Research ArticleEstimation for Two-Dimensional Nonsymmetric CoherentlyDistributed Source in L-Shaped Arrays

Tao Wu ,1 Zhenghong Deng,1 Qingyue Gu,2 and Jiwei Xu1

1School of Automation, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China2School of Marine, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China

Correspondence should be addressed to Tao Wu; taowu_nwpu@126.com

Received 28 December 2017; Revised 11 April 2018; Accepted 13 June 2018; Published 3 September 2018

Academic Editor: Shiwen Yang

Copyright © 2018 TaoWu et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We explore the estimation of a two-dimensional (2D) nonsymmetric coherently distributed (CD) source using L-shaped arrays.Compared with a symmetric source, the modeling and estimation of a nonsymmetric source are more practical. Anonsymmetric CD source is established through modeling the deterministic angular signal distribution function as a summationof Gaussian probability density functions. Parameter estimation of the nonsymmetric distributed source is proposed under anexpectation maximization (EM) framework. The proposed EM iterative calculation contains three steps in each cycle. Firstly, thenominal azimuth angles and nominal elevation angles of Gaussian components in the nonsymmetric source are obtained fromthe relationship of rotational invariance matrices. Then, angular spreads can be solved through one-dimensional (1D) searchingbased on nominal angles. Finally, the powers of Gaussian components are obtained by solving least-squares estimators.Simulations are conducted to verify the effectiveness of the nonsymmetric CD model and estimation technique.

1. Introduction

Although point source models are commonly used in appli-cations such as wireless communications, radar and sonarsystems, distributed source models which have consideredmultipath propagation and the surface features of targetstend to perform better. Position localization estimators basedon distributed source models have proved to be more precisein multipath scenarios compared with point source models.With information of surface features, the distributed sourcemodels have potential application for high-resolution under-water acoustical imaging.

Sources may be classified into coherently and incoher-ently distributed sources [1]. A coherently distributed (CD)source is defined as the signal components arriving from dif-ferent angles within the range of extension that are coherent.For an incoherently distributed (ID) source, these compo-nents are uncorrelated. In this paper, the modeling and esti-mation of a CD source are considered.

Some classical point source estimation techniques havebeen extended to CD sources. DSPE [1], DISPARE [2], and

vec-MUSIC [3] are developed from MUSIC for distributedsources, where parameters are estimated by two-dimensional (2D) spectral searching. In [4], ESPRIT isextended for distributed sources, where the TLS-ESPRITalgorithm is used to estimate nominal angles of sourcesfirstly, and then angular spreads are obtained by one-dimensional (1D) spectral searching. The authors of [5] havedeveloped an efficient DSPE algorithm and proposed gener-alized beamforming estimators in [6]. Generally, the param-eters of CD sources are approximate solutions under theassumption of small angular spreads; some comparativestudies on different methods have been proposed. The per-formance of two beamforming methods is analyzed in [7].In the presence of model errors, the performance of DSPEalgorithm is analyzed in [8], and the performance of MUSICis analyzed in [9]. Considering mismodeling of the spatialdistribution of distributed sources, a robust estimator is pro-posed in [10] by means of exploiting some properties of thegeneralized steering vector in the case of CD sources andthe covariance matrix in the case of ID sources. All thesemethods are based on the 1D distributed source models,

HindawiInternational Journal of Antennas and PropagationVolume 2018, Article ID 5247919, 16 pageshttps://doi.org/10.1155/2018/5247919

which assume that signal sources are in the same plane,where distributed sources are characterized by the nominaldirection of arrival (DOA) and angular spread. However,impinging signals are not generally in the same plane but ina three-dimensional space practically. The 2D distributedsource models are usually described by the nominal azimuthDOA, nominal elevation DOA, azimuth angular spread, andelevation angular spread. Involving more parameters, 2D CDsource estimation problem is more complicated.

Based on DSPE, several spectral searching methods for2D CD sources have been proposed. In [11–15], algorithmsfor exponential CD sources are presented under L-shapedarrays, uniform circular arrays, and nested arrays, whichtransform a four-dimensional parameter searching probleminto a 2D parameter searching problem. The authors of[16] have proposed an estimator for Gaussian CD sourcesunder L-shaped arrays.

Using an array with sensors oriented along three axes ofthe Cartesian coordinates, the authors of [17] have extendedthe matrix pencil method for 2D Gaussian and Laplacian CDsources. The method needs not spectral searching, but itsarray structure is not beneficial to engineering realizationand its concern is DOAs without angular spreads.

In [18], a sequential one-dimensional searching (SOS)algorithm is proposed for Gaussian CD sources using a pairof uniform circular arrays, which first estimates the nominalelevation by the TLS-ESPRIT method, followed by sequentialone-dimensional searching to seek the nominal azimuth. Asthe method presented in [17], the SOS algorithm only dealswith the DOAs.

Several low-complexity algorithms have been presentedin [19–24] utilizing two closely spaced parallel ULAs, tre-ble parallel ULAs, and conformal arrays. It is a commoncharacteristic that though preliminary Taylor approxima-tion to generalized steering vectors, the nominal elevationand azimuth are solved under ESPRIT or modified propa-gator. The authors of [25] have proposed a method usinga centrosymmetric crossed array, where DOAs areobtained through the symmetric properties of the specialarray. These methods need not any spectral searchingand it can deal with unknown angular distribution func-tions. The disadvantage of these algorithms is that angularspreads are not within consideration.

Utilizing two closely spaced parallel ULAs and L-shapedarrays, two estimators for CD noncircular signals are pro-posed [26, 27] which exhibit better accuracy and resolutioncompared with circular signals.

