RANDOM VARI.fl.3LES, WIENER-HOPF FILTE.RING AND CONTROL

FORMULATED IN ABSTRACT SPACES

by

LEONARD JENG TUNG, B.S., M.S. in E.E.

A DISSERTATION

IN

ELECTRICAL ENGINEERING

Submitted to the Grad'jate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

August, 1977

ABV-^.5m r

'B0\ a-1

/frr :'

i

ACKNOWLEDGEMENTS

I am grateful to Dr. Richard Saeks for his helpful discussions

and guidance through the course of this investigation; to Dr. Stanley

R. Liberty, Dr. John F. Walkup, Dr. Marion 0. Hagler and Dr. Thomas G.

McLaughlin for their suggestions and criticisms in the preparation of

this dissertation.

n

TABLE OF CONTENTS

ACKNOWLEDGEMENTS i i

I. INTRODUCTION 1

II. BANACH RESOLUTION SPACE 9

III. FACTORIZATION OF OPERATORS 23

IV. HILBERT SPACE REPRESENTATION THEORY 35

V. REPRODUCING KERNEL RESOLUTION SPACE OF HILBERT SPACE VALUED

RANDOM VARIABLES 49

VI. WIENER-HOPF OPTIMIZATION THEORY 58

VII. CONCLUSION 70

LIST OF REFERENCES 72

m

CHAPTER I

INTRODUCTION

For a long time, it has been a goal of some systems engineers to

combine the concepts of causality with the optimization theorems so that

optimization would lead to a realizable solution. During the past years

causality has been introduced into Hilbert space — the so-called Hil-

bert resolution space , where most of the optimization problems are

formulated. However, not enough effort has been devoted to solving the

optimization problems under the constraint of causality. A typical

problem is the Wiener-Hopf filter. This problem has been thoroughly

ixudied in the time domain. However in the frequency domain, the pro­

blem is solved in a rather intuitive fashion. People simply take the

causal part of an optimal solution obtained without the causality con­

straint and claim it is the optimal solution under the constraint of

causality. It is the purpose of this dissertation to formulate the

problem in an abstract space and attack it by well-developed linear

operator theorems. For comparison, we first look into the classical

Wiener-Hopf filter. Then we discuss the techniques needed for the

formulation — the techniques that will be developed in this disserta-

ticpi, as well as other applications where these techniques could be

used.

2 Classical Wiener-Hopf Filtering

The problem itself is quite simple. Let x(t) denote a signal

process and n(t) denote a noise process. Both processes are stationary

1

and zero-mean. They are statistically independent. However they are

mixed together, i.e. they appear in the form of x(t) + n(t). What we

would like to do is to design a filter, linear and causal, to operate on

x(t) + n(t) such that the error defined as x(t) - y(t), where y(t) is

the output of the filter, has the smallest variance. Let e denote the

error and a denote its variance, then

a = E{e^} = - ^ f.^ {S^(s)[l - H(s)][l - H(-s)]

+ S^(s)H(s)H(-s)} ds,

where S^(s) and S^(s) are the spectral density of x(t) and n(t), respec­

tively, and H(s) is the transfer function of the filter. Rearranging

terms in a, we get

1 J~ ^ = -^;^ /j„ {[Sx(s) + S^(s)]H(s)H(-s) - S^(s)H(s)

- S^(s)H(-s) + S^(s)} ds.

Since S (s) + S (s) is also a spectral density, it is symmetric in the

s-plane and can be factored into one factor having poles and zeros in r

the left-half plane and another having corresponding poles and zeros in

the right-half plane. Thus, it can be written as

S^(s) + S^(s) = F(s)F(-s).

Then

c = - ^ /.^ {[F(s)H(s) - -^ ][F(-s)H(-s) - -^ ] ^ ^ "' F(-s) F(s)

Minimum a occurs when

SJs)S fs) + — } ds

F(s)F(-s)

SJs) H{s) = ^

S,(s)

F(s)F(-s) S^(s) + S^(s)

Unfortunately, H(s) is also symmetric in the s-plane, hence it cannot

represent a causal system. So far, the argument is correct, although it

leads nowhere. In order to get a causal filter that might do the job,

some authors suggest an intuitive approach. Since minimum a occurs when

F(s)H(s) -S (s) x^ ' F(-s)

= 0,

if we simply let

F(s)H(s) = fS,(s)1

F(-s) , the par t ia l f rac t ion of S (s) /F( -s) with a l l

the poles in the l e f t - h a l f plane, then

1 H(s) =

fS,(s)l

F(-s) F(s)

problem left is to prove that

1

does represent a causal system. The only

rs,(s)1

F{s) [H-s)\ , the so-called Wiener-Hopf filter, is actually the

optimal filter. To do so, one ought to prove that those cross terms in

a, such as

^^ f. {H(s)F(s) -fSx(s)1

lF(-s)J }

fS,(s)1

F(-s) V. » ' J

ds, do not contribute R

negative values to a.

The Wiener-Hopf filter is a well-known result. Our purpose here

is not to rederive it but rather to point out what procedures should be

done for the formulation in abstract space.

Pre-formulation of Wiener-Hopf Filtering in Abstract Space

In this section, we will not actually formulate the problem but

rather point out what should be done for the formulation. Referring to

the classical W-H filtering, we find that there are five major consider­

ations. They are

i. Ransom Process

ii. Causality

iii. Factorization

iv. Additive Decomposition

v. Optimization.

These problems are discussed in the following.

(i) Random Process: How should a random process be defined when

it is formulated in an abstract space, and what kinds of spaces are

suitable for our purpose? In order to answer this question, let's first

review what a random process is. A random process can be thought of as

a function of two independent variables, x(t,w), where w is the outcome

of some experiment. When t is fixed, x(t,w) represents a random varia­

ble. When w is fixed, X(T:,W) represents a time function. So if x(t,w)

is an element of some function space, say Lp, when w is fixed, we could

think of the random process as a random variable which takes values in

the function space, L2. Of course, this idea will work only if an

adequate probability measure can be defined over the function space.

Fortunately this kind of probability measure has been defined over

3 4 metric spaces ' . For our purposes, Hilbert space and Banach space are

used, not only because these spaces possess nice properties but also

because stochastic concepts such as "mean" and "variance operator" can

be defined therein. It turns out that the variance operator is "posi­

tive" and "self-adjoint" for a reflexive Banach space valued random

variable, and these are just the properties required for operator

factorization. In chapters IV and V of this dissertation, Banach space

and Hilbert space valued random variables along with their characteris­

tics, especially the reproducing kernel resolution space, are discussed

with the probability measures over these spaces assumed implicitly.

(ii) Causality: We mestioned that causality has been introduced

in Hilbert space — the so-called Hilbert resolution space. In chapter

II, we extend the works done for Hilbert space to Banach space. When a

Banach space is equipped with a resolution of the identity (then it is

called a Banach resolution space), there is a resolution of the identity

over the dual space which is naturally induced by the original resolu­

tion of the identity (by using the adjoint). Since most of our

applications are formulated in reflexive Banach space, emphases are

given to the reflexive Banach resolution space and its dual space with

the induced resolution of the identity. Operator properties such as

6

causality, anti-causality, left- and right-mini phase, maxiphase, are

also defined and discussed in chapter II.

(iii) Factorization: Operator factorization is the fundation of

this dissertation. Because of this, chapter III is totally devoted to

the development of the factorization theorems. Throughout this disser­

tation, the desired form of factorization is either K K* or T* T. In

either form, the operator to be factorized has to be "positive" and

"self-adjoint". These commonly used properties among operators on

Hilbert space can be extended for operators which map reflexive Banach

spaces to their dual spaces. One more thing is demanded for the

factorization: the factor space, from which or to which the factor

operator maps, has to be a Hilbert space. In some applications the

factor space is more important than the factor operator itself. For

example, the reproducing kernel resolution space.

In the first half of chapter III, factorization theorems are

developed. These theorems give miniphase (or maxiphase) factors which

in turn give causal (or anti-causal) inverse operators when an inverse

is guaranteed. The second half of chapter III deals with reproducing

kernel resolution space — a special kind of factor space which is

formed by the factor operator given in the factorization theorem, but

is independent of the factor used in its definition.

(iv) Additive Decomposition: Operator decomposition into causal

part, anti-causal part and memoryless part is treated in Ref.l. For our

purpose, the operators involved are Hilbert-Schmidt operators, which

guarantee the decomposition.

(v) Optimization: As in the classical W-H filtering, we would

like to minimize the second-order statistics of the error. However,

when the W-H filtering is formulated in Hilbert space, the second-order

statistics of the error are manifested in the variance operator. Even

though variance operators are positive and self-adjoint, it is not an

easy task to minimize the variance operator directly, since here we are

dealing with the partial ordering of positive operators. Fortunately,

when the operators involved are nuclear, there is a theorem which allows

us to take the trace of the variance operators and to minimize it, thus

changing the operator minimization problem into a minimization of some

scalar. Moreover, taking the trace of the operator actually kills the

cross terms in the variance operator of the error, since the cross terms

involved are either strictly causal or strictly anti-causal and the

traces of these kinds of operator are zero. This is the job that is not

done in the classical W-H filtering. In chapter VI, we first formulate

the W-H filtering in Hilbert space and solve it by using linear operator

theorems. With the result from the filtering, we then look into the

so-called W-H control. In the context, a feedoack and feedforward

control system is derived by the techniques developed in the W-H

filtering.

Summary

While we have found that factorization theorems are most impor­

tant to the formulation and the solving of the W-H filtering and con­

trol, we have also found two other areas of application -— scattering

operators (the operator version of the scattering variables of the

8

classical network theory) and the reproducing kernel resolution space of

the Banach space valued random variables (Hilbert space valued random

variables are special cases). Hence for the reference of the reader,

we summarize in the following.

