sensors Article

Ambiguity Resolution for Phase-Based 3-D Source Localization under Fixed Uniform Circular Array Xin Chen, Zhen Liu * and Xizhang Wei School of Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, China; [email protected] (X.C.); [email protected] (X.W.) * Correspondence: [email protected]; Tel.: +86-731-8457-5713 Academic Editors: Zhiguo Shi, Yujie Gu and Rongxing Lu Received: 17 March 2017; Accepted: 3 May 2017; Published: 11 May 2017

Abstract: Under fixed uniform circular array (UCA), 3-D parameter estimation of a source whose half-wavelength is smaller than the array aperture would suffer from a serious phase ambiguity problem, which also appears in a recently proposed phase-based algorithm. In this paper, by using the centro-symmetry of UCA with an even number of sensors, the source’s angles and range can be decoupled and a novel algorithm named subarray grouping and ambiguity searching (SGAS) is addressed to resolve angle ambiguity. In the SGAS algorithm, each subarray formed by two couples of centro-symmetry sensors can obtain a batch of results under different ambiguities, and by searching the nearest value among subarrays, which is always corresponding to correct ambiguity, rough angle estimation with no ambiguity is realized. Then, the unambiguous angles are employed to resolve phase ambiguity in a phase-based 3-D parameter estimation algorithm, and the source’s range, as well as more precise angles, can be achieved. Moreover, to improve the practical performance of SGAS, the optimal structure of subarrays and subarray selection criteria are further investigated. Simulation results demonstrate the satisfying performance of the proposed method in 3-D source localization. Keywords: 3-D source localization; phase ambiguity; uniform circular array; subarray grouping; ambiguity searching

1. Introduction Passive source localization using an array of sensors plays an increasingly important role in wireless communication, electronic reconnaissance, sonar and other applications [1–4]. As uniform circular array (UCA) has several advantages over the other array geometries, such as its 360◦ azimuth angle coverage, almost identical beamwidth and additional elevation angle information [5,6], it is widely applied in three-dimensional (3-D) source localization. Currently, in the context of 3-D parameter estimation, many effective solutions mainly focus on two types, which are spectrum-based and phase-based estimators [7–9]. The spectrum-based estimators, such as multiple signal classification (MUSIC) algorithms [8,10], can achieve source localization with high precision, but suffer from a high computational cost. To cope with this problem, a series of phase-based solutions have been recently proposed in [11,12], which are computationally simpler, since multidimensional search is not required. Although the estimation performance of these approaches is acceptable, for fixed UCA, in practice, when the array aperture is larger than source’s half-wavelength they would suffer from a serious phase ambiguity problem, which will lead to inaccuracies in 3-D parameter estimation. In the case of ambiguity resolution under UCA, by utilizing the nonlinear property of pseudo direction of arrival (DOA) estimation, a rotary way to resolve ambiguity for the 3D-MUSIC algorithm was proposed in [13], which can suppress pseudo peaks of spectrum and obtain the actual source’s parameters. Referring to the idea of ambiguity resolution via rotating interferometer [14,15], we have Sensors 2017, 17, 1086; doi:10.3390/s17051086

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recently proposed an improved, unambiguous phase-based algorithm in [16], which performs adaptive rotation to make the sensors form a virtual short baseline. Even so, before and after rotation, as the location of moving sources varies or the two groups of receiving data of multiple frequency-hopping sources can not correspond to the same source, these rotary ways will be inapplicable. In this paper, without rotation, we address unambiguous 3-D source localization by utilizing different ambiguity properties of each subarray divided by a fixed UCA, and each subarray can obtain the actual source’s parameters under a certain ambiguity. A novel method called subarray grouping and ambiguity searching (SGAS) will be proposed. In SGAS, we first note that the ambiguity is mainly induced by angle parameters, and that the separation of the source’s angles and range is necessary as both are interlaced with each other, which causes the difficulty of ambiguity resolution. To resolve this trouble, we refer to [9,12] and find that the source’s range can be counteracted by computing the phase differences of receiving data from centro-symmetric sensors (CSS), thus source’s angles and range could be decoupled. Then, based on the traverse of ambiguity and some algebraic operations, each subarray with the same structure formed by two couples of CCS can obtain a batch of results under different ambiguities, and the rough angle estimation can be achieved by searching the nearest value among subarrays. Finally, to realize source’s range as well as more precise angle estimation, the unambiguous angles obtained by SGAS are utilized to resolve ambiguity in the phase-based algorithm [11] via the approximation of steering vectors between planar wavefront and curved wavefront. Moreover, the optimal structure of subarrays and subarray selection criteria are further investigated to make good use of our approach in practical applications. This paper contributes to the area of 3-D source localization in the following three aspects: 1

2 3

For the source’s 3-D localization, this paper provides a novel ambiguity resolution method to obtain the source’s actual parameters with satisfactory performance and low computational complexity. As the proposed algorithm does not need rotation, it can achieve real-time source localization, thus the applied scope is wider than the rotary way to resolve ambiguity. The optimal structure of subarrays and subarray selection criteria can be exploited in SGAS to improve the performance and practicability of our approach.

The rest of the paper consists of seven sections. Section 2 reviews the model of a narrowband source’s 3-D parameter estimation. Section 3 presents the source’s angle estimation under an unambiguous situation and the method of SGAS. Section 4 introduces the method of phase-based parameter estimation on the basis of SGAS (SGAS-PBPE) to obtain the source’s range and more precise angle estimation. Section 5 discusses the optimum structure of subarray and subarray selection criteria in terms of computational cost and estimation performance. Section 6 carries out simulations to demonstrate the effectiveness of SGAS, theoretical analysis of optimum subarray selection, and to compare estimation performance between SGAS and SGAS-PBPE. Section 7 concludes the whole paper. 2. Signal Modeling Consider a narrowband radiating source impinging on a fixed UCA with radius R and M identical omni-directional sensors uniformly spaced on the circumference in the xy-plane, where M is an even integer. The geometry of the signal model is shown in Figure 1. The source is located at (φ, θ, r ), where the azimuth angle φ ∈ [−π, π ) is measured counterclockwise from the x-axis, the elevation angle θ ∈ [0, π/2) is measured downward from the z-axis, and the range r is measured from the center of the UCA.

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z

 r sin  cos  , r sin  sin  , r cos   rm

m

m 1 R

m

 r



M

Figure 1. Signal model.

Figure 1. Signal model.

