Computational Methods in the Theory of Synthesis of Radio and Acoustic Radiating Systems

Transcription

1 Applied Mathematis Published Olie Marh 3 ( omputatioal Methods i the Theory o Sythesis o Radio ad Aousti Radiatig Systems Petro Sаveko Departmet o Numerial Methods o the Pidstryhah Istitute or Applied Problems o Mehais ad Mathematis Natioal Aademy o Siees o kraie Lviv kraie spo@iapmmlvivua Reeived November 7 ; revised Jauary 3; aepted Jauary 9 3 ABSTRAT A brie review o the works o the author ad his o-authors o the appliatio o oliear aalysis umerial ad aalytial methods or solvig the oliear iverse problems (sythesis problems) or optimizig the dieret types o radiatig systems is preseted i the paper The sythesis problems are ormulated i variatioal statemets ad urther they are redued to researh ad umerial solutio o oliear itegral equatios o Hammerstei type The existee theorems are proo the ivestigatio methods o ouiqueess problem o solutios ad umerial algorithms o idig the optimal solutios are proved Keywords: Noliear Iverse Problems; Sythesis o Radiatig Systems; Noliear Equatios o Hammerstei Type; Brahig o Solutios; Noliear Two-Parameter Spetral Problem; Loalizatio o Solutios; Numerial Methods ad Algorithms; overgee o Iterative Proesses Itrodutio I may pratial appliatios at the optimal desig o various types o radio ad aousti radiatig systems the requiremets are oly to the eergy harateristis o the diretivity o the radiated ield (amplitude diretivity patter (DP) or DP by the power) Thereore there is a eed to approximate real iite utios by modules o oedimesioal or two-dimesioal ad disrete ourier trasorm depedet o the real physial parameters At the same time the absee o requiremets to phase harateristis o ield is used to improve the quality o approximatio o sythesized DP to give Later o the variatioal ormulatios o dieret types o iverse problems i mea-square approah whih i urther are redued to ivestigatio ad umerial solutio o oe-dimesioal or two-dimesioal oliear itegral equatios o the Hammerstei type with separate module ad argumet o desired omplex-valued utio are osidered Nouiqueess ad brahig (or biuratio) o solutios depedet o the hage o the physial parameters haraterizig the radiatig system are harateristi eatures o suh equatios Problems o idig the set o brahig poits (biuratio) are ot ivestigated oliear oe-parameter or two-parameter spetral problems The existee o oeted ompoets o the spetrum whih i the ase o real parameters are o spetral lies is essetial dieree betwee the two-dimesioal ad oe-dimesioal spetral problems The problem o idig the spetral lies is redued to umerial solutio o the auhy problem or a ordiary dieretial equatio o the irst order The degeerate o kerels i liear operators o the Hammerstei type equatios is eature o the sythesis problems o atea arrays It allows to redue oliear two-parameter spetral problems o idig the set o brahig poits o solutios to the orrespodig systems o liear algebrai equatios with oliear ourree o the spetral parameters i the oeiiets o system I the basis o ostrutio o umerial algorithms or idig the optimal solutios are take suh prieples: loalizatio o existig solutios depedet o the value o the physial parameters o the problem by meas the use o umerial methods o solvig the o-liear oe-parameter ad two-parametri spetral problems ad methods o the brahig theory o solutios ostrutio ad justiiatio o the overgee o iterative proesses or umerial idig the various types o existig solutios o basi equatios (equatios o Hammerstei type) aalysis o the eetiveess o oud solutios ormulatio o Problems Basi Equatios o Sythesis I geeral ase the aalysis problems (diret problems)

2 54 o radio (or aousti) radiatig systems are redued to solutio the orrespodig boudary problems o eletrodyamis (aousti) at a give exitatio soures o ields [-4] o the basis o Maxwell s equatios (wave equatio) The diretivity patter is oe o the basi harateristis o the emitted ield o large distaes rom the radiatig system It desribes the properties o the ield i spae depedet o the agular oordiates o a spherial oordiate system I geeral ase the DP is a vetor o omplex-valued utio whih has the orm [56] i i A i A i () Here A A A is liear operator atig rom some utioal Hilbert spae H (the spae o square itegrable utios i the domai V desribig the distributio o extraeous ields (urrets) i volume V ) ito the spae o omplex-valued otiuous utios deied i some domai (or ) [7] The orm ad properties o operators A A are deied by type ad geometry o the radiatig system The set (domai) o values o the operator A A A is alled [89] set or lass o realized diretivity patters This meas that or ay DP rom this lass there exists suh utio o distributio o the urrets (ields) H that realizes this DP ie A I the simplest orm the iverse problem (the sythesis problem) aordig to the presribed amplitude DP a be ormulated as the problem o idig the solutios o oliear operator equatio o the irst kid A () where is a give amplitude DP I staged thus the sythesis problem all three orret oditios o problems by Hadamard [-]: existee o the solutio uiqueess o the solutio otiuous depedee o the solutio o the iput data a be violated simultaeously Violatio o oditio () i the irst plae is oeted with the at that the give DP a ot belog to the lass realized that is to the domai o values o the operator A I other words suh DP a ot be obtaied at ay distributio o ield i the aperture o the radiatig system belogig to the spae H Tryig to rereate the DP just leads to eet o superdiretivity [5] The system beomes resoae ad ritial to hage o parameters oditio () is violated due to the oliearity o the problem Thereore the variatioal ormulatios o problems whih i additio to the requiremets o the basi harateristis o DP also otai requiremets to the distributio utio o the urrets (ields) i the aperture o the radiatig system are osidered At that is required ot omplete oiidee obtaied DP with give but oly the best (i the sese o the seleted riterio) approximatio to it A importat eature o the variatioal ormulatio o sythesis problems is the at that i the optimizatio riterio a itrodue utioals desribig ertai other requiremets to amplitude-phase distributio (APD) o outside exitatio soures Their mea-square deviatio as a rule will be used as the riterio o proximity o amplitudes o the give ad sythesized DP The ase o Liear Polarizatio o Extraeous ield irst we osider the salar ase o problems whe extraeous ields (urrets) i the radiatig system is liearly polarized [73] ad reated by their DP () has oly oe ompoet Let the operator A ats rom some Hilbert utioal spae H L V ito the omplex spae o otiuous ad square itegrable utios i domai (or ) I spae H we itrodue the salar produt ad orm H V P H d V V where P x y z P P d V (3) is a poit o itegratio Alog with the hebyshev orm sup Q (4) Q i the spae we itrodue salar produt ad the ge- erated by it orm ad metri as ollows: g Q g Q dq (5) Note spae is a Baah spae relatively uiorm orm (4) ad it is iomplete spae oerig orm deied aordig to (5) [4] We osider also that the operator A has the izometri property or it is ompletely otiuous Let the give amplitude DP is real positive (oegative) otiuous utio whih dieret rom ozero i some limited losed domai ad idetially equal zero o omplemet Let A is isometri operator that is or ay H ad A H equality is satisied H H A A (6) I [5-7] the sythesis problem o give amplitude

