2017 2nd International Conference on Software, Multimedia and Communication Engineering (SMCE 2017) ISBN: 978-1-60595-458-5
An Efficient Method for the Synthesis of Sparse Planar Antenna Arrays
Hai-ou SHEN
*and Bu-hong WANG
Information and Navigation College, Air Force Engineering University, Xi’an, China
*Corresponding author
Keywords: Sparse antenna arrays, Planar arrays, Pattern synthesis, 2-D unitary matrix pencil (UMP) method.
Abstract. Two-dimensional unitary matrix pencil (2-D UMP) method is utilized for the synthesis of sparse planar arrays in this paper. By virtue of the equivalent matrix pencil obtained with a unitary transformation, 2-D positions including horizontal and vertical coordinates are directly found as the generalized eigenvalues of the real matrix pencils without solving poles. Owing to the utilization of a unitary transformation, the computational complexity can be significantly reduced since only the real computations are involved in the singular value decomposition (SVD) and eigenvalue decomposition (EVD) procedures. A representative numerical experiment is provided to assess the effectiveness and advantages of the proposed method.
Introduction
Planar antenna arrays have many advantages in many applications that require a high resolution in both elevation and azimuth, such as biomedical imaging, satellite communications, radar tracking and remote sensing [1]. Compared with uniformly λ/2-spaced arrays, unequally spaced arrangements with a fewer number of elements bring several advantages in terms of simplifying antenna system and reducing mutual coupling effects. Therefore, the synthesis of sparse arrays with arbitrary desired patterns is a tough but necessary research related to finding the minimum number of antenna elements as well as their corresponding positions and excitations.
As for the synthesis of sparse planar arrays, some methods have been proposed in the literatures [2-5]. Some optimization algorithms have been successfully applied to optimize the element positions in order to achieve a better peak side lobe level performance [2-3]. But for extremely large apertures, stochastic methods are easily trapped in local optimal solutions because of associated non-linear and non-convex constraints. More recently, the non-iterative matrix pencil method capable of accurately estimating the synthesis parameters from a sum of complex exponentials has been introduced to design a sparse planar array [4]. However, this sparse planar array is designed to a separable planar array by a product of two orthogonal sparse linear arrays, which means the best features of this planar array only occur along two principal planes. Another recent approach named compressive sensing matching technique has been utilized to jointly optimize the element positions and excitations with a view to matching the user-defined radiated patterns [5]. Unfortunately, the optimal matching results cannot be easily obtained because multiple control parameters in this solver need to be optimized concurrently.
This paper is aimed at proposing an innovative and computationally efficient method for the design of sparse planar arrays. The outline of this paper is organized as follows. The mathematical formulation for the proposed synthesis method is introduced in Section II. The numerical analysis and assessment are presented in Section III. Finally, the main conclusions are drawn in Section IV.
Mathematical Formulation
0 1
( , ) exp( j ( ))
P
p p p
p
F u v w k x u y v
(1)where j 1,k0=2π/λ is the spatial wavenumber and λ is the wavelength. u=sinθcos𝜑, v=sinθsin𝜑,
where θ and 𝜑 are azimuth and elevation angles, respectively. xp and yp are the abscissa and ordinate of the pth element, wp is the corresponding excitation coefficient. The main objective is to synthesize a new planar array that has the minimum number of elements while radiating the same pattern as a prescribed reference pattern FREF(u,v) with a small tolerance ε. Therefore, the synthesis problem turns out to be that of finding the minimum element count Q (Q<P) as well as the element positions
(xq ,yq)and excitations wq (q=1,…,Q) for the optimal sparse array.
The array factor function is uniformly sampled from u=-1, v=-1 to u=1, v=1, and um=m∆1=m/M
(m=-M,-M+1,…,0,…,M), vn=n∆2=n/N (n=-N,-N+1,…,0,…,N). The sample values can be written as
1
( , ) ( , ) P
m n
m n p p p
p
f m n F u v w a b
(2)where ap=exp(jk0xp∆1) , bp=exp(jk0yp∆2). Set z(s, t)=fREF(s-M,t-N) (s=0,1,…,S-1; t=0,1,…,T-1; S=2M+1, T=2N+1), which are S×T sample values of FREF(u,v). Then an enhanced matrix ZENH can be constructed as follows:
(0) (1) ( - )
(1) (2) ( 1)
( 1) ( ) ( 1)
S K S K ENH
K K S
D E
Z Z Z
Z Z Z
Z =
Z Z Z
(3)
( )
( 1)
( , 0) ( ,1) ( , )
( ,1) ( , 2) ( , 1)
( , 1) ( , ) ( , 1)
0,1,..., 1 s
L T L
z s z s z s T L
z s z s z s T L
z s L z s L z s T
s S
（ ）
Z
(4)
where Z(s) is an Hankel matrix, D=KL and E=(S-K+1)(T-L+1). The choice of the parameters K and L should satisfy the following necessary conditions [6]
( 1)
( 1)
K L P
K L P
E P
(5) The 2-D UMP method is proposed here for the solution of the element positions and excitations. Exploiting a unitary transformation to construct an equivalent real matrix pencil, -tan(k x0 q 1 2) and
0 2
-tan(k yq 2) can be found as the generalized eigenvalues and of
2 1 1
4 3 2
{ } { }
{ } { }
I
I
U U F
U U F (6)
T
[ (1), (1 ),..., (1 ( 1) ), (2), (2 )...,
(2 ( 1) ),..., ( ), ( ),..., ( ( 1) )]
L K L L
K L L L L L K L
P p p p p p
p p p p (7) Let YA denote a A×A unitary matrix
1 T T 1 1 1 j 1 , =2 j 2 j 1
2 , 2 1
2
j
C C
A
C C A A
C C C
A C C
C C C
A A A C -A C - I I
I 0 I
0 0 0 Y Y (8)
where 0 is a C×1 vector whose elements are zero, IC is a C×C identity matrix and ∏C denotes a C×C exchange matrix
0 0 1
0 1 0
1 0 0
C
C C
(9)
H 2
RE D CH E
Z = Y Z Y is the real matrix obtained by the multiplication of the unitary matrices and the centro-Hermitian matrix ZCH = Z[ ENH DZENHE D] 2E . X = Y1 D LH J Y1 D and X = Y2 D KH J Y3 D , where J1 and J3 are called selection matrices and used to select the required rows of a matrix.
