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A deterministic procedure to design a nonperiodic planar array radiating a rotationally symmetric pencil beam pattern with an adjustable sidelobe level is proposed. The elements positions are derived by modifying the peculiar locations of the sunflower seeds in such a way that the corresponding spatial density fits a Taylor amplitude tapering law which guarantees the pattern requirements in terms of beamwidth and sidelobe level. Different configurations, based on a Voronoi cell spatial tessellation of the radiative aperture, are presented, having as a benchmark the requirements for a typical multibeam satellite antenna.

Communication satellites use multiple beam antennas providing downlink and uplink coverages over a field of view for high data rate, multimedia, or mobile personal communication applications. High gain, multiple overlapping spot beams, using both frequency and polarization reuse, provide the needed coverage. In order to generate high gain spot beams, electrically large antenna apertures are required. These apertures may be generated by either reflectors or phased arrays. Phased arrays would be a natural choice to generate multiple beams but up to now the poor efficiency, the high cost, and the deployment complexity of active arrays have been their main drawbacks, limiting their use onboard satellites. These drawbacks are mainly due to the required distributed and tapered power amplification which is inducing poor power efficiency.

Aperiodic arrays with equiamplitude elements permit to
mitigate these limitations and represent a valid alternative to traditional
periodic phased arrays with amplitude tapering. Resorting to aperiodic arrays
with equiamplitude fed elements is particularly effective for the design of
large arrays working in transmission. This
type of antenna architecture is considered extremely promising for achieving a multibeam
coverage on the Earth from a geostationary satellite [

Unequally spaced arrays have several interesting
characteristics and may offer some potential advantages with respect to periodic
arrays [

In terms of limitations, nonperiodic arrays exhibit a reduced aperture efficiency when identical, non-equispaced elements are used. As a consequence, a reduced maximum equivalent isotropically radiated power (EIRP) is obtained if not compensated by an increase of the power radiated by each active chain. Furthermore, implementation constraints, as a nonregular lattice, may jeopardize the use of generic building blocks, with consequences on the costs. This particular drawback may be mitigated by implementing a set of different types of subarrays to fill the whole aperture.

Up to now, sparse and thinned arrays have been rarely used, essentially because of the complexity of their analysis and synthesis with a reduced knowledge, as a consequence of their radiative properties. The main concern in the design of sparse arrays is to find an optimal set of element spacing to meet the array specifications, while assuming a uniform excitation for practical convenience.

The synthesis of aperiodic arrays is a known problem
in the antenna community [

The problem of aperiodic arrays has recently gained a
renewed interest especially for the design of multibeam satellite antennas
[

In this paper, the equiamplitude elements constituting
the aperiodic array are placed on a lattice reproducing the positions of the
sunflower seeds, opportunely adjusted according to a desired amplitude tapering.
This type of lattice is selected essentially because it guarantees a really
good radial and azimuthal spreading in the element positions. As a consequence,
the pattern in the sidelobes and grating lobes region tends having a
plateau-like shape [

An aperiodic planar array with the elements organized
according to a sunflower lattice has been already proposed in [

The hereby proposed sunflower lattice is completely adjustable in order to follow stringent requirements on the beamwidth and the SLL without using any amplitude taper. This planar array can be considered in the design of a transmitting direct radiating array for a satellite communication antenna on a geostationary satellite.

The paper now proceeds as follows. In Section

The antenna radiation pattern of a planar array is
given by

In the following section, a particular spiral configuration will be introduced as a starting point before the space taper is applied.

A well-known spiral is the Fermat one (see Figure

Let us now introduce a normalized element density
function:

The lattice in [

Distribution of the 250 elements in the uniform
sunflower array antenna, as reported in [

It is now clear that the only possibility to control
the SLL as well is by introducing a density taper. In the following section, it
will be demonstrated how, by translating a Taylor amplitude tapering law
[

The spiral aperiodic lattice with a uniform element
density introduced in the previous section is an excellent starting point to
apply a space tapering process. The spreading of the elements in the spiral
arms guarantees an optimal behavior in terms of GL even when the interelement
spacing is larger than

The space taper technique presented here consists of
choosing a reference amplitude distribution whose pattern satisfies the
assigned requirements and emulates it by varying the radiator distance from the
center. Concretely, a Taylor amplitude taper law with a certain SLL and

Distribution of the 250 elements in the tapered sunflower array antenna.

The transmitting antenna considered in this study is
operating in Ka-band (19.7–20.2 GHz) and may have a maximum diameter of 1.3 m.
The starting point considers the circular direct radiating array with
dimensions deemed as sufficient to provide the required maximum gain and
beamwidth. The array must generate 64 spot beams. The total frequency band is
divided into 4 subbands, and each of them being assigned to a set of beams so
that there are no adjacent pencil beams using the same resource. Figure

European multibeam coverage in a 1:4 frequency re-use scheme from a geostationary satellite.

