tensor-based methods for blind spatial signature

Research ArticleTensor-Based Methods for Blind Spatial Signature Estimation inMultidimensional Sensor Arrays

Paulo R B Gomes1 Andreacute L F de Almeida1 Joatildeo Paulo C L da Costa234

Joatildeo C M Mota1 Daniel Valle de Lima2 and Giovanni Del Galdo34

1Department of Teleinformatics Engineering Federal University of Ceara Fortaleza CE Brazil2Department of Electrical Engineering University of Brasılia DF Brasılia Brazil3Institute for Information Technology Ilmenau University of Technology Ilmenau Germany4Fraunhofer Institute for Integrated Circuits IIS Erlangen Germany

Correspondence should be addressed to Paulo R B Gomes paulogtelufcbr

Received 10 September 2016 Accepted 15 January 2017 Published 15 February 2017

Academic Editor Elias Aboutanios

Copyright copy 2017 Paulo R B Gomes et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The estimation of spatial signatures and spatial frequencies is crucial for several practical applications such as radar sonarand wireless communications In this paper we propose two generalized iterative estimation algorithms to the case in which amultidimensional (119877-D) sensor array is used at the receiver The first tensor-based algorithm is an 119877-D blind spatial signatureestimator that operates in scenarios where the sourcersquos covariance matrix is nondiagonal and unknown The second tensor-basedalgorithm is formulated for the case in which the sources are uncorrelated and exploits the dual-symmetry of the covariance tensorAdditionally a new tensor-based formulation is proposed for an 119871-shaped array configuration Simulation results show that ourproposed schemes outperform the state-of-the-art matrix-based and tensor-based techniques

1 Introduction

High resolution parameter estimation plays a fundamentalrole in array signal processing and has practical applicationsin radar sonar mobile communications and seismologyIn light of this several techniques have been developed toincrease the accuracy of the estimated parameters fromwhich we may cite the classical Multiple Signal Classifica-tion (MUSIC) [1] and Estimation of Signal Parameters viaRotational Invariance Technique (ESPRIT) [2] However theirperformance can be further improved by exploiting themultidimensional structure of the data by means of tensormodeling which can include several signal dimensions suchas space time frequency and polarization Tensor decom-positions have been successfully employed in array signalprocessing for parameters estimation since they provide bet-ter identifiability conditions when compared to conventionalmatrix-based methods Another advantage of tensor-basedmethods is the so-called ldquotensor gainrdquo which manifests itselfwith more precise parameter estimates due to the good

noise rejection capability of tensor-based signal processingas shown in [3ndash6]

In regards to tensor-based methods for blind spatialsignatures estimation the Parallel Factor (PARAFAC) anal-ysis decomposition [7] is widely applied due to its well-defined conditions for uniqueness [8] As seen in [9] aniterative technique for PARAFAC decomposition such asTrilinear Alternating Least Squares (TALS) can be applied toestimate the directions of arrival of the sources Closed-formsolutions such as the Standard Tensor ESPRIT (STE) [10]and Closed-Form PARAFAC [11] are also appealing sincethese exploit the multidimensional structure in a noniter-ative fashion Recently in [12] an iterative algorithm wasproposed in a manner similar to Independent ComponentAnalysis (ICA) based on the Orthogonal Procrustes Problem(OPP) and Khatri-Rao factorization [13] for a PARAFACdecomposition with dual-symmetry This solution exploitsthe dual-symmetry property of the data tensor and can beapplied in covariance-based array signal processing tech-niques The method proposed in [14] is based on the Tucker

HindawiInternational Journal of Antennas and PropagationVolume 2017 Article ID 1615962 11 pageshttpsdoiorg10115520171615962

2 International Journal of Antennas and Propagation

decomposition [15] of a fourth-order covariance tensor andwas elaborated for arrays with arbitrary structures wherea priori knowledge about the geometry of the sensor arrayis not required However a limitation of the method in[14] is the necessity of transmitting the same sequence ofsymbols in different time blocks which results in a loss ofspectral efficiency The proposed solution is an algorithm fora multidimensional (119877-D) sensor array in which the differentdimensions of the array are exploited thus dismissing theneed to transmit a repeated sequence as in [14] as will bedetailed later

In this paper two tensor-based approaches to the esti-mation of spatial signatures are presented By using thesignals received on a 119877-D sensor array covariance tensorsare calculated and solutions for correlated and uncorrelatedsources are presented respectively For the former scenarioin which the sourcersquos covariance structure is nondiagonaland unknown the covariance tensor of the received datais formulated as a Tucker decomposition of order 2119877 Sucha formulation yields a generalized Tucker model based 119877-D sensor array processing that deals with arbitrary sourcecovariance structures By assuming uncorrelated sources wethen show that the problem boils down to a PARAFACdecomposition from which a method that exploits the dual-symmetry property of the covariance tensor is derived byconsidering the ideas rooted in [12] For both Tucker andPARAFAC based models the blind estimation of the spatialsignatures is achieved by means of an alternating leastsquares (ALS) algorithm The contributions of this paper aretwofold (i) we propose a covariance-based generalizationof the Tucker decomposition for the blind spatial signatureestimation problem with 119877-D sensor arrays and (ii) weestablish a link between dual-symmetry decompositions andtechniques based on covariance-based array signal process-ing for parameter estimation The performance of the pro-posed algorithms is evaluated by Monte Carlo simulationscorroborating their gains over competing state-of-the-artmatrix-based and tensor-based techniques

The rest of this paper is organized as follows Section 2briefly introduces tensor operations and decompositionsThesignal model for an 119877-D sensor array is then presentedin Section 3 In Section 4 a novel covariance-based tensormodel for the received data is formulated and our blindspatial signature estimation algorithms are formulated InSection 5 an approach for 119871-shaped sensor arrays is proposedThe computational complexity of the proposed methods isanalyzed in Section 6 In Section 7 the advantages and disad-vantages of the proposed methods are discussed Simulationresults are provided in Section 8 and the conclusions aredrawn in Section 9

Notation Scalar values are represented by lowercase letters119886 vectors by bold lowercase letters a matrices by bolduppercase lettersA and tensors by calligraphic lettersAThesymbols 119879 119867 dagger and lowast represent the transpose conjugatetranspose pseudoinverse and complex conjugate operationsrespectively diag(a) operator generates a diagonal matrixfrom a vector a The 119894th row of A isin C119868times119877 is representedby A(119894 ) isin C1times119877 while its 119903th column is represented by

A( 119903) isin C119868times1 vec(A) operator converts A into a vectora isin C119868119877times1 while unvec119868times119877(a) converts a into a 119868 times 119877 matrixD119894(A) stands for a diagonal matrix constructed from the 119894throw of A sdot 119865 stands for the Frobenius norm of a matrixor tensor ldquo∘rdquo operator stands for the vector outer productThe Kronecker product is represented by otimes The Khatri-Raoproduct between the matrices A isin C119868times119877 and B isin C119869times119877represented by is defined as

A B = [A (1) otimes B (1) A (119877) otimes B (119877)] (1)

2 Tensor Preliminaries

In the following we briefly introduce for convenience thebasics on operations involving tensors and tensor decom-positions which refer to [16 17] Firstly we present theTucker decomposition Then the PARAFAC decompositionis introduced and issues involving uniqueness are brieflydiscussed for both cases which will be useful later Then weintroduce the dual-symmetry property for these decomposi-tionsThe basic material presented in this section is exploitedin later sections in the context of our blind spatial signatureestimation problem

21 Basic Tensor Operations Let X isin C1198681timessdotsdotsdottimes119868119873 denote an119873th order tensor (1198941 119894119873)th entry of which is denoted by1199091198941 119894119873 The fibers are the higher-order analogues of matrixrows and columnsThe 119899-mode fibers ofX are vectors of size119868119899 defined by fixing every index but 119894119899The 119899-mode unfoldingoperation denoted by [X](119899) stands for the process ofreordering the elements of X into a matrix by arrangingits 119899-mode fibers to be the columns of the resulting matrixThe 119899-mode product between X and a matrix A isin C119869times119868119899

along of the 119899th mode denoted byXtimes119899A is a tensor of size1198681 times sdot sdot sdot times 119868119899minus1 times 119869 times 119868119899+1 times sdot sdot sdot times 119868119873 obtained by taking theinner product between each 119899-mode fiber and the rows of thematrix A that is [16 17]

