Stochastic Modeling
with Lossless Structures
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ityShiliiliiiiiiill . n i !
-P- -«4Continuous-Time Stochastic Modeling
with Lossless Structures
BIBLIOTHEEK TU Delft P 1725 4472
w i t h L o s s l e s s S t r u c t u r e s
P r o e f s c h r i f t
t e r v e r k r i j g i n g van de graad van d o c t o r i n de t e c h n i s c h e wetenschappen aan de Technische Hogeschool t e D e l f t , op gezag van de r e c t o r m a g n i f i c u s , P r o f . i r . B.P.Th. Veltman
voor een commissie aangewezen door het c o l l e g e van dekanen t e v e r d e d i g e n op d i n s d a g 7 december 1982 te 14.00 uur.
door
Ing W i d y a
and a l s o t o those who made i t p o s s i b l e f o r me t o pursue my s t u d i e s i n the N e t h e r l a n d s .
In t h i s t h e s i s we d e r i v e white n o i s e e x c i t e d modeling f i l t e r s f o r
continuous-time s t a t i o n a r y p r o c e s s e s c o n t a i n i n g a white n o i s e component. The c l a s s o f f i l t e r s we c o n s i d e r have l o s s l e s s s t r u c t u r e s and i n c l u d e both d i s t r i b u t e d and lumped p a r t s .
To b e g i n w i t h we do l i n e a r l e a s t squares p r e d i c t i o n on the given p r o c e s s , so t h a t the p r e d i c t i o n e r r o r i s o r t h o g o n a l t o the f i n i t e p a s t which i s used f o r p r e d i c t i o n . An e q u i v a l e n t f o r m u l a t i o n of the o r t h o g o n a l i t y i s the Fredholm e q u a t i o n from which we d e r i v e a d i f f e r e n t i a l e q u a t i o n f o r the t r a n s f e r f u n c t i o n o f the i n n o v a t i o n f i l t e r . T h i s f u n c t i o n t u r n s out t o be o f minimal phase (outer) type. We embed i t i n t o a l o s s l e s s t r a n s f e r m a t r i x which d e s c r i b e s a l o s s l e s s t r a n s m i s s i o n l i n e . The i n v e r s e o f the i n n o v a t i o n f i l t e r d e s c r i b e s the modeling f i l t e r , the spectrum o f which, due t o o r t h o g o n a l i t y , equals the r e a l p a r t o f the impedance f u n c t i o n o f the t r a n s m i s s i o n l i n e . T h i s impedance f u n c t i o n u n i q u e l y determines a s c a t t e r i n g f u n c t i o n which expresses the r e l a t i o n between the i n c i d e n t and r e f l e c t e d wave o f the l i n e . We use the method o f i n v e r s e s c a t t e r i n g t o s o l v e the modeling problem which f o r c e s the impedance t o be the c a u s a l p a r t o f t h e spectrum o f the observed p r o c e s s .
We t r a n s l a t e t h e d i f f e r e n t i a l e q u a t i o n f o r the p r e d i c t i o n e r r o r (more o f t e n r e f e r e d as the d i f f e r e n t i a l e q u a t i o n f o r the p r e d i c t o r ) t o a
system o f d i f f e r e n t i a l equations f o r the waves i n the t r a n s m i s s i o n l i n e . A p p l y i n g n u m e r i c a l r u l e s on the system we f i n d a s e t o f r e c u r s i v e wave e q u a t i o n s . The r e f l e c t i o n f u n c t i o n which c h a r a c t e r i z e s the t r a n s m i s s i o n l i n e i s s o l v e d n u m e r i c a l l y as a r o o t o f a t h i r d order e q u a t i o n which a r i s e s from t h e wave equations and the c a u s a l i t y c o n d i t i o n of the t r a n s m i s s i o n l i n e . The s e t o f r e c u r s i v e wave e q u a t i o n s and the p r o c e -dure which s o l v e s the r e f l e c t i o n f u n c t i o n together form the b a s i s f o r our i n v e r s e s c a t t e r i n g a l g o r i t h m s . Next we d e r i v e i t e r a t i v e as w e l l as r e c u r s i v e a l g o r i t h m s t o s o l v e the system and demonstrate t h e i r s t a b i l i -t y by v e r i f y i n g -t h e i r performance w i -t h a number o f -t e s -t f u n c -t i o n s . A chapter i s devoted t o the mixed modeling, i . e . when both lumped and d i s t r i b u t e d p a r t s are p r e s e n t . We show how t o s h i f t a lumped s e c t i o n through a d i s t r i b u t e d one. Having t h i s s h i f t through r e l a t i o n we may s e p a r a t e l y determine the lumped and the d i s t r i b u t e d s e c t i o n s d i r e c t l y from the g i v e n d a t a . F i n a l y , we show how, u s i n g t h i s methods, the t r u e cascade i s o b t a i n e d .
I . INTRODUCTION 1
I I . LINEAR PREDICTION AND INVERSE SCATTERING 9
2.1. I n t r o d u c t i o n 9 2.2. The s t o c h a s t i c p r o c e s s y ( t ) 12
2.3. The p r e d i c t i o n problem 16 2.4. The c h a i n s c a t t e r i n g m a t r i x 0(T,p) 21
2.5. The s c a t t e r i n g m a t r i x I(T,p) 26 2.6. The p r e d i c t i o n and modeling f i l t e r 34
2.7. Inverse s c a t t e r i n g 36 2.8. The r e l a t i o n between i n v e r s e s c a t t e r i n g and
p r e d i c t i o n 37 2.9. Approximation o f the p r o c e s s impedance and the
i n v e r s e s c a t t e r i n g theorem 42 2.10. S p e c t r a l approximation problem 48
2.11. References 52
I I I . NUMERICAL IMPLEMENTATION AND ALGORITHMS 55
3.1. I n t r o d u c t i o n 55 3.2. R e l a x a t i o n s o l u t i o n f o r the t r a n s m i s s i o n l i n e 56
3.3. P a r t i t i o n i n g o f the i n v e r s e s c a t t e r i n g problem 60 3.4. Methods t o s o l v e the i n v e r s e s c a t t e r i n g problem 62 3.5. The procedure t o compute the waves a'(T,t) and
b ' ( T , t ) 63 3.6. The n u m e r i c a l implementation 67 3.7. The f i l t e r s t r u c t u r e 69 3.8. The r e c u r s i v e i n v e r s e s c a t t e r i n g a l g o r i t h m s 74 3.9. The £ i t e r a t i v e a l g o r i t h m 79 3.10. The modeling a l g o r i t h m f o r k ( t ) 84 3.11. O r t h o g o n a l i t y r e l a t i o n s 92
3.14. References 101
IV. NUMERICAL RESULTS 103 4.1. I n t r o d u c t i o n 103 4.2. Accuracy o f the computed f u n c t i o n s k ( t ) 106
4.3. The t e s t f u n c t i o n s k ( t ) 111 4.4. A p r e l i m i n a r y t o the i n v e r s e s c a t t e r i n g r e s u l t s 120
4.5. Accuracy o f the waves i n the i n v e r s e s c a t t e r i n g case 122 4.6. The performance o f the i n v e r s e s c a t t e r i n g a l g o r i t h m s 125
4.7. Numerical v e r i f i c a t i o n o f the o r t h o g o n a l i t y 133 4.8. V e r i f i c a t i o n o f the c o n d i t i o n i n g o f the Schur
e q u a t i o n 137 4.9. Numerical r e s u l t s i n n o i s y circumstance 142
V. MIXED MODELING: LUMPED AND DISTRIBUTED 153
5.1. I n t r o d u c t i o n 153 5.2. The D a r l i n g t o n s e c t i o n 155 5.3. A l g e b r a on c o n s t a n t J - u n i t a r y and J - l o s s l e s s m a t r i c e s 159 5.4. S h i f t - t h r o u g h o f two D a r l i n g t o n s e c t i o n s 161 5.5. The i n f i n i t e s i m a l K r e i n s e c t i o n 169 5.6. S h i f t - t h r o u g h o f a D a r l i n g t o n s e c t i o n and an i n f i n i t e s i m a l K r e i n s e c t i o n 171 5.7. S h i f t - t h r o u g h o f a D a r l i n g t o n s e c t i o n and a K r e i n s e c t i o n 176 5.8. Numerical procedure f o r the s h i f t - t h r o u g h problem 182
5.9. D e t e r m i n a t i o n o f the D a r l i n g t o n r e f l e c t i o n c o e f f i c i e n t 191
5.10. The output waves o f the D a r l i n g t o n s e c t i o n 193 5.11. Computing k ( t ) from a cascade o f K r e i n and
D a r l i n g t o n s e c t i o n s 195 5.12. The s t r u c t u r e o f the D a r l i n g t o n s e c t i o n 6 199
APPENDIX A 203
L i s t o f symbols and a b b r e v i a t i o n s 207
I. INTRODUCTION
In t h i s t h e s i s we develop models f o r s t o c h a s t i c p r o c e s s e s of a s p e c i a l type. The p r o c e s s e s we c o n s i d e r are time-continuous and posses a c o v a r i a n c e f u n c t i o n o f the type 6(t) + (t) where 6(t) i s a D i r a c impulse and k^ (t) i s (reasonably) smooth. A speech s i g n a l f o r i n s t a n c e o f t e n produces such a p r o c e s s . The models w i l l c o n s i s t s o f e i t h e r a
(nonuniform) e l e c t r i c a l t r a n s m i s s i o n l i n e or a mixture o f lumped and d i s t r i b u t e d c i r c u i t s , e x c i t e d by white n o i s e . We w i l l show t h a t our problem may be reduced t o an i n v e r s e s c a t t e r i n g problem and we s h a l l develop numerical methods t o s o l v e the case a t hand. The models d e s c r i b e d w i l l be a c c u r a t e i n the "wide s t a t i o n a r y " sense - which would be good enough f o r the c l a s s i c a l speech modeling case.
