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On the radiation propertiesof ADS-thinned dipolearrays
Toshifumi Moriyama1a), Giacomo Oliveri2b), Marco Salucci2,and Enrico Giarola21 Department of Electrical and Electronic Engineering, Nagasaki University,
Nagasaki 852–8521, Japan2 ELEDIA Research Center at DISI, University of Trento,
Via Sommarive 9, 38123 Trento, Italy
Abstract: The results of a full-wave analysis of the performances ofADS-based thinned arrays of wire dipoles are presented. The patternfeatures of the ADS layouts are numerically evaluated by consideringdifferent aperture sizes, lattice spacings, and dipole lengths. The out-comes are that, despite a slight and expected performance worsening,the a-priori PSL bounds deduced for the “ideal” case still provide usefulguidelines for the design of ADS thinned layouts featuring real dipoleelements.Keywords: thinned arrays, almost difference sets, sidelobe controlClassification: Electromagnetic theory
References
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LETTER IEICE Electronics Express, Vol.11, No.16, 1–10
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1 Introduction
Large arrays with low peak sidelobe levels (PSLs) can be obtained by suitablyweighting the signals received/transmitted at each element [1] even thoughwith non-negligible costs, high power consumptions as well as complex feedingnetworks especially when designing wide architectures [2]. A less expensiveand architecturally simpler alternative is represented by thinned arrange-
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IEICE Electronics Express, Vol.11, No.16, 1–10
ments [1] where array elements of uniformly-spaced lattices are turned on andoff to synthesize desired low-sidelobe patterns [1]. The periodic nature of thearising layouts also enables an easier construction compared to other non-regular arrangements such as aperiodic structures [1]. Also for these reasons,array thinning has been deeply investigated and several methodologies havebeen proposed ranging from dynamic programming [3], random approaches [4,5], as well as stochastic tools based on genetic algorithms [1, 6, 7, 8, 9, 10, 11,12, 13], simulated annealing [14], particle swarm optimizations [15, 16, 17],and pattern search [18]. Although very effective, these methodologies exhibitsome drawbacks either in terms of PSL control (specifically, random layouts)or computational costs (namely, stochastic optimization-based techniques)especially when large layouts are at hand. Therefore, analytical techniqueshave been studied for efficiently thinning arrays with well-behaved sidelobeswhatever the aperture at hand [19, 20]. Firstly, binary sequences derived fromdifference set (DSs) have been exploited for designing both linear and planarlayouts [19, 21]. Such an approach has been successively generalized to dealwith a larger set of geometrical configurations thanks to the properties ofalmost difference set (ADSs) [20, 22] and the existence of suitable ADS
construction techniques [23, 24, 25] as well as of sequence repositories [26].The main advantages coming from analytic designs are (a) the availability ofa-priori PSL bounds just knowing the descriptors of the sequences at hand;(b) the numerical efficiency of the synthesis approach; (c) the regular behaviorof the arising pattern sidelobes also well below those achievable with cut-and-try random placements [20, 22]. Thanks to these features, ADSs and DSs havebeen widely applied to thinning and also to deal with other array designproblems [27, 28, 29, 30, 31, 32].
However, to the best of the authors’ knowledge and despite their success-ful exploitations [20, 22], very few studies have been carried out so far on ADS
arrays when composed by non-ideal (i.e., non “point-like isotropic”) radiatorsalthough this is not a trivial issue or even more an academic exercise. Indeed,the pattern features of such analytically-thinned layouts depend on theirthree-level autocorrelation [20], but such a property does not hold true whenreal elements are at hand because of their non-isotropic nature as well as ofthe mutual inter-element electrical interactions that take place in real-worldarchitectures.