Y = Xtimes119899A lArrrArr[Y](119899) = A [X](119899) (2)

22 Tucker Decomposition The Tucker decomposition [15]represents a tensorX isin C1198681timessdotsdotsdottimes119868119873 of order119873 as a multilineartransformation of a core tensor G isin C1198771timessdotsdotsdottimes119877119873 by factormatrices A(119899) = [a(119899)1 a(119899)2 a(119899)119877119899 ] isin C119868119899times119877119899 along eachmode 119899 = 1 2 119873 In scalar form the 119873th order Tuckerdecomposition is given by

1199091198941 1198942119894119873 = 1198771sum1199031=1

1198772sum1199032=1

sdot sdot sdot 119877119873sum119903119873=1

1198921199031 1199032 119903119873119886(1)1198941 1199031119886(2)1198942 1199032 sdot sdot sdot 119886(119873)119894119873119903119873 (3)

where 119886(119899)119894119899 119903119899 is (119894119899 119903119899)th entry of the 119899th mode factor matrixA(119899) isin C119868119899times119877119899 and 1198921199031 1199032119903119873 is (1199031 119903119873)th entry of the coretensor G isin C1198771timessdotsdotsdottimes119877119873 Using 119899-mode product notation theTucker decomposition can be written as

X = Gtimes1A(1)times2A(2) sdot sdot sdot times119873A(119873) (4)

International Journal of Antennas and Propagation 3

which admits the following factorization in terms of thefactor matrices and core tensor[X](119899)

= A(119899) [G](119899) (A(119873) otimes sdot sdot sdotA(119899+1) otimes A(119899minus1) otimes sdot sdot sdotA(1))119879 (5)

In general the Tucker decomposition is not unique thatis there are infinite solutions for A(119899) 119899 = 1 119873 and Gthat yield the same reconstructed version of the data tensorX However in special cases where several elements of thecore tensor are constrained to be equal to zero that is ifthe core tensor has some sparsity the number of solutionsmay be finite and the associated factor matrices and coretensor become unique up to trivial permutations and scalingambiguities [18] The Tucker based methods presented inSections 42 and 51 belong to a special category where uniquesolutions exist

23 PARAFAC Decomposition The PARAFAC decomposi-tion [7] expresses a tensorX isin C1198681timessdotsdotsdottimes119868119873 as a sum of 119877 rank-one tensors that is

X = 119877sum119903=1

a(1)119903 ∘ a(2)119903 ∘ sdot sdot sdot ∘ a(119873)119903 (6)

where 119877 is the number of factors also known as the rank ofthe decomposition and is defined as the minimum numberof rank-one tensors that yieldX exactly

The119873th order PARAFAC decomposition (6) can be seenas a special case of the Tucker decomposition (4) with a coretensor G = I119873119877 and 119877119899 = 119877 for 119899 = 1 119873 Theelements of the 119873th order identity tensor I119873119877 are equal toone when all indices are equal and zero elsewhere Using the119899-mode product notation the PARAFAC decomposition canbe written as

X = I119873119877times1A(1)times2A(2) sdot sdot sdot times119873A(119873) (7)

while the 119899-mode unfolding ofX can be expressed as

[X](119899) = A(119899) (A(119873) sdot sdot sdotA(119899+1) A(119899minus1) sdot sdot sdotA(1))119879 (8)

The119873th order PARAFAC decomposition is unique up topermutation and scaling ambiguities affecting the columns offactors matrices A(119899) isin C119868119899times119877 119899 = 1 119873 if the followingsufficient condition is satisfied [19]

119873sum119899=1

120581A(119899) ge 2119877 + 119873 minus 1 (9)

where 120581A(119899) denotes the Kruskal-rank of A(119899) defined as themaximumvalue120581 such that any subset of120581 columns is linearlyindependent [20]

Throughout this work special attention is given to dual-symmetric tensors The PARAFAC decomposition of a giventensor X isin C1198681timessdotsdotsdottimes1198682119873 of order 2119873 is said to have dual-symmetry if defined as follows

X = I2119873119877times1A(1)times2A(2) sdot sdot sdot times119873A(119873)times119873+1A(1)lowasttimes119873+2A(2)lowast sdot sdot sdot times2119873A(119873)lowast (10)

Note that this definition also applies to Tucker decompositionby simply replacing the identity tensorI2119873119877 by an arbitrarycore tensorG of order 2119873

3 Signal Model

We consider119870 snapshots originating from the superpositionof 119872 far-field narrowband signal sources sampled by a 119877-dimensional sensor array of size1198731times1198732timessdot sdot sdottimes119873119877 where119873119903is the size of the 119903th array dimension 119903 = 1 119877ThematrixX isin C119873times119870 collects the samples received by the sensor arraywhich can be factored as [10]

X = (A(1) A(2) sdot sdot sdot A(119877)) S + V (11)

where

(i) A = A(1) A(2) sdot sdot sdot A(119877) isin C119873times119872 is the spatialsignature matrix of the 119877-D array for 119903 = 1 119877 and119873 = prod119877119903=1119873119903

(ii) A(119903) = [a(119903)1 a(119903)119872 ] isin C119873119903times119872 is the spatial signaturematrix of the 119903th dimension

(iii) a(119903)119898 = [1 119890119895sdot120583(119903)119898 119890119895sdot2120583(119903)119898 sdot sdot sdot 119890119895sdot(119873119903minus1)120583(119903)119898 ]119879 isin C119873119903times1 isthe array response in the 119903th dimension to the 119898thplanar wavefront (119898 = 1 119872) which is function ofthe spatial frequency 120583(119903)119898

(iv) S = [s(1) s(119870)] isin C119872times119870 is the matrix containingthe signal transmitted by the sources

(v) V = [k(1) k(119870)] isin C119873times119870 is the additive whiteGaussian noise (assumed uncorrelated to the sourcesignals)

From (11) the sample covariance matrix R isin C119873times119873 of thesignals received at the sensor array is given by

R ≜ 1119870XX119867 asymp ARsA119867 + 1205902V I

asymp (A(1) sdot sdot sdot A(119877))Rs (A(1) sdot sdot sdot A(119877))119867+ 1205902V I

(12)

where Rs = (1119870)SS119867 is the sample covariance matrix of thesource signals and 1205902V is the noise variance4 Tensor-Based Methods for Blind Spatial

Signature Estimation

In this section we propose two iterative algorithms tosolve the blind spatial signature estimation problem in 119877-Dsensor arrays Initially a novel multidimensional structure isformulated from the covariance matrix of the received dataThen an alternating least squares- (ALS-) based algorithm fora Tucker decomposition of order 2119873 is proposed Finally wederive a link between the method in [12] and a covariance-based blind spatial signature estimation problem


41 Novel Covariance Tensor With the intention of exploitingthe multidimensional structure of the received signal thenoiseless sample covariance matrix (12) given by Ro = R minus1205902V I isin C119873times119873 is interpreted as a multimode unfolding of thenoiseless covariance tensor Ro isin C1198731times1198732timessdotsdotsdottimes119873119877times1198731times1198732timessdotsdotsdottimes119873119877

of order 2119873 defined as

Ro = Rstimes1A(1)times2A(2) sdot sdot sdot times119877A(119877)times119877+1A(1)lowasttimes119877+2A(2)lowast sdot sdot sdot times2119877A(119877)lowast (13)

where Rs is the source covariance tensor which has 2119877dimensions each of size 119872 Note that this tensor is dual-symmetric that is the factor matrix related to (119877 + 119903)thdimension is equal to A(119903)

lowast

and 119873119903 = 119873119877+119903 (119903 = 1 119877)The 119898th frontal slice ofRs is a diagonal matrix whose maindiagonal is given by the119898th column of the covariance matrixRs For instance considering 119877 = 2 for the sake of notationthe following expression satisfies the relationship previouslycited

Rs (119898119898) = diag (Rs (119898)) 119898 = 1 119872 (14)

where the matrix Rs( 119898119898) isin C119872times119872 denotes the 119898thfrontal slice of the covariance tensor Rs obtained by fixingits last two modes The tensor Ro follows a dual-symmetricTucker decomposition of order 2119877 with factor matrices A(119903)

and A(119903)lowast 119903 = 1 119877 and core tensorRs

Considering the case in which the sources are uncorre-lated and have unitary variance we can rewrite (13) as

