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Source Localization over Spherical Microphone Array
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Speech Source Localization over Spherical
Microphone ArrayLalan Kumar
Electrical Engineering Department
Indian Institute of Technology KanpurWISSAP 2015Jan 4-7, 2015
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Presentation Outline
Why Source Localization?
My Research Journey : Uniform Linear Array (ULA) to Spherical MicrophoneArray (SMA)
Spherical Coordinate System
Uniform Linear Array and Uniform Circular Array (UCA)
Data Model in Spatial Domain
MUltiple SIgnal Classfication (MUSIC) and MUSIC-Group delay (MGD) Spec-trum
Near-field Source Localization in Spherical Harmonics (SH) Domain
Data Model in SH Domain
SH-MUSIC, SH-MGD, SH-MVDR
Cramr-Rao Bound Analysis
Experiments on Source Localization
Conclusion
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My Research Journey : ULA to SMA
Spherical Coordinate system
Location of a source is given by r = (r, ), with = (, )
The range (r), elevation () and azimuth () takes values as r (0,), [0, ], [0, 2]
X
Y
Z
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My Research Journey : ULA to SMA
Linear and Planar Arrays
d
M0 M1 M2 M3X
Y
Uniform Linear Array geometry
d
M0 M1
S1
S2
Front back ambiguity in ULA
X
Y
Z
Uniform circular array
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My Research Journey : ULA to SMA
Data Model in Spatial Domain
A sound field of L far-field sources with wavenumber k, is incident on amicrophone array ofI microphones.
In spatial domain, the sound pressure, p(k) = [p1(k), p2(k), . . . , pI(k)]T, iswritten as,
p(k) = V(, k)s(k) + n(k), (1)
V(, k)is I Lsteering matrix,s(k)is L 1vector of signal amplitudes,n(k)is I 1vector of zero mean, uncorrelated sensor noise.
The steering matrixV(, k)is expressed as
V(, k) = [v1(1, k), v2(, k), . . . , vL(, k)], where (2)
vl(l, k) = [ejkTl r1
, ejkTl r2
, . . . , ejkTl rI
]T
(3)
kl = (k sin l cos l, k sin l sin l, k cos l)T, withl =/2for ULA.
ri = ((i 1)d, 0, 0)T for ULA andri = (r cos i, r sin i, 0)T for UCA.
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My Research Journey : ULA to SMA
MUSIC and MUSIC-Group Delay Spectrum for Source Localization
The MUSIC spectrum for source localization is given by
PMUSIC() = 1
vH()Rpns[Rp
ns]Hv()(4)
Rpns
is noise subspace obtained from eigenvalue decomposition of auto-correlation matrix,Rp =E[p(k)p(k)H].
MUSIC-Group delay spectrum is given by
PM GD() = (U
u=1
|arg(v().qu)|2).PMUSIC() (5)
U = I L, is the gradient operator, arg(.)indicates unwrapped phase,andqurepresents theu
th eigenvector of the noise subspace, Rpns.
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My Research Journey : ULA to SMA
MUSIC Magnitude and MUSIC phase Spectrum : ULA and UCA
0 20 40 60 80 100 120 140 160 180050
1000
2000
4000
Ele()
MUSIC
Magnitude
0 10 20 30 40 50 60 70 80 900
0.5
1
Azi()
(a)
0 20 40 60 80 100
120 140 160 180
0
20
40
60
80
100
505
Ele()
MP
0 10 20 30 40 50 60 70 80 900
0.5
1
Azi()
MP
(b)
(a) Spectral magnitude of MUSIC for UCA (top) and ULA (bottom). (b)Spectral phase of MUSIC forUCA (top) and ULA (bottom). Sources at (15,50) and (20,60) for UCA. Sources at 50 and 60
for ULA.
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My Research Journey : ULA to SMA
Group Delay and MUSIC-Group delay Spectrum : ULA and UCA 2
0 20 40 60 80 100 120 140 160 180050
1000
10
20
30
Azi()
Ele()
StandardGroupdelay
0 10 20 30 40 50 60 70 80 900
0.5
1
(a)
0 20 40 60 80 100 120 140 160 180050
1000
2
4x 10
4
Azimuth()
Ele()
MU
SICGroupDelay
0 10 20 30 40 50 60 70 80 900
0.5
1
Azi()
(b)
(a) Standard group delay spectrum of MUSIC for UCA (top) and ULA (bottom) (b) MUSIC-Group
delay spectrum for UCA (top) and ULA (bottom).
