Research Article High Accuracy 2D-DOA Estimation for ...thealphalab.org/papers/High Accuracy 2D-DOA Estimation for Conformal Array Using... · D-DOA estimation of the cylindrical

Research ArticleHigh Accuracy 2D-DOA Estimation for Conformal ArrayUsing PARAFAC

Liangtian Wan1 Weijian Si1 Lutao Liu1 Zuoxi Tian2 and Naixing Feng3

1 Department of Information and Communication Engineering Harbin Engineering University Harbin 150001 China2 Science and Technology on Underwater Test and Control Laboratory Dalian 116013 China3 Institute of Electromagnetics and Acoustics Xiamen University Xiamen 361005 China

Correspondence should be addressed to Weijian Si swj0418263net

Received 26 October 2013 Revised 14 December 2013 Accepted 16 December 2013 Published 16 January 2014

Academic Editor Hon Tat Hui

Copyright copy 2014 Liangtian Wan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Due to the polarization diversity (PD) of element patterns caused by the varying curvature of the conformal carrier the conventionaldirection-of-arrival (DOA) estimation algorithms could not be applied to the conformal array In order to describe the PD ofconformal array the polarization parameter is considered in the snapshot datamodelThe paramount difficulty forDOA estimationis the coupling between the angle information and polarization parameter Based on the characteristic of the cylindrical conformalarray the decoupling between the polarization parameter and DOA can be realized with a specially designed array structure2D-DOA estimation of the cylindrical conformal array is accomplished via parallel factor analysis (PARAFAC) theory To avoidparameter pairing problem the algorithm forms a PARAFAC model of the covariance matrices in the covariance domain Theproposed algorithm can also be generalized to other conformal array structures and nonuniform noise scenario Cramer-Raobound (CRB) is derived and simulation results with the cylindrical conformal array are presented to verify the performance ofthe proposed algorithm

1 Introduction

Conformal arraysmounted on curved surfaces are commonlyapplied in various areas such as radar sonar and wirelesscommunication [1] The conformal antennas could fulfillspecific aerodynamics space-saving elimination of random-induced bore-sight error potential increase in available aper-ture and so on [2 3] Most researches focus on the designof antenna configuration [4ndash6] the transform between thelocal and global coordinate [7ndash10] and the pattern synthesisof conformal array [11ndash15]

The conventional direction-of-arrival (DOA) estimationalgorithms for example the multiple signal classification(MUSIC) [16] and the estimation of signal parameters viarotational invariance techniques (ESPRIT) [17] are notapplicable due to the varying curvature of the conformalarray The ldquoshadow effectrdquo of the conformal array is causedby the metallic shelter leading to the condition that notall elements have the ability to receive the signal As aresult of the incomplete steering vector common DOA

estimation algorithms cannot be used for conformal arrayRecently DOA estimation algorithms for conformal arrayhave been proposed for high resolution [18ndash22] With thehelp of fourth-order cumulant and ESPRIT algorithm a blindDOA estimation algorithm is proposed in [18] Based onthe mathematical technique of geometric algebra 2D-DOAand polarization parameter estimations are completed byexploiting iterative ESPRIT algorithm [19] The state spaceand propagator method are used for joint frequency and 2D-DOA estimations for the cylindrical conformal array [20]However the main drawback of the algorithms in [18ndash20] isthe need of parameter pairing

Parallel factor (PARAFAC) analysis attracted the atten-tion of researchers when it was originally introduced in arraysignal processing in 2000 [21 22] It has been widely usedfor low-rank decomposition of three and higher way arrayIn this paper a high accuracy 2D-DOA estimation algorithmfor the conformal array is proposed using PARAFAC theoryThe proposed algorithmdoes not need parameter pairing andperforms well when the signal-to-noise ratio (SNR) is low

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014 Article ID 394707 14 pageshttpdxdoiorg1011552014394707

2 International Journal of Antennas and Propagation

or in situations where high estimation accuracy is neededSimulation results demonstrate the efficiency and accuracy ofthe proposed algorithm

The rest of this paper is structured as follows Section 2introduces the snapshot data model Section 3 elaborates thedesign of the conformal array structure Section 4 containsthe core contributions of this paper in which the PARAFACmodel is used for high accuracy 2D-DOA estimation Sec-tion 5 extends the proposed algorithm to more generalcases Section 6 analyses the computational complexity of thealgorithm Section 7 derives the Cramer-Rao bound (CRB)Section 8 discusses the array design Section 9 presents thesimulation result Conclusions are drawn in Section 10

2 The Snapshot Data Model

In order to achieve accurate parameter estimation the snap-shot data model of the conformal array must be preciselyestablished The pattern of the elements is different due tothe varying curvature of carriers [8] The mathematic modelof the conformal array constructed in [8 18] is adopted inthis paper The far-field narrowband signal impinging on theconformal array with elevation 120579 and azimuth 120593 is shown inFigure 1(a) The steering vector takes the following form

a (120579 120593) = [ℎ1119890minus1198952120587((p

1∙u)120582)

ℎ2119890minus1198952120587((p

2∙u)120582)

ℎ2119898+2

119890minus1198952120587((p

2119898+2∙u)120582)

]119879

(1)

ℎ119894= (119892

2

119894120579+ 119892

2

119894120593)12

(1198962

120579+ 119896

2

120593)12

cos (120579119894119892119896)

=1003816100381610038161003816119892119894100381610038161003816100381610038161003816100381610038161199011198971003816100381610038161003816 cos (120579119894119892119896)

= g119894∙ p

119897= 119892

119894120579119896120579+ 119892

119894120593119896120593

(2)

where P119894represents the position vector of the 119894th element u

is the propagation vector and 120582 is wavelength of the incidentsignal As shown in Figure 1(b) u

120579and u

120593are orthogonal unit

vectors and 119896120579and 119896

120593are the polarization components of the

incident signal ℎ119894denotes the 119894th elementrsquos response of the

incident signal g119894is the pattern of the 119894th element and p

119897is

the electric field direction of the incident signal 120579119894119892119896

is theangle between vector g

119894and p

119897 (∙)119879 denotes the transpose of

matrix (∙)Assuming that 119903 incident signals impinge on the designed

conformal array the snapshot data model can be representedas

X (119899) = G ∙ AS (119899) + N (119899)

= (G120579∙ A

120579K120579+ G

120593∙ A

120593K120593) S (119899) + N (119899)

= BS (119899) + N (119899)

G120579= [g

120579(1205791 120593

1) g

120579(1205792 120593

2) g

120579(120579119903 120593

119903)]

G120593= [g

120593(1205791 120593

1) g

120593(1205792 120593

2) g

120593(120579119903 120593

119903)]

A120579= [a

120579(1205791 120593

1) a

120579(1205792 120593

2) a

120579(120579119903 120593

119903)]

A120593= [a

120593(1205791 120593

1) a

120593(1205792 120593

2) a

120593(120579119903 120593

119903)]

K120579= diag (119896

1120579 119896

2120579 119896

119903120579)

K120593= diag (119896

1120593 119896

2120593 119896

119903120593)

S (119899) = [1199041(119899) 119904

2(119899) 119904

119903(119899)]

119879

W (119899) = [1199081(119899) 119908

2(119899) 119908

119903(119899)]

119879

(3)

where G denotes the pattern matrix and A denotes themanifold matrix K

120579and K

120593are the diagonal matrices whose

diagonal entries are 1198961120579 119896

2120579 119896

119903120579and 119896

1120593 119896

2120593 119896

119903120593

respectively 119896119894120579and 119896

119894120593are the components of 119894th signalrsquos

polarization parameters along the orthogonal unit vector u120579

and u120593 respectively S(119899) denotes the signal vector W(119899) is

additive Gaussian white noise with the covariance matrix

119864 W (119899)W(119899)119867 = Q = 120590

2I (4)

(∙)119867 denotes conjugate transpose of matrix (∙) Collecting119873

snapshots

X = BS +W (5)

where S is the 119903 times119873 signal matrix andW is the (2119898 + 2) times119873

noise matrixThe definition of the pattern is in the local coordinate

meaning that transformation must be performed from theglobal coordinate to the local coordinate [8] In addition thepolarization parameters couple with the signal parametersthus the decoupling is definitely necessary for the 2D-DOAestimation of the conformal array

3 The Structure of the Array

Since the signalrsquos amplitude of the sensors received willdecrease because of the ldquoshadow effectrdquo the whole conformalarray is divided into three subarrays and each subarray covers120

∘ For the single curvature and symmetry of the cylindereach subarray possesses and remains the same structure aswell as the parameter estimationmechanismThus this paperonly considers one subarray

As shown in Figure 2 the distance between two sensorswhich are in the same intersecting surface is 1205824 and thedistance between two adjacent intersecting surfaces is 5120582

As shown in Figure 2 1 sim 119898 constitute array 1 2 sim 119898 + 1

constitute array 2 119898 + 2 sim 2119898 + 1 constitute array 3 and119898 + 3 sim 2119898 + 2 constitute array 4 (1 sim 2119898 + 2 are sensorsof the array) The distance vector between array 1 and array2 is ΔP

1and 119889

1= |ΔP

1| = 1205824 The distance vector between

array 1 and array 3 is ΔP2and 119889

2= |ΔP

2| = 1205824 as shown

in Figure 3 The sensorsrsquo patterns of the same generatrix are

International Journal of Antennas and Propagation 3

Z

O

X

Y

120579

120593

minusu

(a)

g i

pl

O u120579k120579

gi120593

k120593

gi120579

120579igk

u120593

(b)

Figure 1 (a) The direction vector u impinges on the array (b) The 119894th sensors response

Z

Y

X

m + 1

m

O

2

1m + 2

m + 3

2m + 1

2m + 2

Figure 2 The structure of cylindrical conformal array

identical The sensors 1 sim 119898 + 1 possess the same patterng1 and the sensors119898 + 2 sim 2119898 + 2 possess the same pattern

g2The decoupling between polarization parameter and angle

information can be realized by taking full advantage of thecharacteristic of the array structure

Assume that X1 X

2 X

3 and X

4represent the data that

array 1 array 2 array 3 and array 4 receive respectively Theorigin of the coordinate is the reference point So the receiveddata can be expressed as

X1= BS + N

1

X2= BΨ

1S + N

2

X3= BΨ

2S + N

3

X4= BΨ

1Ψ2S + N

4

(6)

The covariance matrices among the received data are

R1= 119864 X

1X119867

1 = BR

119904B119867 +Q

1

R2= 119864 X

2X119867

1 = BΨ

1R119904B119867 +Q

2

R3= 119864 X

3X119867

1 = BΨ

2R119904B119867 +Q

3

R4= 119864 X

4X119867

1 = BΨ

1Ψ2R119904B119867 +Q

4

(7)

where R119904= diag1199042

1 119904

2

119903 represents the signal covariance

matrix and the noise covariance matrices areQ1ndashQ

4 respec-

tivelyConsider

Ψ1= diag [exp (minus119895120596

11) exp (minus119895120596

12) exp (minus119895120596

1119903)]

(8)

Ψ2= diag [

ℎ3(1205791 120593

1)

ℎ1(1205791 120593

1)exp (minus119895120596

21)

ℎ3(1205792 120593

2)

ℎ1(1205792 120593

2)exp (minus119895120596

22)

ℎ3(120579119903 120593

119903)

ℎ1(120579119903 120593

119903)exp (minus119895120596

2119903)]

(9)

1205961119894= (

2120587

120582)ΔP

1∙ u

119894

= (2120587119889

1

120582)

times [sin (120579ΔP1

) cos (120593ΔP1

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP1

) sin (120593ΔP1

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP1

) cos (120579119894)]

(10)

1205962119894= (

2120587

120582)ΔP

2∙ u

119894

= (2120587119889

2

120582)


) cos (120593ΔP2

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP2

) sin (120593ΔP2

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP2

) cos (120579119894)]

(11)

where 120579ΔP119894

and 120593ΔP119894

represent the elevation and azimuthof the distance vector in the global coordinate The array


OY

Z

Δ

P2

P1

P1

(a)

Y

Xm+1 2m+1

O

ΔP

P

2

P

(b)

Figure 3 (a) The distance vector ΔP1 (b) The distance vector ΔP

2

structure is shown in Figures 2 and 3 in which the distancevectorΔP

1is parallel to the119885-axisThe elevation and azimuth

of ΔP1in the global coordinate are

120579ΔP1

= 0 120593ΔP1

=120587

2 (12)

ΔP2is parallel to the 119883-axis The elevation and azimuth of

ΔP2in the global coordinate are

120579ΔP2

=120587

2 120593

ΔP2

= 0 (13)

In real applications the ideal covariance matrix has to bereplaced by the sample covariance matrix which is obtainedby a finite number of snapshots R

119896(119896 = 1 4) that is

R119896=1

119873

119873

sum

119896=1

X (119899)X(119899)119867 =1

119873XXH

(14)

4 The DOA Estimation Based on PARAFAC

Parallel factor (PARAFAC) analysis is a method of multipledata analysis which has been first introduced in psycho-metrics It has been used in many fields such as statisticsarithmetic complexity and chemometrics In this sectionfirst the PARAFAC theory is introduced briefly secondbased on the PARAFAC model the multiple way array isconstructed and the identifiability of inherently unresolvablesource permutation and scaling ambiguities are analyzedfinally the trilinear alternating least squares (TALS) regres-sion algorithm is used to fit the PARAFAC model thusthe unknown parameters in the PARAFAC model can beestimated

41 Parallel Factor Analysis In this subsection the basicidea of PARAFAC theory is introduced First the trilineardecomposition of the element of the three-way array is elabo-rated Second based on the symmetrical characteristic of the

trilinear decomposition the three-way array is decomposedinto three dimensions

Considering a 119862 times 119863 times 119864 three-way array X (the entry ofthe array denotes 119909

119888119889119890) the trilinear decomposition of 119909

119888 119889119890

can be expressed as [21]

119909119888 119889 119890

=

119872

sum

119901=1

119904119888119901119905119889119901119906119890119901 (15)

where 119888 = 1 119862 119889 = 1 119863 and 119890 = 1 119864 Thethree-way array X is represented as the sum of 119872 rank-1 factor Here the rank of X is defined as the minimumnumber of rank-1 three-way components which are neededto decompose X The vectors sp isin C119862times1 tp isin C119863times1 andup isin C119864times1 are called load vector score vector and factorprofiles respectively

Assume 119872 = 3 an example is given as follows andthus we can understand the idea of trilinear decompositionintuitivelyThe decomposition of three-way array X is shownin Figure 4 it can be seen that three dimensions can bedecomposed

The typical element of the 119862 times 119875 matrix S is defined asS(119888 119901) = 119904

119888119901 the typical element of the 119863 times 119875 matrix T is

defined as T(119888 119901) = 119905119888119901 the typical element of the 119864 times 119875

matrix U is defined as U(119888 119901) = 119906119888119901 In addition a 119863 times 119864

matrix X119888 a 119862 times 119864 matrix X

119889 and a 119862 times 119863 matrix X

119890are

defined The typical element of each matrix is X119888(119889 119890) =

X119889(119888 119890) = X

119890(119888 119889) = 119909

119888119889119890 The three-way matrix X can

be ldquoslicedrdquo along three different dimensions as shown inFigure 4

Consider

X119888= TΛ

119888(S)U119879

119888 = 1 119862

X119889= UΛ

119889(T) S119879 119889 = 1 119863

X119890= SΛ

119890(U)T119879 119890 = 1 119864

(16)


= ++

1u 2u

3

3u

t2t1t

1s 2s 3sX

Figure 4 The decomposition of three-way array

where Λ119888(S) is the diagonal matrix constructed by the 119888th

row of the matrix S According to the definition of Khatri-Rao product X(119862119863times119864) can be written in another form as

X(119862119863times119864)=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

X119888=1

X119888=2

X119888=119862

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

TΛ1(S)

TΛ2(S)

TΛ119862(S)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

U119879

= (S ⊙ T)UT (17)

where ldquo⊙rdquo is used to denote the Khatri-Rao (KR) product| ∙ | stands for the three-way array which is constructed bystacking thematrices into a columnThe definition of Khatri-Rao (KR) product can be found in the appendix The three-way array X can also be expressed in two other forms

X(119863119864times119862)= (T ⊙ U) ST (18)

X(119864119862times119863)= (U ⊙ S)TT

(19)

The reason of decomposition along three different dimen-sions is that the PARAFAC model can be solved by usingTALS algorithm The matrices T U and S can be updatedalternately

42 The Identifiability of the Model In this subsection thePARAFAC model is constructed by the received data ofthe conformal array The decomposition of the PARAFACmodel must be unique if not so the PARAFAC modelcannot be solved Thus some requirements of the matriceswhich construct the model have to be satisfied Theorem 2guarantees the uniqueness decomposition of themodel If thePARAFAC model is constructed by the received data of theconformal array then Theorem 2 can be used to judge theuniqueness decomposition of the model

On the basis of the PARAFAC theory [23] the 119898 times

119898 times 4 three-way array of the cylindrical conformal array isconstructed by (7)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)R ( 2)R ( 3)R ( 4)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R1

R2

R3

R4

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

BR119904B119867

BΨ1R119904B119867

BΨ2R119904B119867

BΨ1Ψ2R119904B119867

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+Q (20)

where R119894(119894 = 1 4) are the sample covariance matrices

respectively and Q represents the noise in real observation

Let C = B119867 based on the definition of Khatri-Rao product(20) can be transformed as

R = (D ⊙ B)C +Q (21)

R119883= (C119879 ⊙D)B119879 +Q

119883 (22)

R119884= (B ⊙ C119879)D119879

+Q119884 (23)

D =[[[

[

Λminus1(R

119904)

Λminus1(Ψ

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

1Ψ2R119904)

]]]

]

(24)

where Λminus1(R119904) is the row vector constructed by the diagonal

entries of the diagonal matrix R119904 [∙] stands for the matrix

Definition 1 (see [21]) Considering a givenmatrixA isin C119862times119875if and only if the matrix A contains at least 119903 but not 119903 + 1linearly independent columns then the rank of matrix A is119903A = Rank(A) = 119903 If any 119896 columns of matrix A are linearlyindependent then the Kruskal rank (119896-rank) is 119896A = 119896 [24]Generally speaking 119896A le 119903A

