Two-Dimensional DOA Estimation for Incoherently Distributed Sources

sensors Article

Two-Dimensional DOA Estimation for Incoherently Distributed Sources with Uniform Rectangular Arrays Tao Wu 1, *, Zhenghong Deng 1, *, Yiwen Li 2 and Yijie Huang 1 1 2

*

School of Automation, Northwestern Polytechnical University, Xi’an 710072, China; [email protected] Science and Technology on Combustion, Thermal-Structure and Internal Flow Laboratory, Northwestern Polytechnical University, Xi’an 710072, China; [email protected] Correspondence: [email protected] (T.W.); [email protected] (Z.D.); Tel.: +86-150-2988-2069 (T.W.)

Received: 22 August 2018; Accepted: 18 October 2018; Published: 23 October 2018







Abstract: Aiming at the two-dimensional (2D) incoherently distributed (ID) sources, we explore a direction-of-arrival (DOA) estimation algorithm based on uniform rectangular arrays (URA). By means of Taylor series expansion of steering vector, rotational invariance relations with regard to nominal azimuth and nominal elevation between subarrays are deduced under the assumption of small angular spreads and small sensors distance firstly; then received signal vectors can be described by generalized steering matrices and generalized signal vectors; thus, an estimation of signal parameters via rotational invariance techniques (ESPRIT) like algorithm is proposed to estimate nominal elevation and nominal azimuth respectively using covariance matrices of constructed subarrays. Angle matching method is proposed by virtue of Capon principle lastly. The proposed method can estimate multiple 2D ID sources without spectral searching and without information of angular power distribution function of sources. Investigating different SNR, sources with different angular power density functions, sources in boundary region, distance between sensors and number of sources, simulations are conducted to investigate the effectiveness of the proposed method. Keywords: incoherently distributed sources; generalized steering matrix; generalized signal vector; rotational invariance relations; angle matching

1. Introduction In most applications of array signal processing, traditional DOA estimation is based on point source models which can simplify calculations. Point source models assume that propagations between sources and receive arrays are straight paths and the spatial characteristics of sources can be ignored. In the field of wireless communication, there are obstacles around sources; the propagations from sources to receive arrays are multipath. In the real surrounding of underwater, on the one hand, there exist paths from seabed and sea surface of backscatters; on the other hand, spatial scatterers of targets cannot be ignored when the distances from targets and receive arrays are short. Considering the spatial scatterers and multipath phenomena, point models cannot characterize sources effectively, which should be described by distributed source models [1]. The signal of a source only propagates from a single direction through a straight path under the assumption of point source model. Distributed sources can be regarded as an assembly of point sources within a spatial distribution. The shape of spatial distribution is related to geometry and surface property of a target in underwater detection for instance. Spatial distribution of a distributed source can be generally modeled as Gaussian or uniform with parameters containing nominal angles and angular spreads. Nominal angles represent the center of targets and angular spreads represent the spatial extension of targets. According to the scatterers coherence of sources, distributed sources can be classified into coherently distributed (CD) sources and incoherently distributed (ID) sources [1]. CD sources suppose Sensors 2018, 18, 3600; doi:10.3390/s18113600

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that scatterers within a source are coherent whereas scatterers of an ID source are assumed to be uncorrelated. In this paper, ID sources are considered. For CD sources, representative achievements of DOA estimation are methods extended from point source models through rotation invariance relations with respect to different array configuration under the small angular spreads assumption [2–10]. As to ID sources, people have presented subspace-based algorithms such as distributed signal parameter estimator (DSPE) [1] and dispersed signal parametric estimation (DSPARE) [11] which are developed from multiple signal classification (MUSIC). Based on uniform linear arrays (ULA), the authors of [12] have extended Capon method for ID sources, where two-dimensional (2D) spectral searches based on high-order matrix inversion are involved. The authors of [13] have proposed a generalized Capon’s method by introducing a matrix pencil of sample covariance matrix and normalized covariance matrix to traditional Capon framework. A robust generalized Capon’s method in [14] has been proposed by supplementing a constraint function using property of covariance matrix to cost function of algorithm in [12], which has better accuracy without a priori knowledge of shape of ID sources. Maximum likelihood (ML) approaches [15–17] have better accuracy but require high computational complexity. The authors of [18] have taken the lead in extending covariance matching estimation techniques (COMET) to DOA estimation of ID sources using ULA. They have converted complex nonlinear optimization to two successive one-dimensional (1D) searches utilizing the extended invariance principle (EXIP). Considering Gaussian and uniform ID sources, number of sensors in experiment of [18] ranges from 4 to 20; they have proved that when sensors is increased to a certain extent with other parameters fixed, estimation tends to be less accurate. Applying Taylor series expansion of steering vector to approximate the array covariance matrix using the central moments of the source, the authors of [19] have elaborated a lower computational COMET for ID sources, which exhibits a good performance at low SNR based on ULA with 11 sensors. The authors of [20] have presented the ambiguity problem of COMET-EXIP algorithm using ULA and proposed an inequality constraint in the original objective functions to solve this problem. The authors of [21] have turned covariance matching problem to retrieve parameters of received covariance by exploiting the annihilating property of linear nested arrays. The authors of [22] have embedded algorithm of [19] into a Kalman filter for DOA tracking problem of ID sources. The aforementioned DOA estimations of ID sources consider the sources and receive arrays are in the same plane where sources are described by 1D model with two parameters: nominal angle and angular spread. Generally, sources are in different plane with receive arrays, which should be modeled as two-dimensional (2D) models with parameters as nominal azimuth angle, nominal elevation angle, azimuth spread, and elevation spread. With more parameters, there have been relatively few studies on estimation of 2D ID sources. The authors of [23] have extended COMET algorithm to 2D scenarios, which separates variables of each source based on alternating projection technique, then formulate equations set of unknown variables. The algorithm from [23] is applicable to any arrays in three-dimensional spaces but requires considerably high computational complexity. In addition, experiments of [23] have only considered a relatively small number of targets. The authors of [24] have proposed a two-stage approach based on rotation invariance relations of generalized steering vector of double parallel uniform linear arrays (DPULA), where nominal elevation is firstly estimated via TLS-ESPRIT, then nominal azimuth is acquired by 1D searching. In order to estimate DOA of 2D ID sources, we propose an algorithm based on URA. Through Taylor series expansions of steering vectors, received signal vectors of arrays can be expressed as a generalized form which is combination of generalized steering matrix and generalized signal vector under the assumption of small distance of sensors and small angular spread. Consequently, the rotation invariance relations of constructed subarrays with respect to nominal azimuth and nominal elevation are derived. Constructed subarrays fully use elements of URA, so the estimation accuracy is improved. Then the nominal azimuth and nominal elevation can be calculated respectively by means of an ESPRIT like algorithm. Thus, estimation of multiple 2D ID sources do not need spectral searching and avoid high computational complexity. Lastly, the angle matching method is proposed according to Capon

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spectral search. Without information of angular power distribution function, the proposed method can detect sources with different angular power distribution functions. 2. Arrays Configuration and Signal Model As shown in Figure 1, the URA consists of M × K sensors on xoz plane. The distance between sensors along the direction of x and z axes is set at d meters. Azimuth θ and elevation ϕ are used to describe the direction of signal in a three dimensional space. There are q 2D ID narrowband sources with nominal angles (θ i , ϕi ) (i = 1, 2, . . . , q) denoting nominal azimuth and nominal elevation of the ith sources impinging into arrays. θ i ∈ [0, π], ϕi ∈ [0, π]. λ is the wavelength of the impinging signal. The additive is considered Sensors 2018, 18, xnoise FOR PEER REVIEW as Gaussian white with zero mean and uncorrelated with sensors. 4 of 18