Implementation of our models c o u l d be e i t h e r by a p h y s i c a l r e a l i z a t i o n o f the t r a n s m i s s i o n systems which we compute, or i f a numerical a p p r o x i mation i s d e s i r e d , by a d i s c r e t i z a t i o n o f i t . We show t h a t a d i s c r e t i -z a t i o n , i f p r o p e r l y executed, belongs t o a c l a s s o f f i l t e r s w i t h v e r y a t t r a c t i v e numerical p r o p e r t i e s , namely " o r t h o g o n a l " f i l t e r s . Our t h e o r y then i s an outgrowth o f some c l a s s i c a l t h e o r i e s : the t h e o r y o f Schur-p a r a m e t r i z a t i o n o f a g i v e n i n Schur-p u t s c a t t e r i n g f u n c t i o n , the theory o f s y n t h e s i s o f a p a s s i v e p o s i t i v e r e a l f u n c t i o n as c o n s i d e r e d i n Network Theory, and the theory o f l e a s t squares e s t i m a t o r s due t o Wiener, L e v i n s o n and K r e i n .
The d i s t r i b u t e d o r t h o g o n a l f i l t e r s d e s c r i b e l o s s l e s s media or t r a n s m i s s i o n l i n e s . The main parameter i n these f i l t e r s i s the r e f l e c t i o n f u n c t i o n of the medium, i . e . the f u n c t i o n e x p r e s s i n g the r e l a t i o n between the i n c i d e n t and r e f l e c t e d waves i n s i d e the medium.
The r e f l e c t i o n f u n c t i o n i s a l s o e x p r e s s a b l e i n terms of the changes o f the c h a r a c t e r i s t i c impedance of the t r a n s m i s s i o n l i n e . One of our g o a l s i s t o s y n t h e s i z e a t r a n s m i s s i o n l i n e by d e t e r m i n i n g i t s r e f l e c t i o n f u n c t i o n from the e s t i m a t e d c o v a r i a n c e f u n c t i o n , such t h a t the r e a l p a r t of the i n p u t impedance Z i s equal t o the spectrum W of the p r o c e s s , i . e . -jCz(jo)) + Z(jco)] = W(joj), when the l i n e i s t e r m i n a t e d w i t h an a p p r o p r i a t e l o a d . An o b s e r v a t i o n o f f i n i t e l e n g t h o f the p r o c e s s i s not s u f f i c i e n t t o determine t h i s l o a d . We s h a l l show, however, t h a t by n e g l e c t i n g t h i s l o a d we s t i l l may generate a p r o c e s s , which i s s t a t i s -t i c a l l y c l o s e -t o -the observed p r o c e s s . Moreover -the s o l u -t i o n w i l l be r e c u r s i v e i n the sense t h a t i f the o b s e r v a t i o n i n t e r v a l i s e n l a r g e d we w i l l o b t a i n a longer t r a n s m i s s i o n l i n e i n which the s h o r t v e r s i o n i s i n c o r p o r a t e d i n the f r o n t . By doing so we s h a l l a l s o show t h a t the approximation improves. The method then has the advantage t h a t recom-p u t i n g a l r e a d y c o n s t r u c t e d recom-p a r t s i s not necessary. Each time we r e c e i v e new i n f o r m a t i o n from the o b s e r v a t i o n s , e.g. i n the form o f (t) i n a subsequent i n t e r v a l , we may determine a p i e c e o f l o s s l e s s t r a n s m i s s i o n l i n e which appends to the c o n s t r u c t e d p a r t . F o l l o w i n g t h i s approach we do not need the complete knowledge of the p r o c e s s , e.g. the measurement o f k j (t) i n the i n t e r v a l ( - 0 0 , 0 0 ) , t o be a b l e t o approximate i t . Moreover, i t i s not even necessary t h a t the o r i g i n a l p r o c e s s corresponds t o a matched t r a n s m i s s i o n l i n e although we approximate i t so. Once we have s y n t h e s i z e d the t r a n s m i s s i o n l i n e (or more g e n e r a l l y a l o s s l e s s two p o r t s ) , we can generate a p r o c e s s a t the l e f t p o r t when white noise i s i n j e c t e d a t the r i g h t p o r t and both e n t r i e s o f the l e f t p o r t are connec-t e d connec-to each o connec-t h e r . T h i s means connec-t h a connec-t a connec-t r a n s f e r f u n c connec-t i o n T^ i s embedded i n the o r t h o g o n a l f i l t e r , with T b e i n g the s p e c t r a l f a c t o r o f the
generated p r o c e s s . We s h a l l show t h a t Tf i s s t a b l e and s t a b l y i n v e r t i b l e .
The i n v e r s e o f T i s an i n n o v a t i o n f i l t e r i . e . a f i l t e r which generates an u n c o r r e l a t e d process from the observed p r o c e s s .
In the p r e d i c t i o n theory one t r i e s t o f i n d a p r e d i c t o r y ( t ) of a
s t o c h a s t i c v a r i a b l e y ( t ) g i v e n the p a s t of the p r o c e s s , e.g. g i v e n y(x) f o r T e t t - T , t ) . A mathematically tractable and widely used method to s o l v e such problem i s the l e a s t squares method ( see f o r i n s t a n c e a "Benchmark Papers"on t h i s s u b j e c t e d i t e d by K a i l a t h [ 1 ] ) . In t h i s case
the p r e d i c t i o n e r r o r C y ( t ) - y ( t ) ] should be s t a t i s t i c a l l y o r t h o g o n a l t o y(-r) f o r T e [ t - T , t ) . This orthogonality condition i s c a l l e d the Wiener c o n d i t i o n . Using the Kolmogorov isomorphisme, i . e . the isomorphism between the s t o c h a s t i c v a r i a b l e s of a second order p r o c e s s and a n a l y t i c f u n c t i o n s e v a l u a t e d with the a p p r o p r i a t e measures, we may i n t e r p r e t the t r a n s f e r f u n c t i o n o f the p r e d i c t o r as the p r o j e c t i o n o f the p r e s e n t on the p a s t i n the Wiener sense. The o r t h o g o n a l i t y
c o n d i t i o n may then be w r i t t e n as a Fredholm e q u a t i o n . By i n c r e a s i n g the o b s e r v a t i o n i n t e r v a l and o b s e r v i n g the change i n the Fredholm e q u a t i o n we s h a l l f i n d a d i f f e r e n t i a l e q u a t i o n f o r the t r a n s f e r f u n c t i o n of the p r e d i c t o r . From t h i s we can d e r i v e a r e l a t e d d i f f e r e n t i a l
e q u a t i o n which t o g e t h e r with the f i r s t one forms a system of f i r s t order p a r t i a l d i f f e r e n t i a l equations d e s c r i b i n g a l o s s l e s s t r a n s m i s s i o n l i n e . We s h a l l choose an i n c i d e n t and a r e f l e c t e d wave such t h a t we f o r c e the i n p u t impedance o f the above mentioned t r a n s m i s s i o n l i n e t o be Z. Using the c a u s a l i t y c o n d i t i o n o f the t r a n s m i s s i o n l i n e we may s o l v e the r e f l e c t i o n f u n c t i o n r e c u r s i v e l y . T h i s procedure may be t r a c e d back to a paper of Schur [2 ] i n 1917 and was i n t r o d u c e d i n the p r e s e n t context i n [ 3 ]. The c a u s a l i t y o f the t r a n s m i s s i o n l i n e i s i n f a c t i m p l i e d by the Fredholm e q u a t i o n , which i s e q u i v a l e n t t o the Wiener c o n d i t i o n . T h i s Schur problem w i l l be s o l v e d n u m e r i c a l l y and r e c u r s i v e l y by c o n s i d e r i n g the d i f f e r e n t i a l equations mentioned b e f o r e as of the p e r t u r b a t i o n type [ 4 ] . The r e s u l t i n g i n t e g r a l equations o f the waves w i l l be d i s c r e t i z e d by a numerical i n t e g r a t i o n r u l e . When the t r a p e z o i d a l r u l e i s a p p l i e d we s h a l l f i n d a l a d d e r f i l t e r of normalized L e v i n s o n type.