In order to ensure the uniqueness of the PARAFACmodelin (21)ndash(23) the sufficient condition must be met

a ( 119894) = a ( 119895) (25)

For any two different incident signals 119904119894(119896) and 119904

119895(119896)

ℎ119900119890119890minus1198952120587((p

119900119890∙u119894)120582)

= ℎ119900119890119890minus1198952120587((p

119900119890∙u119895)120582) (26)

ensures the uniqueness of the PARAFAC model where

119901119900119890= sin (120579

119900119890) cos (120593

119900119890) x

+ sin (120579119900119890) sin (120593

119900119890) y + cos (120579

119900119890) z

(27)

u119894= sin (120579

119894) cos (120593

119894) x

+ sin (120579119894) sin (120593

119894) y + cos (120579

119894) z

(28)

120579119900119890

and 120593119900119890

are the elevation and azimuth of the sensorrsquosposition in the global coordinate 120579

119894and 120593

119894are the elevation

and azimuth of the 119894th incident signalThus (26) is equivalentto

sin (120579119900119890) cos (120593

119900119890) 120588

1

+ sin (120579119900119890) sin (120593

119900119890) 120588

2+ cos (120579

119900119890) 120588

3= 0

1205881= sin (120579

119894) cos (120593

119894) minus sin (120579

119895) cos (120593

119895)

1205882= sin (120579

119894) sin (120593

119894) minus sin (120579

119895) sin (120593

119895)

1205883= cos (120579

119894) minus cos (120579

119895)

(29)

Theorem 2 The sufficient conditions of the PARAFAC modelin (21)ndash(23) require that at least one of 120588

1and 120588

2and 120588

3is not

zero


Proof The 119896-rank of matricesD B and C is 119896D = min (4 119903)119896B = 119903 and 119896C119879 = 119903 respectively The 119896-rank decompositionof the model is unique with 119896D + 119896B + 119896C119879 ge 2119903 + 2 for 119896 ge 2(see Theorem 1 in [21])

43 The Trilinear Alternating Least Squares Regression Algo-rithm The principle of the trilinear alternating least squares(TALS) regression algorithm is to fit the PARAFACmodel inthe noisy observation The idea of TALS is very simple Ineach step only onematrix is updatedTheupdating algorithmis based on the estimated results of the last updating andthe remaining matrices are updated by least squares methodRepeat the step as mentioned above until the algorithmconverges The advantage of TALS algorithm is that theparameter adjustment is not necessary A standard leastsquare problem is solved in each step and the performanceof TALS is good [25] Empirically if the 119896-rank condition(Theorem 2) is satisfied the TALS algorithm can convergeto the global minimum [26] The accelerating convergencetechnique of TALS algorithm can be found in [27 28] A briefintroduction about TALS algorithm can be found in [21]Theapplication details of the TALS algorithm which is used to fitthe PARAFAC model are elaborated as follows

On the basis of noisy observation the problem (21) canbe transformed into solving a least square problem

minDBC

R minus (D ⊙ B)C2119865 (30)

The principle of alternating least squares (ALS) can be usedto fit the problem in (30) In the noiseless condition ALScan be used to solve the matrices D B and C which con-structed the three-way array R The least square estimationof matrix C can be expressed as

C = argminCR minus (D ⊙ B)C2

119865 (31)

Similarly the matrices B andD can be expressed as

B119879 = argminB

10038171003817100381710038171003817R119883minus (C119879 ⊙D)B11987910038171003817100381710038171003817

2

119865

D119879= argmin

D

10038171003817100381710038171003817R119884minus (B ⊙ C119879)D11987910038171003817100381710038171003817

2

119865

(32)

In the iterative procedure givenmatricesB andD thematrixC can be represented as

C = (D ⊙ B)daggerR (33)

The expression of matrices B119879 andD119879 is

B119879 = (C119879 ⊙D)dagger

R119883

D119879= (B ⊙ C119879)

dagger

R119884

(34)

where (∙)dagger denotes the pseudoinverse of matrix (∙)Now the ALS algorithm steps can be summarized as

follows

(1) initialize B(0) isin C119872times119875D(0)isin C4times119875

(2) initialize 120576 gt 0 119896 = 0(3) if 120588(119896+1) minus120588(119896)120588(119896) gt 120576 calculate matricesD B and

C by (35)ndash(37) 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

44 The 2D-DOA Estimation Algorithm The estimators ofmatrices D B and C are obtained by the TALS algorithmwhich is introduced in the last subsection In this section the2D-DOA estimation can be obtained by the matrixD whichis shown as follows

ThematrixD can be acquired by the TALS algorithm 1205961119894

and 1205962119894can be calculated by matrixD

1205961119894= minus

1

2(angle lceil

D2119894

D1119894

rceil + angle lceilD4119894

D3119894

rceil) (35)

where D119895119894represents the 119895th row of D lceil∙rceil stands for the

absolute value operation Because ℎ1and ℎ

3are real numbers

D3119894D

1119894and D

4119894D

2119894are squared to solve the ambiguity

caused by the positive and negative values of ℎ1and ℎ

3

Consider

1205962119894= minus

1

2angle([

ℎ3(120579119894 120593

119894)

ℎ1(120579119894 120593

119894)exp (minus119895120596

2119894)]

2

)

= minus1

2angle (exp (minus1198952120596

2119894))

= minus1

2angle([

D3119894

D1119894

]

2

)

= minus1

2angle([

D4119894

D2119894

]

2

)

(36)

Then

1205962119894= minus

1

4

100381610038161003816100381610038161003816100381610038161003816

angle([D3119894

D1119894

]

2

) + angle([D4119894

D2119894

]

2

)

100381610038161003816100381610038161003816100381610038161003816

(37)

Take (12) (13) into (10) (11) and the elevation 120579119894and azimuth

120593119894of 119894th incident signal can be expressed as

120579119894= arccos(

1205821205961119894

21205871198892

) = arccos(2120596

1119894

120587)

120593119894= arccos(

1205821205962119894

21205871198892sin (120579

119894)) = arccos(

21205962119894

120587 sin (120579119894))

(38)

Based on Theorem 2 the estimators of D B and C havethe same column permutation matrix that is the 119894th columnof the steering matrix B corresponds to the 119894th of the matrixD Thus the elevation and azimuth pair with each otherautomatically

Combining PARAFAC theory with the TALS algorithmthe 2D-DOA estimation for the cylindrical conformal arraycan be summarized as follows

(1) calculate the signalsrsquo covariance matrices received byeach subarray using (7)


Z

Y

X

m + 1

O

m

2m + 1

1

2m + 2

2m + 3

2m + 4

3m + 2

3m + 3

2m + 2

m + 3

Figure 5 The extendable cylindrical conformal array structure

(2) construct the PARAFAC model by (20)ndash(23)

(3) estimate the matrixD using the TALS algorithm

(4) Calculate (36) and (37) using the estimator of matrixD then obtain the estimators of 120596

1119894and 120596

2119894

(5) acquire the elevation and azimuth estimation from(40) and (41)

5 The Extendable Array Structure

The proposed algorithm can be extended to other arraystructures with little modification First some elements areadded in the cylindrical conformal array and then theelements can be arrangedmore flexibly Second the proposedalgorithm is extended to conical conformal array

51 The Extendable Cylindrical Conformal Array First thedesign of cylindrical conformal array is introduced Secondthe PARAFAC model is constructed by the received dataFinally the 2D-DOA estimation is obtained

As shown in Figures 2 and 3 the proposed algorithm isfeasible when the distance vector ΔP

2between array 1 and

array 3 is parallel to119883-axis However the proposed algorithmmust be modified to be used in general cases Arrays 5 andarray 6 are added so that the arrays can be designed moreflexibly The extendable cylindrical conformal array is shownin Figure 5 2119898 + 3 sim 3119898 + 2 is array 5 and 2119898 + 4 sim 3119898 + 3

is array 6 The sensors 2119898 + 3 sim 3119898 + 3 possess the samepattern g

3 The distance vector between array 1 and array 5 is

ΔP3 and 119889

3= |ΔP

3| Under the design in Figure 5 the only

restriction is that arrays are parallel to 119885-axisThe received data of array 5 and array 6 are

X5= BΨ

3S + N

5 (39)

X6= BΨ

1Ψ3S + N

6 (40)

where

Ψ3= diag [

ℎ5(1205791 120593

1)

ℎ1(1205791 120593

1)exp (minus119895120596

31)

ℎ5(1205792 120593

2)

ℎ1(1205792 120593

2)exp (minus119895120596

32)

ℎ5(120579119903 120593

119903)

ℎ1(120579119903 120593

119903)exp (minus119895120596

3119903)]

(41)

1205963119894= (

2120587

120582)ΔP

3∙ u

119894

= (2120587119889

2

120582)


) cos (120593ΔP3

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP2

) sin (120593ΔP2

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP3

) cos (120579119894)]

(42)

The cross-covariance matrices among array 1 array 5 andarray 6 are

R5= 119864 X

5X119867

1 = BΨ

3R119904B119867 +Q

5

R6= 119864 X

6X119867

1 = BΨ

1Ψ3R119904B119867 +Q

6

(43)

where Q5and Q

6are noise covariance matrices Then (20)

is modified into an 119898 times 119898 times 6 three-way array PARAFACmodel

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)R ( 2)R ( 3)R ( 4)R ( 5)R ( 6)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R1

R2

R3

R4

R5

R6

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

BR119904B119867

BΨ1R119904B119867

BΨ2R119904B119867

BΨ1Ψ2R119904B119867

BΨ3R119904B119867

BΨ1Ψ3R119904B119867

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+Q1 (44)

where Q1is the observed noise The form of Khatri-Rao

product is applied in (44)

R = (D ⊙ B)C +Q1 (45)

where

D =

[[[[[[[

[

Λminus1(R

119904)

Λminus1(Ψ

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

1Ψ2R119904)

Λminus1(Ψ

3R119904)

Λminus1(Ψ

1Ψ3R119904)

]]]]]]]

]

(46)

As long as Theorem 2 holds (45) is unique 1205963119894can be

obtained similarly with (37)

1205963119894= minus

1

4

100381610038161003816100381610038161003816100381610038161003816

angle([D5119894

D1119894

]

2

) + angle([D6119894

D2119894

]

2

)

100381610038161003816100381610038161003816100381610038161003816

(47)

after calculating matrixD by ALS algorithm


Assume that Δ11990111

= sin(120579ΔP1

) cos(120593ΔP1

) Δ11990112

=

sin(120579ΔP1

) sin(120593ΔP1

) andΔ11990113= cos(120579

ΔP1

) similar toΔ1199012119894and

Δ1199013119894(119894 = 1 2 3) 120574

1119894= sin(120579

119894) cos(120593

119894) 120574

2119894= sin(120579

119894) sin(120593

119894)

and 1205743119894= cos(120579

119894) solving (10) (11) (42) (36) (37) and (24)

we can obtain

120582

2120587

[[[[[

[

1205961119894

1198891

1205962119894

1198892

1205963119894

1198893

]]]]]

]

= [

[

Δ11990111

Δ11990112

Δ11990113

Δ11990121

Δ11990122

Δ11990123

Δ11990131

Δ11990132

Δ11990133

]

]

[

[

1205741119894

1205742119894

1205743119894

]

]

(48)

The solution of (48) is

[

[

1205741119894

1205742119894

1205743119894

]

]

=120582

2120587

[

[

Δ11990111

Δ11990112

Δ11990113

Δ11990121

Δ11990122

Δ11990123

Δ11990131

Δ11990132

Δ11990133

]

]

minus1 [[[[[

[

1205961119894

1198891

1205962119894

1198892

1205963119894

1198893

]]]]]

]

(49)

1205741 120574

2 and 120574

3can be acquired by solving (49) Two methods

can be used to obtain 120579119894120593

119894The firstmethod can be described

as

120579119894= arccos 120574

3119894

120593119894= arcsin(

1205741119894

cos (120579119894)) or 120593

119894= arcsin(

1205742119894

sin (120579119894))

(50)

The second one is

120593119894= arctan(

1205742119894

1205741119894

)

120579119894= arccos(

1205741119894

cos (120593119894)) or 120579

119894= arcsin(

1205742119894

sin (120593119894))

(51)

52The Extendable Conical Conformal Array In this sectionthe proposed algorithm is extended to conical conformalarray First the design of conical conformal array is intro-duced Second the PARAFAC model is constructed by thereceived data Finally the 2D-DOA estimation is obtainedThe analytic solution of the conical conformal array does notexist Thus the iterative numerical approximation method isused to obtain the optimal solution

Figure 6 applies the proposed algorithm to the conicalconformal array 1 sim 119898 is array 1 2 sim 119898 + 1 is array2 1 sim 2119898 is array 3 and 119898 + 2 sim 2119898 + 1 is array 4The distance vector between array 1 and array 2 is ΔP

1 and

1198891= |ΔP

1| The distance vector between array 3 and array

4 is ΔP2 and 119889

2= |ΔP

2| 1 sim 119898 + 1 possess the same

pattern g1 and 1 sim 2119898 + 1 possess the same pattern g

2

The decoupling between polarization parameter and DOAinformation can be completed by making full use of thegeometry characteristic of the array structure

Z

Y

XO

3m2m

2m + 1

2m + 2

2m + 3

3m + 1

m + 2

1

2

m

m

+ 3 3

m + 1

Figure 6 The extendable conical conformal array structure

The received data from array 1 to array 4 are

X1= B

1S + N

1 (52)

X2= B

1Ψ1S + N

2 (53)

X3= B

2S + N

3 (54)

X4= B

2Ψ2S + N

4 (55)

where N1ndashN

6are the noise matrices received by each array

The cross-covariance matrices among the arrays are

R1= 119864 X

3X119867

1 = B

2R119904B1198671+Q

1

R2= 119864 X

4X119867

1 = B

2Ψ2R119904B1198671+Q

2

R3= 119864 X

3X119867

2 = B

2Ψ119867

1R119904B1198671+Q

3

R4= 119864 X

4X119867

2 = B

2Ψ119867

1Ψ2R119904B1198671+Q

4

(56)

where B1is the manifold matrix of array 1 and array 2 and

B2is the manifold matrix of array 3 and array 4 Q

1ndashQ

4

are the noise covariance matrices received by the arrays Ψ1

and Ψ2possess the same forms as (8) and (9) According to

PARAFAC theory the119898 times 119898 times 4 three-way array for conicalconformal array can be constructed by (56)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)R ( 2)R ( 3)R ( 4)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R1

R2

R3

R4

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

B2R119904B1198671

B2Ψ119867

1R119904B1198671

B2Ψ2R119904B1198671

B2Ψ119867

1Ψ2R119904B1198671

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+Q2

(57)


and Q2is the observed noise Let C = B119867

1 and (57) can be

written in the following form of Khatri-Rao product

R = (D ⊙ B2)C +Q

2

D =

[[[[[[[[

[

Λminus1(R

119904)

Λminus1(Ψ

119867

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

119867

1Ψ2R119904)

]]]]]]]]

]

(58)

The matrix D can be obtained by the ALS algorithm ifthe uniqueness of (57) is guaranteed By using the similarmethod 120596

1119894and 120596

2119894can be represented as

1205961119894=1

2(angle[

[[

D2119894

D1119894

]]]

+ angle[[[

D4119894

D3119894

]]]

) (59)

1205962119894= minus

1

2(angle[

[[

D3119894

D1119894

]]]

+ angle[[[

D4119894

D2119894

]]]

) (60)

1205961119894= (

2120587

120582119894

)ΔP1∙ u

119894

= (2120587119889

1

120582119894

)


) cos (120593ΔP1

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP1

) sin (120593ΔP1

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP1

) cos (120579119894)]

(61)

1205962119894= (

2120587

120582119894

)ΔP2∙ u

119894

= (2120587119889

2

120582119894

)


) cos (120593ΔP2

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP2

) sin (120593ΔP2

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP2

) cos (120579119894)]

(62)

The 2D-DOA estimation can be acquired by solving (61) and(62) The equations are nonlinear and the analytical solutiondoes not exist Thus the iterative numerical approximationmethod is used to obtain the optimal solutionThe nonlinearequations can be written as

1198911(119909

1 119909

2) = 0

1198912(119909

1 119909

2) = 0

(63)

where 1199091= 120579 and 119909

2= 120593 The equation with the same

solution of (63) is expressed as

x = 119891minus1119894(119909

1 119909

2) 119894 = 1 2 (64)

The iterative form can be represented as

x119896+1 = 119865119894(119909

119896

1 119909

119896

2) 119894 = 1 2 (65)

The initial vector is selected as x(0) = [119909(0)1 119909

(0)

2]119879

Calculating(65) until the sequence converges that is x119896 rarr xlowast andregarding xlowast as the iterative solution of (63) we can achievethe 2D-DOA estimation of the incident signal

Assuming that the noise is nonuniform in theorythrough calculating the cross-covariance matrices amongdifferent arrays nonuniform noise could be suppressed

6 The Computational Complexity Analysis

We only focus on the major part which is the number ofmultiplications involved in calculating covariance matricesand the ALS algorithm As mentioned above 119873 119903 and 2119898represent the snapshot source number and sensor numberrespectively The ESPRIT algorithm which is similar to thealgorithm proposed in [18] needs to calculate the eigende-composition of covariance matrices and parameter pairing(the covariance matrix is used instead of the four-ordercumulant in [18]) The eigendecomposition of three 2119898times 2119898

matrices requires about119874(241198983)The algorithm in this paper

uses COMFAC algorithm to fit an 119898 times 119898 times 4 three-wayarray The computational complexity per iteration is 119874(1199033) +119874(4119898

2119903) For the simulation in Section 9 the proposed

algorithm converges in only two iterations Therefore thetotal computation complexity of the proposed algorithm is119874[119870(119903

