Compressed Sensing PARALIND Decomposition-Based Coherent ...downloads.hindawi.com/journals/wcmc/2019/3619818.pdf · Compressed Sensing PARALIND Decomposition-Based Coherent Angle

Research ArticleCompressed Sensing PARALIND Decomposition-BasedCoherent Angle Estimation for Uniform Rectangular Array

Wu Wei1 Xu Le1 Zhang Xiaofei 1 and Li Jianfeng 12

1College of Electronic and Information Engineering Nanjing University of Aeronautics and Astronautics Nanjing 211106 China2College of Computer and Information Hohai University Nanjing 211100 China

Correspondence should be addressed to Li Jianfeng lijianfengtin126com

Received 17 November 2018 Revised 27 March 2019 Accepted 27 May 2019 Published 24 June 2019

Guest Editor Alex da Silva

Copyright copy 2019 WuWei et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper the topic of coherent two-dimensional direction of arrival (2D-DOA) estimation is investigated Our study jointlyutilizes the compressed sensing (CS) technique and the parallel profiles with linear dependencies (PARALIND)model and presentsa 2D-DOA estimation algorithm for coherent sources with the uniform rectangular array Compared to the traditional PARALINDdecomposition the proposed algorithm owns lower computational complexity and smaller data storage capacity due to the processof compression Besides the proposed algorithm can obtain autopaired azimuth angles and elevation angles and can achieve thesame estimation performance as the traditional PARALIND which outperforms some familiar algorithms presented for coherentsources such as the forward backward spatial smoothing-estimating signal parameters via rotational invariance techniques (FBSS-ESPRIT) and forward backward spatial smoothing-propagator method (FBSS-PM) Extensive simulations are provided to validatethe effectiveness of the proposed CS-PARALIND algorithm

1 Introduction

Array signal processing has aroused considerable concerns inrecent decades owing to its extensive engineering applicationin satellite communication radar sonar and some otherfields [1ndash4] In array signal processing spectrum estimationalso known as direction of arrival (DOA) estimation is acrucial issue Till now there are already many neoteric algo-rithms [5ndash8] proposed for DOA estimation with linear arraywhich include estimating signal parameters via rotationalinvariance techniques (ESPRIT) algorithm [5 6] multiplesignal classification (MUSIC) algorithm [7] and propagatormethod (PM) [8] Compared with linear array rectangulararray can measure both azimuth angle and elevation angleand hence 2D-DOA estimation with rectangular array hasmotivated enormous investigations Many traditional DOAestimation methods have already been extended to 2D-DOAestimation [9ndash12] For example the 2D-ESPRIT algorithms[9ndash11] have utilized the invariance property to obtain the 2D-DOA estimation with uniform rectangular array and the 2D-MUSIC [12] algorithm has also proved to be applicable Someother algorithms such as angle estimation with generalized

coprime planar array [13] and 2D-DOA estimation withnested subarrays [14] have better performance for 2D-DOAestimation as well

However in many practical situations the complex prop-agation environment usually leads to the presence of coherentsignals which makes it complicated to obtain the valid DOAestimationTherefore the research on coherent angle estima-tion has gained great significanceThe algorithms mentionedabove are only applicable to noncoherent sources and thecoherent sources will lead to severe invalidity for thesemethods Some other coherent estimationmethods includingtraditional forward spatial smoothing (FSS) or forward back-ward spatial smoothing (FBSS) [15 16] algorithm have goodestimation performance Exploiting coprime multiple-inputmultiple-output (MIMO) radar [17] has proposed the DOAestimation method for mixed coherent and uncorrelatedtargets and [18] obtained the DOA estimation of coherentsources via fourth-order cumulants

The trilinear decomposition namely parallel factor(PARAFAC) technique [19ndash21] has been widely employed toresolve the problem of angular estimation with rectangulararray [22] However the decomposition of PARAFAC model

HindawiWireless Communications and Mobile ComputingVolume 2019 Article ID 3619818 10 pageshttpsdoiorg10115520193619818

2 Wireless Communications and Mobile Computing

fails to work if it contains coherent sources The parallelprofiles with linear dependencies (PARALIND) model [2324] discussed in this paper can be regarded as a generalizationof PARAFAC model and is efficient to solve the problemof coherent DOA estimation where the PARAFAC methodusually cannot present significant results In [25 26] thePARALIND decomposition method has already been suc-cessfully applied to obtain the coherent sources estimationwithMIMO radar and acoustic vector-sensor array Howeverthe traditional algorithms based on PARALIND decomposi-tion involve heavy computational burden and huge capacityconsumption of data storage

Compressed sensing (CS) [27 28] has aroused con-siderable concern which is introduced to areas includingchannel estimation image beamforming and radar [29ndash32] Specifically the angular information of sources canbe structured as a sparse vector and hence the CS tech-nique can be directly utilized [33 34] By applying CStheory many novel parameter estimation algorithms havebeen proposed for different scenarios Reference [35] pro-posed a CS-based angle estimation method for noncircularsources which can be applied to MIMO radar while [3334] combined the CS technique with PARAFAC modeland proposed an estimation algorithm for joint angle andfrequency Based on the CS PARAFAC model a 2D-DOAestimation method was presented in [36] with a uniformrectangular array whereas it works only for noncoherentsources

In this paper we propose a CS-PARALIND algorithmfor coherent sources to extract the DOA estimates with auniform rectangular array by generalizing thework presentedin [36] and the CS theory for PARALIND decompositionWe first construct the received data model which can betransformed to the PARALIND model Then we compressthe data model and perform the PARALIND decompositionon it to achieve the estimation of compressed directionmatrices Finally to acquire the 2D-DOA estimation weformulate a sparse recovery problem which can be solvedby the orthogonal matching pursuit (OMP) method [37]Note that the compression process only compresses thedirectional matrix and the source matrix while the coherentmatrix remains the same Therefore the coherent structureof impinging signals is not destroyed after compressionComparing to [36] we summarize the main contributions ofour research in this paper(1) We construct the received data model of coherentsignals for uniform rectangular array which is suitable for thePARALIND decomposition(2) We generalize the method in [36] and propose thecompressed sensing PARALIND (CS-PARALIND) modelNotably there is no existing report for CS-PARALINDdecomposition so far as we know(3) We develop the CS-PARALIND decompositionmethod to obtain the coherent 2D-DOA estimation foruniform rectangular array and give the detailed descriptionof it

In this paper the proposed algorithm can obtainautopaired 2D-DOA estimation of coherent signals Inaddition the corresponding correlated matrix can also be

Z

Y

XM

N

Subarray 1 Subarray 2 Subarray N

Source

k

k

Figure 1 The structure of uniform rectangular array

obtained The proposed algorithm can attain the sameangle estimation performance as the traditional PARALINDmethod [25] Compared to the FBSS-ESPRIT and the FBSS-PM algorithm our method can achieve better DOA esti-mation performance In addition due to the process ofcompression the proposed algorithm consumes lower com-putational burden and requires limited storage capacity inpractical application The Cramer-Rao bound (CRB) forthe DOA estimation with uniform rectangular array is alsoprovided in this paper A series of simulation results verifythe effectiveness of our approach

The remainder of our paper is organized as followsSection 2 presents the received datamodel of coherent signalswith uniform rectangular array Section 3 depicts the detailedderivation of the proposed CS-PARALIND algorithm aswell as the uniqueness certification and complexity analysisNumerical simulations are exhibited in Section 4 and weconclude this paper in Section 5

Notation oplus ∘ and otimes respectively represent Hadamardproduct KhatrindashRao product and Kronecker product ()119867()119879 ()lowast ()minus1 and ()+ stand for the operations of conjugate-transpose transpose conjugation inverse and pseudoin-verse The Frobenius norm and l0ndashnorm are denoted by ∙119865 and ∙ 0 diag(b) is a diagonal matrix composed ofelements in vector b while diagminus1(A) produces a columnvector composed of the diagonal elements ofmatrixA119863119899(A)denotes a diagonalmatrix which consists of the n-th row ofA119904119901119886119899(H) represents the subspace spanned by the columns ofH V119890119888(A)means stacking the columns of matrix A 120597meanspartial derivatives

