Stochastic Cram ́er-Rao Bound Analysis for DOA Estimation in Spherical Harmonics Domain

8/10/2019 Stochastic Cram er-Rao Bound Analysis for DOA Estimation in Spherical Harmonics Domain

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IEEE SIGNAL PROCESSING LETTERS 1

Stochastic Cramer-Rao Bound Analysis for DOA

Estimation in Spherical Harmonics DomainLalan Kumar, Student Member, IEEE, and Rajesh M Hegde, Member, IEEE

AbstractCramer-Rao bound (CRB) has been formulated inearlier work for linear, planar and 3-D array configurations. Theformulations developed in prior work, make use of the standardspatial data model. In this paper, the existence of CRB for thespherical harmonics data model is first verified. Subsequently, anexpression for stochastic CRB is derived for direction of arrival(DOA) estimation in spherical harmonics domain. The stochasticCRBs for azimuth and elevation are plotted at various Signalto Noise Ratios (SNRs) and snapshots. It is noted that a lowerbound on the CRB is attained at high SNR. A similar observationis made when larger number of snapshots are used.

Index TermsCramer-Rao bound, Spherical microphone ar-

ray, Spherical harmonics

I. INTRODUCTION

After the introduction of higher order spherical microphone

array and associated signal processing in [1, 2], the spherical

microphone array is widely being used for direction of arrival

(DOA) estimation and tracking of acoustic sources [39]. This

is primarily because of the relative ease with which array

processing can be performed in spherical harmonics (SH)

domain without any spatial ambiguity. Cramer-Rao bound

(CRB) places a lower bound on the variance of a unbiased

estimator. It provides a benchmark against which any estimator

is evaluated. Hence, it is of sufficient interest to develop

an expression for Cramer-Rao bound in spherical harmonicsdomain.

In [10], CRB expression was derived for the case of uniform

linear array (ULA) but without using the theory of CRB. This

is addressed in [11], which provides a textbook derivation

for stochastic CRB. Explicit CRBs of azimuth and elevation

are developed in [12, 13] for planar arrays. CRB analysis is

presented for near-field source localization in [14, 15] using

ULA and UCA (Uniform Circular Array) respectively. In

[16], closed-form CRB expressions has been derived for 3-

D array made from ULA branches. However, to the best

of authors knowledge, closed-form expression for CRB in

spherical harmonics domain is not available in literature. In

this paper, an expression for stochastic CRB for spherical arrayis derived in spherical harmonics domain.

This work was funded in part by TCS Research Scholarship Programunder project number TCS/CS/20110191 and in part by DST projectEE/SERB/20130277.

c2014 IEEE. Personal use of this material is permitted. However, permis-sion to use this material for any other purposes must be obtained from theIEEE by sending a request to [email protected].

Lalan Kumar and Rajesh M. Hegde are with the Departmentof Electrical Engineering, Indian Institute of Technology, Kanpur, e-mail:{lalank,rhegde}@iitk.ac.in

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier xxxxx

The rest of the paper is organized as follows. In Section

II, we present the signal processing in spherical harmonics

domain. In Section III, formulation of CRB is given in detail.

In Section IV, a simulation is presented showing the behavior

of CRB at various SNRs and snapshots. This is followed by

conclusion and future scope in Section V.

I I . DECOMPOSITION OFC OMPLEX P RESSURE IN

SPHERICAL H ARMONICSDOMAIN

We consider a spherical microphone array with I identical

and omnidirectional microphones, mounted on the surface ofa sphere with radius r. The position vector ofith microphone

is given by ri = (r sin icos i, r sin isin i, r cos i)T,

where (.)T denotes the transpose. The elevation angle ismeasured down from positive z axis, while the azimuthal

angle is measured counterclockwise from positive x axis.

Let i (i, i) denotes the angular location of theith microphone. A narrowband sound field of L plane-

waves is incident on the array with wavenumber k. The

wavevector corresponding to lth planewave is given by kl =(k sin lcos l, k sin lsin l, k cos l)T. The direction of ar-rival of the lth source is denoted by l (l, l).

