Decoupled Estimation of 2D DOA for Coherently Distributed Sources Using 3D Matrix Pencil Method

8/8/2019 Decoupled Estimation of 2D DOA for Coherently Distributed Sources Using 3D Matrix Pencil Method

1/8


2/8

2 EURASIP Journal on Advances in Signal Processing

x

...

2

2

y

z...

Figure 1: Array geometry.

2. SIGNAL MODEL

Assume that stationary signals impinge on an array of Ksensors from I narrowband far-field sources. The output ofthe sensors of the array is given by

v(t) = Bs(t) + n(t), (1)

where B = [b1,b2, . . . ,bI], s(t) = [s1(t), s2(t), . . . , sI(t)]T,and n(t), respectively, are the K I generalized steeringmatrix formed between the sources and the antenna elementsat the array, the I

1 signal vector transmitted by the source,

and the K1 additive noise vector. bi is the K1 GSV of theith CD source, which is defined as

bi =

a(,)(,; i)d d, (2)

where a(,) is the steering vector for a point source at 2DDOA (,), i = (i, i , i, i ) is a vector whose elements,respectively, are the azimuth DOA, the angular spread of theazimuth DOA, the elevation DOA, and the angular spreadof the elevation DOA of the ith CD source. (, ; i) isthe deterministic angular weighting function of the ith CDsource.

Similar to [4], when the angular spread is small, for asingle CD source narrowband centered in , we can usethe Taylor approximation, the elements of GSV can beapproximately decomposed as

bk() =b(,,,)

k a(,)]k[g(, ,,)k, (3)

where [a(,)]k is the phase part of the GSV, and[g(, ,,)]k is the module part of the GSV.

Consider a three-dimensional array in space as illustratedin Figure 1 with the axes oriented along the Cartesiancoordinates. The distances between the elements are x,y,and z, which are always half of the wavelength.

The number of sensors are A, B, and C satisfying A + B +C = K. Therefore, we have

[a(,)](a,b,c)

= ej((2/)xcoscosa+(2/)y sin cosb+(2/)zsin c),(4)

where a = 1, . . . ,A; b = 1, . . . , B; c = 1, . . . , C; and

[g(, ,,)](a,b,c)Gaussian

= e222 (x sin cosa+ycoscosb)2/2

e222 (xcos sin ay sin sin b+zcosc)2/2(5)

for Gaussian shaped coherently distributed (GCD) source,

[g(,,,)](a,b,c)Laplacian

= 1/1 + 2((x sin cosa + ycoscosb)/)2) (1 + 2((xcos sin a y sin sin b+ zcosc)/)2

(6)

for Laplacian shaped coherently distributed (LCD) source.It is noted that when the angular spread of coherently

distributed source is small, for Gaussian shaped and Lapla-cian shaped distributed source, the angular information andangular spread information could be separated from thesignal pattern. From (3), (4), (5), and (6), we know that theangular spread only affect the module of the received signalbecause MP algorithm extracts the poles from the phase of

signal, the MP algorithm might be used for the estimationof 2D DOA similar as [9] for point source. For differentlyshaped coherently distributed source, the 2D DOA can bedecoupled from the angular spreads by using MP algorithm.So, the MP algorithm can obtain the 2D DOA of coherentlydistributed sources without the prior information of theshape of the angular weighting function. Obviously, when = 0 and = 0, the above model is a point model. Whenthe angular spread increases, the module of the coherentlydistributed sources decreases.

3. MATRIX PENCIL METHOD

Assume the ith coherently distributed source signals arenarrowband centered in i, consider a 3D data cube as

v(a; b; c)

=I

i=1ej((2/ i)xcosicosia+(2/ i)y sin icosib+(2/ i)zsin ic)i

+ w(a, b, c),

(7)

where w(a, b, c) denotes the noise, and

i =g(i)(a,b,c)si(t) = [g(i,i ,i,i )](a,b,c)si(t), (8)


3/8


4/8


0

0.1

0.2

0.3

0.4

RMSEofazimu

thangle(deg)

0 5 10 15 20 25

SNR (dB)

MP for GCD source 1

MP for GCD source 2

CRB for GCD source 1


Figure 2: RMSE of azimuth angle for GCD sources.

where Fij i s a 7 7, (i, j)th block matrix of F. E{} isthe expectation operator, /i is the partial derivative withrespect to the ith element i of , and log is the naturallogarithm. Using (18) in (20), we have

Fi j = 1

2Re

vH

i

v

j

, (21)

where Re() denotes the real part.By using the Fisher information matrix, the Cramer-Rao

bound (CRB) is defined as

var(i) [F1()]ii. (22)

So the variance of the unbiased estimate of the ithparameter is the ith diagonal element of the inverse of theFisher information matrix. Thus, we can compare the RMSE

of the MP algorithm with

var(i) to measure the goodness

of the estimator.

