"Applied Science, Faculty of"@en . "Electrical and Computer Engineering, Department of"@en . "DSpace"@en . "UBCV"@en . "Martin, Alan Desmond"@en . "2011-07-26T19:06:28Z"@en . "1967"@en . "Master of Applied Science - MASc"@en . "University of British Columbia"@en . "A study is made of some of the existing techniques for determining the positions and excitations of the elements in an array, in order to produce a desired radiation pattern. A new method based on numerical quadrature is proposed. Equal-sidelobe patterns are synthesized and a comparison of the methods is made on this basis. Results indicate that the proposed quadrature method is an improvement over the other tested methods in two ways, (i) it produces a more accurately synthesized pattern and (ii) it is simple to compute. It is therefore useful for synthesizing arrays containing\r\nlarge numbers of elements.\r\nHowever, for the particular radiation pattern selected,\r\nnone of the methods gave an improvement over the corresponding\r\nuniformly-spaced array."@en . "https://circle.library.ubc.ca/rest/handle/2429/36310?expand=metadata"@en . "SYNTHESIS METHODS FOR ANTENNA ARRAYS WITH NON-UNIFORMLY-SPACED ELEMENTS by ALAN DESMOND MARTIN B.Sc.(Hons.), U n i v e r s i t y of Leeds, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n t he Department o f E l e c t r i c a l E n g i n e e r i n g We ac c e p t t h i s t h e s i s as con f o r m i n g t o the r e q u i r e d s t a n d a r d Research S u p e r v i s o r Members o f t h e Committee Head of the Department Members of the Department of E l e c t r i c a l E n g i n e e r i n g THE UNIVERSITY OF BRITISH COLUMBIA September, 1967 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t u d y . | f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Depar tment o r by h.iJs r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f 0 e o A ^ C ^ / ^ / A ^ A / A ^ The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada Date ABSTRACT A study i s made of some of the e x i s t i n g t e c h n i q u e s f o r d e t e r m i n i n g the p o s i t i o n s and e x c i t a t i o n s of the elements i n an a r r a y , i n o r d e r t o produce a d e s i r e d r a d i a t i o n p a t t e r n . A new method based on n u m e r i c a l q u a d r a t u r e i s proposed. E q u a l -s i d e l o b e p a t t e r n s a re s y n t h e s i z e d and a comparison of the methods i s made on t h i s b a s i s . R e s u l t s i n d i c a t e t h a t the proposed q u a d r a t u r e method i s an improvement over the o t h e r t e s t e d methods i n two ways, ( i ) i t produces a more a c c u r a t e l y s y n t h e s i z e d p a t t e r n and ( i i ) i t i s s i m p l e t o compute. I t i s t h e r e f o r e u s e f u l f o r s y n t h e s i z i n g a r r a y s con-t a i n i n g . l a r g e numbers of elements. However, f o r the p a r t i c u l a r r a d i a t i o n p a t t e r n s e l e c t -ed, none of the methods gave an improvement over the c o r r e s -ponding u n i f o r m l y - s p a c e d a r r a y . i i TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS v LIST OF SYMBOLS v i i ACKNOWLEDGEMENT i x 1. INTRODUCTION 1 1.1 Beam Scanning 2 1 . 2 P r e v i o u s Work 5 1 . 3 S y n t h e s i s of U n i f o r m l y Spaced A r r a y s .... 7 1 .3.1 S c h e l k u n o f f ' s Method 8 1 .3.2 F o u r i e r S e r i e s Method 10 2. THEORY OF THE SYNTHESIS METHODS 12 2.1 Methods Based on the F o u r i e r S e r i e s R e p r e s e n t a t i o n of the D e s i r e d P a t t e r n ... 13 2.1.1 Unz's Method : 13 2.2 Methods Based on the L i n e Source 14 2.2.1 Ishimaru's Method 14 2.2.2 M a f f e t t ' s Method 18 2 . 3 Methods Based on the U n i f o r m - A r r a y Rep-r e s e n t a t i o n of the D e s i r e d P a t t e r n 19 2 .3 .1 H a r r i n g t o n ' s Method 20 2.3-2 W i l l e y ' s Method 21 3. ARRAY SYNTHESIS USING NUMERICAL QUADRATURE ... 23 3.1 I n t r o d u c t i o n 23 3.2 Quadrature Theory 23 3.3 Gauss-Chebychev Quadrature 25 3.4 A p p l i c a t i o n of Gauss-Chebychev Quad-r a t u r e t o A r r a y S y n t h e s i s 26 4. RESULTS 29 i i i Page 4.1 I n t r o d u c t i o n 29 4.2 Choice of P a t t e r n t o be S y n t h e s i z e d ...... 29 4.3 Unz 's Method 30 4.4 Ishimaru's Method 35 4.5 M a f f e t t ' s Method 40 4.6 H a r r i n g t o n ' s Method 44 4.7 W i l l e y ' s Method 48 4.8 The Quadrature Method 48 4.9 Summary of R e s u l t s 50 4.10 Comparison w i t h U n i f o r m l y - S p a c e d A r r a y ... 55 5. CONCLUSIONS 59 APPENDIX The T a y l o r L i n e Source 61 REFERENCES . 64 i v LIST OF ILLUSTRATIONS A r r a y Geometry Complex Plane f o r 6 Element U n i f o r m l y Spaced A r r a y T y p i c a l Source Number F u n c t i o n Example of Unz's Method A r r a y S y n t h e s i z e d by Unz's Method Unz S y n t h e s i s of P(0) = 32 cos 4 0 U n i f o r m l y - S p a c e d B i n o m i a l A r r a y Non-Uniformly Spaced S y n t h e s i z e d A r r a y . Unz B i n o m i a l P a t t e r n S y n t h e s i s Ishimaru's S y n t h e s i s of 15 dB E q u a l -S i d e l o b e P a t t e r n 15 Element A r r a y by Ishimaru's Method .. 21 Element A r r a y by Ishimaru's Method .. A r r a y C o n f i g u r a t i o n f o r Ishimaru's Method M a f f e t t ' s S y n t h e s i s of 15 dB S i d e l o b e P a t t e r n 15 Element A r r a y by M a f f e t t ' s Method 21 Element A r r a y by M a f f e t t ' s Method ... H a r r i n g t o n ' s S y n t h e s i s of 15 Element A r r a y Impulse F u n c t i o n f o r S i d e l o b e R e d u c t i o n P e r t u r b a t i o n Parameters R e d u c t i o n of S i d e l o b e s by H a r r i n g t o n ' s Method W i l l e y ' s S y n t h e s i s of B i n o m i a l P a t t e r n . v F i g u r e Page 4.17 Quadrature Method S y n t h e s i s of 15 dB S i d e l o b e P a t t e r n 51 4.18 15 Element A r r a y by Quadrature Method . 52 4 .19 21 Element A r r a y by Quadrature Method . 53 4 .20 P r o p e r t i e s of the Methods 54 4 .21 Dolph-Chebychev A r r a y Pattern,- 21 Elements w i t h 15 dB S i d e l o b e L e v e l .... 56 4 .22 Dolph-Chebychev A r r a y P a t t e r n , x 11 Elements w i t h 15 dB S i d e l o b e s 57 v l LIST OF PRINCIPAL SYMBOLS c o n s t a n t s a n o r m a l i z i n g c o n s t a n t n o r m a l i z e d c u r r e n t c o n s t a n t s complex F o u r i e r c o e f f i c i e n t s a c o n s t a n t s p a c i n g the n o r m a l i z e d e l e c t r i c f i e l d s t r e n g t h p a t t e r n of a n o n - u n i f o r m l y spaced a r r a y the n o r m a l i z e d d e s i r e d p a t t e r n a n o r m a l i z e d c u r r e n t d i s t r i b u t i o n = g(x)/w(x) a w e i g h t i n g f a c t o r a n o r m a l i z e d c u r r e n t B e s s e l ' s f u n c t i o n of o r d e r m and argument x an i n t e g e r v a r i a b l e a l i n e s ource a p e r t u r e l e n g t h an i n t e g e r v a r i a b l e ( N - l ) / 2 , f o r odd N or N/2 f o r even N an i n t e g e r v a r i a b l e the number of elements i n an a r r a y or as d e f i n e d a n o r m a l i z e d c u r r e n t d e n s i t y t h an n o r d e r p o l y n o m i a l i n x = (cos 0 - cos 9 ), u n l e s s o t h e r w i s e d e f i n e d o a dummy v a r i a b l e of i n t e g r a t i o n a w e i g h t i n g f u n c t i o n the a p e r t u r e p o s i t i o n v a r i a b l e v i i the p o s i t i o n of the n element the source number f u n c t i o n = exp( j (3d s i n 0) an a p e r t u r e phase c o n s t a n t = 2%/\ the u n i t impulse f u n c t i o n a r e s i d u a l e r r o r term the f r a c t i o n a l change from u n i f o r m s p a c i n g the a n g l e between the l i n e of the a r r a y and a g i v e n d i r e c t i w a velength +K the i moment of the e x c i t a t i o n f u n c t i o n a dummy v a r i a b l e of i n t e g r a t i o n the complement of 9 = (3d cos9 v i i i ACKNOWLEDGEMENT G r a t e f u l acknowledgement i s g i v e n t o the N a t i o n a l R e s e a r c h C o u n c i l of Canada f o r f i n a n c i a l support r e c e i v e d under B l o c k Term Grant A68 and 67-3295 from 1965 t o 1967. The a u t h o r would l i k e t o express h i s a p p r e c i a t i o n t o h i s s u p e r v i s o r , P r o f e s s o r P.K. Bowers, f o r guidance throughout the course of t h i s work. Thanks a re due t o Dr. E.V. Bohn f o r r e a d i n g the manus c r i p t and t o Dr. M.M.Z. K h a r a d l y f o r many h e l p f u l s u g g e s t i o n s . The a u t h o r i s a l s o i n d e b t e d t o Mr. J.E. Lewis and Mr. A. Deczky f o r p r o o f - r e a d i n g and t o Mrs. M. Wein f o r t y p i n g the t h e s i s . IX 1. INTRODUCTION 1 Antenna systems composed of a f a m i l y of i d e n t i c a l i n d i v i -d u a l r a d i a t o r s are termed a r