Low Sidelobes in Array Antennas of Equal Sub-arrays
through Element Phase Control
P. Rocca* (1) and R. L. Haupt (2)
(1) ELEDIA Group - Department of Information Engineering and Computer
Science, University of Trento, Via Sommarive, 14,1-38050 Trento, Italy
(2) Applied Research Laboratory - The Pennsylvania State University, State
College, PA 16801
E-mail: paolo.rocca@disLunitn.it, Web-page: http://www.eledia.ing.unitn.it
Introduction
In the last few decades, many techniques have been proposed for the synthesis of
large array antennas with simplified feed networks. In this framework of radar
applications, the use of spatially contiguous sub-arrays of equal size allows the
reduction of the number of antenna control elements [1], as well as to simplify the
circuit complexity of the feed network. Despite the simplification of the antenna
manufacturing and the costs saving, sub-arraying does not guarantee suitable
radiation performances because of the usually limited control of the radiated
pattern. As a matter of fact, unwanted high secondary lobes are generated in the
far field pattern due to the periodic quantization of the element excitations. In
order to improve the performance of the antenna, while maintaining an easy
design of the array, various approaches have been presented. For instance, let us
consider partially overlapped sub-arrays [2], the combination of tapered element
excitations and sub-array excitations [3] or sub-arrays of random sizes [4]. More
recently, a hybrid approach has been proposed for the joint optimization of both
weights and dimensions of sub-arrays with unequal sizes [5].
To the best of the authors' knowledge and unlike to state-of-the-art strategies, this
work proposes a phase only control strategy starting from the key observation that
in several applications each element of the array has a phase shifter for scanning
the beam towards a desired direction. Therefore it turns profitable to use phase
control for lowering the sidelobes of the radiated pattern to break the periodicity
of the aperture illumination resulting from sub-array weighting and without extra
costs. As a matter of fact, one of the main advantages of such a technique lies in
the possibility to optimize the beam pattern without additional hardware. More
specifically, the optimization of the static mode configuration (broadside array) is
performed to minimize the pattern sidelobes by means of the iterative projection
method (IPM) [6] by considering an array configuration with uniform and equal
sub-arrays having fixed amplitude weights. The IPM is an iterative algorithm in
which the array illumination function is projected onto the corresponding space of
far field patterns and then vice versa, until a desired pattern is synthesized.
978-1-4244-3647-7/09/$25.00 ©2009 IEEE
Mathematical Formulation
Let us consider a linear array of Q spatially-contiguous sub-arrays of equal sizes,
each containing M q elements disposed along the x -axis. The sub-arrays have
fixed amplitude weights wq , q =1,..., Q and the array elements are provided with
phase shifters of values fPn' n = 1,..., N. For the sake of generality, the
N
=Q x M q
elements are isotropic and have the array factor expressed as
f(13) = LLwq £5'nq e J(kndSint9+qJn)
Q N
(1)
q=1 n=1
where k = 2; is the background wavenumber, d is the inter-element spacing, 13
identifies the generic angular direction. Moreover, 8nq
belongs to the q-th sub-array, and 8nq
=1
if the n-th element
= 0 otherwise.
As regards to the optimization of (1), the values fPn' n = 1, ..., N are properly
computed to lower the sidelobe level (SLL) by means of the Iterative Projection
Method [6]. The phase shifter can assume any real value in the range
f{J~E l-180o;180oJ, n=I, ...,N (i being the iteration index of the iterative
fi (13) generated through the optimization
procedure are subject to condition LB( 13) :::; (13 ~ :::; UB( 13), LB( 13) and UB( 13)
optimization), and the patterns
Ip
being amplitude lower and upper bounds, respectively.
The sub-array amplitude weights are fixed to wq
element phases are set to fP~
= 0, n = 1,..., N
= wq ,
q =1,..., Q, while the
at the IPM initialization (Le., i
= 0).
The far field pattern fi(13) is then generated through the Fourier transformation
of the array illumination. Secondly, the pattern is projected onto the admissible
space by constraining fi (13) to LB(13) when fi (13) < LB(13) and to UB(13) when
fi(13) > UB(13) , respectively. Starting from the arising projected field, f;,(13), the
value of the cost function in correspondence with actual solution is computed as
'I'(f{J~,n = 1,...,N)=
Illp (13~ -If;r(13~
2
11
•
(2)
The iteration index is updated (Le., i = i + 1) and a new set of element excitations
is computed. The phase values are updated by performing an inverse Fourier
transformation of f;;1 (13) while the amplitude coefficients are set to w~ = wq ,
q =1,..., Q. The procedure is iterated until the condition
'I'(f{J~,n =1,...,N)< 1] is
satisfied, 1] being a user defined threshold, or a maximum number of user-defined
iterations is achieved.