3 55 DP is ormulated (is ivestigated) as a miimizatio problem i the Hilbert spae H L V o the utioal A (7) haraterizig the value o mea-square deviatio o modules o give ad sythesized DP i domai or the ormulated problem ours [89] Theorem Let the liear operator A: L V is isometri relatively mea-square L V ito metri ad it is otiuous operator rom oerig uiorm metri o the spae ad the give amplitude DP s is a real positive (oegative) otiuous utio i the domai The at least oe poit o absolute miimum o the iarga utioal (7) exists i the spae A e ad a subsequee whih overges weakly to oe o poits o absolute miimum a be seleted rom ay miimizig sequee O the base o the eessary oditio or a extremum o the utioal (zero equality o its Hato diere- tial []) D w w we obtai the equatio with respet to the optimal distributio o exitatio soures iarg A e (8) A where A is the ojugate with A operator Let the set o zeros NA o the operator A osists o oly oe zero elemet The atig o both parts o (8) by operator A we obtai the equivalet to (8) : equatio i spae AA iarg e (9) By solutios o this equatio the optimal distributio o exitatio soures i radiatig system are deied by the ormula iarg A e () rom Theorem ad the properties o utioal ollows orollary Sie utioal is dieretiable i H by Hato it is growig [] ad aordig to Theorem has at least oe poit o absolute miimum the (8) i the spae H ad (9) i the spae have at least oe solutio Lemma Betwee solutios o (8) ad (9) there exists bijetio that is i is the solutio o (8) the A is the solutio o (9) O the otrary i is the solutio o (9) the the orrespodig solutio o (8) is deied by () The possibility o ivestigatio o solutios o sythesis problems usig (8) or (9) ollows rom Lemma Note Equatio (9) is simpler as (8) sie the latter otais the operator expoet Note solutios o sythesis problems aordig to the presribed amplitude DP are determied with preisio i to value e ( is arbitrary ostat) sie i e So i there exists the solutio o Equatios (8) (or (9)) the there is also geerated by its amily o solutios i whih oe solutio dieret rom aother by phase ostat or the uiqueess o desired solutios additioal oditios impose o the utios arg or arg I the ase ompletely otiuous operator desribig DP o radiatig system the smoothig Tikhoov type utioal [] H H I A () whih iludes requiremets as to the mea-square deviatio o DP so to orm o urret is used [8] or the sythesis o various types o ateas The parameter α a be viewed as regularizatio parameter [3] or as a weighig oeiiet by meas o whih a otrol ratio betwee the irst ad seod summad o utioal Theorem Let the liear operator A ats rom the omplex Hilbert spae H L V ito the om- plex spae o otiuous utios ad it is ompletely otiuous ad give DP is real positive (o- egative) ad otiuous utio i (or i ) The at least oe poit o absolute miimum o utioal exists i H L V ad a subsequee whih overges weakly to oe o the poits o absolute miimum a be seleted rom ay miimizig sequee Dieretiatig utioal by Hato ad per- ormig appropriate trasormatios we obtai the equatio [4] A A A exp iarg A () i the spae H Equatio with respet to sythesized DP based o equality A ad () has the orm AA AA exp iarg (3) Lemma similar to Lemma is valid or () ad (3) rom Theorem ad properties o utioal ollows [78]

4 56 orollary Sie dieretiable i H by Hato utioal is growig ad aordig to Theo- rem has at least oe poit o absolute miimum the () i spae H ad (3) i the spae have at least oe by oe solutio The ase o Arbitrary Polarizatio o Exitatio ields osider the more geeral ase whe the exitatio ields (or urrets) i the radiatig system ad geerated by it DP have vetor harater [56] I this ase we set that the operator A is ompletely otiuous ad it ats rom H LVLV LV omplex spae o square itegrable i the domai V vetor-valued utios ito the omplex spae o otiuous utios o the ompat vetor-valued utios equipped by salar produt We itrodue the salar produt ad geerated by it orm i H : x x y y z z x xyz x xyz y xyz y xyz V z xyzz xyz dxyz d d (4) H x x y y z z (5) We deie module o vetor as ollowig: x x y y z z I the spae alog with the hebyshev orm max (6) where we itrodue the salar produt ad geerated by it the mea-square orm ad metri sidd (7) (8) I DP o radiatig system has two ompoets ie it is desribed by () the as the optimizatio riterios are used the ollowig utioals [7]: H (9) A H H A A H () I the utioal () are the give amplitude o ompoets o required DP At that ad utios a be give with aout o existig requiremets to polarizatio harateristis o emitted ield I i the sythesis problem utioal (9) is used as the optimizatio riterio the problem o idig the miimum poits is redued to idig the solutios o the equatio * A A A A A () i the spae H I Equivalet to () equatio with re- spet to vetor DP i spae has the orm AA AA () I this ase the ollowig theorem is valid [7] Theorem 3 Let liear ompletely otiuous operator A ats rom the omplex Hilbert spae H LV LV LV ito the omplex spae o otiuous utios equipped by the salar produt ad give DP is a real positive otiuous utio o the ompat The i H there exists at least oe poit o absolute miimum o utioal ad a subsequee whih overges weakly to oe o the poits o absolute miimum a selet rom ay miimizig sequee or the utioal () is true Theorem 4 At oditios o Theorem 3 utioal i the spae H has at least oe poit o absolute miimum ad subsequee whih overges weakly to oe o poits o absolute miimum a be seleted rom ay miimizig sequee or miimizig o the utioal i the spae H we obtai equatio [7] iarga A A iarga A e e (3)

5 57 Equivalet to (3) equatio with respet to sythesized DP i the spae has the orm iarg iarg AA e e (4) The existee o solutios o (3) i spae H ad (4) i the spae respetively ollows rom eessary oditio o utioal miimum I eessary the weight utio [57] a itrodue i the utioals ad by meas eterig appropriate salar produts ad aet o quality o the approximatio o sythesized ad give DP i a ertai rage o agles 3 Simultaeous Optimizatio o the eometry o Aperture ad Exitatio ields The sythesis problems with optimizatio o geometry o radiatig system are more ompliated lass o problems These problems eed to id a oiguratio o the radiatig system ad amplitude-phase distributio o exitatio ields (urrets) i it [56] Moreover the operator A depeds o two utios: utio desribig the geometry o the system ad amplitude-phase distributio utio o exitatio soures ie A (5) I additio the utio has as rule vetor harater ad the operator A is a oliear oerig the utio Later o we shall osider the sythesis problem o a lat aperture i whih i additio to amplitude-phase distributio (APD) desired is too the utio that desribes the boudary o aperture The basis o the ormulatio o suh problems a be put the utioals (7) () ad () expadig their by orrespodig requiremets to geometry o radiatig system 4 Sythesis Problem o Disrete Radiatig Systems Atea Arrays I may radio egieerig systems atea arrays (AR) have gaied widespread use Atea array is [467] atea whih osists o N idetial (or dierettype) radiators plaig orrespodig way i spae ad they ollate by ommo system o power ad otrol I [8-37] ivestigatios o oliear sythesis problems ad plaar atea arrays aordig to the presribed amplitude DP are preseted To desribe the eletromageti harateristis o atea arrays are used dieret by preisio mathematial models [38-4] I the base o ostrutio o mathematial models is imposed [44] that the exitatio o eah radiator is haraterized by a uique omplex umber I -omplex amplitude o exitatio It s the physial meaig depeds o the type o radiatig system O the base o the liearity o the Maxwell s equatios the omplex exitatio amplitudes eter liearly i the expressio or DP o array that is N I ikxsi osysisizos e (6) where DP -th radiator Vetor I I I i i is vetor I N is alled vetor exitatio o array or vetor o amplitude-phase distributio o urrets i the array I geeral the ostrutio o high-auray mathematial models o array is redued to solvig the orrespodig exterior boudary problem o high-requey eletrodyamis or system o the Maxwell s equatios i multiply-oeted domai [39-4] I the partiular ase where the elemets o the array are ideally leadig talamy aoutig o the mutual iluee is based o the method o idued eletromotive ores (IE) ad it is redued to solutio the orrespodig system o liear itegral equatios [4] I (7) Bwhere is matrix-itegral liear operator; I is omplex- Bvalued vetor distributio utio o surae urrets o radiators; H is vetor-valued utio desribig the outside ields (voltage) whih is eessary to reate i the system o power o array Alloatig i the spae H I ompat lass o solutios (where (7) is orret) solutio o (7) is writte as I Here it is assumed that the orrespodig regularized system I the basis o (8) ormula or DP o array takes the orm B (8) Bexists or the system o (7) The o A B (9) This relatio allows to ormulate the sythesis problem o atea array aordig to the presribed amplitude DP with aout o the mutual iluee o elemets as the problem o idig the vetor miimizig the utioal A H B(3) i spae H At eed to take ito aout the ompoet-wise approximatio o modules o give ad sythesized DP s Note: regularizatio methods [] are the basi methods o ostrutio o the operator uder whih the stable solutios o (7) Bare obtaied ad or equatios whose kerels have a weak eature sel-regularizatio The etity o ostrutio o the operator (at Bexeutio o respetive oditios) osists i reduig the system o itegral equatios o the irst kid to the orrespodig system o itegral redholm equatios o the seod kid that is to orret the problem

