Design of Kronecker Product Beamformers with Cuboid Microphone Arrays

Xuehan WANG(1) , Jacob BENESTY(2) , Gongping HUANG(1) , Jingdong CHEN(1) , and Israel COHEN(3)(1)

Center of Intelligent Acoustics and Immersive Communication, Northwestern Polytechnical University, Xi’an, 710072, China.
E-mail: wangxuehan.123@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(3)

Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 3200003, Israel. E-mail:icohen@ee.technion.ac.ilAbstract
Microphone array beamforming has been widely used in a wide range of acoustic applications. To make it
effective in suppressing noise, yet being able to preserve the ﬁdelity and quality of broadband speech signals ofinterest, the beamformer needs to be designed with high spatial gain, consistent responses at different frequencies, and high robustness against array imperfections. A great deal of efforts have been devoted in the literature
to achieving this goal, among which the Kronecker product beamforming method, developed recently for linear
and rectangular microphone arrays, has demonstrated some interesting properties. In this work, we extend thisapproach to three-dimensional arrays. Focusing on cuboid shape of microphone arrays, we ﬁrst decompose
the global beamforming ﬁlter into a Kronecker product of two sub-beamforming ﬁlters. Algorithms are thendeveloped to design each sub-beamforming ﬁlter so that the global beamformer has a high directivity factor andcan be steered ﬂexibly in the three-dimensional space.

Keywords: Microphone arrays, cuboid arrays, superdirective beamforming, Kronecker product.

1 INTRODUCTION
Microphone arrays can be used to solve many important acoustic problems in a wide range of applications. A
critical component of a microphone array system is the so-called beamforming, which is basically a spatial ﬁlter
applied to the array observation signals to estimate the signal of interest while suppressing noise and interferences [1–3]. While beamforming has been studied for many decades in the ﬁeld of sensor arrays, extending
those methods developed for narrowband applications to microphone arrays is not trivial as microphone array
beamforming needs to suppress noise and meanwhile preserve the ﬁdelity of the speech and acoustic signals ofinterest, which are broadband in nature and their frequencies span from 20 Hz to 20 kHz. Among many beamforming methods developed in the literature [5–10], the Kronecker product beamforming developed recently
has demonstrated some promising results in terms of performance, robustness, and complexity [2, 11–20]. Thismethod basically decomposes the global beamforming ﬁlter into the Kronecker product of two sub-ﬁlters. It has
been demonstrated, based on linear and rectangular microphone arrays, that by optimizing these sub-ﬁlters, theglobal beamforming ﬁlter can be made to achieve high spatial gain and consistent responses over frequencies
with high robustness [2, 21].
In this paper, we extend the Kronecker product beamforming from two-dimensional microphone arrays to a particular type of three-dimensional microphone arrays, i.e., uniform cuboid arrays. We ﬁrst show how to decompose the global beamforming ﬁlter into the Kronecker product of two sub-beamforming ﬁlters: one correspondsto a linear array and the other corresponds to a rectangular grid array. We then derive algorithms on how todesign those two sub-beamforming ﬁlters. It is shown that the developed Kronecker product beamformer can2660

zδzyxδyδx

Figure 1. Illustration of the Kronecker product decomposition of a uniform cuboid array into a uniform rectangular subarray and a uniform linear subarray.
achieve a high directivity factor (DF) with a reasonable white noise gain (WNG), and its beampattern has goodsteering ﬂexibility in the three-dimensional space.

