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R E S E A R C H Open Access

DOA estimation for conformal vector-

sensor array using geometric algebra

Tianzhen Meng, Minjie Wu

*

and Naichang Yuan

Abstract

In this paper, the problem of direction of arrival (DOA) estimation is considered in the case of multiple polarized

signals impinging on the conformal electromagnetic vector-sensor array (CVA). We focus on modeling the manifold

holistically by a new mathematical tool called geometric algebra. Compared with existing methods, the presented

one has two main advantages. Firstly, it acquires higher resolution by preserving the orthogonality of the signal

components. Secondly, it avoids the cumbersome matrix operations while performing the coordinate

transformations, and therefore, has a much lower computational complexity. Simulation results are provided to

demonstrate the effectiveness of the proposed algorithm.

Keywords: Geometric algebra, Conformal array, Electromagnetic vector sensors, DOA estimation

1 Introduction

The direction of arrival (DOA) estimation has received a

strong interest in wireless communication systems such

as radar, sonar, and mobile systems [1]. In this corres-

pondence, the problem of DOA estimation is considered

in the case of multiple polarized signals impinging on

the conformal vector-sensor array (CVA). We name our

target array CVA since it is a conformal array whose ele-

ments are electromagnetic vector sensors. Interest in

this problem can be divided into two topics: (1) con-

formal array and (2) electromagnetic vector sensors.

A conformal antenna is an antenna that conforms to a

prescribed shape. The shape can be some part of an air-

plane, high-speed missile, or other vehicle [2]. Their

benefits include reducing aerodynamic drag, covering

wide angle, space-saving and so on [3, 4]. Nevertheless,

due to the curvature of the bearing surface, the far-field

contribution in the incident direction of one element is

different from that of others [5]. The pattern synthesis

theorem is not available resulting from the fact that the

conformal arrays can no longer be regarded as simple

isotropic ones. In [4], Wang et al. proposed a uniform

method for the element-polarized pattern transform-

ation of arbitrary three-dimensional (3-D) conformal ar-

rays based on Euler rotation. However, the Euler

rotation involves cumbersome matrix transformations,

and therefore, has a considerable computational burden.

Zou et al. analyzed the 3-D pattern of arbitrary con-

formal arrays using geometric algebra in [6]. Neverthe-

less, this mathematical language was not transplanted to

the DOA estimation. In view of this, Wu et al. combined

the geometric algebra with multiple signal classification

(MUSIC), termed as GA-MUSIC, to solve the DOAs for

cylindrical conformal array [7]. It used short dipole as

the element which made the array belong to a scalar

array. In addition, the electromagnetic vector sensors are

not taken into account.

As for the second point, we know the electromagnetic

vector sensor can measure the three components of the

electric field and the three components of the magnetic

field simultaneously. And, considerable studies on the

extensions of traditional array signal processing tech-

niques to the vector sensors are available in literature. In

[8], Nehorai concatenated all the output vectors into a

long vector and derived the Cramer-Rao bound (CRB).

However, the orthogonality of the signal components

was lost in this case. In view of this, a hypercomplex

model for multicomponent signals impinging on vector

sensors was presented in [9]. This model was based on

biquaternions (quaternions with complex coefficients).

Subsequently, Jiang et al. introduced geometric algebra

into the electromagnetic vector-sensor processing field

[10]. However, the model cannot be applied to the

* Correspondence: wmj601@nuaa.edu.cn

College of Electronic Science and Engineering, National University of Defense

Technology, Deya Road 109, Changsha 410073, China

EURASIP Journal on Advances

in Signal Processing

© The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0

International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and

reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to

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Meng et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:64

DOI 10.1186/s13634-017-0503-y

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

conformal array since the pattern is assumed to be a sca-

lar and the same for each element.

In this correspondence, we will combine the electro-

magnetic vector sensors with the conformal array, and

present a unified model based on geometric algebra to

estimate the DOAs. The proposed technique in this

paper is regarded as a generalization of the one pre-

sented in [10] to the case of the conformal arrays. Com-

pared with existing methods, the proposed one has two

main advantages. Firstly, it can give a more accurate esti-

mation by preserving the orthogonality of the signal

components. Secondly, it largely decreases the computa-

tion complexity for the coordinate transformations are

avoided. In addition, it has a strong commonality, that is

to say, it is not limited to any specific conformal array.

The rest of this paper is as follows. In Section 2, some

notations about geometric algebra are briefly introduced,

and on this basis, the manifold for the conformal vector-

sensor array is derived. Section 3 analyzes the computa-

tional burden. Illustrative examples are carried out to

verify the effectiveness of the proposed algorithm in Sec-

tion 4, followed by concluding remarks.

Throughout this correspondence, we use lowercase

boldface letters to denote vectors and uppercase bold-

face letters to represent matrices for notational conveni-

ence. Moreover, the uppercase letters symbolize the

multivectors whenever there is no possibility of confu-

sion. Superscripts “*”,“T”, and “H”represent the conju-

gation, transpose, and conjugate transpose, respectively.

In addition, (⋅)

+

and (⋅)

~

symbolize the conjugate trans-

pose in geometric algebra and the reverse operator, re-

spectively. Finally, ℜmn

3stands for the m×nreal matrix

in 3-D space and E{⋅} denotes the expectation operator.

2 The proposed algorithm

2.1 Some notations about geometric algebra

Geometric algebra is the largest possible associative alge-

bra that integrates all algebraic systems (algebra of com-

plex numbers, matrix algebra, quaternion algebra, etc.)

into a coherent mathematical language [11]. In view of

its widespread usage in subsequent sections, it is worth-

while to review some notations about geometric algebra

before proceeding to the physical problems of interest.

The geometric product is considered as the fundamental

product of geometric algebra, and its definition is as follows

xy ¼x⋅yþx∧yð1Þ

where the wedge symbol “”denotes the outer product

with the properties listed in Table 1.

