On the design of time-domain implementation structure for steerable spherical modal beamformers with arbitrary beampatterns

Applied Acoustics 122 (2017) 146–151

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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust

On the design of time-domain implementation structure for steerable spherical modal beamformers with arbitrary beampatterns Weiyi Ren, Huawei Chen ⇑, Weixia Gao College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China

a r t i c l e

i n f o

Article history: Received 19 December 2016 Received in revised form 21 February 2017 Accepted 22 February 2017 Available online 7 March 2017 Keywords: Spherical modal beamforming Microphone array Broadband beamforming Beam steering Time-domain structure

a b s t r a c t It is noted that the existing time-domain implementation structure for steerable spherical modal beamformers is only applicable to the specific beamformers with rotationally symmetric beampatterns about the look direction, which may limit its applications. To overcome the restriction, this paper presents an alternative time-domain implementation structure for spherical modal beamformers using the Wigner-D function, which enables three-dimensional beam steering with arbitrary patterns. In particular, a necessary condition for guaranteeing only real-valued operations to make the time-domain implementation viable is derived. Design examples are presented to demonstrate the effectiveness of the presented time-domain modal beamformer structure. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Spherical microphone arrays are able to analyze threedimensional (3-D) sound fields effectively and facilitate array signal processing in the spherical harmonics domain, thus have found a variety of applications such as sound field reproduction, beamforming, sound localization, and noise reduction, acoustic absorption measurement, among others [1–5]. Spherical array beamforming in the spherical harmonics domain, also known as spherical modal beamforming, offers several advantages when compared with conventional beamforming in the element-space domain [6]. One of the advantages of spherical modal beamforming is that beampattern design can be decoupled from beampattern steering in the spherical harmonics domain, which results in efficient implementation of steerable modal beamformers in 3-D space. Spherical modal beamformers can be implemented either in the frequency domain or in the time domain. In comparison to the time-domain implementation, the frequency-domain implementation is usually block-based processing with the discrete Fourier transform and it may not be suitable for time-critical speech and audio applications due to its associated time delay [7]. It is noted that the frequency-domain implementation approaches for beam steering of spherical modal beamformers with arbitrary beampatterns have been available in the literature [8,9]. In contrast, however, the existing time-domain implementation approach for ⇑ Corresponding author. E-mail address: [email protected] (H. Chen). http://dx.doi.org/10.1016/j.apacoust.2017.02.013 0003-682X/Ó 2017 Elsevier Ltd. All rights reserved.

steerable spherical modal beamformers is only applicable to the specific beamformers, i.e., the beampatterns should be rotationally symmetric [7], which may limit its applications. By rotationally symmetric, it means that the beampattern is rotationally invariant with respect to the look direction, i.e., the beampattern will not change when rotated at an arbitrary angle along the look direction. In contrast, non-rotationally symmetric implies that the beampattern will change when rotated along the look direction. In some practical applications, however, a non-rotationally symmetric beampattern may be desired. For instance, as noted in [8], the recording of sound sources of interest in an auditorium with a spherical array placed at the seating area requires a mainlobe that is wide along the azimuth dimension but narrow along the elevation dimension to cover the entire stage. Inspired by the Wigner-D function [6], a time-domain implementation structure for spherical modal beamformers which enables 3-D beam steering with arbitrary patterns is developed in this paper. In particular, a necessary condition for guaranteeing only real-valued operations to make the time-domain implementation viable is also derived. The advantage of the proposed structure is that it is applicable to not only rotationally symmetric beampatterns but also to non-rotationally symmetric beampatterns in the time-domain implementation. 2. Spherical modal beamforming The standard spherical ðr; h; /Þ coordinate system is used hereafter, where h and / denote elevation and azimuth angles, respectively [10]. Consider a unit magnitude plane wave with wave

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number k impinging on a sphere of radius a from direction X ¼ ðh; /Þ, where k ¼ 2pf =c with f denoting the frequency and c the speed of sound. Then, the frequency-domain expression of the sound pressure at a point Xs ¼ ðhs ; /s Þ on the sphere surface can be expressed as [11]:

pðka; X; Xs Þ ¼

1 X n X

m bn ðkaÞY m n ðXÞY n ðXs Þ

ð1Þ ð2Þ

where Y m n ðXÞ is the spherical harmonic of order n and degree m, which is defined as follows [13]:

