Two-Dimensional Direction-of-Arrival Estimation of...

Research ArticleTwo-Dimensional Direction-of-Arrival Estimation ofNoncircular Signals in Coprime Planar Array with HighDegree of Freedom

Haiyun Xu DamingWang Size Lin Bin Ba and Yankui Zhang

National Digital System Engineering and Technological Research R amp D Center Zhengzhou 450001 China

Correspondence should be addressed to Bin Ba xidianbabin163com

Received 26 July 2018 Accepted 23 December 2018 Published 2 January 2019

Academic Editor Raffaele Solimene

Copyright copy 2019 Haiyun Xu et al This 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

In estimating the two-dimensional (2D) direction-of-arrival (DOA) using a coprime planar array there are problems of the limiteddegree of freedom (DOF) and high complexity caused by the spectral peak search We utilize the time-domain characteristicsof signals and present a high DOF algorithm with low complexity based on the noncircular signals The paper first analyzes thecovariance matrix and ellipse covariance matrix of the received signals vectorizes these matrices and then constructs the receiveddata of a virtual uniform rectangular array (URA) 2D spatial smoothing processing is applied to calculate the covariance of thevirtual URA Finally the paper presents an algorithm using 2D multiple signal classification and an improved algorithm usingunitary estimating signal parameters via rotational invariance techniqueswhere the latter solves the closed-form solutions ofDOAsreplacing the spectral peak search to reduce the complexityThe simulation experiments demonstrate that the proposed algorithmsobtain the high DOF and enable to estimate the underdetermined signals Furthermore both two proposed algorithms can acquirethe high accuracy

1 Introduction

Direction-of-arrival (DOA) is a significant research field inmany applications such as radar [1] underwater acoustics[2 3] and indoor navigation At present the uniform andnonsparse arrays are widely applied including uniform linearrays (ULAs) [4] uniform rectangular arrays (URAs) [5]and uniform L-shaped arrays [6] The array spacing of thoseis set to no more than half the wavelength of the impingingsignals We then can obtain the accurate parameters usingthe high-resolution algorithms such as multiple signal clas-sification (MUSIC) [7] root-MUSIC [8] estimating signalparameters via rotational invariance techniques (ESPRIT)[9] and propagatormethod (PM) [10]With the developmentof technology the nonsparse array generally needs moresensors to expand array aperture to meet the requirement formore precise location determination However this process-ing makes the systems more complicated and enhances theantenna mutual coupling interference which increases theestimation errors

To acquire both high array degree of freedom (DOF)and precision researchers have begun to examine the signalfeatures A method proposed in [11] constructs a virtual arrayin the frequency domain combined with spacing featuresbased on orthogonal frequency division multiplexing sys-tems Thus that method not only improves the accuracy ofestimated parameters but also estimates underdeterminedsignals (no less than the number of sensors) Furthermorethe sparse and nonuniform array structure has become afocus The nested array in [12] consists of a uniform andsparse array and a uniform and nonsparse array of whichDOF and precision are high However the nested arraycannot avoid the mutual coupling interference among thesensors of the nonsparse array Thus a one-dimensionalcoprime array consisting of two uniform sparse line arrays isintroduced in [13] This weakens the mutual coupling inter-ference and simultaneously constructs a larger aperture withfewer sensors when compared with uniform and nonsparsearrays Moreover the two-dimensional (2D) coprime planararrays are proposed in [14] which can estimate both azimuth

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 3078376 10 pageshttpsdoiorg10115520193078376

2 Mathematical Problems in Engineering

and elevation angles A partial spectral search (PSS) methodis introduced to reduce the complexity caused by 2D-MUSICmentioned in [14] but PSS is still based on peak search whichdoes not significantly reduce the complexity Moreover theDOF is limited by the number of subarray sensors We haveproposed a method in [15] which uses the covariance matrixof coprime planar array to estimate a new covariance matrixwith matrix completion This processing has much improvedDOF bigger than the number of sensors However matrixcompletion can introduce the additional errors so that itsprecision is limited and it costs much complexity All themethods mentioned above do not take some signal featuresinto consideration

There are many kinds of noncircular signals in moderncommunication systems with such as BPSK ASK and AMmodulation The methods which construct extended virtualarrays and improve the accuracy of estimated parametersbased on noncircular signals are proposed in [16 17] Con-sidering the problems that existing methods for estimatingDOAs of general uncorrelated signals using coprime planararrays have huge computational complexity and low DOFwe present an algorithm to estimate 2D DOAs of underde-termined noncircular signals with low complexity The algo-rithm vectorizes the covariance matrix and ellipse covariancematrix of received signals to construct the new received dataof a virtual URA Combining 2D spatial smoothing process-ing and Unitary-ESPRIT we then realize the fast estimationof 2D DOAs Through constructing a virtual URA with morenumber of sensors than that of coprime planar array weimprove the DOF Moreover Unitary-ESPRIT can solve theclosed-form solutions of DOAs replacing the spectral peaksearch to reduce the complexity

The remainder of this paper is arranged as followsSection 2 introduces the model of coprime planar array andSection 3 describes the steps of the algorithm Sections 4 and5 analyze the computational complexity and performance ofthe model respectively to demonstrate the validity of thisalgorithm Section 6 gives the conclusion to this paper

The notations used in this paper are as follows (∙)119879 (∙)lowastand (∙)119867 respectively represent the transposition conjuga-tion and conjugate transposition 119864(∙) denotes the math-ematical expectation diag(∙) expresses the transformationof a vector to a diagonal matrix otimes ∘ and (∙)+ denote theKronecker product Khatri-Rao product and pseudoinverseoperator respectively

2 System Model

Considering the coprime planar array model the arraygeometry is shown in Figure 1 The coprime planar arrayconsists of two URAs Subarray 1 has 1198721 times 1198721 sensors andsubarray 2 has 1198722 times 1198722 sensors where 1198721 and 1198722 arethe coprime integers (generally assuming 1198721 lt 1198722) anddenote the sensor numbers on the 119909 119910 axis Correspondinglythe distance between the two adjacent sensors is 11987221205822 and11987211205822 respectively where the 120582 represents the wavelength ofthe impinging signalsThe subarrays coincide at the origin sothe total number of sensors is 119872 = 11987221 + 11987222 minus 1 Supposethere are 119870 uncorrelated narrowband far-field noncircular

3 2

2

2

k

k

Figure 1 Geometry of coprime planar arraywhen1198721 = 2 and1198722 =3signals impinging on the array with power 12059021 12059022 1205902119870The 119896th signal is located at elevation angle 120579119896 which isdownward from the z-axis and azimuth angle 120593119896 which iscounterclockwise from the x-axis

We define 1198631 and 1198632 as the location set of the subarrays1198631 = 11987111205822 and 1198632 = 11987121205822 where 1198711 = (119898 119899)1198722 0 le119898 119899 le 1198721 minus 1 and 1198712 = (119898 119899)1198721 0 le 119898 119899 le 1198722 minus1 Hence the location set of the coprime planar array isexpressed as 119863 = 1198631 cup 1198632 and we have 119871 = 1198711 cup 1198712 Thereceived signals at the array can be represented as

X (119905) = A (120593 120579) S (119905) +N (119905) (1)

The array manifold is

A (120593 120579) = [a (1205931 1205791) a (1205932 1205792) sdot sdot sdot a (120593119870 120579119870)] (2)

where

a (120593119896 120579119896) = [1198861 (120593119896 120579119896) 1198862 (120593119896 120579119896) sdot sdot sdot 119886119872 (120593119896 120579119896)]119879 (3)

119886119898 (120593119896 120579119896) = 119890minus119895120587 sin 120579119896(119909119898 cos120593119896+119910119898 sin120593119896) (119909119898 119910119898) isin 119871 (4)

The noncircular signal data vector is

S (119905) = [1199041 (119905) 1199042 (119905) sdot sdot sdot 119904119870 (119905)]119879 (5)

where 119905 = 1 2 119869 is the sampling time and 119869 is the numberof snapshots And the noise vector is

N (119905) = [1198991 (119905) 1198992 (119905) sdot sdot sdot 119899119872 (119905)]119879 (6)

where the elements are usually Gaussian random variableswith zero means and variance 1205902119899