Exchanging the order of xand yin (1),and utilizing

the symmetry of the inner product and the anti-

symmetry of the outer product, it follows that

yx ¼x⋅y‐x∧yð2Þ

Combining (1) with (2), we can find that the inner

product and the outer product can be uniformly repre-

sented by the geometric product, that is,

x⋅y¼xy þyx

2ð3Þ

x∧y¼xy ‐yx

2ð4Þ

In general, we call an outer product of kvectors a k-

blade. The value of kis referred to as the grade of the

blade. Specially, the top-grade blade E

n

in an n-dimen-

sional space is called pseudo-scalars. Essentially, blades

are just elements of the geometric algebra. It is noted

that we restrict the discussion to 3-D Euclidean space

[12], that is, a space with an orthonormal basis {e

x

,e

y

,

e

z

}. As shown in Fig. 1, E

3

is the pseudo-scalar, relative

to the origin denoted by O. The three-blade is drawn as

a parallelepiped. The volume depicts the weight of the

three-blade. Nevertheless, blades have no specific shape.

A linear combination of blades with different grades is

called a multivector [13]. Multivectors are the general el-

ements of geometric algebra. Thus, a generic element

can be expressed by

Table 1 Properties of the outer product

Property Meaning

Anti-symmetry (xʌy)=−(yʌx)

Scaling xʌ(γy)=γ(xʌy)

Distributivity xʌ(y+z)=(xʌy)+(xʌz)

Associativity xʌ(yʌz)=(xʌy)ʌz

Fig. 1 The geometry of 3-blade

Meng et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:64 Page 2 of 12

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A¼a0þa1ex∧eyþa2ez∧exþa3ey∧ez

þa4ex∧ey∧ezþa5ezþa6eyþa7exð5Þ

where a

0

,a

1

,…,a

7

are real numbers. For e

x

,e

y

,e

z

are

mutually orthogonal, using the definition of the geomet-

ric product, (5) can be represented by another shape.

A¼a0þa1exy þa2ezx þa3eyz þa4exyz þa5ezþa6eyþa7ex

¼A

hi

0þA

hi

1þA

hi

2þA

hi

3

ð6Þ

where the notation 〈A〉

k

means to select or extract the

grade kpart of Aand the reverse of 〈A〉

k

can be calcu-

lated as follows

Ae

DE

k¼−1ðÞ

kk−1ðÞ

2A

hi

kð7Þ

Thus, the reverse of Ais given

Ae¼A

hi

0þA

hi

1−A

hi

2−A

hi

3ð8Þ

In the discussion up to this point, we can define the

norm of a multivector.

A

kk

¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

AAe

DE

0

r¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

X

3

k¼0

A

hi

kAe

DE

k

DE

0

v

u

u

tð9Þ

We will next introduce the rotor, one of the most im-

portant objects in applications of geometric algebra. As

shown in Fig. 2, vector yis acquired through rotating

the vector xwith θ. The rotation can be regarded as two

consecutive reflections, first in a, then in b. Correspond-

ingly, the expression that reflects xin the line with dir-

ection ais

x0¼axa ¼2axðÞa‐xð10Þ

The expression appears to be strange at first, but it is

actually one of the most important rationales why the

geometric product is so useful.

Similarly, ycan be obtained by reflecting x’in the line

with direction b

y¼bx0b‐1¼baxa‐1b‐1¼baðÞxbaðÞ

‐1¼RxR‐1ð11Þ

As shown in Eq. (11), Ris identified as the rotor. If we

want to rotate a vector counterclockwise by a specific

angle, we only need to apply the rotor to the vector.

And, the rotation must be over twice the angle between

aand b. In Appendix 1, a brief proof is given.

2.2 Complex representation matrix (CRM)

As stated above, we adopt multivector as a generic elem-

ent of geometric algebra. However, the analysis of the

mulitvector and its attendant theory are scarce. In view

of this, we will introduce the CRM since the matrix the-

ories are mature [14]. Similar to the multivector, we con-

struct the matrix in geometric algebra, noted Gmn

3,as

follows

A¼A0þA1exy þA2ezx þA3eyz þA4exyz þA5ezþA6exþA7ex

ð12Þ

where A

0

,A

1

,…,A

7

∈ℜmn

3. Thus, the CRM can be de-

fined as

ψAðÞ¼ A0þA4exyz þA1exyz þA5−A2þA6exyz−A3exyz −A7

A2−A6exyz−A3exyz −A7A0þA4exyz −A1exyz−A5

ð13Þ

Given a matrix A∈Gmn

3and its CRM ψ(A), then the

following equalities stand

A¼P2mψAðÞPþ

2nð14Þ

ψAðÞ¼Q2m

A

A

Q2nð15Þ

where

P2m¼1

21þez

ðÞImezx−ex

ðÞIm

½∈G3m2mð16Þ

Q2m¼1

2

1þez

ðÞImezx−ex

ðÞIm

−ezx−ex

ðÞIm1−ez

ðÞIm

∈G32m2mð17Þ

with I

m

being the identity matrix of dimension

m×m. Properties (14)and(15) can be verified by

direct calculation using Eq. (16)andEq.(17). For e

xyz

is isomorphic to complex imaginary unit j[9],

ψ(A)can be regarded as a complex matrix. Then, all

the operation rules of the complex matrix are applic-

able to ψ(A). Some properties [15] which will be used

in the sequel are listed as follows.

Fig. 2 Rotation of vector x

Meng et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:64 Page 3 of 12

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a) A=B⇔ψ(A)=ψ(B);

b) ψ(A+B)=ψ(A)+ψ(B),ψ(AC)=ψ(A)ψ(C);

c) ψ(A

+

)=ψ

+

(A).

