Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 463828 10 pageshttpdxdoiorg1011552013463828

Research ArticleJoint Estimation of 2D-DOA and FrequencyBased on Space-Time Matrix and Conformal Array

Liang-Tian Wan1 Lu-Tao Liu1 Wei-Jian Si1 and Zuo-Xi Tian2

1 Department of Information and Communication Engineering University of Harbin Engineering Harbin 150001 China2 Science and Technology on Underwater Test and Control Laboratory Dalian 116013 China

Correspondence should be addressed to Lu-Tao Liu liulutaomsncom

Received 16 September 2013 Accepted 18 November 2013

Academic Editors H R Karimi X Yang Z Yu and W Zhang

Copyright copy 2013 Liang-Tian Wan et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Each element in the conformal array has a different pattern which leads to the performance deterioration of the conventional highresolution direction-of-arrival (DOA) algorithms In this paper a joint frequency and two-dimension DOA (2D-DOA) estimationalgorithm for conformal array are proposedThedelay correlation function is used to suppress noise Both spatial and time samplingare utilized to construct the spatial-time matrix The frequency and 2D-DOA estimation are accomplished based on parallel factor(PARAFAC) analysis without spectral peak searching and parameter pairing The proposed algorithm needs only four guidingelements with precise positions to estimate frequency and 2D-DOA Other instrumental elements can be arranged flexibly on thesurface of the carrier Simulation results demonstrate the effectiveness of the proposed algorithm

1 Introduction

Conformal array is the array mounted on the surface ofthe conformal carrier [1] The advantage of the conformalarray includes the alleviation of array weight reduction ofaerodynamic drag reduction of radar cross-section (RCS)and the cover of the whole range of azimuth [2] Thusthe conformal array is of great interest in a variety ofapplications such as radar sonar wireless communicationand seismology

Conventional algorithms for direction-of-arrival (DOA)estimation such asmultiple signal classification (MUSIC) [3]and estimation of signal parameters via rotational invariancetechniques (ESPRIT) [4] are not suitable for conformalarray because of the varying curvature of the conformalcarrier Based on array interpolation [5 6] the manifoldseparation technique was applied to arbitrary array structure[7] The above algorithm possesses good performance whenthe pattern of element is isotropic directional However dueto the effect of the curvature of conformal carrier eachelement of the conformal array has different patterns [8]Thus the algorithm proposed in [7] could not be used forconformal array Not all elements can receive signals becauseof the ldquoshadow effectrdquo caused bymetallic carrierThe subarray

divided technique and interpolation transformation wereadopted for DOA estimation of conformal array But theinterpolation accuracy of virtual manifold transformation isnot high [9 10] Recently DOA estimation algorithms forconformal array have been proposed Based on ESPRIT andspecially designed array structure a blind DOA estimationalgorithmwas presented [11]The iterative ESPRIT algorithmwas proposed for joint DOA and polarization parameter esti-mation [12] Nevertheless the two algorithms need parameterpairing and special array design

By combining the techniques of temporal filtering andspatial beam forming the joint DOA and delay were esti-mated [13] Similarly the proposed approach in [14] was anovel twist of parameter estimation and filtering processesin which two 1D frequencies and one 1D spatial MUSIC wereemployed to estimate theDOAs and frequencies respectivelyTwo approximate maximum likelihood (ML) algorithmswere proposed in the spatially correlated noise scenariofor joint frequency and DOA estimation However MUSICand ML algorithms used in [13ndash15] suffer from tremendouscomputational complexity and are only suitable for lineararray or planar array which could not be used for conformalarray The research of joint frequency and DOA estimation israre for the conformal array

2 The Scientific World Journal

As a useful analysis tool of data arrays parallel factor(PARAFAC) analysis model [16 17] is a generalization oflow-rank matrix decomposition to three-way arrays (TWAs)or multiway arrays (MWAs) which was introduced in arraysignal processing to estimate DOA [18] PARAFAC has beenfirst introduced as a data analysis tool in psychometricswhere it has been used in arithmetic complexity exploratorydata analysis and other fields The PARAFAC model wasdeveloped by Sidiropoulos et al [19] Most parameter esti-mation algorithms for conformal array focus on DOA esti-mation The frequency estimation is not researched as faras we known For 2D-DOA estimation of conformal arraythe algorithms based on ESPRIT suffer from parameterpairing problem The algorithms based on MUSIC sufferfrom tremendous computational complexity In this paper anovel joint frequency and 2D-DOA estimation algorithm forconformal array is proposed The spatial domain samplingand time domain sampling are utilized to construct thespatial-time matrix The delay correlation function is usedto suppress noise Based on PARAFAC the joint frequencyand 2D-DOA estimation are accomplished without param-eter pairing Unlike the special array design in [9ndash12] theproposed algorithm needs only four guiding elements otherinstrumental elements of the array can be arranged flexibly

The organization of this paper is structured as followsSection 2 introduces the structure of the conformal array anddescribes the snapshot data model Section 3 contains thecore contributions of this paper including the constructionof spatial-time matrix and the PARAFAC model used forhigh accuracy frequency and 2D-DOA estimation Section 4presents the simulation result Section 5 summarizes ourconclusions

2 The Structure of the Conformal Array andthe Snapshot Data Model

A curve array antenna mounted on an arbitrary shape carrieris shown in Figure 1 Assuming that there are 119872 elements onthe surface of the carrier the coordinate of each element canbe represented as (119909

119898 119910119898 119911119898) 119898 = 1 2 119872 Four guiding

elements p1 p2 p3 and p

4with precise positions are chosen

and p1is assumed to be the reference element p

119898represents

the position vector of the 119898th element on the surface of thecarrier e

119909 e119910 and e

119911represent the unit vector of 119883-axis 119884-

axis and 119885-axis respectively p119898can be expressed as

p119898

= 119909119898e119909+ 119910

119898e119910+ 119911

119898e119911 (1)

Consider119870narrow far field incident signals impinging onthe conformal array which is shown in Figure 1The elevationand azimuth of 119896th incident signal are denoted as (120579

119896 120593119896)

119896 = 1 2 119870 Thus the general form of the array outputis represented as

119909119898

(119905) =

119870

sum

119896=1

119904119896(119905) 119892

119898exp(minus119895

2120587119891119896

119888p119898

∙ u119896) (2)

where 119904119896(119905) stands for the 119896th incident signal and 119892

119898is the

element pattern which is defined in the119898th local coordinate

Z

XXY

Y

Z

O

O

120579

120579m

120593

120593mp4(x4 y4 z4)

p2(x2 y2 z2)

p1(x1 y1 z1)

p3(x3 y3 z3)

pM(xM yM zM)

(xm ym zm)pm

Figure 1 The structure of the conformal array

119891119896is the frequency of the 119896th incident signal 119906

119896is the

direction vector of the incident signal 119904119896(119905)

Consider

u119896= sin (120579

119896) cos (120593

119896) e119909+sin (120579

119896) sin (120593

119896) e119910+cos (120579

119896) e119911

(3)

The time domain sampling is done for the signal of thearray output The number of the tapped delay line is 119871 Thenthe 119897th tapped delay of the signal output of the 119898th elementcan be represented as

119909119898

(119905 minus 119897120591) =

119870

sum

119896=1

119904119896(119905) 119892

119898exp [minus119895120596

119896(120591119898119896

+ 119897120591)]

+ 119899119898

(119905 minus 119897120591) 0 le 119897 le 119871 minus 1

(4)

where 120591119898119896

represents the time delay between the119898th elementp119898and reference element p

1when the mth element receives

the 119896th incident signal 120591 is the time interval of sampling 119899119898is

the additional white Gaussian noise (AWGN)with zeromeanand variance 120590

2 120596119896= 2120587119891

119896 and 120591

119898119896= (p

119898∙ u

119896)119888

The condition that the same incident signal impinges onthe global coordinate and local coordinate is also shown inFigure 1 The coordinate which is constructed by the originalpoint p

1is the global coordinate The coordinate which is

constructed by the original point p119898is the local coordinate

As shown in Figure 1 the elevation and azimuth of theincident signal in global coordinate and local coordinate aredistinct The pattern function 119892

119898(120579119898 120593119898) is defined in the

local coordinate of each elementThus the pattern in the localcoordinate of each element is different Here the parametercomponent transformation from the global coordinate tothe 119898th local coordinate can be accomplished by using thegeneral Euler rotate method [8] More details can be found in[20]

Due to the ldquoshadow effectrdquo caused by the metal not allelements in the conformal array can receive the signal Thusthe subarray divided technique [9 10] is adopted and thewhole array can be divided into several parts The same

The Scientific World Journal 3

parameter estimation mechanism is used for each part Forthe sake of simplicity it is assumed that all the elements canreceive the signal

3 Joint Frequency and 2D-DOA Estimation

31 The Construction of Spatial-Time Matrix The delay cor-relation function between the 119898th element and 119899th elementof the conformal array is constructed as follows

119877119909119898119909119899

= 119864 [119909119898

(119905) 119909lowast

119899(119905 minus 120591)]

=

119870

sum

119896=1

119864 [119904119896(119905) 119904

lowast

119896(119905 minus 120591)]

times exp [minus119895120596119896(120591119898119896

minus 120591119899119896

minus 120591)]

+ 119864 [119899119898

(119905) 119899lowast

119899(119905 minus 120591)]

=

119870

sum

119896=1

119877119904119896119904119896

(120591) 119892119898119892119899exp [minus119895120596

119896(120591119898119896

minus 120591119899119896

)]

times exp (119895120596119896120591) + 119877

119899119898119899119899

(120591) 120591 gt 0

(5)

where R119904119896119904119896

(120591) = 119864[119904119898(119905)119904

lowast

119899(119905 minus 120591)] represents the delay

correlation function of 119904119896(119905) andR

119899119898119899119899

(120591) = 119864[119899119898(119905)119899

lowast

119899(119905minus120591)]

represents the noise cross-correlation function between the119898th element and 119899th element

On condition that the incident signal is coming fromnarrow far field and the bandwidth of the incident signalis B the delay correlation processing of the data that theelements receive takes a long period of timeHenceR

119904119896119904119896

(120591) =

119864[119904119898(119905)119904

lowast

119899(119905minus120591)] = 0 In themore general case when 120591 ≪ 1119861

the Gaussian white noise is not correlated during the timeinterval 120591 At the same time the variety of the signal envelopecan be neglected Then we can suppress the noise withoutdestroying the signal

Consider

R119899119898119899119899

(120591) = 119864 [119899119898

(119905) 119899lowast

119899(119905 minus 120591)]

= 1205902

120575 (120591) 120575 (119898 minus 119899) = 0

(6)

where the delay correlation function is constructed between1199091(119905 minus 120591) and the data that each element receives Based on

(5) we can obtain

R1199091198981199091

(120591) = 119864 [119909119898

(119905) 119909lowast

1(119905 minus 120591)]

=

119870

sum

119896=1

119877119904119896119904119896

(120591) 1198921198981198921exp [minus119895120596

119896(120591119898119896

minus 1205911119896

)]

times exp (119895120596119896120591) 1 le 119898 le 119872

(7)

Similarly the delay correlation functions are constructedbetween 119909

2(119905 minus 120591) 119909

3(119905 minus 120591) 119909

4(119905 minus 120591) and 119909

1(119905) and the data

that each element receives respectively

Consider

R1199091198981199092

(120591) = 119864 [119909119898

(119905) 119909lowast

2(119905 minus 120591)]

=

119870

sum

119896=1

R119904119896119904119896

(120591) 1198921198981198921exp [minus119895120596

119896(120591119898119896

minus 1205912119896

)]

times1198922

1198921

exp (119895120596119896120591) 1 le 119898 le 119872

R1199091198981199093

(120591) = 119864 [119909119898

(119905) 119909lowast

3(119905 minus 120591)]

=

119870

sum

119896=1

R119904119896119904119896

(120591) 1198921198981198921exp [minus119895120596

119896(120591119898119896

minus 1205913119896

)]

times1198923

1198921

exp (119895120596119896120591) 1 le 119898 le 119872

R1199091198981199094

(120591) = 119864 [119909119898

(119905) 119909lowast

4(119905 minus 120591)]

=

119870

sum

119896=1

R119904119896119904119896

(120591) 1198921198981198921exp [minus119895120596

119896(120591119898119896

minus 1205914119896

)]

times1198924

1198921

exp (119895120596119896120591) 1 le 119898 le 119872

R1199091198981199091

(0) = 119864 [119909119898

(119905) 119909lowast

1(119905)]

=

119870

sum

119896=1

R119904119896119904119896

(0) 1198921198981198921exp [minus119895120596

119896(120591119898119896

minus 1205911119896

)]

+ 1205902I 1 le 119898 le 119872

(8)

The spatial-time matrix is constructed as follows

R119878(120591) = [R

11990411199041

(120591) R11990421199042

(120591) R119904119870119904119870

(120591)]119879

(9)

R119891(120591) = [R

11990911199091

(120591) R11990921199091

(120591) R1199091198981199091

(120591)]119879

(10)

R1(120591) = [R

11990911199092

(120591) R11990921199092

(120591) R1199091198981199092

(120591)]119879

(11)

R2(120591) = [R

11990911199093

(120591) R11990921199093

(120591) R1199091198981199093

(120591)]119879

(12)

R3(120591) = [R

11990911199094

(120591) R11990921199094

(120591) R1199091198981199094

(120591)]119879

(13)

R (0) = [R11990911199091

(0) R11990921199091

(0) R1199091198981199091

(0)]119879

(14)

A = [a1(1205961) a

2(1205962) a

119870(120596119870)] (15)

a119896(120596119896) = [119892

1e(minus119895(120596119896119888)p1∙u119896) 119892

2e(minus119895(120596119896119888)p2∙u119896)

119892119872e(minus119895(120596119896119888)p119872∙u119896)]

119879

(16)

4 The Scientific World Journal

The matrix A is the manifold matrix and a is the steeringmatrix Then (9)ndash(14) can be written as

R119891(120591) = AΦ

119891R119878(120591) (17)

R1(120591) = AΦ

1Φ119891R119878(120591) (18)

R2(120591) = AΦ

2Φ119891R119878(120591) (19)

R3(120591) = AΦ

3Φ119891R119878(120591) (20)

R (0) = AR119878(0) (21)

where

Φ119891

= diag [e(1198951205961120591) e(1198951205962120591) e(119895120596119870120591)]119879

(22)

Φ1= diag [

1198922

1198921

e(119895(1205961119888)(p2minusp1)∙u1)1198922

1198921

e(119895(1205962119888)(p2minusp1)∙u2)

1198922

1198921

e(119895(120596119870119888)(p2minusp1)∙u119870)]119879

(23)

Φ2= diag [

1198923

1198921

e(119895(1205961119888)(p3minusp1)∙u1)1198923

1198921

e(119895(1205962119888)(p3minusp1)∙u2)

1198923

1198921

e(119895(120596119870119888)(p3minusp1)∙u119870)]119879

(24)

Φ3= diag [

1198924

1198921

e(119895(1205961119888)(p4minusp1)∙u1)1198924

1198921

e(119895(1205962119888)(p4minusp1)∙u2)

1198924

1198921

e(119895(120596119870119888)(p4minusp1)∙u119870)]119879

(25)

120578119891119896

= 120596119896120591 = 2120587119891

119896120591 (26)

1205781119896

=120596119896

119888(p2minus p

1) ∙ 119906

119894 (27)

1205782119896

=120596119896

119888(p3minus p

1) ∙ 119906

119894 (28)

1205783119896

=120596119896

119888(p4minus p

1) ∙ 119906

119894 (29)

The incident signal is narrowband thusR119904119896119904119896

(119899120591) = R119904119896119904119896

((119899minus

1)120591) Equation (21) can be written in another form as

R (120591) = AR119878(120591) (30)

Equations (17)ndash(21) possess the analogous form R119891(120591)

R1(120591) R

2(120591) R

3(120591) and R(0) are sampled at a time delay 120591

119904

for119873 (119873 ge 119870) times 120591119904= 119879

119904 2119879

119904 119873119879

119904 and 119879

119904is the time

interval of sampling which should be less than the reciprocalof the highest frequency Thus the aliasing effect would not

happen The pseudosnapshot data matrix can be constructedby using (17)ndash(21) as follows

R119891

= [R (119879119904) R (2119879

119904) R (119873119879

119904)]

R1= [R

1(119879119904) R

1(2119879

119904) R

1(119873119879

119904)]

R2= [R

2(119879119904) R

2(2119879

119904) R

2(119873119879

119904)]

R3= [R

3(119879119904) R

3(2119879

119904) R

3(119873119879

119904)]

R = [R119891(119879119904) R

119891(2119879

119904) R

119891(119873119879

119904)]

(31)

Equation (31) can be written as

R119891

= AΦ119891R119878

R1= AΦ

1Φ119891R119878

R2= AΦ

2Φ119891R119878

R3= AΦ

3Φ119891R119878

R = AR119878

(32)

where

R119878= [R

119878(119879119904) R

119878(2119879

119904) R

119878(119873119879

119904)] (33)

32 PARAFAC The element of a tensor X isin C119862times119863times119864 is 119909119894119895119896

and its 119875-component PARAFAC decomposition is given by

119909119888119889119890

=

119872

sum

119901=1

119904119888119901

119905119889119901

119906119890119901

(34)

where 119888 = 1 119862 119889 = 1 119863 and 119890 = 1 119864 119904119888119901 119905119889119901

and 119906

119890119901stand for the elements of S isin C119862times119875 T isin C119863times119875

and U isin C119864times119875 respectively The typical elements of theX119888

isin C119863times119864 X119889

isin C119864times119862 and X119890

isin C119862times119863are 119883119888(119889 119890) =

119883119889(119888 119890) = 119883

119890(119888 119889) = 119909

119888119889119890 respectivelyThe three way array

(TWA) X is ldquoslicedrdquo in three different directions

X119888= TΛ

119888(S)U119879 119888 = 1 119862

X119889= UΛ

119889(T) S119879 119889 = 1 119863

X119890= SΛ

119890(U)T119879 119890 = 1 119864

(35)

For a given matrix S Λ119888(S) means a diagonal matrix with

the diagonals given by the 119888th row of the matrix The symbolldquo⊙rdquo is used to denote the Khatri-Rao (KR) product As anexample the KR product of two matrices S isin C119862times119875 andT isin C119863times119875 of an identical number of columns is given by

S ⊙ T = [s1otimes t

1 s

119875otimes t

119875] isin C

119862119863times119875

(36)

The Scientific World Journal 5

where ldquootimesrdquo represents the Kronecker product Then thedefinition of Kronecker product of two vectors s isin C119862 andt isin C119863 is given by

s otimes t =

[[[[

[

1199041t

1199042t

119904119862t

]]]]

]

(37)

Based on the definition of KR product X119862119863times119864 can beexpressed as

X119862119863times119864 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

X119888=1

X119888=2

X119888=119862

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

TΛ1(S)

TΛ2(S)

TΛ119862(S)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

U119879 = (S ⊙ T)U119879 (38)

Similarly the matrix X119863119864times119862 and X119864119862times119863 can be expressed as

X119863119864times119862 = (T ⊙ U) S119879

X119864119862times119863 = (U ⊙ S)T119879(39)

Definition 1 (Kruskal rank [18]) The Kruskal rank or k-rankof a given matrix S isin C119862times119875 denoted by krank (S) is saidto be equal to 119903 when every collection of 119903 columns of (S)is linearly independent but there exists a collection of 119903 + 1

linearly dependent columns However the rank of S is said tobe themaximal number of linearly independent columnsThedefinition of rank(S) is more relaxed than that of krank(S)that is krank(S) le rank(S)

Theorem2 (Kruskal theorem [19]) Given a PARAFACmodelX isin C119862times119863times119864 then the decomposition of this PARAFAC modelis unique to permutation and scaling when

119896S + 119896T + 119896U ge 2119875 + 2 (40)

The matrix X is constructed by S T and U as

S = SΠΔ1 T = TΠΔ

2 U = UΠΔ

3 (41)

whereΠ stands for the permutation matrixΔ1Δ2 andΔ

3are

diagonal scaling matrices satisfying

Δ1Δ2Δ3= I (42)

where I is a unit matrix

The119872times119872times5TWAof the conformal array is constructedbased on PARAFAC model

R =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)

R ( 2)

R ( 3)

R ( 4)

R ( 5)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R119891

R1

R2

R3

R

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

AΦ119891R119878

AΦ1Φ119891R119878

AΦ2Φ119891R119878

AΦ3Φ119891R119878

AR119878

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ Q (43)

where Q represents the observed noise in practice In orderto make the decomposition of the PARAFAC unique thefollowing assumption must be satisfied

a119894(120596119894) = a

119895(120596119895) (44)

In other words for two incident signals 119904119894(119905) and 119904

119895(119905) with

different DOAs the following condition must be satisfied

119892119900119890eminus1198952120587119891119894((p119900119890∙u119894)119888) = 119892

119900119890eminus1198952120587119891119895((p119900119890∙u119895)119888) (45)

The k-rank of matrix D A and R119878are 119896

119863= min(5 119870) 119896

119860=

119870 and 119896119877119878

= 119870 respectively Thus when 119870 ge 2 the k-rankdecomposition of the PARAFAC is unique

Based on the definition of KRproduct (43) can bewrittenin three forms as follows

R = (D ⊙ A)R119878+ Q (46)

R119883

= (R119879119878⊙ D)A119879 + Q

119883 (47)

R119884

= (A ⊙ R119879119878)D119879 + Q

119884 (48)

where

D =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Λminus1

(Φ119891)

Λminus1

(Φ1Φ119891)

Λminus1

(Φ2Φ119891)

Λminus1

(Φ3Φ119891)

Λminus1

(1)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(49)

where Λminus1

(Φ119891) means the row vector constructed by the

diagonal elements of the diagonal matrixΦ119891

The matrices D A and R119878can be solved by the trilinear

alternating least squares (TALS) regression algorithm Dueto the existence of the observed noise (46) could be trans-formed as a least square problem as follows

minDAR

119878

10038171003817100381710038171003817R minus (D ⊙ A)R

119878

10038171003817100381710038171003817

2

119865

(50)

Then the least square estimator of R119878can be expressed as

R119878= argmin

R119878

10038171003817100381710038171003817R minus (D ⊙ A)R

119878

10038171003817100381710038171003817

2

119865

(51)

Similarly the solutions ofD and A can be expressed as

A119879 = (R119879119878⊙ D)

dagger

R119883

D119879 = (A ⊙ R119879119878)dagger

R119884

(52)

We can update the matrices D A and R119878alternately until

the algorithm converges the estimators of the three matricescan be acquired In the following experiments the COMFACalgorithm (the TALS regression algorithm is realized inpractice) is used to fit the 119898 times 119898 times 5 TWAThe initializationand fitting of the COMFAC is done in compressed spaceTheTucker 3 three-way model is used in data compression [21]

Now the ALS algorithm steps can be summarized asfollows

6 The Scientific World Journal

(1) Initialize A(0) isin C119872times119870 andD(0) isin C5times119870

(2) Initialize 120576 gt 0 and 119896 = 0

(3) If 120588(119896+1)minus120588(119896)

120588(119896)

gt 120576 calculatematricesAR119878 and

D by (51)-(52) Update just one matrix at each timethen 119896 rarr 119896 + 1

(4) Else 120588(119896+1)minus120588(119896)

120588(119896)

lt 120576 the iteration is terminated

In this paper 120588 stands for A R119878 andD respectively

33 The Joint Parameter Estimation The matrix D can beestimated by TALS regression algorithm then the estimatorof 120596

119891119896is given by

120596119896119894

= angle10038161003816100381610038161003816100381610038161003816

D1119896

D5119896

10038161003816100381610038161003816100381610038161003816

(53)

Because 1199031and 119903

2are real numbers D

2119896D

1119896is squared

to avoid the ambiguity caused by the positive and negativevalues of 119903

1and 119903

2 Then

1205961119896

= minus1

2angle([

1198922(120579119896 120593119896)

1198921(120579119896 120593119896)exp (minus119895120596

1119896)]

2

)

= minus1

2angle (exp (minus1198952120596

1119896))

= minus1

2angle([

D2119896

D1119896

]

2

)

(54)

Similarly

1205962119896

= minus1

2angle([

D3119896

D1119896

]

2

) (55)

1205963119896

= minus1

2angle([

D4119896

D1119896

]

2

) (56)

Taking (26) into (53) the frequency 119891119896of the 119896th incident

signal is expressed as

119891119896=

1

2120587120591angle

10038161003816100381610038161003816100381610038161003816

D1119896

D5119896

10038161003816100381610038161003816100381610038161003816

(57)

Assuming that 1205741119896

= sin(120579119896) cos(120593

119896) 120574

2119896= sin(120579

119896) sin(120593

119896)

and 1205743119896

= cos(120579119896) and solving (27) (28) (29) (55) (56) and

(57) simultaneously we can obtain the equation as follows

119888

2120587119891119896

[[[[[[[

[

1205961119896

1198891

1205962119896

1198892

1205963119896

1198893

]]]]]]]

]

= [

[

1199092minus 119909

11199102minus 119910

11199112minus 119911

1

1199093minus 119909

11199103minus 119910

11199113minus 119911

1

1199094minus 119909

11199104minus 119910

11199104minus 119910

1

]

]

[

[

1205741119896

1205742119896

1205743119896

]

]

(58)

The solution of the equation is represented as

[

[

1205741119896

1205742119896

1205743119896

]

]

=119888

2120587119891119894

[

[

1199092minus 119909

11199102minus 119910

11199112minus 119911

1

1199093minus 119909

11199103minus 119910

11199113minus 119911

1

1199094minus 119909

11199104minus 119910

11199104minus 119910

1

]

]

minus1[[[[[[[

[

1205961119896

1198891

1205962119896

1198892

1205963119896

1198893

]]]]]]]

]

(59)

There are two approaches to solve 120579119896and 120593

119896 The first

approach is

120579119896= arccos120574

3119896

120593119896= arcsin(

1205741119896

cos (120579119896)) or 120593

119896= arcsin(

1205742119896

sin (120579119896))

(60)

The second approach is

120593119896= arctan(

1205742119896

1205741119896

)

120579119896= arccos(

1205741119896

cos (120593119896)) or 120579

119896= arcsin(

1205742119896

sin (120593119896))

(61)