Ro = I2119877119872times1A(1)times2A(2) sdot sdot sdot times119877A(119877)times119877+1A(1)lowasttimes119877+2A(2)lowast sdot sdot sdot times2119877A(119877)lowast (15)

whereI2119877119872 is the identity tensor of order 2119877 in which eachdimension has size119872 In this case the covariance tensorRofollows a dual-symmetric PARAFAC decomposition of order2119877

In general the Tucker decomposition does not imposerestrictions on the core tensor structure which makes thismodel more flexible In the context of this paper thischaracteristic reflects an arbitrary and unknown structure forthe sourcersquos covariance Rs which can also be estimated from(13) In contrast the PARAFAC decomposition (15) denotesa particular case of the Tucker decomposition when thesourcesrsquo signals are uncorrelated and the source covariancematrix is perfectly known (ie diagonal) In practice thismaynot hold

42 ALS-Tucker Algorithm Our goal is to blindly estimatethe spatial signature matrices A(119903) and A(119903)

lowast (119903 = 1 119877)which refer to the different dimensions of the sensor arrayfrom the covariance tensor Ro For the sake of simplicityfrom this point on we consider A(119877+119903) = A(119903)

lowast

In matrix-based notation the Tucker decomposition (13) allows thefollowing factorization in terms of its factormatrices and coretensor in accordance with (5)

[Ro](119903) = A(119903)Δ(119903) (16)

where

Δ(119903) = [Rs](119903)sdot (A(2119877) otimes sdot sdot sdot otimes A(119903+1) otimes A(119903minus1) otimes sdot sdot sdot otimes A(1))119879 (17)

while [Ro](119903) and [Rs](119903) 119903 = 1 2119877 denote the 119899-modeunfolding of the covariance tensor Ro and the core tensorRs respectively

From the matrix unfoldings of Ro an ALS based algo-rithm is formulated to estimate the desired factor matri-ces An estimate of the spatial signature matrix A(119903) (119903 =1 2119877) associated with the 119903th dimension of the covari-ance tensor is obtained by solving the following least squares(LS) problem

A(119903) = argminA(119903)

10038171003817100381710038171003817[Ro](119903) minus A(119903)Δ(119903)100381710038171003817100381710038172119865 (18)

whose analytic solution is given by

A(119903) = [Ro](119903) (Δ(119903))dagger (19)

As discussed in Section 41 the Tucker decompositiondoes not impose restrictions on the structure of the coretensor and its estimation becomes necessary Let Rs be anunknown matrix of arbitrary structure The following LSproblem is formulated from the vectorization of the samplecovariance matrix R

vec (Rs) = argminRs

10038171003817100381710038171003817vec (R) minus (Alowast otimes A) vec (Rs)100381710038171003817100381710038172119865 (20)

from which an estimate of Rs can be obtained as

vec (Rs) = (Alowast otimes A)dagger vec (R) (21)

where A = A(1) A(2) sdot sdot sdot A(119877) isin C119873times119872Since (19) and (21) are nonlinear functions of the param-

eters to be estimated the blind spatial signature estimationproblem can be solved using a classical ALS iterative solution[21 22] The basic idea of the algorithm is to estimate onefactor matrix at each step while the others remain fixedat the values obtained in previous steps This procedureis repeated until convergence The proposed generalizedALS-Tucker algorithm for 119877-D sensor arrays is summarizedin Algorithm 1

In this approach the factor matrices are treated as inde-pendent variables that is the dual-symmetry property ofthe covariance tensor is not exploited In this case a finalestimate of the spatial signature matrix associated with the119903th dimension of the array is given by

A(119903)final = A(119903) + A(119877+119903)lowast

2 (22)

43 ALS-ProKRaft Algorithm In this section a link isestablished between the ALS-ProKRaft algorithm proposedinitially in [12] and blind spatial signature estimation in


(1) Initialize A(119903) isin C119873119903times119872 for 119903 = 2 2119877 and the coretensor Rs randomly

(2) for 119903 = 1 2119877 doAccording to (19) obtain an estimate for the matrix A(119903)A(119903) = [R](119903)(Δ(119903))daggerNote The matrix Δ(119903) described in (17) is updatedby fixing A(119903) calculated previously

end(3) According to (21) obtain an estimate for Rs

vec(Rs) = (Alowast otimes A)daggervec(R)Rs = unvec119872times119872(vec(Rs))

(4) Using (14) reconstruct the core tensor Rs from Rs(5) Repeat Steps (2)ndash(4) until convergence

Algorithm 1 Summary of the ALS-Tucker algorithm

array signal processing The main idea behind this algorithmis to exploit the dual-symmetry property of the PARAFACdecomposition described in (15) Next the ALS-ProKRaftalgorithm is formulated in the context of this work A moredetailed description of the method can be found in theoriginal reference

The multimode unfolding of the PARAFAC decomposi-tion in (15) can be rewritten as

Rmm = (A(1) sdot sdot sdot A(119877)) (A(1) sdot sdot sdot A(119877))119867 (23)

which assumes the following factorization

Rmm = R12mm sdot (R12mm)119867 (24)

where R12mm isin C119873times119872 can be obtained from the singular valuedecomposition of Rmm given by

Rmm = U sdot Σ sdot V119867 (25)

obeying the following structure

R12mm = U[119872] sdot Σ[119872] sdot T119867 = (A(1) sdot sdot sdot A(119877)) (26)

where U[119872] isin C119873times119872 is formed by the first 119872 columns of Uand Σ[119872] isin C119872times119872 is a diagonal matrix which contains the119872dominant singular values of Rmm The matrix T represents aunitary rotation factor

Equation (26) describes an orthogonal Procrustes prob-lem (OPP) [23] in which T is a transformation matrix thatmapsU[119872] sdot Σ[119872] to (A(1) sdot sdot sdot A(119877)) such thatU[119872] sdot Σ[119872] sdotT119867 = (A(1)sdot sdot sdotA(119877)) An efficient estimate forT is obtainedminimizing the Frobenius norm of the residual error

argminT

10038171003817100381710038171003817U[119872] sdot Σ[119872] sdot T119867 minus (A(1) sdot sdot sdot A(119877))10038171003817100381710038171003817119865 (27)

This problem can be solved using a change of basis from thesingular value decomposition of the matrix

(A(1) sdot sdot sdot A(119877))119867 sdot U[119872] sdot Σ[119872] = U119875 sdot Σ119875 sdot V119867119875 (28)

which leads to the following solution [23]

T = U119875 sdot V119867119875 (29)

From (26) and (29) an ALS-based iterative algorithmis formulated to estimate the spatial signature matricesfrom the PARAFAC decomposition (15) Firstly individualestimates for each factor matrix A(119903) 119903 = 1 119877 areobtained by applying the multidimensional LS Khatri-Raofactorization (LS-KRF) algorithm on the composite spatialsignature matrix A = A(1) A(2) sdot sdot sdot A(119877) Then thematrix T is obtained from (29) For more details and accessto the pseudocode of this algorithm we refer the interestedreader to [12] The ALS-ProKRaft algorithm for blind spatialsignature estimation in 119877-D sensor arrays is summarizedin Algorithm 2

Note that when compared with conventional ALS-basedPARAFAC solutions [21] formulated from the unfoldingmatrices (8) the ALS-ProKRaft algorithm becomes preferredsince only half of the factors matrices needs to be estimatedby exploiting the dual-symmetry property of the covariancetensor This generally leads to a fast convergence rate of thealgorithm

5 Spatial Signature Estimation in 119871-ShapedSensor Arrays

In this section the blind spatial signature estimation problemis formulated for 119871-shaped array configuration Consideringthat the receiver array is divided into smaller subarrays theTucker decomposition of a fourth-order tensor is formulatedfrom the sample cross-correlationmatrix of the data receivedby the different subarrays From this multidimensionalstructure the proposed generalized ALS-Tucker algorithmpreviously presented in Section 42 can be used to estimatethe sourcersquos spatial signatures