Figure1.1.Uniform Uniformrectangular rectangulararrays arraysconfiguration. configuration. Figure

TheM M××1 1dimensional dimensionalreceived receivedvector vectorof ofthe thekth M sensors located on to thex-axis, x-axiswhich defined subarray The subarray parallel is as defined as x can be written as 1 subarray xk, can be written as q x x1 (t) = ∑ q α1 (θ, ϕ)si (θ, ϕ, t)dθdϕ + nx1 (t), (1) x k (t ) i==1  α k (θ , φ )si (θ , φ , t )dθdφ + n xk (t ) , (6) i =1

where nx1 (t) denotes the noise vector of subarray. The M × 1 dimensional vector α1 (θ,ϕ) denotes the where αk(θ,φ) the steering xk with respect to point source, steering vectordenotes of subarray x1 with vector respectof tosubarray point source, which can be expressed as which can be written as h iT cos θ sin j2π(ϕ/λ k−1)dcos α1 (θ, ϕ) = e j2πd cos θ sinαϕ/λ ,φe)j2π2d , ·φ/·λ·. , e j2πMd cos θ sin ϕ/λ , (2) ( θ , = α ( θ , φ ) e (7) k 1

The is M the × K complex dimensional received matrix of thedensity URA along x-axis can source be expressed as si (θ,ϕ,t) random angular signal of thethe distributed representing the reflection intensity of the source from angle (θ,ϕ) at the snapshot index t. Unlike a point source, X(t) = [x (t), x2 (t),, xK(t)]T . (8) signal of a distributed source exists not only in1the direction of (θ i ,ϕi ) but also in a spatial distribution around A distributed received source is vector definedofasthe incoherently if si (θ,ϕ,t) oneto direction The(θKi ,ϕ × i ). 1 dimensional K sensors distributed with distance d and from parallel z-axis is uncorrelated with other directions, which can be modeled as a random process as defined as subarray z1 can be written as q

0 0 E [si (θ, ϕ,zt)( ts)i ∗=(θ 0 , ϕ0 ,βt)](θ= (θ, )δ(θ − , φP ) si f(iθ , φϕ; , t )udθdφ + nθ )(δt() ϕ, − ϕ ), 1

  i =1

1

i

z1

(3) (9)

where δ(·) is the Kronecker delta function, Pi is the power of the source, fi (θ,ϕ;ui ) is its normalized where the K × 1 dimensional vector β1(θ,φ) denotes the steering vector of subarray z1 with respect to angular power density function (APDF) reflecting geometry and surface property of a distributed point source, nz1(t) denotes the noise vector of subarray z1. β1(θ,φ) can be expressed as source. APDF is determined by parameter set ui . For a Gaussian ID source, ui = [θi , φi , σθi , σφi , ρ] T

β1 (θ , φ ) = e j 2 πd cos θ sin φ / λ 1, e j 2 πd cos φ / λ , , e j 2 π( K −1) d cos φ / λ  .

(10)

The K × 1 dimensional received vector of the mth subarray parallel to z-axis defined as subarray zm can be written as q

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denoting nominal azimuth, nominal elevation, azimuth spread, elevation spread, and covariance coefficient respectively, APDF can be expressed as f i (θ, ϕ; ui ) =

1√

2πσθi σϕi

 exp − 2σ

1 2 θi σϕi (1− ρ )

1− ρ2



θ − θi σθi

2

− 2ρ (θ −θσiθi)(σϕϕi− ϕi ) +



ϕ − ϕi σϕi

2 

(4)

For a uniform ID source, ui = [θi , φi , σθi , σφi ] denoting nominal azimuth, nominal elevation, azimuth spread and elevation spread respectively. APDF can be expressed as ( f i (θ, ϕ; ui ) =

1 4σθi σϕi

|θ − θi | ≤ σθi and | ϕ − ϕi | ≤ σϕi

0

|θ − θi | ≥ σθi or | ϕ − ϕi | ≥ σϕi

.

(5)

The M × 1 dimensional received vector of the kth subarray parallel to x-axis, which is defined as subarray xk , can be written as q

xk ( t ) =

∑

i =1

x

αk (θ, ϕ)si (θ, ϕ, t)dθdϕ + nxk (t),

(6)

where αk (θ,ϕ) denotes the steering vector of subarray xk with respect to point source, which can be written as αk (θ, ϕ) = α1 (θ, ϕ)e j2π (k−1)d cos ϕ/λ . (7) The M × K dimensional received matrix of the URA along the x-axis can be expressed as X(t) = [x1 (t), x2 (t), · · · , xK (t)] T .

(8)

The K × 1 dimensional received vector of the K sensors with distance d and parallel to z-axis defined as subarray z1 can be written as q

z1 (t) =

∑

i =1

x

β1 (θ, ϕ)si (θ, ϕ, t)dθdϕ + nz1 (t),

(9)

where the K × 1 dimensional vector β1 (θ,ϕ) denotes the steering vector of subarray z1 with respect to point source, nz1 (t) denotes the noise vector of subarray z1 . β1 (θ,ϕ) can be expressed as h iT β1 (θ, ϕ) = e j2πd cos θ sin ϕ/λ 1, e j2πd cos ϕ/λ , · · · , e j2π (K −1)d cos ϕ/λ .

(10)

The K × 1 dimensional received vector of the mth subarray parallel to z-axis defined as subarray zm can be written as q x zm ( t ) = ∑ βm (θ, ϕ)si (θ, ϕ, t)dθdϕ + nzm (t), (11) i =1

where βm (θ,ϕ) denotes the steering vector of subarray zm with respect to point source, which can be written as βm (θ, ϕ) = β1 (θ, ϕ)e j2π (m−1)d cos θ sin ϕ/λ . (12) The K × M dimensional received matrix of the URA along the z-axis can be expressed as Z(t) = [z1 (t), z2 (t), · · · , z M (t)] T .

(13)

3. Proposed Method This section consists of five parts. First of all, generalized steering matrices are obtained from taking the first-order Taylor series expansions of steering vectors under the assumption of small angular spreads and small distance d of the URA. Then, the rotation invariance relations of generalized steering

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matrices within subarrays are derived. In the next part, the received signal vectors are transformed as combinations of generalized signal vectors and generalized steering matrices. Next, based on rotation invariance relations of constructed subarrays, nominal azimuth and nominal elevation can be estimated separately by an ESPRIT like algorithm. Afterwards, angle matching method is proposed based on the Capon principle. Lastly computational procedure is summarized and complexity analysis is analyzed with comparison of two existing methods for 2D ID sources. 3.1. Generalized Steering Matrix Assume that angular spread of distributed sources is small, take the first-order Taylor series expansions of steering vectors of subarrays xk and xk− 1 at (θ i , ϕi ), we have αk−1 (θ, ϕ) ≈ αk−1 (θi , ϕi ) + [αk−1 (θi , ϕi )]0 θ (θ − θi ) + [αk−1 (θi , ϕi )]0 ϕ ( ϕ − ϕi ),

(14)

αk (θ, ϕ) ≈ αk (θi , ϕi ) + [αk (θi , ϕi )]0 θ (θ − θi ) + [αk (θi , ϕi )]0 ϕ ( ϕ − ϕi ),

(15)

where [·]0 θ and [·]0 ϕ denote the first-order partial derivatives of θ and ϕ at (θ i , ϕi ), the following relationship can be obtained from Equation (7) αk (θi , ϕi ) = αk−1 (θi , ϕi )e j2πd cos ϕi /λ ,

(16)

[αk (θi , ϕi )]0 θ = [αk−1 (θi , ϕi )]0 θ e j2πd cos ϕi /λ ,

(17)

[αk (θi , ϕi )]0 ϕ = [αk−1 (θi , ϕi )]0 ϕ e j2πd cos ϕi /λ + (− j2πd sin ϕi /λ)e j2π (k−1)d cos ϕi /λ αk−1 (θi , ϕi ).