The instantaneous pressure amplitude at theith microphone,

can be expressed as [17]

pi(; t) =Ll=1

slt i(l)

+ni(t) (1)

where t = 1, 2, , Ns, with Ns being the snapshots andi(l) is the propagation delay between the reference micro-phone and ith microphone for the lth source impinging from

direction l. ni is uncorrelated sensor noise component. It

is to be noted that the microphone gain for far-field sources

is taken to be unity. Utilizing the identity slt i(l)

=

ejkT

l risl(t) for narrowband assumption, the Equation 1 can

be rewritten as

pi(; t) =Ll=1

vi(l, k)sl(t) +ni(t) (2)

where vi(l, k) = ejkT

l ri , is referred as steering vector

component corresponding to the ith microphone response for

lth source. Rewriting the Equation 2 in matrix form, we have

p(; t) =V(, k)s(t) + n(t). (3)

Taking appropriate Fourier co-efficients of Equation 3, the

spatial data model in frequency domain can be written as

p(; ) = V(, k)s() + n() (4)


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2 IEEE SIGNAL PROCESSING LETTERS

whereis FFT index, V is ILsteering matrix,s is LNssignal matrix and n is I Ns matrix of uncorrelated sensornoise. The noise components are assumed to be circularly

Gaussian distributed with zero mean and covariance matrix

2I, I being the identity matrix. The steering matrix V(, k)is expressed as

V(, k) = [v1,v2, . . . ,vL], where (5)

vl= [ejkTl r1 , ejkTl r2 , . . . , ejkTl rI ]T (6)

wherej is the unit imaginary number. Theith term in Equation

6 refers to pressure due to l th unit amplitude planewave with

wavevector kl at locationri. This may alternatively be written

as [18]

ejkT

l ri =

n=0

nm=n

bn(kr)[Ymn (l)]

Ymn (i) (7)

where bn(kr) is called mode strength.The far-field mode strength bn(kr) for open sphere (virtual

sphere) and rigid sphere is given by

bn(kr) = 4jnjn(kr), for open sphere (8)

= 4jnjn(kr)

jn(kr)

hn(kr)hn(kr)

, rigid sphere (9)

wherejn(kr)is spherical Bessel function of first kind, hn(kr)is spherical Hankel function of first kind and refers to first

derivative. The extra term in far-field mode strength for rigid

sphere accounts for scattered pressure from the surface of the

sphere [19, p. 228]. Figure 1 illustrates mode strength bn as

a function of kr and n for a open sphere. For kr = 0.1,zeroth order mode amplitude is22 dB, while the first order hasamplitude 8 dB. Hence, for order greater than kr, the modestrengthbn decreases significantly. Therefore, the summation

in Equation 7 can be truncated to finite N, called the arrayorder.

101

100

101

120

100

80

60

40

20

0

20

40

kr

bn

(kr)in

dB

n=0

n=4

n=3

n=2

n=1

Fig. 1. Variation of mode strength bn in dB as a function ofkr andn foran open sphere.

The spherical harmonic of order n and degree m, Ymn ()is given by

Ymn () =

(2n+ 1)(nm)!

4(n+m)! Pmn (cos)e

jm (10)

0 n N,n m n

Ymn are solution to the Helmholtz equation [20] and Pmn are

associated Legendre functions. For negative m, Ymn () =

(1)|m|Y|m|n (). The spherical harmonics are used for

spherical harmonics decomposition of a square integrable

function, similar to complex exponentialejt used for decom-

position of real periodic functions.

Substituting Equations 6 and 7 in Equation 5, we have the

expression for steering matrix as

V(, k) =Y()B(kr)YH() (11)

where Y() is I (N+ 1)2 matrix whose ith row is given

as

y(i) = [Y00(i), Y

11 (i), Y

01(i), Y

11(i), . . . , Y

NN(i)].

(12)

The L (N+ 1)2 matrixY() can be expanded on similarlines. The (N+ 1)2 (N+ 1)2 matrix B(kr) is given by

B(kr) =diagb0(kr), b1(kr), b1(kr), b1(kr), . . . , bN(kr)

.

(13)

Having introduced the spherical harmonics, the spherical

harmonics decomposition of the received pressure p(; ), isgiven as [21]

pnm() = 20

0

p(; )[Ymn ()] sin()dd

=I

i=1

aipi(; )[Ynm(i)] (14)

where pnm() are spherical Fourier co-efficients. The spatialsampling of pressure over a spherical microphone array is cap-

tured using sampling weights, ai [22]. Rewriting the Equation

14 in matrix form, we have

pnm(; ) = YH()p(; ) (15)

where pnm(; ) = [p00, p1(1), p10, p11, . . . , pNN]T and

= diag(a1, a2, , aI). Also, under the assumption ofEquation 14, following orthogonality property of spherical

harmonics holds

YH()Y() =I, (16)

where I is(N+ 1)2 (N+ 1)2 identity matrix. SubstitutingEquation 11 in 4, then multiplying both side with YH()and utilizing relations 15,16, we have data model in spherical

harmonics domain as

pnm(; ) =B(kr)YH()s() + nnm() (17)

Multiplying both side of Equation 17 by B

1

(kr), the finalspherical harmonics data model is given by

anm(; ) = YH()s() + znm() (18)

[anm](N+1)2Ns = [YH](N+1)2L[s]LNs+ [znm](N+1)2Ns

(19)

where

znm() = B1(kr)nnm() =n() (20)

and, = B1(kr)YH() (21)

It must be noted that is known for a given array geometry.


