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DOA estimation for conformal vector-sensor array using geometric algebra

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DOA estimation for conformal vector-sensor array using geometric algebra

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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.
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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 Hrepresent 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, mn
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 ¼xyþxyð1Þ
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 ¼xyxyð2Þ
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,
xy¼xy þyx
2ð3Þ
xy¼xy 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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A¼a0þa1exeyþa2ezexþa3eyez
þa4exeyezþa5ezþa6eyþa7exð5Þ
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
k¼1ðÞ
kk1ðÞ
2A
hi
kð7Þ
Thus, the reverse of Ais given
Ae¼A
hi
0þA
hi
1A
hi
2A
hi
3ð8Þ
In the discussion up to this point, we can define the
norm of a multivector.
A
kk
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
AAe
DE
0
r¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
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 ¼2axðÞaxð10Þ
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 xin the line
with direction b
y¼bx0b1¼baxa1b1¼baðÞxbaðÞ
1¼RxR1ð11Þ
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
mn
3. Thus, the CRM can be de-
fined as
ψAðÞ¼ A0þA4exyz þA1exyz þA5A2þA6exyzA3exyz A7
A2A6exyzA3exyz A7A0þA4exyz A1exyzA5

ð13Þ
Given a matrix AGmn
3and its CRM ψ(A), then the
following equalities stand
A¼P2mψAðÞPþ
2nð14Þ
ψAðÞ¼Q2m
A
A

Q2nð15Þ
where
P2m¼1
21þez
ðÞImezxex
ðÞIm
½G3m2mð16Þ
Q2m¼1
2
1þez
ðÞImezxex
ðÞIm
ezxex
ðÞIm1ez
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=Bψ(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
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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
0sinθ
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

eexyz2πuTθk;;φk
ðÞ
r1
ðÞ
=λk
g2θk;;φk

eexyz2πuTθk;;φk
ðÞ
r2
ðÞ
=λk
gMN θk;;φk

eexyz2πuTθk;;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
S¼S1S2SK
½
Tð30Þ
Ny¼Ny1Ny2NyK

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
ξ¼m1ðÞβM1
2β¼mMþ1
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 ¼eezey
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 ¼RmnexRmn1¼exð35Þ
eymn ¼RmneyRmn1¼cos mMþ1
2

β

eysin mMþ1
2

β

ez
ð36Þ
ezmn ¼RmnezRmn1¼sin mMþ1
2

β

eyþcos mMþ1
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 ¼nδðÞexþRsin mMþ1
2

β

eyþRcos mMþ1
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 MKsmaller 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
AþUYn¼02MKð41Þ
where 0
2MK
is a 2 MKrow 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
AþUYn02MKð42Þ
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 cosCsinC
0 sinCcosC
2
43
5
cosD0sinD
01 0
sinD 0 cosD
2
43
5
cosFsinF0
sinFcosF0
001
2
43
5
¼
cosDcosFcosDsinFsinD
cosCsinFcosFsinCsinDcosCcosFþsinCsinDsinFcosDsinC
sinCsinFþcosCcosFsinDcosFsinCcosCsinDsinFcosCcosD
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
1¼Rz1FðÞRy1DðÞRx1CðÞ
¼
cosFsinF0
sinFcosF0
001
2
43
5
cosD0 sinD
010
sinD 0 cosD
2
43
5
10 0
0 cosCsinC
0sinCcosC
2
43
5
¼
cosDcosFcosCsinFcosFsinCsinDsinCsinFþcosCcosFsinD
cosDsinFcosCcosFþsinCsinDsinFcosFsinCcosCsinDsinF
sinDcosDsinCcosCcosD
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
^
θiθi

2þ^
φiφi
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, Qis method [3] and Gaos 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 Qis
method [3] by preserving the orthogonality of the re-
ceived signal components. In addition, the performance
of Gaos 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 ¼baþbað47Þ
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 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
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
baðÞbaðÞ¼babaðÞbaabðÞ
¼babaþabðÞbaðÞ
2baab
¼ba2baðÞbaðÞ
2baaðÞb
¼baðÞ
2bb
¼cos2θ1
¼sin2θ
ð48Þ
Thus, we define the 2-blade E
2
:
E2¼ba
sinθð49Þ
Rcan be further simplified by substituting (49) into (47):
R¼cosθE2sinθð50Þ
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
:
R¼eE2θð51Þ
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 productRaR
1
as shown in (11):
RaR1¼baaa1b1¼bab1ð52Þ
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
Ε¼μH
tð53Þ
∇⋅Ε¼ρ
εð54Þ
Η¼εE
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:
xy¼exyzxyð57Þ
xy¼xyexyz
 ð58Þ
xy¼exyz xexyzy
 ð59Þ
Using Eqs. (57, 58 and 59), the Maxwell equations in
geometric algebra can be expressed as follows:
∇∧ΕþμH
texyz ¼0ð60Þ
∇⋅Ε¼ρ
εð61Þ
∇⋅ Ηexyz