In chapter II, Banach resolution spaces are defined first. Then

causality, anti-causality, miniphase and maxiphase are defined. Mini-

phase and maxiphase are defined in such a way that the causality of the

inverse can be determined once its existence is guaranteed. In chapter

III, factorization theorems are developed. Positive and self-adjoint

operators over reflexive Banach spaces are factorized into factors of

left- and right-miniphase operators and their adjoint. A "unique"

factor space — the so-called reproducing kernel resolution space, is

also discussed in this chapter. In chapter IV, scattering variables

along with Banach space valued random variables are discussed. This

chapter is termed "Hilbert space representation theory" since the pro­

blems formulated more generally in Banach spaces may be transformed into

problems in Hilbert spaces where they are easier to deal with. In

chapter V, we examine Hilbert space valued random variables. A certain

kind of random variable can be approximated by a sequence of random

variables which take values in the RKRS of the variance operator of the

original random variable and have variance operators that approach (

weakly) the identity operator of the RKRS. In chapter VI, Wiener-Hopf

filtering is formulated in Hilbert space and attacked via linear opera­

tor theorems. With the result for the filtering, we then look into the

so-called W-H control of a feedback and feedforward control system.

CHAPTER II

BANACH RESOLUTION SPACE

By a Banach resolution space, we mean a 2-tuple, (B, ^F), where

B is a Banach space (complete normed linear vector space) over the field

of real numbers and gF is the so-called resolution of the identity in B

which will be defined in the following section. By working with Banach

resolution spaces, we will define and discuss properties associated with

operators over the Banach spaces, such as causality, anti-causality,

left- and right-miniphase and maxiphase, etc. All these properties are

for future development of the main work of this dissertation.

Resolution of Identity in Banach Space

Def.II.l. Let B be a Banach space. By a resolution of identity,

DF, in B, we mean a family of bounded linear operators, nF(A), on B

defined for each Borel subset. A, of the real number set R, satisfying

the following conditions:

i- Q^W ^ ^B " "" " ' y operator on B.

ii. 3F(A^) BF(A2) = 3F(AI f! A^), for all A ^ A2 e B(R) = the

set of all Borel subsets of R.

iii. DF(U A.) = E D F ( A . ) , {A.} : finite set of disjoint Borel

subset of R.

iv. ||gF(A)x|| <_ ||x||, for all A e 3(R) and x s B (equivalent

statem.ent: Norm of gF(A) is either 0 or 1).

t See Ref-1 for comparative work in Hilbert space

10

The subscript on the left in the notation, gF, is to notify that

the resolution of identity is defined over the space B, and will be

dropped if no ambiguity would result.

With the resolution of identity defined as above, there are some

properties of it worth mentioning.

Thm.II.l. Let (B, gF) be a Banach resolution space. We have

i. gF(A)x = 0 => X e gF(R - A)[B] = the range of gF(R - A ) , for

all A £ 3(R)

ii. gF($) = 0, where a is the empty set.

iii. gF(A)[B] is complete (closed), for all A e s(R)

Proof: i. x = Ig(x) = gF(R)x = gF(A U (R - A ) ) X

= gF(A)x + gF(R - A)X, for all x £ B. Hence

gF(A)x = 0 => X = gF(R - A ) X £ gF(R - A ) [ B ] .

ii. Ig = gF(R) = gF(R U $) = gF(R) + gF(^) = Ig + gF($),

30 gF($) = 0.

iii. Let {x.} be a Cauchy sequence in nF(A)[B], then there I D

exists X £ B such that x. -)- x. Since

gF(A)x = gF(A)(lim X.) = lim gF(A)x. = lim x. = x,

so X £ gF(A)[B]. (Here we employ the fact that

gF(A)2 = gF(A), for all A £ a(R). Thus x. e gF(A)[B]

implies x. = „F(A)X..) I D I

Working with a Banach resolution space (B, gF), it is natural

and important to ask whether we can define a resolution of identity in

B*, the dual space of B. The following theorem gives us the answer.

Thm.II.2. Let (B, gF) be a Banach resolution space. Then

11

{gF(A)*|A £ 3(R)} is a resolution of identity in B*. This resolution of

identity in B* is called the induced resolution of identity.

Proof: i. (x, gF(A)*y) = (gF(A)x, y ) , for all x £ B, y £ B* and

for all A £ 3(R), so

(x, BF(R)*y) = (gF(R)x, y) = (x, y ) , for all x £ B and

y £ B*. This implies that gF(R)*y = y, for all y £ B*,

i.e. gF(R)* = Ig^.

ii. gF(Ai)* gF(A2)* = [gF(A2) ^.^U^n* = QHA^ n A2)*.

iii. We have

n . n gF(U A .) = z B^^^i^' ^°

BF( 0 A,.)* = [E3F( .)]* = ? 3F(A^)*.

iv. |1BF(A)*X|1 < ||BF(A)*|| lixll = 1IBF(A)|| 11X11

The definition of a resolution of identity can be better under­

stood by examples. Two typical examples are illustrated as follows.

Example 1. L : the Banach space of equivalent classes of

functions that map R to R and satisfy the following inequality.

iZ |f(t)|^ dt < °°, where 1 < p < «.

Define

0 , for t ^ A F(A)f(t) = x(A)f(t) = , for f £ L .

f(t), for t £ A P

It is easy to show that {F(A)|A E 3(R)} is a resolution of identity in

L once the properties of X(A) are explored.

12

Example 2, Let p, q £ R such that 1/p + 1/q = 1. Then L is a r

reflexive Banach space whose dual space is L . For all f £ L , g £ L , ^ P q

define

(f, g) = r f(t)g(t) dt.

Let {F(A)|A £ 3(R)} be the resolution of identity in L as defined in P

Example 1. Then we have

(F(A)f, g) = r [x(A)f(t)]g(t) dt

= /A f(t)g(t) dt = C i^(t)[x(A)g(t)] dt

— 00

= (f, x(A)g)

So {X(A)|A £ 3(R)} is also the induced resolution of identity in L .

Causality of Operators

(1) Causal Operators

Def.II.2. Let (X, wF), (Y, yF) be Banach resolution space.

T: X -> Y, is linear. T is said to be causal, if

t t t t JF x-j = „F ^2 => Y^ "'' 1 " Y^ ''' 2' ^ ^ ^

gF^ = gF(-«, t), B = X, Y.

The definition above is good for e\/ery linear operator. However,

when we deal with linear bounded operators, the following theorem will

come in handy.

Thm,II.3. T: X ^ Y, is bounded and linear. Then the following

statements are equivalent:

i. T is causal.

13

ii. yF^ T = yF^ T y?^, for all t £ R.

iii. llvF'' Til < IIT vF II, for all t £ R

iv. x^^ x = 0 =c> yF^ T X = 0.

V. T ^F^ [X] c yF^ [Y], for all t £ R.

^ * x' t ~ Y' t ^ x'^t' ^ ° ^ ^ t £ R.

vii. yF T wF. = 0, for all t £ R.,

where gF^ = Ig - gF , B = X, Y

Proof: a. (i =^ ii)

For all t £ R,

/^ X = (^F^)^ X = yF^ (/^ x ) , for all x £ X. So,

yF^ T X = yF^ T {yF^ x ) , for all x £ X, i.e.

yF^ T = yF^ T , F ^

b. (ii =^ iii)

t T- M _ I I ct T ct I I ^ 1 I ct I I 1 I T pt F^TII = lUF'-T vF"!! < \ K r ||T ,F Y"- M ! - I IY^ ' X" " - " Y ' " '• ' X

£ IJT ^F^ll' " 01 ^ ^'

c. (iii =^ iv)

F X = 0 ^ T / X = 0 ^ | | T / xjl = 0

14

yF^ T x|| = 0 =^ yF^ T X = 0, for all

t £ R.

d. (iv =^ v)

For all y e T ^F^ [X], there exists x £ X such that

y = T ^F^ X. But yF^ {yF^ x) = 0

=> yF^ T ( F x) = 0

=> "f / t ^ ^ Y^t '--'' ° - Y^t '--'

e. (v => vi)

Since for all x e X, there exists y £ Y such that

^ X^t ^ = Y^t y- ^°

yF^ T j F X = yF^ (yF^ y) = ^F^ Y = T /^ X, for all

X £ X. Hence yF. T „F. = T wF..

f. (vi =^ vii)

/ t T ,F, = /^ {/t T ,F^) = 0.

g. (vii =-> i)

0 = yF^ (x - X2) => yF^ T yF^ (x - X2) = 0

=0 yF^ T (I^ - x^t^^^l " ^2^ " °'

i.e. yF^ T (x - X2) - yF^ T ^F^ (x - X2) = 0. So

15

yF T Cx-j - X2) = 0 and T is causal.

(2) Anti-causal Operators

The definition of anti-causality is quite similar to that of

causality and the corresponding theorem can be proved by a similar

argument. Therefore, the definition as well as the corresponding

theorem are stated as follows without further explanation.

Def.II.3. (X, „F), (Y, yF): Banach resolution spaces. T: X - Y,

is linear. T is said to be anti-causal if

X 't ^1 " X^t ^2 ^ Y 't ^ 1 " Y 't " 2*

Thm.II.4. T: X ^ Y, is linear and bounded, then the following

statements are equivalent:

i. T is anti-causal.

ii. yF^ T = yF^ T ^F^, for all t £ R.

m . ^ Tjl 1 jjT »F.l!, for all t £ R Y' t • 1 I _ 1 I • X' t

iv. x^t " Y^t " " °*

V. T /^ [X] c yF^ [Y], for all t £ R.

vi. T vF^ = vF^ T YF^, for all t £ R.

Vii. yF^ T yF^ = 0, for all t £ R.

(3) Memoryless Operators

With causality and anti-causality defined as above, a special

class of operators can be defined as shown in the next definition.

16

Def.II.4. (X, F), (Y, Y F ) : Banach resolution spaces. T: X Y,

is linear. T is said to be memoryless if T is causal and anti-causal.

As in the cases of causal and anti-causal operators, we have a

corresponding theorem for memoryless operators.

Thm.II.5 T: X ->- Y, is linear and bounded. T is memoryless if

and only if T yF(A) = yF(A) T, for all A £ 3(R).

Proof: a.(^) With T ^ F = yF^ T, we have

Y^t "' x' ~ Y^t x' ^ ~ * ^° " ' anti-causal.

Similarly T F^ = yF^ T implies

Y^ "'" x' t ~ Y' W^t ^ ~ ' ^'^' ^ ^ causal.

b.(<H By causality,

^X^t = Y ^ T x ^ t = Yf^t^(^X-X^')

" Y^t ^ " Y^t ^ X^ •

By an t i -causa l i t y , yF. T wF = 0 . So

T „F. = yF. T, for a l l t £ R. Simi lar ly

T yF = yF T, for a l l t £ R. These imply that

T yFM = yF(A) T, for a l l A £ 3(R).