Here, we assume that the range of the source with a curved wavefront is beyond the Fresnel area, Here, we assume that the of the source a curvedthe wavefront is beyond Fresnel but dissatisfies the condition ofrange far-field. Under thiswith assumption, observation vectorthe of the mth area, but dissatisfies the condition of far-field. Under this assumption, the observation vector of the sensor at time n can be expressed as m th sensor at time n can be expressed as   2π xm (n) = s(n) exp j 2(r − rm (φ, θ, r ))  + wm (n) (1) xm (n)  s  n  exp  j λ r  rm  , , r    wm (n) (1)

 







for m = 1, . . . , M, n = 1, . . . , N. s(n) is assumed to be a zero-mean complex envelope with power for m  1, , M , n  1, , N . s ( n ) is assumed to be a zero-mean complex envelope with power σs2 , and wm (n) is assumed to be a zero-mean white complex Gaussian noise with power σn2 , which is  s 2 , and wmof(ns()n)is. λassumed  n2 , to be a zero-mean white with independent is the wavelength of the source. rm (complex φ, θ, r ) is Gaussian the range noise between thepower mth sensor and theissource, which is of given s ( nby ) .  is the wavelength of the source. r ( ,  , r ) is the range which independent m

q is given by between the mth sensor and the source, which rm (φ, θ, r ) = r 1 + ( R/r )2 −2Rζ m (φ, θ )/r

(2)

2

(2) r ( ,  , r )  r 1  ( R r ) 2 R m ( ,  ) r where ζ m (φ, θ ) = cos ϕm = cosm(γm − φ) sin θ with γm = 2π (m − 1)/M, and the angle ϕm is measured counterclockwise of centre UCA and the Consider a function defined where  ( ,  ) from the angle cos the  line  cos(   )ofsin  with  mth  2sensor.  (m  1) / M , and  asis m

m

m

m

m

n of centre of UCAo1/2 measured counterclockwise from the line and the m th sensor. Consider a f (ιr ) = 1 + ι2r − 2ιr · ζ m (φ, θ ) (3) function defined as 1/2 where ιr = R/r. According to a sufficiently large2 value r relative to (3) f (r )  1  r  2r   m ( , ) R, f (ιr ) can be well approximated by a second-order Taylor series expansion as





According to a sufficiently large value 1 r relative to f (ιr ) ≈ f (0) + f 0 (0)ιr + f 00 (0)ι2r approximated by a second-order Taylor series expansion as 2

where

r  R / r .

Then, rm (φ, θ, r ) can be approximated as

1 f (0)r2  2 2 R 2

R , f (r ) can be well

f (r )  f (0)  f (0)r 

(4)

(4) 

rm (φ, θ, r ) ≈ r − Rζ m (φ, θ ) + 1 − ζ m (φ, θ ) (5) 2r Then, rm ( ,  , r ) can be approximated as Substituting (2) and (5) into (1) yields the approximated expression of observation vector as 2    (5)  r  R m ( ,  ) R R 1 2  m2 ( ,  ) rm ( ,  , r )2πR xm (n) = s(n) exp j ( φ, θ ) + wm ( n ) (6) ζ m (φ, θ ) − 1 − ζ m 2 r λ 2r Substituting and(6) (5)that intothe (1)source’s yields the ofits observation as It can be seen(2) from 3-Dapproximated parameters areexpression included in exponent, vector so we intend





to estimate them by utilizing the phase of receiving data.

 2 R  R  2 xm (n)  s (n) exp  j   m ( , )  1   m ( , )     wm (n) 2r    

(6)

It can be seen from (6) that the source’s 3-D parameters are included in its exponent, so we intend to estimate them by utilizing the phase of receiving data.

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3. Ambiguity Resolution of Source’s Angles As we know, the ambiguity of 3-D parameter estimation mainly focuses on 2-D angle 4 of 16 estimation. However, ambiguity resolution becomes quite difficult as the source’s angles interweave with range. Thus, we first tend to decouple the source’s angles and range via the centro-symmetric configuration of UCA. By of utilizing some algebraic operations, the source’s angles are then obtained 3. Ambiguity Resolution Source’s Angles under an unambiguous situation. Finally, considering an ambiguous situation, subarray grouping As we know, the ambiguity of 3-D parameter estimation mainly focuses on 2-D angle estimation. and ambiguity searching are employed to obtain actual angle parameters. However, ambiguity resolution becomes quite difficult as the source’s angles interweave with range. Thus, we first tend to decouple the source’s angles and range via the centro-symmetric configuration 3.1. Source’s Angle Estimation under Unambiguous Situation of UCA. By utilizing some algebraic operations, the source’s angles are then obtained under an Note thatsituation. ambiguity in parameter estimation is mainly inducedsubarray by anglegrouping parameters. The range unambiguous Finally, considering an ambiguous situation, and ambiguity parameter is considered as a nuisance, which causes difficulty in ambiguity resolution. Hence, we searching are employed to obtain actual angle parameters. should take some measures to separate the source’s angles and range. 3.1. Source’s Angle under Unambiguous Situation According to Estimation the array configuration of UCA with even number sensors, it can be noticed that , and thenin parameter estimation is mainly induced by angle parameters. The  m  MNote ambiguity / 2  that m  Sensors 2017, 17, 1086

range parameter is considered as a nuisance, which causes difficulty in ambiguity resolution. Hence,  m  M the ,  )  angles ) range. (7) /2 (source’s m ( , and we should take some measures to separate According configuration of UCA with even number sensors, it candefined be noticed  1, 2, ,toMthe / 2array for m . By using the property in (7), consider a correlation function as that γm+ M/2 = γm + π, and then ζ (φ, θ ) = −ζ  (7) 4θ )R  m ( φ, Rm  E xm (n) xm mM+/ 2M/2 (n)   s2 exp  j  m ( ,  )    n2 (8)  acorrelation function  for m = 1, 2, . . . , M/2. By using the property in (7), consider defined as



where



 

of Rm is



   4πRreceiving data is denotes the complex conjugate. Assuming noiseless, the phase ∗ 2 that the Rm = E xm (n) xm+ M/2 (n) = σs exp j ζ m (φ, θ ) + σn2 (8) λ

where (·)∗ denotes the complex conjugate. Assuming that the receiving data is noiseless, the phase of um  arg( Rm )  4 R m ( ,  ) /   2 q (9) Rm is um = ( Rm )parameter = 4πRζ m (isφ,counteracted. θ )/λ + 2πq (9) where q is a certain integer and thearg range

q  0 and employ two couples of adjacent Firstly, we consider situation where q is a certain integerthe andunambiguous the range parameter is counteracted. CCS Firstly, to formwesubarray easily. Takingqa=UCA as an example, subarrays M two  8 couples considerfor thededucing unambiguous situation 0 andwith employ of adjacent CCS to after grouping arededucing shown ineasily. FigureTaking 2. Based on trigonometric we can simplify the grouping sum and form subarray for a UCA with M = 8 asfunction, an example, subarrays after are shown in Figure 2. Based on trigonometric function, we can simplify the sum and the difference the difference between um and umod( m1, M /2) , which can be written as between um and umod(m+1,M/2) , which can be written as vm  um  u m ,1 

