N O V E L A C O U S T IC A R R A Y S A N D A R R A Y
P A T T E R N S Y N T H E S IS M E T H O D S
by LIX U E W U
B.Sc., E ast C h irn N orm al U niversity, 1982 M .A .Sc., U niversity of V icto ria, 1989
A D ISSE R TA TIO N S U B M IT T E D IN PA R T IA L F U L F IL L M E N T O F T H E R E Q U IR E M E N T S F O R T H E D E G R E E O F
D O C T O R O F P H IL O S O P H Y in th e D e p a rtm e n t of
A C C L ' P J E D Fit rtric a l an d C o m p u te r E ngineering
FAC ULTY 01 G R A D U A T E STUDIES
We a ccep t th is d isse rta tio n as conform ing to th e s q u i r e d s ta n d a rd
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I)r. A. ? u e lin s k i(7 ^ p e ^ is o r (D e p t, of E lect. Sz C om p. Eng.)
Dr. P. A gathoklis, D e p a rtm e n ta l M em ber, (D e p t, of E lect. Sz C om p. EngA
D r. R. L. Ivirlin, D e p a rtm e n ta l M em ber, (D e p t, of Elect." &; C om p. E ng.)
D r. R. M. C lem ents, O u tsid e M em ber, (D ep t, of P hysics & A stronom y)
y / X ) x . J. S. B ird, E x te rn a l E x am in er, (Sim on F raser U niv ersity )
© LIX U E W U , 1992 U N IV E R S IT Y O F V IC T O R IA
All rights reserved. This dissertation m a y not be reproduced in whole or in part by mimeograph or other means,
11 Supervisor: Dr. A. Zielinski
ABSTRACT
D irectional acoustic beam s a re used in diverse sonar system s. For efficient transm ission of a sonar signal, th e sound energy is pro jected in a narrow b eam . For reduced interference in reception, the sound signal is received from a narrow sp atial sector. T ypically, such beam s have associated sidelobes which adversely affect sonar perform ance.
T h e goal of th is thesis is to propose several novel acoustic array s w hich are capable of gen eratin g desired search-light-type and fan-type beam s w ith g reatly reduced sidelobes. T hese novel acoustic array s have fewer elem ents th a n con ventional arrays of sim ilar perform ance. T h e design of such novel a rra y s is in herently m ore difficult, however, since it involves nonlinear o p tim iz a tio n . Such an o p tim iz a tio n is norm ally co m p u tatio n ally intensive and m ay not be globally convergent.
T his difficulty has been overcom e by new ly developed concepts and associated array p a tte rn synthesis m ethods. A now concept called th e equivalent lin ear array is introduced; a design m eth o d based ou th is concept benefits from ex istin g design techniques developed for linear arrays. T h e equivalent linear a rra y concept is fur th e r developed to lead to a new a n d effective m eth o d for array ra d ia tio n p a tte rn synthesis. A second new concept called th e scale-invariance rad ia tio n p a tte rn is in tro d u ced , and th e subsequent relation betw een two novel arrays is discovered. Using th is concept an angle m ap p in g approach is developed which tran sfo rm s a radiat ion p a tte rn g e n e ra te d by a circular rin g array to th a t of an elliptic ring a r ray. T h is approach takes advantage of m ethodologies developed for th e design of circular ring array s. A th ird concept, c o n stra in t directions, is in tro d u ced ; a su b sequent new ite ra tiv e m eth o d for a rra y p a tte rn synthesis is developed to m ee t th e need in com pact re c e iv in g /tra n sm ittin g a rra y design. W ith th e help of th ese new concepts, th e proposed synthesis m ethods avoid the use of nonlinear o p tim iz a tio n
I l l
tech n iq u es an d m erely req u ire sim ple m a trix o p eratio n s. T h e m ethods can he a p plied to th e p ro b lem s of synthesizing rad ia tio n p a tte rn s of a rra y s w ith a rb itra ry sidelobe envelopes, w ith nonisotropic elem ents, and w ith :i' spacing b e tw een elem en ts. T h e usefulness of th e developed m ethodologies is d e m o n stra ted in various design exam ples. T h e m eth o d s developed provide pow erruI tools not only to design novel aco ustic array s b u t also to design a n te n n a arrays.
E xam iners:
w
Dr. A . Z ie lin sk i, S upervisor (D ep t, of Elect., k C om p. Eng.)
r ?
_
D r. P. A gathoklis, D e p a rtm e n ta l M em ber, (D ept, ot Elect. k C om p. Eng.
' ' f ' ...
D r. R. L. K irlin , D e p a rtm e n ta l M em ber, (D ep t, of Elect, k Com]). Eng.)
D r. R. M. C lem en ts, O u tsid e Meml>er, (D e p t, of Physics k As'
/ f C . J . S. Birch E x te rn a l E x am in er, (Sim on Eraser C niversiiy) / '
^
iv
T ab le o f C o n te n ts
T itle Page i
A bstract ii
Table o f C ontents iv
List o f Tables vii
List o f Figures viii
A bb reviation s x iv
A cknow ledgem ents x v
1 In trodu ction 1
1.1 M o tiv a tio n ... 1 1.1.1 Sonar S y s te m s ... 1 1.1.2 A coustic T ransducers and th e ir R ad iatio n P a tte rn s . . . . 3 1.1.3 Sidelobes in R ad iation P a t t e r n s ... 5 1.2 L ite ra tu re R e v ie w ... 9 1.3 O rganization of th e T h e s i s ... 12
2 N ovel Arrays o f Circular R ing R adiators 15
2.1 I n tr o d u c ti o n ... 15 2.2 An A rray of C ircu lar R ing R a d i a t o r s ... 16 2.3 Design of an A rray of C ircular Ring R a d i a t o r s ... 20 2.3.1 E quivalent L inear A rray of a C ircular Ring R a d ia to r . . . 21
T A B L E O F C m T E N T S v 2.3.2 Design of an A rray of King R adiators by th e Equivalent
L inear A rray Met h o d ... 22
2.4 S u m m a r y ... 32
3 D esig n C onsiderations 36
3.1 I n t r o d u c ti o n ... 36 3.2 M odified Design P r o c e d u r e s ... 37
3.2.1 P rinciples of C ircular Riiig A rray D e s i g n ... 37 3.2.2 Design of C ircu lar R ing A rrays w ith Nonzero (lap s between
Rings ... 3!) 3.2.3 Design of C ircu lar Ring A rrays w ith Reduced N um ber of
Rings ... It) 3.3 A spects of Im p lem en tatio n of A coustic Ring A r r a y s ... 51 3.3.1 Design T able ... 51 3 3.2 Eliect of A rray F in ite Tolerance on the Ream P a tte rn of a
R ing A rray ... 61 3 4 C o m p act R eceiv in g /T ran sm it t ing C o n fig u ra tio n s... (17 3.5 S u m m a r y ... 82
4 N ovel A co u stic Arrays o f E lliptic R ing R adiators 83 4.1 I n t r o d u c t i o n ... <33 4.2 D irec tiv ity F unction of an Elliptic P i s t o n ... 81 4.3 C ircu la r to E lliptic P isto n T r a n s f o r m a t i o n ... 87 4.4 An A rray of E lliptic R ing R a d i a to r s ... % 4.5 Design of an A rray of E lliptic Ring R adiators ... !)!f