(4) Causality of Adjoint and Inverse Operator

The following theorem tells us the relation between the causality

17

of an operator and the causality of its adjoint.

Thm.II.6. (X, ^F), (Y, Y F ) : Banach resolution spaces. T: X ^ Y,

is linear and bounded. And we take the induced resolution of identity

over each X* and Y*. Then

i. T is causal ^=>T* is anti-causal,

ii. T is anti-causal <=>T* is causal.

Proof: The proof is trivial following the fact that

(YF, T / ) * = (^F^)* T* (yF^)*.

In the above theorem, the causality of T* is dependent on the

causality of T. However, when we deal with a special class of Banach

spaces, the so-called reflexive Banach space, we actually have an "if

and only if" condition. The corollary below makes this clear.

Cor. (X, j^F), (Y, yF) and T are as in Thm.II.6. And X, Y are

reflexive, i.e. X** = X, Y** = Y. Then

i. T is causal <=;> T* is anti-causal,

ii. T is anti-causal <=^ T* is causal.

The proof of this corollary is nothing more than identifying

notation, so it is omitted.

As the above theorem deals with the causality of adjoint opera­

tors, so the next theorem deals with the causality of the inverse of

an operator.

Thm.II.7. T: X -> Y, is linear, bounded and invertible. Then

18

i. T' is causal iff YF^ T x = 0 implies „F^ x = 0.

ii. T is anti-causal iff Y^^ T x = 0 implies ^F x = 0.

Proof: The proofs for i and ii are similar, so we only describe

one of them, namely the proof of i.

(=^) T is causal, i.e.

^F^ y = 0 => ^F^ T"^ y = 0. If yF^ T x = 0 then

^F^ T"^ (T x) = 0, i.e. ^F^ x = 0.

(<=) For all y £ Y, there exists x £ X such that y = T x,

i.e. X = T'^ y. With

yF^ y = 0 =^ yF^ T X = 0 ^ ^F^ X = 0

=> wF T" y = 0, so T' is causal.

Although the above condition for causal invertibility is

necessary and sufficient, it is not readily applicable. However, for

those applications in which we are interested, the invertibility is

either guaranteed or it is of no concern. Hence the above theorem is

good enough for our purpose.

(5) Mimiphase and Maxiphase

The definitions of left-, right-miniphase and maxiphase are wery

important for the development of factorization of operators. Hence,

they are emphasized seperately in this section.

19

Def.II.5. T: X -> Y, is linear. T is said to be

i, left-miniphase if yF T x = 0 <=^ „F^ x = 0,

ii. left-maxiphase if yF. T x = 0 <=^ „F. x = 0,

iii. right-miniphase if T[j^F^[X]] = yF^[Y]

iv. right-maxiphase if T[^F^[X]] = yF^[Y].

It is easy to see that being miniphase, left- or right-, implies

being causal while being maxiphase implies being anti-causal. Further­

more, when invertibility is guaranteed, left-miniphase implies the

inverse is causal, while left-maxiphase implies the inverse is anti-

causal. An important property of being maxiphase or miniphase is given

in the following theorem when the spaces involved are reflexive.

Thm.II.8. i. (X, ^F), (Y, yF) are Banach resolution spaces.

(X*, v/F*), (Y*, yF*) are the dual spaces of X and

Y, respectively, with their own induced resolution

of identity,

ii. X, Y are reflexive. T: X - Y, is linear & bounded.

Then

1 . T is left-miniphase i f f T* is right-maxiphase,

2. T is left-maxiphase i f f T* is right-miniphase.

We need a lemma to prove Thm.II .8. The lemma needed is stated

and proved as fo l lows.

Lemma, i . B is a Banach space.

20

ii. C, D are subspaces of B, with C £ D and D is closed.

Then C is dense in D iff D^ = C^, where

C" = {X* e B*|(x, X*) = 0, for all x e C} and

D^ = {y* e B*|(y, y*) = 0, for all y e D}.

Proof: (=^) a. C c D ^ D"" £ C"".

b. For all x* £ C" , (x, x*) = 0 for all x e C. For

all y £ D, there exists {x .} in C such that

x^ -)- y. So

(y, X*) = (lim x^, x*) = lim (x., x*) = 0, i.e.

x* £ D"" and C"" £ D"".

(<=) a. Assume there is a y £ D such that

inf Ijx - y|| f 0 (assume C is not dense in D). X £ C

b. inf ||x - y|| = max (y, x*), (Ref.U) X £ C X* £ C"",

x*||<l

max (y, y*) = 0: y* e D ", |y*||<l

Contradiction.

Proof of Thm.II.8: The proofs for 1 and 2 in Thm.II.8 are similar.

Here we describe one of them, namely the proof of part 1.

(=») We have yF^ T x = 0 <=> j F x = 0.

i. T is causal, so T* is anti-causal. By Thm.II.3,

T * [ ( Y F ^ ) * [ Y * ] ] £ (xF^)*[X*].

ii. For all X £ (T*[(yF'')*[Y*]])\ (X** = X),

21

0 = (x, T* C Y F ^ ) * y*) = (yF^ T x, y * ) , for y* £ Y*.

This implies yF^ T x = 0, so ^ F x = 0, (Y** = Y).

Hence, 0 = ( F x, x*) = (x, ( F )* x*), for all x* £ X*,

so X £ ((j^F^)*[X*])^.

By i and ii, (T*[(YF^)*[Y*]])^= ((xF^)*[X*])\ so

T * [ ( Y F ^ ) * [ Y * ] ] = (xF^)*[X*], by Lemma.

(<=) We have T*[(yF^)*[Y*]] = (j(F^)*[X*]

i. T* is anti-causal, so T is causal. So

j F X = 0 => yF" T X = 0.

ii. If yF^ T X = 0, then

0 = (yF^ T X, y*) = (x, T* (yF^)* y*), for all y* £ Y*.

But for all (x^^)* x*, x* £ X*, there is a sequence {y^*}

in Y* such that T* (yF^)* y.* ^ (^F^)* x*. So

(x, (^F^)* x*) = (x, lim T* (yF^)* y.*)

= lim (x, T* (yF^)* y.*) = 0. So

{yF^ X, X*) = 0, for all x* £ X*, implies ^F^ x = 0.

From the theorem, we notice that the definitions for left- and

22

right-miniphase are equivalent when the operator is invertible. This

statement is also true for left- and right-maxiphase.

CHAPTER III

FACTORIZATION OF OPERATORS

Operator factorization is the fundation of this dissertation; but

not every operator on an arbitrary Banach space can be factorized in the

form desired. Throughout this dissertation, the desired form of

factorization is either k k* or T* T. In either form, we notice that

the operator to be factorized is "self-adjoint". This requirement

presents no problem in Hilbert space. However, "self-adjoint" does not

always make sense for operators on Banach spaces. Furthermore, the

operator has to be "positive". Again, "positive" does not always make

sense in Banach spaces. Fortunately these concepts can be defined in a

rather useful class of Banach spaces — the reflexive Banach spaces, in

which they are defined as follows:

Def.III.l. i. B is a reflexive Banach space.

ii. T: B ^ B*, is linear and bounded.

T is said to be positive if

(x, T x) >_ 0, for all x £ B.

T is said to be self-adjoint if

T* = T.

Note that T*: B**=B - B*, so it makes sense to compare T and T*.

Next, we are going to write dov/n a theorem which is the fundation

for the factorizations to be discussed. This theorem is stated without

proof. Interested readers are referred to Masani's work (9).

23

24

Thm.III.l. Let Q: B -> B*, be a linear, bounded, positive, and

self-adjoint operator, where B is a reflexive Banach space and B* is its

dual. Then there exists a Hilbert space H and a linear bounded operator

K from H to B* such that Q = K K*.^'^'^'^

With this theorem, we can go on to the main subject.

Left- and Right-miniphase

Thm.III.2(left-factorization),

i. (B, F) is a reflexive Banach resolution space,

ii. Q: (B, F) -> (B*, F*), where F* is the induced resolution of

identity in B*. Q is linear, bounded, positive and self-

adjoint.

Then there exists a Hilbert resolution (H, E) and a linear

bounded operator K: (H, E) -> (B*, F*), such that

1. Q = K K*,

2. K is left-miniphase,

3. The factorization is unique up to a memoryless unitary trans­

formation.

Proof: A. i. By Thm.III.l, there exists a Hilbert space H and a

linear bounded operator K from H to B* such that

Q = K K .

ii. Define H = R(K*) and K = Kj^: H - B*, then

t The requirements for a Hilbert resolution space are more restrictive than those for a Banach resolution space, namely E ( A ) * = E(A) is required for a Hilbert resolution space, which condition does not make sense in a Banach resolution space.

25

K*: B**^B -> H,

iii. (K* b, x ) ^ = (b, K x)g = (b, K x)g = (K* b, x)^, for

all b £ B, X e H,

Since K* b, K b e H, so K* b = K b, for all b e B.

iv. K K* b = K K* b = K K* b = Q b, for all b £ B. So

K K* = K K* = Q.

V. Since H = K [B] = K*[B] , so K* has dense range. So

K is 1-1.

B. i. Define H^ = R(K* F^) and let E^ be the orthogonal

projection on H . Then (E^)^ = (E^)* = E^.

ii. Since H becomes R(K*) which is H, as t -> «>, so

:t = T ^rt

^H

iii. When s < t.

lim E^ = I^ (E^ ^ I^ weakly).

H^ = R(K* F^), H^ = R(K* F^) and R(F^) £ R(f^).

Since F^ b = F^ b + F[s,t) b, for all b e B, and

for b' £ R(F^), b' = F^ b', so

F^ b' = F^ (F^ b') + F[s,t) (F^ b') = F^ b' = b'.

i.e. b' £ R(F^). So R(K* F^) £ R(K* F^) and

H^ = R(K* F^) £ R(K* F^) = H^. So E^ E^ = E^ E^ = E^

iv. With E defined for all (-<x>,t), t £ R, E can be

26

extended to 3(R) uniquely to be a spectral measure,

i.e. a resolution of identity.

V. Since E^[H] = H^ = R(K* F^), by d e f i n i t i o n , so K* is

a right-maxiphase (De f . I I . 5 ) . So K is a l e f t -

miniphase (Thm.I I .8) .

