A Maximum-Achievable-Directivity Beamformer with White-Noise-Gain
Constraint for Spherical Microphone Arrays
Xi Chen(1) , Gongping Huang(1) , Jingdong Chen(1) , and Jacob Benesty(2)(1)

Center of Intelligent Acoustics and Immersive Communications, Northwestern Polytechnical University, Xi’an, 710072, China.
E-mail: xichen128@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
In microphone array beamforming, it is desirable to achieve a directive gain as high as possible for maximum
acoustic noise rejection. The well-known superdirective beamformer was developed for this purpose; but it issensitive to array imperfections such as sensors’ self noise, sensor placement errors, mismatch among sensor
responses, etc, which restrict its application in practical systems. To circumvent the robustness issue, we recently
developed a flexible high directivity beamforming method that can achieve a flexible compromise between thedirectivity factor (DF) and the level of white noise gain (WNG) by adjusting the value of a control parameter.
This principle is further extended in this work. We present a beamforming method with spherical microphonearrays. It first determines the maximum order (consequently the largest value of DF) that can be achieved in ananalytical way based on a specified level of WNG that is tolerable by the array in the frequency band of interest.
This order is then used to determine the value of a control parameter. The maximum-achievable-directivity
beamformer is then designed. The advantage of this beamformer over the existing ones is that it is robust toimplement and yet able to achieve the maximum possible DF.
Keywords: Spherical microphone arrays, directivity factor, superdirective beamformer, white noise gain.

1 INTRODUCTION
Microphone array beamforming, which aims at enhancing acoustic and speech signals from a desired lookdirection and attenuating noise and interference from other directions, is an important topic in speech processing [1–4]. Many algorithms have been developed including but not limited to the delay-and-sum beamformer,
the superdirective beamformer [5–7], differential beamformers [8–13], and the modal beamformers [14, 15].
Generally, the beamformer’s performance depends not only on the beamforming algorithm used, but also on thearray geometry. In many practical applications such as teleconferencing and three-dimensional sound recording
and reproduction, spherical microphone arrays are generally preferred because they can have similar responses
across different directions in the three-dimensional space. Among many beamforming methods developed forspherical microphone arrays [2, 15–21], the superdirective beamformer, which maximizes the directivity factor(DF), is effective in suppressing directional noise and reverberation and therefore has great potential to be usedin noisy auditory scenes [2, 22–24]. However, the lack of robustness is a serious issue that prevents this beamformer from being widely used in a wide range of acoustic applications. In our previous work, we proposeda flexible high directivity beamformer, which attempts to a compromise between the DF and the white noisegain (WNG) [25]. In this paper, we extend the idea to design a maximum-achievable-directivity beamformer
with spherical microphone arrays that can achieve the largest possible DF and maintain the WNG larger than apre-specified level.

2752

zpθs , φs q

θr

φyx

Figure 1. Illustration of a spherical microphone array in the Cartesian coordinate system, where (θs , ϕs ) denotes
the source incidence direction.

2 SIGNAL MODEL AND PROBLEM FORMULATION
We consider a spherical microphone array with a radius of r, which consists of M omnidirectional microphones. We assume that the center of the spherical microphone array coincides with the origin of the Cartesian
coordinate system as shown in Fig.1. The azimuth angle ϕ is measured counterclockwise from the x-axis, while
the elevation angle θ is measured downwards from the z-axis. Suppose that we want to steer the beamformer
to the direction (θ, ϕ), the steering vector of length M is written as[]TTT
Td (ω, θ, ϕ) = eȷϖp r1 eȷϖp r2 · · · eȷϖp rM,(1)where the superscript T is the transpose operator, ȷ is the imaginary unit with ȷ2 = −1, ϖ = ωr/c, ω = 2πf
is the angular frequency, f > 0 is the temporal frequency, c is the speed of sound in the air (c = 340 m/s),[]T[]Tp = sin θ cos ϕ sin θ sin ϕ cos θ, and rm = sin θm cos ϕm sin θm sin ϕm cos θm.
Assume that the source signal of interest propagates from the direction (θs , ϕs ). The observation signal vector
can be written as[]Ty (ω) = Y1 (ω) Y2 (ω) · · · YM (ω)= d (ω, θs , ϕs ) X (ω) + v (ω) ,(2)where Ym (ω) , m = 1, 2, . . . , M is the signal observed by the mth microphone, X (ω) is the source signal ofinterest (also called the desired signal), and v (ω) is the additive noise vector.
Beamforming is achieved by applying a complex-valued weight to each microphone signal and then summingup the weighted outputs to obtain an estimate of the desired source signal, i.e.,
Z (ω) =M∑∗
Hm(ω) Ym (ω)