6 58 the utioal [43-45] H A A B H B (3) aalogously to () a be used as optimizig riterio By the desired solutios o this problem the optimal vetor o extraeous voltages o iputs o radiators is determied o the basis o (7) Thus the basi requiremets or sythesis problems o dieret types o radiatig systems aordig to the presribed amplitude DP are ormulated Note that reorded utioals is ot ovex [] ad thereore may have ouique extreme poit I urther the above statemets o problems allow to obtai relatively simple oliear itegral or matrix equatios or the study ad solutio o whih a be applied umerial methods o oliear aalysis methods Itegral equatios method [44] is used widely i suh lasses o problems The method o sythesis o atea arrays rom ylidrial dipoles with aout o mutual iluee is proposed i [43-45] Aalysis o problem o ouiqueess solutios is studied there by meas omputatioal experimets 5 Noliear Sythesis Problem o Radiatig Systems with se o Eergy riterio I spite o the at that rom the amplitude DP is easy to obtai the DP by power N ad vie versa i the mathematial aspet the sythesis problems o give amplitude DP ad give eergy DP N are dieret tasks or example i is the optimal solutio o some variatioal sythesis problem o ampli- tude DP the will ot be the optimal solutio o a similar problem or the give DP O this basis i [46-49] o the operator level are osidered statemets o sythesis problems with use o two types o stabilizig utioals i whih the vetor harater o the eletromageti ields takes ito aout osider the sythesis problem o give eergy DP N Takig ito aout the expressio or DP N A A i the simplest orm this problem a be ormulated as a problem o idig the solutios o oliear operator equatio o the irst kid A A A N (3) where N is a real positive otiuous utio i (at that N max ) whih a ot be- log to the domai o values o the operator A It is kow [] that (3) is severely ill-posed problem I this oetio we osider the problem o best measquare approximatio o the real positive otiuous (i the domai ) utio N by utio ( A R A I H ) x y z ormulate it as miimizig problem o utioal [49] N N N A H H (33) i the spae H I this utioal the irst summad haraterizes mea-square deviatio o give ad sythesized DP by power Seod summad imposes restritios o orm o urrets i the radiatig system Real parameter we shall osider as a weighig multiplier The existee o at least oe poit o miimum utioal N i the spae H states [749] Theorem 5 Let the liear operator A ats rom the spae H ito ad it is ompletely otiuous N is give oegative otiuous the utio max N i at that The i the spae H there exists at least oe poit o absolute miimum o the utioal N ad sub- sequee that overges weakly to oe o the poits o absolute miimum a be seleted rom ay miimizig sequee O the base o eessary oditio o miimum utioal is obtaied the equatio [7] A N A A A A (34) with respet to optimal urrets i the spae H This equatio is a oliear operator equatio havig i the right part liear operator alog with the Hammerstei type operator I the set o zeros NA osists o oly the zero elemet the atig o both parts o (34) by operator A we obtai equatio a equivalet to (34) with respet to sythesized DP i the spae B AA N AA (35) I [49] is show that the utioal N has m -property [5] that is the miimum poit o the u- tioal is iterior poit o some ovex weakly losed set o the spae H O this basis rom Theorem 5 ollows [7] is di- orollary 3 Sie the utioal N

7 59 eretiable i H by Hato has at least oe miimum poit ad m- property the (34) i the spae H ad (35) i the spae have at least by oe solutio Lemma At oditios o Theorem 5 ad limited values o parameter operator N B AA AA (36) is ompat i the spae Sie or elemets relatively ompat set o ormalized spae the strog ad weak overgee oiide [5] the with Theorem 5 ad Lemma ollows orollary 4 I is miimizig sequee o the utioal N overgig weakly to the miimum poit A o- the the sequee to A verges uiormly i 3 About Brahig o Solutios o the Basi Equatios o Sythesis Partial ases Here o the example o salar sythesis problems o liear radiator ad radiatig system with a lat aperture are preseted the results o ivestigatio o ouiqueess problem o solutios orrespodig to these tasks oliear itegral equatios o Hammerstei type depedig o the hage o the physial parameters 3 The ase o a Liear Radiator We put that the liear atea is liear eletri odutor o legth a sizes o ross-setio o whih are muh less tha the wavelegth Due to these limitatios the exitatio urrets i the atea shall have oly diretaxis urret [5] Itrodue the artesia ad oeted with spherial oordiate systems suh that the origi o oordiates oiides with the middle o the atea We diret the axis OZ alog the atea The the urret vetor i a artesia oordiate system will have oly z-ompoet z We shall itrodue the dimesioless oordiates s si si z z a ad parameter kasi (37) oetig the eletri size o atea with agle outside o whih give amplitude DP s idetially zero The the ormula or DP o liear atea takes the orm [7] izs s A z e dz (38) or DP o atea the Parseval equality [8] s d s π z d z (39) is valid that is the operator A is isometri Takig ito aout that is iite utio with ompat arrier ad expressios or the operators A ad A we obtai the expaded orm o (8): i z s iarg ze dzzs z π s exp ds (4) O the basis o (9) (38) we obtai the Hammerstei type equatio oerig optimal DP iarg s s B s K s s; e ds (4) i the spae where K s s; si ss π s s (4) The existee o at least oe solutio o (4) i the spae H L ad (4) i the spae ollows rom orollary We shall preset three importat properties o (4) ) I s is the solutio o (4) the omplexojugate utio s is the solutio o (4) too i ) I s is the solutio o (4) the e s is the solutio o (4) too where β is a arbitrary real ostat 3) or eve utios s oliear operator B whih is i the right part o (4) trasers eve phase DP arg s i eve ad odd i a odd That is the operator B is ivariat with respet to the type o parity o utio arg s Due to this property the existee o ixed poits o the operator B -solutios o (4) is possible i the lasses o eve ad odd phase DP s I [65] is show that (4) has two solutios i the lass o real utios: s s K ss ; d s (43) whih is alled the primary solutio o the irst type ad s s K ss ; sg s s d s (44) is the primary solutio o the seod type Poit s s is determied rom the oditio s or the eve s s that is the solutio (44) is a real odd utio (the orrespodig to it amplitude DP is eve utio) These solutios are eetive oly at small values o parameter With the growth o this parameter there exist brahig poits i at whih more eetive (i the sese value o utioal ) omplex solutios brah-o rom real solutios osider aordig to [76] results o ivestigatio o brahigs o primary solutio o the irst type o solutios o (4) sig the proedure o deomplexiiatio o the spae [4] rom (4) we move to the equivalet system

