Research ArticleTwo-Dimensional Direction of Arrival (DOA) Estimation forRectangular Array via Compressive Sensing Trilinear Model

Huaxin Yu1 Xiaofeng Qiu2 Xiaofei Zhang13 Chenghua Wang1 and Gang Yang1

1 Key Laboratory of Radar Imaging and Microwave Photonics Nanjing University of Aeronautics and AstronauticsMinistry of Education Nanjing 210016 China

2 Institute of Command Information System PLA University of Science and Technology Nanjing 210007 China3 Laboratory of Modern Acoustics of Ministry of Education Nanjing University Nanjing 210093 China

Correspondence should be addressed to Xiaofei Zhang njxnd88126com

Received 15 March 2014 Revised 21 July 2014 Accepted 17 September 2014

Academic Editor Hang Hu

Copyright copy 2015 Huaxin Yu 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

We investigate the topic of two-dimensional direction of arrival (2D-DOA) estimation for rectangular array This paper linksangle estimation problem to compressive sensing trilinear model and derives a compressive sensing trilinear model-based angleestimation algorithm which can obtain the paired 2D-DOA estimationThe proposed algorithm not only requires no spectral peaksearching but also has better angle estimation performance than estimation of signal parameters via rotational invariance techniques(ESPRIT) algorithm Furthermore the proposed algorithm has close angle estimation performance to trilinear decompositionTheproposed algorithm can be regarded as a combination of trilinear model and compressive sensing theory and it brings much lowercomputational complexity and much smaller demand for storage capacity Numerical simulations present the effectiveness of ourapproach

1 Introduction

Array signal processing has received a significant amount ofattention during the last decades due to its wide applicationin radar sonar radio astronomy and satellite communi-cation [1] The direction of arrival (DOA) estimation ofsignals impinging on an antenna array is a fundamentalproblem in array signal processing and many DOA esti-mation methods [2ndash7] have been proposed for its solutionThey contain estimation of signal parameters via rotationalinvariance techniques (ESPRIT) algorithm [2 3] multiplesignal classification (MUSIC) algorithm [4] Root-MUSIC[5] matrix pencil methods [6] and so on Compared withlinear arrays uniform rectangular array can identify two-dimensional DOA (2D-DOA) 2D-DOA estimation withrectangular array has received considerable attention in thefield of array signal processing [7ndash12] ESPRIT algorithmsin [8ndash11] have exploited the invariance property for 2D-DOA estimation in uniform rectangular array Parallel factor

analysis (PARAFAC) in [12] which is also called trilineardecomposition method was proposed for 2D-DOA estima-tion for uniform rectangular array and it has better angleestimation performance than ESPRIT MUSIC algorithm asa subspace method has good angle estimation performanceand matches irregular arrays It has been proved that two-dimensionalMUSIC (2D-MUSIC) algorithm [13] can be usedfor 2D-DOA estimation However the requirement of two-dimensional (2D) spectrum searching renders much highercomputational complexity

Compressive sensing [14 15] has attracted a lot of atten-tion recently and it has been applied to image processingmachine learning channel estimation radar imaging andpenalized regression [16] According to the theory of com-pressive sensing a signal that is sparse in some domaincan be recovered via fewer samples than required by theNyquist sampling theorem The DOAs of sources form asparse vector in the potential signal space and thereforecompressive sensing can be applied to DOA estimation

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2015 Article ID 297572 10 pageshttpdxdoiorg1011552015297572

2 International Journal of Antennas and Propagation

The superresolution property and ability of resolving coher-ent sources can be achieved when we apply it to the sourcelocation [17] Lots of the DOA estimation methods withcompressive sensing just use one snapshot and are verysensitive to the noise Formultiple snapshots ℓ

1-SVDmethod

[16] employed ℓ1norm to enforce sparsity and singular value

decomposition to reduce complexity and sensitivity to noiseand sparse recovery for weighted subspace fitting in [17]improved the ℓ

1-SVDmethod via the weight to the subspace

Compared to matrix decomposition trilinear decompo-sition has a distinctive and attractive feature it is often unique[18ndash22] In the signal processing field trilinear decompo-sition can be regarded as a generalization of ESPRIT andjoint approximate diagonalization [19ndash22] The compressivesensing trilinear model-based algorithm discussed in thispaper can be regarded as a combination of trilinear modeland compressive sensing theory which brings much lowercomputational complexity and much smaller demand forstorage capacity

The framework of compressive sensing for sparse low-rank tensor is proposed in [23] and used for signal detectionand multiple-input-multiple-output radar in [24 25] In thispaper the problem of 2D-DOA estimation for rectangu-lar array is linked to compressive sensing trilinear modelExploiting this link we derive a compressive sensing trilinearmodel-based 2D-DOA estimation algorithm for rectangu-lar array Firstly we compress the received data to get acompressed trilinear model and then obtain the estimatesof compressed direction matrices through performing tri-linear decomposition for the compressed model Finally weformulate a sparse recovery problem through the estimatedcompressed direction matrices and apply the orthogonalmatching pursuit (OMP) [26] to resolve it for 2D-DOA esti-mation Due to compression the proposedmethod hasmuchlower computational complexity than conventional trilineardecomposition method [12] and 2D-MUSIC algorithm andrequires much smaller storage capacity We illustrate that theproposed algorithm has better angle estimation performancethan ESPRIT algorithm Furthermore our algorithm canobtain paired elevation angles and azimuth angles automat-ically We also derive the Cramer-Rao bound (CRB) for 2D-DOA estimation in rectangular array Numerical simulationspresent the effectiveness of our approach

The remainder of this paper is structured as followsSection 2 presents the data model and Section 3 proposesthe compressed sensing trilinear model-based algorithm for2D-DOA estimation in rectangular array In Section 4 thesimulation results are presented to verify improvement ofthe proposed algorithm while the conclusions are drawn inSection 5

Notation Bold lower (upper) case letters are adopted to rep-resent vectors (matrices) (sdot)119879 (sdot)119867 (sdot)lowast (sdot)minus1 and (sdot)+ denotetranspose conjugate transpose conjugate matrix inversionand pseudoinverse operations respectively I

119875stands for a

119875 times 119875 identity matrix 119863119899(A) denotes a diagonal matrix with

the entries of the matrix Arsquos 119899th row on the main diagonalThe 119894th entry of a given column vector g is denoted by g(119894)

⊙ otimes and oplus denote Khatri-Rao product Kronecker productand Hadamard product respectively

If A is a 119901-by-119902 matrix and B is an 119898-by-119899 matrix thenthe Kronecker product A otimes B is the119898119901-by-119899119902 block matrix

A otimes B =

[[[[[[[

[

11988611B 119886

12B sdot sdot sdot 119886

1119902B

11988621B 119886

22B sdot sdot sdot 119886

2119902B

d

1198861199011B 119886

1199012B sdot sdot sdot 119886

119901119902B

]]]]]]]

]

(1)

where 119886119894119895is the (119894 119895) element of the matrix A

IfA is an 119868-by-119865matrix andB is a 119869-by-119865matrix then theKhatri-Rao product A ⊙ B is the 119868119869-by-119865 block matrix

A ⊙ B = [a1otimes b

1 a

119865otimes b

119865] (2)

where a119891and b

119891are the 119891th column of the matricesA and B

respectivelyIf A is an 119868-by-119869 matrix and B is an 119868-by-119869 matrix then

the Hadamard product A oplus B is

A oplus B =[[[[

[

11988611119887111198861211988712sdot sdot sdot 119886

11198691198871119869

11988621119887211198862211988722sdot sdot sdot 119886

21198691198872119869

d

1198861198681119887119868111988611986821198871198682sdot sdot sdot 119886

119868119869119887119868119869

]]]]

]

(3)

where 119886119894119895and 119887

119894119895are the (119894 119895) element of the matrices A and

B respectively

2 Data Model

A rectangular array consisted of119872times119873 elements is shown inFigure 1 where the distance between two adjacent elementsis 119889 We consider signals in the far field in which case thesignal sources are far away enough that the arriving waves areessentially planes over the array We assume that the noise isindependent of the sources It is also assumed that there are119870 noncoherent or independent sources and the number ofsources is preknown 120579

119896and 120601

119896are the elevation angle and

the azimuth angle of the 119896th source respectively We assumethe sources impinge on the array with different DOAs

The received signal of the first subarray in the rectangulararray is x

1(119905) = A

119909s(119905) + n

1(119905) where A

119909= [a

119909(1206011 1205791)

a119909(1206012 1205792) a

119909(120601119870 120579119870)] isin C119872times119870 with a

119909(120601119896 120579119896) = [1

1198901198952120587119889 sin 120579119896 cos120601119896120582 1198901198952120587(119872minus1)119889 sin 120579119896 cos120601119896120582]119879 and 120582 is thewavelength n

1(119905) is the received additive white Gaussian

noise of the first subarray s(119905) isin C119870times1 is the sourcevector The received signal of the 119899th subarray in therectangular array is x

119899(119905) = A

119909Φ119899minus1s(119905) + n

119899(119905) where Φ =

diag(1198901198952120587119889 sin 1205791 sin1206011120582 1198901198952120587119889 sin 120579119870 sin120601119870120582) and n119899(119905) is the

received additive white Gaussian noise of the 119899th subarrayTherefore the received signal of the rectangular array is [27]

x (119905) =[[[[[[

[

x1(119905)

x2(119905)

x119873(119905)

]]]]]]

]

=

[[[[[[[

[

A119909

A119909Φ

A119909Φ119873minus1

]]]]]]]

]

s (119905) +[[[[[[

[

n1(119905)

n2(119905)

n119873(119905)

]]]]]]

]

(4)

International Journal of Antennas and Propagation 3

Z

Y

X

M

N

1SubarraySubarray Subarray

2 N

120579k

120601k

middot middot middot

middot middot middot

middot middotmiddotmiddot middot

middot

middot middot middot

Figure 1 The structure of uniform rectangular array

The signal x(119905) isin C119872119873times1 in (4) can also be denoted by

x (119905) = [A119910⊙ A

119909] s (119905) + n (119905) (5)

where A119910

= [a119910(1206011 1205791) a

119910(1206012 1205792) a

119910(120601119870 120579119870)] with

a119910(120601119896 120579119896) = [1 119890

1198952120587119889 sin 120579119896 sin120601119896120582 1198901198952120587(119873minus1)119889 sin 120579119896 sin120601119896120582]119879n(119905) = [n

1(119905)

119879n2(119905)

119879 n

119873(119905)

119879]119879isin C119872119873times1 ⊙ denotes

Khatri-Rao productAccording to the definition of Khatri-Rao product the

signal in (5) can be rewritten as

x (119905) = [a119910(1206011 1205791) otimes a

119909(1206011 1205791)

a119910(120601119870 120579119870) otimes a

119909(120601119870 120579119870)] s (119905) + n (119905)

(6)

where otimes denotes Kronecker product We collect 119871 samplesand define X = [x(1) x(2) x(119871)] isin C119872119873times119871 which canbe expressed as

X = [A119910⊙ A

119909] S119879 + N =

[[[[

[

X1

X2

X119873

]]]]

]

=

[[[[[

[

A1199091198631(A

119910)

A1199091198632(A

119910)

A119909119863119873(A

119910)

]]]]]

]

S119879 +[[[[

[

N1

N2

N119873

]]]]

]

(7)

where S = [s(1) s(2) s(119871)]119879 isin C119871times119870 is source matrixand N = [n(1)n(2) n(119871)] is the received additive whiteGaussian noise matrixN

119899isin C119872times119871 (119899 = 1 119873) is the noise

matrix Thus X119899isin C119872times119871 in (7) is denoted as

X119899= A

119909119863119899(A

119910) S119879 + N

119899 119899 = 1 2 119873 (8)

Equation (7) can also be denoted with the trilinear model [1828]

119909119898119899119897

=

119870

sum

119896=1

A119909(119898 119896)A

119910(119899 119896) S (119897 119896) + 119899

119898119899119897

119898 = 1 119872 119899 = 1 119873 119897 = 1 119871

(9)

where A119909(119898 119896) is the (119898 119896) element of the matrix A

119909and

similarly for the others 119899119898119899119897

is noise part X119899isin C119872times119871 (119899 =

1 119873) can be regarded as slicing the three-dimensionaldata in a series of slices which is shown in Figure 2 Thereare two more matrix system rearrangements in which wehave Y

119898= S119863

119898(A

119909)A119879

119910+ N119910

119898 119898 = 1 119872 and Z

119897=

A119910119863119897(S)A119879

119909+ N119911

119897 119897 = 1 119871 where N119910

119898and N119911

119897are noise

matrices Then we form the matrices of Y isin C119872119871times119873 andZ isin C119871119873times119872

Y =[[[[[[

[

Y1

Y2

Y119872

]]]]]]

]

= [A119909⊙ S]A119879

119910+ N119910

(10)

Z =[[[[[[

[

Z1

Z2

Z119871

]]]]]]

]

= [S ⊙ A119910]A119879

119909+ N119911

(11)

where

N119910=

[[[[[[

[

N119910

1

N119910

2

N119910

119872

]]]]]]

]

N119911=

[[[[[[

[

N119911

1

N119911

2

N119911

119871

]]]]]]

]

(12)

3 2D-DOA Estimation Based onCompressive Sensing Trilinear Model

We link the problem of 2D-DOA estimation for rectangulararray to compressive sensing trilinear model and derive acompressive sensing trilinear model-based 2D-DOA estima-tion algorithm Firstly we compress the received data to get acompressed trilinear model and then obtain the estimates ofcompressed direction matrices through performing trilineardecomposition for the compressed model Finally we formu-late the sparse recovery problem for 2D-DOA estimation

31 Trilinear Model Compression We compress the three-way data Χ isin C119872times119873times119871 into a smaller three-way data Χ1015840 isinC1198721015840times1198731015840times1198711015840

where1198721015840lt 1198721198731015840

lt 119873 and 1198711015840 lt 119871 The trilinearmodel compression processing is shown in Figure 3 Wedefine the compression matrices as U isin C119872times119872

1015840

V isin C119873times1198731015840

and W isin C119871times119871

1015840

and the compression matrices U V and Wcan be generated randomly or obtained by Tucker3 decom-position [23 29] We can use the Tucker3 decomposition

4 International Journal of Antennas and Propagation

M

Ym isin CLtimesN

L

N

Xn isin CMtimesL

Zl isin CNtimesM

Figure 2 Trilinear model

U isin CMtimesM998400

V isin CNtimesN998400

W isin CLtimesL998400N998400

L998400

M998400

L

M

N

CompressionCompressed

trilinear modelTrilinearmodel

Figure 3 The compression of trilinear model

where tensor is decomposed into the core tensor to obtain thecompression matrices The compression matrices should sat-isfy the restricted isometry property And random GaussianBernoulli and partial Fourier matrices satisfy the restrictedisometry property with number of measurements nearlylinear in the sparsity level [30 31]

Then compressX isin C119872119873times119871 in (7) to a smaller one asX1015840 isinC11987210158401198731015840times1198711015840

X1015840 = (V119879 otimes U119879)XW119879

= (V119879 otimes U119879) [A119910⊙ A

119909] S119879W119879

+ (V119879 otimes U119879)NW119879

(13)

According to the property of Khatri-Rao product [23] weknow

(V119879 otimes U119879) [A119910⊙ A

119909] = (V119879A

119910) ⊙ (U119879A

119909) (14)

Define A1015840119909= U119879A

119909 A1015840

119910= V119879A

119910 and S1015840 = W119879S Equation

(11) is also denoted as

X1015840 = [A1015840119910⊙ A1015840

119909] S1015840119879 + N1015840

(15)

where N1015840= (V119879 otimes U119879)NW119879 X1015840 can be denoted by trilinear

model With respect to (10) and (11) we form the matrices ofY1015840 and Z1015840 according to the compressed data

Y1015840 = [A1015840119909⊙ S1015840]A1015840119879

119910+ N1015840119910

(16)

Z1015840 = [S1015840 ⊙ A1015840119910]A1015840119879

119909+ N1015840119911

(17)

where N1015840119910 and N1015840119911 are the noise part The compressed trilin-ear model may degrade the angle estimation performance

By trilinear model compression the proposed methodhasmuch lower computational complexity than conventionaltrilinear decomposition method and requires much smallerstorage capacity Conventional compressive sensing is tocompress the matrix while our algorithm compresses thethree-dimensional tensor

32 Trilinear Decomposition Trilinear alternating leastsquare (TALS) algorithm is an iterative method forestimating the parameters of a trilinear decomposition[18 28] We concisely show the basic idea of TALS (1)update one matrix each time via LS which is conditionedon previously obtained estimates of the remaining matrices(2) proceed to update the other matrices (3) repeat untilconvergence of the LS cost function [21 22] TALS algorithmis discussed as follows

According to (15) least squares (LS) fitting is

minA1015840119909A1015840119910S101584010038171003817100381710038171003817X1015840 minus [A1015840

119910⊙ A1015840

119909] S101584011987910038171003817100381710038171003817119865 (18)

and LS update for the matrix S1015840 is

S1015840119879 = [A1015840119910⊙ A1015840

119909]+

X1015840 (19)

where A1015840119909and A1015840

119910are previously obtained estimates ofA1015840

119909and

A1015840119910 respectively According to (16) LS fitting is

minA1015840119909A1015840119910S101584010038171003817100381710038171003817Y1015840 minus [A1015840

119909⊙ S1015840]A1015840119879

119910

10038171003817100381710038171003817119865 (20)

and LS update for A1015840119910is

A1015840119879119910= [A1015840

119909⊙ S1015840]

+

Y1015840 (21)

where A1015840119909and S1015840 stand for the previously obtained estimates

of A1015840119909and S1015840 Similarly according to (17) LS fitting is

minA1015840119909A1015840119910S101584010038171003817100381710038171003817Z1015840 minus [S1015840 ⊙ A1015840

119910]A1015840119879

119909

10038171003817100381710038171003817119865 (22)

where Z1015840 is the noisy compressed signal LS update for A1015840119909is

A1015840119879119909= [S1015840 ⊙ A1015840

119910]+

Z1015840 (23)

where A1015840119910and S1015840 stand for the previously obtained estimates

of A1015840119910and S1015840 respectively

Define E = X minus [A1015840119910⊙ A1015840

119909]S1015840119879 where S1015840 A1015840

119910 and A1015840

119909

present the estimates of S1015840A1015840119910 andA1015840

119909 respectivelyThe sum

of squared residuals (SSR) in trilinear model is defined asSSR = sum

119871

119895=1sum119872119873

119894=1|119890119894119895| where 119890

119894119895is the (119894 119895) element of the

matrix E According to (19) (21) and (23) the matrices S1015840A1015840119910 and A1015840

119909are updated with least squares repeatedly until

SSR attain apriorthreshold The we obtain the final estimatesS1015840 A1015840

119910 and A1015840

119909

International Journal of Antennas and Propagation 5

Theorem 1 (see [22]) Considering X1015840119899= A1015840

119909119863119899(A1015840

119910)S1015840119879 119899 =

1 1198731015840 where A1015840

119909isin C119872

1015840times119870 S1015840 isin C119871

1015840times119870 and A1015840

119910isin C119873

1015840times119870 if

119896A1015840119909+ 119896A1015840

119910+ 119896S1015840119879 ge 2119870 + 2 (24)

where 119896A is the 119896-rank of the matrix A [18] then A1015840119909 S1015840 and

A1015840119910are unique up to permutation and scaling of columns

For the different DOAs and independent sources we have119896A1015840119909= min(1198721015840

119870) 119896A1015840119910= min(1198731015840

119870) and 119896S1015840119879 = min(1198711015840 119870)in the trilinear model in this paper and then the inequality in(24) becomes

min (1198721015840 119870) +min (1198731015840

119870) +min (1198711015840 119870)

ge 2119870 + 2

(25)

When 1198721015840ge 119870 1198731015840

ge 119870 and 1198711015840 ge 119870 the identifiablecondition is 1 le 119870 le min(1198721015840

1198731015840)

When 1198721015840le 119870 1198731015840

le 119870 and 1198711015840 ge 119870 the identifiablecondition is max(1198721015840

1198731015840) le 119870 le 119872

1015840+ 119873

1015840minus 2 Hence the

proposed algorithm is effective when119870 le 1198721015840+119873

1015840minus2 and the

maximum number of sources that can be identified is1198721015840+

1198731015840minus 2After the trilinear decomposition we obtain the estimates

of the loading matrices

A1015840119909= A1015840

119909ΠΔ

1+ E

1

S1015840 = S1015840ΠΔ2+ E

2

A1015840119910= A1015840

119910ΠΔ

3+ E

3

(26)

where Π is a permutation matrix and Δ1 Δ

2 Δ

3note for

the diagonal scaling matrices satisfying Δ1Δ2Δ3= I

119870 E

1

E2 and E

3are estimation error matrices After the trilinear

decomposition the estimates of A1015840119909 A1015840

119910 and S1015840 can be

obtained Scale ambiguity and permutation ambiguity areinherent to the trilinear decomposition problem Howeverthe scale ambiguity can be resolved easily by means ofnormalization while the existence of permutation ambiguityis not considered for angle estimation

33 Angle Estimation via Sparse Recovery Use a1015840119909119896

and a1015840119910119896

todenote the 119896th column of estimates A1015840

119909and A1015840

119910 respectively

According to the compression matrices we have

a1015840119909119896= U119879120597

119909119896a119909119896+ n

119909119896 119896 = 1 119870 (27a)

a1015840119910119896= V119879120597

119910119896a119910119896+ n

119910119896 119896 = 1 119870 (27b)

where a119909119896

and a119910119896

are the 119896th column ofA119909A

119910 respectively

n119909119896and n

119910119896are the corresponding noise respectively 120597

119909119896and

120597119909119896

are the scaling coefficients Construct two Vandermondematrices A

119904119909isin C119872times119875 and A

119904119910isin C119873times119875 (119875 ≫ 119872 119875 ≫ 119873)

composed of steering vectors corresponding to each potentialsource location as its columns

A119904119909

= [a1199041199091 a1199041199092 a

119904119909119875]

=

[[[[[

[

1 1 sdot sdot sdot 1

1198901198952120587119889g(1)120582

1198901198952120587119889g(2)120582

1198901198952120587119889g(119875)120582

1198901198952120587(119872minus1)119889g(1)120582

1198901198952120587(119872minus1)119889g(2)120582

sdot sdot sdot 1198901198952120587(119872minus1)119889g(119875)120582

]]]]]

]

(28a)

A119904119910

= [a1199041199101 a1199041199102 a

119904119910119875]

=

[[[[[

[

1 1 sdot sdot sdot 1

1198901198952120587119889g(1)120582

1198901198952120587119889g(2)120582

1198901198952120587119889g(119875)120582

1198901198952120587(119873minus1)119889g(1)120582

1198901198952120587(119873minus1)119889g(2)120582

sdot sdot sdot 1198901198952120587(119873minus1)119889g(119875)120582

]]]]]

]

(28b)

where g is a sampling vector and its 119901th elements is g(119901) =minus1 + 2119901119875 119901 = 1 2 119875 The matrices A