So f a r , we have emphasized t r a n s m i s s i o n l i n e s , b u t the procedure f o r the lumped p a r t s i s s i m i l a r . A new c o n t r i b u t i o n o f our work i s mixed-m o d e l l i n g , i . e . when both d i s t r i b u t e d and lumixed-mped p a r t s a r e comixed-mbined i n a cascade. However, the r e c u r s i v e Schur procedure f o r d e t e r m i n i n g the r e f l e c t i o n f u n c t i o n or c o e f f i c i e n t o f a d i s t r i b u t e d / l u m p e d s e c t i o n may l o s e i t s accuracy due t o accumulation o f computing e r r o r s when r e s i d u a l data i s used t o compute the r e f l e c t i o n s . To a v o i d the above mentioned problem we develop a theory f o r s h i f t i n g a lumped s e c t i o n through a d i s t r i b u t e d one. The r e f l e c t i o n f u n c t i o n and c o e f f i c i e n t o f the
d i s t r i b u t e d and lumped s e c t i o n s w i l l be computed s e p a r a t e l y as i f each o f these s e c t i o n s l i e s a t the f r o n t o f the cascade, so the c a l c u l a t i o n s use d i r e c t l y the given data. T h e r e a f t e r we s h i f t the lumped o r
d i s t r i b u t e d s e c t i o n through t h e other so t h a t i n d i r e c t l y we s y n t h e s i z e the cascade. The Kuroda i d e n t i t i e s known i n the wave d i g i t a l f i l t e r t h e o r y [ 5 ] are examples o f the s h i f t - t h r o u g h procedure.
We l i k e t o p o i n t out two major a s p e c t s o f our approach.
(i) In the modeling problem we model a s t a b l e and s t a b l y i n v e r t i b l e t r a n s f e r f u n c t i o n by imbedding i t i n a l o s s l e s s wave t r a n s m i s s i o n system. I t i s r e c o g n i z e d t h a t o p t i m a l l e a s t squares p r e d i c t i o n i n the e s t i m a t i o n c o n t e x t t r a n s l a t e s i n t o l o s s l e s s n e s s o f the s t r u c t u r e s t o be determined by an i n v e r s e s c a t t e r i n g a l g o r i t h m . ( i i ) We do not e v a l u a t e the t r a n s f e r f u n c t i o n b u t s h a l l r e p r e s e n t i t
by the r e f l e c t i o n f u n c t i o n of the medium i n which i t i s embedded.
Although t h i s work [ see a l s o 6 ] o r i g i n a t e s from the p r e d i c t i o n / modeling problem, i t i s c l o s e l y r e l a t e d t o i n v e r s e s c a t t e r i n g problems i n other f i e l d s such as i n v e r s i o n o f t e l e g r a p h e q u a t i o n , G e l f a n d - L e v i t a n , and S t u r m - L i o u v i l l e problems [7,8,9,4]. F o r i n s t a n c e , our n u m e r i c a l
method o f s o l v i n g the wave equations i n a t r a n s m i s s i o n l i n e f o l l o w s c l o s e l y t h a t o f T i j h u i s i n [ 4 ] . Our p r e d i c t i o n / m o d e l i n g approach d i f f e r s from o t h e r s . K a i l a t h [10] c a l l s the d i f f e r e n t i a l e q u a t i o n s f o r the p r e d i c t o r s the K r e i n L e v i n s o n equations because they are the c o n t i n u o u s
-time c o u n t e r p a r t o f the Levinson r e c u r s i v e equations and was
i n v e s t i g a t e d by K r e i n i n [ 11]. He s o l v e s the p r e d i c t o r d i r e c t l y from the above mentioned equations. Dym - Gohberg [ 12] c o n s i d e r the e x t e n s i o n o f the f u n c t i o n k ^ ( t ) g i v e n i n a bounded i n t e r v a l such t h a t the e x t e n s i o n t o R admits a c a n o n i c a l f a c t o r i z a t i o n . The f u n c t i o n s they c o n s i d e r are m a t r i x v a l u e d f u n c t i o n s ( and a r e not necessary H e r m i t i a n ) . The d i f f e r e n t i a l equations i n v o l v e d are a g e n e r a l i z a t i o n o f the K r e i n -L e v i n s o n e q u a t i o n s ( see a l s o [ 1 0 ] ) . They show t h a t the e x t e n s i o n a l s o maximizes a c e r t a i n entropy i n t e g r a l [ 1 3 ] . The f a c t o r i z a t i o n o f f u n c t i o n s d e f i n e d on the imaginary a x i s was a l s o i n v e s t i g a t e d by K r e i n i n [14,15]. K r e i n proves t h a t the smoothness o f the p r e d i c t o r as a s o l u t i o n o f the Fredholm equation i s o f the same c l a s s as the f u n c t i o n k ^ ( t ) .
In chapter I I we g i v e a comprehensive treatment o f the p r e d i c t i o n theory i n a network t h e o r e t i c context. F i r s t we d e r i v e the K r e i n -L e v i n s o n equations f o r the p r e d i c t o r . We s h a l l show how the p r e d i c t o r w i l l be embedded i n an o r t h o g o n a l ^ m a t r i x . Both l o s s l e s s s c a t t e r i n g and l o s s l e s s c h a i n s c a t t e r i n g m a t r i c e s w i l l be d i c u s s e d . The r e l a t i o n t o the i n v e r s e s c a t t e r i n g problem w i l l be made c l e a r .
In chapter I I I we d e r i v e the r e c u r s i v e and i t e r a t i v e i n v e r s e s c a t t e r i n g a l g o r i t h m s , which c a l c u l a t e the r e f l e c t i o n f u n c t i o n from the c o v a r i a n c e f u n c t i o n , as w e l l as the modeling a l g o r i t h m f o r k ^ ( t ) . The system o f wave d i f f e r e n t i a l equations i s s o l v e d as a p e r t u r b a t i o n problem. Sets of r e c u r s i v e wave e q u a t i o n s w i l l be d e r i v e d from the s o l u t i o n s above. The Schur c o n d i t i o n which r e p r e s e n t e d the c a u s a l i t y o f the l i n e s o l v e s the r e f l e c t i o n f u n c t i o n .
In chapter IV we show the r e s u l t s o f the computer s i m u l a t i o n s . S e v e r a l t e s t f u n c t i o n s are generated by the modeling a l g o r i t h m , and a f t e r b e i n g down-sampled they are used as t e s t i n p u t s t o the i n v e r s e s c a t t e r i n g a l g o r i t h m s . The performance o f the i n v e r s e s c a t t e r i n g a l g o r i t h m s w i l l be v e r i f i e d .
I n c h a p t e r V we d e s c i b e t h e l u m p e d s e c t i o n s , w h i c h we c a l l t h e D a r l i n g t o n s e c t i o n s . We d e v e l o p a n o v e l t h e o r y f o r s h i f t i n g l u m p e d s e c t i o n s t h r o u g h o t h e r l u m p e d s e c t i o n s a n d f o r s h i f t i n g l u m p e d s e c t i o n s t h r o u g h d i s t r i b u t e d o n e s , a n d v i c e v e r s a .
R e f e r e n c e s
1. K a i l a t h T., " L i n e a r L e a s t - S q u a r e s E s t i m a t i o n " , Benchmark P a p e r s i n E l e c t r i c a l E n g i n e e r i n g a n d C o m p u t e r S c i e n c e , Dowden H u t c h i n s o n & R o s s I n c . , 1977 2. S c h u r J . , " U e b e r P o t e n z r e i h e n , d i e i m I n n e r n d e s E i n h e i t s k r e i s e s beschränkt s i n d " , J . für d i e R e i n e und A n g e w a n d t e M a t h e m a t i k , v o l . 147, B e r l i n , 1917, p p . 205 - 2 3 2 .3. D e w i l d e P., V i e i r a A.C., a n d K a i l a t h T.,"On a G e n e r a l i z e d Szegö-L e v i n s o n R e a l i z a t i o n A l g o r i t h m f o r O p t i m a l Szegö-L i n e a r P r e d i c t o r s B a s e d on a N e t w o r k S y n t h e s i s A p p r o a c h " , I E E E CAS-25,no. 9, 1 9 7 8 , p p . 6 6 3 - 6 7 5 4. T y h u i s A.G., " I t e r a t i v e D e t e r m i n a t i o n o f P e r m i t t i v i t y a n d C o n d u c t i -v i t y P r o f i l e s o f a D i e l e c t r i c S l a b i n t h e Time D o m a i n " , I E E E A P - 2 9 , no. 2, 1 9 8 1 , p p . 239 - 245 5. N o u t a R., " S t u d i e s i n wave d i g i t a l f i l t e r t h e o r y a n d d e s i g n " , Ph.D. d i s s e r t a t i o n D e l f t U n i v e r s i t y o f T e c h n o l o g y , N e t w o r k S e c t i o n r e p o r t n o . 8 0 , D e l f t 1978. 6. D e w i l d e P., Fokkema J . , a n d W i d y a I . , " I n v e r s e S c a t t t e r i n g a n d L i n e a r P r e d i c t i o n , t h e C o n t i n u o u s - t i m e c a s e " , P r o c . N a t o A d v a n c e d I n s t , on S t o c h a s t i c S y s t e m s , R o t t e r d a m E r a s m u s U n i v e r s i t y , 1 9 8 0 .
