Minimum-Complexity Failure Correction in Linear Arrays via Compressive Processing
IIEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2020 1
Minimum-Complexity Failure Correction in LinearArrays via Compressive Processing
F. Zardi, G. Oliveri,
Senior Member, IEEE , M. Salucci,
Member, IEEE , and A. Massa,
Fellow, IEEE
Abstract —Given an array with defective elements, failurecorrection ( FC ) aims at finding a new set of weights for theworking elements so that the properties of the original patterncan be recovered. Unlike several FC techniques available inthe literature, which update all the working excitations, theMinimum-Complexity Failure Correction ( MCFC ) problem isaddressed in this paper. By properly reformulating the FC problem, the minimum number of corrections of the wholeexcitations of the array is determined by means of an innovativeCompressive Processing ( CP ) technique in order to afford apattern as close as possible to the original one (i.e., the arraywithout failures). Selected examples, from a wide set of numericaltest cases, are discussed to assess the effectiveness of the proposedapproach as well as to compare its performance with othercompetitive state-of-the-art techniques in terms of both patternfeatures and number of corrections. Index Terms —Element failure correction, phased arrays, com-pressive sensing, Minimum-complexity.
I. I
NTRODUCTION R ECENTLY, there has been an increased interest in phasedarrays comprising hundreds of elements for several ap-plications ranging from in-flight connectivity up to advancedradar technology (see [1]-[5] and the reference therein). In-deed, the cost and the complexity of array systems havereduced thanks to the availability on the market of transmitand receive modules transmit and received modules [6] andthe development of simplified feeding networks [7][8]. Fur-thermore, more and more challenging requirements in manyand relevant civil, commercial, and military applications havefurther stimulated/forced the adoption of large array systems.For instance, in mobile communications, the massive MIMOparadigm is driving the request of more and more antennas
Manuscript received May 0, 2020This work has been partially supported by the Italian Ministry of Ed-ucation, University, and Research within the Program Smart cities andcommunities and Social Innovation (CUP: E44G14000060008) for the ProjectWATERTECH - 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 for the Project Cloaking Metasurfaces for a New Generation ofIntelligent Antenna Systems (MANTLES).F. Zardi, G. Oliveri, M. Salucci, and A. Massa are with the ELE-DIA@UniTN (DISI - University of Trento), Via Sommarive 9, 38123Trento - Italy (e-mail: {francesco.zardi, giacomo.oliveri, marco.salucci, an-drea.massa}@unitn.it)G. Oliveri and A. Massa are also with the ELEDIA Research Center(ELEDIA@L2S - UMR 8506), 3 rue Joliot Curie, 91192 Gif-sur-Yvette -France (e-mail: {giacomo.oliveri, andrea.massa}@l2s.centralesupelec.fr)A. Massa is also with the ELEDIA Research Center (ELEDIA@UESTC -UESTC), School of Electronic Engineering, Chengdu 611731 - China (e-mail:[email protected]. Massa is also with the ELEDIA Research Center (ELE-DIA@TSINGHUA - Tsinghua University), 30 Shuangqing Rd, 100084Haidian, Beijing - China (e-mail: [email protected] built in next-generation mobile phones and base stations[9][10]. On the other hand, phased arrays with large aperturesare investigated in satellite systems to counteract the heavierpath loss at higher frequency bands [11][12]. Moreover, phasedarrays with thousands of elements are going to replace theprevious generation of highly-directive parabolic reflectors inradar meteorology [13][14]. Of course, as the number of trans-mit and receive modules grows, the probability that a failureoccurs also increases and suitable countermeasures need to beenvisaged to prevent the loss of working functionality of thesystem as well as to guarantee its reliability.Failure correction ( FC ) aims at reconfiguring the workingelements of a faulty array to recover (all or the key) patternfeatures afforded by the original whole array to guaranteeconsistent performance of the system. Towards this end, manycorrection techniques, which differ in the problem formulationand/or the solution strategy, have been proposed. First, cor-rection methods involving efficient numerical implementationsthat require few computational resources have been developed.As an example, the FC problem has been formulated in [15]as a minimization one with quadratic constraints, then solvedwith a fast method devoted to minimize the average powerwithin the sidelobe region. Similarly, a correction methodbased on conjugate gradients that reduces the average sidelobeprovided an increment of the mainlobe beamwidth has beenderived in [16]. In [17], the linear least square deviation froma reference pattern has been minimized.Although these approaches give FC solutions in a fast and ef-ficient way, they are not suitable for directly minimizing high-level pattern features such as the Side Lobe Level ( SLL ) or thedirectivity. In order to address such a challenge, a categoryof FC methods based on alternating projection methods andinitially devised for the synthesis of large arrays is availablein the state-of-the-art literature. In [18], the Vector-SpaceProjection algorithm has been employed to jointly optimizedifferent array characteristics concerned with both the radiatedpattern (e.g., the SLL and the total transmitted power) and thearray architecture (e.g., the maximum excitation magnitude).Moreover, Keizer exploited in [19] the invertible Fourier-basedrelation between the array excitations and the correspondingarray factor to compute the corrected weights to recover areference pattern. In [20], the alternate ℓ -norm projectionmethod has been combined with a sparse failure detectionstrategy to correct a failed planar array with N = 289 elements.Reviewing the FC literature, it is worth mentioning the use ofglobal optimization techniques based on Genetic Algorithms( GA s). For instance, Yeo et al. analyzed different matingschemes and reported numerical results for two- and three- 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.2020.3045511Copyright (c) 2021 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, 2020 2 element failures [21]. In [22], a GA-based FC approach hasbeen proposed jointly with an adaptive weighted pattern maskaimed at updating, throughout the optimization process, thepattern constraints along user-defined angular regions. Theoptimization of the SLL , the directivity, and the DynamicRange ( DR ) of the array excitations as well as the numberof corrections has been dealt with in [23] by means of the GA -based minimization of a single-objective cost function.In [24], GA s have been still applied to synthesize the arrayarchitecture that minimizes the failure-probability-weightedpattern deviation from the whole/original one due to the failureof each element of the array. Such an approach provides aviable solution for arrays without reconfiguration capabilities,but deteriorations of the radiated pattern may arise duringnormal operations.Finally, the recovery of the signals not received at the faultyelements of an array has been addressed in [25][26] byassuming the a-priori (although not precise) knowledge of thedirections-of-arrival of plane waves impinging on the array.As a general comment on FC techniques, it is worthwhile topoint out that the complete restoration of the original pattern isgenerally not possible and, almost always, a trade-off solutionin terms of pattern features and array performance is lookedfor. Furthermore, most of the state-of-the-art solutions resultin many - or even all - working elements being reconfigured.However, while the modification of the element excitationscan be done on-line for reconfigurable arrays, it might bedesirable in other array architectures to minimize the numberof corrections so that reducing the maintenance costs and thesystem down-time. Additionally, both the availability and thenumber of spare parts can be limited in some applications(e.g., the on-orbit satellite servicing [27][28]).Within the FC framework, this work addresses the problem offinding the minimum set of corrections for the restoration of auser-defined performance metric in a faulty array. By formu-lating the Minimum-Complexity Failure Correction ( MCFC )problem as a non-deterministic polynomial-time hard ℓ -normminimization one with non-linear constraints, a correctionmethod based on the Compressive Processing ( CP ) paradigm(see [29]-[31] and the reference therein) is then developed.The proposed approach combines a ℓ -norm relaxation of the ℓ -norm and a backtracking strategy for an efficient samplingof the high-dimensional solution space to yield a satisfactorycorrection of the array excitations in a proper amount of time.The key motivations of these choices can be summarized asfollows. First, the practice of approximating the ℓ -norm withthe ℓ -norm is often adopted when exploiting the CP paradigm[32][33]. Indeed, the arising approximated cost function turnsout to be convex so that it can be minimized with any ofthe many available implementations of convex-optimizationmethods. Moreover, theoretical analyses have shown that,under specific conditions, the ℓ -norm minimization is equiv-alent to the ℓ -norm one [34]-[36]. As for the backtrackingalgorithm, it has proven reliable in solving CP problems[37][38]. As a matter of fact, it has been successfully appliedin electromagnetics to the sparse reconstruction of scatterersin through-the-wall imaging [39] and to the optimal signalsampling for bandwidth enhancement in phased arrays [40]. x x x x x x N E xc i t a t i on m agn i t ude xOriginal, w n Faulty, w~ n Corrected, w^ n ( a ) x x x x x x N xFaulty, Ω n Reconfigurable, Λ n Corrected, Λ ^ n ( b ) Figure 1.