51 Cross-Correlation Tensor for L-Shaped Sensor ArraysIn this approach we consider an 119871-shaped sensor arrayequipped with 1198731 + 1198732 minus 1 sensors positioned in the 119909-119911


119894 = 0 Initialize T(119894=0) = I119872(1) According to (25) obtain U[119872] and Σ[119872] from the SVD

of the multimode unfolding matrix Rmm(2) 119894 = 119894 + 1(3) According to (26) obtain estimates for A(119903)

(119894)for 119903 = 1 119877

by applying the multidimensional LS-KRF algorithm onU[119872] sdot Σ[119872] sdot T119867(119894minus1)

(4) According to (29) compute the SVD for the matrix(A(1)(119894)

sdot sdot sdot A(119877)(119894)

)119867 sdot U[119872] sdot Σ[119872] = U119875 sdot Σ119875 sdot V119867119875and obtain T(119894) = U119875 sdot V119867119875

(5) Repeat Steps (2)ndash(4) until convergenceAlgorithm 2 Summary of the ALS-ProKRaft algorithm

x

y

z

N1

N1 minus 1

d(1)

d(2)

120572m

120573m

N2 minus 1

1

2

2

N2

sm (k) m = 1 M

Figure 1 L-shaped array configurationwith1198731+1198732minus1 sensorsThedistance between the sensors in the 119911-axis is 119889(1) while the distancebetween the sensors in the 119909-axis is 119889(2) The 119898th wavefront haselevation and azimuth angles equal to 120572119898 and 120573119898 respectivelyplane as illustrated in Figure 1 Each linear array contains1198731 and 1198732 sensors equally spaced at distances 119889(1) and 119889(2)respectively We consider that each linear array is dividedinto 119875 and 119882 smaller subarrays respectively Each subarraycontains119873(sub)1 = 1198731minus119875+1 and119873(sub)2 = 1198732minus119882+1 sensorsThe signal received at the 119901th subarray for 119901 = 1 119875 isgiven by

X(1)119901 = A(1)s D119901 (Φ(1)) S + V(1)119901 isin C119873(sub)1 times119870 (30)

and the signal received at the 119908th subarray 119908 = 1 119882 isgiven by


where

(i) A(119903)s isin C119873(sub)119903 times119872 is the spatial signature matrix of the

first subarray (or reference subarray) for the 119903th arraydimension 119903 = 1 2

(ii) D119901(Φ(1)) and D119908(Φ(2)) are diagonal matrices whosemain diagonal is given by the 119901th and 119908th row of thematricesΦ(1) isin C119875times119872 andΦ(2) isin C119882times119872 respectively

The rows of Φ(1) and Φ(2) capture the delays suffered by thesignals impinging the 119901th and 119908th subarrays with respectto the reference subarray which are defined based on thefollowing spatial frequencies

120583(1)119898 = 2120587 sdot 119889(1) sdot cos120572119898120582 120583(2)119898 = 2120587 sdot 119889(2) sdot sin120572119898 sdot cos120573119898120582

(32)

where 120572119898 and 120573119898 are the azimuth and elevation angles of the119898th source respectivelyFrom (30) and (31) let us introduce the following

extended data matrices

X(1) = [X(1)1 X(1)119875 ]119879 isin C119873(sub)1 119875times119870

X(2) = [X(2)1 X(2)119882 ]119879 isin C119873(sub)2 119882times119870 (33)

or more compactly

X(119903) = (Φ(119903) A(119903)s ) S + V(119903) 119903 = 1 2 (34)

In contrast with (12) in this approach we shall work withthe following sample cross-correlation matrix

R = (Φ(1) A(1)s )Rs (Φ(2) A(2)s )119867 + 1205902V I (35)

As mentioned in Section 41 we can see that the noiselessterm in (35) denotes a multimode unfolding of the followingcross-correlation tensor of size119873(sub)1 times 119875 times 119873(sub)2 times 119882

Ro = Rstimes1A(1)s times2Φ(1)times3A(2)lowasts times4Φ(2)lowast (36)


where Rs isin C119872times119872times119872times119872 In this modeling the tensor Rofollows a fourth-order Tucker decomposition By analogywith (4) we deduce the following correspondences

(1198681 1198682 1198683 1198684) larrrarr (119873(sub)1 119875119873(sub)2 119882)(1198771 1198772 1198773 1198774) larrrarr 119872(GA(1)A(2)A(3)A(4))

larrrarr (RsA(1)s Φ(1)A(2)lowasts Φ(2)lowast) (37)

The spatial signatures of the sources can be estimatedfrom (36) by using the proposed generalized ALS-Tuckeralgorithm Note that in this case the ALS-Tucker algorithmis simplified to a fourth-order tensor input

For all the previously proposed algorithms the finalestimates for the spatial signaturematrices are obtainedwhenthe convergence is declared A usually adopted criterion forconvergence is defined as |119890(119894) minus 119890(119894minus1)| le 10minus6 where 119890(119894)denotes the residual error of the 119894th iteration defined as

119890(119894) = 10038171003817100381710038171003817R minus R(119894)100381710038171003817100381710038172119865 (38)

whereR = Ro+V is a noisy version ofRV is an additivecomplex-valued white Gaussian noise tensor and R(119894) is thecovariance tensor reconstructed from the estimated factormatrices and core tensor Since the ALS-ProKRaft algorithmexploits the dual-symmetry property of the data tensor theprocedure in (22) is not necessary

52 Estimation of the Spatial Frequencies After the estima-tion of the spatial signatures matrices A(119903)final 119903 = 1 119877 thefinal step is to estimate the spatial frequencies of the sources120583(119903)119898 119898 = 1 119872 The final estimates can be computed fromthe average over the values obtained in each row of A(119903)final asfollows

120583(119903)119898 = 1119873119903 minus 1119873119903sum119899=2

arg A(119903)final (119899119898)119899 minus 1 (39)

6 Computational Complexity

In the following we discuss the computational complexity ofthe iterative ALS-Tucker and ALS-ProKRaft algorithms Forthe sake of simplicity the computational complexity of theproposedmethods is described in terms of the computationalcost of the matrix SVD For a matrix of size 1198681 times 1198682 thenumber of floating-point operations associated with theSVD computation is O(1198681 sdot 1198682 sdot min(1198681 sdot 1198682)) [24] Thecomputational complexity of one Tucker iteration refers tothe cost associated with the SVD used to calculate the matrixpseudoinverses in the least squares problems (18) and (20)The overall computational complexity per iteration of theALS-Tucker algorithm equals the complexity of 2119877 matrixSVDs associated with each estimated factor matrix accordingto (19) plus the complexity of one additional matrix SVDassociated with the estimated core tensor according to (21)

The overall computational cost per iteration of the ALS-ProKRaft algorithm equals the complexity of 119872(119877 minus 1)matrix SVDs associated with the application of the multi-dimensional LS-KRF algorithm in (26) plus the complexityof one additional matrix SVD associated with the update ofthe unknown unitary rotation factor matrix T according to(29)

7 Advantages and Disadvantages ofthe Proposed Methods

In this section we discuss the advantages and disadvantagesof the proposed methods to blind spatial signatures estima-tion in 119877-D sensor arrays As previously stated in Section 43the ALS-ProKRaft algorithm works on the assumption thatthe sample covariance matrix of the sources Rs is perfectlyknown and diagonal However this is only true in theasymptotic casewhen a sufficiently large number of snapshotsis assumed (ie 119870 rarr infin) as well as when the source signalsare perfectly uncorrelated In practice this assumption is notguaranteed On the other hand the ALS-Tucker algorithmpreviously formulated in Section 42 naturally captures anystructure for the sources covariance into the core tensorRsTherefore the assumption of uncorrelated source signals isnot necessary for the ALS-Tucker algorithmmaking it able tooperate in scenarios where the source covariance structure isunknown and arbitrary (nondiagonal) Such scenarios occurfor instance when the sample covariance is computed from alimited number of snapshots

