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Research ArticleComputationally Efficient Ambiguity-Free Two-DimensionalDOA Estimation Method for Coprime Planar ArrayRD-Root-MUSIC Algorithm

Luo Chen123 Changbo Ye 12 and Baobao Li12

1Key Laboratory of Dynamic Cognitive System of Electromagnetic Spectrum Space (Nanjing University of Aeronautics andAstronautics) Ministry of Industry and Information Technology Nanjing 211106 China2College of Electronic and Information Engineering Nanjing University of Aeronautics and Astronautics Nanjing 211106 China3+e 28th Research Institute of China Electronics Technology Group Corporation Nanjing 210007 China

Correspondence should be addressed to Changbo Ye ycbnuaaeducn

Received 28 April 2020 Revised 31 May 2020 Accepted 18 June 2020 Published 10 July 2020

Guest Editor Junpeng Shi

Copyright copy 2020 Luo Chen et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

While the two-dimensional (2D) spectral peak search suffers from expensive computational burden in direction of arrival (DOA)estimation we propose a reduced-dimensional root-MUSIC (RD-Root-MUSIC) algorithm for 2D DOA estimation with coprimeplanar array (CPA) which is computationally efficient and ambiguity-free Different from the conventional 2D DOA estimationalgorithms based on subarray decomposition we exploit the received data of the two subarrays jointly by mapping CPA to the fullarray of the CPA (FCPA) which contributes to the enhanced degrees of freedom (DOFs) and improved estimation performanceIn addition due to the ambiguity-free characteristic of the FCPA the extra ambiguity elimination operation can be avoidedFurthermore we convert the 2D spectral search process into 1D polynomial rooting via reduced-dimension transformationwhich substantially reduces the computational complexity while preserving the estimation accuracy Finally numerical simu-lations demonstrate the superiority of the proposed algorithm

1 Introduction

Two-dimensional direction of arrival (2D DOA) estimationhas been extensively utilized in radar sonar wirelesscommunication and other fields [1ndash3] Numerous DOAestimation algorithms eg multiple signal classification(MUSIC) [4] estimation of signal parameters via rotationalinvariance techniques (ESPRIT) [5ndash7] propagator method(PM) [8] and PARAllel FACtor (PARAFAC) techniquehave been applied to various planar arrays such as uniformplanar arrays (UPAs) [9ndash11] L-shaped arrays [12 13]uniform circular arrays [5] and two parallel linear arrays[14] However the distance between adjacent elements inthese traditional arrays is limited to no larger than half-wavelength to avoid spatial aliasing which bring undesiredserious mutual coupling effect Besides since the DOA es-timation accuracy has positive correlation with the array

aperture the limited interelement spacing brings a negativeimpact on the estimation performance [15 16]

In recent years sparse arrays such as coprime arrays[15 17 18] nested arrays [19] and minimum redundancyarrays [20] have been proposed to tackle this issue As atypical sparse array the coprime arrays have inherent su-periorities over the conventional compact arrays includingenlarged array aperture increased DOFs and reducedmutual coupling [21] employed the traditional 2D-MUSICalgorithm to CPA by exploring the transformation relationbetween true and ambiguous estimates and further pro-posed a 2D partial spectrum search (2D-PSS) method [22]which considerably relieves the computational burden of 2Dtotal spectrum search (TSS) By combining the reduced-dimensional MUSIC (RD-MUSIC) [23] method with thePSS method the reduced-dimension transformation isperformed to further reduce complexity [24] A generalized

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 2794387 10 pageshttpsdoiorg10115520202794387

CPA array structure was designed in [25] based on themechanism of ambiguity elimination method which pro-vides a more flexible array layout and significant increase inDOFs e aforementioned methods [22 24 25] can becategorized as decomposition algorithms which process thereceived data of each subarray separately and then the es-timates are combined to determine the final DOAs whereasthe mutual information between the two subarrays is un-fortunately neglectede root-MUSICmethod to CPAwithlow complexity is applied [26] while the estimation per-formance of this cascade approach depends heavily on theinitial estimates especially at low SNRs An ambiguity-freeMUSIC (AF-MUSIC) method was proposed in [27] wherethe output of two subarrays were stacked and processedjointly to avoid the ambiguous problem Although the entireinformation of CPA is fully exploited the 2D spectrumsearch leads to heavy computational burden

In the above research studies for 2D DOA estimationmethods with CPA they either treat the two subarrays asindividual arrays which suffers performance degradationdue to the loss of mutual information or 2D spectrumsearch is required leading to expensive computationalcost or extra ambiguity elimination process is involvedTo address these issues we propose a computationallyefficient ambiguity-free algorithm via reduced-dimen-sional polynomial rooting technique Specifically we firstmap the CPA into full array of the CPA (FCPA) using anextraction matrix based on the characteristics of arrayconfiguration which enables the sufficient utilization ofthe entire received data of the CPA Meanwhile thenumber of achievable DOFs is enhanced benefiting fromthe utilization of full information as compared to theconventional decomposition algorithms Furthermore wetransform the 2D spectrum search into 2D polynomialroot-finding process and further perform reduced-di-mension transformation to convert the 2D root-findingoperation into two 1D one which substantially reducescomplementation complexity as well as the computationalburden In addition extra ambiguity elimination can beavoided owing to the inherent ambiguity-free charac-teristic of the FCPA

We summarize the major contributions of our workbelow

(1) We construct the FCPA corresponding to CPAwhich processes ambiguity-free characteristic andthereby extra ambiguity elimination operation canbe avoided

(2) We exploit the received data of the two subarraysjointly where improved DOA estimation perfor-mance as well as enhanced achievable DOFs can beachieved

(3) We propose a reduced-dimensional polynomialroot-finding algorithm with CPA for 2D DOA es-timation which transforms the 3D spectrum searchinto 1D polynomial rooting and hence reduces thecomplexity significantly while preserving the esti-mation accuracy

We outline this paper as follows Section 2 introduces thedata model of CPA and its corresponding FCPA e pro-posed algorithm is elaborated in Section 3 and we analyze thecomplexity and DOFs in Section 4 Section 5 provides sim-ulation results to corroborate the effectiveness of the proposedalgorithm and Section 6 concludes this paper

11 Notations Bold uppercase (lowercase) characters rep-resent matrices (vectors) (middot)T (middot)H (middot)minus 1 and (middot)lowast denotethe transpose conjugate transpose inverse and conjugateoperation respectively otimes and ⊙ are Kronecker product andKhatrindashRao product respectively Rank (middot) means the rankof the matrix angle(middot) represents the phase operator det(middot)

denotes the determinant of the matrix

2 Preliminaries

21 Data Model with CPA Assume that K far-field narrow-band uncorrelated signals impinge on the CPA with DOAs(θk ϕk) k 1 2 K where θk and ϕk are the elevationand azimuth angles of the kth signal respectively e CPAconsists of two uniform planar arrays with M1 times M1 and M2 times

M2 sensorse spacing between adjacent elements of subarray1 with M1 times M1 sensors is d1 M2λ2 and subarray 2 withM2 times M2 sensors has the interelement spacing d2 M1λ2whereM1 andM2 are coprime integers and λ is thewavelengthe total number of elements is T M2

1 + M22 minus 1 since the

two subarrays share the same element at the origin Define atransformation as uk sin θk sinϕk and vk sin θk cos ϕk forsimplification A CPA configuration is displayed in Figure 1 asan example where M1 2 M2 3 and T 12

For the ith (i 1 2) subarray the received signal can beexpressed by [22]

Xi AiS + Ni (1)

where S represents the source matrix andS [s1 s2 sK]T isin CKtimesL sk denotes source vector andsk [sk(1) sk(L)]T isin CLtimes1 L is the number of snap-shotsC represents a complex setAi is the steering matrix ofthe ith (i 1 2) subarray and Ai [ayi(u1)otimes axi(v1)

ayi(uK)otimes axi(vK)] isin CM2itimesK ayi(uk) and axi(vk) represent

the steering vectors along the y-axis and x-axis respectivelythe specific forms can be expressed asayi(uk) [1 ej2πdiukλ ej2π(Miminus 1)diukλ]T isin CMitimes1 andaxi(vk) [1 ej2πdivkλ ej2π(Miminus 1)divkλ]T isin CMitimes1 Ni iswhite Gaussian noise with mean value zero and variance σ2of the ith (i 1 2) subarray

e output of the whole CPA can be stacked as [27]