2 Data Model

Assume that there are 119870 far-field narrow-band signals withDOA (120579119896 120593119896) impinging on a uniform rectangular arrayconsisting of119872 times 119873 sensors where 120593119896 is the azimuth angleof the 119896-th signal 120579119896 represents the elevation angle and thedistance between any two adjacent elements is 119889 The sourcesnumber K is known and the noise is additive white Gaussianwhich is assumed to be independent and uncorrelated withreceived signals Figure 1 shows the structure of the array

Wireless Communications and Mobile Computing 3

For the first subarray in the uniform rectangular array thereceived data at 119905-th time can be represented as [36]

x1 (119905) = A119909s0 (119905) + n1 (119905) (1)

where A119909 = [a119909(1205791 1205931) a119909(1205792 1205932) sdot sdot sdot a119909(120579119870 120593119870)]denotes the directional matrix of the first subarray witha119909(120579119896 120593119896) = [1 exp(1198952120587119889 cos120593119896 sin 120579119896120582) sdot sdot sdot exp(1198952120587(119872 minus1)119889 cos120593119896 sin 120579119896120582)]119879 and 119895 = radicminus1 The received noise of thefirst subarray is denoted by n1(119905) and s0(119905) is the119870times1 sourcevector of the K signals Then it follows that the received dataof 119899-th subarray at 119905-th time can be written as [36]

x119899 (119905) = A119909Φ119899minus1s0 (119905) + n119899 (119905) (2)

where Φ = diag(e1198952120587119889 sin 1205791 sin1205931120582 sdot sdot sdot e1198952120587119889 sin 120579119870 sin120593119870120582) andn119899(119905) stands for the received noise of the 119899-th subarrayTherefore the received data of the entire rectangular array canbe denoted by [38]

x (119905) = [[[[[[[

x1 (119905)x2 (119905)x119873 (119905)

]]]]]]]=[[[[[[[[

A119909A119909Φ

A119909Φ119873minus1

]]]]]]]]s0 (119905) +

[[[[[[[

n1 (119905)n2 (119905)n119873 (119905)

]]]]]]] (3)

or more compactly as

x (119905) = (A119910 ∘ A119909) s0 (119905) + n (119905) (4)

where A119910 = [a119910(1205791 1205931) a119910(1205792 1205932) sdot sdot sdot a119910(120579119870 120593119870)] witha119910(120579119896 120593119896) = [1 exp(1198952120587119889 sin120593119896 sin 120579119896120582) sdot sdot sdot exp(1198952120587(119873 minus1)119889 sin120593119896 sin 120579119896120582)] and n(119905) is the noise of the wholerectangular array with n(119905) = [n1(119905)119879n2(119905)119879 n119873(119905)119879]119879

By exploiting 119869 snapshots the received data can bedenoted as [36]

X = [x (1199051) x (1199052) sdot sdot sdot x (119905119869)] (5)

where X stands for the noisy received signals and rewriting(5) in matrix form we obtain [38]

X =[[[[[[[[

X1X2X119873

]]]]]]]]=[[[[[[[[

A1199091198631 (A119910)A1199091198632 (A119910)A119909119863119873 (A119910)

]]]]]]]]S0 +

[[[[[[[

N1N2N119873

]]]]]]]= [A119910 ∘ A119909] S0 + N

(6)

where S0 = [s0(1199051) s0(1199052) sdot sdot sdot s0(119905119869)] isin C119870times119869 and X119899 =A119909119863119899(A119910)S0 + N119899 The noise of J snapshots is N =[n(1199051)n(1199052) sdot sdot sdot n(119905119869)] isin C119872119873times119869

The received data matrix shown in (6) is a traditionalPARAFAC model [19ndash22] the decomposition of which isusually nonunique if there exist coherent signals [19]

Assume that among the K received sources there are 1198701groups of coherent signals then (6) can be represented as [25]

X = [A119910 ∘ A119909] ΓS + N (7)

where S isin C1198701times119869 is the source matrix of 119869 snapshots of1198701 noncoherent signals Γ isin C119870times1198701 is the correspondingcorrelated matrix satisfying ΓS = S0

3 CS-PARALIND Decomposition-BasedAlgorithm

The PARALIND algorithm suffers from expensive computa-tional cost especially in the case of large number of sensorsor snapshots Specifically the angular information of receivedsources can be structured as a sparse vector and hence theCS technique can be directly utilized [33] To counter thisproblem we bring in the CS theory by compressing thereceived signal in (7) to a smaller matrix followed by thePARALINDdecomposition and finally acquire the 2D-DOAestimates by exploiting sparse recovery method

31 Compression We compress the received data X isinC119872times119869times119873 into a smaller matrix X1015840 isin C119872

1015840times1198691015840times1198731015840with threecompression matrices U isin C119872times119872

1015840

V isin C119873times1198731015840

and W isinC119869times119869

1015840

with1198721015840 lt 119872 1198691015840 lt 119869 and1198731015840 lt 119873 which is illustratedin Figure 2

The three compression matrices can be constructed viasome random special matrices eg the random GaussianBernoulli andpartial Fouriermatrices or theTucker3 decom-position introduced in [33 39]

Applying U V and W to (7) we can obtain the com-pressed received data X1015840 isin C(119872

10158401198731015840times1198691015840) as [27]

X1015840 = (V119879 otimes U119879) X(119872119873times119869)W= (V119879 otimes U119879) [A119910 ∘ A119909] ΓSW + (V119879 otimes U119879)NW (8)

And according to the property of KhatrindashRao product [33]we have (V119879 otimes U119879)(A119910 ∘ A119909) = (V119879A119910) ∘ (U119879A119909) and then(8) can be further simplified to

X1015840 = [A1015840119910 ∘ A1015840119909] ΓS1015840 +N1015840 (9)

where A1015840119909(1198721015840times119870) = U119879A119909 A1015840119909

(1198721015840times119870) = U119879A119909 A1015840119910(1198731015840times119870) =

V119879A119910 S1015840(119870times1198691015840) = SW and N1015840 = (V119879 otimes U119879)NW

32 PARALIND Decomposition To transform (9) to PAR-ALIND model we define Y = X1015840

119879

or equivalently

Y = (ΓS1015840)119879 (A1015840119910 ∘ A1015840119909)119879 +N1015840119879 = Y + N1015840119879 (10)

where Y = (ΓS1015840)119879(A1015840119910 ∘ A1015840119909)119879 denotes the noise-freecompressed received matrix


Compression

M

J

N

XX

J

M

NU CCC

isin MtimesM

Visin NtimesN

Wisin JtimesJ

Figure 2 Compression processing

Obviously (10) is equivalent to the PARALIND model[20] and the least fitting for it amounts to [23]

minΓSA1015840119910A1015840119909

100381710038171003817100381710038171003817Y minus (ΓS1015840)119879 (A1015840119910 ∘ A1015840119909)1198791003817100381710038171003817100381710038172119865 (11)

Under noise-free condition (10) can be rewritten as

Y = [Y1Y2 sdot sdot sdot Y1198731015840] = [(ΓS1015840)1198791198631 (A1015840119910)A1015840119909119879 (ΓS1015840)119879sdot 1198632 (A1015840119910)A1015840119909119879 sdot sdot sdot (ΓS1015840)1198791198631198731015840 (A1015840119910)A1015840119909119879] (12)

where Y119899 = (ΓS1015840)119879119863119899(A1015840119910)A1015840119909119879 isin C1198691015840times1198721015840 119899 = 1 2 sdot sdot sdot 1198731015840

After vectoring Y119899 we have [38]

V119890119888 (Y119899) = V119890119888 ((ΓS1015840)119879119863119899 (A1015840119910)A1015840119909119879)= (A1015840119909119863119899 (A1015840119910) otimes S1015840119879) V119890119888 (Γ119879) (13)

Stacking these vectors by columns

[[[[[[[

V119890119888 (Y1)V119890119888 (Y2)V119890119888 (Y1198731015840)

]]]]]]]=[[[[[[[[[

A10158401199091198631 (A1015840119910) otimes S1015840119879

A10158401199091198632 (A1015840119910) otimes S1015840119879A10158401199091198631198731015840 (A1015840119910) otimes S1015840119879

]]]]]]]]]V119890119888 (Γ119879) (14)

Equation (14) also can be denoted as

V119890119888 (Y) = [(A1015840119910 ∘ A1015840119909) otimes S1015840119879] V119890119888 (Γ119879) (15)