8 R

8πRsin( ) cos(  m ,1   ) cos( m ,1 ) vm = um + uε m,1 =  sin(θ ) cos(ρτm,1 − φ) cos(ρυm,1 ) λ

(10) (10)

w  u  u ,1 =− 8πRsin(  ) sin(  m ,1 − m,1))  φ)) sin( wmm = umm − uεmm,1 sin(ρυm,1 λ sin(θ ) sin(ρτm,1

(11) (11)

8 R

where (mm+1, , ρ, = =mm − ε  . m,1 .  m,1=mod mod( 1,M/2 M / )2)  −m,12,υ2m,1,   π/M,  / Mτm,1, =m,1m+mε m,1 where ε m,1 ,1  mm,1 1 1

8

2

2

7

3

8

7

3

4

6 5

6

4

Figure2.2.Subarray Subarraygrouping groupingwith with M M = 8. Figure 8.

5

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Then, by utilizing (10) and (11), a plural can be constructed as zm =

wm vm −j = exp( j(ρτm,1 − φ)) sin(θ ) 8πR/λ · cos(ρυm,1 ) 8πR/λ · sin(ρυm,1 )

(12)

Further, by eliminating the known parts, the plural which only includes source’s angles can be expressed as z0 = z∗m exp( jρτm,1 ) = exp( jφ) sin(θ )

(13)

Finally, the expression of source’s angles is given by φ = angle(z0 )
θ = arcsin z0

(14) (15)

Under the unambiguous assumption, the actual source’s angles can be obtained via the above procedure. In practice, ambiguity always appears in the source’s localization and it is quite significant to investigate effective schemes of ambiguity resolution. Moreover, it should be pointed out that using two couples of adjacent CCS to form a subarray is only considered for ease of deduction, and the different structures of subarray will be discussed in the complementary section. 3.2. Ambiguity Resolution by Using the Method of SGAS To guarantee no phase ambiguity in (9), the condition R ≤ λ/4 is commonly demanded, but when the array has been designed, it would be always invalid, especially for localization of a high-frequency source. For fixed UCA, as each subarray can obtain the actual source’s angles under a certain ambiguity, we employ traverse of ambiguity to make each subarray acquire a batch of results and search the nearest value among subarrays to ascertain actual source’s angles. Firstly, the maximum ambiguity of phase difference um can be computed by D = ceil(2R/λ)

(16)

where ceil(·) denotes upper nearest integer. Then, based on traverse of ambiguity, the phase difference matrix U is expressed as   u1 − 2πD u2 − 2πD ··· u M/2 − 2πD    u1 − 2π ( D − 1) u2 − 2π ( D − 1) · · · u M/2 − 2π ( D − 1)    U= (17) .. .. ..  . . ··· .   u1 + 2πD u2 + 2πD ··· u M/2 + 2πD Finally, similar to (10)–(13), any two phase differences of adjacent columns in (17) could develop a plural, and there are (2D + 1)2 combinations. Thus, we can obtain a plural matrix (PM), which is given by  0  0 z1, 1 z2, · · · z0M/2, 1 1  z0  0 0  1, 2 z2, 2 · · · z M/2, 2   Z= (18) .. .. ..  ..  . . .  .  0 0 z2, · · · z0M/2, µ2 z1, µ2 µ2 where µ = 2D + 1, and z0v,δ in (18) for v = 1, 2, ..., M/2, δ = 1, 2, ..., µ2 is computed by the ξth element of the vth column and the ϑth element of the ςth column in (17), where ξ = ceil(δ/µ), ϑ = mod(δ/µ), ς = mod((v + 1), M/2). The sketch map of computation process is shown in Figure 3.

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  ceil( /  ) ,   mod  /   ,   mod  ( +1)，M / 2  .

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The sketch map of computation 6 of 16

process is shown in Figure 3. -D

1

M /2

2

-D

-D

0

0

0

D

D

D

Figure 3. The sketch map of the plural matrix (PM) computation process. Figure 3. The sketch map of the plural matrix (PM) computation process.

It is worth noting that the value of each column in PM represents a batch of subarray results It is worth noting that the value of each column in PM represents a batch of subarray results under different ambiguities and each column only includes a plural constructed by the actual source’s under different ambiguities and each column only includes a plural constructed by the actual angles. Therefore, the source’s unambiguous angles can be obtained by finding the nearest value source’s angles. Therefore, the source’s unambiguous angles can be obtained by finding the nearest among subarrays. value among subarrays. Firstly, we take the first column in PM as the reference and select an element from the referenced Firstly, we take the first column in PM as the reference and select an element from the column in turn. Then, the sum of minimum difference between the element of referenced column and referenced column in turn. Then, the sum of minimum difference between the element of referenced the element of other columns can be given by column and the element of other columns can be given by ! M/2M /2 0 0 dp = min (19) d ∑ min z1,p z −zk,h z  (19) p



2 1,2,...,  2 k =2k  2h=h1,2,...,µ

1, p

k ,h



for ..., µ2 .2 It for pp=1, 1,2,2,..., . Itisisclear clearthat thatthe theminimum minimumsum sumof ofdifferences differencesamong among columns columns is is brought brought from from actual angles. Finally, by finding the coordinate of minimum value in (19) and its corresponding plural, actual angles. Finally, by finding the coordinate of minimum value in (19) and its corresponding the source’s unambiguous angles can be obtained by plural, the source’s unambiguous angles can be obtained by

 


gg=argmin arg mind pd p

(20) (20)

0 φˆ 0 = angle(z1,g ) ˆ    angle(z  1, g) 0 θˆ0 = arcsin z1,g

(21) (21) (22)

pp





In practical application, the phaseˆdifference  arcsin zu1,mg would be replaced with its estimate (22) N