119904119909and A

119904119910can be

regarded as the completed dictionariesThen (27a)-(27b) canbe expressed as

a1015840119909119896= U119879A

119904119909x119904+ n

119909119896 119896 = 1 119870 (29a)

a1015840119910119896= V119879A

119910119909y119904+ n

119910119896 119896 = 1 119870 (29b)

where x119904and y

119904are sparse The estimates of x

119904and y

119904can be

obtained via 1198970-norm constraint

min 10038171003817100381710038171003817a1015840119909119896minus U119879A

119904119909x119904

10038171003817100381710038171003817

2

2

st 1003817100381710038171003817x11990410038171003817100381710038170 = 1

(30a)

min 10038171003817100381710038171003817a1015840119910119896minus V119879A

119904119910y119904

10038171003817100381710038171003817

2

2

st 1003817100381710038171003817y11990410038171003817100381710038170 = 1

(30b)

where sdot 0denotes the 119897

0-norm x

1199040= 1 that is to say

there is only one nonzero element in the vector x119904 similar

to y1199040= 1 We can use the OMP recovery method [26]

to find the nonzero element in x119904or y

119904 The OMP algorithm

tries to recover the signal by finding the strongest componentin the measurement signal removing it from the signal andsearching the dictionary again for the strongest atom that ispresented in the residual signal [32] We extract the indexof the maximum modulus of elements in x

119904and y

119904 respec-

tively noted as 119901119909and 119901

119910 According to the corresponding

columns in A119904119909

and A119904119910 we obtain g(119901

119909) and g(119901

119910) which

are estimates of sin 120579119896cos120601

119896and sin 120579

119896sin120601

119896 We define

6 International Journal of Antennas and Propagation

119903119896= g(119901

119909) + 119895g(119901

119910) and then the elevation angles and

azimuth angles can be obtained via

120579119896= sinminus1 (abs (119903

119896)) 119896 = 1 119870 (31a)

120601119896= angle (119903

119896) 119896 = 1 119870 (31b)

where abs(sdot) is the modulus value symbol and angle(sdot) is toget the angle of an imaginary number As the columns ofthe estimated matrices A1015840

119909and A1015840

119910are automatically paired

then the estimated elevation angles and azimuth angles canbe paired automatically

34 The Procedures of the Proposed Algorithm Till nowwe have achieved the proposal for the compressive sensingtrilinear model-based 2D-DOA estimation for rectangulararray We show major steps of the proposed algorithm asfollows

Step 1 Form the three-way matrix Χ isin C119872times119873times119871 thencompress the three-way matrix into a much smaller three-way matrix Χ1015840 isin C119872

1015840times1198731015840times1198711015840

via the compression matricesU isin C119872times119872

1015840

V isin C119873times1198731015840

andW isin C119871times1198711015840

Step 2 Perform trilinear decomposition through TALS algo-rithm for the compressed three-way matrix to obtain theestimation of A1015840

119909 A1015840

119910 and S1015840

Step 3 Estimate the sparse vectors

Step 4 Estimate 2D-DOA via (31a)-(31b)

Remark A Because the trilinear decomposition brings thesame permutation ambiguity for the estimatesA1015840

119909A1015840

119910 and S1015840

the estimated elevation angles and azimuth angles are pairedautomatically

Remark B The conventional compressive sensing methodformulates an angle sampling grid for sparse recovery toestimate angles When it is applied to 2D-DOA estimationboth elevation and azimuth angles must be sampled andit results in a two-dimensional sampling problem whichbrings much heavier cost for sparse signal recovery Inthis paper sin 120579

119896cos120601

119896(or sin 120579

119896sin120601

119896) is bundled into a

single variable in the range of minus1 to 1 The bundled variableis sampled for sparse recovery to obtain the estimates ofsin 120579

119896cos120601

119896and sin 120579

119896sin120601

119896 respectively Afterwards the

elevation and azimuth angles are estimated through theestimates of sin 120579

119896cos120601

119896and sin 120579

119896sin120601

119896

Remark C If the number of sources 119870 is unknown it canbe estimated by performing singular value decompositionfor received data matrix X in (7) and finding the numberof largest singular values [33] We also use some lower-complexity algorithm in [34] for estimating the number ofthe sources

Remark D When the coherent sources impinge on the arraywe can use the parallel profiles with linear dependencies

(PARALIND) model [35 36] which is a generalization ofPARAFAC suitable for solving problems with linear depen-dent factors to resolve coherent DOA estimation problem

35 Complexity Analysis and CRB The proposed algorithmhas much lower computational cost than conventional trilin-ear decomposition-based method The proposed algorithmrequires 119874(1198703 + 1198721015840

11987310158401198711015840119870) operations for a iteration while

the trilinear decomposition algorithm needs119874(1198703+119872119873119871119870)operations [28] for a iteration where1198721015840

lt 1198721198731015840lt 119873 and

1198711015840lt 119871We define A = [a

119910(1205791) otimes a

119909(1205791) a

119910(120579119870) otimes a

119909(120579119870)]

According to [37] we can derive the CRB

CRB = 1205902

2119871Re [D119867

Πperp

AD oplus P119879

119904]minus1

(32)

where 119871 denotes the number of samples a119896is the 119896th column

of A and P119904= (1119871)sum

119871

119905=1s(119905)s119867(119905) 1205902 is the noise power

Πperp

A = I119872119873

minus A(A119867A)minus1A119867 and

D = [120597a11205971205791

120597a2

1205971205792

120597a119870

120597120579119870

120597a1

1205971206011

120597a2

1205971206012

120597a119870

120597120601119870

] (33)

The advantages of the proposed algorithm can be presentedas follows

(1) The proposed algorithm can be regarded as a com-bination of trilinear model and compressive sensingtheory and it brings much lower computationalcomplexity and much smaller demand for storagecapacity

(2) The proposed algorithm has better 2D-DOA estima-tion performance than ESPRIT algorithm and closeangle estimation performance to TALS algorithmwhich will be proved by Figures 6-7

(3) The proposed algorithm can achieve paired elevationangles and azimuth angles automatically

4 Numerical Simulations

In the following simulations we assume that the numericalsimulation results converge when the SSR le 10minus8119872 119873 119871and 119870 denote the number of antennas in 119909-axis number ofantennas in 119910-axis samples and sources respectively Andwe compress the parameters 119872 119873 119871 to 1198721015840 1198731015840 and 1198711015840(usually set 119872 = 16 119873 = 20 119871 = 100 and 1198721015840

= 1198731015840=

1198711015840= 5 in numerical simulations) 119889 = 1205822 is considered

in the simulation We present 1000 Monte Carlo simulationsto assess the angle estimation performance of the proposedalgorithm Define root mean squared error (RMSE) as

RMSE = 1119870

119870

sum

119896=1

radic1

1000

1000

sum

119897=1

(120601119896119897minus 120601

119896)2

+ (120579119896119897minus 120579

119896)2

(34)

where 120601119896and 120579

119896denote the perfect elevation angle and

azimuth angle of 119896th source respectively 120601119896119897

and 120579119896119897

are

International Journal of Antennas and Propagation 7

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

Elev

atio

n (d

eg)

Azimuth (deg)

2D-DOA estimation

Figure 4 2D-DOA estimation of our algorithm in SNR = minus10 dB(119873 = 20119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

402D-DOA estimation

Elev

atio

n (d

eg)

Azimuth (deg)

Figure 5 2D-DOAestimation of our algorithm in SNR = 5 dB (119873 =20119872 = 16 119871 = 100 and 119870 = 3)

the estimates of 120601119896and 120579

119896in the 119897th Monte Carlo trail

Assume that there are 3 noncoherent sources located at angles(1206011 1205791) = (5

∘ 10

∘) (120601

2 1205792) = (15

∘ 20

∘) and (120601

3 1205793) =

(40∘ 30

∘)

Figure 4 presents the 2D-DOA estimation of the pro-posed algorithm for uniform rectangular array with119873 = 20119872 = 16 119871 = 100119870 = 3 and SNR = minus10 dB Figure 5 depictsthe 2D-DOA estimation performance with SNR = 5 dBFigures 4-5 illustrate that our algorithm is effective for 2D-DOA estimation

Figure 6 shows the 2D-DOA estimation performancecomparison of the proposed algorithm the ESPRIT algo-rithm the TALS algorithm and the CRB for the uniformrectangular array with119873 = 20119872 = 16 119871 = 100 and 119870 = 3while Figure 7 depicts the 2D-DOA estimation performancecomparison with 119873 = 16 119872 = 16 119871 = 200 and 119870 = 3It is indicated that our algorithm has better angle estimation

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

10minus3

minus10 minus5

ProposedTALS

ESPRITCRB

Figure 6 2D-DOA estimation performance comparison (119873 = 20119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

10minus3

minus10 minus5

ProposedTALS

ESPRITCRB

Figure 7 2D-DOA estimation performance comparison (119873 = 16119872 = 16 119871 = 200 and 119870 = 3)

performance than the ESPRIT algorithm and close angleestimation to TALS algorithm The angle estimation perfor-mance of the proposed algorithm will be further improvedthrough increasing the compressed parameters1198721015840 1198731015840 and1198711015840Figure 8 depicts the 2D-DOA estimation performance

of the proposed algorithm with different value of 119873 (119872 =

16 119871 = 100 and 119870 = 3) while Figure 9 presents the2D-DOA estimation performance of the proposed algorithmwith different value of 119872 It is clearly shown that theangle estimation performance of our algorithm is graduallyimproved with the number of antennas increasing Multiple

8 International Journal of Antennas and Propagation

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

N = 8

N = 16

N = 24

Figure 8 Angle estimation performance of our algorithm withdifferent119873 (119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

M = 8

M = 16

M = 24

Figure 9 Angle estimation performance of our algorithm withdifferent119872 (119873 = 16 119871 = 100 and 119870 = 3)

antennas improve the angle estimation performance becauseof diversity gain

Figure 10 presents 2D-DOA estimation performance ofthe proposed algorithm with different value of 119871 (119873 =

20 119872 = 16 and 119870 = 3) It illustrates that the angleestimation performance becomes better in collaborationwith119871 increasing

5 Conclusions

In this paper we have addressed the 2D-DOA estimationproblem for rectangular array and have derived a compressive

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

L = 50

L = 100

L = 200

Figure 10 Angle estimation performance of our algorithm withdifferent 119871 (119873 = 16119872 = 20 and 119870 = 3)

sensing trilinear model-based 2D-DOA estimation algo-rithm which can obtain the automatically paired 2D-DOAestimateThe proposed algorithm has better angle estimationperformance than ESPRIT algorithm and close angle esti-mation performance to conventional trilinear decompositionmethod Furthermore the proposed algorithm has lowercomputational complexity and smaller demand for storagecapacity than conventional trilinear decomposition methodThe proposed algorithm can be regarded as a combination oftrilinear model and compressive sensing theory and it bringsmuch lower computational complexity and much smallerdemand for storage capacity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by China NSF Grants (6137116961301108 61471191 61471192 and 61271327) Jiangsu PlannedProjects for Postdoctoral Research Funds (1201039C)China Postdoctoral Science Foundation (2012M5210992013M541661) Open Project of Key Laboratory of ModernAcoustics of Ministry of Education (Nanjing University) theAeronautical Science Foundation of China (20120152001)Qing Lan Project Priority Academic Program Developmentof Jiangsu High Education Institutions and the FundamentalResearch Funds for the Central Universities (NS2013024kfjj130114 and kfjj130115)

International Journal of Antennas and Propagation 9

References

[1] HKrim andMViberg ldquoTwodecades of array signal processingresearch the parametric approachrdquo IEEE Signal ProcessingMagazine vol 13 no 4 pp 67ndash94 1996

[2] R Roy and T Kailath ldquoESPRIT-estimation of signal parametersvia rotational invariance techniquesrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 37 no 7 pp 984ndash995 1989

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

[4] D Kundu ldquoModified MUSIC algorithm for estimating DOA ofsignalsrdquo Signal Processing vol 48 no 1 pp 85ndash90 1996

[5] B D Rao and K V S Hari ldquoPerformance analysis of Root-Musicrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 37 no 12 pp 1939ndash1949 1989

[6] N Yilmazer J Koh and T K Sarkar ldquoUtilization of a unitarytransform for efficient computation in thematrix pencilmethodto find the direction of arrivalrdquo IEEE Transactions on Antennasand Propagation vol 54 no 1 pp 175ndash181 2006

[7] Y Chiba K Ichige and H Arai ldquoReducing DOA estimationerror in extended ES-root-MUSIC for uniform rectangulararrayrdquo in Proceedings of the 4th International Congress on Imageand Signal Processing (CISP rsquo11) vol 5 pp 2621ndash2625 October2011

[8] 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

[9] 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

[10] 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

[11] 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

[12] 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

[13] 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

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

[15] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo IEEE Transactions on InformationThe-ory vol 52 no 2 pp 489ndash509 2006

[16] 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

[17] N Hu Z Ye D Xu and S Cao ldquoA sparse recovery algorithmfor DOA estimation using weighted subspace fittingrdquo SignalProcessing vol 92 no 10 pp 2566ndash2570 2012

[18] 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

[19] L de Lathauwer B de Moor and J Vandewalle ldquoComputationof the canonical decomposition by means of a simultaneousgeneralized Schur decompositionrdquo SIAM Journal on MatrixAnalysis and Applications vol 26 no 2 pp 295ndash327 2004

[20] L de Lathauwer ldquoA link between the canonical decompositionin multi-linear algebra and simultaneous matrix diagonaliza-tionrdquo SIAM Journal on Matrix Analysis and Applications vol28 no 3 pp 642ndash666 2006

[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000

[22] N D Sidiropoulos and X Liu ldquoIdentifiability results for blindbeamforming in incoherent multipath with small delay spreadrdquoIEEE Transactions on Signal Processing vol 49 no 1 pp 228ndash236 2001

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

[24] X F Zhang H X Yu J F Li andD Ben ldquoBlind signal detectionfor uniform rectangular array via compressive sensing trilinearmodelrdquo Advanced Materials Research vol 756ndash759 pp 660ndash664 2013

[25] R Cao X Zhang and W Chen ldquoCompressed sensing parallelfactor analysis-based joint angle andDoppler frequency estima-tion for monostatic multiple-inputndashmultiple-output radarrdquo IETRadar Sonar amp Navigation vol 8 no 6 pp 597ndash606 2014

[26] 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

[27] X Zhang F Wang and H Chen Theory and Application ofArray Signal Processing (version 2) National Defense IndustryPress Beijing China 2012

[28] X Zhang J Li H Chen and D Xu ldquoTrilinear decomposition-based two-dimensional DOA estimation algorithm for arbitrar-ily spaced acoustic vector-sensor array subjected to unknownlocationsrdquoWireless Personal Communications vol 67 no 4 pp859ndash877 2012

[29] R Bro N D Sidiropoulos and G B Giannakis ldquoA fast leastsquares algorithm for separating trilinear mixturesrdquo in Proceed-ings of the International Workshop on Independent ComponentAnalysis and Blind Signal Separation pp 289ndash294 January 1999

[30] R A DeVore ldquoDeterministic constructions of compressedsensing matricesrdquo Journal of Complexity vol 23 no 4ndash6 pp918ndash925 2007

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

[32] F Wang and X Zhang ldquoJoint estimation of TOA and DOA inIR-UWB system using sparse representation frameworkrdquo ETRIJournal vol 36 no 3 pp 460ndash468 2014

[33] A Di ldquoMultiple source locationmdasha matrix decompositionapproachrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 33 no 5 pp 1086ndash1091 1985

[34] J Xin N Zheng and A Sano ldquoSimple and efficient nonpara-metric method for estimating the number of signals withouteigendecompositionrdquo IEEE Transactions on Signal Processingvol 55 no 4 pp 1405ndash1420 2007

10 International Journal of Antennas and Propagation

[35] 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

[36] X Zhang M Zhou and J Li ldquoA PARALIND decomposition-based coherent two-dimensional direction of arrival estimationalgorithm for acoustic vector-sensor arraysrdquo Sensors vol 13 no4 pp 5302ndash5316 2013

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

International Journal of

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Active and Passive Electronic Components

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

2 International Journal of Antennas and Propagation

The superresolution property and ability of resolving coher-ent sources can be achieved when we apply it to the sourcelocation [17] Lots of the DOA estimation methods withcompressive sensing just use one snapshot and are verysensitive to the noise Formultiple snapshots ℓ

1-SVDmethod

[16] employed ℓ1norm to enforce sparsity and singular value

decomposition to reduce complexity and sensitivity to noiseand sparse recovery for weighted subspace fitting in [17]improved the ℓ

1-SVDmethod via the weight to the subspace

Compared to matrix decomposition trilinear decompo-sition has a distinctive and attractive feature it is often unique[18ndash22] In the signal processing field trilinear decompo-sition can be regarded as a generalization of ESPRIT andjoint approximate diagonalization [19ndash22] The compressivesensing trilinear model-based algorithm discussed in thispaper can be regarded as a combination of trilinear modeland compressive sensing theory which brings much lowercomputational complexity and much smaller demand forstorage capacity

The framework of compressive sensing for sparse low-rank tensor is proposed in [23] and used for signal detectionand multiple-input-multiple-output radar in [24 25] In thispaper the problem of 2D-DOA estimation for rectangu-lar array is linked to compressive sensing trilinear modelExploiting this link we derive a compressive sensing trilinearmodel-based 2D-DOA estimation algorithm for rectangu-lar array Firstly we compress the received data to get acompressed trilinear model and then obtain the estimatesof compressed direction matrices through performing tri-linear decomposition for the compressed model Finally weformulate a sparse recovery problem through the estimatedcompressed direction matrices and apply the orthogonalmatching pursuit (OMP) [26] to resolve it for 2D-DOA esti-mation Due to compression the proposedmethod hasmuchlower computational complexity than conventional trilineardecomposition method [12] and 2D-MUSIC algorithm andrequires much smaller storage capacity We illustrate that theproposed algorithm has better angle estimation performancethan ESPRIT algorithm Furthermore our algorithm canobtain paired elevation angles and azimuth angles automat-ically We also derive the Cramer-Rao bound (CRB) for 2D-DOA estimation in rectangular array Numerical simulationspresent the effectiveness of our approach

The remainder of this paper is structured as followsSection 2 presents the data model and Section 3 proposesthe compressed sensing trilinear model-based algorithm for2D-DOA estimation in rectangular array In Section 4 thesimulation results are presented to verify improvement ofthe proposed algorithm while the conclusions are drawn inSection 5

Notation Bold lower (upper) case letters are adopted to rep-resent vectors (matrices) (sdot)119879 (sdot)119867 (sdot)lowast (sdot)minus1 and (sdot)+ denotetranspose conjugate transpose conjugate matrix inversionand pseudoinverse operations respectively I

119875stands for a

119875 times 119875 identity matrix 119863119899(A) denotes a diagonal matrix with

the entries of the matrix Arsquos 119899th row on the main diagonalThe 119894th entry of a given column vector g is denoted by g(119894)

⊙ otimes and oplus denote Khatri-Rao product Kronecker productand Hadamard product respectively

If A is a 119901-by-119902 matrix and B is an 119898-by-119899 matrix thenthe Kronecker product A otimes B is the119898119901-by-119899119902 block matrix

A otimes B =

[[[[[[[

[

11988611B 119886

12B sdot sdot sdot 119886

1119902B

11988621B 119886

22B sdot sdot sdot 119886

2119902B

d

1198861199011B 119886

1199012B sdot sdot sdot 119886

119901119902B

]]]]]]]

]

(1)

where 119886119894119895is the (119894 119895) element of the matrix A

IfA is an 119868-by-119865matrix andB is a 119869-by-119865matrix then theKhatri-Rao product A ⊙ B is the 119868119869-by-119865 block matrix

A ⊙ B = [a1otimes b

1 a

119865otimes b

119865] (2)

where a119891and b

119891are the 119891th column of the matricesA and B

respectivelyIf A is an 119868-by-119869 matrix and B is an 119868-by-119869 matrix then

the Hadamard product A oplus B is

A oplus B =[[[[

[

11988611119887111198861211988712sdot sdot sdot 119886

11198691198871119869

11988621119887211198862211988722sdot sdot sdot 119886

21198691198872119869

d

1198861198681119887119868111988611986821198871198682sdot sdot sdot 119886

119868119869119887119868119869

]]]]

]

(3)

where 119886119894119895and 119887

119894119895are the (119894 119895) element of the matrices A and

B respectively

2 Data Model

A rectangular array consisted of119872times119873 elements is shown inFigure 1 where the distance between two adjacent elementsis 119889 We consider signals in the far field in which case thesignal sources are far away enough that the arriving waves areessentially planes over the array We assume that the noise isindependent of the sources It is also assumed that there are119870 noncoherent or independent sources and the number ofsources is preknown 120579

119896and 120601

119896are the elevation angle and

the azimuth angle of the 119896th source respectively We assumethe sources impinge on the array with different DOAs

The received signal of the first subarray in the rectangulararray is x

1(119905) = A

119909s(119905) + n

1(119905) where A

119909= [a

119909(1206011 1205791)

a119909(1206012 1205792) a

119909(120601119870 120579119870)] isin C119872times119870 with a

119909(120601119896 120579119896) = [1

1198901198952120587119889 sin 120579119896 cos120601119896120582 1198901198952120587(119872minus1)119889 sin 120579119896 cos120601119896120582]119879 and 120582 is thewavelength n

1(119905) is the received additive white Gaussian

noise of the first subarray s(119905) isin C119870times1 is the sourcevector The received signal of the 119899th subarray in therectangular array is x

119899(119905) = A

119909Φ119899minus1s(119905) + n

119899(119905) where Φ =

diag(1198901198952120587119889 sin 1205791 sin1206011120582 1198901198952120587119889 sin 120579119870 sin120601119870120582) and n119899(119905) is the

received additive white Gaussian noise of the 119899th subarrayTherefore the received signal of the rectangular array is [27]

x (119905) =[[[[[[

[

x1(119905)

x2(119905)

x119873(119905)

]]]]]]

]

=

[[[[[[[

[

A119909

A119909Φ

A119909Φ119873minus1

]]]]]]]

]

s (119905) +[[[[[[

[

n1(119905)

n2(119905)

n119873(119905)

]]]]]]

]

(4)

International Journal of Antennas and Propagation 3

Z

Y

X

M

N

1SubarraySubarray Subarray

2 N

120579k

120601k

middot middot middot

middot middot middot

middot middotmiddotmiddot middot

middot

middot middot middot

Figure 1 The structure of uniform rectangular array

The signal x(119905) isin C119872119873times1 in (4) can also be denoted by

x (119905) = [A119910⊙ A

119909] s (119905) + n (119905) (5)

where A119910

= [a119910(1206011 1205791) a

119910(1206012 1205792) a

119910(120601119870 120579119870)] with

a119910(120601119896 120579119896) = [1 119890

1198952120587119889 sin 120579119896 sin120601119896120582 1198901198952120587(119873minus1)119889 sin 120579119896 sin120601119896120582]119879n(119905) = [n

1(119905)

119879n2(119905)

119879 n

119873(119905)

119879]119879isin C119872119873times1 ⊙ denotes

Khatri-Rao productAccording to the definition of Khatri-Rao product the

signal in (5) can be rewritten as

x (119905) = [a119910(1206011 1205791) otimes a

119909(1206011 1205791)

a119910(120601119870 120579119870) otimes a

119909(120601119870 120579119870)] s (119905) + n (119905)

