# Irregular Polyomino-Shaped Subarrays for Space-Based Active Arrays

Authors: R. J. Mailloux; S. G. Santarelli; T. M. Roberts; D. Luu
Journal: International Journal of Antennas and Propagation
 (2009)

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## Abstract

This paper presents new results showing the application of polyomino-based subarrays to limited field of view and wideband, wide-angle scanning. This technology can reduce the number of phase controls in arrays used for limited sector coverage or the number of time delay devices for wideband radar or communications, and so can reduce the cost of space-based active arrays. We concentrate on the wideband application. Results are presented by comparing the gain and peak sidelobe results of irregular polyomino subarray-based arrays with those of rectangular subarrays. It is shown that using irregular polyomino subarrays can result in a major decrease in sidelobes while presenting, in most cases, only a few tenths of a dB gain reduction compared to rectangular subarrays.


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## Body

## 1. Introduction

The increasing use of large scanning arrays for space
applications emphasizes a growing need for subarray technology to reduce the
number of controls in systems that scan over limited fields of view (LFOV) or
to minimize the number of time delays for arrays with active phase control at
the element level. These two applications are addressed by the same subarray
technology, and so these will be discussed together initially before presenting
detailed results for the wideband case. Subarray configurations for the two
applications are illustrated in
Figure1.Figure 1
Subarray applications to LFOV and time-delayed wideband
arrays.LFOV applications include synchronous satellites, where the
earth subtends a conical angle of8.7° half angle. Other
satellite systems sometimes scan a high-directivity beam over even smaller
angles to do detailed mapping. In the LFOV configuration shown at left in
Figure 1, there is no phase shift at the element level, but phases are
introduced at the input port to each subarray. The subarray is used as an
oversized element and the gain and pointing accuracy is maintained by phase
control at the subarray input ports. In the time-delayed configuration of
Figure 1, the subarray has phase shifters at the element level and time delays
at each subarray input [1]. The phase shifters scan the subarray beam
direction, but the time delays are required to keep the array pattern
stationary for all frequencies within the frequency band.The subarrays most commonly used for either application are rectangular groups of elements. When placed in a periodic array these radiate discrete sidelobes called quantization lobes at the location of the grating lobes of the array factor.Solid-state modules with phase shift pattern control are now
being used in many space applications, and they can be designed to have good
performance over wide bandwidths but the phenomenon of beam “squint” limits
their use operationally. Beam squint is the property of a phase steered array
that results in frequency-dependent pointing angles. This can result in
severely limited instantaneous bandwidth for large phase-steered arrays. Time-delay-steered arrays do not exhibit this phenomenon.Time delay control of scanning arrays remains an expensive
and often impractical option, because of the difficulty of implementing long
time delays at the element level or switched time delay tree networks. Digital
or optical beam forming networks can introduce time delay at element or
subarray level but for reasons of economy it is often convenient to introduce
time delay at the subarray level. New and projected applications with
requirements for bandwidths up to limits of 10:1 place increasingly severe
restrictions on the size of phase steered arrays or on the subarray technology
that enables the use of relatively few time delay networks.The two cited applications for subarray technology exhibit
similar behavior and suffer the same deficiencies. Quantization lobes occur in
the limited field of view array because the subarrays are arranged periodically
in the array. At broadside (and for a uniform array), the array appears
continuous and no extraneous lobes are radiated. When the array is scanned by
the phase shifters behind the subarrays, the aperture phase is quantized
because the subarrays themselves have a pattern that radiates broadside. The
array factor of this array of subarrays is, however, scanned to the new chosen
scan angle, giving rise to quantization lobes that are only suppressed at
broadside.An upper bound for LFOV scanning in one dimension is that
for idealized subarrays with spacings of dimensionD,  and maximum scan angle θ0, the subarray size D is given by
the following relationship: (1)(Dλ0)sin|θ0|≤0.5. This limit is reached using an idealized subarray pattern
that is pulse-shaped, with a flat top over that range [2].For LFOV, therefore, the quantization lobes occur because
the subarray pattern is stationary, while the array factor is scanned. However,
in the case of using time delays for broadband radiation, the phase shifters in
each subarray do scan the subarray patterns correctly at center frequency, and
the time-delayed array of subarrays remains at the chosen scan angle for all
frequencies. However, away from center frequency, the phase-steered subarray
patterns do not scan to the chosen locationu0;  but “squint” to the
new location  u0(f0/f)  for u0  =  sin θ0. Since the subarray patterns and the
array factor peaks are no longer aligned, this case also radiates quantization
lobes.This usage is indicated on the right side of Figure1, where
it shows a subarray with element level phase shifters and time delay at the
subarray input. For modest bandwidth, an upper bound on the system bandwidth in
this case is again obtained by the idealized case in which the subarray spacing
D is limited by the following equation: (2)Δff0(Dλ0)sin|θ0|≤1.0. Reaching this limit is again only achieved if the subarray
is designed to produce the “ideal” pulse-shaped pattern.Throughout this paper,  the data
that are presented pertain to the wideband scanning case. It should be
understood that the results apply equally well to the LFOV case. The
LFOV case results are readily inferred from (1) and (2) resulting in the
relationship of (3) for scan to some maximum angle  θmax  [3]:(3)Δff0sin(|θ0|)=2sin(|θmax|).We have not discussed the issue of impedance matching
elements over such wide bandwidths. The element lattice periodicity requires
element spacing on the order of 0.5λ at the highest frequency,  and this makes
them very closely spaced at the lowest frequency. This close spacing actually
improves the array wideband impedance match, as discussed in the text by Munk [4].
Vivaldi elements, connected dipole arrays and matched microstrip patch elements,
have all been used in wideband arrays.