7. G o p i n a t h B., a n d S o n d h i M.M., " I n v e r s i o n o f t h e T e l e g r a p h E q u a t i o n a n d S y n t h e s i s o f N o n u n i f o r m L i n e s " , P r o c . I E E E , M a r c h 1971,pp383-392 8. S o n d h i M.M., a n d G o p i n a t h B., " D e t e r m i n a t i o n o f V o c a l T r a c t S h a p e f r o m I m p u l s e R e s p o n s e a t t h e L i p s " , T h e J . o f A c o u s t i c a l S o c . o f Amer., v o l 49, n o . 6 ( 2 ) , 1971, p p 1867 - 1873 9. B u r r i d g e R., "The G e l f a n d L e v i t a n , t h e M a r c h e n k o , a n d t h e G o p i n a t h -S o n d h i I n t e g r a l E q u a t i o n s o f i n v e r s e s c a t t e r i n g t h e o r y , r e g a r d e d i n t h e c o n t e x t o f i n v e r s e i m p u l s e - r e s p o n s e p r o b l e m s " , Wave M o t i o n 2 ( 1 9 8 0 ) , p p 305 - 323 10. K a i l a t h T., L j u n g L . , a n d M o r f M . , " G e n e r a l i z e d K r e i n - L e v i n s o n e q u a t i o n s f o r e f f i c i e n t C a l c u l a t i o n o f F r e d h o l m R e s o l v e n t s o f N o n d i s p l a c e m e n t K e r n e l s " , T o p i c s i n F u n c t i o n a l A n a l y s i s , e d i t e d b y G o h b e r g I . , a n d K a c M., A c a d e m i c P r e s s , NY 1978, p p 169 - 184 11. K r e i n M.G., "The c o n t i n u o u s a n a l o g u e s o f t h e o r e m s o n p o l y n o m i a l s o r t h o g o n a l o n t h e u n i t c i r c l e " , D o k l . A k a d . Nauk. SSSR, v o l 104, 1955, p p 637 - 4 4 0 . 12. Dym H., a n d G o h b e r g I.,"On a n e x t e n s i o n p r o b l e m , g e n e r a l i z e d F o u r i e r a n a l y s i s a n d an e n t r o p y f o r m u l a " , I n t e g r a l e q u a t i o n s a n d O p e r a t o r t h e o r y , V o l . 3/2, B i r k h a u s e r V e r l a g , 1 9 8 0 , p p . 143 - 215 13. B u r g J . P . , "Maximum E n t r o p y S p e c t r a l A n a l y s i s " , Ph.D. d i s s e r t a t i o n , D e p t . o f G e o p h y s i c s S t a n f o r d U n i v e r s i t y , S t a n f o r d , C a l i f . , 1 9 7 5 14. K r e i n M.G., " I n t e g r a l e q u a t i o n s on a h a l f l i n e w i t h K e r n e l s d e p e n -d i n g u p o n t h e -d i f f e r e n c e o f t h e a r g u m e n t s " , E n g l i s h t r a n s l . : Amer. M a t h . S o c . T r a n s l . ( 2 ) 2 2 , 1 9 6 2 , p p . 163 - 288 15. G o h b e r g I . C . , a n d K r e i n M.G., " S y s t e m s o f i n t e g r a l e q u a t i o n s o n a h a l f l i n e w i t h K e r n e l d e p e n d i n g o n t h e d i f f e r e n c e o f a r g u m e n t s " , U s p e h i M a t . Nauk. 1 3 ( 1 9 5 8 ) 3 - 7 2 ( A m e r . M a t h . S o c . T r a n s l . ( 2 ) 1 4 , 1 9 6 0 ) .
I I . LINEAR PREDICTION AND INVERSE SCATTERING
2.1. I n t r o d u c t i o n
In t h i s chapter we i n t r o d u c e l i n e a r p r e d i c t i o n o f continuous-time s t o c h a s t i c p r o c e s s e s and we r e l a t e i t t o the i n v e r s e s c a t t e r i n g problem f o r a p a s s i v e medium. We a l s o prepare the d e r i v a t i o n o f the i n v e r s e s c a t t e r i n g a l g o r i t h m , which w i l l s o l v e the o p t i m a l p r e d i c t i o n problem and forms the content o f the next chapter.
In the p r e d i c t i o n t h e o r y an important problem i s t o f i n d a c a u s a l , s t a b l e and s t a b l y i n v e r t i b l e f i l t e r , namely a p r e d i c t i o n f i l t e r , t h a t generates a white n o i s e l i k e p r o c e s s when i t s i n p u t i s a g i v e n s t o c h a s -t i c p r o c e s s ( f i g u r e 2.1.1). When -t h i s f i l -t e r i s i n v e r -t e d , one f i n d s a modeling f i l t e r which produces a p r o c e s s with s t a t i s t i c a l p r o p e r t i e s c l o s e t o the o r i g i n a l p r o c e s s i f i t i s i n p u t t e d w i t h white n o i s e .
process white noise
On the other hand, the g o a l o f the i n v e r s e s c a t t e r i n g problem i s t o f i n d the parameters which c h a r a c t e r i z e a medium by means o f the r e l a t i o n between the i n c i d e n t and r e f l e c t e d waves a t the medium. I f the medium i s p a s s i v e the s c a t t e r i n g f u n c t i o n S, i . e . the t r a n s f e r f u n c t i o n between the i n c i d e n t and r e f l e c t e d waves, i s bounded [ l , p . 98] ( f i g u r e 2.1.2).
i n c i d e n t wave
• ' • •
D
medium
Fig. 2.1.2 The scattering function.
We s h a l l connect the p r e d i c t i o n and i n v e r s e s c a t t e r i n g problems through the f o l l o w i n g embedding procedure [ 2 ] . The t r a n s f e r f u n c t i o n P o f the p r e d i c t o r w i l l be embedded i n a ' l a r g e r ' o r t h o g o n a l ( l o s s l e s s ) f i l t e r . T h i s embedding i s shown i n f i g u r e 2.1.3. The embedding o f the s c a t t e r i n g f u n c t i o n S i n a l o s s l e s s f i l t e r i s such t h a t t h i s f u n c t i o n S i s an e n t r y o f the o r t h o g o n a l matrix which d e s c r i b e s the f i l t e r . In t h i s case the f i l t e r s h o u l d be matched a t the r i g h t p o r t , i . e . t e r m i n a t e d w i t h a f u l l y a b s o r b i n g load ( f i g u r e 2.1.4). I f the f i l t e r i s mis-matched b u t t e r m i n a t e d w i t h a p a s s i v e l o a d then S i s a f u n c t i o n o f the e n t r i e s o f the o r t h o g o n a l m a t r i x and o f t h e s c a t t e r i n g f u n c t i o n which d e s c r i b e s the l o a d .
In t h i s chapter we s h a l l d e r i v e the c o n d i t i o n s under which the l o s s l e s s f i l t e r s o f the two problems match each o t h e r . Under these c o n d i t i o n s we s o l v e an i n v e r s e s c a t t e r i n g problem w h i l e d o i n g l i n e a r p r e d i c t i o n and v i c e v e r s a . We s h a l l show the e x i s t e n c e and the uniqueness o f the above r e l a t i o n .
1 l o s s l e s s f i l t e r l o s s l e s s
f i l t e r
Fig. 2.1.3 Embedding P(p) in a lossless f i l t e r .
S (p)
l o s s l e s s f i l t e r
f u l l y a b s o r b i n g or p a s s i v e load
Fig. 2.1.4 Embedding S(p) in a lossless f i l t e r .
In the next chapter we s h a l l c o n s t r u c t a medium o r s c a t t e r e r l a y e r by l a y e r w i t h o u t a f f e c t i n g the c o n s t r u c t e d p a r t by means o f the i n v e r s e s c a t t e r i n g method. Such a procedure needs a r e c u r s i v e r e l a t i o n between the wave v a r i a b l e s i n the medium. In the continuous-time case t h i s r e l a t i o n i s a system o f d i f f e r e n t i a l e q u a t i o n s . We s h a l l show, i n t h i s chapter, t h a t a r e c u r s i v e way t o s o l v e l i n e a r p r e d i c t i o n w i l l l e a d t o d i f f e r e n t i a l equations which the o r t h o g o n a l matrix mentioned e a r l i e r obeys. I f t h i s m a t r i x d e s c r i b e s a medium, the waves i n s i d e w i l l s a t i s f y
the same e q u a t i o n s .
T h i s chapter i s o r g a n i z e d as f o l l o w s . In the next s e c t i o n we s h a l l d e s c r i b e the p r o c e s s and d e f i n e some important f u n c t i o n such as the p r o c e s s impedance, e t c . The l i n e a r p r e d i c t i o n w i l l be d e s c r i b e d i n s e c t i o n 2.3. T h e r e i n we s h a l l d i s c u s s the Fredholm equation, the B e l l m a n - K r e i n - S i e g e r t r e l a t i o n , the K r e i n f u n c t i o n P which i s the t r a n s f e r f u n c t i o n o f the p r e d i c t o r , and the K r e i n - L e v i n s o n system o f d i f f e r e n t i a l e q u a t i o n s . From s e c t i o n 2.4. upto s e c t i o n 2.6. we s h a l l d i s c u s s the l o s s l e s s f i l t e r which embeds P. The system o f d i f f e r e n t i a l equations f o r the waves i n a medium w i l l be found i n s e c t i o n 2.7. The r e l a t i o n between the i n v e r s e s c a t t e r i n g and l i n e a r p r e d i c t i o n w i l l be e l a b o r a t e d i n s e c t i o n s 2.8 and 2.9. We s h a l l show t h a t the p r e d i c t o r i s the s o l u t i o n o f an o p t i m i z a t i o n problem i n s e c t i o n 2.10.
2.2. T h e s t o c h a s t i c p r o c e s s y ( t ) .
In this section we shall introduce the process and some important functions such as the process impedance Z(p) and its input-scattering function S(p) - (Z(p)-l)/ (Z(p)+1).
