Richards Antenna Pattern Synthesis Based on Optimization in a Probabilistic Sense

8/14/2019 Richards Antenna Pattern Synthesis Based on Optimization in a Probabilistic Sense

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A N TE N NA P A TT ER N S YN T HE SI S B A SE D O N O PT I MI Z AT IO NI N A P RO BA B IL I ST IC S EN S E

BY

W I L LI A M F O R RE S T R I C HA R D S

B.S., O id D o mi ni o n U n iv er s it y , 1 97 0

THESIS

O m it te d i n p a r ti a l f ul f il lm e nt o f t h e r e q ui r em en t s

~2 d eg r ee o f Ma s te r o f S ci e nc e i n E le ct r ic a l E n gi n ee ri n gi n t he G ra du at e C ol le ge o f t he

- - iv er s it y o f I ll i no is a t U rb a na -C h am p ai gn , 1 97 2

U r ba n a , I l li n o is


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(;,2 \.~~t-t\o5

R ~G(l. aUNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN

THE GRADUATE COLLEGE

August 1972

I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY

SUPERVISION BY WILLIAM FORREST RICHARDS

ENTITLED ANTENNA PATTERN SYNTHESIS BASED ON OPTIMIZATION

IN A PROBABILISTIC SENSE

BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR

THE DEGREE OF__ MA_S_T_E_R_OF_S_C_I_E_N_CE _

1 I1 7- r k);. J~ITICharg~ of Thesis. / G . . , - _ t ; . 'o../~

-------------. . ead of Department

Recommendation concurred int

Committee

on

Final Examinationt

-----------------------

t Required for doctor's degree but not for master's.

D517


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F ILLINOIS

-=..-npaignCampus

- -- ate College

- ~.istration Building

FORMAT APPROVAL

~he Graduate College:

rormat or the thesis submitted by

_ r the degree of Master of Science

=:_artment of Electrical Engineering

William Forrest Richards

is acceptable to the

July 31, 1972

Date

(Signed) ~/.,/2:Z/~~Departmental Representative


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iii

ACKNOWLEDGEMENT

~ e a ut ho r w is he s t o e x pr e ss h is g r at it ud e t o t h e p a ti en t g ui da nc e

- - __ or es so r Y . T . L o i n t he s tu di es t ha t l ed t o, a nd i n t he p re pa ra ti on

- - , t hi s t h e si s. A pp re ci at io n i s a l so e x pr es se d t o P ro fe ss o r A . H . S am eh

i s d i s cu ss io ns w it h t he a u th or o n m a tt er s r el at in g t o n u m er ic al

~,- ysis.

T ha nk s a re e xp r es se d t o P r of es s or O . L . G ad d y f or o bt ai n in g f un ds

= r t yp in g a nd p ub li ca ti on , a nd t o Mr s. B ur ns a nd M i ss A nd er so n f or

' ng . S pe ci a l t ha n ks a re e xp re ss ed t o M rs . B ro wn o f t he G ra du at e C o ll eg e

- ~ t he e x te ns io n g iv en d ur in g t he p er io d o f p er s on al t ra ge dy e xp e ri en ce d

~ h e a u th o r.


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A N E XA MP LE O F T HE S YN TH ES IS O F A T WO -D IM EN SI ON AL P AT TE RN

F UN CT IO N B AS ED O N MI NI MI ZA TI ON O F SO 0 0 0 0 0 21

TABLE OF CONTENTS

APPROXIMATE CALCULATION OF THE DISTRIBUTION FUNCTION

O F E . e 0 (') 0 0 0 0 0 e 0 C J 0 0 0 s 0 0 0 0 0 C; 0 16

iv

3

7

1

49

58

53

55

60

11

Page

A N E XA MP LE O F A MP LI TU DE P AT TE RN S YN TH ES IS .

NOTATION 0 0

STATEMENT OF PROBLEM

CONCLUSIONS 0

MODELING OF ERRORS AND PHILOSOPHIES OF OPTIMIZATION. 0

M IN IM IZ AT IO N O F E { e} 0

~ = T OF REFERENCES

_ ENDIX A 0

-=-_ENDIX B


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4 . R ES UL TS OF O PT IMI ZA TIO N OF s , . 50

3. COMPUTED EIGENVALUES OF G , 42

1 , R ES UL TS O F O PT IM IZ AT IO N A ND S IM UL AT IO N F OR a .1%

AND VARIOUS V ALUES O F M, . . 32

v

Page

L IS T O F T AB LE S

2 . R ES UL TS O F O PT IM IZ AT IO N A ND S IM UL AT IO N F OR M = 10

AND VARIOUS LEVELS OF ERROR. . . 34

- LE


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vi

( a) P ha se F un ct io n o f P at te rn o f F ig ur e 1 9 ( a) , ( b) P ha se

F un ct io n o f P at te rn o f Fi gu re 1 9 ( b) 0 0 0 0 0 > 52

38

9

37

46

40

35

30

41

39

36

27

31

28

29

26

25

22

Page

3 %

10

E dg ew or th S er ie s a nd S im ul at io n f or N = M = 10 , 0=.1%

S yn th es ize d a nd D es ir ed P at te rn F un ct io ns f or (J =10%.

S yn th es ize d a nd D es ir e d P at te rn F un ct io ns f or 0 = 7%

S yn th es ize d a nd D e si re d P at te r n F un ct io ns f or 0

S yn th es ize d a nd D e si re d P at te rn F un ct io ns f o r 0= 5%

L IS T O F F IG UR ES

(a ) P at te rn o f I n it ia l A pp ro xim at io n C or re sp on di ng t o

M in im um o f E qu at io n ( 8) ( K ;; : 0), (b) Pattern of Final Ite'cate

C or re sp on di ng t o M in im um o f Eq ua ti ~n ( 17 ) ( K = 0) . 0 0 51

D is tr ib ut io n F un ct io ns f or S o lu ti on s R eg ul ar ize d t o

V ar io us N oi se L ev el s b ut i n t he P re se nc e o f a n A ct ua l

c r = 5% Le'v'el II () GOO Cl 0 Ii 0 0 0 I!l 0 0 0 0 () 0 47

Mean E'S f or S ol ut io ns R eg ul ar i ze d t o Va ri ou s N oi se

L ev el s b ut i n t h e P r es en ce o f a n A c tu al 0= 5% 0 0 0

S yn th es i ze d a nd D es ir ed P at te r n F un ct io ns f o r 0 = 1%

S yn th es ize d a n d D es ir ed P at te rn F un ct i on s f or 0= 0%

S yn th es iz ed a nd D es ir ed P at te rn F un ct io ns f or M

S yn th es iz ed a nd D es ir ed P at te rn F un ct io ns f or M = 80

S yn th es iz ed a nd D es ir ed P at te rn F un ct io ns f or M = 40

S yn th es iz ed a nd D es ir ed P at te rn F un ct io ns f or M =3 0

S yn th es iz ed a nd De si re d P at te rn F un ct io ns f or M 2 .

S yn th es iz ed a nd D es ir ed P at te rn F un ct io ns f or M = L

A n E x am pl e o f t h e G en er al A nt en na S tr uc tu re U se d i n

Chapter .VI II 0 II GOO 0 0 0 (! II 0

I ll us tr at io n o f T h re e P hi lo so ph ie s o f O p t im iza ti on 0

re

7. S yn th es iz ed a nd D e si re d P at te rn F un ct io ns f or M - 6 .

--.


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1

I . N OT AT IO N

~ e f ol lo wi ng n ot at io n i s u s ed t hr ou gh ou t t hi s r ep or t; t ho se n ot

- ed i n t h e l is t b e lo w a re d ef in ed i n s ub se qu en t c ha pt er s 0

d en ot es a t hr ee -d im en si on al E uc li de an v ec to r; e . g. , A

i s a v ec to r.

** d en ot es c om pl ex c on ju ga ti on o f a c om pl ex o bj ec t; e .g ., A

i s t he c om pl ex c on ju ga te o f A .

d en ot es a u ni t t hr ee -d im en si on al E uc li de an v ec to r i n t h e

*s en se t ha t i f e i s a u ni t v ec to r, t he n e e 10

> d en ot es a 1 x N m a tr ix w ho se e le me nt s m ay b e c om pl ex

n u mb e rs o f v a r ia b le s o r c o mp l ex t h re e -d i me n si o na l

E uc li de an v ec to rs o r v ec to r f un ct io ns ; e . g. ,

. Y : . lJ (1)

1

. Y : . ZJ(l)

z

V> Jl

>

t d en ot es t he H e r mi ti an c on ju ga te ( c on ju ga te t ra ns po se ) o f

a ma tr ix ; e .g ., i f A i s a m at ri x, t he n t he e le me nt i n t h e

ith

r ow a nd j th c ol um n o f A~[At].., is [A]~..~J J~

< denotes (t.

is t he product defined in t he u sual w ay


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k d en ot es th e f re e- sp ac e w av en um be r.o

2.

c o n si d e re d i n t h i s r e p o r t.

i s t h e t im e v a r ia t io n o f a l l t h e f i e ld s a n d c u rr e nt s

denotes 1 = 1 .

i s a p o i nt i n t h e u s u al p o la r c o or d in a te s .

i s t h e. i ma g in ar y p a rt o f Q.

i s t he r ea l p a rt o f Q.

d e no te s t h e s a m pl e m e an o f Q; t ha t i s, i f QI

, Q2

,

a r e r a n do m s a mp le s o f r a nd o m v a r ia b le o r p r oc e ss Q, then

dQ ~s th e e lem en t o f s ol id a ngl e, si n e d e d ~. v

Z is the wa ve i mp eda nc e of fre e sp ac e.o

j

jw te

R e . Q

1 m Q

(r, e, ~)


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3

I I. S TA TE ME NT O F PR OB LE M

-jk r'r0-eE (r , 8 , ~ ) = -j(k Z /4 nr ) f (8 , ~ )

- 0 0 -

I i~ _ o r c u rr e nt s u s ua l ly g i ve r i se t o l a rg e e r ro r s' i n t h e p a t te r n f u nc t io n

t he si z ed , i t a pp ea rs t ha t a sk in g f or t oo m u ch p re ci si on c an d o m o re

- ~ t ha n g o od . L ar ge c ur re nt s a ls o c re at e o th er p ro bl em s s uc h a s h ig h

- a ti ve i n n a tu re , t ha t i s, a s t he m e an c ur re nt s g ro w, t he p ro ba bi li ty

_ es en t i n t he p h ys ic al r ea li za ti on o f t he s yn th es iz ed c ur re nt s a re

~ e ci s el y t h e s y nt h es i z~ d a n te n na m a y p r od u ce a p a t t er n t h at a p pr o xi m at e s

e d e si r ed p a tt e rn v e ry c l os e ly ) t h e l a r ge c u rr e nt s n e ce s sa r y f o r t h i s

~ sh es . H ow e ve r, t hi s i s d o ne a t t h e e x pe ns e o f r e qu ir in g v er ~ l ar ge

I t i s w el l k no wn [ 1] t ha t t he r ad ia ti on e le ct ri c f ie ld c an b e

jk r'o ri (8 , ~ ) = -r x r x II ~ (r ', 8 ', ~ ' ) e 0 dr'd~'

e re ~ (r , 8 , ~ ) i s t he s ou rc e c ur re nt d en si ty .

