesized beampattern at this frequency fits the desired one completely. In the neighboring area of this dip, the error first monotonously decreases before reaching the dip point and then monotonously increases. Based on this error curve, the appropriate frequency band can be determined once the tolerable error level is chosen. The synthesized FI broadband beampatterns in the frequency band ka 2 ½1; 11 are shown in Fig. 2(c). Since they are the optimal ones that can be obtained under the least mean square error criterion, the agreement with the desired beampattern in the given frequency band is perfect and all the levels of mean square error are kept less than 10lgðdÞ ¼ 40dB. The FI property is self-evident, especially in the main lobe area. A lake experiment is conducted to further demonstrate the performance of the proposed approach and the results are analyzed in detail. The experimental uniform circular array consists of 24 omnidirectional hydrophones and its diameter is 3 m. In the experiment, both the circular array and the sound source were placed at a depth of 30 m below the lake surface, and the water depth and sound speed at this location were about 70 m and 1430 m/s, respectively. The sound source was far-field and its distance from the circular array was 53 m. The transmitted signals were rectangular window modulated single-frequency continuous-wave pulses with different frequencies. All the repetition periods of these pulses were 200 ms. The sampling frequency of the time signals was 8192 Hz. The signal-to-noise ratio was high thus the background noise could be ignored. The practical broadband beampatterns at selected frequencies ka ¼ 5.23, 5.63, 6.03, 6.43, 6.84, 7.24, 8.04, 9.25, 9.65, and 10.46 are obtained using the data acquired in the experiment and they are shown in Fig. 3(a). Some deviations between the synthesized and desired beampatterns do exist owing to the errors from characteristics of actual sensors, configuration scattering, channel mismatches, etc. However, the errors of the half-power beamwidth are less than 4 and the sidelobe levels are only raised by about 2 dB, which are acceptable in our application. In comparison with the conventional ones (which are obtained using the delay-and-sum method) in Fig. 3(b), the beampatterns synthesized using this proposed method show better FI property and lower sidelobes. The effectiveness of the proposed approach is thus demonstrated.
Fig. 2. Simulations. (a) Desired beampattern. (b) Levels of the minimum mean square error between the desired and synthesized beampatterns. (c) Synthesized FI broadband beampatterns.
J. Acoust. Soc. Am. 136 (2), August 2014
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Wang et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890213]
Published Online 15 July 2014
Fig. 3. Experimental results. (a) FI broadband beampatterns. (b) Conventional broadband beampatterns.
5. Conclusions The broadband pattern synthesis method of circular sensor arrays is studied in this letter by minimizing the mean square error between the desired and synthesized beampatterns. The accurate solutions are derived by utilizing the properties of circulant matrix. Both the array weighting vector and the synthesized beampattern can be expressed in closed-form when the desired beampattern is properly formulated. The minimum mean square error is also modified to a more concise form and the proper frequency band can be readily determined given the tolerable error level. A more effective method of designing FI beamformer is thus formulated based on these results. The FI broadband beampatterns are conveniently synthesized in both simulation and experiment. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos. 11274253 and 60901076) and the Fundamental Research Funds for the Central Universities (Grant No. 3102014JCQ01008). References and links 1
J. Meyer, “Beamforming for a circular microphone array mounted on spherically shaped objects,” J. Acoust. Soc. Am. 109, 185–193 (2001). 2 S.-C. Chan and C. K. Pun, “On the design of digital broadband beamformer for uniform circular array with frequency invariant characteristics,” in IEEE International Symposium on Circuits and Systems (ISCAS) 2002 (2002), Vol. 1, pp. I-693–I-696. 3 H. Teutsch and W. Kellermann, “Acoustic source detection and localization based on wavefield decomposition using circular microphone arrays,” J. Acoust. Soc. Am. 120, 2724–2736 (2006). 4 E. Tiana-Roig, F. Jacobsen, and E. F. Grande, “Beamforming with a circular microphone array for localization of environmental noise sources,” J. Acoust. Soc. Am. 128, 3535–3542 (2010). 5 Y. L. Ma, Y. X. Yang, Z. Y. He, K. D. Yang, C. Sun, and Y. M. Wang, “Theoretical and practical solutions for high-order superdirectivity of circular sensor arrays,” IEEE Trans. Ind. Electron. 60, 203–209 (2013). 6 F. Wang, V. Balakrishnan, P. Y. Zhou, J. J. Chen, R. Yang, and C. Frank, “Optimal array pattern synthesis using semidefinite programming,” IEEE Trans. Signal Process. 51, 1172–1183 (2003). 7 S. Yan, Y. Ma, and C. Hou, “Optimal array pattern synthesis for broadband arrays,” J. Acoust. Soc. Am. 122, 2686–2696 (2007). 8 G. J. Tee, “Eigenvectors of block circulant and alternating circulant matrices,” Res. Lett. Inf. Math. Sci. 8, 123–142 (2005). 9 Y. Wang, Y. X. Yang, Y. L. Ma, Z. Y. He, and C. Sun, “Design of robust superdirective beamformer for circular sensor arrays,” in Proceedings of OCEANS 2013, San Diego, CA (2013), pp. 1–5. 10 S. Yan and Y. Ma, “A unified framework for designing FIR filters with arbitrary magnitude and phase response,” Digital Signal Process. 14, 510–522 (2004).