3+4119898

2119903)] (119870 stands for the number of iterations) Since

the initialization takes up a large proportion of computationtime in each iteration the proposed algorithm suffers from ahigher computational complexity

7 CRB of Joint Polarization andDOA Estimation

Cramer-Rao bound (CRB) gives a lower bound of unbiasedparameter estimation In this section the multiparameterestimation CRB is derived For the sake of simplicity thesignal covariance matrix R

119904is assumed to be known and the

noise is normalized as one 4119903 parameters are contained in thecovariancematrixR that is 119903 elevation parameters 119903 azimuthparameters and 2119903 polarization parameters respectively Inthis paper the polarization parameters of the incident signalsare assumed to be known Only 2119903 parameters remain tobe estimated The vector parameters to be estimated arerepresented as

k119879 = [1205791 120593

1 1205792 120593

2 120579

119903 120593

119903] (66)


The CRB of joint polarization parameters and angle parame-ters is defined as

119864 [(k minus k) (k minus k)119879] ge CRB

CRB = Fminus1(67)

The 2119903 times 2119903 Fisher information matrix (FIM) for theparameter k is given by

F = [F120579120579 F120579120593

F120593120579

F120593120593

] (68)

where F120579120579

is the block matrix of elevation estimator and F120593120593

is the block matrix of azimuth estimatorThe entry of 119894th rowand 119895th column of FIM is represented as

F119894119895= 119873trace[Rminus1 120597R

120597V119894

Rminus1 120597R120597V

119895

]

= 2119873Re trace [D119894R119904B119867Rminus1BR

119904D119867

119895Rminus1]

+ trace [D119894R119904B119867Rminus1BR

119904D119895Rminus1]

D119894=120597B120597k

119894

(69)

where trace[∙] is the trace of matrix [∙] and 120597R120597V119894is the

partial derivative of matrix R

8 Discussion

The proposed algorithm is able to operate on very irregularshapes which is not following a circular geometry Howeversome requirements have to be satisfied The design of thearray is discussed in two categories the algorithm has ananalytic solution the algorithm does not have an analyticsolution

(1) The Algorithm Has an Analytic Solution Based on thedesign of array in Figure 2 and (12) (13) it can be seenthat the distance vector ΔP

1is perpendicular to ΔP

2 This is

one requirement for obtaining the analytic solution Anotherrequirement is that the distance vector ΔP

1or ΔP

2is parallel

or perpendicular to the coordinate axis Then the analyticsolution can be obtained For example the design of thespherical conformal array is given as follows

The structure of spherical conformal array is shown inFigure 7 The elements are arranged on the surface of thespherical conformal array 1 sim 119898minus1 constitute array 1 2 sim 119898constitute array 2 119898 + 1 sim 2119898 minus 1 constitute array 3 and119898 + 2 sim 2119898 constitute array 4 The elements of the arrayare divided into two subarrays Subarray 1 consists of array1 and array 2 and subarray 2 consists of array 3 and array 4The distance vector between array 1 and array 2 is ΔP

1 The

distance vector between array 3 and array 4 is ΔP2 As shown

in Figures 7 and 8 the distance vector ΔP1is perpendicular

to ΔP2 Also the distance vector ΔP

1is parallel to 119884-axis

(2) The Algorithm Does Not Have an Analytic Solution Theonly requirement is that ΔP

1is not parallel to ΔP

2as shown

Z

YX

O

m2m

m + 2m + 1

1

23

42m minus 1

m minus 1

⋱⋱

Figure 7 The structure of spherical conformal array

in Figure 2 In other words the subarray 1 is not parallel tosubarray 2 as shown in Figure 6 According to (61) and (62)the analytic solution of the proposed algorithmdoes not existThus the iterative numerical approximation method is usedto obtain the optimal solution

For very irregular shapes array if we can find twosubarrays as mentioned above and the two distance vectorsare not parallel to each other then the proposed algorithmcan be used for DOA estimation

9 Simulation Results

In this section we demonstrate the performance of theproposed algorithm via numerical simulation Taking acylindrical conformal array as an example we use the arraystructure in Figure 2 for simulation The number of sensorsis 16 that is 119898 = 8 Without loss of generality 119896

1120579= 05

1198961120593

= 05 1198962120579

= 03 1198962120593

= 07 The sensor patterns are119892119894120579= sin(120579

119895minus 120593

119895) and 119892

119894120593= cos(120579

119895minus 120593

119895) in the global

coordinate 120579119895and 120593

119895are the elevation and azimuth of the

119894th incident signal in the 119895th local coordinate respectivelyThe details regarding how the pattern is transformed fromthe local coordinate to the global coordinate can be foundin [8 20] 500 independent trials are considered Root meansquare error (RMSE) which indicates the performance of theproposed algorithm is defined as

RMSE = radic 1

500

500

sum

119905=1

[(120579119894119905minus 120579)

2

+ (120593119894119905minus 120593)

2

] (70)

where 120579119894119905and 120593

119894119905are the elevation and azimuth estimators of

the 119894th incident signal in the 119905th trial The ESPRIT algorithmproposed in [18] is simulated in the same scenarios (theESPRIT algorithm is used for the covariance matrix ratherthan the four-order cumulant)

For the following experiments theCOMFACalgorithm isused to fit the119898times119898times4 three-way arrayThe initialization andfitting of the COMFAC are done in compressed space TheTucker3 three-way model is used in data compression [29]

In the first experiment we show how the success rateand RMSE changed with different SNR Here a successful


Y

XO

m

m minus 11

2

middot middot middot


ΔP1

(a) The distance vector ΔP1

O

Y

X

m + 2

m + 1

ΔP2

(b) The distance vector ΔP2

Figure 8 The schematic diagram of the distance vector

0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Succ

ess r

ate (

)

PARAFAC source 1 PARAFAC source 2

ESPRIT source 1ESPRIT source 2

minus10 minus5 0 5 10 15 20

(a) The success rate versus SNR

SNR (dB)

RMSE

(deg

)

PARAFAC source 1 PARAFAC source 2 ESPRIT source 1

ESPRIT source 2 CRB source 1 CRB source 2

07

06

05

04

03

02

01

0minus10 minus5 0 10 15 20

(b) The RMSE versus SNR

Figure 9 The estimation performance versus SNR

experiment is defined as the experimentwith estimation errorof less than 1 degreeThe elevation and azimuth angles of twofar-field narrow incident signals are (95∘ 50∘) and (100∘ 60∘)respectively 500 snapshots are used in this experiment Thesuccess rate and RMSE are shown in Figures 9(a) and 9(b)respectively Figure 9(a) shows that the success rate of theproposed PARAFAC algorithm is higher than that of theESPRIT algorithm when the SNR is low In addition theRMSE of the proposed algorithm ismuch smaller than that ofthe ESPRIT algorithm at high SNR As the SNR increases theRMSE of the proposed algorithm approximates to the CRBFour covariance matrices are used to estimate DOA in theproposed algorithm However only two covariance matrices

are used for the ESPRIT algorithm The proposed algorithmutilizes more data information than the ESPRIT algorithmwhich leads to better performance

We plot the curves of success rate and RMSE versusnumber of snapshots in Figures 10(a) and 10(b) SNR is0 dB and 10 dB in Figures 10(a) and 10(b) respectivelyOther simulation conditions are the same as those in thefirst experiment Figure 10(a) shows that the success rate ofthe proposed algorithm is higher than that of the ESPRITalgorithm In particular the success rate of source 2 esti-mated by the ESPRIT algorithm is lower than others Asshown in Figure 10(b) the RMSE of the proposed algorithmis much smaller than that of the ESPRIT algorithm at


100 200 300 400 500 600 700 800 900 100050

55

60

65

70

75

80

85

90

95

100

Number of snapshots

Succ

ess r

ate (

)


ESPRIT source 1 ESPRIT source 2

(a) The success rate versus number of snapshots

100 200 300 400 500 600 700 800 900 10000

002

004

006

008

01

012

014

016

Number of snapshots

RMSE

(deg

)

PARAFAC source 1PARAFAC source 2ESPRIT source 1


(b) The RMSE versus number of snapshots

Figure 10 The estimation performance versus number of snapshots

0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Reso

lutio

n pr

obab

ility

()

PARAFAC source 1PARAFAC source 2


minus10 minus5 0 5 10 15 20

Figure 11 The resolution probability versus SNR

the same number of snapshots The RMSE approaches to theCRB as the number of snapshots increases The RMSE ofsource 2 with ESPRIT algorithm is relatively larger which ismainly caused by the fact that the ESPRIT algorithm onlyuses the autocovariance matrices of the array However theproposed algorithm achieves higher accuracy by using bothautocovariance and cross-covariance matrices of the array

Figure 11 shows the resolution probability versus SNRfor both the proposed PARAFAC algorithm and the ESPRITalgorithm when the sources are closely spaced (6 degreesseparation)The elevation and azimuth of the incident signalsare (100∘ 54∘) and (100∘ 60∘) respectively Other simulation

conditions are same as those in the first experiment Thetwo incident signals are resolved if |120579

1minus 120579

1| |120579

2minus 120579

2| are

smaller than |1205791minus 120579

2|2 Meanwhile |120593

1minus 120593

1| |120593

2minus 120593

2| are


2|2 120579

119894and 120579

119894represent the estimated

and real elevations for 119894th incident signal respectively 120593119894

and 120593119894represent the estimated and real azimuths for 119894th

incident signal respectively [30] Figure 11 shows that theresolution of ESPRIT algorithm outperforms the proposedalgorithm in relatively low SNR (minus2 dB to 2 dB) WhenSNR is greater than 2 dB the proposed algorithm performsbetter According to (25) and Theorem 2 to ensure theuniqueness of the PARAFAC model a( 119894) = a( 119895) must beguaranteed In other words the two incident signals couldnot be too close When the SNR is low the effect ofnoise is fairly remarkable All the above reasons result inthe relatively poor resolution performance of the proposedalgorithm in low SNR Nevertheless it is a good compromisefor having excellent estimation accuracy and robustness tonoise

10 Conclusion

In this paper a novel high accuracy 2D-DOA estimationalgorithm for the conformal array is proposed The ordinaryDOA estimation algorithm cannot be used on conformalarray because of the polarization diversity of the varyingcurvature To avoid parameter pairing problem the algo-rithm forms a PARAFAC model of covariance matricesin the covariance domain to estimate the 2D-DOA Theuniqueness condition of the PARAFAC model is derived Itcan be generalized to other conformal array structure that isconical conformal array and so forth Computer simulationverifies the effectiveness of the proposed algorithm in termsof accuracy and robustness to noise


Appendix

The KR product of two matrices S isin C119862times119875 and T isin C119863times119875 ofan identical number of columns is given by

S ⊙ T = [s1otimes t

1 s

119875otimes t

119875] isin C

119862119863times119875 (A1)

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

1199042t119904119862t

]]]]

]

(A2)

Conflict of Interests

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

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] K M Tsui and S C Chan ldquoPattern synthesis of narrowbandconformal arrays using iterative second-order cone program-mingrdquo IEEE Transactions on Antennas and Propagation vol 58no 6 pp 1959ndash1970 2010

[2] M Comisso and R Vescovo ldquoFast co-polar and cross-polar3D pattern synthesis with dynamic range ratio reduction forconformal antenna arraysrdquo IEEE Transactions on Antennas andPropagation vol 61 pp 614ndash626 2013

[3] W-J Zhao L-W Li E-P Li and K Xiao ldquoAnalysis of radiationcharacteristics of conformal microstrip arrays using adaptiveintegral methodrdquo IEEE Transactions on Antennas and Propaga-tion vol 60 no 2 pp 1176ndash1181 2012

[4] A Elsherbini and K Sarabandi ldquoENVELOP antenna a classof very low profile UWB directive antennas for radar andcommunication diversity applicationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 3 pp 1055ndash1062 2013

[5] B R Piper and N V Shuley ldquoThe design of spherical conformalantennas using customized techniques based on NURBSrdquo IEEEAntennas and Propagation Magazine vol 51 no 2 pp 48ndash602009

[6] J L Gomez-Tornero ldquoAnalysis and design of conformal taperedleaky-wave antennasrdquo IEEE Antennas and Wireless PropagationLetters vol 10 pp 1068ndash1071 2011

[7] T Milligan ldquoMore applications of euler rotationrdquo IEEE Anten-nas and Propagation Magazine vol 41 no 4 pp 78ndash83 1999

[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] L Zou J Laseby and Z He ldquoBeamformer for cylindrical con-formal array of non-isotropic antennasrdquo Advances in Electricaland Computer Engineering vol 11 no 1 pp 39ndash42 2011

[10] L Zou J Lasenby and Z He ldquoBeamforming with distortionlessco-polarisation for conformal arrays based on geometric alge-brardquo IET Radar Sonar andNavigation vol 5 no 8 pp 842ndash8532011

[11] D W Boeringer D H Werner and D W Machuga ldquoA simul-taneous parameter adaptation scheme for genetic algorithmswith application to phased array synthesisrdquo IEEE Transactionson Antennas and Propagation vol 53 no 1 pp 356ndash371 2005

[12] Y Y Bai S Xiao C Liu and B ZWang ldquoOptimisationmethodon conformal array element positions for low sidelobe patternsynthesisrdquo IEEE Transactions on Antennas and Propagation vol61 no 4 pp 2328ndash2332 2013

[13] K Yang Z Zhao J Ouyang Z Nie and Q H Liu ldquoOptimi-sation method on conformal array element positions for lowsidelobe pattern synthesisrdquo IET Microwave vol 6 no 6 pp646ndash652 2012

[14] R Karimizadeh M Hakkak A Haddadi and K ForooraghildquoConformal array pattern synthesis using the weighted alternat-ing reverse projectionmethod consideringmutual coupling andembedded-element pattern effectsrdquo IET Microwave vol 6 no6 pp 621ndash626 2012

[15] L I Vaskelainen ldquoConstrained least-squares optimization inconformal array antenna synthesisrdquo IEEE Transactions onAntennas and Propagation vol 55 no 3 pp 859ndash867 2007

[16] J He M O Ahmad and M N S Swamy ldquoNear-field local-ization of partially polarized sources with a cross-dipole arrayrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 2 pp 859ndash867 2013

[17] X Yuan ldquoEstimating the DOA and the polarization of apolynomial-phase signal using a single polarized vector-sensorrdquoIEEE Transactions on Signal Processing vol 60 no 3 pp 1270ndash1282 2012

[18] 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

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

[20] W J Si L T Wan L T Liu and Z X Tian ldquoFast estimationof frequency and 2-D DOAs for cylindrical conformal arrayantenna using state-space and propagator methodrdquo Progress inElectromagnetics Research vol 137 pp 51ndash71 2013

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

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

[23] J B Kruskal ldquoRank decomposition and uniqueness for 3-wayand Nway arraysrdquo inMultiway Data Analysis pp 8ndash18 1988

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

[25] R Bro ldquoPARAFAC Tutorial and applicationsrdquo Chemometricsand Intelligent Laboratory Systems vol 38 no 2 pp 149ndash1711997


[26] P K Hopke P Paatero H Jia R T Ross and R A Harsh-man ldquoThree-way (PARAFAC) factor analysis examination andcomparison of alternative computational methods as applied toill-conditioned datardquo Chemometrics and Intelligent LaboratorySystems vol 43 no 1-2 pp 25ndash42 1998

[27] R Bro Multi-Way Analysis in the Food Industry ModelsAlgorithms and Applications University of Amsterdam (NL)amp Royal Veterinary and Agricultural University (DK) Amster-dam The Netherlands 1998

[28] H A L Kiers and W P Krijnen ldquoAn efficient algorithm forPARAFAC of three-way data with large numbers of observationunitsrdquo Psychometrika vol 56 no 1 pp 147ndash152 1991

[29] N D Sidiropoulos ldquoCOMFAC Matlab Code for LS Fitting ofthe Complex PARAFAC Model in 3-Drdquo 1998 httpwwweceumnedusimnikoscomfacm

[30] P Stoica and A B Gershman ldquoMaximum-likelihoodDOA esti-mation by data-supported grid searchrdquo IEEE Signal ProcessingLetters vol 6 no 10 pp 273ndash275 1999

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



or in situations where high estimation accuracy is neededSimulation results demonstrate the efficiency and accuracy ofthe proposed algorithm

The rest of this paper is structured as follows Section 2introduces the snapshot data model Section 3 elaborates thedesign of the conformal array structure Section 4 containsthe core contributions of this paper in which the PARAFACmodel is used for high accuracy 2D-DOA estimation Sec-tion 5 extends the proposed algorithm to more generalcases Section 6 analyses the computational complexity of thealgorithm Section 7 derives the Cramer-Rao bound (CRB)Section 8 discusses the array design Section 9 presents thesimulation result Conclusions are drawn in Section 10

2 The Snapshot Data Model

In order to achieve accurate parameter estimation the snap-shot data model of the conformal array must be preciselyestablished The pattern of the elements is different due tothe varying curvature of carriers [8] The mathematic modelof the conformal array constructed in [8 18] is adopted inthis paper The far-field narrowband signal impinging on theconformal array with elevation 120579 and azimuth 120593 is shown inFigure 1(a) The steering vector takes the following form

a (120579 120593) = [ℎ1119890minus1198952120587((p

1∙u)120582)

ℎ2119890minus1198952120587((p

2∙u)120582)

ℎ2119898+2

119890minus1198952120587((p

2119898+2∙u)120582)

]119879

(1)

ℎ119894= (119892

2

119894120579+ 119892

2

119894120593)12

(1198962

120579+ 119896

2

120593)12

cos (120579119894119892119896)

=1003816100381610038161003816119892119894100381610038161003816100381610038161003816100381610038161199011198971003816100381610038161003816 cos (120579119894119892119896)

= g119894∙ p

119897= 119892

119894120579119896120579+ 119892

119894120593119896120593

(2)

where P119894represents the position vector of the 119894th element u

is the propagation vector and 120582 is wavelength of the incidentsignal As shown in Figure 1(b) u