2 SIGNAL MODEL, PROBLEM FORMULATION, AND PERFORMANCE MEASURESLet us consider a uniform cuboid microphone array as illustrated in Fig. 1. We choose the sensor at the top-left
inner corner as the origin of the Cartesian coordinate system. The array consists of Mz parallel uniform rectangular array along z (negative) axis and each rectangular array consists of Mx sensors along the x (negative)
axis and My sensors along the y (negative) axis. The interelement spacing along each axis is denoted as δx ,δy , and δz , respectively.
Assume that, in an anechoic acoustic environment, a far-ﬁeld source signal (plane-wave) propagates from thedirection {θ, φ} at the speed of sound, i.e., c = 340 m/s, and impinges on the cuboid array, where the elevationangle θ (0 ≤ θ ≤ π) is measured downward from the z axis and the azimuth angle φ (0 ≤ φ ≤ 2π) is measured
counterclockwise from the x axis. The steering matrix of size Mx × My corresponding to the rectangular arraycan be written as [2, 7]
Dxy (ω, θ, φ) = dx (ω, θ, φ) dTy (ω, θ, φ) ,(1)wheredx (ω, θ, φ) =dy (ω, θ, φ) =ωδxsin θ cos φ

1 e−j cωδy1 e−j c sin θ sin φ· · · e−j(Mx −1)ωδxsin θ cos φc· · · e−j(My −1)ωδysin θ sin φc

T
T,

(2)
(3)

are the steering vectors associated with the linear subarrays along the x and y axes, respectively, j is theimaginary unit with j2 = −1, ω = 2πf is the angular frequency, and f > 0 is the temporal frequency.
Applying the vectorization operation to (1), we can deduce thatdxy (ω, θ, φ) = vec [Dxy (ω, θ, φ)] = dy (ω, θ, φ) ⊗ dx (ω, θ, φ) ,(4)where ⊗ denotes the Kronecker product operation. Then, the steering matrix of size (Mx My ) × Mz correspond-2661

ing to the cuboid array can be written asωδ
D (ω, θ, φ) = dxy (ω, θ, φ) e−j cz cos θ dxy (ω, θ, φ)· · · e−j(Mz −1)ωδzcos θcdxy (ω, θ, φ)= dxy (ω, θ, φ) dTz (ω, θ) ,(5)wheredz (ω, θ) =ωδz1 e−j c cos θ· · · e−j(Mz −1)ωδzcos θcT(6)is the steering vector of length Mz associated with the linear subarray along the z axis Note that dz (ω, θ) isindependent of the azimuth angle φ . Similarly, applying the vectorization operation to (5) yieldsd (ω, θ, φ) = vec [D (ω, θ, φ)] = dz (ω, θ) ⊗ dxy (ω, θ, φ) .

(7)

The conventional beamforming is performed by applying a complex-valued ﬁlter, h (ω) of length M =Mx My Mz , to all the microphone observations to get an estimate of the desired signal [7]. Assume that thedesired source is incident from the direction {θd , φd }. The distortionless constraint in the desired look direction
is required, i.e.,hH (ω) d (ω, θd , φd ) = 1,(8)where the superscript H is the conjugate-transpose operator.
Three important performance measures are commonly used to evaluate the performance of beamformers: thebeampattern, the WNG, and the DF. The beampattern describes the sensitivity of the beamformer to a planewave; it is deﬁned as [3]
B [h (ω) , θ, φ] = hH (ω) d (ω, θ, φ) .
The WNG evaluates the sensitivity of the array to some of its imperfections; it is deﬁned as [4] H
h (ω) d (ω, θd , φd )2.
W [h (ω)] =hH (ω) h (ω)

(9)

(10)

The DF quantiﬁes how the microphone array performs in the presence of spatially diffuse noise; it is deﬁnedas [4]2|B [h (ω) , θd , φd ]|D [h (ω)] = π  2π2|B [h (ω) , θ, φ]| sin θdφdθ Hh (ω) d (ω, θd , φd )2,=hH (ω) Γ (ω) h (ω)

whereΓ (ω) =

14π

14π0

 π  2π00d (ω, θ, φ) dH (ω, θ, φ) sin θdφdθ.

