Probabilistic Interval Analysis for the Analytic Prediction of the Pattern Tolerance Distribution in Linear Phased Arrays With Random Excitation Errors
IIEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2020 1
Probabilistic Interval Analysisfor the Analytic Prediction of the Pattern ToleranceDistribution in Linear Phased Arrays with RandomExcitation Errors
Paolo Rocca,
Senior Member, IEEE , Nicola Anselmi,
Member, IEEE , Arianna Benoni,and Andrea Massa,
Fellow, IEEE
Abstract —A statistical approach based on the Interval Analysis( IA ) is proposed for the analysis of the effects, on the radiationpatterns radiated by phased arrays, of random errors andtolerances in the amplitudes and phases of the array-elementsexcitations. Starting from the efficient, reliable, and inclusivecomputation of the bounds of the complex-valued interval arraypattern function by means of IA , an analytic method is presentedto yield closed-form expressions for the probability of occurrenceof a user-chosen value of the power pattern or of its featureswithin the corresponding IA -derived bounds. A set of numericalexamples is reported and discussed to assess the reliability ofthe proposed probabilistic interval analysis ( PIA ) method withthe results from Monte Carlo simulations as well as to pointout its effectiveness and potentialities/advantages/efficiency in realapplications of great industrial interest.
Index Terms —Phased Array Antenna, Linear Arrays, Ampli-tude and Phase Excitation Errors, Tolerance Analysis, Proba-bilistic Interval Analysis.
I. I
NTRODUCTION N OWADAYS, the assessment and - even more - the pre-diction of the impact on the antenna array performanceof the errors and/or the tolerances caused by the fabrication
Manuscript received on January XX, 2020; revised on May XX, 2020This work benefited from the networking activities carried out withinthe Project "CYBER-PHYSICAL ELECTROMAGNETIC VISION: Context-Aware Electromagnetic Sensing and Smart Reaction (EMvisioning)" (Grantno. 2017HZJXSZ) funded by the Italian Ministry of Education, University, andResearch within the Program PRIN2017 (CUP: E64I19002530001) and theProject "WATERTECH - Smart Community per lo Sviluppo e l’Applicazionedi Tecnologie di Monitoraggio Innovative per le Reti di Distribuzione Idricanegli usi idropotabili ed agricoli" (Grant no. SCN_00489) funded by the ItalianMinistry of Education, University, and Research within the Program "Smartcities and communities and Social Innovation" (CUP: E44G14000060008).P. Rocca, N. Anselmi, A. Benoni, and A. Massa are with the ELE-DIA@UniTN (DISI - University of Trento), Via Sommarive 9, 38123Trento - Italy (e-mail: {paolo.rocca, nicola.anselmi.1, arianna.benoni, an-drea.massa}@unitn.it).P. Rocca, is also with the ELEDIA Research Center (ELEDIA@XIDIAN- Xidian University),P.O. Box 191, No.2 South Tabai Road, 710071 Xi’an,Shaanxi Province - China (e-mail: [email protected]).A. Massa is also with the ELEDIA Research Center (ELEDIA@L2S -UMR 8506), 3 rue Joliot Curie, 91192 Gif-sur-Yvette - France (e-mail:[email protected]).A. Massa is also with the ELEDIA Research Center (ELEDIA@UESTC- University of Electronic Science and Technology of China), Schoolof Electronic Engineering, 611731 Chengdu - China (e-mail: [email protected]).A. Massa is also with the ELEDIA Research Center (ELE-DIA@TSINGHUA - Tsinghua University), 30 Shuangqing Rd, 100084Haidian, Beijing - China (e-mail: [email protected]). process [1][2], the material defects, and the operation/workingconditions (e.g., electromagnetic couplings [3], mechanicalstrains [4], and thermal drifts [5]) is of great importance.Indeed, phased arrays ( PA s) are and they will be increasinglyused for key commercial and industrial applications [6] (e.g.,5G and next-generation mobile communications, autonomousdriving, and industry 4.0) as an effective technological solu-tion to guarantee high-quality and reliable data links thanksto the synthesis of complex pattern features (e.g., directivemainlobes, low sidelobes, etc.) through a precise control of thearray element excitations. Because of the ever growing numberof systems and scenarios that exploit PA -based architectures, anumber of uncertainties, also totally new, will certainly occurwith unknowns effects. In order to predict the differences in thebehavior of the real/actual implementation from its numericalmodel, several tolerance analysis techniques, based on eitherstatistical or analytic/interval methods, have been proposedsince the origin of PA s to analyze the effects of the deviationsfrom the nominal values of the excitation coefficients. Inthe state-of-the-art literature, statistical approaches [7]-[12]provide simple closed-form expressions for the mean valueof the power pattern and its main features [e.g., mainlobepeak, sidelobe level ( SLL ), and pattern nulls] by exploiting thecentral limit theorem (
CLT ) and small error approximations.Despite the wide application, due to their simplicity andeffectiveness when the underlying hypotheses hold true (e.g.,the case of large enough arrays, such that the
CLT is valid,and small errors ), statistical techniques have not always beenreliable when dealing with small/medium arrays [13]. In orderto numerically estimate the probability distribution functionof the power pattern and of the array features, Monte Carlosimulations [14] have been exploited, as well. Although such amethodological approach does not recur to any approximationand it allows the testing of realistic arrays modeled with veryaccurate computer-aided simulation tools, the intrinsic need torun a statistically meaningful number of simulations turns outto be prohibitive because of the rapidly growing computationalburden due to the huge set of admissible error configurationsalso in case of small arrays. As for the computation of the For example the small phase approximation, e jx ≃ x , can be usedfor relatively small ( x ≪ ) deviations in the phase of the array excitationsor in the position of the array elements [12]. 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.2020.2998924Copyright (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, 2020 2 probability of exceeding a user-chosen
SLL value, improvedstatistical models, devoted to describe the distribution of themagnitude of the array amplitude pattern, have been proposed[15], as well.Recently, analytic techniques based on the arithmetic of inter-vals and on the theory of the Interval Analysis [16][17] havebeen introduced to efficiently compute the tolerance boundsof the radiated power pattern and of the related features ofan antenna affected by errors and uncertainties laying withinknown intervals of values. More specifically, the key featuresof such IA -derived bounds are that they are finite and inclusive,since all the admissible realizations of the power pattern ofthe actual antenna lie within the IA interval, thanks to the IAInclusion Property [16][17]. Moreover, it is worth pointingout that the IA bounds are exact since yielded by extendingthe crisp power pattern function to the corresponding intervalfunction without any approximation and without recurring toan exhaustive (ideally infinite) run of Monte Carlo simulations.Indeed, the only quantities involved in the IA -based maths arethe endpoints of the intervals of the input variables (i.e., theantenna descriptors such as the excitations for the antennaarrays) affected by uncertainties. Thanks to these advantages, IA -based methods have been profitably applied to differentantenna systems and devices such as PA s [18]-[23], reflectorantennas [24][25], reflectarrays, [26] and antenna materials[27] or radomes [28][29]. Moreover, IA -based tolerance anal-ysis techniques have been exploited, jointly with optimizationalgorithms, for the