2Kumar, L.; Tripathy, A.; Hegde, R.M., "Robust Multi-Source Localization Over Planar Arrays Using MUSIC-
Group Delay Spectrum," Signal Processing, IEEE Transactions on , vol.62, no.17, pp.4627,4636, Sept.1, 2014 doi:
10.1109/TSP.2014.2337271
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My Research Journey : ULA to SMA
Application in DSR
Estimate
DOA
Compute
TDOA
Train
FSBDSR
X Y
Z
S1 (40,19) S2(30
,15)
Methods CTM T60
(150ms)
T60
(250ms)
T60
(150ms)
T60
(250ms)
MONCMGD
9.212.98 23.96 11.99 23.58
MUSIC 14.21 26.01 13.78 25.56BS-MUSIC 15.02 27.99 15.22 27.32
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My Research Journey : ULA to SMA3
Spherical Microphone Array (SMA)
The position vector of ith microphone is given asri = (ra, i) where ra isradius of the spherical array andsi = (i, i).
(a)
Near-field
Far-field
(b)
(a) Spherical microphone array : Eigenmike system (b)Near-field and far-field region aroundspherical microphone array. Theith microphone is positioned at riandl
th source atrl.
3Kumar, L.; Singhal, K.; Hegde, R.M., "Robust source localization and tracking using MUSIC-Group delay spectrum
over spherical arrays," CAMSAP 2013, vol., no., pp.304,307, 15-18 Dec. 2013
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Near-field Source Localization in SH Domain
Data Model in Spatial Domain 4
Pressure at theith microphone due to lth source is sl(ti(l))|rirl| withi(l) =|rirl|
c , wherec is speed of sound.
Total pressure atith microphone amounts to be
pi(t) =
Ll=1
sl(t i(l))
|ri rl| +ni(t). (6)
Taking Fourier transform, the Equation6turns out to be
pi(fq) =
Ll=1
ej2fqi(l)
|ri rl| sl(fq) + ni(fq), q=1, ,Q. (7)
4Kumar, L.; Singhal, K.; Hegde, R.M., "Near-field source localization using spherical microphone array," HSCMA 2014,
vol., no., pp.82,86, 12-14 May 2014
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Near-field Source Localization in SH Domain
Data Model in Spatial Domain
Droppingq, the Equation7can be re-written in wavenumber domain as
pi(k) =L
l=1
ejk|rirl|
|ri rl|sl(k) + ni(k). (8)
In matrix form, the final near-field data model in spatial domain can be writ-ten as
p(k) = V()s(k) + n(k) (9)
The steering matrixV()is
V() = [v(1), v(2), . . . , v(L)], where (10)
v(l) = [ejk|r1rl|
|r1 rl|, . . . ,
ejk|rIrl|
|rI rl|]T (11)
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Near-field Source Localization in SH Domain
Data Model in Spherical Harmonics Domain
Monochromatic spherical wave solution for wave equation ejk|r
ir
l|
|rirl| , can bewritten in spherical coordinates as
ejk|rirl|
|ri rl|=
n=0
nm=n
bn(k, ra, rl)Ymn (l)
Ymn (i) (12)
bn(k, ra, rl) is nth order near-field mode strength. It is related to far-field
mode strengthbn(k, ra)as bn(k, ra, rl) =j(n1)kbn(k, ra)hn(krl).
The far-field mode strength for open sphere (virtual sphere) and rigid sphere[1] is given by
bn(k, r) = 4j njn(kr), open sphere (13)
= 4j njn(kr)
jn(kra)
hn(kra)hn(kr)
, rigid sphere. (14)
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Near-field Source Localization in SH Domain
Near-field Criterion for SMA
101
100
101
250
200
150
100
50
0
50
k
Magnitude(dB)
Kmax
n=0
n=1
n=2
n=3
n=4
Nearfield
Farfield
Far-field and near-field mode strength for Eigenmike system. Near-field source is atrl = 1mandorder is varied fromn = 0(top) ton = 4(bottom)
The near-field criteria for spherical array is presented based on similar-ity of near-field mode strength (|bn(k, ra, rl)|) and far-field mode strength(|bn(k, ra)|).