4.5.1 Angle M apping A pproach ... !)!) 4.5.1.1 V alidity of Angle M apping A p p r o a 'd i .... !f!)
4.5.1.2 Design of an A rray of Elliptic Ring R adiators by th e A ngle M apping A p p r o a c h ... 102
T A B L E O E C O N T E N T S vi 1.5.2.1 Equivalent Linear A rray of an E lliptic R ing R a d ia to r 106 1.5.2.2 Design of an A rray of Elliptic Ring R adiators by
th e Equivalent L inear A rray M e t h o d ... 109
1.6 S u m m a r y ... 113
5 Equivalent Linear Array Approach to Array P attern Syn thesis 123 5.1 I n tr o d u c ti o n ... 123
5.2 F o r m u l a ti o n ... 124
5.3 A l g o r i t h m ... 136
5.4 A p p lic a tio n s ... 138
5.5 S u m m a r y ... 153
8 Iterative M eth od for Array P attern Syn thesis 154 6.1 hit r e d u c t i o n ... 154
6.2 F o r m u l a ti o n ... 155
6.3 A l g o r i t h m ... 159
6.1 A p p lic a tio n s ... 161
6.4.1 D olph C hebyshev P a t t e r n s ... 161
6.4.2 M odified D olph-C hebyshev P a tte rn s ... 162
6.4.3 R ad iatio n P a tte r n w ith A rb itra ry Sidelobe Envelope . . . 168
6.1.4 N onuniform ly Spaced A r r a y s ... 175
6.4.5 L inear A rrays w ith N onisotropic E l e m e n t s ... 178
6.5 S u m m ary ... 179
7 Su m m ary and Future R esearch C onsiderations 181 7.1 I n tr o d u c ti o n ... 181
7.2 C o n t r i b u t i o n s ... 181
7.3 Suggestions for F u tu re R e s e a r c h ... 182
A D erivation o f th e O ptim um W eighting C oefficients 184
vn
L ist o f T ab les
2.1 W eighting coefficients of a array of ring rad iato rs for different side lobe suppression l e v e l s ... 35
3.1 W eighting coefficients of a circular ring array for different gap w idths betw een r i n g s ... II 3.2 W eighting coefficients of th e equivalent linear array of a circular
ring ra d ia to r w ith gap of 0.0A for di Hoi out ou ter r a d i i ... 3.3 W eighting coefficients of th e equivalent linear array of a circulm
ring ra d ia to r w ith gap of 0.1 A for different ou ter radii . . .
3.4 W eighting coefficients of th e equivalent, linear array of a eirenlai ring ra d ia to r w ith gap of 0.2A for different ou ter r a d i i ... 3.5 W eighting coefficients of ring array for different pi • on d iam eters
r»s
til
SI
4.1 W eighting coefficients and ring sizes of an array of elliptic ring ra d ia to rs for different sidelobe suppression l e v e l s ... . . . I!5
5.1 W eighting coefficients of a dipole array for different, p ro to ty p e sizes I 12 5.2 C u rren ts and radii of a circular loop array for dilTerent sidelobe
suppression l e v e l s ... 151 5.3 C u rren ts and radii of an a d ju sted circular loop array for different
sidelobe suppression l e v e l s ... 152
6.1 W eighting coefficients for radiation p a tte rn s of big. 6 . 2 ... 167 6.2 W eighting coefficients for radiation p a tte rn s of Fig. 6 . 3 ... 173
V lll
L ist o f F ig u r e s
1.1 ( loom etry of sidescan sonar and to p o g ra p h , of b o tto m , (a) V ertical profile of ship an d “fish” moving into th e page and sonar beam aim ed ath w a rtsh ip . (b) Plan view of raw (uncorrected) sidescan im age... 2 1.2 Typical rad ia tio n p a tte rn of a sidescan sonar, (a) P lan view, (b)
Vertical profile... 4 1.3 T hree-dim ensional rad iatio n p a tte rn of a circular piston tra n sd u c er. 6 1.4 T hree-dim ensional rad iatio n p a tte rn s of a linear a rra y of re c ta n
gular tra n sd u c ers, (a) U niform w eighting of elem ents, (b) Dolph- C hebyshev w eighting w ith p o stu la te d 30 dB sidelobe suppression. 7
2.1 (le o m e try of an a rra y of circular ring rad ia to rs (3 rings shown). . 1.7 2.2 Beam p a tte rn s of a disc rad ia to r w ith r — 5A and its equivalent
linear a rra y tru n c a te d to 20 elem ents (10 te rm s )... 24 2.3 Identity of th e a rra y of disc rad iato rs and th e array of contiguous
ring rad ia to rs, (a) C oncentric disc array, (b) E quivalent stacked ring array, (c) Single layer ring a rra y ... 25 2.4 W eighting coefficient of a tru n c a te d equivalent linear a rra y vs. ra
dius of disc ra d ia to r for different values of N r . (a) Linear scale, (b) S em i-logarithm ic scale... 27 2.5 Size of equivalent linear a rra y vs. radius of disc radiato- for different
L I S T O F F I O F R F S ix 2.6 Beam p a tte rn s of an array of 10 ring rad iato rs com part'd with
C hebyshev p ro to ty p e s (d o tted line), (a) P o stu lated 30 dB side lobe suppression in th e prototype. (1>) P ostulated 10 dB sidt'lobe suppression in th e prototype, (c) P ostulated 50 dB sidt'lobe su p pression in th e p ro to ty p e, (d) P ostulated 00 dB sidelobe su p p res sion in th e p ro to ty p e ... 33
3.1 Front face of an array of ring rad iato rs and its equivalent linear array 38 3.2 R a d ia tio n p a tte rn s of an array of 10 ring rad iato rs w ith various
gap sizes and p o stu la te d 30 dB sidelobe suppression in Dolph- C hebyshev p ro to ty p e . (a) (la p = 0.0A. (I>) (la p 0.05A. (c) (la p = 0.1A. (d) (la p — 0.15A. (e) (la p =; 0.2A. (f) (la p -- 0.25A. 11 3.3 W eighting coeflicieuts as a function of ring gap for a 10-ring array.
(a) P o stu la te d 30 dB sidelobe suppression, (b) P o stu ,ated 10
till
shlelobe suppression, (c) P o stu lated 50 <1B sidelobe suppression. (d) P o stu la te d 60 dB sidelobe suppression... 15 3.4 R elativ e erro r as a function of th e num ber of ring rad iato rs fordifferent p o stu la te d sidelobe suppression levels... 51 3.5 R ad iatio n p a tte rn s of ring arrays w ith rem oved central elem ents. . 53 3.6 R ad ia tio n p a tte rn of a 5-ring array w ith p o stu la te d 26 dB sidelobe
suppression in D olph-C hebyshev p ro to ty p e ... 65 3.7 T hree-dim ensional plot of th e radiation p a tte rn of th e 5 ring array. 66 3.8 R a d ia tio n p a tte rn s of a 10-ring array for different p o stulated side
lobe suppression levels in D olph-C hebyshev p rototype, (a) 5 % w eighting coefficient tolerance w ith p o stu la te d 30 dB sidelobe su p pression. (b) 5 % w eighting coefficient, tolerance w ith po stu lated 40 dB sidelobe suppression, (c) 10 % w eighting coefficient tolerance w ith p o stu la te d 30 dB sidelobe suppression, (d) 10 % weighting coefficient to leran ce w ith postu lated 40 dB sidelobe suppression. . 68
U S '! O F F lC l R E S x 3.9 R adiation p a tte rn s of a 10-ring a rra y p o stu la te d 30 dB sidelobe
suppression in D olph-C hebyshev p ro to ty p e, (a) 5 V< o u ter and inner radii tolerance, (h) 10 ’/ o u ter and inner radii tolerance. . . 72 3.10 R adiation p a tte rn s of an arr;._ of 8 rings an d 1 cen tral piston.
(a) P o stu la ted 30 dB sidelobe suppression, (b) P o stu la ted 40 dB sidelobe suppression... 75 3.11 R adiation p a tte rn s of an a rr a y of 7 rings and 1 cen tral piston.
(a) P o stu la ted 30 dB sidelobe suppression, (b) P o stu la ted 40 dB sidelobe suppression... 76 3.12 R adiation p a tte rn s of an array c f 6 rings and 1 cen tral piston.
(a) p o stu la te d 30 dB sidelobe suppression, (b) p o stu la te d 40 dB sidelobe suppression... 77 3.13 R adiation p a tte rn s of an array of 5 rings an d 1 cen tral piston.
(a) P o stu la ted 30 dB sidelone suppression, (b) P o stu la ted 40 dB sidelobe suppression... 78 3.1 1 R adiation p a tte rn s of an array of 4 rings an d 1 c e n tral piston.
