An Irregular Two-Sizes Square Tiling Method for the Design of Isophoric Phased Arrays
IIEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 1
An Irregular Two-Sizes Square Tiling Method forthe Design of Isophoric Phased Arrays
Paolo Rocca,
Senior Member, IEEE , Nicola Anselmi,
Member, IEEE , Alessandro Polo,
Member, IEEE , and A.Massa,
Fellow, IEEE
Abstract —The design of isophoric phased arrays composedof two-sized square-shaped tiles that fully cover rectangularapertures is dealt with. The number and the positions of thetiles within the array aperture are optimized to fit desiredspecifications on the power pattern features. Towards this end,starting from the derivation of theoretical conditions for thecomplete tileability of the aperture, an ad-hoc coding of theadmissible arrangements, which implies a drastic reductionof the cardinality of the solution space, and their compactrepresentation with a graph are exploited to profitably apply aneffective optimizer based on an integer-coded Genetic Algorithm.A set of representative numerical examples, concerned with state-of-the-art benchmark problems, is reported and discussed to givesome insights on the effectiveness of both the proposed tiledarchitectures and the synthesis strategy.
Index Terms —Phased Array Antenna, Planar Array, SquareTiles, Isophoric Array, Optimization.
I. I
NTRODUCTION N EW space technologies are pushing the research towardsa new generation of antenna systems for space appli-cations to fit more and more stringent requirements in termsof pattern performance, power consumption, and geometricalconstraints [1]. For instance, let us consider the CubeSat mis-sions [1]-[5] based on swarms of cheap miniaturized satellitesfor communications and sensing as well as recent investi-gations on space-based infrastructures for Internet-of-Space(
IoS ) applications [6]. All these applicative scenarios need
Manuscript received on October XX, 2019This work has been partially supported by the Italian Ministry of Education,University, and Research within the Program "Smart cities and commu-nities and Social Innovation" (CUP: E44G14000060008) for the Project"WATERTECH - Smart Community per lo Sviluppo e l’Applicazione diTecnologie di Monitoraggio Innovative per le Reti di Distribuzione Idrica negliusi idropotabili ed agricoli" (Grant no. SCN_00489) and within the ProgramPRIN 2017 (CUP: E64I19002530001) for the Project "CYBER-PHYSICALELECTROMAGNETIC VISION: Context-Aware Electromagnetic Sensingand Smart Reaction (EMvisioning)" (Grant no. 2017HZJXSZ).P. Rocca, N. Anselmi, A. Polo, and A. Massa are with the ELEDIA@UniTN(DISI - University of Trento), Via Sommarive 9, 38123 Trento - Italy (e-mail:{paolo.rocca, nicola.anselmi.1, alessandro.polo.1, andrea.massa}@unitn.it).P. Rocca, is also with the ELEDIA Research Center (ELEDIA@XIDIAN- Xidian University),P.O. Box 191, No.2 South Tabai Road, 710071 Xi’an,Shaanxi Province - China (e-mail: [email protected]).A. Massa is also with the ELEDIA Research Center (ELEDIA@L2S -UMR 8506), 3 rue Joliot Curie, 91192 Gif-sur-Yvette - France (e-mail:[email protected]).A. Massa is also with the ELEDIA Research Center (ELEDIA@UESTC- University of Electronic Science and Technology of China), Schoolof Electronic Engineering, 611731 Chengdu - China (e-mail: [email protected]).A. Massa is also with the ELEDIA Research Center (ELE-DIA@TSINGHUA - Tsinghua University), 30 Shuangqing Rd, 100084Haidian, Beijing - China (e-mail: [email protected]). ( a )( b ) Figure 1. Sketch of ( a ) the MSTA architecture and ( b ) an example of σ S and σ L tile arrangement when S = 2 and L = 6 . high-efficiency, low-profile, and low-cost antennas to be easilypackaged into small payloads and lunched in the space withreduced costs. Moreover, current space programs have shownan increasing interest in array solutions for sensing [7][8]and IoS [9] because of their advanced and attractive func-tionalities. However, conventional (fully-populated) phasedarrays (
FPA s) imply the fabrication of complex beamformingnetworks (
BFN s) with many transmission/receive modules(
TRMs ), generally one for each element of the array, thusresulting unaffordable for space applications. The need oflowering the fabrication costs of phased arrays has induced theantenna research community to investigate alternative architec-tures to fully-populated arrangements for reducing the numberof
TRMs in the
BFN, as they represent the main source of cost[10], while still keeping the positive features of phased arrayantennas. Consequently, unconventional compromise arrayshave been proposed [11] including sparse [12]-[19], thinned[20][21], and clustered architectures [22]-[31]. Among theselatter, tiled layouts recently attracted a large interest being apromising architectural solution for a future mass deployment
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2020.2970088Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 2 of phased arrays. Thanks to their modularity (the array beingcomposed of one or few types of tiles), the production and themaintenance costs can be significantly reduced as well as the
BFN turns out to be simplified since each tile is controlledby a single
TRM . However, designing a tiled array is not atrivial task and it is more and more difficult when dealingwith irregular clusters to mitigate the presence of undesiredgrating lobes and to reduce high quantization lobes [25][29].Moreover, a complete coverage of the antenna aperture ismandatory when challenging directivity requirements must besatisfied. From a functional point of view, both the exact tilingof a finite region - given a limited set of tile shapes - and thegeneration of the set of admissible and complete clusteringsare challenging combinatorial tasks. Fortunately, these topicshave been addressed in several theoretical contributions [32]-[37] in the mathematical literature and useful coverage theo-rems, along with exhaustive generation algorithms, have beenderived. Recently, such a theoretical/algorithmic backgroundhas been fruitfully exploited and profitably customized to theirregular tiling of rectangular and hexagonal array apertureswith domino and diamond shapes [29]-[31], respectively.Besides previous challenges in designing tiled arrays, anotherfundamental need of standard applications of space antennas isthe maximum efficiency of the beam-forming amplifiers sincethe energy budget is extremely tight. Therefore, isophoric ar-rays (i.e., arrays of elements controlled by amplifiers workingat the maximum efficiency) organized in sparse [13][14][16]-[18] or clustered [22][15] configurations have been proposedin the last years by recurring to a density- [13][16]-[18] or anelement-size [14][15] tapering to synthesize desired patternshapes, as well.Dealing with isophoric tiled arrays, this work is concernedwith an innovative synthesis method that guarantees the fullcoverage of the antenna aperture by means of an irregularplacement of uniformly-fed tiles having two square sizes (i.e.,each square size clusters a different number of elementary ra-diators). Thanks to the exploitation of different tile dimensionsand the arising amplitude tile-size tapering of the apertureillumination, an advanced control of the beam-pattern featuresis yielded. To the best of the authors’ knowledge, the mainnovelties of this research activity lie in ( i ) the exploitationof suitable mathematical theorems to solve the problem offully covering rectangular apertures with two square tilesof different sizes; ( ii ) the design of an array architecturecomposed of two square isophoric sub-arrays with differentnumber of elements, denoted as Multi-Square Tiled Array ( MSTA ), to yield an advanced control of the beam-patternfeatures through an amplitude tile-size tapering of the single-element excitations; ( iii ) the definition of an ad-hoc binarycoding of all the possible rows composing the planar array andthe representation of the whole set of complete aperture tilingsas a graph where each node corresponds to one admissiblerow of the array and each different arrangement of the tiles ismapped into a path of connected nodes; ( iv ) the developmentof an innovative Integer-coded GA ( IGA ) for an efficientexploration of the solution space/graph to effectively deal withlarge arrays, as well.The rest of the paper is organized as follows. The mathematical ( a )( b ) Figure 2.