(11)

(12)

03 KRONECKER PRODUCT BEAMFORMERSAs seen from (7), the steering vector of the cuboid array can be decomposed into the Kronecker product of thecounterparts of two subarrays. Consequently, we propose to decompose the global beamforming ﬁlter ash (ω) = hz (ω) ⊗ hxy (ω) ,2662(13)

where hz (ω) is the sub-beamforming ﬁlter of length Mz corresponding to the linear subarray along the z axis
and hxy (ω) is the sub-beamforming ﬁlter of length Mx My corresponding to the rectangular subarray along thex − y plane.
Now, the beampattern of the Kronecker product beamformer can be written asH

B [h (ω) , θ, φ] = [hz (ω) ⊗ hxy (ω)] [dz (ω, θ) ⊗ dxy (ω, θ, φ)] H
= hHz (ω) dz (ω, θ) hxy (ω) dxy (ω, θ, φ)= B [hz (ω) , θ] × B [hxy (ω) , θ, φ] ,(14)where B [hz (ω) , θ] and B [hxy (ω) , θ, φ] are the beampatterns corresponding to the linear and rectangular subarrays, respectively. It can be easily observed that the global beampattern of the cuboid array is the product of
the individual beampatterns of the two sub-beamforming ﬁlters.
The distortionless constraint becomes

 H
(15)
hz (ω) dz (ω, θd ) hH
xy (ω) dxy (ω, θd , φd ) = 1.
Let us set the distortionless constraint for each sub-beamforming ﬁlter asH
hHz (ω) dz (ω, θd ) = hxy (ω) dxy (ω, θd , φd ) = 1,(16)so that the global distortionless constraint in (15) is satisﬁed.
The WNG of the Kronecker product beamformer is deduced as H h (ω) dz (ω, θd ) hH (ω) dxy (ω, θd , φd ) 2zxy
W [h (ω)] =H[hz (ω) ⊗ hxy (ω)] [hz (ω) ⊗ hxy (ω)]
 H

2
 H
hxy (ω) dxy (ω, θd , φd )2
hz (ω) dz (ω, θd )
×=hHhHz (ω) hz (ω)
xy (ω) hxy (ω)= W [hz (ω)] × W [hxy (ω)] ,(17)where W [hz (ω)] and W [hxy (ω)] are the WNGs corresponding to the linear and rectangular subarrays, respectively. It is clear that the WNG of the cuboid array is the product of the WNGs of the two subarrays.
Now, we can design the two sub-beamforming ﬁlters individually, and thereby obtaining the global beamformer[22–24].
Apparently, there are different ways to design the sub-beamforming ﬁlters. For example, we may design a superdirective beamformer for the rectangular subarray, which is obtained by maximizing the DF while considering
the distortionless constraint [2, 25–27]. The result is as follows:
hxy,SD (ω) =where⎡⎢⎢Γxy (ω) = ⎢⎣Γ−1xy (ω) dxy (ω, θd , φd )−1dHxy (ω, θd , φd ) Γxy (ω) dxy (ω, θd , φd )

Γ1 (ω)
Γ2 (ω)..
.

Γ2 (ω)
Γ1 (ω)
..
.

,

⎤
···
ΓMy (ω)· · · ΓMy −1 (ω) ⎥⎥
⎥
..
..
⎦.
.

ΓMy (ω) ΓMy −1 (ω) · · ·(18)(19)Γ1 (ω)

is a symmetric block Toeplitz matrix of size (Mx My ) × (Mx My ) and Γmy (ω) , my = 1, 2, . . . , My are symmetric Toeplitz matrices of size Mx × Mx whose (i, j)th element is deﬁned as⎡ ⎤ω (i − j)2 δx2 + (my − 1)2 δy2⎦,Γmy (ω) ij = sinc ⎣(20)
c2663

Figure 2. Beampatterns of the Kronecker product beamformer with a cuboid array for different values of θd : (a)θd = 0◦ , (b) θd = 30◦ , (c) θd = 45◦ , and (d) θd = 90◦ . The upper subplots show the three-dimensional beampatterns and the lower subplots show the corresponding two-dimensional beampatterns for φ = 0◦ . Conditionsof simulations: Mx = My = 4, Mz = 8, δx = 1 cm, δy = δz = 2 cm, φd = 0◦ , and f = 2 kHz.