(a) P o stu la te d 30 dB sidelobe suppression, (b) P o stu la ted 40 dB sidelobe su p p ression... 79 3.15 T ra n sm ittin g /re c e iv in g two-way rad iatio n p a tte rn s of a 6-ring a r
ray. (a) P o stu la ted 30 dB sidelobe suppression in receiving, (b) P o stu la ted 40 dB sidelobe suppression in receiving... 80
1.1 (le o m e try of an elliptic piston and its equivalent ra d iu s... 85 1.2 R adiation p a tte rn s of an elliptic pisto n tra n sd u c er w ith a — 5A
and b = 0.175a on different planes p e rp e n d icu la r to its surface, (a) c> = 0°. (b) <p = 30°. (c) ^ = 60°. (d) <f> = 90°... 88 1.3 T ransform ations from space angles <f>, 0 to p k to n angle u for an
elliptic piston tran sd u cer, (a) R ad iatio n p a tte rn in th e -‘-dom ain. (b) T ran sfo rm atio n from 9 . r to u. (c) T ransform ation from <j> to r . 92
L I S T O F F I G U R E S xi 4.4 P erspective plots of rad iatio n p a tte rn s of an elliptic piston tra n s
ducer. (a) a — L\, b — ;A. (hi a - 2A, /> — 1A. (c) a — 5A, 6 = 1A... 94 4.5 G eo m etry of an elliptic ring ra d ia to r and its equivalent radii . . . !)7 4.6 Front face of an array of elliptic ring rad ia to rs... !)8 4.7 B earnw idths (3 dB ) 0 of an array of elliptic ring radiators for differ
ent space angle 0 ob tain e d th ro u g h transform ations, (a) R adiation p a tte rn in u-dom ain. (h) T ransform ation from 0 ,r to u. (c) T ra n s fo rm atio n from 0 to r ... 104 4.8 Front face of an elliptic ring ra d ia to r and its equivalent linear arrays. IPS 4.9 R ad ia tio n p a tte rn s of an elliptic ring rad ia to r with a x -■ 4.5A,
«;_i = 4.1 A, bi = 0.2a,-, 6,_i — 0.2«,-_i and its equivalent linear a r rays tru n c a te d to 18 elem ents (9 term s) for different. 0. (a) </> = 0°. (b) 0 = 30°. (c) 0 = 60°. (cl) 0 - 90°... I l l 4.10 (le o m e try of an array of 10 elliptic ring ra d ia to rs ... I l l 4.11 R ad iatio n p a tte rn s of an array of elliptic ring rad iato rs with p o s
tu la te d 30 dB sidelobe suppression for different <l>. (a) r/> = 0°. (I>) 0 = 30°. (c) 0 = 60°. (d) 0 = 90“... 116 4.12 R a d ia tio n p a tte rn s of an array of elliptic ring rad iato rs with pos
tu la te d 40 dB sidelobe suppression for different 0. (a) <j>— 0°. (b)
0 = 30°. (c) 0 = 60°. (cl) 0 = 90°... 118 4.13 T h ree-dim ensional rad iatio n p a tte rn s of an a rra y of elliptic* ring ra
diato rs. (a) P o stu la te d 30 dB sidelobe suppression, (b) P ostulated 40 dB sidelobe su p p ressio n... 120
I 1ST O F F l O l ' l t F S xii 5.2 Kquivalent linear array concept for a 1-element, array, (a) A linear
aw ay >f 1 nonisotropic elem ents weighted by wt. i — —2, —1 ,1 ,2 . (b) Each elem ent is represented by its equivalent linear array tru n cated to 4 elem ents, (e) T he equivalent array of 7 isotropic elem ents lbr th e 4-nonisotropic-elem ent a rra y is o b tain ed by superposition. 132 5.3 R adiation p a tte rn s of a dipole lin ear a rra y w ith w eighting coeffi
cients o b tain e d by th e D olph-C hebyshev m e th o d ... 139 5.4 R adiation p a tte rn s of a dipole and its equivalent linear array. . . . 140 5.5 R adiation p a tte rn s of a 20-dipole array, (a) 15 elem ents in th e
p ro to ty p e, (b) 17 elem ents in th e p ro to ty p e ... 143 5.6 R ad iatio n p a tte rn s of a 20-dipole array, (a) 19 elem ents in th e
p ro to ty p e, (b) 21 elem ents in th e p ro to ty p e ... 144 5.7 R adiation p a tte rn s of a 20-dipole array, (a) 23 elem ents in th e
p ro to ty p e, (b) 25 elem ents in th e p ro to ty p e ... 145 5.S Linear array of 10 equally spaced circular loops each w ith a different
ra d iu s ... 146 5.9 R ad iatio n p a tte rn of a 10-circular-loop a rra y p o stu la te d sidelobe
suppression of 30 d B ... 148 5.10 R ad iatio n p a tte rn s of a 10-circular-loop a rra y for different design
m ethods, (a) P o stu la ted sidelobe suppression of 33 dB . (b) P o stu lated sidelobe suppression of 44 d B ... 149 5.11 R ad iatio n p a tte rn s of a 10-circular-loop a rra y w ith a d ju sted radii
for different design m ethods, (a) P o stu la te d sidelobe suppression of 33 dB . (b) P o stu la ted sidelobe suppression of 44 d B ... 150
xm 6.2 T he evolution of a D olph-C hebyshev p a tte rn with a postulated 20
dB sidelobe suppression, (a) S ta rtin g p a tte rn : uniform weighting array, (b) R esulting p a tte rn after th e first iteratio n , ( c ) R esulting p a tte rn after th e second ite ratio n , (d) R esulting p a tte rn after the th ird ite r a tio n ... 162 6.3 T h e evolution of a m odified D olph-C hebyshev p a tte rn with the
th re e innerm ost pairs of sidelobes a t -15 dB and th e next, six pairs at -30 dB. (a) S ta rtin g p a tte rn : D olph-Chebyshev array with a. 30 dB sidelobe suppression, (b) R esulting p a tte rn after the first, ite ra tio n , (c) R esulting p a tte rn after th e second iteratio n , (d) R esulting p a tte rn after th e th ird ite ra tio n ... 16!) 6.4 R ad ia tio n p a tte rn s of a tran sm it tin g /re c e iv in g array of 20 elem ents
w ith p rescrib ed tra n s m ittin g w eighting coefficients. T h e receiving w eighting coefficients are o b tained by th e proposed ite rativ e al g o rith m . (a) T ra n sm ittin g and receiving p a tte rn s, (b) T ra n sm it tin g /re c e iv in g 2-wa.y rad iatio n p a tte rn of ( a ) ... 176 6.5 R ad ia tio n p a tte rn s of a 20-element, linear array w ith uouuniform
spacing p o stu la te d 30 dB sidelobe suppression for different, sy n th e sis m e th o d s... 17b 6.6 R ad ia tio n p a tte rn s of a linear array of 20 dipole elem ents p ostulated
xiv
A b b re v ia tio n s
2-1) Tw o-dim ensional .‘J-I) Th ree- d i m ension al KLA Kquivalenl linear array
A ck n o w led g m en ts
XV
I w ould like to th a n k m y supervisor. Dr. A. Zielinski of th e D epartm ent of E lectrical and C o m p u te r E ngineering, for his encouragem ent, guidance, and advice d u rin g th e course of th is lvsearch and for his help in th e preparation of th is thesis.
F in an cial assistance received from Dr. A. Zielinski (through th e N atural Sci ences and E ngineering R esearch Council of C an ad a), the U niversity of V ictoria, a n d th e BC A dvanced System s Institute' is gratefully acknowledged.
I am gratefu l to m y p a re n ts for m aking it possible for me to becom e what 1 am . A special an d w arm expression of g ra titu d e is reserved for my son, Ken YVu, for his u n d e rsta n d in g a n d m any sacrifice's.
C h a p te r 1
I n tr o d u c tio n
1.1
M o tiv a tio n
1.1.1
S onar S y ste m s
Sensing th e ocean beyond visual iange requires sonar system s [I]. O b jects such as subm ersibles, ships a n d u n d erw ater beacons rad ia te acoustic energy and passive sonars can be used to d e te rm in e th eir presence and th e ir acoustical characteristics. On th e o th er h a n d , in active sonar a directional beam of svuind in th e form of a sh o rt b u rst of energy is tra n s m itte d . If an object is in the beam , som e of the sound is b a c k sc a tte re d . T h e back scattered sound echo is d etected bv a receiver. T he received signal is usually displayed as a function of travel tim e and direction. T h e com m on d e p th sounder is an exam ple of this kind of sonar. Sidescan som«. uses a fan -ty p e b eam which is narrow in one dim ension ( ~ 1°) and relatively wide ( ~ 30°) in th e o th er. A short pulse is tra n s m itte d along th e beam and the reflected echo from th e b o tto m is sequentially recorded. T h e record represents an echo from a narrow s trip of th e b o tto m . Tin* sonar is continuously m oving forw ard, and th e accu m u latio n of successive* retu rn s produces an overall im age of th e b o tto m (F ig. 1.1).
