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8/10/2019 NEAR-FIELD SOURCE LOCALIZATION USING SPHERICAL MICROPHONE ARRAY
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NEAR-FIELD SOURCE LOCALIZATION USING SPHERICAL MICROPHONE ARRAY
Lalan Kumar, Kushagra Singhal, and Rajesh M Hegde
Indian Institute of Technology Kanpur
{rhegde,lalank}@iitk.ac.in
ABSTRACT
Source localization using spherical microphone arrays has re-
ceived attention due to the ease of array processing in the
spherical harmonics (SH) domain with no spatial ambigu-
ity. In this paper, we address the issue of near-field source
localization using a spherical microphone array. In particu-
lar, three methods that jointly estimate the range and bear-
ing of multiple sources in the spherical array framework, areproposed. Two subspace-based methods called the Spherical
Harmonic MUltiple SIgnal Classification (SH-MUSIC) and
the Spherical Harmonics MUSIC-Group Delay (SH-MGD)
for near field source localization, are first presented. Addi-
tionally, a method for near-field source localization using the
Spherical Harmonic MVDR (SH-MVDR) is also formulated.
Experiments on near-field source localization are conducted
using a spherical microphone array at various SNR. The SH-
MGD is able to resolve closely spaced sources when com-
pared to other methods.
Index Terms MUSIC, Spherical Harmonics, Near-
field, Group delay
1. INTRODUCTION
Spherical microphone array processing has been a growing
area of research in the last decade [1, 2]. This is primarily
because of the relative ease with which array processing can
be performed in the spherical harmonics (SH) domain without
any spatial ambiguity [3].
Various algorithms have been proposed for far-field
source localization using spherical microphone array. Esti-
mation of Signal Parameters via Rotational Invariance Tech-
niques (ESPRIT) [4] algorithm is extended for spherical array
in [5]. Multiple SIgnal Classification (MUSIC) [6] is imple-mented in terms of spherical harmonics in [7]. In [8], room
acoustics analysis is presented using spherical array, based on
SH-MUSIC in frequency domain. All these source localiza-
tion methods deals with planar wavefront of far-field sources.
However, in applications like Close Talk Microphone (CTM),
video conferencing etc, the planar wavefront assumption is
This work was funded by the DST project EE/SERB/20130277. The
author L. Kumar was supported by TCS Research Scholarship Program
TCS/CS/20110191.
no more valid. In [9], design of a low order spherical micro-
phone array is proposed to acquire the sound from near field
sources. Near-field criterion for spherical array is discussed
in [10]. However, spherical array has not been utilized for
near-field source localization. In [11], 2-Dimensional (2D)
MUSIC spectrum is presented for multiple near-field sources
using Uniform Linear Array (ULA). In this work, we propose
3D SH-MUSIC spectrum for range and bearing (elevation,
azimuth) estimation of multiple near-field sources. MVDR[12] and MUSIC-Group Delay (MGD) spectrum [1316]
have also been studied for near-field source localization using
spherical array of microphone. The primary contribution of
this work is in the proposal of novel methods for near-field
source localization in spherical harmonics domain.
The rest of the paper is organized as follows. In Section 2,
signal model in spherical harmonics domain is presented. The
near-field criteria is discussed, followed by the development
of SH-MUSIC, SH-MGD and SH-MVDR methods. The pro-
posed method is evaluated in Section 3. Section 4 concludes
the paper.
2. NEAR-FIELD SOURCE LOCALIZATION USINGSPHERICAL MICROPHONE ARRAY
In this Section, a mathematical derivation of 3-Dimensional
MUSIC spectrum is presented using spherical harmonics for
near-field sources. The SH-MUSIC utilizes the magnitude
spectrum. However, magnitude spectrum suffers from se-
vere environmental conditions like low SNR, reverberation
and closely spaced sources. In [16], a high resolution source
localization based on the MUSIC-Group delay spectrum over
ULA has been proposed. The method is non-trivially ex-
tended for planar arrays in [14, 15] and for spherical array
in [13]. In all these works, far-field source were considered.
In this work, group delay spectrum in spherical harmonicsdomain has been developed for range and bearing estimation.
Beamforming based SH-MVDR is also formulated for near-
field source localization.