robust synthesis of antennas [30][31][32] inorder to obtain antenna designs resilient to uncertainties withinthe considered error intervals, without the need of correctingtheir effects through suitable compensation methods [33]-[35].Thanks to its effectiveness, IA has been also recently appliedwith success to thermal structures [36], composite laminates[37], and the predictions of the structures behaviour [38].By focusing on the tolerance analysis of PA through IA meth-ods, a Cartesian implementation has been firstly introducedto separately evaluate the effects on the array performance ofamplitude [18][19] and phase [13] excitation errors. Analyticclosed-form expressions for the infimum and the supremumof the real and imaginary parts of the interval array factorfunction have been derived in terms of only the endpointsof the excitations errors. Successive extensions have dealtwith calibration errors and mutual coupling effects throughthe circular interval arithmetic and the circular IA in whichthe intervals are represented as circles in the complex planecentered in the nominal value and characterized by an intervalradius proportional to the error tolerance [39]. To mitigate thedependency problem , a Taylor expansion has been used toestimate the sensitivity of the array pattern performance onsmall errors in the excitation amplitudes [21], while a matrix-based IA method has been applied to PA s with errors on theamplitude coefficients [22][23]. The presence of both ampli-tude and phase deviations from the nominal array weights has The dependency problem arises when an interval variable is present morethan once into an expression. Without suitable mathematical manipulations,this causes an overestimation of the bounds of the result of the expressionat hand since each occurrence of the same interval variable is considered asindependent (i.e., a different variable). been addressed in [20] by means of a suitable integration of the IA theory with a Minkowski-Sum procedure ( IA-MS ). Such atechnique has proved to be able to mitigate the overestimationof the power pattern bounds, of both the Cartesian and thecircular IA approaches, due to the “wrapping effect” in theinterval representation of the complex intervals/phasors for thecomputation of the array radiation pattern.Despite the significant advances of IA -based methods overother state-of-the-art competitive alternative for sensitivityanalysis, the key limitation of such analytic/inclusive ap-proaches is that, to date and to the best of the authors’knowledge, no information on the probability of occurrenceof a value within the interval bounds is available, but onlythe upper and the lower limits of the power pattern functionare available. This work is aimed at overcoming such adrawback by presenting a Probabilistic Interval Analysis ( PIA )method that provides, through analytic rules, the probabilitydistribution of the power pattern values and of the array perfor-mance indexes within the corresponding IA -computed bounds.More specifically, starting from the accurate description of thecomplex intervals with the IA-MS method, the informationcoded into the representation of the PA radiation pattern inthe complex plane, as a function of the angular direction andof the nominal array excitations together with their toleranceintervals, is mapped into a probability distribution function( PDF ). The main methodological novelties of this researchstudy over the tolerance methods available in the literaturecomprise ( i ) the introduction of an innovative IA -based strat-egy to give a probabilistic distribution of the occurrence ofthe power pattern values within inclusive, yet finite, boundsfor the tolerance analysis of PA s characterized by arbitrary,but bounded, amplitude and phase excitation errors; ( ii ) thecomplete development of a customized procedure, which isbased on analytic relationships, for the computation of thepower pattern PDF by extracting the probabilistic informationembedded into the complex envelope of the interval functionof the array radiation pattern.The rest of the paper is organized as follows. The
PIA methodis formulated in Sect. II by also detailing the numericalprocedure for computing the
PDF of the power pattern withinthe
IA-MS bounds (Sect. II.A). Section III is devoted tothe numerical analysis and the assessment of the proposedapproach also in comparison with the Monte Carlo method.Finally, some conclusions and final remarks are drawn (Sect.IV). II. M
ATHEMATICAL F ORMULATION
Let us consider a linear phased array of N isotropic elementsuniformly-spaced by d along the x -axis and controlled inamplitude and phase through a set of N amplifiers and N phase shifters, respectively. Let the nominal (complex)excitation of the n -th ( n = 1 , ..., N ) array element be W n = A n e jB n , (1) A n and B n being its nominal (i.e., error/tolerance-free) ampli-tude and phase coefficient, respectively. The far-field pattern This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2020.2998924Copyright (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, 2020 3 x Nn (cid:22) NN (cid:135) ±A (cid:135) ±A v ±B (cid:228) (cid:135) ±A v ±B NN v ±B d z } W W N W Figure 1. Sketch of a linear PA with N elements uniformly-spaced by d along the x -axis. Im n jBn eA supn A infn A supn B infn B > @ n A > @ n B infn A supn A Re Figure 2. Representation in the complex plane of the n -th ( n = 1 , ..., N )complex-valued excitation interval [ W n ] ( [ W n ] , [ A n ] e j [ B n ] ; [ W n ] , h W infn , W supn i ) as a function of the real-valued amplitude [ A n ] ( [ A n ] , h A infn , A supn i ) and phase [ B n ] ( [ B n ] , h B infn , B supn i ) intervals. of the array is mathematically described by the array factor( AF ) AF ( u ) = N X n =1 A n e jB n e j πλ d ( n − u (2)where j = √− is the complex variable, λ is the wavelength,and u = sin ( θ ) is the angular direction (Fig. 1).Let us suppose that the amplifiers and the phase shifters,which determine/control the amplitude and the phase exci-tation weights in the beamforming network, are affected byknown or measurable tolerances (i.e., deviations from thenominal value) such that the actual n -th ( n = 1 , ..., N ) arraycoefficients, e A n and e B n , belong to the interval [16][17] [ A n ][ A n ] , (cid:2) A infn , A supn (cid:3) = (cid:2) A n − ξ infn , A n + ξ supn (cid:3) (3)and [ B n ][ B n ] , (cid:2) B infn , B supn (cid:3) = (cid:2) B n − γ infn , B n + γ supn (cid:3) , (4)respectively ( e A n ∈ [ A n ] , e B n ∈ [ B n ] ), ξ inf/supn and γ inf/supn being the maximum deviations from the corresponding nom-inal values. Accordingly, the n -th ( n = 1 , ..., N ) actualcomplex excitation, f W n , lies within the complex interval [ W n ] given by [ W n ] , [ A n ] e j [ B n ] (5) Figure 3. Representation in the complex plane of the complex-valued intervalfunctions [ G ( u )] ( [ G ( u )] , (cid:2) G inf ( u ) , G sup ( u ) (cid:3) ) and [ AF ( u )] (i.e.,the interval array factor, [ AF ( u )] , (cid:2) AF inf ( u ) , AF sup ( u ) (cid:3) ) along withtheir K partitions, [ G k ( u )] and [ AF k ( u )] ( k = 1 , ..., K ) , so that [ G ( u )] = ∪ Kk =1 [ G k ( u )] , [ AF ( u )] = ∪ Kk =1 [ AF k ( u )] , and [ AF k ( u )] = [ AF ( u )] ∩ [ G k ( u )] , while G infk ( u ) and G supk ( u ) coincide with the circles C k ( u ) and C k +1 ( u ) centered in the origin and having radius r k = (cid:12)(cid:12) AF inf ( u ) (cid:12)(cid:12) +( k − w k ( → G infk ( u ) = r k ) and r k +1 = (cid:12)(cid:12) AF inf ( u ) (cid:12)(cid:12) + kw k ( → G supk ( u ) = r k +1 ), respectively, being r = (cid:12)(cid:12) AF inf ( u ) (cid:12)(cid:12) and r K +1 = | AF sup ( u ) | . as shown in Fig. 2.By substituting (5) into (2), the AF turns out to be a complexinterval function [ AF ( u )] defined as [ AF ( u )] , N X n =1 [ W n ] e j πλ d ( n − u (6)and characterized by an infimum/lower-bound, AF inf ( u ) ,and a supremum/upper-bound, AF sup ( u ) , crisp func-tions of the angular variable u (i.e., [ AF ( u )] , (cid:2) AF inf ( u ) , AF sup ( u ) (cid:3) ).According to the IA-MS method [20], the complex interval [ AF ( u )] is represented through a polygon (Fig. 3), whichis the smallest convex hull of the interval phasor, whosevertices are computed according to the Minkowski Sum( MS ) strategy [40]. Afterwards, the lower and the upperbounds of the real-valued power pattern interval function, [ P ( u )] , | [ AF ( u )] | , are derived by firstly determiningthe bounds of the AF module, | [ AF ( u )] | ( | [ AF ( u )] | , (cid:2)(cid:12)(cid:12) AF inf ( u ) (cid:12)(cid:12) , | AF sup ( u ) | (cid:3) ), as the minimum, (cid:12)(cid:12) AF inf ( u ) (cid:12)(cid:12) ,and the maximum, | AF sup ( u ) | , distance of the correspondingcomplex interval [ AF ( u )] from the origin of the complexplane (Fig. 3). Finally, the bounds of [ P ( u )] are easilyyielded by setting P inf ( u ) = (cid:12)(cid:12) AF inf ( u ) (cid:12)(cid:12) and P sup ( u ) = | AF sup ( u ) | .Although the IA-MS significantly improves the accuracy inpredicting the bounds of [ P ( u )] with respect to [13][18][39],it is worth pointing out that the interval polygon [ AF ( u )] ( u ∈ [ − , ), resulting from the complex-valued intervaloperations, contains more information than the one needed forcomputing [ P ( u )] . For instance, the two complex intervals, 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.2020.2998924Copyright (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, 2020 4 [ AF ( u )] and [ AF ( u )] , with the same AF module interval | [ AF ( u )] | [i.e., | [ AF ( u )] | = | [ AF ( u )] | → (cid:12)(cid:12) AF inf ( u ) (cid:12)(cid:12) = (cid:12)(cid:12) AF inf ( u ) (cid:12)(cid:12) and | AF sup ( u ) | = | AF sup ( u ) | - Fig. 4( a )]turn out to have the same power pattern interval [i.e., [ P ( u )] = [ P ( u )] - Fig. 4( b )], but they identify two dif-ferent regions/shapes in the complex plane [i.e., [ AF ( u )] =[ AF ( u )] - Fig. 4( a )]. The probability that the power patternof a given array with specified AF , AF ( u ) , assumes a value ˆ P ( u ) within [ P ( u )] ( p P ( u ) , Pr n ˆ P ( u ) ∈ [ P ( u )] o ) isexpected to be different for each complex interval [ AF ( u )] and [ AF ( u )] [i.e., p P ( u ) = p P ( u ) - Fig. 4( e )] since itis related to the membership, in the complex plane, of thecorresponding AF sample ˆ AF ( u ) to [ AF ( u )] [e.g., Figs. 4( c )-4( d )] being ˆ P ( u ) , (cid:12)(cid:12)(cid:12) ˆ AF ( u ) (cid:12)(cid:12)(cid:12) .In the following, the information coming from the knowledgeof [ AF ( u )] , but not used for computing [ P ( u )] , is exploitedto analytically predict the power pattern tolerance distribution. A. Probabilistic Interval Analysis (
PIA ) Method
Starting from the complex-valued interval array factor func-tion (6) obtained with the
IA-MS approach [20], the followingprocedure is applied to compute the
PDF associated to theoccurrence of a power pattern value ˆ P ( u ) , along the angulardirection u ( u ∈ [ − , ), within the interval bounds P inf ( u ) and P sup ( u ) . More specifically, the following steps are per-formed: • Step 1 - Identification of the Probability Regions .The interval bounds (cid:12)(cid:12) AF inf ( u ) (cid:12)(cid:12) and | AF sup ( u ) | definein the complex plane an interval ring (or a circle if (cid:12)(cid:12) AF inf ( u ) (cid:12)(cid:12) = 0 ) [ G ( u )] of width w = | AF sup ( u ) | − (cid:12)(cid:12) AF inf ( u ) (cid:12)(cid:12) to which the interval phasor [ AF ( u )] be-longs to (Fig. 3). Such a ring is then partitioned into K uniform annular regions, { [ G k ( u )] ; k = 1 , ..., K } sothat [ G ( u )] = ∪ Kk =1 [ G k ( u )] and the width of each k -th( k = 1 , ..., K ) sub-ring is equal to w k = wK (Fig. 3).Moreover, the lower, G infk ( u ) , and the upper, G supk ( u ) ,bounds of the k -th ( k = 1 , ..., K ) annular region [ G k ( u )] are concentric circles C k ( u ) and C k +1 ( u ) centered in theorigin of the complex plane with radius r k = (cid:12)(cid:12) AF inf ( u ) (cid:12)(cid:12) + ( k − w k (7)and r k +1 = (cid:12)(cid:12) AF inf ( u ) (cid:12)(cid:12) + kw k , (8)respectively, subject to r = (cid:12)(cid:12) AF inf ( u ) (cid:12)(cid:12) and r K +1 = | AF sup ( u ) | . Therefore, the region that identifies the in-terval AF in the complex plane turns out to be subdividedinto K planar sectors, { [ AF k ( u )] ; k = 1 , ..., K } , that bydefinition fit the condition [ AF ( u )] = ∪ Kk =1 [ AF k ( u )] ,the k -th ( k = 1 , ..., K ) sub-domain being the intersectionof the interval AF, [ AF ( u )] , with the k -th annular region [ G k ( u )] [ AF k ( u )] = [ AF ( u )] ∩ [ G k ( u )] (9)where ∩ stands for the intersection operator; Re Im (cid:11) (cid:12) > @ uAF (cid:11) (cid:12) > @ uAF (cid:11) (cid:12) uAF (cid:11) (cid:12) uAF (cid:11) (cid:12) uAF (cid:11) (cid:12) uAF Re (cid:11) (cid:12) (cid:11) (cid:12) uP uAF (cid:11) (cid:12) > @ (cid:11) (cid:12) > @ uPuP (cid:11) (cid:12) (cid:11) (cid:12) uP uAF (cid:11) (cid:12) (cid:11) (cid:12) uP uAF (cid:11) (cid:12) (cid:11) (cid:12) uP uAF ( a ) ( b ) Re Im (cid:11) (cid:12) uAF (cid:11) (cid:12) uAF (cid:11) (cid:12)(cid:11) (cid:12)
2P 1P z Re Im (cid:11) (cid:12) uAF (cid:11) (cid:12) uAF (cid:11) (cid:12)(cid:11) (cid:12)
2P 1P z ( c ) ( d ) inf (u )=P inf (u ) P sup (u )=P sup (u ) P r obab ili t y [ ] P(u ) P(u ) ( e ) Figure 4. Illustrative example of the relationships between ( a ) the arrayfactor interval [ AF ( u )] , ( b ) the power pattern interval [ P ( u )] , and ( c )-( e )the probability of power pattern occurrence, p P ( u ) . • Step 2 - Computation of the PDF of | AF ( u ) | . The PDF of | AF ( u ) | , p | AF | k ( u ) , is analytically determinedas a step-wise function of the K strips in the complexplane. Since it defines the probability that the amplitudeof the array factor assumes a value (cid:12)(cid:12)(cid:12) ˆ AF ( u ) (cid:12)(cid:12)(cid:12) within the k -th ( k = 1 , ..., K ) region [ | AF k ( u ) | ] p | AF | k ( u ) , Pr n(cid:12)(cid:12)(cid:12) ˆ AF ( u ) (cid:12)(cid:12)(cid:12) ∈ [ | AF k ( u ) | ] o , (10)it is related to the ratio between the area of the k -th( k = 1 , ..., K ) annular sector [ AF k ( u )] and that of thewhole region [ AF ( u )] p | AF | k ( u ) = A { [ AF k ( u )] }A { [ AF ( u )] } (11)where A ( · ) stands for the operator returning the area ofthe argument; • Step 3 - Computation of the Occurrence Probabil-ity P ( u ) . Since each sample (cid:12)(cid:12)(cid:12) ˆ AF ( u ) (cid:12)(cid:12)(cid:12) ( (cid:12)(cid:12)(cid:12) ˆ AF ( u ) (cid:12)(cid:12)(cid:12) ∈ [ | AF h ( u ) | ] ; h ∈ [1 , K ] ) maps into a correspondingpower pattern one ˆ P ( u ) ( ˆ P ( u ) ∈ [ | P h ( u ) | ] ; h ∈ 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.2020.2998924Copyright (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, 2020 5 -60-50-40-30-20-10 0 10 -1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] u=sin( θ )cos( φ )IA-MS, InfIA-MS, Sup Nominal ( a ) P r obab ili t y [ % ] u=sin( θ )cos( φ )k=1k=2 k=3k=4 k=5 0 10 20 30 40 50 -1 -0.5 0 0.5 1 P r obab ili t y [ % ] u=sin( θ )cos( φ ) k=1k=2k=3 k=4k=5k=6 k=7k=8k=9 k=10 ( b ) ( d ) -60-50-40-30-20-10 0 10 -1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] u=sin( θ )cos( φ )k=1k=2k=3 k=4k=5Nominal 0 5 10 15 20 25 30 M ean P r obab ili t y ( % ) IA-MS InfIA-MS Sup -60-50-40-30-20-10 0 10 -1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] u=sin( θ )cos( φ ) k=1k=2k=3k=4k=5k=6 k=7k=8k=9k=10Nominal M ean P r obab ili t y ( % ) IA-MS InfIA-MS Sup ( c ) ( e ) Figure 5.