Mathematical Problems in Engineering 3

virtual sensorsreal sensors

minus3 minus2 minus1 0 1 2 3 4minus4x

minus4

minus3

minus2

minus1

0

1

2

3

4

y

Figure 2 The location of virtual sensors and real sensors whensignals are circular and1198721 = 21198722 = 3

Considering (1) the covariance matrix of the receivedsignals is defined as

RX = 119864 [XX119867] = ARSA119867 + 1205902119899I119872 (7)

where RS = 119864[SS119867] = diag(12059021 12059021 1205902119870) Vectorize thecovariance matrix in (7) as

Z1 = V119890119888 (RX) = B1p + 12059021198991119899 (8)

where p = [12059021 12059022 1205902119870]119879 1119899 = V119890119888(I119872) andB1 = [b1 (1205931 1205791) sdot sdot sdot b1 (120593119870 120579119870)]= [alowast (1205931 1205791) otimes a (1205931 1205791) sdot sdot sdot alowast (120593119870 120579119870) otimes a (120593119870 120579119870)]= Alowast ∘ A

(9)

Define the new set as = (119909119898 119910119898) minus (119909119899 119910119899) |(119909119898 119910119898) (119909119899 119910119899) isin 119871Thus the any element of 1198871(120593119896 120579119896) canbe expressed as 119890minus119895120587 sin 120579119896(119909119898 cos120593119896+119910119898 sin120593119896) (119909119898 119910119898) isin Z1 canbe denoted as the received data of a virtual array containingone snapshot whose manifold matrix is defined as B1 And119871 and represent the location set of real array and virtualarray respectively As shown in Figure 2 the virtual array isnot a completed URA because there are holes We can use theconsecutive uniform and nonsparse virtual array to resolveDOAs which avoids the ambiguous results and enhances theDOF but this method loses much array aperture and has notsignificant improvement on DOF

3 DOA Estimation for Noncircular Signals

31 Construct Extended Virtual Array In our paper weassume the received signals as noncircular signals of max-imum noncircular rate [16] The ellipse covariance matrixof noncircular signals is nonzeros where we can obtain theadditional information besides the covariance matrix Hence


minus8

minus6

minus4

minus2

0

2

4

6

8

y


Figure 3 The location of virtual sensors and real sensors whensignals are noncircular and1198721 = 21198722 = 3we calculate the ellipse covariancematrices of received signalsin (1) given by

R1015840X = 119864 [XXT] = AR1015840SA119879 (10)

R10158401015840X = 119864 [XlowastX119867] = AlowastR10158401015840SA119867 (11)

where the ellipse covariance of noise is 0 because noise iscircular R1015840S = 119864[SS119879] and R10158401015840S = 119864[SlowastS119867] And thenoncircular signals S = Slowast so R1015840S = R10158401015840S = RS Vectorizethe ellipse covariance matrix as

Z2 = V119890119888 (R1015840X) = B2p (12)

Z3 = V119890119888 (R10158401015840119883) = B3p (13)

where B2 = A ∘ A and B3 = Alowast ∘ Alowast Thus Z2 and Z3 can bedenoted as the received data of virtual arrays whose manifoldmatrices are defined as B2 and B3 respectively Furthermorecombining the received data of virtual arrays the receiveddata of an extended virtual array is expressed as

Z = [Z1198791 ZT2 ZT3 ]T = Bp + 1205902119899 1119899 (14)

where B = [B1198791 B1198792 B1198793 ]119879 and 1119899 is the noise vector Defineanother two sets 1006704119871 = (119909119898 119910119898)+(119909119899 119910119899) | (119909119898 119910119898) (119909119899 119910119899) isin119871 and S119871= -(119909119898 119910119898) - (119909119899 119910119899) | (119909119898 119910119898) (119909119899 119910119899) isin 119871Hence the location of the extended virtual array is denotedas cup 1006704119871 cup S119871 and the values are shown in Figure 3 Theextended virtual array has the biggest URA size as 119881 times 119881 Asshown in Table 1 using noncircular signals has increased thearray aperture of virtual URAThe received data of the virtualURA after removing the repeated rows is given by

Z = Bp + 1205902119899e (15)


Table 1 Size of virtual URA

Sensors without non-circular signals with non-circular signals1198721 = 21198722 = 3 119881 = 6 119881 = 91198721 = 31198722 = 4 119881 = 8 119881 = 131198721 = 21198722 = 5 119881 = 8 119881 = 131198721 = 31198722 = 5 119881 = 10 119881 = 151198721 = 41198722 = 5 119881 = 12 119881 = 17

111VminusV+1

VminusV+11

V

middot middot middot








Figure 4 Spatial smoothing processing schemewhen = (119881+1)2where B is the manifold matrix of size1198812times119870 and e is a vectorof all zeros except a 1 corresponding to the virtual sensor at(0 0) In practice because of the limited number of snapshotsthe covariance matrix is usually estimated as

RX = 1119869XX119867 (16)

Similarly the ellipse covariance matrices are R1015840X and R10158401015840XHence the real received data of virtual URA is denoted as

Z = Bp + 1205902119899 e (17)

where p and e is the real impinging signals and noise of virtualarray

32 2D Spatial Smoothing Processing We apply the 2D spatialsmoothing processing to received data of the virtual URAThe detailed smoothing scheme is presented in Figure 4

We first assume the smoothing subarray of size timesThereceived data Z11 of the 2 sensors in the upper right cornerof the virtual URA can be expressed as

Z11 = B11p + e11 (18)

where B11 is the subarray manifold matrix of size 2 times119870 andgiven by

B11 = [b11 (1205931 1205791) sdot sdot sdot b11 (120593119870 120579119870)] (19)

Define the set 11987111 = (119909 119910) | (119881 minus 1)2 minus + 1 le 119909 119910 le (119881minus1)2Thus the any element of b11(120593119896 120579119896) can be expressed as119890minus119895120587 sin 120579119896(119909119898 cos120593119896+119910119898 sin120593119896) (119909119898 119910119898) isin 11987111 There are 119881minus + 1overlapping smoothing subarray from upper right corner toupper left corner The received data of the 2 sensors in theupper left corner Z1119881minus+1 can be given by

Z1119881minus+1 = B11D119881minuscol p + e1119881minus+1 (20)

where Dcol = diag (119890minus119895120587 sin 1205791 cos1205931 sdot sdot sdot 119890minus119895120587 sin 120579119870 cos120593119870) Sim-ilarly there are 119881 minus + 1 overlapping smoothing subarrayfrom upper right corner to lower right corner The receiveddata of the 2 sensors in the lower right corner Z119881minus+11 canbe given by

Z119881minus+11 = B11D119881minusrow p + e119881minus+11 (21)

where Drow = diag (119890minus119895120587 sin 1205791 sin1205931 sdot sdot sdot 119890minus119895120587 sin 120579119870 sin120593119870) Thusas for a 119881 times 119881 virtual URA there are total (119881 minus +1)2 overlapping subarrays Hence the smoothing covariancematrix can be calculated by averaging over the correspondingcovariance matrix of the (119881 minus + 1)2 subarrays to obtain

RZ = 1(119881 minus + 1)2119881minus+1sum119903=1

119881minus+1sum119888=1

Z119903119888Z119867

119903119888 (22)

Given (20) and (21) (22) can be rewritten as

RZ = 1(119881 minus + 1)2

sdot 119881minus+1sum119903=1

B1199031(119881minus+1sum119888=1

D119888minus1col Rp (D119888minus1col )H) BH1199031

+ 1(119881 minus + 1)2

119881minus+1sum119903=1

119881minus+1sum119888=1

e119903119888eH119903119888

= 1(119881 minus + 1)2

sdot B11(119881minus+1sum119903=1

119881minus+1sum119888=1

D119903minus1rowD119888minus1col Rp (D119888minus1col )H


sdot (D119903minus1row)H) BH11 + 1

(119881 minus + 1)2sdot 119881minus+1sum119903=1

119881minus+1sum119888=1

e119903119888eH119903119888 = 1

(119881 minus + 1)2 B11RpBH11

+ 1(V minus + 1)2

119881minus+1sum119903=1

119881minus+1sum119888=1

e119903119888eH119903119888

(23)

where Rp = pp119867 and rank(Rp) = 1 Considering the proof in[18] when (119881 minus + 1)2 gt 119870 rank(Rp) = 119870 which satisfiesthe requirement of using subspace algorithms