120579and u

120593are orthogonal unit

vectors and 119896120579and 119896

120593are the polarization components of the

incident signal ℎ119894denotes the 119894th elementrsquos response of the

incident signal g119894is the pattern of the 119894th element and p

119897is

the electric field direction of the incident signal 120579119894119892119896

is theangle between vector g

119894and p

119897 (∙)119879 denotes the transpose of

matrix (∙)Assuming that 119903 incident signals impinge on the designed

conformal array the snapshot data model can be representedas

X (119899) = G ∙ AS (119899) + N (119899)

= (G120579∙ A

120579K120579+ G

120593∙ A

120593K120593) S (119899) + N (119899)

= BS (119899) + N (119899)

G120579= [g

120579(1205791 120593

1) g

120579(1205792 120593

2) g

120579(120579119903 120593

119903)]

G120593= [g

120593(1205791 120593

1) g

120593(1205792 120593

2) g

120593(120579119903 120593

119903)]

A120579= [a

120579(1205791 120593

1) a

120579(1205792 120593

2) a

120579(120579119903 120593

119903)]

A120593= [a

120593(1205791 120593

1) a

120593(1205792 120593

2) a

120593(120579119903 120593

119903)]

K120579= diag (119896

1120579 119896

2120579 119896

119903120579)

K120593= diag (119896

1120593 119896

2120593 119896

119903120593)

S (119899) = [1199041(119899) 119904

2(119899) 119904

119903(119899)]

119879

W (119899) = [1199081(119899) 119908

2(119899) 119908

119903(119899)]

119879

(3)

where G denotes the pattern matrix and A denotes themanifold matrix K

120579and K

120593are the diagonal matrices whose

diagonal entries are 1198961120579 119896

2120579 119896

119903120579and 119896

1120593 119896

2120593 119896

119903120593

respectively 119896119894120579and 119896

119894120593are the components of 119894th signalrsquos

polarization parameters along the orthogonal unit vector u120579

and u120593 respectively S(119899) denotes the signal vector W(119899) is

additive Gaussian white noise with the covariance matrix

119864 W (119899)W(119899)119867 = Q = 120590

2I (4)

(∙)119867 denotes conjugate transpose of matrix (∙) Collecting119873

snapshots

X = BS +W (5)

where S is the 119903 times119873 signal matrix andW is the (2119898 + 2) times119873

noise matrixThe definition of the pattern is in the local coordinate

meaning that transformation must be performed from theglobal coordinate to the local coordinate [8] In addition thepolarization parameters couple with the signal parametersthus the decoupling is definitely necessary for the 2D-DOAestimation of the conformal array

3 The Structure of the Array

Since the signalrsquos amplitude of the sensors received willdecrease because of the ldquoshadow effectrdquo the whole conformalarray is divided into three subarrays and each subarray covers120

∘ For the single curvature and symmetry of the cylindereach subarray possesses and remains the same structure aswell as the parameter estimationmechanismThus this paperonly considers one subarray

As shown in Figure 2 the distance between two sensorswhich are in the same intersecting surface is 1205824 and thedistance between two adjacent intersecting surfaces is 5120582

As shown in Figure 2 1 sim 119898 constitute array 1 2 sim 119898 + 1

constitute array 2 119898 + 2 sim 2119898 + 1 constitute array 3 and119898 + 3 sim 2119898 + 2 constitute array 4 (1 sim 2119898 + 2 are sensorsof the array) The distance vector between array 1 and array2 is ΔP

1and 119889

1= |ΔP

1| = 1205824 The distance vector between

array 1 and array 3 is ΔP2and 119889

2= |ΔP

2| = 1205824 as shown

in Figure 3 The sensorsrsquo patterns of the same generatrix are


Z

O

X

Y

120579

120593

minusu

(a)

g i

pl

O u120579k120579

gi120593

k120593

gi120579

120579igk

u120593

(b)


Z

Y

X

m + 1

m

O

2

1m + 2

m + 3

2m + 1

2m + 2





Assume that X1 X

2 X

3 and X



X1= BS + N

1

X2= BΨ

1S + N

2

X3= BΨ

2S + N

3

X4= BΨ

1Ψ2S + N

4

(6)


R1= 119864 X

1X119867

1 = BR

119904B119867 +Q

1

R2= 119864 X

2X119867

1 = BΨ

1R119904B119867 +Q

2

R3= 119864 X

3X119867

1 = BΨ

2R119904B119867 +Q

3

R4= 119864 X

4X119867

1 = BΨ

1Ψ2R119904B119867 +Q

4

(7)

where R119904= diag1199042

1 119904

2



4 respec-

tivelyConsider


11) exp (minus119895120596

12) exp (minus119895120596

1119903)]

(8)

Ψ2= diag [

ℎ3(1205791 120593

1)

ℎ1(1205791 120593

1)exp (minus119895120596

21)

ℎ3(1205792 120593

2)

ℎ1(1205792 120593

2)exp (minus119895120596

22)

ℎ3(120579119903 120593

119903)

ℎ1(120579119903 120593

119903)exp (minus119895120596

2119903)]

(9)

1205961119894= (

2120587

120582)ΔP

1∙ u

119894

= (2120587119889

1

120582)


) cos (120593ΔP1

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP1

) sin (120593ΔP1

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP1

) cos (120579119894)]

(10)

1205962119894= (

2120587

120582)ΔP

2∙ u

119894

= (2120587119889

2

120582)


) cos (120593ΔP2

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP2

) sin (120593ΔP2

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP2

) cos (120579119894)]

(11)

where 120579ΔP119894

and 120593ΔP119894



OY

Z

Δ

P2

P1

P1

(a)

Y

Xm+1 2m+1

O

ΔP

P

2

P

(b)


2




120579ΔP1

= 0 120593ΔP1

=120587

2 (12)



120579ΔP2

=120587

2 120593

ΔP2

= 0 (13)


119896(119896 = 1 4) that is

R119896=1

119873

119873

sum

119896=1

X (119899)X(119899)119867 =1

119873XXH

(14)







119888 119889119890


119909119888 119889 119890

=

119872

sum

119901=1

119904119888119901119905119889119901119906119890119901 (15)









119890are


X119889(119888 119890) = X

119890(119888 119889) = 119909



Consider

X119888= TΛ

119888(S)U119879

119888 = 1 119862

X119889= UΛ

119889(T) S119879 119889 = 1 119863

X119890= SΛ

119890(U)T119879 119890 = 1 119864

(16)


= ++

1u 2u

3

3u

t2t1t

1s 2s 3sX




X(119862119863times119864)=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

X119888=1

X119888=2

X119888=119862

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

TΛ1(S)

TΛ2(S)

TΛ119862(S)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

U119879

= (S ⊙ T)UT (17)


X(119863119864times119862)= (T ⊙ U) ST (18)

X(119864119862times119863)= (U ⊙ S)TT

(19)





100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)R ( 2)R ( 3)R ( 4)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R1

R2

R3

R4

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

BR119904B119867

BΨ1R119904B119867

BΨ2R119904B119867

BΨ1Ψ2R119904B119867

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+Q (20)




R = (D ⊙ B)C +Q (21)

R119883= (C119879 ⊙D)B119879 +Q

119883 (22)

R119884= (B ⊙ C119879)D119879

+Q119884 (23)

D =[[[

[

Λminus1(R

119904)

Λminus1(Ψ

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

1Ψ2R119904)

]]]

]

(24)





a ( 119894) = a ( 119895) (25)


119895(119896)

ℎ119900119890119890minus1198952120587((p

119900119890∙u119894)120582)

= ℎ119900119890119890minus1198952120587((p

119900119890∙u119895)120582) (26)


119901119900119890= sin (120579

119900119890) cos (120593

119900119890) x

+ sin (120579119900119890) sin (120593

119900119890) y + cos (120579

119900119890) z

(27)

u119894= sin (120579

119894) cos (120593

119894) x

+ sin (120579119894) sin (120593

119894) y + cos (120579

119894) z

(28)

120579119900119890

and 120593119900119890


119894and 120593



sin (120579119900119890) cos (120593

119900119890) 120588

1

+ sin (120579119900119890) sin (120593

119900119890) 120588

2+ cos (120579

119900119890) 120588

3= 0

1205881= sin (120579

119894) cos (120593

119894) minus sin (120579

119895) cos (120593

119895)

1205882= sin (120579

119894) sin (120593

119894) minus sin (120579

119895) sin (120593

119895)

1205883= cos (120579

119894) minus cos (120579

119895)

(29)


1and 120588

2and 120588

3is not

zero





minDBC

R minus (D ⊙ B)C2119865 (30)



119865 (31)


B119879 = argminB

10038171003817100381710038171003817R119883minus (C119879 ⊙D)B11987910038171003817100381710038171003817

2

119865

D119879= argmin

D

10038171003817100381710038171003817R119884minus (B ⊙ C119879)D11987910038171003817100381710038171003817

2

119865

(32)




B119879 = (C119879 ⊙D)dagger

R119883

D119879= (B ⊙ C119879)

dagger

R119884

(34)


follows








1205961119894= minus

1

2(angle lceil

D2119894

D1119894


D3119894

rceil) (35)



3are real numbers

D3119894D

1119894and D

4119894D



3

Consider

1205962119894= minus

1

2angle([

ℎ3(120579119894 120593

119894)

ℎ1(120579119894 120593

119894)exp (minus119895120596

2119894)]

2

)

= minus1


2119894))

= minus1

2angle([

D3119894

D1119894

]

2

)

= minus1

2angle([

D4119894

D2119894

]

2

)

(36)

Then

1205962119894= minus

1

4

100381610038161003816100381610038161003816100381610038161003816

angle([D3119894

D1119894

]

2

) + angle([D4119894

D2119894

]

2

)

100381610038161003816100381610038161003816100381610038161003816

(37)



120579119894= arccos(

1205821205961119894

21205871198892

) = arccos(2120596

1119894

120587)

120593119894= arccos(

1205821205962119894

21205871198892sin (120579

119894)) = arccos(

21205962119894

120587 sin (120579119894))

(38)





Z

Y

X

m + 1

O

m

2m + 1

1

2m + 2

2m + 3

2m + 4

3m + 2

3m + 3

2m + 2

m + 3





1119894and 120596

2119894










ΔP3 and 119889

3= |ΔP



X5= BΨ

3S + N

5 (39)

X6= BΨ

1Ψ3S + N

6 (40)

where

Ψ3= diag [

ℎ5(1205791 120593

1)

ℎ1(1205791 120593

1)exp (minus119895120596

31)

ℎ5(1205792 120593

2)

ℎ1(1205792 120593

2)exp (minus119895120596

32)

ℎ5(120579119903 120593

119903)

ℎ1(120579119903 120593

119903)exp (minus119895120596

3119903)]

(41)

1205963119894= (

2120587

120582)ΔP

3∙ u

119894

= (2120587119889

2

120582)


) cos (120593ΔP3

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP2

) sin (120593ΔP2

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP3

) cos (120579119894)]

(42)


R5= 119864 X

5X119867

1 = BΨ

3R119904B119867 +Q

5

R6= 119864 X

6X119867

1 = BΨ

1Ψ3R119904B119867 +Q

6

(43)

where Q5and Q



10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)R ( 2)R ( 3)R ( 4)R ( 5)R ( 6)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R1

R2

R3

R4

R5

R6

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

BR119904B119867

BΨ1R119904B119867

BΨ2R119904B119867

BΨ1Ψ2R119904B119867

BΨ3R119904B119867

BΨ1Ψ3R119904B119867

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+Q1 (44)



R = (D ⊙ B)C +Q1 (45)

where

D =

[[[[[[[

[

Λminus1(R

119904)

Λminus1(Ψ

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

1Ψ2R119904)

Λminus1(Ψ

3R119904)

Λminus1(Ψ

1Ψ3R119904)

]]]]]]]

]

(46)



1205963119894= minus

1

4

100381610038161003816100381610038161003816100381610038161003816

angle([D5119894

D1119894

]

2

) + angle([D6119894

D2119894

]

2

)

100381610038161003816100381610038161003816100381610038161003816

(47)




= sin(120579ΔP1

) cos(120593ΔP1

) Δ11990112

=

sin(120579ΔP1

) sin(120593ΔP1

) andΔ11990113= cos(120579

ΔP1


Δ1199013119894(119894 = 1 2 3) 120574

1119894= sin(120579

119894) cos(120593

119894) 120574

2119894= sin(120579

119894) sin(120593

119894)

and 1205743119894= cos(120579

119894) solving (10) (11) (42) (36) (37) and (24)

we can obtain

120582

2120587

[[[[[

[

1205961119894

1198891

1205962119894

1198892

1205963119894

1198893

]]]]]

]

= [

[

Δ11990111

Δ11990112

Δ11990113

Δ11990121

Δ11990122

Δ11990123

Δ11990131

Δ11990132

Δ11990133

]

]

[

[

1205741119894

1205742119894

1205743119894

]

]

(48)


[

[

1205741119894

1205742119894

1205743119894

]

]

=120582

2120587

[

[

Δ11990111

Δ11990112

Δ11990113

Δ11990121

Δ11990122

Δ11990123

Δ11990131

Δ11990132

Δ11990133

]

]

minus1 [[[[[

[

1205961119894

1198891

1205962119894

1198892

1205963119894

1198893

]]]]]

]

(49)

1205741 120574

2 and 120574




as

120579119894= arccos 120574

3119894

120593119894= arcsin(

1205741119894

cos (120579119894)) or 120593

119894= arcsin(

1205742119894

sin (120579119894))

(50)

The second one is

120593119894= arctan(

1205742119894

1205741119894

)

120579119894= arccos(

1205741119894

cos (120593119894)) or 120579

119894= arcsin(

1205742119894

sin (120593119894))

(51)



1 and

1198891= |ΔP


4 is ΔP2 and 119889

2= |ΔP



2


Z

Y

XO

3m2m

2m + 1

2m + 2

2m + 3

3m + 1

m + 2

1

2

m

m

+ 3 3

m + 1



X1= B

1S + N

1 (52)

X2= B

1Ψ1S + N

2 (53)

X3= B

2S + N

3 (54)

X4= B

2Ψ2S + N

4 (55)

where N1ndashN



R1= 119864 X

3X119867

1 = B

2R119904B1198671+Q

1

R2= 119864 X

4X119867

1 = B

2Ψ2R119904B1198671+Q

2

R3= 119864 X

3X119867

2 = B

2Ψ119867

1R119904B1198671+Q

3

R4= 119864 X

4X119867

2 = B

2Ψ119867

1Ψ2R119904B1198671+Q

4

(56)



1ndashQ

4




100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)R ( 2)R ( 3)R ( 4)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R1

R2

R3

R4

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

B2R119904B1198671

B2Ψ119867

1R119904B1198671

B2Ψ2R119904B1198671

B2Ψ119867

1Ψ2R119904B1198671

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+Q2

(57)



1 and (57) can be


R = (D ⊙ B2)C +Q

2

D =

[[[[[[[[

[

Λminus1(R

119904)

Λminus1(Ψ

119867

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

119867

1Ψ2R119904)

]]]]]]]]

]

(58)


1119894and 120596


1205961119894=1

2(angle[

[[

D2119894

D1119894

]]]

+ angle[[[

D4119894

D3119894

]]]

) (59)

1205962119894= minus

1

2(angle[

[[

D3119894

D1119894

]]]

+ angle[[[

D4119894

D2119894

]]]

) (60)

1205961119894= (

2120587

120582119894

)ΔP1∙ u

119894

= (2120587119889

1

120582119894

)


) cos (120593ΔP1

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP1

) sin (120593ΔP1

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP1

) cos (120579119894)]

(61)

1205962119894= (

2120587

120582119894

)ΔP2∙ u

119894

= (2120587119889

2

120582119894

)


) cos (120593ΔP2

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP2

) sin (120593ΔP2

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP2

) cos (120579119894)]

(62)


1198911(119909

1 119909

2) = 0

1198912(119909

1 119909

2) = 0

(63)

where 1199091= 120579 and 119909



x = 119891minus1119894(119909

1 119909

2) 119894 = 1 2 (64)


x119896+1 = 119865119894(119909

119896

1 119909

119896

2) 119894 = 1 2 (65)


(0)

2]119879









3+4119898







k119879 = [1205791 120593

1 1205792 120593

2 120579

119903 120593

119903] (66)




CRB = Fminus1(67)


F = [F120579120579 F120579120593

F120593120579

F120593120593

] (68)

where F120579120579



F119894119895= 119873trace[Rminus1 120597R

120597V119894

Rminus1 120597R120597V

119895

]


119904D119867

119895Rminus1]


119904D119895Rminus1]

D119894=120597B120597k

119894

(69)



8 Discussion




2 This is


1or ΔP

2is parallel



1 The







2as shown

Z

YX

O

m2m

m + 2m + 1

1

23

42m minus 1

m minus 1

⋱⋱






1120579= 05

1198961120593

= 05 1198962120579

= 03 1198962120593


119895minus 120593

119895) and 119892

119894120593= cos(120579

119895minus 120593


coordinate 120579119895and 120593



RMSE = radic 1

500

500

sum

119905=1

[(120579119894119905minus 120579)

2

+ (120593119894119905minus 120593)

2

] (70)

where 120579119894119905and 120593






Y

XO

m

m minus 11

2



ΔP1


O

Y

X

m + 2

m + 1

ΔP2



0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Succ

ess r

ate (

)



minus10 minus5 0 5 10 15 20


SNR (dB)

RMSE

(deg

)



07

06

05

04

03

02

01

0minus10 minus5 0 10 15 20







100 200 300 400 500 600 700 800 900 100050

55

60

65

70

75

80

85

90

95

100

Number of snapshots

Succ

ess r

ate (

)




100 200 300 400 500 600 700 800 900 10000

002

004

006

008

01

012

014

016

Number of snapshots

RMSE

(deg

)





