NEW SOLUTIONS TO A LINEAR ANTENNA SYNTHESIS PROBLEM ACCORDING TO THE GIVEN AMPLITUDE PATTERN

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1 UDK 519.6: M. Adriychu Pidstryhach Istitute for Appied Probems of Mechaics ad Mathematics, NASU CAD Departmet, Lviv Poytechic Natioa Uiversity NEW SOLUTIONS TO A LINEAR ANTENNA SYNTHESIS PROBLEM ACCORDING TO THE GIVEN AMPLITUDE PATTERN Adriychu M., 15 Отримано явну форму розв язку задачі амплітудно-фазового синтезу лінійної антени за заданою амплітудною діаграмою спрямованості. Наведено числові результати, які демонструють залежність точності одержаного розв язку від вхідних параметрів задачі. Ключові слова: амплітудно-фазовий синтез антен, амплітудна діаграма, явна форма розв язку. Cosed form of soutio to the probem of iear atea ampitude-phase sythesis accordig to the give ampitude patter is obtaied. The computatioa resuts, reated to depedece of the exactess of soutio o the probem s iput parameters, are discussed. Key words: atea ampitude-phase sythesis, ampitude patter, cosed form of soutio. Itroductio The big umber of papers (e.g. [1,, 4 8, 13]) is devoted to fidig the curret j( x ) if the radiatio patter (RP) ix f( ) = e j( x) dx : = Aj, (1) is give. Here > ad > are fixed umbers. The RP f( ) is a etire fuctio of, f( ) ce, C. By c > we deote various estimatio costats. Equatio (1) for j is a itegra equatio with compact operator A: L (, ) L (, ). The operator A is ijective, its rage R( A): = { f : Aj, j L (, )} is ot cosed. Thus, if fδ f L (, < δ, the f ) δ may be ot i R( A ). I this paper the foowig probem is discussed: Give h ( ), h( ) L (, ), ad δ >, fid j L (, ) such that h ( ) f( ) L (, < δ, δ ) () where f ix ( ) = e j ( x) dx. (3) δ This is a oiear probem. There was o cosed form soutio to this probem. The probem has bee discussed i [1], where the approach was based o a umerica soutio of a correspodig oiear miimizatio probem. Some theoretica aspects of atea sythesis probem were discussed i [4 11], [13]. Durig the ast decade, the approach, based o the poyomias represetatio of soutio to the oiear atea sythesis probem accordig to the prescribed ampitude ad power radiatio patters, was deveoped too (see, e. g., [14, 15]). δ 4

2 Cosed form soutio The approach proposed is quite differet from the oe i [1]. It reduces probem () (3) to a iear probem which is soved i cosed form. Mai theoretica resut was formuated i [16] as respective theorem. Deote ad where A soutio to probem ()-(3) is f = Fj : = e j( x) dx, ix j L (, ), j =, x >, 1 1 ix F f = e f( ) d. π (5) 1 1 jδ ( x) = π( F G ( δ) ( F h), (6) + p 1/ ix 1 p + ( ) = 1, = ( δ ), π 1 1 G e (7) 1 1 >, p > is fixed, ad = ( δ ) is chose so that G( shsds ) ( ) h ( ) < δ. (8) L (, ) The formuas (6), (7) give the expicit form of soutio to probem (), (3). The proposed method is vaid if the iear segmet (, ) is repaced by a mutidimesioa bouded domai D. I this case the origi has to be chose at the gravity ceter of D, that is, at the poit such that xdx =. I this case the fuctio G ( ) is + p i x 1 + p ( ) =, N 1, D D 1 G e dx c (9) where is the dot product of vectors, D is measure (voume) of D, N is the dimesio of the space, ad c, N is the ormaizig costat: im c 1 d = 1. (1) N, < 1 < 1 If D is smooth ad stricty covex the the Fourier trasform of the characteristic fuctio of D 1 N G ( ) = O L (R ) p if p > N. The iformatio reated to the rate of decay is O 1 ( ). Therefore of the Fourier trasform of the characteristic fuctio of a bouded domai D i R N exampe, i [3, 11]. D (4) is give, for Numerica modeig The umerica resuts reated to ivestigatio of the roe of the umber o the quaity of approximatio of the give RP h( ) are show i Tabe 1. The parameters of the probem are the foowig: =., = 3., 1 = + 1.5, p = 1.. The errors of the estimate (8) are give i the secod coum of Tabe 1. The mea-square deviatio (MSD) h( ) f( ), obtaied i the process of sovig the oiear sythesis probem by approach i [1], is preseted i the third coum, ad the square of the 43