In this paper, ADS-based arrays of real elements (i.e., wire dipoles) areanalyzed by modelling the whole antenna architecture through a commercialmethod of moments (MoM ) software package FEKO [33]. The results from aset of numerical simulations concerned with different aperture sizes, dipolelengths, and spacings are discussed to assess the reliability of the ADS-baseddesign guidelines deduced in [20] for ideal radiators. The paper is organized asfollows. ADS thinning is summarized in Sect. 2, while the numerical resultsabout the radiation characteristics when real dipoles are at hand are presentedin Sect. 3. Finally, some conclusions are drawn (Sect. 4).© IEICE 2014
DOI: 10.1587/elex.11.20140569Received June 11, 2014Accepted July 14, 2014Publicized August 1, 2014Copyedited August 25, 2014
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2 Mathematical modelling
Starting from a regular N-sized lattice placed along the z axis with elementsspaced by d wavelengths, the design of an “ideal” broadside-steered thinnedarray with a user-defined PSL, 2, can be formulated as follows [6]
Thinning Problem: Given N, d, and a number of active elements K(K < N), find W ¼ fwðnÞ 2 f0; 1g, n ¼ 1; . . . ; Ng such thatPN
n¼1wðnÞ ¼ K and PSLfWg � �
where
Fig. 1. Experiment #1 [d ¼ 0:5, t ¼ 10�3, � ¼ 0:5] - Plots ofPSL
�W
ð�ÞADS
�and PSLMoM
�W
ð�ÞADS
�versus < for
different l values: (a) N ¼ 52 and (b) N ¼ 100.Beam patterns generated in correspondence with theoptimal shifts �opt and �MoM
opt (l ¼ 0:5) when (c) N ¼52 and (d) N ¼ 100.
Table I. ADS-Based Arrays - PSL values for different lvalues.
N 8 l [6] PSLopt [dB] PSLMoMopt [dB]
0.25 !12.59 !12.6252 0.5 0.5 !12.59 !11.17
1.0 !12.59 !12.45
0.25 !15.70 !15.76100 0.5 0.5 !15.70 !14.31
1.0 !15.70 !15.26
0.25 !21.11 !21.31400 0.5 0.5 !21.11 !19.05
1.0 !21.11 !20.75
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PSLfWg ,max juj<uRfPPðuÞg
PPð0Þ ; ð1Þ
uR being the mainlobe region size defined as in [19, 20] and PPðuÞ is theradiated power pattern [2]
PPðuÞ ¼XN�1
n¼0
wðnÞ expð2�induÞ�����
�����2
ð2Þ
where wðnÞ indicates the status of the n-th lattice element [wðnÞ ¼ 0=1 if then-th element is off/on] and u ¼ sinð�Þ.
The Thinning Problem is solved in ADS-based approaches [20] by per-forming the following iterative procedure: (I) a suitable ðN;K;�; T Þ-ADS,S ¼ sk 2 Z
N; k ¼ 0; . . . ;K � 1� �
, is selected from [26] or constructed accord-ing to [25]. The cyclic shift index < [20] is set to 0 (� ¼ 0) and the followinginitialization is assumed: PSLopt ¼ 0dB and �opt ¼ 0; (II) The array weightsW
ð�ÞADS ¼ �
wð�ÞADSðnÞ; n ¼ 0; . . . ; N � 1
�are determined as follows
wð�ÞADSðnÞ ¼
1 if ðnþ �ÞcN 2 S
0 otherwise
�ð3Þ
where ð�ÞcN is the remainder of the division by N; (III) The peak sidelobelevel PSL
�W
ð�ÞADS
�of the arising arrangement is evaluated through (1). If
PSL�W
ð�ÞADS
� � PSLopt , then PSLopt ¼ PSL�W
ð�ÞADS
�and Wopt
ADS ¼ Wð�ÞADS ,
�opt ¼ �; (IV) If � ¼ N � 1, then the iterations are terminated. Otherwise,< is incremented by 1 (� ! �þ 1) and the procedure is iterated repeatingsteps (II)(III)(IV) until PSL Wopt
ADS
� � � �. At the convergence, the synthe-sized array is given by
WoptADS , W
ð�optÞADS ¼ w
ð�optÞADS ðnÞ; n ¼ 0; . . . ; N � 1
n oð4Þ
where
Fig. 2. Experiment #1 [d ¼ 0:5, t ¼ 10�3, � ¼ 0:5] - Plots ofPSLopt, PSLMoM
opt , and a-priori PSL bounds versus Nwhen l ¼ f0:25; 0:5; 1:0g.