0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Reso

lutio

n pr

obab

ility

()



minus10 minus5 0 5 10 15 20





1minus 120579

1| |120579

2minus 120579

2| are



1minus 120593

1| |120593

2minus 120593

2| are


2|2 120579

119894and 120579





10 Conclusion



Appendix



1 s

119875otimes t

119875] isin C

119862119863times119875 (A1)


s otimes t =[[[[

[

1199041t

1199042t119904119862t

]]]]

]

(A2)



Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Z

O

X

Y

120579

120593

minusu

(a)

g i

pl

O u120579k120579

gi120593

k120593

gi120579

120579igk

u120593

(b)


Z

Y

X

m + 1

m

O

2

1m + 2

m + 3

2m + 1

2m + 2





Assume that X1 X

2 X

3 and X



X1= BS + N

1

X2= BΨ

1S + N

2

X3= BΨ

2S + N

3

X4= BΨ

1Ψ2S + N

4

(6)


R1= 119864 X

1X119867

1 = BR

119904B119867 +Q

1

R2= 119864 X

2X119867

1 = BΨ

1R119904B119867 +Q

2

R3= 119864 X

3X119867

1 = BΨ

2R119904B119867 +Q

3

R4= 119864 X

4X119867

1 = BΨ

1Ψ2R119904B119867 +Q

4

(7)

where R119904= diag1199042

1 119904

2



4 respec-

tivelyConsider


11) exp (minus119895120596

12) exp (minus119895120596

1119903)]

(8)

Ψ2= diag [

ℎ3(1205791 120593

1)

ℎ1(1205791 120593

1)exp (minus119895120596

21)

ℎ3(1205792 120593

2)

ℎ1(1205792 120593

2)exp (minus119895120596

22)

ℎ3(120579119903 120593

119903)

ℎ1(120579119903 120593

119903)exp (minus119895120596

2119903)]

(9)

1205961119894= (

2120587

120582)ΔP

1∙ u

119894

= (2120587119889

1

120582)


) cos (120593ΔP1

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP1

) sin (120593ΔP1

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP1

) cos (120579119894)]

(10)

1205962119894= (

2120587

120582)ΔP

2∙ u

119894

= (2120587119889

2

120582)


) cos (120593ΔP2

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP2

) sin (120593ΔP2

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP2

) cos (120579119894)]

(11)

where 120579ΔP119894

and 120593ΔP119894



OY

Z

Δ

P2

P1

P1

(a)

Y

Xm+1 2m+1

O

ΔP

P

2

P

(b)


2




120579ΔP1

= 0 120593ΔP1

=120587

2 (12)



120579ΔP2

=120587

2 120593

ΔP2

= 0 (13)


119896(119896 = 1 4) that is

R119896=1

119873

119873

sum

119896=1

X (119899)X(119899)119867 =1

119873XXH

(14)







119888 119889119890


119909119888 119889 119890

=

119872

sum

119901=1

119904119888119901119905119889119901119906119890119901 (15)









119890are


X119889(119888 119890) = X

119890(119888 119889) = 119909



Consider

X119888= TΛ

119888(S)U119879

119888 = 1 119862

X119889= UΛ

119889(T) S119879 119889 = 1 119863

X119890= SΛ

119890(U)T119879 119890 = 1 119864

(16)


= ++

1u 2u

3

3u

t2t1t

1s 2s 3sX




X(119862119863times119864)=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

X119888=1

X119888=2

X119888=119862

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

TΛ1(S)

TΛ2(S)

TΛ119862(S)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

U119879

= (S ⊙ T)UT (17)


X(119863119864times119862)= (T ⊙ U) ST (18)

X(119864119862times119863)= (U ⊙ S)TT

(19)





100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)R ( 2)R ( 3)R ( 4)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R1

R2

R3

R4

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

BR119904B119867

BΨ1R119904B119867

BΨ2R119904B119867

BΨ1Ψ2R119904B119867

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+Q (20)




R = (D ⊙ B)C +Q (21)

R119883= (C119879 ⊙D)B119879 +Q

119883 (22)

R119884= (B ⊙ C119879)D119879

+Q119884 (23)

D =[[[

[

Λminus1(R

119904)

Λminus1(Ψ

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

1Ψ2R119904)

]]]

]

(24)





a ( 119894) = a ( 119895) (25)


119895(119896)

ℎ119900119890119890minus1198952120587((p

119900119890∙u119894)120582)

= ℎ119900119890119890minus1198952120587((p

119900119890∙u119895)120582) (26)


119901119900119890= sin (120579

119900119890) cos (120593

119900119890) x

+ sin (120579119900119890) sin (120593

119900119890) y + cos (120579

119900119890) z

(27)

u119894= sin (120579

119894) cos (120593

119894) x

+ sin (120579119894) sin (120593

119894) y + cos (120579

119894) z

(28)

120579119900119890

and 120593119900119890


119894and 120593



sin (120579119900119890) cos (120593

119900119890) 120588

1

+ sin (120579119900119890) sin (120593

119900119890) 120588

2+ cos (120579

119900119890) 120588

3= 0

1205881= sin (120579

119894) cos (120593

119894) minus sin (120579

119895) cos (120593

119895)

1205882= sin (120579

119894) sin (120593

119894) minus sin (120579

119895) sin (120593

119895)

1205883= cos (120579

119894) minus cos (120579

119895)

(29)


1and 120588

2and 120588

3is not

zero





minDBC

R minus (D ⊙ B)C2119865 (30)



119865 (31)


B119879 = argminB

10038171003817100381710038171003817R119883minus (C119879 ⊙D)B11987910038171003817100381710038171003817

2

119865

D119879= argmin

D

10038171003817100381710038171003817R119884minus (B ⊙ C119879)D11987910038171003817100381710038171003817

2

119865

(32)




B119879 = (C119879 ⊙D)dagger

R119883

D119879= (B ⊙ C119879)

dagger

R119884

(34)


follows








1205961119894= minus

1

2(angle lceil

D2119894

D1119894


D3119894

rceil) (35)



3are real numbers

D3119894D

1119894and D

4119894D



3

Consider

1205962119894= minus

1

2angle([

ℎ3(120579119894 120593

119894)

ℎ1(120579119894 120593

119894)exp (minus119895120596

2119894)]

2

)

= minus1


2119894))

= minus1

2angle([

D3119894

D1119894

]

2

)

= minus1

2angle([

D4119894

D2119894

]

2

)

(36)

Then

1205962119894= minus

1

4

100381610038161003816100381610038161003816100381610038161003816

angle([D3119894

D1119894

]

2

) + angle([D4119894

D2119894

]

2

)

100381610038161003816100381610038161003816100381610038161003816

(37)



120579119894= arccos(

1205821205961119894

21205871198892

) = arccos(2120596

1119894

120587)

120593119894= arccos(

1205821205962119894

21205871198892sin (120579

119894)) = arccos(

21205962119894

120587 sin (120579119894))

(38)





Z

Y

X

m + 1

O

m

2m + 1

1

2m + 2

2m + 3

2m + 4

3m + 2

3m + 3

2m + 2

m + 3





1119894and 120596

2119894










ΔP3 and 119889

3= |ΔP



X5= BΨ

3S + N

5 (39)

X6= BΨ

1Ψ3S + N

6 (40)

where

Ψ3= diag [

ℎ5(1205791 120593

1)

ℎ1(1205791 120593

1)exp (minus119895120596

31)

ℎ5(1205792 120593

2)

ℎ1(1205792 120593

2)exp (minus119895120596

32)

ℎ5(120579119903 120593

119903)

ℎ1(120579119903 120593

119903)exp (minus119895120596

3119903)]

(41)

1205963119894= (

2120587

120582)ΔP

3∙ u

119894

= (2120587119889

2

120582)


) cos (120593ΔP3

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP2

) sin (120593ΔP2

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP3

) cos (120579119894)]

(42)


R5= 119864 X

5X119867

1 = BΨ

3R119904B119867 +Q

5

R6= 119864 X

6X119867

1 = BΨ

1Ψ3R119904B119867 +Q

6

(43)

where Q5and Q



10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)R ( 2)R ( 3)R ( 4)R ( 5)R ( 6)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R1

R2

R3

R4

R5

R6

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

BR119904B119867

BΨ1R119904B119867

BΨ2R119904B119867

BΨ1Ψ2R119904B119867

BΨ3R119904B119867

BΨ1Ψ3R119904B119867

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+Q1 (44)



R = (D ⊙ B)C +Q1 (45)

where

D =

[[[[[[[

[

Λminus1(R

119904)

Λminus1(Ψ

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

1Ψ2R119904)

Λminus1(Ψ

3R119904)

Λminus1(Ψ

1Ψ3R119904)

]]]]]]]

]

(46)



1205963119894= minus

1

4

100381610038161003816100381610038161003816100381610038161003816

angle([D5119894

D1119894

]

2

) + angle([D6119894

D2119894

]

2

)

100381610038161003816100381610038161003816100381610038161003816

(47)




= sin(120579ΔP1

) cos(120593ΔP1

) Δ11990112

=

sin(120579ΔP1

) sin(120593ΔP1

) andΔ11990113= cos(120579

ΔP1


Δ1199013119894(119894 = 1 2 3) 120574

1119894= sin(120579

119894) cos(120593

119894) 120574

2119894= sin(120579

119894) sin(120593

119894)

and 1205743119894= cos(120579

119894) solving (10) (11) (42) (36) (37) and (24)

we can obtain

120582

2120587

[[[[[

[

1205961119894

1198891

1205962119894

1198892

1205963119894

1198893

]]]]]

]

= [

[

Δ11990111

Δ11990112

Δ11990113

Δ11990121

Δ11990122

Δ11990123

Δ11990131

Δ11990132

Δ11990133

]

]

[

[

1205741119894

1205742119894

1205743119894

]

]

(48)


[

[

1205741119894

1205742119894

1205743119894

]

]

=120582

2120587

[

[

Δ11990111

Δ11990112

Δ11990113

Δ11990121

Δ11990122

Δ11990123

Δ11990131

Δ11990132

Δ11990133

]

]

minus1 [[[[[

[

1205961119894

1198891

1205962119894

1198892

1205963119894

1198893

]]]]]

]

(49)

1205741 120574

2 and 120574




as

120579119894= arccos 120574

3119894

120593119894= arcsin(

1205741119894

cos (120579119894)) or 120593

119894= arcsin(

1205742119894

sin (120579119894))

(50)

The second one is

120593119894= arctan(

1205742119894

1205741119894

)

120579119894= arccos(

1205741119894

cos (120593119894)) or 120579

119894= arcsin(

1205742119894

sin (120593119894))

(51)



1 and

1198891= |ΔP


4 is ΔP2 and 119889

2= |ΔP



2


Z

Y

XO

3m2m

2m + 1

2m + 2

2m + 3

3m + 1

m + 2

1

2

m

m

+ 3 3

m + 1



X1= B

1S + N

1 (52)

X2= B

1Ψ1S + N

2 (53)

X3= B

2S + N

3 (54)

X4= B

2Ψ2S + N

4 (55)

where N1ndashN



R1= 119864 X

3X119867

1 = B

2R119904B1198671+Q

1

R2= 119864 X

4X119867

1 = B

2Ψ2R119904B1198671+Q

2

R3= 119864 X

3X119867

2 = B

2Ψ119867

1R119904B1198671+Q

3

R4= 119864 X

4X119867

2 = B

2Ψ119867

1Ψ2R119904B1198671+Q

4

(56)



1ndashQ

4




100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)R ( 2)R ( 3)R ( 4)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R1

R2

R3

R4

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

B2R119904B1198671

B2Ψ119867

1R119904B1198671

B2Ψ2R119904B1198671

B2Ψ119867

1Ψ2R119904B1198671

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+Q2

(57)



1 and (57) can be


R = (D ⊙ B2)C +Q

2

D =

[[[[[[[[

[

Λminus1(R

119904)

Λminus1(Ψ

119867

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

119867

1Ψ2R119904)

]]]]]]]]

]

(58)


1119894and 120596


1205961119894=1

2(angle[

[[

D2119894

D1119894

]]]

+ angle[[[

D4119894

D3119894

]]]

) (59)

1205962119894= minus

1

2(angle[

[[

D3119894

D1119894

]]]

+ angle[[[

D4119894

D2119894

]]]

) (60)

1205961119894= (

2120587

120582119894

)ΔP1∙ u

119894

= (2120587119889

1

120582119894

)


) cos (120593ΔP1

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP1

) sin (120593ΔP1

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP1

) cos (120579119894)]

(61)

1205962119894= (

2120587

120582119894

)ΔP2∙ u

119894

= (2120587119889

2

120582119894

)


) cos (120593ΔP2

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP2

) sin (120593ΔP2

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP2

) cos (120579119894)]

(62)


1198911(119909

1 119909

2) = 0

1198912(119909

1 119909

2) = 0

(63)

where 1199091= 120579 and 119909



x = 119891minus1119894(119909

1 119909

2) 119894 = 1 2 (64)


x119896+1 = 119865119894(119909

119896

1 119909

119896

2) 119894 = 1 2 (65)


(0)

2]119879









3+4119898







k119879 = [1205791 120593

1 1205792 120593

2 120579

119903 120593

119903] (66)




CRB = Fminus1(67)


F = [F120579120579 F120579120593

F120593120579

F120593120593

] (68)

where F120579120579



F119894119895= 119873trace[Rminus1 120597R

120597V119894

Rminus1 120597R120597V

119895

]


119904D119867

119895Rminus1]


119904D119895Rminus1]

D119894=120597B120597k

119894

(69)



8 Discussion




2 This is


1or ΔP

2is parallel



1 The







2as shown

Z

YX

O

m2m

m + 2m + 1

1

23

42m minus 1

m minus 1

⋱⋱






1120579= 05

1198961120593

= 05 1198962120579

= 03 1198962120593


119895minus 120593

119895) and 119892

119894120593= cos(120579

119895minus 120593


coordinate 120579119895and 120593



RMSE = radic 1

500

500

sum

119905=1

[(120579119894119905minus 120579)

2

+ (120593119894119905minus 120593)

2

] (70)

where 120579119894119905and 120593






Y

XO

m

m minus 11

2



ΔP1


O

Y

X

m + 2

m + 1

ΔP2



0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Succ

ess r

ate (

)



minus10 minus5 0 5 10 15 20


SNR (dB)

RMSE

(deg

)



07

06

05

04

03

02

01

0minus10 minus5 0 10 15 20







100 200 300 400 500 600 700 800 900 100050

55

60

65

70

75

80

85

90

95

100

Number of snapshots

Succ

ess r

ate (

)




100 200 300 400 500 600 700 800 900 10000

002

004

006

008

01

012

014

016

Number of snapshots

RMSE

(deg

)





0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Reso

lutio

n pr

obab

ility

()



minus10 minus5 0 5 10 15 20





1minus 120579

1| |120579

2minus 120579

2| are



1minus 120593

1| |120593

2minus 120593

2| are


2|2 120579

119894and 120579





10 Conclusion



Appendix



1 s

119875otimes t

119875] isin C

119862119863times119875 (A1)


s otimes t =[[[[

[

1199041t

1199042t119904119862t

]]]]

]

(A2)



Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










OY

Z

Δ

P2

P1

P1

(a)

Y

Xm+1 2m+1

O

ΔP

P

2

P

(b)


2




120579ΔP1

= 0 120593ΔP1

=120587

2 (12)



120579ΔP2

=120587

2 120593

ΔP2

= 0 (13)


119896(119896 = 1 4) that is

R119896=1

119873

119873

sum

119896=1

X (119899)X(119899)119867 =1

119873XXH

(14)







119888 119889119890


119909119888 119889 119890

=

119872

sum

119901=1

119904119888119901119905119889119901119906119890119901 (15)









119890are


X119889(119888 119890) = X

119890(119888 119889) = 119909



Consider

X119888= TΛ

119888(S)U119879

119888 = 1 119862

X119889= UΛ

119889(T) S119879 119889 = 1 119863

X119890= SΛ

119890(U)T119879 119890 = 1 119864

(16)


= ++

1u 2u

3

3u

t2t1t

1s 2s 3sX




X(119862119863times119864)=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

X119888=1

X119888=2

X119888=119862

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

TΛ1(S)

TΛ2(S)

TΛ119862(S)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

U119879

= (S ⊙ T)UT (17)


X(119863119864times119862)= (T ⊙ U) ST (18)

X(119864119862times119863)= (U ⊙ S)TT

(19)





100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)R ( 2)R ( 3)R ( 4)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R1

R2

R3

R4

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

BR119904B119867

BΨ1R119904B119867

BΨ2R119904B119867

BΨ1Ψ2R119904B119867

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+Q (20)




R = (D ⊙ B)C +Q (21)

R119883= (C119879 ⊙D)B119879 +Q

119883 (22)

R119884= (B ⊙ C119879)D119879

+Q119884 (23)

D =[[[

[

Λminus1(R

119904)

Λminus1(Ψ

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

1Ψ2R119904)

]]]

]

(24)





a ( 119894) = a ( 119895) (25)


119895(119896)

ℎ119900119890119890minus1198952120587((p

119900119890∙u119894)120582)

= ℎ119900119890119890minus1198952120587((p

119900119890∙u119895)120582) (26)


119901119900119890= sin (120579

119900119890) cos (120593

119900119890) x

+ sin (120579119900119890) sin (120593

119900119890) y + cos (120579

119900119890) z

(27)

u119894= sin (120579

119894) cos (120593

119894) x

+ sin (120579119894) sin (120593

119894) y + cos (120579

119894) z

(28)

120579119900119890

and 120593119900119890


119894and 120593



sin (120579119900119890) cos (120593

119900119890) 120588

1

+ sin (120579119900119890) sin (120593

119900119890) 120588

2+ cos (120579

119900119890) 120588

3= 0

1205881= sin (120579

119894) cos (120593

119894) minus sin (120579

119895) cos (120593

119895)

1205882= sin (120579

119894) sin (120593

119894) minus sin (120579

119895) sin (120593

119895)

1205883= cos (120579

119894) minus cos (120579

119895)

(29)


1and 120588

2and 120588

3is not

zero





minDBC

R minus (D ⊙ B)C2119865 (30)



119865 (31)