:. i . Let K: (ff, ?) ^ (B*, F*) be another left-miniphase

fac tor iza t ion of Q.

n . Define W on R(K ) by W (K b) = K* b. W is we l l -

defined, for i f K b = K a, then

K K* a = Q a = K K* a = K K* b = Q b = K K* b. But

K is 1-1 , so K* a = K* b.

i i i . For any b £ B,

1|W K b j p = ||K* b j r = (K* b, K* b)^ H . H ^

= (b, K K* b)g = (b, K K* b)g

= (K b, K b)f[ = IJK b j j ^ , H

SO W is isometric on R(K } . Since R(K ) is dense in Oi a.

H, W can be isometrically extended to H.

iv. For.all z £ R(K ), z = K x, for some x £ B, hence

K W z = K W K x = K K * x = K K x = K z . So

K W = K over R(K ). So K W = K over H via continuity

as W extended to H.

V. By the same argument, there exists V: H ->- H, V iso­

metric, such that K V = K. So K W V = K, and hence

27

W V = I,., since K is 1-1. Similarly, V W = Iff.

vi. With K W = K, we have

(F^)* K E^ W = (F^)* K W = (F^)* K = (F^)* K E^

= (F^)* K W E" = (F^)* K E^ W E^,

(K, K are causal). So

(F^)* K (E^ W - E^ W E^) z = 0 for all z £ H, hence

E^ (E^ W - E^ W E^ ) z = 0, (K is left-miniphase),

i.e. E^ W = E^ W E^, so W is causal,

vii. Similarly, V is causal. But W = V* which is anti-

causal (Thm.II.6), so W is memoryless.

Should one notice that in the theorem above that we actually

define a resolution of identity to make the factorization be left-

miniphase, he will not be surprised that other resolutions of identity

can be defined to make the factorization be left-maxiphase. The

following corollary explains just that point and most of the proof is

omitted to avoid repetition.

Cor. i. Q: (B, F) ^ (B*, F*).

ii. Q is linear, bounded, positive and self-adjoint.

Then there exists a Hilbert resolution space (H, E) and a

linear bounded operator K: (H, E) (B*, F*), such that

1. Q = K K*,

2. K is left-maxiphase,

3. The factorization is unique up to a memoryless unitary

transformation.

28

Proof: Define H^ = R(K* F^), where K is as defined in Thm.III.2.

Then the proof follows correspondingly.

Since we are dealing with reflexive Banach space, there is no

necessity of labeling "K" as the factorization. Simply treating K* as

an independent operator, say T, we can call T the factorization. In

order to tell the difference between K and T, we simply call T (which is

K*) the right-factorization, and we have the following theorem.

Thm.III.3(right-factorization).

i. Q: (B, F) ^ (B*, F*).

ii. Q is linear, bounded, positive and self-adjoint.

Then there exists a Hilbert resolution space (H, E) and a linear a. a*

bounded operator T: (B, F) ^ (H, E), such that

1. Q = T* T,

2. T is right-miniphase.

3. The factorization is unique up to a memoryless unitary trans­

formation. Proof: Define T = K*, then T* = K, and define H. = R(T F.). Then

the rest of the proof follows as in Thm.III.2.

Reproducing Kernel Relolution Space (RKRS)

There is a common statement in each theorem of the previous

section, i.e. "The factorization is unique up to a memoryless unitary

transformation". For certain applications such as the study of Banach

space valued random variables, we would like to get rid of the ambiguity

just mentioned. The concept of a reproducing kernel resolution space

29

evolved from theorems of the previous section is just what we want.

First, let's define some notation.

Def.III.2. Q, K, and T are defined as in the previous section.

Q = K K* = T* T. Let H^ = R(K), HQ = R(T*). For x, y in R(K), define

(x, Y)f[ = (K" ^ X, K'^ y)jv . For z, w in R(T*), define

(z, w ) ^ = (T*"*- z, T*"^ w ) ^ . Define

«E^ = K E^ K"'- on H^ and .E. = T* E^ T*"*" on H->, then extend them to

a* 3(R). Note that E corresponds to E in Thm.III.2.

Thm.III.4. (HQ, QE) (the so-called RKRS) defined above is a

Hilbert resolution space which is independent of the factorization

Q = K K* used in its definition. Moreover,

i. (HQ, QE) is unitarily equivalent to (H, E),

ii. R(Q) is dende in HQ,

iii. (F^)* X = 0 iff E^ X = 0, for x e HQ,

iv. K: (H, E) ^ (HQ, Q E ) , is memoryless,

where (H, E) corresponds to (H, E) in Thm.III.2.

Proof: i. HQ = R(K), so f(p is a linear vector space.

ii. By definition, (x, y)'^ = (K"^ x, K"^ y)jf. It is MQ

trivial to show that

30

1. (x, y) = (y, x ) , for a l l x, y £ H ,

2. (x, ay) = a(x, y ) , for a l l a e R,

3. (x, y + z) = (x, y) + (x, z ) , for a l l x, y, z £ H ,

4. (x, x)f[ = (K"^ X, K'^ xh > 0, for a l l x £ H and Q rl ~ W

Since K is l inear, x f 0 implies K" x f 0, So

(x, x) > 0, for X £ HQ and X ^ 0.

So (x, y)u is an inner product over HQ.

iii. Let {x.} be a Cauchy sequence in HQ, then {K" x.} is

Cauchy in H. So K x-j -> z, for some z £ H. But

z = K" K z , SOX. - > K z i n H . Hence H,. is a Hilbert 1 Q

space,

iv. Assume Q = K K* = K' K'*, both K and K' are left-

miniphase factorization of Q on factor spaces (H, E) and

(H', E'), respectively.

1. By Thm.III.2, there exists a memoryless unitary

transformation U: (H, E) (H*, E'), such that

K U = K'. So R(K') = R(K U) = R(K), since U is onto.

Hence HQ is independent of the factorization.

2. (K'"'- z, K'-*- w);if. = ((K U)"^ z, (K U)"^ w)J(. H' ^ " "' "' ' "' "'H

H

= (K"' z, K"'- w)p|, for all

= (U""* K'^ z, U""* K"'- w);^,

z, w £ HQ. SO the inner product is independent of

the factorization.

31

3, K' E'^ K'""- = (KU) ^^ (K U ) " ^ = K U ^ ^ U"^ K^^

= K E^ U U"^ K"^ = K E^ K"^

Therefore, E is also independent of the factoriza­

tion.

V. Define K: H -> HQ by K x = K x, for all x £ Pf. Then ' a,

K is 1-1 and onto and for all x £ H,

IlKxIl^ = IlKxJl! = IJK-L K x i | ^ = llxll! HQ HQ H H

S O K is a unitary mapping. Furthermore, we have

Q E ^ = K E* K"^ on HQ and for all z £ L , there is a

unique x £ H such that z = K x = K x. So

-I ' -1 -I ' -1 x = K '- z = K ' z, i.e. K ^ = K '. Hence

QE = K E K' . This means that (HQ, Q E ) is unitarily

equivalent to (H, E).

vi. R(Q) = R(K K*) = K[R(K*)] = K[R(K*)]

Since R(K*) is dense in H, so K[R(K*)] is dense in H^

(for K is unitary),

vii. 1, K is left-miniphase, so

E^ x = 0 iff (F^)* K X = 0, for X e H.

2. For all z £ HQ, there exists x £ H such that

= K-L

E^ K"''- z = 0 iff (F^)* z = 0, z £ L .

z = K X or X = K"^ z. So

32

3. If (F^)* z = 0, then

Q E ^ z = K E^ K •- z = 0.

And if Q E ^ z = 0, i.e. K E^ K"^ z = 0, then

E^ K"^ z = 0. So (F^)* z = 0.

viii. K: (H, E) -> (HQ, Q E ) , is actually K defined in v. K is

left-miniphase, so ^ x = 0 iff (F^)* K x = 0. Hence

E X = 0 iff Q E ^ K X = 0, for x £ H. But since

K = K, the above equation implies K is also a left-

'v-i miniphase. So K = K'' is causal (by being left-

miniphase), and K is also anti-causal. Therefore, Oi

K is memoryless.

Immediately following from the previous theorem is a rather

interesting result which gives us a sort of "unique" left-factorization,

This result is indicated in the next corollary.

Cor. (HQ, Q E ) is as defined in Thm.III.4. Then there exists a

left-factorization P: (HQ, Q E ) -> (B*, F*), such that

i. P z = z for all z e HQ £ B*,

ii. P* b = Q b, for all b e B.

Proof: By Thm.III.4, we have K: H -> HQ, a memoryless unitary

operator, and K x = K x, for all x e ?f. Define

P = K K"^ HQ -> B*. Then Of a. O;

i. For all z £ HQ, there exists x £ H such that K x = z.

33

2. By Thm.III.4,

Q^^ z = 0 iff (F^)* z = 0, for z £ A'Q. SO

qE z = 0 iff (F^)* P z = 0, for z £ HQ. Hence P is

left-miniphase.

3. P* b = (K K'"")* b = (K""")* K* b = K K* b = K K* b = Q b,

for all b £ B. And P P* b = P (Q b) = Q b.

The "uniqueness" we mentioned is due to the fact that (HQ, QE) is

independent of the factorization.

As in the first section of this chapter, there are corresponding

theorems to Thm.III.4. These theorems are described below, with proofs

only sketched.

Thm.III.5. (HQ, Q E ) , defined at the beginnig of this section, is a

Hilbert resolution space which is independent of the factorization,

Q = T* T, used in its definition. Moreover,

i. (HQ, Q E ) is unitarily equivalent to (H, E),

ii. R(Q) is dense in HQ,

iii. (F^)* X = 0 iff E^ x = 0, for x z H,

iv. T*: (H, E) ^ (HQ, Q E ) , is memoryless.

Proof: i. The proof of (HQ, QE) being a Hilbert resolution space

and being independent of the factorization is

essentially the same as that of Thm.III.4, and

therefore is omitted,

ii. Define T : (H, E) -> (HQ, Q E ) , by T X = T* x, for all

34 '\j-k

X £ H. Then T is 1-1 and onto. For all x £ H,

ll^*x||; = ||T*x||^ = I|{T*)-LT*X||H^ = IIXIIH^.