∗ uˆ m = arg( Rˆ m ), where a finite sample estimate Rˆ m = (1/N ) ∑ xm (n) xm + M/2 ( n ). In addition, In practical application, the phase difference um wouldn=1be replaced with its estimate considering computational cost, not all the columns in PM are Nnecessary to find the coordinate in andRˆthe detailed research will be presented in aˆ subsequent section. uˆm(20),  arg( xm (n) xm  M /2 (n) . In addition, m ) , where a finite sample estimate Rm  (1/ N ) The procedure of SGAS could be summarized as n 1 considering computational cost, not all the columns in PM are necessary to find the coordinate in Step ascertain scope of ambiguity, the maximum ambiguity of phase difference D needs to (20), 1andTo the detailedthe research will be presented in a subsequent section. calculatedoffirstly to the equation Thebe procedure SGASaccording could be summarized as in (16). Step 2 After subarray grouping, the phase difference estimate uˆ m can be computed by using receiving Step 1 data To of ascertain the scope of ambiguity, the maximum ambiguityphase of phase difference ˆ each subarray. To find the unambiguous phase difference, difference matrixD U needs to be calculated firstly according to the equation in (16). including actual phase difference can be obtained by utilizing the traverse of ambiguity. Step 2 subarray grouping, phase difference estimate can beform computed byplural, using uˆm could the unambiguous phase the differences of different subarrays the same Step 3 AsAfter ˆ ˆ receiving data Zofis each subarray. phasetodifference, phase the plural matrix computed basedTo on find U andthe theunambiguous Equations (10)–(13) estimate source’s ˆ angles without ambiguity. difference matrix U including actual phase difference can be obtained by utilizing the Step 4 Astraverse each column of matrix Zˆ includes plural value constructed by the actual angles of the source, of ambiguity. of searching the most similarof plural matrix terms by minimizing of Step 3 the Asmethod the unambiguous phase differences different subarrays could form themagnitude same plural, difference in the Equations (19) and (20) can be utilized to obtain the coordinate corresponding ˆ and ˆ is computed based on U the Equations (10)–(13) to estimate the plural matrix Z to unambiguous angles. The ambiguity-free estimation of the source’s angles (φˆ 0 , θˆ0 ) are then source’s angles without ambiguity. computed by substituting the coordinate into Equations (21) and (22).



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4. 3-D Parameter Estimation of Source It can be noticed that each subarray only utilizes four sensors to obtain the aforementioned angle estimation. According to the analysis of Cramer Rao bound (CRB) in [17], the estimation performance improves as the number of sensors increases, so we can employ more sensors to make the source’s angle estimation more precise. In terms of computational complexity and estimation performance, the phase-based algorithm in [11] has obvious advantages. Considering that the steering vectors of sources with planar wavefronts and curved wavefronts could be well approximated as a sufficiently large value r relative to R, we utilize the unambiguous angle estimation of SGAS to resolve the ambiguity in phase-based 3-D parameter estimation algorithm called SGAS-PBPE, which could acquire the source’s range and more precise angles. Sufficient theoretical results and simulative/experimental ones demonstrate that, the longer the baseline is, the higher the DOA estimation precision will be [15]. The longest baseline is just constructed by CSS, but the source’s range parameter cannot be estimated. Therefore, we select the sub-longest baseline to obtain the source’s 3-D parameter estimation. According to (8) and (9), the phase difference using the sub-longest baseline can be expressed as u0m =

2πR {ψm (φ, θ, r ) − ψm0 (φ, θ, r )} + 2πκ λ

(23)

for m = 1, 2, . . . , M, where m0 = mod(m + M/2 − 1, M ), ψm (φ, θ, r ) = ζ m (φ, θ ) −

R 2r



1 − ζ m (φ, θ )2



and κ is a certain integer. Similarly, the phase difference u0m would be replaced with its estimate N

∗ ( n ). uˆ 0m = arg( Rˆ 0m ), where a finite sample estimate Rˆ 0m = (1/N ) ∑ xm (n) xm 0 n =1

It is clear that the steering vector of a source with planar wavefront lacks the section −2πR2 1 − ζ m (φ, θ )2 /λr compared with the steering vector of a source with curved wavefront. Yet, according to a sufficiently large value r relative to R, these two steering vectors can be well approximated. Thus, the unambiguous phase difference in (23) can be estimated by using the actual angle estimation obtained by SGAS, which is given by 00

um =

2πR  ζ m (φˆ 0 , θˆ0 ) − ζ m (φˆ 0 , θˆ0 ) λ

(24)

Based on (24), we can compute the ambiguity of phase difference in (23) and more precise estimation of unambiguous phase difference can be written as
2πR 000 00 uˆ m = uˆ 0m − 2πκ = uˆ 0m − 2π · round (uˆ 0m − um )/2π = {ψm (φ, θ, r ) − ψm0 (φ, θ, r )} λ

(25)

000

where round(·) denotes rounding off. By observing (25) and referring [11], uˆ m can be reformulated as ˆ 0 = Ab, where U ˆ 0 = [ uˆ 000 uˆ 000 . . . uˆ 000 ]T , matrix A = [ am,1 , am,2 , am,3 , am,4 ] and the form of matrix U 1

 b=

2πR    λ 

2

M

cos(φ) sin(θ ) sin(φ) sin(θ ) R ( 4r ) cos(2φ) sin2 (θ ) R ( 4r ) sin(2φ) sin2 (θ )

    

(26)

where am,1 = cos(γm − γm0 ), am,2 = sin(γm − γm0 ), am,3 = cos(2γm − 2γm0 ), and am,4 = sin(2γm − 2γm0 ). Notice that only the matrix b includes the 3-D parameters of the source, so the optimal solution b h iT −1 ˆ 0 . As a can be estimated by employing the least square (LS) bˆ = bˆ 1 bˆ 2 bˆ 3 bˆ 4 = (AT A) AT U result, from (26), the estimates of (φ, θ, r ) can be obtained.

obtained. 5. Complementary Issues to SGAS In this section, to make good use of the proposed ambiguity resolution algorithm SGAS in 8 of 16 practical application, we will discuss some complementary issues to SGAS, including the optimum structure of the subarray and subarray selection criteria. Sensors 2017, 17, 1086

5. Complementary Issues to SGAS

5.1. Selection of Subarray In this section, to Structure make good use of the proposed ambiguity resolution algorithm SGAS in practical application, we willprocedure discuss some complementary issues to SGAS, structure of by In the algorithm above, the source’s angles are including deducedthe viaoptimum a subarray formed the subarray and subarray selection criteria.

two couples of adjacent CSS. In fact, a subarray formed by two different CSS groups could obtain the estimation of theofsource’s 5.1. Selection Subarrayangles. StructureThe formula in (10) and (11) could be rewritten as

8 the R source’s angles are deduced via a subarray formed by In the algorithm procedure above,  v u u sin(formed  ) cos by two ( m,ldifferent m,l ) could obtain the (27)      ) cos( m m  m ,l two couples of adjacent CSS. In fact, a subarray CSS groups  estimation of the source’s angles. The formula in (10) and (11) could be rewritten as