Illustrative Example ( N = 7 , N F = 1 → η F = 14 %) - Pictorial representation of the excitation set of the “ original ” array, w = { , , , , , , } , of the “ damaged “ array, e w = { , , , , , , } ,and of the “ corrected “ array, b w = { , , , , , , } , along with thecorresponding binary vectors indicating the locations of the failed ele-ments, Ω = { , , , , , , } , of the reconfigurable/admissible elements, Λ = { , , , , , , } ( Λ , − Ω ), and of the corrected elements, b Λ = { , , , , , , } . The main contributions of this paper are: ( a ) a formulation ofthe MCFC problem that allows one to adopt arbitrary metricsfor dealing with any user-chosen array requirements; ( b ) thedefinition of a mathematical/theoretical framework suitable foran effective and reliable application of CP -based techniquesto minimize the number of excitations to be reconfiguredfor recovering the pattern features of the original array; ( c )the introduction of an innovative CP -based correction methodthat combines a ℓ -norm relaxation of the ℓ -norm with abacktracking strategy.The outline of the paper is as follows. The MCFC problem isstated and mathematically formulated in Sect. II. Section IIIdetails the procedural steps of the proposed correction method,while Section IV reports some representative numerical resultsto assess the arising correction performance in a comparativestudy, as well. Concluding remarks are finally drawn (Sect.V). II. P
ROBLEM S TATEMENT
A linear array comprising N isotropic elements, which aredistributed along the x axis at the positions { x n ; n = 1 , ..., N } [Fig. 1( a )], affords the following radiation pattern F ( u ) = N X n =1 w n ψ n ( u ) (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.2020.3045511Copyright (c) 2021 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, 2020 3 where ψ n ( u ) = e jkx n u [ u = sin ( θ ) ] and k is the wavenum-ber ( k = πλ ), while w is the set of excitations, w = { w n ; n = 1 , ..., N } , w n being the working excitation of the n -th element of the “ original “ array (i.e., the whole arraywithout failures). Let us assume that N F radiating elements,whose locations are denoted by the non-null entries of the N -elements binary vector Ω [i.e., Ω n = 1 if the n -th elementof the original array is faulty and Ω n = 0 otherwise - Fig.1( b )], are damaged and they are not reconfigurable anymore.Accordingly, the n -th ( n = 1 , ..., N ) element of the excitationvector of the damaged array, e w [Fig. 1( a )], turns out to be e w n = w n (1 − Ω n ) , (2)while the radiated pattern is given by e F ( u ) , N X n =1 e w n ψ n ( u ) . (3)Although N F excitations of the faulty array cannot be mod-ified, the remaining N C ( N C , N − N F ) elements can stillbe reconfigured to perform the array correction [Fig. 1( b )]. Byindicating with b w [ b w = { b w n ; n = 1 , ..., N } - Fig. 1( a )] theset of array excitations after correction, the associated radiatedfield is b F ( u ) = N X n =1 b w n ψ n ( u ) (4)the weights of the N F failed elements being set to zero (i.e., b w n Ω n = 0 ), while the corrections of the working elements arecoded into the excitation correction vector ∆ w ∆ w , b w − e w . (5)The FC problem at hand can be then formalized with thefollowing statement Minimum-Complexity Failure Correction ( MCFC ) Problem - Given the “ original array” excitationsset, w , the set of N F faulty elements, Ω , and theexcitations of the “ faulty array”, e w , find the optimalcorrection set ∆ w opt so that ∆ w opt = arg min ∆ w n k ∆ w k (cid:12)(cid:12)(cid:12) Φ (cid:16) ∆ w ( k ) (cid:17) ≤ Φ target and ∆ w n Ω n = 0 o (6)where | stands for “subject to”, k·k denotes the ℓ -norm, k ∆ w k is the number of corrections ( d N C ≡k ∆ w k ), and Φ (∆ w ) is a single function measuringthe array performance when ∆ w is applied, while Φ target is the fixed user-defined target value of thearray performance for a reliable working of theradiating system.It is worth pointing out that the choice of Φ (∆ w ) dependson the applicative context and it can take into account clas-sical array parameters (e.g., HPBW , SLL , and Directivity)or high-level system requirements such as the link budgetor capacity. For instance,
Φ (∆ w ) , HP BW (∆ w ) and Φ target , HP BW original can be considered if the beamwidthof the original array
HP BW original is the fundamental feature F = 1 k = k = k = k = || ∆ w|| , CP|| ∆ w|| , CP ( a ) E l e m en t i nde x , n Iteration index, kToy example, N = 4, N F = 1 1 2 3 4 0 1 2 3 ( b ) Figure 2.
Illustrative Example (“ Toy Example ”: N = 4 , N F = 1 → η F =25 %; DC SLL = − [dB], SLL target = − . [dB]) - Evolution of( a ) the ℓ -norm and the ℓ -norm of the excitation correction vector, ∆ w ,and of ( b ) the status (“required/non-required”) of the element corrections, r n / s n ( n = 1 , ..., N ) [”required” correction ( r n = 1 ) → red; “non-required”correction ( s n = 1 ) → green; failed element ( Ω n = 1 ) → gray; “unknown”-status correction ( r n = s n = Ω n = 0 ) → white], versus the CP iterationindex, k . to be recovered. Furthermore, the formulation in (6) can beseamlessly extended to include multiple array characteristicsinto the algorithm by recasting Φ (∆ w ) and Φ target to vec-torial quantities.III. CP -B ASED S OLUTION T ECHNIQUE
Before presenting the CP -based correction method, let us pointout that the direct ℓ -norm minimization of the correctedweights as in (6) is a non-deterministic polynomial-time hardproblem [41]. Indeed, finding the optimal solution, ∆ w opt ,would require testing all possible combinations of d N C ( d N C =1 , ..., N C ) corrections, whose number grows exponentially 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.2020.3045511Copyright (c) 2021 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, 2020 4
Table I
Illustrative Example (“ Toy Example ”: N = 4 , N F = 1 → η F = 25 %; DC SLL = − [ D B],
SLL target = − . [ D B]) - S
TEP - BY - STEP DESCRIPTIONOF THE EVOLUTION OF THE CP VARIABLES THROUGHOUT THE ITERATIVE PROCESS FOR THE CORRECTED - ARRAY SYNTHESIS . k Step n ( k ) least s ( k ) r ( k ) ∆ w ( k ) tr ∆ w ( k ) opt (cid:13)(cid:13)(cid:13) ∆ w ( k ) opt (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ∆ w ( k ) opt (cid:13)(cid:13)(cid:13) Φ (cid:16) ∆ w ( k ) opt (cid:17) − { , , , } { , , , } − {− . , . , . , − . × − } . − .
51 4 { , , , } − {− . , . , . , . } − − − − { , , , } − {− . , . , . , . } {− . , , . , } . − .
51 1 { , , , } − { , , . , } − − − − { , , , } − { , , . , } { , , . , } . − . { , , , } − { , , , } − − − − { , , , } − { , , , } { , , , } − . { , , , } { , , , } { , , . , } { , , . , } . − .
