Density and Element-Size Tapering
for the Design of Arrays with a Reduced Number
of Control Points and High Efficiency
O. M. Bucci*, T. Isernia+, A. F. Morabito+, S. Perna#, D. Pinchera§
*
DIBET, Università Federico II di Napoli, I-80125 Napoli, Italy, and IREA-CNR
[email protected]
+
DIMET, Università Mediterranea di Reggio Calabria, I-89100 Reggio Calabria, Italy
{tommaso.isernia, andrea.morabito}@unirc.it
#
DiT, Università Parthenope di Napoli, I-80133 Napoli, Italy
[email protected]
§
DAEIMI, Università di Cassino, I-03043 Cassino (FR), Italy, and CNIT
[email protected]
Abstract— In the design of Direct Radiating Arrays for
satellite communications, where the constraints are
generally given in terms of a directivity mask, solutions
using a reduced number of uniformly excited radiating
elements are particularly attractive because they allow to
reduce the array cost and weight and maximize the
amplifiers efficiency. In this communication, an innovative
architecture exploiting both a density and an element-size
tapering is presented, as well as two innovative and
computationally effective synthesis strategies. Examples
demonstrating the effectiveness of both the proposed
architectures and of the relative synthesis techniques are
also provided, with reference to an actual problem.
I. INTRODUCTION
In a number of applications, including the synthesis
of Direct Radiating Arrays (DRA) for transmission from
satellites [1,2], one is interested in achieving a directive
behavior of the overall array while reducing as much as
possible the number of amplifiers and phase shifters (i.e.,
control points). Moreover, in order to optimize the
efficiency of the amplifiers, the different radiating
elements should be fed with the same amplitude
(isophoric array).
At present time, in the open literature essentially
three different kinds of architectures have been
considered to address such a problem, i.e.: sparse arrays
[1,3] (i.e., arrays whose uniformly excited elements are
properly located onto a non-regular grid), thinned arrays
[1] (wherein, starting from an otherwise regular array,
the required performances are achieved by properly
withdrawing a certain number of elements), and
‘clustered’ arrays [4] (wherein the overall array is
subdivided into a number of possibly different uniformly
excited subarrays). In all of these architectures, when
dealing with satellites applications, a relatively large
spacing (and hence large radiating elements) can be used
by virtue of the fact that grating or pseudo-grating lobes
will be outside of the Earth cone as seen from the
satellite.
In the first two architectures (i.e., sparse and thinned)
if a single kind of radiating element is adopted while
strict requirements are enforced on the sidelobe level, the
required density tapering on the array aperture will
induce a low aperture efficiency. Such a circumstance
affects, of course, the possibility of getting a high gain,
which is instead a primary goal. On the other side, the
third solution (as well as the second one) is more subject
to the raising of grating lobes if a standard (regular) grid
is assumed before clustering (or thinning). In order to
avoid such a problem, an architecture based on the
exploitation of aperiodic tilings of the plane coupled
with a suitable clustering of the elements has been
recently proposed in [5], which exploits all the available
aperture while avoiding as much as possible the grating
lobes induced from the underlying grid.
In this paper we present a different approach aimed at
achieving the same goals.
It exploits in a joint fashion both a density taper and
a size tapering of the elements. By doing so, one keeps
relatively insensitive to the raising of grating lobes while
exploiting the available aperture much better than in
standard uniformly excited sparse arrays solutions. In
fact, in the synthesis of usual isophoric sparse arrays, the
tapering which is required on the source in order to get
the required far field shaping is realized by means of a
density taper of the (identical) radiating elements [1,3].
If low sidelobes are required, this can induce large
differences in the spacing between the array elements,
leading to a poor illumination efficiency.
A suitable way to overcome this problem is that of
allowing the presence of two or even more different
radiating elements in such a way to guarantee a better
filling of the aperture while still using a nonuniform
reference grid.
To catch the idea, one can assume that the single
elements are aperture antennas. Then, two apertures
carrying the same overall power but having different
dimensions will produce different fields having different
levels on their aperture. Consequently, a smart
positioning of elements of two or more different kinds
can allow to fill the aperture while realizing the desired
tapering on the overall array aperture.