33 2D DOA Estimation According to [19] we apply 2D-MUSIC to the smoothing covariance matrix to estimate 2Dangle Take the eigenvalue decomposition of RZ and obtainthe noise subspace U119867119899 The spatial spectrum function is givenby

119875 (120593 120579) = 100381710038171003817100381710038171003817U119867119899 b11 (120593 120579)100381710038171003817100381710038171003817minus22 (24)

Using 2D-MUSIC can estimate the angles but the spectralpeak search costs much complexity Considering this wepresent an improved algorithm which uses Unitary-ESPRIT[20 21] to replace 2D-MUSIC Unitary-ESPRIT transformsthe complex matrix into real matrix and solves the closed-form solutions of DOAs to reduce the complexity

First define the inverse matrix Π119896 whose counter-diagonal values are 1 while the others are 0 as (Π119896)2 = I119896The unitary matrix is defined to satisfy

Π119896Qlowast = Q (25)

Moreover the matrix can be expressed as

Q2119896 = 1radic2 [ I119896 119895I119896Π119896 minus119895Π119896] (26)

Q2119896+1 = 1radic2 [[[[I119896 O 119895I119896O119879 radic2 O119879

Π119896 O minus119895Π119896]]]]

(27)

We then define the selection matrix as J1 = [I O] and J2 =[O I] and have

K1 = Re Q119867minus1

J2Q119867 (28)

K2 = Im Q119867minus1

J2Q119867 (29)

where Re[∙] and Im[∙] denote as the real part and imaginarypart of the matrix or vector respectively Therefore we define

K1205831 = I otimes K1 (30)

K1205832 = I otimes K2 (31)

K]1 = K1 otimes I (32)


Next we transform RZ to give

R119879 = Re (Q119867

otimesQ119867) RZ (Q119867

otimesQ119867)119867 (34)

where R119879 is a real matrix Taking eigenvalue decompositionof this realmatrix can cost less complexity than RZ and obtainthe noise subspace U119878 At last we combine the last two stepand have

Ψ120583 = (K1205831U119878)+K1205832U119878 (35)

Ψ] = (K]1U119878)+K]2U119878 (36)

Define Ψ = Ψ120583 + 119895Ψ] and calculate its eigenvalue vector 120582We can acquire the estimation

119894 = arcsin(radic(120583119894120587 )2 + (]119894120587 )2) (37)

119894 = arctan ( ]119894120583119894) (38)

where (119894 119894)(119894 = 1 119870) is the estimated value 120583119894 =2 arctan(Re(120582119894)) and ]119894 = 2 arctan(Im(120582119894)) Equations (37)and (38) give the closed-form solutions of DOAs thusavoiding the spectral peak search

34 Algorithm Step Conclusion The main steps of the pro-posed algorithm can be summarized as follows

Step 1 Calculate the received signalsrsquo covariance matrix RX

and ellipse covariance matrices R1015840X R10158401015840X

Step 2 Vectorize those covariance matrices and construct thereceived data of the virtual URA

Step 3 Apply 2D spatial smoothing processing and calculatethe smoothing covariance matrix R119885 via (22)

Step 4 Use Unitary-ESPRIT to estimate (119894 119894) (119894 = 1 119870)4 DOF and ComputationalComplexity Analysis

41 DOFAnalysis TheDOF determines themaximum num-ber of signals that we can estimate directly Compared withthe coprime planar array the URA with the same numberof sensors has much smaller array aperture and DOF is nomore than the 119872 minus 1 but the algorithm in [22] enhancesthe DOF to 2119872 minus 1 based on the noncircular signals TheDOF of PSS presented in [14] is depended on 1198721 As for


URAPSSProposed

20 25 30 35 4015M

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90

DO

F

Figure 5 DOF versus119872

two proposed algorithms the smoothing covariance matrixenables us to resolve DOA estimation of maximum 119870 =(119881 + 1)24 minus 1 signals when = (119881 + 1)2 The detailedvalues are presented in Figure 5 The DOF of PSS is limitedby the number of subarray sensors so it is the lowest Theproposed algorithm has much improved the DOF morethan the number of sensors which means it can estimateunderdetermined signals

42 Complexity Analysis We analyze the computationalcomplexity of the proposed algorithm using 2D-MUSICand improved proposed algorithm using Unitary-ESPRITcomparing them with that of PSS in [14]

The complexity of the proposed algorithm is madeup of four parts covariance matrix estimation calculatingsmoothing covariance matrix eigenvalue decompositionand 2D-MUSICThe complexities of these parts are119874(31198691198722)119874(((119881 minus + 1))2) 119874((2)3) and 119874(22(2 minus 119870)119866120593119866120579)respectively where 119866120593 and 119866120579 represent the number of spec-tral points The complexity of improved proposed algorithmalso has four parts which is only different in fourth partUnitary-ESPRIT with 119874((22)3 + 1198703) Therefore the com-putational complexity of proposed algorithm is 119874(31198691198722 +((119881minus+1))2+6+22(2minus119870)119866120593119866120579) and that of improvedproposed algorithm is119874(31198691198722+((119881minus+1))2+96 +1198703)Moreover the complexity of PSS is 119874(119869(11987241 + 11987242) + 11987261 +11987262 + 119866120593119866120579(1198724111987222 + 1198724211987221 )) For the sake of claritythe computational complexities of all these methods aresummarized in Table 2 We also compare the complexity ofalgorithmmentioned above versus snapshots (119869) the numberof sensors (119872) and the searching step (Δ120579 = Δ120593 where119866120579 = 90Δ120579119866120593 = 360Δ120593) in Figures 6(a)ndash6(c) respectivelyunder the condition that = (119881 + 1)2

As shown in Figure 6 the improved proposed algo-rithm using Unitary-ESPRIT has the smallest complexityThe proposed algorithm using 2D-MUSIC and PSS has

the larger complexity due to the 2D spectral peak searchThe complexity is huge when the searching step is smallTherefore we introduce the Unitary-ESPRIT to solve theclosed-form solutions of DOAs where the searching stepdoes not affect the complexity Compared with the searchingstep the number of snapshots and sensors has the weakerimpact on complexity As a result the improved proposedalgorithm can efficiently reduce the complexity

5 Simulation Results

This section performs the results of simulation experimentscomparing the proposed algorithm using Unitary-ESPRITwith that using 2D-MUSIC and PSS proposed in [14] Tomeasure the accuracy of the algorithms define the root meansquare error (RMSE) as

RMSE = radic 1119876119870119876sum119902=1

10038171003817100381710038171003817120574 minus 119902100381710038171003817100381710038172 (39)

where 119876 represents the number of simulations 120574 and 119902are denoted as the real values and the 119894th estimated valuesrespectively We assume that the impinging signals are BPSKmodulation which are a kind of noncircular signals ofmaximum noncircular rate Moreover we set = (119881 + 1)2Simulation 1 (performance of estimating underdeterminedsignals) Through the analysis of DOF the proposed algo-rithm and improved proposed algorithm can estimate under-determined signals To verify the ability estimating under-determined signals we consider the case of 119870 = 16 witha signal-of-noise (SNR) of 15dB and 119869 = 5000 We set thecoprime planar array as 1198721 = 2 and 1198722 = 3 (119872 = 12) Thedistribution of the estimated values from20 simulations using2D-MUSIC and Unitary-ESPRIT is presented in Figures 7(a)and 7(b) respectively The figure proves that both algorithmscan estimate DOAs of 16 signals which is more than thenumber of sensors

Simulation 2 RMSE comparison of the proposed algorithmimproved proposed algorithm PSS and the URA with thesame number of sensors [22] under different SNRs