It is also worthwhile to note that the following three

properties regarding P

2m

and Q

2m

will be of use in the

forthcoming calculations.

d) P2mPþ

2m¼Im;

e) Pþ

2mP2mψAðÞ¼ψAðÞPþ

2nP2n;

f) Q2m¼Qþ

2m:

2.3 Manifold modeling of vector sensors in the conformal

array

In this subsection, we will combine the electromagnetic

vector sensors with the conformal array, and present a

unified model based on geometric algebra to estimate

the DOAs. To illustrate the versatility of this algorithm,

we consider a M×Ncylindrical conformal array as

shown in Fig. 3. The array contains Nuniformly spaced

rings on the surface. In addition, there are Melectro-

magnetic vector sensors distributed on each ring. The

angle between two consecutive elements on the same

ring is β. In addition, the radius of the cylinder and the

distance between adjacent rings are Rand W,

respectively.

Since the electromagnetic vector sensor consists of six

spatially collocated antennas, the three electric field

components (E

x

,E

y

,E

z

) and the three magnetic field

components (H

x

.H

y

,H

z

) can be measured simultan-

eously. Thus, we can use two multivectors, X

e

and X

h

,to

represent the electric field signal and the magnetic field

signal, respectively.

Xe¼ExexþEyeyþEzezð18Þ

Xh¼HxexþHyeyþHzezð19Þ

Similarly, the noise can be written as

Ne¼NExexþNEy eyþNEz ezð20Þ

Nh¼NHxexþNHy eyþNHz ezð21Þ

Then, the output of single element can be obtained in

the frame of geometric algebra.

Y¼XeþexyzXhþNeþexyz Nhð22Þ

From (22), we see that e

xyz

not only provides a vital

link between electric field components and magnetic

field components, but also offers the possibility to han-

dle the data model in geometric algebra. Due to the lim-

ited length, the relationship between the two fields will

be derived in Appendix 2. In addition, from (18, 19, 20,

and 21), we see that the orthogonality of the signal com-

ponents is reserved. Compared with the conventional

methods, such as the long vector algorithm [8], this or-

thogonality constraint implies stronger relationships be-

tween the signal components. The proof can be seen in

Appendix 3. And it is also the most important advantage

of the output model. Using the Maxwell equations in the

formalism of geometric algebra, Eq. (22) can be written

in another shape.

Y¼1þuðÞSEþNeþexyzNhð23Þ

where

u¼cosφsinθexþsinφsinθeyþcosθezð24Þ

with urepresenting the unit vector of the signal

propagation andS

E

being the complex envelope of the

electric field. In addition, the signal has an elevation

angle θand an azimuth angle φ.Thederivationof

(23) is omitted here and the interested reader will

find more material in [10].

Considering the polarization information, the afore-

mentioned complex envelope, S

E

, can be written as

SE¼ΘhSð25Þ

Fig. 3 The cylindrical conformal array consisting of MxN

electromagnetic vector sensors

Meng et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:64 Page 4 of 12

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

where his the signal polarization vector [16] and can be

described by the auxiliary polarization angle (γ) and the

polarization phase difference (η), that is, h¼

cosγsinγeexyzη

½

T. And Sis the multivector symboliz-

ing the complex envelope of the signal. Moreover, the

parameter Θdenotes the steering vector of the angle

field [17] and is independent of the space location:

Θ¼

−sinφcosθcosφ

cosφcosθsinφ

0−sinθ

2

43

5ð26Þ

Thus, the polarized version of (23) can be expressed as

Y¼1þuðÞΘhS

|ﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄ}

X

þNeþexyzNh

|ﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄ}

Ny

ð27Þ

As stated above, the cylindrical conformal array is

composed of M×Nelements. In addition, suppose that

there are Knarrowband sources impinging on the array.

The manifold of the conformal array as shown in Fig. 3

corresponding to the kth source is

asθk;;φk

¼ask ¼

g1θk;;φk

e−exyz2πuTθk;;φk

ðÞ

r1

ðÞ

=λk

g2θk;;φk

e−exyz2πuTθk;;φk

ðÞ

r2

ðÞ

=λk

⋮

gMN θk;;φk

e−exyz2πuTθk;;φk

ðÞ

rMN

ðÞ

=λk

#

2

6

6

6

4ð28Þ

where g

mn

(θ

k

,φ

k

), m= 1,2,…,M,n= 1,2,…,Nis the elem-

ent pattern in the array global Cartesian coordinate sys-

tem. In subsequent equations the range of mand nis

the same and is omitted. R

mn

and λ

k

are the (m, n)th

element location vector and the kth signal wavelength

respectively. The received signals of the array are the

superposition of the response of each signal, the output

can be expressed as

Y¼X

K

k¼1

ask XkþNy¼Aθk;;φk;;γk;;ηk

SþNyð29Þ

where X

k

is a special case of Xregarding the kth source.

And

S¼S1S2⋯SK

½

Tð30Þ

Ny¼Ny1Ny2⋯NyK

Tð31Þ

with S

k

being the complex envelope of the kth signal.

For notational convenience, we will simply write Ain-

stead of A(θ

k

,φ

k

,γ

k

,η

k

) whenever there is no possibility

of confusion.

Let us refer back to Eq. (28). It is worthwhile to note

that the aforementioned manifold of the conformal

array, a

sk

, is derived under the global coordinate system.

The azimuth and elevation angles are defined in Fig. 3.

In most ready-made algorithms, the element pattern,

g

mn

(θ

k

,φ

k

), is always considered to be identical. Never-

theless, due to the effects of the curvature of conformal

carriers, the above assumption is not satisfied in the cy-

lindrical conformal array.

In what follows, we will use the rotor in geometric al-

gebra to model the conformal array, together with the

vector-sensor array. The most important advantage of

geometric algebra in analyzing conformal arrays is that

we are able to express the geometry and the physics in a

coordinate-free language. As stated above, the rotor can

be used to realize the rotation between the two coordin-

ate systems. Thus, we define the local coordinate system

of the (m,n)th element as shown in Fig. 4.