Test Case 1 ( N = 16 , d = λ/ , ξ = 1 %, γ = 3 [deg]; Taylor pattern: SLL = − [dB], ¯ n = 3 ) - Plot of ( a ) the IA-MS computedbounds of the power pattern interval function, [ P ( u )] , together with the nominal power pattern function, P ( u ) , ( b )( d ) the probability functions { p Pk ( u ) ; k = 1 , ..., K }, and ( c )( e ) the K power pattern probability regions { [ P k ( u )] ; k = 1 , ..., K } with the grey-level representation of the mean probability values{ ¯ p k ; k = 1 , ..., K } when ( b )( c ) K = 5 and ( d )( e ) K = 10 . [1 , K ] ) and ˆ P ( u ) , (cid:12)(cid:12)(cid:12) ˆ AF ( u ) (cid:12)(cid:12)(cid:12) being [ P k ( u )] , h P infk ( u ) , P supk ( u ) i ( k = 1 , ..., K ) where P infk ( u ) = (cid:12)(cid:12)(cid:12) AF infk ( u ) (cid:12)(cid:12)(cid:12) and P supk ( u ) = | AF supk ( u ) | ( k =1 , ..., K ), it turns out that the probability that the powerpattern assumes a value ˆ P ( u ) within the k -th ( k =1 , ..., K ) region [ P k ( u )] p Pk ( u ) , Pr n ˆ P ( u ) ∈ [ P k ( u )] o (12)is equal to (10), p Pk ( u ) = p | AF | k ( u ) , thus its value canbe computed as in (11) and according to the numericalprocedure detailed in Appendix I . Moreover, the meanvalue of the probability of occurrence of the k -th ( k =1 , ..., K ) power pattern region [ P k ( u )] , ¯ p k , can be yieldedas follows ¯ p k = 12 Z +1 u = − p Pk ( u ) du (13)physically indicating the mean probability for the powerpattern to lay within the k -th ( k = 1 , ..., K ) intervalregion.III. N UMERICAL A SSESSMENT AND P ERFORMANCE A NALYSIS
This section is devoted to illustrate the features of the proposed
PIA method as well as to assess its performance in a repre-sentative set of numerical examples concerned with differenterrors/uncertainties and arrays by proposing some comparisonswith Monte Carlo simulations, as well.The first test case (’
Test Case 1 ’) refers to a linear arrayof N = 16 elements uniformly-spaced by d = λ along Table I
Test Case 1 ( N = 16 , d = λ/ , ξ = 1 %, γ = 3 [ DEG ]; T
AYLORPATTERN : SLL = − [ D B], ¯ n = 3 ) - N OMINAL VALUES ANDTOLERANCE INTERVALS OF THE AMPLITUDE AND THE PHASEEXCITATIONS OF THE ARRAY . n A n B n [deg] [ A n ] [ B n ] [deg] ,
16 0 .
365 0 . . , . − . , +3 . ,
15 0 .
422 0 . . , . − . , +3 . ,
14 0 .
522 0 . . , . − . , +3 . ,
13 0 .
646 0 . . , . − . , +3 . ,
12 0 .
773 0 . . , . − . , +3 . ,
11 0 .
881 0 . . , . − . , +3 . ,
10 0 .
960 0 . . , . − . , +3 . , .
000 0 . . , . − . , +3 . the x -axis. The nominal amplitudes and phases of the arrayexcitations, { A n ; n = 1 , ..., N } and { B n ; n = 1 , ..., N }, havebeen set to afford a Taylor pattern steered along broadsidewith SLL = − [dB] and ¯ n = 3 (Tab. I). Moreover,the amplifiers and the phase shifters of the array have beenassumed to be affected by uniformly-distributed amplitude andphase tolerances with maximum deviations equal to ξ = 1 %(i.e., ξ n = ξ ; n = 1 , ..., N ) and γ = 3 [deg] (i.e., γ n = γ ; n = 1 , ..., N ), respectively, with respect to the nominal valuesin Tab. I. Thus, the actual amplitude and phase excitationsturn out to belong to the intervals { [ A n ] ; n = 1 , ..., N } and{ [ B n ] ; n = 1 , ..., N } in Tab. I. Figure 5( a ) shows the boundsof the interval power pattern function, [ P ( u )] , computed withthe IA-MS approach [20], along with the nominal pattern. Asit can be observed, although inclusive, the
IA-MS bounds onlyprovide the information on the worst/largest deviations fromthe nominal pattern, but nothing more in terms of occurrenceof a power pattern value, ˆ P ( u ) , within such bounds. Of 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.2020.2998924Copyright (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, 2020 6
Table III
PIA C OMPUTATIONAL COST ( CPU -T IME ). K N u N u N v γ ∆ t [deg] [sec] Test Case 1
Fig. 5( a ) −
16 [ − ,
1] 501 272 - Figs. 5( b )-5( c ) − ,
1] 501 272 3 45 . Figs. 5( d )-5( e )
10 16 [ − ,
1] 501 272 3 47 . -
20 16 [ − ,
1] 501 272 3 50 . Figs. 6( a )-6( b ) − .
336 1 272 3 9 . × − Figs. 6( c )-6( d )
10 16 − .
336 1 272 3 9 . × − Test Case 2
Figs. 8( a )-8( b ) − ,
1] 501 112 1 7 . Figs. 5( b )-5( c ) − ,
1] 501 272 3 45 . Figs. 8( c )-8( d ) − ,
1] 501 368 5 102 . Figs. 8( e )-8( f ) − ,
1] 501 720 10 414 . Test Case 3
Figs. 9( a )-9( b ) − ,
1] 251 136 3 13 . Figs. 5( b )-5( c ) − ,
1] 501 272 3 45 . Figs. 9( c )-9( d ) − ,
1] 1001 544 3 307 . Figs. 9( e )-9( f ) − ,
1] 1501 1088 3 1407 . Table II
Test Case 1 ( N = 16 , d = λ/ , ξ = 1 %, γ = 3 [ DEG ]; T
AYLORPATTERN : SLL = − [ D B], ¯ n = 3 ) - M EAN PROBABILITY VALUES { ¯ p k ; k = 1 , ..., K } WHEN K = 5 AND K = 10 . K = 5 K = 10 k ¯ p k [%] k ¯ p k [%] k ¯ p k [%] .
76 1 2 .
84 2 6 .
922 21 .
59 3 9 .
84 4 11 .
253 26 .
28 5 12 .
81 6 13 .
474 26 .
19 7 13 .
51 8 12 .
685 16 .
18 9 10 .
41 10 5 . course, the knowledge of bounds that are inclusive is veryimportant for the array designers since they guarantee that noone realization of the power pattern, whatever the tolerancevalue within the amplitude/phase intervals, can lie outside,but some more could be very helpful in planning the trade-off between costs and robustness to tolerances when buildingan array for real applications. Towards this purpose, the PIA method has been then applied to compute the probabilisticdistribution of the values of the actual power pattern within the
IA-MS bounds, p Pk ( u ) ( k = 1 , ..., K ) [Figs. 5( b )]. By settingthe number of strips/region of interest to K = 5 , it turns outthat the one with the highest probability of occurrence is, onaverage, the k ∗ = 3 -th since the mean probability value (13)is equal to ¯ p = 26 . % (Tab. II), while the less probableregion is the first one ( ¯ p = 9 . %). For more immediate andeasier inferences, the mean probabilities values of each k -th( k = 1 , ..., K ) pattern strip, ¯ p k , are represented with a grey-scale color within the corresponding bounds in Figs. 5( c ).Depending on the user-required spatial resolution and thanksto the computational efficiency of the PIA (Tab. III), there areno problems in having a more detailed power pattern tolerancedistribution by considering narrower/finer regions (i.e., greatervalues of K ), the CPU -time being almost independent on K (Tab. III) thanks to the closed-form expressions detailedin Appendix I (see
Case 1 - Case 4 ). As an example, thestep-wise distribution of the power pattern occurrence when K = 10 is reported in Fig. 5( d ), while the correspondingmean probability values are given Tab. II and displayed in Fig. 5( e ). As it can be observed, the comparisons betweenthe probability plots in Fig. 5( d ) and Fig. 5( b ) as well asthose in Fig. 5( e ) and Fig. 5( c ) point out the higher resolutionin the definition of the power pattern PDF within the
IA-MS bounds, P infIA − MS ( u ) and P supIA − MS ( u ) , thanks to theincreased number of K partitions. On the other hand, thevalues of p Pk ( u ) ( k = 1 , ..., K ) reduce as K increases bothon average [see the dynamic of the grey-scale in Fig. 5( e ) vs.Fig. 5( c ) - Tab. II] and whatever the angular direction (i.e., u value) being by definition P Kk =1 p Pk ( u ) = 1 as well as P Kk =1 ¯ p k = 1 . Indeed, since w k ⌋ K =5 = 2 × w k ⌋ K =10 , itturns out that p Ph ( u ) (cid:5) K =5 = p P h − ( u ) (cid:5) K =10 + p P h ( u ) (cid:5) K =10 ( h = 1 , ..., K ; K = 5 ) [Fig. 5( b ) vs. Fig. 5( d )] and ¯ p h ⌋ K =5 = ¯ p h − ⌋ K =10 + ¯ p h ⌋ K =10 ( h = 1 , ..., K ; K = 5 )(Tab. II).Next, the results from PIA have been compared with thosefrom the Monte Carlo method when evaluating ℵ = 10 realizations/configurations of the excitation tolerances withinthe ranges { [ W n ] ; n = 1 , ..., N } defined by the deviations ξ = 1 % and γ = 3 [deg]. For illustrative purposes, Fig. 6( a )and Fig. 6( c ) give the representation in the complex planeof the AF interval function sampled at the angular direction u = − . (i.e., the surface mapping the complex interval [ AF ( u )] in the complex plane) and predicted by the IA-MS method. Moreover, the probability that the actual power patternvalue lies within the k -th region ( k = 1 , ..., K ), p Pk ( u ) ,computed by the PIA when K = 5 [Fig. 6( a )] and K = 10 [Fig. 6( c )] is reported in color within the [ AF ( u )] area,as well. As expected, p Ph (cid:5) K =5 = p P h − (cid:5) K =10 + p P h (cid:5) K =10 (Tab. IV) being [ P h ] ⌋ K =5 = [ P h − ] ⌋ K =10 + [ P h ] ⌋ K =10 ( h = 1 , ..., K ; K = 5 ). Concerning the comparative study,the plot of the PDF of the Monte Carlo simulations isreported in both Fig. 6( b ) and Fig. 6( d ) along with the color-map representation of the K probability regions, [ P k ] , andtheir probability values, p Pk ( k = 1 , ..., K ). More in detail,each k -th ( k = 1 , ..., K ) vertical strip extends from theinfimum, P infk ( P infk , (cid:12)(cid:12)(cid:12) AF infk ( u ) (cid:12)(cid:12)(cid:12) ), to the supremum, The following simplification of the notation is used hereinafter: p Pk ← p Pk ( u ) and [ P k ] ← [ P k ( u )] being p Pk ( u ) , Pr n ˆ P ( u ) ∈ [ P k ( u )] o 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.2020.2998924Copyright (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, 2020 7
Table IV
Test Case 1 ( N = 16 , d = λ/ , ξ = 1 %, γ = 3 [ DEG ]; T
AYLOR PATTERN : SLL = − [ D B], ¯ n = 3 ; u = − . ) - P OWER PATTERN PROBABILITYREGIONS { [ P k ] ; k = 1 , ..., K } AND THEIR PROBABILITY VALUES { p Pk ; k = 1 , ..., K } WHEN K = 5 AND K = 10 . K = 5 K = 10 k p Pk [%] [ P k ] [dB] k p Pk [%] [ P k ] [dB] k p Pk [%] [ P k ] [dB] .