54 1 − − − − { , , . , } . − . with N . On the other hand, one needs to note that, if theoptimal set of correction positions [i.e., the binary vector b Λ = nb Λ n ; n = 1 , ..., N o where b Λ n = 1 if the n -th arrayelement is corrected and b Λ n = 0 , otherwise - Fig. 1( b )] was a-priori known, then the original failure correction problem(6) would reduce to a much simpler synthesis problem aimedat determining the weights of the corrected excitations, b w .Moreover, the proposed approach also takes advantage fromthe fact that, within the framework of ℓ -norm minimizationproblems, the backtracking algorithm has yielded remarkableresults [37][38].In words, the CP method starts from a full set of corrections byiteratively reducing its ℓ -norm. At each iteration ( k being theiteration index, k = 1 , ..., K ), the algorithm uses a heuristic toguess which correction is of least importance. A trial solutionis then generated by removing the least important correctionfrom the current best solution. Subsequently, the ℓ -norm ofthe trial solution is minimized subject to the pattern require-ments. If the requirements are satisfied, the trial solution isstored as the new best solution and the iteration continuesuntil the convergence ( k = k opt ). Otherwise, the algorithm backtracks on its previous guess and the correction previously-removed is restored. Towards this purpose, the algorithm keepstrack throughout the iterations of which corrections are “re-quired” (i.e., removing them the algorithm failed and resultedin backtracking) or “non-required” (i.e., the constraints can bemet without using these corrections) by means of two binaryvectors, r = { r n ; n = 1 , ..., N } and s = { s n ; n = 1 , ..., N } ,whose n -th entry is if the n -th correction is marked asrequired/non-required and , otherwise. Accordingly, the CP nature of the proposed methodology is not related to thecompression of a matrix through singular value decomposi-tion, but rather to the retrieval of the sparsest set of failurecorrections ∆ w opt to comply with the radiation constraints athand. From an algorithmic viewpoint, the procedural steps ofthe CP method look as follows: • Step 0 [ Initialization ] ( k = 0 ) - Reset the backtrackvectors ( r ( k ) (cid:5) k =0 = and s ( k ) (cid:5) k =0 = ) and computethe initial solution, ∆ w ( k ) opt k k =0 ( ∆ w ( k ) opt being the currentbest solution of the CP algorithm at the k -th iteration), as the ℓ -norm solution of the failure correction problem ∆ w ( k ) opt k k =0 = arg min ∆ w n k ∆ w k (cid:12)(cid:12)(cid:12) Φ (∆ w ) ≤ Φ target and ∆ w n Ω n = 0 o (7)where k·k denotes the ℓ -norm. Towards this end, theconstrained minimization problem defined in (7) is solvedwith the interior-point algorithm [42]. This latter tech-nique iteratively defines and solves (via a gradient descentsearch) a sequence of intermediate equality-constrainedoptimization problems which approximate the originalone with increasing accuracy until convergence is reached[42]. More specifically, the procedure is initialized at ∆ w = and the iterations ( i being the interior-pointalgorithm iteration index, i = 1 , ..., I ) are stopped whenat least one of the following conditions holds true: ( a )the number of iterations exceeds I max ( i > I max ),( b ) a step smaller than ξ is attempted, or ( c ) the first-order optimality condition is satisfied within the thresholdvalue ζ , where I max , ξ , and ζ are user-defined controlparameters. • Step 1 [ Least-Important Correction Guess ] - Incrementthe iteration index ( k → k + 1 ) and find the position, n least , of the correction within the last best solution, ∆ w ( k − opt , that is of “least importance”. In the following,this choice is carried out by identifying the correctionhaving the minimal non-zero magnitude (since, accordingto Parseval theorem, it is expected to have the minimumintegral impact on the radiation pattern) and not markedas “required” n least = arg min n ((cid:12)(cid:12)(cid:12) ∆ w ( k − opt,n (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ w ( k − opt,n = 0 and r ( k − n = 0 o . (8)If no such correction is found, then all corrections areeither or marked as “required” and the iteration stops( k = k opt ) by setting the optimal set of excitationcorrections to ∆ w opt ≡ ∆ w ( k − opt . Otherwise, the vectorof “non-required” excitations is updated by adding the n least -th correction ( s ( k ) n = s ( k − n + δ n,n least , δ p,q being 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.2020.3045511Copyright (c) 2021 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, 2020 5 the Kronecker delta) and a trial solution, ∆ w ( k ) tr , isgenerated by removing the n least -th correction from thelast best solution ∆ w ( k ) tr,n = ∆ w ( k − opt,n (1 − δ n,n least ) . (9)It is worth noting that the trial solution ∆ w ( k ) tr has a ℓ -norm value smaller than ∆ w ( k − opt since ∆ w ( k ) tr,n = 0 if n = n least , while ∆ w ( k ) tr,n = ∆ w ( k − opt,n otherwise, but thepattern constraints could be not fit since the array weightshave been changed and, in turn, the pattern, as well; • Step 2 [ Correction Removal Attempt ] - By using ∆ w ( k ) tr as reference configuration, look for a new set of correc-tions, ∆ w ( k ) opt , with minimal ℓ -norm that meets the pat-tern requirements without changing any of the correctionsthat are faulty or marked as “non-required”. Towards thisend, the following constrained minimization problem ∆ w ( k ) opt = arg min ∆ w n k ∆ w k (cid:12)(cid:12)(cid:12) Φ (∆ w ) ≤ Φ target and ∆ w n (cid:16) Ω n + s ( k ) n (cid:17) = 0 o (10)is solved still by means of the interior-point algorithm.If the attempt is successful, then the set of “required”excitations is cleared ( r ( k ) = ) and goto Step 1 .Otherwise, goto
Step 3 ; • Step 3 [ Backtrack ] - If the
Step 2 does not succeed infinding a new best vector, ∆ w ( k ) opt , the CP algorithmbacktracks on the last guess. This means that both thebest correction vector and the vector of “non-required”corrections are kept from the previous iteration ( ∆ w ( k ) opt ≡ ∆ w ( k − opt , s ( k ) ≡ s ( k − ), while the n least -th correctionis marked as required ( r ( k ) n = r ( k − n + δ n,n least ). Goto Step 1 .It is worth noticing that the ℓ -norm of the corrections isnever minimized directly by the interior point algorithm, whichrather operates on the ℓ -norm expression in (10). Moreover,alternative concepts may be used to implement the heuristicfor the selection of the “least important correction” in Step. 1.Indeed, a universal heuristic cannot be a-priori defined sincethe relevance of a correction strongly depends on the user-defined Φ ( · ) . Nevertheless, according to the Parseval theorem,the proposed heuristic (also adopted in [37] in a differentcontext) is expected to provide robust performance whenever Φ ( · ) measures a feature depending on the array pattern shape(such as SLL or HPBW ).In order to detail the step-by-step behaviour of the CP method,let us consider a “toy” example where a simplified low-dimension correction problem is dealt with. More in detail,the scenario at hand is that of an array with N = 4 elementssubject to a failure on the n = 2 element (i.e., N F = 1 and Ω = 1 → N C = 3 ), where the original set of excitations hasbeen chosen so that the radiated pattern is a Dolph-Chebyshev( DC ) one with a SLL value equal to
SLL = − [dB] (i.e., w = { , . , . , } ). Moreover, the metric of the arrayperformance, Φ , has been set to the maximum of the SLL [ SLL ( u ) , × log (cid:16) d P P ( u ) d P P (0) (cid:17) , being P P ( u ) , | F ( u ) | ] E xc i t a t i on m agn i t ude [ no r m a li z ed v a l ue ] x / λ DC tapering, SLL = -15 [dB], N = 16, N F = 3OriginalCorrectedFaulty ( a ) -20-15-10-5 0-1 -0.5 0 0.5 1 P o w e r P a tt e r n , PP ( u ) [ d B ] uDC tapering, SLL = -15 [dB], N = 16, N F = 3OriginalFaultyCorrectedSLL ( b ) Figure 3.