In this paper the first approach is employedIn order to ensure the uniqueness of the estimated

parameters the following condition must be satisfied

1003817100381710038171003817p2 minus p1

1003817100381710038171003817 le119888

(2max (119891119894))

=min (120582

119896)

2

1003817100381710038171003817p3 minus p1

1003817100381710038171003817 le119888

(2max (119891119894))

=min (120582

119896)

2

1003817100381710038171003817p4 minus p1

1003817100381710038171003817 le119888

(2max (119891119894))

=min (120582

119896)

2

(62)

According to Theorem 2 the matrices D A and R119878have the

same permutation matrix that is the 119896th column of matrixD corresponds to the 119896th column of the manifold matrix A

Based on the PARAFAC model and TALS regressionalgorithm the joint frequency and 2D-DOA estimation canbe accomplished The algorithm steps are summarized asfollows

(1) choose four guiding elements and obtain the accuratepositions of the elements

(2) calculate the delay correlation function between thefour elements mentioned above and the whole ele-ments of the conformal array

(3) calculate the matrices R119891(120591) R

1(120591) R

2(120591) R

3(120591) and

R(0) respectively according to (5)(4) calculate the pseudosnapshot data matrix based on

(17)ndash(21) R119891(120591) R

1(120591) R

2(120591) R

3(120591) and R(0) are

sampled at time delay 120591119904 respectively

(5) construct the PARAFAC model based on (43)(6) estimate the matrix D using TALS regression algo-

rithm

The Scientific World Journal 7

(7) obtain12059611989111989612059611198961205962119896 and120596

3119896by using the estimator of

the matrixD and calculating (53)ndash(56)(8) obtain the joint frequency and 2D-DOA estimation

by calculate (57)ndash(59) or (57) (60) and (61)

4 Simulation Results

This section focuses on several sets of simulation results todemonstrate the performance of the proposed algorithmIn all simulation examples 200 Monte Carlo trials areconsidered and the root mean square error (RMSE) is usedas our performance measure which is defined as

RMSE = radic1

200

200

sum

120578=1

[(120579119896120578

minus 120579)2

+ (120593119896120578

minus 120593)2

] (63)

where 120579119894119905and 120593

119894119905are the elevation and azimuth estimators of

the 119896th incident signal in the 120578th trialThe successful rate is defined as the proportion of the

number of the successful experiments to the total numberof the experiments For frequency estimation a successfulexperiment is defined as the experiment with estimationerror of less than 2MHz for DOA estimation a successfulexperiment is defined as the experimentwith estimation errorof less than 2 degree

In order to simplify the transformation from the globalcoordinate to the local coordinate only the cylinder withsingle curvature is considered in the simulations The arraystructure is shown in Figure 2 is only a subarray which onlycovers 120

∘ Then three subarrays can estimate the wholespace The highest frequency of the incident signal is 119891max =

2GHz 120582min = 119888119891max The radius of the cylinder is 5120582min theadjacent element spacing in the same intersecting surface ofthe cylinder is 120582min4 and the adjacent intersecting surfacespacing is 120582min4 120582min stands for the shortest wavelength ofthe incident signal The position of the reference element is1199011= (0 5120582 0)The element pattern is defined in the local coordinate

of each element thus the transformation from the globalcoordinate to the local coordinate of elevation 120579

119896and azimuth

120593119896of the incident signal must be accomplished For the

cylindrical conformal array the corresponding Euler rotationangles are

119863 = minusΘ 119864 = minus1205872 119865 = 0 (64)

respectively 119863 represents the rotation angle based on theright-hand rule in the first rotation and the 119885-axis is therotation axis 119864 represents the rotation angle in the secondrotation and the 119884-axis is the rotation axis and 119865 representsthe rotation angle in the third rotation and the 119883-axis is therotation axis Θ represents the angle between the positionvector of the element and the positive direction of 119883-axis

The unit vector of the 119896th incident signal (120579119896 120593119896) in the

global coordinate can be expressed as

119909 = sin (120579119896) cos (120593

119896) 119910 = sin (120579

119896) sin (120593

119896)

119911 = cos (120579119896)

(65)

Z

X

Y

O

pm

p3

p4

p2p1

Figure 2 The structure of the cylindrical conformal array

Based on the Euler rotation matrix the unit vector inglobal coordinate can be transformed into the local coordi-nate of the 119898th element

Consider

[119909119898

119910119898

119911119898]119879

= 119877 (119863119898 119864119898 119865119898) [119909 119910 119911]

119879

(66)

where

119877 (119863119898 119864119898 119865119898) = [

[

cos119865 sin119865 0

minus sin119865 cos119865 0

0 0 1

]

]

times [

[

cos119864 0 minus sin119864

0 1 0

sin119864 0 cos119864]

]

times [

[

cos119863 sin119863 0

minus sin119863 cos119863 0

0 0 1

]

]

(67)

Then the elevation 120579 and azimuth 120593 of the 119896th incident signalin the local coordinate of the119898th element can be representedas

120593119898119896

= arctan (119910119898119909119898) 120579

119898119896= arccos (119911

119898) (68)

Thus the element pattern transformation from the globalcoordinate to the local coordinate is completed

First the curves of RMSE of frequency and angle and thesuccessful probability of angle in different SNR are plottedin Figure 3 using the proposed algorithm The number ofsnapshots is 200 the number of pseudosnapshots is 100SNR isin [5 30] and the noise is AWGN which is independentof incident signals The sampling frequency is 119891

119904= 1119879

119904=

5GHzThe elevation 120579 azimuth120593 and frequency of these twonarrow incident signals are (100∘ 60∘ 1 GHz) and (95∘ 50∘2 GHz) respectively which are independent of each other

8 The Scientific World Journal

5 10 15 20 25 300

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Succ

ess p

roba

bilit

y (

)

Frequency 1Frequency 2

(a)

5 10 15 20 25 300

05

1

15

2

25

3

35

4

RMSE

(MH

z)

SNR (dB)

Frequency 1Frequency 2

(b)

5 10 15 20 25 3040

50

60

70

80

90

100

SNR (dB)

Succ

ess p

roba

bilit

y (

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(c)

5 10 15 20 25 300

002

004

006

008

01

012

014

016

SNR (dB)

RMSE

(deg

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(d)

Figure 3The RMSE and successful probability with different SNR (a)The successful probability of frequencies (b) the RMSE of frequency(c) the successful probability of angles and (d) the RMSE of angles

The number of elements are 10 including 4 guiding elementsand other 6 instrumental elementsThe element pattern usedin simulation is the lowest order circular patch model [8 22]

119892120579(120579 120593) = 119869

2(120587119889

120582sin 120579) minus 119869

0(120587119889

120582sin 120579)

times (cos120593 minus 119895 sin120593) 0 le 120579 le120587

2

119892120593(120579 120593) = 119869

2(120587119889

120582sin 120579) + 119869

0(120587119889

120582sin 120579)

times cos 120579 (sin120593 minus 119895 cos120593) 0 le 120579 le120587

2

119892120579(120579 120593) = 119892

120593(120579 120593) = 0 otherwise

(69)

where 1198690and 119869

2are the zeroth- and second-order Bessel func-

tions of the first kind The element pattern transformation isachieved with the method proposed above

Figure 3 shows that the RMSE of frequency and angledecreases while the SNR increases meanwhile the successprobability of frequency and angle increase as the SNRincreases The estimated performance of the two frequenciesare very close which can be seen in Figures 3(a) and 3(b)As shown in Figure 3(a) the successful probability is largerthan 80 when the SNR is larger than 15 dB The RMSEof frequency is less than 2MHz when SNR is larger than10 dB as shown in Figure 3(b) In Figure 3(c) the successfulprobability of azimuths is slightly lower than that of theelevations The success probability of the angle is almost

The Scientific World Journal 9

100 200 300 400 500 600 700 80020

30

40

50

60

70

80

90

100

Snapshot number

Succ

ess p

roba

bilit

y (

)

Frequency 1Frequency 2

(a)

100 200 300 400 500 600 700 80008

1

12

14

16

18

2

Snapshot number

RMSE

(MH

z)

Frequency 1Frequency 2

(b)

100 200 300 400 500 600 700 80095

955

96

965

97

975

98

985

99

995

100

Snapshot number (dB)

Succ

ess p

roba

bilit

y (

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(c)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

100 200 300 400 500 600 700 800Snapshot number (dB)

002

004

006

008

01

012

014

016RM

SE (d

eg)

(d)

Figure 4 The RMSE and successful probability with different snapshot number (a) The successful probability of frequencies (b) the RMSEof frequency (c) the successful probability of angles and (d) the RMSE of angles

100 when the SNR reaches 15 dB Figure 3(d) shows thatthe RMSE is smaller than 016

∘ when SNR is larger than5 dB The RMSE of azimuth is larger than that of elevationThe estimation results of frequency and elevation are appliedto estimate azimuth The phenomenon in Figure 3(d) canbe interpreted as the existence of the estimated error offrequency and elevation Next the RMSE of frequency andangle as well as the success probability of angle in differentsnapshot number are depicted in Figure 4 SNR = 10 dBother simulation conditions are identical with the previousexperiment In Figure 4 the RMSE of frequency and angledecreases as the snapshot number increases and the successprobability of frequency and angle rises as the snapshotnumber increases As shown in Figure 4(a) the successfulprobability is larger than 80 when the number of snapshot

is larger than 500 Figure 4(b) shows that the RMSE offrequency 1 is slightly larger than that of frequency 2 Whenthe snapshot number is larger than 500 the RMSE dropsbelow 1MHz meaning that the proposed algorithm achievedthree orders of magnitude reduction Figures 4(c) and 4(d)show that the success probability of azimuth is lower thanthat of the elevation and the RMSE of azimuth are largerthan that of elevationThe reasonwhy this happens is becausethe estimation error of frequency and elevation affects theestimation performance of azimuth

5 Conclusion

Due to the varying curvature of the surface of the conformalcarrier the pattern of each element of the conformal array

10 The Scientific World Journal

is different Thus the conventional algorithms could notbe used for conformal array In this paper a novel jointfrequency and 2D-DOA estimation algorithm with highaccuracy are proposed Both spatial and time samplingare utilized to construct the spatial-time matrix The delaycorrelation function is used to suppress noiseThe PARAFACmodel is used for parameter estimation without parameterpairing Only four elements are needed and the positionsof these elements should be known accurately Other instru-mental elements can be flexibly arranged on the surface ofthe conformal carrier The algorithm proposed in this papercan be extended to estimate 2D-DOA straightly with littlemodification The simulation results verify the effectivenessof the proposed algorithm It can be expected that theproposed algorithm would have an application prospect inthe parameter estimation of conformal array In the futurework we will focus on the application of the proposedalgorithm [23ndash26]

Acknowledgments

This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)

References

[1] L Josefsson and P Persson Conformal Array Antenna Theoryand Design IEEE Press Piscataway NJ USA 2006

[2] W T Li X W Shi Y Q Hei S F Liu and J Zhu ldquoAhybrid optimization algorithmand its application for conformalarray pattern synthesisrdquo IEEE Transactions on Antennas andPropagation vol 58 no 10 pp 3401ndash3406 2010

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

[4] 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

[5] P Hyber M Jansson and B Ottersten ldquoArray interpolation andbias reductionrdquo IEEE Transaction on Signal Processing vol 52no 10 pp 2711ndash2720 2004

[6] P Hyberg M Jansson and B Ottersten ldquoArray interpolationand DOAMSE reductionrdquo IEEE Transactions on Signal Process-ing vol 53 no 12 pp 4464ndash4471 2005

[7] F Belloni A Richter and V Koivunen ldquoDOA estimationvia manifold separation for arbitrary array structuresrdquo IEEETransactions on Signal Processing vol 55 no 10 pp 4800ndash48102007

[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008

[9] P Yang F Yang and Z Nie ldquoDOA estimation using MUSICalgorithm on a cylinderical conformal arrayrdquo in Proceedingsof the IEEE Antennas and Propagation Society InternationalSymposium pp 5299ndash5302 June 2007

[10] P Yang F Yang and Z P Nie ldquoDOA estimation with sub-arraydivided technique and interporlated esprit algorithmon a cylin-drical conformal array antennardquo Progress in ElectromagneticsResearch vol 103 pp 201ndash216 2010

[11] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011

[12] L Zou J Lasenby and Z He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012

[13] Y-Y Wang J-T Chen andW-H Fang ldquoTST-MUSIC for jointDOA-delay estimationrdquo IEEE Transactions on Signal Processingvol 49 no 4 pp 721ndash729 2001

[14] J-D Lin W-H Fang Y-Y Wang and J-T Chen ldquoFSF MUSICfor joint DOA and frequency estimation and its performanceanalysisrdquo IEEE Transactions on Signal Processing vol 54 no 12pp 4529ndash4542 2006

[15] M Djeddou A Belouchrani and S Aouada ldquoMaximum likeli-hood angle-frequency estimation in partially known correlatednoise for low-elevation targetsrdquo IEEE Transactions on SignalProcessing vol 53 no 8 pp 3057ndash3064 2005

[16] R B Cattell ldquolsquoParallel proportional profilesrsquo and other prin-ciples for determining the choice of factors by rotationrdquo Psy-chometrika vol 9 no 4 pp 267ndash283 1944

[17] J D Carroll and J-J Chang ldquoAnalysis of individual differencesin multidimensional scaling via an n-way generalization ofldquoEckart-Youngrdquo decompositionrdquo Psychometrika vol 35 no 3pp 283ndash319 1970

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

[19] 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

[20] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 77ndash83 1999

[21] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fittingof the Complex PARAFAC Model in 3-Drdquo 1998 httpwwwtelecomtucgrsimnikos

[22] J R James P S Hall and C WoodMicrostrip Antenna Theoryand Design Peter Peregrinus New York NY USA 1981

[23] S Yin S Ding and H Luo ldquoReal-time implementation of faulttolerant control system with performance optimizationrdquo IEEETransactions on Industrial Electronics vol 61 no 5 pp 2402ndash2411 2014

[24] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[25] S Yin S Ding A Haghani andH Hao ldquoData-drivenmonitor-ing for stochastic systems and its application on batch processrdquoInternational Journal of Systems Science vol 44 no 7 pp 1366ndash1376 2013

[26] S Yin X Yang and H Karimi ldquoData-driven adaptive observerfor fault diagnosisrdquoMathematical Problems in Engineering vol2012 Article ID 832836 21 pages 2012

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DistributedSensor Networks

International Journal of

2 The Scientific World Journal

As a useful analysis tool of data arrays parallel factor(PARAFAC) analysis model [16 17] is a generalization oflow-rank matrix decomposition to three-way arrays (TWAs)or multiway arrays (MWAs) which was introduced in arraysignal processing to estimate DOA [18] PARAFAC has beenfirst introduced as a data analysis tool in psychometricswhere it has been used in arithmetic complexity exploratorydata analysis and other fields The PARAFAC model wasdeveloped by Sidiropoulos et al [19] Most parameter esti-mation algorithms for conformal array focus on DOA esti-mation The frequency estimation is not researched as faras we known For 2D-DOA estimation of conformal arraythe algorithms based on ESPRIT suffer from parameterpairing problem The algorithms based on MUSIC sufferfrom tremendous computational complexity In this paper anovel joint frequency and 2D-DOA estimation algorithm forconformal array is proposed The spatial domain samplingand time domain sampling are utilized to construct thespatial-time matrix The delay correlation function is usedto suppress noise Based on PARAFAC the joint frequencyand 2D-DOA estimation are accomplished without param-eter pairing Unlike the special array design in [9ndash12] theproposed algorithm needs only four guiding elements otherinstrumental elements of the array can be arranged flexibly

The organization of this paper is structured as followsSection 2 introduces the structure of the conformal array anddescribes the snapshot data model Section 3 contains thecore contributions of this paper including the constructionof spatial-time matrix and the PARAFAC model used forhigh accuracy frequency and 2D-DOA estimation Section 4presents the simulation result Section 5 summarizes ourconclusions

2 The Structure of the Conformal Array andthe Snapshot Data Model

A curve array antenna mounted on an arbitrary shape carrieris shown in Figure 1 Assuming that there are 119872 elements onthe surface of the carrier the coordinate of each element canbe represented as (119909

119898 119910119898 119911119898) 119898 = 1 2 119872 Four guiding

elements p1 p2 p3 and p

4with precise positions are chosen

and p1is assumed to be the reference element p

119898represents

the position vector of the 119898th element on the surface of thecarrier e

119909 e119910 and e

119911represent the unit vector of 119883-axis 119884-

axis and 119885-axis respectively p119898can be expressed as

p119898

= 119909119898e119909+ 119910

119898e119910+ 119911

119898e119911 (1)

Consider119870narrow far field incident signals impinging onthe conformal array which is shown in Figure 1The elevationand azimuth of 119896th incident signal are denoted as (120579

119896 120593119896)

119896 = 1 2 119870 Thus the general form of the array outputis represented as

119909119898

(119905) =

119870

sum

119896=1

119904119896(119905) 119892

119898exp(minus119895

2120587119891119896

119888p119898

∙ u119896) (2)

where 119904119896(119905) stands for the 119896th incident signal and 119892

119898is the

element pattern which is defined in the119898th local coordinate

Z

XXY

Y

Z

O

O

120579

120579m

120593

120593mp4(x4 y4 z4)

p2(x2 y2 z2)

p1(x1 y1 z1)

p3(x3 y3 z3)

pM(xM yM zM)

(xm ym zm)pm

Figure 1 The structure of the conformal array

119891119896is the frequency of the 119896th incident signal 119906

119896is the

direction vector of the incident signal 119904119896(119905)

Consider

u119896= sin (120579

119896) cos (120593

119896) e119909+sin (120579

119896) sin (120593

119896) e119910+cos (120579

119896) e119911

(3)

The time domain sampling is done for the signal of thearray output The number of the tapped delay line is 119871 Thenthe 119897th tapped delay of the signal output of the 119898th elementcan be represented as

119909119898

(119905 minus 119897120591) =

119870

sum

119896=1

119904119896(119905) 119892

119898exp [minus119895120596

119896(120591119898119896

+ 119897120591)]

+ 119899119898

(119905 minus 119897120591) 0 le 119897 le 119871 minus 1

(4)

where 120591119898119896

represents the time delay between the119898th elementp119898and reference element p

1when the mth element receives

the 119896th incident signal 120591 is the time interval of sampling 119899119898is

the additional white Gaussian noise (AWGN)with zeromeanand variance 120590

2 120596119896= 2120587119891

119896 and 120591

119898119896= (p

119898∙ u

119896)119888

The condition that the same incident signal impinges onthe global coordinate and local coordinate is also shown inFigure 1 The coordinate which is constructed by the originalpoint p

1is the global coordinate The coordinate which is

constructed by the original point p119898is the local coordinate

As shown in Figure 1 the elevation and azimuth of theincident signal in global coordinate and local coordinate aredistinct The pattern function 119892

119898(120579119898 120593119898) is defined in the

local coordinate of each elementThus the pattern in the localcoordinate of each element is different Here the parametercomponent transformation from the global coordinate tothe 119898th local coordinate can be accomplished by using thegeneral Euler rotate method [8] More details can be found in[20]

Due to the ldquoshadow effectrdquo caused by the metal not allelements in the conformal array can receive the signal Thusthe subarray divided technique [9 10] is adopted and thewhole array can be divided into several parts The same

The Scientific World Journal 3

parameter estimation mechanism is used for each part Forthe sake of simplicity it is assumed that all the elements canreceive the signal

3 Joint Frequency and 2D-DOA Estimation

31 The Construction of Spatial-Time Matrix The delay cor-relation function between the 119898th element and 119899th elementof the conformal array is constructed as follows

119877119909119898119909119899

= 119864 [119909119898

(119905) 119909lowast

119899(119905 minus 120591)]

=

119870

sum

119896=1

119864 [119904119896(119905) 119904

lowast

119896(119905 minus 120591)]

times exp [minus119895120596119896(120591119898119896

minus 120591119899119896

minus 120591)]

+ 119864 [119899119898

(119905) 119899lowast

119899(119905 minus 120591)]

=

119870

sum

119896=1

119877119904119896119904119896

(120591) 119892119898119892119899exp [minus119895120596

119896(120591119898119896

minus 120591119899119896

)]

times exp (119895120596119896120591) + 119877

119899119898119899119899

(120591) 120591 gt 0

(5)

where R119904119896119904119896

(120591) = 119864[119904119898(119905)119904

lowast

119899(119905 minus 120591)] represents the delay

correlation function of 119904119896(119905) andR

119899119898119899119899

(120591) = 119864[119899119898(119905)119899

lowast

119899(119905minus120591)]

represents the noise cross-correlation function between the119898th element and 119899th element

On condition that the incident signal is coming fromnarrow far field and the bandwidth of the incident signalis B the delay correlation processing of the data that theelements receive takes a long period of timeHenceR

119904119896119904119896

(120591) =

119864[119904119898(119905)119904

lowast

119899(119905minus120591)] = 0 In themore general case when 120591 ≪ 1119861

the Gaussian white noise is not correlated during the timeinterval 120591 At the same time the variety of the signal envelopecan be neglected Then we can suppress the noise withoutdestroying the signal

Consider

R119899119898119899119899

(120591) = 119864 [119899119898

(119905) 119899lowast

119899(119905 minus 120591)]

= 1205902

120575 (120591) 120575 (119898 minus 119899) = 0

(6)

where the delay correlation function is constructed between1199091(119905 minus 120591) and the data that each element receives Based on

(5) we can obtain

R1199091198981199091

(120591) = 119864 [119909119898

(119905) 119909lowast

1(119905 minus 120591)]

=

119870

sum

119896=1

119877119904119896119904119896

(120591) 1198921198981198921exp [minus119895120596

119896(120591119898119896

minus 1205911119896

)]

times exp (119895120596119896120591) 1 le 119898 le 119872

(7)

Similarly the delay correlation functions are constructedbetween 119909

2(119905 minus 120591) 119909

3(119905 minus 120591) 119909

4(119905 minus 120591) and 119909

1(119905) and the data

that each element receives respectively

Consider

R1199091198981199092

(120591) = 119864 [119909119898

(119905) 119909lowast

2(119905 minus 120591)]

=

119870

sum

119896=1

R119904119896119904119896

(120591) 1198921198981198921exp [minus119895120596

119896(120591119898119896

minus 1205912119896

)]

times1198922

1198921

exp (119895120596119896120591) 1 le 119898 le 119872

R1199091198981199093

(120591) = 119864 [119909119898

(119905) 119909lowast

3(119905 minus 120591)]

=

119870

sum

119896=1

R119904119896119904119896

(120591) 1198921198981198921exp [minus119895120596

119896(120591119898119896

minus 1205913119896

)]

times1198923

1198921

exp (119895120596119896120591) 1 le 119898 le 119872

R1199091198981199094

(120591) = 119864 [119909119898

(119905) 119909lowast

4(119905 minus 120591)]

=

119870

sum

119896=1

R119904119896119904119896

(120591) 1198921198981198921exp [minus119895120596

119896(120591119898119896

minus 1205914119896

)]

times1198924

1198921

exp (119895120596119896120591) 1 le 119898 le 119872

R1199091198981199091

(0) = 119864 [119909119898

(119905) 119909lowast

1(119905)]

=

119870

sum

119896=1

R119904119896119904119896

(0) 1198921198981198921exp [minus119895120596

119896(120591119898119896

minus 1205911119896

)]

+ 1205902I 1 le 119898 le 119872

(8)

The spatial-time matrix is constructed as follows

R119878(120591) = [R

11990411199041

(120591) R11990421199042

(120591) R119904119870119904119870

(120591)]119879

(9)

R119891(120591) = [R

11990911199091

(120591) R11990921199091

(120591) R1199091198981199091

(120591)]119879

(10)

R1(120591) = [R

11990911199092

(120591) R11990921199092

(120591) R1199091198981199092

(120591)]119879

(11)

R2(120591) = [R

11990911199093

(120591) R11990921199093

(120591) R1199091198981199093

(120591)]119879

(12)

R3(120591) = [R

11990911199094

(120591) R11990921199094

(120591) R1199091198981199094

(120591)]119879

(13)

R (0) = [R11990911199091

(0) R11990921199091

(0) R1199091198981199091

(0)]119879

(14)

A = [a1(1205961) a

2(1205962) a

119870(120596119870)] (15)

a119896(120596119896) = [119892

1e(minus119895(120596119896119888)p1∙u119896) 119892

2e(minus119895(120596119896119888)p2∙u119896)

119892119872e(minus119895(120596119896119888)p119872∙u119896)]

119879

(16)

4 The Scientific World Journal

The matrix A is the manifold matrix and a is the steeringmatrix Then (9)ndash(14) can be written as

R119891(120591) = AΦ

119891R119878(120591) (17)

R1(120591) = AΦ

1Φ119891R119878(120591) (18)

R2(120591) = AΦ

2Φ119891R119878(120591) (19)

R3(120591) = AΦ

3Φ119891R119878(120591) (20)

R (0) = AR119878(0) (21)

where

Φ119891

= diag [e(1198951205961120591) e(1198951205962120591) e(119895120596119870120591)]119879

(22)

Φ1= diag [

1198922

1198921

e(119895(1205961119888)(p2minusp1)∙u1)1198922

1198921

e(119895(1205962119888)(p2minusp1)∙u2)

1198922

1198921

e(119895(120596119870119888)(p2minusp1)∙u119870)]119879

(23)

Φ2= diag [

1198923

1198921

e(119895(1205961119888)(p3minusp1)∙u1)1198923

1198921

e(119895(1205962119888)(p3minusp1)∙u2)

1198923

1198921

e(119895(120596119870119888)(p3minusp1)∙u119870)]119879

(24)

Φ3= diag [

1198924

1198921

e(119895(1205961119888)(p4minusp1)∙u1)1198924

1198921

e(119895(1205962119888)(p4minusp1)∙u2)

1198924

1198921

e(119895(120596119870119888)(p4minusp1)∙u119870)]119879

(25)

120578119891119896

= 120596119896120591 = 2120587119891

119896120591 (26)

1205781119896

=120596119896

119888(p2minus p

1) ∙ 119906

119894 (27)

1205782119896

=120596119896

119888(p3minus p

1) ∙ 119906

119894 (28)

1205783119896

=120596119896

119888(p4minus p

1) ∙ 119906

119894 (29)