8 R 8πR wm  umv0m=umod( sin( ()ρsin  (φ))m,cos ) m,lsin( ) cos (τm,l − ) m ,l ) um + m u l ,εMm,l/2)=  sin( θ l (ρυ λ



(27) (28)

8πR 0 l  1, 2,w, M 2m−1 umod , (which represents − sin(θthe ) sin(ρspace (τm,l − φof )) sintwo (ρυm,l )couples of(28)CSS, m =/ u m+l,M/2) = λ  mod( / 2) −   m,l  2the, space m,l ofm  couples , 1,mwhich subsequent ,l  mrepresents m,l . Asofthe where l =m1,2,l., .M . , M/2 two CSS, ε m,l = modprocedure (m + l, M/2of ), the

where

 m ,l

τm,l = m + εparameter = m − ε m,l . isAssimilar the subsequent procedure of the source’s parameter source’s angle to (12)–(15), the details are not angle provided here. By m,l − 2, υm,l estimation estimation is similar to (12)–(15), the details are not provided here. By taking phase difference u as 1 taking phase difference u1 as the reference, the different structures of subarray with Mthe 12 reference, the different structures of subarray with M = 12 versus l is shown in Figure 4.

versus l is shown in Figure 4.

1 12

2

11

3

l2 l 1

l 3

10

4

l 5 l4

5

9 8

6 7

Figure 4. Different structureof of aa subarray subarray versus Figure 4. Different structure versusl.

l.

In addition, it is clear that the estimation performance of the source’s angle parameter is not the same for different l. In fact, the selection of the optimal subarray structure, which provides the best estimation performance, is a problem in the performance comparison of array geometry. In Section 1 we have pointed out several advantages of UCA over the other geometries, thus the best estimation performance should be brought by the selection of a subarray whose structure is similar to a UCA with four sensors. Taking M = 12 in Figure 5 as an example, the best selection might be two couples of CSS with l = 3. Meanwhile, we will demonstrate the conclusion in the subsequent simulation section. 5.2. Criteria of Subarray Selection In the practical application of SGAS, it is unnecessary to employ all results of subarrays under different ambiguities, which would cause high computational complexity. Therefore, it is significant to research proper subarray selection in terms of computational cost and effectiveness of ambiguity resolution. A batch of results corresponds to a column in PM. It is clear that two columns in PM are the least which we can employ. Yet, how to properly select two columns in PM to resolve ambiguity effectively needs to be further investigated.

estimation performance, is a problem in the performance comparison of array geometry. In Section 1 we have pointed out several advantages of UCA over the other geometries, thus the best estimation performance should be brought by the selection of a subarray whose structure is similar to a UCA with four sensors. Taking M  12 in Figure 5 as an example, the best selection might be two Sensors 2017, 17, 1086 9 of 16 couples of CSS with l  3 . Meanwhile, we will demonstrate the conclusion in the subsequent simulation section.

Figure 5. Flow chart of the proposed algorithm SGAS-PBPE. Figure 5. Flow chart of the proposed algorithm SGAS-PBPE.

5.2. Criteria of Subarray Selection First, we intend to analyze the selection of columns by employing the structure of a subarray formed two couples of adjacent CSS.itIn ambiguoustosituation, theresults sum and differenceunder um and In theby practical application of SGAS, is an unnecessary employ all of subarrays uε m,1 inambiguities, (10) and (11)which can bewould expressed ashigh computational complexity. Therefore, it is significant different cause to research proper subarray selection in terms of computational cost and effectiveness of ambiguity 00 vm = vm + 2παm (29) resolution. A batch of results corresponds to a column in PM. It is clear that two columns in PM are the 00 wm = wm + 2πβ m (30) least which we can employ. Yet, how to properly select two columns in PM to resolve ambiguity effectively further where −needs 2D ≤ to αmbe , βm ≤ 2D,investigated. and they are certain integer, representing the ambiguities of sum and First, weofintend to analyze selection Based of columns employing the structure a written subarray difference two CCS groups, the respectively. on (12)byand (13), the new plural canofbe as formed by two couples of adjacent CSS.In an ambiguous situation, the sum and difference and u  m λ αm βm 0 0 z = z + + j exp ( jρτ ) (31) m,1 u m ,1 in (10) and (11) can bem expressed 4Ras cos(ρυm,1 ) sin(ρυm,1 ) When |αm | 6= | β m |, the plural value among actual vm  vmcolumns  2 m would be equivalent only under the (29) source’s angles. However, when the ambiguities of one subarray satisfy αχ = β χ , (31) can be simplified as wm  wm  2 m (30)
λαχ exp jρ(τχ,1 − υχ,1 ) λαχ exp{2jρ(mod(χ + 1, M/2) − 1)} 0  ,0   0 where 2 Dz = zinteger, + j representing the ambiguities of sum and (32) χ =mz +m j 2 D , and they are certain 4R sin(ρυχ,1 ) cos(ρυχ,1 ) 4R sin(ρυχ,1 ) cos(ρυχ,1 ) difference of two CCS groups, respectively. Based on (12) and (13), the new plural can be written as And at the same time, if the ambiguities of the adjacent subarray satisfy αχ+1 = − β χ+1 , (31) can  be simplified as m    j
m zm  z  (31)   exp( j  m,1 )   4 R cos( ) sin( ) m m ,1 ,1   λαχ+1 exp jρ(τχ+1,1 + υχ+1,1 ) λαχ+1 exp(2jρχ) z0χ+1 = z0 + j = z0 + j (33) 4R sin(ρυχ+1,1 ) cos(ρυχ+1,1 ) 4R sin(ρυχ+1,1 ) cos(ρυχ+1,1 ) It should be noted that υχ+1,1 = υχ,1 and χ = mod(χ + 1, M/2) − 1 for χ = 1, 2, . . . , M/2 − 1. Satisfying the conditions in (32)–(33), the plural value of adjacent columns would be equivalent, which causes the ineffectiveness of ambiguity resolution for multiple nearest value. Moreover, for a different structure of subarrays, ambiguity resolution would be ineffective under similar conditions. For example, we only employ the first and the third columns in PM to resolve ambiguity with M = 12 and l = 2. Furthermore, we find that the value of ineffective combinations are computed by four CSS groups with a common group. In terms of the aforementioned example, the results in the two ineffective columns are obtained based on a common CSS group constructed by 3th sensor and 9th sensor respectively.