X X1

X21113890 1113891

A1

A21113890 1113891S +

N1

N21113890 1113891 AS + N (2)

where A represents the direction matrix of the whole CPAA [AT

1 AT2 ]T isin C(M2

1+M22)timesK and N denotes the white

Gaussian noise of the whole arrays andN [NT

1 NT2 ]T N1 isin CM2

1timesL N2 isin CM22timesL

2 Mathematical Problems in Engineering

In practice the covariance matrix of X can be calculatedusing L snapshots by

1113954R 1LXXH

(3)

Perform eigenvalue decomposition (EVD) and 1113954R can bedecomposed by

1113954R EsΛsEHs + EnΛnE

Hn (4)

where Λs is a diagonal matrix whose diagonal elements arethe K largest eigenvalues Λn is a diagonal matrix composedof the rest eigenvalues Es represents the signal subspacespanned of the eigenvectors corresponding to the K largesteigenvalues and En is the noise subspace composed of theremaining eigenvectors

22 Full Array of CPA and Extraction e CPA can beextracted from a large nonuniform planar array which canbe denoted by the full array of CPA (FCPA) e sensorlocation of the FCPA can be expressed as

ΩFCPA ΩxFCPAΩyFCPA1113966 1113967 (5)

where ΩxFCPA and ΩyFCPA represent the location sets of x-axis and y-axis respectively ΩxFCPA m1d1 cupm2d2ΩyFCPA m1d1 cupm2d2 0lem1 leM1 minus 1 0lem2 leM2 minus 1and m1 m2 isin Z Accordingly the number of elements inFCPA is TF (M1 + M2 minus 1)2

Figure 2 illustrates the FCPA corresponding to the CPAshown in Figure 1 where ΩxFCPA 0 2 3 4 d andΩyFCPA 0 2 3 4 d d λ2 It can be observed that theFCPA contains all elements of the CPA and has four ad-ditional elements with sensor number 7 10 12 and 13respectively which demonstrates that the CPA can beregarded as an extraction from FCPA

According to the correspondence of the CPA and FCPAwe introduce an extraction matrix G isin 0 1 to characterizethe mapping relation as

A GAF (6)

where G isin Z(M21+M2

2)times(M1+M2minus 1)2 represents the extractionmatrix Z denotes a set of integers AF is the steering matrixof the FCPA AF [aFy(u1)otimesaFx

(v1) aFy(uK)otimesaFx(vK)] isin C(M1+M2minus 1)2timesK aFy(uk) andaFx(vk) represent the steering vector of the FCPA along they-axis and x-axis the specific forms areaFy(uk) [1 ej2πdFy2ukλ ej2πdFy(M1+M2minus1)ukλ]T

aFx(vk) [1 ej2πdFx2vkλ ej2πdFx(M1+M2minus1)vkλ]T respectivelyand dFyi isin ΩyFCPA dFxi isin ΩxFCPA(1le ileM1 + M2 minus 1) de-note the location sets of elements on the y-axis and x-axisrespectively

To demonstrate the extraction more specifically eachelement in the CPA and FCPA is labeled according to theirorder in the steering vectors ie 1 sim M2

1 for the subarray 1and M2

1 + 1 sim M21 + M2

2 for the subarray 2 in CPA1 sim (M1 + M2 minus 1)2 for the FCPA If the ith sensor in theCPA and the jth sensor in the FCPA overlap then gij 1otherwise gij 0 where gij denotes the (i j)th element ofGFor the FCPA given in Figure 2 G is a 13 times 16 matrix with 3columns of all zeros

Definition 1 (extraction efficiency)e extraction efficiencyis the proportion of nonzero elements in the extractionmatrix

e sensor location of CPA can be given by

ΩCPA ΩxCPA 01113864 1113865cup 0ΩyCPA1113966 1113967 (7)

where ΩxCPA m1dcupm2d2 ΩyCPA m1d1cupm2d20lem1 leM1 minus 1 0lem2 leM2 minus 1 and m1 m2 isin Z en weconstruct a uniform planar array that has the same arrayaperture as the CPA with the location set

ΩUPA ΩxUPA 01113864 1113865cup 0ΩyUPA1113966 1113967 (8)

Z

X

YO

θ

φ

S(t)

M1d

Subarray 2 Full array

Subarray 1

15 3

4

6

7

8

9

10 13

11

12

2

M 2d

Figure 1 CPA configuration with M1 2 and M2 3

Z

YO

θ

φ1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16X

S(t)

M1d

M 2d

Figure 2 e FCPA configuration corresponding to CPA

Mathematical Problems in Engineering 3

where ΩxUPA md ΩyUPA md d λ20lemleM1(M2 minus 1) and m isin Z

e locations of the sensors in the FCPA correspondingto the CPA can be expressed as


where ΩxCPA subeΩxFCPA subeΩxUPA and ΩyCPA subeΩyFCPA subeΩyUPArepresent the location set of FCPA on the x- and y-axesrespectively

For the CPA study in this paper the correspondingFCPA can be constructed by the following four forms

ΩxFCPA 0 2 3 4 d

ΩyFCPA 0 2 3 4 d

ΩxFCPA_1 0 2 3 4 d

ΩyFCPA_1 0 1 2 3 4 d

ΩxFCPA_2 0 1 2 3 4 d

ΩyFCPA_2 0 2 3 4 d

ΩxFCPA_3 0 1 2 3 4 d

ΩyFCPA_3 0 1 2 3 4 d

(10)

According to Definition 1 the extraction efficiency of theabove schemes is 00625 005 005 and 004 respectively Itis clearly seen that the FCPA we designed in (5) has thehighest extraction efficiency

3 The Proposed Algorithm

31 2D-MUSIC Algorithm Derive from the orthogonalrelationship between the noise subspace and the steeringvector and the spectral function of CPA can be representedby [26 27]

P(u v) 1

aH(u v)EnEHn a(u v)

(11)

where a(u v) denotes the steering vector of CPAa(u v) [aT

1 (u v) aT2 (u v)]T isin C(M2

1+M22)times1 ai(u v)

ayi(u)otimes axi(v) ayi(u) [1 ej2πdiuλ ej2π(Miminus 1)diuλ]Taxi(v) [1 ej2πdivλ ej2π(Miminus 1)divλ]T (i 1 2) and En isthe noise subspace of CPA

According to (6) we have a(u v) GaF(u v) en (11)can be rewritten as [26 27]

P(u v) 1

aHF (u v)GHEnEH

n GaF(u v)

1aH

F (u v)EFnEHFnaF(u v)

(12)

where aF(u v) is the steering vector of the FCPAaF(u v) aFy(u)otimes aFx(v) isin C(M1+M2minus 1)2times1 aFy(u)

[1 ej2πdFy2uλ e j2πdFy(M1+M2minus1)uλ]T isin C(M1+M2minus 1)times1aFx(v) [1 ej2πdFx2vλ

ej2πdFx(M1+M2minus1)vλ]T isin C(M1+M2minus 1)times1 EFn denotes the noisespace of the FCPA and EFn GHEn

Although the autopaired 2D DOA estimates can beobtained via performing spectral search on (12) it suffersfrom tremendously expensive computational cost To tacklethis issue we first performed reduced-dimension transfor-mation and then exploited 1D polynomial root-findingtechnique to estimate u and v

32 Reduced-Dimensional Polynomial Root-Finding ProcessConstruct the polynomial based on (12) as

V(u v) aFy(u)otimes aFx(v)1113960 1113961HEFnE

HFn EFy(u)otimes aFx(v)1113960 1113961

aHFx(v) aFy(u)otimesIM1+M2minus 11113960 1113961

H

EFnEHFn aFy(u)otimesIM1+M2minus11113960 1113961aFx(v)

aHFx(v)Q(u)aFx(v)

(13)


HFn aFy(u)otimes aFx(v)1113960 1113961

aHFy(u) IM1+M2minus 1otimes aFx(v)1113960 1113961

H

EFnEHFn IM1+M2minus1otimes aFx(v)1113960 1113961aFy(u)

aHFy(u)Q(v)aFy(u)

(14)

where Q(u) [aFy(u)otimes IM1+M2minus 1]HEFnEH

Fn[aFy(u)

otimes IM1+M2minus1] and Q(v) [IM1+M2minus 1otimes aFx(v)]HEFnEHFn

[IM1+M2minus1otimesaFx(v)]According to the relation of rank of the matrix product

the following constraint

0ltRank EFnEHFn1113872 1113873leRank EFn( 1113857 (15)

has to be satisfied and then we have

det EFnEHFn1113966 1113967ne 0 (16)

We can conclude that det Q(u) is nonzero polynomialfrom (16) thusQ(u) is a factor of V(u v) AsQ(u) dependsonly on the variable u the roots of det Q(u) 0 can makethe following equation hold

V(u v) aHFx(v)Q(u)aFx(v) 0 (17)