G e ne r al l y, t h e s y n th e si s p r ob l em i s t o f i nd a r e a l iz a bl e f u nc t io n J

e re ( r, 8 , ~ ) i s t he p oi nt o f o bs er va ti on a nd i ( 8, ~ ) , c a ll e d t h e

a t te r n f u nc t io n , i s g i v en b y

- _ ec is io n w il l l ik el y g iv e r is e t o l a r ge e rr or c ~r re nt s. S in ce l ar ge

- - la rg e e r r or s a ls o g r o ws . F or t hi s t yp e o f e rr or s ( a lt ho ug h i f b ui lt

_ = Iii d - !.II. As E : d ec re as es , i . b ec om es c lo se r t o i .d ' I t a pp ea rs

, 3] t ha t a p at te rn f un ct io n c an b e a pp ro xi ma te d a s p r ec is el y a s o ne

_ = e ls o f c u r re nt s. S up po se t ha t t he e rr or s t ha t w il l i ne vi ta bl y b e

~ c h t ha t t he p at te rn f un ct io n it hu s r ea li z ed i s c lo se i n s o m e s en se t o

- d e s ir e d p a tt e rn f u nc t io n id

. T o gi ve me an in g t o th e te rm c lo se ne ss ,

n o rm i s u s ua l ly d e fi n ed w i t h a c o rr e sp o nd i ng p e rf o rm a nc e i n de x

ax pressed as


11/68

-0 r es tr ic t J to a f in it e d im en si on al l in ea r s pa ce s pa nn ed b y b as is

(3)

(Z)

(1)

(4 )utC>,-l~ i\; , N

.!:!.N ; i e. , l. = J! ul + . + J ~ . ! : !.N' Th e

-1 -1= U Diag [A I ' A Z 'J >

o

IN

a re g iv en b y

E = 1 1 ! u II Z + < JG J> - Z R e ,

C> = f ~ w V > d~ .

G = f V>. I t i s p ar ti cu la rl y

e rn t hu s o bt ai ne d i s a m e mb er o f a f in it e l in ea r n or me d s pa ce Pjk r"r

-r x r x f f ~ e 0- d r' d ~ ', i = 1 , Z ,

4

is a unitary matrix w hich diagonaliz es G and AI: ~ :' : AN > a

t he e ig en va lu es o f G , t he n ( 3 ) m ay b e w r i tt en e qu iv al en tl y a s

e ni en t t o d e fi ne a n i nn er p ro du ct b et we en a ny t wo m e mb er s o f P , i I '

t i on s .! :! .l'uz , .

c l os se s a nd e x tr em e f re qu en cy s en si ti vi t y.

~ c at in g t ha t J > m ay b e s tr on gl y i nf lu en ce d b y t he l as t f ew s ma llo

a s [ iI ' iz ] = f f t iz w d ~, w he re w ee , ~ ) i s p os it iv e f or a ll d ir ec ti on s

: ). W it h t he n or m d ef in ed a s 1 1 . 1 IZ = [ . , . ] t h e p e rf o r ma n ce i n de x

1 1 ! u - il i Z c an b e e xp an de d a s

~ v al ue s o f G a nd t ha t t he l ev el o f c ur re nt s, o f w hi ch < J J > is ao 0

- e

: a p o si t i ve d e fi n i te l ie r mi , ti a n [4] m a t ri x a n d

=


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5

- a su re ,m ay b e l ar ge i f s om e o f t he e ig ne va lu es o f G a r e s ma ll . I nd ee d,

~ ~ c a n be s ho wn [ 5] t h at a s mo re b as is f un ct io ns a re a dd ed t o P ( w hi ch a d ds

r e r ow s a nd c ol um ns t o G ) t ha t t he s ma ll es t e ig en va lu e c an c er ta in ly n ev er

~ c re as e a nd m a y d ec re as e. T hu s, a s r e ma rk ed e ar li er , i f t h e u n av oi da bl e

r r or s m a de i n r e a li z at i on a r e r e la t iv e i n n at u re , t h e s o l ut i on o bt a in e d

( 3) mi ni mi zi ng s m ay n ot r ea ll y b e a n o pt im um s ol ut io n i n a ny p ra ct ic al

sense.

A n um be r o f a t te mp ts h av e b ee n m ad e t o d ef in e a m or e r ea li st ic s en se o f

t im iz at io n t ha n t he m i ni mi za ti on o f s . R ho de s [ 6] m in im iz ed s s u bj ec t

t h e c o n st r ai nt t h at T ay l or ' s s u pe rd i re c ti v e r a ti o [ 7] i s s o me p re -

-etermined constant, y. I n t he s am e s pi ri t, L o [ 8] m ax im iz ed g ai n a nd

- : gn al - to - no i se r a ti o s u bj e ct t o a c on s tr a in t o n Q = < JJ >/ . I n

t h c as e s, b y c o ns t ra i ni n g y or Q, m or e p r ac t ic a l s o lu t io n s w i th l o we r

r r en t l ev e ls t h an t ho s e o b ta i ne d b y s i mp l y m i ni mi z in g s o r m a x im i zi n g

- i n an d s i gn al - to - no i se r a ti o , r e s pe c ti v el y , w e re f o un d . H ow e ve r , t he

~ rm er r eq ui re d t he u s e o f c om pl ic at ed f un ct io ns a nd n ei th er a ns we re d t he

- e s ti on o f e xa ct ly h ow y an d Q s ho ul d b e d et er mi ne d. C ab ay an [ 9]a 2

t he s iz e d l i ne s o ur c e d i st r ib u ti o ns b y m i ni m iz i ng s + a f lu( x ) ! dx-a

e re u ( x ) i s t he s ou rc e c ur re nt d is tr ib ut io n i n a n a pe rt ur e o f 2 a, an d

: s s om e p os i ti v e c on s fa n t c a ll e d t h e " re g ul a ri z at i on p a ra m et e r. " T h is

h o d, w h i ch i s c o m pu t at i on al l y s im p le r t h an m i n im i zi n g s wi t h Q or y

s t ra i ne d , a l so l e ad s t o p r ac ti c al s o lu t io n s w i th c ur r en t l ev e ls

alu(x)1

2dx : ( l/~ ) I Ifdl I. C a ba y an g a ve a m e th o d f o r d e t er m in i ng

-ab ut i t i s o ne w h i ch r eq ui re s a r an do m s im ul at io n w hi ch c an b e c os tl y

u ta ti on al ly f or l ar ge s am pl e s iz es a nd s ub je ct t o q ue st io n f or s ma ll

I e s i ze s. A t hi rd m e th od d ev el op ed i n t hi s p ap er a pp ro ac he s t he p r ob le m

- e c tl y b y a ss um in g t ha t t he e rr or c ur re nt s a re r an do m v ar ia bl es a nd


13/68

6

_:> a ti ve i n n at ur e. U nd er t hi s a s su mp ti on , E : b e co m es a r a nd o m

r i ab le E c h ar a ct e ri z ed b y a d i st r ib ut i on f u nc t io n w h os e v a ri o us

_ p e rt ie s m ay b e o pt im iz e do B y m in im iz i ng E{s}, t h e c o n ce p t o f

_ e gu l ar i za t io n p a ra me t er i s g e ne r al i ze d a n d t h e n e ed f o r s i mu l at i on i s

:> i mi na te d. B y m ak in g u se o f a n a sy mp to ti c s er ie s d er iv ed f ro m t he n o rm al

~ s t ri b ut i on , t h e d i st r ib u ti o n f u nc t io n o f s, F ( E : ; J wh i c h d e p en d s

m e an c u rr e nt s J >, c a n b e a pp r ox i ma t ed . T h is a p pr o x im a te d is t ri b ut i on

y t he n b e u se d f or o pt im iz a ti on o f o th er p ro pe rt ie s o f F ( E : ; J .

7r t he e xa mp le s c on si de re d i n t hi s p ap er , a nu mb er o f s im ul at io ns w er e

r fo rm ed t o c he ck t he t he or y, a nd t he ir r es ul ts a re f ou nd t o b e i n

- c e ll e nt a g re e me n t w i th t h eo r y.


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7

= un ct io n. O n t he o th er h an d, i t w as p oi nt ed o ut i n C ha pt er I I t ha t i f

This

(5)

> 0 w her e Z. = Y. J . .1. 1. 1.

b e n on ne ga ti ve d ef in it e. I t

=

A .. = X J.J~1.J 1.J 1. J

A.,E{oJ> i s a s su me d t o e xi st a nd b e o f t he f or m

'11 a ls o h av e t o be r an do m. T hu s t he p at te rn b ec om es

o p ti m um l li n s o me p r ob a bi li s ti c s en s e.

_ at te rn f un ct io n, t he h i gh er t he c ur re nt l ev el b ec om es . T hu s i t a pp ea rs

O ne o bs er ve s f ro m t he f or m o f (5), t ha t a s t he a mp li tu de s o f t he

u rr en ts g ro w, t he p ro ba bi li ty o f h a vi ng l ar ge e rr or c ur re nt s a ls o

I II . M OD EL IN G O F E RR OR S A ND P HI LO SO PH IE S O F O P TI MI ZA TI ON

A cc ep ti ng t he f ac t t ha t a ny p hy si ca l r ea li za ti on o f a n a nt en na i s

I f t h e c u rr en ts a re r an do m, t he n t he p at te rn f un ct io n t he y p r od uc e

e ve ry th in g c ou ld b e d on e ~ it h p re ci si on , t he b et te r t he m a tc h t o t he d es ir ed

o ro ws . L ar ge e rr or c ur re nt s g iv e r is e t o l a rg e e rr or s i n t h e p at te rn

- ' at ~ c om pr om is e m ig ht b e s t ru ck s uc h t ha t t h e r es ul ti ng a nt en na i s

s e co n d m o me n t m a tr ic e s o f ' oJ > , n o a s su mp t io n h as b e en m a de a s t o t h e p r ec i se

s ho ul d b e n ot ed .t ha t a t t hi s p oi nt ,. e xc ep t f or t he f or m o f t he f ir st a nd

i s t r ue f or a ll

a s r an do m v ar ia bl es J >. F or c on ve ni en ce , t he r an do m e rr or c ur re nt s, o J> ,

T he f ir st m om en t m at ri x o f o j> i s z er o. T he s ec on d m o me nt m at ri x o f

s ub je ct t o e rr or s o f a r an do m n at ur e, t he c ur re nt s J > w i ll b e c on si de re d

a re d ef in ed a s o J~ = J > - J > w he re J > ~ E {J >} a re t he m ea n c ur re nt s.

r eq ui re d n on ne ga ti ve d ef in it e. f or m f or a ll J > i s e a si ly v er if ie d b y

*= L ( Y. J. ) (Y .J .) X . .ij 1. 1. J. J 1.J

Y > s o t ha t A m us t a ls o

' = or m o f t he p r ob a bi l it y l a w w h ic h o j > o be y s.

'=orming


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. o ut t he m in im um o f E . R at he rt o ne m us t c ho os e s om e a t tr ib ut e o f t he

w it h r es pe ct t o J> h ol di ng f ix ed . S uc h a s ch em e w il l b e r e fe rr ed t o

8

(6 )

o f + f.i(St ep ) = = +

p la ci ng i b y i i n t h e d e f in it io n o f c or re sp on di ng t o E q ua ti on ( l) t

E x ce pt f or t he c as e w he re E o f t he v e r ti ca l o pt im iz at io n s ch em e

~5 c ho se n t o be t he r es ul t, E i o f t he h or iz on ta l o pt im iz at io n, t hemn

- _ ut io ns t o t he h or iz on ta l a nd v er ti ca l o pt im iz a ti on s ch em es w i ll b e

p 1i ci t1 y d ef in in g a f un ct io n E = E (J . In t hi s c as e on e w ou ld a tt em pt

f in d a s et o f c u rr en ts J > w hi ch m in im iz es E (J . T hi s s ch em e wi ll b e

l Ie d " ho ri z on ta l o pt im iz a ti on ." A t hi rd m e th od i s t o mi ni mi z e E { E} w i th

_ es pe ct t o J >. T he t hr ee m e th od s a re i ll us tr at ed i n Fi gu re 1 ( a, b , a nd

, r es pe ct iv el y) . I n e a ch o f t h e c a s es i ll us tr at ed i n Fi gu re 1 , t he

. ~ ff er en t. I t i s a ls o d ou bt fu l t ha t t he s ol ut io n t o t h e t hi rd o pt im iz a ti on

s ch em e wo ul d c oi nc id e w i t h t ho se o f t h e f ir st t w o m et ho ds . T hu s, i t a p pe ar s

. .s t ri b ut i on f u nc t io n s a r e l a be l ed b y t h ei r c o rr e sp o nd i ng m e an c u rr e nt s .