## 2. Subarray Technology for Space Applications

The several upper bound equations noted above make it clear
that there can be completely different applications for various subarray sizes.
In the LFOV case, the subarray sizes can vary from about 3 wavelengths for ±10
degrees of scan to about 14 wavelengths for
±2
degrees of scan, and even larger for less scan. The wide instantaneous
bandwidth is required for lower orbiting radars and communication arrays, where
scanning requirements can exceed±20°  and array sizes up to hundreds of thousands of
elements. Subarrays used to insert such time delay can be very large.The technology for producing large subarrays with the
qualities to suppress these quantization lobes depends upon the size of the
subarray. Large subarrays have been constructed by dividing the array into an
irregular group of elements and building special power divider networks for
each subarray [5], or by using so-called dual-transform or overlapped
subarray networks [6, 7]. These networks use a multiple beam feed and a
focusing lens to form overlapping distributions to radiate pulse-shaped
subarrays that approximate the ideal flat-topped subarray radiation pattern.
Such large subarrays formed in this manner can vary from 10 to perhaps 50
elements on a side and so are useful for very limited scan coverage or very
narrow bandwidth scanned performance, as dictated by (1) and (2). Their
advantage, however, is the huge savings in phase or time delay control devices
and resulting cost savings.Small subarrays have been fabricated to produce overlap and
an approximation to the pulse-shaped radiation patterns that have wider but
still limited field of view [8, 9], or to applications involving very wide band
radiation. Other technologies applicable to smaller subarrays include using
simple rectangular subarrays but rotating them [10] or randomly spacing them to
suppress the quantization lobes  to use different-sized rectangular subarrays
that do fill the aperture [11–13], or to use some sort of irregular subarrays
that fill the aperture [14–16]. Of these, the randomly spaced subarrays produce
excellent suppression of extraneous lobes, but radiate with reduced gain in
proportion to the reduced filling of the aperture. The use of unequal size
rectangular subarrays or dissimilar irregular subarrays introduces problems in
excitation of the subarrays because of the
need to use different power divider combinations within the array. The
proposed method that we introduce employs only one type of irregular subarray
within the array, and these are rotated (or rotated and flipped) to completely
fill the array. The result is to form an array with nonperiodically spaced subarray
phase centers but with a tiled (filled) aperture that has full array gain.