In contrast the ALS-Tucker algorithm does not exploitthe dual-symmetry property of the data covariance tensorand all factor matrices need to be estimated as independentvariables However in the ALS-ProKRaft algorithm only halfof the factor matrices are estimated by exploiting the dual-symmetry property of the covariance tensor Therefore theALS-ProKRaft algorithm is more computationally attractivethan the ALS-Tucker algorithm When compared with thestate-of-the-art matrix-based algorithms such as MUSICESPRIT and Propagator Method the proposed tensor-basedalgorithms have the advantage of fully exploiting the multi-dimensional nature of the received signal in less specific sce-narios which leads to more accurate estimates For instancethe ESPRIT algorithm was formulated for sensor arrays thatobey the shift invariance property On the other hand theMUSIC algorithm has high computational complexity due tothe search of parameters in the spatial spectrum

8 Simulation Results

In the following simulation results and performance evalu-ations of the ALS-Tucker and ALS-ProKRaft algorithms for119877-D sensor arrays are presented This section is divided intotwo parts Firstly the results related to Section 4 are presentedand discussed Then the same is done for the 119871-shaped arrayapproach presented in Section 5 Results are obtained froman average of 1000 independent Monte Carlo runs In thefirst part of this section we consider a uniform rectangulararray (URA) positioned on the 119909-119911 planeThe119898th wavefront


SESTEMUSIC

ALS-TuckerALS-ProKRaft

10minus3

10minus2

10minus1

100

RMSE

0 5 10 15 20minus5

SNR (dB)

Figure 2 Total RMSE versus SNR for 119873 = 64 sensors 119870 = 10samples and DoAs 30∘ 55∘ and 45∘ 60∘ for Hadamard sequen-ces

SESTEMUSIC

ALS-ProKRaftALS-Tucker

10minus3

10minus2

10minus1

100

RMSE

0 5 10 15 20minus5

SNR (dB)

Figure 3 Total RMSE versus SNR for 119873 = 64 sensors 119870 = 10samples and DoAs 30∘ 55∘ and 45∘ 60∘ for BPSK sequences

has direction of arrival 120572119898 120573119898119872119898=1 where 120572119898 and 120573119898 areelevation and azimuth angles respectively

In Figures 2 and 3 the performance is measured in termsof the root mean square error (RMSE) of the estimatedspatial frequencies 120583(119903)119898 in terms of Signal to Noise Ratio(SNR)The relations between directions of arrival and spatialfrequencies are given by (32) where 119889(119903) denotes the distancebetween sensors in the 119903th array dimensionwhich is assumedhere equal to 1205822 In Figure 2 we consider Hadamardsequences for the sources while in Figure 3we consider BPSKmodulated sources In both cases we have 119873 = 64 sensors

ALS-Tucker SNR = 10 dBALS-Tucker SNR = 20 dBALS-Tucker SNR = 30 dB

ALS-ProKRaft SNR = 10 dBALS-ProKRaft SNR = 20 dBALS-ProKRaft SNR = 30 dB

50 100 150 200 250 3000Number of iterations

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

101

Nor

mal

ized

estim

atio

n er

ror

Figure 4 Convergence of the ALS-Tucker and ALS-ProKRaft algo-rithms

(ie1198731 and1198732 equal to 8) and the sample covariance matrixof the received data (12) is calculated from a reduced number119870 = 10 of samples The total RMSE is defined as

RMSE = radic119864 119877sum119903=1

119872sum119898=1

(120583(119903)119898 minus 120583(119903)119898 )2 (40)

From Figure 2 it can be seen that both ALS-Tuckerand ALS-ProKRaft algorithms have similar performances interms of RMSE when Hadamard sequences are consideredIn contrast in Figure 3 when BPSK symbols are considereda floor is exhibited by the ALS-ProKraft algorithm for highSNR values This behavior occurs due to the modeling errorsin the core tensor of the PARAFAC decomposition which inturn arises due to the nonorthogonality of the source signalsresulting in a nondiagonal sample covariance matrix of thesources in this case Note that in the ALS-Tucker algorithmthe covariance matrix of the sources possess an arbitrary andunknown structure which discards possible constraints to thesource signals This difference makes the Tucker decompo-sition approach more attractive in those practical scenariosin which source uncorrelatedness is not guaranteed Whencompared tomatrix-based standard ESPRIT (SE) [2] matrix-based MUSIC algorithm to planar array configuration [25]and tensor-based standard ESPRIT (STE) [10] the proposedalgorithms have improved accuracy in all the consideredscenarios

Figure 4 shows the convergence performance of theiterative algorithms In this experiment the median valuesof the normalized estimation error 119890(119894)119873(2119877) are plotted interms of the number of iterations for different SNR It isnoteworthy that the ALS-ProKRaft algorithm has a fasterconvergence compared to the ALS-Tucker algorithm This


PMESPRIT

MUSICALS-Tucker

10minus2

10minus1

100

101

102

RMSE

5 15 20 25 3010SNR (dB)

Figure 5 Total RMSE versus SNR for 119873 = 13 sensors 119870 = 500samples and DoAs 30∘ 45∘ and 50∘ 55∘

behavior is expected since ALS-ProKRaft exploits the dual-symmetry property of the data tensor which results inestimating half asmany factormatrices compared to theALS-Tucker approach

In the second part of this section we consider a 119871-shapedconfiguration array In Figure 5 we set 119873 = 13 sensors(ie 1198731 and 1198732 equal to 7) and 119870 = 500 samples Eachuniform linear array is divided into 119875 = 2 and 119882 = 2subarrays respectively In this experiment the performanceof the proposed ALS-Tucker algorithm is compared to thestate-of-the-art matrix-based methods namely PropagatorMethod (PM) [26]MUSIC [27] and ESPRIT [28] all of themoriginally formulated for 119871-shaped arrays Note that the ALS-Tucker algorithm presents an improved performance overits competitors with more evidenced gains in the low-to-medium SNR range For high SNR values the performanceof the MUSIC method comes close to that of our proposalHowever theALS-Tucker algorithmdispenses any estimationprocedure via bidimensional peak search as occurs withMUSIC being the former more computationally attractive

Figure 6 shows the performance of the ALS-Tucker byassuming different number of sensors In this experimentwe consider 119870 = 500 samples We can observe a betterperformance in terms of RMSE when the number sensorsof the 119871-shaped array is increased This is valid for all thesimulated SNR values

In Figure 7 we analyze the influence of the number 119870of samples on the performance of the ALS-Tucker algorithmThis experiment considers the same parameters as the experi-ment of Figure 5 except the SNRvalue that is assumedfixed at20 dB and the number of samples that varies between 50 and3000 First we can see that the performance of the algorithmsimproves by increasing the number of samples collected bysensor array as expected However similar to Figure 5 the

L-shaped array with 17 sensorsL-shaped array with 13 sensorsL-shaped array with 9 sensors

10minus2

10minus1

100

101

RMSE

10 15 20 25 305SNR (dB)

Figure 6 Total RMSE versus SNR (performance of the ALS-Tuckeralgorithm for different number of sensors)

PMESPRIT

MUSICALS-Tucker

500 1000 1500 2000 2500 30000Number of samples

10minus2

10minus1

100

RMSE

Figure 7 Total RMSE versus number of samples

proposed ALS-Tucker algorithm outperforms the state-of-the-art PM ESPRIT and MUSIC methods

9 Conclusion

In this paper two tensor-based approaches based on theTucker and PARAFAC decompositions have been formulatedto solve the blind spatial signatures estimation problem inmultidimensional sensor arrays First we have proposeda covariance-based generalization of the Tucker decom-position for 119877-D sensor arrays Then a link between theALS-ProKRaft algorithm and covariance-based array signalprocessing for blind spatial signatures estimation has been


established As another contribution we have formulated across-correlation-based fourth-order Tucker decompositionwhich makes the proposed ALS-Tucker algorithm applicablein scenarios composed by 119871-shaped array configurationsThe two proposed tensor methods differ in the structureassumed for the source covariance It is worth pointing outthat in realistic scenarios when the received covariancematrix is calculated from a reduced number of samplesor snapshots the ALS-Tucker algorithm becomes preferablesince it operates with an arbitrary and unknown struc-ture for the covariance of the source signals In contrastwhen the sources can be assumed to be uncorrelated wecan achieve improved performance by exploiting the dual-symmetry property of the covariance tensor whichmakes theALS-ProKRaft algorithm preferable since it provides goodestimation accuracy with a smaller number of ALS iterationsFinally when compared with other state-of-the-art matrix-based and tensor-based techniques the proposed tensor-based iterative algorithms have shown their effectiveness withremarkable gains in terms of estimation error