46 [ − . , − .
76] 1 2 .
15 [ − . , − .
97] 2 5 .
31 [ − . , − . .
59 [ − . , − .
17] 3 8 .
31 [ − . , − .
53] 4 11 .
28 [ − . , − . .
30 [ − . , − .
80] 5 13 .
87 [ − . , − .
33] 6 14 .
43 [ − . , − . .
41 [ − . , − .
38] 7 13 .
91 [ − . , − .
51] 8 13 .
50 [ − . , − . .
25 [ − . , − .
49] 9 12 .
49 [ − . , − .
38] 10 4 .
76 [ − . , − . -0.6-0.4-0.2 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 1 I m ag i na r y a x i s Real axisIA-MS 0 5 10 15 20 25 30 35 P r obab ili t y ( % ) k=1 k=2 k=3 k=4 k=5 O cc u rr en c e s ( % ) Normalized Power Pattern [dB]IA-MS Monte Carlo 0 5 10 15 20 25 30 35 P r obab ili t y ( % ) k=1 k=2 ( a ) ( b ) -0.6-0.4-0.2 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 1 I m ag i na r y a x i s Real axisIA-MS 0 5 10 15 P r obab ili t y ( % ) k=1k=2 ... k=10 O cc u r en c e s ( % ) Normalized Power Pattern [dB]IA Exact-MS Monte Carlo 0 3 6 9 12 15 P r obab ili t y ( % ) k=1 k=2 k=3 ( c ) ( d ) Figure 6.
Test Case 1 ( N = 16 , d = λ/ , ξ = 1 %, γ = 3 [deg];Taylor pattern: SLL = − [dB], ¯ n = 3 ; u = − . ) - Plot of ( a )( c ) thecomplex-valued IA-MS array factor interval, [ AF ( u )] , partitioned into K power pattern probability regions { [ P k ( u )] ; k = 1 , ..., K } with the color-level representation of the probability values { p Pk ( u ) ; k = 1 , ..., K } and( b )( d ) comparison between the PDF s from the Monte Carlo simulations andderived with the
PIA when ( a )( b ) K = 5 and ( c )( d ) K = 10 . P supk ( P supk , | AF supk ( u ) | ), of the corresponding PIA -derived interval power pattern region. It is worth noticing thatthe bell of the Gaussian-like Monte Carlo
PDF is localizedwithin the most probable k -th [ k ∗ = 3 - Fig. 6( b ); k ∗ = 6 -Fig. 6( d )] region predicted with the PIA method [Figs. 6( b )-6( d )] to confirm the reliability of the proposed probabilisticmethod. On the other hand, the interested reader needs totake into account that the PIA prediction has been yieldedwith just a single analytic computation, instead of running ℵ times the computation of the power pattern, carried out inonly few milliseconds ( ∆ t = 9 . × − [sec] - K = 5 ; ∆ t = 9 . × − [sec] - K = 10 ; Tab. III) to derivean inclusive result, while the non-exhaustive Monte Carloestimation run for some hours.A very fruitful and free by-product of the PIA analytic pre-diction of the power pattern tolerance distribution is the easyderivation of the
PDF of the pattern features. With referenceto the intervals of the
SLL and of the beam pattern peak Γ ,whose endpoints are [18] Table V
Test Case 1 ( N = 16 , d = λ/ , ξ = 1 %, γ = 3 [ DEG ]; T
AYLOR PATTERN : SLL = − [ D B], ¯ n = 3 ; K = 5 ) - S IDELOBE LEVEL INTERVALS { [ SLL k ] ; k = 1 , ..., K } AND POWER PATTERN PEAK INTERVALS { [Γ k ] ; k = 1 , ..., K } ALONG WITH ITS PROBABILITIES { p Γ k ; k = 1 , ..., K }. k [ SLL k ] [dB] p Γ k [%] [Γ k ] [dB] − . , − .
25] 18 .
11 [ − . , − . − . , − .
53] 20 .
35 [ − . , − . − . , − .
93] 20 .
44 [ − . , . − . , − .
93] 20 .
52 [0 . , . − . , − .
31] 20 .
58 [0 . , . IA − MS [ − . , − . − [ − . , . SLL inf/sup , P inf/sup ( u max ) − P sup/inf ( u SLL ) (14)where u max is the steering direction and u SLL , arg { max u/ ∈ Ω { P ( u ) }} , Ω being the sidelobe-region outsidethe mainbeam, and Γ inf/sup , P inf/sup ( u max ) (15)respectively, the same definitions can be now extended to each k -th ( k = 1 , ..., K ) probability region of the power pattern.Accordingly, let us define SLL inf/supk , P inf/sup ( u max ) − P sup/infk (cid:0) u SLL ( k ) (cid:1) (16)where u SLL ( k ) , arg n max u/ ∈ Ω k n ˆ P ( u ) oo , while Γ inf/supk , P inf/supk ( u max ) . (17)As for the corresponding PDF s, the probabilityof Γ ( p Γ k , Pr nb Γ ∈ [Γ k ] o ) is given by p Γ k =Pr n ˆ P ( u max ) ∈ [ P k ( u max )] o , while the occurrencedistribution of the SLL , p SLLk , Pr n [ SLL ∈ [ SLL k ] o ,is approximated with the mean probability of the k -th( k = 1 , ..., K ) power pattern strip (i.e., p SLLk ≈ ¯ p k ) since p SLLk → ¯ p k ( K → ∞ ).By applying (16) and (17) to the IA-MS intervals [ SLL ] IA − MS = [ − . , − . [dB] and [Γ] IA − MS =[ − . , . [dB] (Tab. V) of the Test Case 1 (i.e., N = 16 , d = λ/ , ξ = 1 %, γ = 3 [deg]; Taylor pattern: SLL = − [dB], ¯ n = 3 ), the K intervals of the pattern features and theirprobabilities turn out to be those in Tab. V ( K = 5 ). As itcan be noticed, both the peak power and the SLL intervalsperfectly fill the
IA-MS bounds (i.e.,
SLL supK = SLL supIA − MS 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.2020.2998924Copyright (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, 2020 8 P r obab ili t y [ % ] u=sin( θ )cos( φ )k=1k=2 k=3k=4 k=5 0 10 20 30 40 50 -1 -0.5 0 0.5 1 P r obab ili t y [ % ] u=sin( θ )cos( φ )k=1k=2 k=3k=4 k=5 0 10 20 30 40 50 -1 -0.5 0 0.5 1 P r obab ili t y [ % ] u=sin( θ )cos( φ )k=1k=2 k=3k=4 k=5 ( a ) ( c ) ( e ) -60-50-40-30-20-10 0 10 -1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] u=sin( θ )cos( φ )k=1k=2k=3 k=4k=5Nominal 0 5 10 15 20 25 30 M ean P r obab ili t y ( % ) IA-MS InfIA-MS Sup -60-50-40-30-20-10 0 10 -1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] u=sin( θ )cos( φ )k=1k=2k=3 k=4k=5Nominal 0 5 10 15 20 25 30 M ean P r obab ili t y ( % ) IA-MS InfIA-MS Sup -60-50-40-30-20-10 0 10 -1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] u=sin( θ )cos( φ )k=1k=2k=3 k=4k=5Nominal 0 5 10 15 20 25 30 M ean P r obab ili t y ( % ) IA-MS InfIA-MS Sup ( b ) ( d ) ( f ) Figure 8.