The incident signal is narrowband thusR119904119896119904119896

(119899120591) = R119904119896119904119896

((119899minus

1)120591) Equation (21) can be written in another form as

R (120591) = AR119878(120591) (30)

Equations (17)ndash(21) possess the analogous form R119891(120591)

R1(120591) R

2(120591) R

3(120591) and R(0) are sampled at a time delay 120591

119904

for119873 (119873 ge 119870) times 120591119904= 119879

119904 2119879

119904 119873119879

119904 and 119879

119904is the time

interval of sampling which should be less than the reciprocalof the highest frequency Thus the aliasing effect would not

happen The pseudosnapshot data matrix can be constructedby using (17)ndash(21) as follows

R119891

= [R (119879119904) R (2119879

119904) R (119873119879

119904)]

R1= [R

1(119879119904) R

1(2119879

119904) R

1(119873119879

119904)]

R2= [R

2(119879119904) R

2(2119879

119904) R

2(119873119879

119904)]

R3= [R

3(119879119904) R

3(2119879

119904) R

3(119873119879

119904)]

R = [R119891(119879119904) R

119891(2119879

119904) R

119891(119873119879

119904)]

(31)

Equation (31) can be written as

R119891

= AΦ119891R119878

R1= AΦ

1Φ119891R119878

R2= AΦ

2Φ119891R119878

R3= AΦ

3Φ119891R119878

R = AR119878

(32)

where

R119878= [R

119878(119879119904) R

119878(2119879

119904) R

119878(119873119879

119904)] (33)

32 PARAFAC The element of a tensor X isin C119862times119863times119864 is 119909119894119895119896

and its 119875-component PARAFAC decomposition is given by

119909119888119889119890

=

119872

sum

119901=1

119904119888119901

119905119889119901

119906119890119901

(34)

where 119888 = 1 119862 119889 = 1 119863 and 119890 = 1 119864 119904119888119901 119905119889119901

and 119906

119890119901stand for the elements of S isin C119862times119875 T isin C119863times119875

and U isin C119864times119875 respectively The typical elements of theX119888

isin C119863times119864 X119889

isin C119864times119862 and X119890

isin C119862times119863are 119883119888(119889 119890) =

119883119889(119888 119890) = 119883

119890(119888 119889) = 119909

119888119889119890 respectivelyThe three way array

(TWA) X is ldquoslicedrdquo in three different directions

X119888= TΛ

119888(S)U119879 119888 = 1 119862

X119889= UΛ

119889(T) S119879 119889 = 1 119863

X119890= SΛ

119890(U)T119879 119890 = 1 119864

(35)

For a given matrix S Λ119888(S) means a diagonal matrix with

the diagonals given by the 119888th row of the matrix The symbolldquo⊙rdquo is used to denote the Khatri-Rao (KR) product As anexample the KR product of two matrices S isin C119862times119875 andT isin C119863times119875 of an identical number of columns is given by

S ⊙ T = [s1otimes t

1 s

119875otimes t

119875] isin C

119862119863times119875

(36)

The Scientific World Journal 5

where ldquootimesrdquo represents the Kronecker product Then thedefinition of Kronecker product of two vectors s isin C119862 andt isin C119863 is given by

s otimes t =

[[[[

[

1199041t

1199042t

119904119862t

]]]]

]

(37)

Based on the definition of KR product X119862119863times119864 can beexpressed as

X119862119863times119864 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

X119888=1

X119888=2

X119888=119862

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

TΛ1(S)

TΛ2(S)

TΛ119862(S)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

U119879 = (S ⊙ T)U119879 (38)

Similarly the matrix X119863119864times119862 and X119864119862times119863 can be expressed as

X119863119864times119862 = (T ⊙ U) S119879

X119864119862times119863 = (U ⊙ S)T119879(39)

Definition 1 (Kruskal rank [18]) The Kruskal rank or k-rankof a given matrix S isin C119862times119875 denoted by krank (S) is saidto be equal to 119903 when every collection of 119903 columns of (S)is linearly independent but there exists a collection of 119903 + 1

linearly dependent columns However the rank of S is said tobe themaximal number of linearly independent columnsThedefinition of rank(S) is more relaxed than that of krank(S)that is krank(S) le rank(S)

Theorem2 (Kruskal theorem [19]) Given a PARAFACmodelX isin C119862times119863times119864 then the decomposition of this PARAFAC modelis unique to permutation and scaling when

119896S + 119896T + 119896U ge 2119875 + 2 (40)

The matrix X is constructed by S T and U as

S = SΠΔ1 T = TΠΔ

2 U = UΠΔ

3 (41)

whereΠ stands for the permutation matrixΔ1Δ2 andΔ

3are

diagonal scaling matrices satisfying

Δ1Δ2Δ3= I (42)

where I is a unit matrix

The119872times119872times5TWAof the conformal array is constructedbased on PARAFAC model

R =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)

R ( 2)

R ( 3)

R ( 4)

R ( 5)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R119891

R1

R2

R3

R

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

AΦ119891R119878

AΦ1Φ119891R119878

AΦ2Φ119891R119878

AΦ3Φ119891R119878

AR119878

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ Q (43)

where Q represents the observed noise in practice In orderto make the decomposition of the PARAFAC unique thefollowing assumption must be satisfied

a119894(120596119894) = a

119895(120596119895) (44)

In other words for two incident signals 119904119894(119905) and 119904

119895(119905) with

different DOAs the following condition must be satisfied

119892119900119890eminus1198952120587119891119894((p119900119890∙u119894)119888) = 119892

119900119890eminus1198952120587119891119895((p119900119890∙u119895)119888) (45)

The k-rank of matrix D A and R119878are 119896

119863= min(5 119870) 119896

119860=

119870 and 119896119877119878

= 119870 respectively Thus when 119870 ge 2 the k-rankdecomposition of the PARAFAC is unique

Based on the definition of KRproduct (43) can bewrittenin three forms as follows

R = (D ⊙ A)R119878+ Q (46)

R119883

= (R119879119878⊙ D)A119879 + Q

119883 (47)

R119884

= (A ⊙ R119879119878)D119879 + Q

119884 (48)

where

D =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Λminus1

(Φ119891)

Λminus1

(Φ1Φ119891)

Λminus1

(Φ2Φ119891)

Λminus1

(Φ3Φ119891)

Λminus1

(1)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(49)

where Λminus1

(Φ119891) means the row vector constructed by the

diagonal elements of the diagonal matrixΦ119891

The matrices D A and R119878can be solved by the trilinear

alternating least squares (TALS) regression algorithm Dueto the existence of the observed noise (46) could be trans-formed as a least square problem as follows

minDAR

119878

10038171003817100381710038171003817R minus (D ⊙ A)R

119878

10038171003817100381710038171003817

2

119865

(50)

Then the least square estimator of R119878can be expressed as

R119878= argmin

R119878

10038171003817100381710038171003817R minus (D ⊙ A)R

119878

10038171003817100381710038171003817

2

119865

(51)

Similarly the solutions ofD and A can be expressed as

A119879 = (R119879119878⊙ D)

dagger

R119883

D119879 = (A ⊙ R119879119878)dagger

R119884

(52)

We can update the matrices D A and R119878alternately until

the algorithm converges the estimators of the three matricescan be acquired In the following experiments the COMFACalgorithm (the TALS regression algorithm is realized inpractice) is used to fit the 119898 times 119898 times 5 TWAThe initializationand fitting of the COMFAC is done in compressed spaceTheTucker 3 three-way model is used in data compression [21]

Now the ALS algorithm steps can be summarized asfollows

6 The Scientific World Journal

(1) Initialize A(0) isin C119872times119870 andD(0) isin C5times119870

(2) Initialize 120576 gt 0 and 119896 = 0

(3) If 120588(119896+1)minus120588(119896)

120588(119896)

gt 120576 calculatematricesAR119878 and

D by (51)-(52) Update just one matrix at each timethen 119896 rarr 119896 + 1

(4) Else 120588(119896+1)minus120588(119896)

120588(119896)

lt 120576 the iteration is terminated

In this paper 120588 stands for A R119878 andD respectively

33 The Joint Parameter Estimation The matrix D can beestimated by TALS regression algorithm then the estimatorof 120596

119891119896is given by

120596119896119894

= angle10038161003816100381610038161003816100381610038161003816

D1119896

D5119896

10038161003816100381610038161003816100381610038161003816

(53)

Because 1199031and 119903

2are real numbers D

2119896D

1119896is squared

to avoid the ambiguity caused by the positive and negativevalues of 119903

1and 119903

2 Then

1205961119896

= minus1

2angle([

1198922(120579119896 120593119896)

1198921(120579119896 120593119896)exp (minus119895120596

1119896)]

2

)

= minus1

2angle (exp (minus1198952120596

1119896))

= minus1

2angle([

D2119896

D1119896

]

2

)

(54)

Similarly

1205962119896

= minus1

2angle([

D3119896

D1119896

]

2

) (55)

1205963119896

= minus1

2angle([

D4119896

D1119896

]

2

) (56)

Taking (26) into (53) the frequency 119891119896of the 119896th incident

signal is expressed as

119891119896=

1

2120587120591angle

10038161003816100381610038161003816100381610038161003816

D1119896

D5119896

10038161003816100381610038161003816100381610038161003816

(57)

Assuming that 1205741119896

= sin(120579119896) cos(120593

119896) 120574

2119896= sin(120579

119896) sin(120593

119896)

and 1205743119896

= cos(120579119896) and solving (27) (28) (29) (55) (56) and

(57) simultaneously we can obtain the equation as follows

119888

2120587119891119896

[[[[[[[

[

1205961119896

1198891

1205962119896

1198892

1205963119896

1198893

]]]]]]]

]

= [

[

1199092minus 119909

11199102minus 119910

11199112minus 119911

1

1199093minus 119909

11199103minus 119910

11199113minus 119911

1

1199094minus 119909

11199104minus 119910

11199104minus 119910

1

]

]

[

[

1205741119896

1205742119896

1205743119896

]

]

(58)

The solution of the equation is represented as

[

[

1205741119896

1205742119896

1205743119896

]

]

=119888

2120587119891119894

[

[

1199092minus 119909

11199102minus 119910

11199112minus 119911

1

1199093minus 119909

11199103minus 119910

11199113minus 119911

1

1199094minus 119909

11199104minus 119910

11199104minus 119910

1

]

]

minus1[[[[[[[

[

1205961119896

1198891

1205962119896

1198892

1205963119896

1198893

]]]]]]]

]

(59)

There are two approaches to solve 120579119896and 120593

119896 The first

approach is

120579119896= arccos120574

3119896

120593119896= arcsin(

1205741119896

cos (120579119896)) or 120593

119896= arcsin(

1205742119896

sin (120579119896))

(60)

The second approach is

120593119896= arctan(

1205742119896

1205741119896

)

120579119896= arccos(

1205741119896

cos (120593119896)) or 120579

119896= arcsin(

1205742119896

sin (120593119896))

(61)

In this paper the first approach is employedIn order to ensure the uniqueness of the estimated

parameters the following condition must be satisfied

1003817100381710038171003817p2 minus p1

1003817100381710038171003817 le119888

(2max (119891119894))

=min (120582

119896)

2

1003817100381710038171003817p3 minus p1

1003817100381710038171003817 le119888

(2max (119891119894))

=min (120582

119896)

2

1003817100381710038171003817p4 minus p1

1003817100381710038171003817 le119888

(2max (119891119894))

=min (120582

119896)

2

(62)

According to Theorem 2 the matrices D A and R119878have the

same permutation matrix that is the 119896th column of matrixD corresponds to the 119896th column of the manifold matrix A

Based on the PARAFAC model and TALS regressionalgorithm the joint frequency and 2D-DOA estimation canbe accomplished The algorithm steps are summarized asfollows

(1) choose four guiding elements and obtain the accuratepositions of the elements

(2) calculate the delay correlation function between thefour elements mentioned above and the whole ele-ments of the conformal array

(3) calculate the matrices R119891(120591) R

1(120591) R

2(120591) R

3(120591) and

R(0) respectively according to (5)(4) calculate the pseudosnapshot data matrix based on

(17)ndash(21) R119891(120591) R

1(120591) R

2(120591) R

3(120591) and R(0) are

sampled at time delay 120591119904 respectively

(5) construct the PARAFAC model based on (43)(6) estimate the matrix D using TALS regression algo-

rithm

The Scientific World Journal 7

(7) obtain12059611989111989612059611198961205962119896 and120596

3119896by using the estimator of

the matrixD and calculating (53)ndash(56)(8) obtain the joint frequency and 2D-DOA estimation

by calculate (57)ndash(59) or (57) (60) and (61)

4 Simulation Results

This section focuses on several sets of simulation results todemonstrate the performance of the proposed algorithmIn all simulation examples 200 Monte Carlo trials areconsidered and the root mean square error (RMSE) is usedas our performance measure which is defined as

RMSE = radic1

200

200

sum

120578=1

[(120579119896120578

minus 120579)2

+ (120593119896120578

minus 120593)2

] (63)

where 120579119894119905and 120593

119894119905are the elevation and azimuth estimators of

the 119896th incident signal in the 120578th trialThe successful rate is defined as the proportion of the

number of the successful experiments to the total numberof the experiments For frequency estimation a successfulexperiment is defined as the experiment with estimationerror of less than 2MHz for DOA estimation a successfulexperiment is defined as the experimentwith estimation errorof less than 2 degree

In order to simplify the transformation from the globalcoordinate to the local coordinate only the cylinder withsingle curvature is considered in the simulations The arraystructure is shown in Figure 2 is only a subarray which onlycovers 120

∘ Then three subarrays can estimate the wholespace The highest frequency of the incident signal is 119891max =

2GHz 120582min = 119888119891max The radius of the cylinder is 5120582min theadjacent element spacing in the same intersecting surface ofthe cylinder is 120582min4 and the adjacent intersecting surfacespacing is 120582min4 120582min stands for the shortest wavelength ofthe incident signal The position of the reference element is1199011= (0 5120582 0)The element pattern is defined in the local coordinate

of each element thus the transformation from the globalcoordinate to the local coordinate of elevation 120579

119896and azimuth

120593119896of the incident signal must be accomplished For the

cylindrical conformal array the corresponding Euler rotationangles are

119863 = minusΘ 119864 = minus1205872 119865 = 0 (64)

respectively 119863 represents the rotation angle based on theright-hand rule in the first rotation and the 119885-axis is therotation axis 119864 represents the rotation angle in the secondrotation and the 119884-axis is the rotation axis and 119865 representsthe rotation angle in the third rotation and the 119883-axis is therotation axis Θ represents the angle between the positionvector of the element and the positive direction of 119883-axis

The unit vector of the 119896th incident signal (120579119896 120593119896) in the

global coordinate can be expressed as

119909 = sin (120579119896) cos (120593

119896) 119910 = sin (120579

119896) sin (120593

119896)

119911 = cos (120579119896)

(65)

Z

X

Y

O

pm

p3

p4

p2p1

Figure 2 The structure of the cylindrical conformal array

Based on the Euler rotation matrix the unit vector inglobal coordinate can be transformed into the local coordi-nate of the 119898th element

Consider

[119909119898

119910119898

119911119898]119879

= 119877 (119863119898 119864119898 119865119898) [119909 119910 119911]

119879

(66)

where

119877 (119863119898 119864119898 119865119898) = [

[

cos119865 sin119865 0

minus sin119865 cos119865 0

0 0 1

]

]

times [

[

cos119864 0 minus sin119864

0 1 0

sin119864 0 cos119864]

]

times [

[

cos119863 sin119863 0

minus sin119863 cos119863 0

0 0 1

]

]

(67)

Then the elevation 120579 and azimuth 120593 of the 119896th incident signalin the local coordinate of the119898th element can be representedas

120593119898119896

= arctan (119910119898119909119898) 120579

119898119896= arccos (119911

119898) (68)

Thus the element pattern transformation from the globalcoordinate to the local coordinate is completed

First the curves of RMSE of frequency and angle and thesuccessful probability of angle in different SNR are plottedin Figure 3 using the proposed algorithm The number ofsnapshots is 200 the number of pseudosnapshots is 100SNR isin [5 30] and the noise is AWGN which is independentof incident signals The sampling frequency is 119891

119904= 1119879

119904=

5GHzThe elevation 120579 azimuth120593 and frequency of these twonarrow incident signals are (100∘ 60∘ 1 GHz) and (95∘ 50∘2 GHz) respectively which are independent of each other

8 The Scientific World Journal

5 10 15 20 25 300

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Succ

ess p

roba

bilit

y (

)

Frequency 1Frequency 2

(a)

5 10 15 20 25 300

05

1

15

2

25

3

35

4

RMSE

(MH

z)

SNR (dB)

Frequency 1Frequency 2

(b)

5 10 15 20 25 3040

50

60

70

80

90

100

SNR (dB)

Succ

ess p

roba

bilit

y (

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(c)

5 10 15 20 25 300

002

004

006

008

01

012

014

016

SNR (dB)

RMSE

(deg

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(d)

Figure 3The RMSE and successful probability with different SNR (a)The successful probability of frequencies (b) the RMSE of frequency(c) the successful probability of angles and (d) the RMSE of angles

The number of elements are 10 including 4 guiding elementsand other 6 instrumental elementsThe element pattern usedin simulation is the lowest order circular patch model [8 22]

119892120579(120579 120593) = 119869

2(120587119889

120582sin 120579) minus 119869

0(120587119889

120582sin 120579)

times (cos120593 minus 119895 sin120593) 0 le 120579 le120587

2

119892120593(120579 120593) = 119869

2(120587119889

120582sin 120579) + 119869

0(120587119889

120582sin 120579)

times cos 120579 (sin120593 minus 119895 cos120593) 0 le 120579 le120587

2

119892120579(120579 120593) = 119892

120593(120579 120593) = 0 otherwise

(69)

where 1198690and 119869

2are the zeroth- and second-order Bessel func-

tions of the first kind The element pattern transformation isachieved with the method proposed above

Figure 3 shows that the RMSE of frequency and angledecreases while the SNR increases meanwhile the successprobability of frequency and angle increase as the SNRincreases The estimated performance of the two frequenciesare very close which can be seen in Figures 3(a) and 3(b)As shown in Figure 3(a) the successful probability is largerthan 80 when the SNR is larger than 15 dB The RMSEof frequency is less than 2MHz when SNR is larger than10 dB as shown in Figure 3(b) In Figure 3(c) the successfulprobability of azimuths is slightly lower than that of theelevations The success probability of the angle is almost

The Scientific World Journal 9

100 200 300 400 500 600 700 80020

30

40

50

60

70

80

90

100

Snapshot number

Succ

ess p

roba

bilit

y (

)

Frequency 1Frequency 2

(a)

100 200 300 400 500 600 700 80008

1

12

14

16

18

2

Snapshot number

RMSE

(MH

z)

Frequency 1Frequency 2

(b)

100 200 300 400 500 600 700 80095

955

96

965

97

975

98

985

99

995

100

Snapshot number (dB)

Succ

ess p

roba

bilit

y (

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(c)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

100 200 300 400 500 600 700 800Snapshot number (dB)

002

004

006

008

01

012

014

016RM

SE (d

eg)

(d)

Figure 4 The RMSE and successful probability with different snapshot number (a) The successful probability of frequencies (b) the RMSEof frequency (c) the successful probability of angles and (d) the RMSE of angles

100 when the SNR reaches 15 dB Figure 3(d) shows thatthe RMSE is smaller than 016

∘ when SNR is larger than5 dB The RMSE of azimuth is larger than that of elevationThe estimation results of frequency and elevation are appliedto estimate azimuth The phenomenon in Figure 3(d) canbe interpreted as the existence of the estimated error offrequency and elevation Next the RMSE of frequency andangle as well as the success probability of angle in differentsnapshot number are depicted in Figure 4 SNR = 10 dBother simulation conditions are identical with the previousexperiment In Figure 4 the RMSE of frequency and angledecreases as the snapshot number increases and the successprobability of frequency and angle rises as the snapshotnumber increases As shown in Figure 4(a) the successfulprobability is larger than 80 when the number of snapshot

is larger than 500 Figure 4(b) shows that the RMSE offrequency 1 is slightly larger than that of frequency 2 Whenthe snapshot number is larger than 500 the RMSE dropsbelow 1MHz meaning that the proposed algorithm achievedthree orders of magnitude reduction Figures 4(c) and 4(d)show that the success probability of azimuth is lower thanthat of the elevation and the RMSE of azimuth are largerthan that of elevationThe reasonwhy this happens is becausethe estimation error of frequency and elevation affects theestimation performance of azimuth

5 Conclusion

Due to the varying curvature of the surface of the conformalcarrier the pattern of each element of the conformal array

10 The Scientific World Journal

is different Thus the conventional algorithms could notbe used for conformal array In this paper a novel jointfrequency and 2D-DOA estimation algorithm with highaccuracy are proposed Both spatial and time samplingare utilized to construct the spatial-time matrix The delaycorrelation function is used to suppress noiseThe PARAFACmodel is used for parameter estimation without parameterpairing Only four elements are needed and the positionsof these elements should be known accurately Other instru-mental elements can be flexibly arranged on the surface ofthe conformal carrier The algorithm proposed in this papercan be extended to estimate 2D-DOA straightly with littlemodification The simulation results verify the effectivenessof the proposed algorithm It can be expected that theproposed algorithm would have an application prospect inthe parameter estimation of conformal array In the futurework we will focus on the application of the proposedalgorithm [23ndash26]

Acknowledgments

This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)

References

[1] L Josefsson and P Persson Conformal Array Antenna Theoryand Design IEEE Press Piscataway NJ USA 2006

[2] W T Li X W Shi Y Q Hei S F Liu and J Zhu ldquoAhybrid optimization algorithmand its application for conformalarray pattern synthesisrdquo IEEE Transactions on Antennas andPropagation vol 58 no 10 pp 3401ndash3406 2010

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

[4] 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

[5] P Hyber M Jansson and B Ottersten ldquoArray interpolation andbias reductionrdquo IEEE Transaction on Signal Processing vol 52no 10 pp 2711ndash2720 2004

[6] P Hyberg M Jansson and B Ottersten ldquoArray interpolationand DOAMSE reductionrdquo IEEE Transactions on Signal Process-ing vol 53 no 12 pp 4464ndash4471 2005

[7] F Belloni A Richter and V Koivunen ldquoDOA estimationvia manifold separation for arbitrary array structuresrdquo IEEETransactions on Signal Processing vol 55 no 10 pp 4800ndash48102007

[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008

[9] P Yang F Yang and Z Nie ldquoDOA estimation using MUSICalgorithm on a cylinderical conformal arrayrdquo in Proceedingsof the IEEE Antennas and Propagation Society InternationalSymposium pp 5299ndash5302 June 2007

[10] P Yang F Yang and Z P Nie ldquoDOA estimation with sub-arraydivided technique and interporlated esprit algorithmon a cylin-drical conformal array antennardquo Progress in ElectromagneticsResearch vol 103 pp 201ndash216 2010

[11] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011

[12] L Zou J Lasenby and Z He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012

[13] Y-Y Wang J-T Chen andW-H Fang ldquoTST-MUSIC for jointDOA-delay estimationrdquo IEEE Transactions on Signal Processingvol 49 no 4 pp 721ndash729 2001

[14] J-D Lin W-H Fang Y-Y Wang and J-T Chen ldquoFSF MUSICfor joint DOA and frequency estimation and its performanceanalysisrdquo IEEE Transactions on Signal Processing vol 54 no 12pp 4529ndash4542 2006

[15] M Djeddou A Belouchrani and S Aouada ldquoMaximum likeli-hood angle-frequency estimation in partially known correlatednoise for low-elevation targetsrdquo IEEE Transactions on SignalProcessing vol 53 no 8 pp 3057ndash3064 2005

[16] R B Cattell ldquolsquoParallel proportional profilesrsquo and other prin-ciples for determining the choice of factors by rotationrdquo Psy-chometrika vol 9 no 4 pp 267ndash283 1944

[17] J D Carroll and J-J Chang ldquoAnalysis of individual differencesin multidimensional scaling via an n-way generalization ofldquoEckart-Youngrdquo decompositionrdquo Psychometrika vol 35 no 3pp 283ndash319 1970

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

[19] 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

[20] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 77ndash83 1999

[21] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fittingof the Complex PARAFAC Model in 3-Drdquo 1998 httpwwwtelecomtucgrsimnikos

[22] J R James P S Hall and C WoodMicrostrip Antenna Theoryand Design Peter Peregrinus New York NY USA 1981

[23] S Yin S Ding and H Luo ldquoReal-time implementation of faulttolerant control system with performance optimizationrdquo IEEETransactions on Industrial Electronics vol 61 no 5 pp 2402ndash2411 2014

[24] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[25] S Yin S Ding A Haghani andH Hao ldquoData-drivenmonitor-ing for stochastic systems and its application on batch processrdquoInternational Journal of Systems Science vol 44 no 7 pp 1366ndash1376 2013

[26] S Yin X Yang and H Karimi ldquoData-driven adaptive observerfor fault diagnosisrdquoMathematical Problems in Engineering vol2012 Article ID 832836 21 pages 2012

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DistributedSensor Networks

International Journal of

The Scientific World Journal 3

parameter estimation mechanism is used for each part Forthe sake of simplicity it is assumed that all the elements canreceive the signal

3 Joint Frequency and 2D-DOA Estimation

31 The Construction of Spatial-Time Matrix The delay cor-relation function between the 119898th element and 119899th elementof the conformal array is constructed as follows

119877119909119898119909119899

= 119864 [119909119898

(119905) 119909lowast

119899(119905 minus 120591)]

=

119870

sum

119896=1

119864 [119904119896(119905) 119904

lowast

119896(119905 minus 120591)]

times exp [minus119895120596119896(120591119898119896

minus 120591119899119896

minus 120591)]

+ 119864 [119899119898

(119905) 119899lowast

119899(119905 minus 120591)]

=

119870

sum

119896=1

119877119904119896119904119896

(120591) 119892119898119892119899exp [minus119895120596

119896(120591119898119896

minus 120591119899119896

)]

times exp (119895120596119896120591) + 119877

119899119898119899119899

(120591) 120591 gt 0

(5)

where R119904119896119904119896

(120591) = 119864[119904119898(119905)119904

lowast

119899(119905 minus 120591)] represents the delay

correlation function of 119904119896(119905) andR

119899119898119899119899

(120591) = 119864[119899119898(119905)119899

lowast

119899(119905minus120591)]

represents the noise cross-correlation function between the119898th element and 119899th element