(3)

m=1
H

= h (ω) y (ω)= hH (ω) d (ω, θs , ϕs ) X (ω) + hH (ω) v (ω) ,2753

where the superscript ∗ is the complex-conjugate operator, the superscript H is the conjugate-transpose operator,and[]Th (ω) = H1 (ω) H2 (ω) · · · HM (ω)(4)
is the beamforming filter of length M .
The objective of beamforming is to design h (ω) such that Z (ω) is a good estimate of X (ω). Generally, thedistortionless constraint at the desired look direction, i.e.,hH (ω) d (ω, θs , ϕs ) = 1,(5)should be used so that the desired signal can pass through the beamformer without being distorted while noise
and interference from directions other than the desired look direction are attenuated.

3 MAXIMUM-ACHIEVABLE-DIRECTIVITY BEAMFORMER WITH WHITE-NOISEGAIN CONSTRAINT
The beampattern, which depicts the response of the array to the signal incident from the direction (θ, ϕ), isdefined as
B [h (ω) , θ, ϕ] = hH (ω) d (ω, θ, ϕ)=M∑(6)∗(ω) eȷϖp rm .
HmT

m=1

The ideal beampattern is expected to have a unit impulse at the desired look direction and be zero elsewhere,i.e.,
B (θ, ϕ) = δ (cos θ − cos θs ) δ (ϕ − ϕs ) ,(7)where δ (·) denotes the Dirac delta function. In practice, this beampattern is impossible to achieve, so weapproximate it by a series of spherical harmonics up to the N th order [19], i.e.,B (θ, ϕ) ≈ BN (θ, ϕ) =N ∑n∑[ l]∗Yn (θs , ϕs ) Ynl (θ, ϕ) ,(8)n=0 l=−n√whereYnl (θ, ϕ) =2n + 1 (n − l)! l
P (cos θ) eȷlϕ
4π (n + l)! n(9)denotes the spherical harmonics of order n (= 0, 1, 2, . . . , +∞) and degree l (= 0, ±1, . . . , ±n), with (·)! being
the factorial function and Pnl (cos θ) being the associated Legendre function of order n and degree l [26].
One way to design beamformers is to optimize the filter, h (ω), to make the designed beampattern, i.e.,
B [h (ω) , θ, ϕ], as close as possible to BN (θ, ϕ). However, the resulting beamformer generally suffers fromserious white noise amplification at low frequencies. To achieve a flexible compromise between the DF and thelevel of WNG, a flexible high directivity beampattern is defined in [25]1[αBN (θ, ϕ) + (1 − α)BN −1 (θ, ϕ)]κNNn
N −1 n]∗]∗α ∑ ∑ [ l1−α ∑ ∑ [ l=Yn (θs , ϕs ) Ynl (θ, ϕ) +Yn (θs , ϕs ) Ynl (θ, ϕ)κN n=0κN n=0BN,α (θ, ϕ) =l=−n=2754

N ∑n
∑

1
κN n=0l=−nl=−n[]∗ξn Ynl (θs , ϕs ) Ynl (θ, ϕ) ,(10)[ 2]1where α ∈ [0, 1] is a real-valued control parameter, κN = 4π
N + α (2N + 1) is the corresponding normalized factor, ξn = 1, for n = 0, 1, . . . , N − 1, and ξn = α, for n = N . Clearly, if α = 0, BN,α = BN −1 (θ, ϕ),and if α = 1, BN,α = BN (θ, ϕ). As the value of α increases from 0 to 1, the DF of the resulting beamformer
increases; but the noise amplification problem may become more serious.
The WNG, which evaluates the robustness of the beamformer to the array’s imperfection [2, 3], is defined as2