5. NUMERICAL RESULTS

In this section, we provide numerical illustrations of theperformance of the proposed algorithm. We assume all of

the signals are equipower and have the same frequency. Thenumbers of the array elements in x-direction, y-direction,and z-direction are all 10, the distance between adjacentsensors is /2. It is assumed that all the signals impingingon the array with amplitudes Mi = 1 and phases i = 0. Theresults are based on 500 Monte Carlo simulations.

In the first example, we illustrate the performance of MPalgorithm for two GCD source one with 1 = (35,3.5,40,4)and another one with 2 = (25,2.5,20, 2). The pencilparameters are all 4. We compare with the CRB for 2DDOA estimation of Gaussian shaped coherently distributedsource. Figures 2 and 3 show that the RMSE of the estimatorsapproaches the CRB when SNR varies from 0 dB to 25 dB.

0

0.2

0.4

0.6

0.8

RMSEofelevationangle(deg)

0 5 10 15 20 25

SNR (dB)

MP for GCD source 1

MP for GCD source 2



Figure 3: RMSE of elevation angle for GCD sources.

0

0.5

1

1.5

RMSEofazimuthangle(deg)

0 5 10 15 20 25

SNR (dB)

MP for LCD source 1

MP for LCD source 2

CRB for LCD source 1


Figure 4: RMSE of azimuth angle for LCD sources.

The RMS errors for the two GCD sources using MP are allsmaller than 1 degree when the SNR at 0 dB.

In the second example, we illustrate the performance

of MP algorithm for two LCD source, one with 1 =(35,3.5, 40,4) and another one with 2 = (25, 2.5,20,2).The pencil parameters are all 4. Figures 4 and 5 show thatthe RMSE of the estimators approaches the CRB when SNRvaries from 0 dB to 25 dB. The RMS errors for the two LCDsource using MP are smaller than 1 degree when the SNR at0 dB.

In the third example, we first illustrate the performanceof MP algorithm when = 35, = 40, = 4, theangular spread varies, the pencil parameters are all 4,and SNR is 10 dB: it is observed that the variation of RMSEof azimuth angle is rather small even when (tdelta inFigure 6) increases. We then illustrate the performance of MP


5/8

Z. Gaoyi and T. Bin 5

0

0.2

0.4

0.6

0.8

RMSEofelevationangle(deg)

0 5 10 15 20 25

SNR (dB)

MP for LCD source 1

MP for LCD source 2



Figure 5: RMSE of elevation angle for LCD sources.

0.022

0.023

0.024

0.025

0.026

0.027

0.028

RMSEofazimuthangle(deg)

0 1 2 3 4 5

tdelta (deg)

Figure 6: RMSE of azimuth angle versus .

algorithm when = 35, = 40, = 4, the angularspread varies, also the pencil parameters are all 4, andSNR is 10 dB: it is observed that the variation of RMSE ofelevation angle is also rather small even when (fdelta inFigure 7) increases.

Clearly, the MP algorithm provides good estimationaccuracy for estimating the nominal azimuth and elevationDOA of coherently distributed source. Note that becausethe angular information of coherently distributed source isseparated from angular spread information, the estimationof the 2D DOA does not need the information of the shapeof angular weighting function.

6. CONCLUSIONS

In this study, the coherently distributed source with 3Ddata cube is constructed using the Taylor approximation,whereas the angular and the angular spread information is

0.085

0.09

0.095

0.1

0.105

0.11

0.115

RMSEofelevation

angle(deg)

0 1 2 3 4 5

fdelta (deg)

Figure 7: RMSE of elevation angle versus .

separated from the signal pattern. The matrix pencil methodis extended to the estimation of 2D DOA for coherentlydistributed sources without any search. 3D data matrix isconstructed to estimate poles of 3D plane, the azimuthand elevation of each signal could be obtained from thepoles. This method could deal with differently shaped smallangular spread coherently distributed sources without theprior information of the shape of the angular weightingfunction. Computer simulation validated the efficiency ofthe method. The estimation performance of different shapedcoherently distributed source is studied. The RMS errors ofthe estimator have been compared with the CRB to observe

the goodness of the method at low SNR.