On condition that the incident signal is coming fromnarrow far field and the bandwidth of the incident signalis B the delay correlation processing of the data that theelements receive takes a long period of timeHenceR

119904119896119904119896

(120591) =

119864[119904119898(119905)119904

lowast

119899(119905minus120591)] = 0 In themore general case when 120591 ≪ 1119861

the Gaussian white noise is not correlated during the timeinterval 120591 At the same time the variety of the signal envelopecan be neglected Then we can suppress the noise withoutdestroying the signal

Consider

R119899119898119899119899

(120591) = 119864 [119899119898

(119905) 119899lowast

119899(119905 minus 120591)]

= 1205902

120575 (120591) 120575 (119898 minus 119899) = 0

(6)

where the delay correlation function is constructed between1199091(119905 minus 120591) and the data that each element receives Based on

(5) we can obtain

R1199091198981199091

(120591) = 119864 [119909119898

(119905) 119909lowast

1(119905 minus 120591)]

=

119870

sum

119896=1

119877119904119896119904119896

(120591) 1198921198981198921exp [minus119895120596

119896(120591119898119896

minus 1205911119896

)]

times exp (119895120596119896120591) 1 le 119898 le 119872

(7)

Similarly the delay correlation functions are constructedbetween 119909

2(119905 minus 120591) 119909

3(119905 minus 120591) 119909

4(119905 minus 120591) and 119909

1(119905) and the data

that each element receives respectively

Consider

R1199091198981199092

(120591) = 119864 [119909119898

(119905) 119909lowast

2(119905 minus 120591)]

=

119870

sum

119896=1

R119904119896119904119896

(120591) 1198921198981198921exp [minus119895120596

119896(120591119898119896

minus 1205912119896

)]

times1198922

1198921

exp (119895120596119896120591) 1 le 119898 le 119872

R1199091198981199093

(120591) = 119864 [119909119898

(119905) 119909lowast

3(119905 minus 120591)]

=

119870

sum

119896=1

R119904119896119904119896

(120591) 1198921198981198921exp [minus119895120596

119896(120591119898119896

minus 1205913119896

)]

times1198923

1198921

exp (119895120596119896120591) 1 le 119898 le 119872

R1199091198981199094

(120591) = 119864 [119909119898

(119905) 119909lowast

4(119905 minus 120591)]

=

119870

sum

119896=1

R119904119896119904119896

(120591) 1198921198981198921exp [minus119895120596

119896(120591119898119896

minus 1205914119896

)]

times1198924

1198921

exp (119895120596119896120591) 1 le 119898 le 119872

R1199091198981199091

(0) = 119864 [119909119898

(119905) 119909lowast

1(119905)]

=

119870

sum

119896=1

R119904119896119904119896

(0) 1198921198981198921exp [minus119895120596

119896(120591119898119896

minus 1205911119896

)]

+ 1205902I 1 le 119898 le 119872

(8)

The spatial-time matrix is constructed as follows

R119878(120591) = [R

11990411199041

(120591) R11990421199042

(120591) R119904119870119904119870

(120591)]119879

(9)

R119891(120591) = [R

11990911199091

(120591) R11990921199091

(120591) R1199091198981199091

(120591)]119879

(10)

R1(120591) = [R

11990911199092

(120591) R11990921199092

(120591) R1199091198981199092

(120591)]119879

(11)

R2(120591) = [R

11990911199093

(120591) R11990921199093

(120591) R1199091198981199093

(120591)]119879

(12)

R3(120591) = [R

11990911199094

(120591) R11990921199094

(120591) R1199091198981199094

(120591)]119879

(13)

R (0) = [R11990911199091

(0) R11990921199091

(0) R1199091198981199091

(0)]119879

(14)

A = [a1(1205961) a

2(1205962) a

119870(120596119870)] (15)

a119896(120596119896) = [119892

1e(minus119895(120596119896119888)p1∙u119896) 119892

2e(minus119895(120596119896119888)p2∙u119896)

119892119872e(minus119895(120596119896119888)p119872∙u119896)]

119879

(16)

4 The Scientific World Journal

The matrix A is the manifold matrix and a is the steeringmatrix Then (9)ndash(14) can be written as

R119891(120591) = AΦ

119891R119878(120591) (17)

R1(120591) = AΦ

1Φ119891R119878(120591) (18)

R2(120591) = AΦ

2Φ119891R119878(120591) (19)

R3(120591) = AΦ

3Φ119891R119878(120591) (20)

R (0) = AR119878(0) (21)

where

Φ119891

= diag [e(1198951205961120591) e(1198951205962120591) e(119895120596119870120591)]119879

(22)

Φ1= diag [

1198922

1198921

e(119895(1205961119888)(p2minusp1)∙u1)1198922

1198921

e(119895(1205962119888)(p2minusp1)∙u2)

1198922

1198921

e(119895(120596119870119888)(p2minusp1)∙u119870)]119879

(23)

Φ2= diag [

1198923

1198921

e(119895(1205961119888)(p3minusp1)∙u1)1198923

1198921

e(119895(1205962119888)(p3minusp1)∙u2)

1198923

1198921

e(119895(120596119870119888)(p3minusp1)∙u119870)]119879

(24)

Φ3= diag [

1198924

1198921

e(119895(1205961119888)(p4minusp1)∙u1)1198924

1198921

e(119895(1205962119888)(p4minusp1)∙u2)

1198924

1198921

e(119895(120596119870119888)(p4minusp1)∙u119870)]119879

(25)

120578119891119896

= 120596119896120591 = 2120587119891

119896120591 (26)

1205781119896

=120596119896

119888(p2minus p

1) ∙ 119906

119894 (27)

1205782119896

=120596119896

119888(p3minus p

1) ∙ 119906

119894 (28)

1205783119896

=120596119896

119888(p4minus p

1) ∙ 119906

119894 (29)

The incident signal is narrowband thusR119904119896119904119896

(119899120591) = R119904119896119904119896

((119899minus

1)120591) Equation (21) can be written in another form as

R (120591) = AR119878(120591) (30)

Equations (17)ndash(21) possess the analogous form R119891(120591)

R1(120591) R

2(120591) R

3(120591) and R(0) are sampled at a time delay 120591

119904

for119873 (119873 ge 119870) times 120591119904= 119879

119904 2119879

119904 119873119879

119904 and 119879

119904is the time

interval of sampling which should be less than the reciprocalof the highest frequency Thus the aliasing effect would not

happen The pseudosnapshot data matrix can be constructedby using (17)ndash(21) as follows

R119891

= [R (119879119904) R (2119879

119904) R (119873119879

119904)]

R1= [R

1(119879119904) R

1(2119879

119904) R

1(119873119879

119904)]

R2= [R

2(119879119904) R

2(2119879

119904) R

2(119873119879

119904)]

R3= [R

3(119879119904) R

3(2119879

119904) R

3(119873119879

119904)]

R = [R119891(119879119904) R

119891(2119879

119904) R

119891(119873119879

119904)]

(31)

Equation (31) can be written as

R119891

= AΦ119891R119878

R1= AΦ

1Φ119891R119878

R2= AΦ

2Φ119891R119878

R3= AΦ

3Φ119891R119878

R = AR119878

(32)

where

R119878= [R

119878(119879119904) R

119878(2119879

119904) R

119878(119873119879

119904)] (33)

32 PARAFAC The element of a tensor X isin C119862times119863times119864 is 119909119894119895119896

and its 119875-component PARAFAC decomposition is given by

119909119888119889119890

=

119872

sum

119901=1

119904119888119901

119905119889119901

119906119890119901

(34)

where 119888 = 1 119862 119889 = 1 119863 and 119890 = 1 119864 119904119888119901 119905119889119901

and 119906

119890119901stand for the elements of S isin C119862times119875 T isin C119863times119875

and U isin C119864times119875 respectively The typical elements of theX119888

isin C119863times119864 X119889

isin C119864times119862 and X119890

isin C119862times119863are 119883119888(119889 119890) =

119883119889(119888 119890) = 119883

119890(119888 119889) = 119909

119888119889119890 respectivelyThe three way array

(TWA) X is ldquoslicedrdquo in three different directions

X119888= TΛ

119888(S)U119879 119888 = 1 119862

X119889= UΛ

119889(T) S119879 119889 = 1 119863

X119890= SΛ

119890(U)T119879 119890 = 1 119864

(35)

For a given matrix S Λ119888(S) means a diagonal matrix with

the diagonals given by the 119888th row of the matrix The symbolldquo⊙rdquo is used to denote the Khatri-Rao (KR) product As anexample the KR product of two matrices S isin C119862times119875 andT isin C119863times119875 of an identical number of columns is given by

S ⊙ T = [s1otimes t

1 s

119875otimes t

119875] isin C

119862119863times119875

(36)

The Scientific World Journal 5

where ldquootimesrdquo represents the Kronecker product Then thedefinition of Kronecker product of two vectors s isin C119862 andt isin C119863 is given by

s otimes t =

[[[[

[

1199041t

1199042t

119904119862t

]]]]

]

(37)

Based on the definition of KR product X119862119863times119864 can beexpressed as

X119862119863times119864 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

X119888=1

X119888=2

X119888=119862

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

TΛ1(S)

TΛ2(S)

TΛ119862(S)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

U119879 = (S ⊙ T)U119879 (38)

Similarly the matrix X119863119864times119862 and X119864119862times119863 can be expressed as

X119863119864times119862 = (T ⊙ U) S119879

X119864119862times119863 = (U ⊙ S)T119879(39)

Definition 1 (Kruskal rank [18]) The Kruskal rank or k-rankof a given matrix S isin C119862times119875 denoted by krank (S) is saidto be equal to 119903 when every collection of 119903 columns of (S)is linearly independent but there exists a collection of 119903 + 1

linearly dependent columns However the rank of S is said tobe themaximal number of linearly independent columnsThedefinition of rank(S) is more relaxed than that of krank(S)that is krank(S) le rank(S)

Theorem2 (Kruskal theorem [19]) Given a PARAFACmodelX isin C119862times119863times119864 then the decomposition of this PARAFAC modelis unique to permutation and scaling when

119896S + 119896T + 119896U ge 2119875 + 2 (40)

The matrix X is constructed by S T and U as

S = SΠΔ1 T = TΠΔ

2 U = UΠΔ

3 (41)

whereΠ stands for the permutation matrixΔ1Δ2 andΔ

3are

diagonal scaling matrices satisfying

Δ1Δ2Δ3= I (42)

where I is a unit matrix

The119872times119872times5TWAof the conformal array is constructedbased on PARAFAC model

R =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)

R ( 2)

R ( 3)

R ( 4)

R ( 5)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R119891

R1

R2

R3

R

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

AΦ119891R119878

AΦ1Φ119891R119878

AΦ2Φ119891R119878

AΦ3Φ119891R119878

AR119878

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ Q (43)

where Q represents the observed noise in practice In orderto make the decomposition of the PARAFAC unique thefollowing assumption must be satisfied

a119894(120596119894) = a

119895(120596119895) (44)

In other words for two incident signals 119904119894(119905) and 119904

119895(119905) with

different DOAs the following condition must be satisfied

119892119900119890eminus1198952120587119891119894((p119900119890∙u119894)119888) = 119892

119900119890eminus1198952120587119891119895((p119900119890∙u119895)119888) (45)

The k-rank of matrix D A and R119878are 119896

119863= min(5 119870) 119896

119860=

119870 and 119896119877119878

= 119870 respectively Thus when 119870 ge 2 the k-rankdecomposition of the PARAFAC is unique

Based on the definition of KRproduct (43) can bewrittenin three forms as follows

R = (D ⊙ A)R119878+ Q (46)

R119883

= (R119879119878⊙ D)A119879 + Q

119883 (47)

R119884

= (A ⊙ R119879119878)D119879 + Q

119884 (48)

where

D =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Λminus1

(Φ119891)

Λminus1

(Φ1Φ119891)

Λminus1

(Φ2Φ119891)

Λminus1

(Φ3Φ119891)

Λminus1

(1)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(49)

where Λminus1

(Φ119891) means the row vector constructed by the

diagonal elements of the diagonal matrixΦ119891

The matrices D A and R119878can be solved by the trilinear

alternating least squares (TALS) regression algorithm Dueto the existence of the observed noise (46) could be trans-formed as a least square problem as follows

minDAR

119878

10038171003817100381710038171003817R minus (D ⊙ A)R

119878

10038171003817100381710038171003817

2

119865

(50)

Then the least square estimator of R119878can be expressed as

R119878= argmin

R119878

10038171003817100381710038171003817R minus (D ⊙ A)R

119878

10038171003817100381710038171003817

2

119865

(51)

Similarly the solutions ofD and A can be expressed as

A119879 = (R119879119878⊙ D)

dagger

R119883

D119879 = (A ⊙ R119879119878)dagger

R119884

(52)

We can update the matrices D A and R119878alternately until

the algorithm converges the estimators of the three matricescan be acquired In the following experiments the COMFACalgorithm (the TALS regression algorithm is realized inpractice) is used to fit the 119898 times 119898 times 5 TWAThe initializationand fitting of the COMFAC is done in compressed spaceTheTucker 3 three-way model is used in data compression [21]

Now the ALS algorithm steps can be summarized asfollows

6 The Scientific World Journal

(1) Initialize A(0) isin C119872times119870 andD(0) isin C5times119870

(2) Initialize 120576 gt 0 and 119896 = 0

(3) If 120588(119896+1)minus120588(119896)

120588(119896)

gt 120576 calculatematricesAR119878 and

D by (51)-(52) Update just one matrix at each timethen 119896 rarr 119896 + 1

(4) Else 120588(119896+1)minus120588(119896)

120588(119896)

lt 120576 the iteration is terminated

In this paper 120588 stands for A R119878 andD respectively

33 The Joint Parameter Estimation The matrix D can beestimated by TALS regression algorithm then the estimatorof 120596

119891119896is given by

120596119896119894

= angle10038161003816100381610038161003816100381610038161003816

D1119896

D5119896

10038161003816100381610038161003816100381610038161003816

(53)

Because 1199031and 119903

2are real numbers D

2119896D

1119896is squared

to avoid the ambiguity caused by the positive and negativevalues of 119903

1and 119903

2 Then

1205961119896

= minus1

2angle([

1198922(120579119896 120593119896)

1198921(120579119896 120593119896)exp (minus119895120596

1119896)]

2

)

= minus1

2angle (exp (minus1198952120596

1119896))

= minus1

2angle([

D2119896

D1119896

]

2

)

(54)

Similarly

1205962119896

= minus1

2angle([

D3119896

D1119896

]

2

) (55)

1205963119896

= minus1

2angle([

D4119896

D1119896

]

2

) (56)

Taking (26) into (53) the frequency 119891119896of the 119896th incident

signal is expressed as

119891119896=

1

2120587120591angle

10038161003816100381610038161003816100381610038161003816

D1119896

D5119896

10038161003816100381610038161003816100381610038161003816

(57)

Assuming that 1205741119896

= sin(120579119896) cos(120593

119896) 120574

2119896= sin(120579

119896) sin(120593

119896)

and 1205743119896

= cos(120579119896) and solving (27) (28) (29) (55) (56) and

(57) simultaneously we can obtain the equation as follows

119888

2120587119891119896

[[[[[[[

[

1205961119896

1198891

1205962119896

1198892

1205963119896

1198893

]]]]]]]

]

= [

[

1199092minus 119909

11199102minus 119910

11199112minus 119911

1

1199093minus 119909

11199103minus 119910

11199113minus 119911

1

1199094minus 119909

11199104minus 119910

11199104minus 119910

1

]

]

[

[

1205741119896

1205742119896

1205743119896

]

]

(58)

The solution of the equation is represented as

[

[

1205741119896

1205742119896

1205743119896

]

]

=119888

2120587119891119894

[

[

1199092minus 119909

11199102minus 119910

11199112minus 119911

1

1199093minus 119909

11199103minus 119910

11199113minus 119911

1

1199094minus 119909

11199104minus 119910

11199104minus 119910

1

]

]

minus1[[[[[[[

[

1205961119896

1198891

1205962119896

1198892

1205963119896

1198893

]]]]]]]

]

(59)

There are two approaches to solve 120579119896and 120593

119896 The first

approach is

120579119896= arccos120574

3119896

120593119896= arcsin(

1205741119896

cos (120579119896)) or 120593

119896= arcsin(

1205742119896

sin (120579119896))

(60)

The second approach is

120593119896= arctan(

1205742119896

1205741119896

)

120579119896= arccos(

1205741119896

cos (120593119896)) or 120579

119896= arcsin(

1205742119896

sin (120593119896))

(61)

In this paper the first approach is employedIn order to ensure the uniqueness of the estimated

parameters the following condition must be satisfied

1003817100381710038171003817p2 minus p1

1003817100381710038171003817 le119888

(2max (119891119894))

=min (120582

119896)

2

1003817100381710038171003817p3 minus p1

1003817100381710038171003817 le119888

(2max (119891119894))

=min (120582

119896)

2

1003817100381710038171003817p4 minus p1

1003817100381710038171003817 le119888

(2max (119891119894))

=min (120582

119896)

2

(62)

According to Theorem 2 the matrices D A and R119878have the

same permutation matrix that is the 119896th column of matrixD corresponds to the 119896th column of the manifold matrix A

Based on the PARAFAC model and TALS regressionalgorithm the joint frequency and 2D-DOA estimation canbe accomplished The algorithm steps are summarized asfollows

(1) choose four guiding elements and obtain the accuratepositions of the elements

(2) calculate the delay correlation function between thefour elements mentioned above and the whole ele-ments of the conformal array

(3) calculate the matrices R119891(120591) R

1(120591) R

2(120591) R

3(120591) and

R(0) respectively according to (5)(4) calculate the pseudosnapshot data matrix based on

(17)ndash(21) R119891(120591) R

1(120591) R

2(120591) R

3(120591) and R(0) are

sampled at time delay 120591119904 respectively

(5) construct the PARAFAC model based on (43)(6) estimate the matrix D using TALS regression algo-

rithm

The Scientific World Journal 7

(7) obtain12059611989111989612059611198961205962119896 and120596

3119896by using the estimator of

the matrixD and calculating (53)ndash(56)(8) obtain the joint frequency and 2D-DOA estimation

by calculate (57)ndash(59) or (57) (60) and (61)

4 Simulation Results

This section focuses on several sets of simulation results todemonstrate the performance of the proposed algorithmIn all simulation examples 200 Monte Carlo trials areconsidered and the root mean square error (RMSE) is usedas our performance measure which is defined as

RMSE = radic1

200

200

sum

120578=1

[(120579119896120578

minus 120579)2

+ (120593119896120578

minus 120593)2

] (63)

where 120579119894119905and 120593

119894119905are the elevation and azimuth estimators of

the 119896th incident signal in the 120578th trialThe successful rate is defined as the proportion of the

number of the successful experiments to the total numberof the experiments For frequency estimation a successfulexperiment is defined as the experiment with estimationerror of less than 2MHz for DOA estimation a successfulexperiment is defined as the experimentwith estimation errorof less than 2 degree

In order to simplify the transformation from the globalcoordinate to the local coordinate only the cylinder withsingle curvature is considered in the simulations The arraystructure is shown in Figure 2 is only a subarray which onlycovers 120

∘ Then three subarrays can estimate the wholespace The highest frequency of the incident signal is 119891max =

2GHz 120582min = 119888119891max The radius of the cylinder is 5120582min theadjacent element spacing in the same intersecting surface ofthe cylinder is 120582min4 and the adjacent intersecting surfacespacing is 120582min4 120582min stands for the shortest wavelength ofthe incident signal The position of the reference element is1199011= (0 5120582 0)The element pattern is defined in the local coordinate

of each element thus the transformation from the globalcoordinate to the local coordinate of elevation 120579

119896and azimuth

120593119896of the incident signal must be accomplished For the

cylindrical conformal array the corresponding Euler rotationangles are

119863 = minusΘ 119864 = minus1205872 119865 = 0 (64)

respectively 119863 represents the rotation angle based on theright-hand rule in the first rotation and the 119885-axis is therotation axis 119864 represents the rotation angle in the secondrotation and the 119884-axis is the rotation axis and 119865 representsthe rotation angle in the third rotation and the 119883-axis is therotation axis Θ represents the angle between the positionvector of the element and the positive direction of 119883-axis

The unit vector of the 119896th incident signal (120579119896 120593119896) in the

global coordinate can be expressed as

119909 = sin (120579119896) cos (120593

119896) 119910 = sin (120579

119896) sin (120593

119896)

119911 = cos (120579119896)

(65)

Z

X

Y

O

pm

p3

p4

p2p1

Figure 2 The structure of the cylindrical conformal array

Based on the Euler rotation matrix the unit vector inglobal coordinate can be transformed into the local coordi-nate of the 119898th element

Consider

[119909119898

119910119898

119911119898]119879

= 119877 (119863119898 119864119898 119865119898) [119909 119910 119911]

119879

(66)

where

119877 (119863119898 119864119898 119865119898) = [

[

cos119865 sin119865 0

minus sin119865 cos119865 0

0 0 1

]

]

times [

[

cos119864 0 minus sin119864

0 1 0

sin119864 0 cos119864]

]

times [

[

cos119863 sin119863 0

minus sin119863 cos119863 0

0 0 1

]

]

(67)

Then the elevation 120579 and azimuth 120593 of the 119896th incident signalin the local coordinate of the119898th element can be representedas

120593119898119896

= arctan (119910119898119909119898) 120579

119898119896= arccos (119911

119898) (68)

Thus the element pattern transformation from the globalcoordinate to the local coordinate is completed

First the curves of RMSE of frequency and angle and thesuccessful probability of angle in different SNR are plottedin Figure 3 using the proposed algorithm The number ofsnapshots is 200 the number of pseudosnapshots is 100SNR isin [5 30] and the noise is AWGN which is independentof incident signals The sampling frequency is 119891

119904= 1119879

119904=

5GHzThe elevation 120579 azimuth120593 and frequency of these twonarrow incident signals are (100∘ 60∘ 1 GHz) and (95∘ 50∘2 GHz) respectively which are independent of each other

8 The Scientific World Journal

5 10 15 20 25 300

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Succ

ess p

roba

bilit

y (

)

Frequency 1Frequency 2

(a)

5 10 15 20 25 300

05

1

15

2

25

3

35

4

RMSE

(MH

z)

SNR (dB)

Frequency 1Frequency 2

(b)

5 10 15 20 25 3040

50

60

70

80

90

100

SNR (dB)

Succ

ess p

roba

bilit

y (

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(c)

5 10 15 20 25 300

002

004

006

008

01

012

014

016

SNR (dB)

RMSE

(deg

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(d)

Figure 3The RMSE and successful probability with different SNR (a)The successful probability of frequencies (b) the RMSE of frequency(c) the successful probability of angles and (d) the RMSE of angles

The number of elements are 10 including 4 guiding elementsand other 6 instrumental elementsThe element pattern usedin simulation is the lowest order circular patch model [8 22]

119892120579(120579 120593) = 119869

2(120587119889

120582sin 120579) minus 119869

0(120587119889

120582sin 120579)

times (cos120593 minus 119895 sin120593) 0 le 120579 le120587

2

119892120593(120579 120593) = 119869

2(120587119889

120582sin 120579) + 119869

0(120587119889

120582sin 120579)

times cos 120579 (sin120593 minus 119895 cos120593) 0 le 120579 le120587

2

119892120579(120579 120593) = 119892

120593(120579 120593) = 0 otherwise

(69)

where 1198690and 119869

2are the zeroth- and second-order Bessel func-

tions of the first kind The element pattern transformation isachieved with the method proposed above

Figure 3 shows that the RMSE of frequency and angledecreases while the SNR increases meanwhile the successprobability of frequency and angle increase as the SNRincreases The estimated performance of the two frequenciesare very close which can be seen in Figures 3(a) and 3(b)As shown in Figure 3(a) the successful probability is largerthan 80 when the SNR is larger than 15 dB The RMSEof frequency is less than 2MHz when SNR is larger than10 dB as shown in Figure 3(b) In Figure 3(c) the successfulprobability of azimuths is slightly lower than that of theelevations The success probability of the angle is almost

The Scientific World Journal 9

100 200 300 400 500 600 700 80020

30

40

50

60

70

80

90

100

Snapshot number

Succ

ess p

roba

bilit

y (

)

Frequency 1Frequency 2

(a)

100 200 300 400 500 600 700 80008

1

12

14

16

18

2

Snapshot number

RMSE

(MH

z)

Frequency 1Frequency 2

(b)

100 200 300 400 500 600 700 80095

955

96

965

97

975

98

985

99

995

100

Snapshot number (dB)

Succ

ess p

roba

bilit

y (

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(c)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

100 200 300 400 500 600 700 800Snapshot number (dB)

002

004

006

008

01

012

014

016RM

SE (d

eg)

(d)

Figure 4 The RMSE and successful probability with different snapshot number (a) The successful probability of frequencies (b) the RMSEof frequency (c) the successful probability of angles and (d) the RMSE of angles

100 when the SNR reaches 15 dB Figure 3(d) shows thatthe RMSE is smaller than 016

∘ when SNR is larger than5 dB The RMSE of azimuth is larger than that of elevationThe estimation results of frequency and elevation are appliedto estimate azimuth The phenomenon in Figure 3(d) canbe interpreted as the existence of the estimated error offrequency and elevation Next the RMSE of frequency andangle as well as the success probability of angle in differentsnapshot number are depicted in Figure 4 SNR = 10 dBother simulation conditions are identical with the previousexperiment In Figure 4 the RMSE of frequency and angledecreases as the snapshot number increases and the successprobability of frequency and angle rises as the snapshotnumber increases As shown in Figure 4(a) the successfulprobability is larger than 80 when the number of snapshot

is larger than 500 Figure 4(b) shows that the RMSE offrequency 1 is slightly larger than that of frequency 2 Whenthe snapshot number is larger than 500 the RMSE dropsbelow 1MHz meaning that the proposed algorithm achievedthree orders of magnitude reduction Figures 4(c) and 4(d)show that the success probability of azimuth is lower thanthat of the elevation and the RMSE of azimuth are largerthan that of elevationThe reasonwhy this happens is becausethe estimation error of frequency and elevation affects theestimation performance of azimuth