2.1. Signal processing in Spherical Harmonics domain
A spherical microphone array of order N with radius r and
number of sensorsIis considered. A sound field of spherical-
waves with wavenumber k from L near-field sources is in-
cident on the array. The lth source location is denoted by
8/10/2019 NEAR-FIELD SOURCE LOCALIZATION USING SPHERICAL MICROPHONE ARRAY
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rl = (rl,l), where l = (l, l). The elevation angle is measured down from positive z axis, while the azimuthal
angle is measured counterclockwise from positive x axis.
Similarly, the ith sensor location is given by ri = (r,i),where i = (i, i).
In spatial domain, the sound pressure at Imicrophones,
p(k) = [p1(k), p2(k), . . . , pI(k)]T
, is written as
p(k) = V(k)s(k) + n(k) (1)
where V(k) is I L steering matrix, s(k) is L 1 vectorof signal amplitudes, n(k)is I 1 vector of zero mean, un-correlated sensor noise and (.)T denotes the transpose. Thesteering matrix V(k)is expressed as
V(k) = [v1(k),v2(k), . . . ,vL(k)], where (2)
vl(k) = [ejk|r1rl|
|r1 rl| , . . . ,
ejk|rIrl|
|rI rl| ]T (3)
Denoting the acoustic pressure on the surface of the
sphere byp(k,r,,), the Spherical Fourier Transform (SFT)
and its inverse is defined by [17],
pnm(k, r) =
20
0
p(k,r,,)[Ymn (, )] sin()dd
(4)
p(k,r,,) =n=0
nm=n
pnm(k, r)Ymn (, ) (5)
whereYmn (, )is spherical harmonic of order n and degreemdefined in Equation 6, and(.) denotes the complex conju-gate.
Ymn (, ) =(2n+ 1)(nm)!
4(n+m)! Pmn (cos)ejm (6)
It is to be noted that Ymn are solution to the Helmholtz equa-
tion [18] andPmn are associated Legendre function.
The acoustic pressure is sampled by the microphones on
the surface of the sphere. Hence, the SFT in the Equation 4
can be approximated by following summation
pnm(k, r) =
Ii=1
aip(k, r,i)[Ynm(i)] (7)
0 n N,n m n
where ai are the sampling weights [19]. For order limited
pressure function with orderN, Equation 5 can be written as
p(k, r,) =
Nn=0
nm=n
pnm(k, r)Ymn () (8)
The pressure at the ith microphone due to the lth source
isp(k,r,i) = ejk|rirl|
|rirl| and it is given by [20]
ejk|rirl|
|ri rl| =Nn=0
nm=n
bn(k,r,rl)Ymn (l)
Ymn (i) (9)
wherebn(k,r,rl)is the near-field mode strength. It is relatedto far-field mode strengthbn(kr)as [21]
bn(k,r,rl) = j(n1)kbn(kr)hn(krl)where, (10)
bn(kr) = 4jnjn(kr), open sphere (11)
= 4jnjn(kr) jn(kr)hn(kr)
hn(kr), rigid sphere(12)
jn is spherical Bessel function, hn is spherical Hankel func-
tion,j is unit imaginary number and refers to first derivative.
The extra term in far-field mode strength for rigid sphere ac-
counts for scattered pressure from the sphere. The range of
the source is captured in the Hankel function.
101
100
101
150
100
50
0
50
k
Mag
nitude(dB)
FarfieldNearfield
Fig. 1. Plot showing the nature of far-field and near-field
mode strength for rigid sphere. Near-field source is atrl =1mand order is varied from n= 0(top) ton= 4(bottom)
2.2. Near-field criterion in spherical harmonics domain
In general, the boundary between near-field and far-field is
decided by Fraunhofer distances [22]. However, these pa-rameters do not indicate the extent of near-field in spherical
harmonics domain. For spherical array, the near-field crite-
ria is presented in [10] based on similarity of near-field mode
strength (|bn(k,r,rl)|) and far-field mode strength (|bn(kr)|).The two functions start behaving in similar way at krl N,for array of order N. This is illustrated in Figure 1 for rigid
sphere Eigenmike system [23] with rl = 1mand order vary-ing fromn = 0to n = 4. Hence the near-field condition forspherical array becomes
rNFN
k (13)
ButrNF r,r being the radius of the sphere. So the highestwavenumber possible is
kmax=N
r (14)
From Equations 13,14,rNF =rkmax
k (15)
Hence, for a source to be in near-field, the range of the source
should satisfy
r rl rkmax
k (16)
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020
4060
80100
020
4060
80100
0
0.5
1
1.5
2
Azimuth()Elevation()
SHMUSIC
(a)
020
4060
80100
020
4060
80100
0
20
40
60
80
100
Azimuth()Elevation()
SHMGD
(b)
0 10 20 30 40 50 60 70 80 900
0.5
1
0
0.2
0.4
0.6
0.8
1
Azimuth()
Range(m)
SHMUSIC
(c)
0 10 20 30 40 50 60 70 80 900
0.5
1
0
0.2
0.4
0.6
0.8
1
Azimuth()
Range(m)
SHMGD
(d)
Fig. 2. Illustration of Azimuth and Elevation estimation by (a) SH-MUSIC (b)SH-MGD. Illustration of range and azimuth
estimation using (c) SH-MUSIC (d) SH-MGD. The sources are at (0.4m,60,30) and (0.5m,55,35) at SNR 10dB.