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Therefore, we would reach the significant conclusion that the SGAS algorithm needs at least four couples of different CSS, thus, the number of sensors should be no less than eight to guarantee the effectiveness of ambiguity resolution. 6. Experiments In this section, we carry out simulations to demonstrate the satisfying performance of our proposed algorithm and support the theoretical results. A UCA with M = 12 and fixed array radius R = 0.5 m is employed. Three hundred Monte Carlo runs are performed to obtain the probability of precise estimation and the root mean square errors (RMSEs) of the source’s parameter estimation. Simulation experiments are divided into four parts. In Part A, the effectiveness of SGAS is firstly presented. Then, by defining precise estimation as the difference between parameter estimation and actual value being no more than one degree, the probability of precise estimation Ppe = ( Npe /Nmo ) · 100% representing the ratio between the count of precise estimation Npe and Monte Carlo runs Nmo is utilized to illustrate the applied scope of SGAS. In Part B, we mainly research the optimal subarray structure by employing the RMSEs of the source’s angle parameter estimation versus azimuth angle under different structures of subarray. In Part C, considering subarray selection criteria, the computational time of different numbers of subarray versus the source’s frequency is firstly compared, and the effectiveness of the source’s angle estimation under different combinations of two subarrays is then shown to validate theoretic analysis in part B of Section 4. Finally, the comparison of estimation performance between SGAS and SGAS-PBPE is given in Part D. It should be pointed out that by using the SGAS method, the unambiguous source’s angles are utilized to estimate the source’s range via the 1-D MUSIC algorithm, which compares with range estimation using the method of SGAS-PBPE. 6.1. Effectiveness of SGAS Suppose that the source is located at (φ, θ, r ) = (120.5◦ , 20.1◦ , 6.5 m), and the signal-to-noise ratio (SNR) is 10 dB. Each subarray is formed by two couples of adjacent CSS and all subarrays are employed in ambiguity searching. As the scope is wide, the sum of minimum difference between the element of the column referenced and the element of other columns in (19) is partially shown and four typical source frequencies f = 1 GHz, f = 2 GHz, f = 3 GHz, f = 4 GHz are considered in Figure 6. It can be seen that the blue points represent the sum of minimum difference under different combinations of ambiguities corresponding to different values of p in (19), and the red circle represents the actual ambiguities corresponding to the minimum value of the sum in (19), which illustrates the effectiveness of SGAS. In addition, it can also be noted that the number of ambiguity combinations increases as the source’s frequency rises, and the difference between minimum value and other value reduces as the source’s frequency rises. Moreover, to present the applied scope of SGAS, under the corresponding frequencies of source in Figure 6, the probabilities of precise source angle estimation versus SNR and the radius of UCA are given in Figure 7, respectively. As shown in Figure 7a, the probabilities of precise estimation rise as the SNR increases, and all the probabilities under different source frequencies are acceptable when SNR is more than 0 dB. Meanwhile, when compared with a low frequency of source, the probability of precise estimation for a high-frequency source is smaller, especially under low SNR, since the superiority of minimum value reduces as the source’s frequency rises (shown in Figure 6), which causes the ineffectiveness of SGAS more easily for a high-frequency source at low SNR. As shown in Figure 7b, the probabilities of precise estimation decrease as the radius of UCA increases, and all the probabilities under different source’s frequencies are acceptable when the radius is less than 0.7 m. Similarly, it is easier for a high-frequency source to cause the ineffectiveness of the SGAS algorithm under a large radius of UCA. Consequently, the proposed SGAS algorithm is effective and the applied scope including frequency of ambiguity resolution, acceptable SNR and the maximum aperture is comparatively wide.

f  1 GHz , f  2 GHz , f  3 GHz , f  4 GHz are considered in Figure 6. It can be seen that the blue points represent the sum of

partially shown and four typical source frequencies

minimum difference under different combinations of ambiguities corresponding to different values of p in (19), and the red circle represents the actual ambiguities corresponding to the minimum value of the sum in (19), which illustrates the effectiveness of SGAS. In addition, it can also be noted Sensors 2017, that 17, 1086 the number of ambiguity combinations increases as the source’s frequency rises, and the difference between minimum value and other value reduces as the source’s frequency rises.

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(b)

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thanFigure 0.7 m. 6.Similarly, it is easier for a high-frequency to causeofthe ineffectiveness of the SGAS Sum of minimum difference versus differentsource combinations ambiguities under different

Figurealgorithm 6. Sum of minimum difference versus differentthe combinations of ambiguities under different a large UCA. proposed SGAS algorithm f =1 of GHz f =2 GHz ; (c) f =3 GHz ; (d) f =4 GHz . is effective and ; (b)Consequently, source under frequencies. (a)radius sourcethe frequencies. (a) f = 1 GHz; (b) f = 2 GHz; (c) f = 3 GHz; (d) f = 4 GHz. applied scope including frequency of ambiguity resolution, acceptable SNR and the maximum aperture is comparatively wide. Moreover, to present the applied scope of SGAS, under the corresponding frequencies of source in Figure 6, the probabilities of precise source angle estimation versus SNR and the radius of UCA are given in Figure 7, respectively. As shown in Figure 7a, the probabilities of precise estimation rise as the SNR increases, and all the probabilities under different source frequencies are acceptable when SNR is more than 0 dB. Meanwhile, when compared with a low frequency of source, the probability of precise estimation for a high-frequency source is smaller, especially under low SNR, since the superiority of minimum value reduces as the source’s frequency rises (shown in Figure 6), which causes the ineffectiveness of SGAS more easily for a high-frequency source at low SNR. As shown in Figure 7b, the probabilities of precise estimation decrease as the radius of UCA increases, and all the probabilities under different source’s frequencies are acceptable when the radius is less

(a)

(b)

Figure 7. Probability of precise estimation under different source’s frequencies. (a) variational SNR;

Figure 7. Probability of precise estimation under different source’s frequencies. (a) variational SNR; (b) variational radius of uniform circular array (UCA). (b) variational radius of uniform circular array (UCA). 6.2. The Optimal Structure of Subarray In Structure this part, we demonstrate the theoretical analysis in part A of Section 4 by simulation 6.2. The Optimal of will Subarray experiments, and the conclusion that the optimal structure of subarray could be reached.