Then under noise environment V119890119888(Γ119879) can be obtained by

V119890119888 (Γ119879) = [(A1015840119910 ∘ A1015840119909) otimes S1015840119879]+ V119890119888 (Y) (16)

Now we can update the value of Γ via transforming V119890119888(Γ119879)to its original modality

According to (11) the least square (LS) update for S1015840119879 is

S1015840119879 = Y (((A1015840119909 ∘ A1015840119910) Γ)119879)+ (17)

From (12) it follows that the covariance matrix of Y119899 canbe written as

1198731015840sum119899=1

Y119899119867Y119899 = (1198731015840sum

119899=1

Y119899119867 (ΓS1015840)119879119863119899 (A1015840119910))A1015840119909

119879

= A1015840119909lowast(1198731015840sum119899=1

119863119899lowast (A1015840119910) (ΓS1015840)lowast (ΓS1015840)119879119863119899 (A1015840119910))A1015840119909119879

(18)

As A1015840119909 is of full column rank andsum1198731015840119899=1119863119899lowast(A1015840119910)(ΓS1015840)lowast(ΓS1015840)119879119863119899(A1015840119910) is nonsingular [24]we can obtain A1015840119909

lowast via

A1015840119909lowast = (1198731015840sum

119899=1

Y119899119867 (ΓS1015840)119879119863119899 (A1015840119910))

sdot (1198731015840sum119899=1

119863119899lowast (A1015840119910) (ΓS1015840)lowast (ΓS1015840)119879119863119899 (A1015840119910))minus1

= (1198731015840sum119899=1

Y119899119867 (ΓS1015840)119879119863119899 (A1015840119910))

sdot (1198731015840sum119899=1

((ΓS1015840)lowast (ΓS1015840)119879) oplus (A1015840119910119867A1015840119910))minus1

(19)

Substitute Y119899 of (12) into the following

(ΓS1015840)lowast Y119899A1015840119909 = (ΓS1015840)lowast (ΓS1015840)119879119863119899 (A1015840119910)A1015840119909119879A1015840119909 (20)

Taking the diagonal elements of both sides in (20) we have

diagminus1 ((ΓS1015840)lowast Y119899A1015840119909)= ((ΓS1015840)lowast (ΓS1015840)119879)oplus (A1015840119909119879A1015840119909) diagminus1 (119863119899 (A1015840119910))

(21)


By enforcing left multiplication operation of (21) we get

diagminus1 (119863119899 (A1015840119910)) = (((ΓS1015840)lowast (ΓS1015840)119879)oplus (A1015840119909119879A1015840119909))minus1 diagminus1 ((ΓS1015840)lowast Y119899A1015840119909)

(22)

And hence A119910 can be easily obtained form (22)For the PARALIND decomposition above according to

(17) (18) and (19) we repeatedly update the estimation ofA1015840119909 A

1015840119910 S1015840 and Γ until convergence Define the sum of

squared residuals (SSR) of the k-th iteration as 119878119878119877119896 =sum11987310158401198721015840119894=1 sum1198691015840119895=1 |119888119894119895|2where 119888119894119895 is the (119894 119895) element of C = Y minus(ΓS1015840)119879(A1015840119910 ∘ A1015840119909)119879 Define the convergence rate of PARALINDdecomposition as 119878119878119877119903119886119905119890 = (119878119878119877119896 minus 119878119878119877119896minus1)119878119878119877119896minus1 [40]When 119878119878119877119903119886119905119890 is smaller than a certain small value (deter-mined by the noise level) then the above iteration processcan be considered convergent

Based on the PARALIND decomposition the estimatesof A1015840119909 and A1015840119910 can be achieved by

A1015840119909 = A1015840119909ΠΔ119909 +W119909 = U119879A119909ΠΔ119909 +W119909 (23)

A1015840119910 = A1015840119910ΠΔ119910 +W119910 = V119879A119910ΠΔ119910 +W119910 (24)

whereΠ stands for the permutationmatrix andΔ119910 andΔ119909 arethe diagonal scalingmatrix ofA1015840119910 andA

1015840119909 In additionW119909 and

W119910 denote the estimation error

Remark 1 In the proposed algorithm the scale ambiguity canbe eliminated by direct normalization while the permutationambiguity makes no difference in the angle estimation

33 DOA Estimation with Sparsity By utilizing A1015840119909 A1015840

119910 the2D-DOA estimation can be obtained with sparsity Withnoiseless case we use a1015840119909119896 and a1015840119910119896 to denote the 119896-th columnof A1015840119909 A

1015840

119910 respectively From (23) and (24) it follows that [36]

a1015840119909119896 = U119867120588119909119896a119909119896 (25)

a1015840119910119896 = V119867120588119910119896a119910119896 (26)

where a119909119896 and a119910119896 represent the 119896-th column in the direc-tional matrices A119909 and A119910 In addition 120588119909119896 and 120588119910119896 denotethe scaling coefficients

Then we construct two Vandermonde matrices A119904119909 isinC119872times119876 and A119904119910 isin C119873times119876 (119876 gtgt 119872119876 gtgt 119873) the columns ofwhich consist of the steering vectors corresponding to eachpotential source location [36]

A119904119909 = [a1199041199091 a1199041199092 sdot sdot sdot a119904119909119876]

=[[[[[[[[[

1 1 sdot sdot sdot 11198901198952120587119889g(1)120582 1198901198952120587119889g(2)120582 1198901198952120587119889g(119876)120582

1198901198952120587(119872minus1)119889g(1)120582 1198901198952120587(119872minus1)119889g(2)120582 sdot sdot sdot 1198901198952120587(119872minus1)119889g(119876)120582

]]]]]]]]] (27)


=[[[[[[[[[



]]]]]]]]] (28)

where g is a sampling vector with g(119902) = minus1 + 2119902119876 119902 =0 1 119876A119904119909 andA119904119910 can be referred to as the overcompletedictionary for our 2D-DOA estimation [36] Then (25) (26)can be converted to

a1015840119909119896 = U119879A119904119909x119904119896 119896 = 1 sdot sdot sdot 119870 (29)

a1015840119910119896 = V119879A119904119910y119904119896 119896 = 1 sdot sdot sdot 119870 (30)

where x119904119896 and y119904119896 are sparse the estimates ofwhich can be castas an optimization problem subject to 1198970-norm constraint

min 10038171003817100381710038171003817a1015840119909119896 minus U119867A119904119909x119904119896100381710038171003817100381710038172119865 119904119905 1003817100381710038171003817x11990411989610038171003817100381710038170 = 1 (31)

min 10038171003817100381710038171003817a1015840119910119896 minus V119867A119904119910y119904119896100381710038171003817100381710038172119865 119904119905 1003817100381710038171003817y11990411989610038171003817100381710038170 = 1 (32)

x119904119896 and y119904119896 can be gained by the OMP recovery method [37]Denote the maximummodulus of elements in x119904119896 and y119904119896

as 119902119909119896 and 119902119910119896 respectivelyThen the corresponding elementsg(119902119909119896) and g(119902119910119896) inA119904119909 andA119904119910 are just the sin 120579119896 cos120593119896 andsin 120579119896 sin120593119896 estimation Define 119903119896 = g(119902119909119896) + 119895g(119902119910119896) andthe estimates of elevation angles and azimuth angles can beobtained by

119896 = sinminus1 (119886119887119904 (119903119896)) 119896 = 1 2 119870 (33)

119896 = 119886119899119892119897119890 (119903119896) 119896 = 1 2 119870 (34)

where 119886119887119904() outputs the modulus value and 119886119899119892119897119890() com-putes the angular part of a complex number Finally theautopaired estimates of elevation and azimuth angles canbe attained by employing the paired relationship in thedirectionalmatrices obtained by PARALINDdecomposition

34 The Procedure of the Proposed Algorithm Till now theangle estimation of received coherent signals with a uniformrectangular array has been acquired andwe provide themajorsteps as follows


Step 1 Compress the received data X construct the PAR-ALINDmodel Y according to (8) and then initialize the valueof A1015840119909 A

1015840119910 Γ and S1015840

Step 2 According to (16) (17) (19) and (22) repeatedlyupdate the estimates of A1015840119909 A

1015840

119910 S1015840 and Γ according to the

convergence conditions

Step 3 According to (31)-(34) the elevation and azimuthangles can be estimated by A119904119909 and A119904119910