Estimation techniques for distributed sources men-tioned above are performed based on the assumption thatthe spatial distributions of sources are symmetric. Never-theless, scatters are distributed irregularly or nonuni-formly around targets; the distributions of scatters inpractice are generally nonsymmetric. Considering com-plexity, people have paid less attention to nonsymmetricdistributed sources.

Based on the principle that a nonsymmetric probabilitydistribution can be composed of several symmetric distribu-tions, some methods for 1D nonsymmetric ID sources havebeen presented. In [28], a nonsymmetric distribution is

modeled by two Gaussian distributions. The shape of thenonsymmetric distribution would be figured via the variationof the ratio between two Gaussian distributions. In [29], theGaussian mixture model (GMM) is employed to characterizethe 1D nonsymmetric ID sources, and the expectation maxi-mization (EM) algorithm is applied to solve the problem.

To the best of our knowledge, there are no algorithms fora 2D nonsymmetric distributed source. In this paper, we areconcerned on the estimation of a 2D nonsymmetric CDsource. Compared with a symmetric source, the modelingand estimation of a nonsymmetric source are more complex.Through modeling the nonsymmetric deterministic angularsignal distribution function by Gaussian mixture, we havepresented a parameter estimation method under an EMframework based on L-shaped arrays. In general EM frame-works, the maximization step is to maximize the likelihoodfunction to get the best parameters which are hidden inexpectation step results. In our method, the DOA parametersof Gaussian components of the 2D CD source are obtainedthrough two approximate rotational invariance relations. Bysolving least-squares estimators, powers of Gaussian compo-nents are obtained.

2. Signal Model

Figure 1 shows the L-shaped array configuration, which usesthe xoy plane. Each linear array consists of K sensor elementsseparated by d meters, and the two linear arrays share anorigin sensor. Suppose that there is a CD source with anominal azimuth angle θ and a nominal elevation angle ϕ,θ ∈ −π/2, π/2 and ϕ ∈ 0, π/2 . λ is the wavelength of theimpinging signal. The CD source is considered as stationarynarrowband stochastic process.

The K × 1 output vectors of the arrays X and Y can beexpressed as follows:

x t =∬a θ, ϕ s θ, ϕ, t dθdϕ + nx t ,

y t =∬b θ, ϕ s θ, ϕ, t dθdϕ + ny t ,1

where the steering vectors of the arrays X and Y can beexpressed as follows:

a θ, ϕ = 1, ej2πd cos θ sin ϕ/λ,… , ej2π K−1 d cos θ sin ϕ/λ T,

b θ, ϕ = 1, ej2πd sin θ sin ϕ/λ,… , ej2π K−1 d sin θ sin ϕ/λ T

2

s θ, ϕ, t is the angular signal density function of thedistributed source. According to the CD source assumption,s θ, ϕ, t can be expressed as

s θ, ϕ, t = s t f θ, ϕ ; u , 3

where s t is a random process, P = E s t 2 is the power ofthe CD source, f θ, ϕ ; u is the deterministic angular signaldistribution function of the CD source, and u is the parame-ter set of the function.

2 International Journal of Antennas and Propagation

nx t and ny t are the K × 1 additive noise vectors whichcan be combined into

n t =nx t

ny t4

The noise is assumed to be uncorrelated between sensorsand Gaussian white with zero mean

E n t nH t′ = ρI2Kδ t − t′ ,

E n t nT t′ = 0 ∀t, t′,5

where ρ is the noise power, I2K is the 2K × 2K identitymatrix; superscript · T denotes the transpose and · H

denotes the Hermitian transpose. δ · is the Kroneckerdelta function. The signal is assumed to be uncorrelatedwith the noise.

In this study, the deterministic angular signal distributionfunction of the CD source can be modeled as a summation ofGaussian distributions in order to express nonsymmetry, sothe deterministic angular signal distribution function isobtained as

f θ, ϕ ; u = 〠q

i=1wigi θ, ϕ ; ui , 6

where q is the number of Gaussian components. The ithGaussian component is defined as

gi θ, ϕ ; ui =1

2πσθiσϕiexp −0 5

θ − θiσθi

2+

ϕ − ϕiσφi

2

,

7

where ui = θi, ϕi, σθi, σϕi is the parameter set with fourelements denoting the nominal azimuth, nominal eleva-tion, azimuth angular spread, and elevation angularspread, respectively.

To normalize f θ, ϕ ; u , the weighting coefficient wi sat-isfies the following constraint:

〠q

i=1wi = 1 8

The summation of Gaussian distributions in (6) can besubstituted for f θ, ϕ ; u in (3) and (1). The output vectorsof the arrays X and Y can be expressed as follows:

x t = s t 〠q

i=1wi∬a θ, ϕ gi θ, ϕ ; ui dθdϕ + nx t ,

y t = s t 〠q

i=1wi∬b θ, ϕ gi θ, ϕ ; ui dθdϕ + ny t

9

3. Proposed Method

The deterministic angular signal distribution function char-acterized by (6) has 5q unknown parameters. Compared witha symmetric CD source, there are much more parameters tobe estimated. Due to nonlinear and high-dimensional prop-erties, traditional methods such as maximum likelihoodand spectral searching are hard to be used. The EM algorithm[30] is traditionally used to estimate Gaussian mixtureparameters. Here, according to the iterative EM optimizationframework, an iterative algorithm is presented for estimatingthe deterministic angular signal distribution function in (6).

