Optimal Design of Symmetric and Asymmetric Beampatterns with Circular
Microphone Arrays
Xudong ZHAO(1) , Gongping HUANG(1) , Jacob BENESTY(2), and Jingdong CHEN(1)(1)

CIAIC, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China, E-mail: xudong.zhao@mail.nwpu.edu.cn,
gongpinghuang@gmail.com, jingdongchen@ieee.org.

(2)

INRS-EMT, University of Quebec, 800 de la Gauchetiere Ouest, Suite 6900, Montreal, QC H5A 1K6, Canada, E-mail:benesty@emt.inrs.ca.

Abstract
This paper is devoted to the study of the beamforming problem with circular microphone arrays (CMAs). It presents
an approach to the design of beamformers with asymmetric and symmetric frequency-invariant beampatterns. We
ﬁrst discuss how to express a desired target directivity pattern, either symmetric or asymmetric, into a linear weighted
combination of sine and cosine functions as well as circular harmonics of different orders. Using the hypercardioid
pattern as an example, we show how to determine the weighting coefﬁcients by maximizing the directivity factor(DF) with different constraints. Then, by using the Jacobi-Anger expansion, an approximation of the beamformer’s
beampattern is presented. A linear system is subsequently formed by forcing the approximated beampattern to beequal to the target asymmetric or symmetric directivity pattern. The optimal beamforming ﬁlter is ﬁnally determinedby identifying the linear system. In comparison with symmetric beampatterns, the use of asymmetric beampatterns
can offer more ﬂexibility for practical application, such as a high DF or better null positions for interference rejection,
which is demonstrated by simulations.

Keywords: Microphone arrays, circular microphone arrays, frequency-invariant beamforming, differential
microphone arrays, symmetric beampattern, asymmetric beampattern.

1 INTRODUCTION
Frequency-invariant beamforming with small microphone arrays has been extensively studied as it has the great
potential in processing broadband audio and speech signals [1–5]. Several frequency-invariant beamforming
techniques have been developed over the last few decades. Among those, the differential beamformers, which
respond to the differential of the acoustic pressure ﬁeld, have attracted much research and engineering interest[6, 7] as they can achieve frequency-invariant beampatterns and high directivity factors at the same time.
Traditional differential beamformers are designed with a multistage structure [7,8]. However, this structure lacks
ﬂexibility in dealing with the problem of white noise ampliﬁcation, which is inherent to differential microphonearrays (DMAs). Recently, a method to design DMAs in the short-time-Fourier-transform (STFT) domain wasdeveloped [9], which provides a minimum-norm beamforming approach, which basically circumvents the white
noise ampliﬁcation problem by maximizing the white noise gain (WNG) while making the order of the differential beamformer smaller than the number of microphone sensors minus one. This method is then extended tothe design of differential beamformers with circular microphone arrays (CMAs) [11], which have better steering
ﬂexibility as compared to linear microphone arrays (LMAs) [12–14].
Another widely used beamforming method with CMAs is the modal beamformer, which is based on decomposing the sound ﬁeld into a series of circular harmonics and then combining those circular harmonics withappropriate weighting coefﬁcients to form the beamformer’s output [4, 5, 11, 15]. In our recently work, we developed a differential beamforming method with CMAs based on the Jacobi-Anger expansion and showed thatthe modal beamformer is a particular case of this method [12].
However, in all the aforementioned works [11–14], the designed beampatterns are assumed to be symmetric.

2909

The underlying reason for this is that the conventional DMAs were based on LMAs and any DMA pattern
for LMAs is a linear combination of cosine functions of different orders, which is symmetric in nature [7].
Recently, a more general expression of directivity patterns, including both symmetric and asymmetric patterns,
was developed [16]. With such an expression, a beamforming approach were developed based on solving linear equations formed using constraints from the target directivity pattern, which can design beamformers withsymmetric and asymmetric beampatterns.
This paper is an extension of the principle presented in [12, 16]. We develop an approach to the design ofbeamformers with both symmetric and asymmetric beampatterns based on the use of the Jacobi-Anger expansion. We ﬁrst discuss how to express a desired target directivity pattern, either symmetric or asymmetric, intoa linear weighted combination of sine and cosine functions as well as circular harmonics of different orders.
Then, by using the Jacobi-Anger expansion, an approximation of the beamformer’s beampattern is presented. A
linear system is subsequently formed by forcing the approximated beampattern to be equal to a target asymmetric or symmetric directivity pattern. The optimal beamforming ﬁlter is subsequently determined by identifying
the linear system. In comparison with the existing methods that can only design symmetric beampatterns, thedeveloped method has the ﬂexibility in forming either asymmetric or symmetric beampatterns for beneﬁts suchas higher DF, better null steering for interference rejection, etc. In comparison with the method in [16], thedeveloped beamforming approach offers better performance since the resulting beampattern is an optimal approximation of the target beampattern from the least-squares error perspective.