APPENDIX

APPROXIMATION TO THE STEERING VECTOR FORSMALL ANGULAR SPREADS

From (2), we have

[b()] = [b(, ,,)]

=

[a(,)](, ; )dd

=

ej2(xcoscosa+y sin cosb+zsin c)/

( + , + ; )d d,(A.1)

where = (, , , ) characterizes the complex sourcetogether with the angular weighting function(, ; ) whichshows the angular spreading of the source, for instance,the Gaussian shaped angular weighting function can beexpressed as

(, ; ) = 12

e1/2(()2/2 +()2/2 ). (A.2)

For small values of variables = and = , the

functions sin

, cos

, sin

, and cos

can be approximated


6/8


by the first terms in the Taylor series expansions. Using thetrigonometric identity cos( + ) = coscos sin sinand sin( + ) = sin cos + cos sin, we have

ej2(xcos(+)cos(+)a+y sin(+)cos(+)b+zsin(+)c)/

=ej2(xcos(+

)cos(+

)a)/

ej2(y sin(+)cos(+)b)/ ej2(zsin(+)c)/ ej2(x(cos sin )(cos sin )a)/ ej2(y(sin +cos)(cos sin )b)/ ej2(z(sin +cos)c)/

ej2(xcos()cos()a+y sin()cos()b+zsin()c)/

ej2(x sin cosa+ycoscosb)/ ej2(xcos sin ay sin sin b+zcosc)/,

(A.3)

where we assume that 0 and consequentlyej2 sin sin / 1, ej2cos sin / 1. Thus, we can rewrite(A.1) as

b(, , , ) a(, ) g(, , , ), (A.4)

or

[b(, ,,)](a,b,c) [a(,)](a,b,c)[g(, ,,)](a,b,c),(A.5)

where

[g(, ,,)](a,b,c)

= ej2(x sin cosa+ycoscosb)/ ej2(xcos sin ay sin sin b+zcosc)/( + , + ; )d d.

(A.6)

For Gaussian shaped angular weighting function, wehave

[g(, ,,)](a,b,c)

= 12

ej2

(x sin cosa+ycoscosb)/e(2/22 )d ej2(xcos sin ay sin sin b+zcosc)/e(

2/22 )d= e222 (x sin cosa+ycoscosb)2/2

e222 (xcos sin ay sin sin b+zcosc)2/2 ,(A.7)

where the integral formula e

q2x2 ej p(x+)dx = ejp e(p

2/4q2)/q is used.Similarly, when the angular weighting function is Lapla-

cian shaped:

(

,

; ) = 1

2e(

2||/ +2||/), (A.8)

we have

[b(,,,)](a,b,c)

[a(,)](a,b,c)[g(, ,,)](a,b,c) [a(,)](a,b,c)

1

2

ej2

(x sin cosa+ycoscosb)/e(2||/ )d

12

ej2(xcos sin ay sin sin b+zcosc)/

e(

2||/)d [a(,)](a,b,c) 1/1 + 2((x sin cosa + ycoscosb)/)2

1/1 + 2((xcos sin a y sin sin b

+zcosc)/)2

(A.9)

using

0 epx cos(vx + )dx = (p cos v sin )/(p2 + v2),

p > 0.

REFERENCES

[1] P. Zetterberg, Mobile cellular communications with base stationantenna arrays: spectrum efficiency, algorithms and propagationmodels, Ph.D. dissertation, Signals, Sensors, Systems Depart-ment, Royal Institute of Technology, Stockholm, Sweden,1997.

[2] S. Valaee, B. Champagne, and P. Kabal, Parametric local-ization of distributed sources, IEEE Transactions on SignalProcessing, vol. 43, no. 9, pp. 21442153, 1995.

[3] S. Shahbazpanahi, S. Valaee, and M. H. Bastani, Distributedsource localization using ESPRIT algorithm, IEEE Transac-tions on Signal Processing, vol. 49, no. 10, pp. 21692178, 2001.

[4] J. Lee, I. Song, H. Kwon, and S. R. Lee, Low-complexityestimation of 2D DOA for coherently distributed sources,Signal Processing, vol. 83, no. 8, pp. 17891802, 2003.

[5] G. Y. Zhang and T. Bin, Estimation of 2D-DOAs and angularspreads for coherently distributed sources using cumulants,in Proceedings of the 8th IEEE Workshop on Signal Processing

Advances in Wireless Communications (SPAWC 07), pp. 15,Helsinki, Finland, June 2007.