C o n s i d e r a s c a l a r complex-valued zero mean 'wide sense s t a t i o n a r y ' s t o c h a s t i c p r o c e s s y ( t ) , with c o v a r i a n c e f u n c t i o n
R(t) = & (t) + k1 (t)
where k^ (t) i s h e r m i t i a n , i n t e g r a b l e and continuous. F o r the sake o f n o t a t i o n d e f i n e a l s o a c a u s a l f u n c t i o n k ( t ) such t h a t
We s h a l l assume t h a t k ( t ) i s known f o r a l l t o r i n a c e r t a i n i n t e r v a l [ 0 , T ] , f o r i n s t a n c e , by measurements on the p r o c e s s . The D i r a c impulse i n t h e c o v a r i a n c e f u n c t i o n , which r e p r e s e n t s the white n o i s e component o f the p r o c e s s , i s e s s e n t i a l f o r a meaningful i n t e r p r e t a t i o n o f the p r e d i c t i o n problem on s t o c h a s t i c p r o c e s s e s (see a d i s c u s s i o n o f t h i s p o i n t i n K a i l a t h [ 3 ] ) .
I t i s known, see e.g. Wong [4,chapter 3 ] , t h a t the spectrum
W(jui) = R ( t ) e jutd t > 0 f o r a l l u.
Because o f the D i r a c impulse i n the c o v a r i a n c e f u n c t i o n t h i s r e s u l t needs some care i n p r o v i n g . In Wong i t i s o b t a i n e d through i n t r o d u c t i o n of Brownian-motion p r o c e s s e s . Another approach i s t o approximate the
2
white n o i s e p r o c e s s by a sequence o f L p r o c e s s e s as i t i s a l s o done i n Wong and i n Ash and Gardners [5,p.92]. An easy but somewhat l e s s a c c u r a t e way t o g a i n i n s i g h t i n the reason i s the f o l l o w i n g . Suppose, as one may imagine, t h a t the p r o c e s s i n q u e s t i o n has been generated by p a s s i n g white n o i s e through an unknown f i l t e r with impulse response h ( t ) , where h ( t ) decays f a s t enough to s t a t i s f y i n t e g r a b i l i t y c o n d i t i o n s to be imposed l a t e r . We s h a l l r e q u e s t t h a t h ( t ) c o n t a i n s a 6 ( t ) .
The white n o i s e a t the i n p u t i s r e p r e s e n t e d by the d e r i v a t i v e o f the Brownian motion b ( t ) . A sample s i g n a l a t the output i s then o b t a i n e d by c o n v o l u t i o n ,
y ( t ) = h ( t - x ) d b ( x ) ,
1 where the i n t e g r a l w i l l converge i f h e L T a k i n g e x p e c t a t i o n s : R(t,s) = IE{y(t)yli') } ,00 IE{[ h ( t - x ) d b ( T ) ] [ J _oo h ( s - n ) d b ( n ) ]}
For Brownian motion r e s u l t i n g i n white n o i s e we have
lE{db(x)db(n)} = 5 dx,
TTI
where S stands f o r the Kronecker D e l t a f u n c t i o n , xn
Hence
R ( t , s ) = h ( u ) h ( u + ( s - t ) ) d u ,
the w e l l known formula f o r the c o v a r i a n c e o f f i l t e r e d white n o i s e . N o t i c e t h a t R(t,s) may be r e p r e s e n t e d somewhat s l o p p i l y by R ( t - s ) . T a k i n g the F o u r i e r t r a n s f o r m , i t i s now easy t o v e r i f y by d i r e c t i n t e g r a t i o n the c l a s s i c a l r e s u l t
W(joo) = R ( t ) e ~j u td t h ( t ) e "j u td t |2 > 0
However, the p r o p e r t y i s more g e n e r a l l y t r u e : even i f the c o v a r i a n c e f u n c t i o n does n o t r e s u l t from f i l t e r e d white n o i s e , the spectrum w i l l s t i l l be non-negative 1 We r e f e r the reader t o the s p e c i a l i z e d
l i t e r a t u r e i f he i s i n t e r e s t e d i n the s u b j e c t , see e.g. Wong.
Define now the f u n c t i o n
z ( t ) A 6 (t) + 2k (t)
so t h a t
Z(p) = 1 + 2K(p)
The L a p l a c e t r a n s f o r m Z(p) w i l l be c a l l e d the p r o c e s s impedance f o r the f o l l o w i n g reason.
t The necessary and s u f f i c i e n t c o n d i t i o n f o r a s t o c h a s t i c p r o c e s s t o be a f i l t e r e d p r o c e s s i s known as the S z e g o - c o n d i t i o n .
P r o p o s i t i o n 2.2.1
Z(p) i s a p o s i t i v e r e a l (p.r.) f u n c t i o n , i . e .
(i) Z(p) i s a n a l y t i c i n RHP (the open r i g h t h a l f p l a n e ) , ( i i ) Re{ (Z (p) } > 0 i n RHP.
Proof
Z(p) i s a n a l y t i c by the c a u s a l i t y and i n t e g r a b i l i t y of k ( t ) . T h i s i m p l i e s t h a t Re { Z(p) } i s harmonic i n the RHP. Because of the i n t e g r a -b i l i t y o f k ( t ) i t assumes on the imaginary a x i s H c o n t i n u o u s l y the
v a l u e s Re{Z(joi)} = 1 + k(jui) + k(jw) = W(jw) > 0. Hence Re{z(p)} i s
i
bounded i n L and the P o i s s o n i n t e g r a l may be a p p l i e d on i t , see Hoffman [6,p.38]^. Item ( i i ) then f o l l o w s from the maximum modulus theorem.
•
From Z(p) we d e f i n e the s c a t t e r i n g f u n c t i o n S(p) by Cayley t r a n s f o r m
I t i s not true t h a t the f o l l o w i n g p r o p e r t i e s f o r Z(p) have as a consequence t h a t Z(p) i s p . r . :
(i) Z(p) a n a l y t i c i n RHP ( i i ) Re{Z (jto) } > 0
For i n s t a n c e Z(p) = (1+e"^)/(1-e^) s t a t i s f i e s both and i s not p . r . Use o f the P o i s s o n r e p r e s e n t a t i o n i s e s s e n t i a l i n e s t a b l i s h i n g the r e s u l t .
P r o p o s i t i o n 2.2.2
The s c a t t e r i n g f u n c t i o n S(p) i s a bounded f u n c t i o n [ 7 ] , i . e . (i) S(p) i s a n a l y t i c i n RHP,
( i i ) | S (p) | < 1 i n RHP.
Proof
Re{Z(p)} S 0 f o r a l l p i n the RHP, which i m p l i e s t h a t (1+Z(p)) has no z e r o f o r p e RHP. Thus S(p) i s a n a l y t i c due t o the p r o p e r t y mentioned above and the r i n g p r o p e r t y o f a n a l y t i c f u n c t i o n , i . e . sums and
p r o d u c t s o f a n a l y t i c f u n c t i o n s are a n a l y t i c . The v a l u e
I-, , i2 | z ( p ) - l |2 (Re{z}-1)2 + (Im{Z})2 , , . ,
|S(p)| =J — = - <1, because |Re{Z)-l| <
|z(p)+l| (Re{Z}+l)2 + (im{Z})
|Re(z}+l| f o r Re{z} > 0.
. T h e p r e d i c t i o n p r o b l e m
We shall discuss the prediction of the process y(t) defined in section 2.2 given a finite period of its past. Linear least squares prediction leads to a Fredholm equation. From the uniqueness of the Fredholm solution one deduces the Bellman-Krein-Siegert identity and the Krein-Levins on equations. These equations will define the lossless f i l t e r which embeds the prediction transfer function.
In the p r e d i c t i o n problem we wish t o p r e d i c t the s t o c h a s t i c v a r i a b l e y ( t ) g i v e n a f i n i t e p a s t o f the p r o c e s s , i . e . y ( x ) f o r T e [ t - T , t ) . A l i n e a r p r e d i c t o r o f y ( t ) i s n a t u r a l l y o b t a i n e d as f o l l o w s :
y ( T , t ) h ( T , t - T ) y ( T ) d t t-T
where y ( T , t ) i s the p r e d i c t e d value o f y a t time t'based on an i n t e r v a l of l e n g t h T.
The p r e d i c t i o n e r r o r , a l s o c a l l e d the i n n o v a t i o n s [ 3 ] , [ 8 ] , i s then
e ( T , t ) = y ( t ) rt h ( T , t - r ) y ( T ) d T Jt - T 0 [ < 5 ( t - T ) - h ( T , t - i ) ] y ( x ) d T ,
where h ( T , t ) ^ 0 f o r t i CO,T], The p r e d i c t i o n f i l t e r has t h e r e f o r e an impulse response 6 ( t ) - h ( T , t ) .