r a y s . The f i e l d produced by an antenna system as a f u n c t i o n of some s p a t i a l v a r i a b l e , g e n e r a l l y i n one of s e v e r a l p r i n c i p a l p l a n e s , i s c a l l e d the r a d i a t i o n p a t -t e r n , though t h i s name i s a l s o g i v e n t o the curve r e l a t i n g the power r a d i a t e d and the s p a t i a l v a r i a b l e . The main f a c t o r s i n -f l u e n c i n g the r a d i a t i o n p a t t e r n s of such a system are the type of r a d i a t o r s , the d i s t r i b u t i o n and o r i e n t a t i o n of r a d i a t o r s and t h e i r c u r r e n t e x c i t a t i o n i n am p l i t u d e and phase. The t h e o r y of two and t h r e e - d i m e n s i o n a l a r r a y s f o l l o w s r e a d i l y from one-d i m e n s i o n a l a r r a y t h e o r y by the p r i n c i p l e of p a t t e r n m u l t i p l i c -a t i o n and so o n l y l i n e a r a r r a y t h e o r y w i l l be d i s c u s s e d h e r e . Antenna s y n t h e s i s i s the problem of d e t e r m i n i n g the p a r a -meters of an antenna system t h a t w i l l produce a r a d i a t i o n p a t -t e r n which a c c u r a t e l y approximates some d e s i r e d p a t t e r n . F o r the l i n e a r - a r r a y s y n t h e s i s of a g i v e n p a t t e r n , i t i s r e q u i r e d to f i n d , i n . g e n e r a l , the p o s i t i o n s of the r a d i a t o r s a l o n g a l i n e and t h e i r c u r r e n t e x c i t a t i o n s i n magnitude and phase. T h i s p r e s e n t s g r e a t d i f f i c u l t y , however. P r e s e n t p r a c t i c e i s to s i m p l i f y the problem by making two assumptions, ( i ) t h a t the element s p a c i n g i s u n i f o r m and ( i i ) t h a t the phase of the c u r r e n t e x c i t a t i o n i s not a d e s i g n v a r i a b l e . T h i s reduces the parameters t o the magnitudes of the c u r r e n t e x c i t a t i o n s and the co n s t a n t s p a c i n g . The purpose of t h i s work i s ( i ) t o st u d y some of the r e c e n t s y n t h e s i s methods t h a t have the non-uniform 2 s p a c i n g of the elements i n c l u d e d i n the d e s i g n parameters, ( i i ) t o e v a l u a t e the r e l a t i v e u s e f u l n e s s o f these methods and ( i i i ) t o propose a new s y n t h e s i s method based on n u m e r i c a l q u a d r a t u r e . The r a d i a t i o n p a t t e r n of an antenna system depends on the i n d i v i d u a l p a t t e r n s of the elements composing the system. I n the case of a l i n e a r a r r a y the p r i n c i p l e o f p a t t e r n m u l t i p l i c -a t i o n a p p l i e s and so the system p a t t e r n i s the p r o duct of the element p a t t e r n and the a r r a y f a c t o r . T h i s l a t t e r i s the p a t t e r n of a s i m i l a r a r r a y c o n t a i n i n g i s o t r o p i c elements, and i s n e a r l y always the more d i r e c t i v e p a t t e r n . As t h e r e i s no c o n c e p t u a l d i f f i c u l t y i n t a l c i n g the element p a t t e r n i n t o account,, a r r a y t h e o r y i s g e n e r a l l y based on i s o t r o p i c s o u r c e s . Throughout t h i s work the e f f e c t s of mutual c o u p l i n g between the elements of the a r r a y are not c o n s i d e r e d . Some p work on t h i s t o p i c has been done by A l l e n . 1.1 Beam Scanning Antenna r a d i a t i o n p a t t e r n s c o n s i s t i n g e n e r a l , of a main beam and a number of s m a l l e r beams, the s i d e l o b e s . In many a p p l i c a t i o n s s p a t i a l s c a n n i n g of the main beam i s r e q u i r e d . Hhis i s a c h i e v e d by two main t e c h n i q u e s , m e c h a n i c a l and e l e c -t r o n i c s c a n n i n g . The former r e q u i r e s no e l a b o r a t i o n . E l e c -t r o n i c beam s c a n n i n g i n a r r a y s i s r e a l i z e d by a r r a n g i n g t h a t c o n t r i b u t i o n s from the elements are a l l i n phase i n a d i r e c t i o n o t h e r than b r o a d s i d e or e n d f i r e . A u n i f o r m p r o -g r e s s i v e phase s h i f t a l o n g the a p e r t u r e w i l l produce t h i s e f f e c t . When a l l elements of a uniformly-spaced, a r r a y are i n phase, 3 the a r r a y f a c t o r i s ' N E (e) = Y_ I e n j |3ndcos9 1 . 1 n=0 where I i s the magnitude of the c u r r e n t e x c i t a t i o n n \u00C2\u00B0 t h of the n element, d i s the c o n s t a n t s p a c i n g of the elements, 9 i s the a n g l e from the l i n e of the a r r a y . N+l i s the number of elements .. 2JC P = , X b e i n g the wavelength B r o a d s i d e D i r e c t i o n E n d f i r e D i r e c t i o n A p e r t u r e F i g . 1 . 1 A r r a y Geometry t h I f the phase of the n element i s now -na where a i s a c o n s t a n t , then the p a t t e r n becomes N ( 9 ) = \ j e j n ( p d c o s 0 - a) E . n=0 I f 9 Q i s the angle between the a r r a y and the d i r e c t i o n of the main beam, then (3d cos 9 = a o N hence E(9) = Y\" I e ^ d u 1.2 n n=0 where u = cos 9 - cos 9 o The \" v i s i b l e \" r e g i o n of the p a t t e r n l i e s between 0 = + ^ . Eor the b r o a d s i d e d i r e c t i o n t h i s corresponds t o v a l u e s of u i n the range 1 > u ^ -1. The r e g i o n h a v i n g v a l u e s of u i n the range 1 < u 4 2 i s termed the \" s c a n n i n g \" r e g i o n , s i n c e , when the main beam i s scanned from b r o a d s i d e t o e n d f i r e , s i d e -l o b e s from t h i s r e g i o n move i n t o v i s i b l e space. Hence i f an a r r a y i s t o have beam s c a n n i n g out to e n d f i r e , then the s i d e l o b e l e v e l ( d e f i n e d as the h e i g h t of the h i g h e s t s i d e l o b e ) must be c o n t r o l l e d f o r a l l u v a l u e s i n the range -2 4 u 4- 2. I f no s c a n n i n g i s r e q u i r e d then o n l y the s i d e l o b e l e v e l f o r u v a l u e s between -1 and 1 need be c o n t r o l l e d . Thus s c a n n i n g r e q u i r e m e n t s need t o be taken i n t o c o n s i d e r a t i o n when p a t t e r n s y n t h e s i s i s b e i n g attempted. 5 1o 2 P r e v i o u s Work With the advent of r a d i o communications, the Importance of d i r e c t i v e antennas was r e a l i z e d . I t was not u n t i l 1937 however, when W o l f f ^ showed how t o determine the r a d i a t i n g system t h a t would produce a s p e c i f i e d d i r e c t i v e c h a r a c t e r i s t i c , t h a t the f i r s t s i g n i f i c a n t s t e p was t a k e n . I n 1943 S c h e l k u n o f f p u b l i s h e d h i s c l a s s i c paper on the mathematical t h e o r y of l i n e a r a r r a y s i n which he put f o r w a r d a s y n t h e s i s , t e c h n i q u e of a c o m p l e t e l y d i f f e r e n t n a t u r e from t h a t of W o l f f . He a l s o demonstrated t h a t the e f f e c t of u s i n g d i f f e r e n t c u r r e n t e x c i t a t i o n s on elements i n an a r r a y was t o a l t e r b oth the w i d t h of the main beam and the l e v e l of the s i d e l o b e s . The next l o g i c a l s t e p was the o p t i m i z a t i o n of these two parameters, a problem overcome by Dolph^ who u t i l i z e d the e q u a l - r i p p l e p r o p e r t y of Chebychev p o l y n o m i a l s . Dolph (1946) showed t h a t the minimum beam w i d t h f o r a g i v e n s i d e l o b e l e v e l i s o b t a i n e d when a l l the s i d e l o b e s a re of eq u a l h e i g h t , and a l s o gave a method f o r d e t e r m i n i n g the element e x c i t a t i o n s r e q u i r e d t o a t t a i n t h i s optimum. 7 In the decade t h a t f o l l o w e d Woodward and Woodward and Q Lawson produced a s y n t h e s i s method based on the a d d i t i o n of s i n \u00C2\u00A9 terms of the \u00E2\u0080\u0094 Q t y p e , each term h a v i n g i t s main beam i n a d i f f e r e n t d i r e c t i o n . T h i s proved to be a u s e f u l t e c h n i q u e . Work has a l s o been done on a p e r t u r e s c o n t a i n i n g c o n t i n u o u s c u r r e n t s o u r c e s . T a y l o r (1955) showed how t o determine the co n t i n u o u s -c u r r e n t e x c i t a t i o n f u n c t i o n which would have the o p t i m a l 6 p r o p e r t y as d e f i n e d by Dolph. R e a l i z a b i l i t y l i m i t a t i o n s p r e v e n t e d an i d e a l s o l u t i o n but T a y l o r showed how the i d e a l e q u a l - s i d e l o b e p a t t e r n c o u l d be o b t a i n e d a r b i t r a r i l y c l o s e l y . P r e v i o u s t o 1956 a l l d i s c r e t e a r r a y s were assumed t o c o n s i s t of u n i f o r m l y - s p a c e d elements, but i n t h a t y e a r Unz\"^ proposed t h a t the use of a r b i t r a r i l y p o s i t i o n e d elements would be advantageous s i n c e i t g i v e s the a r r a y d e s i g n e r an e x t r a degree of freedom, namely, the element p o s i t i o n s . I n view of t h i s e x t r a degree of freedom Unz suggested t h a t the n o n - u n i f o r m l y - s p a c e d a r r a y needs, i n g e n e r a l , fewer elements i n o r d e r t o a c h i e v e the same performance as an a r r a y w i t h u n i f o r m l y - s p a c e d elements. A l s o i n t h i s paper Unz proposed a s y n t h e s i s method f o r n o n - u n i f o r m l y - s p a c e d a r r a y s based on a P o u r i e r - B e s s e l e x p a n s i o n . I n I960, K i n g et al?