## 3. Irregular Polyomino Subarrays

Figure2 shows an array with subarrays and indicates the array filling with
L-octomino subarrays. It also shows four rotations of an L-tetromino and two
kinds of octominos, an L-octomino and a “point-up” octomino, that we use to
fill the array. In this paper,  we are concerned only with shapes that completely
fill, or “tile” the array area, because the fully filled area has maximum
aperture gain. There are many other polyomino shapes and sizes that can be used
for tiling an array, and the branch of mathematics that is closest to polyomino
tilings is called combinatorics, of which polyominos are a subset.Figure 2
Irregular and rectangular subarrays.A
more basic combinatoric problem may query whether a collection of identical
polyominos could be tiled as to exactly fill a rectangle.  Such collections are called rectifiable. One
researcher, Michael Reid [17], has a compact, permanent link for the
subject [18]. A study limited to
octominoes will yield, in a recognized Wolfram reference [19], a list of 369 distinct
octominos. These few topics are evidence of a concentrated and vibrant area of
mathematical research.Throughout this work, we have dealt exclusively with
polyominos that fully tile the apertures but have allowed some or many
polyominos to extend beyond the boundaries of the aperture, assuming that those
squares outside of the aperture need not be excited.Figure3 shows a group of three-dimensional patterns radiated
by an array of 64 × 64 elements at f/f0 = 1.3 and u0=v0  =  0.5, where u0=sinθ0cosϕ0, v0=sinθ0cosϕ0,
and θ0,  and ϕ0 define the direction of scan. The array has a 40 dB Taylor taper, and therefore
has low side lobes at broadside and/or center frequency. All element patterns
are assumed to have a cosθ angular
dependence. The left pattern in Figure 3 is that of an array with time delay at
every element. This pattern exhibits the design sidelobes of  –40 dB with a
pattern gain of 37.3 dB as determined by integration. The center pattern
corresponds to an array in which the individual elements are grouped into 512
rectangular subarrays (two elements in the elevation plane and four in the
azimuth plane). This pattern has a gain of 37.03 dB (approximately a 0.3 dB loss
due to phase error compared to the left pattern) and has five quantization
lobes. The largest of these lobes is 11.45 dB below the gain at broadside. The
pattern at right is produced by an array of 512 L-octominos and radiates a main
beam with a 36.83 dB gain (only 0.2 dB less than the array of rectangular
subarrays). This pattern has no quantization lobes and has side lobes that are
all at least 26.6 dB below the broadside gain.Radiation patterns (a) atf/f0 = 1.3 for an array with time delay at element level, (b)   at rectangular subarray inputs, and (c) at L-octomino subarray inputs.
(a)(b)(c)These data are typical of results
that have been presented previously to describe this work. In general, the
various polyomino-based arrays have been found to have nearly the same gain but
much smaller sidelobes than arrays of periodic, rectangular subarrays. Note
that these residual side lobes arenot suppressed quantization lobes, since the quantization lobe size remains
constant relative to the main beam as array size is changed.  Rather, we see that the residual side lobes
of the polyomino-based arrays are reduced as the inverse of the array area. Our
experience with arrays of up to 128 elements on a side indicate that the peak
sidelobes continue to decrease by 5 to 6 dB with each quadrupling of array
area.Figure4 addresses the question as to why the
polyomino-based arrays are successful at eliminating the quantization lobes.
Clearly the procedure breaks up the periodicity of the array, and so one would
expect a better result than with contiguous rectangular subarrays, but with
this figure we investigate much suppression due entirely to the quasi-randomness
and how much due to the changes in the subarray pattern amplitude as each is
rotated.Sidelobe structure for18×24 element array of 54 subarrays of rectangular or L-octomino subarrays.
(a)
Array of rectangular subarrays(b)
Array of omnidirectional elements at rectangular subarray phase centers(c)
Array of octomino subarrays. Subarray phase centers are indicated by red dots(d)
Array of omnidirectional elements at octomino subarray phase centersFigure4(a)  shows an array of rectangular subarrays and the
final pattern for an array of rectangular subarrays with f/f0  = 1.3
and scanned to  u0=v0 = 0.5. The elements are 0.5 λ apart in
both planes, and the inter-subarray spacing of the eight element subarrays is λ
in azimuth and 2 λ in elevation. The array has 18 elements in azimuth and 24 in
elevation. The contour levels are set at dB values of −3, −10, −15,  and −20. The
scanned pattern, shown at right in Figure 4(a), has a set of 5 quantization lobes, with
varying degrees of suppression due to the subarray pattern. Figure 4(b) shows the
array factor for the array with the subarray patterns assumed omnidirectional,
as illustrated on the left of Figure 4(b). All the quantization lobes of  Figure 4(a) are present at the same locations, but are now grating lobes unattenuated by
the subarray pattern.  A comparison of
the numerical data shows that the resulting array factor of Figure 4(a) has its
grating lobes suppressed at least 12 dB by the subarray pattern. These peak quantization lobes are at the same
levels for much larger arrays.Figures4(c) and 4(d) present the same situations as Figures 4(a) and 4(b),
but for the case of irregular (point-up)
subarrays Figure 4(c) shows an actual
subarray configuration of “point-up” polyominos
and the pattern for that array. Although the array is small, the figure
again demonstrates significantly lower sidelobes than the array of rectangular
subarrays Figure 4(a). All of the larger sidelobes are in the neighborhood of  –20 dB,
or 8 dB below those of the rectangular subarrays, and they are no longer
located at the quantization lobe angles. The pattern shows that no quantization
lobes remain. Figure 4(d) shows the array factor of the polyomino based array
with the subarray patterns replaced by omnidirectional one. This pattern thus
includes only the effect of the aperiodic grid of phase centers. This pattern
shows that only one peak sidelobe remains at the quantization lobe locations
shown in 4(b), while other peaks do not correspond to the quantization lobe
locations. Note that some of these peaks are greater than –10 dB since there
are no element patterns to suppress them. The peak sidelobes now are distributed
throughout a much wider area of space and our experience has shown that unlike
the data of Figure 4(a), they continue to decrease with array size. Comparing
Figures 4(a) and 4(c) shows that the combination of the rotating subarray patterns
and the phase center displacement produce a much larger role in sidelobe
suppression than phase center displacement alone, but the phase center
displacement is the major contributor to the successful result.


