Progress In Electromagnetics Research B, Vol. 23, 215–228, 2010
ROBUST ADAPTIVE BEAMFORMER USING INTERPO-
LATION TECHNIQUE FOR CONFORMAL ANTENNA
ARRAY
P. Yang, F. Yang, and Z. P. Nie
School of Electronic Engineering
University of Electronic Science and Technology of China (UESTC)
Chengdu 610054, China
B. Li and X. F. Tang
Institute of Electronic Engineering
China Academy of Engineering and Physics
Mianyang 621900, China
Abstract—A novel robust adaptive beamforming method for
conformal array is proposed. By using interpolation technique,
the cylindrical conformal array with directional antenna elements is
transformed to a virtual uniform linear array with omni-directional
elements. This method can compensate the amplitude and mutual
coupling errors as well as desired signal point errors of the conformal
array efficiently. It is a universal method and can be applied to other
curved conformal arrays. After the transformation, most of the existing
adaptive beamforming algorithms can be applied to conformal array
directly. The efficiency of the proposed scheme is assessed through
numerical simulations.
1. INTRODUCTION
Adaptive beamforming with conformal antenna arrays are of interest
for future communication and defense applications [1, 2]. In ideal case,
adaptive arrays can suppress the interference signals and noise while
efficiently keeping the interest signals. However, in practice, there
are all kinds of errors in the system, such as elements amplitude and
phase errors, elements location errors, mutual coupling errors and
Received 15 June 2010, Accepted 14 July 2010, Scheduled 23 July 2010
Corresponding author: P. Yang (yangp001@tom.com).
216 Yang et al.
desired signal point errors, etc. Most traditional adaptive beamforming
algorithms are sensitive to the environment. They often give poor
performance when these errors are taken into consideration [3]. In
past decades, many approaches have been proposed to improve the
robustness. One of the efficient ways among them is to recast the
sample covariance matrix, such as diagonal loading [4, 5]. How to
choose the diagonal loading level based on the information of the
uncertainty of the array steering vector is an open problem. Recently,
many automatic diagonal loading approaches have been developed [6–
10]. These methods have high performance and can be seen as robust
adaptive beamformers.
Compared to uniform linear array (ULA), when conformal array
with a small radius of curvature is used for adaptive signal processing
or beamforming, some important problems must be considered. First,
conformal array has “shadow effect” due to the metallic platforms. It
means that an incident wave comes from a special angle, but not all
of the antenna elements can receive this signal. Second, the radiation
patterns of conformal antennas are always directional. Because each
element has a different normal direction, the maximum radiation point
is different. Hence, for any incident wave, different elements have
different responses. Third, the mutual coupling between the elements
cannot be ignored. It becomes more complex in the situation of
conformal array due to the effects of the platform. It can only be
analyzed by using some numerical methods, such as finite element
method (FEM), finite differential time domain method (FDTD) and
method of moment (MOM). These errors will seriously affect the
performance of traditional beamformers if we do not compensate them.
Unfortunately, most of the existing robust algorithms are based on
the ideal case (i.e., ULA with omni-directional elements, without
mutual coupling). Hence conventional algorithms often have a poor
performance on conformal arrays and can hardly be applied directly.
In order to overcome the problems mentioned above, we propose
a robust, interpolation based adaptive beamforming method to
transform a cylindrical conformal array with directional antenna
elements to a virtual ULA with omni-directional elements. The
transformation can be seen as a procedure of array optimization. In
the process, the mutual coupling effect is taken into consideration.
After the transformation, the conformal array has the character of
ideal ULA. Hence, all of the errors can be suppressed effectively.
Moreover, most existing adaptive beamforming algorithms that can
just be used to ULA can be applied to conformal array after the
transformation. This method is simple and easy for implementation.
It is a universal method and independent of the array configurations.
Progress In Electromagnetics Research B, Vol. 23, 2010 217
Simulation results show that the proposed method achieves a higher
performance in conformal array than conventional methods. This
paper is organized as follows: In Section 2, cylindrical conformal array
beamforming based on interpolation method is proposed in detail. In
Section 3, the performance of the proposed method and conventional
methods are compared by some simulation examples. The new method
is concluded in Section 4.
2. CYLINDRICAL CONFORMAL ARRAY
BEAMFORMING BASED ON INTERPOLATION
METHOD
It has been demonstrated that uniform circular arrays (UCA) have
better performance, especially in azimuth-plane beamforming, than
uniform rectangular arrays (URA). Most UCAs use dipole as the
elements. Dipole antenna has an omni-directional radiation pattern
at φ direction. But for conformal arrays, we cannot use dipole as
the element because of the metallic platform. Microstrip antenna is
a good choice for conformal array elements due to its low profile and
unidirectional radiation characteristics.
2.1. Cylindrical Conformal Array
Consider a cylindrical conformal array with M identical microstrip
antennas. As shown in Fig. 1, these antennas are mounted uniformly
on the surface of the cylinder. The radius of the cylinder is r. At time t,
P (P < M) narrowband signals with azimuth angle φ
i
(i = 1, 2, ..., P )
impinge on the conformal array. The M × 1 receiving data vector of
Figure 1. Cylindrical conformal array with directional antenna
elements.