1[I(D L ) 0(D L ) L](D L ) D 3 [I(D K ) 0(D K ) K](D K ) D
J J (10) The elements excitations wq can be computed as follows once (xq ,yq) are obtained.
1 2
H 1 H H H 1
1 1 1 2 2 2
diag( , ,..., )
( ) ( )
Q
ENH
w w w
W
E E E Z E E E
(11)
Where
T 1 T 1 T T
1 1 1 1
1
2 2 2 2
[ ] [( ) , ( ) ,..., ( ) ]
[ ] [ , ,..., ]
M M M A
D Q
M M M S A
Q E
E B A B A B A
E A B A B A B
(12)
And
1 2
1 2
1 1 1
1 2
1
1 1 1
1 2
1
1 1 1
1
2 2 2
2
diag( , ,..., )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Q
N N N
Q
N N N
Q
N L N L N L
Q L Q
N N N T L
N N N T L
Q
a a a
b b b
b b b
b b b
b b b
b b b
b A B B 1 ( 1) ( ) ( )
N N N T L
Q Q Q T L
b b
Numerical Analysis and Assessment
In this section, a representative experiments are presented to assess the performances of the proposed method for the synthesis of a sparse linear array. Let us take a 12×12 uniform Chebyshev array with λ/2 element spacing (PSLL=-20 dB) as reference.The aperture size is 5.5λ×5.5λ. Figure 1 shows the patterns radiated by using the reference array and the reconstructed array. The cross sections of the patterns are represented in Figure 2. It can be observed that the reference pattern is carefully approximated by the 2-D UMP method, and only minor deviations from the reference appear in the far sidelobe region. The distributions of the excitations are shown in Figure 3. It turns out that the 2-D UMP can achieve a perfect reconstruction of this uniform planar array with a minor matching error of 1.5712×10-4 and an element saving of 50% as compared with the corresponding uniform array.
-1 -0.5
0 0.5
1
-1 -0.5 0 0.5 1 -50 -40 -30 -20 -10 0
u v
N
o
rm
a
lize
d
va
lu
e
(
d
B
)
-1 -0.5
0 0.5 1 -1
-0.5 0 0.5 1 -50 -40 -30 -20 -10 0
u v
N
o
rm
a
lize
d
va
lu
e
(
d
B
)
[image:4.595.96.488.227.359.2](a) (b)
Figure 1. Power patterns of (a) the reference array and (b) the 2-D UMP solution.
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -50
-40 -30 -20 -10 0
u
N
o
rm
a
lize
d
va
lu
e
(
d
B
[image:4.595.205.377.393.513.2]) Reference 2D-UMP
Figure 2. Cross section of the patterns.
-2 -1
0 1
2
-2 -1 0 1 2 0 0.2 0.4 0.6 0.8 1
x() y()
N
o
rm
a
lize
d
e
xci
ta
tio
n
-2 -1
0 1
2
-2 -1 0 1 2 0 0.2 0.4 0.6 0.8 1
x() y()
N
o
rm
a
lize
d
e
xci
ta
tio
n
(a) (b)
Figure 3. Excitations of (a) the reference array and (b) the 2-D UMP solution.
Summary
procedures. A representative numerical experiment has been presented to prove that the number of antenna elements is saved by 50% while providing an excellent pattern matching. In our future works, the further researches will be focused on analyzing the mutual coupling effects on realistic antenna elements.
Acknowledgement
This work was supported by the National Natural Science Foundation of China under Grant 61172148 and 61671465. The authors would like to thank the associate editor and the anonymous reviewers for their helpful comments and suggestions.
References
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[2] B. V. Ha, R. E. Zich, M. Mussetta, P. Pirinoli, Synthesis of sparse planar array using modified Bayesian optimization algorithm. International Conference on Electromagnetics in Advanced Applications (ICEAA). (2013) 1541-1543.
[3] K. Chen, X. Yun, Z. He, C. Han, Synthesis of sparse planar arrays using modified real genetic algorithms. IEEE Transactions on Antenna and Propagation. 55(4) (2007) 1067-1073.
[4] K. Yang, Z. Zhao, Y. Liu, Synthesis of sparse planar arrays with matrix pencil method. International Conference on Computational Problem Solving (ICCP). (2011) 82-85.
[5] F. Viani, G. Oliveri, A. Massa, Compressive sensing pattern matching techniques for synthesizing planar sparse arrays. IEEE Transactions on Antenna and Propagation. 61(9) (2013) 4577-4587.
