Optimal Excitation Matching Strategy for Sub-Arrayed Phased Linear Arrays Generating Arbitrary-Shaped Beams
IIEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 1
Optimal Excitation Matching Strategy forSub-Arrayed Phased Linear Arrays GeneratingArbitrary Shaped Beams
P. Rocca,
Senior Member, IEEE , L. Poli,
Member, IEEE , A. Polo,
Member, IEEE , and A. Massa,
Fellow, IEEE
Abstract —The design of phased arrays able to generate ar-bitrary shaped beams through a sub-arrayed architecture ishere addressed. The synthesis problem is cast in the excitationmatching framework so as to yield clustered phased arraysproviding optimal trade-offs between the complexity of the arrayarchitecture (i.e., the minimum number of control points at thesub-array level) and the matching of a reference pattern. A syn-thesis tool based on the k-means algorithm is proposed for jointlyoptimizing the sub-array configuration and the complex sub-array coefficients. Selected numerical results, including pencilbeams with sidelobe notches and asymmetric lobes as well asshaped main lobes, are reported and discussed to highlight thepeculiarities of the proposed approach also in comparison withsome extensions to complex excitations of state-of-the-art sub-array design methods.
Index Terms —Phased Array, Linear Array, Sub-Arraying,Excitation Matching, K-means Algorithm, Arbitrary ShapedBeams.
I. I
NTRODUCTION P HASED array antennas ( PA s) are a key enabling technol-ogy for modern communications and radar systems [1][2].Thanks to the high speed scanning, the easy reconfiguration,and the multi-function capability, they are suitable for anumber of civil, commercial, and military applications of great Manuscript received August 0, 2019; revised January 0, 2020This 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 ProgramPRIN2017 (CUP: E64I19002530001) for the Project "CYBER-PHYSICALELECTROMAGNETIC VISION: Context-Aware Electromagnetic Sensingand Smart Reaction (EMvisioning)" (Grant no. 2017HZJXSZ), and the Project"Antenne al Plasma - Tecnologia abilitante per SATCOM (ASI.EPT.COM)"funded by the Italian Space Agency (ASI) under Grant 2018-3-HH.0 (CUP:F91I17000020005).P. Rocca, L. Poli, A. Polo, and A. Massa are with the ELEDIA Re-search Center (ELEDIA@UniTN - University of Trento), Via Sommarive 9,38123 Trento - Italy (e-mail: {paolo.rocca, lorenzo.poli, alessandro.polo.1,andrea.massa}@unitn.it)P. Rocca is also with the ELEDIA Research Center (ELEDIA@XIDIAN -Xidian University), 3P.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 -UESTC), School of Electronic Engineering, Chengdu 611731 - 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]) interest including 5G [3][4], small-satellite communications[5], and anti-collision systems for autonomous driving [6].Due to the increasing demand of lower costs towards the mass-market production, while fully-populated PA s ( FPA s) turn outto be unaffordable architectures [7] since each element ofthe array is equipped with a transmit-receive module (
TRM )including amplifiers and phase/time delays, unconventionalphased arrays (
UPA s) characterized by an irregular inter-element spacing and/or the aperiodic sub-arraying and/or thespace-time modulation [8] have led to alternative solutionswith good trade-offs between the complexity (i.e., the costs)and radiation performance.Among
UPA s, irregular sub-arrays have received a widerattention because of the capability of mitigating quantizationand grating lobes, while still yielding a high aperture efficiency[9][10]. When designing an irregular sub-arrayed PA , twosets of degrees-of-freedom ( DoF s) have to be determined:the membership of each array element to a cluster and thevalues of the beam-forming coefficients at the sub-array level.With reference to an N -element array with Q sub-array ports,the cardinality of the solution space of all possible sub-arrayconfigurations amounts to Q N . As a matter of fact, eachelement can be potentially grouped in one of the Q clusters. Itfollows that testing all possible sub-array aggregations is com-putationally unfeasible also for arrays with few elements. Forthis reason, only small-scale clustered arrays have been origi-nally dealt with by iteratively determining the sub-array coef-ficients through the pseudo-inversion of an over-determinedsystem of linear equations for an a-priori given clustering[11]. Afterwards, nature-inspired optimization algorithms havebeen profitably adopted to exploit their effective samplingcapabilities. For instance, the Simulated Annealing [12], theGenetic Algorithms [13][14], and the Differential Evolution[15][16] have been used. Hybrid strategies, which integrateglobal and local optimization techniques, able to synthesizeoptimal beam-forming weights of a clustered layout have beenproposed [17][18], as well. However, despite the advancedexploration features supported by the growing computationalresources and the capacity to prevent local minima (i.e., sub-optimal solutions) related to the non-convexity of the problemwith respect to the sub-array configurations, the use of globaloptimizers is strongly penalized by the slow convergence rate.Therefore, once again, the design of clustered arrangementshas been confined to small and medium size apertures.An effective synthesis tool, named Contiguous PartitionMethod ( CPM ), for synthesizing the cluster layout and 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.2020.2972641Copyright (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 corresponding sub-array coefficients also for large arrays hasbeen proposed in [19]. By casting the clustered array designwithin the