Coverage Theorem ( S = 2 , L = 6 ) - Example of ( a ) a tileableaperture ( M = 8 , N = 14 ) and of ( b ) a non-tileable aperture ( M = 7 , N = 14 ). formulation of the synthesis of tiled arrays is summarized inSect. 2, while Section 3 details the proposed tiling method byincluding the theoretical conditions for the complete tileabilityof the aperture (Sect. 3.1), the coding of the admissibleclustered configurations as well as the mapping of these latterin a suitable minimum-redundancy graph-based representation(Sect. 3.2), and the optimization approach for the definition ofthe optimal layout fitting the user-defined pattern requirements(Sect. 3.3). Representative examples from a wide numericalassessment are reported in Sect. 4 where test cases, alsoconcerned with full-wave modeled realistic radiating elements,are discussed. Eventually, some conclusions and final remarksare drawn (Sect. 5).II. M ATHEMATICAL F ORMULATION
Let us consider a planar phased array composed of M × N elementary radiators placed on a regular rectangular latticewith inter-element distances d x and d y along the x and the y axis, respectively [Fig. 1( a )]. The antenna aperture R iscompletely filled in with Q square tiles having two differentsizes, but each one has a single input/output port. The twotypes of tile, σ S and σ L , contain γ S ( γ S , S × S ) and( γ L , L × L ) radiating elements [ L = 6 , S = 2 - Fig. 1( b )],respectively, L being an integer multiple of S (i.e., ˆ L , LS isan integer number and ˆ L ≥ ) to set an additional degree ofmodularity that further simplifies the array manufacturing (i.e.,the larger tile σ L can be generated by assembling a set of σ S tiles). Moreover, the Q sub-arrays are assumed to have uniform This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2020.2970088Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 3 ( a ) ( b ) ( c ) ( d )( e ) Figure 3.
Solution Coding and Graph Mapping (, M = 4 , N = 7 ; S = 1 , L = 3 ) - ( a ) Coding of the σ S and σ L tiles [Fig. 1( b ) - R physical aperture] in the virtual aperture ˆ R , ( b ) the row dictionary (i.e., the set of H admissible configurations of a row of A ), A , { a h ; h = 1 , ..., H }, ( c ) the solution graph G , ( d ) virtual representation of the tiling configuration (i.e., the tiling mapped in ˆ R ) with element-membershipvector c = { , , , , , , , , , , , , , , , , , , , , , , , , , , , } , and ( e ) its corresponding matrix A (i.e., a set of ˆ M y elements of A ). amplitudes (i.e., α = , = { α q , ; q = 1 , ..., Q }) to avoidany amplitude tapering [18]. The pattern control is yielded bymeans of the phases of the Q clusters, β = { β q ; q = 1 , ..., Q } ,along with the two-level equivalent amplitude distributionwithin the aperture. This latter is due to the different numberof array elements clustered in the σ S and σ L tiles. Indeed,if each q -th ( q = 1 , ..., Q ) tile is fed by the same amountof power (i.e., α q , ), then the equivalent element-levelamplitude excitations, { α ( q ) m,n ; m = 1 , ..., M ; n = 1 , ..., N ; q = 1 , ..., Q }, turn out to be equal to α ( q ) m,n = ( √ γ S if ( x n , y m ) ∈ σ S √ γ L if ( x n , y m ) ∈ σ L ; m = 1 , ..., M ; n = 1 , ..., N (1)being α q = P Mm =1 P Nn =1 α ( q ) m,n δ c m,n q , while the far-fieldarray radiation can be mathematically expressed as E ( θ, φ ; c , β ) = P Mm =1 P Nn =1 e m,n ( θ, φ ) Λ m,n ( θ, φ ) × I m,n ( c ) (2)where e m,n ( θ, φ ) is the active element pattern of the ( m, n ) -th ( m = 1 , ..., M , n = 1 , ..., N ) radiator located at ( x n , y m ) , Λ m,n ( θ, φ ) = e j πλ sin θ ( x n cos φ + y m sin φ ) is the correspondingsteering vector, λ being the wavelength, and I m,n ( c ) is therelated weighting term [ I m,n ( c ) , P Qq =1 α ( q ) m,n δ c m,n q e jβ q ],while ( θ, φ ) are the angular directions ( θ ∈ [0 : 90] [deg], φ ∈ [0 : 360] [deg]). Moreover, c is the tiling vector ( c , { c m,n ∈ [1 : Q ] ; m = 1 , ..., M, n = 1 , ..., N } ) whose ( m, n ) -th entry, c m,n , indicates the membership of the ( m, n ) -th ( m =1 , ..., M ; n = 1 , ..., N ) elementary radiator to the q -th ( q =1 , ..., Q ) tile, δ c m,n q being the Kronecker delta function equalto δ c m,n q = 1 if c m,n = q and δ c m,n q = 0 , otherwise.The synthesis problem at hand then consists in determin-ing the tiling configuration c and the set of phase coeffi-cients, β , so that the radiated power pattern P ( θ, φ ; c , β ) [ P ( θ, φ ; c , β ) , | E ( θ, φ ; c , β ) | ] pointing along ( θ , φ ) ( a )( b ) Figure 4.
Solution Coding and Graph Mapping ( M = 4 , N = 5 ; S = 1 , L = 3 ) - Examples of ( a ) an horizontal overlap and ( b ) a vertical overlapbetween two σ L tiles in the virtual aperture ˆ R . fits user-defined constraints on the sidelobe level ( SLL )[ SLL , max ( θ,φ ) ∈ Ξ { P ( θ,φ ; c , β ) } P ( θ ,φ ; c , β ) , Ξ being the sidelobe region] andon the half-power beamwidth ( HPBW ) along the principalplanes [i.e., the horizontal/azimuth ( φ = 0 [deg]) and thevertical/elevation ( φ = 90 [deg]) planes], HP BW az and HP BW el : SLL ≤ η SLL , HP BW az ≤ η az , and HP BW el ≤ This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2020.2970088Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 4 -13.5-13-12.5-12-11.5-11-10.5-10-9.5-9-8.5-8-7.5-7 0 2 4 6 8 10 12 14 16 18 S LL [ d B ] Sorted Index [x10 ]MS-ETM MS-OTMSLL=-13.06 [dB] Figure 5.
Numerical Assessment ( M = 5 , N = 8 , d = 0 . λ ; S = 1 , L = 2 ) - Value of the SLL of the whole set of U tiling configurations sortedfrom the worst to the best ( Q = 34 : Q S = 32 and Q L = 2 ) and of the best MS-OTM solution ( P = 12 ). η el .Since pointing the main-lobe towards a desired steered di-rection ( θ , φ ) means analytically setting the values of thesub-array phases as follows β q = − πλ ( x q sin θ cos φ + y q sin θ sin φ ) ,q = 1 , ..., Q (3)where ( x q , y q ) are the planar coordinates of the center of the q -th ( q = 1 , ...., Q ) tile given by (cid:18) x q y q (cid:19) = 1 γ q M X m =1 N X n =1 δ c mn q (cid:18) x n y m (cid:19) (4)where γ q = γ S or γ q = γ L if the q -th tile is either σ S or σ L ,respectively, then P ( θ, φ ; c , β ) = P ( θ, φ ; c ) since β = β (c) .The tiling problem can be then reformulated as follows: MSTA Synthesis Problem - Given an aperture R , arectangular phased array composed of M rows, eachcontaining N elements, and two types of isophoricsquare tiles, σ S and σ L , that contain γ S and γ L =ˆ L × γ S ( ˆ L ≥ ) array elements, respectively, find theoptimal R -complete tiling configuration, c opt (i.e.,the optimal arrangement of σ S and σ L sub-arraymodules) so that the pattern-fitting cost function ΦΦ ( c ) , w sl {| SLL [ P ( θ, φ ; c )] − η SLL |×H [ | SLL [ P ( θ, φ ; c )] − η SLL | ] } + w bw {| HP BW az [ P ( θ, φ ; c )] − η az |×H [ | HP BW az [ P ( θ, φ ; c )] − η az | ]+ | HP BW el [ P ( θ, φ ; c )] − η el |×H [ | HP BW el [ P ( θ, φ ; c )] − η el | ] } (5)is minimized (i.e., c opt , arg { min c [Φ ( c )] } ), w sl and w bw being real-valued user-defined weightingcoefficients, while H [ · ] is the Heaviside function.III. M ULTI -S QUARE T ILED A RRAY S YNTHESIS
The aforementioned
MSTA synthesis problem is solved bymeans of an ad-hoc strategy that exploits the recent theoryfrom [37] on the set of admissible complete partitionings of a rectangle by means of two squares of different sizes.More specifically, the proposed tiling method consists of thefollowing logical steps: ( ) Assessment of the condition forthe complete tileability of the rectangular array aperture R with the two tiles σ S and σ L ; ( ) Binary coding of the M rows composing the rectangular array aperture and mapping ofthe whole set of admissible tile-arrangements in a graph; ( )Searching for the optimal tiling, c opt , within the solution graphby means of a computationally-effective optimization algo-rithm called Multi-Square Optimization-Based Tiling Method ( MS-OTM ). Each step will be detailed in the following.