It is noteworthy that only 1D polynomial is involved toachieve the estimates of u Similarly we can get the estimatesof v Consequently the problem of obtaining paired esti-mates of u and v from the 2D polynomial is transformed intotwo 1D root-finding process en we reconstruct (13) and(14) as


det Q(u) det aFy(u)otimes IM1+M2minus 11113960 1113961HEFnE

HFn aFy(u)otimes IM1+M2minus11113960 11139611113882 1113883 0 (18)

det Q(v) det IM1+M2minus 1otimes aFx(v)1113960 1113961HEFnE

HFn IM1+M2minus1otimes aFx(v)1113960 11139611113882 1113883 0 (19)

Define

z1 ej2πduλ

z2 ej2πdvλ

(20)

where d λ2Define the steering vector of UPA that has the same array

aperture as FCPA along the x-axis as aEx(v) [1 ej2πdvλ

ej2πmin M1 M2 (max M1 M2 minus 1)

dvλ]T isin C((min M1 M2 max M1 M2 minus 1)+1)times1 We assume thatM1 ltM2 for simplification and then aEx(v) can be expressedas aEx(v) [1 ej2πdvλ

ej2πM1(M2minus 1)dvλ]T isin C(M1(M2minus 1)+1)times1 Based on the

correspondence between FCPA and UPA with same arrayaperture we have

aFx(v) G1aEx(v) (21)

where G1 isin Z(M1+M2minus1)times(M1(M2minus1)+1) To be specific g1ij 1holds when the ith sensor in the aFx(v) and the jth sensor inthe aEx(v) overlap otherwise gij 0 where gij is the (i j)thelement of G1 For the FCPA displayed in Figure 2 G1 is a4 times 5matrix with one columns of all zeros Similarly we canobtain the relation aFy(u) G1aEy(u)

Correspondingly the steering vectors can be rewritten as

aFy(u) G1aEy(u) G1 1 ej2πduλ

ej2πM1 M2minus 1( )duλ

1113876 1113877T

G1 1 z1 zM1 M2minus1( )11113876 1113877

T

infinFy z1( 1113857 (22)

aFx(v) G1aEx(v) G1 1 ej2πdvλ

ej2πM1 M2minus 1( )dvλ

1113876 1113877T

infin1 1 z2 zM1 M2minus1( )21113876 1113877

T

aFx z2( 1113857 (23)

Without loss of generality substitutingz

M1(M2minus1)1 [aT

Fy(zminus11 )otimes IM1+M2minus 1]

H for [aFy(u)otimesIM1+M2minus 1]H

and substituting zM1(M2minus1)2 [IM1+M2minus 1otimes aT

Fx(zminus12 )]H for

[IM1+M2minus 1otimes aFx(v)]H ie

det Q z1( 11138571113864 1113865 det zM1 M2minus1( )1 aT

Fy zminus111113872 1113873otimes IM1+M2minus 11113960 1113961

HEnE

Hn aFy z

minus111113872 1113873otimes IM1+M2minus11113960 11139611113882 1113883 0 (24)

det Q z2( 11138571113864 1113865 det zM1 M2minus1( )2 IM1+M2minus 1otimes a

TFx z

minus121113872 11138731113960 1113961

HEnE

Hn IM1+M2minus1otimes aFx z

minus121113872 11138731113960 11139611113882 1113883 0 (25)

Considering that det Q(z1)1113864 1113865 and det Q(z2)1113864 1113865 are poly-nomials of even degree 1113954uk and 1113954vi can be obtained from the Kroots distributed closest to the unit circle corresponding to (24)and (25) and the roots are denoted by 1113954z11 1113954z1k 1113954z1K

and 1113954z21 1113954z2i 1113954z2K respectively ie

1113954uk angle 1113954z1k( 1113857λ

2πd1113888 1113889 k 1 K (26)

1113954vi angle 1113954z2i( 1113857λ

2πd1113888 1113889 i 1 K (27)

For the conventional DOA estimation methods withCPA the ambiguity elimination operation is required sincethe interelement spacing in the two subarrays is larger thanhalf-wavelength e FCPA is an unambiguous array whichhas at least one sensor pair with separation no larger thanhalf-wavelength according to (5) and we can obtain the true

DOA estimates directly after the parameter pairing withoutextra ambiguity elimination process

33 Parameter Pairing andDOAEstimation In this part wedetermine the pairing of 1113954uk and 1113954vi since the two root-findingprocedures are conducted separately Construct the costfunction for pairing as

Vki argi1K

min aH1113954uk 1113954vi( 1113857EnE

Hn a 1113954uk 1113954vi( 1113857

(k 1 K)

(28)

where a(1113954uk 1113954vi) represents the steering vector reconstructed1113954uk and 1113954vi which can be obtained according to (1)

For each 1113954uk we can obtain the value of i and k thatminimize Vki(1le ileK) and we define the paired index as iprimeand k Finally the 2D DOAs can be calculated by


1113954θk arcsin

1113954u2k + 1113954v2

iprime

1113969

1113874 1113875 1le kleK (29)

1113954ϕk arctan1113954uk

1113954viprime1113888 1113889 1le kleK (30)

where 1113954viprime is reconstructed by iprime(1le iprime leK)

34+eProcedure of the ProposedAlgorithm We summarizethe major steps of the proposed algorithm as follows

Step1 calculate 1113954R of the received data X and performEVD to obtain the noise space En

Step 2 reconstruct the spectral function P(u v)

according to (6)Step 3 construct the polynomial V(u v) and conductreduced-dimension transformation according to (12)Step 4 calculate 1113954uk and 1113954vi according to (26) and (27)Step 5 perform parameter matching to obtain 1113954θk and 1113954ϕk

according to (28)ndash(30)

4 Performance Analysis

41 Complexity Analysis Herein we compare the compu-tational complexity of the proposed algorithm 2D-PSS [22]RD-MUSIC [23] AF-MUSIC [27] and 2D-ROOT [26]methods in this section For the proposed algorithm cal-culating the covariance matrix requires O T2L1113864 1113865 and thecomplexity of eigenvalue decomposition is O T31113864 1113865 e root-finding operation costs O 2(2M1(M2 minus 1))31113966 1113967 and param-eter matching process requires O K2(T minus K)(T + 1)1113864 1113865Consequently the total complexity isO T2L + T3 + 2(2(M1(M2 minus 1)))3 + K2(T minus K)(T + 1)1113966 1113967

Table 1 lists the total complexity of above algorithmswhere Δ 00001 is the spectral search interval In additionFigure 3 displays the complexity comparison versus numberof sensors where K 2 and L 500 while the complexitycomparison with different number of snapshots is shown inFigure 4 where K 2 M1 2 andM2 3 It is clearly seenthat the proposed algorithm owns the approximate lowcomplexity to the 2D-ROOTmethod which is significantlylower than that of the 2D-PSS RD-MUSIC and AF-MUSICmethods as the spectral search process with heavy com-putational burden is transformed into computationally ef-ficient polynomial root-finding

42 CramerndashRao Bound In this part the derivation ofCramerndashRao Bound (CRB) of the 2D DOA estimation withCPA is given as the performance comparison metric

Define A A1A2

1113890 1113891 where A1 [ay1(θ1ϕ1) otimes ax1(θ1

ϕ1) ay1(θK ϕK)otimes ax1(θKϕK)] and A2 [ay2(θ1 ϕ1)otimes ax2(θ1 ϕ1) ay2(θKϕK)otimes ax2(θK ϕK)] According to[28] the CRB of CPA can be given by

CRB σ2

2LRe DHΠperpADoplus1113954P

T1113876 11138771113882 1113883

minus1 (31)

where D [za1zθ1 zaKzθK za1zϕ1 zaKzϕK]

ΠperpA IM21+M2

2minus1 minus A[AHA]minus1AH 1113954P 1113954Ps

1113954Ps1113954Ps

1113954Ps

1113890 1113891 1113954Ps SSHL

ak denotes the kth column of A and σ2 is the variance of thereceived noise

43 Achievable DOFs As for the conventional decompo-sition-based 2D DOA estimation methods such as RD-MUSIC 2D-PSS and other algorithms the number ofachievable DOFs is min M2

1 minus 1 M22 minus 11113864 1113865 According [26]

M21 + M2

2 minus 2 signals at most can be resolved by utilizingAF-MUSIC algorithm which processes the received data ofthe two subarrays jointly whereas extremely high com-putational complexity is involved due to the 2D spectralpeak search RD root-finding technique is employed in theproposed algorithm to deal with the complexity and theachievable DOFs can be obtained from (15) If u or v doesnot match any of the incident signals we have theconstraint

max M1 M21113864 1113865leRank EFnEHFn1113872 1113873leRank EFn( 1113857 (32)

whereRank (EFn) min (M1 + M2 minus 1)2 M2

1 + M22 minus 1 minus K1113966 1113967

Consequently the maximum number of signals which canbe identified by the proposed algorithm is