2 SIGNAL MODEL AND PERFORMANCE MEASURES
Consider a uniform circular microphone array (UCMA) with radius r, which consists of M omnidirectional
microphones. Suppose that we want to steer the beamformer to the direction θ, the steering vector of length M
is written as [17, 18]dθ (ω) =ej cos(θ−ψ1 )ej cos(θ−ψ2 )· · · ej cos(θ−ψM )

T

,

(1)

where j is the imaginary unit with j2 = −1,  = ωr/c, ω = 2πf is the angular frequency, f > 0 is the
temporal frequency, ψm = 2π(m − 1)/M is the angular position of the mth array element, c is the speed ofsound in the air, and the superscript T is the transpose operator. The interelement spacing is δ = 2r sin (π/M ).
In order to avoid spatial aliasing, it is assumed that this interelement spacing is much smaller than the minimum
acoustic wavelength of interest.
Assume that the source signal of interest (i.e., the desired signal) comes from the direction θs . The objectiveof beamforming is then to design a frequency-invariant beampattern, which is as close as possible to a desired(symmetric or asymmetric) directivity pattern, with its main beam pointing in the direction θs . For that, weapply a beamforming ﬁlter of length M :h(ω) =

H1 (ω) H2 (ω) · · · HM (ω)

T

,

(2)

to the UCMA observation signals.
In order to evaluate the designed beamformer, we will use the three commonly used measures. They are• the beampattern, which describes the sensitivity of the beamformer to plane waves arriving at the arrayfrom different directions, is deﬁned as [1, 9]
Bθ [h (ω)] = hH (ω) dθ (ω) =

M


∗
Hm
(ω)ej cos(θ−ψm ) ,m=1

where the superscript H is the conjugate-transpose operator,

2910

(3)

• the white noise gain (WNG), which reﬂects the robustness of the beamformer, is deﬁned as [9] Hh (ω) dθs (ω)2,
W [h (ω)] =hH (ω) h (ω)(4)• and the directivity factor (DF), which quantiﬁes the signal-to-noise ratio (SNR) gain of the beamformer
in the spherically isotropic noise ﬁeld, is deﬁned as [1, 9, 10] Hh (ω) dθs (ω)2
D [h (ω)] = H,(5)h (ω) Γd (ω) h (ω)
where the elements of the M × M matrix Γd (ω) are given byωδij[Γd (ω)]ij = sincc(6)with i, j = 1, 2, . . . , M , sinc(x) = sin x/x, and π (i − j) δij = 2r sin

M

(7)

being the distance between sensors i and j.

3 DESIRED DIRECTIVITY PATTERN
We know from the array literature that any desired (symmetric or asymmetric) beampattern of order N ≥ 1 canbe written asBθ (aN , bN ) =N


aN,n sin (nθ) +

n=0
T
= aN
ps,θ + bTN pc,θ ,

N


(8)

bN,n cos (nθ)

n=0where aN,n , bN,n, n = 0, 1, . . . , N are real coefﬁcients andT
aN = aN,0 aN,1 · · · aN,N,
T
ps,θ = 0 sin θ · · · sin (N θ),T,bN = bN,0 bN,1 · · · bN,N
T.
pc,θ = 1 cos θ · · · cos (N θ)
The values of the coefﬁcients aN,n , bN,n, n = 0, 1, . . . , N , determine the shape of the directivity pattern. In theparticular case of symmetric beampatterns, i.e., aN = 0, (8) degenerates toBθ (bN ) = bTN pc,θ .