XÞ ¼

Ym n ðh; /Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2n þ 1 ðn  mÞ! m P ðcos hÞejm/ ¼ 4p ðn þ mÞ! n

Z

pðka; X; Xs ÞY m n ðXs ÞdXs

Xs 2S2

R

where the integral

Xs 2S2

dXs ¼

R 2p R p 0

0

ð4Þ

sin hs dhs d/s covers the entire

2

surface of the unit sphere S . In practice, since the number of microphones mounted on a sphere is usually limited, the microphone positions are required to satisfy the following discrete orthonormality condition M X

m sYn ð

a

Xs Þ ¼ dnn0 dmm0

ð5Þ

s¼1

where dnn0 and dmm0 are the Kronecker delta functions, as are realvalued parameters depending on the spatial sampling scheme, and M denotes the number of microphones. Then, (4) can be approximated by its discrete version

pnm ðka; XÞ ¼

M X

as pðka; X; X

m s ÞY n ð

Xs Þ:

ð6Þ

s¼1

Accordingly, the array order is limited to N which satisfies ðN þ 1Þ2 6 M [14]. By the discrete spherical Fourier transform for the sound pressure samples, the beampattern which is the array’s response to a unit input signal from X can be expressed in the spherical harmonics domain as [14]:

Bðf ; XÞ ¼

N X n X

pnm ðka; XÞwnm ðf Þ

ð7Þ

n¼0 m¼n

where wnm ðf Þ are the spherical Fourier coefficients of the array weights wðf ; Xs Þ. 3. Main results The 3-D rotation of the beampattern Bðf ; XÞ can be achieved by using the Wigner-D function and the rotated beampattern can be expressed as [8,6]

Br ðf ; XÞ ¼ Kða; b; cÞBðf ; XÞ ¼

N X n X

bn ðkaÞ

with dm0 m ðbÞ denoting the real-valued Wigner-d function which can be written in terms of the Jacobi polynomial [15]: n dm0 m ðbÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s!ðs þ l þ v Þ! l ½sin ðb=2Þ ½cos ðb=2Þv Pðsl;v Þ ðcos bÞ ¼ 1m0 m ðs þ lÞ!ðs þ v Þ! ð10Þ

l ¼ jm0  mj; v ¼ jm0 þ mj; s ¼ n  ðl þ v Þ=2, and 1m0 m ¼ 1 0 when m P m0 ; 1m0 m ¼ ð1Þmm when m < m0 . where

3.1. Proposed time-domain beamformer structure According to (8) and [8], the structure for a steerable spherical modal beamformer usually consists of three stages: modal transformation, beam steering, and pattern forming. Denote xs ðlÞ ¼ xs ðtÞjt¼lT s as the discrete-time series received at the sth microphone, where T s denotes the sampling interval. Note that the spherical harmonics are independent of frequency. Performing the spherical Fourier transform to xs ðlÞ yields the modal transformation of xs ðlÞ

xnm0 ðlÞ ¼

M X

0

as xs ðlÞY mn ðXs Þ

ð11Þ

N X n X

n X

0

0

m  Dn m0 m ða; b; cÞY n ðXÞwnm ðf Þ

 r bn ðkaÞY m n ðXÞwnm0 ðf Þ

n¼0 m0 ¼n

Denote the real and imaginary parts of xnm0 ðlÞ as

~xnm0 ðlÞ ¼

M X

0

as xs ðlÞRe½Y mn ðXs Þ

ð8Þ

ð12Þ

s¼1

smile x nm0 ðlÞ

¼

M X

0

as xs ðlÞIm½Y mn ðXs Þ

ð13Þ

s¼1

where ReðÞ and ImðÞ stand for the real and imaginary parts, respectively. Then, (11) can be expressed as smile nm0 ðlÞ

xnm0 ðlÞ ¼ ~xnm0 ðlÞ  j x

ð14Þ

With beam steering in the spherical harmonics domain using the Wigner-D function, xnm0 ðlÞ now becomes