8 53 u s Q s s u s v s ds us s K s s; d s u s v s d vs u s v s v s Q s s u s v s s s K s s; d s (45) O the base o the brahig theory o solutios [53] the problem o idig suh values o parameter l (brahig poits) ad all dieret rom s l solutios o system o (45) satisyig the oditios max us l s l s at l (46) max vs l s are osidered oditio (46) meas that it is eessary to id small otiuous solutios wsus s l s vs overgig uiormly to zero at Puttig i (45) l u s s w s v s s (47) l ad expadig the itegrad Q Q i the viiity o the poit l s l i the power series by w ad ad takig ito aout that the utio s is its solutio we obtai system o itegral equatios o Lyapuov-Shmidt type [53] with respet to small solutios ws s : as w s l p m mp mp A s s w s s d s s s sks s; ds s p m mp mp where B s s w s s d s (48) (49) a s d A s s s O the base o the let part o (49) we obtai liear homogeeous itegral equatio s st sks s; ds (5) s or idig the poits o possible brahig o solutios Equatio (5) is a oliear oe-parameter spetral problem oerig parameter It is show i [76] that or a give eve amplitude DP s there exist brahig poits o two types: eigevalues o multipli- ity two orrespod to the irst type eigevalues the multipliity o three to the seod type It is oud i [54] aalytial expressios or eigeutios o (5) ad are obtaied systems o trasedetal equatios or idig the possible brahig poits sig or idig the solutios o brahig equatio the Newto diagram method it is show [53] that two omplex-ojugate betwee themselves solutios whih i the irst approximatio have the orm l l l l 3 i s h O l s s a s s h brah-o rom the real solutio l ig poits o the irst type Here l s (5) s i the brah- l a s are eve by s utios whih are obtaied by meas orrespodig trasormatios l l s s s is the seod eigeutio o l (5) at the poits I [7] it is show also that the brahig-o solutios brah-o too Aalogous ivestigatios are perormed l i brahig poits o the seod type To estimate the eetiveess o dieret types o solutios we osider value o the utioal depedig o the parameter whih it takes o these solutios or example i igure are show the values o the utioal or s The most eetive solutio images evelope whih: o the segmets I orrespods to the primary solutio o II-brahig-o solutios at the poit with odd phase DP o III-solu- tios ad brahig-o solutios rom these o IV-solutios ad brahig-o solutios at the poit with eve phase на V-brahig-o at the poit solutios o the type o VI-solu- tios Thus the aalytial ivestigatios [76554] ad the results o umerial experimets eable to desribe the geeral struture o the solutios o the problem depedeig o hage o the value o the parameter Beause the values o the utioal o some types o solutios i a give iterval o hage o the parameter may be equal the urves show i igure does ot map the ull struture o the existig solutios or greater larity this struture a be represeted shematially as a tree o solutios I igure it is show or the ase o eve DP The primary solutio s is etral The solutio s with odd phase DP arg s brahes irst At the poit solutio

9 log I II III IV V -5 () 4 ( 6 8 ) () () ~ (3) igure The value o the utioal σ o the irst primary ad brahig-o solutios or (s) = () (s) () ~ s ad solutio (s) (3) () () () () (s) (s) (s) ~ () () (s) igure Tree o solutios ~ () VI (s) brahes rom brahig solutio s s eter i a real solutio i a eighborhood o the poit at tedig to At the same poit solutio with eve phase DP s brahes rom primary solutio The solutio o the type s with a odd phase DP brahes at the poit whih is loated diretly behid The solutios o the type s ad the type s orm basi brahes o tree The possible brahig poits o brahig-o solutios are show o these brahes 3 Radiatig System with a lat Aperture 3 Basi Equatios ad Relatios osider aordig with [55-58] the sythesis problems o a lat aperture assumig that orm o aperture S is kow ad a ield has elliptial polarizatio Let the plae i whih aperture is loated oiides with the plae XOY o the artesia oordiate system The the radiated ield i the ar zoe a be represeted by the ormula [6]: ikr k e ER D 4π R where D i r i r x y ikxsios ysisi e dxdy (5) S i r is radial ort o spherial oordiate system i z is a vetor DP o lat aperture S Sie i z utio x y desribes the tagetial ompoet o the eletri vetor E or vetor o urret lowig through the aperture S : x y x y x y i i (53) x x y y Itroduig i a ar zoe speial oordiate system [6] g g g 3 oeted with orts o spherial oordiate system by ormulas g osi sii g sii osi g3 i r (54) eables the vetor sythesis problem to redue to two idepedet salar sythesis problems Obviously the system g g g 3 is orthoormal ad trasormatio (54) is rotatio o spherial oordiate system o a agle aroud the vetor i r At that vetor D i the oordiate system (54) has the orm [6] D os x y (55) where xy Axy xy x y ikxsios ysisi (56) e dxdy S xy or mappigs (56) the Parseval equality [59]: xy xy H xy H A (57) are valid that is operators A x A y are isometri osider the sythesis problem o a lat aperture i whih ompoet-wise deviatio o modules give ad sythesized diagrams is take ito aout As optimizatio riterio we hoose the utioal type g s s g s s s s s s dsds g g g g d d s s s s s s (58)

10 53 where g are modulus o ompoets o the g give amplitude DP g g i losed domai This riterio provides ot oly proximity o modules o give ad sythesized DP s but it allows ertai to iluee o the polarizatio harateristis o the radiated ield [5] O the base o the eessary miimum oditio o the utioal (58) ad the orrespodig trasormatio we obtai system equatios (these equatios is ot oeted betwee themselves) oerig ompoets o sythesized DP: k j g s s B j j g π iarg g ; e j s s g s s K s s s s k ds ds j (59) where j K Q Q π exp i x ss y s s dxdy (6) is a kerel I the ase o retagular aperture the kerel K s s s s ; takes the orm si s s si s s Ks s s s ; s s s s where (6) ka si ka si (6) are real umeri parameters haraterizig the sizes o aperture a a i wavelegths k π is wave umber are agles that haraterize the domai (solid agle) i whih dieret rom the idetity ompoets o amplitude DP g j s s are give Later o we omit idex i (59) ad shall ivestigate the solutios o oe equatio iarg Q Q B Q K Q Q e dq (63) where or redutio o reords we itrodue the ollowig otatios Q s s dq ds ds Thus the sythesis problem o lat radiatig system with arbitrary polarizatio o irradiatio aordig to the presribed amplitude DP is redued to two idepedet ad simpler sythesis problems with liearly polarized ields i the aperture Equatio (63) is a oliear two-dimesioal itegral equatio o the Hammerstei type ad it has ouique solutios Their quality ad properties deped o the orm o aperture S the values o parameters ad properties o give amplitude DP O the base o deomplexiiatio [4] we shall osider the omplex spae as a diret sum o two real spaes o otiuous utios i the domai The elemets o this spae are writte as: uv u Re v Im spaes have the orm: Norms i these u max u Q v max v Q Q max u v Q Equatio (63) i the deomplexiied spae redued to equivalet to it system o equatios (64) is uq d u Q v Q u Q B u v Q K Q Q Q vq d u Q v Q vq B uv Q KQQ Q (65) Deote the losed ovex set o otiuous utios S settig that as M SM SM S : u M S v M us u M u M u SM vs : v M v M v max d M Q K Q Q Q Q osider oe o the properties o the utio expiarg Q that iluded i (6) at Q Obviously exp i arg Q uq ivq Q Q u Q v Q (66) is a otiuous utio i uq Re Q ad vq Im Q are otiuous utios at that exp i arg Q or ay Q I uq ad vq simultaeously the Q is a omplex zero with udeied argumet by deiitio [6 p ] O this basis at uq ad vq we redeie expiarg Q as utio module o whih is equal to oe ad argumet is udeied