3.1. Latent Variable Model.On the basis of the latent variablemodel described by the authors of [31], the incomplete data issimply the set of observations themselves. The many-to-onefunction, which is linear and maps the complete data toincomplete data, can be expressed as follows:

x t = 〠q

i=1xi t ,

y t = 〠q

i=1yi t

10

Define the generalized steering vectors of the completedata for arrays X and Y as follows:

αi ui =∬a θ, ϕ gi θ, ϕ ; ui dθdϕ,

βi ui =∬b θ, ϕ gi θ, ϕ ; ui dθdϕ11

Combine the incomplete data x t and y t into

z t =x t

y t= s t

〠q

i=1wiαi ui

〠q

i=1wiβi ui

+nx t

ny t12

yd

Array X

Array YArra

y X B

Array YA

Array YB

Array XA

x

z

𝜃

𝜙

Figure 1: The L-shaped array configuration.

3International Journal of Antennas and Propagation

The noise is assumed to be distributed in the completedata equally. Combine the complete data xi t and yi t into

zi t =xi tyi t

= s twiαi uiwiβi ui

+1q

nx t

ny t13

Define two data selection matrices as follows:

Α = I K−1 × K−1 ∣ 0 K−1 ×1 ,

B = 0 K−1 ×1 ∣ I K−1 × K−1 ,14

where I K−1 × K−1 is the K − 1 × K − 1 identity matrixand 0 K−1 ×1 is the K − 1 × 1 zero matrix. Divide X intotwo shifted subarrays XA and XB, and divide Y into twoshifted subarrays YA and YB, which are depicted in Figure 1.

The output vectors of the subarrays XA, XB, YA, and YBcan be expressed as

xA t , xB t , yA t , yB t = Ax t , Bx t ,Ay t , By t

15

The complete data of output vectors of the subarrays XA,XB, YA, and YB is obtained as

xAi t , xBi t , yAi t , yBi t = Axi t , Bxi t ,Ayi t , Byi t16

The generalized steering vectors of the complete data forXA, XB, YA, and YB can be written as

αAi ui , αBi ui , βAi ui , βBi ui = Aαi ui , Bαi ui ,Aβi ui , Bβi ui17

Assume that the angular spreads of Gaussian angularsignal distribution in the complete data are small. Theauthors of [19] have proved that for d/λ = 1/2, there existsan approximate rotational invariance relation as follows(see Appendix A):

αAi ui ≈ ejπ cos θi sin ϕiαBi ui ,

βAi ui ≈ ejπ sin θi sin ϕiβBi ui18

3.2. EM Algorithm. The sample covariance matrices of thecomplete data Rxi and Ryi with N snapshots are defined asfollows:

Rxi = 〠N

t=1xi t xi t H,

Ryi = 〠N

t=1yi t yi t H,

19

while the sample covariance matrix Rzi with N snapshots isdefined as

Rzi = 〠N

t=1

xi tyi t

xi tyi t

H

20

The negative logarithm of the likelihood function can besimplified as

L wi, ui = log Rzi + tr R−1zi Rzi , 21

where the covariance matrix of the complete data zi t can beexpressed as

Rzi = E zi t zi t H = Pi

αi uiβi ui

αi uiβi ui

H

+ ρi 22

In formula (22), Pi is the power of the ith complete data.ρi = ρ/q2 is the noise power in the ith complete data. Fromformulas (12) and (13), zi t can be denoted as

zi t =xi tyi t

= s twiαi uiwiβi ui

+1q

x t − 〠q

i=1s t wiαi ui

y t − 〠q

i=1s t wiβi ui

23

As Rzi is a sufficient statistic of wi, θi, ϕi, σi, and Pi, them

th expectation step of the EM algorithm serves to calculate

the expected value of the sufficient statistic as

Rmzi = wm

i2Pm

αi umiβi umi

αi umiβi umi

H

+1q2

Rx − Pm 〠q

i=1wm

i αi umi 〠q

i=1wm

i αHi umi Rxy − Pm 〠q

i=1wm

i αi umi 〠q

i=1wm

i βHi umi

RHxy − Pm 〠

q

i=1wm

i βi umi 〠q

i=1wm

i αHi umi Ry − Pm 〠q

i=1wm

i βi umi 〠q

i=1wm

i βHi umi

,

24

4 International Journal of Antennas and Propagation

where the superscript m denotes the value of variables at themth iteration. The sample covariance matrices Rx, Ry, andRxy with N snapshots are defined as

Rx =1N〠N

t=1x t x t H,

Ry =1N〠N

t=1y t y t H,

Rxy =1N〠N

t=1x t y t H

25

To simplify the estimation, we assume that the azi-muth angular spread and elevation angular spread arethe same; σi is used to replace σθi and σϕi for convenience.The maximization step will minimize the objective func-tion (21) to find the optimal parameters in the m + 1 thiteration. The function (21) can be expressed as

wm+1i , θm+1

i , ϕm+1i , σm+1

i , Pm+1i = argmin

wi ,uilog Rzi + tr R−1

zi Rmzi ,

26

which means exploring the best m + 1 th parameters ofthe ith Gaussian component based on the sample covariancematrix of the complete data Rm

zi obtained in themth iteration.Minimizing (26) is too computationally complex mainlybecause it is nonlinear and involves a high dimensionalmatrix inversion. According to the rotational invariancerelation, we can firstly estimate the nominal azimuth andnominal elevation of the complete data and then solveother parameters successively based on parameters thathave been solved.