Simulations are conducted with 1198721 = 2 1198722 = 3 119876 =100 119869 = 5000 and SNRs from minus5dB to 15dB at 5dB intervalsAnd we set 119870 = 2 Δ120579 = 01∘ and 119870 = 6 Δ120579 = 01∘ TheRMSE results are shown in Figures 8 and 9 respectively

The RMSEs of the two proposed algorithms are lowerthan that of URA with the same number of sensors becausethe virtual URA has a bigger array aperture The figuresshow that there are gaps between the RMSE of two proposedalgorithm and that of PSS The virtual URA has the samesize as the coprime planar array but the PSS uses moreinformation Nevertheless the gap is not big when their arrayapertures are same Compared with proposed algorithm theimproved proposed algorithm obtains the similarly preciseestimations with much less complexity Moreover whenapplied tomultiple signals (119870 = 6) both proposed algorithmsmaintain their accuracy but the PSS fails


PSSProposedImproved Proposed

105

106

107

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109

1010

1011C

ompl

exity

1000 2000 3000 5000 7000 10000500J

(a)

24 33 4012M


104

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1011

1012

1013

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plex

ity

(b)


05 02 01 0051Δ

105

106

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108

109

1010

1011

(c)

Figure 6 Complexity comparison (a) versus 119869 when1198721 = 21198722 = 3 and Δ120579 = 01∘ (b) versus119872 when 119869 = 5000 and Δ120579 = 01∘ (c) versusΔ120579 when1198721 = 21198722 = 3 and 119869 = 5000Table 2 Computational complexity comparison

Algorithm Complexity

The proposed algorithm 119874(31198691198722 + ( (119881 minus + 1))2 + 6 + 22 (2 minus 119870)119866120593119866120579)Improved proposed algorithm 119874(31198691198722 + ( (119881 minus + 1))2 + 96 + 1198703)PSS 119874(119869 (11987241 + 11987242) + 11987261 + 11987262 + 119866120593119866120579 (1198724111987222 + 1198724211987221 ))

Simulation 3 RMSE comparison of the proposed algo-rithm improved proposed algorithm PSS and the URAwith the same number of sensors [22] under differentsnapshots

In Simulation 3 we set the SNR to 15dB1198721 = 21198722 = 3119876 = 100 119870 = 2 and Δ120579 = 01∘ and vary the number of

snapshots to 119869 = [500 1000 2000 3000 5000 7000 10000]The results are presented in Figure 10 The RMSE decreasesas the number of snapshots increase although the decline isnegligible once 119869 gt 5000Simulation 4 RMSE comparison under different SNRs witha bigger array aperture than Simulation 2


10

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70El

evat

ion

(Deg

ree)

minus100 minus50 0 50 100 150minus150Azimuth (Degree)

(a)

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50

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70

Elev

atio

n (D

egre

e)


(b)

Figure 7 Distribution of azimuth and elevation estimation (a) proposed algorithm (b) improved proposed algorithm

URAPSS

ProposedImproved Proposed

10 11 12 13 14 15002003004005006007

0 5 10 15minus5SNR (dB)

0

005

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(Deg

ree)

(a)

10 11 12 13 14 150015

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0040045

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025

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SE (D

egre

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URAPSS


(b)

Figure 8 RMSE comparison under different SNRs with 119870 = 2 (a) azimuth (b) elevation

We conduct this simulation as Simulation 2 with thecondition that 1198721 = 3 1198722 = 4 and 119870 = 2 The resultsare shown in Figure 11 There are two differences in Figure 11compared with Figure 8 One is that the gaps between theRMSE of URA and those of two proposed algorithms arelarger because the array aperture gap between the virtualURA and URA is bigger with the more number of sensorsThe other is that the gaps between the RMSE of PSS andthose of two proposed algorithms are also larger Given theTable 1 the virtual smoothing URA has the array aperture as(1198721+1198722)times(1198721+1198722)while that of the coprime planar arrayis (11987211198722 minus 1198721 + 1) times (11987211198722 minus 1198721 + 1) When1198721 gt 2 thearray aperture of virtual URA is smaller Hence the RMSEsof two proposed algorithms are higher Moreover this meansthat the proposed algorithms improve the DOF with the lossof array aperture and accuracy And the loss can get biggerwith the increase of 1198721 and this is still true when 1198721 gt 4

How to deal with the balance between the DOF enhancementand accuracy improvement is worthy our further research

6 Conclusions

The paper has proposed a 2D DOA estimation algorithmusing a coprime planar array for noncircular signals whichhas much improved the DOF and reduced the complexityThe paper has described the model and the associated algo-rithms and analyzed the DOF and computational complexityof the proposed algorithms in comparison with that ofexisting algorithms Through the theoretical analysis andsimulation experiments we conclude that the proposedalgorithms can obtain the higher accuracy than theURAwiththe same number of sensors where the proposed algorithmsconstruct a more bigger virtual URA aperture based onthe coprime planar array Compared with DOF of PSS as


URAProposedImproved Proposed

10 11 12 13 14 15005006007008009

01011012013014

0

02

04

06

08

1

12RM

SE (D

egre

e)


(a)

10 11 12 13 14 15003004005006007008009

0

01

02

03

04

05

06

07

08

09

RMSE

(Deg

ree)



(b)


URAPSS


5000 6000 7000 8000 9000 100000018

0020022002400260028

003003200340036

2000 3000 4000 5000 6000 7000 8000 9000 100001000J

001002003004005006007008009

01011

RMSE

(Deg

ree)

(a)

5000 6000 7000 8000 9000 100000015

002

0025

003

URAPSS


2000 3000 4000 5000 6000 7000 8000 9000 100001000J

001

002

003

004

005

006

007

008

009RM

SE (D

egre

e)

(b)

Figure 10 RMSE comparison under different snapshots with 119870 = 2 (a) azimuth (b) elevation

10 11 12 13 14 150005

0010015

0020025

0030035

0040045

005

0

005

01

015

02

025

RMSE

(Deg

ree)


URAPSS


(a)

10 11 12 13 14 150005

0010015

0020025

0030035

004

005

01

015

02

025

03

RMSE

(Deg

ree)


URAPSS


(b)



11987221 minus 1 the proposed algorithms has improved DOF biggerthan 119872 which has an ability to estimate underdeterminedsignals Furthermore both two proposed algorithms have theclose RMSEs but the improved algorithm applying Unitary-ESPRIT has greatly reduced the complexity compared withthe algorithm using 2D-MUSIC

Data Availability

No data were used to support this study

Disclosure

The authors claim that the data used in this article areprovided by their simulations and this is developed withoutusing any data in a published article to support their results

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (Grant no 61401513)

References

[1] X Wei Z Liu X Ding and M Fan ldquoSuper-resolutionreconstruction of radar tomographic image based on imagedecompositionrdquo IEEE Geoscience and Remote Sensing Lettersvol 11 no 3 pp 607ndash611 2013

[2] Y L Wang S L Ma G L Liang and Z Fan ldquoChirp SpreadSpectrum of Orthogonal Frequency Division MultiplexingUnderwater Acoustic Communication System Based on Multi-path Diversity Receiverdquo Acta Physica Sinica vol 63 no 4 p044302 2014

[3] Y Zhou Y Wu D Chen and F Tong ldquoCompressed sensingestimation of underwater acoustic MIMO channels based ontemporal joint sparse recoveryrdquoDianzi YuXinxi XuebaoJournalof Electronics and Information Technology vol 38 no 8 pp1920ndash1927 2016

[4] Z Ye J Dai X Xu and X Wu ldquoDOA estimation for uniformlinear array with mutual couplingrdquo IEEE Transactions onAerospace and Electronic Systems vol 45 no 1 pp 280ndash2882009

[5] P Heidenreich A M Zoubir and M Rubsamen ldquoJoint 2-DDOA estimation and phase calibration for uniform rectangulararraysrdquo IEEETransactions on Signal Processing vol 60 no 9 pp4683ndash4693 2012

[6] H Han and P J Zhao ldquoA Two-dimensional Direction FindingEstimation with L-shape Uniform Linear Arraysrdquo in Proceed-ings of the International Symposium on Instrumentation andMeasurement Sensor Network and Automation IEEE pp 779ndash782 Piscataway NJ USA 2014