11 533 Theorem 3 The operator T B B B determied by the ormulas (65) maps a losed ovex set S M o the Baah spae i itsel ad it is ompletely otiuous As the orollary rom the Theorem 3 ollows satisatio o oditios o the Shauder priiple [4 p 4] aordig to whih the operator B B T B has a ixed poit u v T belogig to the set S M This poit is a solutio o a system o (65) ad (63) respetively Easily to be ovied that utio Q Q K Q Q dq (67) is oe o solutios o (63) i the ase o symmetri domai I [5658] it is show that the operator D K Q Q Q dq is positive o the oe o oegative utios o the spae ( ) [6] Aordig to this the operator D leaves ivariat oe that is D Sie the primary solutio D is also oegative utio i the domai To id the brahig lies ad omplex solutios o (63) that brah-o rom the real (primary) solutio Q we shall osider the problem o idig suh a set o parameter values ad all dier- et rom Q solutios o (65) that at satisy the oditios max u Q Q Q max vq Q (68) These oditios idiate the eed to id suh small otiuous i solutios Q vq wq uq Q (69) whih overge uiormly to zero as At that it should take ito aout also the diretio o movemet o vetor to vetor Set ad desired solutios we id i the orm Q u Q Q w Q vq (7) We write the system o oliear itegral equatios o Lyapuov-Shmidt with respet to small solutios w as u Q a Q a Q A QQ w Q Q d Q p q m mpq mpq Q Q Q Q K Q Q dq (7) B Q Q w Q Q d Q p q m mpq mpq (7) Here Ampq Q Q Bmpq Q Q are oeiiets o expasio o itegrad utios o (65) i uiorm overget power series The problem o idig the set o possible brahig poits o solutios o (7) ad (7) is redued [5658] to id the eigevalues o two-dimesioal liear homogeeous itegral equatio Q T Q (73) K QQ QdQ Q at oditio Q Eigeutios o (73) are used [58] at the ostrutio o brahig-o solutios o (7) ad (7) 3 Noliear Two-Parameter Spetral Problem Note that (73) i the geeral ase is a oliear two-parameter spetral problem or the umerial idig the approximate solutios it is eessary to ostrut its digitizatio ad osider the orrespodig problem i iite-dimesioal spaes It should be oted that i the literature i partiular i [663] more attetio is give to the ostrutio o umerial methods or solvig the oliear oe-parameter problems I [64-67] a geeral method or idig the approximate solutios o (73) whih may be appliable to a wide rage o oliear two-parameter spetral problems is proposed Deote the spetral parameters as λ Let E ad V are omplex Baah spaes ad the vetor parameter λ belogs to domai (ope oeted set) o the omplex spae where i i i ii : i r i r is some real ostat osider the operator-utio : EV λ is put i orrespodee operator EV AHere the spae o liear bouded operators [4] is marked as EV LWe shall osider the oliear two-parameter spetral problem o the orm x (74) AA L where to every L

12 PP534 where eessary to id the eigevalues λ x E ad orrespodig eigevetors x suh that Let the Baah spaes E ad give ad also a system operators p : E E suh that E E x Ap E be Po liear bouded px x x E (75) Operators p are alled oetig [468] Note by the priiple o uiorm boudedess [4] with (73) ollows iequality p ost Let i every spae E the elemet x be seleted Writig these elemets i order to irease the umbers we shall orm a sequee x Let elemet x is seleted i eah spae E Writig these elemets i asedig umerial order we orm sequee x Deiitio 3 [68] The sequee x rom x E overges (disrete overges) to x E i Px px E ; we deote x x Deiitio o dieret types o overgee o operators to is give i [68] Later o oly required i AAurther deiitio o stable overgee to is AApreseted Disretizatio o iitial problem (74) the hoie o the spae E ad deiitio o operators p : E E a be dieretially I partiular oe o the approahes to the digitizatio o (74) i the operator-utio Ais desribed by ormula T I Awhere T is a liear otious operator ad I is uique operator i the separable (iiite-dimesioal) Hilbert spae E osists i ollowig Take a arbi- trary omplete orthoormal system o utios x k k i E Eah elemet x E is represeted as a series x x k k k where k x xk is ourier oeiiet o elemet x Sie T is liear otiuous operator atig i separable Hilbert spae it admits the matrix represetatio [69]: T M t jk (76) j k where tjk T xk xj o the ourier oeiiets o elemet At that sequee y T x is obtaied rom the sequee o ourier oeiiets o T elemet x by meas trasormatio matrix Deiitio 3 [68] The sequae A o operators Q A LE V overges stably to A LE V i A A LV E exists at all at that A ost M ad the ollowig oditio (stability oditio) is valid: A sig the matrix represetatio o the operator TM i partiular ase (oerig Equatio (73)) the spetral problem (74) is ormulated as x T I x (77) M M where I M is idetity matrix i the spae o sequees l Thus the operators T λ ad TM λ are equivalet i the sese that they put i orrespodee oe ad the same elemet y E but we obtai the ourier oeiiets o elemet y T λ x as a result o operatio o operator TM λ o elemet x Obviously that the spetrums o these operators oiide that is the spetral problem (77) ad the problem Aare equivalet x T I x Aordig to [468] applyig to the problem (74) other disretizatio methods iludig the ollowig: quadrature (ubature) proesses i the ase o homogeeous itegral equatios ad hagig the derivatives by dieree aalogues i dieretial equatios we obtai the approximatio problems or approximate idig the eigevalues ad eigeutios i iite-dimesioal spaes x (78) AAt that the problem o idig the eigevalues is redued to idig the roots o the -th order determiat ie the roots o the equatio i j det a (79) i j osider the eessary i urther auxiliary oe-parameter spetral problem as a partiular ase o (74) Set that variable i the operator-utio is Aexpressed by some uique dieretiable utio z mappig domai i some subdomai I the simplest ase we put where is a real parameter Itrodue ito osideratio at operator utio z ) Oe-parameter oliear spetral problem A AA(arrowig o operator-utio x (8) A is oeted with it Here to eah value λ z operator z E V Aput i orrespodee Aalogously to (78) we osider approximatig sequee o disretizig problems (8) at Lis z x (8) AThe spetrum o operator-utio as s Suppose that A s Ao (74) holds [5667] s Ais deoted A or spetral

13 L\A 535 Theorem 3 Let the ollowig oditios be satisied: ) operator-utio : EV morphi ad s ; A) operator-utios : EV morphi ad or ay losed bouded set the ollowig iequality max ost is valid; λ A 3) operators EV A E V rators with zero idex or ay L AA A4) spetrum tios 5) λ λ Ar s s Lis holo- Lare holo- are the redholm ope- λ ; Aad a sequee o u- are dieretiable i the domai ; is stable or ay λ AThe the ollowig statemets are true: ) every poit o spetrum s is isolated it A z A is eigevalue o the operator A the iite-dimesioal eigesubspae N ad the Aiite-dimesioal root subspae orrespod to it; ) or eah there exists a sequee A s rom s ; A 3) eah poit λ z poit o the operator-utio A suh that is a spetrum ; A4) i i some small -eighborhood o the poit z λ at all larger ay umber N (orrespodig aordig to deiitio o limit o sequee p )) the sequee o partial derivates z is ozero the i a arbitrarily small -eighborhood o poit z there exists a otiuous dieretiable utio whih is solutio o (89) at that N N N N N N N N N N ad at the poit however little diers rom poit o a spetrum o auxiliary oe-parameter problem (9) N ; that is i some biylidrial domai : there exists a oeted ompoet o spetrum o the operator-utio N are small real ostats) Proo The proo o the theorem is give i [56] ad is based o Theorems ad with [68 pp 68-69] ad the Theorem about existee o impliit utio (see or example [7]) A( I the poits are the eigevalues o (78) ad derivatives i these poits are ozero to id oeted ompoets o the spetrum o this problem o the base o (79) auhy problem [565765] we solve the i a eighborhood o eah poit d d (8) (83) 33 Numerial Algorithms or idig the Possible Brahig Lies o Solutios Retur to idig the solutios o (73) i whih are spetral parameters Let where : i i i i r By diret hek we set that or arbitrary values o the parameters the utio Q Q KQ Q d Q Q (84) is oe o the eigeutios that is there exists a oeted set o the spetrum oiidig with the domai As a result o this the oditio s is ot Asatisied To id aother oeted ompoets o spetrum we exlude eigeutio (73) rom the kerel o itegral equatio amely osider the equatio Q T K Q Q Q dq (85) where Here Q Q Q Q K Q Q K Q Q (86) Q is adjoit with (73) eigeutio o equatio o rom Lemma Shmidt [53 p 3] ollows that rom spetrum o operator T is exluded oheret ompoet oiidig with the domai ad the orrespodig to the utio Q sig to (73) ertai overget ubature proess with oeiiets a j ad odes Q j ad rejetig i it remaider we obtai homogeeous system o liear algebrai equatios (SLAE) u T i M j j i j j where u uq i u a Q Q u i i K (87)