(6)

where otimes denotes Kronecker product We collect 119871 samplesand define X = [x(1) x(2) x(119871)] isin C119872119873times119871 which canbe expressed as

X = [A119910⊙ A

119909] S119879 + N =

[[[[

[

X1

X2

X119873

]]]]

]

=

[[[[[

[

A1199091198631(A

119910)

A1199091198632(A

119910)

A119909119863119873(A

119910)

]]]]]

]

S119879 +[[[[

[

N1

N2

N119873

]]]]

]

(7)

where S = [s(1) s(2) s(119871)]119879 isin C119871times119870 is source matrixand N = [n(1)n(2) n(119871)] is the received additive whiteGaussian noise matrixN

119899isin C119872times119871 (119899 = 1 119873) is the noise

matrix Thus X119899isin C119872times119871 in (7) is denoted as

X119899= A

119909119863119899(A

119910) S119879 + N

119899 119899 = 1 2 119873 (8)

Equation (7) can also be denoted with the trilinear model [1828]

119909119898119899119897

=

119870

sum

119896=1

A119909(119898 119896)A

119910(119899 119896) S (119897 119896) + 119899

119898119899119897

119898 = 1 119872 119899 = 1 119873 119897 = 1 119871

(9)

where A119909(119898 119896) is the (119898 119896) element of the matrix A

119909and

similarly for the others 119899119898119899119897

is noise part X119899isin C119872times119871 (119899 =

1 119873) can be regarded as slicing the three-dimensionaldata in a series of slices which is shown in Figure 2 Thereare two more matrix system rearrangements in which wehave Y

119898= S119863

119898(A

119909)A119879

119910+ N119910

119898 119898 = 1 119872 and Z

119897=

A119910119863119897(S)A119879

119909+ N119911

119897 119897 = 1 119871 where N119910

119898and N119911

119897are noise

matrices Then we form the matrices of Y isin C119872119871times119873 andZ isin C119871119873times119872

Y =[[[[[[

[

Y1

Y2

Y119872

]]]]]]

]

= [A119909⊙ S]A119879

119910+ N119910

(10)

Z =[[[[[[

[

Z1

Z2

Z119871

]]]]]]

]

= [S ⊙ A119910]A119879

119909+ N119911

(11)

where

N119910=

[[[[[[

[

N119910

1

N119910

2

N119910

119872

]]]]]]

]

N119911=

[[[[[[

[

N119911

1

N119911

2

N119911

119871

]]]]]]

]

(12)

3 2D-DOA Estimation Based onCompressive Sensing Trilinear Model

We link the problem of 2D-DOA estimation for rectangulararray to compressive sensing trilinear model and derive acompressive sensing trilinear model-based 2D-DOA estima-tion algorithm Firstly we compress the received data to get acompressed trilinear model and then obtain the estimates ofcompressed direction matrices through performing trilineardecomposition for the compressed model Finally we formu-late the sparse recovery problem for 2D-DOA estimation

31 Trilinear Model Compression We compress the three-way data Χ isin C119872times119873times119871 into a smaller three-way data Χ1015840 isinC1198721015840times1198731015840times1198711015840

where1198721015840lt 1198721198731015840

lt 119873 and 1198711015840 lt 119871 The trilinearmodel compression processing is shown in Figure 3 Wedefine the compression matrices as U isin C119872times119872

1015840

V isin C119873times1198731015840

and W isin C119871times119871

1015840

and the compression matrices U V and Wcan be generated randomly or obtained by Tucker3 decom-position [23 29] We can use the Tucker3 decomposition

4 International Journal of Antennas and Propagation

M

Ym isin CLtimesN

L

N

Xn isin CMtimesL

Zl isin CNtimesM

Figure 2 Trilinear model

U isin CMtimesM998400

V isin CNtimesN998400

W isin CLtimesL998400N998400

L998400

M998400

L

M

N

CompressionCompressed

trilinear modelTrilinearmodel

Figure 3 The compression of trilinear model

where tensor is decomposed into the core tensor to obtain thecompression matrices The compression matrices should sat-isfy the restricted isometry property And random GaussianBernoulli and partial Fourier matrices satisfy the restrictedisometry property with number of measurements nearlylinear in the sparsity level [30 31]

Then compressX isin C119872119873times119871 in (7) to a smaller one asX1015840 isinC11987210158401198731015840times1198711015840

X1015840 = (V119879 otimes U119879)XW119879

= (V119879 otimes U119879) [A119910⊙ A

119909] S119879W119879

+ (V119879 otimes U119879)NW119879

(13)

According to the property of Khatri-Rao product [23] weknow

(V119879 otimes U119879) [A119910⊙ A

119909] = (V119879A

119910) ⊙ (U119879A

119909) (14)

Define A1015840119909= U119879A

119909 A1015840

119910= V119879A

119910 and S1015840 = W119879S Equation

(11) is also denoted as

X1015840 = [A1015840119910⊙ A1015840

119909] S1015840119879 + N1015840

(15)

where N1015840= (V119879 otimes U119879)NW119879 X1015840 can be denoted by trilinear

model With respect to (10) and (11) we form the matrices ofY1015840 and Z1015840 according to the compressed data

Y1015840 = [A1015840119909⊙ S1015840]A1015840119879

119910+ N1015840119910

(16)

Z1015840 = [S1015840 ⊙ A1015840119910]A1015840119879

119909+ N1015840119911

(17)

where N1015840119910 and N1015840119911 are the noise part The compressed trilin-ear model may degrade the angle estimation performance

By trilinear model compression the proposed methodhasmuch lower computational complexity than conventionaltrilinear decomposition method and requires much smallerstorage capacity Conventional compressive sensing is tocompress the matrix while our algorithm compresses thethree-dimensional tensor

32 Trilinear Decomposition Trilinear alternating leastsquare (TALS) algorithm is an iterative method forestimating the parameters of a trilinear decomposition[18 28] We concisely show the basic idea of TALS (1)update one matrix each time via LS which is conditionedon previously obtained estimates of the remaining matrices(2) proceed to update the other matrices (3) repeat untilconvergence of the LS cost function [21 22] TALS algorithmis discussed as follows

According to (15) least squares (LS) fitting is

minA1015840119909A1015840119910S101584010038171003817100381710038171003817X1015840 minus [A1015840

119910⊙ A1015840

119909] S101584011987910038171003817100381710038171003817119865 (18)

and LS update for the matrix S1015840 is

S1015840119879 = [A1015840119910⊙ A1015840

119909]+

X1015840 (19)

where A1015840119909and A1015840

119910are previously obtained estimates ofA1015840

119909and

A1015840119910 respectively According to (16) LS fitting is

minA1015840119909A1015840119910S101584010038171003817100381710038171003817Y1015840 minus [A1015840

119909⊙ S1015840]A1015840119879

119910

10038171003817100381710038171003817119865 (20)

and LS update for A1015840119910is

A1015840119879119910= [A1015840

119909⊙ S1015840]

+

Y1015840 (21)

where A1015840119909and S1015840 stand for the previously obtained estimates

of A1015840119909and S1015840 Similarly according to (17) LS fitting is

minA1015840119909A1015840119910S101584010038171003817100381710038171003817Z1015840 minus [S1015840 ⊙ A1015840

119910]A1015840119879

119909

10038171003817100381710038171003817119865 (22)

where Z1015840 is the noisy compressed signal LS update for A1015840119909is

A1015840119879119909= [S1015840 ⊙ A1015840

119910]+

Z1015840 (23)

where A1015840119910and S1015840 stand for the previously obtained estimates

of A1015840119910and S1015840 respectively

Define E = X minus [A1015840119910⊙ A1015840

119909]S1015840119879 where S1015840 A1015840

119910 and A1015840

119909

present the estimates of S1015840A1015840119910 andA1015840

119909 respectivelyThe sum

of squared residuals (SSR) in trilinear model is defined asSSR = sum

119871

119895=1sum119872119873

119894=1|119890119894119895| where 119890

119894119895is the (119894 119895) element of the

matrix E According to (19) (21) and (23) the matrices S1015840A1015840119910 and A1015840

119909are updated with least squares repeatedly until

SSR attain apriorthreshold The we obtain the final estimatesS1015840 A1015840

119910 and A1015840

119909

International Journal of Antennas and Propagation 5

Theorem 1 (see [22]) Considering X1015840119899= A1015840

119909119863119899(A1015840

119910)S1015840119879 119899 =

1 1198731015840 where A1015840

119909isin C119872

1015840times119870 S1015840 isin C119871

1015840times119870 and A1015840

119910isin C119873

1015840times119870 if

119896A1015840119909+ 119896A1015840

119910+ 119896S1015840119879 ge 2119870 + 2 (24)

where 119896A is the 119896-rank of the matrix A [18] then A1015840119909 S1015840 and

A1015840119910are unique up to permutation and scaling of columns

For the different DOAs and independent sources we have119896A1015840119909= min(1198721015840

119870) 119896A1015840119910= min(1198731015840

119870) and 119896S1015840119879 = min(1198711015840 119870)in the trilinear model in this paper and then the inequality in(24) becomes

min (1198721015840 119870) +min (1198731015840

119870) +min (1198711015840 119870)

ge 2119870 + 2

(25)

When 1198721015840ge 119870 1198731015840

ge 119870 and 1198711015840 ge 119870 the identifiablecondition is 1 le 119870 le min(1198721015840

1198731015840)

When 1198721015840le 119870 1198731015840

le 119870 and 1198711015840 ge 119870 the identifiablecondition is max(1198721015840

1198731015840) le 119870 le 119872

1015840+ 119873

1015840minus 2 Hence the

proposed algorithm is effective when119870 le 1198721015840+119873

1015840minus2 and the

maximum number of sources that can be identified is1198721015840+

1198731015840minus 2After the trilinear decomposition we obtain the estimates

of the loading matrices

A1015840119909= A1015840

119909ΠΔ

1+ E

1

S1015840 = S1015840ΠΔ2+ E

2

A1015840119910= A1015840

119910ΠΔ

3+ E

3

(26)

where Π is a permutation matrix and Δ1 Δ

2 Δ

3note for

the diagonal scaling matrices satisfying Δ1Δ2Δ3= I

119870 E

1

E2 and E

3are estimation error matrices After the trilinear

decomposition the estimates of A1015840119909 A1015840

119910 and S1015840 can be

obtained Scale ambiguity and permutation ambiguity areinherent to the trilinear decomposition problem Howeverthe scale ambiguity can be resolved easily by means ofnormalization while the existence of permutation ambiguityis not considered for angle estimation

33 Angle Estimation via Sparse Recovery Use a1015840119909119896

and a1015840119910119896

todenote the 119896th column of estimates A1015840

119909and A1015840

119910 respectively

According to the compression matrices we have

a1015840119909119896= U119879120597

119909119896a119909119896+ n

119909119896 119896 = 1 119870 (27a)

a1015840119910119896= V119879120597

119910119896a119910119896+ n

119910119896 119896 = 1 119870 (27b)

where a119909119896

and a119910119896

are the 119896th column ofA119909A

119910 respectively

n119909119896and n

119910119896are the corresponding noise respectively 120597

119909119896and

120597119909119896

are the scaling coefficients Construct two Vandermondematrices A

119904119909isin C119872times119875 and A

119904119910isin C119873times119875 (119875 ≫ 119872 119875 ≫ 119873)

composed of steering vectors corresponding to each potentialsource location as its columns

A119904119909

= [a1199041199091 a1199041199092 a

119904119909119875]

=

[[[[[

[

1 1 sdot sdot sdot 1

1198901198952120587119889g(1)120582

1198901198952120587119889g(2)120582

1198901198952120587119889g(119875)120582

1198901198952120587(119872minus1)119889g(1)120582

1198901198952120587(119872minus1)119889g(2)120582

sdot sdot sdot 1198901198952120587(119872minus1)119889g(119875)120582

]]]]]

]

(28a)

A119904119910

= [a1199041199101 a1199041199102 a

119904119910119875]

=

[[[[[

[

1 1 sdot sdot sdot 1

1198901198952120587119889g(1)120582

1198901198952120587119889g(2)120582

1198901198952120587119889g(119875)120582

1198901198952120587(119873minus1)119889g(1)120582

1198901198952120587(119873minus1)119889g(2)120582

sdot sdot sdot 1198901198952120587(119873minus1)119889g(119875)120582

]]]]]

]

(28b)

where g is a sampling vector and its 119901th elements is g(119901) =minus1 + 2119901119875 119901 = 1 2 119875 The matrices A

119904119909and A

119904119910can be

regarded as the completed dictionariesThen (27a)-(27b) canbe expressed as

a1015840119909119896= U119879A

119904119909x119904+ n

119909119896 119896 = 1 119870 (29a)

a1015840119910119896= V119879A

119910119909y119904+ n

119910119896 119896 = 1 119870 (29b)

where x119904and y

119904are sparse The estimates of x

119904and y

119904can be

obtained via 1198970-norm constraint

min 10038171003817100381710038171003817a1015840119909119896minus U119879A

119904119909x119904

10038171003817100381710038171003817

2

2

st 1003817100381710038171003817x11990410038171003817100381710038170 = 1

(30a)

min 10038171003817100381710038171003817a1015840119910119896minus V119879A

119904119910y119904

10038171003817100381710038171003817

2

2

st 1003817100381710038171003817y11990410038171003817100381710038170 = 1

(30b)

where sdot 0denotes the 119897

0-norm x

1199040= 1 that is to say

there is only one nonzero element in the vector x119904 similar

to y1199040= 1 We can use the OMP recovery method [26]

to find the nonzero element in x119904or y

119904 The OMP algorithm

tries to recover the signal by finding the strongest componentin the measurement signal removing it from the signal andsearching the dictionary again for the strongest atom that ispresented in the residual signal [32] We extract the indexof the maximum modulus of elements in x

119904and y

119904 respec-

tively noted as 119901119909and 119901

119910 According to the corresponding

columns in A119904119909

and A119904119910 we obtain g(119901

119909) and g(119901

119910) which

are estimates of sin 120579119896cos120601

119896and sin 120579

119896sin120601

119896 We define

6 International Journal of Antennas and Propagation

119903119896= g(119901

119909) + 119895g(119901

119910) and then the elevation angles and

azimuth angles can be obtained via

120579119896= sinminus1 (abs (119903

119896)) 119896 = 1 119870 (31a)

120601119896= angle (119903

119896) 119896 = 1 119870 (31b)

where abs(sdot) is the modulus value symbol and angle(sdot) is toget the angle of an imaginary number As the columns ofthe estimated matrices A1015840

119909and A1015840

119910are automatically paired

then the estimated elevation angles and azimuth angles canbe paired automatically

34 The Procedures of the Proposed Algorithm Till nowwe have achieved the proposal for the compressive sensingtrilinear model-based 2D-DOA estimation for rectangulararray We show major steps of the proposed algorithm asfollows

Step 1 Form the three-way matrix Χ isin C119872times119873times119871 thencompress the three-way matrix into a much smaller three-way matrix Χ1015840 isin C119872

1015840times1198731015840times1198711015840

via the compression matricesU isin C119872times119872

1015840

V isin C119873times1198731015840

andW isin C119871times1198711015840

Step 2 Perform trilinear decomposition through TALS algo-rithm for the compressed three-way matrix to obtain theestimation of A1015840

119909 A1015840

119910 and S1015840

Step 3 Estimate the sparse vectors

Step 4 Estimate 2D-DOA via (31a)-(31b)

Remark A Because the trilinear decomposition brings thesame permutation ambiguity for the estimatesA1015840

119909A1015840

119910 and S1015840

the estimated elevation angles and azimuth angles are pairedautomatically

Remark B The conventional compressive sensing methodformulates an angle sampling grid for sparse recovery toestimate angles When it is applied to 2D-DOA estimationboth elevation and azimuth angles must be sampled andit results in a two-dimensional sampling problem whichbrings much heavier cost for sparse signal recovery Inthis paper sin 120579

119896cos120601

119896(or sin 120579

119896sin120601

119896) is bundled into a

single variable in the range of minus1 to 1 The bundled variableis sampled for sparse recovery to obtain the estimates ofsin 120579

119896cos120601

119896and sin 120579

119896sin120601

119896 respectively Afterwards the

elevation and azimuth angles are estimated through theestimates of sin 120579

119896cos120601

119896and sin 120579

119896sin120601

119896

Remark C If the number of sources 119870 is unknown it canbe estimated by performing singular value decompositionfor received data matrix X in (7) and finding the numberof largest singular values [33] We also use some lower-complexity algorithm in [34] for estimating the number ofthe sources

Remark D When the coherent sources impinge on the arraywe can use the parallel profiles with linear dependencies

(PARALIND) model [35 36] which is a generalization ofPARAFAC suitable for solving problems with linear depen-dent factors to resolve coherent DOA estimation problem

35 Complexity Analysis and CRB The proposed algorithmhas much lower computational cost than conventional trilin-ear decomposition-based method The proposed algorithmrequires 119874(1198703 + 1198721015840

11987310158401198711015840119870) operations for a iteration while

the trilinear decomposition algorithm needs119874(1198703+119872119873119871119870)operations [28] for a iteration where1198721015840

lt 1198721198731015840lt 119873 and

1198711015840lt 119871We define A = [a

119910(1205791) otimes a

119909(1205791) a

119910(120579119870) otimes a

119909(120579119870)]

According to [37] we can derive the CRB

CRB = 1205902

2119871Re [D119867

Πperp

AD oplus P119879

119904]minus1

(32)

where 119871 denotes the number of samples a119896is the 119896th column

of A and P119904= (1119871)sum

119871

119905=1s(119905)s119867(119905) 1205902 is the noise power

Πperp

A = I119872119873

minus A(A119867A)minus1A119867 and

D = [120597a11205971205791

120597a2

1205971205792

120597a119870

120597120579119870

120597a1

1205971206011

120597a2

1205971206012

120597a119870

120597120601119870

] (33)

The advantages of the proposed algorithm can be presentedas follows

(1) The proposed algorithm can be regarded as a com-bination of trilinear model and compressive sensingtheory and it brings much lower computationalcomplexity and much smaller demand for storagecapacity

(2) The proposed algorithm has better 2D-DOA estima-tion performance than ESPRIT algorithm and closeangle estimation performance to TALS algorithmwhich will be proved by Figures 6-7

(3) The proposed algorithm can achieve paired elevationangles and azimuth angles automatically

4 Numerical Simulations

In the following simulations we assume that the numericalsimulation results converge when the SSR le 10minus8119872 119873 119871and 119870 denote the number of antennas in 119909-axis number ofantennas in 119910-axis samples and sources respectively Andwe compress the parameters 119872 119873 119871 to 1198721015840 1198731015840 and 1198711015840(usually set 119872 = 16 119873 = 20 119871 = 100 and 1198721015840

= 1198731015840=

1198711015840= 5 in numerical simulations) 119889 = 1205822 is considered

in the simulation We present 1000 Monte Carlo simulationsto assess the angle estimation performance of the proposedalgorithm Define root mean squared error (RMSE) as

RMSE = 1119870

119870

sum

119896=1

radic1

1000

1000

sum

119897=1

(120601119896119897minus 120601

119896)2

+ (120579119896119897minus 120579

119896)2

(34)

where 120601119896and 120579

119896denote the perfect elevation angle and

azimuth angle of 119896th source respectively 120601119896119897

and 120579119896119897

are

International Journal of Antennas and Propagation 7

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

Elev

atio

n (d

eg)

Azimuth (deg)

2D-DOA estimation

Figure 4 2D-DOA estimation of our algorithm in SNR = minus10 dB(119873 = 20119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

402D-DOA estimation

Elev

atio

n (d

eg)

Azimuth (deg)

Figure 5 2D-DOAestimation of our algorithm in SNR = 5 dB (119873 =20119872 = 16 119871 = 100 and 119870 = 3)

the estimates of 120601119896and 120579

119896in the 119897th Monte Carlo trail

Assume that there are 3 noncoherent sources located at angles(1206011 1205791) = (5

∘ 10

∘) (120601

2 1205792) = (15

∘ 20

∘) and (120601

3 1205793) =

(40∘ 30

∘)

Figure 4 presents the 2D-DOA estimation of the pro-posed algorithm for uniform rectangular array with119873 = 20119872 = 16 119871 = 100119870 = 3 and SNR = minus10 dB Figure 5 depictsthe 2D-DOA estimation performance with SNR = 5 dBFigures 4-5 illustrate that our algorithm is effective for 2D-DOA estimation

Figure 6 shows the 2D-DOA estimation performancecomparison of the proposed algorithm the ESPRIT algo-rithm the TALS algorithm and the CRB for the uniformrectangular array with119873 = 20119872 = 16 119871 = 100 and 119870 = 3while Figure 7 depicts the 2D-DOA estimation performancecomparison with 119873 = 16 119872 = 16 119871 = 200 and 119870 = 3It is indicated that our algorithm has better angle estimation

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

10minus3

minus10 minus5

ProposedTALS

ESPRITCRB

Figure 6 2D-DOA estimation performance comparison (119873 = 20119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

10minus3

minus10 minus5

ProposedTALS

ESPRITCRB

Figure 7 2D-DOA estimation performance comparison (119873 = 16119872 = 16 119871 = 200 and 119870 = 3)

performance than the ESPRIT algorithm and close angleestimation to TALS algorithm The angle estimation perfor-mance of the proposed algorithm will be further improvedthrough increasing the compressed parameters1198721015840 1198731015840 and1198711015840Figure 8 depicts the 2D-DOA estimation performance

of the proposed algorithm with different value of 119873 (119872 =

16 119871 = 100 and 119870 = 3) while Figure 9 presents the2D-DOA estimation performance of the proposed algorithmwith different value of 119872 It is clearly shown that theangle estimation performance of our algorithm is graduallyimproved with the number of antennas increasing Multiple

8 International Journal of Antennas and Propagation

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

N = 8

N = 16

N = 24

Figure 8 Angle estimation performance of our algorithm withdifferent119873 (119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

M = 8

M = 16

M = 24

Figure 9 Angle estimation performance of our algorithm withdifferent119872 (119873 = 16 119871 = 100 and 119870 = 3)

antennas improve the angle estimation performance becauseof diversity gain

Figure 10 presents 2D-DOA estimation performance ofthe proposed algorithm with different value of 119871 (119873 =

20 119872 = 16 and 119870 = 3) It illustrates that the angleestimation performance becomes better in collaborationwith119871 increasing

5 Conclusions

In this paper we have addressed the 2D-DOA estimationproblem for rectangular array and have derived a compressive