T h e ra d ia tio n p a tte rn of a tra n sm ittin g /re c e iv in g t ransducer plays an im p o r ta n t ro le in sy ste m p erform ance, and therefore it should he designed to suit various needs. O f p a rtic u la r in te rest is th e presence of sidelobes in th e radiation p a tte rn
Sea surface Cable Sidescan fish Acoustic array Acoustic
ray paths Bump on
seafloor Seafloor Direction of fish travel No return First bottom return Return from side of target facing fish Acoustic shadow behind target Gray zone of normal regional backscatter (b)
Figure 1.1: G eo m etry of sidescan sonar and topography of b o tto m , (a) V ertical profile of ship and “fish" m oving in to th e page and sonar beam aim ed a th w a rtsh ip . (b) P la n view of raw (u n co rrected ) sidescan image.
as illu s tra te d in Fig. 1.2. These sidelobes adversely a ile d sonar perform ance since signals a re a d m itte d from sidelobe directions in addition to signals being d etected in th e m ain lobe direction.
1 .1 .2
A c o u stic T ransdu cers and th e ir R a d ia tio n P a tte r n s
T o d a y 's m ost com m on acoustic tran sd u cers are slabs of piezoelectric m aterials t h a t e x p a n d , c o n tra c t, or change shape when electrical voltage's are' applie'el. Since 1950, ceram ic m a te ria ls have been used be'cause of th eir high o u tp u t, m eehaui cal s tre n g th , a n d ab ility to be shapeel. A e-eramic tran sd u cer com m only has a th in co n d u c tin g electro d e covering each face1. A voltage applie'el across the olec- tro d e s causes th e thickness of th e slab to increase; rewe'rsal of the voltage' cause's t:.- thickness to decrease. W hen th e e'le'me'iit is in wate'r, the'se* e'xpansious and c o n tra ctio n s cause sound rad iatio n [2].O ften a prin cip al goal in tra n sd u c er or tran sd u cer array ele'sign is to ae'hieve* a specified ra d ia tio n p a tte rn p(0,(j)), irpre'senting penver density in w atts pen sepiare m e te r, th ro u g h su ita b le geo m etry a n d /o r arrangem ent of sources. The1 spe'cilied p a tte rn fre q u e n tly em bodies th e in te n t to enhance radiation in ce'rtain d iirc tio n s an d to suppress it elsew here. Sonar system s use' such directional transducers. In tra n sm issio n , th e sound energy is projected along a narrow beam to m axim ize o u tp u t in a c e rta in direction. In reception, th e sound signal is received from a narrow sp a tia l sector to reduce interference from o th er sectors. The o th er im p o rta n t use of d ire c tio n a lity is to resolve th e backscattered signals from closely spaced ob jects.
T h e d ire c tio n a l responses of th e tra n sd u c er are the sam e when used as a source an d as a receiver. A useful m easure of this directional response is th e power d ire c tiv ity function, which is th e rad ia te d power d e n si'v p(0,cj>) in the* direction (0, $ ) divided by th e ra d ia te d power density averaged over all directions; th a t is,
Di0d>) = ______________ p ( M ) ______________ (] , ( 1/47Tr2) Jq71 p(0', <t>')r2 sin 0'd0'<lfp'
t
Directionof travel I Distance across track
Horizontal beamwidth Sidelobes Sidescan "fish" Linear array Sidelobes Array beam! pattern Vertical beamwidth Seafloor
F igure 1.2: Typical rad ia tio n p a tte rn of a sidescan sonar, (a) P la n view. V ertical profile.
w here /• is th e d ista n c e from th e tra n sd u c er and is assum ed large enough to ensure th e far field of th e tran sd u cer.
1.1.3
S id e lo b e s in R a d ia tio n P a tte r n s
D irectional acoustic beam s, such as search-light-type or fan-type beam s, are widely used in sonar system s. A search-light-type beam can be generated by a circular piston tra n sd u c e r. T h e three-dim ensional radiation p a tte rn of such a tra n sd u c er is show n in Fig. 1.3. A rec tan g u la r tra n sd u c er can be used to produce a fan-type b eam . T h e th ree-d im en sio n al rad iatio n p a tte rn of such a tran sd u cer is shown in Fig. l.-l(a). However, th e m ajo r problem encountered in these* t ransducers is th e existence of sidelobes in th e rad iatio n p a tte rn s. T he first sidelobe* le*ve*l of a c ircu lar piston is a b o u t 17.6 dB down from the* m ain lobe m axim um , whe*re*d.s the* rec tan g u la r tra n s d u c e r offers only 13.5 dB suppression of its lirst side*le>be> [3].
In fan -ty p e b e a m applications it is possible* to use* a line*ar array of r e c t a n g u l a r tra n sd u c e rs to co n tro l th e rad ia tio n p a tte rn in the* narrow be*am elirc*ction by th e a p p lic atio n of p ro p er weights to each element.. In the broad be*am elire*cl’em, how ever, th e ra d ia tio n p a tte rn is d eterm ined bv th e width of rectan g u lar e*le*ment s and still ex h ib its large first sidelobes (13.5 dB down) as shown in Fig. 1.4(b).
A re c ta n g u la r p lan a r a rra y can Ir* used to re'duce* side*lobe*s for both se*arch lig h t-ty p e and fan ty p e b eam s by su itab ly w eighting its ele*ment,s; howe*ve*r, the* n u m b er of e lem en ts required to accom plish this task is ab o u t M / .V, whe*re* A7, N are th e a rra y dim ensions expressed in m ultiples of half wave)e*ngth [3]. T h is re*sult.s in com plexity, n o t only to th e power d istrib u tio n in transm ission b ut alse> to the* su m m in g netw ork in reception.
T h e goal of th is thesis is to propose several novel acoustic arrays which are* ca pab le of g e n eratin g th e desired beam s with g reatly reduced sieh'lobe* h*ve*ls. The*se* novel acoustic a rra y s require fewer elem ents t han p ian a r arrays to achie*ve* the* same* perform ance. T h e design of such arrays is inhe.*re*nt ly more* difficult, howe*ve*r, since* it involves non lin ear o p tim iz a tio n , a c o m p u ta tio n a lly intensive* te*chnique th a t m ay
6
(a)
F ig u re 1.4: T hree-dim ensional rad iatio n p a tte rn s of a linear array of rectangular tra n sd u c e rs, (a ) U niform w eighting of elem ents, (b) Dolph ( 'hebyshev weighting w ith p o s tu la te d 30 dB sidelobe1 suppression.
8
0
>)9 not be globally convergent [•!]. The diesis presents new concepts and associated array p a tte rn synthesis m ethods to overcom e these difficulties. The new synthesis m eth o d s provide powerful tools not only for the design of novel acoustic arrays b u t also for a n te n n a arrays.
1.2
L ite r a tu r e R e v ie w
In th e m id-1940s, w ith th e advent of high perform ance ceram ic piezoelectric m a terials such as b a riu m tita n a te , it becam e possible to m ake transducer elem ents into sh a p es such as rods, hollow cylinders, hollow spheres, sect ions of spheres, and parab o lo id s [5], Each sh a p e is ch aracterized by a d istinctive directional p attern when u se d as a source o r receiver. Recently developed piezoelectric polym eric m a te ria ls [6], such as K Y N A R , available in th e form of film and wire, now e n able c o n stru ctio n of inexpensive a n d innovative acoustic a n ays. T h e use of arrays in stead of single ra d ia to rs m akes it possible' to tailor radiation p a tte rn s to a l m ost a n y desired shape as well as to increase th e gain and the pow er-handling capability.