(9)Another very useful way to express (8) isNBθ (aN , bN ) =
=

N

1
1 
aN,n ejnθ − e−jnθ +bN,n ejnθ + e−jnθ
2j n=0
2 n=0
N
−1
N
−1
1 
1
1 
1 
aN,i ejiθ −aN,−i ejiθ + bN,0 +bN,i ejiθ +bN,−i ejiθ
2j i=1
2j
2 i=1
2
i=−N

=

N

n=−N

c2N,n ejnθ =Nc∗2N,n e−jnθ = cH
2N pe,θ = Bθ (c2N ) ,

i=−N

(10)

n=−N

2911

where the superscript ∗ is the complex-conjugate operator,⎧
c2N,0 = bN,0⎪⎪⎪⎨
bN,i − jaN,i, i = 1, 2, . . . , N
c2N,i =
,
2
⎪
⎪
b
+
ja
N,i
N,i
⎪
∗
⎩ c2N,−i == c2N,i , i = 1, 2, . . . , N
2

(11)

and
c2N =
pe,θ =

· · · c2N,0· · · c2N,N−jN θ T··· 1 ··· e.

c2N,−N
e

jN θ

T

,

The vector pe,θ of length 2N + 1 is composed of circular harmonics of different orders.
In order to take the important steering information into account, let us rewrite Bθ (c2N ) asH

H
Bθ−θs (c2N ) = cH
2N pe,θ−θs = (Υθs c2N ) pe,θ = c2N,θs pe,θ = Bθ (c2N,θs ) ,(12)Υθs = diag (pe,θs ) = diag ejN θs , . . . , 1, . . . , e−jN θs(13)

c2N,θs = Υθs c2N .

(14)

where

is a diagonal matrix andWe see that our general desired directivity pattern is Bθ (c2N,θs ). In practice, the shape of the (symmetricor asymmetric) beampattern is determined by aN , bN (which are given by design) and the steering angle θs ;then, c2N is obtained from (11) and c2N,θs from (14). Note that we should normalize aN , bN such thatBθs (aN , bN ) = 1 and Bθs (c2N,θs ) = 1.
3.1 Asymmetric Hypercardioid
Now, we give an example of the desired, target directivity pattern: the hypercardioid pattern. The asymmetric
hypercardioid pattern is obtained by maximizing the DF:|Bθ (c2N ) |2
D =  2π s,
20 |Bθ (c2N ) | dθ(15)

Dc2N = g,(16)

with several attenuation constraints:

where
D=
g=




pe,θs
1

g1

· · · pe,θL
T· · · gL
.

pe,θ1

H

,

The linear system of equations (16) includes the distortionless constraint and L  2N additional attenuation
constraints, where 0  gl < 1, l = 1, 2, . . . , L. The optimization problem can be formed as
min cH2N Γp c2N subject to Dc2N = g,
c2N

2912

(17)

where 2πΓp =0pe,θ pHe,θ dθ = 2πdiag {12N +1 }(18)

is a diagonal matrix of size (2N + 1) × (2N + 1). Using the method of Lagrange multipliers, we getHH
c2N,opt = Γ−1
DΓ−1
p D
p D

−1

(19)

g.

Note that without the L extra constraints, the solution will reduce to the traditional symmetric hypercardioid.

4 OPTIMAL DESIGN
It is well known that the mth element of the steering vector, d (ω, θ), can be expressed as∞ej cos(θ−ψm ) ==jn Jn () e−jn(θ−ψm )

n=−∞
∞


(20)

jn Jn () ejn(θ−ψm )n=−∞= J0 () + 2∞


jn Jn () cos [n (θ − ψm )] ,n=1

where Jn () is the Bessel function of the ﬁrst kind [19] and we have J−n () = (−1)n Jn (). The series in(20) is known as the Jacobi-Anger expansion [19].
Substituting (20) into (3), we obtainBθ [h (ω)] =

M


∗
Hm
(ω) ej cos(θ−ψm )

m=1

=

M


∗
Hm
(ω)

m=1

=

∞


ejnθ

n=−∞

∞


jn Jn () ejn(θ−ψm )

n=−∞

 M




∗
jn Jn () e−jnψm Hm
(ω) .