4 Performance Analysis

41 Complexity Analysis of the Proposed Algorithm We ana-lyze the complexity of the proposed algorithm in this sub-section For the proposed algorithm the computational com-plexity is 119874(11987210158401198731015840119872119873119869 + 119869119872101584011987310158401198691015840 + 1198991(119872101584011987310158401198691015840(2119870211987012 +21198701198701 + 2119870 + 1198701) + 11987210158401198731015840(211987012 + 1198702 + 1198701198701 + 2119870) +1198702(11986910158401198731015840 + 1198691015840 + 11987310158402 + 1198731015840 + 1198721015840 + 2) + 119870311987013 + 21198703 +11987013 + 11987011987011198691015840(1198731015840 + 1))) + 119870119875(119872 +119873)) where 1198991 denotes thetime of iterations With regard to the traditional PARALINDdecomposition [23] it has (1198992(119872119873119869(2119870211987012 + 21198701198701 + 2119870 +1198701) + 119872119873(211987012 + 1198702 + 1198701198701 + 2119870) + 1198702(119869119873 + 119869 + 1198732 +119873 + 119872 + 2) + 119870311987013 + 21198703 + 11987013 + 1198701198701119869(119873 + 1)) +21198702(119872+119873)+61198702) where 1198992 represents the time of iterationsIn the comparison of computational complexity we have theparameters of 119872 = 119873 = 10 119870 = 3 and 1198701 = 2 Inaddition assume that 1198991 = 1198992 = 20 119875 = 200 and1198721015840119872 =1198731015840119873 = 1198691015840119869 = 05The comparison of complexity versus 119869 isdepicted in Figure 3 which concludes that the computationalburden of the proposed algorithm is decreased compared tothe traditional PARALIND algorithm

42 CRB Define A = [a119910(1205791 1205931) otimesa119909(1205791 1205931) sdot sdot sdot a119910(120579119870 120593119870) otimes a119909(120579119870 120593119870)] and accordingto [41] we derive the CRB

119862119877119861 = 12059022119869 Re [D119867ΠperpAD oplus P119904119879]minus1 (35)

where 119869 is the number of snapshots and a119896 is the 119896-th column of A 1205902 denotes the noise power ΠperpA =I119872119873minusA(A119867A)minus1A119867 and I119872119873 is an119872119873times119872119873 identitymatrixD = [120597a11205971205791 120597a21205971205792 120597a119870120597120579119870 120597a11205971205931 120597a21205971205932120597a119870120597120593119870] and P119904 = (1119869)sum119869119905=1 s(119905)s119867(119905)43 Advantages We summarize the advantages of the pro-posed algorithm as follows(1) The proposed algorithm possesses lower computa-tional cost and requires smaller data storage due to thecompression operation(2)Theproposed algorithmcan be applied to the coherentsignals Furthermore the corresponding correlated matrixalso can be obtained(3) The proposed algorithm can achieve the same angleestimation performance with the traditional PARALINDmethod and outperforms the FBSS-ESPRIT and FBSS-PMalgorithm

150 200 250 300 3500

1

2

3

4

5

6

7

JSnapshots

Com

plex

ity

CS-PARALINDPARALIND

times107

Figure 3 Comparison of complexity versus different snapshots (J)

5 Simulation Results

Suppose that 119870 = 3 signals with two coherent signals andone noncoherent signal impinge on a rectangular array wherethe correlated matrix is Γ = [ 1 0 10 1 0 ]119879 In the followingsimulations we exploit the rootmean square error (RMSE) toevaluate the DOA estimation performance and it is presentedby

119877119872119878119864 = 1119870119870sum119896=1

radic 1119871119871sum119897=1

(119896119897 minus 120593119896)2 + (119896119897 minus 120579119896)2 (36)

where 119896119897 119896119897 are the estimations of 120593119896 120579119896 in 119897-th simulationand the times of Monte-Carlo simulations are indicated by 119871In the following simulations we set 119871 = 1000

The locations of the three sources are (1205791 1205931) = (15∘ 10∘)(1205792 1205932) = (25∘ 30∘) and (1205793 1205933) = (35∘ 50∘)119872119873 119869 119870 arethe number of rows and columns of the rectangular arraysnapshots and sources respectively And assume that therectangular array is uniform and the distance of any twoadjacent sensors is 1205822Simulation 1 Figure 4 shows the 2D-DOA estimation resultsof our proposed algorithm with SNR=0dB and SNR=20dBrespectively The size of PARALINDmodel is defined as119872times119873times119869The received data matrix has the size of 10times10times200 inthis simulation which becomes 7times7times140(1198721015840times1198731015840times1198691015840) aftercompression Figure 4 indicates that the proposed algorithmis effective for 2D-DOA estimation of coherent sources

Simulation 2 In Figure 5 we exhibit the comparison of theDOA estimation performance among the proposed algo-rithm FSS-ESPRIT FSS-PM FBSS-ESPRIT and FBSS-PMalgorithm and the traditional PARALIND algorithm The


0 10 20 30 40 50 6010

15

20

25

30

35

40

Azimuth (degree)

Elev

atio

n (d

egre

e)

(a)

0 10 20 30 40 50 6010

15

20

25

30

35

40

Azimuth (degree)

elev

atio

n (d

egre

e)

(b)

Figure 4 DOA estimation of the proposed algorithm with (a) SNR=0dB (b) SNR=20dB

0

RMSE

(deg

ree)

5 10 15 20 25SNR (dB)

FSS-PMFSS-ESPRITFBSS-PMFBSS-ESPRITPARALIND

PARALINDCS-CRB

102

100

10minus2

Figure 5 DOA estimation performance comparison

received data matrix has the size of 10 times 10 times 100 and iscompressed to the size of 8 times 8 times 80 It is illustrated clearlyin Figure 5 that the approach algorithm can achieve the sameDOA estimation performance compared with the traditionalPARALIND algorithm and furthermore outperforms theother four algorithms

Simulation 3 TheDOA estimation performance result of theproposed algorithm versus compression ratio is provided inFigure 6 where the compression ratio is 119875 = 1198721015840119872 =1198731015840119873 = 1198691015840119869 In this part the size of the original PARALIND

RMSE

(deg

ree)

P=04P=06P=08

101

100

10minus1

10minus2 0 5 10 15 20 25SNR (dB)

Figure 6 DOA estimation performance under different compres-sion ratio

model is 15 times 15 times 200 with 119875 = 04 119875 = 06 and 119875 = 08 Itis shown in Figure 6 that the estimation performance of ourmethod improves with P getting larger

Simulation 4 The angle estimation performance of the pro-posed algorithm versus 119869 is illustrated in Figure 7 where therectangular array is structured as 119872 = 10 119873 = 10 and1198721015840 = 1198731015840 = 8 1198691015840119869 = 08 It is verified in Figure 7 that theproposed algorithm can obtain improved DOA estimationaccuracy with a larger snapshots

Simulation 5 Figure 8 shows DOA estimation performanceof the proposed algorithm with varied119872 Assume that 119873 =1198731015840 = 10 and 119869 = 1198691015840 = 200 with1198721015840119872 = 23 It is attested


0 5 10 15 20 25

RMSE

(deg

ree)

101

100

10minus1

10minus2

J=100Jrsquo=80J=200Jrsquo=160J=300Jrsquo=240

SNR (dB)

Figure 7 DOA estimation performance under different snapshots

0 5 10 15 20 25SNR (dB)

RMSE

(deg

ree)

M=12Mrsquo=8M=15Mrsquo=10M=18Mrsquo=12

101

100

10minus1

10minus2

Figure 8 DOA estimation performance under differentM

by Figure 8 that the estimation performance of the proposedmethod gets better with the increasing119872

Simulation 6 The estimation performance of the proposedmethod versus 119873 is illustrated in Figure 9 where 119872 =15 119869 = 200 with 1198721015840119872 = 1198691015840119869 = 1198731015840119873 = 08It is indicated obviously that the proposed method obtainsimproved performance with larger119873

6 Conclusions

We jointly utilize the CS theory and the PARALIND decom-position in this paper and propose a CS-PARALIND algo-rithm with a uniform rectangular array to extract the DOAestimates of coherent signals The proposed algorithm canattain autopaired 2D-DOA estimates and benefiting from

0 5 10 15 20 25SNR (dB)

RMSE

(deg

ree)