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�opt ¼ arg min�¼0;...;N�1
PSL Wð�ÞADS
n o� �� ð5Þ
Thanks to the ADS properties, the PSL (i.e., PSLopt ¼ PSL WoptADS
� �) satisfies
the following inequalities [20]
PSLMIN � PSLopt � PSLMAX ð6Þ
where PSLMAX ¼ 0:8488þ1:128 log10Nð Þ K���1þffiffiffiffiffiffiffiffiffiffiffiffiffiT ðN�T Þ
p� � �ðN�1Þ�þK�1þN�T and PSLMIN ¼
K���1�ffiffiffiffiffiffiffiffiffiffiffiffiT ðN�tÞðN�1Þ
qðN�1Þ�þK�1þN�T are the a-priori known bounds since $ and T derive from
the autocorrelation function of the corresponding <-shifted ADS binarysequence equal to [20]
� Wð�ÞADS
� �¼
XN
n¼0wðnÞw½ðnþ �ÞcN � ¼
¼K � ¼ 0
� for T values of � 2 ½1; N � 1��þ 1 elsewhere
8><>: :
ð7Þ
It is worth pointing out that (6) holds true for any ADS-based ideal arrange-ment (i.e., with isotropic point-like sources) provided that N is sufficientlylarge and d < 1 [20]. On the contrary, there is not analytical evidence that it isstill valid when real elements are at hand because of the electromagnetic
Fig. 3. Experiment #2 [l ¼ 0:5, t ¼ 10�3, � ¼ 0:5] - Plotsof PSLopt, PSLMoM
opt , and a-priori PSL boundsversus N when d ¼ f0:3; 0:5; 0:7g (a). Behaviour ofPSL
�W
ð�ÞADS
�and PSLMoM
�W
ð�ÞADS
�versus < for
different d values when (b) N ¼ 52 and (c) N ¼ 100.
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effects caused by the real radiators and since directive element patterns havenot been taken into account when deriving (6) [20]. As a matter of fact,mutual real-element interactions cannot be analytically modelled preventing aclosed-form extension of (6). A full-wave analysis of the whole architecture isthen mandatory to investigate the pattern features and array performanceswhen dealing with non-ideal ADS layouts. Towards this end, a commercialpackage based on theMoM is used to numerically model eachW
ð�ÞADS sequence
of linear dipoles for comparing the corresponding PSL
PSLMoM�W
ð�ÞADS
� ¼ max juj<uR PPMoM ðuÞf gPPð0Þ ; ð8Þ
with that for the ideal case [Eq. (1)].
3 Numerical analysis
In the following, massively-thinned ADS arrangements (� ¼ KN ¼ 0:5, 8 being
the thinning factor) of dipole elements of radius t ¼ 10�3 (in wavelength) havebeen assumed to give some insights about the variations of the patternfeatures in the presence of real elements.