\"\"*\" d i d some n u m e r i c a l s t u d i e s on e m p i r i c a l l y d esigned non-u n i f o r m l y - s p a c e d a r r a y s and demonstrated t h a t some advantages c o u l d be o b t a i n e d over u n i f o r m l y - s p a c e d a r r a y s w i t h r e s p e c t t o the number of elements r e q u i r e d and the s c a n n i n g p r o p e r t i e s . 12 Andreasen (1962) has done s i m i l a r s t u d i e s u s i n g b oth a n a l o g and d i g i t a l computer t e c h n i q u e s and has produced a r r a y s w i t h beam s c a n n i n g over wide a n g l e s , a modest s i d e l o b e l e v e l and a s i g n i f i c a n t r e d u c t i o n i n the number of elements. An o p t i m a l d e s i g n procedure put f o r w a r d i n Andreasen's paper was i n -13 d ependently a p p l i e d by Lo , who p o i n t e d out t h a t , a t b e s t , the s o l u t i o n i s o n l y l o c a l l y optimum. T h i s t e c h n i q u e was, however, used i n the d e s i g n of the U n i v e r s i t y of I l l i n o i s r a d i o telescope\"'\" 4 which appears t o be the o n l y p r a c t i c a l 7 a p p l i c a t i o n of n o n - u n i f o r m l y - s p a c e d a r r a y s t o date. F u r t h e r work on the s y n t h e s i s problem was done by H a r r i n g t o n \" ^ (1961) who used a p e r t u r b a t i o n approach by assuming t h a t the element p o s i t i o n s of the n o n - u n i f o r m l y -spaced a r r a y were not f a r removed from those of some u n i f o r m l y - s p a c e d a r r a y . S a n d l e r ( i 9 6 0 ) and W i l l e y (1962) a l s o conducted t h e i r s y n t h e s i s procedures from the b a s i s of a r e f e r e n c e p a t t e r n produced by a u n i f o r m l y - s p a c e d a r r a y . 18 19 I s h i m a r u , M a f f e t t and o t h e r s c o n s i d e r e d a c o n t i n u o u s -c u r r e n t d i s t r i b u t i o n t o be the source of t h e i r r e f e r e n c e p a t t e r n and d i r e c t e d t h e i r a n a l y t i c a l s y n t h e s i s procedures a c c o r d i n g l y . A g r e a t d e a l of work has been done t h a t i s not 20 r e f e r r e d t o above. N o t a b l y , Lo (a p r o b a b i l i s t i c a p p r o a c h ) , 21 22 23 P o k r o v s k i i et a l . (an \" o p t i m a l \" t h e o r y ) , Unz , Ma and 24 Y a n p o l s k i i ( s y n t h e s i s methods) have c o n t r i b u t e d t o the t h e o r y of the s u b j e c t . B e f o r e d i s c u s s i n g the t h e o r y of some of the non-u n i f o r m l y - s p a c e d a r r a y s y n t h e s i s methods, the s i m p l e r problem of s y n t h e s i z i n g u n i f o r m l y - s p a c e d a r r a y s w i l l be d i s c u s s e d . 1.3 S y n t h e s i s of U n i f o r m l y - S p a c e d A r r a y s There are two b a s i c methods f o r s y n t h e s i z i n g u n i f o r m l y -spaced a r r a y s . The f i r s t , due t o S c h e l k u n o f f , i s based on the f a c t t h a t the space f a c t o r of a l i n e a r a r r a y i s c h a r a c t e r -i z e d c o m p l e t e l y by an a s s o c i a t e d p o l y n o m i a l i n a complex 8 v a r i a b l e , whose range i s the u n i t c i r c l e . The second u t i l i z e s the F o u r i e r s e r i e s r e p r e s e n t a t i o n of the space f a c t o r . / 1.3.1 S c h e l k u n o f f ' s Method The space f a c t o r of a u n i f o r m l y - s p a c e d a r r a y of N elements i s g i v e n by N-1 E(0) = ] T I \u00C2\u00B1 e ^ \u00C2\u00B1 a \u00C2\u00B1 n 0 1=0 where the c o e f f i c i e n t s 1^ may be complex t o a l l o w f o r the p h a s i n g of the elements. A ipdsin0 l e t z = e* N-1 t h e n El. i = 0 i . e . we have expressed the space f a c t o r as a p o l y n o m i a l i n a complex v a r i a b l e z. T h i s p o l y n o m i a l has N-1 r o o t s and can t h e r e f o r e be expressed i n n o r m a l i z e d form, as |E| = |z - z-J | z - z 2 | |z - z N _ x | The l o c u s of z c o r r e s p o n d i n g t o r e a l space i s t h a t p a r t of the u n i t c i r c l e g i v e n by z v a l u e s r a n g i n g from z = 1 t o z = e \u00C2\u00B1 ^ d . The s y n t h e s i s problem then becomes the q u e s t i o n of the l o c a t i o n of the r o o t s of the p o l y n o m i a l i n the complex p l a n e . The z e r o s f o r a u n i f o r m b r o a d s i d e a r r a y as shown P i g . 1.2 Complex Pla n e f o r 6 Element U n i f o r m l y - S p a c e d A r r a y As P moves round the u n i t c i r c l e , the space f a c t o r has n u l l s when P passes through each r o o t and a maximum when P i s a t A. A corresponds t o the c e n t r e of the main beam and the r e g i o n s between the r o o t s c o r r e s p o n d t o the s i d e l o b e s . The s e p a r a t i o n of P^ and P^ i s a measure of the w i d t h of the main beam, so f o r a narrow beam these s h o u l d be c l o s e t o -g e t h e r . However f o r low s i d e l o b e s the r e m a i n i n g r o o t s s h o u l d be s i t u a t e d c l o s e t o g e t h e r , so from the above diagram i t can be seen t h a t by c l u s t e r i n g the r o o t s towards the L.H. plane the s i d e - l o b e l e v e l w i l l be reduced but the main beam w i d t h w i l l be i n c r e a s e d . 10 I n a c l a s s i c paper, Dolph^ d e r i v e d the e x c i t a t i o n f u n c t i o n which y i e l d s the optimum r e l a t i o n s h i p between beam-w i d t h and s i d e - l o b e l e v e l , and t h i s has been the b a s i s f o r many d e s i g n s i n the pa s t twenty y e a r s . 1.3.2 F o u r i e r S e r i e s Method C o n s i d e r an a r r a y of N = 2M + 1 elements; i t s space f a c t o r i s E = Y_ U 1 1.3 2M Ii=0 M D i v i d i n g t hrough by z , which l e a v e s |E| unchanged, I E I = I + I ^ * 1 . . . ! ^ \" I 1.4 T a k i n g the phase r e f e r e n c e a t the c e n t r e of the a r r a y and assuming a p r o g r e s s i v e phase s h i f t from one end of the a r r a y , i t can be seen t h a t s y m m e t r i c a l l y c o r r e s p o n d i n g elements have complex c o n j u g a t e e x c i t a t i o n s , e n a b l i n g p a i r s of terms of eq u a t i o n (1.4) to be added: T \u00E2\u0080\u0094k -p k / k -k\ ., / k -k\ JM-k z + IM+k z = a k ( z + z } + J b k ( z \" z } = 2 a k cos k l j / - 2 b k s i n k^ /\" (where z k=e^ k^) a M El = 2{~1 + X] [ a k c o s k y + ( - \ ) s i n k \ | / ] | 1.5 ^ k=l 11 Any r a d i a t i o n p a t t e r n , f(\"\j/\"), may be expanded as a F o u r i e r s e r i e s . Thus by e q u a t i n g terms i n f ( l ^ ) w i t h those i n e q u a t i o n (1.5) the e x c i t a t i o n d i s t r i b u t i o n r e q u i r e d t o approximate the r a d i a t i o n p a t t e r n can be found. The F o u r i e r s e r i e s r e p r e s e n t a t i o n of the d e s i r e d p a t t e r n , as used h e r e , i s a l s o the b a s i s of the f i r s t of the methods of s y n t h e s i z i n g n o n - u n i f o r m l y spaced a r r a y s , as d e s c r i b e d i n the f o l l o w i n g c h a p t e r . 