: t he t hr ee p os si bl e s ol ut io ns g iv en i n t he i ll us tr at io n, J1

>, J2

>, and

- 3 > a re t he o pt im um s ol ut io ns t o t he f ir stt s e co n d, a n d t h ir d s c he m es ,

=espectively.

_ b ec om es t he r an do m v ar ia bl e g iv en b y

a s " v er ti ca l o pt im iz a ti on ." O ne mi gh t s et F (E ; J = p f or p ( 0, 1) , t hus

s yn th es is p ro bl em s im pl y b y m in im iz in g i n ( 1) . W he n E i s t ho ug ht o f

a s a r an do m v ar ia bl e a s d ef in ed i n ( 6 )t t h en i t i s me an in gl es s t o t al k

~ n ot ed i n C ha pt er l It o n e c la ss ic al ly o bt ai ns a s o lu ti on t o t h e

_ o b ab i1 it y l aw o f E o n w hi ch t o o pt im iz e. F or i ns ta nc et i f F ( ; J =

. ro ba bi 1i ty o f t he e ve nt { E < f or t he g iv en s et o f me an c ur re nt s J >} ,

- e d i s tr ib ut io n f un ct io n o f E , t h e n o n e m i gh t c ho os e t o m ax i mi ze F ( E; J


16/68

E

F i gu re 1 . I ll us tr at io n o f T h re e P hi lo so ph ie s o f O p ti mi za ti o n

9

VERTICAL

OPTIMIZATION

OPTIMIZATION OF E{E}

I

II

I HORIZONTAL

: OPTIMIZATION

II

MIN

(a )

( b)

-E

k= MEAN OF E

REFERRED TO F(E;Jk

E ;J


17/68

10

-: .at these three methods represent fundamentally different philosophies

f o pt im iz at io n. T he c ho ic e o f me th od o ne e mp lo ys a mo ng t he se t hr ee

( an d ma ny o th er s t ha t c an b e d er iv ed ) d ep en ds , a mo ng o th er t hi ng s, o n

- - e a m ou n t o f c o mp u ta t io na l e ff o rt r e qu i re d f r om e a ch m e t ho d.

I t h as b ee n t ac it ly a ss um ed t ha t t he s ol ut io ns e xi st t o t he t hr ee

p ti mi za ti on s ch em es d is cu ss ed a bo ve . I t i s s h ow n i n t h e n ex t s ec ti on

- ha t a s ol ut io n e xi st s t o t h e t h ir d s ch em e. A lt ho ug h n o r ig or ou s p ro of

~ s p o se d f or t he e xi st en ce o f s ol ut io ns t o t he h or iz on ta l a nd v er ti ca l

p t im i za ti o n s c he m es , t h e f ol l ow i ng h e ur i st i c a r gu m en t i s gi v en . F i rs t

f al l, F (E ; J i s bo un de d f ro m a bo ve b y u ni ty i n t he v er ti ca l o pt im iz at io n

. r oc e du r e, a nd E ( J , d ef i ne d i m pl i ci t ly b y F( E ; J = p ,i s b ou nd ed t o

- he l e ft a t l e as t b y z er o i n t h e ho ri zo nt al o pt im iz at io n s ch em e. T hu s,

t he o nl y w a y f or s ol ut io ns t o t he t wo p ro bl em s n ot t o e x is t i s f or t he

_ ev el s o f c u rr en ts t o i n cr ea se w it ho pt b ou nd a s F (E ; J i nc re as es a nd

s (J d ec re as es i n t he f i rs t a n d s e co nd m e th od s, r es pe ct iv el y. H ow ev er ,

s in ce t he e rr or s o ne i s p ro ne t o ma ke g ro w p ro po rt io na te ly t o t he s iz e

o f t he m e an c ur re nt s, m os t s am pl e v al ue s o f E w o ul d e v en t ua l ly b e v e r y

a rg e, i nd ic at in g t ha t F( E; J i s re du ce d f or a f ix ed f in it e E i n t he

e rt ic al s ch em e, a nd t ha t E ( J f or a f i xe d p = F (E ; J i s in cr ea se d

i n t h e h o ri z on t al s ch e me , b o t h b e i ng c o nt r ar y t o h yp o th e si s . T he r ef or e ,

i t a p pe ar s t ha t t he c ur re nt l ev el s a re b ou nd ed a nd h en ce s ol ut io ns e xi st .


18/68

11

(7)

(8)

T a ki ng t he m e< ; l. no f ( 7) ,N.

2 -8 = 1 1! u - il l + 11 0iI I - 2 1 < e . [ !c t - I , 0I ]

E qu at io n ( 6 ) c an b e e xp an de d t o

O f t he t h re e t yp e s o f o p t im iz a ti on d i sc us s ed i n t h e p r ev io us . ch ap t er ,

IV. MINIMIZATION OF E{f;}

E{llofI12

} = E{} = L E { oJ io J j} Gij

i,j

E : = E{E} = 1 1 ! u - I I 12 + < JK J>

m in i mi z at i on , o f E { E} i s t he s im p le st c o mp ut at i on al l y. T h~ c on ce p t. o f

e x pe r im en ta t io n. T hr ee r e la te d v e r si o ns q f t h is t y p e o f o pt im i za ti on a re

r eg ul a ri z at i on , b ut i t g e ne ra l iz e s t he c on ce p t o f r eg u1 a ri z at i qn p a ra me t er

regulariz ation used. by Cabayan llO] i s a n a t ur al r e su lt o f t hi s o pt i mi z at i on

r eg ul a ri z at i on t ha t s ho u ld b e u s ed w i t ho ut t h e n ee d f or n um e ri ca l

presented below .

a nd p ro vi de s a s im pl e a nd d ir ec t w ay o f d et er mi ni ng t he p ro pe r a mo un t o f

added.

p ro ce du re . N ot o nl y d oe s t h is p r oc ed ur e g iv e p hy si ca l m ea ni ng t o

since E{ oJ>} = 0, E{oi} a ls o v a ni sh es a nd , t he re fo re , t he e xp ec ta ti on o f a ny -

t hi ng w hi ch i s l in ea r i n O f w il l a ls o va ni sh . E xp an di ng E { I lo !1 12

},

w hi ch i s t he E : a s d e f in e d i n E qu a ti on ( 1 ) w i th " re g ul ar iz a ti on " < JK J >

N ot ic e t ha t i n t he s pe ci al c as e w he n K . .i s t h e s ca la r m at ri x a I,

( 8 ) r ed u ce s t o e = I I~ - II 12 + a< J J> , w h ic h i s t h e re gu l ar iz e d p er fo r ma nc e


19/68

-a -a

12

(9)

a

J X (x, y) G( x - y) u( x) u( y) * dx dy

a

J X (x , y ) u (x ) u (y )* d xd y < 00, t he m ea n o f t he

-a

a

= J

e ;;: I If d - f 1 12 + aJ~ Iu(x) ]2 dx

-a

2 a a .e

=I

I fd - f lI + J J

K (x , y ) u (x ) u *( y) d xd y,-a -a

k

G(x - y ) = J 0 e (~ )e *( ~) e j( x- y) d ~.

-ko

*E {o u( x) o u( y) } = X (x , y ) u (x ) u (y )* ,

I n o rd er t o e xt en d t he , th eo ry t o t he c as e o f l in e s ou rc es , w hi ch

In t he s pecial case when X (x , y) =ao( x- y) , ( 9) reduces to

where

r an do m i n n a ,t ur e a nd o be y a p r o ba bi li ty l aw w i th a c or re la ti on f un ct io n,

t he m ea n s ou rc e d is tr ib ut io n b e u ( x) , w h ic h g en er at es a p a t te rna

f (~ ) = e (~ ) J e j~ x u (x ) d x, w h er e e (~ ) i s t he n or ma li ze d e le me nt p at te rn ,

-a

and ~ = k sin e with e b ei ng t he a ng le o f d e vi at io n f ro m t he b ro ad si deo

d ir ec ti on . F or a ny a nt en na w ho se c on st ru ct io n e rr or s, o u( x) , ar e

z e ro w i t h e ac h s am pl e a c on ti nu ou s f un ct io n , de fi ne d o n - a < x < a, Let

i nd ex u se d b y C ab ay an ~rlO] i n t he d is cr et e a rr ay c as e.

a

w here for all u, 0 < J

-a

n or m, sq ua re d o f o f i s

Letting K (x , y) = G( x - y) X (x , y ),

w er e a ls o c on si de re d b y C ab ay an , 1 e~ o u( x) b e a r an do m p ro ce ss w it h m ea n


20/68

13

_ i se t o h u ge c u rr e nt s .

f K ( x. y ) u (x ) u ( y) * d x dy i s t h e me an p ow er r ad ia te df

ubstituting (La) i nt o ( 1) .

T he p er fo rm an ce i nd ex m ay b e c al cu la te d f ro m ( I ), ( 8 ). a nd aO) t o b e

~ = Iliul12 - < J( G + K)J> = Iliul12 - < C( G + K )-lC>. (11)

I n v ie w o f t h e p o si t iv e d e fi n it e ne s s o f G a n d t h e s e m id e fi n it e ne s s

I t w as r em ar ke d i n C h ap te r I I t h at t he s ma ll e ig en va lu es o f G c au se

a b ay a n u s ed t o o b ta i n s o lu t io n s f o r s c al a r a . )

J> = (G + K)-l C> (10)

~ J > or e r ro r a p er t ur e f u nc t io n s o u (x ) .

I t a p p e ar s f r om t h e a n a l ys i s a b ov e t h at t h e c o nc e pt o f r e g ul a ri z at i on

- 2f K ( w hi ch f ol lo w s f ro m I I ~ I I : : : .0 ) . a u n iq ue s et o f c ur re nt s

=unction

= un ct io n G a nd o f t he c or re la ti on m at ri x o r f un ct io n X . P hy si ca ll y.

e xp re ss ed i n K a re e ss en ti al ly " cu t- of f" s o t he y c an n o l on ge r g iv e

- h e q u ad r at i c f o rm o f t h e r e g ul a ri z at i on m a tr i x < J KJ > o r o f t h e r e gu l ar i za t io na a

_ fn e s ou rc e c as e c an b e o bt ai ne d b y a s li gh t e xt en si on o f t he m e th od

" p ar a me t er " ( o r m o re p r ec i se l y r e gu l ar i za t io n m a tr i x o r f u nc t io n ) i s a

= ro m E qu at io n ( 10 ) t ha t e ig en va lu es s ma ll er t ha n t he l ev el o f e rr or s

n at ur al p ro pe rt y o f t he a nt en na s tr uc tu re e mb od ie d i n t he m a tr ix o r

a ll t h e d if fi cu lt y i n i n ve rt in g G t o o bt ai n J >. H ow ev er . i t i s c l ea r

- hi ch i s a ga in t he p er fo rm an ce i nd ex u se d b y C ab ay an .

u in im iz es s wh er e C > i s d ef in ed i n E qu at io n ( 2) . ( Th e so lu ti on o f t h e

-a -a

' y a n a n t en na o f t he g iv en s tr uc tu re e x ci te d b y a ll p os si bl e e rr or c ur re nt s


21/68

i s m od if ie d t o

14

(13)

(15)

(16)

(14)

0, is

J > G- l e(J > whereo 0

e(J> = f l!ul y.}>1 I w e e, ~ ) d~.