5 Conclusion

Due to the varying curvature of the surface of the conformalcarrier the pattern of each element of the conformal array

10 The Scientific World Journal

is different Thus the conventional algorithms could notbe used for conformal array In this paper a novel jointfrequency and 2D-DOA estimation algorithm with highaccuracy are proposed Both spatial and time samplingare utilized to construct the spatial-time matrix The delaycorrelation function is used to suppress noiseThe PARAFACmodel is used for parameter estimation without parameterpairing Only four elements are needed and the positionsof these elements should be known accurately Other instru-mental elements can be flexibly arranged on the surface ofthe conformal carrier The algorithm proposed in this papercan be extended to estimate 2D-DOA straightly with littlemodification The simulation results verify the effectivenessof the proposed algorithm It can be expected that theproposed algorithm would have an application prospect inthe parameter estimation of conformal array In the futurework we will focus on the application of the proposedalgorithm [23ndash26]

Acknowledgments

This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)

References

[1] L Josefsson and P Persson Conformal Array Antenna Theoryand Design IEEE Press Piscataway NJ USA 2006

[2] W T Li X W Shi Y Q Hei S F Liu and J Zhu ldquoAhybrid optimization algorithmand its application for conformalarray pattern synthesisrdquo IEEE Transactions on Antennas andPropagation vol 58 no 10 pp 3401ndash3406 2010

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

[4] 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

[5] P Hyber M Jansson and B Ottersten ldquoArray interpolation andbias reductionrdquo IEEE Transaction on Signal Processing vol 52no 10 pp 2711ndash2720 2004

[6] P Hyberg M Jansson and B Ottersten ldquoArray interpolationand DOAMSE reductionrdquo IEEE Transactions on Signal Process-ing vol 53 no 12 pp 4464ndash4471 2005

[7] F Belloni A Richter and V Koivunen ldquoDOA estimationvia manifold separation for arbitrary array structuresrdquo IEEETransactions on Signal Processing vol 55 no 10 pp 4800ndash48102007

[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008

[9] P Yang F Yang and Z Nie ldquoDOA estimation using MUSICalgorithm on a cylinderical conformal arrayrdquo in Proceedingsof the IEEE Antennas and Propagation Society InternationalSymposium pp 5299ndash5302 June 2007

[10] P Yang F Yang and Z P Nie ldquoDOA estimation with sub-arraydivided technique and interporlated esprit algorithmon a cylin-drical conformal array antennardquo Progress in ElectromagneticsResearch vol 103 pp 201ndash216 2010

[11] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011

[12] L Zou J Lasenby and Z He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012

[13] Y-Y Wang J-T Chen andW-H Fang ldquoTST-MUSIC for jointDOA-delay estimationrdquo IEEE Transactions on Signal Processingvol 49 no 4 pp 721ndash729 2001

[14] J-D Lin W-H Fang Y-Y Wang and J-T Chen ldquoFSF MUSICfor joint DOA and frequency estimation and its performanceanalysisrdquo IEEE Transactions on Signal Processing vol 54 no 12pp 4529ndash4542 2006

[15] M Djeddou A Belouchrani and S Aouada ldquoMaximum likeli-hood angle-frequency estimation in partially known correlatednoise for low-elevation targetsrdquo IEEE Transactions on SignalProcessing vol 53 no 8 pp 3057ndash3064 2005

[16] R B Cattell ldquolsquoParallel proportional profilesrsquo and other prin-ciples for determining the choice of factors by rotationrdquo Psy-chometrika vol 9 no 4 pp 267ndash283 1944

[17] J D Carroll and J-J Chang ldquoAnalysis of individual differencesin multidimensional scaling via an n-way generalization ofldquoEckart-Youngrdquo decompositionrdquo Psychometrika vol 35 no 3pp 283ndash319 1970

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

[19] 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

[20] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 77ndash83 1999

[21] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fittingof the Complex PARAFAC Model in 3-Drdquo 1998 httpwwwtelecomtucgrsimnikos

[22] J R James P S Hall and C WoodMicrostrip Antenna Theoryand Design Peter Peregrinus New York NY USA 1981

[23] S Yin S Ding and H Luo ldquoReal-time implementation of faulttolerant control system with performance optimizationrdquo IEEETransactions on Industrial Electronics vol 61 no 5 pp 2402ndash2411 2014

[24] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[25] S Yin S Ding A Haghani andH Hao ldquoData-drivenmonitor-ing for stochastic systems and its application on batch processrdquoInternational Journal of Systems Science vol 44 no 7 pp 1366ndash1376 2013

[26] S Yin X Yang and H Karimi ldquoData-driven adaptive observerfor fault diagnosisrdquoMathematical Problems in Engineering vol2012 Article ID 832836 21 pages 2012

International Journal of

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RotatingMachinery

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 The Scientific World Journal

The matrix A is the manifold matrix and a is the steeringmatrix Then (9)ndash(14) can be written as

R119891(120591) = AΦ

119891R119878(120591) (17)

R1(120591) = AΦ

1Φ119891R119878(120591) (18)

R2(120591) = AΦ

2Φ119891R119878(120591) (19)

R3(120591) = AΦ

3Φ119891R119878(120591) (20)

R (0) = AR119878(0) (21)

where

Φ119891

= diag [e(1198951205961120591) e(1198951205962120591) e(119895120596119870120591)]119879

(22)

Φ1= diag [

1198922

1198921

e(119895(1205961119888)(p2minusp1)∙u1)1198922

1198921

e(119895(1205962119888)(p2minusp1)∙u2)

1198922

1198921

e(119895(120596119870119888)(p2minusp1)∙u119870)]119879

(23)

Φ2= diag [

1198923

1198921

e(119895(1205961119888)(p3minusp1)∙u1)1198923

1198921

e(119895(1205962119888)(p3minusp1)∙u2)

1198923

1198921

e(119895(120596119870119888)(p3minusp1)∙u119870)]119879

(24)

Φ3= diag [

1198924

1198921

e(119895(1205961119888)(p4minusp1)∙u1)1198924

1198921

e(119895(1205962119888)(p4minusp1)∙u2)

1198924

1198921

e(119895(120596119870119888)(p4minusp1)∙u119870)]119879

(25)

120578119891119896

= 120596119896120591 = 2120587119891

119896120591 (26)

1205781119896

=120596119896

119888(p2minus p

1) ∙ 119906

119894 (27)

1205782119896

=120596119896

119888(p3minus p

1) ∙ 119906

119894 (28)

1205783119896

=120596119896

119888(p4minus p

1) ∙ 119906

119894 (29)

The incident signal is narrowband thusR119904119896119904119896

(119899120591) = R119904119896119904119896

((119899minus

1)120591) Equation (21) can be written in another form as

R (120591) = AR119878(120591) (30)

Equations (17)ndash(21) possess the analogous form R119891(120591)

R1(120591) R

2(120591) R

3(120591) and R(0) are sampled at a time delay 120591

119904

for119873 (119873 ge 119870) times 120591119904= 119879

119904 2119879

119904 119873119879

119904 and 119879

119904is the time

interval of sampling which should be less than the reciprocalof the highest frequency Thus the aliasing effect would not

happen The pseudosnapshot data matrix can be constructedby using (17)ndash(21) as follows

R119891

= [R (119879119904) R (2119879

119904) R (119873119879

119904)]

R1= [R

1(119879119904) R

1(2119879

119904) R

1(119873119879

119904)]

R2= [R

2(119879119904) R

2(2119879

119904) R

2(119873119879

119904)]

R3= [R

3(119879119904) R

3(2119879

119904) R

3(119873119879

119904)]

R = [R119891(119879119904) R

119891(2119879

119904) R

119891(119873119879

119904)]

(31)

Equation (31) can be written as

R119891

= AΦ119891R119878

R1= AΦ

1Φ119891R119878

R2= AΦ

2Φ119891R119878

R3= AΦ

3Φ119891R119878

R = AR119878

(32)

where

R119878= [R

119878(119879119904) R

119878(2119879

119904) R

119878(119873119879

119904)] (33)

32 PARAFAC The element of a tensor X isin C119862times119863times119864 is 119909119894119895119896

and its 119875-component PARAFAC decomposition is given by

119909119888119889119890

=

119872

sum

119901=1

119904119888119901

119905119889119901

119906119890119901

(34)

where 119888 = 1 119862 119889 = 1 119863 and 119890 = 1 119864 119904119888119901 119905119889119901

and 119906

119890119901stand for the elements of S isin C119862times119875 T isin C119863times119875

and U isin C119864times119875 respectively The typical elements of theX119888

isin C119863times119864 X119889

isin C119864times119862 and X119890

isin C119862times119863are 119883119888(119889 119890) =

119883119889(119888 119890) = 119883

119890(119888 119889) = 119909

119888119889119890 respectivelyThe three way array

(TWA) X is ldquoslicedrdquo in three different directions

X119888= TΛ

119888(S)U119879 119888 = 1 119862

X119889= UΛ

119889(T) S119879 119889 = 1 119863

X119890= SΛ

119890(U)T119879 119890 = 1 119864

(35)

For a given matrix S Λ119888(S) means a diagonal matrix with

the diagonals given by the 119888th row of the matrix The symbolldquo⊙rdquo is used to denote the Khatri-Rao (KR) product As anexample the KR product of two matrices S isin C119862times119875 andT isin C119863times119875 of an identical number of columns is given by

S ⊙ T = [s1otimes t

1 s

119875otimes t

119875] isin C

119862119863times119875

(36)

The Scientific World Journal 5

where ldquootimesrdquo represents the Kronecker product Then thedefinition of Kronecker product of two vectors s isin C119862 andt isin C119863 is given by

s otimes t =

[[[[

[

1199041t

1199042t

119904119862t

]]]]

]

(37)

Based on the definition of KR product X119862119863times119864 can beexpressed as

X119862119863times119864 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

X119888=1

X119888=2

X119888=119862

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

TΛ1(S)

TΛ2(S)

TΛ119862(S)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

U119879 = (S ⊙ T)U119879 (38)

Similarly the matrix X119863119864times119862 and X119864119862times119863 can be expressed as

X119863119864times119862 = (T ⊙ U) S119879

X119864119862times119863 = (U ⊙ S)T119879(39)

Definition 1 (Kruskal rank [18]) The Kruskal rank or k-rankof a given matrix S isin C119862times119875 denoted by krank (S) is saidto be equal to 119903 when every collection of 119903 columns of (S)is linearly independent but there exists a collection of 119903 + 1

linearly dependent columns However the rank of S is said tobe themaximal number of linearly independent columnsThedefinition of rank(S) is more relaxed than that of krank(S)that is krank(S) le rank(S)

Theorem2 (Kruskal theorem [19]) Given a PARAFACmodelX isin C119862times119863times119864 then the decomposition of this PARAFAC modelis unique to permutation and scaling when

119896S + 119896T + 119896U ge 2119875 + 2 (40)

The matrix X is constructed by S T and U as

S = SΠΔ1 T = TΠΔ

2 U = UΠΔ

3 (41)

whereΠ stands for the permutation matrixΔ1Δ2 andΔ

3are

diagonal scaling matrices satisfying

Δ1Δ2Δ3= I (42)

where I is a unit matrix

The119872times119872times5TWAof the conformal array is constructedbased on PARAFAC model

R =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)

R ( 2)

R ( 3)

R ( 4)

R ( 5)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R119891

R1

R2

R3

R

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

AΦ119891R119878

AΦ1Φ119891R119878

AΦ2Φ119891R119878

AΦ3Φ119891R119878

AR119878

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ Q (43)

where Q represents the observed noise in practice In orderto make the decomposition of the PARAFAC unique thefollowing assumption must be satisfied

a119894(120596119894) = a

119895(120596119895) (44)

In other words for two incident signals 119904119894(119905) and 119904

119895(119905) with

different DOAs the following condition must be satisfied

119892119900119890eminus1198952120587119891119894((p119900119890∙u119894)119888) = 119892

119900119890eminus1198952120587119891119895((p119900119890∙u119895)119888) (45)

The k-rank of matrix D A and R119878are 119896

119863= min(5 119870) 119896

119860=

119870 and 119896119877119878

= 119870 respectively Thus when 119870 ge 2 the k-rankdecomposition of the PARAFAC is unique

Based on the definition of KRproduct (43) can bewrittenin three forms as follows

R = (D ⊙ A)R119878+ Q (46)

R119883

= (R119879119878⊙ D)A119879 + Q

119883 (47)

R119884

= (A ⊙ R119879119878)D119879 + Q

119884 (48)

where

D =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Λminus1

(Φ119891)

Λminus1

(Φ1Φ119891)

Λminus1

(Φ2Φ119891)

Λminus1

(Φ3Φ119891)

Λminus1

(1)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(49)

where Λminus1

(Φ119891) means the row vector constructed by the

diagonal elements of the diagonal matrixΦ119891

The matrices D A and R119878can be solved by the trilinear

alternating least squares (TALS) regression algorithm Dueto the existence of the observed noise (46) could be trans-formed as a least square problem as follows

minDAR

119878

10038171003817100381710038171003817R minus (D ⊙ A)R

119878

10038171003817100381710038171003817

2

119865

(50)

Then the least square estimator of R119878can be expressed as

R119878= argmin

R119878

10038171003817100381710038171003817R minus (D ⊙ A)R

119878

10038171003817100381710038171003817

2

119865

(51)

Similarly the solutions ofD and A can be expressed as

A119879 = (R119879119878⊙ D)

dagger

R119883

D119879 = (A ⊙ R119879119878)dagger

R119884

(52)

We can update the matrices D A and R119878alternately until

the algorithm converges the estimators of the three matricescan be acquired In the following experiments the COMFACalgorithm (the TALS regression algorithm is realized inpractice) is used to fit the 119898 times 119898 times 5 TWAThe initializationand fitting of the COMFAC is done in compressed spaceTheTucker 3 three-way model is used in data compression [21]

Now the ALS algorithm steps can be summarized asfollows

6 The Scientific World Journal

(1) Initialize A(0) isin C119872times119870 andD(0) isin C5times119870

(2) Initialize 120576 gt 0 and 119896 = 0

(3) If 120588(119896+1)minus120588(119896)

120588(119896)

gt 120576 calculatematricesAR119878 and

D by (51)-(52) Update just one matrix at each timethen 119896 rarr 119896 + 1

(4) Else 120588(119896+1)minus120588(119896)

120588(119896)

lt 120576 the iteration is terminated

In this paper 120588 stands for A R119878 andD respectively

33 The Joint Parameter Estimation The matrix D can beestimated by TALS regression algorithm then the estimatorof 120596

119891119896is given by

120596119896119894

= angle10038161003816100381610038161003816100381610038161003816

D1119896

D5119896

10038161003816100381610038161003816100381610038161003816

(53)

Because 1199031and 119903

2are real numbers D

2119896D

1119896is squared

to avoid the ambiguity caused by the positive and negativevalues of 119903

1and 119903

2 Then

1205961119896

= minus1

2angle([

1198922(120579119896 120593119896)

1198921(120579119896 120593119896)exp (minus119895120596

1119896)]

2

)

= minus1

2angle (exp (minus1198952120596

1119896))

= minus1

2angle([

D2119896

D1119896

]

2

)

(54)

Similarly

1205962119896

= minus1

2angle([

D3119896

D1119896

]

2

) (55)

1205963119896

= minus1

2angle([

D4119896

D1119896

]

2

) (56)

Taking (26) into (53) the frequency 119891119896of the 119896th incident

signal is expressed as

119891119896=

1

2120587120591angle

10038161003816100381610038161003816100381610038161003816

D1119896

D5119896

10038161003816100381610038161003816100381610038161003816

(57)

Assuming that 1205741119896

= sin(120579119896) cos(120593

119896) 120574

2119896= sin(120579

119896) sin(120593

119896)

and 1205743119896

= cos(120579119896) and solving (27) (28) (29) (55) (56) and

(57) simultaneously we can obtain the equation as follows

119888

2120587119891119896

[[[[[[[

[

1205961119896

1198891

1205962119896

1198892

1205963119896

1198893

]]]]]]]

]

= [

[

1199092minus 119909

11199102minus 119910

11199112minus 119911

1

1199093minus 119909

11199103minus 119910

11199113minus 119911

1

1199094minus 119909

11199104minus 119910

11199104minus 119910

1

]

]

[

[

1205741119896

1205742119896

1205743119896

]

]

(58)

The solution of the equation is represented as

[

[

1205741119896

1205742119896

1205743119896

]

]

=119888

2120587119891119894

[

[

1199092minus 119909

11199102minus 119910

11199112minus 119911

1

1199093minus 119909

11199103minus 119910

11199113minus 119911

1

1199094minus 119909

11199104minus 119910

11199104minus 119910

1

]

]

minus1[[[[[[[

[

1205961119896

1198891

1205962119896

1198892

1205963119896

1198893

]]]]]]]

]

(59)

There are two approaches to solve 120579119896and 120593

119896 The first

approach is

120579119896= arccos120574

3119896

120593119896= arcsin(

1205741119896

cos (120579119896)) or 120593

119896= arcsin(

1205742119896

sin (120579119896))

(60)

The second approach is

120593119896= arctan(

1205742119896

1205741119896

)

120579119896= arccos(

1205741119896

cos (120593119896)) or 120579

119896= arcsin(

1205742119896

sin (120593119896))

(61)

In this paper the first approach is employedIn order to ensure the uniqueness of the estimated

parameters the following condition must be satisfied

1003817100381710038171003817p2 minus p1

1003817100381710038171003817 le119888

(2max (119891119894))

=min (120582

119896)

2

1003817100381710038171003817p3 minus p1

1003817100381710038171003817 le119888

(2max (119891119894))

=min (120582

119896)

2

1003817100381710038171003817p4 minus p1

1003817100381710038171003817 le119888

(2max (119891119894))

=min (120582

119896)

2

(62)

According to Theorem 2 the matrices D A and R119878have the

same permutation matrix that is the 119896th column of matrixD corresponds to the 119896th column of the manifold matrix A

Based on the PARAFAC model and TALS regressionalgorithm the joint frequency and 2D-DOA estimation canbe accomplished The algorithm steps are summarized asfollows

(1) choose four guiding elements and obtain the accuratepositions of the elements

(2) calculate the delay correlation function between thefour elements mentioned above and the whole ele-ments of the conformal array

(3) calculate the matrices R119891(120591) R

1(120591) R

2(120591) R

3(120591) and

R(0) respectively according to (5)(4) calculate the pseudosnapshot data matrix based on

(17)ndash(21) R119891(120591) R

1(120591) R

2(120591) R

3(120591) and R(0) are

sampled at time delay 120591119904 respectively

(5) construct the PARAFAC model based on (43)(6) estimate the matrix D using TALS regression algo-

rithm

The Scientific World Journal 7

(7) obtain12059611989111989612059611198961205962119896 and120596

3119896by using the estimator of

the matrixD and calculating (53)ndash(56)(8) obtain the joint frequency and 2D-DOA estimation

by calculate (57)ndash(59) or (57) (60) and (61)

4 Simulation Results

This section focuses on several sets of simulation results todemonstrate the performance of the proposed algorithmIn all simulation examples 200 Monte Carlo trials areconsidered and the root mean square error (RMSE) is usedas our performance measure which is defined as

RMSE = radic1

200

200

sum

120578=1

[(120579119896120578

minus 120579)2

+ (120593119896120578

minus 120593)2

] (63)

where 120579119894119905and 120593

119894119905are the elevation and azimuth estimators of

the 119896th incident signal in the 120578th trialThe successful rate is defined as the proportion of the

number of the successful experiments to the total numberof the experiments For frequency estimation a successfulexperiment is defined as the experiment with estimationerror of less than 2MHz for DOA estimation a successfulexperiment is defined as the experimentwith estimation errorof less than 2 degree

In order to simplify the transformation from the globalcoordinate to the local coordinate only the cylinder withsingle curvature is considered in the simulations The arraystructure is shown in Figure 2 is only a subarray which onlycovers 120

∘ Then three subarrays can estimate the wholespace The highest frequency of the incident signal is 119891max =

2GHz 120582min = 119888119891max The radius of the cylinder is 5120582min theadjacent element spacing in the same intersecting surface ofthe cylinder is 120582min4 and the adjacent intersecting surfacespacing is 120582min4 120582min stands for the shortest wavelength ofthe incident signal The position of the reference element is1199011= (0 5120582 0)The element pattern is defined in the local coordinate

of each element thus the transformation from the globalcoordinate to the local coordinate of elevation 120579

119896and azimuth

120593119896of the incident signal must be accomplished For the

cylindrical conformal array the corresponding Euler rotationangles are

119863 = minusΘ 119864 = minus1205872 119865 = 0 (64)

respectively 119863 represents the rotation angle based on theright-hand rule in the first rotation and the 119885-axis is therotation axis 119864 represents the rotation angle in the secondrotation and the 119884-axis is the rotation axis and 119865 representsthe rotation angle in the third rotation and the 119883-axis is therotation axis Θ represents the angle between the positionvector of the element and the positive direction of 119883-axis

The unit vector of the 119896th incident signal (120579119896 120593119896) in the

global coordinate can be expressed as

119909 = sin (120579119896) cos (120593

119896) 119910 = sin (120579

119896) sin (120593

119896)

119911 = cos (120579119896)

(65)

Z

X

Y

O

pm

p3

p4

p2p1

Figure 2 The structure of the cylindrical conformal array

Based on the Euler rotation matrix the unit vector inglobal coordinate can be transformed into the local coordi-nate of the 119898th element

Consider

[119909119898

119910119898

119911119898]119879

= 119877 (119863119898 119864119898 119865119898) [119909 119910 119911]

119879

(66)

where

119877 (119863119898 119864119898 119865119898) = [

[

cos119865 sin119865 0

minus sin119865 cos119865 0

0 0 1

]

]

times [

[

cos119864 0 minus sin119864

0 1 0

sin119864 0 cos119864]

]

times [

[

cos119863 sin119863 0

minus sin119863 cos119863 0

0 0 1

]

]

(67)

Then the elevation 120579 and azimuth 120593 of the 119896th incident signalin the local coordinate of the119898th element can be representedas

120593119898119896

= arctan (119910119898119909119898) 120579

119898119896= arccos (119911

119898) (68)

Thus the element pattern transformation from the globalcoordinate to the local coordinate is completed

First the curves of RMSE of frequency and angle and thesuccessful probability of angle in different SNR are plottedin Figure 3 using the proposed algorithm The number ofsnapshots is 200 the number of pseudosnapshots is 100SNR isin [5 30] and the noise is AWGN which is independentof incident signals The sampling frequency is 119891

119904= 1119879

119904=

5GHzThe elevation 120579 azimuth120593 and frequency of these twonarrow incident signals are (100∘ 60∘ 1 GHz) and (95∘ 50∘2 GHz) respectively which are independent of each other

8 The Scientific World Journal

5 10 15 20 25 300

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Succ

ess p

roba

bilit

y (

)

Frequency 1Frequency 2

(a)

5 10 15 20 25 300

05

1

15

2

25

3

35

4

RMSE

(MH

z)

SNR (dB)

Frequency 1Frequency 2

(b)

5 10 15 20 25 3040

50

60

70

80

90

100

SNR (dB)

Succ

ess p

roba

bilit

y (

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(c)

5 10 15 20 25 300

002

004

006

008

01

012

014

016

SNR (dB)

RMSE

(deg

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(d)

Figure 3The RMSE and successful probability with different SNR (a)The successful probability of frequencies (b) the RMSE of frequency(c) the successful probability of angles and (d) the RMSE of angles

The number of elements are 10 including 4 guiding elementsand other 6 instrumental elementsThe element pattern usedin simulation is the lowest order circular patch model [8 22]

119892120579(120579 120593) = 119869

2(120587119889

120582sin 120579) minus 119869

0(120587119889

120582sin 120579)

times (cos120593 minus 119895 sin120593) 0 le 120579 le120587

2

119892120593(120579 120593) = 119869

2(120587119889

120582sin 120579) + 119869

0(120587119889

120582sin 120579)

times cos 120579 (sin120593 minus 119895 cos120593) 0 le 120579 le120587

2

119892120579(120579 120593) = 119892

120593(120579 120593) = 0 otherwise

(69)

where 1198690and 119869

2are the zeroth- and second-order Bessel func-

tions of the first kind The element pattern transformation isachieved with the method proposed above

Figure 3 shows that the RMSE of frequency and angledecreases while the SNR increases meanwhile the successprobability of frequency and angle increase as the SNRincreases The estimated performance of the two frequenciesare very close which can be seen in Figures 3(a) and 3(b)As shown in Figure 3(a) the successful probability is largerthan 80 when the SNR is larger than 15 dB The RMSEof frequency is less than 2MHz when SNR is larger than10 dB as shown in Figure 3(b) In Figure 3(c) the successfulprobability of azimuths is slightly lower than that of theelevations The success probability of the angle is almost

The Scientific World Journal 9

100 200 300 400 500 600 700 80020

30

40

50

60

70

80

90

100

Snapshot number

Succ

ess p

roba

bilit

y (

)

Frequency 1Frequency 2

(a)

100 200 300 400 500 600 700 80008

1

12

14

16

18

2

Snapshot number

RMSE

(MH

z)

Frequency 1Frequency 2

(b)

100 200 300 400 500 600 700 80095

955

96

965

97

975

98

985

99

995

100

Snapshot number (dB)

Succ

ess p

roba

bilit

y (

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(c)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

100 200 300 400 500 600 700 800Snapshot number (dB)

002

004

006

008

01

012

014

016RM

SE (d

eg)

(d)

Figure 4 The RMSE and successful probability with different snapshot number (a) The successful probability of frequencies (b) the RMSEof frequency (c) the successful probability of angles and (d) the RMSE of angles

100 when the SNR reaches 15 dB Figure 3(d) shows thatthe RMSE is smaller than 016

∘ when SNR is larger than5 dB The RMSE of azimuth is larger than that of elevationThe estimation results of frequency and elevation are appliedto estimate azimuth The phenomenon in Figure 3(d) canbe interpreted as the existence of the estimated error offrequency and elevation Next the RMSE of frequency andangle as well as the success probability of angle in differentsnapshot number are depicted in Figure 4 SNR = 10 dBother simulation conditions are identical with the previousexperiment In Figure 4 the RMSE of frequency and angledecreases as the snapshot number increases and the successprobability of frequency and angle rises as the snapshotnumber increases As shown in Figure 4(a) the successfulprobability is larger than 80 when the number of snapshot

is larger than 500 Figure 4(b) shows that the RMSE offrequency 1 is slightly larger than that of frequency 2 Whenthe snapshot number is larger than 500 the RMSE dropsbelow 1MHz meaning that the proposed algorithm achievedthree orders of magnitude reduction Figures 4(c) and 4(d)show that the success probability of azimuth is lower thanthat of the elevation and the RMSE of azimuth are largerthan that of elevationThe reasonwhy this happens is becausethe estimation error of frequency and elevation affects theestimation performance of azimuth