xrnm ðlÞ ¼

n X m0 ¼n

Dn m0 m ða; b; cÞxnm0 ðlÞ

ð15Þ

In frequency-domain implementation for spherical modal beamformers, a set of complex-valued array weights wnm ðf Þ are employed in the pattern forming stage. In contrast, for timedomain implementation, the complex-valued array weights wnm ðf Þ are replaced by a bank of finite impulse response (FIR) filters with real-valued coefficients such that T

wnm ðf Þ ¼ hnm eðf Þ h

1

ð16Þ 2

L

where hnm ¼ hnm ; hnm ; . . . ; hnm

m0 ¼n

n¼0 m¼n

¼

ð9Þ

s¼1

m0 s ÞY n0 ð

X

n

ð3Þ

where ðÞ! denotes the factorial function, Pm n ðÞ are the associated pffiffiffiffiffiffiffi Legendre functions, and j ¼ 1. The superscript ⁄ denotes complex conjugation. The mode strength bn ðkaÞ for order n is related to both frequencies and sphere configurations [1]. pnm ðka; XÞ are the spherical harmonic coefficients of pðka; X; Xs Þ which are obtained by performing the spherical Fourier transform [12]:

pnm ðka; XÞ ¼

0

Dnm0 m ða; b; cÞ ¼ ejm a dm0 m ðbÞejmc n

n¼0 m¼n

pnm ðka; XÞ ¼ bn ðkaÞY m n ðXÞ

Ym nð

P where wrnm0 ðf Þ ¼ nm¼n Dnm0 m ða; b; cÞwnm ðf Þ are the rotated array weights, Kða; b; cÞ denotes the rotation operation, ða; b; cÞ represents the rotation angle, and Dnm0 m ða; b; cÞ is the Wigner-D function defined as

iT

is the impulse response of the FIR

filter corresponding to the spherical harmonics of order n and degree m, the superscript T denotes the transpose, and L is the tap  T length of each FIR filter. And eðf Þ ¼ 1; ej2pfT s ; . . . ; ejðL1Þ2pfT s . Accordingly, the output time series of the beam-steered beamformers can be expressed as

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W. Ren et al. / Applied Acoustics 122 (2017) 146–151

yr ðlÞ ¼

N X n X n¼0 m¼n

Proof. See the Appendix A. h

xrnm ðlÞ  hnm "

#

Dnm0 ;m ða; b; cÞ ¼ ð1Þm m Dn m0 m ða; b; cÞ [15]. The output time series (17) can be recast as

With the help of Proposition 1, the time-domain implementation structure for steerable spherical modal beamformers with arbitrary beampatterns can be devised as shown in Fig. 1, where a pre-delay element T 0 ¼ ðL  1ÞT s =2 has been incorporated before each FIR filter to compensate for the inherent group delay of FIR filters. It is noted that, given an array order N, the existing time-domain structure tailored for beam steering with rotationally symmetric patterns about the look direction [7] requires N þ 1 FIR filters. In contrast, the present time-domain structure requires ðN þ 1ÞðN þ 2Þ=2 FIR filters, which can be seen as a price paid to implement beam steering with arbitrary patterns. Fortunately, the array order is usually low in practical use, therefore the increasing of FIR filters is not so significant.

yr ðlÞ ¼

3.2. Beampattern and WNG

¼

N X n X

n X

n¼0 m¼n m0 ¼n

Dn m0 m ð

a; b; cÞxnm0 ðlÞ  hnm

ð17Þ

where  denotes the convolution. Note that (17) contains complexvalued operations, due to the presence of the complex-valued Wigner-D function Dnm0 m ða; b; cÞ and xnm0 ðlÞ. In the following, we will reveal a condition under which only real-valued operations are required to enable a time-domain implementation of steerable spherical modal beamformers. m m Note the properties: (1) Y m n ðXÞ ¼ ð1Þ Y n ðXÞ [6]; and (2) 0

("   # N n X X smile Dn00 ða; b; cÞ~xn0 ðlÞ þ 2Re Dnm0 0 ða; b; cÞ ~xnm0 ðlÞ þ j x nm0 ðlÞ  hn0 n¼0