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

L = 50

L = 100

L = 200

Figure 10 Angle estimation performance of our algorithm withdifferent 119871 (119873 = 16119872 = 20 and 119870 = 3)

sensing trilinear model-based 2D-DOA estimation algo-rithm which can obtain the automatically paired 2D-DOAestimateThe proposed algorithm has better angle estimationperformance than ESPRIT algorithm and close angle esti-mation performance to conventional trilinear decompositionmethod Furthermore the proposed algorithm has lowercomputational complexity and smaller demand for storagecapacity than conventional trilinear decomposition methodThe proposed algorithm can be regarded as a combination oftrilinear model and compressive sensing theory and it bringsmuch lower computational complexity and much smallerdemand for storage capacity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by China NSF Grants (6137116961301108 61471191 61471192 and 61271327) Jiangsu PlannedProjects for Postdoctoral Research Funds (1201039C)China Postdoctoral Science Foundation (2012M5210992013M541661) Open Project of Key Laboratory of ModernAcoustics of Ministry of Education (Nanjing University) theAeronautical Science Foundation of China (20120152001)Qing Lan Project Priority Academic Program Developmentof Jiangsu High Education Institutions and the FundamentalResearch Funds for the Central Universities (NS2013024kfjj130114 and kfjj130115)

International Journal of Antennas and Propagation 9

References

[1] HKrim andMViberg ldquoTwodecades of array signal processingresearch the parametric approachrdquo IEEE Signal ProcessingMagazine vol 13 no 4 pp 67ndash94 1996

[2] R Roy and T Kailath ldquoESPRIT-estimation of signal parametersvia rotational invariance techniquesrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 37 no 7 pp 984ndash995 1989

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

[4] D Kundu ldquoModified MUSIC algorithm for estimating DOA ofsignalsrdquo Signal Processing vol 48 no 1 pp 85ndash90 1996

[5] B D Rao and K V S Hari ldquoPerformance analysis of Root-Musicrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 37 no 12 pp 1939ndash1949 1989

[6] N Yilmazer J Koh and T K Sarkar ldquoUtilization of a unitarytransform for efficient computation in thematrix pencilmethodto find the direction of arrivalrdquo IEEE Transactions on Antennasand Propagation vol 54 no 1 pp 175ndash181 2006

[7] Y Chiba K Ichige and H Arai ldquoReducing DOA estimationerror in extended ES-root-MUSIC for uniform rectangulararrayrdquo in Proceedings of the 4th International Congress on Imageand Signal Processing (CISP rsquo11) vol 5 pp 2621ndash2625 October2011

[8] 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

[9] 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

[10] 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

[11] 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

[12] 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

[13] 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

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

[15] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo IEEE Transactions on InformationThe-ory vol 52 no 2 pp 489ndash509 2006

[16] 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

[17] N Hu Z Ye D Xu and S Cao ldquoA sparse recovery algorithmfor DOA estimation using weighted subspace fittingrdquo SignalProcessing vol 92 no 10 pp 2566ndash2570 2012

[18] 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

[19] L de Lathauwer B de Moor and J Vandewalle ldquoComputationof the canonical decomposition by means of a simultaneousgeneralized Schur decompositionrdquo SIAM Journal on MatrixAnalysis and Applications vol 26 no 2 pp 295ndash327 2004

[20] L de Lathauwer ldquoA link between the canonical decompositionin multi-linear algebra and simultaneous matrix diagonaliza-tionrdquo SIAM Journal on Matrix Analysis and Applications vol28 no 3 pp 642ndash666 2006

[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000

[22] N D Sidiropoulos and X Liu ldquoIdentifiability results for blindbeamforming in incoherent multipath with small delay spreadrdquoIEEE Transactions on Signal Processing vol 49 no 1 pp 228ndash236 2001

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

[24] X F Zhang H X Yu J F Li andD Ben ldquoBlind signal detectionfor uniform rectangular array via compressive sensing trilinearmodelrdquo Advanced Materials Research vol 756ndash759 pp 660ndash664 2013

[25] R Cao X Zhang and W Chen ldquoCompressed sensing parallelfactor analysis-based joint angle andDoppler frequency estima-tion for monostatic multiple-inputndashmultiple-output radarrdquo IETRadar Sonar amp Navigation vol 8 no 6 pp 597ndash606 2014

[26] 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

[27] X Zhang F Wang and H Chen Theory and Application ofArray Signal Processing (version 2) National Defense IndustryPress Beijing China 2012

[28] X Zhang J Li H Chen and D Xu ldquoTrilinear decomposition-based two-dimensional DOA estimation algorithm for arbitrar-ily spaced acoustic vector-sensor array subjected to unknownlocationsrdquoWireless Personal Communications vol 67 no 4 pp859ndash877 2012

[29] R Bro N D Sidiropoulos and G B Giannakis ldquoA fast leastsquares algorithm for separating trilinear mixturesrdquo in Proceed-ings of the International Workshop on Independent ComponentAnalysis and Blind Signal Separation pp 289ndash294 January 1999

[30] R A DeVore ldquoDeterministic constructions of compressedsensing matricesrdquo Journal of Complexity vol 23 no 4ndash6 pp918ndash925 2007

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

[32] F Wang and X Zhang ldquoJoint estimation of TOA and DOA inIR-UWB system using sparse representation frameworkrdquo ETRIJournal vol 36 no 3 pp 460ndash468 2014

[33] A Di ldquoMultiple source locationmdasha matrix decompositionapproachrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 33 no 5 pp 1086ndash1091 1985

[34] J Xin N Zheng and A Sano ldquoSimple and efficient nonpara-metric method for estimating the number of signals withouteigendecompositionrdquo IEEE Transactions on Signal Processingvol 55 no 4 pp 1405ndash1420 2007

10 International Journal of Antennas and Propagation

[35] 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

[36] X Zhang M Zhou and J Li ldquoA PARALIND decomposition-based coherent two-dimensional direction of arrival estimationalgorithm for acoustic vector-sensor arraysrdquo Sensors vol 13 no4 pp 5302ndash5316 2013

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

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Active and Passive Electronic Components

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 3

Z

Y

X

M

N

1SubarraySubarray Subarray

2 N

120579k

120601k

middot middot middot

middot middot middot

middot middotmiddotmiddot middot

middot

middot middot middot

Figure 1 The structure of uniform rectangular array

The signal x(119905) isin C119872119873times1 in (4) can also be denoted by

x (119905) = [A119910⊙ A

119909] s (119905) + n (119905) (5)

where A119910

= [a119910(1206011 1205791) a

119910(1206012 1205792) a

119910(120601119870 120579119870)] with

a119910(120601119896 120579119896) = [1 119890

1198952120587119889 sin 120579119896 sin120601119896120582 1198901198952120587(119873minus1)119889 sin 120579119896 sin120601119896120582]119879n(119905) = [n

1(119905)

119879n2(119905)

119879 n

119873(119905)

119879]119879isin C119872119873times1 ⊙ denotes

Khatri-Rao productAccording to the definition of Khatri-Rao product the

signal in (5) can be rewritten as

x (119905) = [a119910(1206011 1205791) otimes a

119909(1206011 1205791)

a119910(120601119870 120579119870) otimes a

119909(120601119870 120579119870)] s (119905) + n (119905)

(6)

where otimes denotes Kronecker product We collect 119871 samplesand define X = [x(1) x(2) x(119871)] isin C119872119873times119871 which canbe expressed as

X = [A119910⊙ A

119909] S119879 + N =

[[[[

[

X1

X2

X119873

]]]]

]

=

[[[[[

[

A1199091198631(A

119910)

A1199091198632(A

119910)

A119909119863119873(A

119910)

]]]]]

]

S119879 +[[[[

[

N1

N2

N119873

]]]]

]

(7)

where S = [s(1) s(2) s(119871)]119879 isin C119871times119870 is source matrixand N = [n(1)n(2) n(119871)] is the received additive whiteGaussian noise matrixN

119899isin C119872times119871 (119899 = 1 119873) is the noise

matrix Thus X119899isin C119872times119871 in (7) is denoted as

X119899= A

119909119863119899(A

119910) S119879 + N

119899 119899 = 1 2 119873 (8)

Equation (7) can also be denoted with the trilinear model [1828]

119909119898119899119897

=

119870

sum

119896=1

A119909(119898 119896)A

119910(119899 119896) S (119897 119896) + 119899

119898119899119897

119898 = 1 119872 119899 = 1 119873 119897 = 1 119871

(9)

where A119909(119898 119896) is the (119898 119896) element of the matrix A

119909and

similarly for the others 119899119898119899119897

is noise part X119899isin C119872times119871 (119899 =

1 119873) can be regarded as slicing the three-dimensionaldata in a series of slices which is shown in Figure 2 Thereare two more matrix system rearrangements in which wehave Y

119898= S119863

119898(A

119909)A119879

119910+ N119910

119898 119898 = 1 119872 and Z

119897=

A119910119863119897(S)A119879

119909+ N119911

119897 119897 = 1 119871 where N119910

119898and N119911

119897are noise

matrices Then we form the matrices of Y isin C119872119871times119873 andZ isin C119871119873times119872

Y =[[[[[[

[

Y1

Y2

Y119872

]]]]]]

]

= [A119909⊙ S]A119879

119910+ N119910

(10)

Z =[[[[[[

[

Z1

Z2

Z119871

]]]]]]

]

= [S ⊙ A119910]A119879

119909+ N119911

(11)

where

N119910=

[[[[[[

[

N119910

1

N119910

2

N119910

119872

]]]]]]

]

N119911=

[[[[[[

[

N119911

1

N119911

2

N119911

119871

]]]]]]

]

(12)

3 2D-DOA Estimation Based onCompressive Sensing Trilinear Model

We link the problem of 2D-DOA estimation for rectangulararray to compressive sensing trilinear model and derive acompressive sensing trilinear model-based 2D-DOA estima-tion algorithm Firstly we compress the received data to get acompressed trilinear model and then obtain the estimates ofcompressed direction matrices through performing trilineardecomposition for the compressed model Finally we formu-late the sparse recovery problem for 2D-DOA estimation

31 Trilinear Model Compression We compress the three-way data Χ isin C119872times119873times119871 into a smaller three-way data Χ1015840 isinC1198721015840times1198731015840times1198711015840

where1198721015840lt 1198721198731015840

lt 119873 and 1198711015840 lt 119871 The trilinearmodel compression processing is shown in Figure 3 Wedefine the compression matrices as U isin C119872times119872

1015840

V isin C119873times1198731015840

and W isin C119871times119871

1015840

and the compression matrices U V and Wcan be generated randomly or obtained by Tucker3 decom-position [23 29] We can use the Tucker3 decomposition

4 International Journal of Antennas and Propagation

M

Ym isin CLtimesN

L

N

Xn isin CMtimesL

Zl isin CNtimesM

Figure 2 Trilinear model

U isin CMtimesM998400

V isin CNtimesN998400

W isin CLtimesL998400N998400

L998400

M998400

L

M

N

CompressionCompressed

trilinear modelTrilinearmodel

Figure 3 The compression of trilinear model

where tensor is decomposed into the core tensor to obtain thecompression matrices The compression matrices should sat-isfy the restricted isometry property And random GaussianBernoulli and partial Fourier matrices satisfy the restrictedisometry property with number of measurements nearlylinear in the sparsity level [30 31]

Then compressX isin C119872119873times119871 in (7) to a smaller one asX1015840 isinC11987210158401198731015840times1198711015840

X1015840 = (V119879 otimes U119879)XW119879

= (V119879 otimes U119879) [A119910⊙ A

119909] S119879W119879

+ (V119879 otimes U119879)NW119879

(13)

According to the property of Khatri-Rao product [23] weknow

(V119879 otimes U119879) [A119910⊙ A

119909] = (V119879A

119910) ⊙ (U119879A

119909) (14)

Define A1015840119909= U119879A

119909 A1015840

119910= V119879A

119910 and S1015840 = W119879S Equation

(11) is also denoted as

X1015840 = [A1015840119910⊙ A1015840

119909] S1015840119879 + N1015840

(15)

where N1015840= (V119879 otimes U119879)NW119879 X1015840 can be denoted by trilinear

model With respect to (10) and (11) we form the matrices ofY1015840 and Z1015840 according to the compressed data

Y1015840 = [A1015840119909⊙ S1015840]A1015840119879

119910+ N1015840119910

(16)

Z1015840 = [S1015840 ⊙ A1015840119910]A1015840119879

119909+ N1015840119911

(17)

where N1015840119910 and N1015840119911 are the noise part The compressed trilin-ear model may degrade the angle estimation performance

By trilinear model compression the proposed methodhasmuch lower computational complexity than conventionaltrilinear decomposition method and requires much smallerstorage capacity Conventional compressive sensing is tocompress the matrix while our algorithm compresses thethree-dimensional tensor

32 Trilinear Decomposition Trilinear alternating leastsquare (TALS) algorithm is an iterative method forestimating the parameters of a trilinear decomposition[18 28] We concisely show the basic idea of TALS (1)update one matrix each time via LS which is conditionedon previously obtained estimates of the remaining matrices(2) proceed to update the other matrices (3) repeat untilconvergence of the LS cost function [21 22] TALS algorithmis discussed as follows

According to (15) least squares (LS) fitting is

minA1015840119909A1015840119910S101584010038171003817100381710038171003817X1015840 minus [A1015840

119910⊙ A1015840

119909] S101584011987910038171003817100381710038171003817119865 (18)

and LS update for the matrix S1015840 is

S1015840119879 = [A1015840119910⊙ A1015840

119909]+

X1015840 (19)

where A1015840119909and A1015840

119910are previously obtained estimates ofA1015840

119909and

A1015840119910 respectively According to (16) LS fitting is

minA1015840119909A1015840119910S101584010038171003817100381710038171003817Y1015840 minus [A1015840

119909⊙ S1015840]A1015840119879

119910

10038171003817100381710038171003817119865 (20)

and LS update for A1015840119910is

A1015840119879119910= [A1015840

119909⊙ S1015840]

+

Y1015840 (21)

where A1015840119909and S1015840 stand for the previously obtained estimates

of A1015840119909and S1015840 Similarly according to (17) LS fitting is

minA1015840119909A1015840119910S101584010038171003817100381710038171003817Z1015840 minus [S1015840 ⊙ A1015840

119910]A1015840119879

119909

10038171003817100381710038171003817119865 (22)

where Z1015840 is the noisy compressed signal LS update for A1015840119909is

A1015840119879119909= [S1015840 ⊙ A1015840

119910]+

Z1015840 (23)

where A1015840119910and S1015840 stand for the previously obtained estimates

of A1015840119910and S1015840 respectively

Define E = X minus [A1015840119910⊙ A1015840

119909]S1015840119879 where S1015840 A1015840

119910 and A1015840

119909

present the estimates of S1015840A1015840119910 andA1015840

119909 respectivelyThe sum

of squared residuals (SSR) in trilinear model is defined asSSR = sum

119871

119895=1sum119872119873

119894=1|119890119894119895| where 119890

119894119895is the (119894 119895) element of the

matrix E According to (19) (21) and (23) the matrices S1015840A1015840119910 and A1015840

119909are updated with least squares repeatedly until

SSR attain apriorthreshold The we obtain the final estimatesS1015840 A1015840

119910 and A1015840

119909

International Journal of Antennas and Propagation 5

Theorem 1 (see [22]) Considering X1015840119899= A1015840

119909119863119899(A1015840

119910)S1015840119879 119899 =

1 1198731015840 where A1015840

119909isin C119872

1015840times119870 S1015840 isin C119871

1015840times119870 and A1015840

119910isin C119873

1015840times119870 if

119896A1015840119909+ 119896A1015840

119910+ 119896S1015840119879 ge 2119870 + 2 (24)

where 119896A is the 119896-rank of the matrix A [18] then A1015840119909 S1015840 and

A1015840119910are unique up to permutation and scaling of columns

For the different DOAs and independent sources we have119896A1015840119909= min(1198721015840

119870) 119896A1015840119910= min(1198731015840

119870) and 119896S1015840119879 = min(1198711015840 119870)in the trilinear model in this paper and then the inequality in(24) becomes

min (1198721015840 119870) +min (1198731015840

119870) +min (1198711015840 119870)

ge 2119870 + 2

(25)

When 1198721015840ge 119870 1198731015840

ge 119870 and 1198711015840 ge 119870 the identifiablecondition is 1 le 119870 le min(1198721015840

1198731015840)

When 1198721015840le 119870 1198731015840

le 119870 and 1198711015840 ge 119870 the identifiablecondition is max(1198721015840

1198731015840) le 119870 le 119872

1015840+ 119873

1015840minus 2 Hence the

proposed algorithm is effective when119870 le 1198721015840+119873

1015840minus2 and the

maximum number of sources that can be identified is1198721015840+

1198731015840minus 2After the trilinear decomposition we obtain the estimates

of the loading matrices

A1015840119909= A1015840

119909ΠΔ

1+ E

1

S1015840 = S1015840ΠΔ2+ E

2

A1015840119910= A1015840

119910ΠΔ

3+ E

3

(26)

where Π is a permutation matrix and Δ1 Δ

2 Δ

3note for

the diagonal scaling matrices satisfying Δ1Δ2Δ3= I

119870 E

1

E2 and E

3are estimation error matrices After the trilinear

decomposition the estimates of A1015840119909 A1015840

119910 and S1015840 can be

obtained Scale ambiguity and permutation ambiguity areinherent to the trilinear decomposition problem Howeverthe scale ambiguity can be resolved easily by means ofnormalization while the existence of permutation ambiguityis not considered for angle estimation

33 Angle Estimation via Sparse Recovery Use a1015840119909119896

and a1015840119910119896

todenote the 119896th column of estimates A1015840

119909and A1015840

119910 respectively

According to the compression matrices we have

a1015840119909119896= U119879120597

119909119896a119909119896+ n

119909119896 119896 = 1 119870 (27a)

a1015840119910119896= V119879120597

119910119896a119910119896+ n

119910119896 119896 = 1 119870 (27b)

where a119909119896

and a119910119896

are the 119896th column ofA119909A

119910 respectively

n119909119896and n

119910119896are the corresponding noise respectively 120597

119909119896and

120597119909119896

are the scaling coefficients Construct two Vandermondematrices A

119904119909isin C119872times119875 and A

119904119910isin C119873times119875 (119875 ≫ 119872 119875 ≫ 119873)

composed of steering vectors corresponding to each potentialsource location as its columns

A119904119909

= [a1199041199091 a1199041199092 a

119904119909119875]

=

[[[[[

[

1 1 sdot sdot sdot 1

1198901198952120587119889g(1)120582

1198901198952120587119889g(2)120582

1198901198952120587119889g(119875)120582

1198901198952120587(119872minus1)119889g(1)120582

1198901198952120587(119872minus1)119889g(2)120582

sdot sdot sdot 1198901198952120587(119872minus1)119889g(119875)120582

]]]]]

]

(28a)

A119904119910

= [a1199041199101 a1199041199102 a

119904119910119875]

=

[[[[[

[

1 1 sdot sdot sdot 1

1198901198952120587119889g(1)120582

1198901198952120587119889g(2)120582

1198901198952120587119889g(119875)120582

1198901198952120587(119873minus1)119889g(1)120582

1198901198952120587(119873minus1)119889g(2)120582

sdot sdot sdot 1198901198952120587(119873minus1)119889g(119875)120582

]]]]]

]

(28b)

where g is a sampling vector and its 119901th elements is g(119901) =minus1 + 2119901119875 119901 = 1 2 119875 The matrices A

119904119909and A

119904119910can be

regarded as the completed dictionariesThen (27a)-(27b) canbe expressed as

a1015840119909119896= U119879A

119904119909x119904+ n

119909119896 119896 = 1 119870 (29a)

a1015840119910119896= V119879A

119910119909y119904+ n

119910119896 119896 = 1 119870 (29b)

where x119904and y

119904are sparse The estimates of x

119904and y

119904can be

obtained via 1198970-norm constraint

min 10038171003817100381710038171003817a1015840119909119896minus U119879A

119904119909x119904

10038171003817100381710038171003817

2

2

st 1003817100381710038171003817x11990410038171003817100381710038170 = 1

(30a)

min 10038171003817100381710038171003817a1015840119910119896minus V119879A

119904119910y119904

10038171003817100381710038171003817

2

2

st 1003817100381710038171003817y11990410038171003817100381710038170 = 1

(30b)

where sdot 0denotes the 119897

0-norm x

1199040= 1 that is to say

there is only one nonzero element in the vector x119904 similar

to y1199040= 1 We can use the OMP recovery method [26]

to find the nonzero element in x119904or y

119904 The OMP algorithm

tries to recover the signal by finding the strongest componentin the measurement signal removing it from the signal andsearching the dictionary again for the strongest atom that ispresented in the residual signal [32] We extract the indexof the maximum modulus of elements in x

119904and y

119904 respec-

tively noted as 119901119909and 119901

119910 According to the corresponding

columns in A119904119909

and A119904119910 we obtain g(119901

119909) and g(119901

119910) which

are estimates of sin 120579119896cos120601

119896and sin 120579

119896sin120601

119896 We define

6 International Journal of Antennas and Propagation

119903119896= g(119901

119909) + 119895g(119901

119910) and then the elevation angles and

azimuth angles can be obtained via

120579119896= sinminus1 (abs (119903

119896)) 119896 = 1 119870 (31a)

120601119896= angle (119903

119896) 119896 = 1 119870 (31b)

where abs(sdot) is the modulus value symbol and angle(sdot) is toget the angle of an imaginary number As the columns ofthe estimated matrices A1015840

119909and A1015840

119910are automatically paired

then the estimated elevation angles and azimuth angles canbe paired automatically

34 The Procedures of the Proposed Algorithm Till nowwe have achieved the proposal for the compressive sensingtrilinear model-based 2D-DOA estimation for rectangulararray We show major steps of the proposed algorithm asfollows

Step 1 Form the three-way matrix Χ isin C119872times119873times119871 thencompress the three-way matrix into a much smaller three-way matrix Χ1015840 isin C119872

1015840times1198731015840times1198711015840

via the compression matricesU isin C119872times119872

1015840

V isin C119873times1198731015840

andW isin C119871times1198711015840

Step 2 Perform trilinear decomposition through TALS algo-rithm for the compressed three-way matrix to obtain theestimation of A1015840

119909 A1015840

119910 and S1015840

Step 3 Estimate the sparse vectors

Step 4 Estimate 2D-DOA via (31a)-(31b)

Remark A Because the trilinear decomposition brings thesame permutation ambiguity for the estimatesA1015840

119909A1015840

119910 and S1015840

the estimated elevation angles and azimuth angles are pairedautomatically

Remark B The conventional compressive sensing methodformulates an angle sampling grid for sparse recovery toestimate angles When it is applied to 2D-DOA estimationboth elevation and azimuth angles must be sampled andit results in a two-dimensional sampling problem whichbrings much heavier cost for sparse signal recovery Inthis paper sin 120579

119896cos120601

119896(or sin 120579

119896sin120601

119896) is bundled into a

single variable in the range of minus1 to 1 The bundled variableis sampled for sparse recovery to obtain the estimates ofsin 120579

119896cos120601

119896and sin 120579

119896sin120601

119896 respectively Afterwards the

elevation and azimuth angles are estimated through theestimates of sin 120579

119896cos120601

119896and sin 120579

119896sin120601

119896

Remark C If the number of sources 119870 is unknown it canbe estimated by performing singular value decompositionfor received data matrix X in (7) and finding the numberof largest singular values [33] We also use some lower-complexity algorithm in [34] for estimating the number ofthe sources

Remark D When the coherent sources impinge on the arraywe can use the parallel profiles with linear dependencies