A d ire c t way of designing a tra n sd u c e r w ith a desired radiation p a tte rn is to co n tro l the se n sitiv ity d istrib u tio n on th e lace of the tran sd u cer. T h is treat, m ent is called shading, a n d the se n sitiv ity d istrib u tio n is calk'd an ap e rtu re . In a classical pap er, Taylor [7] derived a sm ooth a p e rtu re which yields th e m ini m um b e a m w id th for a specified sidelobe level. If the tra n sd u c er can be m ade to v ib ra te in such a way th a t its a p e rtu re function follows a G aussian function, th e n t h e ra d ia tio n p a tte rn is also G aussian and has no sidelobes [b]. Because of these a ttra c tiv e ch a ra c te ristic s, researchers have a tte m p te d to produce an efficient tru n c a te d G aussian tran sd u cer; th e ir efforts are rep o rted in th e lite ra tu re . Von H aselberg and K ia u tk ra m e r [9] used an a rra y of oct agonal tran sd u cers to approx im a te a G aussian function. M artin and B reazeale [10] and Du and B reazeale [1 Ij showed t h a t G aussian b eam s can be o b tain ed by having full plat ing on one face of
10
1.1k* circular piston elem ent and a sm all electrode 011 th e o th er face. Zerwekh and Claus [12] and Claus: and Zerwekh [13] developed a G aussian tra n sd u c er th a t has a num ber of concentric ring electrodes, each driven by a different voltage (piecew ise G aussian) provided by a voltage divided netw ork. All th e above design m eth o d s followed th e approach of e xciting a uniform ly polarized piezoelectric elem ent w ith nonuniform (G aussian) driving voltage and field.
From the practical point of view, however, construction of tran sd u cers w ith a continuously varying sen sitiv ity d istrib u tio n is a com plex task. O ne way to re duce th is com plexity is to ap p ro x im ate the required sensitivity d istrib u tio n on th e tra n s d u c e r’s face w ith a sm all nu m b er of c o n stan t sen sitiv ity segm ents. T h e early work of using discre e c o n stan t sensitivity segm ents to ap p ro x im ate a continuous sensitivity was rep o rted by th e N ational Defense R esearch C o m m itte e [14, 15]; however, th eir work was lim ited to ex p e rim e n tatio n w ith only 2- a n d 3-level dis- trib u tio n s. b a le r, M a rtin and H ickm an [16] considered th e problem of o p tim izing a bi-level sensitivity ra tio betw een th e c e n tral and o u te r segm ents and segm ent size ratio . D rost [17] approached the' problem of optim izing a bi-level sen sitiv ity d istrib u tio n bv coherently adding th e p red icted rad iatio n p a tte rn s of th e tw o lev els. T h e size's of th e segm ents were chosen so th a t th e first positive sidelobe of th e wider segm ent canceled th e first negative sidelobe of th e narrow er one. H e th e n varied th e sen sitiv ity ratio to m inim ize all th e sidelobes. R ecently Zielinski and Wu [18] have used linear op tim izatio n techniques to create arrays of circu lar ring rad ia to rs w ith radial rad iatio n p a tte rn s th a t app ro x im ate th e rad ia tio n p a tte rn s of A/2 spaced linear arrays. M ost recently M cG ehee a n d Jaffe [19] have proposed an o p tim u m tueau sq u are q u an tizer approach to choose the “b e st” segm ent sizes and sen sitiv ity levels for a segm ented tra n sd u c er by m inim izing th e m ean sq u ar° error betw een a desired sen sitiv ity d istrib u tio n and its discrete level a p p ro x im a tion. T h e piecewise a p p ro x im atio n m ay achieve an accep tab le rad ia tio n p a tte rn in a segm ented tra n sd u c e r b u t generally requires a large n u m b er of segm ents.
11 F u rth e rm o re , th e relatio n betw een segm ent size and desired radiation p a tte rn is unknow n.
V arying th e w eighting coefficients of array elem ents provides th e array d e signer w ith th e ab ility to control radiation p a tte rn s. T he designer has th e ability to form a p a tte rn w ith a m ain lobe and sidelobes, to control angular placem ent of th e m ain b e a m , to select b eam sharpness by choosing th e length of the* a r ray, a n d to p ro d u ce a sh ap ed p a tte rn w ithout nulls. All of these* form a basis for p a tte rn synthesis; a large nu m b er of papers have beam elewoteel to this area. Schelkunoff [20] developed th e general co n n p t of the* array polynom ial and th e rela tio n betw een p a tte rn shape and polynom ial z t ros. Doiph [21] ele'te*rmine*el th e a rray w eights for a uniform ly spaced linear array of isotropic elem ents that, yield th e m in im u m b e a m w id th for a specified sidelobe le*ve*l. Villeneuve [22] ele*scribe*d how T ay lo r's m eth o d [7] devedoped for continuous shading can he applied to dis c re te array s. H y n em an [23] a n d H ynem an a n d Johnson refined shaped beam synthesis using p a tte rn zero shifting to m inim ize ripple. Elliot and Stern [25] and O rc h a rd , E lliot, a n d S te rn [26] presen ted additional p a tte rn synthesis techniques for array s. A n o th e r tech n iq u e for p a tte rn synthesis is due to W oodward [27, 2<S], T h is tec h n iq u e is based on th e w eighted superposition of a, num ber of ( s in u ) /« p a tte rn s deflected by different am o u n ts to one or th e o th er side of th e it. - 0 axis, w ith p eak a m p litu d e s su ita b ly chosen. T h e principal m erit of this approach is it s sim plicity; how ever, undesirable ripple betw een sam ple points m akes th e results inferior to those pro d u ced by E llio t’s m ethods. W hite [29] developed a m ethod w hich deals w ith p a tte rn ex trem es ra th e r th a n p a tte rn zeros, and this m ethod m ay b e sim ple to apply in some case. O th e r techniques are described bv Schell an d Ish im aru [30]. All of these techniques, however, are applicable only to arrays consisting of uniform ly spaced elem ents w ith nondirectional radiation p a tte rn s (isotropic elem ents).
12
technique based on a d a p tiv e array theory th a t can be used for arrays consisting of a s e t of elem ents each with an a rb itra ry rad iatio n p a tte rn . T his m eth o d is a num erical technique, a n d however, it does not yield an aly tic solutions for th e required weights.
1.3
O r g a n iz a tio n o f th e T h e s is
This thesis p u ts to g eth er th e research results p ertain in g to two m a jo r them es: Novel Acoustic Arrays and New Ar r ay Pattern Synthesis Methods. T h e following is th e o u tlin e of each ch ap ter.
C H A P T E R 1
T h is ch a p te r introduces th e basic concepts of sonar system s, focusing in p a r ticu lar on directional acoustic beam s and th e existence of sidelobes. T h e devel oped techniques for generating such beam s and th e associated p a tte rn synthesis m ethods are briefly reviewed. T h e m otivation of th e research is addressed. T h e organization of th e thesis is also discussed.
C H A P T E R 2
In th is c h a p te r a novel array of circular ring rad iato rs is proposed w hich gen e rates a sy m m etric search-light-type narrow beam w ith g reatly reduced sidelobes. Such a narrow beam can find several applications rela te d to acoustic rem o te sens ing, te le m e try an d specialized sonars. A design m eth o d is detailed w hich bene- lits from existing design techniques developed for linear arrays. T h e developed m ethodology requires only sim ple m a trix o perations a n d does not involve nonlin ear o p tim iz a tio n . T h e resu lts d e m o n stra te th a t rad iatio n p a tte rn s w ith a rb itra ry sidelobe suppression can be achieved.
C H A P T E R 3
Phis cha.pt.e~ introduces a convenient procedure to design a novel a rra y of circular ring rad ia to rs. Several p ractical aspects associated w ith im p lem en tatio n of such an array are investigated. Specilically, th e im p o rta n t aspect of se n sitiv ity ot th e rad ia tio n p a tte rn to finite tolerances associated w ith a rra y im p lem en tatio n
1:5 is ad d ressed. T h e resu lts are presented in a ta b u la r form convenient for a designer. C h a p te r 3 also describes com pact rec e iv in g /tra n sm ittin g configurations for a pisto n tra n s m ittin g elem ent and a receiving array consisting of th e sam e tra n s m ittin g pisto n su rrounded by several concentric circular rings of piezoelectric film or several concentric hollow cylinders o p eratin g in longitudinal m ode. It is shown th a t such a configuration allows for good sidelobe suppression in th e com bined tra n s m ittin g /re c e iv in g rad ia tio n p a tte rn even for arrays w ith a sm all num ber of elem en ts. A design exam ple shows that a narrow beam of t’° boam w idth with m ore th a n 48 dB sidelobe suppression in tra n sm ittin g /re c e iv in g p a tte rn can be achieved w ith an a rra y consisting of only (i elem ents.