The e

xmn

axis is the same as e

x

axis in the global co-

ordinate system, e

zmn

is perpendicular to the element

surface and e

ymn

is tangent to the surface which can

form a standard Cartesian coordinate system. Then,

transforming the global coordinate into the local one is

equivalent to rotating the global coordinate about e

x

axis. The corresponding rotation angle is

ξ¼m−1ðÞβ−M−1

2β¼m−Mþ1

2

βð32Þ

Substituting e

z

and e

y

for band a, respectively (see

Appendix 1, the exponential form of the rotor), the rotor

is

Fig. 4 The local coordinate of the (m,n)th element

Meng et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:64 Page 5 of 12

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Rmn ¼e‐ez∧ey

ðÞ

ξ

2ð33Þ

Utilizing E

3

=e

x

e

y

e

z

as the pseudo-scalar in 3-D Eu-

clidean space, Eq. (33) can be further simplified

Rmn ¼eE3exξ

2ð34Þ

Through (11), we can acquire the standard orthogonal

basis in the local coordinate. And the specific procedure

can refer to Appendix 4. We herein directly give the

results

exmn ¼RmnexRmn‐1¼exð35Þ

eymn ¼RmneyRmn‐1¼cos m−Mþ1

2

β

ey‐sin m−Mþ1

2

β

ez

ð36Þ

ezmn ¼RmnezRmn‐1¼sin m−Mþ1

2

β

eyþcos m−Mþ1

2

β

ez

ð37Þ

Thus, from (35, 36 and 37), we can obtain the element

pattern, g

mn

(θ

k

,φ

k

).

Up to present, the remaining unknown variable is the

location vector. From Fig. 3, we can obtain its specific

expression

rmn ¼−nδðÞexþRsin m−Mþ1

2

β

eyþRcos m−Mþ1

2

β

ez

ð38Þ

where δmeans the spacing between adjacent rings.

Then, the mainfold of the conformal vector-sensor array,

A, can be obtained.

Since the geometric algebra is introduced in modeling

the manifold, the eigendecomposition is different from

the conventional methods, such as [18]. In fact, similar

to the quaternion case [19], the noncommutativity of the

geometric product leads to two possible eigenvalues,

namely the left and the right eigenvalue. However, in this

paper, we select the right eigenvalue since the right

eigendecomposition of Acan be converted to the right

eigendecomposition of the CRM.

We construct the covariance matrix

RY¼EYY þ

fg

¼AESSþ

fg

Aþþ6σ2IMN ð39Þ

Here, we assume that the noise is identical and uncor-

related from element to element, with covarianceσ

2

.

For R

Y

is a unitary matrix, its eigendecomposition can

be written as

RY¼UYsDYsUþ

YsþUYnDYnUþ

Ynð40Þ

Where U

Ys

is the MNxKmatrix composed of the Kei-

genvectors corresponding to the Klargest eigenvalues of

R

Y

, termed as the signal subspace. U

Yn

represents the

matrix composed of the eigenvectors corresponding to

the 2 M–Ksmaller eigenvalues, i.e., the noise subspace.

According to the principles of the MUSIC algorithm, the

array manifold spans the signal subspace and is orthog-

onal to the noise subspace. In this case, we have

AþUYn¼02M−Kð41Þ

where 0

2M−K

is a 2 M–Krow vector with all elements

equal zero. The proof can be seen in Appendix 5.

In practice,RY¼1

LPL

l¼1YY þ¼UYsDYsUþ

YsþUYnDYn

Uþ

Yn, the maximum likelihood estimation ofR

Y

,isalways

used as the covariance matrix. Among which, Lrepresents

the number of snapshots. In this case, (41) becomes

AþUYn≈02M−Kð42Þ

Up to present, the DOA estimation model of con-

formal vector-sensor array has been established. This is

also the focus of our paper. The contents of constructing

the spatial spectra and searching the peak are omitted

here. The readers can refer to literature [18]. It is worth-

while to note that in introducing the rotor, the spatial lo-

cation of the sensor is not required. Then, the proposed

method can be easily extended to other arrays.

It is also worthwhile to note that {e

x

,e

y

,e

z

}isnotonly

the basis for the multivector in the vector-sensor array,

but also represents the coordinate in the conformal array.

And, it can be used for transformation between the global

and the local coordinates with the help of the rotor. Under

this circumstance, there are some links between those two

arrays. The commonality is one of the motivations for es-

tablishing a unified model to estimate the DOAs.

3 Complexity analysis

To better explain the superiority of the geometric alge-

bra in modeling the conformal vector-sensor array, we

will introduce the computational complexity from the

standpoint of deriving the manifold. And, the computa-

tional burden is evaluated in terms of the number of

multiplications, additions, and transpositions.

To this end, we will briefly introduce the conventional

methods of analysis based on Euler angle in this section.

Generally, the transformations between the element

local coordinates and the array global coordinates may

be implemented by three continuous Euler rotations [3].

The specific rotation matrix can be expressed as

RC;D;FðÞ¼RxCðÞRyDðÞRzFðÞ

¼

10 0

0 cosC−sinC

0 sinCcosC

2

43

5

cosD0−sinD

01 0

sinD 0 cosD

2

43

5

cosF−sinF0

sinFcosF0

001

2

43

5

¼

cosDcosF−cosDsinF−sinD

cosCsinF−cosFsinCsinDcosCcosFþsinCsinDsinF−cosDsinC

sinCsinFþcosCcosFsinDcosFsinC−cosCsinDsinFcosCcosD

2

43

5

ð43Þ

where C,D, and Fare, respectively, three consecutive

Meng et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:64 Page 6 of 12

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Euler rotation angles about e

x

axis, e

y

axis and e

z

axis.