KleM21 + M

22 minus 1 minus max M1 M21113864 1113865 (33)

It is clear that the proposed algorithm can greatly im-prove the DOFs compared with the traditional 2D DOAestimation methods

44 Advantages Based on the above discussion the ad-vantages of the proposed algorithm can be listed as follows

(1) It can achieve superior estimation performance andhigher achievable DOFs than the decompositionmethods owing to the utilization of entire receiveddata Moreover additional ambiguity elimination isno longer required due to the ambiguity-free char-acteristic of the FCPA

(2) It outperforms the AF-MUSIC algorithm RD-MUSIC algorithm 2D-PSS algorithm and 2D-ROOT algorithm in estimation performance

(3) e proposed algorithm owns much lower compu-tational complexity than the RD-MUSIC 2D-PSSand AF-MUSIC methods which is approximately aslow as 2D-ROOT algorithm

5 Simulations

In this section we perform 500 Monte Carlo simulations tovalidate the performance of the proposed algorithm Assume


that K uncorrelated far-field narrowband signals impinge onthe CPA Define root mean square error (RMSE) by

RMSE 1K

1113944

K

k1

1500

1113944

500

i1

1113954ϕki minus ϕk1113872 11138732

+ 1113954θki minus θk1113872 11138732

11139741113972

(34)

where 1113954ϕki and 1113954θki are the estimates of the kth signal in the ithtrial corresponding to the true azimuth ϕk and elevation θkrespectively

51 Scatter Figures e scatter figures of the proposed algo-rithm under K sources are as follows whereK 6 SNR 30 dB L 1000 (θ1 θ2 θ3 θ4 θ5 θ6) (5∘15∘ 25∘ 35∘ 45∘ 55∘) and (ϕ1 ϕ2ϕ3 ϕ4ϕ5 ϕ6) (5∘ 15∘25∘ 35∘ 45∘ 55∘) e CPA is composed of two UPAs withM1 times M1 2 times 2 and M2 times M2 3 times 3 As illustrated inFigure 5 the proposed algorithm can effectively distinguish allsources incident on the CPA

52 RMSE Results Versus Snapshots In this part we presentthe DOA estimation performance of the proposed algorithmwith different number of snapshots in Figure 6 where(θ1 ϕ1) (20∘ 30∘) (θ2ϕ2) (40∘ 50∘) M1 times M1 2 times 2and M2 times M2 3 times 3 e result demonstrates that theestimation accuracy improves as the number of snapshotsincreases owing to the more accurate covariance

53 RMSE Results Versus Number of Sensors Herein weprovide the RMSE results of the proposed algorithm versusnumber of sensors in Figure 7 where (θ1 ϕ1) (20∘ 30∘)(θ2 ϕ2) (40∘ 50∘) and L 200 As the number of arrayelements increases the diversity gain of the receiving an-tenna increases It is illustrated clearly that the increasednumber of sensors leads to improved DOA estimationperformance

54 RMSE Comparison of Different Algorithms Figure 8exhibits the RMSE comparison of the proposed

Table 1 Complexity of different algorithms

Algorithm Complexity

Proposed O T2L + T3 + 2(2(M1(M2 minus 1))2)3 + K2(T minus K)(T + 1)1113966 1113967

2D-PSS O (M41 + M4

2)L + M61 + M6

2 + 4M21(M2

1 minus K)(M22Δ

2) + 4M22(M2

2 minus K)(M21Δ

2)1113864 1113865

RD-MUSIC O(M4

1 + M42)L + M6

1 + M62 + 2(2M3

1(M31 minus K) + M4

1 + M31)Δ

+2(2M32(M3

2 minus K) + M42 + M3

2Δ)1113896 1113897

2D-ROOT O(T2L + T3 + 8(M61 + KM3

1 + M62 + KM3

2))

AF-MUSIC O(T2L + T3 + 8T(T minus K)Δ2)

(2 3) (3 4) (3 5) (4 5)Number of elements (M1 M2)

104

106

108

1010

1012

1014

Com

plex

mul

tiplic

atio

n tim

es

Proposed2D-PSSRD-MUSIC

2D-ROOTAF-MUSIC

Figure 3 Complexity comparison versus number of elements

100 200 300 400 500 600 700 800Number of snapshots

Com

plex

mul

tiplic

atio

n tim

es

ProposedRD-MUSIC2D-PSS

2D-ROOTAF-MUSIC

104

106

108

1010

1012

Figure 4 Complexity comparison versus snapshots


algorithm 2D-ROOT algorithm 2D-PSS algorithm RD-MUSIC and AF-MUSIC algorithm where M1 times M1 2 times 2and M2 times M2 3 times 3 It is indicated explicitly in Figure 8that the proposed algorithm and AF-MUSIC algorithmbenefiting from the utilization of the received data of theentire CPA outperform the decomposition-based RD-MUSIC 2D-PSS and 2D-ROOT algorithms Furthermorethe proposed algorithm yields superior estimation perfor-mance to the AF-MUSIC algorithm as the proposed al-gorithm directly performs two root-finding operations basedon the spectral function (11) while the cascading process inthe AF-MUSIC algorithm may result in performancedegradation

6 Conclusion

In this paper we propose a computationally efficient 2DDOA estimation algorithm for CPA by exploiting the RDpolynomial root-finding technique e proposed algo-rithm first maps CPA into FCPA and exploits the receiveddata of two subarrays jointly where the mutual infor-mation loss is avoided and simultaneously the improvedestimation performance as well as enhanced DOFs can beachieved In particular the FCPA we constructed is anambiguity-free array with high extraction efficiencyFurthermore we convert the 2D total spectral search intoone 1D polynomial root-finding process via reduced-

10 20 30 40 50 60 70 80 90 100Test number

0

10

20

30

40

50

60El

evat

ion

estim

ates

(deg)

(a)

10 20 30 40 50 60 70 80 90 100Test number

0

10

20

30

40

50

60

Azi

mut

h es

timat

es (deg

)

(b)

Figure 5 (a) Elevation estimates of the proposed algorithm (b) Azimuth estimates of the proposed algorithm

0 10 15 20 25SNR (dB)

10ndash2

10ndash1

100

RMSE

(deg)

DOA estimation

L = 100L = 200

L = 300L = 400

5

Figure 6 RMSE performance of the proposed algorithm versus snapshots


dimension transformation which significantly reducesthe computational cost and simultaneously preserves theestimation accuracy Simulations demonstrates the su-periority of the proposed approach in regard to com-plexity achievable DOFs and DOA estimationperformance

Data Availability

edata used to support the findings of this study are availablefrom the corresponding author upon reasonable request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the China NSF (Grant nos61631020 61971218 61601167 and 61371169) Fund ofSonar Technology Key Laboratory (Research on the theoryand algorithm of signal processing for two-dimensionalunderwater acoustics coprime array) and Fund of SONAR

10ndash2

10ndash1

100

RMSE

(deg)

50 100 150 200 250 300 350 400 450 500Snapshots

Different algorithmsrsquo DOA estimation

2D-PSSProposed2D-ROOT

AF-MUSICRD-MUSICCRB

(a)

10ndash2

10ndash1

100

RMSE

(deg)

SNR (dB)



AF-MUSICRD-MUSICCRB

0 5 10 15 20 25

(b)

Figure 8 (a) RMSE comparison of different algorithms versus snapshots (b) RMSE comparison of different algorithms versus SNR

0 5 10 15 20 25SNR (dB)

10ndash3

10ndash2

10ndash1

100

RMSE

(deg)

DOA estimation

M = 2 N = 3M = 3 N = 4M = 3 N = 5

Figure 7 RMSE performance of the proposed algorithm versus number of sensors


Technology Key Laboratory (Range estimation and locationtechnology of passive target via multiple array combination)

References

[1] H Krim and M Viberg ldquoTwo decades of array signal pro-cessing research the parametric approachrdquo IEEE SignalProcessing Magazine vol 13 no 4 pp 67ndash94 1996

[2] L C Godara ldquoApplications of antenna arrays to mobilecommunications performance improvement feasibility andsystem considerationsrdquo Proceedings of the IEEE vol 85 no 7pp 1031ndash1060 1997

[3] L C Godara ldquoApplication of antenna arrays to mobilecommunications beam-forming and direction-of-arrivalconsiderationsrdquo Proceedings of the IEEE vol 85 no 8pp 1195ndash1245 1997