C H A P T E R 4
In th is c h a p te r a novel array of elliptic ring rad iato rs is proposed winch gen e ra te s a fan -ty p e b eam w ith controllable sidelobe level. Such an array can find several ap p licatio n s in specialized sonars such as sidescan sonars, sonars for fish finding a n d stock assessm ent, and obstacle avoidance system s. Tw o possible a p proaches to th e design a re p resented. One approach utilizes a m apping which tra n sfo rm s a ra d ia tio n p a tte rn g en e ra te d by a circular ring array to th a t of an ellip tic ring array. T h is approach takes advantage of m ethodologies developed for th e design of c ircu lar rin g arrays. T he o th er approach uses the concept of an e q u iv alen t lin ear a rra y (E L A ). T h is approach benefits from the existing design tec h n iq u es developed for lin ear array s. It requires only sim ple m atrix operations and does not involve o p tim iz a tio n . T h e design exam ples presented d e m o n stra te th a t a fan -ty p e b eam w ith sidelobes suppressed to m ore th an 40 dB in all possible d ire c tio n s is achievable.
C H A P T E R 5
T h is c h a p te r p resen ts a new and effective m eth o d for array radiation p a tte rn sy nthesis. T h e m eth o d allows th e designer to form ulate th e synthesis of a d e sired p a tte rn as an o p tim iz a tio n problem . T h e proposed solution involves m a td x o p e ra tio n s based on th e linear equivalent array approach, which lead to an easy
14 and effective c o m p u ta tio n . T h e advantage of th is approach is t h a t it requires oidy o rd in ary m a trix o perations instead of nonlinear o p tim ization. Illu stratio n s are presented to highlight th e various aspects of th e m ethod. T he proposed algo rith m can he used to design arrays of uniform ly spaced elem ents w ith nonisotropic and unequal rad ia tio n p a tte rn s.
C H A P T E R 6
In this c h a p te r an ite rativ e m ethod is developed for a rra y p a tte rn synthesis. In th is m ethod th e synthesis problem is form ulated as a w ell-determ ined or over- determ in ed o p tim izatio n problem . T h e problem is th e n sim plified by th e concept of c o n stra in t directions. T h e ite ra tiv e alg o rith m for obtain in g an o p tim u m so lution is presented which does not involve any nonlinear o p tim iz a tio n b u t only requires sim ple m a trix operations. T h e proposed m eth o d can be applied to th e problem s of synthesizing rad ia tio n p a tte rn s of linear arrays w ith a rb itra ry sidelobe envelopes, w ith nonisotropic elem ents, and w ith nonuniform spacing betw een ele m ents. T he resu lts are presented and com pared w ith o th er m eth o d s. It is shown th a t th e proposed ite ra tiv e m eth o d yields superior results.
C H A P T E R 7
A su m m ary of th e im p o rta n t co n trib u tio n s of th e thesis and fu tu re research considerations a re given in th is chapter.
15
C h a p te r 2
N o v e l A rra y s o f C ircu lar R in g
R a d ia to r s
In th is c h a p te r we propose a novel acoustic array of circular ring rad iato rs which can g en e ra te a search -lig h t-ty p e beam w ith greatly reduced sidelobes. We intro duce a new concept called th e tquivalt ill lint ar array and apply it to the design of novel arrays.
2 .1
I n tr o d u c t io n
T h e c irc u la r p isto n is a w idely used p lan a r rad iato r [1]. O ne of tin* a ttra c tiv e fea tu re s of such a ra d ia to r is its sy m m etric, search-light-type radiat ion p a tte rn with th e first sidelobe level of 17.6 dB below the m ain lobe m axim um [B‘2]. However, in som e ap p licatio n s still lower sidelobe levels a n ' desirable. In this c h a p te r we propose an array, consisting of circular ring rad ia to rs, th a t is capable of general ing a su p erio r ra d ia tio n p a tte rn . A design procedure is described which benefits from th e ex istin g design techniques developed for linear arrays. T h e results indi c a te t h a t ra d ia tio n p a tte rn s w ith a rb itra ry sidelobe suppression can be achieved. T h e proposed configuration can readily be im plem ented as a receiving array using piezoelectric film and as a tra n s m ittin g /re c e iv in g array using concentric hollow cylinders o p e ra tin g in lo n g itu d in al m ode.
16
2.2
A n A r ra y o f C ircu la r R in g R a d ia to r s
The d ire c tiv ity function (ra d iatio n p a tte rn ) of a planar, circu lar disc ra d ia to r with rad iu s r , of uniform sensitivity, placed in an infinite rigid baffle, is known to !„■ [33]:
« „ < » ) - r ( 2. , )
(sin 0)1 A
where 0 is th e elevation angle, A is th e w avelength of th e ra d ia te d signal and J i(-) is th e first-order Bessel function of th e first kind. Because of circular sy m m etry the d ire c tiv ity function does not depend on a z im u th angle.
T h e d ire c tiv ity function of a circular ring ra d ia to r can b e readily o b ta in e d by s u b tra c tin g d ire c tiv ity functions of two disc rad ia to rs [34]. In this c h a p te r and ( 'lia p te r 3, we will sim ply call th e circular rin g ra d ia to r a rin g rad ia to r.
W e consider an a rra y form ed by several concentric, contiguous (no gaps be tween rings) ring rad ia to rs. T h e ring rad ia to rs c o n trib u te to th e overall d ire c tiv ity function w ith w eighting coefficients ct. Fig. 2.1 illu strate s an a rra y of th re e rings. Note t h a t we also call th e cen tral, circular po rtio n of th e a rra y a ring.
T h e d ire c tiv ity function of an array w ith N ring rad ia to rs as described above can be w ritten as
m
= { 2 i r r \ j \ ) sin 0 .2 J i [ ( 2 7 r r , / A ) s i n0
]
N f + 1 ] Ci I2 t t r fi—2 I (2 7 rr,/A )sin 0 ? J i [(27rrt_ 1/A )s in ^ l
- 1 (27rr!_ 1/A )sin 0 } {^ >
where r, are o u te r radii of i-th ring, as in d icated in Fig. 2.1.
17
S u m m e r
IS fu n d ion I) to a p p ro x im ate a c e rtain desired function Dj . T he error of approxi m ation r can he defined as
e'2 = \ \ D - I h W 1
= < I) — Dd, 1) — D d > • (2.3)
Here we treat functions as vectors in H ilbert space and use th e norm d enoted by || • || to express th e error function. T he < • > denotes th e inner pro d u ct.
T h e o b jec tiv e of th e design is to d eterm in e th e array p a ra m ete rs as given in Eq. (2.2) th a t m inim ize th e e rro r f . N onlinear o p tim izatio n alg o rith m s can b e used to find th e o p tim u m array p a ra m e te rs. T hese m eth o d s, however, have som e in h eren t lim itatio n s. Specifically, th e n u m b er of elem ents in th e array m u st be d e te rm in e d before each search. In ad d itio n , th e re su lta n t d ire c tiv ity function is a. priori unknow n an d depends on th e selection of th e desired d ire c tiv ity function.
T h e m eth o d proposed here allows for b e tte r control of th e re s u lta n t d ire c tiv ity function by tak in g advantage of som e closed-form solutions. Initially, we confine th e design m eth o d to o p tim ize th e selection of w eights ct- only, b u t la te r on we will e x te n d it to include th e variability of rad ii and th e n u m ber of elem ents in th e array. W ith th is restric tio n th e m in im u m of th e e rro r function given by E q. (2.3) can be o b ta in e d by s e ttin g th e JV-dim ensional g radient vector V to zero, th a t is,
V = ( k 2 d t 2 l T
I k i ’ Ocn = 0. (2.4)
In th e following developm ents it is convenient to m ap angle 0 in to a u-dom ain defined as
ft <1 .
U = y s i n 0 (2.5)
w here d is a c o n sta n t, w hich will b e discussed la te r on.
Eq. (2.4). w ith th e help of Eqs. (2.2), (2.3) and (2.5) can be w ritte n as v
]T V j < l ) i { u ) , l ) j { u ) > = < Ui(u), D ^ u ) > i = l , . . . , A r (2.6)
u> w here D i ( u ) is th e d ire c tiv ity function of th e /-th ring radiator.