3 orm j is give i the ast coum. Oe ca see that the vaue of ifueces strogy the accuracy of the approximatio of the desired diagram. I order to get the error δ of the approximatio which is ess tha 3 1 it is sufficiet to chose aroud 4. The quaity of the approximatio of the give RP h( ) by f( ) for sma is show i Fig. 1. The give RP h( ) is potted by thic soid ie. The modui f( ) for = 1,,5,1 are show with the thi ies, the respective currets j( x ) are show i Fig.. So, the error of estimate (8) for = 1 is equa to.95. The error of the approximatio decreases if icreases, ad its miima vaue i our computatios is equa to ad is attaied at = h(), f () h() =1 = =5 = / Fig. 1. The give h( ) = cos( π / ) ad the obtaied f ( ) RPs The vaues of the MSD, preseted i the third coum, are of the same order as the errors of estimate (8). For the give RP h( ), the icrease of the accuracy of the approximatio does ot force the growth of the orm j of the curret which woud cause practica difficuties (coum 4 i Tabe 1). The quaity of approximatio for h( x) = cos( π / ) at various Tabe 1 Est. (8) MSD jx ( )

4 .6 j(x) =1 = =5 =1.4. x Fig.. The currets jx ( ) for various, h( ) = cos( π / ) The quaity of approximatio to the give RP h ( ) = 1 is show i Tabe. For this h( ) the error of the approximatio is arger tha the error for h( ) = cos( π / ) at the same vaues of. The vaue of δ i estimate (8) at = 4 is two orders greater tha that for the RP h( ) = cos( π / ). Athough the error of estimate (8) ad the MSD is sma, but the differece of the shapes of h( ) ad f ( ) is visibe. I the four ast rows of Tabe the resuts are preseted for = 6. ad = 9.. The error of estimate (8) is amost the same as for = 3., but the vaue of the MSD is ower. This meas a improvemet of the approximatio to the give RP by the shape (compare the dash dot ad dot curves i Fig. 3). The mea-square deviatio at = 1 ad = 4 for = 9. is amost two times ess tha for = h(), f () 1..8 h().6 =3., =5 =3., = =3., =4 =9., =5 / Fig. 3. The give h ( ) = 1 ad the obtaied f ( ) RPs 45

5 The correspodig distributios of the curret j( x ) are show i Fig. 4. For arger the orm of the curret grows. This agrees with the umerica resuts i [1], amey the better approximatio of the give RP eads to the arger orm of the curret. 3. j(x) =3., =5 =3., = =3., =4 =9., =5. 1. x Fig. 4. The currets jx ( ) for severa parameters, h ( ) = 1 The umber of, which is sufficiet for obtaiig the desired error δ, is show i Tabe 3 for =., = 6., 1 = + 1.5, ad p = 1.. The resuts are preseted for the give RPs h ( ) = (cos( π / ) q, with q = 1, 4,8,16,3. To obtai a higher accuracy of approximatio of h it is ecessary to icrease the umber for a q. The quatity for the prescribed δ varies for differet q. The quaity of approximatio for hx ( ) = 1 at various Tabe Est, (8) MSD = 3, = 6, = 9, jx ( ) 1,559,5885,787,3654,371,8314 5,161,19,8967 1,1487,165,918,134,1338,9361 5,645,1141,9456 1,433,173,9458 4,5,11,9515 5,584,715 1,356 4,3,537 1,364 5,565,67 1,6575 4,195,38 1,