II. MAIN RATIONALE OF THE SYNTHESIS PROCEDURES
Based on the considerations above, as well as on
additional skills available, two different fast approaches
have been developed in the framework of an activity
funded by the European Space Agency. In both cases, a
concentric ring array structure is basically looked for.
Also, square radiating elements (of two different sizes)
have been preferred to circular ones in order to hopefully
achieve a better filling of the aperture for a given number
of elements.
The first approach [6] is completely deterministic in
nature, and consists in three different steps.
In the first step, following [7], a circularly
symmetric continuous source fulfilling ‘at best’ the
required directivity and sidelobes constraints is
synthesized. Such an aperture distribution acts both as a
reference and a target for the subsequent steps.
In the second step, a density taper technique is
adopted to compute the values of the radii wherein the
elements will be located as well as the number of
elements per ring. The procedure is an extension to the
2D circularly symmetric case of that presented in [3].
In the third step, the two different kinds of element
are located along the rings according to the criterion of
filling as much as possible the available aperture. As a
consequence, the larger elements will be inserted in
those rings wherein a larger radial spacing is allowed.
Such a deterministic approach shows the attractive
features of being very efficient: it allows generating in a
few seconds layouts of hundreds of elements. In
addition, the overall procedure may take advantage from
the fact that different aperture fields may fulfill the given
radiation constraints [7], and some of them more easily
lend themselves to a discretization into a ring array.
Finally, such a deterministic approach can easily deal
also with geometries different from the concentric ring
array one. In particular, it allows retaining the concentric
ring array structure only in the external part of the
available area, whereas in the most internal part it allows
locating the elements over a regular grid, thus increasing
the aperture efficiency.
In the second approach [8], which can also
explicitly take into account the element patterns, a
hybrid strategy similar in spirit to that proposed in [9] is
adopted.
In particular, the approach is based on the idea of
looking first for an array of continuous ring sources, with
unknown radii and amplitudes. The radii are determined
by a global optimization procedure, while the amplitudes
are determined, for each set of radii, by solving a convex
programming problem. Once the radii and amplitudes of
the ring sources have been determined, the number of
elements on each ring (and hence their location) are
determined on the basis of the available space from one
side, and from the (ring) excitation to be realized from
the other.
III. RESULTS
In order to test the interest of the proposed
architecture and the effectiveness of the devised
procedures, the design goals of [2], as summarized in the
following, have been considered:
D    43.8 dBi
for   0
D    D 0   20 dB
for 1    2
D    D 0   10 dB
for   2
(1)
where D(∙) is the array directivity, 0=0.325° represents
the angle at the Edge Of Coverage (EOC) zone,
1=0.795° marks the EOC of the nearest “iso-color”
(i.e., with the same frequency and polarization) beam,
2=16° represents the (maximum) inside Earth angle,
being the elevation coordinate in a spherical
coordinate system centered at boresight. In addition, a
maximum source of radius 60 λ (being λ the wavelength
in free space) is required, as well as a scanning angle of
s=1.12°.
Two kinds of isophoric uniform square feeds of 4.26
 and 8.26  side, respectively, have been assumed as
radiating elements. These particular sizes have been
chosen according to the sizes of the actual feeds that
could be used for the final implementation of the
suggested solutions, as discussed in the following [2].
Different layouts have been obtained by means of
both the presented approaches.
In particular, the first approach has been applied to
different continuous reference sources and different core
geometries, whereas in the second approach various
layouts having a different number of rings and thus a
different number of overall feeds have been considered.
From the obtained results it turns out, first of all, that
the third constraint in (1) can be easily satisfied; more
critical are instead the two other constraints.
As far as these two latter constraints are concerned,
the best performances have been achieved for those
layouts which employ larger numbers of large feeds, that
is, those layouts with three rings of large feeds. This
circumstance appears very attractive because it allows, at
the same time, reducing the total number of feeds (with
respect to the layouts that present one or two rings with
IV. CONCLUSIONS
Two approaches for the synthesis of equi-amplitude
planar arrays have been presented.