A. Aperture Covering Theorems
Let us model each ( m, n ) -th element of the array with asquare pixel so that the whole aperture R is composed of M × N pixels. The array aperture R can be fully covered by σ S and σ L tiles [Fig. 2( a )] if and only if the conditions • S divides both M and N , namely ( M mod S ) = 0 and ( N mod S ) = 0 , mod being the modulo operation; • M (or N ) is equal or larger than L [ M ≥ L (or N ≥ L )]and N (or M ) is a positive linear combination of S and L [ N = w S + w L (or M = w S + w L ), w and w being real-valued coefficients];jointly hold true. Moreover, if none of the above conditions isverified, the region R is not fully tileable with an arrangementof both σ S and σ L tiles [Fig. 2( b )]. Otherwise, if only the firstcondition holds true, then R can be completely partitioned withonly σ S tiles. B. Solution Coding and Graph-Mapping
Once the tileability of the aperture R has been positivelyassessed, there is the need of a suitable coding for a trialtiling configuration as well as of a compact representation ofthe solution space of the admissible tiling solutions. Towardsthis end, the theory of [37] is here extended to the case oftwo square tiles fitting the condition γ L = ˆ L × γ S ( ˆ L ≥ ).More in detail, a virtual aperture ˆ R of ˆ M [ ˆ M , MS ; ˆ M = 4 -Fig. 3( d )] rows and ˆ N [ ˆ N , NS ; ˆ N = 7 - Fig. 3( d )] columns,thus containing ˆ M × ˆ N pixels, is first defined by rescaling S times the physical aperture R . Accordingly, the smallest tile σ S of dimensions γ S turns out being represented in the virtualaperture by a single-pixel ( γ ˆ S , ˆ S × ˆ S = 1 since ˆ S , SS )virtual tile ˆ σ S [Fig. 3( a )], while the largest one, σ L , is mappedinto an equivalent virtual tile ˆ σ L that contains γ ˆ L ( γ ˆ L , ˆ L × ˆ L )pixels [ ˆ L = 3 - Fig. 3( a )]. Given the virtual aperture, a trial tiling config-uration is then encoded into a binary matrix A = n a ˆ m, ˆ n ∈ [0 ,
1] ; ˆ m = 1 , ..., ˆ M y ; ˆ n = 1 , ..., ˆ N x o ,being ˆ M y , ˆ M − ˆ L + 1 [ ˆ M y = 2 - Fig. 3( d )] and ˆ N x , ˆ N − ˆ L + 1 [ ˆ N x = 5 - Fig. 3( d )], where the value ofthe bits (i.e., the binary entries of the matrix) associated toeach pixels/elements of ˆ R are set according to the following It is worth noticing that R and ˆ R coincide when S = 1 and allcoding/mapping operations can be directly performed on the physical apertureat hand. This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2020.2970088Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 5
Table IN
UMBER OF TILING CONFIGURATIONS , U . ˆ M = ˆ N U . ×
12 60 . ×
16 36 . × tile-placement rules [Fig. 3( a )]: ( a ) the bit corresponding tothe pixel of a virtual tile ˆ σ S is set to zero (i.e., a ˆ m, ˆ n = 0 );( b ) the bit corresponding to the pixel in the bottom-leftcorner of a virtual tile ˆ σ L is set to one (i.e., a ˆ m, ˆ n = 1 ),while the others, still belonging to the same tile ˆ σ L , areset to zero (i.e., a ˆ m, ˆ n = 0 ). It is worth highlighting thatthe binary matrix A has dimensions ˆ M y × ˆ N x instead of ˆ M ( M ≥ ˆ M > ˆ M y ) and ˆ N ( N ≥ ˆ N > ˆ N x ) since thetop ˆ L − rows and the rightmost ˆ L − columns alwayscontain zeros [Fig. 3( d )] as a consequence of the rules ( a )and ( b ). Consequently, there is a significant advantage inpreferring A to c for coding a trial tiling configurationbecause of the drastic reduction of the cardinality of thesolution space from U = Q M × N down to U = 2 ˆ M y × ˆ N x .Successively, let us consider the ˆ M y rows of the matrix A ( A , h a ˆ m ; ˆ m = 1 , ..., ˆ M y i T , a ˆ m , n a ˆ m, ˆ n ; ˆ n = 1 , ..., ˆ N x o being the ˆ m -th ( ˆ m = 1 , ..., ˆ M y ) row of the matrix A ) [Fig.3( e )] to define the space of all the admissible tilings [37].Since each row contains ˆ N x pixels/bits, one would assumethat the row dictionary (i.e., the H admissible configurationsof a row of A ), A , { a h ; h = 1 , ..., H }, has a cardinalityequal to H = 2 ˆ N x , including a h = (i.e., the row containingall zeros, = n a ˆ m, ˆ n ,
0; ˆ n = 1 , ..., ˆ N x o ), but differentlyit is much smaller ( H < ˆ N x ) since all strings that encodean overlap between two virtual tiles must be excluded [Figs.4( a )-4( b )]. In order to avoid an horizontal overlap [Fig. 4( a )],the binary entries of the generic h -th ( h = 1 , ..., H ) elementof A are required to always fit the ’existence conditions ’when a h, ˆ n = 1 ( ˆ n ∈ [1 : ˆ N x ] : a h,i = 0 f or (ˆ n + 1) ≤ i ≤ (cid:16) ˆ n + ˆ L − (cid:17) if ˆ n ∈ h (cid:16) ˆ L − (cid:17)i(cid:16) ˆ n − ˆ L + 1 (cid:17) ≤ i ≤ (cid:16) ˆ n + ˆ L − (cid:17) if ˆ n ∈ h ˆ L : (cid:16) ˆ N x − ˆ L + 1 (cid:17)i(cid:16) ˆ n − ˆ L + 1 (cid:17) ≤ i ≤ (cid:16) ˆ N x − ˆ n (cid:17) if ˆ n ∈ h(cid:16) ˆ N x − ˆ L + 2 (cid:17) : ˆ N x i . (6)According to (6), the row dictionary A [Fig. 3( b )] and theset of nodes of the rows-graph [Fig. 3( c )] are then defined.However, it is not possible to iterate ˆ M y -times the assignmentof a randomly-chosen h -th ( h ∈ [1 : H ] ) element of A , a h ,to a generic ˆ m -th ( ˆ m = 1 , ..., ˆ M y ) row a ˆ m to generate a trialmatrix A (i.e., a trial array tiling) because of possible verticaloverlaps between two ˆ σ L tiles [e.g., Fig. 4( b )]. To avoid suchan unphysical arrangement when filling the matrix A with the elements of the dictionary A , the following ’ feasibilityassignment condition ’ is stated: “ If the ˆ m -th row of A containsan one-bit at the ˆ n -th pixel position (i.e, a ˆ m, ˆ n = 1 , ˆ n ∈ [1 :ˆ N x ] ) and a h, ˆ n = 0 (7) as well as (6) hold