## 4. Power Dividers for Tetrominos and Octominos

In general, one would expect that assembling an array from
irregular subarrays would entail some extra burden in fabrication or assembly.
However, there is little added assembly cost for an array of polyominos that
are all identical but rotated in the plane.
Therefore we have chosen to use polyominos that have 2n elements for n = 2, and 3 for this paper, so that the subarrays can be excited by
lossless power dividers. Other polyominos of larger size “n” can also be used, but larger polyominos have less
bandwidth (or for LFOV they have a more limited scan region), and we have not
investigated these limits. Very few larger polyominos can actually tile
rectangular areas without leaving voids in the surface, and it seems that with
the exception of rectangular shapes one would need to use a multiplicity of
higher-order polyominos to properly tile a rectangular surface. The array
elements themselves are on a fully periodic grid and so the aperture is
unchanged from conventional apertures. Only the feed matrix is constrained to
introduce the aperiodicity needed to eliminate quantization lobes.Figures5, 6, and 7 show the power dividers and feed points (black
rectangles) and a section of the filled arrays corresponding to the shapes
noted in Figure 2 and investigated in this paper. Notice that all power
dividers in any figure are identical, but because of the rotation the
excitation points form an irregular pattern.Figure 5
L-tetromino power divider and array section.Figure 6
L-octomino power divider and array section.Figure 7
Point-up power divider and array section.