2. THEORY OF THE SYNTHESIS METHODS 12 In i t s g e n e r a l form, the s y n t h e s i s problem i s the d e t e r m i n a t i o n of both the complex e x c i t a t i o n s and the element p o s i t i o n s y i e l d i n g an a r r a y f a c t o r , E(0), which a c c u r a t e l y approximates the d e s i r e d p a t t e r n , F(0). The approach to the problem i s determined m a i n l y by the ma t h e m a t i c a l form of the s p e c i f i e d p a t t e r n , F(0), and i t i s t h i s form which - w i l l be used t o c h a r a c t e r i z e the t h r e e b a s i c types of.methods. For an antenna h a v i n g a co n t i n u o u s c u r r e n t d i s t r i -b u t i o n over i t s a p e r t u r e , the f a r f i e l d r a d i a t i o n p a t t e r n i s 25 r e a d i l y o b t a i n e d from the F o u r i e r t r a n s f o r m r e l a t i o n s h i p and as a r e s u l t , many co n t i n u o u s c u r r e n t d i s t r i b u t i o n s y i e l d i n g u s e f u l p a t t e r n s a re known. The f a r f i e l d p a t t e r n of such a con t i n u o u s d i s t r i b u t i o n may be the p a t t e r n r e q u i r e d to-be s y n t h e s i z e d . I n t h i s case we have F(0) = | 2 g ( x ) e ^ x s i n 0 dx 2.1 x l where x r e p r e s e n t s a p o s i t i o n i n the a p e r t u r e , the l i m i t s of the a p e r t u r e b e i n g x-^ and TL^. A l t e r n a t i v e l y , the d e s i r e d p a t t e r n may be g i v e n as t h a t o f a u n i f o r m l y spaced a r r a y , i n the form N F(0) = Y_ I n e ^ n d s i n 0 2.2 n=0 13 A t h i r d form f o r .F(0) i s the complex F o u r i e r s e r i e s whose c o e f f i c i e n t s can be o b t a i n e d f o r any p e r i o d i c f u n c t i o n : M m= -M S y n t h e s i s r e q u i r e s the matching of one of these t h r e e e x p r e s s i o n s t o t h a t of a n o n - u n i f o r m l y spaced a r r a y , ^ \u00E2\u0080\u0094 jBx sm0 E(0) = Z_ I e n 2.4 n=0 n t h where x i s the p o s i t i o n of the n element. n r \u00E2\u0080\u00A2 2.1 Methods Based on the F o u r i e r S e r i e s R e p r e s e n t a t i o n of the D e s i r e d P a t t e r n 1 r~ . ( 2.1.1 Unz's Method The f i r s t n o n - u n i f o r m l y - s p a c e d a r r a y s y n t h e s i s technique was put f o r w a r d by Unz i n 1956 and u t i l i z e s a F o u r i e r - B e s s e l e x p a n s ion t o match e q u a t i o n s (2.3) and ( 2 . 4 ) . 26 The e x p r e s s i o n f o r g e n e r a t i n g B e s s e l f u n c t i o n s i s - , < \u00C2\u00BB m=-a> z m=-<\u00C2\u00BB P u t t i n g t = e ^ , th e n -Kt- )^ = jsin0 and we o b t a i n 00 jzsin0 = y jm0 j ( } Z m m m=-oo U s i n g t h i s r e l a t i o n s h i p e q u a t i o n (2.4) can be t r a n s f o r m e d : \u00C2\u00B1 I e J P X \u00C2\u00BB S l n 0 = \u00C2\u00A3 I \u00C2\u00A3 J (px ) n=0 n=0 m=-oo g i v i n g ^ N B(0) = Z Z : I n J m ( K ) 2.5 m =_a> n - o I t can be seen t h a t t h i s i s the form of e q u a t i o n ^ . 3) i f the s e r i e s i s t r u n c a t e d t o 2M+1 terms. F o r e q u i v a l e n c e we t h e n r e q u i r e I I J ( f l r ) = c , 2.6 n m r n' m ' where the c are the complex F o u r i e r c o e f f i c i e n t s of the m r d e s i r e d p a t t e r n . E q u a t i o n ( 2 . 6 ) r e p r e s e n t s a s e t of n o n - l i n e a r t r a n s c e n d e n t a l e q u a t i o n s which are t o be s o l v e d f o r N+l v a l u e s of the x and/or I . n ' n 2.2 Methods Based on the l i n e - S o u r c e R e p r e s e n t a t i o n of the D e s i r e d P a t t e r n 2.2.1 I s h i m a r u ' s Method A procedure which r e l a t e s e q u a t i o n s (2.l) and (2.4) i s 18 d e s c r i b e d by I s h i m a r u and i s b r i e f l y as f o l l o w s . The n o n - u n i f o r m l y - s p a c e d a r r a y p a t t e r n i s g i v e n i n e q u a t i o n (2.4) as N (0) = X V E(0) = > I e n 2.7 n=l and can be r e w r i t t e n as B(0) f ( n ) 2.8 The P o i s s o n Sum f o r m u l a i s n=l 26 00 00 00 ^ f ( n ) = j f ( v ) e ; j 2 m 3 r v dv 2.9 V i M m *tr% ~ 0 O n= -\u00C2\u00BB m= - 0 0 A p p l y i n g t h i s t o e q u a t i o n 2.8 we o b t a i n E(0) = Y_ m=-oo J 0 (where the i n t e g r a t i o n i s from 0 t o N s i n c e f ( v ) v a n i s h e s f o r v < 0 and v > N). The source p o s i t i o n f u n c t i o n , s = s ( v ) , . g i v e s the t h p o s i t i o n of the n element when v = n and the source number f u n c t i o n , v = u ( s ) , g i v e s the numbering of each element when s i s a t the c o r r e c t p o s i t i o n of the element. s . F i g u r e 2.1 P l o t of a T y p i c a l Source Number F u n c t i o n T r a n s f o r m i n g e q u a t i o n (2.10-) we o b t a i n C O SjJ E(0) = I [ f(s) g e ^ 2 m 7 t v ( s ) d s 2.11 I f e q u a t i o n (.2.7) i s r e w r i t t e n as (0) = ^ Em(0), 2.12 E .m= -\u00C2\u00AB> where E (0) = r N A ( s ) dv e-j ( V r(s)-2m^v(s)) eJSs sin0 d s > s o 2.13 on then the p h y s i c a l s i g n i f i c a n c e of the f o r m u l a t i o n can be seen. (A(s) i s a f u n c t i o n which y i e l d s A , the a m p l i t u d e of the c u r r e n t i n the n^*1 element, at s = s n , and \J/\"(s) i s a f u n c t i o n which y i e l d s \"the phase of the c u r r e n t , at s = s n ) . E q u a t i (2.13) i s s i m i l a r i n form t o e q u a t i o n (2.]) and r e p r e s e n t s the p a t t e r n due t o a co n t i n u o u s c u r r e n t d i s t r i b u t i o n of am p l i t u d e A(s)' 1^ and phase '\|/(s)-2mjrv(s) . Thus the n o n - u n i f o r m l y _ s p a c e d a r r a y p a t t e r n has been reduced t o an i n f i n i t e s e r i e s whose terms correspond t o con-t i n u o u s c u r r e n t d i s t r i b u t i o n s . N o r m a l i z i n g the v a r i a b l e s as f o l l o w s : u = |3asin0 (the a r r a y a p e r t u r e ) x = x ( y ) ( n o r m a l i z e d source p o s i t i o n f u n c t i o n ) y ^ y ( x ) ( n o r m a l i z e d source number f u n c t i o n ) e q u a t i o n s (2.12) and ^.13) become E(u) = ( - l ) m ( N - l } E m ( u ) 2.14 1 7 and E (u) = m A(x) g e dy; -j\"\|/(x)+jmn;N(y-x) j (n+mrfjx dx -1 2.15 The a c t u a l p o s i t i o n of the n^*1 element i s s^ = a x ( y ) n J n F o r odd N, i . e . . N = 2M+1, n y n = gp: , n = 0, + 1 , \u00C2\u00B1 2,...+ M F o r even N, i . e . N = 2M, y n n--r n - M y n n+-g-n - M n > 0 n < . 0 n = + 1, \u00C2\u00B1 2 , . . . +M The t o t a l l e n g t h of the a r r a y i s then L 0 = a [ x ( y M ) - x ( y _ M ) ] I s h i m a r u then argued t h a t the s e r i e s of equation(2.14) i s so r a p i d l y convergent t h a t \u00E2\u0080\u00A2;\u00E2\u0080\u00A2 t o a good a p p r o x i m a t i o n , i t may be t r u n c a t e d a f t e r the f i r s t term, so t h a t E(u) * E Q ( u ) C o n s i d e r i n g the u n i f o r m l y - e x c i t e d b r o a d - s i d e a r r a y , where A(x) = 1 and ~U/(x) = 0, we have 18 E(u) = E Q ( u ) = \ ^ e J' U X dx dx e a x 2.16 which i s the r a d i a t i o n p a t t e r n of a c o n t i n u o u s - s o u r c e d i s t r i b -u t i o n of a m p l i t u d e ^ and z e r o phase s h i f t . Hence i f the con-t i n u o u s s o u r c e d i s t r i b u t i o n g i v i n g the d e s i r e d p a t t e r n i s known, the element p o s i t i o n s i n the e q u i v a l e n t n o n - u n i f o r m l y -spaced a r r a y can be c a l c u l a t e d . 2 . 2 . 2 M a f f e t t ' s M e t h o d 1 9 I f the p a t t e r n t o be s y n t h e s i z e d i s generated by a s y m m e t r i c a l l i n e s ource g(x) of a p e r t u r e l e n g t h L then i n s t e a d of e q u a t i o n (2 .l) we have '2 E(u) = 2 g(x) cos|3ux dx 4) ( u s i n g the n o r m a l i z e d v a r i a b l e u = sin0). D e f i n e a n o r m a l i z e d c u r r e n t d e n s i t y , 2.17 p(x) = j g ( x ) , where A = 2 g ( x ) d x . \u00E2\u0080\u00A2 2.18 '0 then the c u m u l a t i v e c u r r e n t d i s t r i b u t i o n i s p C l )d1s , - x y(x) = I n equation^ 2 . 1 7 ) p u t t i n g x=x(y) , i . e . 2.19 19 dx = ^ dy, and d i f f e r e n t i a t i n g e q u a t i o n (2.19) we o b t a i n j^- = p(x). dx Changing the v a r i a b l e of i n t e g r a t i o n i n (2.17) we get 1 E(u) = 2 A p(x)cos|3ux P H I 2A -1 cos(3ux dy where x=x(y) 2.20 T h i s i s approximated by the t r a p e z o i d a l r u l e t o y i e l d the approximate p a t t e r n F ( u ) . The increment i n y i s chosen t o be , i . e . M+l p o i n t s are chosen a t y Q = :k,j-L,J2 y j y r 1 ' M F(u) = ^jjj ^ cos(3ux.. t h e n n \u00E2\u0080\u00A2> n=l where the x^ are g i v e n by x y n = 2 + n P (S )d^ n=0,l,2 M 2.21 J 0 For some d i s t r i b u t i o n s , e q u a t i o n (2.19) can be i n v e r t e d t o 27 g i v e x^ e x p l i c i t l y ; but i n g e n e r a l i t e r a t i o n t e c h n i q u e s have t o be used. 2 . 3 Methods Based on the U n i f o r m - A r r a y R e p r e s e n t a t i o n of the D e s i r e d P a t t e r n 20 2.3.1 Harrington's Method 15 A p e r t u r b a t i o n approach was used by Harrington who determined the f r a c t i o n a l change from uniform spacing required to synthesize the desired p a t t e r n . For a uniformly spaced and ex c i t e d array of N elements, we have E. u = I E \"n \u00C2\u00B0osC\u00C2\u00BBjJ> > 2.22 n r where / means n N-1 { N even o r ^ , u = pdsin0 E n_l,3,5,... ^ N-1 y~ I N odd |^n=0,2,4 \u00E2\u0084\u00A2 \" n 12 n 0 v For non-uniform spacing a convenient \"base\" separation, 1 n = 0 d, can be chosen and the element spacing expressed as x n = + \u00C2\u00A3 n)d- \u00C2\u00BB i . e . e n i s the f r a c t i o n a l change from uniform spacing. Then F(u) = l I a n c o s { (2 + e n ) u ) n Expanding the cosine and assuming e^a i s small (so that cos e nu - 1 and s i n e nu - e nu) i . e . F(u) = E u - | E V n n E 2 6 m = -M i = l Th i s s e t of n o n - l i n e a r a l g e b r a i c e q u a t i o n s can be s o l v e d f o r N, ithe number of elements r e q u i r e d i n the n o n - u n i f o r m l y -spaced a r r a y , and the p o s i t i o n s , x^, of the N elements. 