true, then any h -th ( h ∈ [1 : H ] ) elementof the dictionary , a h , can be assigned to any row of ( a ) the (cid:16) ˆ L − (cid:17) superior ones when ˆ m ∈ h (cid:16) ˆ M y − ˆ L + 1 (cid:17)i and of ( b ) the (cid:16) ˆ M y − ˆ m (cid:17) ones , from the ( ˆ m + 1) -th row until the (last) one (i.e., the ˆ M y row), when ˆ m ∈ h(cid:16) ˆ M y − ˆ L + 2 (cid:17) : ˆ M y i (i.e., one of the top ˆ L − rows of A ).”Thanks to this formalism and as in [37], the solution spaceof the admissible tilings can be then represented through asolution graph G [Fig. 3( c )] where ( i ) each node corresponds toan element of the row dictionary A that satisfies the existenceconditions and ( ii ) two nodes of G are connected only if thecorresponding rows fit the feasibility assignment condition .Therefore, a matrix A turns out to be equivalent to a pathof ˆ M y nodes in G [Fig. 3( c )] with a node visited more thanone time, as well.Once the matrix A (i.e., a path of connected nodes in thegraph G ) has been determined, the corresponding arrangementof ˆ σ S and ˆ σ L virtual tiles is set in ˆ R according to the tile-placement rules [Fig. 3( b )] so that a bit equal to one in the ( ˆ m, ˆ n ) -th entry of the matrix A [e.g., a , = 1 - Fig. 3( e )]indicates the presence of a tile of larger size ˆ σ L with bottom-left corner at the ( ˆ m, ˆ n ) -th position of the virtual array ˆ R [e.g., a , = 1 - Fig. 3( d )]. Finally, the tiling configurationin the physical aperture R is yielded by “stretching” S timesthe sizes of the virtual tiles of ˆ R along x and y directions.Mathematically, it means setting the membership of the ( p, q ) -th [ p , ˆ m + ( i − × S and q , ˆ n + ( j − × S ; i = 1 , ..., S ; j = 1 , ..., S ] elementary radiator of the array to the samevalue of the membership of the ( ˆ m, ˆ n ) -th ( ˆ m = 1 , ..., ˆ M y ; ˆ n = 1 , ..., ˆ N x ) pixel of the virtual aperture: c p,q = ˆ c ˆ m, ˆ n . (8)Concerning the number of tiling configurations of the arrayaperture, the dimension of the solution space U has beenderived in closed form [37] for the case ˆ L = 2 : U = T G ˆ N − (9)where is the identity vector of length H , T stands for thetranspose operator, and G is the adjacent matrix of the graph G [38] (see Appendix I ). For instance, let us consider the valuesof U for square apertures up to ˆ M = ˆ N = 16 reported in Tab.I. As it can be noticed, the cardinality of the space of clusteredlayouts grows exponentially, thus the use of enumerative/brute-force algorithms testing all the possible arrangements is possi-ble only for very small apertures. Therefore, an optimization-based synthesis strategy is used to deal with large aperture, aswell. This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2020.2970088Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 6 -1.25-0.75-0.25 0.25 0.75 1.25-2 -1 0 1 2 y / λ x/ λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] -1.25-0.75-0.25 0.25 0.75 1.25-2 -1 0 1 2 y / λ x/ λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] -1.25-0.75-0.25 0.25 0.75 1.25-2 -1 0 1 2 y / λ x/ λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] -1.25-0.75-0.25 0.25 0.75 1.25-2 -1 0 1 2 y / λ x/ λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] ( a ) ( b ) ( c ) ( d ) -1.25-0.75-0.25 0.25 0.75 1.25-2 -1 0 1 2 y / λ x/ λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] -1.25-0.75-0.25 0.25 0.75 1.25-2 -1 0 1 2 y / λ x/ λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] -1.25-0.75-0.25 0.25 0.75 1.25-2 -1 0 1 2 y / λ x/ λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] -1.25-0.75-0.25 0.25 0.75 1.25-2 -1 0 1 2 y / λ x/ λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] ( e ) ( f ) ( g ) ( h ) -1.25-0.75-0.25 0.25 0.75 1.25-2 -1 0 1 2 y / λ x/ λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] -1.25-0.75-0.25 0.25 0.75 1.25-2 -1 0 1 2 y / λ x/ λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] -1.25-0.75-0.25 0.25 0.75 1.25-2 -1 0 1 2 y / λ x/ λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] -1.25-0.75-0.25 0.25 0.75 1.25-2 -1 0 1 2 y / λ x/ λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] ( i ) ( l ) ( m ) ( n ) Figure 6.
Numerical Assessment ( M = 5 , N = 8 , d = 0 . λ ; S = 1 , L = 2 ; Q = 34 ) - Two-level distributions of the equivalent element-level amplitudeexcitations { α ( q ) m,n ; m = 1 , ..., M ; n = 1 , ..., N ; q = 1 , ..., Q }, of the global best tiling configurations ( Q S = 32 and Q L = 2 ). -40-35-30-25-20-15-10-5 0 -1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] u=sin( θ )cos( φ ) φ = 0 [deg]ETM-1ETM-2ETM-3 ETM-4ETM-5ETM-6 ETM-7ETM-8ETM-9 ETM-10ETM-11ETM-12 ( a ) -40-35-30-25-20-15-10-5 0 -1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] v=sin( θ )sin( φ ) φ = 90 [deg]ETM-1ETM-2ETM-3 ETM-4ETM-5ETM-6 ETM-7ETM-8ETM-9 ETM-10ETM-11ETM-12 ( b ) Figure 7.
Numerical Assessment ( M = 5 , N = 8 , d = 0 . λ ; S = 1 , L = 2 ; Q = 34 : Q S = 32 , Q L = 2 ) - Cuts of the normalized power patternalong the principal planes, ( a ) φ = 0 [deg] and ( b ) φ = 90 of the global besttiling configurations in Fig. 6. -18.6-18.5-18.4-18.3-18.2-18.1-18-17.9-17.8 1 10 100 1000 F i t ne ss Φ ( k ) [ d B ] Iteration Index, kIGA-1IGA-2IGA-3 IGA-4IGA-5IGA-6 IGA-7IGA-8IGA-9 IGA-10
Figure 8.
Numerical Assessment ( M = 15 , N = 20 , d = 0 . λ ; S = 1 , L = 2 ; P = 42 , ρ c = 0 . , ρ m = 0 . , K = 10 ; O = 10 ) - Behaviourof the fitness of the best individual, Φ optk , versus the iteration index k ( k =1 , ..., K ) for each o -th ( o = 1 , ..., O ) MS-OTM ( IGA -based) run.