Numerical Assessment ( Test Case : N = 16 , N F = 3 → η F ≈ %; DC SLL = − [dB]) - Plot of ( a ) the excitations and of ( b ) thepower patterns of the original, the faulty, and the corrected arrays. within the angular set Θ ( Θ = { u = ± . , u = ± . } ) Φ (∆ w ) = max u ∈ Θ { SLL ( u ) } (11)with a target value equal to Φ target ≡ SLL target = − . [dB].Figure 2 and Table I summarize the application of the CP algorithm to the “toy” problem at hand. At the initialization ( k = 0) , the solution ∆ w (0) opt = {− . , . , . , − . × − } [Tab. I - Fig. 2( a )] has been found by performingthe ℓ -norm minimization (7). Successively ( k = 1 ), first thecorrection at the n = 4 array position has been identified asthe least important ( n least = 4 ), since it has the smallest non-zero magnitude within ∆ w ( k − opt . Thus, a trial solution hasbeen generated by re-setting the n least -th correction of ∆ w (0) opt ( ∆ w (1) tr = {− . , . , . , . } ) and updating the “nonrequired” vector s (1) [ s (1)4 = 1 - Fig. 2( b )] ( Step 1 ). Afterward(
Step 2 ), since ∆ w (1) tr already satisfies the pattern requirements[ Φ(∆ w (1) tr ) ≤ Φ target ], the ℓ -norm minimization (7) hasnot been necessary and the current best solution has beenupdated to the trial one [ ∆ w ( k ) opt ≡ ∆ w (1) tr - Fig. 2( a )] bycontinuing the iterative process with the Step 1 , once again.At the next iteration ( k = 2 ), the correction at n = 1 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.2020.3045511Copyright (c) 2021 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, 2020 6 been guessed to be the least important one ( n least = 1 )and s (2) has been updated by setting s (2)1 = 1 [Fig. 2( b )].Moreover, a trial solution has been generated accordingly( ∆ w (2) tr = { . , . , . , . } - Tab. I) to start the ℓ -norm minimization of the Step 2 . A new solution has beengenerated, ∆ w (2) = {− . , . , . , . } , that fits thepattern requirements [ Φ(∆ w (2) ) ≤ Φ target ], then the currentbest solution has been updated [ ∆ w (2) opt ≡ ∆ w (2) - Fig. 2( a )].When running the iteration k = 3 , since s (3)1 = s (3)4 = 1 and Ω = 1 [Fig. 2( b )], the location of the least importantcorrection has been mandatorily set to n least = 3 . However,removing the correction at n = 3 in ∆ w (2) opt meant nocorrection to be done (i.e., ∆ w (3) tr = { . , . , . , . } ), thusthe generation of an unreliable pattern [ Φ(∆ w (3) tr ) > Φ target ].Therefore, the algorithm has backtracked on its decision byremoving the correction on the n = 3 element of the arrayfrom the non-required corrections ( s (3)3 = 0 ) and marking itas required [ r (3)3 = 1 - Fig. 2( b )]. The algorithm reachedthe last iteration ( k = 4 ), but halted at the Step 1 becauseall corrections are either zero or marked as “required” ( → ∆ w opt ≡ ∆ w (3) opt and d N C = 1 ).IV. N UMERICAL A SSESSMENT AND V ALIDATION
This Section describes a representative set of numerical bench-marks that have been performed to assess the effectiveness aswell as the limitations of the CP -based correction also withrespect to reference methods in the state-of-the-art literature[21][23].For comparison purposes, but without loss of generality, thefunction Φ has been set to the SLL of the radiated pattern(11) in the sidelobe angular region Θ ( BW target ) , BW target being - unless mentioned otherwise - the mainlobe beamwidthof the pattern radiated by an array with N = N C elements.More specifically, the mainlobe beamwidth, BW , is definedas the angular range for which P P ( u ) ≥ SLL target . Finally,the target threshold Φ target has been generally chosen equal tothe SLL of the original whole array ( Φ target ≡ SLL target = SLL ). A. Numerical Assessment
The first test case (
Test Case - Tab. II) deals with an arraywhose dimension allows one the computation of the optimalsolution to the
MCFC problem (6) through an exhaustiveprocess in a limited amount of time. It is concerned witha N = 16 elements Dolph-Chebyshev ( DC ) tapered uniformlinear array that radiates a pattern with an SLL equal to
SLL = − [dB]. The failures have been assumed at N F = 3 ( → η F = 18 . %, η F , N F N ) element locations( Ω = Ω = Ω = 1 ) [Fig. 3( a )] so that the SLL of the faultyarray increases up to ] SLL = − . [dB] [Fig. 3( b )].By comparing the results from the exhaustive search and the CP method (Fig. 4), it turns out that they are coincident with b N C = 3 ( → η C ≡ η F , η C , N C N ) excitations being corrected It means that the constrained minimization problem in (7) is solved foreach value of d N C ( d N C = 1 , ..., N C ), that is P N C d N C =1 (cid:18) N C d N C (cid:19) times. F = 3 k = k = k = k = k = k = k = || ∆ w|| , Full|| ∆ w|| , CP|| ∆ w|| , CP ( a ) E l e m en t i nde x , n Iteration index, kDC tapering, SLL = -15 [dB], N = 16, N F = 3 ( b ) Figure 4.
Numerical Assessment ( Test Case : N = 16 , N F = 3 → η F ≈ %; DC SLL = − [dB]) - Evolution of ( a ) the ℓ -norm andthe ℓ -norm of the excitation correction vector, ∆ w , and of ( b ) the status(“required/non-required”) of the element corrections, r n / s n ( n = 1 , ..., N ),versus the CP iteration index, k . Table II Numerical & Comparative Assessment - T
EST CASES DESCRIPTORS . Test Case N Faulty Element η F Indexes { , , } .
752 50 { , , } .
03 32 { , } .
254 32 { , , } . [Fig. 3( a ) and Fig. 4( a )] and a radiation pattern having thesame (rather smaller) SLL of the original array [Fig. 3( b ) andTab. III], while maintaining the beamwidth within BW target =14 . [deg] (i.e., the beamwidth of a DC -tapered uniform lineararray of N = N C elements), despite the use of only b η ′ C =23 . % ( b η ′ C , d N C N C ) of the admissible corrections, N C . TableIII summarizes the outcomes by also reporting the values ofboth the ℓ - ( ≡ b N C ) and the ℓ -norm of the correction vector. 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.2020.3045511Copyright (c) 2021 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, 2020 7 E xc i t a t i on m agn i t ude [ no r m a li z ed v a l ue ] x/ λ DC tapering with SLL = -25 [dB], N = 50, N F = 3OriginalFaultyCorrected ( a ) -30-25-20-15-10-5 0-1 -0.5 0 0.5 1 P o w e r P a tt e r n , PP ( u ) [ d B ] uDC tapering with SLL = -25 [dB], N = 50, N F = 3OriginalFaultyCorrectedSLL ( b ) Figure 5.
Comparative Assessment ( Test Case N = 50 , N F = 3 → η F = 6 %; DC SLL = − [dB]; SLL target = − . [dB]) - Plotof ( a ) the excitations and of ( b ) the power patterns of the original, the faulty,and corrected arrays either in [23] or synthesized with the CP method.Table III Numerical Assessment ( Test Case : N = 16 , N F = 3 → η F ≈ %; DC SLL = − [ D B]) - P
ERFORMANCE INDEXES . SLL [dB] BW [deg] k ∆ w k k ∆ w k Target − . . − − Original Array − .
00 11 . − − Faulty Array − .
19 11 . − − Corrected Array − .
02 14 .
15 1 .
31 3
To analyze the behavior of the CP correction method morein detail, Figure 4 shows the evolution of the norms of thecurrent best solution, w ( k ) opt [Fig. 4( a )], as well as of thebacktrack vectors [Fig. 4( b )] versus the iteration index k .According to the CP guidelines and starting from the initialsetup ( r ( k ) (cid:5) k =0 = and s ( k ) (cid:5) k =0 = ), the iterative processmarks each array element as “required” or “non-required”until all elements, unless the faulty ones, are labeled [Fig.4( b )]. Unless the beginning, each time a correction turns outto unnecessary ( s ( k ) n = 1 ), the number of corrections, d N C k k ( d N C k k ≡ (cid:13)(cid:13)(cid:13) ∆ w ( k ) opt (cid:13)(cid:13)(cid:13) ), decreases by a unit, while (cid:13)(cid:13)(cid:13) ∆ w ( k ) opt (cid:13)(cid:13)(cid:13) seems unaffected since, when the ℓ -norm minimization (7)takes place, most corrections already have a very small value( | ∆ w n | ≈ ), thus removing them does not bring a visiblevariation. E xc i t a t i on m agn i t ude [ no r m a li z ed v a l ue ] x/ λ DC tapering with SLL = -25 [dB], N = 50, N F = 3OriginalFaulty Corrected[Rodriguez 2000] ( a ) -30-25-20-15-10-5 0-1 -0.5 0 0.5 1 P o w e r P a tt e r n , PP ( u ) [ d B ] uDC tapering with SLL = -25 [dB], N = 50, N F = 3OriginalFaultyCorrected[Rodriguez 2000] ( b ) Figure 6.