Competing Interests

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

Acknowledgments

The authors would like to thank the Fundacao Cearensede Apoio ao Desenvolvimento Cientıfico e Tecnologico(FUNCAP) the Coordenacao de Aperfeicoamento de Pes-soal de Nıvel Superior (CAPES) under the PVE Grant no888810303922013-01 the productivity Grant no 3039052014-0 the postdoctoral scholarship abroad (PDE) no2076442015-2 and the Program Science without BordersAerospace Technology supported by CNPq and CAPES forthe postdoctoral scolarship abroad (PDE) no 2076442015-2

References

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

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

[3] J P C L da Costa F Roemer M Haardt and R T de Sousa JrldquoMulti-dimensional model order selectionrdquo EURASIP Journalon Advances in Signal Processing vol 2011 article 26 2011

[4] F Roemer Advanced algebraic concepts for efficient multi-channel signal processing [PhD thesis] Ilmenau University ofTechnology Ilmenau Germany 2013

[5] J P C L DaCosta K LiuH Cheung So S SchwarzMHaardtand F Romer ldquoMultidimensional prewhitening for enhancedsignal reconstruction and parameter estimation in colorednoise with Kronecker correlation structurerdquo Signal Processingvol 93 no 11 pp 3209ndash3226 2013

[6] J P C L da Costa Parameter Estimation Techniques for Multi-Dimensional Array Signal Processing Shaker Publisher AachenGermany 2010

[7] R Harshman ldquoFoundations of the PARAFAC proceduremodels and conditions for an explanatory multimodal factoranalysisrdquo UCLAWorking Papers in Phonetics vol 16 no 10 pp1ndash84 1970

[8] R A Harshman and M E Lundy ldquoUniqueness proof fora family of models sharing features of Tuckerrsquos three-modefactor analysis and PARAFACCANDECOMPrdquo PsychometrikaA Journal of Quantitative Psychology vol 61 no 1 pp 133ndash1541996

[9] 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

[10] M Haardt F Roemer and G Del Galdo ldquoHigher-order SVD-based subspace estimation to improve the parameter estimationaccuracy in multidimensional harmonic retrieval problemsrdquoIEEE Transactions on Signal Processing vol 56 no 7 pp 3198ndash3213 2008

[11] J P C L Da Costa F Roemer M Weis and M HaardtldquoRobust R-D parameter estimation via closed-formPARAFACrdquoin Proceedings of the International ITG Workshop on SmartAntennas (WSA rsquo10) pp 99ndash106 Bremen Germany February2010

[12] MWeis F Roemer M Haardt and P Husar ldquoDual-symmetricparallel factor analysis using procrustes estimation and Khatri-Rao factorizationrdquo in Proceedings of the 20th EUSIPCO Euro-pean Signal Processing Conference Bucharest Romania August2012

[13] F Roemer and M Haardt ldquoTensor-based channel estimation(TENCE) for two-way relaying withmultiple antennas and spa-tial reuserdquo in Proceedings of the IEEE International Conferenceon Acoustics Speech and Signal Processing (ICASSP rsquo09) TaipeiTaiwan April 2009

[14] P R B Gomes A L F de Almeida and J P C L daCostal ldquoFourth-order tensor method for blind spatial signatureestimationrdquo in Proceedings of the IEEE International Conferenceon Acoustics Speech and Signal Processing (ICASSP rsquo14) pp2992ndash2996 IEEE Florence Italy May 2014

[15] L R Tucker ldquoImplications of factor analysis of three-waymatrices for measurement of changerdquo in Problems inMeasuringChange C W Harris Ed pp 122ndash137 University of WisconsinPress 1963

[16] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009

[17] A Cichocki D Mandic L De Lathauwer et al ldquoTensordecompositions for signal processing applications from two-way to multiway component analysisrdquo IEEE Signal ProcessingMagazine vol 32 no 2 pp 145ndash163 2015

[18] J M F Ten Berge and A K Smilde ldquoNon-triviality andidentification of a constrained Tucker3 analysisrdquo Journal ofChemometrics vol 16 no 12 pp 609ndash612 2002

[19] N D Sidiropoulos and R Bro ldquoOn the uniqueness of multilin-ear decomposition of N-way arraysrdquo Journal of Chemometricsvol 14 no 3 pp 229ndash239 2000

[20] 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


[21] R Bro Multi-way analysis in the food industry models algo-rithms and applications [PhD thesis] University of AmsterdamAmsterdam The Netherlands 1998

[22] A Smilde R Bro and P Geladi Multi-Way Analysis Applica-tions in the Chemical Sciences John Wiley amp Sons 2004

[23] P H Schonemann ldquoA generalized solution of the orthogonalProcrustes problemrdquo Psychometrika vol 31 no 1 pp 1ndash10 1966

[24] G H Golub and C F Van Loan Matrix Computations JohnsHopkins Studies in the Mathematical Sciences Johns HopkinsUniversity Press Baltimore Md USA 3rd edition 1996

[25] S Sekizawa ldquoEstimation of arrival directions using MUSICalgorithm with a planar arrayrdquo in Proceedings of the IEEEInternational Conference onUniversal Personal Communications(ICUPC rsquo98) pp 555ndash559 Florence Italy October 1998

[26] N Tayem and H M Kwon ldquoL-shape 2-dimensional arrivalangle estimation with propagator methodrdquo IEEE TransactionsonAntennas and Propagation vol 53 no 5 pp 1622ndash1630 2005

[27] H Changuel F Harabi and A Gharsallah ldquo2-L-shape two-dimensional arrival angle estimation with a classical subspacealgorithmrdquo in Proceedings of the International Symposium onIndustrial Electronics (ISIE rsquo06) pp 603ndash607Montreal CanadaJuly 2006

[28] X Zhang X Gao and W Chen ldquoImproved blind 2D-directionof arrival estimation with L-shaped array using shift invariancepropertyrdquo Journal of Electromagnetic Waves and Applicationsvol 23 no 5-6 pp 593ndash606 2009

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



decomposition [15] of a fourth-order covariance tensor andwas elaborated for arrays with arbitrary structures wherea priori knowledge about the geometry of the sensor arrayis not required However a limitation of the method in[14] is the necessity of transmitting the same sequence ofsymbols in different time blocks which results in a loss ofspectral efficiency The proposed solution is an algorithm fora multidimensional (119877-D) sensor array in which the differentdimensions of the array are exploited thus dismissing theneed to transmit a repeated sequence as in [14] as will bedetailed later

In this paper two tensor-based approaches to the esti-mation of spatial signatures are presented By using thesignals received on a 119877-D sensor array covariance tensorsare calculated and solutions for correlated and uncorrelatedsources are presented respectively For the former scenarioin which the sourcersquos covariance structure is nondiagonaland unknown the covariance tensor of the received datais formulated as a Tucker decomposition of order 2119877 Sucha formulation yields a generalized Tucker model based 119877-D sensor array processing that deals with arbitrary sourcecovariance structures By assuming uncorrelated sources wethen show that the problem boils down to a PARAFACdecomposition from which a method that exploits the dual-symmetry property of the covariance tensor is derived byconsidering the ideas rooted in [12] For both Tucker andPARAFAC based models the blind estimation of the spatialsignatures is achieved by means of an alternating leastsquares (ALS) algorithm The contributions of this paper aretwofold (i) we propose a covariance-based generalizationof the Tucker decomposition for the blind spatial signatureestimation problem with 119877-D sensor arrays and (ii) weestablish a link between dual-symmetry decompositions andtechniques based on covariance-based array signal process-ing for parameter estimation The performance of the pro-posed algorithms is evaluated by Monte Carlo simulationscorroborating their gains over competing state-of-the-artmatrix-based and tensor-based techniques

The rest of this paper is organized as follows Section 2briefly introduces tensor operations and decompositionsThesignal model for an 119877-D sensor array is then presentedin Section 3 In Section 4 a novel covariance-based tensormodel for the received data is formulated and our blindspatial signature estimation algorithms are formulated InSection 5 an approach for 119871-shaped sensor arrays is proposedThe computational complexity of the proposed methods isanalyzed in Section 6 In Section 7 the advantages and disad-vantages of the proposed methods are discussed Simulationresults are provided in Section 8 and the conclusions aredrawn in Section 9