5 Conclusion

Due to the varying curvature of the surface of the conformalcarrier the pattern of each element of the conformal array

10 The Scientific World Journal

is different Thus the conventional algorithms could notbe used for conformal array In this paper a novel jointfrequency and 2D-DOA estimation algorithm with highaccuracy are proposed Both spatial and time samplingare utilized to construct the spatial-time matrix The delaycorrelation function is used to suppress noiseThe PARAFACmodel is used for parameter estimation without parameterpairing Only four elements are needed and the positionsof these elements should be known accurately Other instru-mental elements can be flexibly arranged on the surface ofthe conformal carrier The algorithm proposed in this papercan be extended to estimate 2D-DOA straightly with littlemodification The simulation results verify the effectivenessof the proposed algorithm It can be expected that theproposed algorithm would have an application prospect inthe parameter estimation of conformal array In the futurework we will focus on the application of the proposedalgorithm [23ndash26]

Acknowledgments

This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)

References

[1] L Josefsson and P Persson Conformal Array Antenna Theoryand Design IEEE Press Piscataway NJ USA 2006

[2] W T Li X W Shi Y Q Hei S F Liu and J Zhu ldquoAhybrid optimization algorithmand its application for conformalarray pattern synthesisrdquo IEEE Transactions on Antennas andPropagation vol 58 no 10 pp 3401ndash3406 2010

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

[4] 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

[5] P Hyber M Jansson and B Ottersten ldquoArray interpolation andbias reductionrdquo IEEE Transaction on Signal Processing vol 52no 10 pp 2711ndash2720 2004

[6] P Hyberg M Jansson and B Ottersten ldquoArray interpolationand DOAMSE reductionrdquo IEEE Transactions on Signal Process-ing vol 53 no 12 pp 4464ndash4471 2005

[7] F Belloni A Richter and V Koivunen ldquoDOA estimationvia manifold separation for arbitrary array structuresrdquo IEEETransactions on Signal Processing vol 55 no 10 pp 4800ndash48102007

[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008

[9] P Yang F Yang and Z Nie ldquoDOA estimation using MUSICalgorithm on a cylinderical conformal arrayrdquo in Proceedingsof the IEEE Antennas and Propagation Society InternationalSymposium pp 5299ndash5302 June 2007

[10] P Yang F Yang and Z P Nie ldquoDOA estimation with sub-arraydivided technique and interporlated esprit algorithmon a cylin-drical conformal array antennardquo Progress in ElectromagneticsResearch vol 103 pp 201ndash216 2010

[11] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011

[12] L Zou J Lasenby and Z He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012

[13] Y-Y Wang J-T Chen andW-H Fang ldquoTST-MUSIC for jointDOA-delay estimationrdquo IEEE Transactions on Signal Processingvol 49 no 4 pp 721ndash729 2001

[14] J-D Lin W-H Fang Y-Y Wang and J-T Chen ldquoFSF MUSICfor joint DOA and frequency estimation and its performanceanalysisrdquo IEEE Transactions on Signal Processing vol 54 no 12pp 4529ndash4542 2006

[15] M Djeddou A Belouchrani and S Aouada ldquoMaximum likeli-hood angle-frequency estimation in partially known correlatednoise for low-elevation targetsrdquo IEEE Transactions on SignalProcessing vol 53 no 8 pp 3057ndash3064 2005

[16] R B Cattell ldquolsquoParallel proportional profilesrsquo and other prin-ciples for determining the choice of factors by rotationrdquo Psy-chometrika vol 9 no 4 pp 267ndash283 1944

[17] J D Carroll and J-J Chang ldquoAnalysis of individual differencesin multidimensional scaling via an n-way generalization ofldquoEckart-Youngrdquo decompositionrdquo Psychometrika vol 35 no 3pp 283ndash319 1970

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

[19] 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

[20] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 77ndash83 1999

[21] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fittingof the Complex PARAFAC Model in 3-Drdquo 1998 httpwwwtelecomtucgrsimnikos

[22] J R James P S Hall and C WoodMicrostrip Antenna Theoryand Design Peter Peregrinus New York NY USA 1981

[23] S Yin S Ding and H Luo ldquoReal-time implementation of faulttolerant control system with performance optimizationrdquo IEEETransactions on Industrial Electronics vol 61 no 5 pp 2402ndash2411 2014

[24] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[25] S Yin S Ding A Haghani andH Hao ldquoData-drivenmonitor-ing for stochastic systems and its application on batch processrdquoInternational Journal of Systems Science vol 44 no 7 pp 1366ndash1376 2013

[26] S Yin X Yang and H Karimi ldquoData-driven adaptive observerfor fault diagnosisrdquoMathematical Problems in Engineering vol2012 Article ID 832836 21 pages 2012

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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

The Scientific World Journal 5

where ldquootimesrdquo represents the Kronecker product Then thedefinition of Kronecker product of two vectors s isin C119862 andt isin C119863 is given by

s otimes t =

[[[[

[

1199041t

1199042t

119904119862t

]]]]

]

(37)

Based on the definition of KR product X119862119863times119864 can beexpressed as

X119862119863times119864 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

X119888=1

X119888=2

X119888=119862

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

TΛ1(S)

TΛ2(S)

TΛ119862(S)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

U119879 = (S ⊙ T)U119879 (38)

Similarly the matrix X119863119864times119862 and X119864119862times119863 can be expressed as

X119863119864times119862 = (T ⊙ U) S119879

X119864119862times119863 = (U ⊙ S)T119879(39)

Definition 1 (Kruskal rank [18]) The Kruskal rank or k-rankof a given matrix S isin C119862times119875 denoted by krank (S) is saidto be equal to 119903 when every collection of 119903 columns of (S)is linearly independent but there exists a collection of 119903 + 1

linearly dependent columns However the rank of S is said tobe themaximal number of linearly independent columnsThedefinition of rank(S) is more relaxed than that of krank(S)that is krank(S) le rank(S)

Theorem2 (Kruskal theorem [19]) Given a PARAFACmodelX isin C119862times119863times119864 then the decomposition of this PARAFAC modelis unique to permutation and scaling when

119896S + 119896T + 119896U ge 2119875 + 2 (40)

The matrix X is constructed by S T and U as

S = SΠΔ1 T = TΠΔ

2 U = UΠΔ

3 (41)

whereΠ stands for the permutation matrixΔ1Δ2 andΔ

3are

diagonal scaling matrices satisfying

Δ1Δ2Δ3= I (42)

where I is a unit matrix

The119872times119872times5TWAof the conformal array is constructedbased on PARAFAC model

R =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)

R ( 2)

R ( 3)

R ( 4)

R ( 5)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R119891

R1

R2

R3

R

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

AΦ119891R119878

AΦ1Φ119891R119878

AΦ2Φ119891R119878

AΦ3Φ119891R119878

AR119878

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ Q (43)

where Q represents the observed noise in practice In orderto make the decomposition of the PARAFAC unique thefollowing assumption must be satisfied

a119894(120596119894) = a

119895(120596119895) (44)

In other words for two incident signals 119904119894(119905) and 119904

119895(119905) with

different DOAs the following condition must be satisfied

119892119900119890eminus1198952120587119891119894((p119900119890∙u119894)119888) = 119892

119900119890eminus1198952120587119891119895((p119900119890∙u119895)119888) (45)

The k-rank of matrix D A and R119878are 119896

119863= min(5 119870) 119896

119860=

119870 and 119896119877119878

= 119870 respectively Thus when 119870 ge 2 the k-rankdecomposition of the PARAFAC is unique

Based on the definition of KRproduct (43) can bewrittenin three forms as follows

R = (D ⊙ A)R119878+ Q (46)

R119883

= (R119879119878⊙ D)A119879 + Q

119883 (47)

R119884

= (A ⊙ R119879119878)D119879 + Q

119884 (48)

where

D =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Λminus1

(Φ119891)

Λminus1

(Φ1Φ119891)

Λminus1

(Φ2Φ119891)

Λminus1

(Φ3Φ119891)

Λminus1

(1)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(49)

where Λminus1

(Φ119891) means the row vector constructed by the

diagonal elements of the diagonal matrixΦ119891

The matrices D A and R119878can be solved by the trilinear

alternating least squares (TALS) regression algorithm Dueto the existence of the observed noise (46) could be trans-formed as a least square problem as follows

minDAR

119878

10038171003817100381710038171003817R minus (D ⊙ A)R

119878

10038171003817100381710038171003817

2

119865

(50)

Then the least square estimator of R119878can be expressed as

R119878= argmin

R119878

10038171003817100381710038171003817R minus (D ⊙ A)R

119878

10038171003817100381710038171003817

2

119865

(51)

Similarly the solutions ofD and A can be expressed as

A119879 = (R119879119878⊙ D)

dagger

R119883

D119879 = (A ⊙ R119879119878)dagger

R119884

(52)

We can update the matrices D A and R119878alternately until

the algorithm converges the estimators of the three matricescan be acquired In the following experiments the COMFACalgorithm (the TALS regression algorithm is realized inpractice) is used to fit the 119898 times 119898 times 5 TWAThe initializationand fitting of the COMFAC is done in compressed spaceTheTucker 3 three-way model is used in data compression [21]

Now the ALS algorithm steps can be summarized asfollows

6 The Scientific World Journal

(1) Initialize A(0) isin C119872times119870 andD(0) isin C5times119870

(2) Initialize 120576 gt 0 and 119896 = 0

(3) If 120588(119896+1)minus120588(119896)

120588(119896)

gt 120576 calculatematricesAR119878 and

D by (51)-(52) Update just one matrix at each timethen 119896 rarr 119896 + 1

(4) Else 120588(119896+1)minus120588(119896)

120588(119896)

lt 120576 the iteration is terminated

In this paper 120588 stands for A R119878 andD respectively

33 The Joint Parameter Estimation The matrix D can beestimated by TALS regression algorithm then the estimatorof 120596

119891119896is given by

120596119896119894

= angle10038161003816100381610038161003816100381610038161003816

D1119896

D5119896

10038161003816100381610038161003816100381610038161003816

(53)

Because 1199031and 119903

2are real numbers D

2119896D

1119896is squared

to avoid the ambiguity caused by the positive and negativevalues of 119903

1and 119903

2 Then

1205961119896

= minus1

2angle([

1198922(120579119896 120593119896)

1198921(120579119896 120593119896)exp (minus119895120596

1119896)]

2

)

= minus1

2angle (exp (minus1198952120596

1119896))

= minus1

2angle([

D2119896

D1119896

]

2

)

(54)

Similarly

1205962119896

= minus1

2angle([

D3119896

D1119896

]

2

) (55)

1205963119896

= minus1

2angle([

D4119896

D1119896

]

2

) (56)

Taking (26) into (53) the frequency 119891119896of the 119896th incident

signal is expressed as

119891119896=

1

2120587120591angle

10038161003816100381610038161003816100381610038161003816

D1119896

D5119896

10038161003816100381610038161003816100381610038161003816

(57)

Assuming that 1205741119896

= sin(120579119896) cos(120593

119896) 120574

2119896= sin(120579

119896) sin(120593

119896)

and 1205743119896

= cos(120579119896) and solving (27) (28) (29) (55) (56) and

(57) simultaneously we can obtain the equation as follows

119888

2120587119891119896

[[[[[[[

[

1205961119896

1198891

1205962119896

1198892

1205963119896

1198893

]]]]]]]

]

= [

[

1199092minus 119909

11199102minus 119910

11199112minus 119911

1

1199093minus 119909

11199103minus 119910

11199113minus 119911

1

1199094minus 119909

11199104minus 119910

11199104minus 119910

1

]

]

[

[

1205741119896

1205742119896

1205743119896

]

]

(58)

The solution of the equation is represented as

[

[

1205741119896

1205742119896

1205743119896

]

]

=119888

2120587119891119894

[

[

1199092minus 119909

11199102minus 119910

11199112minus 119911

1

1199093minus 119909

11199103minus 119910

11199113minus 119911

1

1199094minus 119909

11199104minus 119910

11199104minus 119910

1

]

]

minus1[[[[[[[

[

1205961119896

1198891

1205962119896

1198892

1205963119896

1198893

]]]]]]]

]

(59)

There are two approaches to solve 120579119896and 120593

119896 The first

approach is

120579119896= arccos120574

3119896

120593119896= arcsin(

1205741119896

cos (120579119896)) or 120593

119896= arcsin(

1205742119896

sin (120579119896))

(60)

The second approach is

120593119896= arctan(

1205742119896

1205741119896

)

120579119896= arccos(

1205741119896

cos (120593119896)) or 120579

119896= arcsin(

1205742119896

sin (120593119896))

(61)

In this paper the first approach is employedIn order to ensure the uniqueness of the estimated

parameters the following condition must be satisfied

1003817100381710038171003817p2 minus p1

1003817100381710038171003817 le119888

(2max (119891119894))

=min (120582

119896)

2

1003817100381710038171003817p3 minus p1

1003817100381710038171003817 le119888

(2max (119891119894))

=min (120582

119896)

2

1003817100381710038171003817p4 minus p1

1003817100381710038171003817 le119888

(2max (119891119894))

=min (120582

119896)

2

(62)

According to Theorem 2 the matrices D A and R119878have the

same permutation matrix that is the 119896th column of matrixD corresponds to the 119896th column of the manifold matrix A

Based on the PARAFAC model and TALS regressionalgorithm the joint frequency and 2D-DOA estimation canbe accomplished The algorithm steps are summarized asfollows

(1) choose four guiding elements and obtain the accuratepositions of the elements

(2) calculate the delay correlation function between thefour elements mentioned above and the whole ele-ments of the conformal array

(3) calculate the matrices R119891(120591) R

1(120591) R

2(120591) R

3(120591) and

R(0) respectively according to (5)(4) calculate the pseudosnapshot data matrix based on

(17)ndash(21) R119891(120591) R

1(120591) R

2(120591) R

3(120591) and R(0) are

sampled at time delay 120591119904 respectively

(5) construct the PARAFAC model based on (43)(6) estimate the matrix D using TALS regression algo-

rithm

The Scientific World Journal 7

(7) obtain12059611989111989612059611198961205962119896 and120596

3119896by using the estimator of

the matrixD and calculating (53)ndash(56)(8) obtain the joint frequency and 2D-DOA estimation

by calculate (57)ndash(59) or (57) (60) and (61)

4 Simulation Results

This section focuses on several sets of simulation results todemonstrate the performance of the proposed algorithmIn all simulation examples 200 Monte Carlo trials areconsidered and the root mean square error (RMSE) is usedas our performance measure which is defined as

RMSE = radic1

200

200

sum

120578=1

[(120579119896120578

minus 120579)2

+ (120593119896120578

minus 120593)2

] (63)

where 120579119894119905and 120593

119894119905are the elevation and azimuth estimators of

the 119896th incident signal in the 120578th trialThe successful rate is defined as the proportion of the

number of the successful experiments to the total numberof the experiments For frequency estimation a successfulexperiment is defined as the experiment with estimationerror of less than 2MHz for DOA estimation a successfulexperiment is defined as the experimentwith estimation errorof less than 2 degree

In order to simplify the transformation from the globalcoordinate to the local coordinate only the cylinder withsingle curvature is considered in the simulations The arraystructure is shown in Figure 2 is only a subarray which onlycovers 120

∘ Then three subarrays can estimate the wholespace The highest frequency of the incident signal is 119891max =

2GHz 120582min = 119888119891max The radius of the cylinder is 5120582min theadjacent element spacing in the same intersecting surface ofthe cylinder is 120582min4 and the adjacent intersecting surfacespacing is 120582min4 120582min stands for the shortest wavelength ofthe incident signal The position of the reference element is1199011= (0 5120582 0)The element pattern is defined in the local coordinate

of each element thus the transformation from the globalcoordinate to the local coordinate of elevation 120579

119896and azimuth

120593119896of the incident signal must be accomplished For the

cylindrical conformal array the corresponding Euler rotationangles are

119863 = minusΘ 119864 = minus1205872 119865 = 0 (64)

respectively 119863 represents the rotation angle based on theright-hand rule in the first rotation and the 119885-axis is therotation axis 119864 represents the rotation angle in the secondrotation and the 119884-axis is the rotation axis and 119865 representsthe rotation angle in the third rotation and the 119883-axis is therotation axis Θ represents the angle between the positionvector of the element and the positive direction of 119883-axis

The unit vector of the 119896th incident signal (120579119896 120593119896) in the

global coordinate can be expressed as

119909 = sin (120579119896) cos (120593

119896) 119910 = sin (120579

119896) sin (120593

119896)

119911 = cos (120579119896)

(65)

Z

X

Y

O

pm

p3

p4

p2p1

Figure 2 The structure of the cylindrical conformal array

Based on the Euler rotation matrix the unit vector inglobal coordinate can be transformed into the local coordi-nate of the 119898th element

Consider

[119909119898

119910119898

119911119898]119879

= 119877 (119863119898 119864119898 119865119898) [119909 119910 119911]

119879

(66)

where

119877 (119863119898 119864119898 119865119898) = [

[

cos119865 sin119865 0

minus sin119865 cos119865 0

0 0 1

]

]

times [

[

cos119864 0 minus sin119864

0 1 0

sin119864 0 cos119864]

]

times [

[

cos119863 sin119863 0

minus sin119863 cos119863 0

0 0 1

]

]

(67)

Then the elevation 120579 and azimuth 120593 of the 119896th incident signalin the local coordinate of the119898th element can be representedas

120593119898119896

= arctan (119910119898119909119898) 120579

119898119896= arccos (119911

119898) (68)

Thus the element pattern transformation from the globalcoordinate to the local coordinate is completed

First the curves of RMSE of frequency and angle and thesuccessful probability of angle in different SNR are plottedin Figure 3 using the proposed algorithm The number ofsnapshots is 200 the number of pseudosnapshots is 100SNR isin [5 30] and the noise is AWGN which is independentof incident signals The sampling frequency is 119891

119904= 1119879

119904=

5GHzThe elevation 120579 azimuth120593 and frequency of these twonarrow incident signals are (100∘ 60∘ 1 GHz) and (95∘ 50∘2 GHz) respectively which are independent of each other

8 The Scientific World Journal

5 10 15 20 25 300

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Succ

ess p

roba

bilit

y (

)

Frequency 1Frequency 2

(a)

5 10 15 20 25 300

05

1

15

2

25

3

35

4

RMSE

(MH

z)

SNR (dB)

Frequency 1Frequency 2

(b)

5 10 15 20 25 3040

50

60

70

80

90

100

SNR (dB)

Succ

ess p

roba

bilit

y (

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(c)

5 10 15 20 25 300

002

004

006

008

01

012

014

016

SNR (dB)

RMSE

(deg

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(d)

Figure 3The RMSE and successful probability with different SNR (a)The successful probability of frequencies (b) the RMSE of frequency(c) the successful probability of angles and (d) the RMSE of angles

The number of elements are 10 including 4 guiding elementsand other 6 instrumental elementsThe element pattern usedin simulation is the lowest order circular patch model [8 22]

119892120579(120579 120593) = 119869

2(120587119889

120582sin 120579) minus 119869

0(120587119889

120582sin 120579)

times (cos120593 minus 119895 sin120593) 0 le 120579 le120587

2

119892120593(120579 120593) = 119869

2(120587119889

120582sin 120579) + 119869

0(120587119889

120582sin 120579)

times cos 120579 (sin120593 minus 119895 cos120593) 0 le 120579 le120587

2

119892120579(120579 120593) = 119892

120593(120579 120593) = 0 otherwise

(69)

where 1198690and 119869

2are the zeroth- and second-order Bessel func-

tions of the first kind The element pattern transformation isachieved with the method proposed above

Figure 3 shows that the RMSE of frequency and angledecreases while the SNR increases meanwhile the successprobability of frequency and angle increase as the SNRincreases The estimated performance of the two frequenciesare very close which can be seen in Figures 3(a) and 3(b)As shown in Figure 3(a) the successful probability is largerthan 80 when the SNR is larger than 15 dB The RMSEof frequency is less than 2MHz when SNR is larger than10 dB as shown in Figure 3(b) In Figure 3(c) the successfulprobability of azimuths is slightly lower than that of theelevations The success probability of the angle is almost

The Scientific World Journal 9

100 200 300 400 500 600 700 80020

30

40

50

60

70

80

90

100

Snapshot number

Succ

ess p

roba

bilit

y (

)

Frequency 1Frequency 2

(a)

100 200 300 400 500 600 700 80008

1

12

14

16

18

2

Snapshot number

RMSE

(MH

z)

Frequency 1Frequency 2

(b)

100 200 300 400 500 600 700 80095

955

96

965

97

975

98

985

99

995

100

Snapshot number (dB)

Succ

ess p

roba

bilit

y (

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(c)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

100 200 300 400 500 600 700 800Snapshot number (dB)

002

004

006

008

01

012

014

016RM

SE (d

eg)

(d)

Figure 4 The RMSE and successful probability with different snapshot number (a) The successful probability of frequencies (b) the RMSEof frequency (c) the successful probability of angles and (d) the RMSE of angles

100 when the SNR reaches 15 dB Figure 3(d) shows thatthe RMSE is smaller than 016

∘ when SNR is larger than5 dB The RMSE of azimuth is larger than that of elevationThe estimation results of frequency and elevation are appliedto estimate azimuth The phenomenon in Figure 3(d) canbe interpreted as the existence of the estimated error offrequency and elevation Next the RMSE of frequency andangle as well as the success probability of angle in differentsnapshot number are depicted in Figure 4 SNR = 10 dBother simulation conditions are identical with the previousexperiment In Figure 4 the RMSE of frequency and angledecreases as the snapshot number increases and the successprobability of frequency and angle rises as the snapshotnumber increases As shown in Figure 4(a) the successfulprobability is larger than 80 when the number of snapshot

is larger than 500 Figure 4(b) shows that the RMSE offrequency 1 is slightly larger than that of frequency 2 Whenthe snapshot number is larger than 500 the RMSE dropsbelow 1MHz meaning that the proposed algorithm achievedthree orders of magnitude reduction Figures 4(c) and 4(d)show that the success probability of azimuth is lower thanthat of the elevation and the RMSE of azimuth are largerthan that of elevationThe reasonwhy this happens is becausethe estimation error of frequency and elevation affects theestimation performance of azimuth

5 Conclusion

Due to the varying curvature of the surface of the conformalcarrier the pattern of each element of the conformal array

10 The Scientific World Journal

is different Thus the conventional algorithms could notbe used for conformal array In this paper a novel jointfrequency and 2D-DOA estimation algorithm with highaccuracy are proposed Both spatial and time samplingare utilized to construct the spatial-time matrix The delaycorrelation function is used to suppress noiseThe PARAFACmodel is used for parameter estimation without parameterpairing Only four elements are needed and the positionsof these elements should be known accurately Other instru-mental elements can be flexibly arranged on the surface ofthe conformal carrier The algorithm proposed in this papercan be extended to estimate 2D-DOA straightly with littlemodification The simulation results verify the effectivenessof the proposed algorithm It can be expected that theproposed algorithm would have an application prospect inthe parameter estimation of conformal array In the futurework we will focus on the application of the proposedalgorithm [23ndash26]

Acknowledgments

This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)

References

[1] L Josefsson and P Persson Conformal Array Antenna Theoryand Design IEEE Press Piscataway NJ USA 2006

[2] W T Li X W Shi Y Q Hei S F Liu and J Zhu ldquoAhybrid optimization algorithmand its application for conformalarray pattern synthesisrdquo IEEE Transactions on Antennas andPropagation vol 58 no 10 pp 3401ndash3406 2010

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

[4] 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

[5] P Hyber M Jansson and B Ottersten ldquoArray interpolation andbias reductionrdquo IEEE Transaction on Signal Processing vol 52no 10 pp 2711ndash2720 2004

[6] P Hyberg M Jansson and B Ottersten ldquoArray interpolationand DOAMSE reductionrdquo IEEE Transactions on Signal Process-ing vol 53 no 12 pp 4464ndash4471 2005

[7] F Belloni A Richter and V Koivunen ldquoDOA estimationvia manifold separation for arbitrary array structuresrdquo IEEETransactions on Signal Processing vol 55 no 10 pp 4800ndash48102007

[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008

[9] P Yang F Yang and Z Nie ldquoDOA estimation using MUSICalgorithm on a cylinderical conformal arrayrdquo in Proceedingsof the IEEE Antennas and Propagation Society InternationalSymposium pp 5299ndash5302 June 2007

[10] P Yang F Yang and Z P Nie ldquoDOA estimation with sub-arraydivided technique and interporlated esprit algorithmon a cylin-drical conformal array antennardquo Progress in ElectromagneticsResearch vol 103 pp 201ndash216 2010

[11] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011

[12] L Zou J Lasenby and Z He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012

[13] Y-Y Wang J-T Chen andW-H Fang ldquoTST-MUSIC for jointDOA-delay estimationrdquo IEEE Transactions on Signal Processingvol 49 no 4 pp 721ndash729 2001

[14] J-D Lin W-H Fang Y-Y Wang and J-T Chen ldquoFSF MUSICfor joint DOA and frequency estimation and its performanceanalysisrdquo IEEE Transactions on Signal Processing vol 54 no 12pp 4529ndash4542 2006

[15] M Djeddou A Belouchrani and S Aouada ldquoMaximum likeli-hood angle-frequency estimation in partially known correlatednoise for low-elevation targetsrdquo IEEE Transactions on SignalProcessing vol 53 no 8 pp 3057ndash3064 2005

[16] R B Cattell ldquolsquoParallel proportional profilesrsquo and other prin-ciples for determining the choice of factors by rotationrdquo Psy-chometrika vol 9 no 4 pp 267ndash283 1944

[17] J D Carroll and J-J Chang ldquoAnalysis of individual differencesin multidimensional scaling via an n-way generalization ofldquoEckart-Youngrdquo decompositionrdquo Psychometrika vol 35 no 3pp 283ndash319 1970