Comparative Assessment ( Test Case N = 50 , N F = 3 → η F = 6 %; DC SLL = − [dB]; SLL target = − . [dB]) - Plotof ( a ) the excitations and of ( b ) the power patterns of the original, the faulty,and corrected arrays either in [23] or synthesized with the CP method. B. Comparative Assessment
As for the comparative study with competitive state-of-the-artmethods, the “
Test Case ” refers to a benchmark examplefrom [23]. More specifically, a N = 50 elements linear hasbeen considered with N F = 3 ( η F = 6 %) faulty elements asindicated in Tab. II and shown in Fig. 5( a ) where the arrayexcitations are reported. Due to the failures, the SLL of thearray worsens from
SLL = − [dB] up to ] SLL = − . [dB] [Fig. 5( b )]. When setting the target SLL to the same valueyielded in [23] with a minimum of corrections, it turns outthat the CP solution recovers the target features by perturbingonly b η ′ C = 4 . % of the non-defective elements (vs. b η ′ C =10 . % in [23] - Tab. IV), the number of corrections ( d N C =2 ) being smaller than the number of faulty elements, as well.When reducing the threshold down to SLL target = − . [dB], the number of compensating elements increases to b N C = 10 [vs. b N [ Rodriguez C = 12 - Fig. 6( a )] witha value of the amplitude distribution coefficient ( DR , max n =1 ,...,N (cid:12)(cid:12)(cid:12) I n I n ± (cid:12)(cid:12)(cid:12) ) better than the original one ( DR CP =1 . vs. DR = 3 . ) as for the GA -based correction( DS [ Rodriguez = 1 . ) (Tab. IV). It is also worth pointingout that the CPU -time for the correction process takes less than minutes (i.e., ∆ t = 75 [sec], while ∆ t = 108 [sec] for thecase with SLL target = − . [dB]) on a standard laptop 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.2020.3045511Copyright (c) 2021 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, 2020 8
Table IV
Comparative Assessment ( Test Case N = 50 , N F = 3 → η F = 6 %; DC SLL = − [ D B];
SLL target = − . [ D B]) - P
ERFORMANCEINDEXES . SLL [dB] BW [deg] DR k ∆ w k k ∆ w k η C [%] b η ′ C [%] Target − . . − − − − − Original Array −
25 5 .
18 3 . − − − − Faulty Array − .
51 5 .
25 3 . − − . . Corrected Array − . .
60 1 .
58 1 .
36 2 94 . . [Rodriguez 2000] − . n.a. n.a. n.a. . . Target − .
50 5 . − − − − − Original Array − .
00 5 .
18 3 . − − − − Faulty Array − .
51 5 .
25 3 . − − . . Corrected Array − .
50 5 .
53 1 .
56 2 .
45 10 94 . . [Rodriguez 2000] − .
47 5 .
76 1 .
29 2 .
67 12 94 . . E xc i t a t i on m agn i t ude [ no r m a li z ed v a l ue ] x/ λ DC Tapering with SLL = -35 [dB], N = 32, N F = 2OriginalFaulty Corrected[Yeo 1999] ( a ) -40-35-30-25-20-15-10-5 0 -1 -0.5 0 0.5 1 P o w e r P a tt e r n , PP ( u ) [ d B ] uDC Tapering with SLL = -35 [dB], N = 32, N F = 2OriginalFaultyCorrected[Yeo 1999] ( b ) Figure 7.
Comparative Assessment ( Test Case N = 32 , N F = 2 → η F = 6 . %; DC SLL = − [dB]) - Plot of ( a ) the excitations and of( b ) the power patterns of the original, the faulty, and corrected arrays eitherin [21] or synthesized with the CP method. computer.Still for comparison purposes, the CP method is applied to the“ Test Case ” (Tab. II) drawn from [21] where the authorsproposed a FC method not aimed at reducing the number ofcorrections (i.e., d N C [ Y eo = N C ), but devoted to yieldthe optimal results in terms of SLL and beamwidth for thecorrected array. In [21], the original array was a N = 32 -elements linear array with a Dolph-Chebyshev tapering and SLL = − [dB], while N F = 2 ( → η F = 6 . %) failureshave been applied ( Ω = Ω = 1 ) causing a performancedowngrade of about [dB] in the SLL (i.e., ] SLL = − . [dB]). E xc i t a t i on m agn i t ude [ no r m a li z ed v a l ue ] x/ λ DC Tapering with SLL = -35 [dB], N = 32, N F = 3OriginalFaulty Corrected[Yeo 1999] ( a ) -40-35-30-25-20-15-10-5 0 -1 -0.5 0 0.5 1 P o w e r P a tt e r n , PP ( u ) [ d B ] uDC Tapering, SLL = -35 [dB], N = 32, N F = 3OriginalFaultyCorrected[Yeo 1999] ( b ) Figure 8.
Comparative Assessment ( Test Case N = 32 , N F = 3 → η F ≈ . %; DC SLL = − [dB]; SLL target = − . [dB]) - Plotof ( a ) the excitations and of ( b ) the power patterns of the original, the faulty,and corrected arrays either in [21] or synthesized with the CP method.Table V Comparative Assessment ( Test Case N = 32 , N F = 2 → η F = 6 . %; DC SLL = − [ D B];
Test Case SLL target = − . [ D B]) -P
ERFORMANCE INDEXES . SLL [dB]
HPBW [deg] k ∆ w k k ∆ w k η C [%] b η ′ C [%] Target − . − − − − − Original Array − .
00 4 . − − − − Faulty Array − .
11 4 . − − .
75 0 . Corrected Array − .
00 4 .
74 2 .
17 17 93 .
75 56 . [Yeo 1999] − .
78 4 .
77 3 .
57 30 93 .
75 100 . Target − . − − − − − Original Array − .
00 4 . − − − − Faulty Array − .
24 4 . − − .
625 0 . Corrected Array − .
28 5 .
34 3 .
86 20 90 .
625 68 . [Yeo 1999] − .
28 5 .
36 5 .
62 29 90 .
625 100 . Figure 7( a ) shows the original and the faulty excitations ofthe array along with the corrected ones reported in [21] orsynthesized with the CP method. As it can be noted, this latteruses only b N C = 17 out of the N C = 30 working elements( → b η ′ C = 56 . % - Tab. V) to fit the target requirements( SLL target = − [dB]) by also slightly improving both thehalf-power beamwidth and the SLL of the corrected array in[21] [
HP BW CP = 4 . [deg] vs. HP BW [ Y eo = 4 . [deg] and SLL CP = − . [dB] vs. SLL [ Y eo = − . [dB] - Tab. V and Fig. 7( b )]. 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.2020.3045511Copyright (c) 2021 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, 2020 9 η F DC Tapering, N = 50, SLL = -25 [dB] η ^ C η C η F ( a ) η F DC Tapering, N = 100, SLL = -25 [dB] η ^ C η C η F ( b ) R e l a t i v e C o rr e c t i on R a t e , η ^ ’ C Failure Rate, η F DC Tapering, SLL = -25 [dB]N = 50N = 100 ( c ) Figure 9.
Performance Analysis ( DC SLL = − [dB]) - Plots of thepercentage of failed, η F , reconfigurable, η C , and corrected, b η C , elementsversus the failure rate, η F in ( a ) a medium ( N = 50 ) and in ( b ) a large( N = 100 ) array; ( c ) Behavior of the relative correction rate, b η ′ C , versus η F . In the next test case (“
Test Case
4” - Tab. II), a failure hasbeen added to the previous array configuration and it stillcomes from [21]. More specifically, the failure rate has beenincreased up to η F ≈ . % with another faulty element at x = x (i.e., Ω = 1 ) that causes a further worsening of the SLL to ] SLL = − . [dB] with respect to the whole array( SLL = − [dB]). For a fair comparison, since the correctionmethod in [21] yielded an SLL equal to
SLL [ Y eo = − . [dB], the same value has been maintained as targetfor the CP -based correction ( SLL target ≡ SLL [ Y eo ).Once again, the CP method was successful in fitting therequirement ( SLL CP = SLL target - Tab. V) with a reducednumber of elements [ b N C = 20 - Fig. 8( a ) and Tab. V], eventhough higher than the “ Test Case ” ( b η ′ C ⌋ N F =3 > b η ′ C ⌋ N F =2 - Tab. V) because of the presence of one more fault element.Moreover, the CP array affords a narrower beam than that ofthe corrected array in [21] [ HP BW CP < HP BW [ Y eo E l e m en t i nde x , n Iteration index, kDC tapering, SLL = -25 [dB], N = 100, η F = 8% 0 20 40 60 80 100 0 20 40 60 80 100 E l e m en t i nde x , n Iteration index, kDC tapering, SLL = -25 [dB], N = 100, η F = 12% ( a ) ( b ) E l e m en t i nde x , n Iteration index, kDC tapering, SLL = -25 [dB], N = 100, η F = 16% ( c ) Figure 10.