The first experiment is aimed at evaluating the PSL behaviour of halfwavelength spaced (d ¼ 0:5) array of dipoles with different lengths l.Fig. 1(a) shows the plots of PSL
�W
ð�ÞADS
�and PSLMoM
�W
ð�ÞADS
�as a func-
tion of < when dealing with arrays of N ¼ 52 elementary radiators [26]. Asit can be noticed [Fig. 1(a)], the presence of real elements significantlymodifies the ideal-ADS PSL behavior as further pointed out by the changeof the optimal shift index �opt from the “ideal” case to the “real elements”one [�opt ¼ 26 vs. �MoM
opt , arg min�¼0;...;N�1 PSLMoM�W
ð�ÞADS
�� �h i¼ 15 when
l ¼ 0:5 - Fig. 1(a)] and the corresponding PSL value (PSLopt ¼ �12:59 vs.PSLMoM
opt , PSLMoM�W
�MoMoptð Þ
ADS
� ¼ �11:17 when l ¼ 0:5 - Table I). Unlike theideal case, the power pattern radiated by the dipole array turns out to be non-symmetric [Fig. 1(c)] because of the electromagnetic coupling effects amongthe dipoles displaced in asymmetric fashion according to the originating ADS
binary sequence [20]. Similar conclusions hold true when dealing with largerarrangements [e.g., N ¼ 100 - Figs. 1(b) and 1(d)], as well.
The comparisons among the PSLopt of the “ideal” and the “real” cases fordifferent lattice dimensions, N, (Fig. 2) show that (a) PSLopt and PSLMoM
opt arequite close whatever N and l (e.g., PSLopt ¼ �12:59 vs. PSLMoM
opt ¼ �12:62
when l ¼ 0:25 andN ¼ 52 - Table I) even though generally PSLopt < PSLMoMopt
(e.g., PSLopt ¼ �15:70 vs. PSLMoMopt ¼ �15:26 when l ¼ 1:0 and N ¼ 100 -
Table I); (b) although derived in the ideal case, Equation (6) provides reliableguidelines also for the PSL prediction in real arrays in the whole set ofanalyzed scenarios (Fig. 2).
The next experiment (Experiment #2) is concerned with lattices withd 6¼ 0:5. Figs. 3(b) and 3(c) compares the behaviours of PSL
�W
ð�ÞADS
�and
PSLMoM�W
ð�ÞADS
�as a function of < when d 2 f0:3; 0:5; 0:7g being l ¼ 0:5. The
same apertures (i.e., N ¼ 52 and N ¼ 100) of Fig. 1 have been considered foradding some more to the comparative assessment of Experiment #1. As it can
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IEICE Electronics Express, Vol.11, No.16, 1–10
be observed, a smaller lattice spacing than d ¼ 0:5 affects the PSL of idealADS layouts (see for example, Fig. 3(b) - N ¼ 52) because of the reduction ofthe visible range [2]. In some cases, the PSL decreases [e.g., � ¼ 38 - Fig. 3(b)],while increasing d above 0.5 does not affect the PSL of ideal ADS arrays [e.g.,Fig. 3(c) - N ¼ 100] because of the periodicity of the sidelobes entering thevisible range [2]. As expected, the same does not hold true when consideringreal layouts of linear dipoles [Figs. 3(b)–3(c)]. Whatever the value of d, PSLopt
modifies (e.g., PSLopt ¼ �15:86 vs. PSLMoMopt ¼ �13:80 when d ¼ 0:3 and N ¼
100 - Table II) due to the variations in the arising inter-element electro-magnetic interactions. Moreover, the optimal cyclic shift index also dependson d [e.g., �MoM
opt ¼ 44 for d ¼ 0:3 vs. �MoMopt ¼ 24 when d ¼ 0:7 and N ¼ 52 -
Fig. 3(b)]. However, the value of PSLMoMopt still lies within the bounds stated in
(6) whatever N [N 2 ½52; 400� - Fig. 3(a)].In order to further generalize and conform the previous outcomes for wider
arrays, the case of an aperture ofN ¼ 400 positions has been analyzed more indetail (Experiment #3 - Fig. 4). As an illustrative result, the PSL plots versus< in correspondence with various dipole lengths l [d ¼ 0:5 - Fig. 4(a)] or dvalues [l ¼ 0:5 - Fig. 4(b)] are reported. It is confirmed that also for large arrays�opt usually depends on d and l [e.g., �MoM
opt ¼ 26 when d ¼ 0:3 vs. �MoMopt ¼ 149
when d ¼ 0:7 - Fig. 4(b)]; (b) �MoMopt 6¼�opt [e.g., �opt ¼ 121 vs. �MoM
opt ¼ 40
when l ¼ d ¼ 0:5 - Fig. 4(a)]; (c) PSLopt < PSLMoMopt (e.g., PSLopt ¼ �21:11
Fig. 4. Experiment #3 [N ¼ 400, t ¼ 10�3, � ¼ 0:5] - Be-haviour of PSL
�W
ð�ÞADS
�and PSLMoM
�W
ð�ÞADS
�ver-
sus < when (a) l ¼ f0:25; 0:5; 1:0g - d ¼ 0:5 and (b)l ¼ 0:5 - d ¼ f0:3; 0:5; 0:7g. Beam patterns gener-ated in correspondence with the optimal shifts �optand �MoM
opt being d ¼ l ¼ 0:5 (c).