The sample covariance matrix of the complete data xi tand yi

t can be obtained as follows:

Rmxi =

IK×K 0K×K0K×K 0K×K

Rmzi ,

Rmyi =

0K×K 0K×K0K×K IK×K

Rmzi ,

27

where IK×K is the K × K identity matrix and 0K×K is the K× K zero matrix. By means of eigendecomposition of thecovariance matrices Rm

xi and Rmyi , we obtain

Rmxi = cEm

xi Emxi

H + ρiNmxi Nm

xiH,

Rmyi = cEm

yi Emyi

H+ ρiNm

yi Nmyi

H,

28

where c is a constant, Emxi and Em

yi are the K-dimensional

eigenvectors of Rmxi and Rm

yi corresponding to the largest

eigenvalues, respectively, while Nmxi and Nm

yi are the K ×K − 1 matrices representing noise subspace.

Let EmxAi and Em

xBi be the upper K − 1 and the lowerK − 1 elements of Em

xi , respectively, corresponding to thesubarrays XA and XB; Em

yAi and EmyBi are the upper K − 1

and the lower K − 1 elements of Emyi , respectively, corre-

sponding to the subarrays YA and YB. There exist rela-tionships as follows:

Emxi =

the f irst row

EmxBi

=EmxAi

the last row

=the f irst row

αBi uic′ =

αAi uithe last row

c′,

Emyi =

the f irst row

EmyBi

=EmyAi

the last row

=the f irst row

βBi uic″ =

βAi uithe last row

c″,

29

where c′ and c″ are constants. Let hi = cos θi sin ϕi and vi =sin θi sin ϕi.

From formula (29), we have

hm+1i = ln

11× K−1 × EmxBi /Em

xAi

jπ K − 1,

vm+1i = ln

11× K−1 × EmyBi

/EmyAi

jπ K − 1,

30

where I1× K−1 is the 1 × K − 1 identity matrix and (./)denotes the element-wise division operation. The nominalazimuth and nominal elevation of the complete data in them + 1 th maximization step can be obtained from

ϕm+1i = arcsin hm+1

i2 + vm+1

i2 ,

θm+1i = arctan

vm+1i

hm+1i

31

According to the orthogonality of the subspace, σi can beobtained from one-dimensional searching

σm+1i = arg min αi θm+1

i , ϕm+1i , σi

H, βi θm+1i , ϕm+1

i , σiH

Nmxi

Nmyi

Nmxi

Nmyi

H

αi θm+1i , ϕm+1

i , σiH, βi θm+1

i , ϕm+1i , σi H H

32

5International Journal of Antennas and Propagation

Under the condition that the parameters nominal azi-muth θi, nominal elevation ϕi, angular spread σi are solved,

the generalized steering vectors of complete data can beexpressed as follows (see Appendix B):

The least-squares fits of the theoretical and samplecovariance can be expressed as

U = Rzi − Piri ui − ρiI2K2F, 34

where

ri ui =αi uiβi ui

αi uiβi ui

H

35

Differentiating (34) with respect to Pi and ϱi setting theresults to zero yield the following equation:

ρi =tr Rzi tr ri ui ri ui ∗ − tr ri ui R

∗zi tr ri ui

2Ktr ri ui ri ui ∗ − tr ri ui tr ri ui, 36

Pi =2Ktr ri ui R

∗zi − tr ri ui tr Rzi

2Ktr ri ui ri ui ∗ − tr ri ui tr ri ui, 37

where superscript · ∗ denotes the conjugate operation andtr · is the trace of matrix.

Let

γm+1τ =

Pm+1i

Pmi+1

, τ = i when i < q 38

From Pi = P/w2i and formula (8), we can get

wm+1i =

∏q−1τ=i γ

m+1τ

∏q−1τ=1γ

m+1τ +∏q−1

τ=2γm+1τ +⋯ + γm+1

q−1 + 1  1 ≤ i < q ,

1

∏q−1τ=1γ

m+1i +∏q−1

τ=2γm+1i +⋯ + γm+1

q−1 + 1  i = q

39

At last, the power of the CD source can be obtained from

Pm+1 =Pm+1i

wm+1i

2 40

3.3. Complexity Analysis and Comparison. Now, we analyzethe computational complexity of the proposed method incomparison with DSPE [16] and TLS-ESPRIT [19] whichare two estimators for 2D symmetric CD sources under L-shaped arrays. The DSPE method estimates DOAs and

angular spreads through two 2D spectral searches; its compu-tation cost is mostly made of three parts: the calculation ofthe sample covariance matrix, the eigendecomposition ofthe matrix, and 2D searching. TLS-ESPRIT [19] is a methodof DOA estimation; its computational cost mostly consists ofthe estimation and eigendecomposition of a 2K × 2K samplecovariance matrix.

The stopping criterion of the EM algorithm is all param-eters no longer changing [30, 31]. We define the iterativevariation of all parameters as

δ = 14q

εm+1μ − εmμεmμ

, μ = 1, 2,… , 4q, 41

where εmμ represents a parameter value in the mth iteration.When δ reaches a sufficiently small value, all parameterscan be considered keeping stable. The computation cost ofthe proposed method in one EM circle mainly consists ofthree parts: the calculation of the 2K × 2K sample covariancematrix, the eigendecomposition of the matrix, and 1Dsearching. Assume that M is the EM iteration number.Table 1 shows the main computational cost of threemethods. From Table 1, we can clearly obtain that thecomputational cost of the estimation for a nonsymmetricdistributed source is significantly higher than that of asymmetric distributed source.

Now, our algorithm can be summarized as follows.