W [h (ω)] =hH (ω) d (ω, θs , ϕs ).
hH (ω) h (ω)(11)

In this paper, we propose a maximum-achievable-directivity (MAD) beamformer, which can obtain the maximum
possible directivity while maintains a specified level of WNG for robustness. This problem can be described as{
B [hN,α (ω), θ, ϕ] = BN,α (θ, ϕ)hN,α (ω) = arg max W [h (ω)] s.t.
,(12)
W [hN,α (ω)] ≥ W0h(ω)
which can be equivalently expressed [assuming that the distortionless constraint is fulfilled in BN,α (θ, ϕ)] as{
B [hN,α (ω), θ, ϕ] = BN,α (θ, ϕ)
H,hN,α (ω) = arg min h (ω)h(ω) s.t.
(13)
W [hN,α (ω)] ≥ W0h(ω)
where W0 denotes the minimum level of WNG.
Using the spherical harmonic expansion, we haveTeȷϖp rm =∞ ∑n∑]∗[4πȷn Jn (ϖ) × Ynl (θm , ϕm ) Ynl (θ, ϕ) ,(14)n=0 l=−nwhere Jn (ϖ) is the nth-order spherical bessel functions of the first kind. Substituting (14) into (6), we deducethat
B [h(ω), θ, ϕ] =M∑∗Hm(ω)eȷϖp rmT(15)m=1≈M∑∗Hm(ω)

m=1=N ∑n∑[]∗4πȷn Jn (ϖ) × Ynl (θm , ϕm ) Ynl (θ, ϕ)

n=0 l=−nN ∑n∑Ynl (θ, ϕ) 4πȷn Jn (ϖ) ×n=0 l=−nM∑[]∗∗
Hm(ω) Ynl (θm , ϕm ) .

m=1

Comparing the result in (15) and the target beampattern BN,α (θ, ϕ) in (10), we get the following relationship:∗4π [ȷn Jn (ϖ)]

M∑

Hm (ω) Ynl (θm , ϕm ) =m=1

ξn lY (θs , ϕs ) .
κN n(16)

It follows immediately thatYh (ω) = η (ϖ) ,where

Y00 (θ1 , ϕ1 )
 Y1−1 (θ1 , ϕ1 ) Y10 (θ1 , ϕ1 )
Y =  Y 1 (θ1 , ϕ1 )
 1

..
.
YNN (θ1 , ϕ1 )Y00 (θ2 , ϕ2 ) · · ·Y1−1 (θ2 , ϕ2 ) · · ·Y10 (θ2 , ϕ2 ) · · ·Y11 (θ2 , ϕ2 ) · · ·..
..
.
.
N
YN (θ2 , ϕ2 ) · · ·(17)
Y00 (θM , ϕM )Y1−1 (θM , ϕM ) 
Y10 (θM , ϕM ) 
Y11 (θM , ϕM ) 

..
.

(18)

YNN (θM , ϕM )2755

Figure 2. Beampatterns of the MAD beamformer with different values of W0 : (a) W0 = −40 dB, (b) W0 =−30 dB, (c) W0 = −20 dB, and (d) W0 = −10 dB. Conditions of simulations: M = 32, r = 3 cm,
f = 1 kHz, and (θs , ϕs ) = (90◦ , 115◦ ).
is a matrix of size (N + 1)2 × M , and1 [ ξ0 Y00 (θs ,ϕs )η (ϖ) =
4πJ0∗ (ϖ)κN

ξ1 Y1−1 (θs ,ϕs )4π[ȷJ1 (ϖ)]∗···N
ξN YN(θs ,ϕs )4π[ȷN JN (ϖ)]∗

is a vector of length (N + 1)2 .
Generally, M ≥ (N + 1)2 , the minimum-norm solution of (17) is()−1hN,α (ω) = YH YYH
η (ϖ) ,which gives the N th-order MAD beamformer.
The corresponding WNG of the N th-order MAD beamformer is[]2M N 2 + (2N + 1) αW[hN,α (ω)] = (2N +1).
∑N −1
α2 + n=0 (2n+1)
J 2 (ϖ)J 2 (ϖ)
N]T
(19)