[7] P Pal and P P Vaidyanathan ldquoCoprime sampling and themusicalgorithmrdquo in Proceedings of the IEEE Digital Signal ProcessingWorkshop and IEEE Signal Processing Education Workshop(DSPSPE rsquo11) pp 289ndash294 IEEE Sedona Ariz USA January2011

[8] D Zhang Y Zhang G Zheng C Feng and J Tang ldquoImprovedDOA estimation algorithm for co-prime linear arrays usingroot-MUSIC algorithmrdquo IEEE Electronics Letters vol 53 no 18pp 1277ndash1279 2017

[9] C Zhou and J Zhou ldquoDirection-of-arrival estimation withcoarray ESPRIT for coprime arrayrdquo Sensors vol 17 no 8 2017

[10] B Byun and D Yoo ldquoImproved Direction of Arrival EstimationBased on Coprime Array and Propagator Method by NoisePower Spectral Density Estimationrdquo The Journal of AdvancedNavigation Technology vol 20 no 4 pp 367ndash373 2017

[11] B Ba G-C Liu T Li Y-C Lin and Y Wang ldquoJoint for Timeof Arrival andDirection of Arrival Estimation Algorithm Basedon the Subspace of Extended Hadamard Productrdquo Acta PhysicaSinica vol 64 no 7 p 078403 2015

[12] P Pal and P P Vaidyanathan ldquoNested arrays a novel approachto array processing with enhanced degrees of freedomrdquo IEEETransactions on Signal Processing vol 58 no 8 pp 4167ndash41812010

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

[14] Q Wu F Sun P Lan G Ding and X Zhang ldquoTwo-Dimensional Direction-of-Arrival Estimation for Co-PrimePlanar Arrays A Partial Spectral Search Approachrdquo IEEESensors Journal vol 16 no 14 pp 5660ndash5670 2016

[15] H Xu Y Zhang B Ba D Wang and X Li ldquoTwo-DimensionalDirection-of-Arrival Fast Estimation of Multiple Signals withMatrix Completion Theory in Coprime Planar Arrayrdquo Sensorsvol 18 no 6 p 1741 2018

[16] Z-L Dai W-J Cui B Ba and Y-K Zhang ldquoTwo-dimensionalDirection-of-arrival Estimation of Coherently DistributedNoncircular Signals via Symmetric Shift Invariancerdquo ActaPhysica Sinica vol 66 no 22 p 220701 2017

[17] W-J Cui Z-L Dai B Ba and H Lu ldquoFast DOA Estimation ofDistributedNoncircular Sources byCross-correlation SamplingDecompositionrdquo Journal of Electronics Information Technologyvol 40 no 5 pp 1226ndash1233 2018

[18] T-J Shan M Wax and T Kailath ldquoOn spatial smoothingfor direction-of-arrival estimation of coherent signalsrdquo IEEETransactions on Signal Processing vol 33 no 4 pp 806ndash8111985

[19] Y-M 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 Process-ing) vol 140 no 1 pp 37ndash42 1993

[20] M Haardt and J A Nossek ldquoUnitary ESPRIT how to obtainincreased estimation accuracy with a reduced computationalburdenrdquo IEEE Transactions on Signal Processing vol 43 no 5pp 1232ndash1242 1995

[21] M D Zoltowski M Haardt and C P Mathews ldquoClosed-form2-D angle estimation with rectangular arrays in element spaceor beamspace via unitary ESPRITrdquo IEEE Transactions on SignalProcessing vol 44 no 2 pp 316ndash328 1996

[22] R Li X Shi L Chen et al ldquoThe non-circular MUSIC methodfor uniform rectangular arraysrdquo in Proceedings of the 2010International Conference on Microwave and Millimeter WaveTechnology ICMMT 2010 pp 1390ndash1393 May 2010

Hindawiwwwhindawicom Volume 2018

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Volume 2018

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


and elevation angles A partial spectral search (PSS) methodis introduced to reduce the complexity caused by 2D-MUSICmentioned in [14] but PSS is still based on peak search whichdoes not significantly reduce the complexity Moreover theDOF is limited by the number of subarray sensors We haveproposed a method in [15] which uses the covariance matrixof coprime planar array to estimate a new covariance matrixwith matrix completion This processing has much improvedDOF bigger than the number of sensors However matrixcompletion can introduce the additional errors so that itsprecision is limited and it costs much complexity All themethods mentioned above do not take some signal featuresinto consideration

There are many kinds of noncircular signals in moderncommunication systems with such as BPSK ASK and AMmodulation The methods which construct extended virtualarrays and improve the accuracy of estimated parametersbased on noncircular signals are proposed in [16 17] Con-sidering the problems that existing methods for estimatingDOAs of general uncorrelated signals using coprime planararrays have huge computational complexity and low DOFwe present an algorithm to estimate 2D DOAs of underde-termined noncircular signals with low complexity The algo-rithm vectorizes the covariance matrix and ellipse covariancematrix of received signals to construct the new received dataof a virtual URA Combining 2D spatial smoothing process-ing and Unitary-ESPRIT we then realize the fast estimationof 2D DOAs Through constructing a virtual URA with morenumber of sensors than that of coprime planar array weimprove the DOF Moreover Unitary-ESPRIT can solve theclosed-form solutions of DOAs replacing the spectral peaksearch to reduce the complexity

The remainder of this paper is arranged as followsSection 2 introduces the model of coprime planar array andSection 3 describes the steps of the algorithm Sections 4 and5 analyze the computational complexity and performance ofthe model respectively to demonstrate the validity of thisalgorithm Section 6 gives the conclusion to this paper

The notations used in this paper are as follows (∙)119879 (∙)lowastand (∙)119867 respectively represent the transposition conjuga-tion and conjugate transposition 119864(∙) denotes the math-ematical expectation diag(∙) expresses the transformationof a vector to a diagonal matrix otimes ∘ and (∙)+ denote theKronecker product Khatri-Rao product and pseudoinverseoperator respectively

2 System Model

Considering the coprime planar array model the arraygeometry is shown in Figure 1 The coprime planar arrayconsists of two URAs Subarray 1 has 1198721 times 1198721 sensors andsubarray 2 has 1198722 times 1198722 sensors where 1198721 and 1198722 arethe coprime integers (generally assuming 1198721 lt 1198722) anddenote the sensor numbers on the 119909 119910 axis Correspondinglythe distance between the two adjacent sensors is 11987221205822 and11987211205822 respectively where the 120582 represents the wavelength ofthe impinging signalsThe subarrays coincide at the origin sothe total number of sensors is 119872 = 11987221 + 11987222 minus 1 Supposethere are 119870 uncorrelated narrowband far-field noncircular

3 2

2

2

k

k

Figure 1 Geometry of coprime planar arraywhen1198721 = 2 and1198722 =3signals impinging on the array with power 12059021 12059022 1205902119870The 119896th signal is located at elevation angle 120579119896 which isdownward from the z-axis and azimuth angle 120593119896 which iscounterclockwise from the x-axis

We define 1198631 and 1198632 as the location set of the subarrays1198631 = 11987111205822 and 1198632 = 11987121205822 where 1198711 = (119898 119899)1198722 0 le119898 119899 le 1198721 minus 1 and 1198712 = (119898 119899)1198721 0 le 119898 119899 le 1198722 minus1 Hence the location set of the coprime planar array isexpressed as 119863 = 1198631 cup 1198632 and we have 119871 = 1198711 cup 1198712 Thereceived signals at the array can be represented as

X (119905) = A (120593 120579) S (119905) +N (119905) (1)

The array manifold is

A (120593 120579) = [a (1205931 1205791) a (1205932 1205792) sdot sdot sdot a (120593119870 120579119870)] (2)

where

a (120593119896 120579119896) = [1198861 (120593119896 120579119896) 1198862 (120593119896 120579119896) sdot sdot sdot 119886119872 (120593119896 120579119896)]119879 (3)

119886119898 (120593119896 120579119896) = 119890minus119895120587 sin 120579119896(119909119898 cos120593119896+119910119898 sin120593119896) (119909119898 119910119898) isin 119871 (4)