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4 IEEE SIGNAL PROCESSING LETTERS

0 2.5 5 7.5 10 12.5 15 17.5 200

1

2

3

4

5x 10

3

SNR(dB)

CRB

CRB()

CRB()

(a)

50 75 100 125 150 175 200 225 250 275 3000

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

4

Snapshots

CRB

CRB()

CRB()

(b)

Fig. 2. Variation of CRB for elevation () and azimuth () estimation (a) at various SNR with 300 snapshots, (b) with varying snapshots at SNR 20dB.Source is located at (20, 50).

of Equation 37 in Equation 34, the FIM elements can now be

written as

Fr,s = 2Retr(x) +tr(y)

= 2Re

tr(YHsRsYR

1a YHRs YrR

1a

)

+tr(YHsRsYR1aYHrRsYR

1a )

(38)

Utilizing the relations in Equation 30-31,

Fr,s = 2Retr(YHeseTsRsYR1a YHRsereTr YR1a )+tr(YHese

TsRsYR

1aYH ere

TrRsYR

1a )

= 2Re

eTsRsYR

1a YHRsere

Tr YR

1aYHes

+eTsRsYR1aYH ere

TrRsYR

1aYHes

(39)

Hence the FIM can finally be written as

F = 2Re

(RsYR1a YHRs)

T (YR1aYH)

+(RsYR1aYH )

T (RsYR1aYH)

(40)

where denotes Hadamard product. The Hadamard productof two matrix are defined as

(X Z)rs (X)rs(Z)rs. (41)

Similar to Equation 40, the other block of FIM with only one

parameter vector, F can be written as

F = 2Re

(RsYR1a YHRs)

T (YR1aYH )

+(RsYR1aYH )

T (RsYR1aYH )

. (42)

F and F can be expressed in the similar way. The Fisher

Information matrix is finally given by

F =

F FF F

.

Now the closed-form CRB can be computed using Equation

27. IV. SIMULATION R ESULTS

Simulation results are presented in this section to observe

the behavior of the stochastic CRB at various SNRs and

snapshots. An Eigenmike microphone array [26] was used

for this purpose. It consists of 32 microphones embedded

in a rigid sphere of radius 4.2 cm. The order of the array

was taken to be N = 3. The signal and noise are takento be Gaussian distributed with zero mean. A source with

DOA(20, 50) is considered in this simulation. Two sets ofexperiments were conducted with 500 independent trials. Inthe first set, experiments were conducted for 300 snapshots,

at various SNRs. In the second set of experiments, CRB was

computed for various snapshots at SNR of20dB. The CRBfor azimuth and elevation is plotted in Figure 2.

V. CONCLUSION ANDF UTURE S COPE

Stochastic Cramer-Rao bound analysis for azimuth and

elevation estimation of far-field sources is presented using

a spherical microphone array. The spherical harmonics data

model is used for this purpose. The Cramer-Rao bound is

derived by direct application of the CRB theory. The CRB forfar-field azimuth and elevation estimation is also illustrated

at various SNRs and snapshots. CRB for range and bearing

estimation of near-field source using spherical harmonics is

currently being developed. Non-matrix closed-form expression

for conditional and unconditional data model will also be

addressed in future work.

APPENDIX

COMPUTING THED ERIVATIVE OFS PHERICAL H ARMONICS

In this Appendix, we detail the steps involved in the

computation of the derivative ofYnm. From Equations 10 and

12, the vector derivative Y can be found using

Ymn (s)s

=jmYmn (s). (43)

Computing Y involves differentiation of the associated Leg-endre function. The derivative of associated Legendre polyno-

mial can be expressed as [27]

Pmn (z)

z =

1

z2 1[znPmn (z) (m+n)P

mn1(z)] (44)

For z = cos , the derivative becomes

Pmn (cos )

=

1

sin [n cos Pmn (cos )(m+n)P

mn1(cos )].

(45)

Utilizing the property of Legendre polynomial, the Equation

45 can be rewritten as

Pmn (cos )

= 1

sin [(nm+ 1)Pmn+1(cos ) (n+ 1) cosP

mn (cos )].

(46)

Now, Y can be computed by using the following equation.

Ymn (r)

r=

(2n+ 1)(nm)!

4(n+m)! ejmr

1

sin r

.[(nm+ 1)Pmn+1(cos r) (n+ 1) cos rPmn (cos r)] (47)


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Stochastic Cram ́er-Rao Bound Analysis for DOA Estimation in Spherical Harmonics Domain