þ

n X

m0 ¼1

("

~ Dn 0m ða; b; cÞxn0 ðlÞ þ

  n  X smile ~ Dn m0 m ða; b; cÞ xnm0 ðlÞ  j x nm0 ðlÞ

Now, we derive the beampattern and white noise gain (WNG) for the proposed time-domain beamformer structure, which are useful in beamformer design. By (8), (16), and (19), and reconsidering the pre-delay element, the beampattern for the proposed beamformer structure can be derived as

m0 ¼1

m¼1

 # smile þð1Þm Dnm0 ;m ða; b; cÞ ~xnm0 ðlÞ þ j x nm0 ðlÞ  hnm " þ ð1Þm Dn0m ða; b; cÞ~xn0 ðlÞ þ

  n  X smile ~ Dn m0 ;m ða; b; cÞ xnm0 ðlÞ  j x nm0 ðlÞ

m0 ¼1

m

þð1Þ

Dnm0 m ð

))  # smile a; b; cÞ ~xnm0 ðlÞ þ j x nm0 ðlÞ  hn;m

ð18Þ

Br ðf ; XÞ ¼

N X n n X X T ej2pfT 0 hnm eðf Þ Q n m0 m ða; b; cÞpnm0 ðka; XÞ

where

hn;m ¼ ð1Þm hnm

ð19Þ

N X 

C n00 ða; b; cÞ~xn0 ðlÞ

n¼0

# n  X smile C nm0 0 ða; b; cÞ~xnm0 ðlÞ þ Snm0 0 ða; b; cÞ x nm0 ðlÞ  hn0

þ

BWNG ¼

m0 ¼1

þ

n X 

BWNG ¼ P

m0 ¼1

ð20Þ which contains only real-valued operations, where

8 n D ða; b; cÞ; > > > 00 n  > > < 2Re D0m ða; b; cÞ ; n

C m0 m ða; b; cÞ ¼ 2Re Dn 0 ða; b; cÞ; > m0 >

>

 > > : 2Re Dnm0 m ða; b; cÞ þ ð1Þm 2Re Dnm0 ;m ða; b; cÞ ;

4p=M P 2 :  n l  n D ð a ; b; c Þh  0 0 nm  m ¼n m¼n m m

l¼1

ð24Þ

Pn n n Further consider that ~ ~ ða; b; cÞ ¼ dmm m0 ¼n Dm0 m ða; b; cÞDm0 m [15], by using (19) the BWNG can be reduced to

) # n  X smile C nm0 m ða; b; cÞ~xnm0 ðlÞ þ Snm0 m ða; b; cÞ x nm0 ðlÞ  hnm

N n¼0

n

4p=M o Pn 2 khn0 k þ 2 m¼1 khnm k 2

ð25Þ

where kk is the Euclidean norm. Note that the BWNG is independent of the beampattern’s look direction.

m0 ¼ 0; m ¼ 0 m0 ¼ 0; m 2 ½1; n m0 2 ½1; n; m ¼ 0

ð21Þ

m0 2 ½1; n; m 2 ½1; n

with n 2 ½0; N , and

4. Design examples

8

 < 2Im Dnm0 0 ða; b; cÞ ; m¼0 n

 Sm0 m ða; b; cÞ ¼ : 2Im Dnm0 m ða; b; cÞ  ð1Þm 2Im Dnm0 ;m ða; b; cÞ ; m 2 ½1; n ð22Þ

with m0 2 ½0; n; n 2 ½0; N  .

m¼0

Dnm0 m ða; b; cÞ þ ð1Þm Dnm0 ;m ða; b; cÞ; m 2 ½1; n:

PN PL Pn n¼0

C n0m ða; b; cÞ~xn0 ðlÞ

m¼1

þ

Dnm0 0 ða; b; cÞ;