The noncircular signal data vector is

S (119905) = [1199041 (119905) 1199042 (119905) sdot sdot sdot 119904119870 (119905)]119879 (5)

where 119905 = 1 2 119869 is the sampling time and 119869 is the numberof snapshots And the noise vector is

N (119905) = [1198991 (119905) 1198992 (119905) sdot sdot sdot 119899119872 (119905)]119879 (6)

where the elements are usually Gaussian random variableswith zero means and variance 1205902119899




minus4

minus3

minus2

minus1

0

1

2

3

4

y



RX = 119864 [XX119867] = ARSA119867 + 1205902119899I119872 (7)


Z1 = V119890119888 (RX) = B1p + 12059021198991119899 (8)


(9)





minus8

minus6

minus4

minus2

0

2

4

6

8

y



R1015840X = 119864 [XXT] = AR1015840SA119879 (10)



Z2 = V119890119888 (R1015840X) = B2p (12)

Z3 = V119890119888 (R10158401015840119883) = B3p (13)


Z = [Z1198791 ZT2 ZT3 ]T = Bp + 1205902119899 1119899 (14)


Z = Bp + 1205902119899e (15)




111VminusV+1

VminusV+11

V










RX = 1119869XX119867 (16)


Z = Bp + 1205902119899 e (17)




Z11 = B11p + e11 (18)









119881minus+1sum119888=1

Z119903119888Z119867

119903119888 (22)


RZ = 1(119881 minus + 1)2


B1199031(119881minus+1sum119888=1


+ 1(119881 minus + 1)2

119881minus+1sum119903=1

119881minus+1sum119888=1

e119903119888eH119903119888

= 1(119881 minus + 1)2

sdot B11(119881minus+1sum119903=1

119881minus+1sum119888=1





119881minus+1sum119888=1

e119903119888eH119903119888 = 1

(119881 minus + 1)2 B11RpBH11

+ 1(V minus + 1)2

119881minus+1sum119903=1

119881minus+1sum119888=1

e119903119888eH119903119888

(23)



119875 (120593 120579) = 100381710038171003817100381710038171003817U119867119899 b11 (120593 120579)100381710038171003817100381710038171003817minus22 (24)



Π119896Qlowast = Q (25)




Π119896 O minus119895Π119896]]]]

(27)



J2Q119867 (28)


J2Q119867 (29)


K1205831 = I otimes K1 (30)

K1205832 = I otimes K2 (31)




R119879 = Re (Q119867

otimesQ119867) RZ (Q119867

otimesQ119867)119867 (34)


Ψ120583 = (K1205831U119878)+K1205832U119878 (35)

Ψ] = (K]1U119878)+K]2U119878 (36)


119894 = arcsin(radic(120583119894120587 )2 + (]119894120587 )2) (37)

119894 = arctan ( ]119894120583119894) (38)










URAPSSProposed

20 25 30 35 4015M

0

10

20

30

40

50

60

70

80

90

DO

F









RMSE = radic 1119876119870119876sum119902=1

10038171003817100381710038171003817120574 minus 119902100381710038171003817100381710038172 (39)







105

106

107

108

109

1010

1011C

ompl

exity

1000 2000 3000 5000 7000 10000500J

(a)

24 33 4012M


104

105

106

107

108

109

1010

1011

1012

1013

Com

plex

ity

(b)


05 02 01 0051Δ

105

106

107

108

109

1010

1011

(c)








10

20

30

40

50

60

70El

evat

ion

(Deg

ree)


(a)

10

20

30

40

50

60

70

Elev

atio

n (D

egre

e)


(b)


URAPSS


10 11 12 13 14 15002003004005006007


0

005

01

015

02

025

03

035

04

045

RMSE

(Deg

ree)

(a)

10 11 12 13 14 150015

0020025

0030035

0040045

0050055

0

005

01

015

02

025

03

035

04RM

SE (D

egre

e)


URAPSS


(b)




6 Conclusions




10 11 12 13 14 15005006007008009

01011012013014

0

02

04

06

08

1

12RM

SE (D

egre

e)


(a)

10 11 12 13 14 15003004005006007008009

0

01

02

03

04

05

06

07

08

09

RMSE

(Deg

ree)



(b)


URAPSS


5000 6000 7000 8000 9000 100000018

0020022002400260028

003003200340036

2000 3000 4000 5000 6000 7000 8000 9000 100001000J

001002003004005006007008009

01011

RMSE

(Deg

ree)

(a)

5000 6000 7000 8000 9000 100000015

002

0025

003

URAPSS


2000 3000 4000 5000 6000 7000 8000 9000 100001000J

001

002

003

004

005

006

007

008

009RM

SE (D

egre

e)

(b)


10 11 12 13 14 150005

0010015

0020025

0030035

0040045

005

0

005

01

015

02

025

RMSE

(Deg

ree)


URAPSS


(a)

10 11 12 13 14 150005

0010015

0020025

0030035

004

005

01

015

02

025

03

RMSE

(Deg

ree)


URAPSS


(b)




Data Availability


Disclosure




Acknowledgments


References






























Journal of












Journal of







Volume 2018






Volume 2018











minus4

minus3

minus2

minus1

0

1

2

3

4

y



RX = 119864 [XX119867] = ARSA119867 + 1205902119899I119872 (7)


Z1 = V119890119888 (RX) = B1p + 12059021198991119899 (8)


(9)





minus8

minus6

minus4

minus2

0

2

4

6

8

y



R1015840X = 119864 [XXT] = AR1015840SA119879 (10)



Z2 = V119890119888 (R1015840X) = B2p (12)

Z3 = V119890119888 (R10158401015840119883) = B3p (13)


Z = [Z1198791 ZT2 ZT3 ]T = Bp + 1205902119899 1119899 (14)


Z = Bp + 1205902119899e (15)




111VminusV+1

VminusV+11

V










RX = 1119869XX119867 (16)


Z = Bp + 1205902119899 e (17)




Z11 = B11p + e11 (18)









119881minus+1sum119888=1

Z119903119888Z119867

119903119888 (22)


RZ = 1(119881 minus + 1)2


B1199031(119881minus+1sum119888=1


+ 1(119881 minus + 1)2

119881minus+1sum119903=1

119881minus+1sum119888=1

e119903119888eH119903119888

= 1(119881 minus + 1)2

sdot B11(119881minus+1sum119903=1

119881minus+1sum119888=1





119881minus+1sum119888=1

e119903119888eH119903119888 = 1

(119881 minus + 1)2 B11RpBH11

+ 1(V minus + 1)2

119881minus+1sum119903=1

119881minus+1sum119888=1

e119903119888eH119903119888

(23)



119875 (120593 120579) = 100381710038171003817100381710038171003817U119867119899 b11 (120593 120579)100381710038171003817100381710038171003817minus22 (24)



Π119896Qlowast = Q (25)




Π119896 O minus119895Π119896]]]]

(27)



J2Q119867 (28)


J2Q119867 (29)


K1205831 = I otimes K1 (30)

K1205832 = I otimes K2 (31)




R119879 = Re (Q119867

otimesQ119867) RZ (Q119867

otimesQ119867)119867 (34)


Ψ120583 = (K1205831U119878)+K1205832U119878 (35)

Ψ] = (K]1U119878)+K]2U119878 (36)


119894 = arcsin(radic(120583119894120587 )2 + (]119894120587 )2) (37)

119894 = arctan ( ]119894120583119894) (38)










URAPSSProposed

20 25 30 35 4015M

0

10

20

30

40

50

60

70

80

90

DO

F









RMSE = radic 1119876119870119876sum119902=1

10038171003817100381710038171003817120574 minus 119902100381710038171003817100381710038172 (39)







105

106

107

108

109

1010

1011C

ompl

exity

1000 2000 3000 5000 7000 10000500J

(a)

24 33 4012M


104

105

106

107

108

109

1010

1011

1012

1013

Com

plex

ity

(b)


05 02 01 0051Δ

105

106

107

108

109

1010

1011

(c)








10

20

30

40

50

60

70El

evat

ion

(Deg

ree)


(a)

10

20

30

40

50

60

70

Elev

atio

n (D

egre

e)