Step1. Determine the number of components q, and initializeP, wi, θi, ϕi, and σi i = 1, 2,… , q .

Step2. Repeat the following substeps for M times which is agiven sufficiently large number or until the iterative variationof all parameters reaches the condition δ ≤ 0 01.

(1) Compute the sample covariance matrices of completedata Rm

zi , Rmxi , and Rm

yi using (24) and (27).

(2) Find the eigenvectors Emxi and Em

yi corresponding tothe largest eigenvalues through the eigendecomposi-tion of Rm

xi and Rmyi .

(3) Calculate the nominal azimuth θm+1i , nominal eleva-

tion ϕm+1i , and angular spread σm+1

i from (30), (31),and (32).

αi ui k = ejπ k−1 cos θi sin ϕi e−0 5 πσi k−1 sin θi sin ϕi2+ πσi k−1 cos θi cos ϕi

2,

βi ui k = ejπ k−1 sin θi sin ϕi e−0 5 πσi k−1 cos θi sin ϕi2+ πσi k−1 sin θi cos ϕi

233

6 International Journal of Antennas and Propagation

(4) Estimate the power of each component Pm+1i using

(37), calculate the weight of each component wm+1i

using (39), and get the power of the distributedsource from (40).

(5) Repeat substeps 1 to 4 for i = 1, 2,… , q.

It is noteworthy that the initial positions of the Gaussiancomponents can be obtained by existing algorithms for asymmetric CD source and set uniformly around the guessedvalues. The distribution of a 2D nonsymmetric CD source isunknown so the true number of Gaussian components q isunknown; q needs to be set at a reasonable value initially.In the next section, the influence of initial parameters on esti-mation performance will be discussed.

4. Numerical Results

In this section, four simulation experiments are conducted toverify the effectiveness of the proposed technique. We con-sider the array geometry as depicted in Figure 1 with K = 4sensors in both the x-axis and y-axis. d/λ is set at 1/2. SNRis defined as

SNR = 10 logPρ

42

We use the root-mean-squared error (RMSE) to evaluatethe estimation performance. The RMSE of the nominalangles is defined as

RMSEa =1Mc

〠Mc

ς

θς− θ

2+

1Mc

〠Mc

ς

φς − ϕ2, 43

where θςand ϕ

ςare the estimated nominal azimuth and esti-

mated nominal elevation of the CD source, respectively. Thesuperscript ς denotes the estimated parameters from the ςthMonte Carlo run. Mc is the number of Monte Carlo simula-tions which is set at 500. Additionally, θ and ϕ are the truenominal azimuth and nominal elevation, respectively.

The nominal angle in the proposed algorithm is definedas the value corresponding to the maximum point of thedeterministic angular signal distribution function, whichcan be obtained by the partial derivative of the function.

To examine the difference between the estimated andtrue nonsymmetric distributed source, the estimation ofnominal angles is only part of the problem; in addition, the

spatial distribution of the source should be emphasized com-pared with the estimation of a symmetric distributed source.To evaluate the estimation of the spatial distribution, theRMSE of the distributed function is defined as

RMSEf =1Mc

〠Mc

ς∬ f θ, ϕ ; uς − f θ, ϕ ; u 2dθdϕ, 44

where f θ, ϕ ; uς is the estimated deterministic angular sig-nal distribution function from the ςth Monte Carlo run. Inthis section, we compare the performance of the proposedalgorithm with DSPE [16] and TLS-ESPRIT [19] for a con-structed 2D nonsymmetric CD source with the deterministicangular signal distribution function as

f θ, ϕ ; u = 0 2g 40, 40, 2 5 + 0 2g 45, 40, 2 5+ 0 2g 50, 40, 2 5 + 0 3g 40, 45, 2 5+ 0 1g 40, 50, 3 ,

45

where g a1, a2, a3 denotes a Gaussian component

g a1, a2, a3 =1

2πa23exp 0 5

θ − a1a3

2+

ϕ − a2a3

2

46

Figure 2 shows the constructed nonsymmetric angularsignal distribution function which needs to be estimated.

Table 1: Computational complexity of different methods.

Method

Calculationof the samplecovariancematrix

Eigendecomposition Searching Total

DSPE o 4NK2 o 8K3 o 24K3 o 4NK2 + o 8K3 + o 24K3

TLS-ESPRIT o 4NK2 o 8K3 o 4NK2 + o 8K3

Proposed o 4MNK2 o 8MK3 o M 8K2 + 2K − 1 2K − 1 o 4MNK2 + o 8MK3 + o M 8K2 + 2K − 1 2K − 1

0.01

0.005

PDF

020

3040

5060

70 7060 𝜃 (deg)𝜙 (deg)

5040

3020

Figure 2: Probability density function (PDF) of the constructednonsymmetric angular signal distribution.