N=10Nrsquo=8N=15Nrsquo=12N=20Nrsquo=16

101

100

10minus1

10minus2

Figure 9 DOA estimation performance under different N

the compression procedure lower computational cost andsmaller demand for storage capacity can be achieved Exten-sive simulations corroborate that the proposed approachobtains superior estimation performance compared to thetraditional PARALIND algorithm and significantly outper-forms the FBSS-PM and FBSS- ESPRIT algorithm

Data Availability

The simulation data used to support the findings of this studyare included within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by China NSF Grants (6137116961601167) the Fundamental Research Funds for the CentralUniversities (NT2019013) Graduate Innovative Base (labo-ratory) Open Funding of Nanjing University of AeronauticsandAstronautics (kfjj20170412) and Shandong ProvinceNSFGrants (ZR2016FM43)

References

[1] X Y Yu and M Q Zhou ldquoA DOA estimation algorithm inOFDMradar-communication system for vehicular applicationrdquoAdvanced Materials Research vol 926-930 pp 1853ndash1856 2014

[2] J Li X Zhang andH Chen ldquoImproved two-dimensional DOAestimation algorithm for two-parallel uniform linear arraysusing propagator methodrdquo Signal Processing vol 92 no 12 pp3032ndash3038 2012


[3] X F Zhang R A Cao and M Zhou ldquoNoncircular-PARAFACfor 2D-DOA estimation of noncircular signals in arbitrarilyspaced acoustic vector-sensor array subjected to unknownlocationsrdquo EURASIP Journal on Advances in Signal Processingvol 2013 no 1 article no 107 2013

[4] X Zhang L Xu D Xu et al ldquoDirection of departure (DOD)and direction of arrival (DOA) estimation inMIMO radar withreduced-dimension MUSICrdquo IEEE Communications Lettersvol 14 no 12 pp 1161ndash1163 2010

[5] R Roy and T Kailath ldquoESPRITmdashestimation of signal parame-ters via rotational invariance techniquesrdquo IEEE Transactions onSignal Processing vol 37 no 7 pp 984ndash995 1989

[6] F Gao and A B Gershman ldquoA generalized ESPRIT approach todirection-of-arrival estimationrdquo IEEE Signal Processing Lettersvol 12 no 3 pp 254ndash257 2005

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

[8] S Marcos A Marsal and M Benidir ldquoThe propagator methodfor source bearing estimationrdquo Signal Processing vol 42 no 2pp 121ndash138 1995

[9] T Filik and T E Tuncer ldquo2-D paired direction-of-arrivalangle estimation with two parallel uniform linear arraysrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 6 pp 3269ndash3279 2011

[10] Y-Y Wang and S-C Huang ldquoAn ESPRIT-based algorithm for2D-DOA estimationrdquo IEICE Transactions on Fundamentals ofElectronics Communications andComputer Sciences vol E94Ano 9 pp 1847ndash1850 2011

[11] C P Mathews M Haardt and M D Zoltowski ldquoPerformanceanalysis of closed-form ESPRIT based 2-D angle estimator forrectangular arraysrdquo IEEE Signal Processing Letters vol 3 no 4pp 124ndash126 1996

[12] Y Hua ldquoA pencil-MUSIC algorithm for finding two-dimensional angles and polarizations using crossed dipolesrdquoIEEE Transactions on Antennas and Propagation vol 41 no 3pp 370ndash376 1993

[13] W Zheng X Zhang and H Zhai ldquoA generalized coprimeplanar array geometry for two-dimensional DOA estimationrdquoIEEE Communications Letters vol 99 p 1 2017

[14] Y-Y Dong C-X Dong Y-T Zhu G-Q Zhao and S-Y LiuldquoTwo-dimensional DOA estimation for L-shaped array withnested subarrays without pair matchingrdquo IET Signal Processingvol 10 no 9 pp 1112ndash1117 2016

[15] 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

[16] S U Pillai and B H Kwon ldquoForwardbackward spatialsmoothing techniques for coherent signal identificationrdquo IEEETransactions on Signal Processing vol 37 no 1 pp 8ndash15 1989

[17] SQin YD Zhang andMGAmin ldquoDOAestimation ofmixedcoherent and uncorrelated targets exploiting coprime MIMOradarrdquo Digital Signal Processing vol 61 pp 26ndash34 2017

[18] Y Hu Y Liu and XWang ldquoDoa estimation of coherent signalson coprime arrays exploiting fourth-order cumulantsrdquo Sensorsvol 17 no 4 article no 682 2017

[19] J B Kruskal ldquoThree-way arrays rank and uniqueness of trilin-ear decompositions with application to arithmetic complexityand statisticsrdquo Linear Algebra and its Applications vol 18 no 2pp 95ndash138 1977

[20] R Cao X Zhang and C Wang ldquoReduced-dimensionalPARAFAC-based algorithm for joint angle and doppler fre-quency estimation in monostatic MIMO radarrdquo Wireless Per-sonal Communications vol 80 no 3 pp 1231ndash1249 2014

[21] N D Sidiropoulos L De Lathauwer X Fu K Huang E EPapalexakis andC Faloutsos ldquoTensor decomposition for signalprocessing and machine learningrdquo IEEE Transactions on SignalProcessing vol 65 no 13 pp 3551ndash3582 2017

[22] N D Sidiropoulos R Bro and G B Giannakis ldquoParallel factoranalysis in sensor array processingrdquo IEEETransactions on SignalProcessing vol 48 no 8 pp 2377ndash2388 2000

[23] R Bro R A Harshman N D Sidiropoulos and M E LundyldquoModeling multi-way data with linearly dependent loadingsrdquoJournal of Chemometrics vol 23 no 7-8 pp 324ndash340 2009

[24] M Bahram and R Bro ldquoA novel strategy for solving matrixeffect in three-way data using parallel profiles with lineardependenciesrdquo Analytica Chimica Acta vol 584 no 2 pp 397ndash402 2007

[25] C Chen X F Zhang and D Ben ldquoCoherent angle estimationin bistatic multi-input multi-output radar using parallel profilewith linear dependencies decompositionrdquo IET Radar Sonar andNavigation vol 7 no 8 pp 867ndash874 2013

[26] X F Zhang M Zhou and J F Li ldquoA PARALINDdecomposition-based coherent two-dimensional directionof arrival estimation algorithm for acoustic vector-sensorarraysrdquo Sensors vol 13 no 4 p 5302 2013

[27] D L Donoho ldquoCompressed sensingrdquo IEEE Transactions onInformation Theory vol 52 no 4 pp 1289ndash1306 2006

[28] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 52 no 2 pp489ndash509 2006

[29] D Malioutov M Cetin and A S Willsky ldquoA sparse signalreconstruction perspective for source localization with sensorarraysrdquo IEEE Transactions on Signal Processing vol 53 no 8pp 3010ndash3022 2005

[30] X Yang S A Shah A Ren et al ldquoWandering pattern sensingat S-bandrdquo IEEE Journal of Biomedical and Health Informaticsvol 22 no 6 pp 1863ndash1870 2018

[31] J-Q Wu W Zhu and B Chen ldquoCompressed sensing tech-niques for altitude estimation in multipath conditionsrdquo IEEETransactions on Aerospace and Electronic Systems vol 51 no 3pp 1891ndash1900 2015

[32] C-R Tsai andA-YWu ldquoStructured random compressed chan-nel sensing for millimeter-wave large-scale antenna systemsrdquoIEEE Transactions on Signal Processing vol 66 no 19 pp 5096ndash5110 2018

[33] N D Sidiropoulos and A Kyrillidis ldquoMulti-way compressedsensing for sparse low-rank tensorsrdquo IEEE Signal ProcessingLetters vol 19 no 11 pp 757ndash760 2012

[34] T Nishimura Y Ogawa and T Ohgane ldquoDOA estimationby applying compressed sensing techniquesrdquo in Proceedingsof the 2014 IEEE International Workshop on ElectromagneticsApplications and Student Innovation Competition IEEE iWEM2014 pp 121-122 Japan August 2014

[35] C Guang and L Qi ldquoCompressed sensing-based angle estima-tion for noncircular sources in MIMO radarrdquo in Proceedingsof the 4th International Conference on Instrumentation andMeasurement Computer Communication and Control IMCCC2014 pp 40ndash44 China September 2014


