Source Localization over Spherical Microphone Array

8/10/2019 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

Speech Source Localization over Spherical Microphone Array 1 / 28 WISSAP 2015
http://fullscreen/http://fullscreen/http://fullscreen/http://fullscreen/


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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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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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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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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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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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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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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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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 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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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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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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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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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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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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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 : 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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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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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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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

[email protected]

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]

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Source Localization over Spherical Microphone Array