EL158 J. Acoust. Soc. Am. 136 (2), August 2014
Wang et al.: Broadband pattern synthesis
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 155.33.16.124 On: Fri, 28 Nov 2014 02:16:59
[http://dx.doi.org/10.1121/1.4890213]
Published Online 15 July 2014
Broadband pattern synthesis for circular sensor arrays Yong Wang, Yixin Yang,a) Yuanliang Ma, Zhengyao He, and YuKang Liu School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an, 710072, China [email protected], [email protected], [email protected], [email protected], [email protected]
Abstract: The solutions of pattern synthesis are derived for circular sensor arrays based on the criterion of minimizing the mean square error between the desired and synthesized beampatterns. Specifically, the optimal weighting vector, the output beam, and the minimum mean square error are all expressed in closed-form exactly when the desired beampattern is properly formulated. These results provide a more effective and convenient scheme for designing practical frequency-invariant beamformers. Simulations and experimental results demonstrate the performance of the proposed approach. C 2014 Acoustical Society of America V
PACS numbers: 43.60.Fg, 43.30.Wi [DC] Date Received: March 20, 2014 Date Accepted: July 2, 2014
1. Introduction Circular sensor arrays are widely used in many fields and the performance of this type of array has been studied extensively.1–5 Among different aspects of array signal processing, array pattern synthesis has been an active research area. One primary idea of pattern synthesis is to design the weighting vector such that certain errors (e.g., a mean square error) between the desired and synthesized beampatterns are minimized under some given constraints. Approaches for designing the weighting vector have been proposed by various researchers6,7 and most of them can be applied to arrays with arbitrary geometry. However, for circular arrays, these methods are not optimal and the advantages of this array configuration are not fully exploited. By contrast, the concept of phase mode was used in Refs. 1–4 for circular arrays, which provides a convenient method for achieving the desired beampattern. Unfortunately, the obtained beampattern is not accurate owing to the spatial sampling and series truncation limitation of phase mode theory. Actually, the optimal analytical solutions can be achieved for circular arrays by minimizing the mean square error between the desired and synthesized beampatterns. These solutions are derived accurately in this letter and a more effective scheme for designing practical broadband frequency-invariant (FI) beamformers is thus formulated. The performance of this approach is shown in simulations and experimental results. 2. Theoretical solutions Consider the circular sensor array shown in Fig. 1. The radius is a and M omnidirectional sensors are equally spaced on the x-y plane. An incident plane-wave with unit-magnitude impinges from ðh; /Þ. The manifold vector is expressed as Pðh; /Þ ¼ ½p0 ðh; /Þ; p1 ðh; /Þ; …; pM1 ðh; /ÞT with mth element being pm ðh; /Þ ¼ exp½ika sin h cos pffiffiffiffiffiffithe ffi ð/ /m Þ, where rm ¼ a, i ¼ 1, k ¼ 2p=k is the wavenumber, k denotes the wavelength, a)
Author to whom correspondence should be addressed.
J. Acoust. Soc. Am. 136 (2), August 2014
C 2014 Acoustical Society of America EL153 V
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 155.33.16.124 On: Fri, 28 Nov 2014 02:16:59
Wang et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890213]
Published Online 15 July 2014
Fig. 1. Coordinates of circular sensor array.