## 5. Peak Sidelobe Behavior of Tetrominos and
Octominos

Peak sidelobes are not readily predicted analytically for
arrays of irregular subarrays and so numerical results have been used to relate
them to the rectangular subarray case. The peak sidelobes depend upon scan
angle and, for the wideband case, frequency offset ratio “r”, but once these
parameters are chosen,  the peak sidelobes decrease monotonically with array size
and very nearly like a constant times 1/N for a large N-element planar array.
The array gain increases with array area modified by the phase error introduced
within each subarray.The bandwidth of polyomino subarrays is primarily determined
by gain falloff with scan, since there are no quantization lobes, and the peak
sidelobes continue to decrease monotonically with array size. At the highest-frequency the element spacing is chosen to be spaced a half wavelength apart to
avoid the element level grating lobes.Figure8 shows five sets of gain curves for square arrays as
the array size is varied. These, and all other gain curves shown in this paper,
were computed by integrating the array pattern using an assumed cosθ element
pattern. The figure compares the gain of arrays with time delay at the subarray
level for rectangular or octomino subarrays. The gain of arrays with time delay
at the element level is given as the solid curve. This represents the ideal
array gain and is higher than any subarrayed case because there is no phase
error.Figure 8
Gain comparison of arrays with time delay at element levels versus time
delay at rectangular and L-octomino subarraysr =  f/f0  =1.3 and 1.82.The several time-delayed
subarray cases plotted include one with about 2:1 bandwidth (r=  f/f0  = 1.3) for the upper-band limit, and one with about 10:1 bandwidth (r = 1.82 upper-frequency) and the design low-frequency of r  = 0.182 (lower-frequency band not
shown). We have not shown the lower-frequency results for this ultra wideband
case because at this lower limit the subarray spacing is so small that there
are no quantization lobes in real space. Unlike the narrower band situations
where there are quantization lobes at both high and low-frequency extremes,
here the subarray imposes no low-frequency limit; only the element spacing and
impedance considerations provide any low-frequency limitation. In the cases
discussed, the low-frequency could be chosen as low as the element design would
allow, with no limitation imposed by the subarraying technique.The curves shown in Figure8 include the solid curve with
full element level time delay and four curves with time delay at the input port
to each subarray. The dashed curves are for rectangular subarrays (2 × 4
elements), one with high-frequency r = 1.3 (bandwidths r = f/f0 = 0.7  to
1.3) and one with much higher bandwidth (r = f/f0 =  0.182 to 1.82, a 10:1 bandwidth).The solid curve at the top of the figure is the gain using time
delay at each element. These integrated data are almost exactly equal to the
usual large array approximation for an aperture with aperture efficiencyεa:(4)G=4πAλ2εacosθ.Notice that this single curve is the array gain for both
frequency offsets;r = 1.3 and 1.82, because the element spacing is set to 0.5 λ at each frequency, so the array size is the same in normalized terms and
the gain is also the same. The curves for time delay at the subarray input
ports are shown dashed for rectangular subarrays and are approximately 0.27 dB
below the time delayed array data at r = 1.3 and 1.45 dB below the curve at
r = 1.82. The added loss is caused by the incremental phase error across the
8-element subarrays.  The two dotted
curves of Figure 8 give the corresponding gain for the L-octomino subarray
based arrays, and lie approximately 0.2 dB below the curves for the rectangular subarrays, owing to the slightly
greater phase error in the polyomino subarrays.Figure9 shows the corresponding peak sidelobe levels for
the subarrayed cases of Figure 8. The peak sidelobe level for the element level
time delayed array is not shown because it is just the idealized Taylor pattern
with –40 dB peak sidelobes. In this figure the
dashed patterns give the peak sidelobe level of arrays of rectangular (2 × 4)
subarrays and show constant levels of  ‒11.4 dB for r
= 1.3, and −5.75 dB below the
broadside gain (or only 4.23 dB below the projected area gain at u0=v0  =   0.50) for r  = 1.82. The quantization lobe level is independent