(18)

If d/λ  0.5, the second term of the right side of the Equation (18) can be ignored, so Equation (18) can be written as [αk (θi , ϕi )]0 ϕ ≈ [αk−1 (θi , ϕi )]0 ϕ e j2πd cos ϕi /λ . (19) In the point source model case, steering vectors is used to describe response of an array. Nevertheless, as for distribute sources, generalized steering vectors or generalized steering matrices [5,24] represent response of arrays. Define M × 3q dimensional generalized steering matrix of subarray x1 as [A11 , A12 , A13 ]. A11 , A12 , and A13 are expressed as    A11 = [α1 (θ1 , ϕ1 ), α1 (θ2 , ϕ2 ), · · · , α1 (θq , ϕq )]  A12 = h[α1 (θ1 , ϕ1 )]0 θ , [α1 (θ2 , ϕ2 )]0 θ , · · · , [α1 (θq , ϕq )]0 θ i .   A = [α (θ , ϕ )]0 , [α (θ , ϕ )]0 , · · · , [α (θ , ϕ )]0 q ϕ 13 1 1 1 ϕ 1 2 2 ϕ 1 q

(20)

Define the generalized steering matrix of subarray xk as [Ak1 , Ak2 , Ak3 ],    Ak1 = [αk (θ1 , ϕ1 ), αk (θ2 , ϕ2 ), · · · , αk (θq , ϕq )]  Ak2 = h[αk (θ1 , ϕ1 )]0 θ , [αk (θ2 , ϕ2 )]0 θ , · · · , [αk (θq , ϕq )]0 θ i .   A = [α (θ , ϕ )]0 , [α (θ , ϕ )]0 , · · · , [α (θ , ϕ )]0 q ϕ k3 k 1 1 ϕ k 2 2 ϕ k q

(21)

From Equations (16), (17) and (19) we can obtain the generalized steering matrix of xk subarray as

[Ak1 , Ak2 , Ak3 ] ≈ [A11 Φkz−1 , A12 Φkz−1 , A13 Φkz−1 ],

(22)

where Φz is rotation invariance operator, which can be written as Φz = diag(e j2πd cos ϕ1 /λ , e j2πd cos ϕ2 /λ , · · · , e j2πd cos ϕq /λ ).

(23)

Take the first-order Taylor series expansions of steering vectors of subarrays zm and zm− 1 at (θ i , ϕi ), we have

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βm−1 (θ, ϕ) ≈ βm−1 (θi , ϕi ) + [βm−1 (θi , ϕi )]0 θ (θ − θi ) + [βm−1 (θi , ϕi )]0 ϕ ( ϕ − ϕi ),

(24)

βm (θ, ϕ) ≈ βm (θi , ϕi ) + [βm (θi , ϕi )]0 θ (θ − θi ) + [βm (θi , ϕi )]0 ϕ ( ϕ − ϕ).

(25)

Form Equation (12) we have βm (θ, ϕ) = βm−1 (θi , ϕi )e j2πd cos θi sin ϕi /λ ,

(26)

[βm (θi , ϕi )]0 θ = [βm−1 (θi , ϕi )]0 θ e j2πd cos θi sin ϕi /λ + (− j2πd sin θi sin ϕi /λ)e j2πd cos θi sin ϕi /λ βm−1 (θi , ϕi ),

(27)

[βm (θi , ϕi )]0 ϕ = [βm−1 (θi , ϕi )]0 ϕ e j2πd cos θi sin ϕi /λ + ( j2πd cos θi cos ϕi /λ)e j2πd cos θi sin ϕi /λ βm−1 (θi , ϕi ).

(28)

If d/λ  0.5, the second term of the right side of the Equations (27) and (28) can be ignored, so Equations (27) and (28) can be written as

[βm (θi , ϕi )]0 θ ≈ [βm−1 (θi , ϕi )]0 θ e j2πd cos θi sin ϕi /λ ,

(29)

[βm (θi , ϕi )]0 ϕ ≈ [βm−1 (θi , ϕi )]0 ϕ e j2πd cos θi sin ϕi /λ .

(30)

Define the M × 3q dimensional generalized steering matrix of subarray z1 as [B11 , B12 , B13 ]; B11 , B12 , and B13 are written as    B11 = [β1 (θ1 , ϕ1 ), β1 (θ2 , ϕ2 ), · · · , β1 (θq , ϕq )]  B12 = h[β1 (θ1 , ϕ1 )]0 θ , [β1 (θ2 , ϕ2 )]0 θ , · · · , [β1 (θq , ϕq )]0 θ i .   B = [β (θ , ϕ )]0 , [β (θ , ϕ )]0 , · · · , [β (θ , ϕ )]0 q ϕ 13 1 1 1 ϕ 1 2 2 ϕ 1 q

(31)

Define the generalized steering matrix of subarray z1 as [Bm1 , Bm2 , Bm3 ],    Bm1 = [βm (θ1 , ϕ1 ), βm (θ2 , ϕ2 ), · · · , βm (θq , ϕq )]  Bm2 = h[βm (θ1 , ϕ1 )]0 θ , [βm (θ2 , ϕ2 )]0 θ , · · · , [βm (θq , ϕq )]0 θ i .   B = [β (θ , ϕ )]0 , [β (θ , ϕ )]0 , · · · , [β (θ , ϕ )]0 q ϕ m3 m 1 1 ϕ m 2 2 ϕ m q

(32)

From Equations (26), (29), and (30), the generalized steering matrix of zm [Bm1 , Bm2 , Bm3 ] can be expressed as −1 −1 −1 [Bm1 , Bm2 , Bm3 ] ≈ [B11 Φm , B12 Φm , B13 Φm ], (33) x x x where Φx is rotation invariance operator, which can be written as Φ x = diag(e j2πd cos θ1 sin ϕ1 /λ , e j2πd cos θ2 sin ϕ2 /λ , · · · , e j2πd cos θq sin ϕq /λ ).

(34)

3.2. Generalized Signal Vector ¯

¯

¯

¯

¯

¯

¯

Define generalized signal vector as s = [s1 , s2 , s3 ]H ; s1 , s2 and s3 are written as  ¯    s1 = [ρ10 , ρ20 , · · · , ρk0 ] ¯

s2 = [ρ1θ , ρ2θ , · · · , ρkθ ] ,    ¯ s3 = [ρ1ϕ , ρ2ϕ , · · · , ρkϕ ] where

 s   ρi0 = s si (θ, ϕ, t)dθdϕ ρiθ = (θ − θi )si (θ, ϕ, t)dθdϕ .   ρ = s ( ϕ − ϕ )s (θ, ϕ, t)dθdϕ iϕ i i

(35)

(36)

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Substitute Taylor series expansions (15) into Equation (6), we obtain q

xk ( t )

≈ ∑

s

i =1 q s

+∑

i =1 q

q

αk (θi , ϕi )si (θ, ϕ, t)dθdϕ+ ∑

i =1

s

[αk (θi , ϕi )]0 θ (θ − θi )si (θ, ϕ, t)dθdϕ

[αk (θi , ϕi )]0 ϕ ( ϕ − ϕi )si (θ, ϕ, t)dθdϕ + nxk (t) q

q

i =1

i =1

(37)

= ∑ αk (θi , ϕi )ρi0 + ∑ [αk (θi , ϕi )]0 ρiθ + ∑ [αk (θi , ϕi )]0 ϕ ρiϕ + nxk (t) i =1

Thus, received vector of subarray xk can be expressed as a combination of the generalized signal vector and generalized steering matrix ¯

xk (t) ≈ [Ak1 , Ak2 , Ak3 ]s + nxk (t).