So T is a unitary mapping. With T defined as above,

it is routine to verify the rest of the theorem. But

we need to note that T*: (H, E) -. (H^, ^E) in iv, is a.*

actually the T defined above.

Cor. (H^, -.E) defined as above, then there is a right-factoriza-

tion (miniphase) of Q, q: (B, F) -> (H^, ^E), such that

i. q X = Q X, for all x e B,

ii. q* z = z, for all z £ Hp, c B*. '\AI

Proof: By Thm. 111.5, T : H -> Hp,, is memoryless and uni tary.

Define q = T T: (B, F) -> (H^, ^E). Then

i . q X = ?* T X = T* (T x) = Q x, for a l l x £ B.

i i . q* z = (T T)* z = T* T z = T (T z) = z, for a l l

z e H, , since T is uni tary,

i i i . By Thm. I l l . 5 ,

^E^ X = 0 i f f (F . ) * X = 0, for X £ H. So

^E, X = 0 i f f (F . ) * q* X = 0, for X e H. So q* is

left-maxiphase, i . e . q is right-miniphase.

i v , q* q b = ( t * T)* (T* T) b = T* T T* T b = T* T b = Q b

for a l l b e B.

CHAPTER IV

HILBERT SPACE REPRESENTATION THEORY

So far, we have been dealing with basic concepts and fundamental

theorems. From now no, applications will be our prime concern. In this

chapter, two special areas, scattering operators generalization of

the scattering variables and Banach space valued random variables,

are discussed. Both areas fit in nicely with the theorems developed in

previous chapters. Although at this moment we don't have complete

theorems for these general problems, the exploration of the problems

itself suits its own purpose.

Scattering Operator

Classically in network analysis, network variables, such as

voltage and current, are assumed to be in Hilbert space. Although the

scattering variable is a very useful tool in network analysis, it is not

clear what is the significance of the "characteristic impedance" which

transforms voltages and currents into objects in a different Hilbert

space. Situations have occurred where we have to assume network

variables to be in Banach spaces. If the same idea for the scattering

variable works here, the function of the characteristic impedance should

be the transforming of network variables in Banach space into elements

in Hilbert space. Theoretically, it is much easier to work with Hilbert

space. Therefore the significance of the characteristic impedance lies

in changing a problem that is more general in formulation into a problem

that is easier to handle. In this section, we are going to generalize

35

36

the idea of scattering variables in Banach spaces with the help of the

factorization theorems developed in chapter III.

The reason for the generalization of the scattering variable

fitting in with the factorization theorems is due to the "duality"

between voltage and current. Thinking of the impedance Z, or the

transfer function, as a operator from a current space to a voltage space,

the product V-I, where V denotes voltage and I current, simulates the

role of a linear functional over the current space. Thus, the voltage V

can be as well thought of as a linear functional. Similarly the

admittance can assume the same role as the impedance. In that case, I

is a linear functional over the voltage space. What model could be

better than a reflexive Banach space to fit the structure just described?

Therefore, the current space and the voltage space are chosen to be a

reflexive Banach space and its dual.

Since the impedance operator is the inverse of the admittance

operator and vice versa, operators involved in the formulation of

scattering operator are invertible. The following theorems describe the

effect of invertibility on the factorization theorems.

Thm.IV.l. Let Z: (B, F) -> (B*, F*), where B is a reflexive

Banach space, be positive, causal, invertible, linear and bounded. Let

M = 1/2-(Z + Z*); then M is also positive, invertible, and furthermore

self-adjoint. Then

i. There exists a Hilbert resolution space (H, E) and a left-

factorization of M, KQ: (H, E) ^ (B*, F*), such that Kg is

left-miniphase and invertible.

37

ii. There exists a Hilbert resolution space (H, E) and a right-

factorization of M, T Q : (B, F) -> (H, E), such that Tg is

right-miniphase and invertible.

Proof: The existence of the left- and right-factorization follows

from Thm.III.2 and Thm.III.3. All we have to show is the

invertibility of Kg and TQ. We have

M = KQ KQ* = T Q * TQ. M is onto. So KQ, T Q * are onto,

but they are also 1-1. Therefore KQ, TQ* are invertible.

T Q * invertible implies TQ invertible.

Note here that we use the same Hilbert space for left- and right-

factorization. This can be justified from the proofs of Thm.III.2 and

Thm.III.3.

Thm.IV.2. Let M be defined as above. If K: (H, E) ^ (B*, F*),

is a linear bounded operator such that

i. K is 1-1, onto and causal,

ii. M = K K*,

then there exists a linear bounded operator U: (H, E) ^ (H, E), such that

a. K = KQ U,

b. U is causal and unitary.

Proof: Simply let U = Kg^ K, then

U* = K* ( K Q ^ ) * = K* (Kg*)"^ and

U* U = K* (Kg*)'^ Kg^ K = K* (Kg Kg*)"' K = K* M"' K

= K* (K K*)'"' K = K* (K*)""* K""* K = I.

Similarly U U* = I, so U* = U"\

38

Thm.IV.3. Let M be defined as above. If T: (B, F) ^ (H, ^ ) , is a

linear bounded operator such that

i. T is 1-1, onto and causal,

ii. M = T* T,

then there exists W: (H, ^) -v (H, JP) , such that

a. T = W Tg,

b. W is causal and unitary.

Proof: Simply let W = T Tg , then the proof follows.

The significance of Thm.IV.2 and Thm.IV.3 is that they enable us

to choose causal and unitary operators U & W as desired in order to make

the factorization satisfy some requirements. Unfortunatley, the

existence and the way to find these U & W are currently beyond our reach,

even though it is trivial to do so in the classical case, i.e. the

scattering variable. With this in mind, let us now formulate the

problem.

Consider the following network:

\

V V , 0 ^

y a ^0 ^0* . '1 ' i 0

a. n-port network series loaded b. optimal matching situation

From the network diagram,

V = (Z^ + Z^*) I. = (Z^ + Z^) I^

= * 0 h = V + z„ r. a 0 a

39

Define V^ = V^ - V^ and I^ = -{T - i ), then we have V = Z I . a I r 0 r

Define I^ = S I., where S is called the current-basis scattering

operator, then

r-h -a I„=I-. - I . = 1. - ( Z ^ . Z ^ ) - ! (Z^.Z^*) I.

SO

^ ~ ^B " ^^0 " L^" ^^0 " V ^ ' " "^ B ' ^ identity mapping

on current space B. Now let K, T be the factorizations of 1/2'(Z + Z *) 0 0

as defined in Thm.IV.2 and Thm.IV.3, i.e.

M = 1/2-(ZQ + ZQ*) = K K* = T* T.

Define a = K* I ., b = T I and b = S a, where S is the so-called scattering operator. Then I = S^ I. implies that

r 1 '

T""' b = S^ (K*)""" a, so b = T S^ (K*)'"' a. Hence

S = T S^ (K*)"'' = T (K*)""" - 2 T (ZQ + Z^)"^ K

= C - 2 T Y K, where C = T (K*)"^ and Y = (Z + Z,)"^. a ' ' a ^ 0 L'

In order to have a causal scattering operator S, we need a causal

C. However, C = T (K*)" which is not causal in general. By Thm.IV.2

and Thm.IV.3,

C = W Tg (Kg*)" U, where Kg, Tg denote the left- and right-

factorization respectively. Therefore the requirement for the selection

of U & W is to make C causal.

Similarly, consider the following network, an n-port network

parallel loaded by an n-port network.

40

Y • 0

I. ^ 1

. fig ' o

a. parallel loaded n-port b. optimal matching situation

Equations as follows can be easily verified.

def def Ir - -(la-Ii)' \ - \-''v

V - S v., where S is the so-called voltage-basis scattering

operator.

S^ = {\ + Y^)-l (Y^* - Y^) = -IB* + 1, i \ * Y^*). where

h-^\'\^''-def def a ^i' Q* v., b - P V , where P, Q are the factorizations of

1/2*(Y + Y *) = P* P = Q Q* as mentioned in Thm.IV.2 and Thm.IV.3. 0 0

b ^^^ S a, where S is the scattering operator,

-1 . S = -p ( Q * ) " ' + 2 P Z Q = -D + 2 P Z Q , where D = P ( Q * ) : -1

Similarly, in order to have a causal S, we must have a causal D.

However, D = P (Q*)"^ = W Pg (QQ*)"^ U*, where Pg, QQ is the right- and

41

left-factorization of 1/2'(YQ + Y^*). Therefore the requirement for the

selection of W' a U' is to make D causal.

One of the most useful properties of scattering variable is that

it gives a measure of the optimal transducer power gain. To see that

this property still holds for scattering operators, let us consider the

"power" entering the load network. For the series loaded network,

I = -I + I. = - T"^ b + (K*)"^ a a r 1

V = V +V. = Z T"''b + Z * (K*)'"' a. a r 1 0 0

So the power entering the load is given by:

(a' ) B = " ^ * ) " ^ " " ' ' 0 " V ^ *'" '! a' 'a'B ''•' ' - • -' "0 0 ' • B

//,^j.\-l - -7 + /'>/*\-l ^\ ^T"l K 7 T"' h\ = ((K*)-^ a. Z * (K*)-'' a)g - (r^ b, Z^ T"'' b)

+ ((K*)-l a, Z^ T-l b)B - (T-1 b, Z^* (K*)'^ a)^

= (a, K-1 Z^* {K*)-l a)^ - (b, (T*)-1 Z^ r^ b)^

+ (a, K-l Z^ T-1 b)^ - (K-l Z^ T-1 b, a)^

= V2(a, K-1 Z^*(K*) a)j. + 1/2(K-1 Z^* (K*)"! a. a)^

- l/2(b. (T*)-1 Z^ T-1 b)^ - 1/2((T*)-1 Z^ 1-1 b. b)^

= (a. V2-K-1 (2^* + 3^) (K*)-l a)^

- (b. V2-(T*)-1 ( y + z^) T-1 b)^

= (a. K-1 K K* (K*)-1 a), - (b. (T*)-^ T* T V^ b)^

42

= (a. a)^ - (b, b)^

= (a, a)^ - (S a, S a)^

= (a, a)^ - (a, S* S a)^

= (a, (I^ - S* S) a)^.

Above equation indicates that for a passive network, i.e.