Test Case 2 ( N = 16 , d = λ/ , ξ = 1 %; Taylor pattern: SLL = − [dB], ¯ n = 3 ; K = 5 ) - Plot of ( a )( c )( e ) the probability functions { p Pk ( u ) ; k = 1 , ..., K }, and ( b )( d )( f ) the K power pattern probability regions { [ P k ( u )] ; k = 1 , ..., K } with the grey-level representation of the mean probabilityvalues { ¯ p k ; k = 1 , ..., K } when ( a )( b ) γ = 1 [deg], ( c )( d ) γ = 5 [deg], and ( e )( f ) γ = 10 [deg]. -40-35-30-25-20-15 1 2 3 4 5 8 12 16 20 24 28 S LL [ d B ] p k S LL [ % ] Region Index, kSLL infIA-MS
SLL supIA-MS
SLL supk
SLL infk ( a ) -0.15-0.1-0.05 0 0.05 0.1 0.15 1 2 3 4 5 8 12 16 20 24 28 Γ [ d B ] p k Γ [ % ] Region Index, k Γ infIA-MS Γ supIA-MS Γ supk Γ infk ( b ) Figure 7.
Test Case 1 ( N = 16 , d = λ/ , ξ = 1 %, γ = 3 [deg]; Taylorpattern: SLL = − [dB], ¯ n = 3 ; K = 5 ) - Bounds of ( a ) the sidelobelevel intervals { [ SLL k ] ; k = 1 , ..., K } and ( b ) the power pattern peakintervals { [Γ k ] ; k = 1 , ..., K } along with their probability values, { p SLLk ; k = 1 , ..., K } and { p Γ k ; k = 1 , ..., K }. and SLL inf = SLL infIA − MS ; Γ supK = Γ supIA − MS and Γ inf =Γ infIA − MS - Tab. V), but only those of Γ are adjacent [i.e., Γ suph = Γ infh +1 ( h = 1 , ..., K − )] and mutually exclusive (i.e., [Γ h ] ∩ [Γ k ] = ∅ ; h , k = 1 , ..., K ; h = k ) since u max is a fixeddirection in (17), while the angular direction of the highestsidelobe in the k -th region, u SLL ( k ) , in (16) can vary. Onthe other hand, a key outcome is that Fig. 7 and the valuesin Tab. V give faithful indications on the most/less probablevalues of the actual SLL [Fig. 7( a )] and Γ [Fig. 7( b )] withinthe IA-MS bounds. For instance, the
SLL most probably hasa value in the range
SLL infk k k =3 ≤ SLL ≤ SLL supk ⌋ k =3 ( SLL infk k k =3 = − . [dB] and SLL supk ⌋ k =3 = − . [dB]) since p SLL ≥ p SLLh ( h = 1 , ..., K ) being p SLL = 26 . %, while the lowest probability arises for the ( k = 1 )-th interval of the admissible SLL values [Fig. 7( a )] being p SLL = 9 . %. Such an analysis clearly highlights that,thanks to the PIA technique, more information than thatcoded into the
IA-MS endpoints/bounds can be drawn fromthe knowledge of [ AF ( u )] . As a matter of fact and unlike PIA , standard IA -based methods only provides the worst casebounds, which are generally characterized by a low or very-low probability of occurrence.In order to further assess the PIA method as well as itsperformance, different values of the uniform error toleranceson the phase excitations have been considered (’
Test Case 2 ’),while keeping the uniform amplitude deviation fixed to ξ = 1 % as well as the array and the radiated pattern (Taylor beam: SLL = − [dB] and ¯ n = 3 - Tab. I). The probabilisticdistributions of the values of the actual power pattern withinthe IA-MS bounds, p Pk ( u ) , u ∈ [ − , ( k = 1 , ..., K ; K = 5 ),are shown on the left column of Fig. 8 when γ = 1 [deg] 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.2020.2998924Copyright (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, 2020 9 P r obab ili t y [ % ] u=sin( θ )cos( φ )k=1k=2 k=3k=4 k=5 0 10 20 30 40 50 -1 -0.5 0 0.5 1 P r obab ili t y [ % ] u=sin( θ )cos( φ )k=1k=2 k=3k=4 k=5 0 10 20 30 40 50 -1 -0.5 0 0.5 1 P r obab ili t y [ % ] u=sin( θ )cos( φ )k=1k=2 k=3k=4 k=5 ( a ) ( c ) ( e ) -60-50-40-30-20-10 0 10 -1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] u=sin( θ )cos( φ )k=1k=2k=3 k=4k=5Nominal 0 5 10 15 20 25 30 M ean P r obab ili t y ( % ) IA-MS InfIA-MS Sup -60-50-40-30-20-10 0 10 -1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] u=sin( θ )cos( φ )k=1k=2k=3 k=4k=5Nominal 0 5 10 15 20 25 30 M ean P r obab ili t y ( % ) IA-MS InfIA-MS Sup -60-50-40-30-20-10 0 10 -1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] u=sin( θ )cos( φ )k=1k=2k=3 k=4k=5Nominal 0 5 10 15 20 25 30 M ean P r obab ili t y ( % ) IA-MS InfIA-MS Sup ( b ) ( d ) ( f ) Figure 9.
Test Case 3 ( d = λ/ , ξ = 1 %, γ = 3 [deg]; Taylor pattern: SLL = − [dB], ¯ n = 3 ; K = 5 ) - Plot of ( a )( c )( e ) the probability functions{ p Pk ( u ) ; k = 1 , ..., K }, and ( b )( d )( f ) the K power pattern probability regions { [ P k ( u )] ; k = 1 , ..., K } with the grey-level representation of the meanprobability values { ¯ p k ; k = 1 , ..., K } for an array of ( a )( b ) N = 8 , ( c )( d ) N = 32 , and ( e )( f ) N = 64 elements. [Fig. 8( a )], γ = 5 [deg] [Fig. 8( c )], and γ = 10 [deg] [Fig.8( e )]. Moreover, the mean probability value (13) of each k -th( k = 1 , ..., K ) pattern strip, ¯ p k ⌋ γ ( γ ∈ { , , } [deg]), isreported with the grey-level representation in Fig. 8( b ) ( γ = 1 [deg]), Fig. 8( d ) ( γ = 5 [deg]), and Fig. 8( f ) ( γ = 10 [deg])as well as in Tab. VI where, for completeness, the intervalsand the probabilities of the SLL , { [ SLL k ] ; k = 1 , ..., K }, andof the power pattern peak, { [Γ k ] ; k = 1 , ..., K } are given,as well. Both figures and tables indicate that, on average andanalogously to the first test case, the most probable regionturns out to be the central one ( k ∗ = 3 ), whatever the phasetolerance γ ( ¯ p ⌋ γ =1 [deg] = 26 . %, ¯ p ⌋ γ =5 [deg] = 27 . %, and ¯ p ⌋ γ =10 [deg] = 28 . % - Tab. VI). Moreover, theregions close to the IA-MS bounds (i.e., k = 1 and k = K )are, once again, the less probable ones. Such an outcomefurther points out a key feature of the PIA as well as itsadvantage over standard, even though inclusive, IA -based toolsfor tolerance analysis, which are unable to give information onthe occurrences of the actual power patter within the intervalbounds.As for the computational costs, the values in Tab. III in-dicate that the CPU -time ∆ t depends on the width of thephase tolerance, γ , since the extension of A { [ AF ( u )] } growsproportionally. Indeed, more vertexes are needed for the IA-MS operations [20] in the case of larger deviations from thenominal excitations, then more triangles are used to partitionthe interval [ AF ( u )] for computing p k ( u ) ( k = 1 , ..., K ) [Fig.A.2]. For the sake of completeness, the values of the numberof vertexes ( N v ) used for each u sample and of the number ofsamples ( N u ) used to discretize the interval u = [ − , arereported in Tab. III.The last example (’ Test Case 3 ’) is aimed at analyzing the
Table VI
Test Case 2 ( N = 16 , d = λ/ , ξ = 1 %; T AYLOR PATTERN : SLL = − [ D B], ¯ n = 3 ; K = 5 ) - S IDELOBE LEVEL INTERVALS { [ SLL k ] ; k = 1 , ..., K } AND POWER PATTERN PEAK INTERVALS { [Γ k ] ; k = 1 , ..., K } ALONG WITH THEIR PROBABILITIES { p SLLk ; k = 1 , ..., K } AND { p Γ k ; k = 1 , ..., K } WHEN γ ∈ { , , } [ DEG ]. k ¯ p k [ SLL k ] p Γ k [Γ k ] [%] [dB] [%] [dB] γ = 1 [deg] .