C. Multi-Square Optimization-Based Tiling Method (
MS-OTM ) The goal of using an heuristic optimization approach forminimizing (5) is dual. First, the reduction of the number offunctional evaluation with respect to an exhaustive search toenable the synthesis of large apertures, which are generallyconsidered in realistic applications. Second, the convergence- with high probability - to the optimal tiling (i.e., c opt = arg { min c u [Φ ( c u ) ; u = 1 , ..., U ] } ) or to a solutionfitting the requirements within a user-defined tolerance η [i.e., c opt : Φ ( c opt ) ≤ η ]. Towards this end, the proposed MS-OTM is based on an integer-coded Genetic Algorithm(
IGA ) where a trial tiling is coded into an integer string t = n t ˆ m ∈ [1 : H ] ; ˆ m = 1 , ..., ˆ M y o whose ˆ m -th entry is the This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2020.2970088Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 7 index of an element of the row dictionary A [Fig. 3( b )] as wellas a node of the solution graph G [Fig. 3( c )]. More specifically,the following procedural steps are performed: • Step 0:
Virtual Aperture Set-up - If
S > , then rescale S times the size of the aperture R to yield the virtualaperture ˆ R (i.e., M → ˆ M and N → ˆ N ) as well as thearea of the two-sizes tiles (i.e., σ S → ˆ σ S and σ L → ˆ σ L ); • Step 1:
Solution Space Generation -
Generate the rowdictionary A with H elements of length ˆ N x that satisfythe existence condition s (6). Execute the IGA -based iter-ative ( k being the iteration index) optimization loop: • Step 2:
Population Initialization ( k = 0 ) - Randomlygenerate P individuals, { t ( p ) k k k =0 ; p = 1 , ..., P }, asfollows n t ( p )ˆ m,k k k =0 = r ( ˆ m ) ; ˆ m = 1 , ..., ˆ M y o ; p = 1 , ..., P (10) r ( ˆ m ) being an integer number randomly-selected withinthe interval [1 : H ] with uniform probability. If the rowsof the p -th ( p ∈ [1 : P ] ) individual do not satisfy the feasibility assignment condition (i.e., the ˆ M y nodes codedin t ( p ) k are not connected by an edge in the solution graph G ), then replace it with another one - still randomly-generated - until the initial population is completed withall P feasible tiling configurations; • Step 3:
Fitness Evaluation - Define the tiling matrix A ( p ) k ( p = 1 , ..., P ) from the individual t ( p ) k ( p = 1 , ..., P ) bysetting each ˆ m -th ( ˆ m = 1 , ..., ˆ M y ) row as follows a ˆ m ⌋ k = a t ( p )ˆ m,k . (11)Generate the corresponding trial clustering c ( p ) k ( p =1 , ..., P ) on the physical aperture R by first positioningthe ˆ σ S and ˆ σ L virtual tiles in the virtual aperture ˆ R according to the tile-placement rules and then mappingthe arising membership vector b c ( p ) k on R by applying (8).Afterwards, set the fitness of each p -th ( p = 1 , ..., P ) trialsolution, Φ ( p ) k , to the value of (5) Φ ( p ) k = Φ (cid:16) c ( p ) k (cid:17) (12) β ( p ) k being computed through (3). If k = 0 , then go toStep 5; • Step 4:
Reproduction Cycle - Apply the roulette-wheel selection, the single-point crossover with probability ρ c ,and the mutation with probability ρ m [39] - properlycustomized to deal with integer-coded chromosomes - togenerate a new population, { t ( p ) k ; p = 1 , ..., P }. For each p -th integer string t ( p ) k check whether the feasibility as-signment condition holds true, otherwise update the newtrial solution by re-applying the IGA operators. Enforcethe elitism mechanism [39] to keep the best solutionfound so far within the current k -th population ( c bestk =arg n min c ( p ) l h Φ (cid:16) c ( p ) l (cid:17) ; p = 1 , ..., P i ; l = 1 , ..., k o ); • Step 5:
Convergence Check - Stop the iterative processif the maximum number of iterations K is reached (i.e., k = K ) or the fitness of the best individual Φ optk ( Φ optk , -3.75-2.25-0.75 0.75 2.25 3.75-5 -4 -3 -2 -1 0 1 2 3 4 5 y / λ x/ λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] -3.75-2.25-0.75 0.75 2.25 3.75-5 -4 -3 -2 -1 0 1 2 3 4 5 y / λ x/ λ - π - π /2 0 π /2 π E xc i t a t i on P ha s e [ r ad ] ( a ) ( b ) -3.75-2.25-0.75 0.75 2.25 3.75-5 -4 -3 -2 -1 0 1 2 3 4 5 y / λ x/ λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] -3.75-2.25-0.75 0.75 2.25 3.75-5 -4 -3 -2 -1 0 1 2 3 4 5 y / λ x/ λ - π - π /2 0 π /2 π E xc i t a t i on P ha s e [ r ad ] ( c ) ( d ) -5 -4 -3 -2 -1 0 1 2 3 4 5x/ λ -3.75-2.25-0.75 0.75 2.25 3.75 y / λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] -5 -4 -3 -2 -1 0 1 2 3 4 5x/ λ -3.75-2.25-0.75 0.75 2.25 3.75 y / λ - π - π /2 0 π /2 π E xc i t a t i on P ha s e [ deg ] ( e ) ( f ) Figure 9.
Numerical Assessment ( M = 15 , N = 20 , d = 0 . λ , ( θ , φ ) =(8 , [deg]; S = 1 , L = 2 ) - Distribution of the element-level ( a )( c )( e )amplitude and ( b )( d )( f ) phase excitations of ( a )( b ) the MS-OTM ( P = 42 , ρ c = 0 . , ρ m = 0 . , K = 10 ) optimized MSTA ( Q = 150 : Q L = 50 , Q S = 100 ), ( c )( d ) the reference FPA ( SLL = − [dB]), and ( e )( f ) the D-OTM optimized ( P = 326 , ρ c = 0 . , ρ m = 0 . , K = 10 ) DTA ( Q = 150 : Q H = 82 , Q V = 68 ). min p =1 ,..,P n Φ ( p ) k o ) is below η , then output c opt ( c opt = c bestk ) and β opt (3). Otherwise, update the iteration index( k ← k + 1 ) and go to Step 3.IV. N UMERICAL V ALIDATION
The proposed isophoric multi-square tiled architecture andthe
MS-OTM synthesis strategy are analyzed and assessed ina representative set of numerical examples concerned withdifferent apertures and design constraints. More specifically,the
MS-OTM approach is firstly validated in a small arraybenchmark case to prove its effectiveness and reliability toconverge towards the global-optimum/best-solution of the op-timization/synthesis problem at hand. Then, a comparison withdifferent tiling layouts composed of domino tiles with non-isophoric feeding [29] is presented to give some indicationson the pros and the cons of adopting an isophoric multi-squaretiling of the array aperture. Afterwards, two realistic antennadesigns for space applications are reported and discussed.The first example deals with a rectangular aperture hosting M × N = 5 × isotropic [i.e., e m,n ( θ, φ ) = 1 ] elementsspaced by d x = d y = 0 . λ , while the two square tiles, σ s and σ l , contain γ S = 1 × and γ L = 2 × elements (i.e., S = 1 and L = 2 ) so that the physical aperture R coincideswith the virtual one ˆ R . As for the goal of the synthesis, it hasbeen chosen to minimize the SLL of the radiated power pattern
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2020.2970088Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 8 ( a ) ( b ) -40-35-30-25-20-15-10-5 0-1 -0.5 0 0.5 1 N o r m a li z ed P o w e r [ d B ] u=sin( θ )cos( φ )u=0.5Reference Domino Tiles Square Tiles -40-35-30-25-20-15-10-5 0-1 -0.5 0 0.5 1 N o r m a li z ed P o w e r [ d B ] v=sin( θ )sin( φ )v=0.5Reference Domino Tiles Square Tiles ( c ) ( d ) Figure 10.