and the superscript T indicates the transpose. /m ¼ mb is the azimuth angle of the mth sensor and b ¼ 2p=M. The beampattern is Bðh; /Þ ¼ wH ðh0 ; /0 ÞPðh; /Þ;
(1)
where w is the weighting vector, ðh0 ; /0 Þ is the preset steering direction, and the superscript H indicates the Hermitian transpose. In this letter, only the horizontal beampatterns are desired and there is h ¼ p=2. The pattern synthesis method is to find the least mean square error approximation to the desired beampattern Bd ð/Þ. The mean square error is defined as ð 1 2p F ð/ÞjBd ð/Þ Bð/Þj2 d/; (2) d ¼ hF ð/Þ½Bd ð/Þ Bð/Þ; Bd ð/Þ Bð/Þi ¼ 2p 0 where F ð/Þ is a positive real weighting function, assigning more or less importance to certain angles, and F ð/Þ ¼ 1 will be assumed Ð 2p in the following parts of this letter. The expression h;i is given by hx; yi ¼ ð2pÞ1 0 xð/Þy ð/Þd/; where the superscript asterisk indicates complex conjugation. The mean square error can be written as a quadratic function d ¼ wH qw wH q qH w þ n;
(3)
where q ¼ hPð/Þ; Pð/Þi ¼
1 2p
ð 2p
Pð/Þ PH ð/Þd/;
(4)
0
q ¼ hPð/Þ; Bd ð/Þi;
(5)
n ¼ hBd ð/Þ; Bd ð/Þi:
(6)
The optimal weighting vector minimizing the mean square error can thus be derived as w ¼ q1 q:
(7)
It is noted that the matrix q is equivalent to the noise covariance matrix in a cylindrically isotropic noise field and its elements are qmm0 ¼ qs ¼ J0 ðkDrs Þ;
EL154 J. Acoust. Soc. Am. 136 (2), August 2014
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Wang et al.: Broadband pattern synthesis
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Wang et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890213]
Published Online 15 July 2014
where J0 ðÞ is the 0th-order cylindrical Bessel function, Drs ¼ 2a sinðsb=2Þ is the spacing between the mth and m0 th sensors, and s ¼ jm m0 j. It is easily obtained that qs ¼ qMs which means q is a circulant matrix and its eigenvector and eigenvalue are ismb vm ¼ M 1=2 ½1 eimb …eiðM1Þmb T , km ¼ RM1 , respectively.8 In addition, the eigens¼0 qs e vectors corresponding to different eigenvalues are mutually orthogonal. Equation (7) can be rewritten as " # M1 X 1 w¼ vm vH (9) m hPð/Þ; Bd ð/Þi; k m m¼0 where the inverse matrix of q is applied. Substitution of Eq. (9) into Eq. (1) yields the modified beampattern which is Bð/Þ ¼ xH Eð/Þ ¼
M1 X
xm Em ð/Þ;
(10)
m¼0
where Eð/Þ ¼ ½E0 ð/Þ; E1 ð/Þ; …; EM1 ð/ÞT is the modified manifold vector and its elements are 1=2 Em ð/Þ ¼ vH m Pð/Þ ¼ M
M1 X
eismb ps ð/Þ
(11)
s¼0
and x ¼ ½x0 ; x1 ; …; xM1 T is the modified weighting vector with elements xm ¼ hEm ð/Þ; Bd ð/Þi=km :
(12)
Combining Eqs. (1) and (10) gives w ¼ Vx; where V ¼ ½v0 ; v1 ; …; vM1 . This equation shows that the classical weighting vector w can be easily obtained from the modified weighting vector x. From Eq. (11), it can be inferred that 0 ðm 6¼ m0 Þ; H H hEm ð/Þ; Em0 ð/Þi ¼ vm qvm0 ¼ km0 vm vm0 ¼ (13) km ðm ¼ m0 Þ: Substitution of Eq. (7) into Eq. (3) yields the minimum mean square error which is d ¼ n qH q1 q ¼ n
M1 X
jhEm ð/Þ; Bd ð/Þij2 =km :
(14)
m¼0
It is noteworthy that hEm ð/Þ; Bd ð/Þi ¼ vH m q; which means it is not necessary to calculate hEm ð/Þ; Bd ð/Þi one by one and the computation complexity is only determined by q. 3. Broadband FI beamforming In this section, it is found that the solutions of broadband FI pattern synthesis can be all expressed in closed-form when the desired beampattern is formulated in an appropriate form. Since the purpose is to achieve a FI beampattern over a range of frequencies, the desired beampattern can be obtained using an M-element uniform circular array with a reference radius ar at a single reference frequency. The general desired beampattern is formulated as9 Bd ðkr ar ; /Þ ¼
M 1 X
xm ðkr ar ÞEm ðkr ar ; /Þ;
(15)
m¼0
where kr denotes the reference wavenumber, and this expression is similar to Eq. (10). The elements of the weighting vector x have the property xMm ¼ ð1Þm xm when the
J. Acoust. Soc. Am. 136 (2), August 2014
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Wang et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890213]
Published Online 15 July 2014
element number is even and they can be calculated according to the desired requirements.9 Substitution of Eq. (15) into Eq. (12) gives xm ðkaÞ ¼
M1 X
xm0 ðkr ar ÞhEm ðka; /Þ; Em0 ðkr ar ; /Þi=km ¼
m0 ¼0
M1 X
vm0 Þ=km ; xm0 ðkr ar ÞðvH mq
m0 ¼0
(16) where q ¼
1 2p
ð 2p
Pðka; /ÞPH ðkr ar ; /Þd/:
(17)
0
And its elements can be derived as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ¼ J0 ð ðkr ar Þ2 þ ðkaÞ2 2kr ar ka cosðsbÞÞ: q
(18)
Ms and q is also a circulant matrix. Therefore, Eq. (16) can s ¼ q It is obvious that q be simplified to be xm ðkaÞ ¼
k m xm ðkr ar Þ km
(19) PM1
s eismb . by employing Eq. (13). In Eq. (19), k m is the eigenvalue of q and k m ¼ s¼0 q It is seen that each entry of the actual weighting vector at another frequency is the corresponding entry of the desired weighting vector multiplying by a coefficient, and the final synthesized beampattern is rewritten as Bðka; /Þ ¼