Notation Scalar values are represented by lowercase letters119886 vectors by bold lowercase letters a matrices by bolduppercase lettersA and tensors by calligraphic lettersAThesymbols 119879 119867 dagger and lowast represent the transpose conjugatetranspose pseudoinverse and complex conjugate operationsrespectively diag(a) operator generates a diagonal matrixfrom a vector a The 119894th row of A isin C119868times119877 is representedby A(119894 ) isin C1times119877 while its 119903th column is represented by

A( 119903) isin C119868times1 vec(A) operator converts A into a vectora isin C119868119877times1 while unvec119868times119877(a) converts a into a 119868 times 119877 matrixD119894(A) stands for a diagonal matrix constructed from the 119894throw of A sdot 119865 stands for the Frobenius norm of a matrixor tensor ldquo∘rdquo operator stands for the vector outer productThe Kronecker product is represented by otimes The Khatri-Raoproduct between the matrices A isin C119868times119877 and B isin C119869times119877represented by is defined as

A B = [A (1) otimes B (1) A (119877) otimes B (119877)] (1)

2 Tensor Preliminaries

In the following we briefly introduce for convenience thebasics on operations involving tensors and tensor decom-positions which refer to [16 17] Firstly we present theTucker decomposition Then the PARAFAC decompositionis introduced and issues involving uniqueness are brieflydiscussed for both cases which will be useful later Then weintroduce the dual-symmetry property for these decomposi-tionsThe basic material presented in this section is exploitedin later sections in the context of our blind spatial signatureestimation problem

21 Basic Tensor Operations Let X isin C1198681timessdotsdotsdottimes119868119873 denote an119873th order tensor (1198941 119894119873)th entry of which is denoted by1199091198941 119894119873 The fibers are the higher-order analogues of matrixrows and columnsThe 119899-mode fibers ofX are vectors of size119868119899 defined by fixing every index but 119894119899The 119899-mode unfoldingoperation denoted by [X](119899) stands for the process ofreordering the elements of X into a matrix by arrangingits 119899-mode fibers to be the columns of the resulting matrixThe 119899-mode product between X and a matrix A isin C119869times119868119899

along of the 119899th mode denoted byXtimes119899A is a tensor of size1198681 times sdot sdot sdot times 119868119899minus1 times 119869 times 119868119899+1 times sdot sdot sdot times 119868119873 obtained by taking theinner product between each 119899-mode fiber and the rows of thematrix A that is [16 17]

Y = Xtimes119899A lArrrArr[Y](119899) = A [X](119899) (2)

22 Tucker Decomposition The Tucker decomposition [15]represents a tensorX isin C1198681timessdotsdotsdottimes119868119873 of order119873 as a multilineartransformation of a core tensor G isin C1198771timessdotsdotsdottimes119877119873 by factormatrices A(119899) = [a(119899)1 a(119899)2 a(119899)119877119899 ] isin C119868119899times119877119899 along eachmode 119899 = 1 2 119873 In scalar form the 119873th order Tuckerdecomposition is given by

1199091198941 1198942119894119873 = 1198771sum1199031=1

1198772sum1199032=1

sdot sdot sdot 119877119873sum119903119873=1

1198921199031 1199032 119903119873119886(1)1198941 1199031119886(2)1198942 1199032 sdot sdot sdot 119886(119873)119894119873119903119873 (3)

where 119886(119899)119894119899 119903119899 is (119894119899 119903119899)th entry of the 119899th mode factor matrixA(119899) isin C119868119899times119877119899 and 1198921199031 1199032119903119873 is (1199031 119903119873)th entry of the coretensor G isin C1198771timessdotsdotsdottimes119877119873 Using 119899-mode product notation theTucker decomposition can be written as

X = Gtimes1A(1)times2A(2) sdot sdot sdot times119873A(119873) (4)






X = 119877sum119903=1








119873sum119899=1

120581A(119899) ge 2119877 + 119873 minus 1 (9)





3 Signal Model



where







R ≜ 1119870XX119867 asymp ARsA119867 + 1205902V I


(12)









lowast


Rs (119898119898) = diag (Rs (119898)) 119898 = 1 119872 (14)









lowast


[Ro](119903) = A(119903)Δ(119903) (16)

where




A(119903) = argminA(119903)

10038171003817100381710038171003817[Ro](119903) minus A(119903)Δ(119903)100381710038171003817100381710038172119865 (18)


A(119903) = [Ro](119903) (Δ(119903))dagger (19)


vec (Rs) = argminRs







A(119903)final = A(119903) + A(119877+119903)lowast

2 (22)













Rmm = R12mm sdot (R12mm)119867 (24)







argminT





T = U119875 sdot V119867119875 (29)









(119894)for 119903 = 1 119877






x

y

z

N1

N1 minus 1

d(1)

d(2)

120572m

120573m

N2 minus 1

1

2

2

N2

sm (k) m = 1 M





where






(32)





or more compactly

X(119903) = (Φ(119903) A(119903)s ) S + V(119903) 119903 = 1 2 (34)


R = (Φ(1) A(1)s )Rs (Φ(2) A(2)s )119867 + 1205902V I (35)









119890(119894) = 10038171003817100381710038171003817R minus R(119894)100381710038171003817100381710038172119865 (38)



120583(119903)119898 = 1119873119903 minus 1119873119903sum119899=2

arg A(119903)final (119899119898)119899 minus 1 (39)










SESTEMUSIC


10minus3

10minus2

10minus1

100

RMSE

0 5 10 15 20minus5

SNR (dB)


SESTEMUSIC


10minus3

10minus2

10minus1

100

RMSE

0 5 10 15 20minus5

SNR (dB)







10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

101

Nor

mal

ized

estim

atio

n er

ror



RMSE = radic119864 119877sum119903=1

119872sum119898=1

(120583(119903)119898 minus 120583(119903)119898 )2 (40)




PMESPRIT

MUSICALS-Tucker

10minus2

10minus1

100

101

102

RMSE

5 15 20 25 3010SNR (dB)







10minus2

10minus1

100

101

RMSE

10 15 20 25 305SNR (dB)


PMESPRIT

MUSICALS-Tucker


10minus2

10minus1

100

RMSE



9 Conclusion




Competing Interests


Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














X = 119877sum119903=1








119873sum119899=1

120581A(119899) ge 2119877 + 119873 minus 1 (9)





3 Signal Model



where







R ≜ 1119870XX119867 asymp ARsA119867 + 1205902V I


(12)









lowast


Rs (119898119898) = diag (Rs (119898)) 119898 = 1 119872 (14)









lowast


[Ro](119903) = A(119903)Δ(119903) (16)

where




A(119903) = argminA(119903)

10038171003817100381710038171003817[Ro](119903) minus A(119903)Δ(119903)100381710038171003817100381710038172119865 (18)


A(119903) = [Ro](119903) (Δ(119903))dagger (19)


vec (Rs) = argminRs







A(119903)final = A(119903) + A(119877+119903)lowast

2 (22)













Rmm = R12mm sdot (R12mm)119867 (24)







argminT





T = U119875 sdot V119867119875 (29)









(119894)for 119903 = 1 119877






x

y

z

N1

N1 minus 1

d(1)

d(2)

120572m

120573m

N2 minus 1

1

2

2

N2

sm (k) m = 1 M





where






(32)





or more compactly

X(119903) = (Φ(119903) A(119903)s ) S + V(119903) 119903 = 1 2 (34)


R = (Φ(1) A(1)s )Rs (Φ(2) A(2)s )119867 + 1205902V I (35)









119890(119894) = 10038171003817100381710038171003817R minus R(119894)100381710038171003817100381710038172119865 (38)



120583(119903)119898 = 1119873119903 minus 1119873119903sum119899=2

arg A(119903)final (119899119898)119899 minus 1 (39)










SESTEMUSIC


10minus3

10minus2

10minus1

100

RMSE

0 5 10 15 20minus5

SNR (dB)


SESTEMUSIC


10minus3

10minus2

10minus1

100

RMSE

0 5 10 15 20minus5

SNR (dB)







10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

101

Nor

mal

ized

estim

atio

n er

ror



RMSE = radic119864 119877sum119903=1

119872sum119898=1

(120583(119903)119898 minus 120583(119903)119898 )2 (40)