(38)

The MK × 1 received vector of the URA along the x-axis can be express as   ¯ T  X(t) = [x1 (t) T , x2 (t) T , · · · , xK (t) T ] ≈  

A11 , A11 Φz ,

A12 , A13 A12 Φz , A13 Φz ··· A11 ΦzK −1 , A12 ΦzK −1 , A13 ΦzK −1

 ¯  s + n X ( t ), 

(39)

where nX (t) is the noise vector of the URA along the x-axis, which can be expressed as h iT nX (t) = nx1 (t) T , nx2 (t) T , · · · , nxK (t) T .

(40)

Similarly, the received vector of subarray zm can be expressed as ¯

zm (t) ≈ [Bm1 , Bm2 , Bm3 ]s + nz1 (t).

(41)

Supposing different ID sources are uncorrelated there exist following relations within generalized signal vectors (see Appendix A)   Pi    PM i θi ∗ E[ρil ρin ]=  Pi M ϕi    0

l l l l

=n=0 =n=θ , =n=ϕ 6= n

(42)

where Mθi , M ϕi can be express as Mθi =

x

(θ − θi )2 pi (θ, ϕ, t)dθdϕ,

(43)

M ϕi =

x

( ϕ − ϕi )2 pi (θ, ϕ, t)dθdϕ.

(44)

According to the assumption that different ID sources are uncorrelated, so we have ¯ ¯H

E [ss ] = diag(Λ, Mθ Λ, M ϕ Λ),

(45)

where Λ, Mθ , and M ϕ can be written as  2 2 2   Λ = diag(σs1 , σs2 , · · · , σsq ) Mθ = diag( Mθ1 , Mθ2 , · · · , Mθq ) .   M = diag( M , M , · · · , M ) ϕ ϕq ϕ2 ϕ1

(46)

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3.3. Nominal Angles Estimation We construct two subarrays X1 and X2 along the direction of the x-axis. X1 is constituted by arrays from x1 to xK− 1 ; while X2 contains arrays from x2 to xK . Thus, X1 (t) is M(K − 1) × 1 dimensional ¯

received vector of X1 also equals a vector containing elements from 1 to MK − M row of X(t); whereas ¯

X2 (t) is M(K − 1) × 1 dimensional received vector of X2 containing elements from M + 1 to MK of X(t). X1 (t) and X2 (t) can be written as (

 ¯ T X1 (t) = [x1 (t) T , x2 (t) T , · · · , xK −1 (t) T ] = A1x1 , A1x2 , A1x3 s + nX1 (t) ,  ¯ T X2 (t) = [x2 (t) T , x2 (t) T , · · · , xK (t) T ] = A2x1 , A2x2 , A2x3 s + nX2 (t)

(47)

nX1 (t) is noise vector of X1 containing elements from 1 to MK − M row of nX (t), nX2 (t) is noise vector of   X2 containing elements from M to MK row of nX (t). A1x1 , A1x2 , A1x3 is the generalized steering matrix of subarray X1 (t), which can be expressed as  h



i   A1x1 , A1x2 , A1x3 =  

A11 , A11 Φz , A11 ΦzK −2 ,

A12 , A12 Φz , ··· A12 ΦzK −2 ,

A13 A13 Φz A13 ΦzK −2

   , 

(48)

 A2x1 , A2x2 , A2x3 is the generalized steering matrix of subarray X2 (t). From Equation (22) we can obtain h

i h i A2x1 , A2x2 , A2x3 ≈ A1x1 Φz , A1x2 Φz , A1x3 Φz ,

(49)

which proves the rotational invariance relation between the two constructed subarrays X1 and X2 . According to the ESPRIT principle, combing the vector X1 (t) and X2 (t), we obtain a vector with rotational invariance property as " X12 (t) =

X1 (t) X2 (t)

#

¯

= [Ax1 , Ax2 , Ax3 ]s + nX12 (t),

(50)

where the generalized steering matrix of the combination X12 (t) can be expressed as "

[Ax1 , Ax2 , Ax3 ] =

A1x1 , A1x2 , A1x3 A2x1 , A2x2 , A2x3

# .

(51)

The combination of noise vector can be expressed as " nX12 (t) =

nX1 (t) nX2 (t)

# .

(52)

ESPRIT framework is based on rotational invariance property of signal subspace, which can be derived from the rotational invariance property of the received signal vector. Signal subspace can be acquired by eigendecomposition of the covariance matrix of received vector X12 (t) which can be expressed as H R12 (53) x = E [X12 (t)X12 (t)]. R12 x can be replaced by sample covariance matrix with N snapshots as 1 N H ˆ 12 R ( t ). X12 (t) X12 x = N t∑ =1

(54)

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From Equations (45) and (46) we find that eigenvalues of covariance matrix of received signal vector consist of three parts Λ, Mθ Λ and M ϕ Λ. The three parts all has q elements. Each part corresponds to respective eigenvectors. Under the assumption of small angular spread, we can obtain Mθi < 1 and M ϕi < 1. Therefore, subspace spanned by eigenvectors corresponding to the largest q eigenvalues is equal to subspace spanned by Ax1 . Suppose Ex is 2M(K − 1) × q dimensional matrix with columns as the eigenvectors of the covariance matrix R12 x corresponding to the q largest eigenvalues. Accordingly, there exists a q × q nonsingular matrix T satisfying the following relation " Ex =

A1x1 A1x1 Φz

# (55)

T.

Let Ex1 and Ex2 denote matrices selecting upper and lower MK − M rows of Ex (

Ex1 = A1x1 T . Ex2 = A2x1 T

(56)

So we have Ex2 = Ex1 T −1 Φz T.

(57)

Ω x = T −1 Φz T.

(58)

Let Thus, the eigenvalues of Ωx is elements of Φz . Ωx can be obtained as Ω x = E+ x1 Ex2 ,

(59)

where (•)+ denotes pseudo-inverse operator, then the nominal elevation of the sources can be get from ϕi = arccos

angle(ηi ) i = 1, 2, · · · , q, 2πd/λ

(60)

where η i is the ith eigenvalues of Ωx , angle(•) denotes argument of complex variable. Similarly, we construct two subarrays Z1 and Z2 along the direction of z-axis. Z1 is constituted by arrays from z1 to zM− 1 ; while Z2 contains arrays from z2 to zM . Z1 (t) is K(M − 1) × 1 dimensional ¯

receive vector of Z1 containing elements from 1 to K(M − 1) row of Z(t); whereas Z2 (t) is K(M − 1) ¯

× 1 dimensional received vector of Z2 containing elements from K + 1 to MK row of Z(t). nZ1 (t) and nZ2 (t) have the similar definition as nX1 (t) and nX2 (t). The received vectors Z1 (t) and Z2 (t) can be expressed as (

 ¯ T Z1 (t) = [z1 (t) T , z2 (t) T , · · · , z M−1 (t) T ] = B1z1 , B1z2 , B1z3 s + nZ1 (t) ,  ¯ T Z2 (t) = [z2 (t) T , z2 (t) T , · · · , z M (t) T ] = B2z1 , B2z2 , B2z3 s + nZ2 (t)

(61)

  where B1z1 , B1z2 , B1z3 is the generalized steering matrix of Z1 (t), which can be expressed as  h

i   B1z1 , B1z2 , B1z3 =  

B11 , B11 Φ x , B11 Φ xM−2 ,

B12 , B13 B12 Φ x , B13 Φ x ··· B12 Φ xM−2 , B13 Φ xM−2

   . 