excitation-matching framework and exploiting theFisher’s grouping theory [20], the cardinality of the synthesisproblem has been reduced to the size of the space of the so-called contiguous partitions [19] equal to the binomial (cid:0) N − Q − (cid:1) . Thanks to a suitable representation of the solution space, firstthrough a non-complete binary tree and then as a direct acyclicgraph [21], where each path codes a feasible/contiguous par-tition, the design of large sub-arrayed arrays has been madeaffordable. Such an approach has been further improved interms of efficiency and success rate in reaching the optimalsolution [22] with a customized integration of an Ant ColonyOptimization. Furthermore, the design of sub-array architec-tures supporting multiple patterns has been addressed [23], aswell. In all these works, the CPM has been used to synthesizethe sub-array amplitudes of PA s along with the clusteringconfiguration. More recently, the approach has been extendedto optimize the sub-array phases [24][25]. However, eitheramplitude or phase excitations have been considered so far , but never both quantities together. Indeed, although the CPM guarantees the minimum - in the least-square sense - matchingof the set of ideal/reference (i.e., independent for each element)excitations, these latter must be real-valued so that they canbe ordered along a line [20].This work presents a novel and optimal strategy for designingsub-array PA s that goes beyond such a limitation to allowthe excitation-matching, still optimal in the least-square sense,of complex-valued reference weights and, thus, to enablethe synthesis of sub-arrays affording arbitrary-shaped beampatterns [26]-[28]. Towards this end, first two extensions ofthe real-valued CPM to complex coefficients are presented tohighlight the key concept of contiguous partition as well as thelimitations of its more simple customizations and the need ofa more proper definition of contiguity in the complex plane.Then, the k-means algorithm is exploited for clustering thearray elements, while the sub-array coefficients are analyticallyderived [19][25]. The main motivation of using the k-means procedure is that it is a natural consequence of the Fisher’stheory [20] for two-dimensional/complex domains as well asthe high convergence rate [29].The remaining of the paper is organized as follows. Themathematical formulation of the clustered synthesis is reportedin Sect. 2, where the proposed design technique is alsodescribed still within the excitation-matching framework. Arepresentative numerical analysis is then (Sect. 3) carried outthrough the presentation and the discussion of a set of selectedsynthesis results concerned with sub-arrayed arrays generatingpencil beams with sidelobe notches and asymmetric lobes aswell as shaped main lobes. Eventually, some conclusions andfinal remarks are drawn (Sect. 4).II. M
ATHEMATICAL F ORMULATION
Let us consider a linear array of N elements equally-spacedby d along the x -axis to be grouped into Q ( Q < N ) sub-arrays, each containing N q ( q = 1 , ..., Q ) elements, so that P Nn =1 N q = N (Fig. 1). For beam-forming purposes, each q -th ( q = 1 , ..., Q ) sub-array is fed by a TRM composed by an ϕ ϕ q ϕ Q α α q α Q P Figure 1. Sketch of the sub-arrayed architecture with sub-array level onlycomplex-valued excitations. amplifier and a phase shifter providing a complex excitation I q (Fig. 1). The mathematical expression of the array factorof the beam generated at sub-array level turns out to be AF ( θ ) = Q X q =1 I q N X n =1 δ c n q e jk ( n − d sin θ (1)where I q = α q e jϕ q , α q and ϕ q being the amplitude andthe phase coefficients of the q -th ( q = 1 , ..., Q ) sub-array,respectively. Moreover, k = πλ is the wavenumber, λ beingthe wavelength, θ is the angle measured from broadside, and δ c n q is the Kronecker delta function equal to δ c n q = 1 if c n = q and δ c n q = 0 , otherwise. The integer vector c = { c n ∈ N | ≤ c n ≤ Q : n = 1 , ..., N } univocally describes themembership of the n -th ( n = 1 , ..., N ) array element to the q -th ( q = 1 , ..., Q ) cluster.According to this mathematical description, the synthesisproblem at hand can be stated as follows: Sub-Arraying Synthesis Problem for ArbitraryShaped Beams Generation - Given a set of complexexcitation coefficients, { v n ; n = 1 , ..., N }, generat-ing a reference arbitrary-shaped pattern AF ref ( θ ) = N X n =1 v n e jk ( n − d sin θ , (2)to determine the optimal clustering of thearray elements into Q disjoint sub-arrays, c opt = { c optn ; n = 1 , ..., N } , and the valuesof the complex-valued sub-array excitations, I opt = (cid:8) I optq ; q = 1 , ..., Q (cid:9) , so that the beamgenerated at sub-array level (1) is close as much aspossible to the reference one (2).Towards this aim, the synthesis problem is here reformulatedas on optimization one in which the DoF s (i.e., the member-ship vector, c , and the sub-array level beam-forming excitationvector, I ) are set to the values minimizing the following pattern-matching cost function Φ ( c , I ) = 1 π Z π/ θ = − π/ (cid:12)(cid:12) AF ref ( θ ) − AF ( θ ; c , I ) (cid:12)(cid:12) dθ . (3) 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.2020.2972641Copyright (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
Re{·} N (cid:77) Im{·} v v n v v n v N v N v N v (cid:145) n v (cid:145) v v (cid:16) N v v v (cid:145) ( a ) EA-CPM : EP-CPM :: 1 5 N -1 2 n
3 4 N nv n (cid:16) : 1 3 2 n
4 5
N N -1 nv n (cid:16)(cid:145) ( b ) Figure 2. Reference excitations, { v n ; n = 1 , ..., N }, ( a ) in the complexplain and ( b ) ordered on a list according to the EA-CPM and the