[4] M Wax T J Tie-Jun Shan and T Kailath ldquoSpatio-temporalspectral analysis by eigenstructure methodsrdquo IEEE Transac-tions on Acoustics Speech and Signal Processing vol 32 no 4pp 817ndash827 1984

[5] C P Mathews and M D Zoltowski ldquoEigen-structure tech-niques for 2-D angle estimation with uniform circular arraysrdquoIEEE Transactions on Signal Processing vol 42 no 9pp 3295ndash3306 1994

[6] H Saarnisaari ldquoTLS-ESPRIT in a time delay estimationrdquo inProceedings of IEEE Vehicular Technology Conferencepp 1619ndash1623 Phoenix AZ USA May 1997

[7] G Xu S D Silverstein R H Roy et al ldquoBeamspace espritrdquoIEEE Transactions on Signal Processing vol 42 no 2pp 349ndash356 1994

[8] N Tayem and H M Kwon ldquoL-shape 2-dimensional arrivalangle estimation with propagator methodrdquo IEEE Transactionson Antennas and Propagation vol 53 no 5 pp 1622ndash16302005

[9] W Zhang W Liu J Wang and S Wu ldquoComputationallyefficient 2-D doa estimation for uniform rectangular arraysrdquoMultidimensional Systems and Signal Processing vol 25 no 4pp 847ndash857 2014

[10] YM Chen J H Lee and C C Yeh ldquoTwo-dimensional angle-of-arrival estimation for uniform planar arrays with sensorposition errorsrdquo IEE Proceedings F Radar and Signal Pro-cessing vol 140 no 1 pp 37ndash42 1993

[11] P Heidenreich A M Zoubir and M Rubsamen ldquoJoint 2-Ddoa estimation and phase calibration for uniform rectangulararraysrdquo IEEE Transactions on Signal Processing vol 60 no 9pp 4683ndash4693 2012

[12] Y Hua T K Sarkar and D DWeiner ldquoAn l-shaped array forestimating 2-d directions of wave arrivalrdquo IEEE Transactionson Antennas and Propagation vol 39 no 2 pp 143ndash1461991

[13] S O Al-Jazzar D C Mclernon and M A Smadi ldquoSVD-based joint azimuthelevation estimation with automaticpairingrdquo Signal Processing vol 90 no 5 pp 1669ndash1675 2010

[14] H Chen C-P Hou Q Wang L Huang and W-Q YanldquoCumulants-based toeplitz matrices reconstruction methodfor 2-d coherent doa estimationrdquo IEEE Sensors Journalvol 14 no 8 pp 2824ndash2832 2014

[15] P P Vaidyanathan and P Pal ldquoSparse sensing with co-primesamplers and arraysrdquo IEEE Transactions on Signal Processingvol 59 no 2 pp 573ndash586 2011

[16] R Schmidt ldquoMultiple emitter location and signal parameterestimationrdquo IEEE Transactions on Antennas and Propagationvol 34 no 3 pp 276ndash280 1986

[17] P Pal and P P Vaidyanathan ldquoCoprime sampling and themusic algorithmrdquo in Proceedings of the 2011 Digital SignalProcessing and Signal Processing Education Meeting (DSPSPE) Sedona AZ USA January 2011

[18] S Qin Y D Zhang and M G Amin ldquoGeneralized coprimearray configurations for direction-of-arrival estimationrdquo IEEETransactions on Signal Processing vol 63 no 6 pp 1377ndash1390 2015

[19] P Pal and P P Vaidyanathan ldquoNested arrays a novel ap-proach to array processing with enhanced degrees of free-domrdquo IEEE Transactions on Signal Processing vol 58 no 8pp 4167ndash4181 2010

[20] A Moffet ldquoMinimum-redundancy linear arraysrdquo IEEETransactions on Antennas and Propagation vol 16 no 2pp 172ndash175 1968

[21] J Shi G Hu X Zhang F Sun and H Zhou ldquoSparsity-basedtwo-dimensional doa estimation for coprime array fromsum-difference coarray viewpointrdquo IEEE Transactions onSignal Processing vol 65 no 21 pp 5591ndash5604 2017

[22] Q Wu F Sun P Lan G Ding and X Zhang ldquoTwo-di-mensional direction-of-arrival estimation for co-prime planararrays a partial spectral search approachrdquo IEEE SensorsJournal vol 16 no 14 pp 5660ndash5670 2016

[23] X Zhang L Xu L Xu and D Xu ldquoDirection of departure(dod) and direction of arrival (doa) estimation in mimo radarwith reduced-dimension musicrdquo IEEE CommunicationsLetters vol 14 no 12 pp 1161ndash1163 2010

[24] W Zheng X Zhang and Z Shi ldquoTwo-dimensional directionof arrival estimation for coprime planar arrays via a com-putationally efficient one-dimensional partial spectral searchapproachrdquo IET Radar Sonar amp Navigation vol 11 no 10pp 1581ndash1588 2017

[25] W Zheng X Zhang and H Zhai ldquoGeneralized coprimeplanar array geometry for 2-d doa estimationrdquo IEEE Com-munications Letters vol 21 no 5 pp 1075ndash1078 2017

[26] D Zhang Y Zhang G Zheng et al ldquoTwo-dimensional di-rection of arrival estimation for coprime planar arrays viapolynomial root finding techniquerdquo IEEE Access vol 6 2018

[27] W Zheng X Zhang L Xu et al ldquoUnfolded coprime planararray for 2d direction of arrival estimation an aperture-augmented perspectiverdquo IEEE Access vol 6 pp 22744ndash227532018

[28] P Stoica and A Nehorai ldquoPerformance study of conditionaland unconditional direction-of-arrival estimationrdquo IEEETransactions on Acoustics Speech and Signal Processingvol 38 no 10 pp 1783ndash1795 1990


CPA array structure was designed in [25] based on themechanism of ambiguity elimination method which pro-vides a more flexible array layout and significant increase inDOFs e aforementioned methods [22 24 25] can becategorized as decomposition algorithms which process thereceived data of each subarray separately and then the es-timates are combined to determine the final DOAs whereasthe mutual information between the two subarrays is un-fortunately neglectede root-MUSICmethod to CPAwithlow complexity is applied [26] while the estimation per-formance of this cascade approach depends heavily on theinitial estimates especially at low SNRs An ambiguity-freeMUSIC (AF-MUSIC) method was proposed in [27] wherethe output of two subarrays were stacked and processedjointly to avoid the ambiguous problem Although the entireinformation of CPA is fully exploited the 2D spectrumsearch leads to heavy computational burden

In the above research studies for 2D DOA estimationmethods with CPA they either treat the two subarrays asindividual arrays which suffers performance degradationdue to the loss of mutual information or 2D spectrumsearch is required leading to expensive computationalcost or extra ambiguity elimination process is involvedTo address these issues we propose a computationallyefficient ambiguity-free algorithm via reduced-dimen-sional polynomial rooting technique Specifically we firstmap the CPA into full array of the CPA (FCPA) using anextraction matrix based on the characteristics of arrayconfiguration which enables the sufficient utilization ofthe entire received data of the CPA Meanwhile thenumber of achievable DOFs is enhanced benefiting fromthe utilization of full information as compared to theconventional decomposition algorithms Furthermore wetransform the 2D spectrum search into 2D polynomialroot-finding process and further perform reduced-di-mension transformation to convert the 2D root-findingoperation into two 1D one which substantially reducescomplementation complexity as well as the computationalburden In addition extra ambiguity elimination can beavoided owing to the inherent ambiguity-free charac-teristic of the FCPA

We summarize the major contributions of our workbelow

(1) We construct the FCPA corresponding to CPAwhich processes ambiguity-free characteristic andthereby extra ambiguity elimination operation canbe avoided

(2) We exploit the received data of the two subarraysjointly where improved DOA estimation perfor-mance as well as enhanced achievable DOFs can beachieved

(3) We propose a reduced-dimensional polynomialroot-finding algorithm with CPA for 2D DOA es-timation which transforms the 3D spectrum searchinto 1D polynomial rooting and hence reduces thecomplexity significantly while preserving the esti-mation accuracy

We outline this paper as follows Section 2 introduces thedata model of CPA and its corresponding FCPA e pro-posed algorithm is elaborated in Section 3 and we analyze thecomplexity and DOFs in Section 4 Section 5 provides sim-ulation results to corroborate the effectiveness of the proposedalgorithm and Section 6 concludes this paper