B119879 = argminB

10038171003817100381710038171003817R119883minus (C119879 ⊙D)B11987910038171003817100381710038171003817

2

119865

D119879= argmin

D

10038171003817100381710038171003817R119884minus (B ⊙ C119879)D11987910038171003817100381710038171003817

2

119865

(32)




B119879 = (C119879 ⊙D)dagger

R119883

D119879= (B ⊙ C119879)

dagger

R119884

(34)


follows








1205961119894= minus

1

2(angle lceil

D2119894

D1119894


D3119894

rceil) (35)



3are real numbers

D3119894D

1119894and D

4119894D



3

Consider

1205962119894= minus

1

2angle([

ℎ3(120579119894 120593

119894)

ℎ1(120579119894 120593

119894)exp (minus119895120596

2119894)]

2

)

= minus1


2119894))

= minus1

2angle([

D3119894

D1119894

]

2

)

= minus1

2angle([

D4119894

D2119894

]

2

)

(36)

Then

1205962119894= minus

1

4

100381610038161003816100381610038161003816100381610038161003816

angle([D3119894

D1119894

]

2

) + angle([D4119894

D2119894

]

2

)

100381610038161003816100381610038161003816100381610038161003816

(37)



120579119894= arccos(

1205821205961119894

21205871198892

) = arccos(2120596

1119894

120587)

120593119894= arccos(

1205821205962119894

21205871198892sin (120579

119894)) = arccos(

21205962119894

120587 sin (120579119894))

(38)





Z

Y

X

m + 1

O

m

2m + 1

1

2m + 2

2m + 3

2m + 4

3m + 2

3m + 3

2m + 2

m + 3





1119894and 120596

2119894










ΔP3 and 119889

3= |ΔP



X5= BΨ

3S + N

5 (39)

X6= BΨ

1Ψ3S + N

6 (40)

where

Ψ3= diag [

ℎ5(1205791 120593

1)

ℎ1(1205791 120593

1)exp (minus119895120596

31)

ℎ5(1205792 120593

2)

ℎ1(1205792 120593

2)exp (minus119895120596

32)

ℎ5(120579119903 120593

119903)

ℎ1(120579119903 120593

119903)exp (minus119895120596

3119903)]

(41)

1205963119894= (

2120587

120582)ΔP

3∙ u

119894

= (2120587119889

2

120582)


) cos (120593ΔP3

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP2

) sin (120593ΔP2

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP3

) cos (120579119894)]

(42)


R5= 119864 X

5X119867

1 = BΨ

3R119904B119867 +Q

5

R6= 119864 X

6X119867

1 = BΨ

1Ψ3R119904B119867 +Q

6

(43)

where Q5and Q



10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)R ( 2)R ( 3)R ( 4)R ( 5)R ( 6)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R1

R2

R3

R4

R5

R6

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

BR119904B119867

BΨ1R119904B119867

BΨ2R119904B119867

BΨ1Ψ2R119904B119867

BΨ3R119904B119867

BΨ1Ψ3R119904B119867

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+Q1 (44)



R = (D ⊙ B)C +Q1 (45)

where

D =

[[[[[[[

[

Λminus1(R

119904)

Λminus1(Ψ

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

1Ψ2R119904)

Λminus1(Ψ

3R119904)

Λminus1(Ψ

1Ψ3R119904)

]]]]]]]

]

(46)



1205963119894= minus

1

4

100381610038161003816100381610038161003816100381610038161003816

angle([D5119894

D1119894

]

2

) + angle([D6119894

D2119894

]

2

)

100381610038161003816100381610038161003816100381610038161003816

(47)




= sin(120579ΔP1

) cos(120593ΔP1

) Δ11990112

=

sin(120579ΔP1

) sin(120593ΔP1

) andΔ11990113= cos(120579

ΔP1


Δ1199013119894(119894 = 1 2 3) 120574

1119894= sin(120579

119894) cos(120593

119894) 120574

2119894= sin(120579

119894) sin(120593

119894)

and 1205743119894= cos(120579

119894) solving (10) (11) (42) (36) (37) and (24)

we can obtain

120582

2120587

[[[[[

[

1205961119894

1198891

1205962119894

1198892

1205963119894

1198893

]]]]]

]

= [

[

Δ11990111

Δ11990112

Δ11990113

Δ11990121

Δ11990122

Δ11990123

Δ11990131

Δ11990132

Δ11990133

]

]

[

[

1205741119894

1205742119894

1205743119894

]

]

(48)


[

[

1205741119894

1205742119894

1205743119894

]

]

=120582

2120587

[

[

Δ11990111

Δ11990112

Δ11990113

Δ11990121

Δ11990122

Δ11990123

Δ11990131

Δ11990132

Δ11990133

]

]

minus1 [[[[[

[

1205961119894

1198891

1205962119894

1198892

1205963119894

1198893

]]]]]

]

(49)

1205741 120574

2 and 120574




as

120579119894= arccos 120574

3119894

120593119894= arcsin(

1205741119894

cos (120579119894)) or 120593

119894= arcsin(

1205742119894

sin (120579119894))

(50)

The second one is

120593119894= arctan(

1205742119894

1205741119894

)

120579119894= arccos(

1205741119894

cos (120593119894)) or 120579

119894= arcsin(

1205742119894

sin (120593119894))

(51)



1 and

1198891= |ΔP


4 is ΔP2 and 119889

2= |ΔP



2


Z

Y

XO

3m2m

2m + 1

2m + 2

2m + 3

3m + 1

m + 2

1

2

m

m

+ 3 3

m + 1



X1= B

1S + N

1 (52)

X2= B

1Ψ1S + N

2 (53)

X3= B

2S + N

3 (54)

X4= B

2Ψ2S + N

4 (55)

where N1ndashN



R1= 119864 X

3X119867

1 = B

2R119904B1198671+Q

1

R2= 119864 X

4X119867

1 = B

2Ψ2R119904B1198671+Q

2

R3= 119864 X

3X119867

2 = B

2Ψ119867

1R119904B1198671+Q

3

R4= 119864 X

4X119867

2 = B

2Ψ119867

1Ψ2R119904B1198671+Q

4

(56)



1ndashQ

4




100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)R ( 2)R ( 3)R ( 4)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R1

R2

R3

R4

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

B2R119904B1198671

B2Ψ119867

1R119904B1198671

B2Ψ2R119904B1198671

B2Ψ119867

1Ψ2R119904B1198671

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+Q2

(57)



1 and (57) can be


R = (D ⊙ B2)C +Q

2

D =

[[[[[[[[

[

Λminus1(R

119904)

Λminus1(Ψ

119867

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

119867

1Ψ2R119904)

]]]]]]]]

]

(58)


1119894and 120596


1205961119894=1

2(angle[

[[

D2119894

D1119894

]]]

+ angle[[[

D4119894

D3119894

]]]

) (59)

1205962119894= minus

1

2(angle[

[[

D3119894

D1119894

]]]

+ angle[[[

D4119894

D2119894

]]]

) (60)

1205961119894= (

2120587

120582119894

)ΔP1∙ u

119894

= (2120587119889

1

120582119894

)


) cos (120593ΔP1

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP1

) sin (120593ΔP1

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP1

) cos (120579119894)]

(61)

1205962119894= (

2120587

120582119894

)ΔP2∙ u

119894

= (2120587119889

2

120582119894

)


) cos (120593ΔP2

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP2

) sin (120593ΔP2

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP2

) cos (120579119894)]

(62)


1198911(119909

1 119909

2) = 0

1198912(119909

1 119909

2) = 0

(63)

where 1199091= 120579 and 119909



x = 119891minus1119894(119909

1 119909

2) 119894 = 1 2 (64)


x119896+1 = 119865119894(119909

119896

1 119909

119896

2) 119894 = 1 2 (65)


(0)

2]119879









3+4119898







k119879 = [1205791 120593

1 1205792 120593

2 120579

119903 120593

119903] (66)




CRB = Fminus1(67)


F = [F120579120579 F120579120593

F120593120579

F120593120593

] (68)

where F120579120579



F119894119895= 119873trace[Rminus1 120597R

120597V119894

Rminus1 120597R120597V

119895

]


119904D119867

119895Rminus1]


119904D119895Rminus1]

D119894=120597B120597k

119894

(69)



8 Discussion




2 This is


1or ΔP

2is parallel



1 The







2as shown

Z

YX

O

m2m

m + 2m + 1

1

23

42m minus 1

m minus 1

⋱⋱






1120579= 05

1198961120593

= 05 1198962120579

= 03 1198962120593


119895minus 120593

119895) and 119892

119894120593= cos(120579

119895minus 120593


coordinate 120579119895and 120593



RMSE = radic 1

500

500

sum

119905=1

[(120579119894119905minus 120579)

2

+ (120593119894119905minus 120593)

2

] (70)

where 120579119894119905and 120593






Y

XO

m

m minus 11

2



ΔP1


O

Y

X

m + 2

m + 1

ΔP2



0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Succ

ess r

ate (

)



minus10 minus5 0 5 10 15 20


SNR (dB)

RMSE

(deg

)



07

06

05

04

03

02

01

0minus10 minus5 0 10 15 20







100 200 300 400 500 600 700 800 900 100050

55

60

65

70

75

80

85

90

95

100

Number of snapshots

Succ

ess r

ate (

)




100 200 300 400 500 600 700 800 900 10000

002

004

006

008

01

012

014

016

Number of snapshots

RMSE

(deg

)





0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Reso

lutio

n pr

obab

ility

()



minus10 minus5 0 5 10 15 20





1minus 120579

1| |120579

2minus 120579

2| are



1minus 120593

1| |120593

2minus 120593

2| are


2|2 120579

119894and 120579





10 Conclusion



Appendix



1 s

119875otimes t

119875] isin C

119862119863times119875 (A1)


s otimes t =[[[[

[

1199041t

1199042t119904119862t

]]]]

]

(A2)



Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










= ++

1u 2u

3

3u

t2t1t

1s 2s 3sX




X(119862119863times119864)=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

X119888=1

X119888=2

X119888=119862

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

TΛ1(S)

TΛ2(S)

TΛ119862(S)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

U119879

= (S ⊙ T)UT (17)


X(119863119864times119862)= (T ⊙ U) ST (18)

X(119864119862times119863)= (U ⊙ S)TT

(19)





100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)R ( 2)R ( 3)R ( 4)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R1

R2

R3

R4

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

BR119904B119867

BΨ1R119904B119867

BΨ2R119904B119867

BΨ1Ψ2R119904B119867

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+Q (20)




R = (D ⊙ B)C +Q (21)

R119883= (C119879 ⊙D)B119879 +Q

119883 (22)

R119884= (B ⊙ C119879)D119879

+Q119884 (23)

D =[[[

[

Λminus1(R

119904)

Λminus1(Ψ

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

1Ψ2R119904)

]]]

]

(24)





a ( 119894) = a ( 119895) (25)


119895(119896)

ℎ119900119890119890minus1198952120587((p

119900119890∙u119894)120582)

= ℎ119900119890119890minus1198952120587((p

119900119890∙u119895)120582) (26)


119901119900119890= sin (120579

119900119890) cos (120593

119900119890) x

+ sin (120579119900119890) sin (120593

119900119890) y + cos (120579

119900119890) z

(27)

u119894= sin (120579

119894) cos (120593

119894) x

+ sin (120579119894) sin (120593

119894) y + cos (120579

119894) z

(28)

120579119900119890

and 120593119900119890


119894and 120593



sin (120579119900119890) cos (120593

119900119890) 120588

1

+ sin (120579119900119890) sin (120593

119900119890) 120588

2+ cos (120579

119900119890) 120588

3= 0

1205881= sin (120579

119894) cos (120593

119894) minus sin (120579

119895) cos (120593

119895)

1205882= sin (120579

119894) sin (120593

119894) minus sin (120579

119895) sin (120593

119895)

1205883= cos (120579

119894) minus cos (120579

119895)

(29)


1and 120588

2and 120588

3is not

zero





minDBC

R minus (D ⊙ B)C2119865 (30)



119865 (31)


B119879 = argminB

10038171003817100381710038171003817R119883minus (C119879 ⊙D)B11987910038171003817100381710038171003817

2

119865

D119879= argmin

D

10038171003817100381710038171003817R119884minus (B ⊙ C119879)D11987910038171003817100381710038171003817

2

119865

(32)




B119879 = (C119879 ⊙D)dagger

R119883

D119879= (B ⊙ C119879)

dagger

R119884

(34)


follows








1205961119894= minus

1

2(angle lceil

D2119894

D1119894


D3119894

rceil) (35)



3are real numbers

D3119894D

1119894and D

4119894D



3

Consider

1205962119894= minus

1

2angle([

ℎ3(120579119894 120593

119894)

ℎ1(120579119894 120593

119894)exp (minus119895120596

2119894)]

2

)

= minus1


2119894))

= minus1

2angle([

D3119894

D1119894

]

2

)

= minus1

2angle([

D4119894

D2119894

]

2

)

(36)

Then

1205962119894= minus

1

4

100381610038161003816100381610038161003816100381610038161003816

angle([D3119894

D1119894

]

2

) + angle([D4119894

D2119894

]

2

)

100381610038161003816100381610038161003816100381610038161003816

(37)



120579119894= arccos(

1205821205961119894

21205871198892

) = arccos(2120596

1119894

120587)

120593119894= arccos(

1205821205962119894

21205871198892sin (120579

119894)) = arccos(

21205962119894

120587 sin (120579119894))

(38)





Z

Y

X

m + 1

O

m

2m + 1

1

2m + 2

2m + 3

2m + 4

3m + 2

3m + 3

2m + 2

m + 3





1119894and 120596

2119894










ΔP3 and 119889

3= |ΔP



X5= BΨ

3S + N

5 (39)

X6= BΨ

1Ψ3S + N

6 (40)

where

Ψ3= diag [

ℎ5(1205791 120593

1)

ℎ1(1205791 120593

1)exp (minus119895120596

31)

ℎ5(1205792 120593

2)

ℎ1(1205792 120593

2)exp (minus119895120596

32)

ℎ5(120579119903 120593

119903)

ℎ1(120579119903 120593

119903)exp (minus119895120596

3119903)]

(41)

1205963119894= (

2120587

120582)ΔP

3∙ u

119894

= (2120587119889

2

120582)


) cos (120593ΔP3

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP2

) sin (120593ΔP2

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP3

) cos (120579119894)]

(42)


R5= 119864 X

5X119867

1 = BΨ

3R119904B119867 +Q

5

R6= 119864 X

6X119867

1 = BΨ

1Ψ3R119904B119867 +Q

6

(43)

where Q5and Q



10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)R ( 2)R ( 3)R ( 4)R ( 5)R ( 6)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R1

R2

R3

R4

R5

R6

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

BR119904B119867

BΨ1R119904B119867

BΨ2R119904B119867

BΨ1Ψ2R119904B119867

BΨ3R119904B119867

BΨ1Ψ3R119904B119867

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+Q1 (44)



R = (D ⊙ B)C +Q1 (45)

where

D =

[[[[[[[

[

Λminus1(R

119904)

Λminus1(Ψ

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

1Ψ2R119904)

Λminus1(Ψ

3R119904)

Λminus1(Ψ

1Ψ3R119904)

]]]]]]]

]

(46)



1205963119894= minus

1

4

100381610038161003816100381610038161003816100381610038161003816

angle([D5119894

D1119894

]

2

) + angle([D6119894

D2119894

]

2

)

100381610038161003816100381610038161003816100381610038161003816

(47)




= sin(120579ΔP1

) cos(120593ΔP1

) Δ11990112

=

sin(120579ΔP1

) sin(120593ΔP1

) andΔ11990113= cos(120579

ΔP1


Δ1199013119894(119894 = 1 2 3) 120574

1119894= sin(120579

119894) cos(120593

119894) 120574

2119894= sin(120579

119894) sin(120593

119894)

and 1205743119894= cos(120579

119894) solving (10) (11) (42) (36) (37) and (24)

we can obtain

120582

2120587

[[[[[

[

1205961119894

1198891

1205962119894

1198892

1205963119894

1198893

]]]]]

]

= [

[

Δ11990111

Δ11990112

Δ11990113

Δ11990121

Δ11990122

Δ11990123

Δ11990131

Δ11990132

Δ11990133

]

]

[

[

1205741119894

1205742119894

1205743119894

]

]

(48)


[

[

1205741119894

1205742119894

1205743119894

]

]

=120582

2120587

[

[

Δ11990111

Δ11990112

Δ11990113

Δ11990121

Δ11990122

Δ11990123

Δ11990131

Δ11990132

Δ11990133

]

]

minus1 [[[[[

[

1205961119894

1198891

1205962119894

1198892

1205963119894

1198893

]]]]]

]

(49)

1205741 120574

2 and 120574




as

120579119894= arccos 120574

3119894

120593119894= arcsin(

1205741119894

cos (120579119894)) or 120593

119894= arcsin(

1205742119894

sin (120579119894))

(50)

The second one is

120593119894= arctan(

1205742119894

1205741119894

)

120579119894= arccos(

1205741119894

cos (120593119894)) or 120579

119894= arcsin(

1205742119894

sin (120593119894))

(51)



1 and

1198891= |ΔP


4 is ΔP2 and 119889

2= |ΔP



2


Z

Y

XO

3m2m

2m + 1

2m + 2

2m + 3

3m + 1

m + 2

1

2

m

m

+ 3 3

m + 1



X1= B

1S + N

1 (52)

X2= B

1Ψ1S + N

2 (53)

X3= B

2S + N

3 (54)

X4= B

2Ψ2S + N

4 (55)

where N1ndashN



R1= 119864 X

3X119867

1 = B

2R119904B1198671+Q

1

R2= 119864 X

4X119867

1 = B

2Ψ2R119904B1198671+Q

2

R3= 119864 X

3X119867

2 = B

2Ψ119867

1R119904B1198671+Q

3

R4= 119864 X

4X119867

2 = B

2Ψ119867

1Ψ2R119904B1198671+Q

4

(56)