EP-CPM . By substituting (1) and (2) in (3), it follows that
Φ ( c , I ) = π R π/ θ = − π/ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)P Nn =1 v n − Q X q =1 δ c n q I q ! × e jk ( n − d sin θ (cid:12)(cid:12) dθ. (4)Thus, it turns out that the optimization of (3) is equivalentto the minimization of the following excitation-matching costfunction Ψ ( c , I ) = 1 N N X n =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v n − Q X q =1 δ c n q I q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (5)since all the remaining terms in (4) are function of neither c nor I . It is worth pointing out that (5) is a typical example ofan over-determined problem with N (complex) values to beapproximated, in the least-square sense, by Q ( Q < N ) ones.On the other hand, let us notice that the optimal value of the q -th ( q = 1 , ..., Q ) sub-array coefficient, I q , for a fixed clusteringconfiguration c that minimizes (5) is the arithmetic mean ofthe reference excitations of the elements belonging to the same q -th ( q = 1 , ..., Q ) cluster of the sub-arrayed architecture: I q ( c ) = P Nn =1 δ c n q v n N q . (6)As a consequence, the synthesis can be performed by onlylooking for the best clustering configuration, c opt ( c opt , arg { min c [Ψ ( c , I )] } ), since the sub-array weights come outas a free by product through (6) (i.e., Ψ ( c , I ) → Ψ ( c ) ).As for the minimization of Ψ ( c ) , two methodological ap-proaches are presented in the following. The former (Sect. ) is the most simple extension of the CPM -based approachespresented in [19] and [25], while the latter (Sect. ) is basedon the generalization of the contiguity concept in clusteringto the complex excitation plane. A. Complex-Extended CPM (E-CPM)
The extended
CPM lies within the Fisher’s grouping theory[20] and it requires the creation of an ordered list of (real)values. Once the list is defined, the Border Element Method(
BEM ), which has been proposed in [19] and further used in[25], is exploited for efficiently sampling the solution space ofthe possible clustering configurations (i.e., the contiguous par-titions of the list of reference excitations). More specifically,the
E-CPM works as follows: • Step 0 -
Ordered-list Creation - Given the set of N complex values of the reference vector v = { v n ; n = 1 , ..., N } [Fig. 2( a )], define the real-valued list L = { ℓ n ; n = 1 , ..., N } by ordering the reference excita-tions according to their magnitude (i.e., l = min n {| v n |} and l N = max n {| v n |} ) [19]. If two (e.g., the m -th and the p -th elements, being m , p ∈ [1, N ] and m < p ) or more elements have the same magnitude,then order them according to their phase values (i.e., l m = min { ∠ v m ; ∠ v p } and l m +1 = max { ∠ v m ; ∠ v p } subject to | v m | = | v p | ) [Fig. 2( b )].Alternatively, unlike the previous amplitude-ordered ver-sion of the E-CPM (namely, the
EA-CPM ), the phase-ordered
E-CPM ( EP-CPM ) defines the list L by or-dering the coefficients according to their phase (i.e., l = min n { ∠ v n } and l N = max n { ∠ v n } ) [25], whilethey are ranked according to their magnitudes when thephase values are identical (i.e., l m = min {| v m | ; | v p |} and l m +1 = max {| v m | ; | v p |} subject to ∠ v m = ∠ v p )[Fig. 2( b )]; • Step 1 -
Initialization - Once L is defined, set the initialsub-array configuration, c ( t ) ( t = 0 , t being the iterationindex), by randomly selecting Q − cut points among the N − admissible ones among the N sorted coefficients{ ℓ n ; n = 1 , ..., N }; • Step 2 -
BEM -Based Solution-Space Sampling - Exe-cute the following steps: – Step 2.1 -
Sub-Array Coefficients Definition - Forthe t -th trial sub-array configuration, c ( t ) , computethe optimal sub-array excitations I ( t ) through (6); – Step 2.2 -
Cost Function Evaluation - Compute thedistance between the Q sub-array coefficients andthe N reference ones by evaluating the excitationmatching metric Ψ ( t ) = Ψ (cid:0) c ( t ) (cid:1) (5); – Step 2.3 -
Solution Update - Compare the currentexcitation-matching value Ψ ( t ) with the best costfunction value found so far, Ψ ( t − opt ( Ψ ( t − opt , min h =0 ,...,t − (cid:8) Ψ ( h ) (cid:9) ). If Ψ ( t ) < Ψ ( t − opt , then set Ψ ( t ) opt = Ψ ( t ) and update the current best clusteringand the corresponding excitation vector: c ( t ) opt ← c ( t ) and I ( t ) opt ← I ( t ) ; – Step 2.4 -
Convergence Check - Stop and go to Step3 if the maximum number of iterations T max has 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.2020.2972641Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
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Re{·}
Im{·} I Q I ( t ) I q I ( t ) I ( t ) I Re{·}
Im{·} (cid:534) ( t ) (cid:534) (cid:534) ( t ) (cid:534) (cid:534) ( t ) (cid:534) q (cid:534) ( t ) (cid:534) q I Q I ( t ) (cid:534) ( t ) (cid:534) NQ I q I ( t ) I ( t ) I Re{·}
Im{·} I ( t +1) I q I I ( t +1) I Q I ( t +1) ( a ) ( b ) ( c ) Figure 3. Diagrams related to ( a ) Step 2.1 - Distance Computation, ( b ) Step 2.2 - Element Clustering, and ( c ) Step 2.3 - Centroids Update of the KMM . been reached ( t ≥ T max ) or the stationary condition (cid:12)(cid:12)(cid:12) T stat Ψ ( t − opt − P T stat +1 h =2 Ψ ( t − h ) opt (cid:12)(cid:12)(cid:12) Ψ ( t ) opt ≤ η (7)holds true, T stat and η being user-defined parameterssetting the number of iterations of the window forchecking the stationary condition (7) and the mini-mum threshold for the decrease of the optimal valueof the cost function, respectively. Otherwise, go toStep 2.5; – Step 2.5 -