Performance Analysis ( N = 100 ; DC SLL = − [dB]) - Evolution of the backtracking vectors versus the CP iteration index when ( a ) η F = 8 %, ( b ) η F = 12 %, and ( c ) η F = 16 %. -25-24-23-22-21-20-19 0 5 10 15 20 25 30 35 40 S LL [ d B ] Num. Corrections, N^ C DC tapering with SLL = -25 [dB], N = 50, η F = 16%CorrectedFaultyOriginal Figure 11.
Performance Analysis ( DC SLL = − [dB]; N = 50 ; η F =16 %) - Behavior of the side lobe level of the corrected array, [ SLL , versusthe number of corrections applied, b N C . - Fig. 8( b ) and Tab. V]. As far as the computational burden isconcerned, the corrections for both cases (i.e., “ Test Case
Test Case CP -basedapproach in around [min] (i.e., ∆ t T est = 47 [sec] and ∆ t T est = 45 [sec]).
C. Performance Analysis
In order to assess the potentialities and the limitations of theproposed approach, a performance analysis has been carriedout by varying the “correction scenario” at hand, but keepingalways the array size from medium to large.The first test case of this section deals with the dependence ofthe CP performance on the failure rate η F . More in detail, twoarrays ( N = N and N = M × N , being N = 50 and M = 2 )with DC tapering and SLL = − [dB] have been considered,while the failure rate has been varied within the range η F ∈ 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.2020.3045511Copyright (c) 2021 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, 2020 10
Table VI
Performance Analysis ( DC SLL = − [ D B]) - T EST CASES DESCRIPTORS AND PERFORMANCE INDEXES . N Faulty Element Indexes
SLL ] SLL SLL target
SLL CP BW ] BW BW target BW CP b N C η C [%] b η ′ C [%] { , } − . − . − . − . .
31 5 .
51 5 .
54 5 .
54 4 96 8 . { , , , } − . − . − . − . .
31 5 .
72 5 .
78 5 .
78 7 92 15 . { , , , , , } − . − . − . − . .
31 5 .
92 6 .
05 6 .
05 21 88 47 . { , , , , , , , } − . − . − . − . .
31 6 .
08 6 .
35 6 .
35 37 84 88 . { , , , } − . − . − . − . .
63 2 .
73 2 .
74 2 .
74 4 96 4 . { , , , , , , , } − . − . − . − . .
63 2 .
83 2 .
86 2 .
86 5 92 5 . { , , , , , , , , , , , } − . − . − . − . .
63 2 .
93 2 .
99 2 .
99 13 88 14 . { , , , , , , , , , , , , , , , } − . − . − . − . .
63 3 .
00 3 .
14 3 .
14 14 84 16 . -35-30-25-20-15-10-5 0-1 -0.5 0 0.5 1 P o w e r P a tt e r n , PP ( u ) [ d B ] uDC tapering with SLL = -25 [dB], N = 25, η F = 12%OriginalFaultyCorrectedSLL -40-30-20-10 0-0.2 -0.1 0 -35-30-25-20-15-10-5 0-1 -0.5 0 0.5 1 P o w e r P a tt e r n , PP ( u ) [ d B ] uDC tapering with SLL = -25 [dB], N = 50, η F = 12%OriginalFaultyCorrectedSLL -40-30-20-10 0-0.1 -0.05 0 ( a ) ( b ) -35-30-25-20-15-10-5 0-1 -0.5 0 0.5 1 P o w e r P a tt e r n , PP ( u ) [ d B ] uDC tapering with SLL = -25 [dB], N = 100, η F = 12%OriginalFaultyCorrectedSLL -40-30-20-10 0-0.05 -0.025 0 -35-30-25-20-15-10-5 0-1 -0.5 0 0.5 1 P o w e r P a tt e r n , PP ( u ) [ d B ] uDC tapering with SLL = -25 [dB], N = 150, η F = 12%OriginalFaultyCorrectedSLL -40-30-20-10 0-0.04 -0.02 0 ( c ) ( d ) -35-30-25-20-15-10-5 0-1 -0.5 0 0.5 1 P o w e r P a tt e r n , PP ( u ) [ d B ] uDC tapering with SLL = -25 [dB], N = 500, η F = 12%OriginalFaultyCorrectedSLL -40-30-20-10 0-0.01 0 ( e ) Figure 12.
Performance Analysis ( η F = 12 %; DC SLL = − [dB]) - Plots of the power patterns of the original, the faulty, and the correctedarrays when ( a ) N = 25 , ( b ) N = 50 , ( c ) N = 100 , ( d ) N = 150 , and ( e ) N = 500 . { p × η ; p = 1 , ..., P } ( η = 2 % and P = 3 ). The descriptiveparameters of the benchmarks at hand are reported in Tab.VI along with the indexes of the correction outcomes. It isworth pointing out that the locations of the faulty elements(Tab. VI) have been chosen so that ] SLL k η F N ≈ ] SLL k η F N (i.e.,elements of the same “importance” turn out to be switched offin both N - and N -elements arrays for a fixed value of η F ).Towards this end, all faulty arrangements share the condition Ω = Ω = 1 . Moreover, the following “ failure rule ” hasbeen applied symmetrically to the array center, n f being thefailure index ( f = 1 , ..., F ). If n f < N then n f − j ⌋ M × N = M × n f ⌋ N − j , else ( n f > N ) n f − j ⌋ M × N = M × n f ⌋ N + j where j = 0 , ..., P × N F .As it can be observed in Figs. 9( a )-9( b ), the number ofcorrections increases ( η C ) with the failure rate, η F , but moresignificantly for the medium array ( N -elements) than for thelarger one ( N -elements). As a matter of fact, ≤ η C ⌋ N ≤ %, while ≤ η C ⌋ N ≤ %. Such an outcome is mademore evident in Fig. 9( c ) where the behavior of b η ′ C versusthe failure rate η F for the two array sizes is shown. It turnsout that, for the medium array, more than % of the N C reconfigurable elements undergo corrections when η F = 16 %, while less than % are needed when the array is N -elements wide. As expected, the larger the array the easier isthe correction provided to have at disposal a suitable correctionprocedure able to deal with high-dimensional solution spacesin a reasonable amount of time. As for this latter item, the CP -method needs at most ∆ t = 25 . [min] for solving thecorrection problems resumed in Tab. VI. For completeness,Figure 10 gives some insights on the iterative process for thesynthesis of the CP -corrected array by showing the evolutionof the backtracking vectors [Fig. 4( b )] for different failurepercentages, η F , but always referring to the arrangement with N elements.The next example is aimed at assessing the trade-off betweenthe performance of the corrected array and the number ofcorrections required. The behaviour of [ SLL as a function b N C when dealing with the previous N = 50 , SLL = − [dB], η F = 16% benchmark configuration shows that ( i ) fewcorrections are sufficient to considerably reduce the recovered SLL (i.e., ] SLL − [ SLL k b N C =1 ≈ . [dB] - Fig. 11), andthat ( ii ) the required number of corrections decreases rapidlyif slightly lower radiation performance are accepted (e.g., [ SLL k b N C =11 ≈ − . [dB] vs. [ SLL k b N C =1 ≈ − . [dB]- Fig. 11).The last test case deals with the effectiveness of the proposedapproach when varying the array size. Towards this end, thenumber of elements of an array affording a Dolph-Chebyshevpattern with SLL = − [dB] has been changed withinthe set of values N ∈ { , , , , } . Additionally,the analysis has been repeated for different failure rates (i.e., η F ∈ { , , } %) with the locations of the faulty elementsas in Tab. VII. Figure 12 plots the pattern of the correctedarray along with those of the original array and of the failedone when η F = 12 %. Whatever the dimension of the array[ N = 25 - Fig. 12( a ); N = 50 - Fig.12( b ); N = 100 -Fig.12( c ); N = 150 - Fig. 12( d ); N = 500 - Fig. 12( e )], the CP correction method successfully recovers the original SLL and it fits the beamwidth target, BW target , which is slightlywider than that of the original array, but always smaller thanthat of the damaged array (see the insets in Fig. 12 and Tab.VII), with a limited number of corrections, which is always
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.2020.3045511Copyright (c) 2021 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, 2020 11
Table VII
Performance Analysis ( DC SLL = − [ D B]) - T EST CASES DESCRIPTORS AND PERFORMANCE INDEXES . N Faulty Element Indexes
SLL ] SLL SLL target
SLL CP BW ] BW BW target BW CP b N C η C [%] b η ′ C [%] { } − . − . − . − . .