(20)

(21)

nClearly, the WNG of the beamformer is a decreasing function of α and satisfies
MN4M (N + 1)4∑N (2n+1) ≤ W[hN,α (ω)] ≤ ∑N −1 (2n+1) .
n=0 Jn2 (ϖ)(22)

n=0 Jn2 (ϖ)

In practice, we should first determine the order N and the value of the parameter α according to W0 . Assume
that the maximum designable order we want to achieve is NM , the order N and the parameter α can bedetermined as follows.

2756

Figure 3. DFs and WNGs of the MAD beamformer as a function of the frequency, f : (a) DF and (b) WNG.
Conditions of simulation: M = 32 and r = 3 cm.
∑NM (2n+1)• If W0 < M (NM + 1)4 /( n=0
Jn2 (ϖ) ), we simply choose N = NM and α = 1.
[]
∑NM (2n+1)• If W0 ∈ M (NM + 1)4 /( n=0J 2 (ϖ) ), M , we first determine the order N asn

N = argN′

M (N ′ + 1)4′N∑
(2n + 1)n=0

≤ W0 ≤

M N ′4′N∑−1,(23)(2n + 1)Jn2 (ϖ)n=0

Jn2 (ϖ)and then determine the value of the parameter α according to the following equation:[]2M N 2 + (2N + 1) α= W0 ,∑N −1(2N +1) 2α + n=0 (2n+1)
J 2 (ϖ)J 2 (ϖ)(24)nNwhose solution is√[]∑N −1 (2n+1)+1)
2 (2N +1)
2−W−WM(2N+1)
M N (2N + 1) + W0 M N 4 (2N2200n=0 J 2 (ϖ)
J (ϖ)
J (ϖ)
4α=NN+12W0 J2N
2 (ϖ) − M (2N + 1)
Nn.

(25)

With the determined order N and the value of the parameter α, the MAD beamformer is subsequently obtained
according to (20), which is robust to implement and can achieve the maximum possible DF with a specified
level of WNG.

4 SIMULATIONSIn this section, we evaluate the performance of the MAD beamformer. We consider a spherical microphonearray with a radius of r = 3 cm, which consists of 32 omnidirectional microphones. The maximum designable
order of the MAD beamformer is NM = 4. The desired steering direction is (90◦ , 115◦ ).

2757

Figure 2 plots the beampatterns of the MAD beamformer for W0 ∈ {−40 dB, −30 dB, −20 dB, −10 dB},
f = 1 kHz, where the upper subplot shows the beampatterns in the three-dimensional space and the lower
subplot shows the corresponding two-dimensional beampatterns relative to ϕ. It is seen that the beampatternvaries with the value of W0 and a larger value of W0 corresponds to a narrower main beam.
Figure 3 plots the DFs and the WNGs of the MAD beamformer as a function of the frequency. It is clearlyseen that the designed beamformer can make the WNG larger than the specified level over the frequency bandof interest. It is also seen the DF is improved with the increase of the frequency, and the maximum DF is
10 log10 (NM + 1)2 ≈ 13.98 dB, which corresponds to the fourth-order MAD beamformer.

5 CONCLUSIONSIn this paper, we developed a MAD beamformer with spherical microphone arrays. It first determines the orderof the beamformer based on the minimum level of WNG that is tolerable by the beamformer in the frequency
band of interest. This order is then used to determine the value of a control parameter with a specified levelof WNG. Both the order and the control parameter are determined in an analytic form. The MAD beamformer
is subsequently designed, which is able to achieve the maximum possible DF while keeping the WNG larger
than the specified level over the frequency band of interest. This MAD beamformer is robust to implement inpractice.