(PARALIND) model [35 36] which is a generalization ofPARAFAC suitable for solving problems with linear depen-dent factors to resolve coherent DOA estimation problem

35 Complexity Analysis and CRB The proposed algorithmhas much lower computational cost than conventional trilin-ear decomposition-based method The proposed algorithmrequires 119874(1198703 + 1198721015840

11987310158401198711015840119870) operations for a iteration while

the trilinear decomposition algorithm needs119874(1198703+119872119873119871119870)operations [28] for a iteration where1198721015840

lt 1198721198731015840lt 119873 and

1198711015840lt 119871We define A = [a

119910(1205791) otimes a

119909(1205791) a

119910(120579119870) otimes a

119909(120579119870)]

According to [37] we can derive the CRB

CRB = 1205902

2119871Re [D119867

Πperp

AD oplus P119879

119904]minus1

(32)

where 119871 denotes the number of samples a119896is the 119896th column

of A and P119904= (1119871)sum

119871

119905=1s(119905)s119867(119905) 1205902 is the noise power

Πperp

A = I119872119873

minus A(A119867A)minus1A119867 and

D = [120597a11205971205791

120597a2

1205971205792

120597a119870

120597120579119870

120597a1

1205971206011

120597a2

1205971206012

120597a119870

120597120601119870

] (33)

The advantages of the proposed algorithm can be presentedas follows

(1) The proposed algorithm can be regarded as a com-bination of trilinear model and compressive sensingtheory and it brings much lower computationalcomplexity and much smaller demand for storagecapacity

(2) The proposed algorithm has better 2D-DOA estima-tion performance than ESPRIT algorithm and closeangle estimation performance to TALS algorithmwhich will be proved by Figures 6-7

(3) The proposed algorithm can achieve paired elevationangles and azimuth angles automatically

4 Numerical Simulations

In the following simulations we assume that the numericalsimulation results converge when the SSR le 10minus8119872 119873 119871and 119870 denote the number of antennas in 119909-axis number ofantennas in 119910-axis samples and sources respectively Andwe compress the parameters 119872 119873 119871 to 1198721015840 1198731015840 and 1198711015840(usually set 119872 = 16 119873 = 20 119871 = 100 and 1198721015840

= 1198731015840=

1198711015840= 5 in numerical simulations) 119889 = 1205822 is considered

in the simulation We present 1000 Monte Carlo simulationsto assess the angle estimation performance of the proposedalgorithm Define root mean squared error (RMSE) as

RMSE = 1119870

119870

sum

119896=1

radic1

1000

1000

sum

119897=1

(120601119896119897minus 120601

119896)2

+ (120579119896119897minus 120579

119896)2

(34)

where 120601119896and 120579

119896denote the perfect elevation angle and

azimuth angle of 119896th source respectively 120601119896119897

and 120579119896119897

are

International Journal of Antennas and Propagation 7

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

Elev

atio

n (d

eg)

Azimuth (deg)

2D-DOA estimation

Figure 4 2D-DOA estimation of our algorithm in SNR = minus10 dB(119873 = 20119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

402D-DOA estimation

Elev

atio

n (d

eg)

Azimuth (deg)

Figure 5 2D-DOAestimation of our algorithm in SNR = 5 dB (119873 =20119872 = 16 119871 = 100 and 119870 = 3)

the estimates of 120601119896and 120579

119896in the 119897th Monte Carlo trail

Assume that there are 3 noncoherent sources located at angles(1206011 1205791) = (5

∘ 10

∘) (120601

2 1205792) = (15

∘ 20

∘) and (120601

3 1205793) =

(40∘ 30

∘)

Figure 4 presents the 2D-DOA estimation of the pro-posed algorithm for uniform rectangular array with119873 = 20119872 = 16 119871 = 100119870 = 3 and SNR = minus10 dB Figure 5 depictsthe 2D-DOA estimation performance with SNR = 5 dBFigures 4-5 illustrate that our algorithm is effective for 2D-DOA estimation

Figure 6 shows the 2D-DOA estimation performancecomparison of the proposed algorithm the ESPRIT algo-rithm the TALS algorithm and the CRB for the uniformrectangular array with119873 = 20119872 = 16 119871 = 100 and 119870 = 3while Figure 7 depicts the 2D-DOA estimation performancecomparison with 119873 = 16 119872 = 16 119871 = 200 and 119870 = 3It is indicated that our algorithm has better angle estimation

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

10minus3

minus10 minus5

ProposedTALS

ESPRITCRB

Figure 6 2D-DOA estimation performance comparison (119873 = 20119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

10minus3

minus10 minus5

ProposedTALS

ESPRITCRB

Figure 7 2D-DOA estimation performance comparison (119873 = 16119872 = 16 119871 = 200 and 119870 = 3)

performance than the ESPRIT algorithm and close angleestimation to TALS algorithm The angle estimation perfor-mance of the proposed algorithm will be further improvedthrough increasing the compressed parameters1198721015840 1198731015840 and1198711015840Figure 8 depicts the 2D-DOA estimation performance

of the proposed algorithm with different value of 119873 (119872 =

16 119871 = 100 and 119870 = 3) while Figure 9 presents the2D-DOA estimation performance of the proposed algorithmwith different value of 119872 It is clearly shown that theangle estimation performance of our algorithm is graduallyimproved with the number of antennas increasing Multiple

8 International Journal of Antennas and Propagation

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

N = 8

N = 16

N = 24

Figure 8 Angle estimation performance of our algorithm withdifferent119873 (119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

M = 8

M = 16

M = 24

Figure 9 Angle estimation performance of our algorithm withdifferent119872 (119873 = 16 119871 = 100 and 119870 = 3)

antennas improve the angle estimation performance becauseof diversity gain

Figure 10 presents 2D-DOA estimation performance ofthe proposed algorithm with different value of 119871 (119873 =

20 119872 = 16 and 119870 = 3) It illustrates that the angleestimation performance becomes better in collaborationwith119871 increasing

5 Conclusions

In this paper we have addressed the 2D-DOA estimationproblem for rectangular array and have derived a compressive

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

L = 50

L = 100

L = 200

Figure 10 Angle estimation performance of our algorithm withdifferent 119871 (119873 = 16119872 = 20 and 119870 = 3)

sensing trilinear model-based 2D-DOA estimation algo-rithm which can obtain the automatically paired 2D-DOAestimateThe proposed algorithm has better angle estimationperformance than ESPRIT algorithm and close angle esti-mation performance to conventional trilinear decompositionmethod Furthermore the proposed algorithm has lowercomputational complexity and smaller demand for storagecapacity than conventional trilinear decomposition methodThe proposed algorithm can be regarded as a combination oftrilinear model and compressive sensing theory and it bringsmuch lower computational complexity and much smallerdemand for storage capacity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by China NSF Grants (6137116961301108 61471191 61471192 and 61271327) Jiangsu PlannedProjects for Postdoctoral Research Funds (1201039C)China Postdoctoral Science Foundation (2012M5210992013M541661) Open Project of Key Laboratory of ModernAcoustics of Ministry of Education (Nanjing University) theAeronautical Science Foundation of China (20120152001)Qing Lan Project Priority Academic Program Developmentof Jiangsu High Education Institutions and the FundamentalResearch Funds for the Central Universities (NS2013024kfjj130114 and kfjj130115)

International Journal of Antennas and Propagation 9

References

[1] HKrim andMViberg ldquoTwodecades of array signal processingresearch the parametric approachrdquo IEEE Signal ProcessingMagazine vol 13 no 4 pp 67ndash94 1996

[2] R Roy and T Kailath ldquoESPRIT-estimation of signal parametersvia rotational invariance techniquesrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 37 no 7 pp 984ndash995 1989

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

[4] D Kundu ldquoModified MUSIC algorithm for estimating DOA ofsignalsrdquo Signal Processing vol 48 no 1 pp 85ndash90 1996

[5] B D Rao and K V S Hari ldquoPerformance analysis of Root-Musicrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 37 no 12 pp 1939ndash1949 1989

[6] N Yilmazer J Koh and T K Sarkar ldquoUtilization of a unitarytransform for efficient computation in thematrix pencilmethodto find the direction of arrivalrdquo IEEE Transactions on Antennasand Propagation vol 54 no 1 pp 175ndash181 2006

[7] Y Chiba K Ichige and H Arai ldquoReducing DOA estimationerror in extended ES-root-MUSIC for uniform rectangulararrayrdquo in Proceedings of the 4th International Congress on Imageand Signal Processing (CISP rsquo11) vol 5 pp 2621ndash2625 October2011

[8] 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

[9] 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

[10] 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

[11] 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

[12] 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

[13] 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

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

[15] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo IEEE Transactions on InformationThe-ory vol 52 no 2 pp 489ndash509 2006

[16] 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

[17] N Hu Z Ye D Xu and S Cao ldquoA sparse recovery algorithmfor DOA estimation using weighted subspace fittingrdquo SignalProcessing vol 92 no 10 pp 2566ndash2570 2012

[18] 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

[19] L de Lathauwer B de Moor and J Vandewalle ldquoComputationof the canonical decomposition by means of a simultaneousgeneralized Schur decompositionrdquo SIAM Journal on MatrixAnalysis and Applications vol 26 no 2 pp 295ndash327 2004

[20] L de Lathauwer ldquoA link between the canonical decompositionin multi-linear algebra and simultaneous matrix diagonaliza-tionrdquo SIAM Journal on Matrix Analysis and Applications vol28 no 3 pp 642ndash666 2006

[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000

[22] N D Sidiropoulos and X Liu ldquoIdentifiability results for blindbeamforming in incoherent multipath with small delay spreadrdquoIEEE Transactions on Signal Processing vol 49 no 1 pp 228ndash236 2001

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

[24] X F Zhang H X Yu J F Li andD Ben ldquoBlind signal detectionfor uniform rectangular array via compressive sensing trilinearmodelrdquo Advanced Materials Research vol 756ndash759 pp 660ndash664 2013

[25] R Cao X Zhang and W Chen ldquoCompressed sensing parallelfactor analysis-based joint angle andDoppler frequency estima-tion for monostatic multiple-inputndashmultiple-output radarrdquo IETRadar Sonar amp Navigation vol 8 no 6 pp 597ndash606 2014

[26] 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

[27] X Zhang F Wang and H Chen Theory and Application ofArray Signal Processing (version 2) National Defense IndustryPress Beijing China 2012

[28] X Zhang J Li H Chen and D Xu ldquoTrilinear decomposition-based two-dimensional DOA estimation algorithm for arbitrar-ily spaced acoustic vector-sensor array subjected to unknownlocationsrdquoWireless Personal Communications vol 67 no 4 pp859ndash877 2012

[29] R Bro N D Sidiropoulos and G B Giannakis ldquoA fast leastsquares algorithm for separating trilinear mixturesrdquo in Proceed-ings of the International Workshop on Independent ComponentAnalysis and Blind Signal Separation pp 289ndash294 January 1999

[30] R A DeVore ldquoDeterministic constructions of compressedsensing matricesrdquo Journal of Complexity vol 23 no 4ndash6 pp918ndash925 2007

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

[32] F Wang and X Zhang ldquoJoint estimation of TOA and DOA inIR-UWB system using sparse representation frameworkrdquo ETRIJournal vol 36 no 3 pp 460ndash468 2014

[33] A Di ldquoMultiple source locationmdasha matrix decompositionapproachrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 33 no 5 pp 1086ndash1091 1985

[34] J Xin N Zheng and A Sano ldquoSimple and efficient nonpara-metric method for estimating the number of signals withouteigendecompositionrdquo IEEE Transactions on Signal Processingvol 55 no 4 pp 1405ndash1420 2007

10 International Journal of Antennas and Propagation

[35] 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

[36] X Zhang M Zhou and J Li ldquoA PARALIND decomposition-based coherent two-dimensional direction of arrival estimationalgorithm for acoustic vector-sensor arraysrdquo Sensors vol 13 no4 pp 5302ndash5316 2013

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

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Submit your manuscripts athttpwwwhindawicom

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

4 International Journal of Antennas and Propagation

M

Ym isin CLtimesN

L

N

Xn isin CMtimesL

Zl isin CNtimesM

Figure 2 Trilinear model

U isin CMtimesM998400

V isin CNtimesN998400

W isin CLtimesL998400N998400

L998400

M998400

L

M

N

CompressionCompressed

trilinear modelTrilinearmodel

Figure 3 The compression of trilinear model

where tensor is decomposed into the core tensor to obtain thecompression matrices The compression matrices should sat-isfy the restricted isometry property And random GaussianBernoulli and partial Fourier matrices satisfy the restrictedisometry property with number of measurements nearlylinear in the sparsity level [30 31]

Then compressX isin C119872119873times119871 in (7) to a smaller one asX1015840 isinC11987210158401198731015840times1198711015840

X1015840 = (V119879 otimes U119879)XW119879

= (V119879 otimes U119879) [A119910⊙ A

119909] S119879W119879

+ (V119879 otimes U119879)NW119879

(13)

According to the property of Khatri-Rao product [23] weknow

(V119879 otimes U119879) [A119910⊙ A

119909] = (V119879A

119910) ⊙ (U119879A

119909) (14)

Define A1015840119909= U119879A

119909 A1015840

119910= V119879A

119910 and S1015840 = W119879S Equation

(11) is also denoted as

X1015840 = [A1015840119910⊙ A1015840

119909] S1015840119879 + N1015840

(15)

where N1015840= (V119879 otimes U119879)NW119879 X1015840 can be denoted by trilinear

model With respect to (10) and (11) we form the matrices ofY1015840 and Z1015840 according to the compressed data

Y1015840 = [A1015840119909⊙ S1015840]A1015840119879

119910+ N1015840119910

(16)

Z1015840 = [S1015840 ⊙ A1015840119910]A1015840119879

119909+ N1015840119911

(17)

where N1015840119910 and N1015840119911 are the noise part The compressed trilin-ear model may degrade the angle estimation performance

By trilinear model compression the proposed methodhasmuch lower computational complexity than conventionaltrilinear decomposition method and requires much smallerstorage capacity Conventional compressive sensing is tocompress the matrix while our algorithm compresses thethree-dimensional tensor

32 Trilinear Decomposition Trilinear alternating leastsquare (TALS) algorithm is an iterative method forestimating the parameters of a trilinear decomposition[18 28] We concisely show the basic idea of TALS (1)update one matrix each time via LS which is conditionedon previously obtained estimates of the remaining matrices(2) proceed to update the other matrices (3) repeat untilconvergence of the LS cost function [21 22] TALS algorithmis discussed as follows

According to (15) least squares (LS) fitting is

minA1015840119909A1015840119910S101584010038171003817100381710038171003817X1015840 minus [A1015840

119910⊙ A1015840

119909] S101584011987910038171003817100381710038171003817119865 (18)

and LS update for the matrix S1015840 is

S1015840119879 = [A1015840119910⊙ A1015840

119909]+

X1015840 (19)

where A1015840119909and A1015840

119910are previously obtained estimates ofA1015840

119909and

A1015840119910 respectively According to (16) LS fitting is

minA1015840119909A1015840119910S101584010038171003817100381710038171003817Y1015840 minus [A1015840

119909⊙ S1015840]A1015840119879

119910

10038171003817100381710038171003817119865 (20)

and LS update for A1015840119910is

A1015840119879119910= [A1015840

119909⊙ S1015840]

+

Y1015840 (21)

where A1015840119909and S1015840 stand for the previously obtained estimates

of A1015840119909and S1015840 Similarly according to (17) LS fitting is

minA1015840119909A1015840119910S101584010038171003817100381710038171003817Z1015840 minus [S1015840 ⊙ A1015840

119910]A1015840119879

119909

10038171003817100381710038171003817119865 (22)

where Z1015840 is the noisy compressed signal LS update for A1015840119909is

A1015840119879119909= [S1015840 ⊙ A1015840

119910]+

Z1015840 (23)

where A1015840119910and S1015840 stand for the previously obtained estimates

of A1015840119910and S1015840 respectively

Define E = X minus [A1015840119910⊙ A1015840

119909]S1015840119879 where S1015840 A1015840

119910 and A1015840

119909

present the estimates of S1015840A1015840119910 andA1015840

119909 respectivelyThe sum

of squared residuals (SSR) in trilinear model is defined asSSR = sum

119871

119895=1sum119872119873

119894=1|119890119894119895| where 119890

119894119895is the (119894 119895) element of the

matrix E According to (19) (21) and (23) the matrices S1015840A1015840119910 and A1015840

119909are updated with least squares repeatedly until

SSR attain apriorthreshold The we obtain the final estimatesS1015840 A1015840

119910 and A1015840

119909

International Journal of Antennas and Propagation 5

Theorem 1 (see [22]) Considering X1015840119899= A1015840

119909119863119899(A1015840

119910)S1015840119879 119899 =

1 1198731015840 where A1015840

119909isin C119872

1015840times119870 S1015840 isin C119871

1015840times119870 and A1015840

119910isin C119873

1015840times119870 if

119896A1015840119909+ 119896A1015840

119910+ 119896S1015840119879 ge 2119870 + 2 (24)

where 119896A is the 119896-rank of the matrix A [18] then A1015840119909 S1015840 and

A1015840119910are unique up to permutation and scaling of columns

For the different DOAs and independent sources we have119896A1015840119909= min(1198721015840

119870) 119896A1015840119910= min(1198731015840

119870) and 119896S1015840119879 = min(1198711015840 119870)in the trilinear model in this paper and then the inequality in(24) becomes

min (1198721015840 119870) +min (1198731015840

119870) +min (1198711015840 119870)

ge 2119870 + 2

(25)

When 1198721015840ge 119870 1198731015840

ge 119870 and 1198711015840 ge 119870 the identifiablecondition is 1 le 119870 le min(1198721015840

1198731015840)

When 1198721015840le 119870 1198731015840

le 119870 and 1198711015840 ge 119870 the identifiablecondition is max(1198721015840

1198731015840) le 119870 le 119872

1015840+ 119873

1015840minus 2 Hence the

proposed algorithm is effective when119870 le 1198721015840+119873

1015840minus2 and the

maximum number of sources that can be identified is1198721015840+

1198731015840minus 2After the trilinear decomposition we obtain the estimates

of the loading matrices

A1015840119909= A1015840

119909ΠΔ

1+ E

1

S1015840 = S1015840ΠΔ2+ E

2

A1015840119910= A1015840

119910ΠΔ

3+ E

3

(26)

where Π is a permutation matrix and Δ1 Δ

2 Δ

3note for

the diagonal scaling matrices satisfying Δ1Δ2Δ3= I

119870 E

1

E2 and E

3are estimation error matrices After the trilinear

decomposition the estimates of A1015840119909 A1015840

119910 and S1015840 can be

obtained Scale ambiguity and permutation ambiguity areinherent to the trilinear decomposition problem Howeverthe scale ambiguity can be resolved easily by means ofnormalization while the existence of permutation ambiguityis not considered for angle estimation

33 Angle Estimation via Sparse Recovery Use a1015840119909119896

and a1015840119910119896

todenote the 119896th column of estimates A1015840

119909and A1015840

119910 respectively

According to the compression matrices we have

a1015840119909119896= U119879120597

119909119896a119909119896+ n

119909119896 119896 = 1 119870 (27a)

a1015840119910119896= V119879120597

119910119896a119910119896+ n

119910119896 119896 = 1 119870 (27b)

where a119909119896

and a119910119896

are the 119896th column ofA119909A

119910 respectively

n119909119896and n

119910119896are the corresponding noise respectively 120597

119909119896and

120597119909119896

are the scaling coefficients Construct two Vandermondematrices A

119904119909isin C119872times119875 and A

119904119910isin C119873times119875 (119875 ≫ 119872 119875 ≫ 119873)

composed of steering vectors corresponding to each potentialsource location as its columns

A119904119909

= [a1199041199091 a1199041199092 a

119904119909119875]

=

[[[[[

[

1 1 sdot sdot sdot 1

1198901198952120587119889g(1)120582

1198901198952120587119889g(2)120582

1198901198952120587119889g(119875)120582

1198901198952120587(119872minus1)119889g(1)120582

1198901198952120587(119872minus1)119889g(2)120582

sdot sdot sdot 1198901198952120587(119872minus1)119889g(119875)120582

]]]]]

]

(28a)

A119904119910

= [a1199041199101 a1199041199102 a

119904119910119875]

=

[[[[[

[

1 1 sdot sdot sdot 1

1198901198952120587119889g(1)120582

1198901198952120587119889g(2)120582

1198901198952120587119889g(119875)120582

1198901198952120587(119873minus1)119889g(1)120582

1198901198952120587(119873minus1)119889g(2)120582

sdot sdot sdot 1198901198952120587(119873minus1)119889g(119875)120582

]]]]]

]

(28b)

where g is a sampling vector and its 119901th elements is g(119901) =minus1 + 2119901119875 119901 = 1 2 119875 The matrices A

119904119909and A

119904119910can be

regarded as the completed dictionariesThen (27a)-(27b) canbe expressed as

a1015840119909119896= U119879A

119904119909x119904+ n

119909119896 119896 = 1 119870 (29a)

a1015840119910119896= V119879A

119910119909y119904+ n

119910119896 119896 = 1 119870 (29b)

where x119904and y

119904are sparse The estimates of x

119904and y

119904can be

obtained via 1198970-norm constraint

min 10038171003817100381710038171003817a1015840119909119896minus U119879A

119904119909x119904

10038171003817100381710038171003817

2

2

st 1003817100381710038171003817x11990410038171003817100381710038170 = 1

(30a)

min 10038171003817100381710038171003817a1015840119910119896minus V119879A

119904119910y119904

10038171003817100381710038171003817

2

2

st 1003817100381710038171003817y11990410038171003817100381710038170 = 1

(30b)

where sdot 0denotes the 119897

0-norm x

1199040= 1 that is to say

there is only one nonzero element in the vector x119904 similar

to y1199040= 1 We can use the OMP recovery method [26]

to find the nonzero element in x119904or y

119904 The OMP algorithm

tries to recover the signal by finding the strongest componentin the measurement signal removing it from the signal andsearching the dictionary again for the strongest atom that ispresented in the residual signal [32] We extract the indexof the maximum modulus of elements in x

119904and y

119904 respec-

tively noted as 119901119909and 119901

119910 According to the corresponding

columns in A119904119909

and A119904119910 we obtain g(119901

119909) and g(119901

119910) which

are estimates of sin 120579119896cos120601

119896and sin 120579

119896sin120601

119896 We define

6 International Journal of Antennas and Propagation

119903119896= g(119901

119909) + 119895g(119901

119910) and then the elevation angles and

azimuth angles can be obtained via

120579119896= sinminus1 (abs (119903

119896)) 119896 = 1 119870 (31a)

120601119896= angle (119903

119896) 119896 = 1 119870 (31b)

where abs(sdot) is the modulus value symbol and angle(sdot) is toget the angle of an imaginary number As the columns ofthe estimated matrices A1015840

119909and A1015840

119910are automatically paired

then the estimated elevation angles and azimuth angles canbe paired automatically

34 The Procedures of the Proposed Algorithm Till nowwe have achieved the proposal for the compressive sensingtrilinear model-based 2D-DOA estimation for rectangulararray We show major steps of the proposed algorithm asfollows