[36] H X Yu X F Qiu X F Zhang et al ldquoTwo-dimensionaldirection of arrival (DOA) estimation for rectangular array viacompressive sensing trilinear modelrdquo International Journal ofAntennas amp Propagation vol 2015 10 pages 2016

[37] J A Tropp and A C Gilbert ldquoSignal recovery from randommeasurements via orthogonal matching pursuitrdquo IEEE Trans-actions on Information Theory vol 53 no 12 pp 4655ndash46662007

[38] X F Zhang F Wang and H W Chen Theory and Applicationof Array Signal Processing National Defense Industry PressBeijing China 2012

[39] S Li and X F Zhang ldquoStudy on the compressed matrices incompressed sensing trilinear modelrdquo Applied Mechanics andMaterials vol 556-562 pp 3380ndash3383 2014

[40] L Xu R Wu X Zhang and Z Shi ldquoJoint two-dimensionalDOA and frequency estimation for L-Shaped array via com-pressed sensing PARAFAC methodrdquo IEEE Access vol 6 pp37204ndash37213 2018

[41] P Stoica andANehorai ldquoPerformance study of conditional andunconditional direction-of-arrival estimationrdquo IEEE Transac-tions on Signal Processing vol 38 no 10 pp 1783ndash1795 1990

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fails to work if it contains coherent sources The parallelprofiles with linear dependencies (PARALIND) model [2324] discussed in this paper can be regarded as a generalizationof PARAFAC model and is efficient to solve the problemof coherent DOA estimation where the PARAFAC methodusually cannot present significant results In [25 26] thePARALIND decomposition method has already been suc-cessfully applied to obtain the coherent sources estimationwithMIMO radar and acoustic vector-sensor array Howeverthe traditional algorithms based on PARALIND decomposi-tion involve heavy computational burden and huge capacityconsumption of data storage

Compressed sensing (CS) [27 28] has aroused con-siderable concern which is introduced to areas includingchannel estimation image beamforming and radar [29ndash32] Specifically the angular information of sources canbe structured as a sparse vector and hence the CS tech-nique can be directly utilized [33 34] By applying CStheory many novel parameter estimation algorithms havebeen proposed for different scenarios Reference [35] pro-posed a CS-based angle estimation method for noncircularsources which can be applied to MIMO radar while [3334] combined the CS technique with PARAFAC modeland proposed an estimation algorithm for joint angle andfrequency Based on the CS PARAFAC model a 2D-DOAestimation method was presented in [36] with a uniformrectangular array whereas it works only for noncoherentsources

In this paper we propose a CS-PARALIND algorithmfor coherent sources to extract the DOA estimates with auniform rectangular array by generalizing thework presentedin [36] and the CS theory for PARALIND decompositionWe first construct the received data model which can betransformed to the PARALIND model Then we compressthe data model and perform the PARALIND decompositionon it to achieve the estimation of compressed directionmatrices Finally to acquire the 2D-DOA estimation weformulate a sparse recovery problem which can be solvedby the orthogonal matching pursuit (OMP) method [37]Note that the compression process only compresses thedirectional matrix and the source matrix while the coherentmatrix remains the same Therefore the coherent structureof impinging signals is not destroyed after compressionComparing to [36] we summarize the main contributions ofour research in this paper(1) We construct the received data model of coherentsignals for uniform rectangular array which is suitable for thePARALIND decomposition(2) We generalize the method in [36] and propose thecompressed sensing PARALIND (CS-PARALIND) modelNotably there is no existing report for CS-PARALINDdecomposition so far as we know(3) We develop the CS-PARALIND decompositionmethod to obtain the coherent 2D-DOA estimation foruniform rectangular array and give the detailed descriptionof it

In this paper the proposed algorithm can obtainautopaired 2D-DOA estimation of coherent signals Inaddition the corresponding correlated matrix can also be

Z

Y

XM

N

Subarray 1 Subarray 2 Subarray N

Source

k

k

Figure 1 The structure of uniform rectangular array

obtained The proposed algorithm can attain the sameangle estimation performance as the traditional PARALINDmethod [25] Compared to the FBSS-ESPRIT and the FBSS-PM algorithm our method can achieve better DOA esti-mation performance In addition due to the process ofcompression the proposed algorithm consumes lower com-putational burden and requires limited storage capacity inpractical application The Cramer-Rao bound (CRB) forthe DOA estimation with uniform rectangular array is alsoprovided in this paper A series of simulation results verifythe effectiveness of our approach

The remainder of our paper is organized as followsSection 2 presents the received datamodel of coherent signalswith uniform rectangular array Section 3 depicts the detailedderivation of the proposed CS-PARALIND algorithm aswell as the uniqueness certification and complexity analysisNumerical simulations are exhibited in Section 4 and weconclude this paper in Section 5

Notation oplus ∘ and otimes respectively represent Hadamardproduct KhatrindashRao product and Kronecker product ()119867()119879 ()lowast ()minus1 and ()+ stand for the operations of conjugate-transpose transpose conjugation inverse and pseudoin-verse The Frobenius norm and l0ndashnorm are denoted by ∙119865 and ∙ 0 diag(b) is a diagonal matrix composed ofelements in vector b while diagminus1(A) produces a columnvector composed of the diagonal elements ofmatrixA119863119899(A)denotes a diagonalmatrix which consists of the n-th row ofA119904119901119886119899(H) represents the subspace spanned by the columns ofH V119890119888(A)means stacking the columns of matrix A 120597meanspartial derivatives

2 Data Model

Assume that there are 119870 far-field narrow-band signals withDOA (120579119896 120593119896) impinging on a uniform rectangular arrayconsisting of119872 times 119873 sensors where 120593119896 is the azimuth angleof the 119896-th signal 120579119896 represents the elevation angle and thedistance between any two adjacent elements is 119889 The sourcesnumber K is known and the noise is additive white Gaussianwhich is assumed to be independent and uncorrelated withreceived signals Figure 1 shows the structure of the array



x1 (119905) = A119909s0 (119905) + n1 (119905) (1)


x119899 (119905) = A119909Φ119899minus1s0 (119905) + n119899 (119905) (2)


x (119905) = [[[[[[[

x1 (119905)x2 (119905)x119873 (119905)

]]]]]]]=[[[[[[[[

A119909A119909Φ

A119909Φ119873minus1

]]]]]]]]s0 (119905) +

[[[[[[[

n1 (119905)n2 (119905)n119873 (119905)

]]]]]]] (3)


x (119905) = (A119910 ∘ A119909) s0 (119905) + n (119905) (4)





X =[[[[[[[[

X1X2X119873

]]]]]]]]=[[[[[[[[

A1199091198631 (A119910)A1199091198632 (A119910)A119909119863119873 (A119910)

]]]]]]]]S0 +

[[[[[[[

N1N2N119873

]]]]]]]= [A119910 ∘ A119909] S0 + N

(6)




X = [A119910 ∘ A119909] ΓS + N (7)






1015840

V isin C119873times1198731015840


1015840




10158401198731015840times1198691015840) as [27]



X1015840 = [A1015840119910 ∘ A1015840119909] ΓS1015840 +N1015840 (9)

where A1015840119909(1198721015840times119870) = U119879A119909 A1015840119909

(1198721015840times119870) = U119879A119909 A1015840119910(1198731015840times119870) =



119879

or equivalently

Y = (ΓS1015840)119879 (A1015840119910 ∘ A1015840119909)119879 +N1015840119879 = Y + N1015840119879 (10)



Compression

M

J

N

XX

J

M

NU CCC

isin MtimesM

Visin NtimesN

Wisin JtimesJ



minΓSA1015840119910A1015840119909

100381710038171003817100381710038171003817Y minus (ΓS1015840)119879 (A1015840119910 ∘ A1015840119909)1198791003817100381710038171003817100381710038172119865 (11)







[[[[[[[

V119890119888 (Y1)V119890119888 (Y2)V119890119888 (Y1198731015840)

]]]]]]]=[[[[[[[[[

A10158401199091198631 (A1015840119910) otimes S1015840119879


]]]]]]]]]V119890119888 (Γ119879) (14)


V119890119888 (Y) = [(A1015840119910 ∘ A1015840119909) otimes S1015840119879] V119890119888 (Γ119879) (15)


V119890119888 (Γ119879) = [(A1015840119910 ∘ A1015840119909) otimes S1015840119879]+ V119890119888 (Y) (16)