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IEICE Electronics Express, Vol.11, No.16, 1–10
vs. PSLMoMopt ¼ �19:05 when d ¼ l ¼ 0:5, N ¼ 400 - Table II). Moreover, the
optimal pattern is asymmetrical for real layouts [e.g., l ¼ d ¼ 0:5 - Fig. 4(c)] asfor smaller arrays. Although the use of real radiators usually causes a worsen-ing of the PSLopt compared to the ideal case [Tables I–II], it is also worthhighlighting that such a negative effect is actually counter-balanced by anincreased directivity defined as
D ¼ maxu2½�1;1�fPPðuÞg12
R 1�1 PPðuÞ du
ð9Þ
as detailed in Figs. 5(a)–5(b), despite the mutual interactions among thedipoles. More specifically, longer dipoles results in greater D values [Fig. 5(a)].Analogously, larger spacings yield higher directivities whatever N as for idealarrays [Fig. 5(b)].
4 Conclusions
In this paper, ADS thinned layouts have been investigated by considering realradiators. Towards this end, ADS arrays of wire dipoles have been simulatedby means a commercial software package to take into account all the arisingelectromagnetic interactions. Different element spacings, aperture sizes, and
Table II. ADS-Based Arrays - PSL values for different dvalues.
N 8 d [6] PSLopt [dB] PSLMoMopt [dB]
0.3 !13.29 !12.2452 0.5 0.5 !12.59 !11.17
0.7 !12.59 !11.31
0.3 !15.86 !13.80100 0.5 0.5 !15.70 !14.31
0.7 !15.70 !13.63
0.3 !21.37 !18.09400 0.5 0.5 !21.11 !19.05
0.7 !21.11 !18.69
Fig. 5. [t ¼ 10�3, � ¼ 0:5] - Directivity values of ADS-basedarrays versus N when (a) l ¼ f0:25; 0:5; 1:0g - d ¼0:5 and (b) l ¼ 0:5 - d ¼ f0:3; 0:5; 0:7g.
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wire lengths have been analyzed to assess the pattern features of ADS-basedarrays in non-ideal conditions. The results from such a numerical validationhave pointed out that (a) unlike the ideal case, the optimal layout for a givenADS also depends on the length l and inter-element distance d of the dipoles;(b) the optimal PSL values of ADS layouts usually turn out higher when realdipoles are at hand; (c) unlike the ideal isotropic case, optimal ADS patternsare generally asymmetric; (d) the directivity of ADS-based arrays increases ifreal dipoles are considered whatever d and l; (e) the PSL bounds (6) analyti-cally deduced in [20] for ideal arrays turn out reliable also when consideringdipole layouts. Future works will be aimed at considering different elementaryradiators as well as at investigating how to explicitly gain from the presence ofreal elements in synthesizing ADS-based array architectures.
Acknowledgments
This work was supported in part by a Grant-in-Aid for Scientific Research (C)Number 25420411.
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