1ndashQ

4




100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)R ( 2)R ( 3)R ( 4)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R1

R2

R3

R4

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

B2R119904B1198671

B2Ψ119867

1R119904B1198671

B2Ψ2R119904B1198671

B2Ψ119867

1Ψ2R119904B1198671

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+Q2

(57)



1 and (57) can be


R = (D ⊙ B2)C +Q

2

D =

[[[[[[[[

[

Λminus1(R

119904)

Λminus1(Ψ

119867

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

119867

1Ψ2R119904)

]]]]]]]]

]

(58)


1119894and 120596


1205961119894=1

2(angle[

[[

D2119894

D1119894

]]]

+ angle[[[

D4119894

D3119894

]]]

) (59)

1205962119894= minus

1

2(angle[

[[

D3119894

D1119894

]]]

+ angle[[[

D4119894

D2119894

]]]

) (60)

1205961119894= (

2120587

120582119894

)ΔP1∙ u

119894

= (2120587119889

1

120582119894

)


) cos (120593ΔP1

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP1

) sin (120593ΔP1

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP1

) cos (120579119894)]

(61)

1205962119894= (

2120587

120582119894

)ΔP2∙ u

119894

= (2120587119889

2

120582119894

)


) cos (120593ΔP2

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP2

) sin (120593ΔP2

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP2

) cos (120579119894)]

(62)


1198911(119909

1 119909

2) = 0

1198912(119909

1 119909

2) = 0

(63)

where 1199091= 120579 and 119909



x = 119891minus1119894(119909

1 119909

2) 119894 = 1 2 (64)


x119896+1 = 119865119894(119909

119896

1 119909

119896

2) 119894 = 1 2 (65)


(0)

2]119879









3+4119898







k119879 = [1205791 120593

1 1205792 120593

2 120579

119903 120593

119903] (66)




CRB = Fminus1(67)


F = [F120579120579 F120579120593

F120593120579

F120593120593

] (68)

where F120579120579



F119894119895= 119873trace[Rminus1 120597R

120597V119894

Rminus1 120597R120597V

119895

]


119904D119867

119895Rminus1]


119904D119895Rminus1]

D119894=120597B120597k

119894

(69)



8 Discussion




2 This is


1or ΔP

2is parallel



1 The







2as shown

Z

YX

O

m2m

m + 2m + 1

1

23

42m minus 1

m minus 1

⋱⋱






1120579= 05

1198961120593

= 05 1198962120579

= 03 1198962120593


119895minus 120593

119895) and 119892

119894120593= cos(120579

119895minus 120593


coordinate 120579119895and 120593



RMSE = radic 1

500

500

sum

119905=1

[(120579119894119905minus 120579)

2

+ (120593119894119905minus 120593)

2

] (70)

where 120579119894119905and 120593






Y

XO

m

m minus 11

2



ΔP1


O

Y

X

m + 2

m + 1

ΔP2



0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Succ

ess r

ate (

)



minus10 minus5 0 5 10 15 20


SNR (dB)

RMSE

(deg

)



07

06

05

04

03

02

01

0minus10 minus5 0 10 15 20







100 200 300 400 500 600 700 800 900 100050

55

60

65

70

75

80

85

90

95

100

Number of snapshots

Succ

ess r

ate (

)




100 200 300 400 500 600 700 800 900 10000

002

004

006

008

01

012

014

016

Number of snapshots

RMSE

(deg

)





0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Reso

lutio

n pr

obab

ility

()



minus10 minus5 0 5 10 15 20





1minus 120579

1| |120579

2minus 120579

2| are



1minus 120593

1| |120593

2minus 120593

2| are


2|2 120579

119894and 120579





10 Conclusion



Appendix



1 s

119875otimes t

119875] isin C

119862119863times119875 (A1)


s otimes t =[[[[

[

1199041t

1199042t119904119862t

]]]]

]

(A2)



Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













minDBC

R minus (D ⊙ B)C2119865 (30)



119865 (31)


B119879 = argminB

10038171003817100381710038171003817R119883minus (C119879 ⊙D)B11987910038171003817100381710038171003817

2

119865

D119879= argmin

D

10038171003817100381710038171003817R119884minus (B ⊙ C119879)D11987910038171003817100381710038171003817

2

119865

(32)




B119879 = (C119879 ⊙D)dagger

R119883

D119879= (B ⊙ C119879)

dagger

R119884

(34)


follows








1205961119894= minus

1

2(angle lceil

D2119894

D1119894


D3119894

rceil) (35)



3are real numbers

D3119894D

1119894and D

4119894D



3

Consider

1205962119894= minus

1

2angle([

ℎ3(120579119894 120593

119894)

ℎ1(120579119894 120593

119894)exp (minus119895120596

2119894)]

2

)

= minus1


2119894))

= minus1

2angle([

D3119894

D1119894

]

2

)

= minus1

2angle([

D4119894

D2119894

]

2

)

(36)

Then

1205962119894= minus

1

4

100381610038161003816100381610038161003816100381610038161003816

angle([D3119894

D1119894

]

2

) + angle([D4119894

D2119894

]

2

)

100381610038161003816100381610038161003816100381610038161003816

(37)



120579119894= arccos(

1205821205961119894

21205871198892

) = arccos(2120596

1119894

120587)

120593119894= arccos(

1205821205962119894

21205871198892sin (120579

119894)) = arccos(

21205962119894

120587 sin (120579119894))

(38)





Z

Y

X

m + 1

O

m

2m + 1

1

2m + 2

2m + 3

2m + 4

3m + 2

3m + 3

2m + 2

m + 3





1119894and 120596

2119894










ΔP3 and 119889

3= |ΔP



X5= BΨ

3S + N

5 (39)

X6= BΨ

1Ψ3S + N

6 (40)

where

Ψ3= diag [

ℎ5(1205791 120593

1)

ℎ1(1205791 120593

1)exp (minus119895120596

31)

ℎ5(1205792 120593

2)

ℎ1(1205792 120593

2)exp (minus119895120596

32)

ℎ5(120579119903 120593

119903)

ℎ1(120579119903 120593

119903)exp (minus119895120596

3119903)]

(41)

1205963119894= (

2120587

120582)ΔP

3∙ u

119894

= (2120587119889

2

120582)


) cos (120593ΔP3

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP2

) sin (120593ΔP2

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP3

) cos (120579119894)]

(42)


R5= 119864 X

5X119867

1 = BΨ

3R119904B119867 +Q

5

R6= 119864 X

6X119867

1 = BΨ

1Ψ3R119904B119867 +Q

6

(43)

where Q5and Q



10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)R ( 2)R ( 3)R ( 4)R ( 5)R ( 6)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R1

R2

R3

R4

R5

R6

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

BR119904B119867

BΨ1R119904B119867

BΨ2R119904B119867

BΨ1Ψ2R119904B119867

BΨ3R119904B119867

BΨ1Ψ3R119904B119867

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+Q1 (44)



R = (D ⊙ B)C +Q1 (45)

where

D =

[[[[[[[

[

Λminus1(R

119904)

Λminus1(Ψ

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

1Ψ2R119904)

Λminus1(Ψ

3R119904)

Λminus1(Ψ

1Ψ3R119904)

]]]]]]]

]

(46)



1205963119894= minus

1

4

100381610038161003816100381610038161003816100381610038161003816

angle([D5119894

D1119894

]

2

) + angle([D6119894

D2119894

]

2

)

100381610038161003816100381610038161003816100381610038161003816

(47)




= sin(120579ΔP1

) cos(120593ΔP1

) Δ11990112

=

sin(120579ΔP1

) sin(120593ΔP1

) andΔ11990113= cos(120579

ΔP1


Δ1199013119894(119894 = 1 2 3) 120574

1119894= sin(120579

119894) cos(120593

119894) 120574

2119894= sin(120579

119894) sin(120593

119894)

and 1205743119894= cos(120579

119894) solving (10) (11) (42) (36) (37) and (24)

we can obtain

120582

2120587

[[[[[

[

1205961119894

1198891

1205962119894

1198892

1205963119894

1198893

]]]]]

]

= [

[

Δ11990111

Δ11990112

Δ11990113

Δ11990121

Δ11990122

Δ11990123

Δ11990131

Δ11990132

Δ11990133

]

]

[

[

1205741119894

1205742119894

1205743119894

]

]

(48)


[

[

1205741119894

1205742119894

1205743119894

]

]

=120582

2120587

[

[

Δ11990111

Δ11990112

Δ11990113

Δ11990121

Δ11990122

Δ11990123

Δ11990131

Δ11990132

Δ11990133

]

]

minus1 [[[[[

[

1205961119894

1198891

1205962119894

1198892

1205963119894

1198893

]]]]]

]

(49)

1205741 120574

2 and 120574




as

120579119894= arccos 120574

3119894

120593119894= arcsin(

1205741119894

cos (120579119894)) or 120593

119894= arcsin(

1205742119894

sin (120579119894))

(50)

The second one is

120593119894= arctan(

1205742119894

1205741119894

)

120579119894= arccos(

1205741119894

cos (120593119894)) or 120579

119894= arcsin(

1205742119894

sin (120593119894))

(51)



1 and

1198891= |ΔP


4 is ΔP2 and 119889

2= |ΔP



2


Z

Y

XO

3m2m

2m + 1

2m + 2

2m + 3

3m + 1

m + 2

1

2

m

m

+ 3 3

m + 1



X1= B

1S + N

1 (52)

X2= B

1Ψ1S + N

2 (53)

X3= B

2S + N

3 (54)

X4= B

2Ψ2S + N

4 (55)

where N1ndashN



R1= 119864 X

3X119867

1 = B

2R119904B1198671+Q

1

R2= 119864 X

4X119867

1 = B

2Ψ2R119904B1198671+Q

2

R3= 119864 X

3X119867

2 = B

2Ψ119867

1R119904B1198671+Q

3

R4= 119864 X

4X119867

2 = B

2Ψ119867

1Ψ2R119904B1198671+Q

4

(56)



1ndashQ

4




100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)R ( 2)R ( 3)R ( 4)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R1

R2

R3

R4

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

B2R119904B1198671

B2Ψ119867

1R119904B1198671

B2Ψ2R119904B1198671

B2Ψ119867

1Ψ2R119904B1198671

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+Q2

(57)



1 and (57) can be


R = (D ⊙ B2)C +Q

2

D =

[[[[[[[[

[

Λminus1(R

119904)

Λminus1(Ψ

119867

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

119867

1Ψ2R119904)

]]]]]]]]

]

(58)


1119894and 120596


1205961119894=1

2(angle[

[[

D2119894

D1119894

]]]

+ angle[[[

D4119894

D3119894

]]]

) (59)

1205962119894= minus

1

2(angle[

[[

D3119894

D1119894

]]]

+ angle[[[

D4119894

D2119894

]]]

) (60)

1205961119894= (

2120587

120582119894

)ΔP1∙ u

119894

= (2120587119889

1

120582119894

)


) cos (120593ΔP1

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP1

) sin (120593ΔP1

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP1

) cos (120579119894)]

(61)

1205962119894= (

2120587

120582119894

)ΔP2∙ u

119894

= (2120587119889

2

120582119894

)


) cos (120593ΔP2

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP2

) sin (120593ΔP2

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP2

) cos (120579119894)]

(62)


1198911(119909

1 119909

2) = 0

1198912(119909

1 119909

2) = 0

(63)

where 1199091= 120579 and 119909



x = 119891minus1119894(119909

1 119909

2) 119894 = 1 2 (64)


x119896+1 = 119865119894(119909

119896

1 119909

119896

2) 119894 = 1 2 (65)


(0)

2]119879









3+4119898







k119879 = [1205791 120593

1 1205792 120593

2 120579

119903 120593

119903] (66)




CRB = Fminus1(67)


F = [F120579120579 F120579120593

F120593120579

F120593120593

] (68)

where F120579120579



F119894119895= 119873trace[Rminus1 120597R

120597V119894

Rminus1 120597R120597V

119895

]


119904D119867

119895Rminus1]


119904D119895Rminus1]

D119894=120597B120597k

119894

(69)



8 Discussion




2 This is


1or ΔP

2is parallel



1 The







2as shown

Z

YX

O

m2m

m + 2m + 1

1

23

42m minus 1

m minus 1

⋱⋱






1120579= 05

1198961120593

= 05 1198962120579

= 03 1198962120593


119895minus 120593

119895) and 119892

119894120593= cos(120579

119895minus 120593


coordinate 120579119895and 120593



RMSE = radic 1

500

500

sum

119905=1

[(120579119894119905minus 120579)

2

+ (120593119894119905minus 120593)

2

] (70)

where 120579119894119905and 120593






Y

XO

m

m minus 11

2



ΔP1


O

Y

X

m + 2

m + 1

ΔP2



0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Succ

ess r

ate (

)



minus10 minus5 0 5 10 15 20


SNR (dB)

RMSE

(deg

)



07

06

05

04

03

02

01

0minus10 minus5 0 10 15 20







100 200 300 400 500 600 700 800 900 100050

55

60

65

70

75

80

85

90

95

100

Number of snapshots

Succ

ess r

ate (

)




100 200 300 400 500 600 700 800 900 10000

002

004

006

008

01

012

014

016

Number of snapshots

RMSE

(deg

)





0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Reso

lutio

n pr

obab

ility

()



minus10 minus5 0 5 10 15 20





1minus 120579

1| |120579

2minus 120579

2| are



1minus 120593

1| |120593

2minus 120593

2| are


2|2 120579

119894and 120579





10 Conclusion



Appendix



1 s

119875otimes t

119875] isin C

119862119863times119875 (A1)


s otimes t =[[[[

[

1199041t

1199042t119904119862t

]]]]

]

(A2)



Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Z

Y

X

m + 1

O

m

2m + 1

1

2m + 2

2m + 3

2m + 4

3m + 2

3m + 3

2m + 2

m + 3





1119894and 120596

2119894










ΔP3 and 119889

3= |ΔP



X5= BΨ

3S + N

5 (39)

X6= BΨ

1Ψ3S + N

6 (40)

where

Ψ3= diag [

ℎ5(1205791 120593

1)

ℎ1(1205791 120593

1)exp (minus119895120596

31)

ℎ5(1205792 120593

2)

ℎ1(1205792 120593

2)exp (minus119895120596

32)

ℎ5(120579119903 120593

119903)

ℎ1(120579119903 120593

119903)exp (minus119895120596

3119903)]

(41)

1205963119894= (

2120587

120582)ΔP

3∙ u

119894

= (2120587119889

2

120582)


) cos (120593ΔP3

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP2

) sin (120593ΔP2

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP3

) cos (120579119894)]

(42)


R5= 119864 X

5X119867

1 = BΨ

3R119904B119867 +Q

5

R6= 119864 X

6X119867

1 = BΨ

1Ψ3R119904B119867 +Q

6

(43)

where Q5and Q



10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)R ( 2)R ( 3)R ( 4)R ( 5)R ( 6)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R1

R2

R3

R4

R5

R6

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

BR119904B119867

BΨ1R119904B119867

BΨ2R119904B119867

BΨ1Ψ2R119904B119867

BΨ3R119904B119867

BΨ1Ψ3R119904B119867

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+Q1 (44)



R = (D ⊙ B)C +Q1 (45)

where

D =

[[[[[[[

[

Λminus1(R

119904)

Λminus1(Ψ

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

1Ψ2R119904)

Λminus1(Ψ

3R119904)

Λminus1(Ψ

1Ψ3R119904)

]]]]]]]

]

(46)



1205963119894= minus

1

4

100381610038161003816100381610038161003816100381610038161003816

angle([D5119894

D1119894

]

2

) + angle([D6119894

D2119894

]

2

)

100381610038161003816100381610038161003816100381610038161003816

(47)




= sin(120579ΔP1

) cos(120593ΔP1

) Δ11990112

=

sin(120579ΔP1

) sin(120593ΔP1

) andΔ11990113= cos(120579

ΔP1


Δ1199013119894(119894 = 1 2 3) 120574

1119894= sin(120579

119894) cos(120593

119894) 120574

2119894= sin(120579

119894) sin(120593

119894)

and 1205743119894= cos(120579

119894) solving (10) (11) (42) (36) (37) and (24)

we can obtain

120582

2120587

[[[[[

[

1205961119894

1198891

1205962119894

1198892

1205963119894

1198893

]]]]]

]

= [

[

Δ11990111

Δ11990112

Δ11990113

Δ11990121

Δ11990122

Δ11990123

Δ11990131

Δ11990132

Δ11990133

]

]

[

[

1205741119894

1205742119894

1205743119894

]

]

(48)


[

[

1205741119894

1205742119894

1205743119894

]

]

=120582

2120587

[

[

Δ11990111

Δ11990112

Δ11990113

Δ11990121

Δ11990122

Δ11990123

Δ11990131

Δ11990132

Δ11990133

]

]

minus1 [[[[[

[

1205961119894

1198891

1205962119894

1198892

1205963119894

1198893

]]]]]

]

(49)

1205741 120574

2 and 120574




as

120579119894= arccos 120574

3119894

120593119894= arcsin(

1205741119894

cos (120579119894)) or 120593

119894= arcsin(

1205742119894

sin (120579119894))

(50)

The second one is

120593119894= arctan(

1205742119894

1205741119894

)

120579119894= arccos(

1205741119894

cos (120593119894)) or 120579

119894= arcsin(

1205742119894

sin (120593119894))

(51)



1 and

1198891= |ΔP


4 is ΔP2 and 119889

2= |ΔP



2


Z

Y

XO

3m2m

2m + 1

2m + 2

2m + 3

3m + 1

m + 2

1

2

m

m

+ 3 3

m + 1



X1= B

1S + N

1 (52)

X2= B

1Ψ1S + N

2 (53)

X3= B

2S + N

3 (54)

X4= B

2Ψ2S + N

4 (55)

where N1ndashN



R1= 119864 X

3X119867

1 = B

2R119904B1198671+Q

1

R2= 119864 X

4X119867

1 = B

2Ψ2R119904B1198671+Q

2

R3= 119864 X

3X119867

2 = B

2Ψ119867

1R119904B1198671+Q

3

R4= 119864 X

4X119867

2 = B

2Ψ119867

1Ψ2R119904B1198671+Q

4

(56)