14 536 The presee o suh values o parameters whih are the solutios o the equatio det T I (88) M is eessary oditio o the existee dieret rom zero solutios o (87) We osider (88) as the problem o idig the impliitly give utio reduig it to the auhy problem (8) ad (83) Puttig i (85) we shall osider the auxiliary oeparameter spetral problem Q T K Q Q Q dq solutios o whih we use as iitial oditios i the auhy problem (83) orrespodig this equatio SLAE has the orm u T u (89) a Q Q u i Ki M j j i j j ad the problem o idig the eigevalues o this system is redued to idig the roots o the equatio det T I or the umerial solu- M tio o the auhy problem (8) ad (83) are used the Ruge-Kutta ad Adams methods We shall preset umerial examples o idig the solutios o (73) or two give amplitude DP s I igures 3 ad 4 are show spetral lies o (73) orrespodig the give DP s s ad give DP whih is deied by the ormula: s s s s s s s s s s - (9) Note that to eah poit o the spetral lies give i these igures orrespod the eigeutios o (73) with the harateristi properties or eah lie or example below are show the eigeutios that orrespod to poits o itersetio o the spetral lies ad (igures 3 ad 4) with the beam 8 34 Variatioal Approah to Solutio o the Noliear Spetral Problems I [77] alog with the impliit utios method a variatioal approah to solutio o the oliear oeparameter ad two-parameter spetral problems o id- ig the eigevalues ad eigeelemets u L o equatio T u u (9) i the real Hilbert spae L T : L L or the ase whe is a liear positive deiite igure 3 The possible brahig lies o solutios o system s s o (65) or = =8 igure 4 The possible brahig lies o solutios o system s s whih is deied by (9) o (65) or sel-adjoit operator oliearly depedig o the parameters is proposed Variatioal problem is ormulated as the problem o idig suh values o parameter * ad suh utios u L o whih utioal u Tu u L T uu T uu (9) beomes miimum The equivalee o the spetral problem (9) ad put it i orrespodee o variatioal problem (9) is proved Based o the method o geeralized oordiate deset iterative proess or the umerial idig oe o the eigevalues ad the orrespodig eigeutio o (9) is suggested Loal overgee is proved Example o use o a variatioal approah to idig the eigevalues ad eigeutios o (73) is show i ig- s s ad i igure 6 or the ase ure 5 or whe the utio s s is deied by (9) Later o the eigeutios o (73) orrespodig to eigevalues belogig to urves 3 illustrated i igure 5(b) are show rom the aalysis o the igures we see that the eigeutios s s utios s s are odd by argumet s ad are odd by both argumets

15 537 oud by umerial method orm ad properties o eigeutios i the possible brahig poits are used to determie o the properties o brahig-o i these poits o solutios o oliear systems o (65) (a) (b) igure 5 Normalized eigeutios o (73) orrespodig to the eigevalues: (а) (с = с = 53754); (b) (с = с = 9587) (a) (b) igure 6 Normalized eigeutios o (73) orrespodig to the eigevalues: (a) (с = с = 74775); (b) (с = с = ) 33 About Brahig o Solutios i the ase o a lat Aperture I [ ] usig the oud brahig lies ad eigeutios the aalytial ivestigatios o brahig o the primary solutio o the irst type o (65) or the ase whe the the kerel K s s s s ; has the orm (6) ad the multipliity o eigevalues o the li- ear Equatio (73) at the brahig poits is two are preseted The study o solutios o (65) is realized o the beam belogig to the domai Let be eigevalue o (73) We l assig to parameter the small disturbae ad osider the problem o idig all dieret rom s s solutios o (65) whih at satisy the oditios max u Q Q Q max vq Q The system o (65) by meas o expadig the itegrad utios is redued to the orrespodig system o Lyapuov-Shmidt equatios similar to (7) ad (7) Desired solutios are oud i the orm u Q Q w Q Q vq As a result we obtai [74] that at the poits rom the primary solutio s s brah-o two omplex-ojugate solutios havig i the irst approximatio the orm s s a s s s s s s h s s 3 s s i h O (93) The imagiary part beig determied by the properties o eigeutios s s utios arg s s obtaied o the base o (93) de-

16 538 termie the properties o the phase DP ad APD o the ield i aperture The properties obtaied i the irst approximatio o solutios agree with umerial results or example i igure 7 are show the values o the utioal at s s whih it takes o the primary (urve ) ad brahig-o (urves 3 4) solutios o the beam 8 Note that o the seg- 5 met the brahig-o solutios with a odd phase DP arg s s to whih the osymmetri amplitude-phase distributio o the ield i aperture orre- 5 spods are the most eetive O the segmet 8 the most eetive is the solutio o 4 with properties arg s s arg s s arg s s arg s s The symmetri but omplex APD o the ield i aperture orrespods to it rom the aalysis o igure 7 ollows that the same eiiey o the sythesis a be ahieved o the brahig-o solutios at smaller sizes o aperture ad smaller values o parameters tha o the primary solutio The liear size o aperture a be dereased by the amout or at realizatio o brahig-o solutio Numerial examples o sythesis o give uelshaped amplitude DP deied i the domai by (9) are give i igures 8 ad 9 The brahig lies o solutios o (63) or this DP are show i igure 4 The give DP ad optimum sythesized DP are preseted i igures 8(a) ad (b) respetively at The optimum amplitude distributio o the ield i a aperture x y whih reates give i igure 8(b) the sythesized DP is show i igure 9 rom the aalysis o these igures we see that the symmetri amplitude DP (igure 8(b)) a be reated by dieret distributios o the ield i aperture o radiatig system iludig real ad osymmetri distributio (igure 9) (a) (b) igure 8 The presribed (а) ad sythesized (b) DP igure 7 The values o the utioal at ( s s) igure 9 Amplitude distributio o ield i aperture whih reates DP give i igure 8(b)