A c c o r d i n g t o the c l a s s i c a l Wiener theory, see e.g. K a i l a t h [ 3 ] , the f i l t e r has t o be chosen so t h a t e(T,t) i s s t a t i s t i c a l l y o r t h o g o n a l t o y(x) f o r T e C t - T , t ) , i . e .
t i e ( T , t ) y T r ) } = 0 f o r T C [t-T,t)
2 (Note t h a t the white n o i s e may be c o n s i d e r e d as the l i m i t o f L,
p r o c e s s e s , so t h a t normal e s t i m a t i o n c a l c u l u s can be used). The above mentioned o r t h o g o n a l i t y r e l a t i o n i s the continuous-time v e r s i o n o f the
d i s c r e t e Yule Walker equation and i s e q u i v a l e n t t o a Predholm type e q u a t i o n . The l a t t e r can be seen as f o l l o w s . For T e C t - T , t ) , we have:
E { e ( T , t ) y < T ) ) = E { y ( t ) y ( x ) } - E{ Ch ( T , t - n ) y ( n ) d n 3 y ( T ) ) t - T
k j ( t - x )
t-T
h ( T , t - n ) [ 6 ( n - T ) + k j ( n - t ) ]dri
The o r t h o g o n a l i t y c o n d i t i o n leads t o the f o l l o w i n g Fredholm e q u a t i o n :
rT
The f u n c t i o n h ( T , t ) i s the boundary value o f the Fredholm r e s o l v e n t (see e.g. the work o f K a i l a t h [ 9 ] , [ 1 0 ] and Dym [11]), and i s o f the same c l a s s , i . e . h has the same smoothness, as the f r e e term k^
(see K r e i n [ 1 2 ] ) .
P r o p o s i t i o n _ 2 . 3 . 1
I f k j ( t ) has the above p r o p e r t i e s , i . e . h e r m i t i a n , i n t e g r a b l e and a l s o continuous, then we have the f o l l o w i n g B e l l m a n - K r e i n - S i e g e r t (BKS) i d e n t i t y
— h(T,t) = -h(T,T)h(T,T-t)
Proof
Gohberg-Krein, see e.g. i n [11, p. 191], have proved the i d e n t i t y f o r the Fredholm r e s o l v e n t . I t a l s o assures the c o n t i n u i t y o f h ( T , t ) i n t and the continuous d i f f e r e n t i a b i l i t y i n T when k ^ ( t ) i s c o n t i n u o u s . An easy way t o d e r i v e the i d e n t i t y i s the f o l l o w i n g . D i f f e r e n t i a t i n g the Fredholm e q u a t i o n .
h(T,T)kj (tT) -T
^-~h(T,T))k1[t-T)dT - -^h{T,t) = 0.
By argument t r a n s f o r m a t i o n T - t + t o f the o r i g i n a l Fredholm e q u a t i o n and m u l t i p l i c a t i o n w i t h - h ( T , T ) , we f i n d : - h ( T , T ) k1( T - t ) + h(T,T) 0 f o r t e [ 0 , T ) , o r : h(T,x)k (T-t-x)dx + h(T,T)h(T,T-t) 0 - h ^ T j k j (T-t) + h(T,T) h(T,T-x)k.(-t+x)dx + h(T,T)h(T,T-t) = 0 0
I f the s o l u t i o n o f the Fredholm e q u a t i o n i s unique then the i d e n t i t y i s proved because k ^ ( t ) i s h e r m i t i a n .
The e x i s t e n c e of a s o l u t i o n h(T,t) i s i m p l i e d by c o n s t r u c t i o n , i n the p r e d i c t i o n c o n t e x t . The uniqueness f o l l o w s from the c l a s s i c a l p r o j e c t i o n
2
theory o f L f u n c t i o n s . Independently from the s t o c h a s t i c p r e d i c t i o n i n t e r p r e t a t i o n , Gohberg-Feldman [13,p.100] g i v e a n i c e p r o o f f o r the uniqueness o f a s o l u t i o n h(T,t) g i v e n an i n t e g r a b l e and h e r m i t i a n k j ( t ) .
•
K r e i n [14] has i n v e s t i g a t e d the t r a n s f e r f u n c t i o n o f the p r e d i c t i o n f i l t e r . We s h a l l c a l l i t the K r e i n f u n c t i o n . D e f i n i t i o n 2.3.2 The K r e i n f u n c t i o n o f the f i r s t k i n d fT Y >0 = 1 - H(T,p) P(T,p) A 1 h ( T , t ) e Ptd t
Introduce now the p a r a h e r m i t i a n conjugate F^(p) of F(p) as the pseudomeromorphic e x t e n s i o n of F (j(o) t o the complex p l a n e , i . e .
F (p) A F <-P>
Define a l s o an 'upper s t a r1 o p e r a t i o n as f o l l o w s :
P*(T,p) A e p TP (T,p)
N o t i c e t h a t the i n v e r s e Laplace transform o f P (T,p) i s c a u s a l .
The K r e i n - L e v i n s o n e q u a t i o n s f o l l o w almost immediately from the BKS i d e n t i t y . P r o p o s i t i o n 2.3.3 d e f i n e s these e q u a t i o n s .
P r o p o s i t i o n 2.3.3
*
The K r e i n f u n c t i o n s P(T,p) and P (T,p) s a t i s f y the f o l l o w i n g K r e i n -L e v i n s o n system o f d i f f e r e n t i a l equations P*(T,p) 1 f -p -h(T,T) 1 f P*(T,p) J L 3T P(T,p) -h(T,T) 0 P(T,p) J \ J With the i n i t i a l c o n d i t i o n P(0,p) = P (0,p) = 1. P r o o f P*(T,p) = e p TP j T , p ) = e P T[ 1 - h ( T , t ) e+ p td t ] -pT h ( T , T - t ) e * d t The i n i t i a l c o n d i t i o n s a r e indeed P(0,p) = P (0,p) = 1. D i f f e r e n t i a t i n g P(T,p) w i l l g i v e P(T,p) = -h(T,T)e p T h ( T , t ) ) e Ptd t - h ( T , T ) C e ~p T h ( T , T - t ) e 'P td t ]
by the BKS i d e n t i t y . T h i s p r o v e s the second e q u a t i o n .
3 * W P (T,p) £ ( e -p TP , ( T , p ) ) -pP*(T,p) + e p TC - h ( T , T ) e pT 3 T h ( T , t ) . e p td t ] -pF (T,p) - h ( T , T ) [ l - h ( T , T - t ) eP ( t"T )d t ] -pP (T,p) - h(T,T)P(T,p) T h i s proves the f i r s t e q u a t i o n .
2.4. T h e c h a i n s c a t t e r i n g m a t r i x 9 ( T , p )
A chain scattering matrix will be derived from
the Krein-Levins on equations. Some relevant •properties of such a matrix will be discussed.
To prepare f o r the d e r i v a t i o n o f the c h a i n s c a t t e r i n g m a t r i x we g i v e the f o l l o w i n g d e f i n i t i o n .
D e f i n i t i o n 2.4.1.
The K r e i n f u n c t i o n o f the second k i n d , Q(T,p), i s d e f i n e d as the s o l u t i o n o f J L 3T Q*(T,p) -Q(T,p) —h (T, T) -h(T,T) Q*(T,p) -Q(T,P) (2.4.1) With the i n i t i a l c o n d i t i o n Q(0,p) = Q (0,p) = 1.
T h i s f u n c t i o n i s an e n t i r e f u n c t i o n of the same type as P ( T , p ) . There
fT
e x i s t s a f u n c t i o n h ^ ( T , t ) such t h a t Q(T,p) = 1 -0
h1( T , t ) e p td t .
The combination of both K r e i n f u n c t i o n s i n P and Q l e a d s t o the f o l l o w i n g system o f d i f f e r e n t i a l equations J)_ dT ' P*(T,p) Q*(T,p) P(T,p) -Q(T,p) -h (T,T) -h(T,T) P (T,p) Q (T,p) P(T,P) -Q(T,p) (2.4.2) w i t h an i n i t i a l c o n d i t i o n m a t r i x A 1 1 1 -1
The n o r m a l i z e d form, i . e . the system which has the i d e n t i t y m a t r i x as i n i t i a l c o n d i t i o n , can be d e r i v e d t o be 3T 0(T,p) = X(T,p)0(T,p) (2.4.3) where 6(T,p) "I (P*(T,p) + Q*(T,p)) (P(T,p) - Q(T,p)) I (P*(T,p) - Q*(T,p)) (P(T,p) + Q(T,p)) and the g e n e r a t o r X(T,p) = -h (T,T) -h(T,T)
T h i s system has a unique s o l u t i o n o f p r o d u c t - ( m u l t i p l i c a t i v e - ) i n t e g r a l form (see e.g. Gantmacher [ 1 5 , c h . l 4 ] or S c h l e s i n g e r [ 1 6 ] ) . The s o l u t i o n i s symbolised as f o l l o w s :
0(T,p) = X ( t , p ) d t
P r o p o s i t i o n 2.4.2
The generator X(T,p) i s J-skew on the i m a g i n a i r y a x i s and J-semi-n e g a t i v e i J-semi-n the opeJ-semi-n r i g h t h a l f p l a J-semi-n e , i . e .
(i) X(T,j(D)J + JX~(T,j(d) = 0
( i i ) X ( T , p ) J + JX~(T,p) < 0 f o r p e RHP
P r o o f : X ( T , p ) J + JX (T,p)= 2 -Re{p} 0 0 0 0 I on U < 0 i n RHP. P r o p o s i t i o n 2.4.3 (T,p)Q(T,p) + P(T,p)Q^(T,p) = 2 P r o o f :
The determinant o f the m a t r i x 0(T,p) i s det0(T,p) = -|[P*(T,p)Q(T,p) + P(T,p)Q*(T,p)]. By the J a c o b i i d e n t i t y [ l 5 , c h . l 4 ] , we have T t r a c e X ( t , p ) d t det0(T,p) = c e ' ° w h i l e c = det0(O,p) = 1. Thus, -pT -pT = c e = e , -kp*(T,p)Q(T,p) + P(T,p)Q*(T,p) ] = e "p T. T h i s completes the p r o o f . •
Note: the determinant det0(T,p) = e f o f o r lp[ f °°. Thus 0(T,p) i s i n v e r t i b l e i n the whole open complex p l a n e .