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

[19] 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

[20] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 77ndash83 1999

[21] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fittingof the Complex PARAFAC Model in 3-Drdquo 1998 httpwwwtelecomtucgrsimnikos

[22] J R James P S Hall and C WoodMicrostrip Antenna Theoryand Design Peter Peregrinus New York NY USA 1981

[23] S Yin S Ding and H Luo ldquoReal-time implementation of faulttolerant control system with performance optimizationrdquo IEEETransactions on Industrial Electronics vol 61 no 5 pp 2402ndash2411 2014

[24] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[25] S Yin S Ding A Haghani andH Hao ldquoData-drivenmonitor-ing for stochastic systems and its application on batch processrdquoInternational Journal of Systems Science vol 44 no 7 pp 1366ndash1376 2013

[26] S Yin X Yang and H Karimi ldquoData-driven adaptive observerfor fault diagnosisrdquoMathematical Problems in Engineering vol2012 Article ID 832836 21 pages 2012

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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

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DistributedSensor Networks

International Journal of

6 The Scientific World Journal

(1) Initialize A(0) isin C119872times119870 andD(0) isin C5times119870

(2) Initialize 120576 gt 0 and 119896 = 0

(3) If 120588(119896+1)minus120588(119896)

120588(119896)

gt 120576 calculatematricesAR119878 and

D by (51)-(52) Update just one matrix at each timethen 119896 rarr 119896 + 1

(4) Else 120588(119896+1)minus120588(119896)

120588(119896)

lt 120576 the iteration is terminated

In this paper 120588 stands for A R119878 andD respectively

33 The Joint Parameter Estimation The matrix D can beestimated by TALS regression algorithm then the estimatorof 120596

119891119896is given by

120596119896119894

= angle10038161003816100381610038161003816100381610038161003816

D1119896

D5119896

10038161003816100381610038161003816100381610038161003816

(53)

Because 1199031and 119903

2are real numbers D

2119896D

1119896is squared

to avoid the ambiguity caused by the positive and negativevalues of 119903

1and 119903

2 Then

1205961119896

= minus1

2angle([

1198922(120579119896 120593119896)

1198921(120579119896 120593119896)exp (minus119895120596

1119896)]

2

)

= minus1

2angle (exp (minus1198952120596

1119896))

= minus1

2angle([

D2119896

D1119896

]

2

)

(54)

Similarly

1205962119896

= minus1

2angle([

D3119896

D1119896

]

2

) (55)

1205963119896

= minus1

2angle([

D4119896

D1119896

]

2

) (56)

Taking (26) into (53) the frequency 119891119896of the 119896th incident

signal is expressed as

119891119896=

1

2120587120591angle

10038161003816100381610038161003816100381610038161003816

D1119896

D5119896

10038161003816100381610038161003816100381610038161003816

(57)

Assuming that 1205741119896

= sin(120579119896) cos(120593

119896) 120574

2119896= sin(120579

119896) sin(120593

119896)

and 1205743119896

= cos(120579119896) and solving (27) (28) (29) (55) (56) and

(57) simultaneously we can obtain the equation as follows

119888

2120587119891119896

[[[[[[[

[

1205961119896

1198891

1205962119896

1198892

1205963119896

1198893

]]]]]]]

]

= [

[

1199092minus 119909

11199102minus 119910

11199112minus 119911

1

1199093minus 119909

11199103minus 119910

11199113minus 119911

1

1199094minus 119909

11199104minus 119910

11199104minus 119910

1

]

]

[

[

1205741119896

1205742119896

1205743119896

]

]

(58)

The solution of the equation is represented as

[

[

1205741119896

1205742119896

1205743119896

]

]

=119888

2120587119891119894

[

[

1199092minus 119909

11199102minus 119910

11199112minus 119911

1

1199093minus 119909

11199103minus 119910

11199113minus 119911

1

1199094minus 119909

11199104minus 119910

11199104minus 119910

1

]

]

minus1[[[[[[[

[

1205961119896

1198891

1205962119896

1198892

1205963119896

1198893

]]]]]]]

]

(59)

There are two approaches to solve 120579119896and 120593

119896 The first

approach is

120579119896= arccos120574

3119896

120593119896= arcsin(

1205741119896

cos (120579119896)) or 120593

119896= arcsin(

1205742119896

sin (120579119896))

(60)

The second approach is

120593119896= arctan(

1205742119896

1205741119896

)

120579119896= arccos(

1205741119896

cos (120593119896)) or 120579

119896= arcsin(

1205742119896

sin (120593119896))

(61)

In this paper the first approach is employedIn order to ensure the uniqueness of the estimated

parameters the following condition must be satisfied

1003817100381710038171003817p2 minus p1

1003817100381710038171003817 le119888

(2max (119891119894))

=min (120582

119896)

2

1003817100381710038171003817p3 minus p1

1003817100381710038171003817 le119888

(2max (119891119894))

=min (120582

119896)

2

1003817100381710038171003817p4 minus p1

1003817100381710038171003817 le119888

(2max (119891119894))

=min (120582

119896)

2

(62)

According to Theorem 2 the matrices D A and R119878have the

same permutation matrix that is the 119896th column of matrixD corresponds to the 119896th column of the manifold matrix A

Based on the PARAFAC model and TALS regressionalgorithm the joint frequency and 2D-DOA estimation canbe accomplished The algorithm steps are summarized asfollows

(1) choose four guiding elements and obtain the accuratepositions of the elements

(2) calculate the delay correlation function between thefour elements mentioned above and the whole ele-ments of the conformal array

(3) calculate the matrices R119891(120591) R

1(120591) R

2(120591) R

3(120591) and

R(0) respectively according to (5)(4) calculate the pseudosnapshot data matrix based on

(17)ndash(21) R119891(120591) R

1(120591) R

2(120591) R

3(120591) and R(0) are

sampled at time delay 120591119904 respectively

(5) construct the PARAFAC model based on (43)(6) estimate the matrix D using TALS regression algo-

rithm

The Scientific World Journal 7

(7) obtain12059611989111989612059611198961205962119896 and120596

3119896by using the estimator of

the matrixD and calculating (53)ndash(56)(8) obtain the joint frequency and 2D-DOA estimation

by calculate (57)ndash(59) or (57) (60) and (61)

4 Simulation Results

This section focuses on several sets of simulation results todemonstrate the performance of the proposed algorithmIn all simulation examples 200 Monte Carlo trials areconsidered and the root mean square error (RMSE) is usedas our performance measure which is defined as

RMSE = radic1

200

200

sum

120578=1

[(120579119896120578

minus 120579)2

+ (120593119896120578

minus 120593)2

] (63)

where 120579119894119905and 120593

119894119905are the elevation and azimuth estimators of

the 119896th incident signal in the 120578th trialThe successful rate is defined as the proportion of the

number of the successful experiments to the total numberof the experiments For frequency estimation a successfulexperiment is defined as the experiment with estimationerror of less than 2MHz for DOA estimation a successfulexperiment is defined as the experimentwith estimation errorof less than 2 degree

In order to simplify the transformation from the globalcoordinate to the local coordinate only the cylinder withsingle curvature is considered in the simulations The arraystructure is shown in Figure 2 is only a subarray which onlycovers 120

∘ Then three subarrays can estimate the wholespace The highest frequency of the incident signal is 119891max =

2GHz 120582min = 119888119891max The radius of the cylinder is 5120582min theadjacent element spacing in the same intersecting surface ofthe cylinder is 120582min4 and the adjacent intersecting surfacespacing is 120582min4 120582min stands for the shortest wavelength ofthe incident signal The position of the reference element is1199011= (0 5120582 0)The element pattern is defined in the local coordinate

of each element thus the transformation from the globalcoordinate to the local coordinate of elevation 120579

119896and azimuth

120593119896of the incident signal must be accomplished For the

cylindrical conformal array the corresponding Euler rotationangles are

119863 = minusΘ 119864 = minus1205872 119865 = 0 (64)

respectively 119863 represents the rotation angle based on theright-hand rule in the first rotation and the 119885-axis is therotation axis 119864 represents the rotation angle in the secondrotation and the 119884-axis is the rotation axis and 119865 representsthe rotation angle in the third rotation and the 119883-axis is therotation axis Θ represents the angle between the positionvector of the element and the positive direction of 119883-axis

The unit vector of the 119896th incident signal (120579119896 120593119896) in the

global coordinate can be expressed as

119909 = sin (120579119896) cos (120593

119896) 119910 = sin (120579

119896) sin (120593

119896)

119911 = cos (120579119896)

(65)

Z

X

Y

O

pm

p3

p4

p2p1

Figure 2 The structure of the cylindrical conformal array

Based on the Euler rotation matrix the unit vector inglobal coordinate can be transformed into the local coordi-nate of the 119898th element

Consider

[119909119898

119910119898

119911119898]119879

= 119877 (119863119898 119864119898 119865119898) [119909 119910 119911]

119879

(66)

where

119877 (119863119898 119864119898 119865119898) = [

[

cos119865 sin119865 0

minus sin119865 cos119865 0

0 0 1

]

]

times [

[

cos119864 0 minus sin119864

0 1 0

sin119864 0 cos119864]

]

times [

[

cos119863 sin119863 0

minus sin119863 cos119863 0

0 0 1

]

]

(67)

Then the elevation 120579 and azimuth 120593 of the 119896th incident signalin the local coordinate of the119898th element can be representedas

120593119898119896

= arctan (119910119898119909119898) 120579

119898119896= arccos (119911

119898) (68)

Thus the element pattern transformation from the globalcoordinate to the local coordinate is completed

First the curves of RMSE of frequency and angle and thesuccessful probability of angle in different SNR are plottedin Figure 3 using the proposed algorithm The number ofsnapshots is 200 the number of pseudosnapshots is 100SNR isin [5 30] and the noise is AWGN which is independentof incident signals The sampling frequency is 119891

119904= 1119879

119904=

5GHzThe elevation 120579 azimuth120593 and frequency of these twonarrow incident signals are (100∘ 60∘ 1 GHz) and (95∘ 50∘2 GHz) respectively which are independent of each other

8 The Scientific World Journal

5 10 15 20 25 300

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Succ

ess p

roba

bilit

y (

)

Frequency 1Frequency 2

(a)

5 10 15 20 25 300

05

1

15

2

25

3

35

4

RMSE

(MH

z)

SNR (dB)

Frequency 1Frequency 2

(b)

5 10 15 20 25 3040

50

60

70

80

90

100

SNR (dB)

Succ

ess p

roba

bilit

y (

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(c)

5 10 15 20 25 300

002

004

006

008

01

012

014

016

SNR (dB)

RMSE

(deg

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(d)

Figure 3The RMSE and successful probability with different SNR (a)The successful probability of frequencies (b) the RMSE of frequency(c) the successful probability of angles and (d) the RMSE of angles

The number of elements are 10 including 4 guiding elementsand other 6 instrumental elementsThe element pattern usedin simulation is the lowest order circular patch model [8 22]

119892120579(120579 120593) = 119869

2(120587119889

120582sin 120579) minus 119869

0(120587119889

120582sin 120579)

times (cos120593 minus 119895 sin120593) 0 le 120579 le120587

2

119892120593(120579 120593) = 119869

2(120587119889

120582sin 120579) + 119869

0(120587119889

120582sin 120579)

times cos 120579 (sin120593 minus 119895 cos120593) 0 le 120579 le120587

2

119892120579(120579 120593) = 119892

120593(120579 120593) = 0 otherwise

(69)

where 1198690and 119869

2are the zeroth- and second-order Bessel func-

tions of the first kind The element pattern transformation isachieved with the method proposed above

Figure 3 shows that the RMSE of frequency and angledecreases while the SNR increases meanwhile the successprobability of frequency and angle increase as the SNRincreases The estimated performance of the two frequenciesare very close which can be seen in Figures 3(a) and 3(b)As shown in Figure 3(a) the successful probability is largerthan 80 when the SNR is larger than 15 dB The RMSEof frequency is less than 2MHz when SNR is larger than10 dB as shown in Figure 3(b) In Figure 3(c) the successfulprobability of azimuths is slightly lower than that of theelevations The success probability of the angle is almost

The Scientific World Journal 9

100 200 300 400 500 600 700 80020

30

40

50

60

70

80

90

100

Snapshot number

Succ

ess p

roba

bilit

y (

)

Frequency 1Frequency 2

(a)

100 200 300 400 500 600 700 80008

1

12

14

16

18

2

Snapshot number

RMSE

(MH

z)

Frequency 1Frequency 2

(b)

100 200 300 400 500 600 700 80095

955

96

965

97

975

98

985

99

995

100

Snapshot number (dB)

Succ

ess p

roba

bilit

y (

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(c)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

100 200 300 400 500 600 700 800Snapshot number (dB)

002

004

006

008

01

012

014

016RM

SE (d

eg)

(d)

Figure 4 The RMSE and successful probability with different snapshot number (a) The successful probability of frequencies (b) the RMSEof frequency (c) the successful probability of angles and (d) the RMSE of angles

100 when the SNR reaches 15 dB Figure 3(d) shows thatthe RMSE is smaller than 016

∘ when SNR is larger than5 dB The RMSE of azimuth is larger than that of elevationThe estimation results of frequency and elevation are appliedto estimate azimuth The phenomenon in Figure 3(d) canbe interpreted as the existence of the estimated error offrequency and elevation Next the RMSE of frequency andangle as well as the success probability of angle in differentsnapshot number are depicted in Figure 4 SNR = 10 dBother simulation conditions are identical with the previousexperiment In Figure 4 the RMSE of frequency and angledecreases as the snapshot number increases and the successprobability of frequency and angle rises as the snapshotnumber increases As shown in Figure 4(a) the successfulprobability is larger than 80 when the number of snapshot

is larger than 500 Figure 4(b) shows that the RMSE offrequency 1 is slightly larger than that of frequency 2 Whenthe snapshot number is larger than 500 the RMSE dropsbelow 1MHz meaning that the proposed algorithm achievedthree orders of magnitude reduction Figures 4(c) and 4(d)show that the success probability of azimuth is lower thanthat of the elevation and the RMSE of azimuth are largerthan that of elevationThe reasonwhy this happens is becausethe estimation error of frequency and elevation affects theestimation performance of azimuth

5 Conclusion

Due to the varying curvature of the surface of the conformalcarrier the pattern of each element of the conformal array

10 The Scientific World Journal

is different Thus the conventional algorithms could notbe used for conformal array In this paper a novel jointfrequency and 2D-DOA estimation algorithm with highaccuracy are proposed Both spatial and time samplingare utilized to construct the spatial-time matrix The delaycorrelation function is used to suppress noiseThe PARAFACmodel is used for parameter estimation without parameterpairing Only four elements are needed and the positionsof these elements should be known accurately Other instru-mental elements can be flexibly arranged on the surface ofthe conformal carrier The algorithm proposed in this papercan be extended to estimate 2D-DOA straightly with littlemodification The simulation results verify the effectivenessof the proposed algorithm It can be expected that theproposed algorithm would have an application prospect inthe parameter estimation of conformal array In the futurework we will focus on the application of the proposedalgorithm [23ndash26]

Acknowledgments

This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)

References

[1] L Josefsson and P Persson Conformal Array Antenna Theoryand Design IEEE Press Piscataway NJ USA 2006

[2] W T Li X W Shi Y Q Hei S F Liu and J Zhu ldquoAhybrid optimization algorithmand its application for conformalarray pattern synthesisrdquo IEEE Transactions on Antennas andPropagation vol 58 no 10 pp 3401ndash3406 2010

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

[4] 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

[5] P Hyber M Jansson and B Ottersten ldquoArray interpolation andbias reductionrdquo IEEE Transaction on Signal Processing vol 52no 10 pp 2711ndash2720 2004

[6] P Hyberg M Jansson and B Ottersten ldquoArray interpolationand DOAMSE reductionrdquo IEEE Transactions on Signal Process-ing vol 53 no 12 pp 4464ndash4471 2005

[7] F Belloni A Richter and V Koivunen ldquoDOA estimationvia manifold separation for arbitrary array structuresrdquo IEEETransactions on Signal Processing vol 55 no 10 pp 4800ndash48102007

[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008

[9] P Yang F Yang and Z Nie ldquoDOA estimation using MUSICalgorithm on a cylinderical conformal arrayrdquo in Proceedingsof the IEEE Antennas and Propagation Society InternationalSymposium pp 5299ndash5302 June 2007

[10] P Yang F Yang and Z P Nie ldquoDOA estimation with sub-arraydivided technique and interporlated esprit algorithmon a cylin-drical conformal array antennardquo Progress in ElectromagneticsResearch vol 103 pp 201ndash216 2010

[11] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011

[12] L Zou J Lasenby and Z He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012

[13] Y-Y Wang J-T Chen andW-H Fang ldquoTST-MUSIC for jointDOA-delay estimationrdquo IEEE Transactions on Signal Processingvol 49 no 4 pp 721ndash729 2001

[14] J-D Lin W-H Fang Y-Y Wang and J-T Chen ldquoFSF MUSICfor joint DOA and frequency estimation and its performanceanalysisrdquo IEEE Transactions on Signal Processing vol 54 no 12pp 4529ndash4542 2006

[15] M Djeddou A Belouchrani and S Aouada ldquoMaximum likeli-hood angle-frequency estimation in partially known correlatednoise for low-elevation targetsrdquo IEEE Transactions on SignalProcessing vol 53 no 8 pp 3057ndash3064 2005

[16] R B Cattell ldquolsquoParallel proportional profilesrsquo and other prin-ciples for determining the choice of factors by rotationrdquo Psy-chometrika vol 9 no 4 pp 267ndash283 1944

[17] J D Carroll and J-J Chang ldquoAnalysis of individual differencesin multidimensional scaling via an n-way generalization ofldquoEckart-Youngrdquo decompositionrdquo Psychometrika vol 35 no 3pp 283ndash319 1970

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

[19] 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

[20] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 77ndash83 1999

[21] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fittingof the Complex PARAFAC Model in 3-Drdquo 1998 httpwwwtelecomtucgrsimnikos

[22] J R James P S Hall and C WoodMicrostrip Antenna Theoryand Design Peter Peregrinus New York NY USA 1981

[23] S Yin S Ding and H Luo ldquoReal-time implementation of faulttolerant control system with performance optimizationrdquo IEEETransactions on Industrial Electronics vol 61 no 5 pp 2402ndash2411 2014

[24] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[25] S Yin S Ding A Haghani andH Hao ldquoData-drivenmonitor-ing for stochastic systems and its application on batch processrdquoInternational Journal of Systems Science vol 44 no 7 pp 1366ndash1376 2013

[26] S Yin X Yang and H Karimi ldquoData-driven adaptive observerfor fault diagnosisrdquoMathematical Problems in Engineering vol2012 Article ID 832836 21 pages 2012

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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RotatingMachinery

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Advances inOptoElectronics

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Volume 2014

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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

The Scientific World Journal 7

(7) obtain12059611989111989612059611198961205962119896 and120596

3119896by using the estimator of

the matrixD and calculating (53)ndash(56)(8) obtain the joint frequency and 2D-DOA estimation

by calculate (57)ndash(59) or (57) (60) and (61)

4 Simulation Results

This section focuses on several sets of simulation results todemonstrate the performance of the proposed algorithmIn all simulation examples 200 Monte Carlo trials areconsidered and the root mean square error (RMSE) is usedas our performance measure which is defined as

RMSE = radic1

200

200

sum

120578=1

[(120579119896120578

minus 120579)2

+ (120593119896120578

minus 120593)2

] (63)

where 120579119894119905and 120593

119894119905are the elevation and azimuth estimators of

the 119896th incident signal in the 120578th trialThe successful rate is defined as the proportion of the

number of the successful experiments to the total numberof the experiments For frequency estimation a successfulexperiment is defined as the experiment with estimationerror of less than 2MHz for DOA estimation a successfulexperiment is defined as the experimentwith estimation errorof less than 2 degree

In order to simplify the transformation from the globalcoordinate to the local coordinate only the cylinder withsingle curvature is considered in the simulations The arraystructure is shown in Figure 2 is only a subarray which onlycovers 120

∘ Then three subarrays can estimate the wholespace The highest frequency of the incident signal is 119891max =

2GHz 120582min = 119888119891max The radius of the cylinder is 5120582min theadjacent element spacing in the same intersecting surface ofthe cylinder is 120582min4 and the adjacent intersecting surfacespacing is 120582min4 120582min stands for the shortest wavelength ofthe incident signal The position of the reference element is1199011= (0 5120582 0)The element pattern is defined in the local coordinate

of each element thus the transformation from the globalcoordinate to the local coordinate of elevation 120579

119896and azimuth

120593119896of the incident signal must be accomplished For the

cylindrical conformal array the corresponding Euler rotationangles are

119863 = minusΘ 119864 = minus1205872 119865 = 0 (64)

respectively 119863 represents the rotation angle based on theright-hand rule in the first rotation and the 119885-axis is therotation axis 119864 represents the rotation angle in the secondrotation and the 119884-axis is the rotation axis and 119865 representsthe rotation angle in the third rotation and the 119883-axis is therotation axis Θ represents the angle between the positionvector of the element and the positive direction of 119883-axis

The unit vector of the 119896th incident signal (120579119896 120593119896) in the

global coordinate can be expressed as

119909 = sin (120579119896) cos (120593

119896) 119910 = sin (120579

119896) sin (120593

119896)

119911 = cos (120579119896)

(65)

Z

X

Y

O

pm

p3

p4

p2p1

Figure 2 The structure of the cylindrical conformal array

Based on the Euler rotation matrix the unit vector inglobal coordinate can be transformed into the local coordi-nate of the 119898th element

Consider

[119909119898

119910119898

119911119898]119879

= 119877 (119863119898 119864119898 119865119898) [119909 119910 119911]

119879

(66)

where

119877 (119863119898 119864119898 119865119898) = [

[

cos119865 sin119865 0

minus sin119865 cos119865 0

0 0 1

]

]

times [

[

cos119864 0 minus sin119864

0 1 0

sin119864 0 cos119864]

]

times [

[

cos119863 sin119863 0

minus sin119863 cos119863 0

0 0 1

]

]

(67)

Then the elevation 120579 and azimuth 120593 of the 119896th incident signalin the local coordinate of the119898th element can be representedas

120593119898119896

= arctan (119910119898119909119898) 120579

119898119896= arccos (119911

119898) (68)

Thus the element pattern transformation from the globalcoordinate to the local coordinate is completed

First the curves of RMSE of frequency and angle and thesuccessful probability of angle in different SNR are plottedin Figure 3 using the proposed algorithm The number ofsnapshots is 200 the number of pseudosnapshots is 100SNR isin [5 30] and the noise is AWGN which is independentof incident signals The sampling frequency is 119891

119904= 1119879

119904=

5GHzThe elevation 120579 azimuth120593 and frequency of these twonarrow incident signals are (100∘ 60∘ 1 GHz) and (95∘ 50∘2 GHz) respectively which are independent of each other

8 The Scientific World Journal

5 10 15 20 25 300

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Succ

ess p

roba

bilit

y (

)

Frequency 1Frequency 2

(a)

5 10 15 20 25 300

05

1

15

2

25

3

35

4

RMSE

(MH

z)

SNR (dB)

Frequency 1Frequency 2

(b)

5 10 15 20 25 3040

50

60

70

80

90

100

SNR (dB)

Succ

ess p

roba

bilit

y (

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(c)

5 10 15 20 25 300

002

004

006

008

01

012

014

016

SNR (dB)

RMSE

(deg

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(d)

Figure 3The RMSE and successful probability with different SNR (a)The successful probability of frequencies (b) the RMSE of frequency(c) the successful probability of angles and (d) the RMSE of angles

The number of elements are 10 including 4 guiding elementsand other 6 instrumental elementsThe element pattern usedin simulation is the lowest order circular patch model [8 22]

119892120579(120579 120593) = 119869

2(120587119889

120582sin 120579) minus 119869

0(120587119889

120582sin 120579)

times (cos120593 minus 119895 sin120593) 0 le 120579 le120587

2

119892120593(120579 120593) = 119869

2(120587119889

120582sin 120579) + 119869

0(120587119889

120582sin 120579)

times cos 120579 (sin120593 minus 119895 cos120593) 0 le 120579 le120587

2

119892120579(120579 120593) = 119892

120593(120579 120593) = 0 otherwise

(69)

where 1198690and 119869

2are the zeroth- and second-order Bessel func-

tions of the first kind The element pattern transformation isachieved with the method proposed above

Figure 3 shows that the RMSE of frequency and angledecreases while the SNR increases meanwhile the successprobability of frequency and angle increase as the SNRincreases The estimated performance of the two frequenciesare very close which can be seen in Figures 3(a) and 3(b)As shown in Figure 3(a) the successful probability is largerthan 80 when the SNR is larger than 15 dB The RMSEof frequency is less than 2MHz when SNR is larger than10 dB as shown in Figure 3(b) In Figure 3(c) the successfulprobability of azimuths is slightly lower than that of theelevations The success probability of the angle is almost

The Scientific World Journal 9

100 200 300 400 500 600 700 80020

30

40

50

60

70

80

90

100

Snapshot number

Succ

ess p

roba

bilit

y (

)

Frequency 1Frequency 2

(a)

100 200 300 400 500 600 700 80008

1

12

14

16

18

2

Snapshot number

RMSE

(MH

z)

Frequency 1Frequency 2

(b)

100 200 300 400 500 600 700 80095

955

96

965

97

975

98

985

99

995

100

Snapshot number (dB)

Succ

ess p

roba

bilit

y (

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(c)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

100 200 300 400 500 600 700 800Snapshot number (dB)

002

004

006

008

01

012

014

016RM

SE (d

eg)

(d)

Figure 4 The RMSE and successful probability with different snapshot number (a) The successful probability of frequencies (b) the RMSEof frequency (c) the successful probability of angles and (d) the RMSE of angles

100 when the SNR reaches 15 dB Figure 3(d) shows thatthe RMSE is smaller than 016