T he p er fo rm an ce i nd ~x d ef in ed i n Eq ua ti on ( 7) re qu ir es t ha t t he

W he n c on si de ri ng c ur re nt s a s r an do m v ar ia bl es , p ro ce ed in g e xa ct ly

J > c an b e d et er mi ne d b y m in im iz in g (1 4) by s om e i te ra ti ve s ch em e.o

p ha se a nd p ol ar iz at io n o f ~(e, ~) b e s pe ci fie d. A mor e p ra ct ica l

p ro bl em i s t o tr y t o ma tc h Iii to I~I. U nf or tu na te ly , t hi s l e ad s t o

c or re sp on di ng t o J d ef in ed i n E qu at io n (3 ).o

e xt re me ly c om pl ic at ed c om pu ta ti on s a re e nc ou nt er ed w hi ch f or s im pl e e rr or

c ho se n t o b e !u = l!uI' . ilIii, in w hi ch c as e (8 ) m us t b e m in im iz ed b y

Replacing !u i n E qua tio n (1 ) b y l!uI . (i I Iii),

a s b ef or e, t he r an do m p er fo rm an ce i nd ex f or a mp li tu de s yn th es is b ec om es

f or t he s olu tio ns . In t hi s c ase t he p er fo rm an ce i nde x g iv en i n Equa tio n (1)

T he p er fo rm an ce i nd ex f or t he u nr eg ul ar iz ed c as e, K

f un ct io ns . T o av oi d t he se d if fi cu lt ie s, t he d es ir ed p at te rn f un ct io n i s

n on qu ad ra ti c p er fo rm an ce i nd ic es , w hi ch i n t ur n l ea d t o n on li ne ar e qu at io ns

U nf or tu na te ly , i f on e a t t em pt s t o c al cu la te E{E } a s d ef in ed i n (1 6),a

m od el s r ed uc e t o e va lu at io n o f i n te gr al s o f c o nf lu en t h yp er ge om et ri c


22/68

15

i ter ati on. T he so lut ion o bt ain ed b y mi ni miz ing (8) is n ot ex pec ted to b e

t he s am e a s o r p ro duc e a s s mal l a n E{s } a s t ha t o bt ai ne d b y m in im iz in ga

(18)

(17)

G(J>(G + K)J>

" Z = II I~ I - If I 1 12

+

with G(J> given by (15).

G ab ay an . T he c ur re nt s m in im iz in g ( 17 ) sa ti sf y

w hi ch a ga in i s a g e ne ra li za ti on o f t he p er fo rm an ce i nd ex u se d b y

E{ s } ; h ow ev er , i t s e em s r ea so na bl e t o e xp ec t t ha t m in im iz in g ( 8) w ou lda

yi eld a b ett er s ol uti on i n t h e se nse of lo wer E{s } t ha n o ne o bt ai ns f ora

an a rbitrary choice of t he p hase and p olarization of~. For the g iven

ch oic e o f ~, (8) can b e wri tte n eq uiv ale ntl y a s


23/68

16

V . A PP RO X IM AT E C AL CU L AT IO N O F TH E D IS TR IB U TI ON F UN CT I ON O F E

(19)

-E { R e . o J > I m < o J } = - A

R I

oj > + ~J> w here ~J> = J> - J >o

< XG X > + Eo

[ R e. < oj , - I m R e . < o J } = 1\.R R'E{1 YnoJ> Im


24/68

17

i = 1, 2, N.

necessary to approximate F(E; J ,

(22)

(21)

(20)

lIJ"> = TlIJ>

vN

b e t he e ig en va lu es o f A I. T he re fo re , i f

s = + Eo

T

variables with means

where X"> are a set of independent,normally distributed, complex,random

With these assumptions, a convenient representation of s c an b e

E quation ( 21 ) shows that the distribution function of s is a

generalized noncentral X2

distribution shifted by E. Although theo

usual x 2 d is tr ib ut io n f un ct io n i s k n ow n t o b e g iv en b y t he i nc om pl et e

then

A I a nd le t v I' v2

d. f h 1 d" f X" 1 1

a n v ar ~a nc es0

t e rea an ~mag~nary p arts0

i equa to2

Vi '

g am ma f un ct io n [ 11 ], f o r t he m o re g en er al d is tr ib ut io n a t h a nd , i t

X" > = TX> where

i n g e ne ra l, n ot d ia go na l. L et W b e a u ni ta ry m at ri x w hi ch d ia go na li ze s

m at ri x o f t h e s qu ar e r o ot s o f t h e e ig en va lu es o f G . T he re fo re ,

= < XI X' >. T he c ov ar ia nc e m at ri x o f X '> i s A ' = S AS t w hi ch i s,

derived through the following transformations. Let X'> = SX> where

S = DU t, U i s a u ni ta ry m at ri x t ha t d ia go na li ze s G , a nd D i s a d ia go na l

O n t he o th er h an d, t he s ec on d a ss um pt io n c on si de ra bl y r ed uc es t he w or k

t he 2 N v ar ia bl es i n 8 J> t o b e e xp re ss ed i n t he N X N c om pl ex m a tr ix A .

d is pe ns e w it h t hi s a ss um pt io n. I t a l lo ws t he c ov ar ia nc e i nf or ma ti on a bo ut

c ri ti ca l; w it h a l it tl e m o re w or k u si ng 2 N X 2 N ma tr ic es , o ne c ou ld


25/68

1 8

E d ge w or th s er i es [ 1 1 ].

(25)

(24)

(23)

jv.t)]}1

2, 3, kk-1 . 2 .

v . (v .+ k\,~ J '.'I > ) ,i' ..i 1

N

Ii=l

N

= ( k - 1 ) ! Ii=l

'\ = E{E} =

N -1 N 2= [ T Ii

=l ( 1 - j v.t )] ex p {j t [ + II~J~I /(11 0 i= l 1

T h e c ha r ac t er i st i c f u nc t io n o f E i s g i v en b y

t h e s y s te m, w h ic h m ay b e d on e s i mp l y o n a h i gh - sp e ed d i gi t al c o mp u te r ,

q ue st io n. O ne p ra ct ic al w ay o f a n sw er in g t hi s q ue st io n i s t o s im ul at e

a n d c o mp a re t h e r e s ul ts o f t he s im u la t io n t o t he a pp r ox i ma t e d i st r ib u ti o n.

S in ce t he E dg ew or th s er ie s i s a n a sy mp to ti c s er ie s f or N ~ 00 and

a gr e e wi t h t h e a p pr o xi m at e d d i st r ib u ti o n f u nc t io n t o w i th i n a f e w p er c en t ,

S in c e o n e i s d e a li n g w it h c h an ce a n yw a y, i f t h e r e s ul t s o f t h e s i m u la t io n

m ay n ot c on ve rg e, t he n u mb er o f t er ms t o be u se d i s a n i mp or ta nt

a s d er i ve d i n Ap p en d ix A . T h e c h a ra c te r is t ic f un c ti o n ~ ( t ) ( wh i ch

a n a s y m pt o ti c s e ri e s d e ri v ed f r om t h e n or ma l d i st r ib u ti o n c a ll ed t h e

t o t he d i st ri bu ti on f un ct io n o f E . T he re i s s uc h a f or mu la w hi ch i s

~ o r e q u iv al e nt l y o n t he m o me n ts o f E , wh ic h y ie l ds a n a p p ro x im a ti o n

t he d i st ri bu ti on f un ct io n o f E . I n v ie w o f t h is , o ne m ig ht a sk i f a

s im p le f o rm u la e x is t s w h os e p a ra m et e rs d e pe n d o n t h e c u mu 1 an ts o f

t h e f r e qu en c y f un c ti o n o f F ( ; J) a nd t h e c u mu 1 an ts o f ~ ( t) c h ar a ct e ri z e

m ig h t b e i n ve r te d b y a pp l ic a ti o n o f t he f a st F o u r ie r t r an s fo r m t o g i ve

w i t h i t s c o r r e sp o n di n g c u m u1 a n t s

r e as o n, a n a p p r ox i ma ti o n t o t h e d i st r ib u ti o n f un c ti o n o f E i s s o u gh t .

a pp ea rs t ha t n o s uc h s im pl e r ep re se nt at io n i s kn ow n [ 1 2] . F or t hi s


26/68

1 9

t he n t h e a p p ro xi ma te d is tr ib ut io n f un ct io n s ho ul d b e g oo d e no ug h f or

p r ac t ic a l e n gi n ee r in g p u rp o se s . T h is w a s d o ne i n t h e e xa m pl e s

c o ns i de r ed b e lo w a n d i t w as f o un d t h at t h e f o l lo w in g a p pr o xi m at e

r e p r es e n ta t i o n f o r F ( e ; J w a s a d e qu a t e

where

HZ (z)Z

- 1z

H3

(z)3

- 3zz

HS(z)5 3

= z - 1 0 z + lSz

a r e H e r mi t e p o l yn o m ia l s ,

i s t h e c o ef f ic i en t o f s k ew n es s ,

i s t h e c o ef f ic i en t o f e x ce s s, ~ ( z) i s t h e n o r ma l d i st r ib u ti o n w i th

dz e ro m e an a n d u n it v a ri a nc e , ~ ( z) = d z ~ ( z) i s t h e f r eq u en c y f u nc t io n

of Hz),

z = (e - E)/cr

i s t h e s t an d ar d iz e d v a ri a bl e , a n d f i na l ly ,

i s t h e s t an d ar d d e vi a ti o n o f s . I n o b t ai n in g t h e c o e ff i ci e nt o f ~ ( z )


27/68

20

dn

i n ( 26 ), u se w as m ad e o f t he f or mu la , - - - ~ (z ) = ( _l )n H ( z) ~( z) [ 11 ].dzn n

S in ce ( 26 ) is a n a sy mp t ot ic f o rm ul a, o ne s ho ul d n o t b e d is tu r be d b y t he

f ac t t ha t t he " ta il s" o f t h e d i st ri bu t io n m ay b e s li gh tl y n eg at iv e

o n t he l ef t o r s li gh tl y g re at er t ha n u ni ty o n t he r ig ht .