78 [ − . , − .
00] 19 .
64 [ − . , − . .
27 [ − . , − .
72] 19 .
98 [ − . , − . .
00 [ − . , − .
60] 20 .
06 [ − . , . .
77 [ − . , − .
61] 20 .
14 [0 . , . .
19 [ − . , − .
72] 20 .
18 [0 . , . IA − MS − [ − . , − . − [ − . , . γ = 5 [deg] .
38 [ −∞ , − .
41] 15 .
36 [ − . , − . .
90 [ − . , − .
38] 21 .
02 [ − . , − . .
19 [ − . , − .
86] 21 .
12 [ − . , . .
83 [ − . , − .
36] 21 .
22 [0 . , . .
70 [ − . , − .
42] 21 .
28 [0 . , . IA − MS − [ −∞ , − . − [ − . , . γ = 10 [deg] .
64 [ −∞ , − .
66] 10 .
51 [ − . , − . .
83 [ − . , − .
64] 19 .
22 [ − . , − . .
01 [ − . , − .
12] 23 .
19 [ − . , − . .
97 [ − . , − .
62] 23 .
47 [ − . , . .
55 [ − . , − .
68] 23 .
61 [0 . , . IA − MS − [ −∞ , − . − [ − . , . PIA method when dealing with different linear arrays. Towardsthis end, different array sizes, ranging from N = 8 up to N = 64 elements, have been considered by still keeping anuniform half-wavelength inter-element spacing ( d = λ ) andthe uniform excitation tolerances to ξ = 1 % and γ = 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.2020.2998924Copyright (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, 2020 10
Table VII
Test Case 3 ( d = λ/ , ξ n = 1 %, γ = 3 [ DEG ]; T
AYLOR PATTERN : SLL = − [ D B], ¯ n = 3 ; K = 5 ) - S IDELOBE LEVEL INTERVALS { [ SLL k ] ; k = 1 , ..., K } AND POWER PATTERN PEAK INTERVALS { [Γ k ] ; k = 1 , ..., K } ALONG WITH THEIR PROBABILITIES { p SLLk ; k = 1 , ..., K } AND { p Γ k ; k = 1 , ..., K } WHEN N ∈ { , , } . k ¯ p k [ SLL k ] p Γ k [Γ k ] [%] [dB] [%] [%] N = 81 11 .
28 [ − . , − .
58] 18 .
01 [ − . , − . .
92 [ − . , − .
47] 20 .
37 [ − . , − . .
13 [ − . , − .
18] 20 .
46 [ − . , . .
62 [ − . , − .
24] 20 .
55 [0 . , . .
05 [ − . , − .
55] 20 .
61 [0 . , . IA − MS − [ − . , − . − [ − . , . N = 321 8 .
17 [ − . , − .
54] 18 .
06 [ − . , − . .
83 [ − . , − .
74] 20 .
36 [ − . , − . .
07 [ − . , − .
11] 20 .
45 [ − . , . .
05 [ − . , − .
09] 20 .
54 [0 . , . .
88 [ − . , − .
45] 20 .
59 [0 . , . IA − MS − [ − . , − . − [ − . , . N = 641 6 .
56 [ − . , − .
62] 18 .
12 [ − . , − . .
01 [ − . , − .
79] 20 .
35 [ − . , − . .
87 [ − . , − .
15] 20 .
43 [ − . , . .
48 [ − . , − .
12] 20 .
52 [0 . , . .
08 [ − . , − .
47] 20 .
58 [0 . , . IA − MS − [ − . , − . − [ − . , . [deg]. When applying the PIA method ( K = 5 ) to the IA-MS bounds of [ AF ( u )] , the PDF of the power pattern (Fig. 9)has been computed with a non-optimized code on a standardlaptop PC having . GHz
CPU and GB RAM in an amountof time ∆ t that significantly depends on N (Tab. III) dueto the IA-MS computation of (6). Indeed, since the
IA-MS calculation of [ AF ( u )] requires the sum of N interval phasorsfor each u -direction and the sampling rate of the angular range u ∈ [ − , is proportional to N , the number of vertexesof the surface mapping the [ AF ( u )] function in the complexplane grows with O (cid:0) N (cid:1) . This implies a higher cost of the PIA prediction when increasing the array size (i.e., N ) withrespect to the widening of the γ interval (Tab. III). As forthe plots in Fig. 9, it can be observed that the variations of p Pk ( u ) ( k = 1 , ..., K ) versus the angular direction u , withinthe range u ∈ [ − , , increase as the array enlarges. Forexample, let us compare the behavior of p P ( u ) in Fig. 9( a )and Fig. 9( e ). Such an outcome is expected since the condition p Pk ( u ) ≈ p Pk ( u + ∆ u ) ( ∆ u being a very small increment, ∆ u ≪ ) holds true when ˆ P ( u ) ≈ ˆ P ( u + ∆ u ) , but thislatter relation turns out to be more and more unfulfilled as N grows. Concerning the most probable power pattern region(right column in Fig. 9 and Tab. VII), the highest value of ¯ p k occurs in the center of the partitioning of the IA-MS bounds(i.e., k ∗ = 3 ) when N ≤ , while the actual power patternvalue tends to fall nearer to the IA-MS upper bound ( k ∗ → K )for larger arrays (e.g., k ∗ ⌋ N =64 = 4 ).For the sake of completeness, the K intervals of the patternfeatures (i.e., SLL and Γ ) along with their probabilities arereported in Tab. VII, as well. IV. C ONCLUSION
A probabilistic IA -based approach has been proposed to dealwith the tolerance analysis of linear phased arrays affected byarbitrary, but bounded, uncertainties/errors in both amplitudeand phase array excitations. More specifically, an analyticmethod has been described to derive the closed-form ex-pression of the probability distribution of the power patterntolerance within the reliable, yet inclusive, interval boundsyielded with the IA-MS technique [20].From a methodological viewpoint, the main novelties andadvantages, over the existing state-of-the-art techniques, of theproposed
PIA method can be summarized in the followingones: • the inclusion, for the first time to the best of the authors’knowledge, of a probabilistic information within theworst-case IA bounds to provide the antenna engineersa more informative, but still inclusive and analytic, pre-diction on the expected PA performance; • the development of a computationally-efficient and exact(i.e., without any approximation) technique for extractingthe probabilistic information coded into the representationin the complex plane of the envelope of the complexinterval function of the array radiation pattern.The numerical assessment has provided evidence of the ef-fectiveness, the reliability, and the computational efficiency ofthe PIA method in dealing with different error tolerances inthe array excitations as well as array dimensions.Future works, beyond the scope of this paper, will be aimedat extending the proposed method to large arrays, also con-sidering planar and conformal layouts with uniform and non-uniform/sparse element arrangements. Moreover, further ad-vances will also consider the application of the
PIA methodto the analysis of other pattern features (e.g., directivity, gain)and its integration into an optimization tool for the robust PA synthesis. Appendix I
The term