∘ when SNR is larger than5 dB The RMSE of azimuth is larger than that of elevationThe estimation results of frequency and elevation are appliedto estimate azimuth The phenomenon in Figure 3(d) canbe interpreted as the existence of the estimated error offrequency and elevation Next the RMSE of frequency andangle as well as the success probability of angle in differentsnapshot number are depicted in Figure 4 SNR = 10 dBother simulation conditions are identical with the previousexperiment In Figure 4 the RMSE of frequency and angledecreases as the snapshot number increases and the successprobability of frequency and angle rises as the snapshotnumber increases As shown in Figure 4(a) the successfulprobability is larger than 80 when the number of snapshot

is larger than 500 Figure 4(b) shows that the RMSE offrequency 1 is slightly larger than that of frequency 2 Whenthe snapshot number is larger than 500 the RMSE dropsbelow 1MHz meaning that the proposed algorithm achievedthree orders of magnitude reduction Figures 4(c) and 4(d)show that the success probability of azimuth is lower thanthat of the elevation and the RMSE of azimuth are largerthan that of elevationThe reasonwhy this happens is becausethe estimation error of frequency and elevation affects theestimation performance of azimuth

5 Conclusion

Due to the varying curvature of the surface of the conformalcarrier the pattern of each element of the conformal array

10 The Scientific World Journal

is different Thus the conventional algorithms could notbe used for conformal array In this paper a novel jointfrequency and 2D-DOA estimation algorithm with highaccuracy are proposed Both spatial and time samplingare utilized to construct the spatial-time matrix The delaycorrelation function is used to suppress noiseThe PARAFACmodel is used for parameter estimation without parameterpairing Only four elements are needed and the positionsof these elements should be known accurately Other instru-mental elements can be flexibly arranged on the surface ofthe conformal carrier The algorithm proposed in this papercan be extended to estimate 2D-DOA straightly with littlemodification The simulation results verify the effectivenessof the proposed algorithm It can be expected that theproposed algorithm would have an application prospect inthe parameter estimation of conformal array In the futurework we will focus on the application of the proposedalgorithm [23ndash26]

Acknowledgments

This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)

References

[1] L Josefsson and P Persson Conformal Array Antenna Theoryand Design IEEE Press Piscataway NJ USA 2006

[2] W T Li X W Shi Y Q Hei S F Liu and J Zhu ldquoAhybrid optimization algorithmand its application for conformalarray pattern synthesisrdquo IEEE Transactions on Antennas andPropagation vol 58 no 10 pp 3401ndash3406 2010

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

[4] 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

[5] P Hyber M Jansson and B Ottersten ldquoArray interpolation andbias reductionrdquo IEEE Transaction on Signal Processing vol 52no 10 pp 2711ndash2720 2004

[6] P Hyberg M Jansson and B Ottersten ldquoArray interpolationand DOAMSE reductionrdquo IEEE Transactions on Signal Process-ing vol 53 no 12 pp 4464ndash4471 2005

[7] F Belloni A Richter and V Koivunen ldquoDOA estimationvia manifold separation for arbitrary array structuresrdquo IEEETransactions on Signal Processing vol 55 no 10 pp 4800ndash48102007

[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008

[9] P Yang F Yang and Z Nie ldquoDOA estimation using MUSICalgorithm on a cylinderical conformal arrayrdquo in Proceedingsof the IEEE Antennas and Propagation Society InternationalSymposium pp 5299ndash5302 June 2007

[10] P Yang F Yang and Z P Nie ldquoDOA estimation with sub-arraydivided technique and interporlated esprit algorithmon a cylin-drical conformal array antennardquo Progress in ElectromagneticsResearch vol 103 pp 201ndash216 2010

[11] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011

[12] L Zou J Lasenby and Z He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012

[13] Y-Y Wang J-T Chen andW-H Fang ldquoTST-MUSIC for jointDOA-delay estimationrdquo IEEE Transactions on Signal Processingvol 49 no 4 pp 721ndash729 2001

[14] J-D Lin W-H Fang Y-Y Wang and J-T Chen ldquoFSF MUSICfor joint DOA and frequency estimation and its performanceanalysisrdquo IEEE Transactions on Signal Processing vol 54 no 12pp 4529ndash4542 2006

[15] M Djeddou A Belouchrani and S Aouada ldquoMaximum likeli-hood angle-frequency estimation in partially known correlatednoise for low-elevation targetsrdquo IEEE Transactions on SignalProcessing vol 53 no 8 pp 3057ndash3064 2005

[16] R B Cattell ldquolsquoParallel proportional profilesrsquo and other prin-ciples for determining the choice of factors by rotationrdquo Psy-chometrika vol 9 no 4 pp 267ndash283 1944

[17] J D Carroll and J-J Chang ldquoAnalysis of individual differencesin multidimensional scaling via an n-way generalization ofldquoEckart-Youngrdquo decompositionrdquo Psychometrika vol 35 no 3pp 283ndash319 1970

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

[19] 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

[20] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 77ndash83 1999

[21] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fittingof the Complex PARAFAC Model in 3-Drdquo 1998 httpwwwtelecomtucgrsimnikos

[22] J R James P S Hall and C WoodMicrostrip Antenna Theoryand Design Peter Peregrinus New York NY USA 1981

[23] S Yin S Ding and H Luo ldquoReal-time implementation of faulttolerant control system with performance optimizationrdquo IEEETransactions on Industrial Electronics vol 61 no 5 pp 2402ndash2411 2014

[24] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[25] S Yin S Ding A Haghani andH Hao ldquoData-drivenmonitor-ing for stochastic systems and its application on batch processrdquoInternational Journal of Systems Science vol 44 no 7 pp 1366ndash1376 2013

[26] S Yin X Yang and H Karimi ldquoData-driven adaptive observerfor fault diagnosisrdquoMathematical Problems in Engineering vol2012 Article ID 832836 21 pages 2012

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 The Scientific World Journal

5 10 15 20 25 300

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Succ

ess p

roba

bilit

y (

)

Frequency 1Frequency 2

(a)

5 10 15 20 25 300

05

1

15

2

25

3

35

4

RMSE

(MH

z)

SNR (dB)

Frequency 1Frequency 2

(b)

5 10 15 20 25 3040

50

60

70

80

90

100

SNR (dB)

Succ

ess p

roba

bilit

y (

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(c)

5 10 15 20 25 300

002

004

006

008

01

012

014

016

SNR (dB)

RMSE

(deg

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(d)

Figure 3The RMSE and successful probability with different SNR (a)The successful probability of frequencies (b) the RMSE of frequency(c) the successful probability of angles and (d) the RMSE of angles

The number of elements are 10 including 4 guiding elementsand other 6 instrumental elementsThe element pattern usedin simulation is the lowest order circular patch model [8 22]

119892120579(120579 120593) = 119869

2(120587119889

120582sin 120579) minus 119869

0(120587119889

120582sin 120579)

times (cos120593 minus 119895 sin120593) 0 le 120579 le120587

2

119892120593(120579 120593) = 119869

2(120587119889

120582sin 120579) + 119869

0(120587119889

120582sin 120579)

times cos 120579 (sin120593 minus 119895 cos120593) 0 le 120579 le120587

2

119892120579(120579 120593) = 119892

120593(120579 120593) = 0 otherwise

(69)

where 1198690and 119869

2are the zeroth- and second-order Bessel func-

tions of the first kind The element pattern transformation isachieved with the method proposed above

Figure 3 shows that the RMSE of frequency and angledecreases while the SNR increases meanwhile the successprobability of frequency and angle increase as the SNRincreases The estimated performance of the two frequenciesare very close which can be seen in Figures 3(a) and 3(b)As shown in Figure 3(a) the successful probability is largerthan 80 when the SNR is larger than 15 dB The RMSEof frequency is less than 2MHz when SNR is larger than10 dB as shown in Figure 3(b) In Figure 3(c) the successfulprobability of azimuths is slightly lower than that of theelevations The success probability of the angle is almost

The Scientific World Journal 9

100 200 300 400 500 600 700 80020

30

40

50

60

70

80

90

100

Snapshot number

Succ

ess p

roba

bilit

y (

)

Frequency 1Frequency 2

(a)

100 200 300 400 500 600 700 80008

1

12

14

16

18

2

Snapshot number

RMSE

(MH

z)

Frequency 1Frequency 2

(b)

100 200 300 400 500 600 700 80095

955

96

965

97

975

98

985

99

995

100

Snapshot number (dB)

Succ

ess p

roba

bilit

y (

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(c)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

100 200 300 400 500 600 700 800Snapshot number (dB)

002

004

006

008

01

012

014

016RM

SE (d

eg)

(d)

Figure 4 The RMSE and successful probability with different snapshot number (a) The successful probability of frequencies (b) the RMSEof frequency (c) the successful probability of angles and (d) the RMSE of angles

100 when the SNR reaches 15 dB Figure 3(d) shows thatthe RMSE is smaller than 016

∘ when SNR is larger than5 dB The RMSE of azimuth is larger than that of elevationThe estimation results of frequency and elevation are appliedto estimate azimuth The phenomenon in Figure 3(d) canbe interpreted as the existence of the estimated error offrequency and elevation Next the RMSE of frequency andangle as well as the success probability of angle in differentsnapshot number are depicted in Figure 4 SNR = 10 dBother simulation conditions are identical with the previousexperiment In Figure 4 the RMSE of frequency and angledecreases as the snapshot number increases and the successprobability of frequency and angle rises as the snapshotnumber increases As shown in Figure 4(a) the successfulprobability is larger than 80 when the number of snapshot

is larger than 500 Figure 4(b) shows that the RMSE offrequency 1 is slightly larger than that of frequency 2 Whenthe snapshot number is larger than 500 the RMSE dropsbelow 1MHz meaning that the proposed algorithm achievedthree orders of magnitude reduction Figures 4(c) and 4(d)show that the success probability of azimuth is lower thanthat of the elevation and the RMSE of azimuth are largerthan that of elevationThe reasonwhy this happens is becausethe estimation error of frequency and elevation affects theestimation performance of azimuth

5 Conclusion

Due to the varying curvature of the surface of the conformalcarrier the pattern of each element of the conformal array

10 The Scientific World Journal

is different Thus the conventional algorithms could notbe used for conformal array In this paper a novel jointfrequency and 2D-DOA estimation algorithm with highaccuracy are proposed Both spatial and time samplingare utilized to construct the spatial-time matrix The delaycorrelation function is used to suppress noiseThe PARAFACmodel is used for parameter estimation without parameterpairing Only four elements are needed and the positionsof these elements should be known accurately Other instru-mental elements can be flexibly arranged on the surface ofthe conformal carrier The algorithm proposed in this papercan be extended to estimate 2D-DOA straightly with littlemodification The simulation results verify the effectivenessof the proposed algorithm It can be expected that theproposed algorithm would have an application prospect inthe parameter estimation of conformal array In the futurework we will focus on the application of the proposedalgorithm [23ndash26]

Acknowledgments

This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)

References

[1] L Josefsson and P Persson Conformal Array Antenna Theoryand Design IEEE Press Piscataway NJ USA 2006

[2] W T Li X W Shi Y Q Hei S F Liu and J Zhu ldquoAhybrid optimization algorithmand its application for conformalarray pattern synthesisrdquo IEEE Transactions on Antennas andPropagation vol 58 no 10 pp 3401ndash3406 2010

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

[4] 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

[5] P Hyber M Jansson and B Ottersten ldquoArray interpolation andbias reductionrdquo IEEE Transaction on Signal Processing vol 52no 10 pp 2711ndash2720 2004

[6] P Hyberg M Jansson and B Ottersten ldquoArray interpolationand DOAMSE reductionrdquo IEEE Transactions on Signal Process-ing vol 53 no 12 pp 4464ndash4471 2005

[7] F Belloni A Richter and V Koivunen ldquoDOA estimationvia manifold separation for arbitrary array structuresrdquo IEEETransactions on Signal Processing vol 55 no 10 pp 4800ndash48102007

[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008

[9] P Yang F Yang and Z Nie ldquoDOA estimation using MUSICalgorithm on a cylinderical conformal arrayrdquo in Proceedingsof the IEEE Antennas and Propagation Society InternationalSymposium pp 5299ndash5302 June 2007

[10] P Yang F Yang and Z P Nie ldquoDOA estimation with sub-arraydivided technique and interporlated esprit algorithmon a cylin-drical conformal array antennardquo Progress in ElectromagneticsResearch vol 103 pp 201ndash216 2010

[11] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011

[12] L Zou J Lasenby and Z He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012

[13] Y-Y Wang J-T Chen andW-H Fang ldquoTST-MUSIC for jointDOA-delay estimationrdquo IEEE Transactions on Signal Processingvol 49 no 4 pp 721ndash729 2001

[14] J-D Lin W-H Fang Y-Y Wang and J-T Chen ldquoFSF MUSICfor joint DOA and frequency estimation and its performanceanalysisrdquo IEEE Transactions on Signal Processing vol 54 no 12pp 4529ndash4542 2006

[15] M Djeddou A Belouchrani and S Aouada ldquoMaximum likeli-hood angle-frequency estimation in partially known correlatednoise for low-elevation targetsrdquo IEEE Transactions on SignalProcessing vol 53 no 8 pp 3057ndash3064 2005

[16] R B Cattell ldquolsquoParallel proportional profilesrsquo and other prin-ciples for determining the choice of factors by rotationrdquo Psy-chometrika vol 9 no 4 pp 267ndash283 1944

[17] J D Carroll and J-J Chang ldquoAnalysis of individual differencesin multidimensional scaling via an n-way generalization ofldquoEckart-Youngrdquo decompositionrdquo Psychometrika vol 35 no 3pp 283ndash319 1970

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

[19] 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

[20] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 77ndash83 1999

[21] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fittingof the Complex PARAFAC Model in 3-Drdquo 1998 httpwwwtelecomtucgrsimnikos

[22] J R James P S Hall and C WoodMicrostrip Antenna Theoryand Design Peter Peregrinus New York NY USA 1981

[23] S Yin S Ding and H Luo ldquoReal-time implementation of faulttolerant control system with performance optimizationrdquo IEEETransactions on Industrial Electronics vol 61 no 5 pp 2402ndash2411 2014

[24] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[25] S Yin S Ding A Haghani andH Hao ldquoData-drivenmonitor-ing for stochastic systems and its application on batch processrdquoInternational Journal of Systems Science vol 44 no 7 pp 1366ndash1376 2013

[26] S Yin X Yang and H Karimi ldquoData-driven adaptive observerfor fault diagnosisrdquoMathematical Problems in Engineering vol2012 Article ID 832836 21 pages 2012

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

The Scientific World Journal 9

100 200 300 400 500 600 700 80020

30

40

50

60

70

80

90

100

Snapshot number

Succ

ess p

roba

bilit

y (

)

Frequency 1Frequency 2

(a)

100 200 300 400 500 600 700 80008

1

12

14

16

18

2

Snapshot number

RMSE

(MH

z)

Frequency 1Frequency 2

(b)

100 200 300 400 500 600 700 80095

955

96

965

97

975

98

985

99

995

100

Snapshot number (dB)

Succ

ess p

roba

bilit

y (

)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

(c)

Elevation 1Elevation 2

Azimuth 1Azimuth 2

100 200 300 400 500 600 700 800Snapshot number (dB)

002

004

006

008

01

012

014

016RM

SE (d

eg)

(d)

Figure 4 The RMSE and successful probability with different snapshot number (a) The successful probability of frequencies (b) the RMSEof frequency (c) the successful probability of angles and (d) the RMSE of angles

100 when the SNR reaches 15 dB Figure 3(d) shows thatthe RMSE is smaller than 016

∘ when SNR is larger than5 dB The RMSE of azimuth is larger than that of elevationThe estimation results of frequency and elevation are appliedto estimate azimuth The phenomenon in Figure 3(d) canbe interpreted as the existence of the estimated error offrequency and elevation Next the RMSE of frequency andangle as well as the success probability of angle in differentsnapshot number are depicted in Figure 4 SNR = 10 dBother simulation conditions are identical with the previousexperiment In Figure 4 the RMSE of frequency and angledecreases as the snapshot number increases and the successprobability of frequency and angle rises as the snapshotnumber increases As shown in Figure 4(a) the successfulprobability is larger than 80 when the number of snapshot

is larger than 500 Figure 4(b) shows that the RMSE offrequency 1 is slightly larger than that of frequency 2 Whenthe snapshot number is larger than 500 the RMSE dropsbelow 1MHz meaning that the proposed algorithm achievedthree orders of magnitude reduction Figures 4(c) and 4(d)show that the success probability of azimuth is lower thanthat of the elevation and the RMSE of azimuth are largerthan that of elevationThe reasonwhy this happens is becausethe estimation error of frequency and elevation affects theestimation performance of azimuth

5 Conclusion

Due to the varying curvature of the surface of the conformalcarrier the pattern of each element of the conformal array

10 The Scientific World Journal

is different Thus the conventional algorithms could notbe used for conformal array In this paper a novel jointfrequency and 2D-DOA estimation algorithm with highaccuracy are proposed Both spatial and time samplingare utilized to construct the spatial-time matrix The delaycorrelation function is used to suppress noiseThe PARAFACmodel is used for parameter estimation without parameterpairing Only four elements are needed and the positionsof these elements should be known accurately Other instru-mental elements can be flexibly arranged on the surface ofthe conformal carrier The algorithm proposed in this papercan be extended to estimate 2D-DOA straightly with littlemodification The simulation results verify the effectivenessof the proposed algorithm It can be expected that theproposed algorithm would have an application prospect inthe parameter estimation of conformal array In the futurework we will focus on the application of the proposedalgorithm [23ndash26]

Acknowledgments

This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)

References

[1] L Josefsson and P Persson Conformal Array Antenna Theoryand Design IEEE Press Piscataway NJ USA 2006

[2] W T Li X W Shi Y Q Hei S F Liu and J Zhu ldquoAhybrid optimization algorithmand its application for conformalarray pattern synthesisrdquo IEEE Transactions on Antennas andPropagation vol 58 no 10 pp 3401ndash3406 2010

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

[4] 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

[5] P Hyber M Jansson and B Ottersten ldquoArray interpolation andbias reductionrdquo IEEE Transaction on Signal Processing vol 52no 10 pp 2711ndash2720 2004

[6] P Hyberg M Jansson and B Ottersten ldquoArray interpolationand DOAMSE reductionrdquo IEEE Transactions on Signal Process-ing vol 53 no 12 pp 4464ndash4471 2005

[7] F Belloni A Richter and V Koivunen ldquoDOA estimationvia manifold separation for arbitrary array structuresrdquo IEEETransactions on Signal Processing vol 55 no 10 pp 4800ndash48102007

[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008

[9] P Yang F Yang and Z Nie ldquoDOA estimation using MUSICalgorithm on a cylinderical conformal arrayrdquo in Proceedingsof the IEEE Antennas and Propagation Society InternationalSymposium pp 5299ndash5302 June 2007

[10] P Yang F Yang and Z P Nie ldquoDOA estimation with sub-arraydivided technique and interporlated esprit algorithmon a cylin-drical conformal array antennardquo Progress in ElectromagneticsResearch vol 103 pp 201ndash216 2010

[11] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011

[12] L Zou J Lasenby and Z He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012

[13] Y-Y Wang J-T Chen andW-H Fang ldquoTST-MUSIC for jointDOA-delay estimationrdquo IEEE Transactions on Signal Processingvol 49 no 4 pp 721ndash729 2001

[14] J-D Lin W-H Fang Y-Y Wang and J-T Chen ldquoFSF MUSICfor joint DOA and frequency estimation and its performanceanalysisrdquo IEEE Transactions on Signal Processing vol 54 no 12pp 4529ndash4542 2006

[15] M Djeddou A Belouchrani and S Aouada ldquoMaximum likeli-hood angle-frequency estimation in partially known correlatednoise for low-elevation targetsrdquo IEEE Transactions on SignalProcessing vol 53 no 8 pp 3057ndash3064 2005

[16] R B Cattell ldquolsquoParallel proportional profilesrsquo and other prin-ciples for determining the choice of factors by rotationrdquo Psy-chometrika vol 9 no 4 pp 267ndash283 1944

[17] J D Carroll and J-J Chang ldquoAnalysis of individual differencesin multidimensional scaling via an n-way generalization ofldquoEckart-Youngrdquo decompositionrdquo Psychometrika vol 35 no 3pp 283ndash319 1970

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

[19] 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

[20] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 77ndash83 1999

[21] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fittingof the Complex PARAFAC Model in 3-Drdquo 1998 httpwwwtelecomtucgrsimnikos

[22] J R James P S Hall and C WoodMicrostrip Antenna Theoryand Design Peter Peregrinus New York NY USA 1981

[23] S Yin S Ding and H Luo ldquoReal-time implementation of faulttolerant control system with performance optimizationrdquo IEEETransactions on Industrial Electronics vol 61 no 5 pp 2402ndash2411 2014

[24] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[25] S Yin S Ding A Haghani andH Hao ldquoData-drivenmonitor-ing for stochastic systems and its application on batch processrdquoInternational Journal of Systems Science vol 44 no 7 pp 1366ndash1376 2013

[26] S Yin X Yang and H Karimi ldquoData-driven adaptive observerfor fault diagnosisrdquoMathematical Problems in Engineering vol2012 Article ID 832836 21 pages 2012

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 The Scientific World Journal

is different Thus the conventional algorithms could notbe used for conformal array In this paper a novel jointfrequency and 2D-DOA estimation algorithm with highaccuracy are proposed Both spatial and time samplingare utilized to construct the spatial-time matrix The delaycorrelation function is used to suppress noiseThe PARAFACmodel is used for parameter estimation without parameterpairing Only four elements are needed and the positionsof these elements should be known accurately Other instru-mental elements can be flexibly arranged on the surface ofthe conformal carrier The algorithm proposed in this papercan be extended to estimate 2D-DOA straightly with littlemodification The simulation results verify the effectivenessof the proposed algorithm It can be expected that theproposed algorithm would have an application prospect inthe parameter estimation of conformal array In the futurework we will focus on the application of the proposedalgorithm [23ndash26]

Acknowledgments

This work was supported in part by the National ScienceFoundation of China under Grant 61201410 and in part byFundamental Research Focused on Special Fund Project ofthe Central Universities (Program no HEUCF130804)

References

[1] L Josefsson and P Persson Conformal Array Antenna Theoryand Design IEEE Press Piscataway NJ USA 2006

[2] W T Li X W Shi Y Q Hei S F Liu and J Zhu ldquoAhybrid optimization algorithmand its application for conformalarray pattern synthesisrdquo IEEE Transactions on Antennas andPropagation vol 58 no 10 pp 3401ndash3406 2010

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

[4] 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

[5] P Hyber M Jansson and B Ottersten ldquoArray interpolation andbias reductionrdquo IEEE Transaction on Signal Processing vol 52no 10 pp 2711ndash2720 2004

[6] P Hyberg M Jansson and B Ottersten ldquoArray interpolationand DOAMSE reductionrdquo IEEE Transactions on Signal Process-ing vol 53 no 12 pp 4464ndash4471 2005

[7] F Belloni A Richter and V Koivunen ldquoDOA estimationvia manifold separation for arbitrary array structuresrdquo IEEETransactions on Signal Processing vol 55 no 10 pp 4800ndash48102007

[8] B-H Wang Y Guo Y-L Wang and Y-Z Lin ldquoFrequency-invariant pattern synthesis of conformal array antenna with lowcross-polarisationrdquo IETMicrowaves Antennas and Propagationvol 2 no 5 pp 442ndash450 2008

[9] P Yang F Yang and Z Nie ldquoDOA estimation using MUSICalgorithm on a cylinderical conformal arrayrdquo in Proceedingsof the IEEE Antennas and Propagation Society InternationalSymposium pp 5299ndash5302 June 2007

[10] P Yang F Yang and Z P Nie ldquoDOA estimation with sub-arraydivided technique and interporlated esprit algorithmon a cylin-drical conformal array antennardquo Progress in ElectromagneticsResearch vol 103 pp 201ndash216 2010

[11] Z-S Qi Y Guo and B-H Wang ldquoBlind direction-of-arrivalestimation algorithm for conformal array antenna with respectto polarisation diversityrdquo IET Microwaves Antennas and Prop-agation vol 5 no 4 pp 433ndash442 2011

[12] L Zou J Lasenby and Z He ldquoDirection and polarizationestimation using polarized cylindrical conformal arraysrdquo IETSignal Processing vol 6 no 5 pp 395ndash403 2012

[13] Y-Y Wang J-T Chen andW-H Fang ldquoTST-MUSIC for jointDOA-delay estimationrdquo IEEE Transactions on Signal Processingvol 49 no 4 pp 721ndash729 2001

[14] J-D Lin W-H Fang Y-Y Wang and J-T Chen ldquoFSF MUSICfor joint DOA and frequency estimation and its performanceanalysisrdquo IEEE Transactions on Signal Processing vol 54 no 12pp 4529ndash4542 2006

[15] M Djeddou A Belouchrani and S Aouada ldquoMaximum likeli-hood angle-frequency estimation in partially known correlatednoise for low-elevation targetsrdquo IEEE Transactions on SignalProcessing vol 53 no 8 pp 3057ndash3064 2005

[16] R B Cattell ldquolsquoParallel proportional profilesrsquo and other prin-ciples for determining the choice of factors by rotationrdquo Psy-chometrika vol 9 no 4 pp 267ndash283 1944

[17] J D Carroll and J-J Chang ldquoAnalysis of individual differencesin multidimensional scaling via an n-way generalization ofldquoEckart-Youngrdquo decompositionrdquo Psychometrika vol 35 no 3pp 283ndash319 1970

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

[19] 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

[20] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 77ndash83 1999

[21] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fittingof the Complex PARAFAC Model in 3-Drdquo 1998 httpwwwtelecomtucgrsimnikos

[22] J R James P S Hall and C WoodMicrostrip Antenna Theoryand Design Peter Peregrinus New York NY USA 1981

[23] S Yin S Ding and H Luo ldquoReal-time implementation of faulttolerant control system with performance optimizationrdquo IEEETransactions on Industrial Electronics vol 61 no 5 pp 2402ndash2411 2014

[24] S Yin S Ding A Haghani H Hao and P Zhang ldquoA com-parison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[25] S Yin S Ding A Haghani andH Hao ldquoData-drivenmonitor-ing for stochastic systems and its application on batch processrdquoInternational Journal of Systems Science vol 44 no 7 pp 1366ndash1376 2013

[26] S Yin X Yang and H Karimi ldquoData-driven adaptive observerfor fault diagnosisrdquoMathematical Problems in Engineering vol2012 Article ID 832836 21 pages 2012

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VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

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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