Numerical Assessment ( M = 15 , N = 20 , d = 0 . λ , ( θ , φ ) =(8 , [deg]) - Color-maps of the normalized power pattern radiated by ( a )the MS-OTM optimized ( P = 42 , ρ c = 0 . , ρ m = 0 . , K = 10 ) MSTA ( S = 1 , L = 2 ; Q = 150 : Q L = 50 , Q S = 100 ) and ( b ) the D-OTM optimized ( P = 326 , ρ c = 0 . , ρ m = 0 . , K = 10 ) DTA ( Q = 150 : Q H = 82 , Q V = 68 ). Cuts of the normalized power patterns ofthe MSTA / DTA / FPA along ( c ) the u = 0 . and ( d ) the v = 0 . planes. when pointing along broadside. Accordingly, the weightingcoefficients and the SLL constraint in (5) have been set to w sl = 1 , w bw = 0 , and η SLL = 0 , respectively.Since the number of elements of the row dictionary A is equalto H = 34 , the cardinality of the solution space turns outto be computationally tractable [ U = 16334 (9)]. Thus, thewhole set of U admissible aperture-tilings has been generatedand an exhaustive enumerative evaluation has been performedto identify the global optimum tiling. More specifically, theMulti-Square Exhaustive Tiling Method ( MS-ETM ) [40] hasbeen used to exhaustively explore the solution graph G bycomputing the fitness of all array partitionings, as well. Onan Intel 1.6GHz i5CPU, 8GB of RAM computer platform,the generation of all admissible tilings required less than seconds, while the evaluation of the corresponding costfunction values took [h:min] by sampling the angular ( u, v ) domain with × samples ( u , sin θ cos φ , u , sin θ sin φ ). Figure 5 shows the fitness of the array layoutssorted from the worst to the best. This latter correspondsto a SLL equal to
SLL opt = − . [dB] and it hasbeen reached by different tiling configurations, all having Q = 34 tiles: Q S = 32 of type σ S and Q L = 2 of type σ L . The most representative ones (i.e., the remaining bestarrangements can be obtained from these by either a clockwiseor a counter-clockwise rotation of [deg]) are reported inFig. 6 by plotting the two-level distribution of the equivalentelement-level amplitude excitations { α ( q ) m,n ; m = 1 , ..., M ; n = 1 , ..., N ; q = 1 , ..., Q }. For completeness, the cuts ofthe power pattern along the principal planes, φ = 0 [deg][Fig. 7( a )] and φ = 90 [deg] [Fig. 7( b )], are given in Fig. 7.The proposed MS-OTM approach has been then run to evaluate
Table II
Numerical Assessment ( M = 5 , N = 8 , d = 0 . λ ; S = 1 , L = 2 ) - SLL S TATISTICS . Min Max. Avg. Var.
SuccessRate[dB] [dB] [dB] - [%]
MS-ETM − . − . − .
57 0 . − MS-OTM - P = 4 − . − . − .
51 0 .
24 42
MS-OTM - P = 8 − . − . − .
83 0 .
18 78
MS-OTM - P = 12 − . − . − .
06 0 .
00 100 the successful rate of the
IGA to converge to one among thebest solutions. More in detail, three different population sizeswith P = {
1; 2; 3 } × V individuals, V ( V , ˆ M y , ˆ M y = 4 )being the number of problem unknowns, have been taken intoaccount and the other IGA parameters have been set to K =100 (maximum number of iterations), ρ c = 0 . (crossoverprobability), ρ m = 0 . (mutation probability) [39]. Due to thestochastic nature of GA s, O = 100 different optimizations havebeen executed for each setup of the control parameters and theresults are summarized in Tab. II. As expected, the successrate increases with the population size and the optimizationprocess always converges to low values of the cost function(worst/max SLL = − . [dB] - Tab. II) within the wholeset of admissible values (Fig. 5). When using a populationof P = 12 (i.e., P = 3 × V ), the MS-OTM is always ableto find an optimal solution (Tab. II) with P × K = 1200 evaluations of the cost function. Such an amount correspondsto the . of the cardinality of the solution space, U , and itpoints out the non-negligible saving of CPU -time with respectto the
MS-ETM without loss of optimization performance.The second example is devoted to compare the
MSTA ar-chitecture, synthesized with the
MS-OTM approach, withthe
Domino-Tiled Array ( DTA ) layout designed with theoptimization-based tiling method (
D-OTM ) presented in [29]where single-shaped domino-like tiles were used. Towards thisend, a rectangular array of M × N = 15 × isotropic elementsspaced by d x = d y = 0 . λ has been taken into account.As in the previous test case, square tiles with S = 1 and L = 2 as well as the minimization of the SLL (i.e.,
Φ ( c ) , SLL [ P ( θ, φ ; c )] × H [ | SLL [ P ( θ, φ ; c )] | ] ), but along thepointing direction ( θ , φ ) = (8 , [deg], have been chosento complete the definition of the benchmark. Moreover, themaximum number of sub-arrays has been limited to M × N (i.e., Q ≤ M × N ) for a fair comparison between the MSTA arrangement and the
DTA one [29].Concerning the
MSTA design, since the number of nodes inthe solutions graph G turns out to be equal to H = 10964 ,the solution space has a dimension, U > , which isout of the computational feasibility of the EM-ETM , thus thesynthesis process has been performed only with the
MS-OTM .Accordingly, a set of O = 10 different IGA runs have beenexecuted by setting the
IGA parameters as in the first example(i.e., P = 3 × V , ρ c = 0 . , and ρ m = 0 . ) except for thenumber of iterations ( K = 10 vs. ), because of the widerdimension of the aperture at hand and the corresponding higherdimensionality of the solution space. This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2020.2970088Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 9
Figure 11. Sketches of a deployable implementation of a × MSTA on a6-Units Cube-Sat payload [5]. Table III
Numerical Assessment ( M = 15 , N = 20 , d = 0 . λ , ( θ , φ ) = (8 , [ DEG ]; S = 1 , L = 2 ) - P ATTERN AND LAYOUT FEATURES . SLL D HP BW az HP BW el T RM [dB] [dBi] [deg] [deg] -
FPA − .
00 27 .
69 5 .
44 7 .
34 300
MSTA − .
54 28 .
76 5 .
60 7 .
50 150
DTA − .
12 27 .
11 5 .
49 7 .