In this part, the theoretical analysis in partItAcanofbeSection 4 by simulation We we take will three demonstrate structures of subarray with M  12 as an example. seen in Figure 8 thatand for the source’s azimuth angle, estimation using different lengths baseline would experiments, thedifferent conclusion that the optimal structure of subarray couldofbe reached. cause distinguishing so with the source’s azimuth estimation sensitive to Figure the We take three structuresperformance, of subarray M = 12 as an angle example. It canis be seen in 8 that structure of subarray. Therefore, to search the optimal structure of subarray, the RMSEs of the for the different source’s azimuth angle, estimation using different lengths of baseline would cause source’s angle parameter estimation versus azimuth angle under different structures of subarray are distinguishing performance, theinsource’s angle estimation frequency is sensitive to theis structure investigated, which are so given Figure 9. azimuth Suppose that the unambiguous of source  of subarray.f  Therefore, to search the optimal structure of subarray, the RMSEs of the angle 100 MHz , the fixed source’s elevation angle is   50 , the source’s azimuth angle variessource’s from   parameter estimation angle different of optional subarray are investigated, 180 to 180versus , and allazimuth the subarrays areunder utilized. It can be structures found that the structure of which are subarray given incorresponding Figure 9. Suppose that the f =is100 MHz, M  12 is l  1,frequency 2, 3, 4, 5 of  14 andsource for M is to CCS space for unambiguous the fixed source’s elevation angle is θ =structure 50◦ , the source’s from 180◦ to 180◦ , l  1, 2, 3, 4, 5, 6 . The subarray with M  12azimuth and spaceangle is similar to a− UCA l  3 varies l  3, 4ofare  14 and space both similar to four sensors and the subarray structure and all the with subarrays are utilized. It can be foundwith thatMthe optional structure subarray corresponding a UCA sensors. Note both changes of RMSEs with to CCS space forwith M four = 12 is l = 1, that 2, 3, 4,of5the and for M = 14 is l =l 1,3 2,versus 3, 4,azimuth 5, 6. angle The subarray in Figure 9a,b are the slightest, and the estimation performance of the source’s angles is satisfactory. structure with M = 12 and space l = 3 is similar to a UCA with four sensors and the subarray structure Similarly, the change of RMSEs with l = 3,4 versus azimuth angle in Figure 9c–d are almost slightest, and the estimation performance of the source’s angles is also satisfactory. Thus, the simulation results indicate that the conclusion in part A of Section 4 is correct. 1

2

1

1

3

baseline

source

source’s angle parameter estimation versus azimuth angle under different structures of subarray are investigated, which are given in Figure 9. Suppose that the unambiguous frequency of source is

f  100 MHz , the fixed source’s elevation angle is   50 , the source’s azimuth angle varies from 180 to 180 , and all the subarrays are utilized. It can be found that the optional structure of Sensors 2017, 17, 1086 of 16 is subarray corresponding to CCS space for M  12 is l  1, 2, 3, 4, 5 and for M 1214 l  1, 2, 3, 4, 5, 6 . The subarray structure with M  12 and space l  3 is similar to a UCA with = 14 and space l =subarray 3, 4 are both similar to aM UCA four sensors. that both of theto l  3,Note 4 are  14with and space both similar withMfour sensors and the structure with

changes RMSEs with l = Note 3 versus angle in Figure 9a,b arewith the slightest, and the estimation a UCA of with four sensors. thatazimuth both of the changes of RMSEs azimuth angle l  3 versus performance of are the the source’s angles is the satisfactory. Similarly, the change RMSEsangles with lis= satisfactory. 3, 4 versus in Figure 9a,b slightest, and estimation performance of the of source’s azimuth angle Figureof9c,d are almost andazimuth the estimation ofare the almost source’sslightest, angles Similarly, thein change RMSEs with lslightest, = 3,4 versus angle performance in Figure 9c–d isand alsothe satisfactory. Thus, the simulation results indicate that the conclusion in part A of Section 4 estimation performance of the source’s angles is also satisfactory. Thus, the simulation isresults correct. indicate that the conclusion in part A of Section 4 is correct. 1

1

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baseline

3

4

10 source

9

8

7

7

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Figure 8. Sketch of source’s azimuthangle angle estimation estimation under structures of subarray withwith Figure 8. Sketch mapmap of source’s azimuth underthree three structures of subarray M (a)  12 . (a) (c) l  3 . l l1=; (b) l l 2=; 3. M = 12. l= 1; (b) 2; (c)

(a)

(c)

(b)

(d)

Figure 9. Root mean square errors (RMSEs) of the source’s angle parameters versus azimuth angle.

Figure 9. Root mean square errors (RMSEs) of the source’s angle parameters versus azimuth angle. (a) Azimuth angle ( M  12) ; (b) Elevation angle ( M  12) ; (c) Azimuth angle ( M  14) ; (d) (a) Azimuth angle ( M = 12); (b) Elevation angle ( M = 12); (c) Azimuth angle ( M = 14); (d) Elevation ( M  14) . Elevation angle ( M = 14angle ).

6.3. The Criteria of Subarray Selection In this part, the complementary issues to SGAS in part B of Section 4 will be demonstrated in simulation experiments, and the conclusion that the criteria of subarray selection could be obtained is discussed. For subarray selection criteria, the computational time using a different number of columns in PM versus source’s frequency is first compared. The results in Figure 10 indicate that the computational time increases as columns are added, especially for the angle parameter estimation of

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coordinates of minimum value in (19). However, other combinations can obtain In this part, the complementary issues to SGAS in part B of Sectionof4 two will columns be demonstrated in precise parameter estimation. Under the space and , ambiguity resolution with the l  2 l  3 simulation experiments, and the conclusion that the criteria of subarray selection could be obtained combinations is discussed. of the same two columns or two columns whose space equal to l in Figure 11b–c are also ineffective forselection similar criteria, reasons,the and other combinations could performnumber well. Inofaddition, For subarray computational time using a different columns under in PM the space resolution with the of the same columns or and l  5is, ambiguity l  4 frequency versus source’s first compared. The results in combinations Figure 10 indicate that thetwo computational two columns spaceare equal to 2especially and 1, respectively, ineffective, which isofdifferent from the time increases whose as columns added, for the angleisparameter estimation a high-frequency above situations but corresponds to the theory in part B of Section 4. By observing, we can further source. It is clear that employing only two columns in PM has advantages over other selections in find that all the ineffective combinations columns in thetwo fivecolumns situationsfor areeffective computed by four terms of computational cost. Then, howoftotwo properly employ ambiguity CSS groupsneeds with to at be least a common group. resolution further researched.