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] M. Brandstein and D. Ward, Microphone Arrays: Signal Processing Techniques and Applications. Springer,
2001.
[2] 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.
[3] J. Benesty, J. Chen, and Y. Huang, Microphone Array Signal Processing. Berlin, Germany: Springer-Verlag,
2008.
[4] J. Benesty, I. Cohen, and J. Chen, Array Processing: Kronecker Product Beamforming. Berlin, Germany:Springer-Verlag, 2019.
[5] G. W. Elko, “Superdirectional microphone arrays,” in Acoustic Signal Processing for Telecommunication,
pp. 181–237, Springer, 2000.
[6] H. Cox, R. M. Zeskind, and T. Kooij, “Practical supergain,” IEEE Trans. Acoust., Speech, Signal Process.,
vol. 34, pp. 393–398, Jun. 1986.
[7] H. Cox, R. Zeskind, and M. Owen, “Robust adaptive beamforming,” IEEE Trans. Acoust., Speech, Signal
Process., vol. 35, pp. 1365–1376, Oct. 1987.
[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.

2758

[10] J. Benesty, J. Chen, and I. Cohen, Design of Circular Differential Microphone Arrays. Berlin, Germany:Springer-Verlag, 2015.
[11] 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, pp. 1140–1153, May
2017.
[12] G. Huang, J. Chen, and J. Benesty, “On the design of differential beamformers with arbitrary planar microphone array,” J. Acoust. Soc. Am., vol. 144, pp. 3024–3035, Jul. 2018.
[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, pp. 2305–2318, Dec. 2018.
[14] J. Meyer and G. Elko, “A highly scalable spherical microphone array based on an orthonormal decomposition of the soundfield,” in Proc. IEEE ICASSP, 2002, pp. II–1781.
[15] S. Yan, H. Sun, U. P. Svensson, X. Ma, and J. M. Hovem, “Optimal modal beamforming for sphericalmicrophone arrays,” IEEE Trans. Acoust., Speech, Signal Process., vol. 19, pp. 361–371, Feb. 2011.
[16] J. Meyer and G. Elko, “Spherical harmonic modal beamforming for an augmented circular microphonearray,” in Proc. IEEE ICASSP, 2008, pp. 5280–5283.
[17] D. N. Zotkin, R. Duraiswami, and N. A. Gumerov, “Sound field decomposition using spherical microphonearrays,” in Proc. IEEE ICASSP, 2008, pp. 277–280.
[18] B. Rafaely, “Phase-mode versus delay-and-sum spherical microphone array processing,” IEEE Signal Process. Lett., vol. 12, no. 10, pp. 713–716, 2005.
[19] B. Rafaely, Fundamentals of Spherical Array Processing. Berlin, Germany: Springer-Verlag, 2015.
[20] S. Yan, Y. Ma, and C. Hou, “Optimal array pattern synthesis for broadband arrays,” J. Acoust. Soc. Am.,
vol. 122, no. 5, pp. 2686–2696, 2007.
[21] H. Sun, W. Kellermann, E. Mabande, and K. Kowalczyk, “Localization of distinct reflections in rooms
using spherical microphone array eigenbeam processing,” J. Acoust. Soc. Am., vol. 131, pp. 2828–2840,
Apr. 2012.
[22] C. Li, J. Benesty, G. Huang, and J. Chen, “Subspace superdirective beamformers based on joint diagonalization,” in Proc. IEEE ICASSP, 2016, pp. 400–404.
[23] G. Huang, J. Benesty, and J. Chen, “Superdirective beamforming based on the Krylov matrix,” IEEE/ACM
Trans. Audio, Speech, Lang. Process., vol. 24, pp. 2531–2543, Dec. 2016.
[24] S. Doclo and M. Moonen, “Superdirective beamforming robust against microphone mismatch,” IEEE Trans.
Acoust., Speech, Signal Process., vol. 15, no. 2, pp. 617–631, 2007.
[25] G. Huang, J. Chen, and J. Benesty, “A flexible high directivity beamformer with spherical microphonearrays,” J. Acoust. Soc. Am., vol. 143, pp. 3024–3035, May 2018.
[26] E. G. Williams, Fourier Acoustics: Sound Radiation and Nearfield Acoustical holography. New York: Academic, 1999.

2759