Step 1 Form the three-way matrix Χ isin C119872times119873times119871 thencompress the three-way matrix into a much smaller three-way matrix Χ1015840 isin C119872

1015840times1198731015840times1198711015840

via the compression matricesU isin C119872times119872

1015840

V isin C119873times1198731015840

andW isin C119871times1198711015840

Step 2 Perform trilinear decomposition through TALS algo-rithm for the compressed three-way matrix to obtain theestimation of A1015840

119909 A1015840

119910 and S1015840

Step 3 Estimate the sparse vectors

Step 4 Estimate 2D-DOA via (31a)-(31b)

Remark A Because the trilinear decomposition brings thesame permutation ambiguity for the estimatesA1015840

119909A1015840

119910 and S1015840

the estimated elevation angles and azimuth angles are pairedautomatically

Remark B The conventional compressive sensing methodformulates an angle sampling grid for sparse recovery toestimate angles When it is applied to 2D-DOA estimationboth elevation and azimuth angles must be sampled andit results in a two-dimensional sampling problem whichbrings much heavier cost for sparse signal recovery Inthis paper sin 120579

119896cos120601

119896(or sin 120579

119896sin120601

119896) is bundled into a

single variable in the range of minus1 to 1 The bundled variableis sampled for sparse recovery to obtain the estimates ofsin 120579

119896cos120601

119896and sin 120579

119896sin120601

119896 respectively Afterwards the

elevation and azimuth angles are estimated through theestimates of sin 120579

119896cos120601

119896and sin 120579

119896sin120601

119896

Remark C If the number of sources 119870 is unknown it canbe estimated by performing singular value decompositionfor received data matrix X in (7) and finding the numberof largest singular values [33] We also use some lower-complexity algorithm in [34] for estimating the number ofthe sources

Remark D When the coherent sources impinge on the arraywe can use the parallel profiles with linear dependencies

(PARALIND) model [35 36] which is a generalization ofPARAFAC suitable for solving problems with linear depen-dent factors to resolve coherent DOA estimation problem

35 Complexity Analysis and CRB The proposed algorithmhas much lower computational cost than conventional trilin-ear decomposition-based method The proposed algorithmrequires 119874(1198703 + 1198721015840

11987310158401198711015840119870) operations for a iteration while

the trilinear decomposition algorithm needs119874(1198703+119872119873119871119870)operations [28] for a iteration where1198721015840

lt 1198721198731015840lt 119873 and

1198711015840lt 119871We define A = [a

119910(1205791) otimes a

119909(1205791) a

119910(120579119870) otimes a

119909(120579119870)]

According to [37] we can derive the CRB

CRB = 1205902

2119871Re [D119867

Πperp

AD oplus P119879

119904]minus1

(32)

where 119871 denotes the number of samples a119896is the 119896th column

of A and P119904= (1119871)sum

119871

119905=1s(119905)s119867(119905) 1205902 is the noise power

Πperp

A = I119872119873

minus A(A119867A)minus1A119867 and

D = [120597a11205971205791

120597a2

1205971205792

120597a119870

120597120579119870

120597a1

1205971206011

120597a2

1205971206012

120597a119870

120597120601119870

] (33)

The advantages of the proposed algorithm can be presentedas follows

(1) The proposed algorithm can be regarded as a com-bination of trilinear model and compressive sensingtheory and it brings much lower computationalcomplexity and much smaller demand for storagecapacity

(2) The proposed algorithm has better 2D-DOA estima-tion performance than ESPRIT algorithm and closeangle estimation performance to TALS algorithmwhich will be proved by Figures 6-7

(3) The proposed algorithm can achieve paired elevationangles and azimuth angles automatically

4 Numerical Simulations

In the following simulations we assume that the numericalsimulation results converge when the SSR le 10minus8119872 119873 119871and 119870 denote the number of antennas in 119909-axis number ofantennas in 119910-axis samples and sources respectively Andwe compress the parameters 119872 119873 119871 to 1198721015840 1198731015840 and 1198711015840(usually set 119872 = 16 119873 = 20 119871 = 100 and 1198721015840

= 1198731015840=

1198711015840= 5 in numerical simulations) 119889 = 1205822 is considered

in the simulation We present 1000 Monte Carlo simulationsto assess the angle estimation performance of the proposedalgorithm Define root mean squared error (RMSE) as

RMSE = 1119870

119870

sum

119896=1

radic1

1000

1000

sum

119897=1

(120601119896119897minus 120601

119896)2

+ (120579119896119897minus 120579

119896)2

(34)

where 120601119896and 120579

119896denote the perfect elevation angle and

azimuth angle of 119896th source respectively 120601119896119897

and 120579119896119897

are

International Journal of Antennas and Propagation 7

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

Elev

atio

n (d

eg)

Azimuth (deg)

2D-DOA estimation

Figure 4 2D-DOA estimation of our algorithm in SNR = minus10 dB(119873 = 20119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

402D-DOA estimation

Elev

atio

n (d

eg)

Azimuth (deg)

Figure 5 2D-DOAestimation of our algorithm in SNR = 5 dB (119873 =20119872 = 16 119871 = 100 and 119870 = 3)

the estimates of 120601119896and 120579

119896in the 119897th Monte Carlo trail

Assume that there are 3 noncoherent sources located at angles(1206011 1205791) = (5

∘ 10

∘) (120601

2 1205792) = (15

∘ 20

∘) and (120601

3 1205793) =

(40∘ 30

∘)

Figure 4 presents the 2D-DOA estimation of the pro-posed algorithm for uniform rectangular array with119873 = 20119872 = 16 119871 = 100119870 = 3 and SNR = minus10 dB Figure 5 depictsthe 2D-DOA estimation performance with SNR = 5 dBFigures 4-5 illustrate that our algorithm is effective for 2D-DOA estimation

Figure 6 shows the 2D-DOA estimation performancecomparison of the proposed algorithm the ESPRIT algo-rithm the TALS algorithm and the CRB for the uniformrectangular array with119873 = 20119872 = 16 119871 = 100 and 119870 = 3while Figure 7 depicts the 2D-DOA estimation performancecomparison with 119873 = 16 119872 = 16 119871 = 200 and 119870 = 3It is indicated that our algorithm has better angle estimation

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

10minus3

minus10 minus5

ProposedTALS

ESPRITCRB

Figure 6 2D-DOA estimation performance comparison (119873 = 20119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

10minus3

minus10 minus5

ProposedTALS

ESPRITCRB

Figure 7 2D-DOA estimation performance comparison (119873 = 16119872 = 16 119871 = 200 and 119870 = 3)

performance than the ESPRIT algorithm and close angleestimation to TALS algorithm The angle estimation perfor-mance of the proposed algorithm will be further improvedthrough increasing the compressed parameters1198721015840 1198731015840 and1198711015840Figure 8 depicts the 2D-DOA estimation performance

of the proposed algorithm with different value of 119873 (119872 =

16 119871 = 100 and 119870 = 3) while Figure 9 presents the2D-DOA estimation performance of the proposed algorithmwith different value of 119872 It is clearly shown that theangle estimation performance of our algorithm is graduallyimproved with the number of antennas increasing Multiple

8 International Journal of Antennas and Propagation

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

N = 8

N = 16

N = 24

Figure 8 Angle estimation performance of our algorithm withdifferent119873 (119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

M = 8

M = 16

M = 24

Figure 9 Angle estimation performance of our algorithm withdifferent119872 (119873 = 16 119871 = 100 and 119870 = 3)

antennas improve the angle estimation performance becauseof diversity gain

Figure 10 presents 2D-DOA estimation performance ofthe proposed algorithm with different value of 119871 (119873 =

20 119872 = 16 and 119870 = 3) It illustrates that the angleestimation performance becomes better in collaborationwith119871 increasing

5 Conclusions

In this paper we have addressed the 2D-DOA estimationproblem for rectangular array and have derived a compressive

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

L = 50

L = 100

L = 200

Figure 10 Angle estimation performance of our algorithm withdifferent 119871 (119873 = 16119872 = 20 and 119870 = 3)

sensing trilinear model-based 2D-DOA estimation algo-rithm which can obtain the automatically paired 2D-DOAestimateThe proposed algorithm has better angle estimationperformance than ESPRIT algorithm and close angle esti-mation performance to conventional trilinear decompositionmethod Furthermore the proposed algorithm has lowercomputational complexity and smaller demand for storagecapacity than conventional trilinear decomposition methodThe proposed algorithm can be regarded as a combination oftrilinear model and compressive sensing theory and it bringsmuch lower computational complexity and much smallerdemand for storage capacity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by China NSF Grants (6137116961301108 61471191 61471192 and 61271327) Jiangsu PlannedProjects for Postdoctoral Research Funds (1201039C)China Postdoctoral Science Foundation (2012M5210992013M541661) Open Project of Key Laboratory of ModernAcoustics of Ministry of Education (Nanjing University) theAeronautical Science Foundation of China (20120152001)Qing Lan Project Priority Academic Program Developmentof Jiangsu High Education Institutions and the FundamentalResearch Funds for the Central Universities (NS2013024kfjj130114 and kfjj130115)

International Journal of Antennas and Propagation 9

References

[1] HKrim andMViberg ldquoTwodecades of array signal processingresearch the parametric approachrdquo IEEE Signal ProcessingMagazine vol 13 no 4 pp 67ndash94 1996

[2] R Roy and T Kailath ldquoESPRIT-estimation of signal parametersvia rotational invariance techniquesrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 37 no 7 pp 984ndash995 1989

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

[4] D Kundu ldquoModified MUSIC algorithm for estimating DOA ofsignalsrdquo Signal Processing vol 48 no 1 pp 85ndash90 1996

[5] B D Rao and K V S Hari ldquoPerformance analysis of Root-Musicrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 37 no 12 pp 1939ndash1949 1989

[6] N Yilmazer J Koh and T K Sarkar ldquoUtilization of a unitarytransform for efficient computation in thematrix pencilmethodto find the direction of arrivalrdquo IEEE Transactions on Antennasand Propagation vol 54 no 1 pp 175ndash181 2006

[7] Y Chiba K Ichige and H Arai ldquoReducing DOA estimationerror in extended ES-root-MUSIC for uniform rectangulararrayrdquo in Proceedings of the 4th International Congress on Imageand Signal Processing (CISP rsquo11) vol 5 pp 2621ndash2625 October2011

[8] 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

[9] 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

[10] 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

[11] 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

[12] 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

[13] 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

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

[15] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo IEEE Transactions on InformationThe-ory vol 52 no 2 pp 489ndash509 2006

[16] 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

[17] N Hu Z Ye D Xu and S Cao ldquoA sparse recovery algorithmfor DOA estimation using weighted subspace fittingrdquo SignalProcessing vol 92 no 10 pp 2566ndash2570 2012

[18] 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

[19] L de Lathauwer B de Moor and J Vandewalle ldquoComputationof the canonical decomposition by means of a simultaneousgeneralized Schur decompositionrdquo SIAM Journal on MatrixAnalysis and Applications vol 26 no 2 pp 295ndash327 2004

[20] L de Lathauwer ldquoA link between the canonical decompositionin multi-linear algebra and simultaneous matrix diagonaliza-tionrdquo SIAM Journal on Matrix Analysis and Applications vol28 no 3 pp 642ndash666 2006

[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000

[22] N D Sidiropoulos and X Liu ldquoIdentifiability results for blindbeamforming in incoherent multipath with small delay spreadrdquoIEEE Transactions on Signal Processing vol 49 no 1 pp 228ndash236 2001

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

[24] X F Zhang H X Yu J F Li andD Ben ldquoBlind signal detectionfor uniform rectangular array via compressive sensing trilinearmodelrdquo Advanced Materials Research vol 756ndash759 pp 660ndash664 2013

[25] R Cao X Zhang and W Chen ldquoCompressed sensing parallelfactor analysis-based joint angle andDoppler frequency estima-tion for monostatic multiple-inputndashmultiple-output radarrdquo IETRadar Sonar amp Navigation vol 8 no 6 pp 597ndash606 2014

[26] 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

[27] X Zhang F Wang and H Chen Theory and Application ofArray Signal Processing (version 2) National Defense IndustryPress Beijing China 2012

[28] X Zhang J Li H Chen and D Xu ldquoTrilinear decomposition-based two-dimensional DOA estimation algorithm for arbitrar-ily spaced acoustic vector-sensor array subjected to unknownlocationsrdquoWireless Personal Communications vol 67 no 4 pp859ndash877 2012

[29] R Bro N D Sidiropoulos and G B Giannakis ldquoA fast leastsquares algorithm for separating trilinear mixturesrdquo in Proceed-ings of the International Workshop on Independent ComponentAnalysis and Blind Signal Separation pp 289ndash294 January 1999

[30] R A DeVore ldquoDeterministic constructions of compressedsensing matricesrdquo Journal of Complexity vol 23 no 4ndash6 pp918ndash925 2007

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

[32] F Wang and X Zhang ldquoJoint estimation of TOA and DOA inIR-UWB system using sparse representation frameworkrdquo ETRIJournal vol 36 no 3 pp 460ndash468 2014

[33] A Di ldquoMultiple source locationmdasha matrix decompositionapproachrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 33 no 5 pp 1086ndash1091 1985

[34] J Xin N Zheng and A Sano ldquoSimple and efficient nonpara-metric method for estimating the number of signals withouteigendecompositionrdquo IEEE Transactions on Signal Processingvol 55 no 4 pp 1405ndash1420 2007

10 International Journal of Antennas and Propagation

[35] 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

[36] X Zhang M Zhou and J Li ldquoA PARALIND decomposition-based coherent two-dimensional direction of arrival estimationalgorithm for acoustic vector-sensor arraysrdquo Sensors vol 13 no4 pp 5302ndash5316 2013

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

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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Navigation and Observation

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DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 5

Theorem 1 (see [22]) Considering X1015840119899= A1015840

119909119863119899(A1015840

119910)S1015840119879 119899 =

1 1198731015840 where A1015840

119909isin C119872

1015840times119870 S1015840 isin C119871

1015840times119870 and A1015840

119910isin C119873

1015840times119870 if

119896A1015840119909+ 119896A1015840

119910+ 119896S1015840119879 ge 2119870 + 2 (24)

where 119896A is the 119896-rank of the matrix A [18] then A1015840119909 S1015840 and

A1015840119910are unique up to permutation and scaling of columns

For the different DOAs and independent sources we have119896A1015840119909= min(1198721015840

119870) 119896A1015840119910= min(1198731015840

119870) and 119896S1015840119879 = min(1198711015840 119870)in the trilinear model in this paper and then the inequality in(24) becomes

min (1198721015840 119870) +min (1198731015840

119870) +min (1198711015840 119870)

ge 2119870 + 2

(25)

When 1198721015840ge 119870 1198731015840

ge 119870 and 1198711015840 ge 119870 the identifiablecondition is 1 le 119870 le min(1198721015840

1198731015840)

When 1198721015840le 119870 1198731015840

le 119870 and 1198711015840 ge 119870 the identifiablecondition is max(1198721015840

1198731015840) le 119870 le 119872

1015840+ 119873

1015840minus 2 Hence the

proposed algorithm is effective when119870 le 1198721015840+119873

1015840minus2 and the

maximum number of sources that can be identified is1198721015840+

1198731015840minus 2After the trilinear decomposition we obtain the estimates

of the loading matrices

A1015840119909= A1015840

119909ΠΔ

1+ E

1

S1015840 = S1015840ΠΔ2+ E

2

A1015840119910= A1015840

119910ΠΔ

3+ E

3

(26)

where Π is a permutation matrix and Δ1 Δ

2 Δ

3note for

the diagonal scaling matrices satisfying Δ1Δ2Δ3= I

119870 E

1

E2 and E

3are estimation error matrices After the trilinear

decomposition the estimates of A1015840119909 A1015840

119910 and S1015840 can be

obtained Scale ambiguity and permutation ambiguity areinherent to the trilinear decomposition problem Howeverthe scale ambiguity can be resolved easily by means ofnormalization while the existence of permutation ambiguityis not considered for angle estimation

33 Angle Estimation via Sparse Recovery Use a1015840119909119896

and a1015840119910119896

todenote the 119896th column of estimates A1015840

119909and A1015840

119910 respectively

According to the compression matrices we have

a1015840119909119896= U119879120597

119909119896a119909119896+ n

119909119896 119896 = 1 119870 (27a)

a1015840119910119896= V119879120597

119910119896a119910119896+ n

119910119896 119896 = 1 119870 (27b)

where a119909119896

and a119910119896

are the 119896th column ofA119909A

119910 respectively

n119909119896and n

119910119896are the corresponding noise respectively 120597

119909119896and

120597119909119896

are the scaling coefficients Construct two Vandermondematrices A

119904119909isin C119872times119875 and A

119904119910isin C119873times119875 (119875 ≫ 119872 119875 ≫ 119873)

composed of steering vectors corresponding to each potentialsource location as its columns

A119904119909

= [a1199041199091 a1199041199092 a

119904119909119875]

=

[[[[[

[

1 1 sdot sdot sdot 1

1198901198952120587119889g(1)120582

1198901198952120587119889g(2)120582

1198901198952120587119889g(119875)120582

1198901198952120587(119872minus1)119889g(1)120582

1198901198952120587(119872minus1)119889g(2)120582

sdot sdot sdot 1198901198952120587(119872minus1)119889g(119875)120582

]]]]]

]

(28a)

A119904119910

= [a1199041199101 a1199041199102 a

119904119910119875]

=

[[[[[

[

1 1 sdot sdot sdot 1

1198901198952120587119889g(1)120582

1198901198952120587119889g(2)120582

1198901198952120587119889g(119875)120582

1198901198952120587(119873minus1)119889g(1)120582

1198901198952120587(119873minus1)119889g(2)120582

sdot sdot sdot 1198901198952120587(119873minus1)119889g(119875)120582

]]]]]

]

(28b)

where g is a sampling vector and its 119901th elements is g(119901) =minus1 + 2119901119875 119901 = 1 2 119875 The matrices A

119904119909and A

119904119910can be

regarded as the completed dictionariesThen (27a)-(27b) canbe expressed as

a1015840119909119896= U119879A

119904119909x119904+ n

119909119896 119896 = 1 119870 (29a)

a1015840119910119896= V119879A

119910119909y119904+ n

119910119896 119896 = 1 119870 (29b)

where x119904and y

119904are sparse The estimates of x

119904and y

119904can be

obtained via 1198970-norm constraint

min 10038171003817100381710038171003817a1015840119909119896minus U119879A

119904119909x119904

10038171003817100381710038171003817

2

2

st 1003817100381710038171003817x11990410038171003817100381710038170 = 1

(30a)

min 10038171003817100381710038171003817a1015840119910119896minus V119879A

119904119910y119904

10038171003817100381710038171003817

2

2

st 1003817100381710038171003817y11990410038171003817100381710038170 = 1

(30b)

where sdot 0denotes the 119897

0-norm x

1199040= 1 that is to say

there is only one nonzero element in the vector x119904 similar

to y1199040= 1 We can use the OMP recovery method [26]

to find the nonzero element in x119904or y

119904 The OMP algorithm

tries to recover the signal by finding the strongest componentin the measurement signal removing it from the signal andsearching the dictionary again for the strongest atom that ispresented in the residual signal [32] We extract the indexof the maximum modulus of elements in x

119904and y

119904 respec-

tively noted as 119901119909and 119901

119910 According to the corresponding

columns in A119904119909

and A119904119910 we obtain g(119901

119909) and g(119901

119910) which

are estimates of sin 120579119896cos120601

119896and sin 120579

119896sin120601

119896 We define

6 International Journal of Antennas and Propagation

119903119896= g(119901

119909) + 119895g(119901

119910) and then the elevation angles and

azimuth angles can be obtained via

120579119896= sinminus1 (abs (119903

119896)) 119896 = 1 119870 (31a)

120601119896= angle (119903

119896) 119896 = 1 119870 (31b)

where abs(sdot) is the modulus value symbol and angle(sdot) is toget the angle of an imaginary number As the columns ofthe estimated matrices A1015840

119909and A1015840

119910are automatically paired

then the estimated elevation angles and azimuth angles canbe paired automatically

34 The Procedures of the Proposed Algorithm Till nowwe have achieved the proposal for the compressive sensingtrilinear model-based 2D-DOA estimation for rectangulararray We show major steps of the proposed algorithm asfollows

Step 1 Form the three-way matrix Χ isin C119872times119873times119871 thencompress the three-way matrix into a much smaller three-way matrix Χ1015840 isin C119872

1015840times1198731015840times1198711015840

via the compression matricesU isin C119872times119872

1015840

V isin C119873times1198731015840

andW isin C119871times1198711015840

Step 2 Perform trilinear decomposition through TALS algo-rithm for the compressed three-way matrix to obtain theestimation of A1015840

119909 A1015840

119910 and S1015840

Step 3 Estimate the sparse vectors

Step 4 Estimate 2D-DOA via (31a)-(31b)

Remark A Because the trilinear decomposition brings thesame permutation ambiguity for the estimatesA1015840

119909A1015840

119910 and S1015840

the estimated elevation angles and azimuth angles are pairedautomatically

Remark B The conventional compressive sensing methodformulates an angle sampling grid for sparse recovery toestimate angles When it is applied to 2D-DOA estimationboth elevation and azimuth angles must be sampled andit results in a two-dimensional sampling problem whichbrings much heavier cost for sparse signal recovery Inthis paper sin 120579

119896cos120601

119896(or sin 120579

119896sin120601

119896) is bundled into a

single variable in the range of minus1 to 1 The bundled variableis sampled for sparse recovery to obtain the estimates ofsin 120579

119896cos120601

119896and sin 120579

119896sin120601

119896 respectively Afterwards the

elevation and azimuth angles are estimated through theestimates of sin 120579

119896cos120601

119896and sin 120579

119896sin120601

119896

Remark C If the number of sources 119870 is unknown it canbe estimated by performing singular value decompositionfor received data matrix X in (7) and finding the numberof largest singular values [33] We also use some lower-complexity algorithm in [34] for estimating the number ofthe sources

Remark D When the coherent sources impinge on the arraywe can use the parallel profiles with linear dependencies

(PARALIND) model [35 36] which is a generalization ofPARAFAC suitable for solving problems with linear depen-dent factors to resolve coherent DOA estimation problem

35 Complexity Analysis and CRB The proposed algorithmhas much lower computational cost than conventional trilin-ear decomposition-based method The proposed algorithmrequires 119874(1198703 + 1198721015840

11987310158401198711015840119870) operations for a iteration while

the trilinear decomposition algorithm needs119874(1198703+119872119873119871119870)operations [28] for a iteration where1198721015840

lt 1198721198731015840lt 119873 and

1198711015840lt 119871We define A = [a

119910(1205791) otimes a

119909(1205791) a

119910(120579119870) otimes a

119909(120579119870)]

According to [37] we can derive the CRB

CRB = 1205902

2119871Re [D119867

Πperp

AD oplus P119879

119904]minus1

(32)

where 119871 denotes the number of samples a119896is the 119896th column

of A and P119904= (1119871)sum

119871

119905=1s(119905)s119867(119905) 1205902 is the noise power

Πperp

A = I119872119873

minus A(A119867A)minus1A119867 and

D = [120597a11205971205791

120597a2

1205971205792

120597a119870

120597120579119870

120597a1

1205971206011

120597a2

1205971206012

120597a119870

120597120601119870

] (33)