When the microphones are uniformly or near-uniformly distributed over a sphere [6], the broadband WNG (BWNG) [7] for the proposed time-domain beamformer structure, which characterizes the robustness of beamformers against sensor imperfections, can be derived as

then (18) can be reduced to

yr ðlÞ ¼

(

Q nm0 m ða; b; cÞ ¼

Proposition 1. If the FIR coefficients satisfy

ð23Þ

m0 ¼n

n¼0 m¼0

Design examples are shown in this section to illustrate the effectiveness of the proposed beamformer structure in comparison to the existing structure [7]. We consider a rigid spherical array of radius 4.2 cm with M ¼ 36 microphones whose positions satisfy the spherical t-design [16]. The array order is chosen as N ¼ 4, the tap length of the FIR filters is set to L ¼ 65, and the frequency band of interest is specified as f 2 ½500; 5000 Hz. The constrained

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W. Ren et al. / Applied Acoustics 122 (2017) 146–151

Fig. 1. Proposed time-domain implementation structure for steerable spherical modal beamformers capable of forming arbitrary patterns.

optimization method similar to [17] is used to design the beamformer weight vector via solving the optimization problem

    min Br ðf ; XÞ  Bd ðf ; XÞ s:t: Br ðf ; X0 Þ ¼ 4p=M BWNG1 6 e

ð26Þ

where Bd ðf ; XÞ denotes the desired beampattern, which is defined as: Bd ðf ; XÞ ¼ 4p=M in the passband and Bd ðf ; XÞ ¼ 0 in the stopband. The passband region is hp 2 ½80 ; 100 ; /p 2 ½140 ; 220 , and the stopband region is hs1 2 ½60 ; 120 ; /s1 2 ½0 ; 120  [ ½240 ; 360 ; hs2 2 ½0 ; 60  [½120 ; 180 ; /s2 2 ½0 ; 360 . Note that the desired beampattern is non-rotationally symmetric about the look direction, since the passband width along the azimuth dimension is wider than that along the elevation dimension. Let the beampattern’s look direction be

360 0.8

0.6

0.2

0.7

0.5

0

0.4

120

0.3 0.2

60

0.1 0

60

θ

120

z

−0.4 −1 −0.8 −0.6 −0.4 −0.2 x

0.4

0.3

0

0.2

0 −0.2 −0.4 y

0.2

0.4

0.2

0.2

0.1

−0.4

0.9

0.8

0.4

0.7

0.4

0.7

0.2

0.6

0

0.5

x

0.3 0.2 0

0.2

(d)

0 −0.2 −0.4 −0.6 −0.8 y

0.1

z

0.6

−0.2 −1 −0.8 −0.6 −0.4 −0.2

0.2

0 y

−0.2

−0.4

0.9

0.8

0.4

0.1 0.4

(c)

0.6

0

0.3

−0.2

(b)

0.5

0.5

0

0.4

180

0.2

0.7 0.6

0.5

(a)

z

0.8

0.6

−0.2

0.6

0.9

0.2

z

φ

180

0

0.4

0.7

240

0.4

0.9

0.8

300

0.4

−0.2 0 −0.2 0 −0.2 −0.4 −0.4 −0.6 −0.6 y −0.8 −1 −0.8 x

0.2

0.3 0.2 0.1

(e)

Fig. 2. Beampatterns of the existing time-domain beamformer at f ¼ 3000 Hz. (a) Beampattern as a function of elevation and azimuth angles, with the look direction X0 ¼ ð90 ; 180 Þ. (b) Balloon plot of the beampattern shown in (a). (c) Same as (b), but viewed from the look direction. (d) Beampattern steered to ð60 ; 240 Þ. (e) Same as (d), but viewed from the new look direction.

150

W. Ren et al. / Applied Acoustics 122 (2017) 146–151 360 0.8

0.6

0.2

0.7

0.5

0

0.4

120

0.3 0.2

60

0.1 0

60

θ

120

z

180

x

0

0 −0.2 −0.4 y

0.2

0.2

0.4

0.5

0

0 −0.2 −0.4 −0.6 −0.8 y

0.2

0 y

−0.2

−0.4

0.7 0.6

0.2 0

0.3

−0.2

0.1

0.1 0.4

0.8

0.4

0.4 0.2

0.2

−0.4

(c)

0.6

0.6

0.2

x

0.2

0.1

0.9

z

z

0.7

0.3

−0.2

0.2

0.9

0

0.4

0.3

(b)