7International Journal of Antennas and Propagation

In the first example, we investigate the performance ofthe proposed method as well as a comparison with DSPEand TLS-ESPRIT. Figures 3(a) and 3(b) display the estima-tion of RMSEa by the proposed algorithm, DSPE, as well asTLS-ESPRIT, while Figures 4(a) and 4(b) show the estima-tion of RMSEf. Figure 3(a) and 4(a) investigate the influenceof SNR on performance with the number of snapshots N =200 while Figure 3(b) and 4(b) show the influence of thenumber of snapshots with SNR = 15 dB. The EM iterationnumberM is set at 200. It can be observed that all algorithmsperform better as the number of snapshots or SNR increases.Clearly, the proposed algorithm provides better performancethan other estimators with regard to RMSEa and especially toRMSEf. From Figures 4(a) and 4(b), we can find that thereexists a big error of RMSEf by DSPE and TLS-ESPRIT. Itcan be concluded that utilizing traditional symmetric estima-tors for a nonsymmetric distributed source is invalid.Figure 5 displays the estimation of the deterministic angular

�e proposedDSPETLE-ESPRIT

0

RMSE

a (d

eg)

10−1

100

101

5 10 15 20SNR (dB)

25 30

(a)

�e proposedDSPETLE-ESPRIT

RMSE

a (d

eg)

10−1

100

101

0 500400300200 600 700Number of snapshots

900800 1000

(b)

Figure 3: (a) RMSEa estimated by the three methods versus SNR; (b) RMSEa estimated by the three methods versus number of snapshots.

�e proposedDSPETLS-ESPRIT

010−1

100

101

5 10 15 20SNR (dB)

25 30

RMSE

f

(a)

�e proposedDSPETLS-ESPRIT

RMSE

f

10−1

100

101

0 500400300200 600 700Number of snapshots

900800 1000

(b)

Figure 4: (a) RMSEf estimated by the three methods versus SNR; (b) RMSEf estimated by the three methods versus number of snapshots.

0.01

0.005

PDF

020

3040

5060

70 7060 𝜃 (deg)𝜙 (deg)

5040

3020

Figure 5: The estimation of nonsymmetric angular signaldistribution function by the proposed method.

8 International Journal of Antennas and Propagation

Component AComponent B

Component CComponent D

60

55

50

45

40

𝜃 (d

eg)

350 20 40 60 80 100

Number of iteration120 140 160 180 200

(a)

Component AComponent B

Component CComponent D

60

55

50

45

40

𝜙 (d

eg)

350 20 40 60 80 100

Number of iteration120 140 160 180 200

(b)

6

5

4

3

2

Ang

ular

spre

ad (d

eg)

1

0

Component AComponent B

Component CComponent D

0 20 40 60 80 100Number of iteration

120 140 160 180 200

(c)

10.90.80.70.60.5

Wei

ght

0.40.30.20.1

0

Component AComponent B

Component CComponent D

0 20 40 60 80 100Number of iteration

120 140 160 180 200

(d)

Figure 6: (a) The changing process of central azimuths in EM iterations; (b) the changing process of central elevations in EM iterations; (c)the changing process of angular spreads in EM iterations; (d) the changing process of weights in EM iterations.

10987654321

RMSE

a (d

eg)

00 100 200 300 400 500

Number of iteration600 700 800 900 1000

1 component2 components3 components

4 components5 components6 components

(a)

1 component2 components3 components

4 components5 components6 components

54.5

43.5

32.5

21.5

10.5

0

RMSE

f

0 100 200 300 400 500Number of iteration

600 700 800 900 1000

(b)

Figure 7: (a) RMSEa estimated by different numbers of Gaussian components; (b) RMSEf estimated by different numbers of Gaussiancomponents.

9International Journal of Antennas and Propagation

60

55

B D

A C

8

7

6

5

4

3

2

1

50

× 10−3

45

40

35

3030 35 40 45

𝜃 (deg)

𝜙 (d

eg)

50 55 60

Figure 8: Squares of taking initial positions for each component.

250

200

150

100

50Num

ber o

f ite

ratio

n

038 40 42 44 46 48 38

4042

4446

48

180

160

140

120

100

80

60

𝜙 (deg)

𝜃 (deg)

(a)

38

1

0.8

0.6

0.4Wei

ght

0.2

040 42

4446 48 38

4042

4446

0.52

0.51

0.5

0.49

0.48

0.47

0.46

0.45

0.44

48

𝜙 (deg)𝜃 (deg)

(b)

0.65

0.64

0.63

0.62

0.61

0.6

0.59

0.58

0.57380

0.2

0.4RMSE

f 0.6

0.8

1

4042

4446

48 3840

4244

4648

𝜙 (deg)𝜃 (deg)

(c)

Figure 9: (a) Number of iteration versus initial position for component A; (b) weight versus initial position for component A; (c) RMSEfversus initial position for component A.

10 International Journal of Antennas and Propagation

signal distribution function by the proposed algorithm. Theresult of the proposed method reflects spatial nonsymmetricdistribution of the source and is closer to the given distrib-uted source.

In the second example, we examine the changing pro-cesses of Gaussian components during the EM iterations.The initial nominal azimuth and nominal elevation of com-ponents A, B, C, and D are (43°, 43°), (43°, 53°), (53°, 43°),and (53°, 53°), respectively. The initial angular spreads areset at 1°. The number of snapshots is set at 200 and SNR is15 dB. The changing processes of parameters in each compo-nent are shown in Figure 6. All parameters converge tocertain values after sufficient iterations. As shown inFigure 6(d), a small weight 0.040 is developed by componentD, which indicated that the nominal angles of the Gaussiancomponent outside the scope of the distributed source makeonly a small contribution to the final result.

In the third example, we examine the relationshipbetween the number of Gaussian components and estima-tion performance. The number of snapshots is set at 200and SNR is 15 dB. Figure 7 shows that the error is largewhen there is only one Gaussian component. It can befound that RMSEf and RMSEa decrease as Gaussian com-ponents increase. Both the final RMSEf and final RMSEachange slightly as the number of Gaussian componentschanges from 3 to 6, and the convergence is markedlyslow, which shows that increasing the number of Gaussiancomponents will improve accuracy of estimation. Never-theless, the effect will not be significantly improved asthe number exceeds a certain extent.