1ndashQ

4




100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)R ( 2)R ( 3)R ( 4)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R1

R2

R3

R4

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

B2R119904B1198671

B2Ψ119867

1R119904B1198671

B2Ψ2R119904B1198671

B2Ψ119867

1Ψ2R119904B1198671

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+Q2

(57)



1 and (57) can be


R = (D ⊙ B2)C +Q

2

D =

[[[[[[[[

[

Λminus1(R

119904)

Λminus1(Ψ

119867

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

119867

1Ψ2R119904)

]]]]]]]]

]

(58)


1119894and 120596


1205961119894=1

2(angle[

[[

D2119894

D1119894

]]]

+ angle[[[

D4119894

D3119894

]]]

) (59)

1205962119894= minus

1

2(angle[

[[

D3119894

D1119894

]]]

+ angle[[[

D4119894

D2119894

]]]

) (60)

1205961119894= (

2120587

120582119894

)ΔP1∙ u

119894

= (2120587119889

1

120582119894

)


) cos (120593ΔP1

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP1

) sin (120593ΔP1

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP1

) cos (120579119894)]

(61)

1205962119894= (

2120587

120582119894

)ΔP2∙ u

119894

= (2120587119889

2

120582119894

)


) cos (120593ΔP2

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP2

) sin (120593ΔP2

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP2

) cos (120579119894)]

(62)


1198911(119909

1 119909

2) = 0

1198912(119909

1 119909

2) = 0

(63)

where 1199091= 120579 and 119909



x = 119891minus1119894(119909

1 119909

2) 119894 = 1 2 (64)


x119896+1 = 119865119894(119909

119896

1 119909

119896

2) 119894 = 1 2 (65)


(0)

2]119879









3+4119898







k119879 = [1205791 120593

1 1205792 120593

2 120579

119903 120593

119903] (66)




CRB = Fminus1(67)


F = [F120579120579 F120579120593

F120593120579

F120593120593

] (68)

where F120579120579



F119894119895= 119873trace[Rminus1 120597R

120597V119894

Rminus1 120597R120597V

119895

]


119904D119867

119895Rminus1]


119904D119895Rminus1]

D119894=120597B120597k

119894

(69)



8 Discussion




2 This is


1or ΔP

2is parallel



1 The







2as shown

Z

YX

O

m2m

m + 2m + 1

1

23

42m minus 1

m minus 1

⋱⋱






1120579= 05

1198961120593

= 05 1198962120579

= 03 1198962120593


119895minus 120593

119895) and 119892

119894120593= cos(120579

119895minus 120593


coordinate 120579119895and 120593



RMSE = radic 1

500

500

sum

119905=1

[(120579119894119905minus 120579)

2

+ (120593119894119905minus 120593)

2

] (70)

where 120579119894119905and 120593






Y

XO

m

m minus 11

2



ΔP1


O

Y

X

m + 2

m + 1

ΔP2



0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Succ

ess r

ate (

)



minus10 minus5 0 5 10 15 20


SNR (dB)

RMSE

(deg

)



07

06

05

04

03

02

01

0minus10 minus5 0 10 15 20







100 200 300 400 500 600 700 800 900 100050

55

60

65

70

75

80

85

90

95

100

Number of snapshots

Succ

ess r

ate (

)




100 200 300 400 500 600 700 800 900 10000

002

004

006

008

01

012

014

016

Number of snapshots

RMSE

(deg

)





0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Reso

lutio

n pr

obab

ility

()



minus10 minus5 0 5 10 15 20





1minus 120579

1| |120579

2minus 120579

2| are



1minus 120593

1| |120593

2minus 120593

2| are


2|2 120579

119894and 120579





10 Conclusion



Appendix



1 s

119875otimes t

119875] isin C

119862119863times119875 (A1)


s otimes t =[[[[

[

1199041t

1199042t119904119862t

]]]]

]

(A2)



Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











= sin(120579ΔP1

) cos(120593ΔP1

) Δ11990112

=

sin(120579ΔP1

) sin(120593ΔP1

) andΔ11990113= cos(120579

ΔP1


Δ1199013119894(119894 = 1 2 3) 120574

1119894= sin(120579

119894) cos(120593

119894) 120574

2119894= sin(120579

119894) sin(120593

119894)

and 1205743119894= cos(120579

119894) solving (10) (11) (42) (36) (37) and (24)

we can obtain

120582

2120587

[[[[[

[

1205961119894

1198891

1205962119894

1198892

1205963119894

1198893

]]]]]

]

= [

[

Δ11990111

Δ11990112

Δ11990113

Δ11990121

Δ11990122

Δ11990123

Δ11990131

Δ11990132

Δ11990133

]

]

[

[

1205741119894

1205742119894

1205743119894

]

]

(48)


[

[

1205741119894

1205742119894

1205743119894

]

]

=120582

2120587

[

[

Δ11990111

Δ11990112

Δ11990113

Δ11990121

Δ11990122

Δ11990123

Δ11990131

Δ11990132

Δ11990133

]

]

minus1 [[[[[

[

1205961119894

1198891

1205962119894

1198892

1205963119894

1198893

]]]]]

]

(49)

1205741 120574

2 and 120574




as

120579119894= arccos 120574

3119894

120593119894= arcsin(

1205741119894

cos (120579119894)) or 120593

119894= arcsin(

1205742119894

sin (120579119894))

(50)

The second one is

120593119894= arctan(

1205742119894

1205741119894

)

120579119894= arccos(

1205741119894

cos (120593119894)) or 120579

119894= arcsin(

1205742119894

sin (120593119894))

(51)



1 and

1198891= |ΔP


4 is ΔP2 and 119889

2= |ΔP



2


Z

Y

XO

3m2m

2m + 1

2m + 2

2m + 3

3m + 1

m + 2

1

2

m

m

+ 3 3

m + 1



X1= B

1S + N

1 (52)

X2= B

1Ψ1S + N

2 (53)

X3= B

2S + N

3 (54)

X4= B

2Ψ2S + N

4 (55)

where N1ndashN



R1= 119864 X

3X119867

1 = B

2R119904B1198671+Q

1

R2= 119864 X

4X119867

1 = B

2Ψ2R119904B1198671+Q

2

R3= 119864 X

3X119867

2 = B

2Ψ119867

1R119904B1198671+Q

3

R4= 119864 X

4X119867

2 = B

2Ψ119867

1Ψ2R119904B1198671+Q

4

(56)



1ndashQ

4




100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R ( 1)R ( 2)R ( 3)R ( 4)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

R1

R2

R3

R4

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

B2R119904B1198671

B2Ψ119867

1R119904B1198671

B2Ψ2R119904B1198671

B2Ψ119867

1Ψ2R119904B1198671

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+Q2

(57)



1 and (57) can be


R = (D ⊙ B2)C +Q

2

D =

[[[[[[[[

[

Λminus1(R

119904)

Λminus1(Ψ

119867

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

119867

1Ψ2R119904)

]]]]]]]]

]

(58)


1119894and 120596


1205961119894=1

2(angle[

[[

D2119894

D1119894

]]]

+ angle[[[

D4119894

D3119894

]]]

) (59)

1205962119894= minus

1

2(angle[

[[

D3119894

D1119894

]]]

+ angle[[[

D4119894

D2119894

]]]

) (60)

1205961119894= (

2120587

120582119894

)ΔP1∙ u

119894

= (2120587119889

1

120582119894

)


) cos (120593ΔP1

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP1

) sin (120593ΔP1

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP1

) cos (120579119894)]

(61)

1205962119894= (

2120587

120582119894

)ΔP2∙ u

119894

= (2120587119889

2

120582119894

)


) cos (120593ΔP2

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP2

) sin (120593ΔP2

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP2

) cos (120579119894)]

(62)


1198911(119909

1 119909

2) = 0

1198912(119909

1 119909

2) = 0

(63)

where 1199091= 120579 and 119909



x = 119891minus1119894(119909

1 119909

2) 119894 = 1 2 (64)


x119896+1 = 119865119894(119909

119896

1 119909

119896

2) 119894 = 1 2 (65)


(0)

2]119879









3+4119898







k119879 = [1205791 120593

1 1205792 120593

2 120579

119903 120593

119903] (66)




CRB = Fminus1(67)


F = [F120579120579 F120579120593

F120593120579

F120593120593

] (68)

where F120579120579



F119894119895= 119873trace[Rminus1 120597R

120597V119894

Rminus1 120597R120597V

119895

]


119904D119867

119895Rminus1]


119904D119895Rminus1]

D119894=120597B120597k

119894

(69)



8 Discussion




2 This is


1or ΔP

2is parallel



1 The







2as shown

Z

YX

O

m2m

m + 2m + 1

1

23

42m minus 1

m minus 1

⋱⋱






1120579= 05

1198961120593

= 05 1198962120579

= 03 1198962120593


119895minus 120593

119895) and 119892

119894120593= cos(120579

119895minus 120593


coordinate 120579119895and 120593



RMSE = radic 1

500

500

sum

119905=1

[(120579119894119905minus 120579)

2

+ (120593119894119905minus 120593)

2

] (70)

where 120579119894119905and 120593






Y

XO

m

m minus 11

2



ΔP1


O

Y

X

m + 2

m + 1

ΔP2



0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Succ

ess r

ate (

)



minus10 minus5 0 5 10 15 20


SNR (dB)

RMSE

(deg

)



07

06

05

04

03

02

01

0minus10 minus5 0 10 15 20







100 200 300 400 500 600 700 800 900 100050

55

60

65

70

75

80

85

90

95

100

Number of snapshots

Succ

ess r

ate (

)




100 200 300 400 500 600 700 800 900 10000

002

004

006

008

01

012

014

016

Number of snapshots

RMSE

(deg

)





0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Reso

lutio

n pr

obab

ility

()



minus10 minus5 0 5 10 15 20





1minus 120579

1| |120579

2minus 120579

2| are



1minus 120593

1| |120593

2minus 120593

2| are


2|2 120579

119894and 120579





10 Conclusion



Appendix



1 s

119875otimes t

119875] isin C

119862119863times119875 (A1)


s otimes t =[[[[

[

1199041t

1199042t119904119862t

]]]]

]

(A2)



Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1 and (57) can be


R = (D ⊙ B2)C +Q

2

D =

[[[[[[[[

[

Λminus1(R

119904)

Λminus1(Ψ

119867

1R119904)

Λminus1(Ψ

2R119904)

Λminus1(Ψ

119867

1Ψ2R119904)

]]]]]]]]

]

(58)


1119894and 120596


1205961119894=1

2(angle[

[[

D2119894

D1119894

]]]

+ angle[[[

D4119894

D3119894

]]]

) (59)

1205962119894= minus

1

2(angle[

[[

D3119894

D1119894

]]]

+ angle[[[

D4119894

D2119894

]]]

) (60)

1205961119894= (

2120587

120582119894

)ΔP1∙ u

119894

= (2120587119889

1

120582119894

)


) cos (120593ΔP1

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP1

) sin (120593ΔP1

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP1

) cos (120579119894)]

(61)

1205962119894= (

2120587

120582119894

)ΔP2∙ u

119894

= (2120587119889

2

120582119894

)


) cos (120593ΔP2

) sin (120579119894) cos (120593

119894)

+ sin (120579ΔP2

) sin (120593ΔP2

) sin (120579119894) sin (120593

119894)

+ cos (120579ΔP2

) cos (120579119894)]

(62)


1198911(119909

1 119909

2) = 0

1198912(119909

1 119909

2) = 0

(63)

where 1199091= 120579 and 119909



x = 119891minus1119894(119909

1 119909

2) 119894 = 1 2 (64)


x119896+1 = 119865119894(119909

119896

1 119909

119896

2) 119894 = 1 2 (65)


(0)

2]119879









3+4119898







k119879 = [1205791 120593

1 1205792 120593

2 120579

119903 120593

119903] (66)




CRB = Fminus1(67)


F = [F120579120579 F120579120593

F120593120579

F120593120593

] (68)

where F120579120579



F119894119895= 119873trace[Rminus1 120597R

120597V119894

Rminus1 120597R120597V

119895

]


119904D119867

119895Rminus1]


119904D119895Rminus1]

D119894=120597B120597k

119894

(69)



8 Discussion




2 This is


1or ΔP

2is parallel



1 The







2as shown

Z

YX

O

m2m

m + 2m + 1

1

23

42m minus 1

m minus 1

⋱⋱






1120579= 05

1198961120593

= 05 1198962120579

= 03 1198962120593


119895minus 120593

119895) and 119892

119894120593= cos(120579

119895minus 120593


coordinate 120579119895and 120593



RMSE = radic 1

500

500

sum

119905=1

[(120579119894119905minus 120579)

2

+ (120593119894119905minus 120593)

2

] (70)

where 120579119894119905and 120593






Y

XO

m

m minus 11

2



ΔP1


O

Y

X

m + 2

m + 1

ΔP2



0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Succ

ess r

ate (

)



minus10 minus5 0 5 10 15 20


SNR (dB)

RMSE

(deg

)



07

06

05

04

03

02

01

0minus10 minus5 0 10 15 20







100 200 300 400 500 600 700 800 900 100050

55

60

65

70

75

80

85

90

95

100

Number of snapshots

Succ

ess r

ate (

)




100 200 300 400 500 600 700 800 900 10000

002

004

006

008

01

012

014

016

Number of snapshots

RMSE

(deg

)





0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Reso

lutio

n pr

obab

ility

()



minus10 minus5 0 5 10 15 20





1minus 120579

1| |120579

2minus 120579

2| are



1minus 120593

1| |120593

2minus 120593

2| are


2|2 120579

119894and 120579





10 Conclusion



Appendix



1 s

119875otimes t

119875] isin C

119862119863times119875 (A1)


s otimes t =[[[[

[

1199041t

1199042t119904119862t

]]]]

]

(A2)



Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












CRB = Fminus1(67)


F = [F120579120579 F120579120593

F120593120579

F120593120593

] (68)

where F120579120579



F119894119895= 119873trace[Rminus1 120597R

120597V119894

Rminus1 120597R120597V

119895

]


119904D119867

119895Rminus1]


119904D119895Rminus1]

D119894=120597B120597k

119894

(69)



8 Discussion




2 This is


1or ΔP

2is parallel



1 The







2as shown

Z

YX

O

m2m

m + 2m + 1

1

23

42m minus 1

m minus 1

⋱⋱






1120579= 05

1198961120593

= 05 1198962120579

= 03 1198962120593


119895minus 120593

119895) and 119892

119894120593= cos(120579

119895minus 120593


coordinate 120579119895and 120593



RMSE = radic 1

500

500

sum

119905=1

[(120579119894119905minus 120579)

2

+ (120593119894119905minus 120593)

2

] (70)

where 120579119894119905and 120593






Y

XO

m

m minus 11

2



ΔP1


O

Y

X

m + 2

m + 1

ΔP2



0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Succ

ess r

ate (

)



minus10 minus5 0 5 10 15 20


SNR (dB)

RMSE

(deg

)



07

06

05

04

03

02

01

0minus10 minus5 0 10 15 20







100 200 300 400 500 600 700 800 900 100050

55

60

65

70

75

80

85

90

95

100

Number of snapshots

Succ

ess r

ate (

)




100 200 300 400 500 600 700 800 900 10000

002

004

006

008

01

012

014

016

Number of snapshots

RMSE

(deg

)





0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Reso

lutio

n pr

obab

ility

()



minus10 minus5 0 5 10 15 20





1minus 120579

1| |120579

2minus 120579

2| are



1minus 120593

1| |120593

2minus 120593

2| are


2|2 120579

119894and 120579





10 Conclusion



Appendix



1 s

119875otimes t

119875] isin C

119862119863times119875 (A1)


s otimes t =[[[[

[

1199041t

1199042t119904119862t

]]]]

]

(A2)



Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Y

XO

m

m minus 11

2



ΔP1


O

Y

X

m + 2

m + 1

ΔP2



0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Succ

ess r

ate (

)



minus10 minus5 0 5 10 15 20


SNR (dB)

RMSE

(deg

)



07

06

05

04

03

02

01

0minus10 minus5 0 10 15 20







100 200 300 400 500 600 700 800 900 100050

55

60

65

70

75

80

85

90

95

100

Number of snapshots

Succ

ess r

ate (

)




100 200 300 400 500 600 700 800 900 10000

002

004

006

008

01

012

014

016

Number of snapshots

RMSE

(deg

)





0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Reso

lutio

n pr

obab

ility

()



minus10 minus5 0 5 10 15 20





1minus 120579

1| |120579

2minus 120579

2| are



1minus 120593

1| |120593

2minus 120593

2| are


2|2 120579

119894and 120579





10 Conclusion



Appendix



1 s

119875otimes t

119875] isin C

119862119863times119875 (A1)


s otimes t =[[[[

[

1199041t

1199042t119904119862t

]]]]

]

(A2)



Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










100 200 300 400 500 600 700 800 900 100050

55

60

65

70

75

80

85

90

95

100

Number of snapshots

Succ

ess r

ate (

)




100 200 300 400 500 600 700 800 900 10000

002

004

006

008

01

012

014

016

Number of snapshots

RMSE

(deg

)





0

10

20

30

40

50

60

70

80

90

100

SNR (dB)

Reso

lutio

n pr

obab

ility

()



minus10 minus5 0 5 10 15 20





1minus 120579

1| |120579

2minus 120579

2| are



1minus 120593

1| |120593

2minus 120593

2| are


2|2 120579

119894and 120579





10 Conclusion



Appendix



1 s

119875otimes t

119875] isin C

119862119863times119875 (A1)


s otimes t =[[[[

[

1199041t

1199042t119904119862t

]]]]

]

(A2)



Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Appendix



1 s

119875otimes t

119875] isin C

119862119863times119875 (A1)


s otimes t =[[[[

[

1199041t

1199042t119904119862t

]]]]

]

(A2)



Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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