0.9
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64
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80
88
96
104
112
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64
Element Index, n
72
80
88
96
104
112
120
128
Element Index, n
(a) Fixed amplitude coefficients.
(b) Optimized phase values.
Figure 1 - Array element excitations.
Numerical Results
For illustrative purposes, let us consider the following representative test case
concerned with a uniform linear array characterized by Q = 8 sub-arrays, each
= 16 elements, such that the array aperture has N = 128 elements
equally spaced of d =
The array is supposed to be symmetric with respect to
one having M q
+.
the center of the antenna and only half elements are considered in the synthesis
procedure. The sub-array amplitude weights are chosen sampling the Taylor
distribution with SLL = -30dB and n = 4 [7]. Accordingly, the element
amplitude coefficients tum out to be those shown in Fig. 1(a) for half of the array.
The IPM-optimized phase values are shown in Fig. l(b), while the corresponding
pattern is reported in Fig. 2(a) superimposed on the pattern with only amplitude
sub-array weighting. It is worth noting that the SLL reduction is almost 4dB.
Moreover, both the directivity D along the boresight direction as well as the main
lobe beamwidth, BW , are close to the case when no-phase control (NPC) is used.
As a matter of fact, it is D 1PM = 20.1dB vs. D NPC = 20.3dB and BW IPM = 1.05
= 1.01 [deg], respectively. Finally, the beam is steered towards
the direction tJ = 30°. For completeness, Figure 2(b) shows both the broadside
[deg] vs. BW NPC
(static) pattern and the steered pattern.
Conclusions
In this work, a procedure for reducing the level of the sidelobes of large array
antennas made of equal contiguous sub-arrays has been presented. Starting from
the assumption that phase shifters are already present at each array element to
steer the beam towards a desired direction, a phase control strategy based on the
IPM has been profitably exploited to avoid the presence of high secondary lobes
without the need of any additional hardware component. The results show the
efficiency of the proposed approach.
No Phase 0 ptirnization -
Relative Power Pattern [dB]
0
0
30~ ...-·······-
..-.....-.• _30 0
L...--.....!.-----'-----'----'-----'------L..-----'---""':""""'"'If'P"
""""'"+=---+---+---+---+---+---+---+---+---I - 90
0
(a) Power patterns obtained with the IPM and without phase control.
Broadside Beam -
Relative Power Pattern [dB]
0
0
__ .-_...0. _..-.. --._._.._~.
30~ --······-
.............\
","..••• ,,_0'
/
••••••
//
"-'
\ . .-
.
,._0"'.....
-
::::\..-.,.
'.
\'.'\
....,.,.•..•.
(b) Broadside and steered power patterns.
Figure 2.
References
[1] R. J. Mailloux, Phased Array Antenna Handbook. Artech Hause, Boston, MA,
2005.
[2] R. J. Mailloux, "A low-sidelobe partially overlapped constrained feed network
for time-delayed subarrays," IEEE Trans. Antennas Propag., vol. 49, no. 2,
pp. 280-29, Feb. 2001.
[3] R. L. Haupt, "Reducing grating lobes due to subarray amplitude tapering,"
IEEE Trans. Antennas Propag., vol. 33, no. 8, pp. 846-850, Aug. 1985.
[4] A. P. Goffer, M. Kam, and P. Herczfeld, "Design of phased arrays in terms of
random subarrays," IEEE Trans. Antennas Propag., vol. 42, no. 6, pp. 820826, Jun. 1994.
[5] R. L. Haupt, "Optimized weighting of uniform subarrays of unequal sizes,"
IEEE Trans. Antennas Propag., vol. 55, no. 4, pp. 1207-1210, Apr. 2007.
[6] O. M. Bucci, G. Franceschetti, G. Mazzarella, and G. Panariello, "Intersection
approach to array pattern synthesis," Proc. lEE, pt. H, vol. 137, no. 6, pp.
349-357, Dec. 1990.
[7] R. S. Elliott, Antenna Theory and Design. Wiley-Interscience, IEEE Press,
2003.