^H " * ^ ^^^^ ^^ ^ positive operator. And S* S = In for a lossless

network. The same result can be obtained for the parallel network.

Banach Space Valued Random Variables

One way to think of a random process is to consider it as a

random variable which takes values in a function space. Of course, we

have to use an adequate probability measure to make the idea work.

Fortunately, this kind of measure has been defined for metric spaces '.

Among all kinds of metric space, Hilbert space has the nicest properties.

Hence, Hilbert space valued random variable is the kind that has been

12 studied most . However there are problems where a random process

considered as a Hilbert space valued random variable does not lead to a

satisfactory result. Consequently this kind of random process has to be

considered as a random variable over spaces other than Hilbert space.

The next choice is naturally Banach space. In this section, we first

define stochastic properties such as "me'an" and "variance operator" of

a Banach space valued random variable assuming the existence of such

random variables. We then look into the factorization of the variance

operator and the characteristic that might be achieved via the

43

factorization, i.e. the RKRS. Interestingly enough, the RKRS of a

Banach space valued random variable is a Hilbert space. This seems to

be a nice result, but there are obstacles for further application. All

these will be discussed in the following.

(1) Variance Operator

Probability measure on Banach space is a rather complicated

matter. In the sequel, we implicitly assume its existence as indicated

by the symbol of expectation value, E{*}. However we will not entangle

ourselves in the probability measure itself.

Let p, TT denote finitely additive random variables taking values

in a reflexive Banach space B. Assume

i. E{|(p, x*)|} < ->, for all x* £ B* laj

ii. E{(p, X*)} is continuous in x*.

Then there exists a unique m £ B such that

E{(p, X*)} = (m^, X*)

For E{(p, X*)} is a continuous linear functional on B*, so it is an

element of B** = B. m is termed as the mean of the random variable p.

The mean has the following properties:

i. m = m + ni » p+7r P TT

ii. IK'pll < E^l|p|l>» iii. L: B - B, is bounded and linear, then

m. = L m .

As in most stochastic processes, mean is not our prime concern.

In the sequel we thus assume that all random variables have zero mean.

For the definition of variance operator, we have to assume the follow-

44

ing:

i. E{|(p, x*)(7r, y*) |} < «, for all x*, y* £ B*, .. , (b)

ii. E{(p, X*)(TT, y*)} is continuous in x* and y*.

It is easy to show that condition (b) implies condition (a). Further­

more, we have the following lemma to facilitate the definition of the

variance operator.

Lemma. A continuous bilinear functional, (xjy), on a Banach

space B is also bounded (i.e. there exists M e R such that

|(xly)|/(||x||-||y||) < M, for all x, y £ B).

Proof: Assume not, then for all i £ N, there exists x., y. £ B uch that

l l=<ill-l iyil l

ef ine x.

u. = '

• > . "• •

V . = i

' ^ T | | X i l l ' y^\\y^\

Then u. - 0 and v. ->- 0. However

(u. v.J = i 1-

' ' i-||Xil|-||yill i

contradiction.

Now if we fix y*, then E{(p, X*)(7T, y*)} is a bounded linear

functional on B* (so an element of B** = B). This implies that there

exists a unique p_^. £ B such that

E{(p, x*)(7T, y*)} = (Py*, X*), for all x* £ B*.

45

Define a mapping Q : B* - B by

It can be easily proved that Q is linear. Moreover, Q is bounded,

since

\\%^y*\\ = i iPv*i i = sup i^^(p> ^*)i-^ v*)>i ^ x * llx*i

. < s u p M | x * | h | | Y * | | = , | . . , , | | x * | |

The operator Q^^ is termed the covariance operator of the random

variables p and TT. Covariance operators satisfy the following

conditions: i. .Q(LP)(KTT) " "^ where L and K are bounded and

linear operators on B.

ii. Define Q = Q ; then p PP

Q = Q + Q +Q + Q .

Q is cal led the variance operator of p.

i i i . Q = Q * , in par t i cu la r , Q* = Q . PIT ^TTp » '' * ^p ^ p

iv. Qp is positive; for, (Q^ y*, y*) = E{(p, y*)^} o.

(2) Reproducing Kernel Resolution Space (RKRS)

As mentioned in the previous section, a reflexive Banach space

valued random variable has a variance operator Q which is positive and

self-adjoint. Then Thm.Ill,2 and Thm.III.4 come into the picture and

we have the following: There exists a Hilbert resolution space (H, E)

and a left-factorization (miniphase) K: (H, E) -> (B, F) such that

Q = K K*, and also the RKRS, (H , E), which corresponds to (H^, nh in P P P H H

Thm,III,4. In this section the resolution of identity does not play a

46

vital role. However for future applications, we include it. Since we

have R(Q^) c H^ £ B, one of the natural questions to ask is whether the

random variable takes values only in H , and, if it does, what can we p

say about the original random var-iable. The answer to the first part of

the question is no, and we will give a counterexample in the next

chapter. However, if we happen to have the random variable taking

values in H , we would have the following properties. First define

BQ = R I K T , where K: H - B. Then K"*": B^ ^ H and we have

(K"^)*: H -. B* Consider E{(p, x) (p, y) }, for x, y in H . u P p p

E{(p, x) (p, y) } = E{(K"'- p, K"^ x)jK'^ p, K'^ y)^} p p n n

= E{(p, (K"4* K"^ x)g (p, (K-4* K"L y)g }. 0 0

Note: (K"^)* K"*" x and (K"^)"* K"^ y are elements of B*, i.e., the linear

functionals on B . By the Habn-Banach theorem, there exist x* and y* in

B* such that

x*|g = (K"4* K"^ X,

& y*|R = (K'4* K"L y, B 0

So (p, (K"4-^ K"^ x)g = (p, x*)g

0

& (p. (K'^)* K-^ y)g = (p, y*)g. 0

Therefore,

E{(p, x)p(p, y)p} = E{(p, x*)g(p, y*)3}

= (Q X*, y * ) . P ^

47

= (K K* X*, y*)g

= (K* X*, K* y*)^.

What is K* y? We claim it is K'^ y. For:

jjK* y* - K-L yj j^ = (K* y* - K' - y, K* y* - K'^ y)^

= (K* y*, K* y*)^;^ - 2(K* y*, K'^ y)^ + (K-^ y, K'^ y)^;

however,

(K* y*, K-L y)^ = (K K-L y, y*)^

= (y» y*)g, since y £ R(K),

= (y, (K-4* K-i- y)g '0

'H

and

= (K-L y, K-L y)^,-

(K* y*, K* y*)^ = (Q y* y*)^.

But Q y*£R(Q ) c R(K), so p P ^

(K* y*, K* y*)^ = (Q y*, (K'^* K'^ y) ii p ' O

= (K-L K K* y*, K-L y)^

= (K* y*, K-L y ) ^

. = (K-L y, K-Ly)^.

Hence ||K* y* - r ^ y M ' = 0, i.e. K* y* = K"^ y. Similarly,

K* X* = K-^ X. Going back to E{(P, x)p(p, y)^}, we see that

E{(p, x)p(p, y)p} = (Qp X*, y*)g

= (K* x*, K* y*)^

= (K-L X, K-^ y ) ^

= (x, y)p.

Furthermore, E{(p, x)^} = E{(p, x*)3} = 0. The above results show not only that p can be considered as a

48

zero-mean random variable over H , by satisfying condition (b) in the

previous section, but also that p has the identity operator on H as its

variance operator. Unfortunately, these results are overshadowed by the

prerequisite that p takes values in H only.

CHAPTER V

REPRODUCING KERNEL RESOLUTION SPACE OF HILBERT

SPACE VALUED RANDOM VARIABLES

In the last chapter, one problem was left unsolved for Banach

space valued random variables. We mentioned there that the random

variable may not take values only in the RKRS of its variance operator,

hence the characterization of a random variable by the RKRS of its

variable operator needs further justification. Here we will give a

counterexample to make the point, then examine some remedies for random

variables on Hilbert space. The result is that a certain kind of random

variable can be approximated by a sequence of random variables which

take values in the RKRS of the variance operator of the original random

variable and have variance operators that approach (weakly) the identity

operator of the RKRS.

Since the counterexample involves random variables on Hilbert

space, there are some special properties of this kind of random variable

that are generally not true for Banach space valued random variables.

These properties are summerized in the following theorems.

Thm.V.l. Let P be a random variable which takes values in Hil­

bert space H, is zero-mean and has a variance operator Q. Then p takes

values in R(Q} with probability 1.

12 Proof: i. By definition ,

E{(p, x)(p, y)} = (x, Qy), for all x, y in H.

i i . Let p = p i + P2, where pi = P p, P is the project ion

49 CTEXAS T r .. ,

50

mapping on R(Q), and P2 = P - PI = (I - P) p.

iii. 0 = E{(p, x)} = E{(pi + P2, x)} = E{(pi, x)}

+ E{(p2, x)}

But E{(pi, x)} = E{(P p, x)} = E{(p, P x)} = 0, so

E{(P2» x)} = 0.

iv. E{(pi, x)^} = E{(P p, x)2} = E{(p, P x)2} = (P X, Q P x)

= (x, P Q P x) = (x, Q P x) = (P Q X, x)

= (Q X, x) = E{(p, x)2}

E{(pi. x)(po, x)} = E{(P p, x)([I - P] p, x)}

= E{(p, P x)(p, [I - P] x)}

(P X, Q [I - P] x) = 0

But E{(p, x)2} = E{(pi, x)2} + 2E{(pi, x)(p2, x)}

+ E{(p2, x)2}

Hence 0 = E{(p2, x)^}, for all x £ H. So P2 = 0 with

probility 1, i.e. p = pi = P p with probability 1.

Therefore p takes values in R(Q) with probability 1.

Thm.V.2. With p, Q as defined in Thm.V.l, let Q be a compact

operator. Therefore Q = z A,^(p, <\>^)^^, where \^ is the eigenvalue and

(|). is the corresponding eigenvector with the properties that

ii. X. > 0, for all k, since Q is positive,

iii. lim x. = 0,

TV, {(J). }°° isan orthonormal system, ^ 1

Then p = z a . , where a^ = (p, <,,^), with probability 1.