© ( T ^ j O ^ ^ p ) = 0^(T,p) J0(T,p) = J
P r o o f :
by d i r e c t computations.
P r o p o s i t i o n 2.4.5 The m a t r i x 8(T,p) i s J - l o s s l e s s , i . e . (i) J - 0 ( T , p ) J 0 ~ ( T , p ) = 0 on n ( i i ) J - 0(T,p)J0~(T,p) > 0 f o r p e RHP P r o o f : (i) i s i m p l i e d by c o r o l l a r y 2.4.4.
A p r o o f o f ( i i ) can be found i n Potapov [ 1 7 ] . An easy but l e s s a c c u r a t e way t o see t h i s i s as f o l l o w s (see a l s o Gantmacher [ 1 5 ] ) .
0(T,p) = eX(t' P)dt A l i m n e 3
•"0 h-H) j=T/h
where j h > t_. > ( j - l ) h and II denotes the m a t r i x product w i t h j = T/h the most l e f t f a c t o r . Define now:
X ( t . , p ) h - X(t.,p) 9. A e 3 = e n 3 and 1 0 (T,p) A n 6. = j = T / h . , 1 w i t h n = — . h ^ We w i l l show t h a t 9. J 6 . < J f o r p e RHP. 3,,n 3 ,n 1 2 9. = I +.X(t.,p)h + — X ( t . , p ) X ( t . , p ) h +... J,n j z. J j L e f t and r i g h t m u l t i p l i c a t i o n s w i t h J g i v e 1 2 J 9 . J = I + J X ( t . , p ) J h + — J X ( t . , p ) J J X ( t . , p ) J h + l . n 1 2! 2 j J X ( t . , p ) J h = e 3
Thus, J X ( t . , p ) J h X ( t . , p ) h 6. J 9 ~ = J e 3 e = J e J e j [ x ( t . . ,p) j + JX ( tj (p ) ] h -Re{p} 0 0 0 2h by p r o p o s i t i o n 2.4.2. < J f o r p i n RHP By i n d u c t i o n eN( T, p ) j 0N( T ,P) = 6Tnin . . . e2 > n e1 > n J 0 ^ &2 ? p . . . ^ < J S J f o r p i n RHP. The l i m i t Q(T,p) i s t h e r e f o r e J - l o s s l e s s C o r o l l a r y _ 214 . 6 ^ (i) J - 0(T,p) J0~(T,p) < 0 f o r p i n LHP ( i i ) J - 0 (T,p)J0(T,p) > 0 p i n RHP = 0 on U < 0 p i n LHP
In the s c a t t e r i n g theory the m a t r i x 0(T,p) i s c a l l e d a c h a i n s c a t t e r i n g m a t r i x (CSM). A CSM i s d e f i n e d as a two p o r t network ( f i g u r e 2.4.1),
A(T,p)
B(T,p)
= 6(T,p)
A(0,p)
B(0,p)
where A(0,p) and B(T,p) denote the Laplace transforms of the i n c i d e n t waves a t the l e f t and r i g h t p o r t r e s p e c t i v e l y . S i m i l a r l y , B(0,p) and A(T,p) are the transforms of the r e f l e c t e d waves a t the l e f t and r i g h t p o r t r e s p e c t i v e l y .
a ( 0 , t ) '
b(0,t)-a ( T , t )
b ( T , t )
Fig. 2.4.1. The chain scattering matrix as a twoport.
In the next s e c t i o n we s h a l l show t h a t the p h y s i c a l s c a t t e r e r d e f i n e d by the CSM 0(T,p) i s l o s s l e s s . T h i s w i l l f o l l o w from the p r o p e r t y o f J - l o s s l e s s n e s s o f 0.
. T h e s c a t t e r i n g m a t r i x £ ( T , p )
Connected with the CSM Q(T,p) there is a scattering matrix l(T,p) which is lossless. The input impedance and scattering function of 1(T,p) will be discussed.
' A(T,p)' A(0,p)
£(T,p) (2.5.1)
B(0,p) B(T,p)
2 2 We assume t h a t A(0,ju) and B(0,ju)) are i n L (du) or i n L
a(0,t) contains a Dirac impulse.
dii) , 2 1+0) i f Lemma_2_. 5^1 The 2,2-entry o f a J - l o s s l e s s m a t r i x 0(T,p) i s b o u n d e d - i n v e r t i b l e , i . e . 0 *(T,p) i s a bounded f u n c t i o n . Proof : • 12 I 12 From p r o p o s i t i o n 2.4.5 ( i i ) we f i n d 18^21 - 1 + I I ^o r p i n R H P -I — 1 1 -1 Thus, 1922 I - 1 i n the r i g h t h a l f plane. The a n a l y t i c i t y of i s
i m p l i e d by the e n t i r e n e s s p r o p e r t y o f 6 2 2 ' • Lemma 2.5.2 £(T,p) = [U0(T,p) + UX] [ U + UX0 ( T , p ) ] 1 [U - 0 ( T , p ) UX] ^ G C T ^ J U - ux] (2.5.2) w i t h 1 0 0 0 and U = I - U.
P r o o f : ' A(0,p) 1 = U ' A(0,p) 1 1 + U ' A(T,p) B(T,p) B(0,p) B(T,p) [ U + U 0 ( T , p ) ] A(0,p) B(0,p) A(T,p) = U A(T,p) 1 + U A(0,p) B(0,p) I B(T,p) B(0,p) = CuO(T,p) + U1] f A(0,p) B(0,p)
From the d e f i n i t i o n o f E ( T , p ) , we have [U0(T,p) + U1] = £(T,p)[u + UlQ{T,
The r i g h t f a c t o r Cu + U^OCTjp)] i s i n v e r t i b l e by lemma 2.5.1. Thus E(T,p) = Cu0(T,p) + t / X u + UX0 ( T , p ) ]- 1. T h i s proves the f i r s t
p a r t o f the lemma. The p r o o f o f the second p a r t i s s i m i l a r .
P r o p o s i t i o n _ 2 ^ 5 ^ 3
I f the CSM Q(T,p) i s J - l o s s l e s s , then the s c a t t e r i n g m a t r i x E(T,p) i s l o s s l e s s , i . e . (i) £(T,ju)r~(T,ju) = I (2.5.3) ( i i ) I - % (T,p) £~(T,p) > 0 f o r p e RHP P r o o f : i - z i ~ = i - (u - 0 ux)_ 1( O u - u1) (uo~ - u1) (u - uxe ~ )- 1, by lemma 2.5.2 = (u - 0 ux)- 1[ ( u - 0ux) (u - ux0~) - (Ou - ux) (u0~ - ux) ] ( u -yx0 ~ ) ' = (u - 0 u1) "1C ( u + 0ux0 ~ ) - (0u0~+ ux) ] ( u - ux0 ~ )_ 1 = (U - 0 ux)_ 1[ j - 0 J 0 ~ ] ( U - ux0 ~ )_ 1 = M~(J - 0J0 )M, w i t h M = (U - UX0 ~ )_ 1 J - l o s s l e s s n e s s p r o p e r t y o f 0, we have f i n a l l y - EZ~ = 0 on H - EE~ > 0 f o r p e RHP • Corollary_2_.5_.4 I - E(T,p)E~(T,p) < 0 f o r p e LHP (2.5.4)
The p r o p e r t y dJQ^ = 0^J0 = J i m p l i e s E^E = EE^ = 1 (2.5.5)
A d d i t i o n a l p r o p e r t i e s o f the m a t r i c e s Z(T,p) and 0(T,p) may be found i n the l i t e r a t u r e , see e.g. Dewilde - Dym [ 1 8 ] .
By lemma 2.5.2 and OjO* = J we e a s i l y f i n d t h a t E(T,p) 11 01 2G2 202 1 -Q2 2 G2 1 ° 1 20M .-1 22 (2.5.6) o r 11* 4 21* 21 (2.5.7)
N o t i c e t h a t i s b o u n d e d - i n v e r t i b l e because o f the f o l l o w i n g reason. By c o r o l l a r y 2.4.6 we have J - 9 J6 < 0 f o r p e LHP. T h i s i m p l i e s t h a t
1 - 0J J @ J j + 02 1G2 1 < 0 f o r p e LHP. Thus 1 < 9^ 7 ° 1 1for P £ LHP" By argument t r a n s f o r m a t i o n p' = -p we have:
>u( T , - p ' ) 91 1( T , - p ' ) > 1 f o r every p' e RHP.
But t h i s i s e q u i v a l e n t t o 9,, 8,, S I f o r p' e RHP. Hence, indeed
^ 11* 11* e
| 9 j ^ | i 1 i n the r i g h t h a l f p l a n e .