Sub-Array Configuration Update -Update the iteration index ( t ← t + 1 ) and definethe new sub-array configuration, c ( t ) , by changingthe position of at least one of the Q − cut points ofthe previous sub-array partitioning, c ( t − , accordingto the BEM procedure [19]. Then, go to Step 2.1; • Step 3 -
Sub-Arrayed Array Design - Set c opt = c ( t ) opt and I opt = I ( t ) opt .Although benefiting from the outcomes of the Fisher’s group-ing theory [20] and the effectiveness of the BEM , the
E-CPM is a sub-optimal method since it casts the original complex-valued clustering problem into a real-valued one. In otherwords, ordering a set of complex values in terms of theiramplitudes/phases is equivalent to project the correspondingrepresentative points from a plane to a line, thus reducing thedimensionality (cardinality) of the solution space, , but alsointroducing an approximation. In order to fully deal with the nature of the clustering problem at hand, still keeping theprinciples of the Fisher’s grouping theory, a second innovativemethod based on a customization of the k-means algorithm isdescribed hereinafter (Sect. ). B. K-Means Method (KMM)
The minimization of the excitation matching metric (5),which is obtained with the association of N array elements to Q sub-arrays and the computation of the corresponding sub-array centroids through (6), can be mathematically classifiedas an unsupervised learning problem of divisive clustering [30]where a complex space of representative solution points has to be partitioned into Q regions. Towards this end, the k-means algorithm is here customized according to the followingprocedural steps: • Step 1 -
Centroids Initialization - Initialize the centroids I ( t ) ( t = 0 , t being the iteration index) to Q randomly-chosen complex-valued reference excitations among the N available ones, { v n ; n = 1 , ..., N } [Fig. 3( a )]; • Step 2 -
KMM -Based Solution-Space Sampling - Exe-cute the following steps: – Step 2.1 - Distance Computation - For each n -th ( n = 1 , ..., N ) reference coefficient, compute theEuclidean distance, γ ( t ) nq , from the q -th ( q = 1 , ..., Q )centroid of the set I ( t ) , I ( t ) q [Fig. 3( a )]: γ ( t ) nq = (cid:13)(cid:13)(cid:13) v n − I ( t ) q (cid:13)(cid:13)(cid:13) = (cid:20)(cid:16) Re { v n } − Re n I ( t ) q o(cid:17) − (cid:16) Im { v n } − Im n I ( t ) q o(cid:17) (cid:21) (8) Re {·} and Im {·} being the real and the imaginaryparts, respectively; – Step 2.2 - Element Clustering - Associate each n -th ( n = 1 , ..., N ) array element to the q -th ( q =1 , ..., Q ) cluster (i.e., c ( t ) n = q ) whose centroid hasthe minimum distance (8) q = arg n min j ∈ [1 , Q ] h(cid:13)(cid:13)(cid:13) v n − I ( t ) j (cid:13)(cid:13)(cid:13)io (9)to create the t -th sub-array membership vector c ( t ) [Fig. 3( b )]; – Step 2.3 -
Centroids Update - Update the iterationindex ( t ← t + 1 ) and compute the optimal (inthe least-square sense) sub-array centroid vector I ( t ) related to the clustering arrangement c ( t − with (6)[Fig. 3( c )]; – Step 2.4 -
Convergence Check - Stop and go toStep 3 if the maximum number of iterations T max has been reached ( t ≥ T max ) or I ( t ) = I ( t − .Otherwise, go to Step 2.1; 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.2020.2972641Copyright (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 -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( q ) Referencek-means, t=0k-means, t=1k-means, t=3k-means, t=5 ( a ) E xc i t a t i on M agn i t ude x / l Referencek-means, t=0k-means, t=1k-means, t=3k-means, t=5 ( b ) -10-8-6-4-2 0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 E xc i t a t i on P ha s e [ deg ] x / l Referencek-means, t=0k-means, t=1k-means, t=3k-means, t=5 ( c ) Figure 4.
Asymmetric Sidelobes Pencil Beam Pattern ( N = 16 , d = 0 . λ , Q = 4 ) - Plot of the ( a ) power pattern, ( b ) the excitation amplitudes, and( c ) the excitation phases of the KMM sub-arrayed solution at the iterations t = { , , , } along with the reference ones. • Step 3 -
Sub-Arrayed Array Design - Set c opt = c ( t ) and I opt = I ( t ) .It is worth pointing out that the execution of the steps fromStep 2.1 up to Step 2.3 is equivalent to the minimization ofthe excitation matching metric, which turns out to be implicitlyperformed, without (5) being explicitly evaluated at each t -thiteration. III. N UMERICAL R ESULTS
Representative results are presented and discussed in thisSection to analyze the behavior of the proposed complex-valued clustering method and to assess its performance alsoin comparison with the
E-CPM .The first example deals with the design of a sub-arrayed arraywith N = 16 , d = λ , and Q = 4 radiating a pattern closeas much as possible to the reference one shown in Fig. 4( a )and characterized by asymmetric sidelobes. More in detail,the reference pattern has monotonically decreasing sidelobeson one side of the main beam ( < u ≤ , u = sin θ )from a level of − [dB] down to − [dB] along the end-fire direction [Fig. 4( a )]. On the other side ( − ≤ u < ), o o o -0.10.1 Re {(cid:215)} Im {(cid:215)} k-meansReference o o o -0.10.1 Re {(cid:215)} Im {(cid:215)} q=1 q=2 q=3 q=4
0 2 4 6 8x/ l q=1 q=2 q=3 q=4 ( a ) ( e ) o o o -0.10.1 Re {(cid:215)} Im {(cid:215)} k-meansReference o o o -0.10.1 Re {(cid:215)} Im {(cid:215)} q=1 q=2 q=3 q=4
0 2 4 6 8x/ l q=1 q=2 q=3 q=4 ( b ) ( f ) o o o -0.10.1 Re {(cid:215)} Im {(cid:215)} k-meansReference o o o -0.10.1 Re {(cid:215)} Im {(cid:215)} q=1 q=2 q=3 q=4
0 2 4 6 8x/ l q=1 q=2 q=3 q=4 ( c ) ( g ) o o o -0.10.1 Re {(cid:215)} Im {(cid:215)} k-meansReference o o o -0.10.1 Re {(cid:215)} Im {(cid:215)} q=1 q=2 q=3 q=4
0 2 4 6 8x/ l q=1 q=2 q=3 q=4 ( d ) ( h ) Figure 5.