M 1 X km k m¼0 m
xm ðkr ar ÞEm ðka; /Þ:
(20)
The coefficient k m =km actually offsets the difference between Em ðkr ar ; /Þ and Em ðka; /Þ for achieving the FI beampatterns, cf. Eqs. (15) and (20), and the desired and synthesized beampatterns will be equal when ka ¼ kr ar . The classical weighting vector can be obtained using w ¼ Vx and the FI beampatterns can also be synthesized by employing Eq. (1). It is evident that these final synthesized FI beampatterns are optimal in terms of their minimum mean square errors with the desired beampattern, and there are no approximations involved during these derivations. P 2^ Substitution of Eq. (15) into Eq. (6) produces n ¼ M1 m¼0 jxm ðkr ar Þj k m ; where ^ k m is the eigenvalue of the circulant matrix q at the reference frequency, and the elements of q should always be kept similar to Eq. (8) despite how the weighting vector of the desired beampattern is obtained. The minimum mean square error in Eq. (14) will be correspondingly changed to d¼
M1 X
jxm ðkr ar Þj2 ð^k m jk m j2 =km Þ:
(21)
m¼0
It is observed that the mean square error is in closed-form and it is only dependent on the desired weighting vector and the eigenvalues of the related circulant matrixes. Additionally, the method described in this letter can be easily implemented in time-domain and the coefficients of finite impulse response filters can be obtained using the method presented in Ref. 10. 4. Simulation computations and experimental results The desired beampattern is obtained using a 24-element uniform circular array with kr ar ¼ 6 and the desired values of x0 x12 are listed in Table 1. Note that other
EL156 J. Acoust. Soc. Am. 136 (2), August 2014
Wang et al.: Broadband pattern synthesis
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Wang et al.: JASA Express Letters
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Published Online 15 July 2014
Table 1. Entries of the weighting vector for desired beampattern. x0 0.0554 x7 0.0700i
x1 0.0675i
x2 0.0734
x3 0.0258i
x4 0.0434
x5 0.0433i
x8 0.0576
x9 0.0298i
x10 0.0106
x11 0.0029i
x12 0.0015
x6 0.0578
entries x13 x23 can be calculated by employing the relation xMm ¼ ð1Þm xm .9 As shown in Fig. 2(a), all the sidelobe levels of the desired beampattern are kept under 15 dB. The uniform circular array of interest also consists of 24 elements. In Fig. 2(b), the minimum mean square error curve computed using Eq. (21) has a dip at ka ¼ 6, which means the synthesized beampattern at this frequency fits the desired one completely. In the neighboring area of this dip, the error first monotonously decreases before reaching the dip point and then monotonously increases. Based on this error curve, the appropriate frequency band can be determined once the tolerable error level is chosen. The synthesized FI broadband beampatterns in the frequency band ka 2 ½1; 11 are shown in Fig. 2(c). Since they are the optimal ones that can be obtained under the least mean square error criterion, the agreement with the desired beampattern in the given frequency band is perfect and all the levels of mean square error are kept less than 10lgðdÞ ¼ 40dB. The FI property is self-evident, especially in the main lobe area. A lake experiment is conducted to further demonstrate the performance of the proposed approach and the results are analyzed in detail. The experimental uniform circular array consists of 24 omnidirectional hydrophones and its diameter is 3 m. In the experiment, both the circular array and the sound source were placed at a depth of 30 m below the lake surface, and the water depth and sound speed at this location were about 70 m and 1430 m/s, respectively. The sound source was far-field and its distance from the circular array was 53 m. The transmitted signals were rectangular window modulated single-frequency continuous-wave pulses with different frequencies. All the repetition periods of these pulses were 200 ms. The sampling frequency of the time signals was 8192 Hz. The signal-to-noise ratio was high thus the background noise could be ignored. The practical broadband beampatterns at selected frequencies ka ¼ 5.23, 5.63, 6.03, 6.43, 6.84, 7.24, 8.04, 9.25, 9.65, and 10.46 are obtained using the data acquired in the experiment and they are shown in Fig. 3(a). Some deviations between the synthesized and desired beampatterns do exist owing to the errors from characteristics of actual sensors, configuration scattering, channel mismatches, etc. However, the errors of the half-power beamwidth are less than 4 and the sidelobe levels are only raised by about 2 dB, which are acceptable in our application. In comparison with the conventional ones (which are obtained using the delay-and-sum method) in Fig. 3(b), the beampatterns synthesized using this proposed method show better FI property and lower sidelobes. The effectiveness of the proposed approach is thus demonstrated.