The two functions start behaving in similar way at krl N, for array oforderNas shown in the Figure.
Hence, near-field condition for spherical array turns out to be rN F N
k and
ra rl N
k [2].
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Near-field Source Localization in SH Domain
Spherical Harmonics
Ymn represents spherical harmonic of ordern and degreem given by
Ymn (, ) =
(2n+ 1)(n m)!
4(n+m)!Pmn (cos)e
jm. (15)
0 n N, n m n
wherePmn are the associated Legendre function.
Spherical harmonics plot : Y00, Y01, Y
11
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Near-field Source Localization in SH Domain
Spherical Fourier Transform
Assuming continuous distribution of pressure, the spherical Fourier trans-form (SFT) of received pressurepc(k ,r ,,)at (r ,,), is given as [3]
pnm(k, r) =
20
0
pc(k ,r ,,)[Ymn (, )]
sin()dd (16)
Rewriting Equation16for discrete microphone array
pnm(k, r) =I
i=1
aipi(k, r, i)[Ynm(i)] (17)
In matrix form for alln and m, we have
pnm(k, r) = YH()p(k, r, ) (18)
where = diag(a1, a2, , aI)is matrix of sampling weights.
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Near-field Source Localization in SH Domain
Data Model in Spherical Harmonics Domain
Substituting the expression for pressure from Equation 12in Equation11,the steering matrix in Equation10can be written as
V() = Y()[B(r1)yH(1), , B(rL)y
H(L)] (19)
Y()is I (N+ 1)2 matrix. A particularith row vector can be written as
y(i) = [Y00(i), Y
11 (i), Y
01(i), Y
11(i), . . . , Y
NN(i)]. (20)
The(N+ 1)2 (N+ 1)2 matrixB(rl)is given by
B(rl) = diag(b0(k, ra, rl), b1(k, ra, rl), b1(k, ra, rl), b1(k, ra, rl), . . ,bN(k, ra, rl))(21)
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Near-field Source Localization in SH Domain
Data Model in Spherical Harmonics Domain
Substituting (19) in (9), multiplying both side byYH()and utilizing Equa-tion17, the data model becomes
pnm(k, r) = YH()Y()[B(r1)y
H(1), , B(rL)yH(L)]s(k) + nnm(k)
(22)
Orthogonality of spherical harmonics under spatial sampling suggests [4],YH()Y() =I.
The data model in spherical harmonics domain turns out to be
pnm(k) = [B(r1)yH(1), , B(rL)y
H(L)]s(k) + nnm(k). (23)
Re-writing the data model in more compact way, we have
pnm(k) = Vnm(r, )s(k) + nnm(k) (24)
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Near-field Source Localization in SH Domain
SH-MUSIC, SH-MGD, SH-MVDR
The near-field spherical harmonics MUSIC spectrum can now be written as
PSHMUSIC(rs, s) = 1
vnmHRpns[Rp
ns]Hvnm(25)
The Spherical Harmonics MUSIC-Group delay (SH-MGD) spectrum is com-
puted as
PSHM GD(rs, s) = (U
u=1
|arg(vnmH.qu)|
2).PM M (26)
The SH-MVDR spectrum for near-field source localization, is written as
PMV DR(rs, s) = 1
y(s)BHRp1ByH(s)
(27)
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Near-field Source Localization in SH Domain
020
4060
80
0.04
0.06
0.08
0.10
0.5
1
Elevation()
Range(m)
SHMUSIC
x : 60Y : 0.06z : 0.66
x : 55Y : 0.08z : 1
(a)
020
4060
80
0.04
0.06
0.08
0.10
0.5
1
Elevation()
Range(m)
SHMGD
x : 55Y : 0.08z : 1
x : 60Y : 0.06z : 0.71
(b)
020
4060
80
0.04
0.06
0.08
0.10
0.5
1
Elevation()Range(m)
SHMVDR
x : 55Y : 0.08z : 1 x : 60
Y : 0.06z : 0.96
(c)
020
4060
80
020
4060
80
0
0.5
1
Elevation()
Azimuth()
SHMUSIC
x : 55Y : 40z : 1
x : 60Y : 30z : 0.71
(d)
020
4060
80
020
4060
80
0
0.5
1
Elevation()
Azimuth()
SHMGD
x : 55Y : 40z : 1 x : 60
Y : 30z : 0.8
(e)
020
4060
80
020
4060
80
0
0.5
1
Elevation()
Azimuth()
SHMVDR
x : 55Y : 40z : 1
x : 60Y : 30z : 0.96
(f)
The sources are at (0.06m,60,30) and (0.08m,55,40) with SNR 10dB.