T h e o p tim u m eoeilieients c, which m inim ize th e error function can theiofore be o b tain e d as th e solution to a set of .V sim ultaneous I* near equations.
W e propose here a different, approach which utilizes the existing design m oth ods for linear arrays a n d allows for b e tte r control of the resultant directiv ity function. F u rth erm o re, th e m eth o d allows for variability of th e ring w idth as an ad d itio n a l possible step to reduce the ap p ro x im atio n o n o r w '/' ‘ resorting to non lin ear equ atio n s. T h e m eth o d , however, requires that- we restrict the dcsivod d ire c tiv ity fu n ctio n to th e following form
7 rd tv (I
A i ( “ ) = X / u ’' & ('“ ); — r - " - “T (2 -7)
i=i A A
w here
(pi{u) = co s((‘2?’ — 1)»). (2.N)
It can b e shown th a t th e functions e o s((2 /— 1 )u); i — 1 , . . . , oo, form an orthogonal set sp anning th e H ilb ert space 'H. T he desired directiv ity function l),i(u) is a v ecto r in th e subspace M of th e Hilbert, space H . S , ace M is spanned by a finite n u m b er of basis functions <pi(u), i. — Due to th e n a tu re o! th e d iie c tiv ity function of a ring ra d ia to r (th a t is, we can only expand a Bessel fu n ctio n by an infinite n u m b er of basis function <j>i(u)), l),(u) is in W-spac<* but not, in A f-su b sp ace a n d th erefore cannot be expressed exactly as a linear com bination of M basis functions. It can, however, be best ap proxim ated (in m inim um error e sense) by its p ro je c tio n in to A d-subspace.
W e first consider th e d ire c tiv ity function of th e largest ring rad ia to r l ) t{u) -
D n ( u ) . Since D jv(u) is in th e H -space, it can be expressed as
Dn[u) - ] T Wj%s<t>3{u). (2.9)
j =i
34
2 0 T he p ro jectio n of I ) x ( u ) into subspace M (for orthogonal basis) can be found by sim ply tru n c a tin g th e above expansion to M term s. T h u s th e p ro jection of I)n(u) into subspace M is
M
I )n(u) = Y l wh ^ (P M ) - (2.10)
j= i
R epeating th e above p rocedure for all D i{ u) , i = Ar — 1 , . . . , 1, we can o b tain a series of p rojections I)i(u) of Di (u) in th e subspaces M i spanned by basis j = 1 , . . . . M — N + T herefore, we have N p rojections in th e subspace AT, am ong th em th e re are M p rojections which a re linearly in d ep en d en t. By linearly com bining those linearly indep en d en t M projections, we can form th e desired di rectivity function in th e subspace M w ith o u t i* Producing any a d d itio n al error. It follows th a t th e m inim um n u m b er of ring rad ia to rs necessary to produce M linear independent p rojections is N = M . Using this approach, we can m o n ito r th e e rro r introduced by tru n c a tio n which depends on th e w idth of each ring rad ia to r. By varying this w id th we can reduce th e tru n c a tio n erro r to an accep tab le value.
2 .3
D e s ig n o f a n A r r a y o f C ircu la r R in g R a d i
a to r s
Based on th e previous considerations we propose here a design tech n iq u e for an array of ring ra d ia to rs which closely ap p roxim ates th e prescribed rad ia tio n p a t tern. T h e technique takes advantage of th e th eory developed for linear arrays. Since linear array design is a m a tu re and w ell-developed field, th e tech n iq u e offers obvious advantages.
21
2.3.1
E q u ivalen t Linea r A rray o f a C ircular R in g R ad ia
to r
A linear array w ith ‘2A (even) point elem ents spared uniform ly by </ lias th e d ire c tiv ity function given by [32]
,V
A ( « ) = 1 ] « ’./ c o s( ( ‘- j - 1 )h) (2. 11)
j =i
w here {i/.q} are th e w eighting coefficients and u is <leliued in Eq. (2.5). Here, p a ra m e te r d has a p a rtic u la r m eaning as th e spacing between array elem ents. Eq. (2.11) has th e form which coincides w ith that, postu lated by Kqs. (2.7) and
(2.8).
T h e d ire c tiv ity function of a ring ra d ia to r w ith o u ter and inner radii /•, and r ,_ i, respectively, can be w ritte n as
„ . .J i[(2 ri/d)u] ,.J|[(2 r,_ |/u )(/]
D r {u) = 7Tnd— — - 7T/",- |<1—— — ’ J ( 2 12)
u u
A ssum ing a linear a rra y of infinite lengt h, it is possible*, by the proper selection of w eighting coefficients W j , to ob tain the directiv ity function of th e linear array
equal to th a t of a ring ra d ia to r, th a t is
D r(u) = Y , wi cos((‘4? - t)« ); E '.ld )
i= i A X
W e call such an a rra y th e equivalent linear array to a ring rad iato r. Sim ilarly, an eq u iv alen t lin ear a rra y can be derived for a disc rad ia to r with d ire c tiv ity function D c given by Eq. (2.1).
As n o te d earlier th e functions cos((2 / - l)u ) are orthogonal in th e w-domain, n am ely th e ir in n er p ro d u cts, defined as
< cos((2f - l)u ),c o s((2 > - l)tt) > = — I ' cos((2z - 1 )m) n 7T Jo
2 2 c o s((2 / — l)u ) d u
= I 1 f° r *' = j (2 14)
| 0 oth erw ise ’
when* n is any p ositive integer.
I lie in n er p ro d u ct definition given by Eq. (2.14) co n strain s th e u-dom ain to range [0, ;/7r/2]. T h is im plies th a t th e c o n sta n t d defined in Eq. (2.5) m u st be eq ual to n A/2. In o rd er to avoid th e g ratin g lobes in th e lin ear a rra y w ith d ire c tiv ity function given by Eq. (2.11), th e c o n sta n t d has to be e q u al to A/2. For this reason we su b seq u en tly set n — 1.
A pp ly in g th e in n er p ro d u c t defined by Eq. (2.14) to Eq. (2.13) we o b ta in
'X>
Y ^ u'j < <'<>s((‘2 / - l)u ),c o s ((2 f — l ) u ) > = < D r (« ),c o s ((2 i — l) u ) > j =o
(2.15)
which leads to th e ex p licit expression for wj:
4 f ?
wj — — D r (a) co s((2 j — 1 ) u)du, j = l , . . . , o o . (2.16) 7T J O
2 .3 .2
D e sig n o f an A rray o f R in g R a d ia to r s b y t h e E q u iv
a len t L inear A rray M e th o d
T h e ecpiivalent linear a rra y m eth o d for th e design of a rra y of rin g ra d ia to rs is a tech n iq u e in w hich all ring ra d ia to rs are rep re sen te d b y a series of equivalent linear arrays. T h e d irectio n al response of th e a rra y of ring ra d ia to rs is th e n th e su m of the d ire c tiv ity fu n ctio n s of all equivalent lin e a r arrays. T h e eq u iv alen t lin ear a rra y of a rin g ra d ia to r has in prin cip le an in fin ite n u m b er of elem en ts a n d associated w eighting coefficients. In general, th e a m p litu d e of th ese coefficients decreases for e le m en ts far away from th e ce n tre of th e array. It is th e re fo re possible to tru n c a te th e equivalent linear a rra y to a fin ite n u m b er of elem en ts by d isregarding
2 3 e lem en ts w ith sm all w eighting coefficients. Ail equivalent linear a rra y can also he found for a disc array, i "ause of th e n a tu re of Bessel function th e w eighting coefficients of a linear a rra y equivalent to a disc ra d ia to r are sm all for elem ents far nvay from th e a rra y center. For a rin g ra d ia to r a s u b tra c tio n , however, is im o lv e d in th e expression of th e d ire c tiv ity function. B ecause of th is su b tra c tio n th e d o m in a n t w eighting coefficients of th e equivalent linear array becom e m uch sm a lle r th a n t h a t of th e disc ra d ia to r. T h is causes th e difficulty in tru n c a tio n since th e ra tio b etw een w eighting coefficients is an im p o rta n t m easure d u rin g th e tru n c a tin g . F u rth e rm o re , th e c o m p u ta tio n a l round off e rro r is doubled due to th e su b tra c tio n . It is for th is reason th a t we in itially carry out th e design using an a rra y of disc ra d ia to rs. We th e n tra n sfo rm such a disc a rra y into tin* array of contiguous rin g ra d ia to rs having th e sam e rad ia tio n p a tte rn . S im ilar observations hold for array s th a t is th e w eighting coefficients of a linear a rra y equivalent to a disc a rra y decrease m uch m ore ra p id ly th a n those of a ring array. T h is in tu n i in tro d u c es less e rro r w hen th e lin e a r equivalent a rra y is tru n c a te d . T ru n c atio n of th e eq u iv alen t lin e a r a rra y is p erform ed using th e p ro je c tin g process as described in S ectio n 2.2. T h e tru n c a te d eq u iv alen t linear array of a ring ra d ia to r is th e p ro je c tio n of its d ire c tiv ity fu n ctio n Di (u ) in to th e subspace M i .