0.4

0.5

0

0.4

−0.4 −1 −0.8 −0.6 −0.4 −0.2

0.8

0.7 0.6

0.5

(a)

−0.2 −1 −0.8 −0.6 −0.4 −0.2

0.8

0.6

−0.2

0.6

0.9

0.2

z

φ

180

0

0.4

0.7

240

0.4

0.9

0.8

300

0.5 0.4

0 −0.2 0 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 y −0.8 −1 x

(d)

0.3 0.2

0.2 0.1

(e)

Fig. 3. Beampatterns of the proposed time-domain beamformer at f ¼ 3000 Hz. (a) Beampattern as a function of elevation and azimuth angles, with the look direction X0 ¼ ð90 ; 180 Þ. (b) Balloon plot of the beampattern shown in (a). (c) Same as (b), but viewed from the look direction. (d) Beampattern steered to ð60 ; 240 Þ, where further rotation of 30 about the new look direction is conducted. (e) Same as (d), but viewed from the new look direction.

X0 ¼ ð90 ; 180 Þ, which means a ¼ 180 ; b ¼ 90 and c ¼ 0 in Br ðf ; XÞ in (23). The BWNG constraint is included in the constraints to ensure the robustness of the beamformer, where e is set to 100:5 , which implies BWNG P 5 dB. By solving (26), we can obtain the weight vectors for the proposed time-domain beamformer and the existing time-domain beamformer [7] with their beampatterns orientated toward X0 ¼ ð90 ; 180 Þ. Fig. 2(a)–(c) display the beampattern at f ¼ 3000 Hz for the existing time-domain beamformer. Therein, Fig. 2(a) shows the beampattern on the elevation-azimuth plane, Fig. 2(b) shows the balloon plot of the beampattern in Fig. 2(a), and (c) is the same as Fig. 2(b) but viewed from the look direction. In Figs. 2(d) and (e), the steered beampatterns with its new look direction at X0 ¼ ð60 ; 240 Þ are shown. As we can see from Fig. 2, although the existing time-domain beamformer can achieve beam steering, its beampattern is actually rotationally symmetric rather than non-rotationally symmetric. Therefore, the existing time-domain beamformer structure is not applicable to the design scenarios where a non-rotationally symmetric pattern is preferred. In Fig. 3(a)–(c), we plot the beampattern at f ¼ 3000 Hz with the proposed time-domain implementation structure. By changing the rotation angles into a ¼ 240 ; b ¼ 60 and c ¼ 30 in the proposed time-domain beamformer structure, the rotated beampattern is obtained as presented in Fig. 3(d) and (e). The look direction now becomes X0 ¼ ð60 ; 240 Þ with the beampattern further counterclockwise rotated 30 about the new look direction

compared with its counterpart shown in Fig. 3(b). It can be seen from Fig. 3 that, the resulting beampattern using the proposed time-domain implementation structure is indeed shown to be non-rotationally symmetric as desired. Moreover, using the proposed structure, this non-rotationally symmetric pattern can also be rotated to arbitrary direction.

5. Conclusion In this paper, a time-domain structure for spherical modal beamformers which enables 3-D beam steering with arbitrary patterns has been proposed. To construct the time-domain structure, a necessary condition for guaranteeing only real-valued operations to make the time-domain implementation viable is derived. The beampattern and WNG for the proposed time-domain beamformer structure are also discussed. A simulation example has been presented to demonstrate the effectiveness of the proposed timedomain beamformer structure.

Acknowledgement This work was supported by the National Natural Science Foundation of China under Grant No. 61471190.