The initial position parameters may be set around theguessed values; there will be initial Gaussian componentsoutside the distributed source inevitably. To increase therobustness of the algorithm, the number of Gaussian

250

200

150

100

50Num

ber o

f ite

ratio

n

038 40 42 44 46 48 48

5052

5456

58

180

170

160

150

140

130

120

110

100

90

𝜙 (deg)

𝜃 (deg)

(a)

Wei

ght

00.20.40.60.8

1

3840

4244

4648 48

5052

5456

58

0.240.220.20.180.160.140.120.10.080.060.04

𝜙 (deg)𝜃 (deg)

(b)

1.5

1.4

1.2

1.2

1.1

1

0.9

0.8

0.7

0.6380

0.5

RSM

E f

1

1.5

2

40 42 44 46 484850

5254

5658

𝜙 (deg)𝜃 (deg)

(c)

Figure 10: (a) Number of iteration versus initial position for component B; (b) weight versus initial position for component B; (c) RMSEfversus initial position for component B.

11International Journal of Antennas and Propagation

components should be selected from a larger range; mean-while, computing cost is a matter of balance.

In the fourth example, we examine the relationshipbetween the initial positions of Gaussian componentsand estimation results. The initial positions of componentA, B, C, and D are considered taking values within squareswhere the centers are set at (43°, 43°), (43°, 53°), (53°, 43°),and (53°, 53°). Initial weights are all set at 0.25. Eachsquare takes 400 values uniformly. Figure 8 shows theinvestigated region of initial positions, where the purpleregion is the projection of the constructed nonsymmetricangular signal distribution function (45) on the θ-ϕ plane,and the color bar represents measurement of PDF. Weinvestigate the influence of the initial position of a compo-nent on the number of iterations, final weight of the com-ponent, and RMSEf while initial positions of othercomponents are set at centers. It should be noted thatweights and positions of components would be changing

during EM iterations, which have been shown in the sec-ond example. We focus on the sensitivity of initial posi-tions to the final results; the weight and RMSEf can bepersuasive. The stopping criterion of the EM algorithm isthe iterative variation δ ≤ 0 01.

Figure 9(a) shows that initial positions of component Ain the lower left part of the square have fewer iterations.The initial positions closer to the distributed source tendto have faster convergence rates. As shown inFigure 9(b), weights change slightly in the whole region.It can be concluded that the contribution of componentA to the final PDF is stable when the initial positionsare set in the investigated range. Figure 9(c) shows RMSEfschange at a low level and in a small range, which meansestimations maintain good accuracy with initial positionsin the square.

Figure 10(a) shows that initial positions of component Bin the lower left part and upper right part have fewer

250

200

150

100

50Num

ber o

f ite

ratio

n

048 50 52 54 56 58 38

4042

4446

48

180

170

160

150

140

130

120

110

100

90

𝜙 (deg)𝜃 (deg)

(a)

1

0.8

0.6

0.4

0.2

Wei

ght

050

4852

5456

58 4850

5254

5658

0.05

0.048

0.046

0.044

0.042

0.04

0.038

0.036

0.034

0.032

0.03𝜙 (deg)

𝜃 (deg)

(b)

1

0.8

0.6

0.4

0.2

RMSE

f

0

0.68

0.66

0.64

0.62

0.6

0.58

0.5650

4852

5456

58 4850

5254

5658

𝜙 (deg)𝜃 (deg)

(c)

Figure 11: (a) Number of iteration versus initial position for component C; (b) weight versus initial position for component C; (c) RMSEfversus initial position for component C.

12 International Journal of Antennas and Propagation

iterations. The lower left part has faster convergence rate dueto closer distances from initial positons to the distributedsource. The upper right part shows that the convergence ratesincrease when the distances from the initial positions to thedistributed source exceed a certain degree. The weights inthe upper right part decrease significantly in Figure 10(b),which implies that component B with initial positions in thispart has little contribution to estimation result; consequently,RMSEfs increase remarkably in Figure 10(c).

From Figure 11, we can find that the influence of initialpositions of component C on the estimation results is similarto that of component B in respective regions. The lower leftpart which is close to the distributed source and the upperright part which is far from the distributed source have feweriterations. As the similar region of component B, the upperright part of componentC has small weights and large RMSEfs.

Figure 12 shows that numbers of iterations, weights ofcomponent D, and the distribution errors vary indistinctively

in the whole range. No matter where the initial position istaken for component D, it is far away from the distributedsource. From the example of components A, B, C, and D,it can be drawn that although the initial positions closer todistributed source have faster convergence rates, theweights and RMSEfs tend to be stable until the distancesfrom the initial positions to the distributed source exceeda certain degree. When the distances exceed a robustrange, the convergence rates increase, the weights decrease,and RMSEfs increase remarkably. The estimation of a 2Dnonsymmetric CD source by the proposed algorithmshows robustness with regard to initial position of theGaussian component.

5. Conclusions

In this study, we have considered the problem of estimating anonsymmetric 2D CD source under L-shaped arrays. The

250

200

150

100

50Num

ber o

f ite

ratio

n

05048

52 54 56 58 4850

5254

5658

185

180

175

170

165

160

155

150𝜙 (deg)

𝜃 (deg)

(a)

1

0.8

0.6

0.4

0.2

Wei

ght

050

4852

5456

58 4850

5254

5658

0.05

0.048

0.046

0.044

0.042

0.04

0.038

0.036

0.034

0.032

0.03𝜙 (deg)

𝜃 (deg)

(b)

1

0.8

0.6

0.4

0.2

RMSE

f

0

0.68

0.66

0.64

0.62

0.6

0.58

0.5650

4852

5456

58 4850

5254

5658

𝜙 (deg)𝜃 (deg)

(c)

Figure 12: (a) Number of iteration versus initial position for component D; (b) weight versus initial position for component D; (c) RMSEfversus initial position for component D.