U si ng ( 26 ) n ot o nl y ma ya n a pp ro xi ma ti on t o F(e; J b e di sp la ye d,

b ut ( 26 ) m ay a ls o b e u se d i n o th er o pt im iz at io n s ch em es . S in ce t he

m e an c ur re nt s, J >, a pp ea r i n a h ig hl y n on li ne a r m an ne r i n F(e; J ,

t he o nl y h o pe o f v e rt ic al ly o pt i mi zi ng F i s t o e mp lo y s o me i te ra ti ve

s ch em e. A c on ju ga te g ra di en t m e th od s uc h a s t ha t d ue t o D av id on c ou ld

b e a pp l ie d, b ut n ot w i th ou t s om e d i ff ic ul ty . E ve ry t er m a n d f a c to r

e xc ep t f or c on s ta nt s i n ( 26 ) a re n on l in ea r f un ct io ns o fJ > d e fi ne d

i mp li c it ly t hr o ug h t h e e i ge nv al ue s a nd e ig en ve c to rs o f A ' . T h us ,

a l th ou gh o ne c o ul d w o r k t hr ou gh t he v a r io us d ef i ni ti on s a n d o b ta in t h e

gradient of F(e; J, it would b e a f ai rl y t ed io us e xe rc is e. I t i s

m or e d if fi c ul t c om pu ta ti o na ll y t o a pp l y a g ra di en t m et h od t o

h or iz o nt al o pt im iz a ti on s in ce i t r eq ui re s t he i nv e rs e f un ct io n o f F(e; J =p

A lt ho ug h t he h or iz on ta l a n d v e rt i ca l o pt im iz at io n w il l n o t b e c on si de re d

h er e, a n d i n s pi t e of : t he a pp ar en t n um er ic a l c om pl ex it y o f t he ir i mp le me n -

t at io n, t he y d o s ee m t o r ep re se nt a m o r e d es ir ab l e p hi l os op hy o f o p ti mi za ti on

A ft er a ll , i f o n e i s o nl y g oi ng t o b ui ld o ne a nt en na , m in im iz in g t he m e an

E d oe s n o t s e em t o ma ke a s m uc h s en se a s m ax im iz i ng t he p ro ba bi li ty t ha t

t he a nt en na w il l h av e a n E l es s t ha n a g iv en a cc ep ta bl e e, It would

seem that minimizing e f or a g iv en l ev el o f c on fi de nc e, p = F(e; J ,

i s e v e n m o re d es ir a bl e t ha n v e r ti ca l o p ti mi za ti on , s i nc e i t d o e s n ot r eq ui r e

a n a p ri o ri c h oi ce o f a r ea s on ab le e. I n a n y c as e, i t i s i nt er es ti ng t o

h av e t he d is tr i bu ti on s o f p r o ba bl e e r ro rs , a nd t he y a re i nc lu de d i n s om e

of the specific examples considered below.


28/68

21

: un c ti o ns c a n b e t a ke n a s t h e s c al a r f u nc t io n s

o= 9 0 p la ne , a ng le s

o < 8 < II '

{

s ec 8

se c II '

2 M jpicos(


29/68

F ig ur e 2 . A n E x am pl e o f t h e G e ne ra l A nt en na S tr uc tu re U se d i n C ha pt er V I

RAY 2

x

RAY 3

zRAY 6

RA Y 4

. . . .

22

,""

PLANAR ANTENNA USING 4 RINGS AND 6 RAYS

RAY 5


30/68

23

T ha t i s ,

T h us Z [ 0l

, 0 2] i s th e me an o f

T I /2 2 T If f l. 1

2s i n e d ~ d e .

o 0

-1= (G + K ) C >

2

I I 1 12

= (1/8TIM)

a n d c u r re n ts g iv en b y E qu a ti on ( 1 0) us i ng K1 2

_ eg ul ar i ze d. T o t h is e n d t he f o ll o wi ng n o ta ti o n i s i nt ro du c ed . L e t

s o me o f t he e x am pl e s i n t hi s s ec t io n w er e d e li be r at el y i m pr op e rl y

I n o rd er t o s e e t he e ff ec t o f o n r e g u l ar i z ed o p ti m u m s o lu t i on s ,

2m a tr ix d ef i ne d i n E q u at io n ( 5 ) is 2 o I , a n d t he c o rr es p on di n g r eg ul a ri za ti o n

a tr ix i s K0

= 202

Diag[Gl l

, G22

, , GNN

].

n d if fe re nt r in gs a re a ss um ed t o be u nc or re la te d. F or s uc h a mo de l t he

a lc ul a te d b y n u m er ic al q ua dr a tu re , a t l e as t t he i n te gr a l o v er ~ c an b e

F or t he se e x am pl e s, t he n o r m w a s d ef i ne d b y

a rg et o ve r a l ar g e a ng ul a r r eg i on [ 1 3] .

F o r a l l t he e x am pl e s c on s id er e d, t h e e r ro rs a re a ss u me d t o o b e y t h e

o ne i n c l os ed f or m. D et ai ls o f t h ei r e va lu at io n a re g iv en i n Ap pe nd ix B o

i m a gi n ar y p a r ts o f oj . a r e i n d ep e n de n t, n o r ma l ly d i s tr i b ut e d, r a nd o m1

a ri a bl es b o th h a vi ng m ea n z er o a n d s t a nd ar d d ev ia t io ns o f 0 1 J. I. Errors1

? or t un at el y , G c a n b e c o mp ut e d i n c lo s ed f or m a nd a l th ou g h C > m u st b e

:: > 900n ee d n ot b e c on si de re d. ) T he m o ti va ti on f or t hi s t yp e o f p ro bl em

. tt h J > g i ve n b y J >

; ol lo wi n g m od el . F o r s i mp li c it y, t he e r ro rs o n e a c h o f t he 2 M e l em en ts

n t h e ith

r in g a re i de n ti ca l a n d a r e g iv en b y oj., w he re t he r ea l a nd1

: 8 t o c r e at e a p a t te rn t h at w i ll p ro v id e a u n i fo r m i l lu mi na t io n o f a

' ~[ 0l ' 0 2] b e E c o mp u t ed f r om E q ua t i on ( 8 ) u s i ng r e gu l a ri z at i o n m a tr i x


31/68

24

f or a n a ct ua l e rr or l ev el o f c r1

o f a n a nt en na o pt im iz ed f or a n e rr or

l e ve l o f c rZ

'

h ' d f N = 1 0 1 ( 1 2) 1 0 - i - 1 ,Patterns were synt eSlze or , Pi = 2T I -

700, a nd a n um be r o f d i ff er en t v al ue s o f M a nd c r, T h e e v ol u ti o n o f

p a t te r n s s y n th e s iz e d b y m i n i m i zi n g E f o r M = 1 , 2 , 3 , 4, 6 , 8 , 1 0 wi th

0 .1 p er ce nt i s d is pl ay ed i n Fi gu re s 3 t h ro ug h 9 . T he f ig ur es a re

a ll p ol ar p lo ts o f t he m e an p at te rn f un ct io ns i n v ar io us p la ne s s up er -

i mp os ed o n p l ot s o f t h e d e si re d p at te rn . T he p lo ts i n t h e f i rs t a n d

s e co n d q u ad r ar t ts a r e, r e sp e ct i ve l y, o f f ( B, 0 ) a n d f ( B, ~~/Z), b ot h o f

w h ic h a r e E - p 1 an e p a tt e rn s . T h e p 1 0 f fii n t h e l o w er t w o q u ad r an t s o f t h e s e

f ig ur es a re o f f ( O, ~ ) wh ic h i s a n H -p 1 an e p at te rn . N ot ic e t ha t a s

M i n cr e as e s, t h e p a tt e rn b e co m es m o re n e ar l y c i rc u la r ly s y mm e tr i c,

F o r e a ch o f t h e e x a mp l es l i st e d a b o v e, 1 0 0 i n d e pe n de n t r a nd o m

s am pl es o f o j> w e re d ra wn a nd c or re sp on di ng s am pl es o f E a n d q =

w e re g e ne r at e d. T o p e r fo r m t h is s i mu l at i on , i n de p en d en t r a nd o m s a mp l es

o f n o rm a l d i st r ib u ti o ns w i th m e an s o f z e ro a n d s t an d ar d d e vi a ti o ns

o f 5 p er ce nt I J, I ( f o r i = 1 , Z , , N ), we re s im ul at ed a s fo ll ow s,1

B y u s in g t he l aw o f l ar ge n um be rs , i t c a n b e s ho wn t ha t a r a nd om v ar ia bl e

f o rm e d b y a d di n g a n u m be r o f i n de p en d en t u n if o rm l y d i st r ib u te d r a nd o m

v a ri a bl e s i s a pp r ox i ma t el y n o rm a ll y d i st r ib u te d . I n t h is c a s e , t w e lv e

u n i fo r m " p s eu d o -r a n d om " v a r ia b l e s ( f o r me d by t h e m u l t ip l i c at i v e c o n gr u e n ti a 1

m et ho d [ 1 4] ) w er e u se d: . W it h t he s am pl es o f o j> c o mp ut ed i n t h is w a y,

t h e r e al a n d i m a gi n ar y p a rt s o f e a c h c o mp o ne n t w e r e i n de p en d en t a n d

t h er e w as n o c o rr e la t io n b e tw e en d i ff e re n t c o mp o ne n ts o f o j >.

L is te d i n T ab le 1 a re t he p r ob ab il it y m ea ns s[ 5 p er ce nt , 0 .1 p er ce nt ]

and q = E {q }, t he c or re sp on di ng s am pl e m e an s o f E a n d q d e n ot ed b y < E> a nd

< q >, r e sp e ct i ve l y, s [ O, l p e rc e nt , 0 . 1 p e r c en t ], a n d t h e m ea n c u rr e nt s , J > ,


32/68

25

M = I

e p =0 PLANE

.5 1.0 1.5 2.0 2.5

.5

1.0 8 = 0 PLANE

1.5

2.0

F ig ur e 3 . S yn th es iz ed a nd D es ir ed P at te rn F un ct io ns f or M = 1

e p = 900

PLANE

2.5 2.0 1.5 1.0 .5


33/68

26

M=2

1.0 I.5 2.0 2.5

e p =0 PLANE

1 .5

1.0 e = 0

0

PLANE

2.0

2.5

1 .0 .5



34/68

27

M =3

e p = 0 PLANE

.5 1.0 1.5 20 2.5

.5

\.0 e = 0 PLANE

1 .5

2.0

2.5

F ig ur e 5 . S yn th es iz ed a nd D e si re d P at te rn F un ct io ns f or M = 3

e p = 30 PLAN E

2.5 2.0 1.5 1.0 .5


35/68

28

o

e p = 0 PLANE

. 5 1.0 1 . 5 2.0 2 . 5

. 5

1.0 8 = 00

PLANE

1.5

2.0

2.5

M=4

o

e p = 22.5 PLANE


2.5 2.0 1.5 1.0 .5


36/68

29

e p = 0 PLANE

.5 1.0 1 . 5 2.0 2.5

.5

1.0 e = 0 PLANE

1.5

2.0

2.5.

M=6

e p = 15PLANE


2.5 2.0 1.5 1.0 .5


37/68

e p = 00

PLANE

30

2.5 2.0 1 . 5 1 . 0 . 5 . 5 1 . 0 1 . 5 2.0 2 . 5

. 5

1 . 00

e = a PLANE

1 . 5

2.0

2 . 5

M =8

: 1 : : : :1 1 1 1 1 1

: : : :

F ig ur e 8 . S yn th es iz ed a nd De si re d P at te rn F un ct io ns f or M = 8


38/68

31

M = 10

e p = 0 PLANE

.5 /.0 1.5 2.0 2.5

.5

1.0 e = 0 PLANE

1 . 5

2.0

2.5


e p = gO PLANE

2.5 2.0 1.5 1.0 .5


39/68

T AB LE 1 . R ES UL T S O F O PT IM IZ A TI ON A ND S IM UL AT I ON F OR (J ::i: 0 1% AN D V A RI OU S V AL UE S O F M .

M 1 2 3 4 6 8 10 *

J1 .5387 -.1557 -.1651 -.3509 -.8022 -.3383 -.3845

J2 1.473 1.353 1.012 .9788 1.184 .5907 .5878

J3 -.5773 -.7060 -.1293 .5093 1. 465 .7157 .7846J

4 -.8400 -.7632 -.9258 -1. 866 -3.427 -1. 541 -1.693J

5 1.195 1.135 1.187 2.070 2.376 .9383 1.020

,

J6 -.1516 -.1264, .06008 -.3540 1.615 1.149 1.249J

7 1.105 1.093 1.488 - 1 . 2 4 5 ' -4.726 -2.530 1-2.912J

8 1.450 1.421 1.81,2 1.593 4.937 2.431 2.994J

9 -.9182 - .8805 1.103 -.9550 -2.813 -1. 228 1-1.719J

10 .2572 .2401 .3019 .2464 .7428 .2946 .4984

t [ 5 % , .1%1 .04066 .01952 .01651 .02149 .1028 .02533 .04215

.04168 .01921 .01649 .0-2161 .1041 .02476 .04323

q .01464 .01230 .01451 .02077.