W. Ren et al. / Applied Acoustics 122 (2017) 146–151

151

Appendix A Substituting (19) into (18) yields

y r ð lÞ ¼

("   # N n X X smile Dn00 ða; b; cÞ~xn0 ðlÞ þ 2Re Dnm0 0 ða; b; cÞ ~xnm0 ðlÞ þ j x nm0 ðlÞ  hn0 n¼0

þ2

n X

m0 ¼1

(

   n  X

 smile Re Dn0m ða; b; cÞ ~xn0 ðlÞ þ Re Dnm0 m ða; b; cÞ ~xnm0 ðlÞ þ j x nm0 ðlÞ

m¼1

m0 ¼1



m

Dnm0 ;m ð

   smile a; b; cÞ ~xnm0 ðlÞ þ j x nm0 ðlÞ  hnm

þð1Þ Re (" # N n  X X

n 

n smile n ~ ~ D00 ða; b; cÞxn0 ðlÞ þ 2 Re Dm0 0 ða; b; cÞ xnm0 ðlÞ  Im Dm0 0 ða; b; cÞ x nm0 ðlÞ  hn0 ¼ n¼0

þ2

n X m¼1

"

m0 ¼1

n hh

i X



 Re Dn0m ða; b; cÞ ~xn0 ðlÞ þ Re Dnm0 m ða; b; cÞ þ ð1Þm Re Dnm0 ;m ða; b; cÞ ~xnm0 ðlÞ m0 ¼1

  h



ismile   Im Dnm0 m ða; b; cÞ þ ð1Þm Im Dnm0 ;m ða; b; cÞ x nm0 ðlÞ  hnm

Furthering substituting (21) and (22) into (A.1), we arrive at (20).

References [1] Meyer J, Elko GW. A highly scalable spherical microphone array based on an orthonormal decomposition of the soundfield. In: IEEE international conference on acoustics, speech and signal processing (ICASSP 2002), vol. II; 2002. p. 1781–4. [2] Abhayapala TD, Ward DB. Theory and design of high order sound field microphones using spherical microphone array. In: IEEE international conference on acoustics, speech and signal processing (ICASSP 2002), vol. II; 2002. p. 1949–52. [3] Gover BN, Ryan JG, Stinson MR. Measurement of directional properties of reverberant sound fields in rooms using a spherical microphone array. J Acoust Soc Am 2004;116(4):2138–48. [4] Li X, Yan S, Ma X, Hou C. Spherical harmonics MUSIC versus conventional MUSIC. Appl Acoust 2011;72:646–52. [5] Rathsam J, Rafaely B. Analysis of absorption in situ with a spherical microphone array. Appl Acoust 2015;89:273–80. [6] Rafaely B. Fundamentals of spherical array processing. Springer topics in signal processing. Springer; 2015.

ðA:1Þ

[7] Yan S, Sun H, Ma X, Svensson UP, Hou C. Time-domain implementation of broadband beamformer in spherical harmonics domain. IEEE Trans Audio Speech Lang Process 2011;19(5):1221–30. [8] Rafaely B, Kleider M. Spherical microphone array beam steering using WignerD weighting. IEEE Signal Process Lett 2008;15:417–20. [9] Atkins J. Robust beamforming and steering of arbitrary beam patterns using spherical arrays. In: IEEE workshop applicat. signal process. audio acoust. (WASPAA); 2011. p. 237–40. [10] Arfken G, Weber HJ. Mathematical methods for physicists. 5th ed. San Diego, CA: Academic; 2001. [11] Williams EG. Fourier acoustics: sound radiation and nearfield acoustical holography. New York: Academic; 1999. [12] Driscoll JR, Healy DM. Computing Fourier transforms and convolutions on the 2-sphere. J Adv Appl Math 1994;15:202–50. [13] Yan S, Sun H, Svensson UP, Ma X, Hovem JM. Optimal modal beamforming for spherical microphone arrays. IEEE Trans Audio Speech Lang Process 2011;19 (2):361–71. [14] Rafaely B. Analysis and design of spherical microphone arrays. IEEE Trans Speech Audio Process 2005;13(1):135–43. [15] Varshalovich DA, Moskalev AN, Khersonskii VK. Quantum theory of angular momentum. 1st ed. Singapore: World Scientific; 1988. [16] Hardin RH, Sloane NJA. McLaren’s improved snub cube and other new spherical designs in three dimensions. Discrete Comput Geom 1996;15:429–41. [17] Li Z, Duraiswami R. Flexible and optimal design of spherical microphone arrays for beamforming. IEEE Trans Audio Speech Lang Process 2007;15(2):702–14.