13International Journal of Antennas and Propagation

method we proposed is developed by modeling the nonsym-metric deterministic angular signal distribution function asGaussian mixture. Parameters of a nonsymmetric CD sourceare more than those of a symmetric CD source. Parameterestimation based on an iterative EM framework has beenintroduced in detail. The computational cost of estimationfor a nonsymmetric CD source is much higher comparedwith that of a symmetric CD source. For the sake of evaluat-ing the estimation, two indicators are defined, and onereflects nominal angles and another for spatial distribution.Simulation results indicated that the proposed method is

effective and robust for the estimation of a nonsymmetricCD source.

Appendix

A. Appendix A

Change the variables (θi + θ) for θ and ϕi + ϕ for ϕ, where θand ϕ are the small deviations of θi and ϕi. Thus, cos θ sin ϕand sin θ sin ϕ can be approximated by the first term in theTaylor series expansions

Considering θ and ϕ are small variables, so we have

e−jπ cos θi cos ϕiϕ−sin θi sin ϕiθ ≈ 1 48

Thus, we can obtain following relationship

αAi ui ≈ ejπ cos θi sin ϕiαBi ui 49

Similarly,

Consider the following relationship:

e−jπ sin θi cos ϕiϕ+cos θi sin ϕiθ ≈ 1 51So we have

βAi ui ≈ ejπ sin θi sin ϕiβBi ui 52

B. Appendix B

αAi ui k =∬gi θ, ϕ ; ui ej2πd k−1 cos θ sin ϕ/λdθdϕ

≈∬gi θ, ϕ ; ui ejπ k−1 cos θi sin ϕi+cos θi cos ϕiϕ−sin θi sin ϕiθ dθdϕ

= ejπ k−1 cos θi sin ϕi∬gi θ, ϕ ; ui ejπ k−1 cos θi cos ϕiϕ−sin θi sin ϕiθ dθdϕ,

αBi ui k =∬gi θ, ϕ ; ui ej2πd k−2 cos θ sin ϕ/λdθdϕ

≈∬gi θ, ϕ ; ui ejπ k−2 cosθi sin ϕi+cos θi cos ϕiϕ−sin θi sin ϕiθ dθdϕ

= ejπ k−2 cos θi sin ϕi∬gi θ, ϕ ; ui e−jπ cos θi cos ϕiϕ−sin θi sin ϕiθ ejπ k−1 cos θi cos ϕiϕ−sin θi sin ϕiθ dθdϕ

47

βAi ui k =∬gi θ, ϕ ; ui ej2πd k−1 sin θ sin ϕ/λdθdϕ

≈∬gi θ, ϕ ; ui ejπ k−1 sin θi sin ϕi+sin θi cos ϕiϕ+cos θi sin ϕiθ dθdϕ

= ejπ k−1 sin θi sin ϕi∬gi θ, ϕ ; ui ejπ k−1 sin θi cos ϕiϕ+cos θi sin ϕiθ dθdϕ,

βBi ui k =∬gi θ, ϕ ; ui ej2πd k−2 sin θ sin ϕ/λdθdϕ

≈∬gi θ, ϕ ; ui ejπ k−2 sin θi sin ϕi+sin θi cos ϕiϕ+cos θi sin ϕiθ dθdϕ

= ejπ k−2 sin θi sin ϕi∬gi θ, ϕ ; ui e−jπ sin θi cos ϕiϕ+cos θi sin ϕiθ ejπ k−1 sin θi cos ϕiϕ+cos θi sin ϕiθ dθdϕ

50

14 International Journal of Antennas and Propagation

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Chinese National ScienceFoundation under Grant no. 61471299.

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αi ui k =∬ 12πσ2i

e−0 5 θ−θiσi

2+ ϕ−ϕi

σi

2

ejπ k−1 cos θ sin ϕdθdϕ

≈∬ 12πσ2i

e−0 5 θ

σi

2+ ϕ

σi

2

ejπ k−1 cos θi sin ϕi+cos θi cos ϕiϕ−sin θi sin ϕiθ dθdϕ

= ejπ k−1 cos θi sin ϕi∬ 12πσ2i

e−0 5 θ

σi

2+ ϕ

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= ejπ k−1 cos θi sin ϕi e−0 5 πσi k−1 sin θi sin ϕi2+ πσi k−1 cos θi cos ϕi

2,

βi ui k =∬ 12πσ2i

e−0 5 θ‐θiσi

2+ ϕ−ϕi

σi

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ejπ k−1 sin θ sin ϕdθdϕ

≈∬ 12πσ2i

e−0 5 θ

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σi

2

ejπ k−1 sin θi sin ϕi+sin θi cos ϕiϕ+cos θi sin ϕiθ dθdϕ

= ejπ k−1 sin θk sin ϕk∬ 12πσ2i

e−0 5 θ

σi

2+ ϕ

σi

2−2jπ k−1 sin θi cos ϕiϕ+cos θi sin ϕiθ

dθdϕ

= ejπ k−1 sin θk sin ϕk e−0 5 πσi k−1 cos θi sin ϕi2+ πσi k−1 sin θi cos ϕi

2

53

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16 International Journal of Antennas and Propagation

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