A { [ AF k ( u )] } at the numerator of (11) isthe portion of [ AF ( u )] inside the k -th ring [ G k ( u )] , A { [ AF ( u )] ∩ C k ( u ) } [Fig. A.1( c )] A { [ AF k ( u )] } , A { [ AF ( u )] ∩ [ G k ( u )] } (18)given by the difference between the sub-region of [ AF ( u )] inside the upper circle C k +1 ( u ) , A { [ AF ( u )] ∩ C k +1 ( u ) } [Fig. A.1( a )] and that bounded by the lower circle C k ( u ) , A { [ AF ( u )] ∩ C k ( u ) } [Fig. A.1( b )] A { [ AF k ( u )] } = A { [ AF ( u )] ∩ C k +1 ( u ) }−A { [ AF ( u )] ∩ C k ( u ) } . (19) 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.2020.2998924Copyright (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, 2020 11 Re Im (cid:11) (cid:12) > @ uAF (cid:11) (cid:12) > @ (cid:11) (cid:12) ^ ‘ uuAF CA (cid:252) (cid:11) (cid:12) u k C (cid:11) (cid:12) u C Re Im (cid:11) (cid:12) > @ uAF (cid:11) (cid:12) > @ (cid:11) (cid:12) ^ ‘ uuAF k CA (cid:252) (cid:11) (cid:12) u k C (cid:11) (cid:12) u C ( a ) ( b ) Re Im (cid:11) (cid:12) > @ uAF (cid:11) (cid:12) > @ (cid:11) (cid:12) > @ ^ ‘ uuAF k GA (cid:252) (cid:11) (cid:12) u k C (cid:11) (cid:12) u C ( c ) Figure A.1. Illustrative sketches of ( a ) the sub-region of [ AF ( u )] in-side the upper circle C k +1 ( u ) with area A { [ AF ( u )] ∩ C k +1 ( u ) } , ( b )the sub-region of [ AF ( u )] inside the upper circle C k ( u ) with area A { [ AF ( u )] ∩ C k ( u ) } , and ( c ) the portion of [ AF ( u )] inside the k -th ring [ G k ( u )] having area ( ) u t ( ) u (cid:11) (cid:12) u Im Re ( ) u T (cid:11) (cid:12) > @ uAF T ( ) ( ) u (cid:252) u kt C (cid:11) (cid:12) u C (cid:11) (cid:12) u C (cid:11) (cid:12) u k C Figure A.2. Illustrative sketch in the complex plane of the partition of [ AF ( u )] into T triangles, { ∆ t ( u ) ; t = 1 , ..., T }. In order to compute the intersection areas between [ AF ( u )] and the circles C k +1 ( u ) and C k ( u ) , a triangulation of [ AF ( u )] is exploited by applying the Delaunay algorithm[40][41] to the vertexes used by the IA-MS operations [20]to determine (6). More specifically, the interval [ AF ( u )] is partitioned into T triangles, { ∆ t ( u ) ; t = 1 , ..., T }(Fig. A.2), and the generic h -th ( h = 1 , ..., K + 1 ) term A { [ AF ( u )] ∩ C h ( u ) } is then yielded as the sum of all thecommon regions between the T triangles and the circle C h ( u ) A { [ AF ( u )] ∩ C h ( u ) } = T X t =1 A { C h ( u ) ∩ ∆ t ( u ) } (20) v v v C Im Re ’ e e e (cid:11) (cid:12) SA v v v C Im Re ’ e e e v v v C Im Re ’ e a a e e S (cid:11) (cid:12) SA ( a ) ( b ) ( c ) v v v C Im Re ’ e a e e a a a S S (cid:11) (cid:12) SA v v v C Im Re ’ e a e e a a a a a S S S (cid:11) (cid:12) SA v v v C Im Re ’ e a e e a F S (cid:11) (cid:12) SA ( d ) ( e ) ( f ) v v v C Im Re ’ e a e e a a a F S S (cid:11) (cid:12) SA v v v C Im Re ’ e a e a e S F (cid:11) (cid:12) SA v v v C Im Re ’ e e e (cid:11) (cid:12) SA ( g ) ( h ) ( i ) Figure A.3. Illustrative sketches for the computation of the area of thecrossing region S ( S , C ∩ ∆ ) between a generic circle C ( C ∈{ C h ( u ) ; h = 1 , ..., K + 1 } ) of radius r ( r ∈ { r h ; h = 1 , ..., K + 1 } )and a triangle ∆ ( ∆ ∈ { ∆ t ( u ) ; t = 1 , ..., T } ) of anti-clockwise orderedvertexes { v j ; j = 1 , ..., } with edges { e j ; j = 1 , ..., }. Case 1 - Allvertexes of ∆ are outside the circle ( v j / ∈ C ; j = 1 , ..., ) and there are notedge intersections with C ( e j ∩ C = ∅ ; j = 1 , ..., ) ( a )( b ). Case 2 - Allvertexes of ∆ are outside the circle C and there is/are ( c ) one or ( d ) two or( e ) three edges intersections. Case 3 - ( f )( h ) One or ( g ) two vertexes of ∆ fall within the circle C . Case 4 - All three vertexes of ∆ lie within the circle C ( i ). where A { C h ( u ) ∩ ∆ t ( u ) } is calculated by considering themutual position of the t -th triangle ∆ t ( u ) and the cir-cumference C h ( u ) and according to the rules pictoriallysummarized in Fig. A.3. As for these latter, the guidelinefor computing the crossing area S ( S , C ∩ ∆ ) betweena generic circle C ( C ∈ { C h ( u ) ; h = 1 , ..., K + 1 } ) ofradius r ( r ∈ { r h ; h = 1 , ..., K + 1 } ) and a triangle ∆ ( ∆ ∈ { ∆ t ( u ) ; t = 1 , ..., T } ) of anti-clockwise ordered ver-texes { v j ; j = 1 , ..., } with edges { e j ; j = 1 , ..., } , is thatof checking the relative positions of the vertexes and edgeswith respect to C . More specifically, depending on the numberof vertexes that lie within the circle C and the number ofintersections between C and ∆ , the following cases arise: • Case 1 - If all the vertexes are outside the circle ( v j / ∈ C ; j = 1 , ..., ) and there are not edge intersections with C ( e j ∩ C = ∅ ; j = 1 , ..., ) [Figs. A.3( a )-A.3( b )], the valueof A ( S ) is equal to: ( a ) the whole circle surface A ( C ) , A ( S ) = πr , if the triangle includes the whole circle[Fig. A.3( a )], ( b ) zero, A ( S ) = 0 , otherwise, the trianglebeing outside the circle [Fig. A.3( b )]; • Case 2 - If all the vertexes are outside the circle ( v j / ∈ C ; j = 1 , ..., ) and one edge intersects the circle of C intwo points a and a [Fig. A.3( c )], the intersection region S is a circular surface delimited by the chord a a and 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.2020.2998924Copyright (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, 2020 12 by the circular arc with endpoints a and a [Fig. A.3( c )- yellow area]. In this case, the value of A ( S ) is givenby A ( S ) = r arctan a a q r − ( a a ) ! − a a × q r − ( a a ) . (21)Differently, if two or all edges intersect the circle [Figs.A.3( d )-A.3( e )], S is obtained as the difference betweenthe circle area A ( C ) and the areas { A ( S i ) ; i = 1 , ..., I }of the circular surfaces { S i ; i = 1 , ..., I }, which arecomputed as in (21), A ( S ) = A ( C ) − I X i =1 A ( S i ) (22)where I indicates the number of circular surfaces and itis equal to I = 2 [Fig. A.3( d )] or I = 3 [Fig. A.3( e )] ifthe circumference is intersected by two or three edges,respectively; • Case 3 - If either one or two vertexes are inside thecircle [Figs. A.3( f )-A.3( h )], A ( S ) is yielded as the sumof the area of the circular surface S [Fig. A.3( f )-A.3( h )],which is computed through (21), and that bounded by thepolygon F whose vertexes are the M intersection points{ a m ; m = 1 , ..., M } between the circle C and the edgesof the triangle ∆ . The area of F is calculated by meansof the the Gauss’s formula [42] as follows A ( F ) = (cid:12)(cid:12)(cid:12)P M − m =1 ℜ ( a m ) ℑ ( a m +1 )+ ℜ ( a M ) ℑ ( a ) − P M − m =1 ℜ ( a m +1 ) ℑ ( a m ) −ℜ ( a ) ℑ ( a M ) | (23)where ℜ ( · ) and ℑ ( · ) stand for the real and the imaginaryparts, respectively; • Case 4 - If all the three vertexes lie within the circle,then the intersection surface S turns out out be equal tothe triangle ∆ and A ( S ) = A (∆) [Fig. A.3( i )] being A (∆) = p χ ( χ − e ) ( χ − e ) ( χ − e ) (24)where χ = e + e + e is the half of the length of triangleperimeter.The procedure is iterated for all T triangles against the K + 1 circles to compute the areas of all K planar sectors,{ A { [ AF k ( u )] } ; k = 1 , ..., K } as well as the whole surfaceof [ AF ( u )] such A { [ AF ( u )] } = Σ Kk =1 A { [ AF k ( u )] } (25)to be used in (12).A CKNOWLEDGEMENTS
A. Massa wishes to thank E. Vico for her never-endinginspiration, support, guidance, and help. R
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