83 11 .
15 11 . . . { , } − . − . − . − . .
83 11 .
69 11 . . . { , , } − . − . − . − . .
83 12 .
43 12 . . . { , } − . − . − . − . .
31 5 .
47 5 .
54 5 .
54 4 96 8 . { , , , } − . − . − . − . .
31 5 .
74 5 .
78 5 .
78 7 92 15 . { , , , , , } − . − . − . − . .
31 6 .
15 6 .
05 6 .
05 8 88 18 . { , , , } − . − . − . − . .
63 2 .
71 2 .
74 2 .
74 3 96 3 . { , , , , , , , } − . − . − . − . .
63 2 .
84 2 .
86 2 .
86 6 92 6 . { , , , , , , , , , , , } − . − . − . − . .
63 3 .
06 2 .
99 2 .
99 6 88 6 . { , , , , , } − . − . − . − . .
75 1 .
80 1 .
82 1 .
82 3 96 2 . { , , , , , , , , , , , } − . − . − . − . .
75 1 .
89 1 .
90 1 .
90 5 92 3 . { , , , , , , , , , , , , , , , , , } − . − . − . − . .
75 2 .
04 1 .
99 1 .
99 7 88 5 . , − . − . − . − . .
52 0 .
54 0 .
54 0 .
54 2 96 0 . , ∪ [81 , − . − . − . − . .
52 0 .
56 0 .
57 0 .
57 5 92 1 . , ∪ [21 , ∪ [81 , − . − . − . − . .
52 0 .
61 0 .
59 0 .
59 5 88 1 . R e l a t i v e C o rr e c t i on R a t e , η ^ ’ C Array Size, NDC tapering, SLL = -25 [dB] η F = 4% η F = 8% η F = 12% Figure 13.
Performance Analysis ( DC SLL = − [dB]) - Behavior of therelative correction rate, b η ′ C , versus N for different values of the failure rate( η F ∈ { , , } %). below the b η ′ C = 60 % of the admissible set of reconfigurableelements (Fig. 13). As expected, the percentage of correctionsgrows with the failure rate η F , but, coherently with the resultsin Fig. 9( c ), it decreases as the array size, N , increases sincemore degrees-of-freedom are available for the correction.As regards the numerical efficiency of the proposed approachwhen large numbers of elements are at hand, the plot of ∆ t ( CP ) obtained in these latter examples (Fig. 14) show that,regardless of η F , the computation time increases polynomiallywith N even though the MCFC solution space size growsexponentially (Fig. 14). This latter result points out the effec-tiveness and the efficiency (e.g., ∆ t ( CP )min k η F =8% = 24 . [s], ∆ t ( CP ) max k η F =8% = 3 . × [sec] - Fig. 14) of the CP methoddespite its non-optimized software implementation, whichturns out reliable even when dealing with high-dimensional MCFC problems including hundreds of array elements (e.g., N = 500 - Fig. 14). Such a conclusion is also consistent withthe well-known efficiency of CP -based approaches in electro-magnetics [31], which usually outperform bare evolutionaryoptimization techniques (such as GA [21][23]) in terms ofsolution speed thanks to their capability to effectively leverageon the a-priori knowledge on the solution sparsity [31].
0 100 200 300 400 500 C o m pu t a t i on T i m e , ∆ t ( C P ) [ s e c ] Array Size, NDC tapering, SLL = -25 [dB] η F = 4% η F = 8% η F = 12% Figure 14.
Performance Analysis ( DC SLL = − [dB]) - CP methodcomputation times, ∆ t ( CP ) , as a function of the array size, N . V. C
ONCLUSIONS AND F INAL R EMARKS
A method to address the
MCFC problem in linear arrayshas been presented. Within the CP framework, such an ap-proach integrates a ℓ -norm relaxation of the ℓ -norm with abacktracking strategy to synthesize the minimum number ofcorrections of the damaged array for recovering the originalpattern features in a limited amount of time when dealingwith large apertures, as well. The proposed correction methodhas been assessed in different scenarios and comparisons withreference state-of-the-art techniques have been performed.The numerical assessment has shown that • the CP method positively compares with competitivealternatives in terms of both number of corrections andrecovered pattern features and/or fitted user-requirements; • the number of required corrections is lower than that ofthe reconfigurable array elements and, often, it is smallerthan the number of failures; • the CP method faithfully, reliably, and efficiently per-forms in correcting small, medium, and large array withhundreds of elements.Thanks to its effectiveness in dealing with high-dimensionsolution spaces, future research studies - beyond the scopeof the present paper - will consider the extension of the CP correction method to two- and three-dimensional arrays. 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.2020.3045511Copyright (c) 2021 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, 2020 12 A CKNOWLEDGMENTS
A. Massa wishes to thank E. Vico for her never-endinginspiration, support, guidance, and help.R
EFERENCES[1] A. H. Aljuhani, T. Kanar, S. Zihir, and G. M. Rebeiz, “A scalabledual-polarized 256-element Ku-band phased-array SATCOM receiverwith ± ◦ beam scanning,” .,Philadelphia, PA, 2018, pp. 1203-1206.[2] B. Rupakula, S. Zihir, and G. M. Rebeiz, “Low complexity 54-63-GHztransmit/receive 64- and 128-element 2-D-scanning phased-arrays onmultilayer organic substrates with 64-QAM 30-Gbps data rates,” IEEETrans. Microw. Theory Tech. , vol. 67, no. 12, pp. 5268-5281, Dec. 2019.[3] G. M. Rebeiz and L. M. Paulsen, “Advances in SATCOM phased arraysusing silicon technologies,” ,Honololu, HI, 2017, pp. 1877-1879.[4] T. Takahashi, K. Yamamoto, T. Sakamoto, H. Suzuki, H. Arai, H. Joba,T. Okura, and H. Tsuji, “Wide-angle beam steering AESA with three-dimensional stacked PCB for Ka-band in-flight connectivity,” , Waltham, MA, USA,2019, pp. 1-5.[5] M. D. Conway, D. D. Russel, A. Morris, and C. Parry, “Multifunctionphased array radar advanced technology demonstrator nearfield testresults,” , Oklahoma City, OK, 2018, pp. 1412-1415.[6] J. S. Herd and M. D. Conway, “The evolution to modern phased arrayarchitectures,”
IEEE Proc ., vol. 104, no. 3, pp. 519-529, Mar. 2016.[7] P. Rocca, M. A. Hannan, L. Poli, N. Anselmi, and A. Massa, “Optimalphase-matching strategy for beam scanning of sub-arrayed phased ar-rays,”
IEEE Trans. Antennas Propag ., vol. 67, no. 2, pp. 951-959, Feb.2019.[8] N. Anselmi, G. Gottardi, G. Oliveri, and A. Massa, “A total-variationsparseness-promoting method for the synthesis of contiguously clusteredlinear arrays,”
IEEE Trans. Antennas Propag ., vol. 67, no. 7, pp. 4589-4601, Jul. 2019.[9] A. Puglielli, A. Townley, G. LaCaille, V. Milovanovic, P. Lu, K.Trotskovsky, A. Whitcombe, N. Narevsky, G. Wright, T. Courtade, E.Alon, B. Nikolic, and A. M. Niknejad, “Design of energy- and cost-efficient massive MIMO arrays,”