(62)

  Form Equation (33), the generalized steering matrix of Z2 (t) B2z1 , B2z2 , B2z3 can be obtained as h

i h i B2z1 , B2z2 , B2z3 ≈ B1z1 Φ x , B1z2 Φ x , B1z3 Φ x ,

(63)

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which proves the rotational invariance relation between the two constructed subarrays Z1 and Z2 . Combing the vector Z1 (t) and Z2 (t), we obtain a vector with rotational invariance property as " Z12 (t) =

Z1 (t) Z2 (t)

#

¯

= [z1 , z2 , z3 ]s + nZ12 (t),

(64)

where the generalized steering matrix of Z12 (t) can be expressed as "

[Bz1 , Bz2 , Bz3 ] =

B1z1 , B1z2 , B1z3 B2z1 , B2z2 , B2z3

# .

(65)

The combination of noise vector of Z1 (t) and Z2 (t) can be expressed as " nZ12 (t) =

nZ1 (t) nZ2 (t)

# .

(66)

The covariance matrices of received signal Z12 (t) can be expressed as H R12 z = E [Z12 ( t )Z12 ( t )].

(67)

which can be replaced by sample covariance matrix with N snapshots as 1 N H ˆ 12 R = Z12 (t) Z12 ( t ). z N t∑ =1

(68)

Suppose Ez is 2K(M − 1) × q matrix with columns as the eigenvectors of the covariance matrix R12 corresponding to the q largest eigenvalues. As the same as Ex , Ez is the same subspace spanned by z Bz1 . Accordingly, there exists a q × q nonsingular matrix Q satisfying the following relation " Ez =

B11 B11 Φ x

# Q.

(69)

Let Ez1 and Ez2 denote matrices selecting upper and lower MK − K rows of Ez (

Ez1 = B1z1 Q . Ez2 = B1z1 Q

(70)

So we have Ez2 = Ez1 Q−1 Φx Q.

(71)

Ωz = Q−1 Φ x Q.

(72)

Let Thus, the eigenvalues of Ωz is elements of Φx . Ωz can be obtained as Ω z = E+ z1 Ez2 .

(73)

Then the nominal azimuth of the sources can be get from θi = arccos where µi is the ith eigenvalues of Ωz .

angle(µi ) i = 1, 2, · · · , q, 2πd/λ sin ϕi

(74)

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3.4. Angle Matching Method After all the eigenvalues of Ωx and Ωz are calculated, we need to match the right θ i to right ϕi . We can obtain cost function by applying Capon principle to subarray x1 firstly. Then angle matching can be obtained by substituting all the possible pairs into the cost function. The generalized steering matrix of subarray x1 is [A11 , A12 , A13 ], the Capon principle with regard to the subarray x1 can be expressed as minw H R1x w subject to w H [A11 , A12 , A13 ] = 1.

(75)

where R1x is the covariance matrix of subarray x1 . Equation (75) can be solved through minimization of Lagrange function as follows n o L(θ, ϕ) = w H R1x w + λ w H [A11 , A12 , A13 ] − 1 ,

(76)

Take the derivative of the Equation (76) with regard to w and set the result equal to 0, we obtain R1x w = −[A11 , A12 , A13 ]λ,

(77)

Considering constraint condition, we obtain n o −1 −1 λ = − [A11 , A12 , A13 ] H (R1x ) [A11 , A12 , A13 ] ,

(78)

The optimal vector wopt can be obtained as follows wopt = −

(R1x )

−1

[A11 , A12 , A13 ]

H

[A11 , A12 , A13 ] (R1x )

−1

[A11 , A12 , A13 ]

,

(79)

.

(80)

Then, the cost function can be expressed as V (θ, ϕ) =

1 H

[A11 , A12 , A13 ] (R1x )

−1

[A11 , A12 , A13 ]

R1x can be replaced by sample covariance matrix with N snapshots as N ˆ 1x = 1 ∑ x1 (t) x1H (t). R N t =1

(81)

Suppose that all eigenvalues of Ωx and Ωz are calculated. Now we summarize the angle matching procedure as follows: Step 1: Calculate nominal elevation ϕi (i = 1, 2, . . . , q) from Equation (60). Select one elevation  angle ϕˆ i = ϕi form set ϕ1 , ϕ2 , · · · , ϕq at random. Superscript ˆ denotes the angle already determined.  Substitute ϕˆ i to the eigenvalues set of Ωz : angle(µ1 ), angle(µ2 ), · · · , angle(µq ) and calculate q matching azimuth angles θ j ( j = 1, 2, · · · , q) from Equation (74). So we get q possible pairs (θ j , ϕˆ i ) ( j = 1, 2, · · · , q). Step 2: As the right parameters of sources can meet the objective function in Equation (75). Substitute (θ j , ϕˆ i ) ( j = 1, 2, · · · , q) into cost function (80). Choose the azimuth θ j making V (θ j , ϕˆ i ) reach the maximum as the right azimuth angle matching to ϕˆ i , which is labeled as θˆi .  ˆ Step 3: Delete ϕˆ i from set ϕ1 , ϕ2 , · · · , ϕq meanwhile delete eigenvalue e j2πd/λ cos θi sin ϕˆ i from set { angle(µ1 ), angle(µ2 ), · · · , angle(µq ) . Step 4: Set q = q − 1. Step 5: Repeat steps 1 to 4.

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All the nominal azimuth and nominal elevation can be matched after (q + 2)(q − 1)/2 times calculation. 3.5. Computational Procedure and Complexity Analysis Now, our algorithm can be summarized as follows ˆ 12 ˆ 12 Step 1: Compute sample covariance matrices R x and Rz using Equations (54) and (68). Step 2: Find the eigenvectors Ex and Ez corresponding to the q largest eigenvalues through ˆ 12 ˆ 12 eigendecomposition of R x and Rz . Divide Ex into Ex1 , Ex2 and divide Ez into Ez1 , Ez2 . Step 3: Obtain Ωx and Ωz from Equations (59) and (73), calculate eigenvalues µi and ηi (i = 1, 2, . . . , q) through eigendecomposition of Ωx and Ωz . Step 4: Calculate nominal azimuth θ i and nominal elevation ϕi from Equations (60) and (74). ˆ 1x form Equation (81) and take the angle Step 5: Compute sample covariance matrix R matching procedure. We analyze the computational complexity of the proposed method in comparison with COMET [23] which can be applied for URA and Zhou’s algorithm [24] which uses DPULA for 2D ID sources. COMET [23] is a method under alternating projection algorithm framework, its computational cost mostly consists of the calculation of the sample covariance matrix which needs O(NM2 K2 ) and the alternating projection technique with respect to cost functions which is O(M4 K4 + 2M2 K2 ). Zhou’s algorithm uses TLS-ESPRIT to calculate nominal elevation and 1D searching to find nominal azimuth. Computational cost of Zhou’s algorithm [24] mainly contains calculation of the sample covariance matrix O(4NM2 ), eigendecomposition and inversion of the sample covariance matrix O(16M3 ) and 1D searching O(8M3 ). Computational cost of the proposed algorithm mainly contains calculation of the ˆ 12 ˆ 12 sample covariance matrix O[N(MK − K)2 + N(MK − M)2 ], eigendecomposition of R x and Rz O[8 3 3 3 (MK − K) + 8(MK − M) ], eigendecomposition of Ωx and Ωz O(q ) and angle matching O(q2 ). The main computational complexity of the three algorithms are shown in Table 1. Table 1. Computational complexity of different methods Method

Total

COMET algorithm [23] Zhou’s algorithm [24] Proposed algorithm

O(NM2 K2 ) + O(M4 K4 + 2M2 K2 ) O(4NM2 ) + O(16M3 ) + O(8M3 ) 2 O[N(MK − K) + N(MK − M)2 ] + O[8(MK − K)3 + 8(MK − M)3 ] + O(q3 ) + O(q2 )