11 Notations Bold uppercase (lowercase) characters rep-resent matrices (vectors) (middot)T (middot)H (middot)minus 1 and (middot)lowast denotethe transpose conjugate transpose inverse and conjugateoperation respectively otimes and ⊙ are Kronecker product andKhatrindashRao product respectively Rank (middot) means the rankof the matrix angle(middot) represents the phase operator det(middot)

denotes the determinant of the matrix

2 Preliminaries

21 Data Model with CPA Assume that K far-field narrow-band uncorrelated signals impinge on the CPA with DOAs(θk ϕk) k 1 2 K where θk and ϕk are the elevationand azimuth angles of the kth signal respectively e CPAconsists of two uniform planar arrays with M1 times M1 and M2 times

M2 sensorse spacing between adjacent elements of subarray1 with M1 times M1 sensors is d1 M2λ2 and subarray 2 withM2 times M2 sensors has the interelement spacing d2 M1λ2whereM1 andM2 are coprime integers and λ is thewavelengthe total number of elements is T M2

1 + M22 minus 1 since the

two subarrays share the same element at the origin Define atransformation as uk sin θk sinϕk and vk sin θk cos ϕk forsimplification A CPA configuration is displayed in Figure 1 asan example where M1 2 M2 3 and T 12

For the ith (i 1 2) subarray the received signal can beexpressed by [22]

Xi AiS + Ni (1)

where S represents the source matrix andS [s1 s2 sK]T isin CKtimesL sk denotes source vector andsk [sk(1) sk(L)]T isin CLtimes1 L is the number of snap-shotsC represents a complex setAi is the steering matrix ofthe ith (i 1 2) subarray and Ai [ayi(u1)otimes axi(v1)

ayi(uK)otimes axi(vK)] isin CM2itimesK ayi(uk) and axi(vk) represent

the steering vectors along the y-axis and x-axis respectivelythe specific forms can be expressed asayi(uk) [1 ej2πdiukλ ej2π(Miminus 1)diukλ]T isin CMitimes1 andaxi(vk) [1 ej2πdivkλ ej2π(Miminus 1)divkλ]T isin CMitimes1 Ni iswhite Gaussian noise with mean value zero and variance σ2of the ith (i 1 2) subarray

e output of the whole CPA can be stacked as [27]

X X1

X21113890 1113891

A1

A21113890 1113891S +

N1

N21113890 1113891 AS + N (2)

where A represents the direction matrix of the whole CPAA [AT

1 AT2 ]T isin C(M2

1+M22)timesK and N denotes the white

Gaussian noise of the whole arrays andN [NT

1 NT2 ]T N1 isin CM2

1timesL N2 isin CM22timesL



1113954R 1LXXH

(3)



Hn (4)







A GAF (6)







1 + 1 sim M21 + M2







Z

X

YO

θ

φ

S(t)

M1d


Subarray 1

15 3

4

6

7

8

9

10 13

11

12

2

M 2d


Z

YO

θ

φ1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16X

S(t)

M1d

M 2d








ΩxFCPA 0 2 3 4 d

ΩyFCPA 0 2 3 4 d

ΩxFCPA_1 0 2 3 4 d

ΩyFCPA_1 0 1 2 3 4 d

ΩxFCPA_2 0 1 2 3 4 d

ΩyFCPA_2 0 2 3 4 d

ΩxFCPA_3 0 1 2 3 4 d

ΩyFCPA_3 0 1 2 3 4 d

(10)




P(u v) 1

aH(u v)EnEHn a(u v)

(11)






P(u v) 1

aHF (u v)GHEnEH

n GaF(u v)

1aH


(12)









H


aHFx(v)Q(u)aFx(v)

(13)




H


aHFy(u)Q(v)aFy(u)

(14)


Fn[aFy(u)






det EFnEHFn1113966 1113967ne 0 (16)









Define

z1 ej2πduλ

z2 ej2πdvλ

(20)












1113876 1113877T

G1 1 z1 zM1 M2minus1( )11113876 1113877

T

infinFy z1( 1113857 (22)



1113876 1113877T

infin1 1 z2 zM1 M2minus1( )21113876 1113877

T

aFx z2( 1113857 (23)


M1(M2minus1)1 [aT




Fx(zminus12 )]H for




HEnE

Hn aFy z



TFx z

minus121113872 11138731113960 1113961

HEnE


minus121113872 11138731113960 11139611113882 1113883 0 (25)



1113954uk angle 1113954z1k( 1113857λ

2πd1113888 1113889 k 1 K (26)

1113954vi angle 1113954z2i( 1113857λ

2πd1113888 1113889 i 1 K (27)




Vki argi1K

min aH1113954uk 1113954vi( 1113857EnE

Hn a 1113954uk 1113954vi( 1113857

(k 1 K)

(28)




1113954θk arcsin

1113954u2k + 1113954v2

iprime

1113969

1113874 1113875 1le kleK (29)


1113954viprime1113888 1113889 1le kleK (30)











Define A A1A2



CRB σ2


T1113876 11138771113882 1113883

minus1 (31)


ΠperpA IM21+M2


1113954Ps1113954Ps

1113954Ps

1113890 1113891 1113954Ps SSHL




M21 + M2




1 + M22 minus 1 minus K1113966 1113967


KleM21 + M







5 Simulations




RMSE 1K

1113944

K

k1

1500

1113944

500

i1

1113954ϕki minus ϕk1113872 11138732

+ 1113954θki minus θk1113872 11138732

11139741113972

(34)









2D-PSS O (M41 + M4

2)L + M61 + M6

2 + 4M21(M2

1 minus K)(M22Δ

2) + 4M22(M2

2 minus K)(M21Δ

2)1113864 1113865

RD-MUSIC O(M4

1 + M42)L + M6

1 + M62 + 2(2M3

1(M31 minus K) + M4

1 + M31)Δ

+2(2M32(M3

2 minus K) + M42 + M3

2Δ)1113896 1113897

2D-ROOT O(T2L + T3 + 8(M61 + KM3

1 + M62 + KM3

2))



104

106

108

1010

1012

1014

Com

plex

mul

tiplic

atio

n tim

es


2D-ROOTAF-MUSIC



Com

plex

mul

tiplic

atio

n tim

es


2D-ROOTAF-MUSIC

104

106

108

1010

1012




6 Conclusion


10 20 30 40 50 60 70 80 90 100Test number

0

10

20

30

40

50

60El

evat

ion

estim

ates

(deg)

(a)

10 20 30 40 50 60 70 80 90 100Test number

0

10

20

30

40

50

60

Azi

mut

h es

timat

es (deg

)

(b)


0 10 15 20 25SNR (dB)

10ndash2

10ndash1

100

RMSE

(deg)

DOA estimation

L = 100L = 200

L = 300L = 400

5




Data Availability




Acknowledgments


10ndash2

10ndash1

100

RMSE

(deg)

50 100 150 200 250 300 350 400 450 500Snapshots



AF-MUSICRD-MUSICCRB

(a)

10ndash2

10ndash1

100

RMSE

(deg)

SNR (dB)



AF-MUSICRD-MUSICCRB

0 5 10 15 20 25

(b)


0 5 10 15 20 25SNR (dB)

10ndash3

10ndash2

10ndash1

100

RMSE

(deg)

DOA estimation

M = 2 N = 3M = 3 N = 4M = 3 N = 5




References































1113954R 1LXXH

(3)



Hn (4)







A GAF (6)







1 + 1 sim M21 + M2







Z

X

YO

θ

φ

S(t)

M1d


Subarray 1

15 3

4

6

7

8

9

10 13

11

12

2

M 2d


Z

YO

θ

φ1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16X

S(t)

M1d

M 2d








ΩxFCPA 0 2 3 4 d

ΩyFCPA 0 2 3 4 d

ΩxFCPA_1 0 2 3 4 d

ΩyFCPA_1 0 1 2 3 4 d

ΩxFCPA_2 0 1 2 3 4 d

ΩyFCPA_2 0 2 3 4 d

ΩxFCPA_3 0 1 2 3 4 d

ΩyFCPA_3 0 1 2 3 4 d

(10)




P(u v) 1

aH(u v)EnEHn a(u v)

(11)






P(u v) 1

aHF (u v)GHEnEH

n GaF(u v)

1aH


(12)









H


aHFx(v)Q(u)aFx(v)

(13)




H


aHFy(u)Q(v)aFy(u)

(14)


Fn[aFy(u)






det EFnEHFn1113966 1113967ne 0 (16)









Define

z1 ej2πduλ

z2 ej2πdvλ

(20)












1113876 1113877T

G1 1 z1 zM1 M2minus1( )11113876 1113877

T

infinFy z1( 1113857 (22)



1113876 1113877T

infin1 1 z2 zM1 M2minus1( )21113876 1113877

T

aFx z2( 1113857 (23)


M1(M2minus1)1 [aT




Fx(zminus12 )]H for




HEnE

Hn aFy z



TFx z

minus121113872 11138731113960 1113961

HEnE


minus121113872 11138731113960 11139611113882 1113883 0 (25)