Asymmetric Sidelobes Pencil Beam Pattern ( N = 16 , d = 0 . λ , Q = 4 ) - Representation of ( a )-( d ) the reference, { v n ; n = 1 , ..., N }, and the KMM sub-array, { I ( t ) q ; q = 1 , ..., Q }, excitations in the complex plane and( e )-( h ) layout of the KMM clustered array at ( a )( e ) the initialization ( t = 0 ),( b )( f ) the first iteration ( t = 1 ), ( c )( g ) the third iteration ( t = 3 ), and ( d )( h )the final iteration ( t = 5 ). a sidelobe depression of [dB] with respect to a sidelobelevel ( SLL ) of − [dB] is present within the angular range − . ≤ u ≤ − . [Fig. 4( a )]. The reference set of excitations, v , has been computed by means of a Convex Programmingoptimization technique [31] and it is indicated with crosses × in the polar plots of Figs. 5( a )-5( d ), while Figures 4( b )-4( c )show the corresponding amplitudes and phases, respectively.Since the KMM is a local/deterministic searching method, itsperformance depends on the initialization, thus a set of R = 50 independent runs has been executed by considering differentstarting solutions. The evolution of the Q sub-array excitationsfor a representative run converging to the median value ofthe excitation matching metric (5) among the R executionsis shown in Fig. 5. More specifically, the circles indicate thevalues of the Q clustered excitations, while the crosses withthe same color denote the reference weights used in (6) for thecomputation of the corresponding q -th ( q = 1 , ..., Q ) centroid I ( t ) q at the initial [ t = 0 - Fig. 5( a )], the convergence [ t = 5 - Fig. 5( d )], and two intermediate [ t = 1 - Fig. 5( b ); t = 3 -Fig. 5( c )] iterations. Moreover, the memberships of the arrayelements, which is coded into the vector c ( t ) , at the iterations t = { , , , } are shown in Figs. 5( e )-( h ). According to the KMM implementation described in Sect. II .B , the centroids(i.e., the coefficients) of the Q sub-arrays are randomly chosenat the initial iteration ( t = 0 ) among the reference excitations. 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.2020.2972641Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
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0 0.25 0.5 0.75 10 o o o o o o o o o o o o Re {(cid:215)} Im {(cid:215)} k-meansReference
0 0.25 0.5 0.75 10 o o o o o o o o o o o o Re {(cid:215)} Im {(cid:215)} q=1 q=2 q=3 q=4 q=5 q=6 q=7 q=8 q=9 q=10 q=11 q=12
0 0.25 0.5 0.75 10 o o o o o o o o o o o o Re {(cid:215)} Im {(cid:215)} EA-CPMReference
0 0.25 0.5 0.75 10 o o o o o o o o o o o o Re {(cid:215)} Im {(cid:215)} q=1 q=2 q=3 q=4 q=5 q=6 q=7 q=8 q=9 q=10 q=11 q=12
0 0.25 0.5 0.75 10 o o o o o o o o o o o o Re {(cid:215)} Im {(cid:215)} EP-CPMReference
0 0.25 0.5 0.75 10 o o o o o o o o o o o o Re {(cid:215)} Im {(cid:215)} q=1 q=2 q=3 q=4 q=5 q=6 q=7 q=8 q=9 q=10 q=11 q=12 ( a ) ( b ) ( c )
0 2 4 6 8x/ l q=1q=2 q=3q=4 q=5q=6 q=7q=8 q=9q=10 q=11q=12
0 2 4 6 8x/ l q=1q=2 q=3q=4 q=5q=6 q=7q=8 q=9q=10 q=11q=12
0 2 4 6 8x/ l q=1q=2 q=3q=4 q=5q=6 q=7q=8 q=9q=10 q=11q=12 ( d ) ( e ) ( f ) Figure 8.
Cosecant-Squared Beam Pattern ( N = 17 , d = 0 . λ , Q = 12 ) - Representation of ( a )-( c ) the reference, { v n ; n = 1 , ..., N }, and the sub-array,{ I ( t ) q ; q = 1 , ..., Q }, excitations in the complex plane and ( d )-( f ) layouts of the clustered arrays synthesized with the ( a )( d ) KMM , ( b )( e ) EA-CPM , and ( c )( f ) EP-CPM . E xc i t a t i on M a t c h i ng , Y ( · - ) P a tt e r n M a t c h i ng , F ( · - ) Iteration Index, t YF Figure 6.