of array size because the lobe samples the subarray pattern at the same point
for any size array. The two solid curves in Figure 9 are the peak sidelobes for
arrays of L-octomino subarrays for the two frequencies noted. Note that the improvement
in comparison with the rectangular subarray results is nearly the same at both
frequencies. Figure 10 shows this added sidelobe reduction to vary from about
10.5 dB for arrays 32 elements on a side to about 15.5 dB for 64 element
arrays, but more interestingly this sidelobe reduction is almost the same
whether the rectangular subarray peak sidelobes are −11.4 dB (for r  =
1.3) or −5.75 (for the r =
1.82). This indicates that these limited results can be expected to be useful
for estimating behavior of the arrays at other frequencies. The rectangular
subarray data of Figure 9 are very simple to compute, but using the result of
Figure 10, one can obtain a good estimate of peak sidelobe level for nearly any
frequency range that one would want to estimate. In addition, we have found it
a general rule for all the cases we have tested, that quadrupling the array size
results in at least a 5 dB increase in this sidelobe reduction factor, so one
can extrapolate these data to get results for various sizes as well as various
frequencies.Figure 9
Peak sidelobe level of rectangular and L-octomino subarraysr =  f/f0
= 1.3 (moderate bandwidth) and r = 1.82 (wide band).Figure 10
Sidelobe reduction factor (max rectangular subarray quantization lobe level—largest octomino sidelobe level) dB for moderate and wide band signals.Figures11–13 show similar results for L-tetrominos when
these are operated over about twice the bandwidth. The normalized upper
frequency ranged from r =  1.6 to 2.64. Figure 11 shows the gain of an array of
the 4 element subarrays, comparing rectangular (square) 2 × 2 subarrays with
the L-tetromino subarrays. The difference in gain between the four element
rectangular subarrays and the L-tetromino subarrays is about 0.2 dB maximum at the r = 1.6 frequency limit,
and about 0.8 dB at the higher r = 2.64 frequency limit. Figure 12
shows the sidelobe behavior of the arrays at both frequencies, and Figure 13
compares the sidelobe reduction for these cases. Here the reduction varies over
a wider range than the octomino case but the slope of the curve is
approximately the same for larger arrays.
Even for the wideband r  =  2.64 case the sidelobe reduction is within 2 dB of
the lower bandwidth cases, again indicating the utility of these curves in generalizing  to other frequencies and array sizes.Figure 11
Gain comparison of arrays with time delay at element levels versus time delay at
rectangular and L-tetromino subarraysr = f/f0  = 1.6, r =  1.9 and r = 2.64  (wide band).Figure 12
Peak sidelobe level of rectangular and L-tetromino subarraysr  = f/f0
= 1.6 (moderate bandwidth), r = 1.9 and r = 2.64  (wide band).Figure 13
Sidelobe reduction factor (max rectangular subarray quantization
lobe level—largest octomino sidelobe level) dB for moderate and wide
band signals.


## 6. Conclusion

This paper has presented new results showing the application
of polyomino-shaped subarrays to large space-based arrays that scan over a
limited field of view with modest bandwidth, or over wider bandwidth and wide
scan angles. In both cases, subarray technology may be necessary to reduce cost
by decreasing the number of phase shifters or time delay devices.The paper compares the use of irregular polyominos versus the
corresponding rectangular subarrays and shows that with only few tenths of a dB
gain reduction the 8-element polyomino subarrays can reduce the peak sidelobes
by more than 15 dB for 64× 64 element arrays. Potentially much lower sidelobes
can be obtained for larger arrays since sidelobe reduction improves by 5-6 dB
for each quadrupling of array size. Results for smaller (tetromino) subarrays
are in accord and provide even wider bandwidth.Finally, it is shown that this sidelobe reduction is
primarily a function of array size, not the frequency of operation. This leads
to the useful conclusion that by focusing attention on the sidelobe reduction
itself, it is possible to estimate peak sidelobe levels for tetromino and
octomino arrays directly from the available rectangular subarray data.



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*Source: 956524-2009-03-30.xml*