IEEE Proc. , vol. 104, no. 3, pp. 586-606, Mar. 2016.[10] Y. Wu, C. Xiao, Z. Ding, X. Gao, and S. Jin, “A survey on MIMOtransmission with finite input signals: technical challenges, advances,and future trends,”
IEEE Proc. , vol. 106, no. 10, pp. 1779-1833, Oct.2018.[11] Y. Rahmat-Samii and A. C. Densmore, “Technology trends and chal-lenges of antennas for satellite communication systems,”
IEEE Trans.Antennas Propag ., vol. 63, no. 4, pp. 1191-1204, Apr. 2015.[12] H. Al-Saedi, W. M. Abdel-Wahab, S. M. Raeis-Zadeh, E. H. M. Alian,A. Palizban, A. Ehsandar, N. Ghafarian, G. Chen, S. Rasti Boroujeni,M. Nezhad-Ahmadi, and S. Safavi-Naeini, “An integrated circularlypolarized transmitter active phased-array antenna for emerging Ka-bandsatellite mobile terminals,”
IEEE Trans. Antennas Propag
IEEE Trans. Geosci. Remote Sens ., vol. 55, no.5, pp. 2827-2841, May 2017.[15] M. H. Er and S. K. Hui, “Beamforming in presence of element failure,”
Electron. Lett. , vol. 27, no. 3, pp. 273-275, 31 Jan. 1991.[16] T. J. Peters, “A conjugate gradient-based algorithm to minimize thesidelobe level of planar arrays with element failures,”
IEEE Trans.Antennas Propag ., vol. 39, no. 10, pp. 1497-1504, Oct. 1991.[17] Y. Chen and I. Tsai, “Detection and correction of element failures usinga cumulative sum scheme for active phased arrays,”
IEEE Access , vol.6, pp. 8797-8809, 2018.[18] Y. Yang and H. Stark, “Design of self-healing arrays using vector-spaceprojections,”
IEEE Trans. Antennas Propag ., vol. 49, no. 4, pp. 526-534,Apr. 2001. [19] W. P. M. N. Keizer, “Element failure correction for a large monopulsephased array antenna with active amplitude weighting,”
IEEE Trans.Antennas Propag ., vol. 55, no. 8, pp. 2211-2218, Aug. 2007.[20] M. D. Migliore, D. Pinchera, M. Lucido, F. Schettino, and G. Panariello,“A sparse recovery approach for pattern correction of active arrays inpresence of element failures,”
IEEE Antennas Wireless Propag. Lett. ,vol. 14, pp. 1027-1030, 2015.[21] B.-K. Yeo and Y. Lu, “Array failure correction with a genetic algorithm,”
IEEE Trans. Antennas Propag ., vol. 47, no. 5, pp. 823-828, May 1999.[22] J. Han, S. Lim, and N. Myung, “Array antenna TRM failure compen-sation using adaptively weighted beam pattern mask based on GeneticAlgorithm,”
IEEE Antennas Wireless Propag. Lett. , vol. 11, pp. 18-21,2012.[23] J. A. Rodriguez, F. Ares, E. Moreno, and G. Franceschetti, “Geneticalgorithm procedure for linear array failure correction,”
Electron. Lett. vol. 36, no. 3, pp. 196-198, 3 Feb. 2000.[24] S. A. Mitilineos, S. C. A. Thomopoulos, and C. N. Capsalis, “On arrayfailure mitigation with respect to probability of failure using constant ex-citation coefficients and a Genetic Algorithm,”
IEEE Antennas WirelessPropag. Lett. , vol. 5, pp. 187-190, 2006.[25] R. J. Mailloux, “Array failure correction with a digitally beamformedarray,”
IEEE Trans. Antennas Propag ., vol. 44, no. 12, pp. 1543-1550,Dec. 1996.[26] M. Steyskal and R. J. Mailloux, “Generalization of a phased array errorcorrection method,”
Proc. IEEE Antennas Propagat. Soc. Int. Symp .1996 Digest, Baltimore, MD, USA, 1996, pp. 506-509, vol. 1.[27] A. Nanjangud, P. C. Blacker, S. Bandyopadhyay, and Y. Gao, “Roboticsand AI-enabled on-orbit operations with future generation of smallsatellites,”
IEEE Proc. , vol. 106, no. 3, pp. 429-439, Mar. 2018.[28] G. Roesler, P. Jaffe, and G. Henshaw, “Orbital mechanics,”
IEEESpectrum , vol. 54, no. 3, pp. 44-50, Mar. 2017.[29] G. Oliveri and A. Massa, “Bayesian compressive sampling for patternsynthesis with maximally sparse non-uniform linear arrays,”
IEEE Trans.Antennas Propag ., vol. 59, no. 2, pp. 467-481, Feb. 2011.[30] G. Oliveri, P. Rocca, and A. Massa, “Reliable diagnosis of largelinear arrays - A bayesian compressive sensing approach,”
IEEE Trans.Antennas Propagat. , vol. 60, no. 10, pp. 4627-4636, Oct. 2012.[31] G. Oliveri, M. Salucci, N. Anselmi, and A. Massa, “Compressive sensingas applied to inverse problems for imaging: theory, applications, currenttrends, and open challenges,”
IEEE Antennas Propag. Mag. , vol. 59, no.5, pp. 34-46, Oct. 2017.[32] M. E. Davies and R. Gribonval, “Restricted isometry constants where ℓ p sparse recovery can fail for ≪ p ≤ ,” IEEE Trans. Inf. Theory ,vol. 55, no. 5, pp. 2203-2214, May 2009.[33] M. Wang, W. Xu, and A. Tang, “On the performance of sparse recoveryvia ℓ p -minimization (0 < p ≤ ,” IEEE Trans. Inf. Theory , vol. 57,no. 11, pp. 7255-7278, Nov. 2011.[34] D. L. Donoho and X. Huo, “Uncertainty principles and ideal atomicdecomposition,”
IEEE Trans. Inf. Theory , vol. 47, no. 7, pp. 2845-2862,Nov. 2001.[35] E. J. Candes and T. Tao, “Near-optimal signal recovery from randomprojections: universal encoding strategies?,”
IEEE Trans. Inf. Theory ,vol. 52, no. 12, pp. 5406-5425, Dec. 2006.[36] J. Peng, S. Yue, and H. Li, “
NP/CMP
Equivalence: a phenomenonhidden among sparsity models ℓ minimization and ℓ p minimizationfor information processing,” IEEE Trans. Inf. Theory , vol. 61, no. 7, pp.4028-4033, Jul. 2015.[37] H. Huang and A. Makur, “Backtracking-based matching pursuit methodfor sparse signal reconstruction,”
IEEE Signal Process. Lett ., vol. 18,no. 7, pp. 391-394, Jul. 2011.[38] F. Li, S. Hong, Y. Gu, and L. Wang, “An optimization-oriented algorithmfor sparse signal reconstruction,”
IEEE Signal Process. Lett ., vol. 26, no.3, pp. 515-519, Mar. 2019.[39] G. Li and R. J. Burkholder, “Hybrid matching pursuit for distributedthrough-wall radar imaging,”
IEEE Trans. Antennas Propag ., vol. 63,no. 4, pp. 1701-1711, Apr. 2015.[40] R. Blanco and M. Manteghi, “Phased array bandwidth enhancementusing a novel sampling scheme,”
IEEE Trans. Antennas Propag ., vol.62, no. 4, pp. 1983-1990, Apr. 2014.[41] E. J. Candes and T. Tao, “Decoding by linear programming,”
IEEETrans. Inf. Theory , vol. 51, no. 12, pp. 4203-4215, Dec. 2005.[42] R. A. Waltz, J. L. Morales, J. Nocedal, and D. Orban, “An interioralgorithm for nonlinear optimization that combines line search and trustregion steps,”
Math. Program ., vol. 107, no. 3, pp. 391-408, Nov. 2005.., vol. 107, no. 3, pp. 391-408, Nov. 2005.