þεE
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;tτðÞ ð64Þ
Hr;tðÞ¼H0;tτðÞ ð65Þ
where r=r
x
e
x
+r
y
e
y
+r
z
e
z
, and τis the so-called trans-
mission delay,
τ¼ur
cð66Þ
with crepresenting the propagation speed and udenot-
ing the signal propagation as defined in (24). Let
EtðÞ¼E0;tðÞ¼ExtðÞexþEytðÞeyþEztðÞezð67Þ
HtðÞ¼H0;tðÞ¼HxtðÞexþHytðÞeyþHztðÞezð68Þ
Then,
Er;tðÞ¼EtτðÞ ð69Þ
Hr;tðÞ¼HtτðÞ ð70Þ
For plane waves, we know
¼u
c
tð71Þ
Substituting Eqs. (67, 68, 69, 70 and 71) into Maxwell
equations, we have
u
cE
tðÞþμH
tðÞexyz ¼0ð72Þ
u
cE
tðÞ¼ρ
εð73Þ
u
cH
tðÞexyz

þεE
tðÞ¼0ð74Þ
u
cH
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
cE
tðÞ¼0ð76Þ
Combining (72) and (76), we have
ffiffi
μ
ε
rH
tðÞexyz ¼uE
tðÞ ð77Þ
where ffiffiμ
ε
qdenotes the intrinsic impedance of the
medium.
Integrating (77) with respect to time t,
ffiffi
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
e1Xe2¼0ð81Þ
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
e1xe2¼0ð87Þ
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 ¼RmnexRmn1
¼e
E3ex
ξ
2exe
E3ex
ξ
2
¼cos ξ
2þE3exsin ξ
2
0
@1
Aexcos ξ
2E3exsin ξ
2
0
@1
A
¼excos ξ
2þE3sin ξ
2
0
@1
Acos ξ
2E3exsin ξ
2
0
@1
A
¼excos2ξ
2exE3excos ξ
2sin ξ
2þE3sin ξ
2cos ξ
2E3E3exsin2ξ
2
¼excos2ξ
2exexeyezexcos ξ
2sin ξ
2þexeyezsin ξ
2cos ξ
2exeyezexeyezexsin2ξ
2
¼excos2ξ
2eyezexcos ξ
2sin ξ
2þexeyezsin ξ
2cos ξ
2exexeyezeyezexsin2ξ
2
¼excos2ξ
2exeyezsin ξ
2cos ξ
2þexeyezsin ξ
2cos ξ
2eyezeyezexsin2ξ
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
ψþAðÞUψn¼02K2MKðÞ ð92Þ
Where UψnG32M2MKðÞ
composed of the eigenvectors
corresponding to the 2 MKsmaller eigenvalues of
ψ(R
Y
). Using the property (d), Eq. (92) is equivalent to
the following equation.
P2KPþ
2KψþAðÞUψn¼02K2MKðÞ ð93Þ
By means of the property (e), Eq. (93) can be further
expressed as
ψþAðÞPþ
2MP2MUψn¼02K2MKðÞ ð94Þ
Taking the multiplication on the left byP
2K
, we have
AþUYn¼0K2MKðÞ ð95Þ
Where UYn¼P2MUψnG3M2MKðÞ
is composed of the ei-
genvectors corresponding to the 2 MKsmaller 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).
Authorscontributions
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.
PublishersNote
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
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1.
2.
3.
4.
5.
6.
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... In view of the paucity of research on calculations with multi-vector, the Complex Representation Matrix (CRM) [20] is introduced because of the mature matrix theories. Consider a matrix A ∈ G m×n 3 , the CRM is defined by Ψ(A) ...
... Since e 2 123 = −1 and e 123 commutes with all elements in G 3 , one can identify it with the complex imaginary unit j [20], and so we can view Ψ(A) given in (8) as a complex matrix. ...
... In other words, there are two possible eigenvalues, namely the left and the right eigenvalue for G 3 matrix. In the proposed algorithm, the right eigenvalue is selected because the right ED of G 3 matrix can be converted to the right ED of its CRM [20]. ...
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Direction-of-arrival (DOA) estimation plays an important role in array signal processing, and the Estimating Signal Parameter via Rotational Invariance Techniques (ESPRIT) algorithm is one of the typical super resolution algorithms for direction finding in an electromagnetic vector-sensor (EMVS) array; however, existing ESPRIT algorithms treat the output of the EMVS array either as a “long vector”, which will inevitably lead to loss of the orthogonality of the signal components, or a quaternion matrix, which may result in some missing information. In this paper, we propose a novel ESPRIT algorithm based on Geometric Algebra (GA-ESPRIT) to estimate 2D-DOA with double parallel uniform linear arrays. The algorithm combines GA with the principle of ESPRIT, which models the multi-dimensional signals in a holistic way, and then the direction angles can be calculated by different GA matrix operations to keep the correlations among multiple components of the EMVS. Experimental results demonstrate that the proposed GA-ESPRIT algorithm is robust to model errors and achieves less time complexity and smaller memory requirements.
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Geometric algebra (GA) is an efficient tool to deal with hypercomplex processes due to its special data structure. In this article, we introduce the affine projection algorithm (APA) in the GA domain to provide fast convergence against hypercomplex colored signals. Following the principle of minimal disturbance and the orthogonal affine subspace theory, we formulate the criterion of designing the GA-APA as a constrained optimization problem, which can be solved by the method of Lagrange Multipliers. Then, the differentiation of the cost function is calculated using geometric calculus (the extension of GA to include differentiation) to get the update formula of the GA-APA. The stability of the algorithm is analyzed based on the mean-square deviation. To avoid ill-posed problems, the regularized GA-APA is also given in the following. The simulation results show that the proposed adaptive filters, in comparison with existing methods, achieve a better convergence performance under the condition of colored input signals.