1113954uk angle 1113954z1k( 1113857λ

2πd1113888 1113889 k 1 K (26)

1113954vi angle 1113954z2i( 1113857λ

2πd1113888 1113889 i 1 K (27)




Vki argi1K

min aH1113954uk 1113954vi( 1113857EnE

Hn a 1113954uk 1113954vi( 1113857

(k 1 K)

(28)




1113954θk arcsin

1113954u2k + 1113954v2

iprime

1113969

1113874 1113875 1le kleK (29)


1113954viprime1113888 1113889 1le kleK (30)











Define A A1A2



CRB σ2


T1113876 11138771113882 1113883

minus1 (31)


ΠperpA IM21+M2


1113954Ps1113954Ps

1113954Ps

1113890 1113891 1113954Ps SSHL




M21 + M2




1 + M22 minus 1 minus K1113966 1113967


KleM21 + M







5 Simulations




RMSE 1K

1113944

K

k1

1500

1113944

500

i1

1113954ϕki minus ϕk1113872 11138732

+ 1113954θki minus θk1113872 11138732

11139741113972

(34)









2D-PSS O (M41 + M4

2)L + M61 + M6

2 + 4M21(M2

1 minus K)(M22Δ

2) + 4M22(M2

2 minus K)(M21Δ

2)1113864 1113865

RD-MUSIC O(M4

1 + M42)L + M6

1 + M62 + 2(2M3

1(M31 minus K) + M4

1 + M31)Δ

+2(2M32(M3

2 minus K) + M42 + M3

2Δ)1113896 1113897

2D-ROOT O(T2L + T3 + 8(M61 + KM3

1 + M62 + KM3

2))



104

106

108

1010

1012

1014

Com

plex

mul

tiplic

atio

n tim

es


2D-ROOTAF-MUSIC



Com

plex

mul

tiplic

atio

n tim

es


2D-ROOTAF-MUSIC

104

106

108

1010

1012




6 Conclusion


10 20 30 40 50 60 70 80 90 100Test number

0

10

20

30

40

50

60El

evat

ion

estim

ates

(deg)

(a)

10 20 30 40 50 60 70 80 90 100Test number

0

10

20

30

40

50

60

Azi

mut

h es

timat

es (deg

)

(b)


0 10 15 20 25SNR (dB)

10ndash2

10ndash1

100

RMSE

(deg)

DOA estimation

L = 100L = 200

L = 300L = 400

5




Data Availability




Acknowledgments


10ndash2

10ndash1

100

RMSE

(deg)

50 100 150 200 250 300 350 400 450 500Snapshots



AF-MUSICRD-MUSICCRB

(a)

10ndash2

10ndash1

100

RMSE

(deg)

SNR (dB)



AF-MUSICRD-MUSICCRB

0 5 10 15 20 25

(b)


0 5 10 15 20 25SNR (dB)

10ndash3

10ndash2

10ndash1

100

RMSE

(deg)

DOA estimation

M = 2 N = 3M = 3 N = 4M = 3 N = 5




References



































ΩxFCPA 0 2 3 4 d

ΩyFCPA 0 2 3 4 d

ΩxFCPA_1 0 2 3 4 d

ΩyFCPA_1 0 1 2 3 4 d

ΩxFCPA_2 0 1 2 3 4 d

ΩyFCPA_2 0 2 3 4 d

ΩxFCPA_3 0 1 2 3 4 d

ΩyFCPA_3 0 1 2 3 4 d

(10)




P(u v) 1

aH(u v)EnEHn a(u v)

(11)






P(u v) 1

aHF (u v)GHEnEH

n GaF(u v)

1aH


(12)









H


aHFx(v)Q(u)aFx(v)

(13)




H


aHFy(u)Q(v)aFy(u)

(14)


Fn[aFy(u)






det EFnEHFn1113966 1113967ne 0 (16)









Define

z1 ej2πduλ

z2 ej2πdvλ

(20)












1113876 1113877T

G1 1 z1 zM1 M2minus1( )11113876 1113877

T

infinFy z1( 1113857 (22)



1113876 1113877T

infin1 1 z2 zM1 M2minus1( )21113876 1113877

T

aFx z2( 1113857 (23)


M1(M2minus1)1 [aT




Fx(zminus12 )]H for




HEnE

Hn aFy z



TFx z

minus121113872 11138731113960 1113961

HEnE


minus121113872 11138731113960 11139611113882 1113883 0 (25)



1113954uk angle 1113954z1k( 1113857λ

2πd1113888 1113889 k 1 K (26)

1113954vi angle 1113954z2i( 1113857λ

2πd1113888 1113889 i 1 K (27)




Vki argi1K

min aH1113954uk 1113954vi( 1113857EnE

Hn a 1113954uk 1113954vi( 1113857

(k 1 K)

(28)




1113954θk arcsin

1113954u2k + 1113954v2

iprime

1113969

1113874 1113875 1le kleK (29)


1113954viprime1113888 1113889 1le kleK (30)











Define A A1A2



CRB σ2


T1113876 11138771113882 1113883

minus1 (31)


ΠperpA IM21+M2


1113954Ps1113954Ps

1113954Ps

1113890 1113891 1113954Ps SSHL




M21 + M2




1 + M22 minus 1 minus K1113966 1113967


KleM21 + M







5 Simulations




RMSE 1K

1113944

K

k1

1500

1113944

500

i1

1113954ϕki minus ϕk1113872 11138732

+ 1113954θki minus θk1113872 11138732

11139741113972

(34)









2D-PSS O (M41 + M4

2)L + M61 + M6

2 + 4M21(M2

1 minus K)(M22Δ

2) + 4M22(M2

2 minus K)(M21Δ

2)1113864 1113865

RD-MUSIC O(M4

1 + M42)L + M6

1 + M62 + 2(2M3

1(M31 minus K) + M4

1 + M31)Δ

+2(2M32(M3

2 minus K) + M42 + M3

2Δ)1113896 1113897

2D-ROOT O(T2L + T3 + 8(M61 + KM3

1 + M62 + KM3

2))



104

106

108

1010

1012

1014

Com

plex

mul

tiplic

atio

n tim

es


2D-ROOTAF-MUSIC



Com

plex

mul

tiplic

atio

n tim

es


2D-ROOTAF-MUSIC

104

106

108

1010

1012




6 Conclusion


10 20 30 40 50 60 70 80 90 100Test number

0

10

20

30

40

50

60El

evat

ion

estim

ates

(deg)

(a)

10 20 30 40 50 60 70 80 90 100Test number

0

10

20

30

40

50

60

Azi

mut

h es

timat

es (deg

)

(b)


0 10 15 20 25SNR (dB)

10ndash2

10ndash1

100

RMSE

(deg)

DOA estimation

L = 100L = 200

L = 300L = 400

5




Data Availability




Acknowledgments


10ndash2

10ndash1

100

RMSE

(deg)

50 100 150 200 250 300 350 400 450 500Snapshots



AF-MUSICRD-MUSICCRB

(a)

10ndash2

10ndash1

100

RMSE

(deg)

SNR (dB)



AF-MUSICRD-MUSICCRB

0 5 10 15 20 25

(b)


0 5 10 15 20 25SNR (dB)

10ndash3

10ndash2

10ndash1

100

RMSE

(deg)

DOA estimation

M = 2 N = 3M = 3 N = 4M = 3 N = 5




References


































Define

z1 ej2πduλ

z2 ej2πdvλ

(20)












1113876 1113877T

G1 1 z1 zM1 M2minus1( )11113876 1113877

T

infinFy z1( 1113857 (22)



1113876 1113877T

infin1 1 z2 zM1 M2minus1( )21113876 1113877

T

aFx z2( 1113857 (23)


M1(M2minus1)1 [aT




Fx(zminus12 )]H for




HEnE

Hn aFy z



TFx z

minus121113872 11138731113960 1113961

HEnE


minus121113872 11138731113960 11139611113882 1113883 0 (25)



1113954uk angle 1113954z1k( 1113857λ

2πd1113888 1113889 k 1 K (26)

1113954vi angle 1113954z2i( 1113857λ

2πd1113888 1113889 i 1 K (27)




Vki argi1K

min aH1113954uk 1113954vi( 1113857EnE

Hn a 1113954uk 1113954vi( 1113857

(k 1 K)

(28)




1113954θk arcsin

1113954u2k + 1113954v2

iprime

1113969

1113874 1113875 1le kleK (29)


1113954viprime1113888 1113889 1le kleK (30)











Define A A1A2



CRB σ2


T1113876 11138771113882 1113883

minus1 (31)