The advantages of the proposed algorithm can be presentedas follows

(1) The proposed algorithm can be regarded as a com-bination of trilinear model and compressive sensingtheory and it brings much lower computationalcomplexity and much smaller demand for storagecapacity

(2) The proposed algorithm has better 2D-DOA estima-tion performance than ESPRIT algorithm and closeangle estimation performance to TALS algorithmwhich will be proved by Figures 6-7

(3) The proposed algorithm can achieve paired elevationangles and azimuth angles automatically

4 Numerical Simulations

In the following simulations we assume that the numericalsimulation results converge when the SSR le 10minus8119872 119873 119871and 119870 denote the number of antennas in 119909-axis number ofantennas in 119910-axis samples and sources respectively Andwe compress the parameters 119872 119873 119871 to 1198721015840 1198731015840 and 1198711015840(usually set 119872 = 16 119873 = 20 119871 = 100 and 1198721015840

= 1198731015840=

1198711015840= 5 in numerical simulations) 119889 = 1205822 is considered

in the simulation We present 1000 Monte Carlo simulationsto assess the angle estimation performance of the proposedalgorithm Define root mean squared error (RMSE) as

RMSE = 1119870

119870

sum

119896=1

radic1

1000

1000

sum

119897=1

(120601119896119897minus 120601

119896)2

+ (120579119896119897minus 120579

119896)2

(34)

where 120601119896and 120579

119896denote the perfect elevation angle and

azimuth angle of 119896th source respectively 120601119896119897

and 120579119896119897

are

International Journal of Antennas and Propagation 7

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

Elev

atio

n (d

eg)

Azimuth (deg)

2D-DOA estimation

Figure 4 2D-DOA estimation of our algorithm in SNR = minus10 dB(119873 = 20119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

402D-DOA estimation

Elev

atio

n (d

eg)

Azimuth (deg)

Figure 5 2D-DOAestimation of our algorithm in SNR = 5 dB (119873 =20119872 = 16 119871 = 100 and 119870 = 3)

the estimates of 120601119896and 120579

119896in the 119897th Monte Carlo trail

Assume that there are 3 noncoherent sources located at angles(1206011 1205791) = (5

∘ 10

∘) (120601

2 1205792) = (15

∘ 20

∘) and (120601

3 1205793) =

(40∘ 30

∘)

Figure 4 presents the 2D-DOA estimation of the pro-posed algorithm for uniform rectangular array with119873 = 20119872 = 16 119871 = 100119870 = 3 and SNR = minus10 dB Figure 5 depictsthe 2D-DOA estimation performance with SNR = 5 dBFigures 4-5 illustrate that our algorithm is effective for 2D-DOA estimation

Figure 6 shows the 2D-DOA estimation performancecomparison of the proposed algorithm the ESPRIT algo-rithm the TALS algorithm and the CRB for the uniformrectangular array with119873 = 20119872 = 16 119871 = 100 and 119870 = 3while Figure 7 depicts the 2D-DOA estimation performancecomparison with 119873 = 16 119872 = 16 119871 = 200 and 119870 = 3It is indicated that our algorithm has better angle estimation

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

10minus3

minus10 minus5

ProposedTALS

ESPRITCRB

Figure 6 2D-DOA estimation performance comparison (119873 = 20119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

10minus3

minus10 minus5

ProposedTALS

ESPRITCRB

Figure 7 2D-DOA estimation performance comparison (119873 = 16119872 = 16 119871 = 200 and 119870 = 3)

performance than the ESPRIT algorithm and close angleestimation to TALS algorithm The angle estimation perfor-mance of the proposed algorithm will be further improvedthrough increasing the compressed parameters1198721015840 1198731015840 and1198711015840Figure 8 depicts the 2D-DOA estimation performance

of the proposed algorithm with different value of 119873 (119872 =

16 119871 = 100 and 119870 = 3) while Figure 9 presents the2D-DOA estimation performance of the proposed algorithmwith different value of 119872 It is clearly shown that theangle estimation performance of our algorithm is graduallyimproved with the number of antennas increasing Multiple

8 International Journal of Antennas and Propagation

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

N = 8

N = 16

N = 24

Figure 8 Angle estimation performance of our algorithm withdifferent119873 (119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

M = 8

M = 16

M = 24

Figure 9 Angle estimation performance of our algorithm withdifferent119872 (119873 = 16 119871 = 100 and 119870 = 3)

antennas improve the angle estimation performance becauseof diversity gain

Figure 10 presents 2D-DOA estimation performance ofthe proposed algorithm with different value of 119871 (119873 =

20 119872 = 16 and 119870 = 3) It illustrates that the angleestimation performance becomes better in collaborationwith119871 increasing

5 Conclusions

In this paper we have addressed the 2D-DOA estimationproblem for rectangular array and have derived a compressive

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

L = 50

L = 100

L = 200

Figure 10 Angle estimation performance of our algorithm withdifferent 119871 (119873 = 16119872 = 20 and 119870 = 3)

sensing trilinear model-based 2D-DOA estimation algo-rithm which can obtain the automatically paired 2D-DOAestimateThe proposed algorithm has better angle estimationperformance than ESPRIT algorithm and close angle esti-mation performance to conventional trilinear decompositionmethod Furthermore the proposed algorithm has lowercomputational complexity and smaller demand for storagecapacity than conventional trilinear decomposition methodThe proposed algorithm can be regarded as a combination oftrilinear model and compressive sensing theory and it bringsmuch lower computational complexity and much smallerdemand for storage capacity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by China NSF Grants (6137116961301108 61471191 61471192 and 61271327) Jiangsu PlannedProjects for Postdoctoral Research Funds (1201039C)China Postdoctoral Science Foundation (2012M5210992013M541661) Open Project of Key Laboratory of ModernAcoustics of Ministry of Education (Nanjing University) theAeronautical Science Foundation of China (20120152001)Qing Lan Project Priority Academic Program Developmentof Jiangsu High Education Institutions and the FundamentalResearch Funds for the Central Universities (NS2013024kfjj130114 and kfjj130115)

International Journal of Antennas and Propagation 9

References

[1] HKrim andMViberg ldquoTwodecades of array signal processingresearch the parametric approachrdquo IEEE Signal ProcessingMagazine vol 13 no 4 pp 67ndash94 1996

[2] R Roy and T Kailath ldquoESPRIT-estimation of signal parametersvia rotational invariance techniquesrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 37 no 7 pp 984ndash995 1989

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

[4] D Kundu ldquoModified MUSIC algorithm for estimating DOA ofsignalsrdquo Signal Processing vol 48 no 1 pp 85ndash90 1996

[5] B D Rao and K V S Hari ldquoPerformance analysis of Root-Musicrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 37 no 12 pp 1939ndash1949 1989

[6] N Yilmazer J Koh and T K Sarkar ldquoUtilization of a unitarytransform for efficient computation in thematrix pencilmethodto find the direction of arrivalrdquo IEEE Transactions on Antennasand Propagation vol 54 no 1 pp 175ndash181 2006

[7] Y Chiba K Ichige and H Arai ldquoReducing DOA estimationerror in extended ES-root-MUSIC for uniform rectangulararrayrdquo in Proceedings of the 4th International Congress on Imageand Signal Processing (CISP rsquo11) vol 5 pp 2621ndash2625 October2011

[8] 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

[9] 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

[10] 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

[11] 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

[12] 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

[13] 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

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

[15] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo IEEE Transactions on InformationThe-ory vol 52 no 2 pp 489ndash509 2006

[16] 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

[17] N Hu Z Ye D Xu and S Cao ldquoA sparse recovery algorithmfor DOA estimation using weighted subspace fittingrdquo SignalProcessing vol 92 no 10 pp 2566ndash2570 2012

[18] 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

[19] L de Lathauwer B de Moor and J Vandewalle ldquoComputationof the canonical decomposition by means of a simultaneousgeneralized Schur decompositionrdquo SIAM Journal on MatrixAnalysis and Applications vol 26 no 2 pp 295ndash327 2004

[20] L de Lathauwer ldquoA link between the canonical decompositionin multi-linear algebra and simultaneous matrix diagonaliza-tionrdquo SIAM Journal on Matrix Analysis and Applications vol28 no 3 pp 642ndash666 2006

[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000

[22] N D Sidiropoulos and X Liu ldquoIdentifiability results for blindbeamforming in incoherent multipath with small delay spreadrdquoIEEE Transactions on Signal Processing vol 49 no 1 pp 228ndash236 2001

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

[24] X F Zhang H X Yu J F Li andD Ben ldquoBlind signal detectionfor uniform rectangular array via compressive sensing trilinearmodelrdquo Advanced Materials Research vol 756ndash759 pp 660ndash664 2013

[25] R Cao X Zhang and W Chen ldquoCompressed sensing parallelfactor analysis-based joint angle andDoppler frequency estima-tion for monostatic multiple-inputndashmultiple-output radarrdquo IETRadar Sonar amp Navigation vol 8 no 6 pp 597ndash606 2014

[26] 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

[27] X Zhang F Wang and H Chen Theory and Application ofArray Signal Processing (version 2) National Defense IndustryPress Beijing China 2012

[28] X Zhang J Li H Chen and D Xu ldquoTrilinear decomposition-based two-dimensional DOA estimation algorithm for arbitrar-ily spaced acoustic vector-sensor array subjected to unknownlocationsrdquoWireless Personal Communications vol 67 no 4 pp859ndash877 2012

[29] R Bro N D Sidiropoulos and G B Giannakis ldquoA fast leastsquares algorithm for separating trilinear mixturesrdquo in Proceed-ings of the International Workshop on Independent ComponentAnalysis and Blind Signal Separation pp 289ndash294 January 1999

[30] R A DeVore ldquoDeterministic constructions of compressedsensing matricesrdquo Journal of Complexity vol 23 no 4ndash6 pp918ndash925 2007

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

[32] F Wang and X Zhang ldquoJoint estimation of TOA and DOA inIR-UWB system using sparse representation frameworkrdquo ETRIJournal vol 36 no 3 pp 460ndash468 2014

[33] A Di ldquoMultiple source locationmdasha matrix decompositionapproachrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 33 no 5 pp 1086ndash1091 1985

[34] J Xin N Zheng and A Sano ldquoSimple and efficient nonpara-metric method for estimating the number of signals withouteigendecompositionrdquo IEEE Transactions on Signal Processingvol 55 no 4 pp 1405ndash1420 2007

10 International Journal of Antennas and Propagation

[35] 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

[36] X Zhang M Zhou and J Li ldquoA PARALIND decomposition-based coherent two-dimensional direction of arrival estimationalgorithm for acoustic vector-sensor arraysrdquo Sensors vol 13 no4 pp 5302ndash5316 2013

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

International Journal of

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Active and Passive Electronic Components

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RotatingMachinery

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

6 International Journal of Antennas and Propagation

119903119896= g(119901

119909) + 119895g(119901

119910) and then the elevation angles and

azimuth angles can be obtained via

120579119896= sinminus1 (abs (119903

119896)) 119896 = 1 119870 (31a)

120601119896= angle (119903

119896) 119896 = 1 119870 (31b)

where abs(sdot) is the modulus value symbol and angle(sdot) is toget the angle of an imaginary number As the columns ofthe estimated matrices A1015840

119909and A1015840

119910are automatically paired

then the estimated elevation angles and azimuth angles canbe paired automatically

34 The Procedures of the Proposed Algorithm Till nowwe have achieved the proposal for the compressive sensingtrilinear model-based 2D-DOA estimation for rectangulararray We show major steps of the proposed algorithm asfollows

Step 1 Form the three-way matrix Χ isin C119872times119873times119871 thencompress the three-way matrix into a much smaller three-way matrix Χ1015840 isin C119872

1015840times1198731015840times1198711015840

via the compression matricesU isin C119872times119872

1015840

V isin C119873times1198731015840

andW isin C119871times1198711015840

Step 2 Perform trilinear decomposition through TALS algo-rithm for the compressed three-way matrix to obtain theestimation of A1015840

119909 A1015840

119910 and S1015840

Step 3 Estimate the sparse vectors

Step 4 Estimate 2D-DOA via (31a)-(31b)

Remark A Because the trilinear decomposition brings thesame permutation ambiguity for the estimatesA1015840

119909A1015840

119910 and S1015840

the estimated elevation angles and azimuth angles are pairedautomatically

Remark B The conventional compressive sensing methodformulates an angle sampling grid for sparse recovery toestimate angles When it is applied to 2D-DOA estimationboth elevation and azimuth angles must be sampled andit results in a two-dimensional sampling problem whichbrings much heavier cost for sparse signal recovery Inthis paper sin 120579

119896cos120601

119896(or sin 120579

119896sin120601

119896) is bundled into a

single variable in the range of minus1 to 1 The bundled variableis sampled for sparse recovery to obtain the estimates ofsin 120579

119896cos120601

119896and sin 120579

119896sin120601

119896 respectively Afterwards the

elevation and azimuth angles are estimated through theestimates of sin 120579

119896cos120601

119896and sin 120579

119896sin120601

119896

Remark C If the number of sources 119870 is unknown it canbe estimated by performing singular value decompositionfor received data matrix X in (7) and finding the numberof largest singular values [33] We also use some lower-complexity algorithm in [34] for estimating the number ofthe sources

Remark D When the coherent sources impinge on the arraywe can use the parallel profiles with linear dependencies

(PARALIND) model [35 36] which is a generalization ofPARAFAC suitable for solving problems with linear depen-dent factors to resolve coherent DOA estimation problem

35 Complexity Analysis and CRB The proposed algorithmhas much lower computational cost than conventional trilin-ear decomposition-based method The proposed algorithmrequires 119874(1198703 + 1198721015840

11987310158401198711015840119870) operations for a iteration while

the trilinear decomposition algorithm needs119874(1198703+119872119873119871119870)operations [28] for a iteration where1198721015840

lt 1198721198731015840lt 119873 and

1198711015840lt 119871We define A = [a

119910(1205791) otimes a

119909(1205791) a

119910(120579119870) otimes a

119909(120579119870)]

According to [37] we can derive the CRB

CRB = 1205902

2119871Re [D119867

Πperp

AD oplus P119879

119904]minus1

(32)

where 119871 denotes the number of samples a119896is the 119896th column

of A and P119904= (1119871)sum

119871

119905=1s(119905)s119867(119905) 1205902 is the noise power

Πperp

A = I119872119873

minus A(A119867A)minus1A119867 and

D = [120597a11205971205791

120597a2

1205971205792

120597a119870

120597120579119870

120597a1

1205971206011

120597a2

1205971206012

120597a119870

120597120601119870

] (33)

The advantages of the proposed algorithm can be presentedas follows

(1) The proposed algorithm can be regarded as a com-bination of trilinear model and compressive sensingtheory and it brings much lower computationalcomplexity and much smaller demand for storagecapacity

(2) The proposed algorithm has better 2D-DOA estima-tion performance than ESPRIT algorithm and closeangle estimation performance to TALS algorithmwhich will be proved by Figures 6-7

(3) The proposed algorithm can achieve paired elevationangles and azimuth angles automatically

4 Numerical Simulations

In the following simulations we assume that the numericalsimulation results converge when the SSR le 10minus8119872 119873 119871and 119870 denote the number of antennas in 119909-axis number ofantennas in 119910-axis samples and sources respectively Andwe compress the parameters 119872 119873 119871 to 1198721015840 1198731015840 and 1198711015840(usually set 119872 = 16 119873 = 20 119871 = 100 and 1198721015840

= 1198731015840=

1198711015840= 5 in numerical simulations) 119889 = 1205822 is considered

in the simulation We present 1000 Monte Carlo simulationsto assess the angle estimation performance of the proposedalgorithm Define root mean squared error (RMSE) as

RMSE = 1119870

119870

sum

119896=1

radic1

1000

1000

sum

119897=1

(120601119896119897minus 120601

119896)2

+ (120579119896119897minus 120579

119896)2

(34)

where 120601119896and 120579

119896denote the perfect elevation angle and

azimuth angle of 119896th source respectively 120601119896119897

and 120579119896119897

are

International Journal of Antennas and Propagation 7

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

Elev

atio

n (d

eg)

Azimuth (deg)

2D-DOA estimation

Figure 4 2D-DOA estimation of our algorithm in SNR = minus10 dB(119873 = 20119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

402D-DOA estimation

Elev

atio

n (d

eg)

Azimuth (deg)

Figure 5 2D-DOAestimation of our algorithm in SNR = 5 dB (119873 =20119872 = 16 119871 = 100 and 119870 = 3)

the estimates of 120601119896and 120579

119896in the 119897th Monte Carlo trail

Assume that there are 3 noncoherent sources located at angles(1206011 1205791) = (5

∘ 10

∘) (120601

2 1205792) = (15

∘ 20

∘) and (120601

3 1205793) =

(40∘ 30

∘)

Figure 4 presents the 2D-DOA estimation of the pro-posed algorithm for uniform rectangular array with119873 = 20119872 = 16 119871 = 100119870 = 3 and SNR = minus10 dB Figure 5 depictsthe 2D-DOA estimation performance with SNR = 5 dBFigures 4-5 illustrate that our algorithm is effective for 2D-DOA estimation

Figure 6 shows the 2D-DOA estimation performancecomparison of the proposed algorithm the ESPRIT algo-rithm the TALS algorithm and the CRB for the uniformrectangular array with119873 = 20119872 = 16 119871 = 100 and 119870 = 3while Figure 7 depicts the 2D-DOA estimation performancecomparison with 119873 = 16 119872 = 16 119871 = 200 and 119870 = 3It is indicated that our algorithm has better angle estimation

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

10minus3

minus10 minus5

ProposedTALS

ESPRITCRB

Figure 6 2D-DOA estimation performance comparison (119873 = 20119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

10minus3

minus10 minus5

ProposedTALS

ESPRITCRB

Figure 7 2D-DOA estimation performance comparison (119873 = 16119872 = 16 119871 = 200 and 119870 = 3)

performance than the ESPRIT algorithm and close angleestimation to TALS algorithm The angle estimation perfor-mance of the proposed algorithm will be further improvedthrough increasing the compressed parameters1198721015840 1198731015840 and1198711015840Figure 8 depicts the 2D-DOA estimation performance

of the proposed algorithm with different value of 119873 (119872 =

16 119871 = 100 and 119870 = 3) while Figure 9 presents the2D-DOA estimation performance of the proposed algorithmwith different value of 119872 It is clearly shown that theangle estimation performance of our algorithm is graduallyimproved with the number of antennas increasing Multiple

8 International Journal of Antennas and Propagation

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

N = 8

N = 16

N = 24

Figure 8 Angle estimation performance of our algorithm withdifferent119873 (119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

M = 8

M = 16

M = 24

Figure 9 Angle estimation performance of our algorithm withdifferent119872 (119873 = 16 119871 = 100 and 119870 = 3)

antennas improve the angle estimation performance becauseof diversity gain

Figure 10 presents 2D-DOA estimation performance ofthe proposed algorithm with different value of 119871 (119873 =

20 119872 = 16 and 119870 = 3) It illustrates that the angleestimation performance becomes better in collaborationwith119871 increasing

5 Conclusions

In this paper we have addressed the 2D-DOA estimationproblem for rectangular array and have derived a compressive

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

L = 50

L = 100

L = 200

Figure 10 Angle estimation performance of our algorithm withdifferent 119871 (119873 = 16119872 = 20 and 119870 = 3)

sensing trilinear model-based 2D-DOA estimation algo-rithm which can obtain the automatically paired 2D-DOAestimateThe proposed algorithm has better angle estimationperformance than ESPRIT algorithm and close angle esti-mation performance to conventional trilinear decompositionmethod Furthermore the proposed algorithm has lowercomputational complexity and smaller demand for storagecapacity than conventional trilinear decomposition methodThe proposed algorithm can be regarded as a combination oftrilinear model and compressive sensing theory and it bringsmuch lower computational complexity and much smallerdemand for storage capacity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by China NSF Grants (6137116961301108 61471191 61471192 and 61271327) Jiangsu PlannedProjects for Postdoctoral Research Funds (1201039C)China Postdoctoral Science Foundation (2012M5210992013M541661) Open Project of Key Laboratory of ModernAcoustics of Ministry of Education (Nanjing University) theAeronautical Science Foundation of China (20120152001)Qing Lan Project Priority Academic Program Developmentof Jiangsu High Education Institutions and the FundamentalResearch Funds for the Central Universities (NS2013024kfjj130114 and kfjj130115)

International Journal of Antennas and Propagation 9

References

[1] HKrim andMViberg ldquoTwodecades of array signal processingresearch the parametric approachrdquo IEEE Signal ProcessingMagazine vol 13 no 4 pp 67ndash94 1996

[2] R Roy and T Kailath ldquoESPRIT-estimation of signal parametersvia rotational invariance techniquesrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 37 no 7 pp 984ndash995 1989

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

[4] D Kundu ldquoModified MUSIC algorithm for estimating DOA ofsignalsrdquo Signal Processing vol 48 no 1 pp 85ndash90 1996

[5] B D Rao and K V S Hari ldquoPerformance analysis of Root-Musicrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 37 no 12 pp 1939ndash1949 1989

[6] N Yilmazer J Koh and T K Sarkar ldquoUtilization of a unitarytransform for efficient computation in thematrix pencilmethodto find the direction of arrivalrdquo IEEE Transactions on Antennasand Propagation vol 54 no 1 pp 175ndash181 2006

[7] Y Chiba K Ichige and H Arai ldquoReducing DOA estimationerror in extended ES-root-MUSIC for uniform rectangulararrayrdquo in Proceedings of the 4th International Congress on Imageand Signal Processing (CISP rsquo11) vol 5 pp 2621ndash2625 October2011

[8] 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

[9] 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

[10] 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

[11] 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

[12] 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

[13] 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

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

[15] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo IEEE Transactions on InformationThe-ory vol 52 no 2 pp 489ndash509 2006

[16] 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

[17] N Hu Z Ye D Xu and S Cao ldquoA sparse recovery algorithmfor DOA estimation using weighted subspace fittingrdquo SignalProcessing vol 92 no 10 pp 2566ndash2570 2012

[18] 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

[19] L de Lathauwer B de Moor and J Vandewalle ldquoComputationof the canonical decomposition by means of a simultaneousgeneralized Schur decompositionrdquo SIAM Journal on MatrixAnalysis and Applications vol 26 no 2 pp 295ndash327 2004

[20] L de Lathauwer ldquoA link between the canonical decompositionin multi-linear algebra and simultaneous matrix diagonaliza-tionrdquo SIAM Journal on Matrix Analysis and Applications vol28 no 3 pp 642ndash666 2006

[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000

[22] N D Sidiropoulos and X Liu ldquoIdentifiability results for blindbeamforming in incoherent multipath with small delay spreadrdquoIEEE Transactions on Signal Processing vol 49 no 1 pp 228ndash236 2001

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

[24] X F Zhang H X Yu J F Li andD Ben ldquoBlind signal detectionfor uniform rectangular array via compressive sensing trilinearmodelrdquo Advanced Materials Research vol 756ndash759 pp 660ndash664 2013