23 3. ARRAY SYNTHESIS USING NUMERICAL QUADRATURE 3.1 I n t r o d u c t i o n I n c h a p t e r 2 the t h e o r y behind some e x i s t i n g t e c h n i q u e s of a r r a y s y n t h e s i s was i n v e s t i g a t e d . I n t h i s c h a p t e r a n o v e l procedure i s proposed. The b a s i s of the method i s n u m e r i c a l q u a d r a t u r e , a t e c h n i q u e f o r e v a l u a t i n g d e f i n i t e i n t e g r a l s . 3.2 Quadrature Theory The problem of f i n d i n g the n u m e r i c a l v a l u e of the i n t e g r a l of a f u n c t i o n of one v a r i a b l e , because of i t s geo-m e t r i c a l meaning, i s o f t e n c a l l e d q u a d r a t u r e . C o n s i d e r an i n t e g r a l of the form b P(x) dx a T h i s might be approximated by d i v i d i n g the i n t e r v a l i n t o n eq u a l segments and e v a l u a t i n g P(x) a t n e q u a l l y spaced v a l u e s of x, one w i t h i n each segment. T h i s g i v e s the a p p r o x i m a t i o n b n F ( x ) dx = ^ ^ ~ F(x.) 3.1' More g e n e r a l l y the x.. c o u l d be i r r e g u l a r l y spaced. Then each of the P(x.) would be m u l t i p l i e d by a w e i g h t i n g f a c t o r , H., and the i n t e g r a l c o u l d be w r i t t e n i n the form b n P ( x ) d x = H. P ( x . ) + A , 3 . 2 r _ ... 24 where A i s a r e s i d u a l e r r o r term. S i n c e t h e r e a r e 2n unknowns i n the q u a d r a t u r e sum i t might be sus p e c t e d t h a t the H. and x. c o u l d be chosen such t h a t A be z e r o f o r a l l p o l y n o m i a l s of 28 or d e r 2 n - l or l e s s . K r y l o v has shown t h a t t h i s i s i n f a c t the case. The q u a d r a t u r e methods t h a t have t h i s degree of a c c u r a c y f o r p o l y n o m i a l s a re termed Gaussian. F o r Gaussian q u a d r a t u r e , a s e t of 2n e q u a t i o n s i n the 2n unknown c o n s t a n t s can be o b t a i n e d by s u b s t i t u t i n g P(x) = x \ k = 0,1,... 2 n - l i n t o e q u a t i o n (3-2) and s e t t i n g A=0. I n s t e a d of c o n s i d e r i n g the s e t of the Xy i t i s co n v e n i e n t t o c o n s i d e r the p o l y n o m i a l , P n ( x ) , which has thes-e as r o o t s . I t i s a l s o c o n v e n i e n t t o c o n s i d e r the i n t e g r a n d of e x p r e s s i o n (3\u00C2\u00ABl) t o be broken up i n t o two f a c t o r s , w(x) and f ( x ) . The qua d r a t u r e e x p r e s s i o n then becomes b w(x) f ( x ) dx * X! A k f ( x k } J k = l .3.3 a and the c o e f f i c i e n t s A^ . are g i v e n by A k = a *b P (x) w(x) n \u00E2\u0080\u0094 r dx (x-x, )P X (xn ) v k y n x k 24 U t i l i z i n g the C h r i s t o f f e l - D a r b o u x i d e n t i t y y i e l d s A, = - r \" \u00E2\u0080\u0094 - where a i s the k a P V (x, )P , ( x j n n n k 7 n+1 k n 30 c o e f f i c i e n t of x i n P ( x ) . Use of the r e c u r s i o n r e l a t i o n s h i p s n * f o r orthonormal p o l y n o m i a l s a l t e r s t h i s e x p r e s s i o n t o 25 a -, V = \u00E2\u0080\u0094 J 1 3-4 k ^ - 1 <: < xk> Fn-l<*k> 3 . 3 Gauss-Chebychev Quadrature By a l i n e a r t r a n s f o r m a t i o n , the l i m i t s (a,b) of the r e g i o n of i n t e g r a t i o n can be tr a n s f o r m e d i n t o an chosen segment of t h e x - a x i s . I n o r d e r t o make use of the symmetry of the nodes x^. and of the c o e f f i c i e n t s A^ ., the st a n d a r d segment w i l l be t a k e n t o be (-1, l ) . By j u d i c i o u s s e l e c t i o n of the p o l y -n o m i a l t y p e , the c o e f f i c i e n t s A^ can be g r e a t l y s i m p l i f i e d . The weight f u n c t i o n t h a t i s o r t h o g o n a l t o the Chebychev p o l y n o m i a l s i s w(x) = ( l - x ) U s i n g \"the Chebychev p o l y n o m i a l s i n e q u a t i o n (3-4) i t can be r e a d i l y shown t h a t the A^ . are. c o n s t a n t , i . e . A, = \u00E2\u0080\u0094 , k = 1, . . . . n k n ' ' t h The x^., b e i n g the r o o t s of the n o r d e r Chebychev p o l y n o m i a l , are g i v e n by x k = cos ( 2 k n 1 ) % Thus the complete Gauss-Chebychev qu a d r a t u r e f o r m u l a i s 1 n - ^ L - d i = i V f ( c o s ( 2 | ^ ) ^ ) 3 .5 - x k = l 2 6 T h i s w i l l he i l l u s t r a t e d by means of an example, the e v a l u a t i o n of r1 8 I = j dx 1 - x '-1 ' U s i n g n=4 i n e q u a t i o n (3.5) T ^ z 8 8 3^ 8 5jt 8 7jt I 4 cos Q + cos + cos ^-Q + cos J-g - 0.834 The a c t u a l v a l u e of the i n t e g r a l i s I = j c o s 8 9 d\u00C2\u00A9 = 0.859 The use of a 5-point quadrature w i l l y i e l d the r e s u l t x j t 8 it 833t 8 5 j t 8 1% I - 5\" c o s io + c o s 10 + c o s 10 + c o s 10 + + c o s 8 |g = 0.859 3.4 A p p l i c a t i o n of G-auss-Chebychev Quadrature t o A r r a y S y n t h e s i s The p a t t e r n of a s y m m e t r i c a l l y e x c i t e d l i n e source i n a n o r m a l i z e d a p e r t u r e i s g i v e n by I g(x) cos ux dx , where g(x) i s the n o r m a l i z e d c u r r e n t - d i s t r i b u t i o n f u n c t i o n . P u t t i n g G-(x) = g(x) A/1-X 2 , we o b t a i n I. 2. ~2 1 P(u) = 1 ( l - x ^ ) G(x) cos ux dx. -1 27 T h i s i s of the form 1 1 2 ~ 2 w(x) f ( x ) d x , w i t h w(x) = ( l - x ) b e i n g the -1 weight f u n c t i o n of Chebychev p o l y n o m i a l s . Hence the quadrature f o r m u l a i s P(u) ~ ^ A k f ( x k ) , where f ( x ) = k = l t h G-(x)cos U X and the x-^ are the r o o t s of the n o r d e r Chebychev p o l y n o m i a l . As s t a t e d p r e v i o u s l y (and shown by K r y l o v ) the c o e f f i c i e n t s a r e a l l e q u a l t o ^ and so n '(u) - ^ X & ( x k ^ c o s u x k 5 , 6 k = l T h i s i s of the same form as the r a d i a t i o n p a t t e r n of a n o n - u n i f o r m l y - s p a c e d symmetric a r r a y w i t h the elements p o s i t i o n e d a t + x^ and h a v i n g n o r m a l i z e d e x c i t a t i o n s g i v e n by ~ G(x k)\u00C2\u00AB t h Thus the e x c i t a t i o n of the k element, 1^ ., i s g i v e n by \ = ^ J 1 - x / 5.7 By t h i s procedure the p a t t e r n of a c o n t i n u o u s l i n e source has been t r a n s f o r m e d i n t o t h a t of a n o n - u n i f o r m l y - s p a c e d a r r a y , e n a b l i n g the element p o s i t i o n s and c u r r e n t e x c i t a t i o n s t o be r e a d i l y c a l c u l a t e d . I t i s w o r t h n o t i n g t h a t though the r e s u l t i n g a r r a y i s non-u n i f o r m l y - s p a c e d , t h i s s p a c i n g does not depend on the d e s i r e d pat-t e r n but o n l y on the number of elements i n the a r r a y . I n t h i s r e s p e c t t h i s method i s s i m i l a r t o the u n i f o r m l y - s p a c e d a r r a y which the s p a c i n g s are p r e a s s i g n e d . 4. RESULTS 4.1 I n t r o d u c t i o ^ T h i s c h a p t e r b e g i n s by s p e c i f y i n g a c e r t a i n antenna p a t t e r n t o be s y n t h e s i z e d . Each of the v a r i o u s methods of Chapters 2 and 3 i s t h e n a p p l i e d t o the s y n t h e s i s of this* p a t t e r n and the s u c c e s s of the methods i n a p p r o x i m a t i n g the p a t t e r n i s examined. 4.2 Choice of P a t t e r n t o be S y n t h e s i z e d I n o r d e r t o be a b l e t o make a c r i t i c a l comparison of the v a r i o u s methods, i t i s d e s i r a b l e \" -( i ) t h a t the p a t t e r n t o be s y n t h e s i z e d can be math-e m a t i c a l l y r e p r e s e n t e d as the p a t t e r n of a l i n e s o u r c e , as the p a t t e r n of a u n i f o r m l y - s p a c e d a r r a y and as a F o u r i e r s e r i e s sum ( i i ) t h a t p a r a m e t r i c v a r i a t i o n s of the p a t t e r n f u n c t i o n w i l l g i v e r i s e t o a wide range of p a t t e r n forms. \u00E2\u0080\u00A2 The p a t t e r n chosen was an e q u a l - s i d e l o b e p a t t e r n . Dolph has determined the element e x c i t a t i o n s of a u n i f o r m l y - s p a c e d a r r a y which w i l l g i v e t h i s p a t t e r n , and h i s work has been g e n e r a l i z e d by o t h e r s . The c o n t i n u o u s c u r r e n t source c o u n t e r -p a r t of t h i s a r r a y has been determined by T a y l o r . However, the l i n e s ource p r o d u c i n g t h i s i d e a l p a t t e r n w i t h c o n s t a n t l e v e l s i d e l o b e s f o r a l l v a l u e s of n, i s u n r e a l i z a b l e i n p r a c t i c e s i n c e the remote s i d e l o b e s do not decay. T a y l o r r e s o l v e d t h i s problem by making the p a t t e r n have eq u a l s i d e -s i n ix l o b e s up t o a p o i n t and then c a u s i n g them t o decay as \u00E2\u0080\u0094 - \u00E2\u0080\u0094 . 