ΠperpA IM21+M2


1113954Ps1113954Ps

1113954Ps

1113890 1113891 1113954Ps SSHL




M21 + M2




1 + M22 minus 1 minus K1113966 1113967


KleM21 + M







5 Simulations




RMSE 1K

1113944

K

k1

1500

1113944

500

i1

1113954ϕki minus ϕk1113872 11138732

+ 1113954θki minus θk1113872 11138732

11139741113972

(34)









2D-PSS O (M41 + M4

2)L + M61 + M6

2 + 4M21(M2

1 minus K)(M22Δ

2) + 4M22(M2

2 minus K)(M21Δ

2)1113864 1113865

RD-MUSIC O(M4

1 + M42)L + M6

1 + M62 + 2(2M3

1(M31 minus K) + M4

1 + M31)Δ

+2(2M32(M3

2 minus K) + M42 + M3

2Δ)1113896 1113897

2D-ROOT O(T2L + T3 + 8(M61 + KM3

1 + M62 + KM3

2))



104

106

108

1010

1012

1014

Com

plex

mul

tiplic

atio

n tim

es


2D-ROOTAF-MUSIC



Com

plex

mul

tiplic

atio

n tim

es


2D-ROOTAF-MUSIC

104

106

108

1010

1012




6 Conclusion


10 20 30 40 50 60 70 80 90 100Test number

0

10

20

30

40

50

60El

evat

ion

estim

ates

(deg)

(a)

10 20 30 40 50 60 70 80 90 100Test number

0

10

20

30

40

50

60

Azi

mut

h es

timat

es (deg

)

(b)


0 10 15 20 25SNR (dB)

10ndash2

10ndash1

100

RMSE

(deg)

DOA estimation

L = 100L = 200

L = 300L = 400

5




Data Availability




Acknowledgments


10ndash2

10ndash1

100

RMSE

(deg)

50 100 150 200 250 300 350 400 450 500Snapshots



AF-MUSICRD-MUSICCRB

(a)

10ndash2

10ndash1

100

RMSE

(deg)

SNR (dB)



AF-MUSICRD-MUSICCRB

0 5 10 15 20 25

(b)


0 5 10 15 20 25SNR (dB)

10ndash3

10ndash2

10ndash1

100

RMSE

(deg)

DOA estimation

M = 2 N = 3M = 3 N = 4M = 3 N = 5




References






























1113954θk arcsin

1113954u2k + 1113954v2

iprime

1113969

1113874 1113875 1le kleK (29)


1113954viprime1113888 1113889 1le kleK (30)











Define A A1A2



CRB σ2


T1113876 11138771113882 1113883

minus1 (31)


ΠperpA IM21+M2


1113954Ps1113954Ps

1113954Ps

1113890 1113891 1113954Ps SSHL




M21 + M2




1 + M22 minus 1 minus K1113966 1113967


KleM21 + M







5 Simulations




RMSE 1K

1113944

K

k1

1500

1113944

500

i1

1113954ϕki minus ϕk1113872 11138732

+ 1113954θki minus θk1113872 11138732

11139741113972

(34)









2D-PSS O (M41 + M4

2)L + M61 + M6

2 + 4M21(M2

1 minus K)(M22Δ

2) + 4M22(M2

2 minus K)(M21Δ

2)1113864 1113865

RD-MUSIC O(M4

1 + M42)L + M6

1 + M62 + 2(2M3

1(M31 minus K) + M4

1 + M31)Δ

+2(2M32(M3

2 minus K) + M42 + M3

2Δ)1113896 1113897

2D-ROOT O(T2L + T3 + 8(M61 + KM3

1 + M62 + KM3

2))



104

106

108

1010

1012

1014

Com

plex

mul

tiplic

atio

n tim

es


2D-ROOTAF-MUSIC



Com

plex

mul

tiplic

atio

n tim

es


2D-ROOTAF-MUSIC

104

106

108

1010

1012




6 Conclusion


10 20 30 40 50 60 70 80 90 100Test number

0

10

20

30

40

50

60El

evat

ion

estim

ates

(deg)

(a)

10 20 30 40 50 60 70 80 90 100Test number

0

10

20

30

40

50

60

Azi

mut

h es

timat

es (deg

)

(b)


0 10 15 20 25SNR (dB)

10ndash2

10ndash1

100

RMSE

(deg)

DOA estimation

L = 100L = 200

L = 300L = 400

5




Data Availability




Acknowledgments


10ndash2

10ndash1

100

RMSE

(deg)

50 100 150 200 250 300 350 400 450 500Snapshots



AF-MUSICRD-MUSICCRB

(a)

10ndash2

10ndash1

100

RMSE

(deg)

SNR (dB)



AF-MUSICRD-MUSICCRB

0 5 10 15 20 25

(b)


0 5 10 15 20 25SNR (dB)

10ndash3

10ndash2

10ndash1

100

RMSE

(deg)

DOA estimation

M = 2 N = 3M = 3 N = 4M = 3 N = 5




References































RMSE 1K

1113944

K

k1

1500

1113944

500

i1

1113954ϕki minus ϕk1113872 11138732

+ 1113954θki minus θk1113872 11138732

11139741113972

(34)









2D-PSS O (M41 + M4

2)L + M61 + M6

2 + 4M21(M2

1 minus K)(M22Δ

2) + 4M22(M2

2 minus K)(M21Δ

2)1113864 1113865

RD-MUSIC O(M4

1 + M42)L + M6

1 + M62 + 2(2M3

1(M31 minus K) + M4

1 + M31)Δ

+2(2M32(M3

2 minus K) + M42 + M3

2Δ)1113896 1113897

2D-ROOT O(T2L + T3 + 8(M61 + KM3

1 + M62 + KM3

2))



104

106

108

1010

1012

1014

Com

plex

mul

tiplic

atio

n tim

es


2D-ROOTAF-MUSIC



Com

plex

mul

tiplic

atio

n tim

es


2D-ROOTAF-MUSIC

104

106

108

1010

1012




6 Conclusion


10 20 30 40 50 60 70 80 90 100Test number

0

10

20

30

40

50

60El

evat

ion

estim

ates

(deg)

(a)

10 20 30 40 50 60 70 80 90 100Test number

0

10

20

30

40

50

60

Azi

mut

h es

timat

es (deg

)

(b)


0 10 15 20 25SNR (dB)

10ndash2

10ndash1

100

RMSE

(deg)

DOA estimation

L = 100L = 200

L = 300L = 400

5




Data Availability




Acknowledgments


10ndash2

10ndash1

100

RMSE

(deg)

50 100 150 200 250 300 350 400 450 500Snapshots



AF-MUSICRD-MUSICCRB

(a)

10ndash2

10ndash1

100

RMSE

(deg)

SNR (dB)



AF-MUSICRD-MUSICCRB

0 5 10 15 20 25

(b)


0 5 10 15 20 25SNR (dB)

10ndash3

10ndash2

10ndash1

100

RMSE

(deg)

DOA estimation

M = 2 N = 3M = 3 N = 4M = 3 N = 5




References































6 Conclusion


10 20 30 40 50 60 70 80 90 100Test number

0

10

20

30

40

50

60El

evat

ion

estim

ates

(deg)

(a)

10 20 30 40 50 60 70 80 90 100Test number

0

10

20

30

40

50

60

Azi

mut

h es

timat

es (deg

)

(b)


0 10 15 20 25SNR (dB)

10ndash2

10ndash1

100

RMSE

(deg)

DOA estimation

L = 100L = 200

L = 300L = 400

5




Data Availability




Acknowledgments


10ndash2

10ndash1

100

RMSE

(deg)

50 100 150 200 250 300 350 400 450 500Snapshots



AF-MUSICRD-MUSICCRB

(a)

10ndash2

10ndash1

100

RMSE

(deg)

SNR (dB)



AF-MUSICRD-MUSICCRB

0 5 10 15 20 25

(b)


0 5 10 15 20 25SNR (dB)

10ndash3

10ndash2

10ndash1

100

RMSE

(deg)

DOA estimation

M = 2 N = 3M = 3 N = 4M = 3 N = 5




References































Data Availability




Acknowledgments


10ndash2

10ndash1

100

RMSE

(deg)

50 100 150 200 250 300 350 400 450 500Snapshots



AF-MUSICRD-MUSICCRB

(a)

10ndash2

10ndash1

100

RMSE

(deg)

SNR (dB)



AF-MUSICRD-MUSICCRB

0 5 10 15 20 25

(b)


0 5 10 15 20 25SNR (dB)

10ndash3

10ndash2

10ndash1

100

RMSE

(deg)

DOA estimation

M = 2 N = 3M = 3 N = 4M = 3 N = 5




References































References






























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