Asymmetric Sidelobes Pencil Beam Pattern ( KMM ; N = 16 , d = 0 . λ , Q = 4 ) - Behavior of the excitation matching, Ψ ( t ) , and thepattern matching, Φ ( t ) , metrics versus the iteration index, t . In this run, for instance, they have been set to I (0)1 = v , I (0)2 = v , I (0)3 = v , and I (0)4 = v [Fig. 5( a )]. After associatingeach n -th ( n = 1 , ..., N ) reference excitation to the closer q -th ( q = 1 , ..., Q ) centroid (Step ) [Fig. 5( a )], the sub-arrayconfiguration turns out to be as in Fig. 5( e ), then the sub-array excitations are updated ( t = 1 ) through (6). The newpositions of the centroids in the complex plane are shown inFig. 5( b ) and the corresponding clustering is given in Fig.5( f ). The clustering process stops after t = 5 iterations whenthe stationary condition I ( t ) = I ( t − is reached. The optimalvalues of the sub-array excitations and the final clustering areshown in Fig. 5( d ) and Fig. 5( h ), respectively.While the iterative procedure minimizes the excitation match-ing cost function (5), the beam generated at sub-array levelbetter and better approximates the reference one as shownby the plots of the synthesized power patterns, P P ( t ) ( θ ) ( P P ( θ ) , | AF ( θ ) | ) and of the reference one, P P ref ( θ ) , -4 -3 -2 -1
0 500 1000 1500 2000 2500 3000 3500 4000 4500 E xc i t a t i on M a t c h i ng , Y Solution IndexEA-CPM (overall)EP-CPM (overall)EA-CPM (best)EP-CPM (best)k-means (best)
Figure 7.
Cosecant-Squared Beam Pattern ( N = 17 , d = 0 . λ , Q = 12 )- Excitation matching values for the whole set of admissible solutions of the EA-CPM and the
EP-CPM methods along with the optimal ones from the
KMM , the
EA-CPM , and the
EP-CPM . in Fig. 4( a ). The same holds true for the corresponding sub-array amplitude [Fig. 4( b )] and phase [Fig. 4( c )] excitations.This behavior is also quantitatively highlighted in Fig. 6 sinceboth the excitation matching Ψ ( t ) and the pattern matching Φ ( t ) indexes monotonically decrease with the iteration thusconfirming the effectiveness of the proposed method.The second example is mainly devoted to perform a firstcomparison of the KMM with the
E-CPM when synthesiz-ing a shaped beam. Towards this end, the cosecant-squaredpattern published in [32] has been selected as reference (see“
Reference ” in Fig. 9). It has been generated by a fully-populated array of N = 17 elements equally-spaced by d = λ and it exhibits a sidelobe level equal to SLL ref = − . [dB]. The clustered array to be designed is supposed to have Q = 12 sub-arrays, that is saving of TRM s with respectto the fully-populated architecture. For comparison purposes,the best result from the
KMM , reached times within R = 50 independent runs, is considered along with 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.2020.2972641Copyright (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
0 0.25 0.5 0.75 10 o o o o o o o o o o o o Re {(cid:215)} Im {(cid:215)} k-meansReference
0 0.25 0.5 0.75 10 o o o o o o o o o o o o Re {(cid:215)} Im {(cid:215)} q=1 q=2 q=3 q=4 q=5 q=6 q=7 q=8 q=9 q=10 q=11 q=12 q=13 q=14 q=15 q=16 q=17 q=18 q=19 q=20 q=21 q=22 q=23 q=24
0 0.25 0.5 0.75 10 o o o o o o o o o o o o Re {(cid:215)} Im {(cid:215)} EA-CPMReference
0 0.25 0.5 0.75 10 o o o o o o o o o o o o Re {(cid:215)} Im {(cid:215)} q=1 q=2 q=3 q=4 q=5 q=6 q=7 q=8 q=9 q=10 q=11 q=12 q=13 q=14 q=15 q=16 q=17 q=18 q=19 q=20 q=21 q=22 q=23 q=24
0 0.25 0.5 0.75 10 o o o o o o o o o o o o Re {(cid:215)} Im {(cid:215)} EP-CPMReference
0 0.25 0.5 0.75 10 o o o o o o o o o o o o Re {(cid:215)} Im {(cid:215)} q=1 q=2 q=3 q=4 q=5 q=6 q=7 q=8 q=9 q=10 q=11 q=12 q=13 q=14 q=15 q=16 q=17 q=18 q=19 q=20 q=21 q=22 q=23 q=24 ( a ) ( b ) ( c )
0 2 4 6 8 10 12 14 16x/ l q=1q=2q=3 q=4q=5q=6 q=7q=8q=9 q=10q=11q=12 q=13q=14q=15 q=16q=17q=18 q=19q=20q=21 q=22q=23q=24
0 2 4 6 8 10 12 14 16x/ l q=1q=2q=3 q=4q=5q=6 q=7q=8q=9 q=10q=11q=12 q=13q=14q=15 q=16q=17q=18 q=19q=20q=21 q=22q=23q=24
0 2 4 6 8 10 12 14 16x/ l q=1q=2q=3 q=4q=5q=6 q=7q=8q=9 q=10q=11q=12 q=13q=14q=15 q=16q=17q=18 q=19q=20q=21 q=22q=23q=24 ( d ) ( e ) ( f ) Figure 11.