51

Proof: i . E{(p - " a .6 . , x ) } = E{(p, x ) } - z (4)., x) E{a.} I l l 1 1 1

= 0 - 2 (({,., x) E{(p, ({).)} 1 T 1

= 0, fo r a l l n.

i i . E{(p - ? a.(|,., x)2} 1 T T

= F{(p, x)-2} - 2E{(p, x)(z a.<{>., x ) } + E{(? a A., X ) 2 } i l l 1 ' '

= (x , Qx)' - 2 z ((j)., x) E{(p, x ) . a . }

+ z z ((()., x)(c{)., x) E{a.a.} I l l 0 T J

But E{(p, x) a.} = E{(p, x ) (p , cj,.)} = (x , Q ^,)

= X. (x , <|).) 1 1

and E{a.a.} = E{(p, (j).)(p, <^.)} = {<^., Q <{>.) = ^.<5... 1 J 1 3 1 J 1 TJ

So E{(p - z a.<j,., x)2} = ? x . ( x , ci>.)2 - 2 ? x . ( x , <D J ^ 3 1 1 1 1 1 I T T

' + Z X . ( X , ({,.)2 1 1 1

= z x . ( x , <},.)2 - z x . ( x , c )2 1 1 1 1 1 1

X ^ 0 , a s n -> 00.

n+] = z x . ( x , * . ) < X | |>^l l

n+i T 1 "+^

Bv i and i i , we conclude that p = z a 4, with probabi l i ty ^ 1 ' '

1.

Now we are ready for the counterexample.

Exp. Let {d)., i|;.}7 be an orthonormal system in Hilbert space H.

Let p = z (l/n)'COs(2nTra))-4) + " (1/n) •sin(2n7ra)) .^^, where ^ is a real

52

valued random variable uniformly distributed in [0,1]. Then we have

i. IIPII^ = z (l/n2) cos2(2niTa)) + z (l/n^) sinHZmt^) 1 1

= z (l/n2)[cos2(2mTU)) + sin^iZn-ni^)]-1

= z (l/n^) < oo 1

ii. E{(p, x)} = E{ z (1/n) cos(2mTw) (({.„, x) 1 "

+ z (1/n) sin(2mTa)) (i „, x)

But E{cos(2mra))} = /o cos(2mT(o) do) = 0 , for a l l n,

and E{sin(2mT(o)} = /Q cos(2mTu)) dw = 0 , for a l l n.

So E{(p, x ) } = 0 , for a l l x £ H.

i i i . E{(p, x ) ( p , y )}

= E{[Z (1/n) cos(2n7Ta)) (<!>_, x) 1

+ z (1 /n) 3in(2nTra)) {i>^. x ) ] 1

= E{ z z (1/nm) cos(2mTa)) cos(2m7Ta)) ((t> , x)(<{)j , y) 1 1 + z z (1/nm) cos(2mTa)) sin(2m7ra)) (c{> , x)(j;^, y)

1 1 + z z (1/nm) sin(2mTa3) cos(2mTTa)) (ip^, x)((t>^, y)

+ z z (1/nm) sin(2mra)) sin(2m7ra)) (^„, x)(i|^^, y ) }

But E C s i n u L ) sin(2m^a>)} = /J s in(2nu. ) sin(2mTra>) do)

E{sin(2mra}) cos(2miTa))} = 0

E{cos(2mTa)) cos(2mTTa))} = 1/2 5^^

53

So •E{(p, x)(p, y)}

= Z (l/2n2)(<,>^, x)(4)^, y) + z (l/2n2)(i|;^, x)(i|. , y)

= (Z (l/2n2)(^^, x)^^, y) + (z (l/2n2)(i|.^, x)^^, y)

= (z (l/2n2)(4)^, x)^^ + z (l/2n2)(i|,^, x) ^ ^ , y)

def = (Q X, y)

Hence Q x = z (l/2n2)(4, x)<i>^ + z (l/2n2)(i|; x)i|;

Q (j). = Z (l/2n2)(cj,^, 4,.)4) + z (l/2n2)(i|,^, ^.)^i,^

= (l/2i^)

Q il;. = z (l/2n2)(4,^, ^.)<^^ + z (l/2n2)(^^, ^ . ) ^ ^

= (l/2i2)

Hence l/2i2 is the eigenvalue and {(f>., i|;} are the

corresponding orthonormal eigenvectors,

iv. We claim that p takes values on R(Q) - R(Q) with probabili­

ty 1. Since R(Q) = R(K K*) c R(K), where

K = z (l/n/2)(4)^, • )c{>n + I (l/nv^(^n, • )i>^> we are

actually going to prove that p takes values on R(Q) - R(K).

Here is how:

For all 0) £ [0,1],

p(a3) = Z (1/n) cos(2n™) (j) + z (1/n) sin(2n™) ij;

and ||p(a))||2 = z l/n2.

If there exists z in R(K) such that Q M = z, for some ,

then there exists x £ H such that p(a)) = z = K x

= z 0/n/Tli^^, x)4>„ + z (l/n/2)(i{,^, x)^.^

This implies that

54

(4>.j, x) = v i(({)., z) and (;/;., x) = ^i{^-, z)

For x is in H, z (cj, x)2 + f ( ., x)2 has to be finite. 1 » 1 1

But z = p(a)), (<|> , z) = ( 1 / i ) cos(2i7TU)) and

{^T> z) = ( 1 / i ) Sin(2i7ra)).

So Z [ i^ i(<{,. , Z)]2 + f [ ^ i ( ^ . , z) ]2 1 » 1 1 0 0

= z 2 cos2(2i7Tu)) + z 2 sin2(2iTTa>) 1 1

This series does not converge, so contradiction occurs.

With Thm.V.l, we conclude that p takes values in R(Q) - R(Q)

with probability 1.

Thm.V.2 gives us a pretty good idea how to approximate a random

variable. But there is a requirement that the random variable must

have a compact variance operator. The following is what we found:

i. With p, Q as defined in Thm.V.l & 2, let Q be compact.

Then by Thm.V.2, we have p = z (p, ({)..)(j)- with probability

1, where {(j).} are the orthonormal eigenvectors of Q.

ii. Define P„ = z (P, <^JC|). = P„ P, where P„ is the projection n ] 1 1 n II

on the subspace spanned by {<p^, (^2^ ""» 'n'*

p is zero-mean and n

E{(p„, x)(p^, y)} = E{(P„ p, x){P^ p, y)}

= E{(p, ?^ x)(p, P^ y)}

= (P„ X. Q P„ y)

= {P„x. ?x.{*.. P„y)6.)

= (P„ X. E x^(P^*,..y)*i)

= (p„ X, S \MA< y)i'i) n 1 ' ' '

55

n 1

n = (x , P z X.(<!)., y)(|,.)

= (x , z X.(<!,., y)^.) = (x, P Q y)

def = (x , Q„ y) 'n

n • So Q^y = Z X.(c{,., y)c(,. = P Q y .

i i i . E{(p-Pn, x ) (p-p^ , y ) } = E{ ( [ I -P^ ]p , x ) ( [ I -P^ ]p , y ) }

= ([I-P^]x, Q[I-P^]y)

= (x, Q y) - (x , Q P y) - (Pp ^ ' Q ^)

+ (Pn ^ ' Q PR ^ '

= (x , Q y) - (x , Q Pn y) - (x, P Q y)

+ (x , P Q P y)

But Q P y = P Q y = P Q P y = Z X.((j>., y)<^..

So E{(p-p^, x ) (p-p^ , y ) } = (x, Q y) - (x, P Q y)

= (x , [Q - P^ Q]y) = (x , [Q - Q^]y)

i v . IKQ - Q j x I P = II z x . ( * . , x)c^.| j^ 11 n - n I I « n+1

= n! i ^ i ^ * i ' ""^^ - ^ n + i * n | i ( * i ' ^^^ 1 V i H ^ M ^

So | | (Q„ - Q ) x | | / | | x | l < /x^^, fo r a l l x f 0.

So I IQ - Q||2 < X T -> 0, as n -> «>, i . e . Q -> Q uniformly. ' ' n ' ' — n+i n

n " V. For p„ = z (p , (j),)cf). = z X.(p/X. , 6.)(|)., p„ takes values in

n 1 • ' 1 • I I I II

R(Q) c_ R(K) only, where K is the factor izat ion of Q. p^ is

also zero-mean in HQ.

Consider def , ,

^n " ^ ^ ^ ^ n ' ^^Q^^n' ^^Q^' ^^"^ ''^^ x, y £ H^ = R(K)

= E{(K-L p^, K-L x)^(K-"- p^, K-L y)^}

56

= E{(p^, (K"h* K"^ x)^(p^, (K"L)* K"'- y)^}

= ((K"^)*K"L X, Q^ (K"4* K - S ) H

= (K"L X, K"^ P^ Q (K"'-)* K"^ y)^

= (K"^ X, K"^ P„ K K* (K"^)* K"'- y)n

= (K"^ X, K"^ P^ K (K'*- K)* K"^ y)^

= (K"^ X, K"^ P^ K K"^ y)^

= (K"^ X, K"^ P^ y)^, for y £ R(K)

= (X, P, y)Q

Note that (K"*")* K"^ x is actually an element in R(K) £ H.

Furthermore, since Q^ " Q uniformly,

Qp (K"^)* K"*- y -> Q (K'^)* K"^ y.

So (P y, X)Q = ((K"4* K"*- X, Q^ (K-4* K"L y)H

^ ((K"^)* K" X, Q (K"4* K" y)H

= ((K"^)* K"^ X, K K* (K"^)* K"^ y)^

= (K"*- X, K"^ K K* (K"^)* K"^ y)^

= (K"^ X, (K"*- K)* K"^ y), H

(K"^ X, K"^ y)^

57

= (x, y)Q, for all x, y £ HQ.

This means that

W ^ "" \ '' ' ^ ' The above results simply indicate that although we cannot

directly consider a random variable as a "white noise" in the RKRS of

its variance operator — by having the identity mapping as its

variance operator, in some situations we may as well approximate the

random variable by a sequence of random variables that take values only

in the RKRS and have variance operators approaching the identity

mapping weakly, i.e., a sequence of random variables "approaching the

white noise".