PMESPRIT

MUSICALS-Tucker

10minus2

10minus1

100

101

102

RMSE

5 15 20 25 3010SNR (dB)







10minus2

10minus1

100

101

RMSE

10 15 20 25 305SNR (dB)


PMESPRIT

MUSICALS-Tucker


10minus2

10minus1

100

RMSE



9 Conclusion




Competing Interests


Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














lowast


Rs (119898119898) = diag (Rs (119898)) 119898 = 1 119872 (14)









lowast


[Ro](119903) = A(119903)Δ(119903) (16)

where




A(119903) = argminA(119903)

10038171003817100381710038171003817[Ro](119903) minus A(119903)Δ(119903)100381710038171003817100381710038172119865 (18)


A(119903) = [Ro](119903) (Δ(119903))dagger (19)


vec (Rs) = argminRs







A(119903)final = A(119903) + A(119877+119903)lowast

2 (22)













Rmm = R12mm sdot (R12mm)119867 (24)







argminT





T = U119875 sdot V119867119875 (29)









(119894)for 119903 = 1 119877






x

y

z

N1

N1 minus 1

d(1)

d(2)

120572m

120573m

N2 minus 1

1

2

2

N2

sm (k) m = 1 M





where






(32)





or more compactly

X(119903) = (Φ(119903) A(119903)s ) S + V(119903) 119903 = 1 2 (34)


R = (Φ(1) A(1)s )Rs (Φ(2) A(2)s )119867 + 1205902V I (35)









119890(119894) = 10038171003817100381710038171003817R minus R(119894)100381710038171003817100381710038172119865 (38)



120583(119903)119898 = 1119873119903 minus 1119873119903sum119899=2

arg A(119903)final (119899119898)119899 minus 1 (39)










SESTEMUSIC


10minus3

10minus2

10minus1

100

RMSE

0 5 10 15 20minus5

SNR (dB)


SESTEMUSIC


10minus3

10minus2

10minus1

100

RMSE

0 5 10 15 20minus5

SNR (dB)







10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

101

Nor

mal

ized

estim

atio

n er

ror



RMSE = radic119864 119877sum119903=1

119872sum119898=1

(120583(119903)119898 minus 120583(119903)119898 )2 (40)




PMESPRIT

MUSICALS-Tucker

10minus2

10minus1

100

101

102

RMSE

5 15 20 25 3010SNR (dB)







10minus2

10minus1

100

101

RMSE

10 15 20 25 305SNR (dB)


PMESPRIT

MUSICALS-Tucker


10minus2

10minus1

100

RMSE



9 Conclusion




Competing Interests


Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




















Rmm = R12mm sdot (R12mm)119867 (24)







argminT





T = U119875 sdot V119867119875 (29)









(119894)for 119903 = 1 119877






x

y

z

N1

N1 minus 1

d(1)

d(2)

120572m

120573m

N2 minus 1

1

2

2

N2

sm (k) m = 1 M





where






(32)





or more compactly

X(119903) = (Φ(119903) A(119903)s ) S + V(119903) 119903 = 1 2 (34)


R = (Φ(1) A(1)s )Rs (Φ(2) A(2)s )119867 + 1205902V I (35)









119890(119894) = 10038171003817100381710038171003817R minus R(119894)100381710038171003817100381710038172119865 (38)



120583(119903)119898 = 1119873119903 minus 1119873119903sum119899=2

arg A(119903)final (119899119898)119899 minus 1 (39)










SESTEMUSIC


10minus3

10minus2

10minus1

100

RMSE

0 5 10 15 20minus5

SNR (dB)


SESTEMUSIC


10minus3

10minus2

10minus1

100

RMSE

0 5 10 15 20minus5

SNR (dB)







10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

101

Nor

mal

ized

estim

atio

n er

ror



RMSE = radic119864 119877sum119903=1

119872sum119898=1

(120583(119903)119898 minus 120583(119903)119898 )2 (40)




PMESPRIT

MUSICALS-Tucker

10minus2

10minus1

100

101

102

RMSE

5 15 20 25 3010SNR (dB)







10minus2

10minus1

100

101

RMSE

10 15 20 25 305SNR (dB)


PMESPRIT

MUSICALS-Tucker


10minus2

10minus1

100

RMSE



9 Conclusion




Competing Interests


Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












(119894)for 119903 = 1 119877






x

y

z

N1

N1 minus 1

d(1)

d(2)

120572m

120573m

N2 minus 1

1

2

2

N2

sm (k) m = 1 M





where






(32)





or more compactly

X(119903) = (Φ(119903) A(119903)s ) S + V(119903) 119903 = 1 2 (34)


R = (Φ(1) A(1)s )Rs (Φ(2) A(2)s )119867 + 1205902V I (35)









119890(119894) = 10038171003817100381710038171003817R minus R(119894)100381710038171003817100381710038172119865 (38)



120583(119903)119898 = 1119873119903 minus 1119873119903sum119899=2

arg A(119903)final (119899119898)119899 minus 1 (39)










SESTEMUSIC


10minus3

10minus2

10minus1

100

RMSE

0 5 10 15 20minus5

SNR (dB)


SESTEMUSIC


10minus3

10minus2

10minus1

100

RMSE

0 5 10 15 20minus5

SNR (dB)







10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

101

Nor

mal

ized

estim

atio

n er

ror



RMSE = radic119864 119877sum119903=1

119872sum119898=1

(120583(119903)119898 minus 120583(119903)119898 )2 (40)




PMESPRIT

MUSICALS-Tucker

10minus2

10minus1

100

101

102

RMSE

5 15 20 25 3010SNR (dB)







10minus2

10minus1

100

101

RMSE

10 15 20 25 305SNR (dB)


PMESPRIT

MUSICALS-Tucker


10minus2

10minus1

100

RMSE



9 Conclusion




Competing Interests


Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















119890(119894) = 10038171003817100381710038171003817R minus R(119894)100381710038171003817100381710038172119865 (38)



120583(119903)119898 = 1119873119903 minus 1119873119903sum119899=2

arg A(119903)final (119899119898)119899 minus 1 (39)










SESTEMUSIC


10minus3

10minus2

10minus1

100

RMSE

0 5 10 15 20minus5

SNR (dB)


SESTEMUSIC


10minus3

10minus2

10minus1

100

RMSE

0 5 10 15 20minus5

SNR (dB)







10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

101

Nor

mal

ized

estim

atio

n er

ror



RMSE = radic119864 119877sum119903=1

119872sum119898=1

(120583(119903)119898 minus 120583(119903)119898 )2 (40)




PMESPRIT

MUSICALS-Tucker

10minus2

10minus1

100

101

102

RMSE

5 15 20 25 3010SNR (dB)







10minus2

10minus1

100

101

RMSE

10 15 20 25 305SNR (dB)


PMESPRIT

MUSICALS-Tucker


10minus2

10minus1

100

RMSE



9 Conclusion




Competing Interests


Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










SESTEMUSIC


10minus3

10minus2

10minus1

100

RMSE

0 5 10 15 20minus5

SNR (dB)


SESTEMUSIC


10minus3

10minus2

10minus1

100

RMSE

0 5 10 15 20minus5

SNR (dB)







10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

101

Nor

mal

ized

estim

atio

n er

ror



RMSE = radic119864 119877sum119903=1

119872sum119898=1

(120583(119903)119898 minus 120583(119903)119898 )2 (40)




PMESPRIT

MUSICALS-Tucker

10minus2

10minus1

100

101

102

RMSE

5 15 20 25 3010SNR (dB)







10minus2

10minus1

100

101

RMSE

10 15 20 25 305SNR (dB)


PMESPRIT

MUSICALS-Tucker


10minus2

10minus1

100

RMSE



9 Conclusion




Competing Interests


Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










PMESPRIT

MUSICALS-Tucker

10minus2

10minus1

100

101

102

RMSE

5 15 20 25 3010SNR (dB)







10minus2

10minus1

100

101

RMSE

10 15 20 25 305SNR (dB)


PMESPRIT

MUSICALS-Tucker


10minus2

10minus1

100

RMSE



9 Conclusion




Competing Interests


Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Competing Interests


Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









tensor-based methods for blind spatial signature

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