The matrices R

x

(C), R

y

(D), and R

z

(F) are the corre-

sponding Euler rotation matrices. It is noted that two

successive Euler rotations are usually adequate to deal

with the cylindrical conformal array. The third Euler ro-

tation matrix is added here to cope with some irregular

or special conformal arrays. Additionally, we know the

rotation matrix is invertible from Eq. (43). Consequently,

taking the inversion with respect to R(C,D,F), we have

RC;D;FðÞ

−1¼Rz−1FðÞRy−1DðÞRx−1CðÞ

¼

cosFsinF0

−sinFcosF0

001

2

43

5

cosD0 sinD

010

−sinD 0 cosD

2

43

5

10 0

0 cosCsinC

0−sinCcosC

2

43

5

¼

cosDcosFcosCsinF−cosFsinCsinDsinCsinFþcosCcosFsinD

−cosDsinFcosCcosFþsinCsinDsinFcosFsinC−cosCsinDsinF

−sinD−cosDsinCcosCcosD

2

43

5

ð44Þ

Combining Eq. (43) with Eq. (44), it is not hard to find

that

RC;D;FðÞ

−1¼RC;D;FðÞ

Tð45Þ

Thus, R(C,D,F) is the so-called orthogonal matrix. In

this case, transforming the local coordinate into the glo-

bal one is equivalent to imposing the transposition/in-

version with respect to the above rotation matrix. If we

model the conformal array based on the Euler angle,

three matrix multiplications and one matrix transpos-

ition are required for each element.

In fact, the matrix operations are essentially the multi-

plications and the additions between elements. To quan-

tify this, the amounts of multiplications and additions of

the two methods (i.e., the proposed method and the Eu-

ler angle method) are calculated, respectively. The corre-

sponding results are shown in Table 2. We assume that

one matrix transposition is considered as one multiplica-

tion or addition operation. And it is obvious that the

multiplication between two 3 × 3 matrices involves

9 × 3 multiplications and 9 × 2 additions. For conveni-

ence, the multiplication and the addition are collectively

referred to as the operation. Then, Eq. (43) contains

2 × 9 × 3 + 2 × 9 × 2 operations. For the conformal

array consisting of M×Nelectromagnetic vector sen-

sors, the transformation between different coordinates

involves 91 × 6 × MN operations. Compared with the

Euler rotation angle, the proposed method effectively

avoids the cumbersome matrix transformations. From

Eqs. (35, 36, and 37), we know e

ymn

and e

zmn

are inde-

pendent of e

x

. In addition, e

xmn

can be obtained directly

from Eq. (35) without extra operations. Thus, Eqs. (35,

36 and 37), can be expressed as a 2 × 2 matrix. While

using the rotor to establish the array manifold, the com-

putational process is equivalent to a 2 × 2 matrix multi-

plied by a 2 × 1 vector. In this case, the operations for

each element involve four multiplications and two

additions. The total amount of operations is

6×6×MN. Thus, the geometric algebra-based method

significantly decreases the computational burden.

In general, the Euler rotation and its matrix represen-

tation cannot intuitively exhibit the complete procedure.

In addition, as the configuration of the conformal array

becomes more irregular and complex, the level of com-

plexity involved in the transformations and the number

of calculations required increases largely.

4 Simulation results

In this section, Monte-Carlo simulation experiments are

used to verify the effectiveness of the proposed algo-

rithm. The array structure is shown in Fig. 3. Among

which, we select Mand Nas 4 and 4, respectively. The

angle between two consecutive elements on the same

ring, β, is 5°. The number of snapshots, L, is 200. Under

these premises, 200 independent simulation experiments

are carried out. The root mean square error (RMSE) is

utilized as the performance measure and is defined as

RMSE ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1

200 X

200

i¼1

^

θi−θi

2þ^

φi−φi

ðÞ

2

v

u

u

tð46Þ

where θi;;φi

fg

are the estimates of elevation angles and

azimuth angles, respectively, at the ith run.

Provided that there are three polarized signals can be

received. The incident angles are (10°, 15°), (35°, 40°),

and (60°, 35°), respectively. The corresponding

polarization auxiliary angles and the polarization phase

differences are (15°, 25°), (30°, 45°), and (50°, 65°). Fig-

ure 5 shows the simulation results of the proposed algo-

rithm. The position of the spectrum peak represents the

possible DOA. Intuitively, the estimation accuracy of the

proposed algorithm is high.

To better demonstrate the performance of the pro-

posed method, Qi’s method [3] and Gao’s algorithm [20]

are included for comparison. We study the performance

with a varying SNR from 0 to 30 dB. Without loss of

generality, we select the first source (T1) and the second

source (T2), respectively, to verify it. Figure 6 shows the

RMSE versus SNR with the snapshots being 200. It can

be seen that the proposed method outperforms the Qi’s

method [3] by preserving the orthogonality of the re-

ceived signal components. In addition, the performance

of Gao’s algorithm is also worse than the proposed one.

Two main reasons lead to this difference. Firstly, the

proposed method imposes stronger constraints between

the components of the signals. Secondly, the conformal

array in [20] essentially belongs to the scalar array from

the standpoint of elements while the conformal vector-

sensor array presented in this paper belongs to the vec-

tor array. And the vector array contains more signal

Meng et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:64 Page 7 of 12

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

information compared with the scalar array. Moreover,

in contrast to those two algorithms, the proposed one

effectively avoids the cumbersome matrix transforma-

tions, and therefore, has a much lower computational

complexity. It is noted that, for the statistical data have

certain randomness, the simulation curve in Fig. 6 is not

smooth.

Figure 7 illustrates the RMSE versus the number of

snapshots with the SNR fixed at 10 dB. Compared with

Fig. 6, we can draw similar conclusions. In particular, if we

pick the points with snapshots being 300 and 600, respect-

ively, we may find that the corresponding RMSEs are

0.5805 and 0.2902. This means the former value is nearly

twice as much as the latter one. In fact, these improve-

ments can be predicted from the derivation of CRB. The

specific derivation process can refer to literature [21]. The

number of snapshots can be extracted from the Fisher in-

formation matrix. Moreover, the CRB is found as the

element of the inverse of that matrix. So, we can conclude

that the RMSE is inversely proportional to L.

To better demonstrate the computational efficiency,

the specific operations, such as the multiplications, addi-

tions, and transpositions, are simulated in Fig. 8. The

value of the x-axis (or the abscissa) represents the product

of Mand N. It can be seen that the multiplications take

up the most resources. Compared with Euler rotation an-

gles, the proposed method reduces the computation by

one order of magnitude. Thus, the proposed algorithm

provides the possibility for real-time processing.