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Near-field Source Localization in SH Domain
Near-field Source Localization : 4D Scatter Plot for source at (0.06m,60,30)
57 58
5960
6162
63
27
2829
3031
3233
0.058
0.059
0.06
0.061
0.062
0.063
Elevation()
X: 60
Y: 30
Z: 0.06
Azimuth()
Range(m)
0.2
5.72
11.2
16.7
22.3
27.8
33.3
38.8
44.3
49.8
55.3
X : 60
Y : 30Z : 0.06
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Near-field Source Localization in SH Domain
Cramr-Rao Bound Analysis
The unknown parameter vector is = [rT T T]T withr = [r1 rL]T =[1 L]
T and = [1 L]T.
Based on Cramr-Rao bound (CRB) expression for far-field case5, CRB ex-pression for near-field case, can be obtained using following Fisher infor-mation matrix (FIM) elements
Fr = 2Re
(RsVHnmR
1p VnmRs)
T (VHnmrR1p Vnm
)
+ (RsVHnmR
1p Vnmr)
T (RsVHnmR
1p Vnm
)
(28)
F = 2Re
(RsVHnmR
1p VnmRs)
T (VHnmR1p Vnm
)
+ (RsVHnmR
1p Vnm
)T (RsVHnmR
1p Vnm
) (29) Other block of FIM can be written in similar way.
5Kumar, L.; Hegde, R.M., "Stochastic Cramr-Rao Bound Analysis for DOA Estimation in Spherical Harmonics Do-
main," Signal Processing Letters, IEEE , vol.22, no.8, pp.1030-1034, Aug. 2015 doi: 10.1109/LSP.2014.2381361
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Near-field Source Localization in SH Domain
Cramr-Rao Bound Analysis
10 7.5 5 2.5 0 2.5 5 7.5 100
0.5
1
1.5
2x 10
6
SNR (dB)
CRB
CRB(r)
CRB()
CRB()
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Near-field Source Localization in SH Domain
Experiments on Source Localization
RMSE was found for sources at (0.06,30,45) and (0.08,40,50) for100iter-ation.
Comparison of the RMSE in(r, )at known.
SNR (dB) S SH-MGD SH-MUSIC SH-MVDR-10
S1 (0.001,0.4) (0.001,0.2449) (0.013,2.97)S2 (0,0.4243) (0.001,0.2) (0.007,2.05)
-5 S1 (4.47e-04,0) (2.0e-04,0) (0.0028,1.0)
S2 (4.9e-04,0) (0,0.1414) (0.0018,0.7071)
0 S1 (0,0) (0,0) (0.001,0)
S2 (0,0) (0,0) (4.0e-04,0)
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Conclusion
MUSIC-Group delay based source localization has been presented for ULA,UCA and SMA.
Near-field source localization for simultaneous estimation of range and bear-ing, has been utilized for the fist time.
Experiments on source localization is presented as RMSE.
Near-field array processing using sparse recovery technique in SH domain,will be dealt with in future.
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Thank You
Lalan Kumar
http://home.iitk.ac.in/~lalank/
http://localhost/var/www/apps/conversion/tmp/scratch_8/[email protected]://home.iitk.ac.in/~lalank/http://home.iitk.ac.in/~lalank/http://localhost/var/www/apps/conversion/tmp/scratch_8/[email protected]