(21)

m=1

If we limit the expansion to the order ±N , Bθ [h (ω)] can be approximated by

 M
N


jnθ
n
−jnψm ∗
ej Jn () e
Hm (ω)
Bθ,2N [h (ω)] ≈
n=−N

=

N


m=1

ejnθ jn Jn () ψ Tn h∗ (ω) ,

(22)

n=−N

where
ψn =



e−jnψ1e−jnψ2· · · e−jnψM

T

.

(23)

Expression (22) is the best approximation of the beampattern from a least-squares error perspective.

2913

Now, by forcing the beampattern from the beamforming ﬁlter h (ω) to the desired directivity pattern, i.e.,
Bθ,2N [h (ω)] = Bθ (c2N,θs ) ,(24)Ψ∗ h∗ (ω) = J () c2N,θs(25)

Ψh (ω) = J∗ () Υ∗θs c∗2N ,

(26)

1
1
1
,...,
,..., Nj−N J−N ()J0 ()
j JN ()(27)

we ﬁnd thator, equivalently,

where


J () = diagis a (2N + 1) × (2N + 1) diagonal matrix and
Ψ=



ψ −N

· · · ψ0· · · ψNH

(28)

is a (2N + 1) × M matrix. Since ΨH is a full-column rank matrix, we can take the minimum-norm solutionof (26), which gives−1

hMN (ω) = ΨH ΨΨHJ∗ () Υ∗θs c∗2N .

(29)

The beamformer hMN (ω) maximizes the WNG [9, 20]. So, it is also called the maximum WNG beamformer.

5 SIMULATIONSIn this section, we study the performance of the developed beamformer through simulations. We consider a
UCMA with M = 8 and r = 1.5 cm. The desired directivity pattern is chosen as the second-order hypercardioidwith two nulls at θ1 = 90◦ and θ2 = 120◦ and the desired look direction is set to θs = 0◦ . Then, the coefﬁcients
vector c2N,opt is computed according to (19), where for the asymmetric second-order hypercardioid is
c2N,opt =0.1736 − 0.0208j 0.1976 + 0.0448j 0.2576 0.1976 − 0.0448j 0.1736 + 0.0208j

T

,

(30)

and for the symmetric second-order hypercardioid is
c2N,opt =

 1
6

1
6

1
3

1
6

1 T
.
6

(31)

Substituting all the parameters into (29), we obtain the minimum-norm beamforming ﬁlter. Figure 1 (a.1) and(a.2) plot the beampatterns as a function of the frequency. It is seen that in the studied frequency range from0 to 4000 Hz, the beampatterns of the developed beamformer do not change much with frequency for boththe asymmetric and symmetric cases. In both cases, the designed beampatterns are very close to the respective
target directivity patterns.
Figure 1 (b.1) and (b.2) plot the DF and the WNG of the developed beamformer, both as a function of thefrequency. It is seen that the asymmetric and symmetric hypercardioid beamformers have similar WNG but theasymmetric hypercardioid beamformer has higher DF, which shows the advantage of asymmetric beampatterns.
In practical applications, if we need to place nulls at some directions and avoid nulls at some other directions
for interference rejection, using asymmetric beampatterns may offer much more ﬂexibility than using symmetric
ones.

2914




G%



G%



G%



G%



G%



G%



G%






  











G%
























G%



G%









  



G%









G%




Figure 1. Performance of the designed beamformers: (a.1) asymmetric beampattern, (a.2) symmetric beampattern, (b.1) DF, and (b.2) WNG. Conditions of simulation: M = 8 and r = 1.5 cm.

6 CONCLUSIONSIn this paper, we studied the problem of beamforming with CMAs and showed how to design beamformers
with symmetric and asymmetric frequency-invariant beampatterns. We started by deﬁning the symmetric andasymmetric directivity patterns from the Fourier series expansion. Then, an asymmetric hypercardioid pattern
was obtained by maximizing the DF with different attenuation constraints. By using the Jacobi-Anger expansion, the beamformer’s beampattern and the desired, target directivity pattern were related, based on which a
linear system is formed and the beamforming ﬁlter was derived by solving the linear system. The developed
method can form beamformers with both symmetric and asymmetric frequency-invariant beampatterns, which
gives much ﬂexibility for real applications such as achieving high DF, placing nulls in proper positions, etc.