Flat-top Pattern ( N = 32 , d = 0 . λ , Q = 24 ) - Representation of ( a )-( c ) the reference, { v n ; n = 1 , ..., N }, and the sub-array, { I ( t ) q ; q = 1 , ..., Q }, excitations in the complex plane and ( d )-( f ) layouts of the clustered arrays synthesized with the ( a )( d ) KMM , ( b )( e ) EA-CPM , and ( c )( f ) EP-CPM . -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( q ) ReferenceEA-CPMEP-CPMk-means Figure 9.
Cosecant-Squared Beam Pattern ( N = 17 , d = 0 . λ , Q = 12 ) -Power pattern of the clustered arrays synthesized with the KMM , the
EA-CPM ,and the
EP-CPM together with the reference one. optimal solutions of the
EA-CPM and the
EP-CPM . Indeed,the size of the solution space of the
CPM being equal to (cid:0) N − Q − (cid:1) = 4368 contiguous partitions, all the admissible CPM solutions can be exhaustively generated, then the optimal onecan be found by means of a simple enumerative procedure. Thevalues of the excitation matching index (5) for every
CPM -based solution, ordered from the worst (i.e., Ψ ⌋ worstEA − CP M =1 . × − and Ψ ⌋ worstEP − CP M = 1 . × − ) to the best (i.e., Ψ ⌋ bestEA − CP M = 4 . × − and Ψ ⌋ bestEP − CP M = 1 . × − )ones, are given in Fig. 7. On the same plot, the excitationmatching value of the best synthesized KMM design (i.e., Ψ ⌋ bestKMM = 8 . × − ) is reported, as well. As it canbe observed, the KMM outperforms all other methods sinceit reaches a solution with the minimum excitation matchingvalue (Fig. 7). Moreover, it is worthwhile to highlight thatsuch a
KMM clustered arrangement [Figs. 8( a ) and 8( d )] doesnot coincide with any of the contiguous partitions of the list ofthe sorted (amplitude/phase) reference excitations of the EA-CPM [Figs. 8( b ) and 8( e )] or the EP-CPM [Figs. 8( c ) and8( f )].For completeness, Figure 9 shows the reference power pattern, -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( q ) ReferenceEA-CPMEP-CPMk-means ( a ) E xc i t a t i on M a t c h i ng , Y ( · - ) Number of Subarrays, QEA-CPMEP-CPMk-means ( b ) Figure 10.
Flat-top Pattern ( N = 32 , d = 0 . λ , ≤ Q ≤ ) - Plotof the ( a ) power pattern of the sub-arrayed solutions synthesized with the KMM , the
EA-CPM , and the
EP-CPM methods when Q = 24 along with thereference one and ( b ) behavior of the excitation matching error, Ψ opt , of thesynthesized solutions versus the number of sub-arrays, Q . P P ref ( θ ) , and those synthesized at sub-array level withthe KMM , P P
KMM ( θ ) , and the two CPM -based imple-mentations [i.e.,
P P EA − CP M ( θ ) and P P EP − CP M ( θ ) ]. Asexpected from the values of the excitation matching index inFig. 7, the KMM and the
EP-CPM approximate the referencepattern better than the
EA-CPM . For instance, the shape ofthe main lobe as well as the ripples of the secondary lobes arecloser to the reference.
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.2020.2972641Copyright (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 -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( q ) Referencek-means ( a ) -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( q ) Referencek-means ( b )
0 0.25 0.5 0.75 10 o o o o o o o o o o o o Re {(cid:215)} Im {(cid:215)} k-meansReference
0 0.25 0.5 0.75 10 o o o o o o o o o o o o Re {(cid:215)} Im {(cid:215)} q=1 q=2 q=3 q=4 q=5 q=6 q=7 q=8
0 0.25 0.5 0.75 10 o o o o o o o o o o o o Re {(cid:215)} Im {(cid:215)} k-meansReference
0 0.25 0.5 0.75 10 o o o o o o o o o o o o Re {(cid:215)} Im {(cid:215)} q=1 q=2 q=3 q=4 q=5 q=6 q=7 q=8 q=9 q=10 q=11 q=12 q=13 q=14 q=15 q=16 q=17 q=18 q=19 q=20 q=21 q=22 q=23 q=24 q=25 q=26 q=27 q=28 q=29 q=30 q=31 q=32 ( c ) ( d )
0 2 4 6 8x/ l q=1q=2 q=3q=4 q=5q=6 q=7q=8
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32x/ l q=1q=2q=3 q=4q=5q=6 q=7q=8q=9 q=10q=11q=12 q=13q=14q=15 q=16q=17q=18 q=19q=20q=21 q=22q=23q=24 q=25q=26q=27 q=28q=29q=30 q=31q=32 ( e ) ( f ) Figure 12.
Steered Pencil Beam Pattern ( d = 0 . λ , Q = N/ , θ = − [deg]) - Plot of the ( a )( b ) power pattern of the clustered solutions together withthe reference ones, representation of ( c )( d ) the reference, { v n ; n = 1 , ..., N },and the sub-array, { I ( t ) q ; q = 1 , ..., Q }, excitations in the complex plane, and( e )( f ) layouts of the clustered arrays synthesized with the KMM when ( a )( c )( e ) N = 16 and ( b )( d )( f ) N = 64 . The third example is concerned with a synthesis problem ofhigher complexity for which the exhaustive evaluation of allpossible contiguous partitions is unfeasible. As a matter offact, (cid:0) N − Q − (cid:1) = 84672315 when Q = 12 and, thus, the CPM exploits the
BEM for the solution space sampling. More indetail, a fully-populated array of N = 32 λ -spaced elementsradiating a flat-top beam pattern with SLL ref = − . [dB]and having maximum main lobe ripples equal to . [dB]has been taken into account [” Reference ” in Fig. 10( a )]. Byvarying the number of sub-arrays in the range ≤ Q ≤ ,the KMM always outperforms the
CPM- based methods sinceit provides the best matching index whatever the number ofclusters [Fig. 10( b )]. For illustrative purposes, the sub-arrayweights [Figs. 11( a )-11( c )] and the sub-array arrangements[Figs. 11( d )-11( f )] of the best solution yielded with the KMM
Table I
Steered Pencil Beam Pattern ( KMM - N = { , , , } , d = 0 . λ , Q = N/ , θ = − [ DEG ]) - V
ALUES OF THE EXCITATION MATCHINGINDEX , Ψ opt , THE PATTERN MATCHING INDEX , Φ opt , THE
SLL , ALONGWITH THE COMPUTATIONAL COST , ∆ τ . Φ opt Ψ opt SLL [ dB ] ∆ τ [ sec ] N = 16 5 . × − . × − − .