17 539 4 Sythesis o Disrete Radiatig Systems Atea Arrays (AA) The ivestigatios o oliear sythesis problems o liear ad plaar atea arrays (AA) aordig to the presribed amplitude DP are preseted partially i [ ] I the basis o ostrutio o mathematial models it is assumed [67] that the exitatio o eah radiator is haraterized by a sigle omplex umber I - omplex amplitude o exitatio the physial meaig o whih depeds o the type o radiatig system Takig ito aout the liearity o Maxwell s equatios the omplex amplitudes o exitatio eter liearly i the expressio or DP o array that is N I ikxsi osysisizos e (94) Here tor DP o -th radiator Vetor I I I i i is a ve- I N is alled the vetor o exitatio o array or vetor o amplitude-phase distributio o urrets i the array Suh ormulatio o DP o array is used i the sythesis problems with regard or mutual iluee o radiators [7 93] Thus the problem o idig the utios is redued to solutio o the orrespodig boudary problem o eletrodyamis i multiply oeted domais [4394] The method o itegral equatios [44] is used widely i suh lasses o problems The sythesis method o atea arrays with ylidrial dipoles with aout o mutual iluee is proposed i [934] Aalysis o ouiqueess problem o solutios is studied there by meas o omputatioal experimets I the problems o aalysis ad sythesis o atea arrays with may elemets is used simpliied mathematial model o AA [56] It is assumed [6] that AA osists o N idetial ad idetially orieted i spae radiators ad vetor DP o radiators are idetial or all R emitters ie N mula (94) or DP o lat AA takes the orm R R M N ix ms yms I e M m m Here s si os si or- (95) s si si si are the geeralized agular oordiates kdx si kdy si are dimesioless umerial parameters haraterizig the distae betwee the radiators ad the domai (solid agle) i whih the required amplitude DP s s is give Sie i (95) oly the seod multiplier depeds o the vetor APD o exitatio urrets i the array: M N ix ms yms e M m m AI I (96) oly the sythesis problem o ator o AA is osiderd utio s s is π -periodi by argumet s ad π -periodi by s We osider also (96) as the atio o the operator A rom a iite-dimesioal N spae H I (N is umber o radiators) ito the iite-dimesioal subspae o the spae H P where P is the domai orrespodig to the period o array Let the amplitude DP s s be give i the domai P ad o the set P is idetially \equal to zero The sythesis problem is to miimize the utioal [35] mi IHI P I AI (97) The basi o sythesis equatios o multiplier o AA have the orm iarg AI I A e (98) the equatio oerig APD o urrets i AA where A is ojugate with A operator ad iarg Q Q B Kar Q Q Q e dq (99) is equatio oerig o sythesized DP Here Q s s dq dsds ; Kar Q Q is the kerel the orm o whih depeds o the distributio o elemets i AA I partiular i the ase o a retagular array with umber o elemets N N M M the kerel Kar Q Q is writte as Kar Q Q sin ss sin s s () si s s si s s To id the possible brahig lies o solutios o (99) a liear homogeeous itegral equatio Q T ar Q K Q Q Q Q dq is obtaied where Q (99) P () is a primary solutio o ar is degeerate osequetly Equatio () is redued to the orrespodig homogeeous SLAE what i a speial ase o retagular array has the orm Note that the kerel K QQ M M kl kl mm m m M x a x k M M l M M () oeiiets o this system deped oliearly o the spetral parameters ad o the give amplitude

18 54 DP I [3565] the oditios are determied ad the existee theorem o oeted ompoets o the spetrum o () is proved To id the spetral lies the impliit utio method (8) ad (83) is used osider the umerial results o idig the solutios o the brahig lies i the sythesis problems o a plae equidistat atea array with radiators or two give i the domai s s: s s amplitude DPs s s osπs siπs (igure ) ad s s siπs si πs (igure ) whih are obtaied by solvig o () ad () The presribed ad sythesized amplitude DPs (with phase DP odd by argumet s ) at 5 5 are show i igure ad 3 respetively The amplitude ad phase distributios o urrets i the array o orrespodig sythesized DP are give i igure 4 rom the aalysis o this igure we see that osymmetri Y-diretio distributio o urrets i the array orms symmetrial amplitude DP igure The presribed DP igure The possible brahig lies o solutios o (99) s s os s si s or igure The possible brahig lies o solutios o (99) s s si s si s or igure 3 The sythesized DP 5 Numerial Methods o Solutio o the Basi Sythesis Equatios The above results show that the oliear sythesis problems aordig to the presribed amplitude DP ad give eergy DP have ouique solutios Appliatio o the methods o brahig theory o solutios to oliear itegral equatios allows to determie the quatity o existig solutios to id solutios i the irst approximatio ad to determie their quality harateristis To id the omplete solutios o these equatios umerial methods [ ] are applied The deied properties o solutios obtaied by aalytial ivestigatios make it possible to hoose the iitial approximatio havig the basi properties o the desired solutios ad they are plaed i ertai eighborhoods o omplete solu-

19 54 (a) (b) igure 4 The optimal amplitude (a) ad phase (b) distributios o urrets i the array tios oditioally the proess o umerial solutio o sythesis problem a be divided ito two stages The irst o them is desribed partially above ad it osists i idig the poits (lies) o brahig ad determiatio o types o existigs solutios depedig o the value o physial parameters The seod stage osists i solvig the basi sythesis equatios by iterative methods 5 Numerial Solutio o Sythesis Equatios orrespodig to utioal σ As a example o the salar problem we osider iterative proess o solvig the equatio o type (9) i the base o whih we put the suessive approximatios method [775] iarg e 3 AA (3) Obviously the suessive approximatios method (3) is equivalet to the ollowig iterative proess iarg e A (4) A 3 I [75] it is show that the sequees ad geerated by iterative proess (4) are relaxatioal or utioal Relaxatio properties o (4) states Theorem 5 The sequee is geerated by the iterative proess (4) it is relaxatio or utioal ad the values whih it takes o orm a overget umerial sequee ormulate also the properties o the operator B eterig i (4) that omplemet the properties o - 3 solutios o (4) preseted i Setio 3 Theorem 5 Noliear operator B deied by (4) ats i the spae o otiuous omplexvalued utios it is a ompat ad maps set S : M it ito itsel where M that is BS M max s K s s ; d s s (5) M SM rom the proved theorem ollows i partiular the ollowig at Sie the solutios o (4) are ixed poits o the operator B rom the relatio BSM SM ollows that all solutios o this equatio belog to the set BSM SM I additio is valid [75] orollary 5 I the sequee whih is geerated by the iterative proess (4) is miimizig or the utioal the rom a be seleted a subsequee k overgig uiormly to the miimum poit o the utioal Note that Theorem 5 ad orollary 5 are exteded to the ase o sythesis problem o a lat aperture with use o equatio o the type (63) 5 Numerial Solutio o Sythesis Equatios orrespodig to utioal I the base o ostrutio o iterative proesses o solvig the oliear operator equatios o the type () ad (3) we put impliit sheme o the suessive approximatios method [776] I a geeral ase the iterative proess o solutio o () has the orm E A A expiarg A A (6) where E is a idetity operator atig i the spae H L V The impliit sheme o iteratio proess or (3) with

20 54 respet to sythesized DP 3 has the orm similar to (6) E AA AA exp i arg (7) Note that the impliit shemes (6) ad (7) are haraterized by the at that liear operator equatio is solved o every iteratio step I additio the questio o solvability o (6) ad (7) appears to whih a positive aswer gives a theorem about the solvability o the utioal equatio o the seod kid o the type x Wx y (8) i the Baah spae X where W is a liear ompat operator [69] Theorem 53 [69] I order that (8) have the solutio at a arbitrary y X it is eessary ad suiietly that homogeeous equatio x Wx have a uique solutio (obviously that x ) or a sequee obtaied by (6) is valid Theorem 54 Let A: LV L be a ompletely otiuous operator be a otiuous real oegative utio i ad at there exists the iverse operator E A A i additio the dimesio o the spae o zeros NA The the sequee geerated by the iterative proess (6) is a miimizig or the utioal grad H A exp i arg A AA H i the spae H We deote the operator i right part o (6) as: or the operator D E AA A exp i arg A (9) () D is valid Lemma 5 Let A: H H be a ompletely otiuous operator The the operator D deied by () is ompat ad it trasers ay bouded set r : r H ito its relatively ompat part at A r H 3 ommet 5 Sie the set o otiuous utios i the domai is a dese everywhere i the spae L ad ompletio o spae o otiuous utios as the or- malized spae (with orm x xx ) oiides with the Hilbertia spae L [4] justiiatio o the overgee o (7) oerig equatio o the type () a be osidered i the Hilbertia spae H L ( or ) Thus it is show that there is true orollary 5 I grad is operator otiuous i some eighborhood H o the poit the rom Theorem 5 ad Lemma 5 ollows that the subsequee k overges to some solutio o () by the orm o the spae H i Depedet o the hoie o iitial approah the suessive approximatios (6) a overge to the solutios o various types [56-58] 53 Numerial Solutio o Sythesis Problems with se o the Eergy riterio irst we shall osider the iterative proess o solutio the equatio o type (34) i the Hilbertia spae H LV LV LV uder ertai restritios o the parameter This equatio is writte as * * E A N A A A A () where E: H H is a idetity operator A: H is ompletely otiuous operator We deote A H Note that the salar produt ad the orrespodig orm i the spae H H are deied by (4) ad (5) ad the hebyshev ad mea-square orms i the spae are itrodued by ormulas (6)-(8) Heeorth we shall osider ompletio o the spae relatively to the orm [4] whih is the Baah spae ad oiides with the Hilbertia spae H L L the orm i whih we shall deote by symbol L We assume that A : H H is a ompletely otiuous operator ad i the spae the domai o its values R A is a set o otiuous utios Takig ito aout the equality A we shall osider the expressio NA i () as a operator o multipliatio by the utio N : N N () atig i the spae H where N is real oegative otiuous utio o the ompat i additio N Obviously that () is a liear bouded operator ad N I H H * A N A the there exists the iverse operator * E A A N the orm o whih satisies the ie- quality [4]