.1026 .02519 .04205

:01566 .01199 .01450 .02090 .1039 .02463 .4314

[ , 1% , . 1% ] .02603 .007220 .002001 7 .2 11 1 0- ~ 2 .3 23 1 0.l

i 1 . 4 23 1 0 .~ 1 .0 9 1 0 ~

S. D. .007343 .006610 .008473 .01181 .06200 .01658 .02771

Y1 1.226 1.333 1.460 1 . : j 3 5 1.467 1.583 1.594Y2 2.546 2.951, 3.504 2.876 3.531 4.009 4.075

I 3.030 2.879 3.185 3.781 8.909 4.411 5.217

* 2 50 0 s am pl es u se d WN


40/68

33

t o f i nd t he u ni ta ry m at ri x R s uc h t ha t

C'>, (28)

w he re J > = R J' >. A lt ho ug h t hi s i s a n i n ef fi ci en t w ay t o s o lv e a s ys te m

T he m et ho d u se d t o s o lv e t he s ys te m o f e qu at io ns , ( G + K)J> = C > w a s

E s in ce t he d ia go na l e le me nt s o f G a re a ll d if fe re nt .

c o nc e pt o f a s c al a r r e gu l ar i za t io n p a ra m et e r i s i n su f fi c ie n t t o mi n im i ze

n ot e t ha t e ve n f or t he s im pl e e rr or m od el u se d i n t h is e xa mp le , t he

a = 5 p e rc en t, i n a gr ee me nt w it h t he t he or y p r es en te d i n C h a pt er I V . A l so

to a= 5 p e r ce n t i n t h i s c a se , e ; T5 p e rc e nt , a] t ak es o n i ts m i ni mu m v al ue w he n

f or t he p ro pe r a mo un t o f re gu la ri za ti on , n am el y t ha t w hi ch c or re sp on ds

E[5 percent, a], w i th i t s c o r r es p on d in g < s > , a g a in s t a. N o ti c e t h at .

T ab le 2 ar e d is pl ay ed i n Fi gu re s 1 0 t hr ou gh 1 5. F ig ur e 1 6 i s a p l ot o f

I n o rd er t o t es t t he t he or y d ev el op ed i n p r ev io us c ha pt er s, s ol ut io ns

a g iv en a bo ve . T he p at te rn f un Tt io ns c or re sp on di ng t o t he c u r re nt s i n

p er fo rm ed f or e ac h c as e. S .D ., Y l' Y 2' E [5 p e rc en t, a], q,,,da, a],

m ea n c ur re nt s J . ' s an d < JJ >1 /2 a re l is te d i n T a bl e 2 f or t he v al ue ~ o f~

o bt ai ne d. S im ul at io ns a s d es cr ib ed a bo ve w i th s am p le s iz es o f 1 00 0 we re

o n e p r o pe r ly r e gu l ar i ze d a n d f i v e i m p ro p er l y r e gu l ar i ze d s o lu t io n s w e re

s im ul at io ns , a n a ct ua l e rr or l ev el o f a = 5 pe rc en t w as u se d. T hu s,

5 p er ce nt , 7 p er ce nt , a nd 1 0 p er ce nt w er e o bt ai ne d. H ow ev er , i n t he

f o r M = 1 0 r e gu la ri ze d t o e r ro r l e ve ls o f a = 0 , 1 p er ce nt , 3 p e rc en t,

p ro ba bi li ty m ea ns . O ne a ls o n ot ic es t ha t a s t h e nu mb er o f r a ys i nc re as es ,

s h ow s c l os e a g re e me n t b e tw e en s a mp l e a v er a ge s a n d t h ei r c o rr e sp o nd i ng

E [ .l p e rc e nt , . 1 p e rc e nt ] d e cr e as e s.

skewness (y1 ) a nd e xc es s (y2 ) o f S a re a ls o l is te d i n T ab le 1 . T he t ab le

a nd I < JJ > . T he s ta nd ar d d ev ia ti on ( S. D. ) an d t h e c oe ff ic ie nt s o f


41/68

TABLE 2. RESULTS OF OPTIMIZATION AND SIMULATION FOR M = 10 AND VARIOUS LEVELS OF ERROR.

0 0% 1% 3%5% 7% 10 %

J1 2.790 .02200 .04127 .04502.04585 .04557

J2 -4.989 .1808 .1693

.1657 .1626 _.1574

J3 -.9968 .04477 -.003275

-.01517 -002019 -.02464

J4 -3.809 -.09016 -.02591

-.01559 -.01535 -.02045

J5 4.613 .1006 .1112

.1143 01123 .1061

J6 -.5083 2124 .1232 .

.1001 .09032 .08179

J7,- 4.136 -.1607 -.08305-.07037 -.06700 -.06599

J ' --.5.853 -.06149 -.02768 -.01155

...008039 -.0098048

J9 - 4 .0 64 . .2824 .1751

.1386 .1241 .1126

JlO 1.346 -.1377 -.06183 -.03736

-.02643. -.01538

d5%; 6] .4718 5 .0 03 1 0- ~3 .9 1 5 10 -

4 3 .8 52 10 -1 1 3 .8 91 1 0-

4 4 . 14 7 10 -4

.4662. 5 .0 07 1 0 -' 3 .9 03 1 0- ' 3 .8 87 10 -l 3 .8 98 1 0-4 4 . 08 2 1 0. . :q

q .4718 3 .6 31 1 0- '2 .4 01 1 0- ' 2 .2 44 10 -

l2 .1 56 1 0 -' " 2 .0 24 1 0-

4

< q> .... .4662 3 .6 34 10 -1 1 2 .3 88 1 0-

4 2 .2 78 1 0 -1 1 2 .1 61 10 -

l 1 . 9 7 5 1 0 ": ' ' '

;[0, 0] 6 .7 91 1 0 -~ 1 . 51 8 1 0 -1 1 . -~ 3 .8 52 1 0- 0 4

..:.~1 .0 22 1 0- ' ;

: . 2. 37 9 1 0 5 .9 61. 10

S. D. .3647 2 .1 68 10 -1 1

1 .6 73 10 -l 1 .6 60 10 -

1 1 1 . 6 39 1 0- ' 1 1 .6 09 1 0 " :- 4

Y1 1.865 1.386' 1.728 1.8041.826 1.841

Y2 5.394 3.087 4.7685.116 5.216 5.278

/ 11. 81 .4763 .3174.2806 .2657 .2514

w~


42/68

35

(j'= 00/ 0

o

e p = 0 PLANE

.5 1.0 1.5 2.0

1.0 8 = 00

PLANE

15

2.0

2.5

.5

4 > = 90

PLANE

F ig ur e 1 0. S yn th es iz ed a nd D e si re d P at te rn F un ct io ns f or c r = 0%

2.5 2.0 1.5 1.0 .5


43/68

F ig ur e 1 1. S yn th es iz ed a nd D e si re d P at te rn F un ct io ns f or 0 = 1%

36

(J = I 0 / 0

/.0 1.5 2.0 2.5

o

8 = a PLANE

e p = 00

PLANE

/. 5

.5

1 .0

2.0

2.5

.5

e p = 90

PLANE

2.5 2.0 1.5 1 .0 .5


44/68

37

(J = = 3

0

/ 0

1.0 1.5 2.0 2.5

e p = = 00

PLANE

.5

.5

1.0 e = = 00

PLANE

1 .5

2.0

2.5

F ig ur e 1 2. S yn th es iz ed a nd D e si re d P at te rn F un ct io ns f or c r = 3%

e p = = 90

PLANE

2.5 2.0 1.5 1.0 .5


45/68

38

o

8 = 0 PLANE/.0

.5 /.0 1.5 2.0 2.5

1 .5

.5

2.0

2.5

e p = 90

PLANE

F ig ur e 1 3 . S yn t he si ze d a nd D es i re d P at te rn F un ct io n s f or a = 5%

2.5 2.0 1.5 /.0 .5


46/68

39

(J = 7

0/ 0

o

8 = 0 PLANE

.5 1.0 1.5 2.0 2.5

1 .0

1 .5

2.0

2.5

.5

o

e p = 9 PLANE

F ig ur e 1 4. S yn th es iz ed a nd D es ir ed P at te rn F un ct io ns f or a = 7%

2.5 2.0 1.5 1.0 .5


47/68

40

1 .0 1 .5 2.0 2.5.5

e p = 9 PLANE

2.5 2.0 1.5 1.0 .5

.5

1.0 e = 0 PLANE

1 .5

2.0

2.5

0- = 10 %

F ig ur e 1 5. S yn th es iz ed a nd D e si re d P at te rn F un ct io ns f or 0= 10 %


48/68

41

5

4

3

2

oo

F ig ur e 1 6. M ea n E'S f or S ol ut io ns R eg ul ar iz ed t o V ar io us N oi se L ev el s

b ut i n t he P re se nc e o f a n A ct ua l c r = 5 % L evel


49/68

42

o f l i n ea r e qu at i on s, i t d o es a vo id s om e o f t he n u m er ic a l d if f ic ul t ie s

o f C ho l es ky d ec om p os it i on a nd G au ss i an e l im in at i on f o r i ll -c on d it io n ed

"m at ri ce s. T he c om pu te d e ig en va lu es o f G a re l is te d i n T ab le 3 be lo w.

T AB LE 3 . C OM PU TE D E IG EN VA LU ES O F G.

~'I 7.742 A 60.01619

A2

0.9582 A7

1.779 1 0 -4

1 .3

= 0.4329 A8

= 4. 918 10 -7

1 .

4

0.1579 A

9

= -1.907. 10 -8

A S 0.04106 A lO="" - 1.98 7. 10 -

8

T he l as t t wo e nt ri es o f T a bl e 3 c a n b e c on si de re d a s n ot hi ng l es s t ha n

n on se ns e s in ce G i s a p os it iv e d ef in it e m at ri x a nd , h en ce , i ts e ig en -

t he s ma ll es t e ig en va lu es o f G , i t i s h ig hl y d ou bt fu l t ha t t he c ur re nt s

o f G, t he c or re sp on di ng f ir st -o rd er v ar ia ti on i n i ts e ig en va lu e; ;V

h av e o ve rw h el m~ d i ts s m a ll es t e ig en v al ue s . O f c ou r se , w h en a s u ff ic i en t

For an error oG .in.):the computation.

a m ou nt o f r e gu la ri z at io n i s a d de d, t he n t he p r o bl em d is a pp ea rs s in ce t h e

s en s e t h at t he n um er i ca l e r ro rs i n c o m pu ta ti o n o f G a nd i t s e i g en v al ue s

I n f a ct t he s o lu ti on o bt a in ed f or 0= 0 i s r ea ll y r eg ul ar iz ed i n t h e

i n T a bl e 2 c or re sp on di ng t o 0= 0 c an b e a ny th in g e ve n c lo se t o J >.o

S in c e t h e c ur re n ts o bt a in ed f ro m E qu a ti on (4) a r e s t ro ng l y d ep en de n t o n

v al ue s m u st b e p os it i ve .

i t s a c t ua l v al ue i s n o t m u ch g r ea te r t ha n t h e. le ve l of m a ch in e . p re ci s io n;

is 0 1 . "':, where u> is a corresponding norm al ized eigenv ector

of G. In view of .this one cannot hope to calculate A accuratel y if


50/68

then

43

s ma ll es t e ig en va lu e t ha t a pp ea rs i n E q u at io n ( 28 ) i s o f t he o rd er o f

02

w hi ch c an b e m u ch g re at er t ha n t he l ev el o f n u me ri ca l e rr or s.