S1015840119879 = Y (((A1015840119909 ∘ A1015840119910) Γ)119879)+ (17)


1198731015840sum119899=1

Y119899119867Y119899 = (1198731015840sum

119899=1

Y119899119867 (ΓS1015840)119879119863119899 (A1015840119910))A1015840119909

119879

= A1015840119909lowast(1198731015840sum119899=1


(18)


lowast via

A1015840119909lowast = (1198731015840sum

119899=1

Y119899119867 (ΓS1015840)119879119863119899 (A1015840119910))

sdot (1198731015840sum119899=1


= (1198731015840sum119899=1

Y119899119867 (ΓS1015840)119879119863119899 (A1015840119910))

sdot (1198731015840sum119899=1


(19)





(21)




(22)






A1015840119909 = A1015840119909ΠΔ119909 +W119909 = U119879A119909ΠΔ119909 +W119909 (23)

A1015840119910 = A1015840119910ΠΔ119910 +W119910 = V119879A119910ΠΔ119910 +W119910 (24)


1015840119909 In additionW119909 and





1015840


a1015840119909119896 = U119867120588119909119896a119909119896 (25)

a1015840119910119896 = V119867120588119910119896a119910119896 (26)




=[[[[[[[[[



]]]]]]]]] (27)


=[[[[[[[[[



]]]]]]]]] (28)





min 10038171003817100381710038171003817a1015840119909119896 minus U119867A119904119909x119904119896100381710038171003817100381710038172119865 119904119905 1003817100381710038171003817x11990411989610038171003817100381710038170 = 1 (31)

min 10038171003817100381710038171003817a1015840119910119896 minus V119867A119904119910y119904119896100381710038171003817100381710038172119865 119904119905 1003817100381710038171003817y11990411989610038171003817100381710038170 = 1 (32)



119896 = sinminus1 (119886119887119904 (119903119896)) 119896 = 1 2 119870 (33)

119896 = 119886119899119892119897119890 (119903119896) 119896 = 1 2 119870 (34)





1015840119910 Γ and S1015840


1015840









150 200 250 300 3500

1

2

3

4

5

6

7

JSnapshots

Com

plex

ity

CS-PARALINDPARALIND

times107




119877119872119878119864 = 1119870119870sum119896=1

radic 1119871119871sum119897=1

(119896119897 minus 120593119896)2 + (119896119897 minus 120579119896)2 (36)





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PARALINDCS-CRB

102

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RMSE

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101

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6 Conclusions


0 5 10 15 20 25SNR (dB)

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ree)


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10minus2



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Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




x1 (119905) = A119909s0 (119905) + n1 (119905) (1)


x119899 (119905) = A119909Φ119899minus1s0 (119905) + n119899 (119905) (2)


x (119905) = [[[[[[[

x1 (119905)x2 (119905)x119873 (119905)

]]]]]]]=[[[[[[[[

A119909A119909Φ

A119909Φ119873minus1

]]]]]]]]s0 (119905) +

[[[[[[[

n1 (119905)n2 (119905)n119873 (119905)

]]]]]]] (3)


x (119905) = (A119910 ∘ A119909) s0 (119905) + n (119905) (4)





X =[[[[[[[[

X1X2X119873

]]]]]]]]=[[[[[[[[

A1199091198631 (A119910)A1199091198632 (A119910)A119909119863119873 (A119910)

]]]]]]]]S0 +

[[[[[[[

N1N2N119873

]]]]]]]= [A119910 ∘ A119909] S0 + N

(6)




X = [A119910 ∘ A119909] ΓS + N (7)






1015840

V isin C119873times1198731015840


1015840




10158401198731015840times1198691015840) as [27]



X1015840 = [A1015840119910 ∘ A1015840119909] ΓS1015840 +N1015840 (9)

where A1015840119909(1198721015840times119870) = U119879A119909 A1015840119909

(1198721015840times119870) = U119879A119909 A1015840119910(1198731015840times119870) =



119879

or equivalently

Y = (ΓS1015840)119879 (A1015840119910 ∘ A1015840119909)119879 +N1015840119879 = Y + N1015840119879 (10)



Compression

M

J

N

XX

J

M

NU CCC

isin MtimesM

Visin NtimesN

Wisin JtimesJ



minΓSA1015840119910A1015840119909

100381710038171003817100381710038171003817Y minus (ΓS1015840)119879 (A1015840119910 ∘ A1015840119909)1198791003817100381710038171003817100381710038172119865 (11)







[[[[[[[

V119890119888 (Y1)V119890119888 (Y2)V119890119888 (Y1198731015840)

]]]]]]]=[[[[[[[[[

A10158401199091198631 (A1015840119910) otimes S1015840119879


]]]]]]]]]V119890119888 (Γ119879) (14)


V119890119888 (Y) = [(A1015840119910 ∘ A1015840119909) otimes S1015840119879] V119890119888 (Γ119879) (15)


V119890119888 (Γ119879) = [(A1015840119910 ∘ A1015840119909) otimes S1015840119879]+ V119890119888 (Y) (16)



S1015840119879 = Y (((A1015840119909 ∘ A1015840119910) Γ)119879)+ (17)


1198731015840sum119899=1

Y119899119867Y119899 = (1198731015840sum

119899=1

Y119899119867 (ΓS1015840)119879119863119899 (A1015840119910))A1015840119909

119879

= A1015840119909lowast(1198731015840sum119899=1


(18)


lowast via

A1015840119909lowast = (1198731015840sum

119899=1

Y119899119867 (ΓS1015840)119879119863119899 (A1015840119910))

sdot (1198731015840sum119899=1


= (1198731015840sum119899=1

Y119899119867 (ΓS1015840)119879119863119899 (A1015840119910))

sdot (1198731015840sum119899=1


(19)





(21)




(22)






A1015840119909 = A1015840119909ΠΔ119909 +W119909 = U119879A119909ΠΔ119909 +W119909 (23)

A1015840119910 = A1015840119910ΠΔ119910 +W119910 = V119879A119910ΠΔ119910 +W119910 (24)


1015840119909 In additionW119909 and





1015840


a1015840119909119896 = U119867120588119909119896a119909119896 (25)

a1015840119910119896 = V119867120588119910119896a119910119896 (26)




=[[[[[[[[[



]]]]]]]]] (27)


=[[[[[[[[[



]]]]]]]]] (28)





min 10038171003817100381710038171003817a1015840119909119896 minus U119867A119904119909x119904119896100381710038171003817100381710038172119865 119904119905 1003817100381710038171003817x11990411989610038171003817100381710038170 = 1 (31)

min 10038171003817100381710038171003817a1015840119910119896 minus V119867A119904119910y119904119896100381710038171003817100381710038172119865 119904119905 1003817100381710038171003817y11990411989610038171003817100381710038170 = 1 (32)



119896 = sinminus1 (119886119887119904 (119903119896)) 119896 = 1 2 119870 (33)

119896 = 119886119899119892119897119890 (119903119896) 119896 = 1 2 119870 (34)





1015840119910 Γ and S1015840


1015840









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7

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Com

plex

ity

CS-PARALINDPARALIND

times107




119877119872119878119864 = 1119870119870sum119896=1

radic 1119871119871sum119897=1

(119896119897 minus 120593119896)2 + (119896119897 minus 120579119896)2 (36)





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(b)


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PARALINDCS-CRB

102

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10minus2




RMSE

(deg

ree)

P=04P=06P=08

101

100

10minus1

10minus2 0 5 10 15 20 25SNR (dB)






0 5 10 15 20 25

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(deg

ree)

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ree)


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6 Conclusions


0 5 10 15 20 25SNR (dB)

RMSE

(deg

ree)


101

100

10minus1

10minus2



Data Availability




Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Compression

M

J

N

XX

J

M

NU CCC

isin MtimesM

Visin NtimesN

Wisin JtimesJ



minΓSA1015840119910A1015840119909

100381710038171003817100381710038171003817Y minus (ΓS1015840)119879 (A1015840119910 ∘ A1015840119909)1198791003817100381710038171003817100381710038172119865 (11)







[[[[[[[

V119890119888 (Y1)V119890119888 (Y2)V119890119888 (Y1198731015840)

]]]]]]]=[[[[[[[[[

A10158401199091198631 (A1015840119910) otimes S1015840119879


]]]]]]]]]V119890119888 (Γ119879) (14)