Figure 3. Beampatterns of the Kronecker product beamformer with a cuboid array for different values of φd : (a)φd = 180◦ , (b) φd = 210◦, (c) φd = 240◦ , and (d) φd = 270◦ . The upper subplots show the three-dimensional
beampatterns and the lower subplots show the two-dimensional beampatterns for θ = 90◦ . Conditions of simulations: Mx = My = 4, Mz = 8, δx = 1 cm, δy = δz = 2 cm, θd = 90◦ , and f = 2 kHz.
with sinc(x) = sin x/x, and the subscript SD stands for “superdirective.”2664

18161412108
0.5

11.5

22.5

33.5

4Figure 4. Performance of the Kronecker product beamformer with a cuboid array for different values of θd : (a)DF and (b) WNG. Conditions of simulations: Mx = My = 4, Mz = 8, δx = 1 cm, δy = δz = 2 cm, andφd = 0◦ .
Superdirective beamformers generally have low WNG at low frequencies, leading to the so-called white noise
ampliﬁcation problem. To deal with this problem, let us consider the beamformer that has the largest WNG forthe linear subarray, which corresponds to the delay-and-sum beamformer:
hz,DS (ω) =1dz (ω, θd ) .
Mz

(21)

Thus, the global beamformer is obtained ash (ω) = hz,DS (ω) ⊗ hxy,SD (ω) .

(22)This beamformer can achieve a high value of DF while maintaining a reasonable level of WNG for robustness.
Meanwhile, it has advantages in computational efﬁciency since the dimension of the matrix to invert is Mx My
instead of Mx My Mz [2]. Moreover, the designed beampattern can be steered to different desired directions inthe three-dimensional space as will be shown in the following section.

4 SIMULATIONSIn this section, we examine the performance of the Kronecker product beamformer through simulations. We
consider a uniform cuboid array with Mx = 4, My = 4, Mz = 8, δx = 1 cm, δy = 2 cm, and δz = 2 cm.
Figure 2 plots the beampatterns of the proposed beamformer for θd ∈ {0◦ , 30◦ , 45◦ , 90◦ }, φd = 0◦ , andf = 2 kHz, where the upper subplots show the three-dimensional beampatterns and the lower subplots show
the two-dimensional beampatterns for φ = 0◦ . Figure 3 plots the beampatterns of the proposed beamformerfor φd ∈ {180◦ , 210◦ , 240◦, 270◦}, θd = 90◦ , and f = 2 kHz, where the lower subplots show the twodimensional beampatterns for θ = 90◦ . Clearly, the Kronecker product beamformer can form a main beam todifferent directions, which implies that the proposed beamforming method has good steering ﬂexibility in thethree-dimensional space.
Figure 4 plots the WNG and the DF of the Kronecker product beamformer as a function of frequency, f , forθd ∈ {0◦ , 45◦ , 90◦ }, and φd = 0◦ . It is clearly seen that the proposed beamformer can achieve a high DF while
maintaining a reasonable level of WNG. It is also observed that the DF increases while the WNG decreases as
θd increases.

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5 CONCLUSIONS
We have proposed a Kronecker product beamforming method for cuboid microphone arrays. This method ﬁrst
decomposes the uniform cuboid array into linear and rectangular subarrays so that the steering vector of thecuboid array is written as a Kronecker product of the steering vectors of the two subarrays. The global beamforming ﬁlter is, in a similar manner, also decomposed as a Kronecker product of two sub-beamforming ﬁlters.
As such, we can design the two sub-beamforming ﬁlters individually. In this way, the design of the global
beamforming ﬁlter becomes very ﬂexible since each sub-beamforming ﬁlter can be designed using different
methods such as the superdirective beamformer, delay-and-sum beamformer, differential beamformer, and adaptive beamformer. Then we demonstrated a design of a global beamforming ﬁlter by combining superdirective
and delay-and-sum beamformers. Simulation results veriﬁed that the designed beamformer can achieve high DF
and a reasonable level of WNG while facilitating good steering ﬂexibility in the three-dimensional space.

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 and 2514/17), the NSFC key program (grantNo. 61831019), the NSFC Distinguished Young Scientists Fund (grant No. 61425005), and the ISF Grant
576/16.

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