A lt ho ug h J > i n i ts el f i s o f n o p ra ct ic al i nt er es t, J > a s w el lo 0

a s t he e ig en va lu es o f G e nt er i nt o t he f or mu la s n ec es sa ry f or t he

c al cu la ti on o f t he m o me nt s o f ~ t hr ou gh E qu at io ns ( 13 ), ( 20 ), a n d

( 22 ). T hi s r a is es t he q u es ti on : H ow c an t he m om en ts a nd a p p ro xi ma te

d is tr ib ut io n f un ct io n o f ~ b e c al cu la te d b y t he m e th od o f C ha pt er V

a c cu r at e ly ? I n de e d, i f s o me o f t h e c a l cu l at e d e i ge n va l ue s a r e n eg a ti v e,

h ow c an t he m et ho d o f C ha pt er V e ve n b e a pp li ed i n vi ew o f t he f or m

o f D ? I t i s d e mo ns tr at ed b el ow t ha t i n s p it e o f t he se n um er ic al

d i ff i cu l ti e s, m e an i ng f ul r e su l ts c a n s t il l b e o b ta i ne d .

F ro m t he d ef in it io n o f U i n C h ap te r V , i t f ol lo ws t ha t i f X> = U Y> ,_ _ r _

< XG X> ' " L A I y 1 2, w he re r i s t he ' ef fe ct iv e ra nk " o fG , t he e ig en va lu esk=l k k

o f w hi ch a re a ss um ed t o b e i n d e sc en di ng o rd er . C er ta in ly r m us t b e c ho se n

s m al l e n ou g h t o e x cl u de a n y e i ge n va l ue s e r ro n eo u sl y c a lc u la t ed t o b e n e ga t iv e .

A l th o ug h s m al l p o si t iv e e i ge n va l ue s m a y a l so b e e r ro n eo u s, . o n e m u st t a ke c a re

n ot t o. ex cl ud e t oo .m an y e ig en va lu es s in ce t he y a pp ea r a s s qu ar e t oo ts i n T i n

E q ua t io n ( 2 0) W i th . .t he a pp ro xi ma ti on g iv en a bo ve ( wh ic h o f c ou rs e i s t o b e

i nt er pr et ed i n a p ro ba bi li st ic s en se ), t he S m at ri x b ec om es t he r X N m a t ri x

1 .1/

2 o . . .01

0

S 1 .1/

2 o . . 0 Ut.2

0 ' . ~ . 1 .1/2 o . 0r

A l = S A St

a nd i ts u n it ar y m at ri x W ar e n ow r X r m at ri ce s a nd t he

t ra ns fo rm at io n m at ri x T b ec om es a n r X N m a tr ix . F ro m E qu at io ns ( 4)

a n d ( 2 0) , t l J" > d e p en d s o n


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d i st r ib u ti o n f u nc t io n o f E t o b e c o mp u te d a cc u ra te l y. T h e d i s pl a ce m en t

C' >

4 4

1 0 , M = 1 0 , a nd c r = .1 percent.

-1A l

-1

0 . . 0 1 .. 2 0

Q Qt

U.u .-1. Ar. . . .

0 00

t he s ma ll er e ig en va lu es o f G . F or tu na te ly t he m ea n o f E c an b e

s em i de fi n it e w i th r a nk r h ol d s.

T hi s w as d on e f or t he c as e w he re N

t [,-1/2 -1/2= W Diag h

1'.~2 '

h ow s ma ll r s ho ul d b e a nd h ow g oo d t he a pp ro xi ma ti on s m ad e a bo ve a re

S in ce o ne h as n o w a y o f k n o wi ng w ha t t he s ma ll er i na cc ur at e e ig en -

X . . . , l

N

v al ue s o f G a ct ua ll y a re , t he b e st a nd p er ha ps t he o nl y w a y t o d e te rm in e

a pp ea rs t o b e s im ul at io n u si ng t he r ep re se nt at io n o f E g i ve n i n ( 19 ).

o f t h e o rd er o f ma ch in e p re ci si on a nd t he e rr or s i n t h e c om pu ta ti on o f

the calculated J > m a y b e c o ns i de r ab ly i n e r ro r , LU"> computed byo

on J >. I t i s d if fi cu lt t o s t ud y t he s en si ti vi ty s t o e rr or s i n 6 Go a

t he s ec on d a nd h ig he r m om en ts o f E a nd h en ce t he s ha pe o f t h e

o f t h e d i st ri bu ti on f un ct io n d ep en ds o n s w hi ch , i n t u rn , d e pe nd so

w h ic h e xc l ud e s t h e q ue s ti o na b le e i ge n va l ue s. T he r ef o re , a l th ou g h

( 22 ) w it h t he T g iv en a bo ve c an b e c om pu te d a cc ur at el y. T hi s a ll ow s

c o mp u te d f ro m ( 1 1) w h ic h d o es n o t i nv o lv e J >. T h us a l l t h e m om e nt s\ a

o f E c an b e c om pu te d a cc ur at el y i f t h e a ss um pt io n t ha t G i s v er y n ea rl y

TJ > =Wto


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45

T he r es ul ts d is pla ye d i n F ig ur e 1 7 s ho w e xce ll ent a gr ee me nt b etw ee n

t he a pp ro xi ma te d is tr ib ut io n f un ct io n a nd t h e s im ul at io n. F or

t hi s e xa mp le , t wo i nd ep en de nt s im ul at io ns w it h s am pl e s iz es o f 2 5 00 w er e

p er for me d. F or a ll t he ex am ple s co ns ide re d i n th is ch ap ter , th e

E dg ew or th s er ie s o f E q ua ti on ( 26 ) wa s u se d t o a pp ro xi ma te t he d is tr ib ut io n

f unc ti on. T he cu mu lan ts o f t he d i str ib uti on f un cti on o f E f ro m w hi ch t he

p ar ame te rs of ( 26) a re c al cu la ted we re c om put ed u si ng t he m at rix T a s

m od if ie d a bo ve b y e xc lu di ng o nl y t he e rr on eo us n eg at iv e e ig en va lu es

of G . It was found that E calculated by (8) and the formula, Eo + E{}=Xl

a s g iv en b y E q ua ti on (2 4) , ag re ed t o a t le as t t hr ee s ig nif ic ant di gi ts fo r

examples considered.

T he d is tr ibu ti on f un ct ion s f or t he e xa mp le s w he re N = 1 0, M = 1 0,

c r = 0 , 1 p e rc en t, 3 pe rc en t, 5 p e rc en t, 7 p er ce nt , a nd 1 0 p e rc en t( co ns id er ed

p re vi ou sl y) ar e p lo tt ed w it h t he ir c or re sp on di ng s im ul at io ns i n F ig ur e 1 8.

T he f ig ur e i nd ic at es t ha t t he d is tr ib ut io n c or re sp on di ng t o t he s ol ut io n

r eg ul ar iz ed t o t he a ct ua l c r = 5 p er ce nt er ro r l ev el pr es ent n ot o nl y

h as t he m ini mu m me an b ut g en er all y h as t he m os t d es ir ab le p ro per ti es o f

a ll t he d i st ri bu ti on s p lo tt ed . T he f ig ur e a ls o i nd ic at es t ha t a lt ho ug h,

in v ie w o f t he s ca le of E , t he re i s ve ry l itt le p ra cti ca l d if fe ren ce

b et we en t he d is tr ib ut io n f un ct io ns , i t do es a pp ea r t ha t i t i s b et te r

t o " o ve r- re gu la ri ze " s li gh tl y t ha n " un de r- re gu la ri ze " i f t he e xa ct

n atu re o f t he p ro ba bi lit y l aw o f oj> i s n ot k no wn .

I n a ll o f t he ex am ple s c on sid er ed i n t his c ha pte r, t he p ha se

fu nc tio n of f d( e, ~ ) w as c ho sen to b e u ni ty w hi ch l ea d t o r ea l s ol uti on s.

I t is n ow sh ow n t ha t: f or t he a nt en na s tr uc tu re u se d i n t hi s ch ap te r

a ll s ta ti on ar y p oi nt s J> of (17) a re real. Moreover, it is shown that

al l s ol ut ion s o bta in ed in t hi s c ha pt er a re s ta ti on ar y p oin ts o f (1 7) . D ue

t o t h e s ym me tr y o f t he a nt en na , v> an d c on se que nt ly G are r e al . E qu at io n ( 15 )

c an b e w ri tt en e qu iv al en tl y a s


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--~ --~-----

F(E)

1.0

46

E DG EW ORT H S ER IE S

G SIMULATION

.9

. 8

.7

.6

.5

.4

.3

. 2

.1

oo .01 .02 .03 .04

X

.05

E

.06 .07 .08

x

.09

oX

.10

Figure 17. Edgeworth S eries and S imulation for N = M = 10 , a . 1 %


54/68

00

x 10-

8

0* . 8 . 0

76

SIMULA TIONS

X 0" = .01

. (J = .03

o (J=

.05A (J = .07

.0"=.10

5E

432

F ig u re 1 8 . D i st r ib u ti o n F u nc t io n s f o r So lu t io n s R eg u la r iz e d t o V ar i ou s

N oi se L ev el s b ut i n t he P r es en ce o f a n A ct ua l a = 5% L evel

F (E )

1 .0

0.1

0.3

0.2

0.4

0.7

0.6

0.9

0.8


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48

G(J> = { J I ~ I v> - HJ >

w he re t he n o rm al iz at io n c on st an t u se d i n t he d ef in it io n o f 1 1 ' 1 1 2 is

i nc lu de d. Li ke G , H i s a po si ti ve d ef in it e r ea l s ym me tr ic m at ri x.

S ta ti on ar y p oi nt s o f ( 17 ) s at is fy ( IS ) w hi ch i s e qu iv al en t t o

(G + K) J> HJ>.

B ut t hi s i s a n e ig en va lu e e qu at io n w it h u ni t e ig en va lu e a nd r ea l

eigenvector J> . T he re fo re , i f a s ta ti on ar y p oi nt o f ( 17 ) e xi st s

then J> is rea l. Th e p lo ts of th e p atte rn s obt ain ed i n t hi s s ect io n

sh ow t ha t f( 8, ~ ) = > O. E vid ent ly , f or t hes e pa tte rns , G>

c al cu la te d f ro m ( 15 ) i s i d en ti ca l t o G > c al cu la te d b y ( 2) , w hi ch

i mp li es t ha t t he c or re sp on di ng s ol ut io n c ur re nt s J~:a r e a l so s t at i on a ry

p oi nts o f (17 ). Any oth er s ta tio na ry poi nts of (17 ) w oul d ha ve to

g iv e ris e to a p at ter n wi th nu ll s in t he p att ern ot her th an at 8 = O.

I f a ny s uc h p oi nt s a ct ua ll y e xi st ~i t s ee ms v er y d ou bt fu l t ha t t he y

w ou ld p ro du ce a n E f ro m (17 ) wh ic h i s l ess t ha n th at pro du ced by th e

s ol ut io ns f ou nd i n t hi s s ec ti on . T hu s, a lt ho ug h i t i s n ot p ro ve n,

i nt ui ti on l ea ds o ne t o s us pe ct t ha t t he s ol ut io ns o bt ai ne d i n t h is

s ec ti on m ay a ls o p ro du ce g lo ba ll y m in im um E d ef in ed i n ( 17 ).