From Table 1, we can conclude that the computational cost of the proposed algorithm is higher than Zhou’s algorithm [24] when K > 2 and lower than COMET algorithm [23] when the two algorithm use same number of sensors. 4. Results and Discussion In this section, the effectiveness of the proposed algorithm is investigated through five simulation experiments. The array configuration is shown in Figure 1. Root mean squared error (RMSE) is applied for the evaluation of the performance of estimation. RMSEθ and RMSE ϕ denote the RMSE of nominal azimuth and nominal elevation respectively, which can be expressed as v u u 1 Mc 2 RMSEθ = t (θˆς − θ ) , ∑ Mc ς

(82)

v u u 1 Mc RMSE ϕ = t ( ϕˆ ς − ϕ)2 , Mc ∑ ς

(83)

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ˆς where MC is the number of Monte Carlo simulations. θˆiςς and φ ς i are the estimated nominal azimuth where MC is the number of Monte Carlo simulations. θˆi and ϕˆ i are the estimated nominal azimuth ςthMonte andnominal nominalelevation elevationof ofith ithsource sourceininςth MonteCarlo Carlosimulation. simulation. and Inthe thefirst firstexperiment, experiment,we weexamine examinethe theperformance performanceof ofthe theproposed proposedmethod methodvesus vesusSNR SNRwith with In regard to two Gaussian ID source with parameters [30°, 45°, 2°, 2°, 0.5] and [50°, 45°, 2°, 2°, 0.5]. The ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ regard to two Gaussian ID source with parameters [30 , 45 , 2 , 2 , 0.5] and [50 , 45 , 2 , 2 , 0.5]. sources are supposed to have the same power. Snapshots number is set at 200, MC = 100, M = K = 8. The sources are supposed to have the same power. Snapshots number is set at 200, MC = 100, θ and RMSEφ curves with SNR varying from −5 dB to 30 dB. The d= =K λ/10. 2a,b show RMSE M = 8. Figure d = λ/10. Figure 2a,b show RMSEθ and RMSE ϕ curves with SNR varying from −5 dB to 30 dB. figures also show the estimation ofof the algorithm [24] [24] using using The figures also show the estimation theCOMET COMET[23] [23]using usingthe the URA, URA, Zhou’s Zhou’s algorithm andxx2 and and the the Cramer–Rao Cramer–Rao lower lower bound bound (CRLB). (CRLB). As As shown shown from from Figure figures2a,b, 2a and2b, arraysxx1 and arrays when when SNR 1 2 SNR changes from −5 dB to 0 dB, COMET [23] presents better performance than the proposed changes from −5 dB to 0 dB, COMET [23] presents better performance than the proposed algorithm. algorithm. As SNRRMSE increases, RMSEθ and RMSEφ of all algorithms decrease. The proposed algorithm As SNR increases, θ and RMSE ϕ of all algorithms decrease. The proposed algorithm has better has better performance than other algorithms with SNR ranging 10 dB to 30it dB. it can be performance than other algorithms with SNR ranging from 10 dB from to 30 dB. Thus, can Thus, be concluded concluded that our method has a good performance when SNR is at high levels . Estimation of our that our method has a good performance when SNR is at high levels. Estimation of our method is based method is based onsubspace acquiringthrough signal subspace through eigendecomposition of covariance matrix of on acquiring signal eigendecomposition of covariance matrix of received vector. received vector. The accuracy of eigendecomposition would deteriorate at low SNR, while COMET The accuracy of eigendecomposition would deteriorate at low SNR, while COMET [23] separates the [23] separates noisethrough and signal power matching through covariance matching fitting matrix by alternating noise and signalthe power covariance fitting matrix by alternating projection technique projection technique firstly, as a result, perform better at low SNR. firstly, as a result, perform better at low SNR.

(a)

(b)

Figure2.2.(a) (a)RMSE RMSEθθ estimated estimated by by three three algorithms algorithmsfor for2D 2D ID ID sources sources vesus vesusSNR; SNR;(b) (b)RMSE RMSEϕφestimated estimated Figure bythree threealgorithms algorithmsfor for2D 2DID IDsources sourcesvesus vesusSNR. SNR. by

In Inthe thesecond secondexperiment, experiment, we we investigate investigate the the estimation estimation of of source source near near the the boundary boundary region. region. Consider and aauniform uniformIDIDsource source respectively. Both sources angular spread= Consider aa Gaussian and respectively. Both sources havehave angular spread σθ = σφ Covariance =2◦ . Covariance coefficients of Gaussian are setSNR at 0.5. SNR is dB set and at 15snapshots dB and =2°. coefficients of Gaussian sourcessources are set at 0.5. is set at 15 snapshots Md==Kλ/10. = 8, dFigure = λ/10.3aFigure shows RMSEθascurves as nominal θ curves nominal azimuth number isnumber 200, MCis=200, 100,MC M = 100, K = 8, shows3aRMSE ◦ ◦ ◦ azimuth changing 0 with to 180nominal with nominal elevation at 20 ,Figure while Figure 3b shows φ curves changing from 0° from to 180° elevation fixed atfixed 20°, while 3b shows RMSERMSE ϕ curves as nominal elevation changing from 0◦ towith 180◦ nominal with nominal azimuth 20◦can . Asbe can be as nominal elevation changing from 0° to 180° azimuth fixed fixed at 20°.atAs seen, seen, RMSE RMSE markedly near boundaryregion. region.Generally, Generally, the the error of θ and RMSE φ increase markedly near thethe boundary of the the both both RMSE ϕ increase θ and boundary boundaryregion regionestimated estimatedby byour ourmethod methodisisacceptable. acceptable. In Inthe thethird thirdexperiment, experiment,we weexamine examinethe theperformance performanceof ofthe theproposed proposedmethod methodwith withregard regardto to estimation estimation of of sources sourceswith withdifferent differentAPDFs APDFssimultaneously. simultaneously. Consider Consider four four ID ID sources sources with withsame same ◦ , 60◦ , power [30◦ , 45°, 45◦ , 2°, 2◦ ,2°, 2◦ ,0.5], 0.5],[50°, [50◦45°, , 45◦1.5°, , 1.5◦3°, , 3◦0.5], , 0.5], [5060°, powerand and parameters parameters of of sources sources are set as [30°, [50°, 2°, ◦ , 2.5◦ ], and [70◦ , 60◦ , 1.5◦ , 3.5◦ ]. The first and second sources are Gaussian whereas the third and 22.5°], and [70°, 60°, 1.5°, 3.5°]. The first and second sources are Gaussian whereas the third and fourth fourth sources are uniform. The experiment setatSNR at 15 dB; number of snapshots 200.= MC sources are uniform. The experiment set SNR 15 dB; number of snapshots is 200.isMC 100, =M100, =K M = dK ==λ/10. 8, d =Figure λ/10. 4Figure shows the estimated of four which sources, which that indicate that the = 8, shows4the estimated result of result four sources, indicate the proposed proposed method can sources estimatewith sources with different APDFs effectively. method can estimate different APDFs effectively.