218 Yang et al.
the array is given by
x(t) = C[F(φ) · A(φ)S(t)] + n(t) (1)
where S(t) (including the desired signal and the interferences) is the
P × 1 vector whose ith element denotes the ith signal. A is the M × P
steering matrix, whose ith (i = 1, 2, . . . , P ) column is the steering
vector of the ith signal.
A = [a(φ
1
), a(φ
2
), · · · , a(φ
P
)] (2)
a
m
(φ
i
) = e
jkr cos(φ
i
−β
m
)
, (m = 1, 2, . . . , M ) (3)
where β
m
= 2π(m − 1)/M and k is the wave number. The M × 1
vector n(t) represents additive white noise. F is the M × P radiation
pattern matrix whose m, ith elements denote the response of the mth
(m = 1, 2, . . . , M ) antenna to the ith (i = 1, 2, . . . , P ) signal.
F = [f (φ
1
), f(φ
2
), . . . , f(φ
P
)] (4)
f(φ
i
) = [f (φ
i
− β
1
), f(φ
i
− β
2
), ..., f(φ
i
− β
M
)]
T
(5)
where f(φ
i
− β
m
) is the radiation pattern of the antenna element. (·)
T
denotes the transpose. If the elements are point sources, f(φ
i
− β
m
) =
1. From Fig. 1, we can see that for the conformal array, different
elements have different responses to the signal. Assume that the
direction of arrive (DOA) of the signal is φ = 90
◦
, then element 5
has the strongest response, while elements 1 and 9 can hardly receive
this signal. Of course those elements behind the cylinder (element 10
to M) have zero response. C is the M ×M circular symmetrical matrix
[11–14] which represents the mutual coupling of the array
C = Z
L
(Z + Z
L
I)
−1
(6)
where Z
L
is the load impedance (usually 50 ohms), and Z is the M ×M
mutual impedance matrix. I is the M × M unit matrix. Once Z is
known, the mutual coupling C of the array can be determined by (6).
The impedance matrix is difficult to determine analytically, but easy
to get through S parameter measurement or numerical methods. Also,
the radiation pattern matrix F can be determined in the same way.
The minimum variance distortionless response (MVDR) beam-
former is to maintain the distortionless response to the desired signal
while minimizing the output power
min
w
w
H
Rw subject to w
H
a(φ
0
) = 1 (7)
where w is the M × 1 beamformer weight vector, and (·)
H
denotes
the conjugate transpose. R = E{x(t)x(t)
H
} is the array covariance
matrix, and E{·} denotes the statistical expectation. a(φ
0
) is the
Progress In Electromagnetics Research B, Vol. 23, 2010 219
steering vector of desired signal with the form of Equation (3), and φ
0
is the desired signal’s DOA. The solution of this problem in the finite
sample case is so called sample matrix inverse (SMI) beamformer and
given by
w = µ
−1
ˆ
R
−1
a(φ
0
) (8)
where µ = a(φ
0
)
H
ˆ
R
−1
a(φ
0
) is a constant;
ˆ
R =
K
P
j=1
x(t)x(t)
H
/K is the
sample estimate of R; K is the number of snapshots, respectively. The
output signal of the cylindrical conformal array can be written as
y(t) = w
H
x(t) (9)
It is clear to see from Equation (8) that the effects of mutual
coupling and directional radiation pattern have been included in
sample covariance matrix
ˆ
R but not in the static steering vector a(φ
0
).
If we use Equation (8) as the weight vector of conformal array without
any compensation, big errors will be introduced. These errors will
change not only the depth of the nulls but also their locations, which
will result in a poor performance to the beamformer. These errors
can be compensated by recasting Equation (8). Here we propose an
alternative method by using the interpolation technique.
2.2. Interpolation Technique for Robust Adaptive
Beamforming
Interpolation technique has been widely used for DOA estimation [15–
18] but rare for adaptive beamforming [19, 20]. The main idea of
interpolation method is dividing the field of view of the array into
L sectors. The size of the sectors depends on the array geometry and
desired accuracy. For example, there is a signal, whose DOA is in
the sector Φ (Φ ∈ [φ
1
, φ
2
]), where φ
1
and φ
2
are the left and right
boundaries of this sector. Let ∆φ as the interpolation step, then Φ
can be represented as
Φ = [φ
1
, φ
1
+ ∆φ, φ
1
+ 2∆φ, · · · , φ
1
+ n∆φ, φ
2
] (10)
the ∆φ is determined by the desired accuracy. In this sector, the real
array manifold is
CF·A = C[f (φ
1
)·a(φ
1
), f(φ
1
+∆φ)·a(φ
1
+∆φ), · · · , f(φ
2
)·a(φ
2
)] (11)
In order to transform the real conformal array to a virtual ULA with
omni-directional elements, we can construct a virtual ULA in the same
sector Φ, whose steering matrix is
¯
A = [¯a(φ
1
), ¯a(φ
1
+ ∆φ), · · · , ¯a(φ
1
+ n∆φ), ¯a(φ
2
)] (12)