So f a r , we have d e r i v e d the s c a t t e r i n g m a t r i x and gave some o f i t s
p r o p e r t i e s . To give a p h y s i c a l meaning, we s h a l l now make a detour i n t o the network theory. We s h a l l i n v e s t i g a t e , i n t h i s p a r t , some energy c o n s t r a i n t s . The two p o r t , d e p i c t e d i n f i g u r e 2.4.1, i s a r e a l i z a t i o n o f the s c a t t e r i n g m a t r i x o r e q u i v a l e n t l y o f the c h a i n s c a t t e r i n g m a t r i x . At the l e f t and r i g h t p o r t , r e s p e c t i v e l y , the energy absorbed by the system up t o time t i s d e f i n e d as
i 12 , ,2 E (t) = [ | a ( 0 , t ) | - , | b ( 0 , t ) | ] d t and (2.5.8) r t C | b ( T , t ) |2 - | a ( T , t ) |2] d t , CO
r e s p e c t i v e l y . Remark: we have assumed t h a t the waves c a r r y f i n i t e energy and hence are square i n t e g r a b l e . A necessary c o n d i t i o n f o r a s c a t t e r e r t o be p a s s i v e i s t h a t f o r a l l f i n i t e t we have
l o s s l e s s . A fundamental theorem i n network theory i s the f o l l o w i n g , see e.g. Newcomb [ l , p . 9 8 ] .
Theorem 2.5.6
A necessary and s u f f i c i e n t c o n d i t i o n f o r a s c a t t e r e r to be time
i n v a r i a n t , l i n e a r and p a s s i v e i s t h a t i t possesses a s c a t t e r i n g m a t r i x S(p) such t h a t I - S (p)S(p) > 0 f o r p i n the r i g h t h a l f p l a n e .
Thus, i n our case, the s c a t t e r i n g m a t r i x I(T,p) d e s c r i b e s a p a s s i v e s c a t t e r e r . T h i s s c a t t e r e r i s even l o s s l e s s due t o the u n i t a r y of E (T, joo) .
Given the K r e i n f u n c t i o n s P ( T , p ) , Q(T,p) and the CSM 0(T,p), we can w r i t e the s c a t t e r i n g m a t r i x i n the f o l l o w i n g form.
E q (t) + ET( t ) > 0. I f i n a d d i t i o n EQ (°°) + E^, (°°) = 0 the s c a t t e r e r i s P(T,p)+Q(T,p) P (T,p)-Q (T,p) P(T,p)+Q(T,p) E(T,p) = (2.5.9) P(T,p)-Q(T,p) P(T,p)+Q(T,p) P(T,p)+Q(T,p) 2
P r o p o s i t i o n 2.5.7
The f u n c t i o n a (T,p) = -
p
!I'
P?
, ? i s a bounded f u n c t i o n .P r o o f :
The a n a l y t i c i t y o f a ^ i n the r i g h t h a l f p l a n e i s i m p l i e d by the e n t i r e n e s s p r o p e r t y o f P and Q (lemma 2.5.1) and the r i n g p r o p e r t y o f a n a l y t i c f u n c t i o n s .
From the l o s s l e s s n e s s o f £(T,p), we have ° 2 1a2 1 + ° 2 2a2 2 - * or
| a2 1l * 1. P r o p o s i t i o n 2.5.8 The Cayley t r a n s f o r m o f a ^ (T,p), i . e . z (P) A 1 + ^ L ( T ' P ) - & ^ ZTlP; = 1 - 02 1( T , p ) P(T,p) ' i s a p . r . f u n c t i o n .
T h i s f u n c t i o n Z^(p) i s c a l l e d the i n p u t impedance o f the s c a t t e r e r . T h i s i s indeed the case when the s c a t t e r e r i s matched a t the r i g h t p o r t .
P r o p o s i t i o n 2.5.9
The p r e d i c t i o n t r a n s f e r f u n c t i o n P(T,p) i s an o u t e r f u n c t i o n , (see Rudin [19,p.370] f o r the d e f i n i t i o n o f o u t e r f u n c t i o n s . )
P r o o f :
P ( T , j u ) c L2([0,<»), d(°2 ) by c o n s t r u c t i o n . Hence P(T,p) e H2
1+0)
P = F.G. The f u n c t i o n F denotes the i n n e r [see 19] and G the o u t e r f u n c t i o n . From the p . r . p r o p e r t y o f ZT( p ) (or e q u i v a l e n t l y p r o p o s i t i o n 2.4.3) we know t h a t P c o n t a i n s no zero i n the open r i g h t h a l f p l a n e . T h i s i m p l i e s t h a t F does not c o n t a i n a Blaschke f a c t o r . F c o u l d o n l y
-kp
be o f the form e . But we have a l s o l i m P(T,p) = 1. Thus F = 1 and p-x»
P i s o u t e r .
•
-1 2 Thus we have t h a t P(T,p) i s s t a b l y i n v e r t i b l e and P e H . Some o f the p r o p e r t i e s o f Z^ are s t a t e d i n the f o l l o w i n g p r o p o s i t i o n .
P r o p o s i t i o n 2^5.10 (i) l i m ZT(jto) = 1 u-*co ( i i ) j (Z (j(0) + ZT( j o i ) ) 1 P(T, joi)P(T, ju) ( i i i ) |- (ZT(j(0) + ZT(ja>) ) > 0 (the n o r m a l i z a t i o n ) (the f a c t o r i z a t i o n ) ( p o s i t i v i t y ) P r o o f :
(i) l i m Z (ju>) = l i m QjT >JM| = 1 by the Riemann-Lebesgue lemma.
urK» T J u-w P(T,joi)
F(T,ju>)
= by p r o p o s i t i o n 2.4.3. P(T, JU))P(T, jü))
( i i i ) R e { ZT( j u ) } > 0 because P(T,p) i s o u t e r .
C o r o l l a r y 2.5*11
The i n v e r s e t r a n s f o r m o f the impedance Z^(p) i s g i v e n by zT( t ) = 6(t) + 2 kT( t ) f o r a smooth and c a u s a l kT.
2.6. T h e p r e d i c t i o n a n d m o d e l i n g f i l t e r s
In this section we shall elaborate on the prediction and modeling filters mentioned in the introduction of this chapter.
In the p r e v i o u s s e c t i o n s we have embedded the s c a l a r p r e d i c t i o n f i l t e r P(T,p) i n an o r t h o g o n a l twoport d e s c r i b e d by the s c a t t e r i n g m a t r i x
I(T,p) or e q u i v a l e n t l y the CSM 0 ( T , p ) . I f we take as l e f t i n p u t o f
T
0(T,p) the v e c t o r [1 1] we e a s i l y see t h a t the output a t the r i g h t * T *
p o r t w i l l be [P P] . Both P(T,p) and P (T,p) d e s c r i b e c a u s a l t r a n s f e r f u n c t i o n s . The s c a t t e r e r connected as a p r e d i c t i o n f i l t e r i s shown i n f i g u r e 2.6.1. Thus, i f a t the l e f t the p r o c e s s y ( t ) d e f i n e d i n s e c t i o n 2.2 i s i n p u t t e d , then the spectrum of the output wave b ( T , t ) i s
|p(T,joi) Z —+ Z (joi) | ^ rpj^g Spe ct r u m i s ' f l a t ' , i . e . o f white n o i s e
type, i f ZT i s an approximant o f Z. In the next s e c t i o n s we s h a l l
*
d i s c u s s t h i s approximation. The t r a n s f e r f u n c t i o n P (T,p) i n f i g u r e 2.6.1. r e p r e s e n t s the so c a l l e d backward p r e d i c t o r [ 8 ] , i . e . the p r e d i c t i o n o f y ( t - T ) g i v e n y ( t ) f o r T e ( t - T , t ] .
0(T,p)
P (T,p)
P(T,p)
Fig. 2.6.1 The scatterer as a prediction f i l t e r
The modeling f i l t e r i s the i n v e r s e o f the p r e d i c t i o n f i l t e r . I f we take B(T,p) as i n p u t o f the s c a t t e r i n g m a t r i x £(T,p) and connect the output B(0,p) i n t o A(0,p) ( f i g u r e 2.6.2), then we s h a l l f i n d a t r a n s f e r P 1( T , p ) .
I f t h i s i n p u t b ( T , t ) i s a white n o i s e p r o c e s s , the output p r o c e s s has as c o v a r i a n c e f u n c t i o n 6 ( t ) + k (t) + k (-t) .
Fig. 2.6.2 The scatterer as a modeling f i l t e r .
For f u t u r e purposes (see e.g. c h a p t e r I I I and IV) we l i k e to model the f u n c t i o n k ( t ) , i . e . the smooth p a r t of the c o v a r i a n c e f u n c t i o n . T h i s ^ - m o d e l i n g g i v e s another c o n f i g u r a t i o n on the c o n n e c t i o n s of the s c a t t e r e r . T h i s i s shown i n f i g u r e 2.6.3.
V P ) + 1
I f A(0,p) = r and B(T,p) = 0 i n the s c a t t e r i n g m a t r i x
V P ) - I
convention then B(0,p) = — — . B(T,p) = 0 means matching a s c a t t e r e r a t the r i g h t p o r t . Thus, i f we i n j e c t A(0,p) and B(0,p) of the above mentioned form i n t o 0(T,p), then the s c a t t e r e r i s matched i n a n a t u r a l way a t the r i g h t p o r t . A(0,p)=l+KT(p) B(0,p)=KT(p) Z(T,p) •A(T,p) B(T,p)=0