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(a) (a)

14 of 18 14 of of 18 18 14

(b) (b)

◦ to ◦ while ◦ ; (b) Figure 3. 3. (a) (a) RMSE RMSEθθθ estimated estimated with with θ changing from from 00° 0° to 180 180° while ϕ φ is isfixed fixedat at20 20°; (b)RMSE RMSEϕφφ Figure 180° φ is fixed at 20°; (b) RMSE estimated with θθ changing ◦ ◦ ◦ estimated with φ changing from 0° to 180° while θ is fixed at 20°. estimated with φ 180° while θ 20°.. ϕ changing from 00° to 180 θ is fixed at 20

Figure 4. 4. Estimated for 2D 2D ID ID sources sources with with different different angular angular power power density densityfunctions. functions. Figure Estimated results results for

In wewe examine the performance of theof proposed methodmethod vesus the distance In the thefourth fourthexperiment, experiment, we examine the performance performance of the proposed proposed method vesus the the fourth experiment, examine the the vesus the d between adjacent sensors. As the rotational invariance relations of generalize steering matrices are distance dd between between adjacent adjacent sensors. sensors. As As the the rotational rotational invariance invariance relations relations of of generalize generalize steering steering distance obtained under the small distance d assumption, necessary to theto of dthe to matrices are are obtained obtained under the small small distance it assumption, is necessary necessary toinfluence investigate the matrices under the distance dd is assumption, itit investigate is investigate the estimation. Four sources with the same parameters as the sources in third example are set to influence of d to the estimation. Four sources with the same parameters as the sources in third influence of d to the estimation. Four sources with the same parameters as the sources in third be estimated as to the varying from λ/20 to λ/2. from SNRλ/20 is setto 15 SNR dB and snapshots is 200, example are set set to be bed estimated estimated as the the varying from λ/20 toatλ/2. λ/2. SNR is set set at 15 15 dB dBnumber and snapshots snapshots example are as dd varying is at and MC = 100, =K = 8.== Figure RMSE RMSE and uniform sources θ and andGaussian RMSEφφ of of both Gaussian and number is M 200, MC 100, M M5==shows K == 8. 8. that Figure shows that RMSE ϕ of θboth RMSE both Gaussian and number is 200, MC 100, K Figure 55 shows RMSE θ and that increase with the distance d from λ/20 to λ/2. When d is λ/2, RMSE and RMSE of Gaussian sources and RMSE RMSEφφ of of uniform sources sources increase increase with with the the distance distance dd from from λ/20 λ/20 to to λ/2. λ/2. When When is λ/2, λ/2, RMSE RMSE ϕ θθand uniform θ dd is reach 0.62sources and 0.44, those of uniform reach 0.79 and 0.67,reach which areand still0.67, satisfactory results. Gaussian sources reach 0.62 and 0.44, 0.44,sources those of of uniform sources reach 0.79 and 0.67, which are are still Gaussian reach 0.62 and those uniform sources 0.79 which still It can be concluded that the algorithm shows satisfactory with small distance satisfactory results. It It can can beproposed concluded that the the proposed proposed algorithmperformance shows satisfactory satisfactory performance satisfactory results. be concluded that algorithm shows performance between adjacent sensors. with small small distance distance between between adjacent adjacent sensors. sensors. with According d dis, the higher estimation accuracy is. Accordingto tothe theresult resultof ofthird thirdexample, example,the thesmaller smallerdistance distance d is, is, the higher estimation accuracy According to the result of third example, the smaller distance the higher estimation accuracy However, in the same channel the frequency increases, wavelength decreases. High frequency detection is. However, in the same channel the frequency increases, wavelength decreases. High frequency is. However, in the same channel the frequency increases, wavelength decreases. High frequency needs smaller d than low frequency detection to attain the same Actually, accuracy. when the detection needs smaller than low low frequency detection to estimation attain the theaccuracy. same estimation estimation accuracy. detection needs smaller dd than frequency detection to attain same distance is small, of sensors accuracy will get worse, which would also deteriorate the Actually,dwhen when theinstallation distance dd is isaccuracy small, installation installation accuracy of sensors sensors will get worse, worse, which would would Actually, the distance small, of will get which estimation accuracy. Thus, installation accuracy and the distance d is a matter of balance especially in also deteriorate the estimation accuracy. Thus, installation accuracy and the distance d is a matter of also deteriorate the estimation accuracy. Thus, installation accuracy and the distance d is a matter of high frequency detection. the field of low frequency underwater detection, of detection, sonar can balance especially in high highIn frequency detection. In the the field field of of low low frequency frequency frequency underwater detection, balance especially in frequency detection. In underwater be down toof HZ,can which means canmeans reach wavelength 14.5 m approximately, on this condition d/λ frequency of100 sonar can be down down towavelength 100 HZ, HZ, which which means wavelength can reach reach 14.5 14.5 m approximately, approximately, frequency sonar be to 100 can m can reach a small d/λ value practically. on this this condition d/λ can can reach aa small small value value practically. practically. on condition reach In the the fifth fifth experiment, experiment, we we investigate investigate the the performance performance of of the the proposed proposed algorithm algorithm versus versus the the In number of sources and sensors. Theoretically, if M = K, the proposed algorithm can estimate M(M number of sources and sensors. Theoretically, if M = K, the proposed algorithm can estimate M(M −− Utilizing double double parallel parallel arrays arrays which which 1) different different ID ID sources sources and and COMET COMET [23] [23] can can estimate estimate M M22.. Utilizing 1)

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In the fifth experiment, we investigate the performance of the proposed algorithm versus the number sources sensors. Theoretically, if M = K, the proposed algorithm can estimate Sensors 2018, 18,of x FOR PEER and REVIEW 15 of 18 M(M − 1) different ID sources and COMET [23] can estimate M2 . Utilizing double parallel arrays contains only 2M sensors can estimate M sources simultaneously. We We consider seven URA with M which contains only 2M sensors can estimate M sources simultaneously. consider seven URA with is a theoretical upperupper limit of the varying from 4 to410. TheThe totaltotal number of sources is setisatset M2atwhich M varying from to 10. number of sources M2 which is a theoretical limit ◦, − ◦ + (b − ◦ ]. Allare trail. (a,The b)th(a, source is set atis[20° + (a + (b All1)10 sources Gaussian and have of theThe trail. b)th source set at [20−◦1)10°, + (a −20° 1)10 201)10°]. sources are Gaussian =2°.σθCovariance samehave power. SNR is 15 dB, ofnumber snapshots 200, MC is = 100, = λ/10, and same power. SNRnumber is 15 dB, of is snapshots 200,dMC = 100, d== λ/10, = σφ =2◦ . Covariance of sources Gaussianare sources setWhen at 0.5. all When all sources are estimated successfully, coefficients coefficients of Gaussian set atare 0.5. sources are estimated successfully, the the difference between estimators is accuracy which be described by RMSE. When all sources difference between estimators is accuracy which cancan be described by RMSE. When not not all sources are are estimated successfully, RMSE cannot differentiateperformance performanceofofestimators. estimators. Thus, Thus, an an indicator estimated successfully, RMSE cannot differentiate reflecting the thenumber number of sources detected is needed to measure the different reflecting of sources detected is needed to measure the performance of differentofestimators. q performance 2 estimators. Estimation is regarded as effective when the estimated angles satisfying Estimation is regarded as effective when the estimated angles satisfying (θˆi − θi ) + ( ϕˆ i − ϕi )2 ≤ 5◦ . 2 where Nd is number of source detection as Nd/M ( θˆ −detection θ ) 2 + ( φˆprobability − φ ) 2 ≤ as5°. Define N Define /M2 where N isprobability number of source estimated effectively. So in theory i