[6] A. Zoubir and Y. Wang, Efficient DSPE algorithm forestimating the angular parameters of coherently distributedsources, Signal Processing, vol. 88, no. 4, pp. 10711078, 2008.

[7] Y. B. Hua, Estimating two-dimensional frequencies by matrixenhancement and matrix pencil, IEEE Transactions on SignalProcessing, vol. 40, no. 9, pp. 22672280, 1992.

[8] Y. B. Hua and T. K. Sarkar, Matrix pencil method forestimating parameters of exponentially damped/undampedsinusoids in noise, IEEE Transactions on Acoustics, Speech, andSignal Processing, vol. 38, no. 5, pp. 814824, 1990.

[9] N. Yilmazer, R. Fernandez-Recio, and T. K. Sarkar, Matrixpencil method for simultaneously estimating azimuth andelevation angles of arrival along with the frequency of the


7/8

Z. Gaoyi and T. Bin 7

incoming signals, Digital Signal Processing, vol. 16, no. 6, pp.796816, 2006.

[10] N. Yilmazer, J. Koh, and T. K. Sarkar, Utilization of a unitarytransform for efficient computation in the matrix pencilmethod to find the direction of arrival, IEEE Transactions on

Antennas and Propagation, vol. 54, no. 1, pp. 175181, 2006.


8/8

PhotographTurismedeBarcelona/J.Trulls

PreliminarycallforpapersThe 2011 European Signal Processing Conference (EUSIPCO2011) is thenineteenth in a series of conferences promoted by the European Association for

Signal Processing (EURASIP, www.eurasip.org). This year edition will take placein Barcelona, capital city of Catalonia (Spain), and will be jointly organized by theCentre Tecnolgic de Telecomunicacions de Catalunya (CTTC) and theUniversitat Politcnica de Catalunya (UPC).

EUSIPCO2011 will focus on key aspects of signal processing theory and

OrganizingCommitteeHonoraryChairMiguelA.Lagunas(CTTC)

GeneralChair

Ana

I.

Prez

Neira

(UPC)GeneralViceChairCarlesAntnHaro(CTTC)

TechnicalProgramChairXavierMestre(CTTC)

Technical Pro ram CoChairsapp c a o ns as s e e ow. ccep ance o su m ss ons w e ase on qua y,relevance and originality. Accepted papers will be published in the EUSIPCOproceedings and presented during the conference. Paper submissions, proposalsfor tutorials and proposals for special sessions are invited in, but not limited to,the following areas of interest.

Areas of Interest

Audio and electroacoustics.

Design, implementation, and applications of signal processing systems.

JavierHernando(UPC)MontserratPards(UPC)

PlenaryTalksFerranMarqus(UPC)YoninaEldar(Technion)

SpecialSessionsIgnacioSantamara(UnversidaddeCantabria)MatsBengtsson(KTH)

Finances

Multimedia signal processing and coding. Image and multidimensional signal processing. Signal detection and estimation. Sensor array and multichannel signal processing. Sensor fusion in networked systems. Signal processing for communications. Medical imaging and image analysis.

Nonstationary, nonlinear and nonGaussian signal processing.

TutorialsDanielP.Palomar(HongKongUST)BeatricePesquetPopescu(ENST)

PublicityStephanPfletschinger(CTTC)MnicaNavarro(CTTC)

PublicationsAntonioPascual(UPC)CarlesFernndez(CTTC)

Procedures to submit a paper and proposals for special sessions and tutorials willbe detailed at www.eusipco2011.org. Submitted papers must be cameraready, nomore than 5 pages long, and conforming to the standard specified on the

EUSIPCO 2011 web site. First authors who are registered students can participatein the best student paper competition.

ImportantDeadlines:

IndustrialLiaison&ExhibitsAngelikiAlexiou(UniversityofPiraeus)AlbertSitj(CTTC)

InternationalLiaison

JuLiu

(Shandong

University

China)

JinhongYuan(UNSWAustralia)TamasSziranyi(SZTAKIHungary)RichStern(CMUUSA)RicardoL.deQueiroz(UNBBrazil)

Webpage:www.eusipco2011.org

roposa s orspec a sess ons ec

Proposalsfortutorials 18Feb 2011

Electronicsubmissionoffullpapers 21Feb 2011Notificationofacceptance 23May 2011

Submissionofcamerareadypapers 6Jun 2011

Documents

Decoupled Estimation of 2D DOA for Coherently Distributed Sources Using 3D Matrix Pencil Method