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Due to the variable curvature of the conformal carrier, the pattern of each element has a different direction. The traditional method of analyzing the conformal array is to use the Euler rotation angle and its matrix representation. However, it is computationally demanding especially for irregular array structures. In this paper, we present a novel algorithm by combining the geometric algebra with Multiple Signal Classification (MUSIC), termed as GA-MUSIC, to solve the direction of arrival (DOA) for cylindrical conformal array. And on this basis, we derive the pattern and array manifold. Compared with the existing algorithms, our proposed one avoids the cumbersome matrix transformations and largely decreases the computational complexity. The simulation results verify the effectiveness of the proposed method.
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Non-circular properties of non-circular signals can be used to improve the performance of the direction-of-arrival (DOA) estimation. However, most ready-made algorithms are not applicable to the general case in which both non-circular and circular signals exist. In this paper, we present a novel DOA estimation algorithm for mixed signals, namely MS-MUSIC (Mixed Signals-Multiple Signals Classification), which can deal with the two kinds of signals simultaneously. And on this basis, we derive the Cramer-Rao Lower Bound (CRLB) of the azimuth and elevation estimation. The effectiveness of the algorithm is confirmed by the simulation results. Meanwhile, it acquires higher accuracy than the traditional algorithms.
Conference Paper
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This is the first book on geometric algebra that has been written especially for the computer science audience. When reading it, you should remember that geometric algebra is fundamentally simple, and fundamentally simplifying. That simplicity will not always be clear; precisely because it is so fundamental, it does basic things in a slightly different way and in a different notation. This requires your full attention, notably in the beginning, when we only seem to go over familiar things in a perhaps irritatingly different manner. The patterns we uncover, and the coordinate-free way in which we encode them, will all pay off in the end in generally applicable quantitative geometrical operators and constructions.
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This book aims to disseminate geometric algebra as a straightforward mathematical tool set for working with and understanding classical electromagnetic theory. It's target readership is anyone who has some knowledge of electromagnetic theory, predominantly ordinary scientists and engineers who use it in the course of their work, or postgraduate students and senior undergraduates who are seeking to broaden their knowledge and increase their understanding of the subject. It is assumed that the reader is not a mathematical specialist and is neither familiar with geometric algebra or its application to electromagnetic theory. The modern approach, geometric algebra, is the mathematical tool set we should all have started out with and once the reader has a grasp of the subject, he or she cannot fail to realize that traditional vector analysis is really awkward and even misleading by comparison. Professors can request a solutions manual by email: [email protected] /* */ © 2011 the Institute of Electrical and Electronics Engineers, Inc.
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The authors propose a novel direction-of-arrival (DOA) estimation method for signals that are uncorrelated, partially coherent or fully coherent in the presence of multipath propagation. First, uncorrelated and coherent signals are distinguished by the rotational invariance techniques as well as the property of the moduli of eigenvalues. The DOAs of the uncorrelated signals are then estimated based on their related eigenvalues. Finally, the singular value decomposition of virtual steering vectors is used to estimate the DOAs of the coherent signals while avoiding the cross-term effects. The effectiveness and efficiency of the proposed method are demonstrated by the simulation results.