(b)


URAPSS


10 11 12 13 14 15002003004005006007


0

005

01

015

02

025

03

035

04

045

RMSE

(Deg

ree)

(a)

10 11 12 13 14 150015

0020025

0030035

0040045

0050055

0

005

01

015

02

025

03

035

04RM

SE (D

egre

e)


URAPSS


(b)




6 Conclusions




10 11 12 13 14 15005006007008009

01011012013014

0

02

04

06

08

1

12RM

SE (D

egre

e)


(a)

10 11 12 13 14 15003004005006007008009

0

01

02

03

04

05

06

07

08

09

RMSE

(Deg

ree)



(b)


URAPSS


5000 6000 7000 8000 9000 100000018

0020022002400260028

003003200340036

2000 3000 4000 5000 6000 7000 8000 9000 100001000J

001002003004005006007008009

01011

RMSE

(Deg

ree)

(a)

5000 6000 7000 8000 9000 100000015

002

0025

003

URAPSS


2000 3000 4000 5000 6000 7000 8000 9000 100001000J

001

002

003

004

005

006

007

008

009RM

SE (D

egre

e)

(b)


10 11 12 13 14 150005

0010015

0020025

0030035

0040045

005

0

005

01

015

02

025

RMSE

(Deg

ree)


URAPSS


(a)

10 11 12 13 14 150005

0010015

0020025

0030035

004

005

01

015

02

025

03

RMSE

(Deg

ree)


URAPSS


(b)




Data Availability


Disclosure




Acknowledgments


References






























Journal of












Journal of







Volume 2018






Volume 2018











111VminusV+1

VminusV+11

V










RX = 1119869XX119867 (16)


Z = Bp + 1205902119899 e (17)




Z11 = B11p + e11 (18)









119881minus+1sum119888=1

Z119903119888Z119867

119903119888 (22)


RZ = 1(119881 minus + 1)2


B1199031(119881minus+1sum119888=1


+ 1(119881 minus + 1)2

119881minus+1sum119903=1

119881minus+1sum119888=1

e119903119888eH119903119888

= 1(119881 minus + 1)2

sdot B11(119881minus+1sum119903=1

119881minus+1sum119888=1





119881minus+1sum119888=1

e119903119888eH119903119888 = 1

(119881 minus + 1)2 B11RpBH11

+ 1(V minus + 1)2

119881minus+1sum119903=1

119881minus+1sum119888=1

e119903119888eH119903119888

(23)



119875 (120593 120579) = 100381710038171003817100381710038171003817U119867119899 b11 (120593 120579)100381710038171003817100381710038171003817minus22 (24)



Π119896Qlowast = Q (25)




Π119896 O minus119895Π119896]]]]

(27)



J2Q119867 (28)


J2Q119867 (29)


K1205831 = I otimes K1 (30)

K1205832 = I otimes K2 (31)




R119879 = Re (Q119867

otimesQ119867) RZ (Q119867

otimesQ119867)119867 (34)


Ψ120583 = (K1205831U119878)+K1205832U119878 (35)

Ψ] = (K]1U119878)+K]2U119878 (36)


119894 = arcsin(radic(120583119894120587 )2 + (]119894120587 )2) (37)

119894 = arctan ( ]119894120583119894) (38)










URAPSSProposed

20 25 30 35 4015M

0

10

20

30

40

50

60

70

80

90

DO

F









RMSE = radic 1119876119870119876sum119902=1

10038171003817100381710038171003817120574 minus 119902100381710038171003817100381710038172 (39)







105

106

107

108

109

1010

1011C

ompl

exity

1000 2000 3000 5000 7000 10000500J

(a)

24 33 4012M


104

105

106

107

108

109

1010

1011

1012

1013

Com

plex

ity

(b)


05 02 01 0051Δ

105

106

107

108

109

1010

1011

(c)








10

20

30

40

50

60

70El

evat

ion

(Deg

ree)


(a)

10

20

30

40

50

60

70

Elev

atio

n (D

egre

e)


(b)


URAPSS


10 11 12 13 14 15002003004005006007


0

005

01

015

02

025

03

035

04

045

RMSE

(Deg

ree)

(a)

10 11 12 13 14 150015

0020025

0030035

0040045

0050055

0

005

01

015

02

025

03

035

04RM

SE (D

egre

e)


URAPSS


(b)




6 Conclusions




10 11 12 13 14 15005006007008009

01011012013014

0

02

04

06

08

1

12RM

SE (D

egre

e)


(a)

10 11 12 13 14 15003004005006007008009

0

01

02

03

04

05

06

07

08

09

RMSE

(Deg

ree)



(b)


URAPSS


5000 6000 7000 8000 9000 100000018

0020022002400260028

003003200340036

2000 3000 4000 5000 6000 7000 8000 9000 100001000J

001002003004005006007008009

01011

RMSE

(Deg

ree)

(a)

5000 6000 7000 8000 9000 100000015

002

0025

003

URAPSS


2000 3000 4000 5000 6000 7000 8000 9000 100001000J

001

002

003

004

005

006

007

008

009RM

SE (D

egre

e)

(b)


10 11 12 13 14 150005

0010015

0020025

0030035

0040045

005

0

005

01

015

02

025

RMSE

(Deg

ree)


URAPSS


(a)

10 11 12 13 14 150005

0010015

0020025

0030035

004

005

01

015

02

025

03

RMSE

(Deg

ree)


URAPSS


(b)




Data Availability


Disclosure




Acknowledgments


References






























Journal of












Journal of







Volume 2018






Volume 2018











119881minus+1sum119888=1

e119903119888eH119903119888 = 1

(119881 minus + 1)2 B11RpBH11

+ 1(V minus + 1)2

119881minus+1sum119903=1

119881minus+1sum119888=1

e119903119888eH119903119888

(23)



119875 (120593 120579) = 100381710038171003817100381710038171003817U119867119899 b11 (120593 120579)100381710038171003817100381710038171003817minus22 (24)



Π119896Qlowast = Q (25)




Π119896 O minus119895Π119896]]]]

(27)



J2Q119867 (28)


J2Q119867 (29)


K1205831 = I otimes K1 (30)

K1205832 = I otimes K2 (31)




R119879 = Re (Q119867

otimesQ119867) RZ (Q119867

otimesQ119867)119867 (34)


Ψ120583 = (K1205831U119878)+K1205832U119878 (35)

Ψ] = (K]1U119878)+K]2U119878 (36)


119894 = arcsin(radic(120583119894120587 )2 + (]119894120587 )2) (37)

119894 = arctan ( ]119894120583119894) (38)










URAPSSProposed

20 25 30 35 4015M

0

10

20

30

40

50

60

70

80

90

DO

F









RMSE = radic 1119876119870119876sum119902=1

10038171003817100381710038171003817120574 minus 119902100381710038171003817100381710038172 (39)







105

106

107

108

109

1010

1011C

ompl

exity

1000 2000 3000 5000 7000 10000500J

(a)

24 33 4012M


104

105

106

107

108

109

1010

1011

1012

1013

Com

plex

ity

(b)


05 02 01 0051Δ

105

106

107

108

109

1010

1011

(c)








10

20

30

40

50

60

70El

evat

ion

(Deg

ree)


(a)

10

20

30

40

50

60

70

Elev

atio

n (D

egre

e)


(b)


URAPSS


10 11 12 13 14 15002003004005006007


0

005

01

015

02

025

03

035

04

045

RMSE

(Deg

ree)

(a)

10 11 12 13 14 150015

0020025

0030035

0040045

0050055

0

005

01

015

02

025

03

035

04RM

SE (D

egre

e)


URAPSS


(b)




6 Conclusions




10 11 12 13 14 15005006007008009

01011012013014

0

02

04

06

08

1

12RM

SE (D

egre

e)


(a)

10 11 12 13 14 15003004005006007008009

0

01

02

03

04

05

06

07

08

09

RMSE

(Deg

ree)



(b)


URAPSS


5000 6000 7000 8000 9000 100000018

0020022002400260028

003003200340036

2000 3000 4000 5000 6000 7000 8000 9000 100001000J

001002003004005006007008009

01011

RMSE

(Deg

ree)