30 The t r a n s i t i o n p o i n t a t which the s i d e l o b e s s t a r t t o decay can be p l a c e d a t w i l l so t h a t the i d e a l p a t t e r n can be approached a r b i -t r a r i l y c l o s e l y . T h i s t r a n s i t i o n p o i n t can be p l a c e d o u t s i d e the v i s i b l e and s c a n n i n g r e g i o n s of the p a t t e r n , t h u s e f f e c t i v e l y p r o -d u c i n g an e q u a l - s i d e l o b e p a t t e r n . The t h e o r y behind T a y l o r ' s l i n e s ource i s c o m p l i c a t e d and a summary i s g i v e n i n Appendix I . Thus the e q u a l - s i d e l o b e p a t t e r n s a t i s f i e s the f i r s t c r i t e r i o n above. By v a r y i n g the parameter c o n t r o l l i n g t he s i d e l o b e l e v e l , p a t t e r n s can be o b t a i n e d r a n g i n g from t h e . b i n o m i a l a r r a y w i t h no s i d e l o b e s , a l l the way t o the i n t e r f e r o m e t e r case w i t h s i d e l o b e s -e q u a l i n h e i g h t t o the main beam, thus s a t i s f y i n g the second c r i t e r i o n . 4.3 Unz's Method Unz's method s y n t h e s i z e s b oth the p o s i t i o n s and e x c i t -a t i o n s o f the elements and r e q u i r e s the s o l u t i o n of the f o l l o w i n g s e t of e q u a t i o n s : I o J - M ( p x o ) + I l J - M ( p x l ) + I N J - M ( P x N ) = \u00C2\u00B0-M *o J o ( P x o } + \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 W ^ V = \u00C2\u00B0o J o J M ( | 3 x o ) + VM^1^ I N J M ( P x N ) = \u00C2\u00B0M Where the X^ a r e the element p o s i t i o n s , the 1^ are t h e i r e x c i t -a t i o n s and the are the F o u r i e r c o e f f i c i e n t s of the d e s i r e d r a d i a t i o n p a t t e r n . I n g e n e r a l , the s o l u t i o n of t h i s s e t i s ext r e m e l y d i f f i c u l t . To g a i n i n s i g h t i n t o the s o l u t i o n , a si m p l e p a t t e r n , F(0) = 32 cos^0, was s y n t h e s i z e d u s i n g 5 31 elements i n a 2\ a p e r t u r e . S i n c e the chosen p a t t e r n has s y m m e t r i c a l F o u r i e r c o e f f i c i e n t s the r e s u l t i n g a r r a y w i l l he symmetric about the c e n t r e element. T h i s reduces the unknowns t o one s p a c i n g ( t h a t of the i n n e r p a i r of elements) and t h r e e e x c i t a t i o n s , i . e . I-L^ ( p x x ) + I 2 J 4 ( 2 ^ ) - 1 = 0 I1J2 ((3x1) + I 2 J 2 ( 2 * ) - 4 = 0 I l J o (P*!* + I 2 J o ( 2 3 r ) - 6 + i I Q = 0 S o l u t i o n s can be o b t a i n e d g r a p h i c a l l y as shown i n f i g . 4.1a. F i g . 4.1a Example of Unz's Method 4.1b A r r a y S y n t h e s i z e d by Unz's Method I t i s of i n t e r e s t to note t h a t one v a l u e of x^ corresponds t o I 2 = 0. In i h a t case the o u t e r elements a r e redundant. T h i s happens when px-^ = 2.9; i . e . x^ = 0.47\. The c e n t r a l element then has t w i c e the c u r r e n t e x c i t a t i o n of the o u t e r elements. S e v e r a l a r r a y c o n f i g u r a t i o n s were t a k e n from F i g . 4.1; the a r r a y y i e l d i n g the b e s t p a t t e r n i s shown i n F i g . 4.1b. I t s 0 05 W t\u00C2\u00A3 2.0 ^ F i g . 4.2 Unz Synthesis of F(0) = 32 cos 40 33 p a t t e r n and the d e s i r e d p a t t e r n are shown i n F i g . 4.2. Agreement between the two p a t t e r n s i s good over the range p l o t t e d , i . e . the v i s i b l e and s c a n n i n g r e g i o n s . A f a r more d i f f i c u l t problem i s the s y n t h e s i s of the p a t t e r n F(0) = 2 1 (^ cos cos 0), which i s the p a t t e r n of an 11 element b i n o m i a l a r r a y i n a 5/V a p e r t u r e . The e q u a t i o n s reduce t o X 3 J 2 n ( P x 3 ) + I 2 J 2 n ( | 3 x 2 ) + I l J 2 n ( p x l ) = * C 2 n ' n > 1 I - \u00C2\u00A7 + J Q ( 0 x 5 ) + I 2 J Q ( p x 2 ) + I 1 J 0 ( P x 1 ) = i C Q Approximate s o l u t i o n s were found by g r a p h i c a l means and f i n a l v a l u e s were reached by c o r r e c t i n g t h e s e . The s y n t h e s i z e d a r r a y i s shown i n F i g . 4-3 and the c o r r e s p o n d i n g p a t t e r n s are shown i n F i g . 4.4. X \ h. h X (a) *\u00E2\u0080\u0094-\u00E2\u0080\u0094*\u00E2\u0080\u0094-\u00E2\u0080\u0094*\u00E2\u0080\u0094-\u00E2\u0080\u0094x\u00E2\u0080\u0094-\u00E2\u0080\u0094*\u00E2\u0080\u0094-\u00E2\u0080\u0094* * * * * * I to AS \10 210 251 (b) Zl F i g . 4.3 (a) U n i f o r m l y - S p a c e d B i n o m i a l A r r a y and (b) Non-Uniformly-Spaced S y n t h e s i z e d A r r a y . F o r each p a t t e r n , the magnitude of the a r r a y f a c t o r i s p l o t t e d a g a i n s t u = cos 9 - cos 9 , f o r u i n the range 0 .^ u< 2, which r e p r e s e n t s the complete v i s i b l e and s c a n n i n g regions.. F i g . 4.4 Unz B i n o m i a l P a t t e r n S y n t h e s i s 35 The correspondence of the two p a t t e r n s i s e x c e l l e n t throughout the v i s i b l e r e g i o n ( w i t h i n 1 % ) , but d e t e r i o r a t e s r a p i d l y i n the s c a n n i n g r e g i o n . Thus the p a t t e r n due t o the a r r a y of F i g . 4 \u00C2\u00BB3a i s an e x c e l l e n t a p p r o x i m a t i o n t o t h a t of the b i n o m i a l a r r a y of F i g . 4.3a i f no beam s c a n n i n g i s r e q u i r e d and- a l s o saves f o u r elements. However the computation i n v o l v e d i n p r o d u c i n g t h i s a r r a y i s v e r y l e n g t h y and becomes p r o h i b i t i v e f o r l a r g e r a r r a y s . 4.4 I s h i m a r u 1 s Method As i n d i c a t e d i n e q u a t i o n (2.16) t h i s method assumes t h a t the e x c i t a t i o n s are u n i f o r m and r e q u i r e s the s o l u t i o n f o r x of an e q u a t i o n of the form y = f ( x ) , f o r each of the elements. The s o l u t i o n c o u l d be o b t a i n e d by i n v e r t i n g t h i s e q u a t i o n , but a Newton-Raphson i t e r a t i o n procedure i s used here. Con-vergence i s good and r e s u l t s a re r e a d i l y o b t a i n e d f o r s m a l l a r r a y s . F o r a r r a y s of more t h a n about t h i r t y elements convergence i s much s l o w e r . A r r a y s were s y n t h e s i z e d t o produce 15 dB s i d e l o b e s u s i n g from 11 t o 21 elements. U s u a l l y o n l y the s i d e l o b e envelope i s of importance when c o n s i d e r i n g the s i d e l o b e r e g i o n and so o n l y the envelopes of the s i x s y n t h e s i z e d a r r a y s are shown i n F i g . 4.5, which shows the \" v i s i b l e r e g i o n \" f o r an a r r a y of a p e r t u r e 10X. The f u l l p a t t e r n s f o r 15 and 21 elements are shown i n F i g s . 4.6 and 4.7, f o r the complete v i s i b l e r e g i o n and s c a n n i n g r e g i o n of an a r r a y of a p e r t u r e 10\. The element 36 F i g . 4.5 I s h i m a r u ' s S y n t h e s i s o f 15 dB E q u a l - S i d e l o b e Pattern F i g . 4.6 15 element A r r a y by Ishimaru's Method 0 OJS 1.0 1.5 2.0 Fig. 4.7 21 element Array by Ishimaru's Method 39 arrangements i n these two a r r a y s a re shown i n F i g . 4.8. ARRAY CENTRE LINE 1 5 element a r r a y K K * # \u00E2\u0080\u0094 K r X fe&A -72> .77X -8fcA I'O+X -Z4X 21 element a r r a y \u00E2\u0080\u00A2 47\ -48X -5lX -6\"4\ \u00E2\u0080\u00A2 63X 73X -SoX 'CuX * * if X X * * ' X * X I F i g . 4.8 A r r a y C o n f i g u r a t i o n s f o r Ishimaru's Method The beamwidths of the r e s u l t i n g p a t t e r n s a re i n v e r y c l o s e agreement w i t h the T a y l o r p a t t e r n beamwidth and the c l o s e - i n s i d e l o b e s are a t the designed l e v e l . I n the case of the 15 element a r r a y , d e t e r i o r a t i o n of the p a t t e r n s e t s i n a f t e r the f i r s t t h r e e or f o u r s i d e l o b e s ; the i n c r e a s e d s i d e l o b e s t h e r e a f t e r a re u n a c c e p t a b l e . As would be expected, the a p p r o x i m a t i o n improves as the number of elements i n c r e a s e such t h a t w i t h 21.elements i n the b r o a d s i d e c o n f i g u r a t i o n , \u00E2\u0080\u00A2 the a r r a y p a t t e r n i s w i t h i n a few p e r c e n t of the d e s i r e d p a t t e r n out t o 61\u00C2\u00B0 from b r o a d s i d e . However, i f the o n l y requirement i s t h a t a l l s i d e l o b e s be below the designed l e v e l , then the main beam produced by t h i s a r r a y may be scanned t o + 48\u00C2\u00B0 from the b r o a d s i d e d i r e c t i o n w i t h o u t the appearance of s i d e l o b e s l a r g e r than 15 dB. A f u r t h e r i n t e r p r e t a t i o n of F i g . 