6 The power q i the fuctio (cos( / ) q π correspods to the RPs with differet widths at the eve h ( ) =.5. The vaue of h ( ) f( ) h ( is miima at q = 16. Aso, this f ( ) has the ) =.5 smaest side obes outside of the iterva i compariso with the vaues of the side obes at other vaues of q. Such RP is caed as optima [], ad it ca be created easier i compariso with other RPs. This eads to the miima vaue of which is ecessary to obtai the desired error δ. Number of ecessary to attai the give vaue of δ for various h( ) h( ) δ =,1 δ =,1 δ =,1 h( ) = (cos( π / ) = 4 = 86 = 3 8 h ( ) = (cos( π / )) = 15 = 5 = 4 16 h ( ) = (cos( π / )) = 1 = 44 = h ( ) = (cos( π / )) = 9 = 46 = h ( ) = (cos( π / )) = 13 = 55 = 18 Tabe 3 Cocusio The cosed form of soutio of the iear atea sythesis probem by the ampitude RP is give. The iitia oiear probem is reduced to a iear oe. The soutio of this iear probem is preseted i cosed form, see formua (). The umerica resuts demostrate the high accuracy of the proposed method. The approach ca be used aso for sovig the mutidimesia atea sythesis probems. Acowedgemet Author thas to Prof. Aexader G. Ramm, Kasas State Uiversity (USA), for creative discussios reated to topic of paper. Refereces 1. M. Adriychu Atea sythesis by ampitude radiatio patter / M. Adriychu, N. Voitovich, P. Saveo, V. Tach. K.: Nau. Duma, Baais C. A. Atea theory: aaysis ad desig / C. A. Baais. 3rd ed., Hoboe. New Jersey: Wiey-Itersciece, Iosevich A. Decay of the Fourier trasform: aaytic ad geometric aspects / A. Iosevich, E. Lifyad. Birhäuser, Base, Miovich B. Theory of atea sythesis / B. Miovich, V. Yaovev. M.: Sov. Radio, Ramm A. G. Optima soutio of the probem of iear atea sythesis / A. G. Ramm // Soviet Physics Doady Vo. 13, Ramm A. G. O atea sythesis theory / A. G. Ramm // Coectio "Ateas". M.: Izd. Svjaz, Ramm A. G. Some oiear probems i the theory of atea sythesis / A. G. Ramm // Soviet Physics Doady Vo. 14, Ramm A. G. Optima soutio of the iear atea sythesis probem / A. G. Ramm // Radiofisia Ramm A. G. Approximatio by etire fuctios / A. G. Ramm // Mathematics, Izv. Vusov Ramm A. G. Siga estimatio from icompete data / A. G. Ramm // J. Math. Aa. App Ramm A. G. The Rado trasform ad oca tomography / A. G. Ramm, A. I. Katsevich // CRC Press, Boca Rato Rudi W. Rea ad compex aaysis / W. Rudi // McGraw Hi. New Yor, Stutzma W. Atea theory ad desig / W. Stutzma, G. Thiee, Wiey, Lodo, Voitovich N. N. Atea sythesis by ampitude radiatio patter ad modified phase probem, i boo B. Z. Katseeebaum / N. N. Voitovich // Eectromagetiv fieds: restrictio ad approximatio, Wiey-WCH, Beri,. 15. Buatsy O. O. Phase optomizatio probems appicatio i wave fied theory / O. O. Buatsy, B. Z. Katseeebaum, Y. P. Topoyu, N. N. Voitovich, Wiey-WCH, Beri, Ramm A. G. Atea sythesis by the moduus of the diagram / A. G. Ramm, M. I. Adriychu. (to be pubished). 47