In the last 3 rows of Table

Mission requirements.

Number of spots | 64 |

Spot diameter | |

Inter-spot distance | |

Rx band | |

Tx band | |

Frequency reuse | 1:4 |

EOC gain | |

SLL
in the first | |

SLL
in the first | |

SLL
in the first |

In Figure

Array Factor, two different

The locations provided by the space taper process (see
Figure

The circular aperture with a maximum radius of

The positions of the phase centers of the subarrays
for

The Voronoi tessellation consisting of the cells enclosing the chosen phase centers.

Subarray allocation and aperture subdivision
corresponding to the Voronoi tessellation in Figure

To obtain the total radiation pattern, each radiation
pattern of the subarray

Since the Voronoi cell shapes are close to circular ones, the subarray patterns result to be almost rotationally symmetric. This is an important property when the beam is scanned.

In Figure

Array pattern for the configuration depicted in Figure

Array pattern for the configuration depicted in Figure

With this method, the entire surface available is used maintaining at the same time a very small number of controls (one for each subarray).

In this case, a more technology-oriented approach is considered: the aperture is filled as much as possible with predefined hexagonal subarrays. A limited number of these subarrays is selected as a compromise in order to keep the complexity and the cost limited while offering good performances. Four subarrays with different sizes have been selected and used to fill the array aperture. All the subarrays have a hexagonal shape and consist of 2, 3, 4, or 5 rings of elements surrounding the central one on a regular triangular lattice. The patches used in the subarrays are the same as the ones described in the previous subsection.

The procedure consists of computing for each cell the
radius of the maximum circle that can be inscribed in it. According to this
value, the best hexagonal sub-array among the four available is selected and
placed in the cell (see Figure

Hexagonal subarray positions and dimension after postprocessing.

In Figure

With the first approach, the results were exceeding
the requirements but the physical implementation of the array would be too
demanding since every subarray is different and has to be designed and tested
individually. With the second approach proposed here, only 4 subarrays need to
be generated and moreover, the feeding network will be easier to implement. The boresight radiation pattern for this configuration is depicted in Figure

Array pattern for the configuration depicted in Figure

Array pattern for the configuration depicted in Figure

The Fermat spiral and its associated coordinate system.

A deterministic procedure to design aperiodic planar arrays which guarantees the control of SLL, GL, and beamwidth without using any amplitude tapering has been introduced. Starting from an array characterized by a uniform spatial density of the elements, the density function has been modified in order to fit a reference amplitude tapering. The design technique has been applied for the preliminary design of a Direct Radiating Array for a multibeam satellite communication mission.

Spirals are one of the most common regular shapes in
nature: from the snail shell to the sunflower seed placement, to the Milky way
arms. Different kinds of spirals are known in literature. Using a spiral
placement for the elements of a planar array guarantees a good spreading of the
energy associated to the side and grating lobes. Furthermore, a spiral lattice
permits obtaining a quite uniform filling of a given aperture compared to other
planar lattices like the ones organized in rings. A well-known spiral is the
Fermat spiral (Figure

This spiral is quite often found in nature. In
particular, there are leafs and seeds whose positions can be obtained by
sampling a Fermat spiral equation, that is,

The Fibonacci sequence is known since 1202 d.C.,
thanks to Leonardo son of Bonaccio from Pisa and his book Liber Abaci. This
sequence has been widely analyzed and applied in different fields: from the
description of particular plants to computer science, from crystallography to
electrical engineering. By solving the Fibonacci quadratic equation [

The second type of spirals employed in this study is the
Fibonacci one, namely, a particular kind of logarithmic spiral, where the ratio
between radii evaluated at each

Sunflower array configuration: the elements are numbered starting from the center. In this case, 5 clockwise Fibonacci spirals appear, red line, and 8 anticlockwise, in blue line. The interval of identificative numbers between elements on the same Fibonacci spiral is always equal to the number of spirals occurring, in this figure notice 22 − 14 = 14 − 6 = 8 and 24 − 19 = 19 − 14 = 5.

Assume the case when a continuous, strictly positive,
rotationally symmetric,

By now invoking the mean function theorem, the area of
the annular ring in the denominator of (

In order to prevent possible above unit values of the
discrete amplitude density

Some remarks are due with respect to the hereby
discussed choice for a (normalized) discrete amplitude density function.
Firstly, in view of the correspondence between

Secondly, the determination of the radii