5 Conclusions

In this correspondence, we combine the electromagnetic

vector sensors with the conformal array, and present a

unified model based on geometric algebra to estimate

the DOAs. Compared with existing methods, the pro-

posed one has two main advantages. Firstly, it can give a

more accurate estimation by preserving the orthogonal-

ity of the signal components. Secondly, it avoids the

cumbersome matrix operations while performing the co-

ordinate transformations, and therefore, has a much

lower computational complexity. In addition, it has a

strong commonality, that is to say, it is not limited to

any specific conformal array. The simulation results ver-

ify the effectiveness of the proposed method.

6 Appendix 1

6.1 Here we will give a brief proof to demonstrate that

the rotation must be over twice the angle between aand

b

To proceed further, we rewrite Raccording to the defin-

ition of the geometric product:

R¼ba ¼b⋅aþb∧að47Þ

Here, we consider the case that the vectors are unit

length. This assumption is reasonable, because the basic

vectors of the Cartesian coordinate system satisfy it as

well. The geometric product of bʌaitself is:

Table 2 The computational complexity of the proposed method and Euler angle

Multiplications Additions Transpositions Operations

Euler angle 2 × 9 × 3 × 6 × MN 2×9×2×6×MN 6×MN 91 × 6 × MN

Proposed method 4 × 6 × MN 2×6×MN 06×6×MN

Fig. 5 The spatial spectrum of the proposed algorithm Fig. 6 RMSE versus SNR with the snapshots being 200

Meng et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:64 Page 8 of 12

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b∧aðÞb∧aðÞ¼ba‐b⋅aðÞb⋅a‐abðÞ

¼b•abaþabðÞ‐b⋅aðÞ

2‐baab

¼b⋅a2b⋅aðÞ‐b⋅aðÞ

2‐baaðÞb

¼b⋅aðÞ

2‐bb

¼cos2θ‐1

¼‐sin2θ

ð48Þ

Thus, we define the 2-blade E

2

:

E2¼b∧a

sinθð49Þ

Rcan be further simplified by substituting (49) into (47):

R¼cosθ‐E2sinθð50Þ

The expression is similar to the polar decomposition

of a complex number with the unit imaginary replaced

by the 2-blade E

2

. (50) can also be written as the expo-

nentials of E

2

:

R¼e‐E2θð51Þ

This formalism is more useful for the log-space of ro-

tors is linear. We split xinto a part (x

p

) parallel to bʌ

a-plane and a part (x

o

) orthogonal to bʌa-plane. Then,

x

o

is not affected by the application R. And we infer that

the rotation must be in the bʌa-plane. As stated above,

the rotation consists of two successive reflections which

are orthogonal (angle-preserving) transformations. Thus,

it allows us to pick any vector in the bʌa-plane to de-

termine the angle. Without loss of generality, we choose

vector a, and construct the “sandwich product”RaR

−1

as shown in (11):

RaR‐1¼baaa‐1b‐1¼bab‐1ð52Þ

where bab

−1

is the reflection of ain b. From this it is

clear that the rotation must be over twice the angle be-

tween aand b, since the angle between aand bab

−1

is

twice the angle between aand b. The negative signature

in (51) represents the rotation direction.

7 Appendix 2

7.1 We demonstrate how e

xyz

links the electric field with

the magnetic field

First, let us refer back to the famous Maxwell equations

described by the vector algebra are

∇Ε¼‐μ∂H

∂tð53Þ

∇⋅Ε¼ρ

εð54Þ

∇Η¼ε∂E

∂tð55Þ

∇⋅Η¼0ð56Þ

where E=E

x

e

x

+E

y

e

y

+E

z

e

z

and H=H

x

e

x

+H

y

e

y

+H

z

e

z

are, respectively, the electric and magnetic fields. In

addition, the parameters ε,μ,andρsymbolize the

Fig. 7 RMSE versus snapshots with the SNR fixed at 10 dB

Fig. 8 The computational efficiency (a) the proposed method (b) the Euler rotation method

Meng et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:64 Page 9 of 12

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

permittivity, the permeability, and the density of the

source charges, respectively.

It is worthwhile to give some results before proceeding

to the physical problems of interest:

x∧y¼exyzxyð57Þ

xy¼‐x⋅yexyz

ð58Þ

x⋅y¼‐exyz x∧exyzy

ð59Þ

Using Eqs. (57, 58 and 59), the Maxwell equations in

geometric algebra can be expressed as follows:

∇∧Εþμ∂H

∂texyz ¼0ð60Þ

∇⋅Ε¼ρ

εð61Þ

∇⋅ Ηexyz

þε∂E

∂t¼0ð62Þ

∇∧ Ηexyz

¼0ð63Þ

Since the arriving signals are assumed to be far-field,

so the signals received on different positions differ only

by a transmission delay, that is,

Er;tðÞ¼E0;t‐τðÞ ð64Þ

Hr;tðÞ¼H0;t‐τðÞ ð65Þ

where r=r

x

e

x

+r

y

e

y

+r

z

e

z

, and τis the so-called trans-

mission delay,

τ¼u⋅r

cð66Þ

with crepresenting the propagation speed and udenot-

ing the signal propagation as defined in (24). Let

EtðÞ¼E0;tðÞ¼ExtðÞexþEytðÞeyþEztðÞezð67Þ

HtðÞ¼H0;tðÞ¼HxtðÞexþHytðÞeyþHztðÞezð68Þ

Then,

Er;tðÞ¼Et‐τðÞ ð69Þ

Hr;tðÞ¼Ht‐τðÞ ð70Þ

For plane waves, we know

∇¼u

c

∂

∂tð71Þ

Substituting Eqs. (67, 68, 69, 70 and 71) into Maxwell

equations, we have

u

c∧E

tðÞþμH

tðÞexyz ¼0ð72Þ

u

c⋅E

tðÞ¼ρ

εð73Þ

u

c⋅H

tðÞexyz

þεE

tðÞ¼0ð74Þ

u

c∧H

tðÞexyz

¼0ð75Þ

where EtðÞ¼dE

dt and HtðÞ¼dH

dt .