The first approach is completely deterministic,
whereas in the second approach a hybrid strategy,
involving the solution of convex and non-convex
optimization problems with a low number of unknowns,
is adopted. Both the proposed methods provide a proper
density tapering in order to get the required far field
shaping, but they also allow one to exploit at best the
available aperture by carrying out an element-size
tapering of the radiating elements.
Central Beam
SLL on Earth [dB]
Scanned Beam
EOC Directivity [dBi]
Scanned Beam
SLL on Earth [dB]
Number of feeds
TABLE I
S UMMARY OF THE ACHIEVED PERFORMANCES
Central Beam
EOC Directivity [dBi]
larger feeds). This can be clearly observed in Fig. 1,
where the performances of a layout obtained via the
second approach is presented. Employment of only 145
elements allows satisfying all the constraints listed in (1)
with a margin on the directivity of 0.9dB on the central
beam, to deal with the aperture and ohmic losses of the
feeds. Better performances can be reached by employing
a larger number of feeds, as in the layout shown in Fig.2,
where 210 elements are employed (see again Table I),
and a margin of 1.7dB respect to the constraint on the
directivity of the central beam is achieved.
Of course, above listed performances degrade when
we pass to the analysis of the scanned beam, but the
constraints listed in (1) are satisfied by the same layouts
also for the scanned beam. In particular, it has been
observed that the scanning process introduces losses in
the EOC of the order of 0.1÷0.3 dB (which checks with
scanning losses due to the element pattern, which, in
turn, is related to the element size) and a severe
degradation on the SLL.
Better performances for the SLL of the scanned beam
can be achieved by employing geometries different from
those shown in Figs.1 and 2. To show this, we present in
Figs. 3 and 4 two layouts obtained via the first approach.
In both the cases a smaller number of rings with large
feeds has been employed. In particular, in Fig.3 it is
shown a 200 elements layout characterized by only one
ring of large feeds, whereas in Fig.4 it is shown a 234
elements layout characterized by two rings of large
feeds. In both the cases the EOC directivity for the
central beam is slightly better than those achieved with
the layout of Fig. 1 (see again Table I), which is
characterized by a much smaller number of feeds. On the
other side, the SLL constraints for the scanned beam are
satisfied by both the solutions of Figs. 3 and 4 with a
greater margin respect to the array geometries of Figs.1
and 2 (see again Table I). In particular it is interesting
the comparison between the layouts of Figs.2 and 3,
which are characterized by a similar number of feeds.
The layout 2 allows achieving higher EOC directivities
for both the central and the scanned beam, at expenses of
the SLL, which, especially for the scanned beam, is
better for the layout 3.
Layout 1
44.7
-21.1
44.4
-20.0
145
Layout 2
45.5
-21.3
45.3
-20.1
210
Layout 3
45.0
-21.4
44.9
-21.0
200
Layout 4
45.2
-24.8
45.00
-21.9
234
Fig. 1 Layout 1 (second approach): the geometry of the array and the
ϕ-cuts for the directivity.
The first approach can easily deal also with
geometries different from the concentric ring array one.
On the other side, the second approach, as compared
with the first one, allows to directly taking into account
the pattern of the employed feeds into the optimization
process. It has been shown that both the proposed
methods provide layouts characterized by directivity
patterns that satisfy realistic satellite project
requirements. In particular, by making use of uniform
square aperture with isophoric uniform square feeds of
4.26 λ and 8.26 λ side, a number of about 150 elements
disposed over a concentric has been found to be
sufficient to fulfill the listed constraints.
Fig. 2 Layout 2 (second approach): the geometry of the array and the
ϕ-cuts for the directivity.
Fig.4 Layout 4 (first approach): the geometry of the array and the ϕcuts for the directivity.
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Fig.3 Layout 3 (first approach): the geometry of the array and the ϕcuts for the directivity.
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