35 150
Figure 8 shows the evolution of the fitness of the bestindividual, Φ optk ( k = 1 , ..., K ) for each o -th ( o = 1 , ..., O )simulation. It can be noticed that the values of the SLL at theconvergence ( k = K ) are very close to that of the best run(i.e. GA -8, Φ opt = − . [dB]) when the MSTA is composedof Q L = 50 tiles of type σ L and Q S = 100 tile of type σ S ( Q = 150 ), while the equivalent element-level amplitudes,{ α ( q ) m,n ; m = 1 , ..., M ; n = 1 , ..., N ; q = 1 , ..., Q }, and phases,{ β ( q ) m,n = β q (3); m = 1 , ..., M ; n = 1 , ..., N ; q = 1 , ..., Q },are distributed as in Fig. 9( a ) and Fig. 9( b ), respectively, toafford the power pattern shown in Fig. 10( a ). As expected, thelarger tiles are located at the edges of the aperture, while thesmaller clusters turn out to be at the center to yield a taperingof the amplitude distribution over the antenna aperture so that SLL = − . [dB] and the directivity is equal to D = 28 . [dBi] (Tab. III).As for the DTA , the binary GA -based D-OTM [29] has beenapplied with the same parameter setup of the
MS-OTM , butusing a larger population (i.e., P = 326 ) because of the greaterlength of the binary-encoded chromosomes since here thenumber of unknowns amounts to V DT A = 747 (vs. V MST A =14 ). Moreover, the non-isophoric sub-array feeding of the
DTA has been determined through [41] by approximating the exci-tations [Figs. 9( c )-9( d )] of a reference FPA of M × N elementsaffording a beam steered towards ( θ , φ ) = (8 , [deg] with SLL = − [dB]. The resulting domino-tiles arrangementand the corresponding amplitude and phase distributions areshown in Figs. 9( e )-( f ), while the radiated power pattern isplotted in Fig. 10( b ). By comparing the pattern descriptorsin Tab. III, it is worth noting that the domino-tiled arrayaffords higher secondary lobes ( SLL
DT A = − . [dB] vs. SLL
MST A = − . [dB] - Tab. III) as well as a lowerdirectivity ( D DT A = 27 . [dBi] vs. D MST A = 28 . [dBi]- Tab III). Indeed, the use of clusters of radiating elementsand the beam broadening cause a decrease of directivity whenfocusing the beam out of the broadside direction which in thiscase amounts to D loss = 0 . [dBi], being D MST A = 28 . [dBi] the directivity of the broadside directed beam when -25-20-15-10-5 0 5 10 15 20 25-25 -20 -15 -10 -5 0 5 10 15 20 25 y / λ x/ λ N o r m a li z ed E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] ( a ) ( b ) -50-40-30-20-10 0-1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] v=sin( θ )sin( φ ) φ = 90 [deg]Ideal MSTAReal MSTAMask -25-20-15-10-5 0-0.1 -0.05 0 0.05 0.1 -50-40-30-20-10 0-1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] u=sin( θ )cos( φ ) φ = 0 [deg]Ideal MSTAReal MSTAMask -25-20-15-10-5 0-0.1 -0.05 0 0.05 0.1 ( c ) ( d ) Figure 12.
Numerical Assessment ( M = 90 , N = 90 , d = 0 . λ , ( θ , φ ) =(0 . , . [deg]; S = 6 , L = 12 ; Q = 129 : Q L = 32 , Q S = 97 ) - Plots of( a ) the equivalent element-level amplitude distribution and of the normalizedpower pattern radiated ( b ) in the whole angular range ( − ≤ u ≤ , − ≤ v ≤ ) and along ( c ) the φ = 90 [deg] and ( d ) the φ = 0 [deg] planes.Table IV Numerical Assessment ( M = 90 , N = 90 , d = 0 . λ , ( θ , φ ) = (0 . , . [ DEG ]) - P
ATTERN FEATURES . SLL HP BW az HP BW el G [dB] [deg] [deg] [dBi] Requirements < − . < . < . > . Ideal MSTA − .
17 1 .
17 1 .
17 43 . Real MSTA − .
04 1 .
16 1 .
17 43 . ( θ , φ ) = (0 , [deg]. For the sake of completeness, the cutsof the DTA / MSTA power patterns along the principal planescrossing the main lobe are reported in Fig. 10( c ) [ u = 0 . ]and Fig. 10( d ) [ v = 0 . ].The synthesis problem addressed in the third example has beenstated starting from the design guidelines for a precipitationradar system used in a recent CubeSat scientific mission [5].More specifically, the requirements @ . [GHz] were: ( a ) G > [dBi], G being the broadside beam peak gain; ( b ) HP BW < . [deg] along both azimuth and elevation; (c) SLL < − [dB] (Tab. IV). Accordingly, a square aperture R of M × N = 90 × λ -spaced elements has been considered(i.e., R extends over an area of . λ × . λ ) for a twofoldreason. On the one hand, to assure an ideal directivity of D ≃ . [dBi] when uniformly-tapering the excitations of an arrayof isotropic sources. On the other, to keep the whole array size(i.e., × Units) suitable for a deployable implementation ona 6-Unit cube-sat payload [5] as shown in Fig. 11. Moreover,the number of elementary radiators of the two
MSTA tiles hasbeen chosen equal to γ S = 6 × and γ L = 12 × (i.e., S = 6 , L = 12 → b L = 2 ; V , ˆ M y = 14 ), respectively.Just running the MS-OTM O = 10 times, a solution with costfunction value equal to Φ opt = 4 . × − has been foundwhose radiated power pattern [Fig. 12( b )] has SLL = − . This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2020.2970088Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 10 ( a ) ( c )( b ) Figure 13.
Numerical Assessment ( Real MSTA ; M = 90 , N = 90 , d =0 . λ , ( θ , φ ) = (0 . , . [deg]; S = 6 , L = 12 ; Q = 129 : Q L = 32 , Q S = 97 ) - Plots of ( a ) the embedded element pattern of the real elementary-radiator in the presence of ( b ) × equal neighbouring array-elements andof ( c ) the normalized power pattern radiated by the MSTA . [dB] and HP BW az = HP BW el = 1 . [deg], while thedirectivity turns out to be D = 43 . [dB] (Tab. IV). Theoptimized tiling configuration contains Q = 129 tiles ( Q L =32 and Q S = 97 ) as shown in Fig. 12( a ) where the equivalentelement-level amplitudes are reported. For completeness, thetop-view representation of the power pattern and its behavioralong the two principal cuts are shown in Fig. 12( b ) and Figs.12( c )-( d ), respectively.The MSTA layout in Fig. 12( a ) has been also validated ina realistic implementation when non-ideal radiating elementsare used. The antenna has been assumed to be an array oflinearly polarized pin-fed circular patches working in the Ka-band around . [GHz] with patch diameter equal to . [mm], feed offset set to . [mm], and a substrate of thickness . [mm] having tan δ = 1 . × − and a relative permittivity ǫ = 2 . . The mutual coupling effects as well as other elec-tromagnetic phenomena among the array elements have beenmodeled by computing the embedded element pattern e ( θ, φ ) [Fig. 13( a )], which has been set identical for all radiators[i.e., e mn ( θ, φ ) = e ( θ, φ ) ; m = 1 , ..., M ; n = 1 , ..., N ],in the presence of two rings of neighboring elements [Fig.13( b )]. The full-wave simulated radiation pattern is given inFig. 13( c ), while the values of the power pattern descriptors arereported in Tab. IV for a quick comparison with those fromthe ideal array and the project requirements. The outcomesare very positive since the application requirements are stillsatisfied (Tab. IV) and the pattern deviations from the idealcase are minimal [Fig. 13( c ) vs. Fig. 12( b ); Figs. 12( c )-12( d )]especially around the main beam zone as pointed out in theinsets of Fig. 12( c ) [ − . ≤ v ≤ . ( φ = 90 [deg])] and Fig.12( d ) [ − . ≤ u ≤ . ( φ = 0 [deg])].The last test case deals with the synthesis of an isophoric ( a )( b ) Figure 14.