From (75), we know uis in the plane HtðÞexyz .In

addition, combining the geometric implication of the

inner product with Eq. (74), we know EtðÞis the orthog-

onal complement of uin the plane Ht

ðÞ

exyz . That is to

say, EtðÞand uare orthogonal in that plane. Thus, the

parameter ρin (73) equals to zero. Then, Eqs. (72, 73)

are equivalent to Eqs. (74, 75). We rewrite (73),

u

c⋅E

tðÞ¼0ð76Þ

Combining (72) and (76), we have

ﬃﬃﬃ

μ

ε

rH

tðÞexyz ¼uE

tðÞ ð77Þ

where ﬃﬃμ

ε

qdenotes the intrinsic impedance of the

medium.

Integrating (77) with respect to time t,

ﬃﬃﬃ

μ

ε

rH

tðÞexyz ¼uE

tðÞþqð78Þ

where qis a constant and equals to zero in the far-field

assumption.

Up to this point, we have derived how e

xyz

links the

electric field with the magnetic field.

8 Appendix 3

8.1 We show the orthogonality constraint implies

stronger relationships between the signal components

Consider two electric field multivectors X

e1

,X

e2

, with

their expressions given by

Xe1¼Ex1exþEy1eyþEz1ezð79Þ

Xe2¼Ex2exþEy2eyþEz2ezð80Þ

By imposing the orthogonality for the two

multivectors

Xe2;Xe1

hi

g¼XH

e1⋅Xe2¼0ð81Þ

where 〈⋅〉

g

represents the inner product in geometric

algebra.

We can get the following relationships between the

signal components:

Meng et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:64 Page 10 of 12

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

EH

x1Ex2þEH

y1Ey2þEH

z1Ez2¼0ð82Þ

Ex2EH

y1¼Ey2EH

x1ð83Þ

Ey2EH

z1¼Ez2EH

y1ð84Þ

Ex2EH

z1¼Ez2EH

x1ð85Þ

However,forlongvectoralgorithm,themultivec-

tors are replaced by the vectors x

e1

,x

e2

,and

correspondingly,

xe1¼"Ex1

Ey1

Ez1#;xe2¼"Ex2

Ey2

Ez2#ð86Þ

Similarly,imposingtheorthogonalityforthetwovectors

xe2;xe1

hi

v¼xH

e1⋅xe2¼0ð87Þ

where 〈⋅〉

v

denotes the inner product between two vectors.

We can get the same result as in (82). However,

Eqs. (83, 84 and 85) cannot be obtained. In other

words, using geometric algebra to model the output

imposes stronger constraints between the compo-

nents of the vector sensor array.

9 Appendix 4

9.1 The detailed calculation procedures of Eqs. (35, 36 and

37), are given as follows. Since the derivations of Eqs. (36,

37) are similar to that of Eq. (35), we will take Eq. (35) as an

example. And the other two equations can be obtained

similarly. The derivation of (35) is

exmn ¼RmnexRmn‐1

¼e

E3ex

ξ

2exe

‐E3ex

ξ

2

¼cos ξ

2þE3exsin ξ

2

0

@1

Aexcos ξ

2‐E3exsin ξ

2

0

@1

A

¼excos ξ

2þE3sin ξ

2

0

@1

Acos ξ

2‐E3exsin ξ

2

0

@1

A

¼excos2ξ

2‐exE3excos ξ

2sin ξ

2þE3sin ξ

2cos ξ

2‐E3E3exsin2ξ

2

¼excos2ξ

2‐exexeyezexcos ξ

2sin ξ

2þexeyezsin ξ

2cos ξ

2‐exeyezexeyezexsin2ξ

2

¼excos2ξ

2‐eyezexcos ξ

2sin ξ

2þexeyezsin ξ

2cos ξ

2‐exexeyezeyezexsin2ξ

2

¼excos2ξ

2‐exeyezsin ξ

2cos ξ

2þexeyezsin ξ

2cos ξ

2‐eyezeyezexsin2ξ

2

¼excos2ξ

2þexsin2ξ

2¼ex

ð88Þ

10 Appendix 5

10.1 We will verify the rationality of Eq. (41)

Using property (b) and Eq. (39), we can obtain the CRM

ofR

Y

, that is

ψRY

ðÞ¼ψAðÞψRs

ðÞψAþ

ðÞþ6σ2IMN

¼ψAðÞψRs

ðÞψþAðÞþ6σ2IMN

ð89Þ

where

Rs¼ESSþ

fg ð90Þ

SinceR

s

is full rank, it is easy to obtain

rank ψAðÞψRs

ðÞψþAðÞ

fg

¼Kð91Þ

According to the principle of MUSIC, we have

ψþAðÞUψn¼02K2M−KðÞ ð92Þ

Where Uψn∈G32M2M−KðÞ

composed of the eigenvectors

corresponding to the 2 M–Ksmaller eigenvalues of

ψ(R

Y

). Using the property (d), Eq. (92) is equivalent to

the following equation.

P2KPþ

2KψþAðÞUψn¼02K2M−KðÞ ð93Þ

By means of the property (e), Eq. (93) can be further

expressed as

ψþAðÞPþ

2MP2MUψn¼02K2M−KðÞ ð94Þ

Taking the multiplication on the left byP

2K

, we have

AþUYn¼0K2M−KðÞ ð95Þ

Where UYn¼P2MUψn∈G3M2M−KðÞ

is composed of the ei-

genvectors corresponding to the 2 M–Ksmaller eigen-

values of R

Y

. Thus, Eq. (41) holds.

Acknowledgements

The authors would like to thank the anonymous reviewers for the

improvement of this paper.

Funding

This project was supported by the National Natural Science Foundation of

China (Grant No.61302017).