2.3. The Spherical Harmonics MUSIC (SH-MUSIC)
spectrum for near-field source localization
This section presents formulation of the proposed SH-MUSIC
spectrum for near-field source localization. Substituting theexpression for pressure from Equation 9 in Equation 3, the
steering matrix in Equation 2 can be written as
V(k) = Y()[B(r1)yH(1), ,B(rL)y
H(L)] (17)
whereY() is I (N+ 1)2 matrix. A particularith rowvector can be written as
y(i) = [Y00(i), Y
11 (i), Y
01(i), Y
11(i), . . . , Y
NN(i)]
(18)
and y(l)is1 (N+ 1)2 vector with similar structure as in
Equation 18 with angle l,l = 1, 2, , L. The(N+ 1)2
(N+ 1)2 matrix B(rl)is given by
B(rl) = diag(b0(k,r,rl), b1(k,r,rl), b1(k,r,rl),
b1(k,r,rl), . . . , bN(k,r,rl)) (19)
Dependency ofB(rl) on k and r is dropped for notationalsimplicity. Substituting (17) in (1), multiplying both side by
YH() and utilizing Equation 7, the data model becomes
pnm(k, r) = YH()Y()[B(r1)y
H(1), ,
B(rL)yH(L)]s(k) + nnm(k) (20)
where = diag(a1, a2, , aI), consists of samplingweights used in Equation 7 and
pnm= [p00, p1(1), p10, p11, , pNN]T. (21)
The orthogonality of spherical harmonics under spatial sam-pling suggests [19]
YH()Y() =I. (22)
Hence, the data model finally becomes
pnm(k, r) = [B(r1)yH(1), ,B(rL)y
H(L)]s(k)
+ nnm(k) (23)
where B(rs)yH(s) is taken to be look-up steering vector.
The 3-Dimensional MUSIC spectrum in spherical harmonics
domain can now be written as
PMUSIC(rs,s) = 1
y(s)BHSNSpnm[SNSpnm
]HByH(s)(24)
The search is performed over rs as in Equation 16 and over
s with(0 s , 0 s 2). SNSpnm
is noise sub-
space obtained from eigenvalue decomposition of autocorre-
lation matrix, Spnm, defined as
Spnm =E[pnm(k, r)pnm(k, r)H] (25)
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The denominator of the MUSIC spectrum tends to zero when
(rs,s)corresponds to source location owing to orthogonal-ity between noise eigenvector and steering vector. Hence, a
peak is obtained in MUSIC spectrum.
2.4. Near-field source localization using Spherical Har-
monic MUSIC-Group Delay (SH-MGD) spectrum
The SH-MUSIC utilizes the magnitude ofy(s)BHSNSpnm as
it is clear from Equation 24 . The phase spectrum of MUSIC
is utilized in [1316] for robust source localization. A sharp
change in unwrapped phase is seen at the Direction of Arrival
(DOA) [14, 16]. Hence, the negative differentiation of un-
wrapped phase spectrum (Group delay) results in peak at the
DOAs. In practice, abrupt changes can occur in the phase due
to small variations in the signal caused by microphone cali-
bration errors. Hence, the group delay spectrum sometimes
may have spurious peaks. The product of MUSIC and Group
delay spectra, called MUSIC-Group delay, removes such spu-
rious peaks and gives high resolution estimation. The Spher-
ical Harmonics MUSIC-Group delay (SH-MGD) spectrum iscomputed as
PMGD(rs,s) = (
Uu=1
|arg(y(s)BH.qu)|
2).PMUSIC
(26)
whereU= (N+ 1)2 L, is the gradient operator,arg(.)indicates unwrapped phase, and qu represents theu
th eigen-
vector of the noise subspace, SNSpnm. The first term within(.)is the group delay spectrum. The gradient is taken with re-
spect to(rs, s, s).Figure 2 illustrates the performance of SH-MUSIC and
SH-MGD for range and bearing estimation using sphericalmicrophone array. The simulation was done considering open
sphere with two closely spaced sources at (0.4m,60,30),(0.5m,55,35) and SNR 10dB. Figure 2(a) and 2(b) showplots corresponding to elevation and azimuth estimation. It is
clear that SH-MGD exhibits higher resolving power. Plots in
Figure 2(c) and 2(d) show range and azimuth of the sources.