Fig. 2. Simulations. (a) Desired beampattern. (b) Levels of the minimum mean square error between the desired and synthesized beampatterns. (c) Synthesized FI broadband beampatterns.
J. Acoust. Soc. Am. 136 (2), August 2014
Wang et al.: Broadband pattern synthesis EL157
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 155.33.16.124 On: Fri, 28 Nov 2014 02:16:59
Wang et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890213]
Published Online 15 July 2014
Fig. 3. Experimental results. (a) FI broadband beampatterns. (b) Conventional broadband beampatterns.
5. Conclusions The broadband pattern synthesis method of circular sensor arrays is studied in this letter by minimizing the mean square error between the desired and synthesized beampatterns. The accurate solutions are derived by utilizing the properties of circulant matrix. Both the array weighting vector and the synthesized beampattern can be expressed in closed-form when the desired beampattern is properly formulated. The minimum mean square error is also modified to a more concise form and the proper frequency band can be readily determined given the tolerable error level. A more effective method of designing FI beamformer is thus formulated based on these results. The FI broadband beampatterns are conveniently synthesized in both simulation and experiment. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos. 11274253 and 60901076) and the Fundamental Research Funds for the Central Universities (Grant No. 3102014JCQ01008). References and links 1
J. Meyer, “Beamforming for a circular microphone array mounted on spherically shaped objects,” J. Acoust. Soc. Am. 109, 185–193 (2001). 2 S.-C. Chan and C. K. Pun, “On the design of digital broadband beamformer for uniform circular array with frequency invariant characteristics,” in IEEE International Symposium on Circuits and Systems (ISCAS) 2002 (2002), Vol. 1, pp. I-693–I-696. 3 H. Teutsch and W. Kellermann, “Acoustic source detection and localization based on wavefield decomposition using circular microphone arrays,” J. Acoust. Soc. Am. 120, 2724–2736 (2006). 4 E. Tiana-Roig, F. Jacobsen, and E. F. Grande, “Beamforming with a circular microphone array for localization of environmental noise sources,” J. Acoust. Soc. Am. 128, 3535–3542 (2010). 5 Y. L. Ma, Y. X. Yang, Z. Y. He, K. D. Yang, C. Sun, and Y. M. Wang, “Theoretical and practical solutions for high-order superdirectivity of circular sensor arrays,” IEEE Trans. Ind. Electron. 60, 203–209 (2013). 6 F. Wang, V. Balakrishnan, P. Y. Zhou, J. J. Chen, R. Yang, and C. Frank, “Optimal array pattern synthesis using semidefinite programming,” IEEE Trans. Signal Process. 51, 1172–1183 (2003). 7 S. Yan, Y. Ma, and C. Hou, “Optimal array pattern synthesis for broadband arrays,” J. Acoust. Soc. Am. 122, 2686–2696 (2007). 8 G. J. Tee, “Eigenvectors of block circulant and alternating circulant matrices,” Res. Lett. Inf. Math. Sci. 8, 123–142 (2005). 9 Y. Wang, Y. X. Yang, Y. L. Ma, Z. Y. He, and C. Sun, “Design of robust superdirective beamformer for circular sensor arrays,” in Proceedings of OCEANS 2013, San Diego, CA (2013), pp. 1–5. 10 S. Yan and Y. Ma, “A unified framework for designing FIR filters with arbitrary magnitude and phase response,” Digital Signal Process. 14, 510–522 (2004).
EL158 J. Acoust. Soc. Am. 136 (2), August 2014
Wang et al.: Broadband pattern synthesis
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