Authors’contributions

Tianzhen MENG conceived the basic idea and designed the numerical

simulations. Minjie WU analyzed the simulation results. Naichang YUAN

refined the whole manuscript. All authors read and approved the final

manuscript.

Competing interests

The authors declare that they have no competing interests.

Publisher’sNote

Springer Nature remains neutral with regard to jurisdictional claims in

published maps and institutional affiliations.

Meng et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:64 Page 11 of 12

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

Received: 31 March 2017 Accepted: 6 September 2017

References

1. Lu Gan, Xiaoyu Luo. Direction-of-arrival estimation for uncorrelated and

coherent signals in the presence of multipath propagation, IET Microwaves,

Antennas & Propagation, 2013, vol. 7, no. 9, pp. 746-753, DOI: https://doi.

org/10.1049/iet-map.2012.0659

2. Lars Josefsson, Patrik Persson. Conformal Array Antenna Theory and Design,

IEEE Press Series on Electromagnetic Wave Theory, 2006, ISBN:978–0–471-

46584-3

3. Z. -S. Qi., Y., Guo., B. -H. Wang. Blind direction-of-arrival estimation algorithm

for conformal array antenna with respect to polarisation diversity, IET

Microwaves, Antennas & Propagation, 2011, vol. 5, no. 4, pp. 433-442, DOI:

https://doi.org/10.1049/iet-map.2010.0166

4. B. H. Wang, Y. Guo, Y. L. Wang, Y. Z. Lin. Frequency-invariant pattern

synthesis of conformal array antenna with low corss-polarisation, IET

Microwaves, Antennas & Propagation, 2008, vol. 2, no. 5, pp. 442-450, ISSN:

1751-8725, DOI:https://doi.org/10.1049/iet-map:20070258

5. P. Alinezhad, S.R. Seydnejad, D. Abbasi-Moghadam, DOA estimation in

conformal arrays based on the nested array principles. Digit. Signal Process.

50, 191–202 (2016). https://doi.org/10.1016/j.dsp.2015.12.009

6. L. Zou, J. Lasenby, Z. He, Pattern analysis of conformal array based on geometric

algebra, IET Microwaves, Antennas & Propagation, 2011, vol. 5, no. 10, p. 1210-

1218, ISSN: 1751-8725, DOI: https://doi.org/10.1049/iet-map.2010.0588

7. W.U. Minjie, Z.H.A.N.G. Xiaofa, H.U.A.N.G. Jingjian, Y.U.A.N. Naichang, DOA

estimation of cylindrical conformal array based on geometric algebra. Int J

Antennas Propagation 2016,1–9 (2016). https://doi.org/10.1155/2016/7832475

8. A. Nehorai, E. Paldi, Vector-sensor array processing for electromagnetic

source localization. IEEE Trans. Signal Process. 42(2), 376–398 (1994). https://

doi.org/10.1109/78.275610

9. N.L. Bihan, S. Miron, J. Mars, MUSIC algorithm for vector-sensors array using

biquaternions. IEEE Trans. Signal Process. 55(9), 4523–4533 (2007). https://

doi.org/10.1109/TSP.2007.896067

10. J.F. Jiang, J.Q. Zhang, Geometric algebra of euclidean 3-space for

electromagnetic vector-sensor array processing, part I: modeling. IEEE Trans.

Antennas Propag. 58(12), 3961–3973 (2010). https://doi.org/10.1109/TAP.

2010.2078468

11. Venzo De Sabbata, Bidyut Kumar Datta. Geometric Algebra and Applications

to Physics. CRC Press, 2007, ISBN: 978-1-58488-772-0

12. J.W. Arthur, Understanding geometric algebra for electromagnetic theory (IEEE

Press, New Jersey, 2011) ISBN: 978-1-118-07854-3

13. Dietmar Hildenbrand. Foundations of geometric algebra computing, 2013,

Springer, ISBN: 978–3–642-31793-4, DOI: https://doi.org/10.1007/978-3-642-

31794-1

14. Leo Dorst, Daniel Fontijne, Stephen Mann. Geometric Algebra for Computer

Science, 2007, Morgan Kaufmann, ISBN: 978-0-12-369465-2

15. John Snygg. A New Approach to Differential Geometry Using Clifford’s

Geometric Algebra, Birkhäuser, 2010, New York, ISBN: 978-0-8176-8282-8

16. DESCHAMPS G. A. Techniques for handling elliptically polarized waves with

special reference to antennas: part II-geometrical representation of the

polarization of a plane electromagnetic wave. Proc. IRE, 1951, vol. 39, p. 540-

544. DOI:https://doi.org/10.1109/JRPROC.1951.233136

17. W.U. Minjie, Y.U.A.N. Naichang, DOA estimation in solving mixed non-

circular and circular incident signals based on the circular array. Prog.

Electromagn. Res. M 53, 141–151 (2017). https://doi.org/10.2528/

PIERM16092105

18. R. Schmidt, Multiple emitter location and signal parameter estimation. IEEE

Trans. Antennas Propag. AP-34(3), 276–280. ISSN: 0018-926X (1986). https://

doi.org/10.1109/TAP.1986.1143830

19. S. Miron, N.L. Bihan, J.I. Mars, Quaternion-MUSIC for vector-sensor array

processing. IEEE Trans. Signal Process. 54(4), 1218–1229 (2006). https://doi.

org/10.1109/TSP.2006.870630

20. Ziyang Gao, Yong Xiao. Direction of arrival estimation for conformal arrays

with diverse polarizations, 2015 12

th

IEEE International Conference on

Electronic Measurement & Instruments, pp. 439–442. DOI: https://doi.org/10.

1109/ICEMI.2015.7494229

21. P. Stoica, A. Nehorai, MUSIC, maximum likelihood, and Cramer-Rao Bound.

IEEE TRANS. ACOUSTICS SPEECH SIGNAL PROCESSING. 37(5), 720–741. ISSN:

0096-3518 (1989). https://doi.org/10.1109/29.17564

Meng et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:64 Page 12 of 12

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1.

2.

3.

4.

5.

6.

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