i

i

i

d

d

the detection probability proposed algorithm is (M − 1)/M, is 1, Zhou’s algorithm [24] estimated effectively. So of in the theory the detection probability of theCOMET proposed algorithm is (M − 1)/M, is 1/M. COMET is 1, Zhou’s algorithm [24] is 1/M. As can be seen from Figure Figure 6, 6, the the estimation estimation probability probability of zhou’s algorithm [24] is consistent consistent with ofof thethe trail, so so areare thethe estimation probabilities of our and with theoretical theoreticalvalue valuewithin withinallallrange range trail, estimation probabilities of method our method COMET with M < 6. When M ≥ 7, our method perform better than COMET, which means using 7 ×7 and COMET with M < 6. When M ≥ 7, our method perform better than COMET, which means using or arrays our method can estimated more sources thanthan COMET if there areare large number of 7 ×larger 7 or larger arrays our method can estimated more sources COMET if there large number sources that need Involving eigendecomposition of high dimensional matrices, effectiveness of sources that estimating need estimating Involving eigendecomposition of high dimensional matrices, of our method the number becomes large. For instance, if M = 8 andiftotal effectiveness ofdeteriorates our method as deteriorates as of thesources number of sources becomes large. For instance, M= source is 64,source the covariance of received vector R12 is 112 × 112 dimensional, Ω and Ω isx is 112 × 112 dimensional, 8 and total is 64, the matrix covariance matrix of received vector x zΩ z 64 × 64 dimensional, which all need eigendecomposition. The deterioration of COMET [23] is also and Ωz is 64 × 64 dimensional, which all need eigendecomposition. The deterioration of COMET [23] closely related related to the number of sources. COMET [23] separates unknown variables of eachof source is also closely to the number of sources. COMET [23] separates unknown variables each based alternating projectionprojection technique,technique, and then formulate sets of unknown variables. In sourceon based on alternating and then equation formulate equation sets of unknown the separating process, 64 inversion operations of a 64 × 64 dimensional matrix are executed on the variables. In the separating process, 64 inversion operations of a 64 × 64 dimensional matrix are condition M= 8 and total source 64.total Consequently, theConsequently, errors of the separating deliver to executed on the condition M = 8 isand source is 64. the errorsprocess of the separating equations set and the validity estimation. process deliver to affect equations set and of affect the validity of estimation.

Figure 5. RMSE estimated for 2D ID sources with different distance distance d. d.

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Figure 6. 6. Detection Detection probability probability by by three three algorithms Figure algorithms with with different different numbers numbers of of sources sources and and sensors. sensors.

5. Conclusions 5. Conclusions In this paper, we propose an estimator for 2D ID sources using URA. The received signal vectors In this paper, we propose an estimator for 2D ID sources using URA. The received signal vectors can be transformed as combinations of generalized signal vectors and generalized steering matrices can be transformed as combinations of generalized signal vectors and generalized steering matrices through Taylor series expansions. By virtue of the rotational invariance relations of generalized steering through Taylor series expansions. By virtue of the rotational invariance relations of generalized matrices within subarrays, an ESPRIT like algorithm are introduced in detail, then angle matching steering matrices within subarrays, an ESPRIT like algorithm are introduced in detail, then angle method based on Capon principle is proposed. Simulations demonstrate the effectiveness of the matching method based on Capon principle is proposed. Simulations demonstrate the effectiveness proposed method with regard to sources with different APDF, distance d between sensors, sources of the proposed method with regard to sources with different APDF, distance d between sensors, in boundary region. Simulation also indicates that our method has better performance when SNR is sources in boundary region. Simulation also indicates that our method has better performance when at high level on the condition that the quantity of sources is relatively small and can estimate more SNR is at high level on the condition that the quantity of sources is relatively small and can estimate sources in sufficient sources circumstance using large dimensional arrays. more sources in sufficient sources circumstance using large dimensional arrays. Author Contributions: Conceptualization, Z.D.; Methodology, T.W.; Software, T.W.; Writing—Original Draft Author Contributions: Conceptualization, Methodology, T.W.; Software, T.W.; Writing—Original Draft Preparation, T.W. and Y.L.; Writing—ReviewZ.D.; & Editing, Y.H. Preparation, T.W. and Y.L.; Writing—Review & Editing, Y.H. Funding: This research was funded by the National Natural Science Foundation of China grant number 61471299 and 51776222. Funding: This research was funded by the National Natural Science Foundation of China grant number 61471299 and 51776222. Acknowledgments: The authors would like to thank editorial board and reviewers for the improvement of this paper. Acknowledgments: The authors would like to thank editorial board and reviewers for the improvement of this Conflicts paper. of Interest: The authors declare no conflict of interest.

Conflicts ofA Interest: The authors declare no conflict of interest. Appendix ∗ ] can be expressed as Take Gaussian distribution described by (4) for example, if l = n = 0, E [ρil ρin Appendix A s ∗ ] = by Take Gaussian distribution l =dθdϕ n = 0 E [ρilρin∗ ] can be expressed as E[ρi0described ρi0 E (4) [s (for θ, ϕ,example, t)si∗ (θ, ϕ,ift)] s i . (A1) = Pi f i (θ, ϕ; u)dθdϕ ∗ E[ρi 0ρi∗0= ] =P E [ s ( θ , φ , t ) s ( θ , φ , t )] dθdφ i i i (A1) = Pi  fi (θ, φ; u)dθdφ . ∗ ] can be expressed as If l = n = θ, E[ρil ρin = P i

s ∗] = P ρiθ E[(θas− θi )si (θ, ϕ, t)(θ − θi )si∗ (θ, ϕ, t)]dθdϕ i If l = n = θ, E[ρilρEin∗ []ρiθcan be expressed s , = Pi (θ − θi )2 f i (θ, ϕ, t)dθdϕ ∗ ∗ E[ρiθρiθ=] =PiPM θi E[(θ − θi )si (θ, φ, t )(θ − θi )si (θ, φ, t )]dθdφ i  where Mθi

where

=

s

=

2 σθi

= Pi  (θ − θi )2 fi (θ, φ, t)dθdφ ,   2  2  2 =exp Pi M−θi ( θ − θi ) θ − θi 1 √ − 2ρ (θ −θi )( ϕ− ϕi ) + ϕ− ϕi dθdϕ

2πσθi σϕi

1− ρ2

2σθi σϕi (1−ρ2 )

σθi

σθi σϕi

σϕi

(A2)

(A2) .

(A3)

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2 . Similarly, E[ρiϕ ρiϕ ] = Pi M ϕi and M ϕi = σϕi If l 6= n, there exists six cases, l = 0 with n = θ or ϕ, l = θ with n = 0 or ϕ and l = ϕ with n = 0 or θ. ∗ ] can be expressed as Take l = θ with n = ϕ and l = 0 with n = ϕ for example, E [ρil ρin

s ∗ ]= P E[ρiθ ρiϕ E[(θ − θi )si (θ, ϕ, t)( ϕ − ϕi )si∗ (θ, ϕ, t)]dθdϕ i s = Pi (θ − θi )( ϕ − ϕi )f i (θ, ϕ, t)dθdϕ       s (θ −θi )( ϕ− ϕi ) , (θ −θi )( ϕ− ϕi ) ϕ − ϕi 2 θ − θi 2 1 √ 2 exp − = − 2ρ + dθdϕ 2 σ σ σ σ 2σ σ (1−ρ ) 2πσθi σϕi

1− ρ

θi ϕi

θi

θi ϕi

(A4)

ϕi

=0 s ∗ ]= P E[ρiθ ρiϕ E[si (θ, ϕ, t)( ϕ − ϕi )si∗ (θ, ϕ, t)]dθdϕ i s = Pi ( ϕ − ϕi ) f i (θ, ϕ, t)dθdϕ      s . (θ −θi )( ϕ− ϕi ) ϕ − ϕi 2 ( ϕ − ϕi ) θ − θi 2 1 √ 2 exp − − 2ρ + dθdϕ = 2 σ σ σ σ 2σ σ (1−ρ ) 2πσθi σϕi

1− ρ

θi ϕi

θi

θi ϕi

(A5)

ϕi

=0

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