[25] R Cao X Zhang and W Chen ldquoCompressed sensing parallelfactor analysis-based joint angle andDoppler frequency estima-tion for monostatic multiple-inputndashmultiple-output radarrdquo IETRadar Sonar amp Navigation vol 8 no 6 pp 597ndash606 2014

[26] 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

[27] X Zhang F Wang and H Chen Theory and Application ofArray Signal Processing (version 2) National Defense IndustryPress Beijing China 2012

[28] X Zhang J Li H Chen and D Xu ldquoTrilinear decomposition-based two-dimensional DOA estimation algorithm for arbitrar-ily spaced acoustic vector-sensor array subjected to unknownlocationsrdquoWireless Personal Communications vol 67 no 4 pp859ndash877 2012

[29] R Bro N D Sidiropoulos and G B Giannakis ldquoA fast leastsquares algorithm for separating trilinear mixturesrdquo in Proceed-ings of the International Workshop on Independent ComponentAnalysis and Blind Signal Separation pp 289ndash294 January 1999

[30] R A DeVore ldquoDeterministic constructions of compressedsensing matricesrdquo Journal of Complexity vol 23 no 4ndash6 pp918ndash925 2007

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

[32] F Wang and X Zhang ldquoJoint estimation of TOA and DOA inIR-UWB system using sparse representation frameworkrdquo ETRIJournal vol 36 no 3 pp 460ndash468 2014

[33] A Di ldquoMultiple source locationmdasha matrix decompositionapproachrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 33 no 5 pp 1086ndash1091 1985

[34] J Xin N Zheng and A Sano ldquoSimple and efficient nonpara-metric method for estimating the number of signals withouteigendecompositionrdquo IEEE Transactions on Signal Processingvol 55 no 4 pp 1405ndash1420 2007

10 International Journal of Antennas and Propagation

[35] 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

[36] X Zhang M Zhou and J Li ldquoA PARALIND decomposition-based coherent two-dimensional direction of arrival estimationalgorithm for acoustic vector-sensor arraysrdquo Sensors vol 13 no4 pp 5302ndash5316 2013

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

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 7

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

Elev

atio

n (d

eg)

Azimuth (deg)

2D-DOA estimation

Figure 4 2D-DOA estimation of our algorithm in SNR = minus10 dB(119873 = 20119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

402D-DOA estimation

Elev

atio

n (d

eg)

Azimuth (deg)

Figure 5 2D-DOAestimation of our algorithm in SNR = 5 dB (119873 =20119872 = 16 119871 = 100 and 119870 = 3)

the estimates of 120601119896and 120579

119896in the 119897th Monte Carlo trail

Assume that there are 3 noncoherent sources located at angles(1206011 1205791) = (5

∘ 10

∘) (120601

2 1205792) = (15

∘ 20

∘) and (120601

3 1205793) =

(40∘ 30

∘)

Figure 4 presents the 2D-DOA estimation of the pro-posed algorithm for uniform rectangular array with119873 = 20119872 = 16 119871 = 100119870 = 3 and SNR = minus10 dB Figure 5 depictsthe 2D-DOA estimation performance with SNR = 5 dBFigures 4-5 illustrate that our algorithm is effective for 2D-DOA estimation

Figure 6 shows the 2D-DOA estimation performancecomparison of the proposed algorithm the ESPRIT algo-rithm the TALS algorithm and the CRB for the uniformrectangular array with119873 = 20119872 = 16 119871 = 100 and 119870 = 3while Figure 7 depicts the 2D-DOA estimation performancecomparison with 119873 = 16 119872 = 16 119871 = 200 and 119870 = 3It is indicated that our algorithm has better angle estimation

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

10minus3

minus10 minus5

ProposedTALS

ESPRITCRB

Figure 6 2D-DOA estimation performance comparison (119873 = 20119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

10minus3

minus10 minus5

ProposedTALS

ESPRITCRB

Figure 7 2D-DOA estimation performance comparison (119873 = 16119872 = 16 119871 = 200 and 119870 = 3)

performance than the ESPRIT algorithm and close angleestimation to TALS algorithm The angle estimation perfor-mance of the proposed algorithm will be further improvedthrough increasing the compressed parameters1198721015840 1198731015840 and1198711015840Figure 8 depicts the 2D-DOA estimation performance

of the proposed algorithm with different value of 119873 (119872 =

16 119871 = 100 and 119870 = 3) while Figure 9 presents the2D-DOA estimation performance of the proposed algorithmwith different value of 119872 It is clearly shown that theangle estimation performance of our algorithm is graduallyimproved with the number of antennas increasing Multiple

8 International Journal of Antennas and Propagation

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

N = 8

N = 16

N = 24

Figure 8 Angle estimation performance of our algorithm withdifferent119873 (119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

M = 8

M = 16

M = 24

Figure 9 Angle estimation performance of our algorithm withdifferent119872 (119873 = 16 119871 = 100 and 119870 = 3)

antennas improve the angle estimation performance becauseof diversity gain

Figure 10 presents 2D-DOA estimation performance ofthe proposed algorithm with different value of 119871 (119873 =

20 119872 = 16 and 119870 = 3) It illustrates that the angleestimation performance becomes better in collaborationwith119871 increasing

5 Conclusions

In this paper we have addressed the 2D-DOA estimationproblem for rectangular array and have derived a compressive

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

L = 50

L = 100

L = 200

Figure 10 Angle estimation performance of our algorithm withdifferent 119871 (119873 = 16119872 = 20 and 119870 = 3)

sensing trilinear model-based 2D-DOA estimation algo-rithm which can obtain the automatically paired 2D-DOAestimateThe proposed algorithm has better angle estimationperformance than ESPRIT algorithm and close angle esti-mation performance to conventional trilinear decompositionmethod Furthermore the proposed algorithm has lowercomputational complexity and smaller demand for storagecapacity than conventional trilinear decomposition methodThe proposed algorithm can be regarded as a combination oftrilinear model and compressive sensing theory and it bringsmuch lower computational complexity and much smallerdemand for storage capacity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by China NSF Grants (6137116961301108 61471191 61471192 and 61271327) Jiangsu PlannedProjects for Postdoctoral Research Funds (1201039C)China Postdoctoral Science Foundation (2012M5210992013M541661) Open Project of Key Laboratory of ModernAcoustics of Ministry of Education (Nanjing University) theAeronautical Science Foundation of China (20120152001)Qing Lan Project Priority Academic Program Developmentof Jiangsu High Education Institutions and the FundamentalResearch Funds for the Central Universities (NS2013024kfjj130114 and kfjj130115)

International Journal of Antennas and Propagation 9

References

[1] HKrim andMViberg ldquoTwodecades of array signal processingresearch the parametric approachrdquo IEEE Signal ProcessingMagazine vol 13 no 4 pp 67ndash94 1996

[2] R Roy and T Kailath ldquoESPRIT-estimation of signal parametersvia rotational invariance techniquesrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 37 no 7 pp 984ndash995 1989

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

[4] D Kundu ldquoModified MUSIC algorithm for estimating DOA ofsignalsrdquo Signal Processing vol 48 no 1 pp 85ndash90 1996

[5] B D Rao and K V S Hari ldquoPerformance analysis of Root-Musicrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 37 no 12 pp 1939ndash1949 1989

[6] N Yilmazer J Koh and T K Sarkar ldquoUtilization of a unitarytransform for efficient computation in thematrix pencilmethodto find the direction of arrivalrdquo IEEE Transactions on Antennasand Propagation vol 54 no 1 pp 175ndash181 2006

[7] Y Chiba K Ichige and H Arai ldquoReducing DOA estimationerror in extended ES-root-MUSIC for uniform rectangulararrayrdquo in Proceedings of the 4th International Congress on Imageand Signal Processing (CISP rsquo11) vol 5 pp 2621ndash2625 October2011

[8] 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

[9] 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

[10] 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

[11] 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

[12] 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

[13] 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

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

[15] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo IEEE Transactions on InformationThe-ory vol 52 no 2 pp 489ndash509 2006

[16] 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

[17] N Hu Z Ye D Xu and S Cao ldquoA sparse recovery algorithmfor DOA estimation using weighted subspace fittingrdquo SignalProcessing vol 92 no 10 pp 2566ndash2570 2012

[18] 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

[19] L de Lathauwer B de Moor and J Vandewalle ldquoComputationof the canonical decomposition by means of a simultaneousgeneralized Schur decompositionrdquo SIAM Journal on MatrixAnalysis and Applications vol 26 no 2 pp 295ndash327 2004

[20] L de Lathauwer ldquoA link between the canonical decompositionin multi-linear algebra and simultaneous matrix diagonaliza-tionrdquo SIAM Journal on Matrix Analysis and Applications vol28 no 3 pp 642ndash666 2006

[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000

[22] N D Sidiropoulos and X Liu ldquoIdentifiability results for blindbeamforming in incoherent multipath with small delay spreadrdquoIEEE Transactions on Signal Processing vol 49 no 1 pp 228ndash236 2001

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

[24] X F Zhang H X Yu J F Li andD Ben ldquoBlind signal detectionfor uniform rectangular array via compressive sensing trilinearmodelrdquo Advanced Materials Research vol 756ndash759 pp 660ndash664 2013

[25] R Cao X Zhang and W Chen ldquoCompressed sensing parallelfactor analysis-based joint angle andDoppler frequency estima-tion for monostatic multiple-inputndashmultiple-output radarrdquo IETRadar Sonar amp Navigation vol 8 no 6 pp 597ndash606 2014

[26] 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

[27] X Zhang F Wang and H Chen Theory and Application ofArray Signal Processing (version 2) National Defense IndustryPress Beijing China 2012

[28] X Zhang J Li H Chen and D Xu ldquoTrilinear decomposition-based two-dimensional DOA estimation algorithm for arbitrar-ily spaced acoustic vector-sensor array subjected to unknownlocationsrdquoWireless Personal Communications vol 67 no 4 pp859ndash877 2012

[29] R Bro N D Sidiropoulos and G B Giannakis ldquoA fast leastsquares algorithm for separating trilinear mixturesrdquo in Proceed-ings of the International Workshop on Independent ComponentAnalysis and Blind Signal Separation pp 289ndash294 January 1999

[30] R A DeVore ldquoDeterministic constructions of compressedsensing matricesrdquo Journal of Complexity vol 23 no 4ndash6 pp918ndash925 2007

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

[32] F Wang and X Zhang ldquoJoint estimation of TOA and DOA inIR-UWB system using sparse representation frameworkrdquo ETRIJournal vol 36 no 3 pp 460ndash468 2014

[33] A Di ldquoMultiple source locationmdasha matrix decompositionapproachrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 33 no 5 pp 1086ndash1091 1985

[34] J Xin N Zheng and A Sano ldquoSimple and efficient nonpara-metric method for estimating the number of signals withouteigendecompositionrdquo IEEE Transactions on Signal Processingvol 55 no 4 pp 1405ndash1420 2007

10 International Journal of Antennas and Propagation

[35] 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

[36] X Zhang M Zhou and J Li ldquoA PARALIND decomposition-based coherent two-dimensional direction of arrival estimationalgorithm for acoustic vector-sensor arraysrdquo Sensors vol 13 no4 pp 5302ndash5316 2013

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

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 International Journal of Antennas and Propagation

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

N = 8

N = 16

N = 24

Figure 8 Angle estimation performance of our algorithm withdifferent119873 (119872 = 16 119871 = 100 and 119870 = 3)

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

M = 8

M = 16

M = 24

Figure 9 Angle estimation performance of our algorithm withdifferent119872 (119873 = 16 119871 = 100 and 119870 = 3)

antennas improve the angle estimation performance becauseof diversity gain

Figure 10 presents 2D-DOA estimation performance ofthe proposed algorithm with different value of 119871 (119873 =

20 119872 = 16 and 119870 = 3) It illustrates that the angleestimation performance becomes better in collaborationwith119871 increasing

5 Conclusions

In this paper we have addressed the 2D-DOA estimationproblem for rectangular array and have derived a compressive

0 5 10 15SNR (dB)

RMSE

(deg

)

101

100

10minus1

10minus2

minus10 minus5

L = 50

L = 100

L = 200

Figure 10 Angle estimation performance of our algorithm withdifferent 119871 (119873 = 16119872 = 20 and 119870 = 3)

sensing trilinear model-based 2D-DOA estimation algo-rithm which can obtain the automatically paired 2D-DOAestimateThe proposed algorithm has better angle estimationperformance than ESPRIT algorithm and close angle esti-mation performance to conventional trilinear decompositionmethod Furthermore the proposed algorithm has lowercomputational complexity and smaller demand for storagecapacity than conventional trilinear decomposition methodThe proposed algorithm can be regarded as a combination oftrilinear model and compressive sensing theory and it bringsmuch lower computational complexity and much smallerdemand for storage capacity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by China NSF Grants (6137116961301108 61471191 61471192 and 61271327) Jiangsu PlannedProjects for Postdoctoral Research Funds (1201039C)China Postdoctoral Science Foundation (2012M5210992013M541661) Open Project of Key Laboratory of ModernAcoustics of Ministry of Education (Nanjing University) theAeronautical Science Foundation of China (20120152001)Qing Lan Project Priority Academic Program Developmentof Jiangsu High Education Institutions and the FundamentalResearch Funds for the Central Universities (NS2013024kfjj130114 and kfjj130115)

International Journal of Antennas and Propagation 9

References

[1] HKrim andMViberg ldquoTwodecades of array signal processingresearch the parametric approachrdquo IEEE Signal ProcessingMagazine vol 13 no 4 pp 67ndash94 1996

[2] R Roy and T Kailath ldquoESPRIT-estimation of signal parametersvia rotational invariance techniquesrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 37 no 7 pp 984ndash995 1989

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

[4] D Kundu ldquoModified MUSIC algorithm for estimating DOA ofsignalsrdquo Signal Processing vol 48 no 1 pp 85ndash90 1996

[5] B D Rao and K V S Hari ldquoPerformance analysis of Root-Musicrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 37 no 12 pp 1939ndash1949 1989

[6] N Yilmazer J Koh and T K Sarkar ldquoUtilization of a unitarytransform for efficient computation in thematrix pencilmethodto find the direction of arrivalrdquo IEEE Transactions on Antennasand Propagation vol 54 no 1 pp 175ndash181 2006

[7] Y Chiba K Ichige and H Arai ldquoReducing DOA estimationerror in extended ES-root-MUSIC for uniform rectangulararrayrdquo in Proceedings of the 4th International Congress on Imageand Signal Processing (CISP rsquo11) vol 5 pp 2621ndash2625 October2011

[8] 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

[9] 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

[10] 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

[11] 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

[12] 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

[13] 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

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

[15] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo IEEE Transactions on InformationThe-ory vol 52 no 2 pp 489ndash509 2006

[16] 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

[17] N Hu Z Ye D Xu and S Cao ldquoA sparse recovery algorithmfor DOA estimation using weighted subspace fittingrdquo SignalProcessing vol 92 no 10 pp 2566ndash2570 2012

[18] 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

[19] L de Lathauwer B de Moor and J Vandewalle ldquoComputationof the canonical decomposition by means of a simultaneousgeneralized Schur decompositionrdquo SIAM Journal on MatrixAnalysis and Applications vol 26 no 2 pp 295ndash327 2004

[20] L de Lathauwer ldquoA link between the canonical decompositionin multi-linear algebra and simultaneous matrix diagonaliza-tionrdquo SIAM Journal on Matrix Analysis and Applications vol28 no 3 pp 642ndash666 2006

[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000

[22] N D Sidiropoulos and X Liu ldquoIdentifiability results for blindbeamforming in incoherent multipath with small delay spreadrdquoIEEE Transactions on Signal Processing vol 49 no 1 pp 228ndash236 2001

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

[24] X F Zhang H X Yu J F Li andD Ben ldquoBlind signal detectionfor uniform rectangular array via compressive sensing trilinearmodelrdquo Advanced Materials Research vol 756ndash759 pp 660ndash664 2013

[25] R Cao X Zhang and W Chen ldquoCompressed sensing parallelfactor analysis-based joint angle andDoppler frequency estima-tion for monostatic multiple-inputndashmultiple-output radarrdquo IETRadar Sonar amp Navigation vol 8 no 6 pp 597ndash606 2014

[26] 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

[27] X Zhang F Wang and H Chen Theory and Application ofArray Signal Processing (version 2) National Defense IndustryPress Beijing China 2012

[28] X Zhang J Li H Chen and D Xu ldquoTrilinear decomposition-based two-dimensional DOA estimation algorithm for arbitrar-ily spaced acoustic vector-sensor array subjected to unknownlocationsrdquoWireless Personal Communications vol 67 no 4 pp859ndash877 2012

[29] R Bro N D Sidiropoulos and G B Giannakis ldquoA fast leastsquares algorithm for separating trilinear mixturesrdquo in Proceed-ings of the International Workshop on Independent ComponentAnalysis and Blind Signal Separation pp 289ndash294 January 1999

[30] R A DeVore ldquoDeterministic constructions of compressedsensing matricesrdquo Journal of Complexity vol 23 no 4ndash6 pp918ndash925 2007

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

[32] F Wang and X Zhang ldquoJoint estimation of TOA and DOA inIR-UWB system using sparse representation frameworkrdquo ETRIJournal vol 36 no 3 pp 460ndash468 2014

[33] A Di ldquoMultiple source locationmdasha matrix decompositionapproachrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 33 no 5 pp 1086ndash1091 1985

[34] J Xin N Zheng and A Sano ldquoSimple and efficient nonpara-metric method for estimating the number of signals withouteigendecompositionrdquo IEEE Transactions on Signal Processingvol 55 no 4 pp 1405ndash1420 2007

10 International Journal of Antennas and Propagation

[35] 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

[36] X Zhang M Zhou and J Li ldquoA PARALIND decomposition-based coherent two-dimensional direction of arrival estimationalgorithm for acoustic vector-sensor arraysrdquo Sensors vol 13 no4 pp 5302ndash5316 2013

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

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 9

References

[1] HKrim andMViberg ldquoTwodecades of array signal processingresearch the parametric approachrdquo IEEE Signal ProcessingMagazine vol 13 no 4 pp 67ndash94 1996

[2] R Roy and T Kailath ldquoESPRIT-estimation of signal parametersvia rotational invariance techniquesrdquo IEEE Transactions onAcoustics Speech and Signal Processing vol 37 no 7 pp 984ndash995 1989

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

[4] D Kundu ldquoModified MUSIC algorithm for estimating DOA ofsignalsrdquo Signal Processing vol 48 no 1 pp 85ndash90 1996

[5] B D Rao and K V S Hari ldquoPerformance analysis of Root-Musicrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 37 no 12 pp 1939ndash1949 1989

[6] N Yilmazer J Koh and T K Sarkar ldquoUtilization of a unitarytransform for efficient computation in thematrix pencilmethodto find the direction of arrivalrdquo IEEE Transactions on Antennasand Propagation vol 54 no 1 pp 175ndash181 2006

[7] Y Chiba K Ichige and H Arai ldquoReducing DOA estimationerror in extended ES-root-MUSIC for uniform rectangulararrayrdquo in Proceedings of the 4th International Congress on Imageand Signal Processing (CISP rsquo11) vol 5 pp 2621ndash2625 October2011

[8] 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

[9] 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

[10] 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

[11] 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

[12] 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

[13] 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

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

[15] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo IEEE Transactions on InformationThe-ory vol 52 no 2 pp 489ndash509 2006

[16] 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

[17] N Hu Z Ye D Xu and S Cao ldquoA sparse recovery algorithmfor DOA estimation using weighted subspace fittingrdquo SignalProcessing vol 92 no 10 pp 2566ndash2570 2012

[18] 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

[19] L de Lathauwer B de Moor and J Vandewalle ldquoComputationof the canonical decomposition by means of a simultaneousgeneralized Schur decompositionrdquo SIAM Journal on MatrixAnalysis and Applications vol 26 no 2 pp 295ndash327 2004

[20] L de Lathauwer ldquoA link between the canonical decompositionin multi-linear algebra and simultaneous matrix diagonaliza-tionrdquo SIAM Journal on Matrix Analysis and Applications vol28 no 3 pp 642ndash666 2006

[21] N D Sidiropoulos G B Giannakis and R Bro ldquoBlindPARAFAC receivers forDS-CDMAsystemsrdquo IEEETransactionson Signal Processing vol 48 no 3 pp 810ndash823 2000

[22] N D Sidiropoulos and X Liu ldquoIdentifiability results for blindbeamforming in incoherent multipath with small delay spreadrdquoIEEE Transactions on Signal Processing vol 49 no 1 pp 228ndash236 2001

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

[24] X F Zhang H X Yu J F Li andD Ben ldquoBlind signal detectionfor uniform rectangular array via compressive sensing trilinearmodelrdquo Advanced Materials Research vol 756ndash759 pp 660ndash664 2013

[25] R Cao X Zhang and W Chen ldquoCompressed sensing parallelfactor analysis-based joint angle andDoppler frequency estima-tion for monostatic multiple-inputndashmultiple-output radarrdquo IETRadar Sonar amp Navigation vol 8 no 6 pp 597ndash606 2014

[26] 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

[27] X Zhang F Wang and H Chen Theory and Application ofArray Signal Processing (version 2) National Defense IndustryPress Beijing China 2012

[28] X Zhang J Li H Chen and D Xu ldquoTrilinear decomposition-based two-dimensional DOA estimation algorithm for arbitrar-ily spaced acoustic vector-sensor array subjected to unknownlocationsrdquoWireless Personal Communications vol 67 no 4 pp859ndash877 2012

[29] R Bro N D Sidiropoulos and G B Giannakis ldquoA fast leastsquares algorithm for separating trilinear mixturesrdquo in Proceed-ings of the International Workshop on Independent ComponentAnalysis and Blind Signal Separation pp 289ndash294 January 1999

[30] R A DeVore ldquoDeterministic constructions of compressedsensing matricesrdquo Journal of Complexity vol 23 no 4ndash6 pp918ndash925 2007

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

[32] F Wang and X Zhang ldquoJoint estimation of TOA and DOA inIR-UWB system using sparse representation frameworkrdquo ETRIJournal vol 36 no 3 pp 460ndash468 2014

[33] A Di ldquoMultiple source locationmdasha matrix decompositionapproachrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 33 no 5 pp 1086ndash1091 1985

[34] J Xin N Zheng and A Sano ldquoSimple and efficient nonpara-metric method for estimating the number of signals withouteigendecompositionrdquo IEEE Transactions on Signal Processingvol 55 no 4 pp 1405ndash1420 2007

10 International Journal of Antennas and Propagation

[35] 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

[36] X Zhang M Zhou and J Li ldquoA PARALIND decomposition-based coherent two-dimensional direction of arrival estimationalgorithm for acoustic vector-sensor arraysrdquo Sensors vol 13 no4 pp 5302ndash5316 2013

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

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 International Journal of Antennas and Propagation

[35] 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

[36] X Zhang M Zhou and J Li ldquoA PARALIND decomposition-based coherent two-dimensional direction of arrival estimationalgorithm for acoustic vector-sensor arraysrdquo Sensors vol 13 no4 pp 5302ndash5316 2013

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

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of