4.7 i s t h a t by r e d u c i n g the s i z e of the a r r a y a p e r t u r e from 10\ t o 7.5\, an a r r a y can 40 be o b t a i n e d whose main beam may be scanned t o w i t h i n + 90\u00C2\u00B0 of the b r o a d s i d e d i r e c t i o n w i t h o u t the appearance of l o b e s l a r g e r t h a n 15 dB. The c o s t of t h i s , however, i s an i n c r e a s e i n the w i d t h of the main beam. T h i s use of a p e r t u r e r e d u c t i o n , f o r e x t e n d i n g the c o n s t a n t s i d e l o b e r e g i o n a t the expense of the main beam w i d t h , can be a p p l i e d t o a l l the p a t t e r n s t h a t f o l l o w . An attempt was made t o reduce the secondary maximum which appears i n the s c a n n i n g r e g i o n of the p r e c e d i n g p a t t e r n s by c o n s i d e r i n g the second term i n the s e r i e s e x p a n sion of e q u a t i o n 2.14- However t h i s gave o n l y a v e r y s l i g h t improve-ment . 4.5 M a f f e t t ' s Method The procedure i s s i m i l a r t o the p r e v i o u s method but the convergence of the e q u a t i o n s f o r the x v a l u e s i s somewhat s l o w e r . The p a t t e r n envelopes f o r a r r a y s of from 11 to 21 elements a r e shown i n P i g . 4.9, over the v i s i b l e r e g i o n c o r r e s p o n d i n g t o a 10\ a p e r t u r e , a n d two f u l l p a t t e r n s are shown i n P i g s . 4.10 and 4.11. The improvement ' w i t h i n c r e a s i n g number of elements i s e a s i l y seen. The r e s u l t s o b t a i n e d by t h i s method show l i t t l e , i f any, improve-ment over the p r e v i o u s method. I t can be seen t h a t b o t h of the p r e c e d i n g methods f a i l t o g i v e a good a p p r o x i m a t i o n t o the d e s i r e d p a t t e r n throughout the v i s i b l e and s c a n n i n g r e g i o n s even when u s i n g 21 elements, as i n the c o r r e s p o n d i n g u n i f o r m l y - s p a c e d a r r a y . Thus t h e r e i s no element s a v i n g i n these c a s e s . 41 F i g . 4.9 Maffett's Synthesis of 15 dB Sidelobe Pattern F i g . 4.10 15 element A r r a y by M a f f e t t ' s Method Fig. 4.11 21 element Array toy Maffett ' s Method 44 4.6 H a r r i n g t o n ' s Method The u n d e r l y i n g assumption of t h i s method i s t h a t the i n t e r - e l e m e n t s p a c i n g s d e v i a t e o n l y s l i g h t l y from the u n i f o r m case; hence the v a l u e s of e c a l c u l a t e d from e q u a t i o n (2.23) s h o u l d he such t h a t d. A 15 element a r r a y was s y n t h e s i z e d t o g i v e 15 dB equa l s i d e l o b e s and the r e s u l t i n g p a t t e r n i s shown i n P i g . 4.12. The v a l u e s of e c o r r -esponding t o t h i s a r r a y do not a l l s a t i s f y the above c r i t e r i o n , e n \u00C2\u00AB d; i n f a c t f o r some v a l u e s of n, y d. The r e s t r i c t i o n on the e i m p l i e s a r e s t r i c t i o n on the n * p a t t e r n t o be s y n t h e s i z e d . F o r s m a l l \u00C2\u00A3 n the d i f f e r e n c e between the p a t t e r n t o be s y n t h e s i z e d and the un p e r t u r b e d p a t t e r n , i . e . the p a t t e r n of the u n i f o r m l y - s p a c e d , u n i f o r m l y \u00E2\u0080\u0094 e x c i t e d a r r a y , must-not be too l a r g e . I t f o l l o w s from t h i s t h a t s i d e l o b e l e v e l s s i g n i f i c a n t l y l o w e r than the 13.2 dB of the u n i f o r m a r r a y cannot be o b t a i n e d by t h i s method. A problem more s u i t e d t o t h i s method i s the r e d u c t i o n of the f i r s t few s i d e l o b e s of the u n i f o r m a r r a y , r a t h e r than E -F(u) a complete p a t t e r n s y n t h e s i s . The term u \u00E2\u0080\u0094 i n e q u a t i o n (2.23) r e p r e s e n t s the n o r m a l i z e d d i f f e r e n c e between the d e s i r e d p a t t e r n and the p a t t e r n of the u n i f o r m a r r a y . F o r r e d u c i n g a number of s i d e l o b e s , t h i s f u n c t i o n may be regarded as the sum of a s e r i e s of i m p u l s e s p o s i t i o n e d a t the c o r r e s p o n d i n g s i d e -l o b e s . T h i s i s shown i n F i g . 4.13 f o r the r e d u c t i o n of the f i r s t two s i d e l o b e s . F i g . 4.12 Harrington's Synthesis of 15 element Array 46 P i g . 4.13 Impulse F u n c t i o n s f o r S i d e l o b e R e d u c t i o n t h I f t h e r e are K im p u l s e s and the k has a h e i g h t of E -P(n) n a^ and i s s i t u a t e d a t u=u^, th e n \u00E2\u0080\u0094 \u00E2\u0080\u0094 may be approximated E - F k ^ u - ^ 1 ^ \u00E2\u0080\u0094 \u00E2\u0080\u0094 - - ^ a k 6 ( u - u k ) , where \u00C2\u00A3 i s k=l the u n i t i m p u l s e f u n c t i o n . A p p l y i n g t h i s t o e q u a t i o n (2.23) we o b t a i n K N V ~ S I N n uk e = r > a. ~n re / k u, kTi k An example of the r e d u c t i o n of the f i r s t f o u r s i d e l o b e s i s g i v e n . The v a l u e s of e are t a b u l a t e d i n F i g . 4.14 and the complete p a t t e r n i s shown i n F i g . 4.15. A f u r t h e r use of t h i s method i s f o r the e l i m i n a t i o n of the g r a t i n g l o b e s F i g . 4.15 Reduction of Sidelobes by Harrington's Method 48 a s s o c i a t e d w i t h u n i f o r m l y .spaced a r r a y s . rv 2 4- (o 8 10 12 1 + 16 18 2 0 - 0 / 7 6 -0-5(8 -0 &03 -0-326 0-/S3 F i g . 4.14 P e r t u r b a t i o n Parameters 4.7 W i l l e y ' s Method I n t h i s method the e x c i t a t i o n s are assumed u n i f o r m and the s e t of e q u a t i o n s r e p r e s e n t e d by e q u a t i o n (2.26) i s s o l v e d f o r the element p o s i t i o n s . As w i t h Unz's method the s o l u t i o n s a re v e r y d i f f i c u l t t o o b t a i n . The p a t t e r n of an 11 element b i n o m i a l a r r a y was s y n t h e s i z e d u s i n g 7 elements i n a 3-5A. a p e r t u r e . The r e s u l t i n g p a t t e r n i s shown i n F i g . 4.16. A r r a y s of more than 7 elements c o u l d not be s y n t h e s i z e d by t h i s method due t o the d i f f i c u l t y of s o l v i n g e q u a t i o n (2.26). 4.8 The Quadrature Method S y n t h e s i s by t h i s procedure i s v e r y c o n v e n i e n t numeric-a l l y s i n c e the element p o s i t i o n s are g i v e n by the r o o t s of 1.0 0.8 0.5 H 0.4 H ,Willey ^Binomial 0.5 1.0 7.5 F i g . 4.16 W i l l e y ' s Synthesis of Binomial Pattern 50 the Chebychev p o l y n o m i a l and the c u r r e n t e x c i t a t i o n s a re o b t a i n e d by e v a l u a t i n g a f u n c t i o n a t these r o o t s . Thus t h e r e are no i t e r a t i o n s for s o l u t i o n s of e q u a t i o n s r e q u i r e d . The p r o p e r t i e s of the Chebychev p o l y n o m i a l s g i v e the s y n t h e s i z e d a r r a y s an i n v e r s e space t a p e r , i . e . the elements tend t o be c l o s e r t o g e t h e r a t the ends o f the a r r a y than they a r e i n the c e n t r e . T h i s appears t o be a c h a r a c t e r i s t i c of methods t h a t s t a r t from a l i n e - s o u r c e p a t t e r n . P a t t e r n s have been o b t a i n e d f o r a r r a y s of 11 t o 21 elements; t h e i r s i d e l o b e envelopes are shown i n F i g . 4.17 and the a c t u a l p a t t e r n s of the 15 and 21 element a r r a y s are shown i n F i g s . 4.18 and 4.19 r e s p e c t i v e l y . 4.9 Summary of R e s u l t s A t a b l e d e m o n s t r a t i n g the p r o p e r t i e s of the d i f f e r e n t methods i s shown i n F i g . 4.20 and an e v a l u a t i o n of t h e i r r e s u l t s f o l l o w s . \"\" : . Unz's method, s t a r t i n g from the F o u r i e r s e r i e s f o r m u l a t i o n o f the d e s i r e d p a t t e r n , has g i v e n good r e s u l t s f o r s m a l l a r r a y s , though c o m p u t a t i o n a l l y the method i s t e d i o u s . For a r r a y s of more th a n about seven elements the s o l u t i o n of the s e t of e q u a t i o n s (2.6) becomes e x c e e d i n g l y d i f f i c u l t . The two methods based on the u n i f o r m - a r r a y r e p r e s e n t a t i o n of the d e s i r e d p a t t e r n a re g e n e r a l l y u n s a t i s f a c t o r y . W i l l e y ' s method becomes i n c r e a s i n g l y d i f f i c u l t f o r l a r g e r a r r a y s and H a r r i n g t o n ' s method f a i l s i n s y n t h e s i z i n g p a t t e r n s f a r removed from the p a t t e r n of a u n i f o r m l y - e x c i t e d , u n i f o r m l y -spaced a r r a y . 51 UO F i g . 4.17 Quadrature Method Synthesis of 15 dB Sidelobe Pattern 0 0.5 10 /.5 2.0 F i g . 