V119890119888 (Y) = [(A1015840119910 ∘ A1015840119909) otimes S1015840119879] V119890119888 (Γ119879) (15)


V119890119888 (Γ119879) = [(A1015840119910 ∘ A1015840119909) otimes S1015840119879]+ V119890119888 (Y) (16)



S1015840119879 = Y (((A1015840119909 ∘ A1015840119910) Γ)119879)+ (17)


1198731015840sum119899=1

Y119899119867Y119899 = (1198731015840sum

119899=1

Y119899119867 (ΓS1015840)119879119863119899 (A1015840119910))A1015840119909

119879

= A1015840119909lowast(1198731015840sum119899=1


(18)


lowast via

A1015840119909lowast = (1198731015840sum

119899=1

Y119899119867 (ΓS1015840)119879119863119899 (A1015840119910))

sdot (1198731015840sum119899=1


= (1198731015840sum119899=1

Y119899119867 (ΓS1015840)119879119863119899 (A1015840119910))

sdot (1198731015840sum119899=1


(19)





(21)




(22)






A1015840119909 = A1015840119909ΠΔ119909 +W119909 = U119879A119909ΠΔ119909 +W119909 (23)

A1015840119910 = A1015840119910ΠΔ119910 +W119910 = V119879A119910ΠΔ119910 +W119910 (24)


1015840119909 In additionW119909 and





1015840


a1015840119909119896 = U119867120588119909119896a119909119896 (25)

a1015840119910119896 = V119867120588119910119896a119910119896 (26)




=[[[[[[[[[



]]]]]]]]] (27)


=[[[[[[[[[



]]]]]]]]] (28)





min 10038171003817100381710038171003817a1015840119909119896 minus U119867A119904119909x119904119896100381710038171003817100381710038172119865 119904119905 1003817100381710038171003817x11990411989610038171003817100381710038170 = 1 (31)

min 10038171003817100381710038171003817a1015840119910119896 minus V119867A119904119910y119904119896100381710038171003817100381710038172119865 119904119905 1003817100381710038171003817y11990411989610038171003817100381710038170 = 1 (32)



119896 = sinminus1 (119886119887119904 (119903119896)) 119896 = 1 2 119870 (33)

119896 = 119886119899119892119897119890 (119903119896) 119896 = 1 2 119870 (34)





1015840119910 Γ and S1015840


1015840









150 200 250 300 3500

1

2

3

4

5

6

7

JSnapshots

Com

plex

ity

CS-PARALINDPARALIND

times107




119877119872119878119864 = 1119870119870sum119896=1

radic 1119871119871sum119897=1

(119896119897 minus 120593119896)2 + (119896119897 minus 120579119896)2 (36)





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PARALINDCS-CRB

102

100

10minus2




RMSE

(deg

ree)

P=04P=06P=08

101

100

10minus1

10minus2 0 5 10 15 20 25SNR (dB)






0 5 10 15 20 25

RMSE

(deg

ree)

101

100

10minus1

10minus2


SNR (dB)


0 5 10 15 20 25SNR (dB)

RMSE

(deg

ree)


101

100

10minus1

10minus2




6 Conclusions


0 5 10 15 20 25SNR (dB)

RMSE

(deg

ree)


101

100

10minus1

10minus2



Data Availability




Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





(22)






A1015840119909 = A1015840119909ΠΔ119909 +W119909 = U119879A119909ΠΔ119909 +W119909 (23)

A1015840119910 = A1015840119910ΠΔ119910 +W119910 = V119879A119910ΠΔ119910 +W119910 (24)


1015840119909 In additionW119909 and





1015840


a1015840119909119896 = U119867120588119909119896a119909119896 (25)

a1015840119910119896 = V119867120588119910119896a119910119896 (26)




=[[[[[[[[[



]]]]]]]]] (27)


=[[[[[[[[[



]]]]]]]]] (28)





min 10038171003817100381710038171003817a1015840119909119896 minus U119867A119904119909x119904119896100381710038171003817100381710038172119865 119904119905 1003817100381710038171003817x11990411989610038171003817100381710038170 = 1 (31)

min 10038171003817100381710038171003817a1015840119910119896 minus V119867A119904119910y119904119896100381710038171003817100381710038172119865 119904119905 1003817100381710038171003817y11990411989610038171003817100381710038170 = 1 (32)



119896 = sinminus1 (119886119887119904 (119903119896)) 119896 = 1 2 119870 (33)

119896 = 119886119899119892119897119890 (119903119896) 119896 = 1 2 119870 (34)





1015840119910 Γ and S1015840


1015840









150 200 250 300 3500

1

2

3

4

5

6

7

JSnapshots

Com

plex

ity

CS-PARALINDPARALIND

times107




119877119872119878119864 = 1119870119870sum119896=1

radic 1119871119871sum119897=1

(119896119897 minus 120593119896)2 + (119896119897 minus 120579119896)2 (36)





0 10 20 30 40 50 6010

15

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elev

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(b)


0

RMSE

(deg

ree)

5 10 15 20 25SNR (dB)


PARALINDCS-CRB

102

100

10minus2




RMSE

(deg

ree)

P=04P=06P=08

101

100

10minus1

10minus2 0 5 10 15 20 25SNR (dB)






0 5 10 15 20 25

RMSE

(deg

ree)

101

100

10minus1

10minus2


SNR (dB)


0 5 10 15 20 25SNR (dB)

RMSE

(deg

ree)


101

100

10minus1

10minus2




6 Conclusions


0 5 10 15 20 25SNR (dB)

RMSE

(deg

ree)


101

100

10minus1

10minus2



Data Availability




Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




1015840119910 Γ and S1015840


1015840









150 200 250 300 3500

1

2

3

4

5

6

7

JSnapshots

Com

plex

ity

CS-PARALINDPARALIND

times107




119877119872119878119864 = 1119870119870sum119896=1

radic 1119871119871sum119897=1

(119896119897 minus 120593119896)2 + (119896119897 minus 120579119896)2 (36)





0 10 20 30 40 50 6010

15

20

25

30

35

40

Azimuth (degree)

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atio

n (d

egre

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(a)

0 10 20 30 40 50 6010

15

20

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40

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elev

atio

n (d

egre

e)

(b)


0

RMSE

(deg

ree)

5 10 15 20 25SNR (dB)


PARALINDCS-CRB

102

100

10minus2




RMSE

(deg

ree)

P=04P=06P=08

101

100

10minus1

10minus2 0 5 10 15 20 25SNR (dB)






0 5 10 15 20 25

RMSE

(deg

ree)

101

100

10minus1

10minus2


SNR (dB)


0 5 10 15 20 25SNR (dB)

RMSE

(deg

ree)


101

100

10minus1

10minus2




6 Conclusions


0 5 10 15 20 25SNR (dB)

RMSE

(deg

ree)


101

100

10minus1

10minus2



Data Availability




Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0 10 20 30 40 50 6010

15

20

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Elev

atio

n (d

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(a)

0 10 20 30 40 50 6010

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40

Azimuth (degree)

elev

atio

n (d

egre

e)

(b)


0

RMSE

(deg

ree)

5 10 15 20 25SNR (dB)


PARALINDCS-CRB

102

100

10minus2




RMSE

(deg

ree)

P=04P=06P=08

101

100

10minus1

10minus2 0 5 10 15 20 25SNR (dB)






0 5 10 15 20 25

RMSE

(deg

ree)

101

100

10minus1

10minus2


SNR (dB)


0 5 10 15 20 25SNR (dB)

RMSE

(deg

ree)


101

100

10minus1

10minus2




6 Conclusions


0 5 10 15 20 25SNR (dB)

RMSE

(deg

ree)


101

100

10minus1

10minus2



Data Availability




Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0 5 10 15 20 25

RMSE

(deg

ree)

101

100

10minus1

10minus2


SNR (dB)


0 5 10 15 20 25SNR (dB)

RMSE

(deg

ree)


101

100

10minus1

10minus2




6 Conclusions


0 5 10 15 20 25SNR (dB)

RMSE

(deg

ree)


101

100

10minus1

10minus2



Data Availability




Acknowledgments


References














































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia











RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Documents

Compressed Sensing PARALIND Decomposition-Based Coherent ...downloads.hindawi.com/journals/wcmc/2019/3619818.pdf · Compressed Sensing PARALIND Decomposition-Based Coherent Angle