The high resolution of MGD is due to additive property of
group delay spectrum. The additive property is proved math-
ematically in our earlier work for ULA [16] and UCA [15].
While this is valid for spherical array also, the mathematical
proof is being developed.
2.5. The Spherical Harmonics MVDR (SH-MVDR) spec-
trum for range and bearing estimation
The conventional MVDR minimizes the contribution of inter-
ference impinging on the array from a DOA = s, while itmaintains certain gain in look direction s. On the similar
lines, the SH-MVDR spectrum for near-field source localiza-
tion, can be written as
PMVDR(rs,s) = 1
y(s)BHS1pnmBy
H(s)(27)
3. PERFORMANCE EVALUATION
The proposed methods, SH-MUSIC, SH-MGD and SH-
MVDR are evaluated by conducting experiments on source
localization. The estimated range and bearing are tabulated at
various SNRs.
The proposed algorithm was tested in a room with dimen-sions,7.3m 6.2m 3.4m. An Eigenmike microphone ar-ray [23] was used for the simulation. It consists of 32 mi-
crophones embedded in a rigid sphere of radius 4.2 cm. The
order of the array was taken to be N = 4. The source local-ization experiments are conducted at various SNR.
3.1. Experiments on source localization
Two sets of experiments were conducted. For the first experi-
ment, two closely spaced narrowband sources were placed in
near-field region at (0.4m,60,30) and (0.4m,65,35). Therange of the sources was kept fixed at0.4m. The experiments
were conducted at SNR 0dB and 8dB. The additive noise isassumed to be zero mean Gaussian distributed. The mean
estimation for azimuth and elevation is presented in the first
part of the Table 1. In the second experiment, the sources
were positioned at (0.4m,60,30) and (0.5m,65,35). Therange and the azimuth were estimated at SNR 5dB and 10dB,
considering fixed elevation. The result shown in Table 1 is
obtained from300 independent Monte Carlo trials. It is clearthat SH-MGD performs reasonably better than SH-MUSIC.
Both of these methods outperform MVDR.
Table 1. Localization experiments, Set 1 : SNR 0dB, 8dB for
fixed range. Set 2 : SNR 5dB, 10dB for fixed elevation
SNR S SH-MGD SH-MUSIC MVDR
0dB S1 (60.46,29.82) (60.04,30.02) (58.35,29.22)
S2 (65.01,34.94) (65.00,35.00) (63.67,34.19)
8dB S1 (60.00,29.96) (60.00,29.99) (61.15,29.33)
S2 (65.00,35.00) (65.00,35.00) (63.65,34.43)
5dB S1 (0.416,29.91) (0.429,30.11) (0.367,29.26)
S2 (0.548,34.91) (0.560,34.49) (0.541,33.28)
10dB S1 (0.409,30.00) (0.410,30.00) (0.406,30.06)
S2 (0.510,35.00) (0.514,35.00) (0.548,33.40)
4. CONCLUSION
In this work, 3-Dimensional SH-MUSIC, SH-MGD and SH-MVDR are proposed for near-field source localization. Since
the phase spectrum of MUSIC is more robust to noise, the SH-
MGD indicates higher resolution. The proof of additive prop-
erty of group delay in the spherical harmonics domain is cur-
rently being developed. The detailed relative performance of
SH-MUSIC and SH-MGD for closely spaced sources under
reverberation will be addressed in future work. The Cramer-
Rao bound for spherical harmonics is being developed for the
performance analysis of the proposed methods.
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