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array of five isotropic radiators located at

49

V II. A N EXAMPL E OF AMPL ITUDE P ATTERN S YNTHESIS

I t wa s s ho wn a t t he e nd o f C h ap te r V I t ha t t he s ol ut io ns f ou nd

to minimize the quadratic performance index of (8) at least make the

nonquadratic index of (17) stationary and possibly minimum. This

type of behavior will occur for structures with certain symmetry. For

structures with no such symmetry, the stationary points of (17) will,

in general, be complex.

To illustrate how the pattern one obtains is improved by allowing

t he p ha se t o b e f re e, t he f ol lo wi ng e xa mp le i s i n cl ud ed . A l in ea r

z1 .0= ( ~2 i - (l)i]/k

2 a

on the z-axis is used to synthesize the one-dimensional (circularly

symmetric)pattern

o

s ec e

For this problem, Vo = ex p (jzo co s e ] and G. = sin ( z. - z.)/( zo zo)'1. 1. 1.J 1. J 1. J

A n initial solution, J >, was obtained without regularizationa

f ro m ( 3) w it h C > c a lc ul at ed f ro m ( 2) . W it h t hi s a s 'a starting.point

Davidon' s method was applied to E of (17) w it h K = O . T he g ra di en t

o f t he p er fo rm an ce i nd ex i s e qu al t o

where

' I T

n> f V> (1 - fd!l-I) sin ede + KJ >o

was calculated by S impson' s rule.


57/68

50

F ig u re 1 9( a ) d is p la ys t h e pa t te r n o f t he i n it i al s o lu t io n w hi ~ e

1 9 (b ) i s t h e p a tt e rn f u nc t io n o f t h e s ol u ti on . f ou n d b y i t er a ti o n.

F ig ur es 2 0( a) a nd 2 0( b) a re t he r es pe ct iv e p ha se f un ct io ns . F ig ur e 1 9( b)

s h ow s a c o ns i de r ab le i ~ pr o ve m en t o v er 1 9 (a ) , a n d t h is i m pr o ve m en t

c om es w i th t he a dd it io na l b en ef it o f a l ow er E uc li di an n or m o f

J >. T he r e su lt s a re s um ma ri ze d i n T a bl e 4 b el ow .

T AB LE 4 . R ES UL TS O F O PT IM IZ AT IO N O F e:.

I n it ia l J > F i n al I t e ra t e J>

Jl

1.662 LJ.3. 87 .752008.43

J2

2.421038.6 1.525 ~9.74II .

J3

2.40504.66 1.075 L1l4.3

J4

1.802040.3 1.175 ! . . : : J 6 . 3 5 I

,

.5800U1.98 .7238047.9,

J5

e : .5457 .19940

1/2 4.237 2 . 4 4 0


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(a)

.5

( b)

1.01 .5

F ig ur e 1 9. ( a) P at te rn o f In it ia l A pp ro xi ma ti on C or re sp on di ng t o M in im um o f

E qu at io n ( 8) ( K = 0)( b) Pa tt er n o f F i na l I te ra te C or re sp on di ng t o M in im um o f

E qu at io n ( 17 ) ( K = 0)

lJl

~


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52

160 180

140 160 \80

(b)

(0)

e (DEGREES)

e (DEGREES)

PHASE (DEGREES)

180

150120

90

60

30

o-30

-60-90

-120

-\50

-180

180

150

120

90

60

30

o-30

-60

-90

-120

-150

- 180

Figure 20. (a) Phase Function of P attern of F igure 19 (a)

( b) P ha se F un ct io n o f Pa tt er n o f F ig ur e 1 9 ( b)


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53

V III. C ONC LU SIONS

It has been demonstrated that the problem of antenna synthesis may

be approached realistically by considering the excitation of antennas

a s r a nd om a ro un d s om e me an v al ue . B y s o d o in g, t he c ri te ri on o f c lo se ne ss

of the synthesi~ed and desired patterns, s, becomes a random variable.

Three philosophies of optimization of such a random variable were given.

I t w as s ho wn t ha t m in im iz at io n o f t he m e an o f E leads very naturally

to a generalization of the concept of regularization and to a simple

and direct mehtod of computing the proper "amount" of regularization.

It should be remembered that to apply this method only the knowledge of

the second moments of the errors is required. By making some additional

assumptions on the probability law of the errors, it was demonstrated

that the distribution function of E could be approximated by the

Edgeworth series. S uch an approximation could then be used to apply

the horizontal (i.e., minimizing s for a given probability) and vertical

(i.e., maximizing probability for ~ not exceeding a given value)

optimization schemes. Finally, a random performance index suitable for

amplitude pattern synthesis was included.

S everal examples were considered, all of which involved the synthesis

of three-dimensional, circularly symmetric pattern functions. Most involved

planar arrays of vertical Hertzian dipoles. In these examples, considerable

simplification was obtained by taking advantage of the high symmetry

properties of the antenna array. To demonstrate the theory, several

improperly regularized solutions and one properly regularized solution

w er e o bt ai ne d. I t w as f ou nd i n t h es e e xa mp le s t ha t b y c ho os in g t he

regularization matrix properly, ErE} is indeed' minimiZed ..' In addition,

simulations of the errors were perfor,med' for each solution


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54

o bt ai ne d. F ~g ur el 6a nd T ab le s 1 a n d 2 d em on st ra te t he c lo se a gr ee me nt

b et we en t he or y a nd t he r es ul ts o f t h es e s im ul at io ns .

T he d is tr ib ut io n f un ct io n o f s w as c om pu te d b y t he E dg ew or th s er ie s

fo r a n umb er of ex amp le s. T hi s w as ac com pl is he d in s pi te o f s eve re

n um er ic al e rr or s i n t he s ma ll er e ig en va lu es o f G ( an d c on se qu en tl y i n t he

u n re g ul a ri z ed s o lu ti o n, J b y c on si de ri ng G a s e ss en ti al ly a p os it iv eo

s em id ef in it e m at ri x. A n um be r o f s i mu la ti on s o f t he d is tr ib ut io n

f un ct io ns w er e p er fo rm ed ( Fi gu re s 17 a nd 1 8) . I n e ac h c as e, t he

d is tr ib ut io n f un ct io n a pp ro xi ma te d b y t h e E d ge wo rt h s er ie s a nd t he s im ul at ed

d is tr ib ut io n a gr ee d q ui te w el l.

A n e xa mp le o f am pl it ud e p at te rn s yn th es is w as i nc lu de d. T hi s e xa mp le

d em on st ra te d t ha t c o ns id er ab le i mp ro ve me nt i n t he a mp li tu de p at te rn

f un ct io n o ve r t ha t r ea li ze d b y mi ni mi z~ ng ( 8) is o b ta in ed ( at th e

e xp en se o f r eq ui ri ng i te ra ti on ) w he n o ne m i ni mi ze s E o f E qu at io n ( 17 ).


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55

APPENDIX A.

1.0 Calculation of the Characteristic Function of a Generalized Noncentral

X2 Distribution

The characteristic function of a random variable E i s d ef in ed a s

-2Consider the random variable X formed by squaring the normal random

v ar ia bl e X w it h t he m ea n y a nd t he v a ri an ce ~ Vo The characteristic

-2function of X is

-2n(t) E{eijtX}= 1/;;:;;. fo o e xp [jtx2 - ( x - y )2 /v ] d x = [11 ~ jvt]-l

_00

r 00

'll/l-1TV-/-(-l---jV-t-) J , [ ex p[ J

2x _ ]1(1 - jvt)

[ ( 1 ~ jv t) ]

dx)-' exp [j"2t/(1 - jvt)]

B y c om pa ri ng t he f ac to r i n b r ac es , { .} , w it h a n or ma l d is tr ib ut io n w it h

1variance of 2 v /( l - j vt ) a nd m e an y j( l - jvt ), i t i s e vi de nt t ha t { .} = 1

an d

n (t ) e xp [ j ]1 2tj (1- j vt) ]/ Il - j vt .

N ex t c on si de r t he r an do m v ar ia bl e ~ 2

X l' X2

, , XN

are independent, normally distributed, complex, random

t he in de pen de nce o f Xl' , XN,

variables

imaginary

w i t h m e an ~ ]11 '1

p ar ts o f 2vI '

Y N a nd v a ri an ce s o f t he r ea l a nd1, 2 v

N' r es pe ct iv el y. I n v ie w o f

t he c ha ra ct er is ti c f un ct io n o f ~ 2 i s

- 2 - . 2t he p ro du ct o f t he c ha ra ct er is ti c f un ct io ns o f t he ( Re X l) , (1 m Xl) ,

- 2 - 2(R e X

N) , (1 m X

N) , an d i s g iv en by


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56

By Shl'ftl'ngX-2 d f' d' th' h b has e lne In e prevlous paragrap y t e constant

So' Equation (23) results

2.0 Calculation of the Cumulants of a Generalized Noncentral x2

Distribution

The cumulants of a characteristic function,~(t), are defined as

,-k dk

v = J --- ~n ~(t)k dtk

With n(t) defined in the first paragraph

t= O

k 1, 2,

~n n(t) -I~n (1 - jvt) + j~2t/(1 - jvt)

-1j

d.Q ,n n

dt

v 2 2.2 + V V yt---+ j-----1 - jvt 2

(1 - jvt)

v2 2

2 + 2V v + j2 1 (1 - jvt)2

2 2V v t

(1 - jvt)3

which is proven by

k-1 d .-k d~n nj dt J k

dt


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57

,

+ j (k +l ) k! t vk,}vl (1 - jvt)k+2

Setting t 0,

a re t he c um ul an ts o f n e t) . I t f ol lo ws f ro m t hi s t ha t t he c um ul an ts o f

~ l j :I (t )ar e

N

vk = ( k - l ) ! Li=l

kk-l I 12[ v. + k v. ]J.]1. 1. 1.

f ro m w h ic h E qu at io ns ( 24 ) a nd ( 25 ) m ay b e d er iv ed .


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A P PE ND I X B .

1 .0 C al cu la ti on o f G

2MV . =I exp [ jPi cos (~ - ~ k) sin e] sin e

1. k= l

2'IT'IT/2 *G. , = f f V ,V , s in e d e d ~/ 8' IT M

1.J 0 0 1.J

58

G ..1. J

where

an d

= ta n-l[(P .1.

U si ng f or mu la 9 .1 ,2 1 o f [15],

si n si n

1 'IT/2. '"-- f e Jz co s ~ d ~ = J (z )2 'IT 0 0

where J i s t h e B es se l f un ct io n o f t h e f ir st k in d o f z er o o rd er ,o

2M 2M 'IT/2

G, . = IL f J (p, 'k .Q, s in e ) s in3e d e/ 4 M.1 .J k =l ,Q ,= l 0 0 1. J

U si ng f or mu la 1 1. 4 .1 0 o f [15],


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59

, an d f or mu la s i n C ha pt er 1 0 o f [ 1 5] ,

2M 2M

G. . L L 8 (p i.k ) 14M~J k = l = 1 J

where

8(p) = sin(p)(l/p - l/p3) + cos (p)/p2.

F in al ly , t ak in g a dv an ta ge o f t he d eg en er ac y i n P ij k ~'

M

G . . = L 8(P"k) + -2l

[8(P.+ p.)+ 8(p. - P.)]~J k= 2 ~J ~ J ~ J

where

( To a p pl y t he f or mu la t o G . . , n ot e t ha t a n a pp li ca ti on o f L 'H os pi ta lt s~~

2r ul e s ho ws t ha t 8 (0 ) = 3 .)

2 .0 C al cu la ti on o f C >

C =i

2T I T I/2

J J

a af ee ) V . s in e d e d ~/ 8T IM

~

1 2M 2TI T I/2

=- L JJ e xp [ j P. c os ( ~-

8/14/2019 Richards

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Richards Antenna Pattern Synthesis Based on Optimization in a Probabilistic Sense