53 0 . N = 32 3 . × − . × − − .
41 0 . N = 48 2 . × − . × − − .
98 0 . N = 64 1 . × − . × − − .
77 0 . [Fig. 11( a ) and Fig. 11( d )] , the EA-CPM [Fig. 11( b ) and Fig.11( e )], and the EP-CPM [Fig. 11( c ) and Fig. 11( f )] are shownwhen Q = 24 , while the reference power pattern together withthe synthesized ones are reported in Fig. 10( a ). Among the R = 50 runs, the success rate of the KMM in converging tothe best solution varied in these cases between and ,with a drastic reduction with respect to the previous exampledue to the much higher cardinality of the solution space.The efficiency of the KMM when designing larger arrays isassessed next by setting the reference beam to a Taylor patternwith
SLL ref = − [dB] and n = 7 pointing along thedirection θ = − [deg]. Four different apertures having N = { , , , } elements and inter-element spacing d = λ have been considered, while keeping constant the ratio QN to . Table I summarizes the outcomes of this analysisby reporting the values of the excitation matching, the patternmatching, and the SLL for the best
KMM clustered solutionrunning the code R = 50 times for each array size, N , witha success rate decreasing with the dimension of the solutionspace from for N = 16 to for N = 64 . Although onlyhalf TRM s have been used as compared to the fully-populatedlayout, the matching with the reference pattern improves as N is getting larger and larger since both Φ opt and Ψ opt monotonically decrease and the arising SLL better and betterapproximates the reference one. For illustrative purposes, the
KMM solutions for N = 16 and N = 64 are shown in Fig.12. Besides the power patterns [Figs. 12( a )-12( b )], the sub-array coefficients along with the reference excitations [Figs.12( c )-12( d )] and the clustering of the array elements [Figs.12( e )-12( f )] are reported for both array dimensions [ N = 16 - Fig. 12( c ) and Fig. 12( e ); N = 64 - Fig. 12( d ) and Fig.12( f )]. As for the average CPU -time ∆ τ to synthesize a sub-arrayed arrangement, less than . [sec] are required on a . GHz
PC with GB of RAM executing a non-optimizedcode whatever the aperture at hand (Tab. I).IV. C ONCLUSIONS
The design of sub-arrayed PA s generating an arbitrary-shapedpattern has been addressed. Towards this aim, an innovativesynthesis method, benefiting from the previously publishedexcitation matching strategies and exploiting a clusteringtechnique suitable for complex-valued excitations, has beenproposed. The synthesis problem has been reformulated as anoptimization one in which a customized version of the k-means has been used for defining the sub-array configuration of 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.2020.2972641Copyright (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 array elements, while the complex-valued sub-array weightshave been yielded in closed form as the arithmetic meansof the reference excitations generating the target pattern andbelonging to the same cluster.The main methodological advances of this work with respect tothe state-of-the-art can be summarized in the following ones: • the theoretical formulation of the sub-array synthesisproblem in the excitation-matching framework for ef-fectively dealing with the sub-array level generation ofarbitrary-shaped beams by extending the theory devel-oped in [19] and [25]; • the introduction of an innovative and ad-hoc approachbased on the k-means for solving the synthesis problemat hand.From the numerical assessment, the proposed KMM designmethod proved: • to overcome the limitations of the CPM -based methodsby enabling the retrieval of sub-array configurations notachievable with the
EA-CPM and the
EP-CPM ; • to provide a high convergence rate and a significantcomputational efficiency regardless the number of arrayelements and sub-arrays; • to enable the sub-array level synthesis of arbitrary-shapedbeams, including pencil beams with asymmetric sidelobesas well as shaped main lobes (e.g., cosecant-square andflat-top beams).However, it is important to observe that there is a trade-offbetween the capability of the proposed approach in matchingthe reference pattern and the complexity of the arising feedingnetwork, characterized by sub-arrays also containing elementsnon-physically contiguous in the array aperture.Future works, outside the scope and objectives of this paper,will be concerned with the development of a constrainedversion of the proposed approach guaranteeing the design ofsub-arrays of physically contiguous elements, the integrationof the proposed method with some power pattern synthesistechnique taking advantage of the multiple solutions existingfor the same power pattern shape in case of uniformly spacedantenna arrays, the design of clustered planar and conformalarrays as well as of non-uniformly spaced arrays, and the studyof a global synthesis strategy instead of the use of a local onewhich, although very effective and robust, has performancestill depending on the initialization.A CKNOWLEDGEMENTS
A. Massa wishes to thank E. Vico for her never-endinginspiration, support, guidance, and help.R
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