ACKNOWLEDGEMENTS
This work was supported in part by the National Science Foundation of China (NSFC) and the Israel Science Foundation (ISF) joint research program (grant No. 61761146001), the NSFC key program (grant No.
61831019), the NSFC Distinguished Young Scientists Fund (grant No. 61425005).

REFERENCES[1] G. W. Elko and J. Meyer, “Microphone arrays,” in Springer Handbook of Speech Processing (J. Benesty,
M. M. Sondhi, and Y. Huang, eds.), ch. 48, pp. 1021–1041, Berlin, Germany: Springer-Verlag, 2008.

2915

[2] J. Benesty, J. Chen, and Y. Huang, Microphone Array Signal Processing. Berlin, Germany: Springer-Verlag,
2008.
[3] J. Benesty, I. Cohen, and J. Chen, Array Processing: Kronecker Product Beamforming, vol. 18. Berlin,
Germany: Springer-Verlag, 2019.
[4] J. Meyer, “Beamforming for a circular microphone array mounted on spherically shaped objects,” J. Acoust.
Soc. Am., vol. 109, pp. 185–193, Jan. 2001.
[5] S. Yan and Y. Ma, “Robust supergain beamforming for circular array via second-order cone programming,”
Appl. Acoust., vol. 66, no. 9, pp. 1018–1032, 2005.
[6] G. W. Elko and A.-T. N. Pong, “A steerable and variable ﬁrst-order differential microphone array,” in Proc.
IEEE ICASSP, 1997, vol. 1, pp. 223–226.
[7] G. W. Elko, “Superdirectional microphone arrays,” in Acoustic Signal Processing for Telecommunication,
pp. 181–237, Springer, 2000.
[8] G. W. Elko, “Differential microphone arrays,” in Audio Signal Processing for Next-Generation Multimedia
Communication Systems, pp. 11–65, Springer, 2004.
[9] J. Benesty and J. Chen, Study and Design of Differential Microphone Arrays. Berlin, Germany: SpringerVerlag, 2012.
[10] L. L. Beranek, Acoustics. Woodbury, NY: Acoustic Society of America, 1986.
[11] J. Benesty, J. Chen, and I. Cohen, Design of Circular Differential Microphone Arrays. Berlin, Germany:
Springer-Verlag, 2015.
[12] G. Huang, J. Benesty, and J. Chen, “On the design of frequency-invariant beampatterns with uniform
circular microphone arrays,” IEEE/ACM Trans. Audio, Speech, Lang. Process., vol. 25, no. 5, pp. 1140–
1153, 2017.
[13] G. Huang, J. Chen, and J. Benesty, “Insights into frequency-invariant beamforming with concentric circular
microphone arrays,” IEEE/ACM Trans. Audio, Speech, Lang. Process., vol. 26, no. 12, pp. 2305–2318, 2018.
[14] G. Huang, J. Chen, and J. Benesty, “On the design of robust steerable frequency-invariant beampatterns
with concentric circular microphone arrays,” in Proc. IEEE ICASSP, 2018, pp. 506–510.
[15] S. Yan, “Optimal design of modal beamformers for circular arrays,” J. Acoust. Soc. Am., vol. 138, no. 4,
pp. 2140–2151, 2015.
[16] Y. Buchris, I. Cohen, and J. Benesty, “Frequency-domain design of asymmetric circular differential microphone arrays,” IEEE/ACM Trans. Audio, Speech, Lang. Process., vol. 26, no. 4, pp. 760–773, 2018.
[17] H. V. Trees, Optimum Array Processing: Part IV of Detection, Estimation, and Modulation theory. New York,
NY: John Wiley & Sons, Inc., 2002.
[18] R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays. Raleigh, NC: SciTech, 2004.
[19] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1970.
[20] J. Chen, J. Benesty, and C. Pan, “On the design and implementation of linear differential microphonearrays,” J. Acoust. Soc. Am., vol. 136, no. 6, pp. 3097–3113, 2014.

2916