A s an illu s tra tio n of th e above process, Fig. 2.2 shows rad ia tio n p a tte rn s of a disc ra d ia to r w ith r — 5A and its eq u iv alen t linear a rra y tru n c a te d to 20 elem ents sp aced by d — A/2 (corresponding to 10 term s in th e d ire c tiv ity fu nction). We can observe on ly a sm all erro r (at h igher o rd er sidelobes) in tro d u ced by this tru n c a tio n . It h a s been found th a t a sim ilar tru n c a tio n perform ed on an equivalent lin e a r array to a ring ra d ia to r leads to m uch ’arger error.
T o in tro d u c e th e tra n sfo rm a tio n , we consider an array of stacked discs as show n in Fig. 2.3(a). For th e sake of exposition all concentric discs (and rings) are show n in a disassem bled form . A disc ra d ia to r can be th o u g h t of as an assem bly of con cen tric, contiguous ring ra d ia to rs. 'H us leads to an equivalent sta c k e d ring a rra y shown in disassem bled form in Fig. 2.3(b). Finally, we c o n stru c t
re la ti ve po w er (d B) 24 -10 -20 Disk Radiator -30 Equvalent Array -40 0 -50 angle 0 (degrees)
F igure 2.2: B eam p a tte rn s of a disc ra d ia to r w ith r = 5A and its equivalent lines array tru n c a te d to 20 elem ents (10 term s).
2 5
F ig u re 2.3: Id e n tity of th e array of disc rad iato rs and th e array of contiguous ring ra d ia to rs, (a) C oncentric disc array, (b) E quivalent stacked ring array, (c) Single layer rin g array.
26
an eq u iv alen t, single layer concentric ring a rra y w ith su ita b le w eighting coefficients as shew n in Fig. 2.3(c). In general, for a given v ector of w eighting coefficient w d associated w ith a stacked disc a rra y shown in Fig. 2.3(a), th e v ector of w eighting coefficients c r for an equivalent single layer ring a rra y showrn in Fig. 2.3(c) is given by
c,. = A w ,; (2.17)
w here th e elem ents of m a trix A a re
f 1 for i < j . .
(l,'j ~ { 0 otherw ise. ( - 18)
T h e following a re th e design ste p s for o b ta in in g th e w eighting coefficients for I lie a rra y of contiguous ring rad iato rs:
1. Design a linear a rra y of p o in t ra d ia to rs w hich pro d u ces a d esirab le form of th e d ire c tiv ity function (for ex am ple, associated w ith a D olph-C hebyshev a rra y ). T h e TV-dimensional colum n v ecto r of w eighting coefficients w ;, sp a c ing d, an d size N of th e a rra y are therefo re given [3]. W e call such an a rra y r prototype.
2. C onsider a disc ra d ia to r a n d d e te rm in e its rad iu s in th e follow steps:
a ) Select an a rb itra ry in itia l value of radius r for th e disc.
b ) F ind th e equivalent lin ear a rra y w ith spacing d o f th is disc. Such an a rra y has in principle an infinite n u m b er of e lem en ts (a n d asso ciated w eighting coefficients). However, it can b e tru n c a te d to a fin ite size N r w ith o u t an ap p reciab le e rro r in its ra d ia tio n p a tte r n . T h is can b e done by n eglecting coefficients w ith a m p litu d e s less th a n a c e rta in sm all p e rc e n ta g e of th e larg est one. W e d e n o te th is tru n c a tin g th re sh o ld by f . For a given size N r of a tru n c a te d lin ear a rra y th e n o rm alized
2 7 1
0.8
0.6
* 0.40.2
0radius of disc radiator, rf k (a)
radius of disc radiator, rf k (b)
F ig u re 2.4: W eighting coefficient of a tru n c a te d equivalent linear a rra y vs. radius of d isc ra d ia to r for different values of N r. (a) L inear scale, (h) S em i-logarithm ic scale.
28
w eighting coefficient \ w \ r+i / w i | is sm all and oscillates u n til th e radius of a disc ra d ia to r reaches a c e rta in critical value. F u rth e r increase in th e rad iu s leads to a rap id increase in coefficient values, as illu s tra te d in Fig. 2.4 a) (lin ear scale) and Fig. 2.4b (sem i-logarithm ic scale) for different values of N r . T h e above in fo rm atio n can b e used to select th e p ro p er value of th e tru n c a tin g th re sh o ld 6. Fig. 2.5(a)-(c) shows N r as a fu n ctio n of rad iu s r for different choices of tru n c a tio n th re sh o ld 6. We see th a t for a given size N r of tru n c a te d lin e a r a rra y th e re is a c e rtain to le ran c e in corresponding rad iu s r of a disc ra d ia to r w hich satisfies a specified tru n c a tio n th resh o ld S. T h e describ ed m e th o d of choosing th e disc rad iu s allows us to m o n ito r in d ire c tly (th ro u g h th e a m p litu d e s of discard ed coefficients) th e e rro r in tro d u c e d by tru n c a tio n .
c ) V ary rad iu s r u n til N T — N for a c e rtain r = rjy. Form th e N - dim ensional colum n vector of w eighting coefficients w,v:
Wjv =
Wi,N U ' 2 , N
. U ’N , N .
(2.19)
3. R educe th e rad iu s r = ryv of th e disc ra d ia to r u n til its eq u iv a len t linear a rra y has th e size of N — 1 for a c e rta in r = r^v-i < t'n- T h e a u g m e n ted v ecto r of w eighting coefficients w n _ i is th e n form ed as
w . v _ i =
m , N - i U’2,N-1
(2.20)
eq ui v ale n t li ne ar ar ra y si ze N r eq u iv al en t li ne ar ar ra y si ze N 25) 10 8 5= 1% 6 4 44 -■H* 2 0
radius o f disc radiator, r/X
(a) 8 44 -8 = 2% 6 4 4 4
-2
0 3 4 5 2 1 0radius of disc radiator, t/X
(b)
F ig u re 2.5: Size of equivalent lin ear a rra y vs. radius of disc ra d ia to r for different tru n c a tio n th re sh o ld 6. (a) 8 = 1%. (b) 6 = 2%. (c) 6 =
equ iv al en t li ne ar ar ra y si ze N 3 0 u 5 = 3% + +
radius of disc radiator, r
I X
(c)
:u a n d form th e subsequent a u g m en ted vectors of w eighting coellicieuts:
w iV_ 2 = l<’2 ,.V - 2 « ’. V - 2 ,.V - 2 0 0 ( T 2 I) W jV- 3 = « ’2 ,JV - 3 *<’A ' - 3 ,.Y - 3 0 0 0 (2.2 2) e tc .
5. F orm th e m a trix of th e w eighting coefficients W as
W = [w j, w 2 • • • wn]- (2.2 2)
6. Solve th e m a trix eq u atio n
W w d = w / (2.2 1)
to o b ta in th e v ecto r of w eighting coefficients w,; of disc rad ia to rs. Since th e vectors W j, w 2, . . . , a re p ro jectio n s of th e d ire c tiv ity functions in th e subspaces th e y are linearly in d ep e n d e n t. T h u s th e m a trix W lias full rank a n d th e m a trix e q u a tio n (2.22) has an unique solution.
7. T h e v e c to r of w eighting coefficients of ring ra d ia to rs c r is t hen found a c co rding to Eq. (2.17) by th e m a trix eq u atio n :