21 543 * E A NA (3) * A NA I this ase Equatio () we shall write as D * * E A N A A A (4) Here we shall show that the solutio o (4) a be obtaied as a limit o suessive approximatios o the iterative proess [6]: t td (5) where t is some ixed umber with the iterval I additio suessive approximatios a overges to dieret solutios o (4) depedig o the hoie o the iitial approximatio To determie the oditios ad to justiy overgee o (5) we shall use the Theorem 4 with [6 p 68] aordig to whih: i oexpadig operator W overts a losed ovex set o stritly ovex Baah spae X ito its ompat part the suessive approximatios x tx t W x where t is ay ixed umber rom the iterval overges to some solutio o the equatio x x at some x Sie the Hilbertia spae H is stritly ovex Baah spae (see [6 p 67]) the to satisy o the oditios o this theorem oerig (4) it is suiietly to show that a losed ovex set S r exists i the spae H where the operator D is oexpadig ad ompletely otiuous I additio there is suh relatio D S S r r Satisiatio o these oditios results rom lemmas proved i [749] Lemma 5 Let A: H H be a liear ompletely otiuous operator ad the domai o its values * RA is a set o otiuous utios A NA The D is a oexpadig operator o S L V where r S r H ANA 3 r that is or ay is satisied D : r 3 A Sr the iequality D H H (6) (7) Lemma 53 Let A: H H be a liear ompletely otiuous operator ad the domai o its values * R A is a set o otiuous utios A N A The D : H H deied by (4) is a ompletely otiuous operator or whih the relatio D S S ( S is a losed ovex set deied by r r r (6)) is satisied 54 Numerial Solutio o Sythesis Problems with Optimizatio o eometry o Radiatig System I this setio we shall osider the sythesis problem o a lat aperture aordig to the presribed amplitude DP or the ase whe the orm o aperture ad amplitudephase distributio o the ield (urrets) i it is optimized simultaeously limitig by the ase o liear polarizatio [5677] We shall osider a speial ase whe the ield i the aperture is liearly polarized alog oe o the oordiate axes ad DP has oly oe ompoet We itrodue iside o aperture the polar oordiate system: x r os y rsi Let be a utio o the boudary o aperture S The DP s s whih is ormed by amplitude-phase distributio o the ield i the r is give by the ormula [76] aperture s s A gj π ikrsos ssi r e rdrd (8) j x; j y j Later o we omit the idex i deiitio o Let the give amplitude DP s s be dieret rom ideti- al zero i some limited losed domai ad it is idetially equal to zero at s s problem o simultaeous sythesis o the aperture shape S ad amplitude-phase distributio o the ield i it is osidered as the problem o idig the utios miimizig the utioal r ad \ The s s s s dsds π (9) s s dsds rdrd \ i whih the irst two summads desribe the measquare deviatio o modules o give ad sythesized DP s i spae ad the third oe imposes restritios o the square o aperture S We shall osider the parameter as a weight oeiiet We itrodue ito osideratio the ollowig utioal spaes: H L S is a spae o square itegrable omplex utios i the domai S H L π is a spae o square itegrable real u-

22 544 tios o the segmet π H L is a spae o square itegrable omplex utios i the domai Salar produts ad geerated by it orms we shall itrodue as ollows: π r r rdrd H H π d H H s s s s dsds H H () Takig ito aout the itrodued orms the last summad i (9) ad Parseval s equality have the orm π π rr dd d H π H H k O this base the utioal is preseted as: H H H π k () We shall osider the iterative proess o umerial miimizatio o () I it base we shall put the ideas similar as at miimizatio o utios o two variables by a oordiate deset method Let V be a miimum poit o the utioal ad V be a iitial approximatio hose rom some eighborhood o the poit V We shall deote by S the iitial shape o aperture that is de- sribed by the utio Substitute i () ad osider its restritio i the spae H : () rom the eessary oditio o the utioal miimum we obtai equatio o type (9) Numerially we solve it by suessive approximatios method give i pt 5: iarg Q Q k π Q K Q Q; k e dq (3) As a result we id the utio Q ad obtai the irst approximatio o the solutio by the ormula o type () We shall pass to idig the utio that desribes the boudary o aperture S We ix the utio extedig its aalytially aordig to () to the plae XOY i () ad osider the utioal whih depeds oly o the utio With the eessary miimum oditio: grad g g where g is a arbitrary elemet o the spae H we obtai the equatio B k π Q iarg Q i k Q e e dq (4) k π whih is a oliear utioal equatio with respet to the utio We shall id umerially solutios o (4) usig the Newto-Katorovih method [69]: where B B (5) (6) B is the partial rehet derivates o opera- tor B by the utio We assume that Equatio (5) is a liear itegral equatio o the orm L π M d (7) B whih equatio o the seod kid at L a be redued to the redholm L Solvig (7) we id the irst approximatio or the utio that desribes a boudary o aperture o the radiatig system otiuig idig i tur the approximatios o u- tios ad we obtai the sequee that is relaxatioal or (9) I more detail the problem o hoie o iitial approximatios ad justiiatio o relaxatio or utioal is give i [75657] irst we shall osider the umerial results o sythesis o lat aperture with optimizatio o its geometry I igures 5 ad 6 the examples o sythesis o amplitude DPs whih i ross setio have quasi-square ad

23 545 igure 5 Sythesis DP with square otour: (a) Sythesized DP with square otour; (b) The optimal shape o aperture quasi-triagular shapes are give The optimal shapes o apertures are give there too Note that the problems o suh lass arise i partiular at the sythesis o otour DPs o ixed ad variable orms or satellite atea systems eeded or uiorm irradiatio o a give territorial zoe rom the board o artiiial satellite where multi-beam ateas are used ote [77-8] I multibeam atea (MBA) has a radiatig aperture o irular shape ad partial beams i the ross setio have the shape o a irle ad ouiorm distributio o radiated eergy iside o the setio the o the jutio o three eighborig rays with a irular ross setio the so-alled ritial zoes (igure 7) with low level o radiated eergy our Oe o the possible ways o solutio o this problem is passage to alterative orms o apertures that o the base o the optimal APD will orm rays that have retagular triagular or hexagoal shapes ad lose to ostat (iside o otour) oeiiet o direted atio i retagular ross setio Obviously that o the base o suh partial beams it is easy to sythesize give summary DP without ritial igure 6 Sythesis o DP with triagular otour: (а) Sythesized DP with quasi-triagular otour; (b) Optimal shape o aperture ritial zoe igure 7 ritial zoes with low level o radiated eergy zoes Below the results o sythesis o triagle-beam otour DP with partial beams with irular (igure 8(a)) ad quasi-retagular (igure 8(b)) otours are preseted rom the aalysis o the igures we see that i the summary DP whih is obtaied o the base o quasisquare o otours ritial zoes are abset ad variatio o radiated eergy iside o the otour does ot exeed db