(a)

5000 6000 7000 8000 9000 100000015

002

0025

003

URAPSS


2000 3000 4000 5000 6000 7000 8000 9000 100001000J

001

002

003

004

005

006

007

008

009RM

SE (D

egre

e)

(b)


10 11 12 13 14 150005

0010015

0020025

0030035

0040045

005

0

005

01

015

02

025

RMSE

(Deg

ree)


URAPSS


(a)

10 11 12 13 14 150005

0010015

0020025

0030035

004

005

01

015

02

025

03

RMSE

(Deg

ree)


URAPSS


(b)




Data Availability


Disclosure




Acknowledgments


References






























Journal of












Journal of







Volume 2018






Volume 2018









URAPSSProposed

20 25 30 35 4015M

0

10

20

30

40

50

60

70

80

90

DO

F









RMSE = radic 1119876119870119876sum119902=1

10038171003817100381710038171003817120574 minus 119902100381710038171003817100381710038172 (39)







105

106

107

108

109

1010

1011C

ompl

exity

1000 2000 3000 5000 7000 10000500J

(a)

24 33 4012M


104

105

106

107

108

109

1010

1011

1012

1013

Com

plex

ity

(b)


05 02 01 0051Δ

105

106

107

108

109

1010

1011

(c)








10

20

30

40

50

60

70El

evat

ion

(Deg

ree)


(a)

10

20

30

40

50

60

70

Elev

atio

n (D

egre

e)


(b)


URAPSS


10 11 12 13 14 15002003004005006007


0

005

01

015

02

025

03

035

04

045

RMSE

(Deg

ree)

(a)

10 11 12 13 14 150015

0020025

0030035

0040045

0050055

0

005

01

015

02

025

03

035

04RM

SE (D

egre

e)


URAPSS


(b)




6 Conclusions




10 11 12 13 14 15005006007008009

01011012013014

0

02

04

06

08

1

12RM

SE (D

egre

e)


(a)

10 11 12 13 14 15003004005006007008009

0

01

02

03

04

05

06

07

08

09

RMSE

(Deg

ree)



(b)


URAPSS


5000 6000 7000 8000 9000 100000018

0020022002400260028

003003200340036

2000 3000 4000 5000 6000 7000 8000 9000 100001000J

001002003004005006007008009

01011

RMSE

(Deg

ree)

(a)

5000 6000 7000 8000 9000 100000015

002

0025

003

URAPSS


2000 3000 4000 5000 6000 7000 8000 9000 100001000J

001

002

003

004

005

006

007

008

009RM

SE (D

egre

e)

(b)


10 11 12 13 14 150005

0010015

0020025

0030035

0040045

005

0

005

01

015

02

025

RMSE

(Deg

ree)


URAPSS


(a)

10 11 12 13 14 150005

0010015

0020025

0030035

004

005

01

015

02

025

03

RMSE

(Deg

ree)


URAPSS


(b)




Data Availability


Disclosure




Acknowledgments


References






























Journal of












Journal of







Volume 2018






Volume 2018










105

106

107

108

109

1010

1011C

ompl

exity

1000 2000 3000 5000 7000 10000500J

(a)

24 33 4012M


104

105

106

107

108

109

1010

1011

1012

1013

Com

plex

ity

(b)


05 02 01 0051Δ

105

106

107

108

109

1010

1011

(c)








10

20

30

40

50

60

70El

evat

ion

(Deg

ree)


(a)

10

20

30

40

50

60

70

Elev

atio

n (D

egre

e)


(b)


URAPSS


10 11 12 13 14 15002003004005006007


0

005

01

015

02

025

03

035

04

045

RMSE

(Deg

ree)

(a)

10 11 12 13 14 150015

0020025

0030035

0040045

0050055

0

005

01

015

02

025

03

035

04RM

SE (D

egre

e)


URAPSS


(b)




6 Conclusions




10 11 12 13 14 15005006007008009

01011012013014

0

02

04

06

08

1

12RM

SE (D

egre

e)


(a)

10 11 12 13 14 15003004005006007008009

0

01

02

03

04

05

06

07

08

09

RMSE

(Deg

ree)



(b)


URAPSS


5000 6000 7000 8000 9000 100000018

0020022002400260028

003003200340036

2000 3000 4000 5000 6000 7000 8000 9000 100001000J

001002003004005006007008009

01011

RMSE

(Deg

ree)

(a)

5000 6000 7000 8000 9000 100000015

002

0025

003

URAPSS


2000 3000 4000 5000 6000 7000 8000 9000 100001000J

001

002

003

004

005

006

007

008

009RM

SE (D

egre

e)

(b)


10 11 12 13 14 150005

0010015

0020025

0030035

0040045

005

0

005

01

015

02

025

RMSE

(Deg

ree)


URAPSS


(a)

10 11 12 13 14 150005

0010015

0020025

0030035

004

005

01

015

02

025

03

RMSE

(Deg

ree)


URAPSS


(b)




Data Availability


Disclosure




Acknowledgments


References






























Journal of












Journal of







Volume 2018






Volume 2018









10

20

30

40

50

60

70El

evat

ion

(Deg

ree)


(a)

10

20

30

40

50

60

70

Elev

atio

n (D

egre

e)


(b)


URAPSS


10 11 12 13 14 15002003004005006007


0

005

01

015

02

025

03

035

04

045

RMSE

(Deg

ree)

(a)

10 11 12 13 14 150015

0020025

0030035

0040045

0050055

0

005

01

015

02

025

03

035

04RM

SE (D

egre

e)


URAPSS


(b)




6 Conclusions




10 11 12 13 14 15005006007008009

01011012013014

0

02

04

06

08

1

12RM

SE (D

egre

e)


(a)

10 11 12 13 14 15003004005006007008009

0

01

02

03

04

05

06

07

08

09

RMSE

(Deg

ree)



(b)


URAPSS


5000 6000 7000 8000 9000 100000018

0020022002400260028

003003200340036

2000 3000 4000 5000 6000 7000 8000 9000 100001000J

001002003004005006007008009

01011

RMSE

(Deg

ree)

(a)

5000 6000 7000 8000 9000 100000015

002

0025

003

URAPSS


2000 3000 4000 5000 6000 7000 8000 9000 100001000J

001

002

003

004

005

006

007

008

009RM

SE (D

egre

e)

(b)


10 11 12 13 14 150005

0010015

0020025

0030035

0040045

005

0

005

01

015

02

025

RMSE

(Deg

ree)


URAPSS


(a)

10 11 12 13 14 150005

0010015

0020025

0030035

004

005

01

015

02

025

03

RMSE

(Deg

ree)


URAPSS


(b)




Data Availability


Disclosure




Acknowledgments


References






























Journal of












Journal of







Volume 2018






Volume 2018










10 11 12 13 14 15005006007008009

01011012013014

0

02

04

06

08

1

12RM

SE (D

egre

e)


(a)

10 11 12 13 14 15003004005006007008009

0

01

02

03

04

05

06

07

08

09

RMSE

(Deg

ree)



(b)


URAPSS


5000 6000 7000 8000 9000 100000018

0020022002400260028

003003200340036

2000 3000 4000 5000 6000 7000 8000 9000 100001000J

001002003004005006007008009

01011

RMSE

(Deg

ree)

(a)

5000 6000 7000 8000 9000 100000015

002

0025

003

URAPSS


2000 3000 4000 5000 6000 7000 8000 9000 100001000J

001

002

003

004

005

006

007

008

009RM

SE (D

egre

e)

(b)


10 11 12 13 14 150005

0010015

0020025

0030035

0040045

005

0

005

01

015

02

025

RMSE

(Deg

ree)


URAPSS


(a)

10 11 12 13 14 150005

0010015

0020025

0030035

004

005

01

015

02

025

03

RMSE

(Deg

ree)


URAPSS


(b)




Data Availability


Disclosure




Acknowledgments


References






























Journal of












Journal of







Volume 2018






Volume 2018










Data Availability


Disclosure




Acknowledgments


References






























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Two-Dimensional Direction-of-Arrival Estimation of...

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