Figure under different different number number of of columns columns in in PM. PM. Figure 10. 10. Computational Computational time time versus versus source’s source’s frequency frequency under

Considering that only two columns are used in PM with M = 12 and all space l of CSS groups are analyzed, the effectiveness of precise estimation with f = 3 GHz under different combinations of two columns is shown in Figure 11. It can be seen from Figure 11 that the abscissa and ordinate of squares show the number of employed column combination in PM and the black squares represent the combinations which are ineffective on the precise estimation of the source’s angles, while the other white squares represent the effective combinations. In Figure 11a, under the space l = 1 of CSS groups, it can be seen that the ambiguity resolution with the combination of same two columns or adjacent two columns in PM would be ineffective as SGAS has multiple coordinates of minimum value in (19). However, other combinations of two estimation. (a) (b) columns can obtain precise parameter (c) Under the space l = 2 and l = 3, ambiguity resolution with the combinations of the same two columns or two columns whose space equal to l in Figure 11b,c are also ineffective for similar reasons, and other combinations could perform well. In addition, under the space l = 4 and l = 5, ambiguity resolution with the combinations of the same two columns or two columns whose space equal to 2 and 1, respectively, is ineffective, which is different from the above situations but corresponds to the theory in part B of Section 4. By observing, we can further find that all the ineffective combinations of two columns in the five situations are computed by four CSS groups with at least a common group. As a result, it could be demonstrated that only two subarrays are enough to accomplish the source’s angle parameter estimation unambiguously, by avoiding the ineffective combinations. It can be concluded that two subarrays whose structure is similar to a UCA with four sensors, formed by different CSS groups, practical applications of SGAS. (d) would be the best selection in (e) Therefore, the least number of sensors employed by the SGAS technology is eight. Figure 11. Effectiveness of precise estimation versus selection of columns in PM. (a) (c) l  3 ; (d) l  4 ; (e) l  5 .

l  1 ; (b) l  2 ;

As a result, it could be demonstrated that only two subarrays are enough to accomplish the source’s angle parameter estimation unambiguously, by avoiding the ineffective combinations. It can be concluded that two subarrays whose structure is similar to a UCA with four sensors, formed by different CSS groups, would be the best selection in practical applications of SGAS. Therefore, the least number of sensors employed by the SGAS technology is eight.

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Figure 10. Computational time versus source’s frequency under different number of columns in PM.

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Figure 11. 11. Effectiveness Effectiveness of ofof columns in in PM. (a)(a)l l= 1 ;1;(b) Figure ofprecise preciseestimation estimationversus versusselection selection columns PM. (b)ll =22;; (c) l = 3; (d) l = 4; (e) l = 5.

6.4. Comparison of Source’s 3-Dl Parameter Estimation Performance between SGAS and SGAS-PBPE (c) l  3 ; (d) l  4 ; (e) 5.

In this part, we will carry out simulations by utilizing RMSEs to compare a source’s 3-D As a estimation result, could be demonstrated that only two subarrays are enough to accomplish the 6.4. Comparison of it Source’s 3-D Parameter Estimation Performance between SGAS and SGAS-PBPE parameter performance between SGAS and SGAS-PBPE. The other simulation conditions source’s angle parameter estimation unambiguously, by avoiding the ineffective combinations. 3 GHz , and are similar thewe experiment in part A exceptby that the source’s frequency is set at f 3-D In thisto part, will carry utilizing RMSEs to compare source’s parameter It can be concluded that out twosimulations subarrays whose structure is similar to a aUCA with four sensors, the SNR is varied frombetween 0 dB toSGAS 20 dB. examine the range estimate performance of estimation andTo SGAS-PBPE. Thesource’s otherin simulation are similar to formed byperformance different CSS groups, would be the best selection practicalconditions applications of SGAS. SGAS-PBPE, we compare it with a 1-D MUSIC estimator based on unambiguous source’s angle the experiment in part A except that theemployed source’s frequency is settechnology at f = 3 GHz, and the SNR is varied Therefore, the least number of sensors by the SGAS is eight. parameters SAGS. the It should berange pointed out that the searchofstep size of thewe 1-D MUSICit from 0 dB toobtained 20 dB. Toby examine source’s estimate performance SGAS-PBPE, compare estimator for the source’s range is .  r  0.01 m with a 1-D MUSIC estimator based on unambiguous source’s angle parameters obtained by SAGS. The RMSEs of the source’s parameters given Figure estimator 12a–c, respectively. As shown It should be pointed out that the3-D search step sizeare of the 1-DinMUSIC for the source’s rangeinis Figure 12a,b, ∆r = 0.01 m. it is clear that the RMSEs of the source’s angle parameter estimation decrease monotonically with and3-D theparameters SGAS-PBPEare method estimationAs accuracy The RMSEs of the the SNR, source’s given provides in Figure improved 12a–c, respectively. shown comparaed to theitSGAS method. Note that of Figure 12c, similar to parameter source’s angle estimation, the in Figure 12a,b, is clear that the RMSEs the source’s angle estimation decrease RMSE of source’s range estimation also decline as SNR increases, and the estimation performance of monotonically with the SNR, and the SGAS-PBPE method provides improved estimation accuracy SGAS-PBPE is analogous to the MUSIC estimator. comparaed to the SGAS method. Note that Figure 12c, similar to source’s angle estimation, the RMSE of As arange whole, simulation resultsasindicate that the performance ofofSGAS-PBPE source’s estimation also decline SNR increases, andestimation the estimation performance SGAS-PBPE outperforms the method of SGAS, which is consistent with the analysis given in Section 4. is analogous to the MUSIC estimator.

(a)

(b)

(c)

Figure Figure12. 12.RMSEs RMSEsof ofsource’s source’s3-D 3-Dparameter parameterestimation estimationversus versusSNR. SNR.(a) (a)Azimuth Azimuthangle; angle;(b) (b)Elevation Elevation angle; (c) Range. angle; (c) Range.

7. Conclusions In this paper, a novel ambiguity resolution algorithm is proposed for a source’s 3-D parameter estimation using UCA. Instead of collecting twice from different times, our approach utilizes subarray grouping and ambiguity searching by only one group of data, and so can adapt to

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As a whole, simulation results indicate that the estimation performance of SGAS-PBPE outperforms the method of SGAS, which is consistent with the analysis given in Section 4. 7. Conclusions In this paper, a novel ambiguity resolution algorithm is proposed for a source’s 3-D parameter estimation using UCA. Instead of collecting twice from different times, our approach utilizes subarray grouping and ambiguity searching by only one group of data, and so can adapt to fixed-frequency and frequency-hopping signals; which means it could be applied more broadly in practice than the rotary way to resolve ambiguity. Moreover, theoretical analysis and simulation experiments demonstrate the optimum structure of subarrays and subarray selection criteria, to make good use of the method. Simulation results show the effectiveness in ambiguity resolution and the satisfactory estimation performance of our approach. However, some drawbacks also exist in our proposed solution, including that the number of sensors is no less than eight and must be even. Acknowledgments: This work is supported in part by the National Natural Science Foundation of China under Grant 61271442, 61401481 and the Natural Science Foundation of Hunan under Grant 2017JJ3368. The authors wish to thank the anonymous reviewers for their valuable comments on improving this paper. Author Contributions: The main idea was proposed by Xin Chen and Zhen Liu; Xizhang Wei conceived and designed the experiments; Xin Chen wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.

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