4.18 15 element A r r a y by Quadrature Method Fig. 4.19 21 element Array by Quadrature Method v PROPERTY METHOD ELEMENT NUMBER RESTRICTION REFERENCE PATTERN FORMULATION SYNTHESIZED PARAMETERS RESULTING ELEMENT EXCITATION MATHEMATICAL TECHNIQUE FOR SOLUTION COMPUTING FEASABIUTY FOR LARGE ARRAYS RANGE OF U FOR WHICH SIDELOBES ARE WITHIN 1% OF DESIGN LEVEL FOR 21 ELEMENT ARRAY UNZ NONE FOURIER SERIES SUM POSITIONS AND EXCITATIONS NON-UNIFORM SOLUTION OF SET OF SIMULTANEOUS EQUATIONS VERY POOR IMPRACTICAL TO SYNTHESIZE f ARRAYS OF MORE THAN ABOUT SEVEN ELEMENTS J HARRINGTON NUMBER OF ELEMENTS IN UNPERTURBED ARRAY UNIFORM ARRAY PATTERN POSITIONS UNIFORM NUMERICAL EVALUATION OF AN INTEGRAL FAIR WILLEY NONE UNIFORM ARRAY PATTERN POSITIONS UNIFORM SOLUTION OF SET OF SIMULTANEOUS EQUATIONS VERY POOR ISHIMARU NONE SOURCE POSITIONS UNIFORM ITERATION OF SINGLE EQUATION GOOD 0 ( U 4 0-75 MAFrETT i NONE LINE SOURCE PATTERN POSITIONS QUASI-UNIFORM ITERATION OF SINGLE EQUATION GOOD 0 * U * 0*0 QUADRATURE NONE LINE SOURCE PATTERN POSITIONS AND EXCITATIONS NON-UNIFORM EVALUATION OF A FUNCTION VERY GOOD 0 4 U * 1-20 F i g . 4.20 P r o p e r t i e s o f t h e Methods 55 The methods s t a r t i n g from the l i n e - s o u r c e f o m u l a t i o n , i . e . w i t h the d e s i r e d p a t t e r n i n the form of an i n t e g r a l , a r e much e a s i e r t o a p p l y t o l a r g e r a r r a y s . The methods y i e l d q u a l i t a t i v e l y s i m i l a r r e s u l t s i n t h a t the a p p r o x i m a t i o n i s e x c e l l e n t from the main beam out t o a p o i n t , a f t e r which the s i d e l o b e s d e p a r t r a p i d l y from the e q u a l - s i d e l o b e d e s i g n . Of t h e s e t h r e e methods the q u a d r a t u r e method g i v e s the l a r g e s t r e g i o n of good a p p r o x i m a t i o n , w h i l e s t i l l o f f e r i n g no element s a v i n g over the u n i f o r m l y - s p a c e d a r r a y . 4.10 Comparison w i t h U n i f o r m l y - S p a c e d A r r a y The q u e s t i o n has been posed as t o whether i t might be advantageous t o use a n o n - u n i f o r m l y - s p a c e d -array i n p l a c e of a u n i f o r m l y - s p a c e d a r r a y . The p a t t e r n of a u n i f o r m l y - s p a c e d a r r a y , a 21 element Dolph-Chebychev a r r a y i n a 10\ a p e r t u r e , i s shown i n F i g . 4.21 f o r a s i d e l o b e l e v e l of 15 dB. The main beam produced by t h i s a r r a y may be scanned t o w i t h i n h a l f a beamwidth of e n d - f i r e w i t h o u t an i n c r e a s e i n the s i d e l o b e . l e v e l . None of the s y n t h e s i s t e c h n i q u e s have produced a r r a y s w i t h the same beamwidth and s c a n n i n g c a p a b i l i t i e s . I f beam s c a n n i n g i s not r e q u i r e d . j i . e . o n l y the v i s i b l e r e g i o n of the p a t t e r n need be c o n s i d e r e d , then the q u a d r a t u r e method w i l l produce a 17 element a r r a y i n a 10\ a p e r t u r e w i t h e q u a l - l e v e l s i d e l o b e s . However, i f no s c a n n i n g i s r e q u i r e d , the number of elements i n the Dolph-Chebychev a r r a y can be decreased u n t i l the second main beam has moved i n t o the edge of the v i s i b l e r e g i o n . The p a t t e r n of an 11 element a r r a y (one wavelength s p a c i n g ) i s shown i n F i g . 4.22. Thus i t would appear t h a t no element F i g . 4 .21 Dolph-Chebychev Array Pat t e r n , 21 elements with 15 dB Sidelobe L e v e l F i g ; 4.22 Dolph-Chebychev A r r a y P a t t e r n , 11 elements w i t h 15 dB S i d e l o b e s 58 s a v i n g can be made u s i n g these s y n t h e s i s methods. 27 Snover et a l . have designed a r r a y s by a t r i a l and e r r o r method' which they c l a i m t o be an improvement on the Dolph-Chebychev a r r a y f o r a g i v e n number of elements and a g i v e n s i d e l o b e l e v e l . However, the s i d e l o b e l e v e l s which they quote f o r t h e i r a r r a y s h o l d o n l y i n the v i s i b l e r e g i o n and t h e r e f o r e s h o u l d be compared w i t h a Dolph-Chebychev a r r a y of one wavelength s p a c i n g , not h a l f - w a v e l e n g t h s p a c i n g as t h e y have done. I f t h i s i s done than t h e r e i s no s a v i n g of elements. 59 5. CONCLUSIONS The p r e c e d i n g c h a p t e r s have examined the t h e o r y of f i v e e x i s t i n g methods f o r the s y n t h e s i s of n o n - u n i f o r m l y - s p a c e d a r r a y s . The methods were c a t e g o r i z e d a c c o r d i n g t o the mat h e m a t i c a l f o r m u l a t i o n t h a t the r e q u i r e d p a t t e r n t a k e s i . e . e i t h e r a F o u r i e r s e r i e s sum, the p a t t e r n of a u n i f o r m l y - s p a c e d a r r a y or the p a t t e r n of a l i n e s o u r c e . I n the course of t h i s e x a m i n a t i o n a new s y n t h e s i s t e c h n i q u e f a l l i n g i n t o the l a s t of the g i v e n c a t e g o r i e s was d e v i s e d . P r e v i o u s workers have made l i t t l e s t u d y o f the u s e f u l n e s s of the e x i s t i n g methods. As a r e s u l t , the p r a c t i c a l v a l u e of n o n - u n i f o r m l y \u00E2\u0080\u0094 s p a c e d a r r a y s i s i n some doubt. Furthermore, some i n v a l i d comparisons w i t h u n i f o r m l y - s p a c e d a r r a y s have r e s u l t e d i n m i s l e a d i n g c l a i m s . A comparison o f a l l these methods has demonstrated t h a t the new t e c h n i q u e based on n u m e r i c a l q u a d r a t u r e i s s u p e r i o r i n two r e s p e c t s . F i r s t l y i t produces a more a c c u r a t e s y n t h e s i s of the e q u a l - s i d e l o b e type of p a t t e r n , and s e c o n d l y i t i s the s i m p l e s t method t o use. T h i s second c h a r a c t e r i s t i c makes the method e s p e c i a l l y u s e f u l f o r h a n d l i n g a r r a y s w i t h l a r g e numbers of elements. D e s p i t e t h i s , no improvement over the u n i f o r m l y - s p a c e d a r r a y was demonstrated s i n c e Dolph-Chebychev a r r a y s c o u l d always be found which used fewer elements f o r the same p a t t e r n . T h i s might be due i n p a r t t o the p a r t i c u l a r c h o i c e of the p a t t e r n t o be s y n t h e s i z e d , s i n c e a u n i f o r m l y - s p a c e d a r r a y can 60 produce an e q u a l - s i d e l o b e p a t t e r n e x a c t l y , whereas a n o n - u n i f o r m l y spaced a r r a y cannot. For l a r g e r a r r a y s s y n t h e s i z e d by the quad-r a t u r e method, i t was found t h a t t h e i r performance improved as the \"-number of elements i n c r e a s e d . T h i s may i n d i c a t e t h a t an improvement over the u n i f o r m l y spaced a r r a y might be a c h i e v e d when e x t r e m e l y l a r g e a r r a y s are b e i n g c o n s i d e r e d . The v a l u e of n o n - u n i f o r m l y \u00E2\u0080\u0094 s p a c e d a r r a y s i s c l e a r l y i n doubt.' 'However no f i n a l v e r d i c t i s p o s s i b l e u n t i l a t r u e o p t i m i z a t i o n procedure i s found: one which a d j u s t s the p o s i t -i o n s , the a m p l i t u d e s and the phases of the elements t o a c h i e v e the b e s t c o m b i n a t i o n of beamwidth and s i d e l o b e l e v e l . T h i s problem has r e s i s t e d the e f f o r t s of many worker s , and e a r l i e r a t tempts by t h i s a u t h o r were no more s u c c e s s f u l . 61 APPENDIX I 1. The T a y l o r L i n e Source T h i s i s based on the e q u a l - r i p p l e p r o p e r t y of the Chebychev p o l y n o m i a l s (as i s i t s d i s c r e t e source c o u n t e r p a r t , the Dolph-Chebychev a r r a y ) . T a y l o r combined two p o l y n o m i a l s t o g i v e the form of F i g . A . l , The f u n c t i o n i s W 2 N = T N ( B - a 2 z 2 ) itA 1 where a i s a c o n s t a n t , B = cosh , A = \u00E2\u0080\u0094 a r c c o s h i j , b e i n g the h e i g h t of the main beam. S i n c e TJJ(Z) = cos(N a r c cos z) the z e r o s of are g i v e n by 1 T-r, /nit it > Z n = \u00E2\u0080\u0094 a L C O s ( ~ ~ . 2 N ' L e t t i n g the o r d e r , N, tend t o i n f i n i t y but k e e p i n g the p o s i t i o n 6 2 of the f i r s t z e r o f i x e d , the z e r o s become s n = \u00C2\u00B1 [ A 2 + (n - i ) 2 ] The c o r r e s p o n d i n g space f a c t o r has u n i t y a m p l i t u d e s i d e l o b e s and a main beam a m p l i t u d e , rj_ , and i s F ( z ) = C cos 3 t(z -A ) cosh ata o r , p u t t i n g C = cosh rca, F ( z ) = cos [ j t ( z 2 - A 2) J S i n c e the r e m o t e ' s i d e l o b e s do not decay, t h i s ' i d e a l ' space f a c t o r i s u n r e a l i z a b l e . I f the z s c a l e i s s t r e t c h e d by a f a c t o r "Thesis/Dissertation"@en . "10.14288/1.0103266"@en . "eng"@en . "Electrical and Computer Engineering"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "Synthesis methods for antenna arrays with non-uniformly-spaced elements"@en . "Text"@en . "http://hdl.handle.net/2429/36310"@en .