Numerical Assessment ( Real MSTA ; d = 0 . λ ) - Plot of ( a ) theembedded element pattern of the real elementary-radiator in the presence of( b ) × equal neighbouring square open-ended waveguides. phased array for satellite communications with the samerequirements of [14][22], but exploiting a square apertureinstead of a circular one. More in detail, the maximumantenna size, namely its extension along the diagonal, hasbeen constrained to be smaller than λ @ . [GHz].Moreover, the array has been designed to scan the beam withinthe range ≤ θ ≤ . [deg] with a directivity patternfitting the following conditions: ( i ) D ( θ, φ ) ≥ . [dBi]for θ ≤ θ ≤ θ EoC , θ EoC being the angle at the edge-of-coverage (
EoC ) direction with respect to boresight; ( ii ) D ( θ, φ ) ≤ D (cid:0) θ + θ EoC , φ (cid:1) − [dB] (i.e., SLL ( θ, φ ) ≤− [dB], being SLL ( θ, φ ) the SLL evaluated with respectto the
EoC angle) when ( θ + θ ) ≤ θ ≤ ( θ + θ ) , θ and θ being the EoC angle of the nearest “iso-color” beam [22]and the (maximum) inside Earth angle [22]; ( iii ) D ( θ, φ ) ≤ D (cid:0) θ + θ EoC , φ (cid:1) − [dB] (i.e., SLL ( θ, φ ) ≤ − [dB])for θ ≥ θ . Towards this end, the MS-OTM process has beencarried out by customizing (5) so that
Φ ( c ) , max ( θ ,φ ) ∈ ζ { SLL [ P ( θ , φ ; c )] } (13)where ζ , { ≤ θ ≤ . [deg]; ≤ φ ≤ [deg]}.The array has been assumed to be arranged on a lattice of N × M = 78 × positions spaced by d x = d y = 0 . λ (i.e., an aperture R of dimensions . λ × . λ havingmaximum diagonal dimension of about . λ , then smallerthan λ ), while square open-ended waveguides of size . λ along the x and y axis, whose embedded elementpattern - when considering the electromagnetic interactions oftwo rings of neighboring elements [Fig. 14( b )] - is shown inFig. 14( a ), have been chosen as radiating elements.By changing different values of L and S and setting the IGA population size to P = 3 × V , the best trade-off (requirement-fitting vs. MSTA -complexity, Q ) solution synthesized by the MS-OTM turned out to be that with S = 3 and L = 9 ( → ˆ L = 3 ) shown in Fig. 15( a ). Such a MSTA contains Q = 372 This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2020.2970088Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 11 -33-22-11 0 11 22 33 -33 -22 -11 0 11 22 33 y / λ x/ λ N o r m a li z ed E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] ( a ) ( b ) -50-40-30-20-10 0-1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] v=sin( θ )sin( φ ) φ = 90 [deg]Real MSTAMask -50-40-30-20-10 0-1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] u=sin( θ )cos( φ ) φ = 0 [deg]Real MSTAMask ( c ) ( d ) Figure 15.
Numerical Assessment ( Real MSTA ; M = 78 , N = 78 , d =0 . λ ; S = 3 , L = 9 ; Q = 372 : Q L = 38 , Q S = 334 ) - Plots of ( a ) theequivalent element-level amplitude distribution and of the normalized powerpattern ( b ) in the whole angular range ( − ≤ u ≤ , − ≤ v ≤ ) andalong ( c ) the φ = 90 [deg] and ( d ) the φ = 0 [deg] planes when steering thebeam along ( θ , φ ) = (0 . , . [deg]. tiles ( Q L = 38 and Q S = 334 ), which means a savingof .
89 %
TRM s with respect to a
FPA . The correspondingpower pattern when the beam is steered at ( θ , φ ) = (0 , [deg] and its principal cuts are given in Fig. 15( b ) and Figs.15( c )-( d ), respectively. Moreover, the values of the sub-arrayphases [Figs. 16( a )-( b )] and the pattern cuts [Figs. 16( c )-( d )]when steering the main lobe towards ( θ , φ ) = (1 . , . [deg] [Fig. 16( a ) and Fig. 16( c )] and ( θ , φ ) = (1 . , . [deg] [Fig. 16( b ) and Fig. 16( d )] are reported, as well. Besidesthe pictorial proofs of the fitting of the project requirements[Figs. 15( c )-15( d ); Figs. 16( c )-16( d )], the reliability of thesynthesized MSTA is also quantitatively confirmed by thenumerical values of the pattern features. As a matter of fact,both the maximum value of
SLL ( θ , φ ) and the minimumvalue of D (cid:0) θ + θ EoC , φ (cid:1) within the angular region ζ fulfillthe project constraints (i.e., max ( θ ,φ ) ∈ ζ { SLL ( θ , φ ) } = − . [dB] and min ( θ ,φ ) ∈ ζ (cid:8) D (cid:0) θ + θ EoC , φ (cid:1)(cid:9) ≥ . [dBi]). As for the maximum directivity loss with respect tothe broadside case, it turns out to be D loss = 0 . [dBi],being D = 46 . [dBi] the peak directivity of the broadsidedirected beam and D = 46 . [dBi] the peak directivity ofthe beam directed towards ( θ , φ ) = (1 . , . [deg].V. C ONCLUSIONS AND R EMARKS
This work has addressed the synthesis of rectangular phasedarrays tiled with two-sized square tiles, each one controlledwith a single isophoric amplifier and a single phase shifter,fitting user-defined constraints on the radiation pattern. Aninnovative tile-size tapering technique has been introducedand exploited to control the
SLL by optimizing the tilingconfiguration. -33-22-11 0 11 22 33 -33 -22 -11 0 11 22 33 y / λ x/ λ - π - π /2 0 π /2 π E xc i t a t i on P ha s e [ r ad ] -33-22-11 0 11 22 33 -33 -22 -11 0 11 22 33 y / λ x/ λ - π - π /2 0 π /2 π E xc i t a t i on P ha s e [ r ad ] ( a ) ( b ) -50-40-30-20-10 0-1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] u=sin( θ )cos( φ ) φ = 0 [deg]Real MSTAMask -50-40-30-20-10 0-1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] v=sin( θ )sin( φ ) φ = 90 [deg]Real MSTAMask ( c ) ( d ) Figure 16.
Numerical Assessment ( Real MSTA ; M = 78 , N = 78 , d =0 . λ ; S = 3 , L = 9 ; Q = 372 : Q L = 38 , Q S = 334 ) - Plots of ( a )( b ) theequivalent element-level phase distribution and of the cuts of the normalizedpower pattern along ( c ) the φ = 0 [deg] and ( d ) the φ = 90 [deg] planeswhen steering the beam towards ( a )( c ) ( θ , φ ) = (1 . , . [deg] and( b )( d ) ( θ , φ ) = (1 . , . [deg]. The following main methodological novelties with respect tothe state-of-the-art methods have been proposed: • the design of rectangular isophoric phased arrays com-posed of two square tiles of different sizes to yield anadvanced control of the beam-pattern features through anamplitude tile-size tapering of the single-element excita-tions; • the exploitation of mathematical theorems from thecombinatorial theory to provide the conditions for thecomplete tileability of a rectangular aperture with tilescomposed of two square clusters of elementary radiators; • a suitable integer coding of the admissible tilings thatallows a drastic reduction of the cardinality of the solutionspace; • the compact representation of the solution space throughan acyclic graph; • the development of an innovative IGA for an efficientexploration of the solution space/graph to effectively dealwith large arrays, as well.Representative examples from a wide set of numerical experi-ments concerned with ideal as well as real radiating elementshave been discussed. From such a numerical assessment, thefollowing outcomes can be drawn: • the IGA -based
MS-OTM guarantees a high success rateto converge towards solutions equal or close to the globaloptimum ones; • the proposed tile-size tapering technique and the related MSTA architecture turn out to be effective in the controlof the pattern features to fit real-problem specificationsand requirements such as those from space (sensing andcommunications) applications.Future research activities, beyond the scope of the current
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2020.2970088Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 12 work, will be aimed at improving the computational efficiencyof the solution space sampling by exploiting combinatorialoptimizer intrinsically designed for exploring graphs-modeledspaces [42]. Moreover, the whole synthesis framework willbe extended to planar arrays having arbitrary aperture shapes,not fully tileable apertures, as well as multiple ( > )square sizes to evaluate the trade-off between modularity-maintenance/manufacturing costs and array performance. Appendix I
The adjacency matrix of the solution graph G is the matrix G , { g ij ; i, j = 1 , ..., H } of size H × H whose entries areequal to g ij = 1 when the node a i is connected to the node a j and g ij = 0 , otherwise [38].A CKNOWLEDGEMENTS
A. Massa wishes to thank E. Vico for her never-endinginspiration, support, guidance, and help.R
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