CHAPTER VI

WIENER-HOPF OPTIMIZATION THEORY

In the first part of this chapter, we formulate the classical

Wiener-Hopf filtering in Hilbert spaces. With the techniques devel­

oped for this filtering, we then look into a special control problem,

the so-called Wiener-Hopf control. By choosing an adequate optimiza­

tion criterion, this kind of control fits nicely into the method of

Wiener-Hopf filtering. In both cases we deal with the optimization of

positive operators, and because of this, we designate this chapter as

"Wiener-Hopf Optimization Theory".

Wiener-Hopf Filtering

Before we formulate the Wiener-Hopf filtering in Hilbert spaces,

we have to assume certain requirements to facilitate the formulation.

For comparison, let X be a random variable denoting the signal and n

be a random variable denoting the noise. Both random variables take

values in the Hilbert resolution space (H, E) and they have the

following properties:

i. E{(X, z)2}andE{(n, z)^} are finite, for all z £ H.

ii. X, n are zero-mean and statistically independent, i.e.

Qxn = %X - 0-iii. Qy, Q denote the variance operators of X and n, respec-

tively. As such, they are positive, self-adjoint, bounded

and linear. We assume that

a. Qy is Hilbert-Schm.idt,

58

59

b. Q„ is positive definite,

0- Qx " n ^ onto.

Let T denote the filter operator. T has to be causal and

Hilbert-Schmidt. Let Y denote the output of the filter. Define

e = X - Y, then e = X - T(n + X) and Q^ = (I - T)Qy{l - T)* + T Q^ T*.

Since Q^ is a function of T, we will write Q as Q (T) to indicate the

dependence of T. Our assignment is to find an optimal filter T such ^ 0

that Qg(TQ) is minimal, i.e. for any Qg(T) 1 Qg(TQ) we have

Qe(T) =Qe(T^).

Solution:

i. Qg = (I - T) Qx (I - T ) * + T Q^ T*

= T (Q, + Q,) T* - T Qx - Qx T* . Q^

Since Q is positive definite and Q„ is positive, Qw + Q^

is positive definite, hence 1-1.

ii. Let F: (H, E) ->• (H, E) be the left factorization of

Qw + Q^ such that

a. Qx + Q, = F F*

b. E^ X = 0 iff E^ F X = 0, for all t £ R.

This can be done because Qx + Qp is positive definite,

self-adjoint and onto. F can be taken as the square root

of Qx + Q,.

iii. Since Qy + Q„ is onto, so for all y e H, there exists A n

X £ H such that (Qx + Qn) ^ " y- ^°

y = F F* X = F(F* x), i.e. F is onto.

F is also 1-1, since it is left-miniphase. So F" exists

60

and is causal (also follows the property of being left-

miniphase).

iv. Qg = T F F* T* - T Qx - Qx T* + Qx

= [T F - Qx (F*)-^][F* T* - F""' Qx] + Qx

- Qx ( ^ * ) " ^ Qx

= [T F - Qx (F*)-"'][T F - Qx (F*)"^]* + Qx

- Qx (F F*)-^ Qx.

Since Qx and Qx (F F*)"^ Qx are positive and independent

of T, therefore, to find the minimal of Q is the same as e

to find the minimal of [T F - Qx (F*)"^][T F - Q (F*)"^]*,

denoted as Q(T).

V. Lemma 1. For Hilbert-Schmidt operators, the additive

decomposition always exists and each component is again 15 Hilbert-Schmidt . The proof of this lemma can be found

in the reference indicated.

vi. Lemma 2. For any self-adjoint, nuclear operator pair A and

B, with A and B comparable, then

A = B iff Tr[A] = Tr[B].

Proof: a. The lemma is equivalent to saying that for any

positive, nuclear and self-adjoint operator A,

A = 0 iff Tr[A] = 0.

b. {=>) trivial.

61

c. (<=) Tr[A] = z X.(A) = 0,^^ where X.(A) denotes

the i-th largest eigenvalues of A. But

^.|(A) > 0, for i, since A is positive and self-

adjoint. So X .(A) = 0, for all i. This implies

the radius of the spectrum of A is zero, so

I|A|| = 0, for A is self-adjoint. So A = 0.

vii. Qx is Hilbert-Schmidt and (F*)"^ is bounded, so

Qx (F*)"^ is also Hilbert-Schmidt^^. Let

Qx (F*) = C + A, where A is the strictly anti-causal

part of Qx (F*)"^ and C is the causal part of Qx {F*)"\

viii. Q(T) = [T F - Qx (F*)'"'][T F - Qx (F*)""^]*

= [T F - C - A][T F - C - A]*

= [T F - C][T F - C]* - [T F - C] A* - A [T F - C]*

+ A A*

ix. Lemma 3. The trace of any strictly causal (strictly anti-

causal) nuclear operator is zero .

Proof: The radius of the spectrum of a strictly causal

(strictly anti-causal) nuclear operator, say T,

is zero . Therefore,

Tr[T] = z X.(T) = 0. 1 T

X. Lemma 4. The composite operator of any two Hilbert-Schmidt

22 operators is nuclear .

Lemma 5. The strictly causal (strictly anti-causal)

operators form a two-sided ideal in the space of causal

62

(anti-causal) operators^.

xi. Now let's consider the trace of Q(T) (Q(T) is nuclear by

Lemma 4).

Tr[Q(T)] = Tr[(T F - C)(T F - C)*] - Tr[(T F - C) A*]

- Tr[A (T F - c)*]+Tr[A A*]

Since A is strictly anti-causal. A* is strictly causal.

T F - C is causal, so (T F - C)* is anti-causal. So

(T F - C) A* as strictly causal and A (T F - C)* is

strictly anti-causal by Lemma 5. So

Tr[(T F - C) A*] = 0 = Tr[A (T F - C)*] by Lemma 5. Hence

Tr[Q(T)] = Tr[(T F - C)(T F - C)*] + Tr[A A*].

Since (T F - C)(T F - C)* and A A* are both positive, their

traces are also positive. So

min Tr[Q(T)] = Tr[A A*], when T = C F"\ a causal

operator,

xii. Claim that when T = C F' » Q^ is minimal. For we first

point out that Q is minimal whenever Q(T) is minimal.

Second for any causal, Hilbert-Schmidt T such that

Q(T) 1 Q(C F"^), we have

Tr[Q(T)] £ Tr[Q(C F " ^ ] * by Lemma 2.

But Tr[Q{C F"b] is minimum. SO

Tr[Q(C F"^)] = Tr[Q(T)]. Hence

Q(C F"b = Q(T) by Lemma 2.

At th is po in t , the operator version of the Wiener-Hopf f i l t e r i n g

63

is completed. However, the requirement that certain operators be

Hilbert-Schmidt seems too restrictive. Being Hilbert-Schmidt is

required for the formation of a nuclear Q(T) in the variance operator

so that the trace of the Q(T) makes sense. Furthermore, the "if and

only if" statement of Lemma 2 enables us to minimize the trace in

order to find the optimal filter. Taking the memoryless part of the

variance operator has a similar effect on the variance operator as just

taking the trace. Intuitively, the scheme used previously might also

work here once the additive decomposition is guaranteed. Unfortunate­

ly, there is no counterpart for Lemma 2, i.e. a positive, self-adjoint

operator having zero memoryless part is not necessarily a zero

operator. Therefore, in order to optimize the operator (filter) over

a broader class of operators — operators that can be decomposed

additively (for example, S in the space of compact operators) , we

have to compromise the optimization criterion. The compromise is made

by minimizing the memoryless part of the variance operator instead of

the operator itself. With this modification, the problem can be

solved in the same fashion once the additive decomposition is

guaranteed by working in S and it leads to the same solution.

Wiener-Hopf Control

Fig.l on the next page shows a control system suggested by

Yula . The center branch represents the plant P together with the

controller C, while the upper and the lower branch represent the

feedforward and the feedback compensators, respectively.

u ^ C

T m

(r

Fig.l. A Feedback and Feedforward Control System.

64

In the diagram, u is the input to the system and y is the output,

"d" represents the load disturbance. "£" is the noise picked up by

the feedforward sensor, "m" is the noise picked up by the feedback

sensor. The only unknown is the controller C. Our goal is to

construct an optimal C such that a certain sum of the variance opera­

tors is minimal. However, before we formulate the problem and solve

it, we must assume the following:

i. Signal u, i, d and m are random variables that take values

in Hilbert resolution spaces (H, ^ E ) , (G, ^E), (D, ^E) and

(M, i^E), respectively,

ii. P: K -> H, P^: D -> H, L: D -> H, L^: G -> H, F: H -> H and

F • M -> H, are linear and bounded. C: H - K is linear. 0

Here (K, j E) is also a Hilbert resolution space.

65

iii. E{(u, x)2}. E{(£, g)2}, E((d, 2)2} ^ ^ E{(m, w)2} are

finite, for all x £ H, g £ G, z £ D and w £ M.

E{(u, Xi)(u, X2)}, Eiii, gi)(z, g^)}, E{(d, Zi)(d, Z2)}

and E{(m, Wi)(m, W2)} are continuous in x, and x^ of H, g^

and g2 of G, zi and Z2 of D and w^ and W2 of M, respec­

tively.

iv. Random variables u, £, d and m are zero-mean and statisti­

cally independent."^

V- Qy> Qo> Q j and Q ^ denote the variance operators of u, Ji, d

and m, respectively. As such, they are positive, self-

adjoint, bounded and linear. We assume that

a. Qjj + Pj Q^ P^* is Hilbert-Schmidt, where

Pd = ^ Po ^ L.

- % ^ ^0 ^m V ^ ^0 \ L,* + P^ Q^ P^* is positive

definite and onto,

vi. Def.: The plant P and the feedback compensator F form an

admissible pair if (a), both are causal, (b). there exists

a controller C (causal and linear) such that (I^ + F P C)"

exist and P C is Hilbert-Schmidt.

vii. P is onto.

Problem: For an admissible pair P and F, find an optimal

controller C (over the class of operators that make P and F admissible)

t Random variables u and i are independent if the covariance operator

Q /. ^ = 0, for all linear bounded operator A: G ^ H.

66

such that Qg + Q,p . is minimal, where e = u - y and r is the input to s '

the plant, while P^ is a linear and bounded mapping from K to H which

transforms the random variable r such that P r takes values in H and

Q(p r) " positive operator over H, so compatible to Q . The

operator P^ will be specified later.

Solution:

i. From Fig.l,

y = P r + P d 0

v = F y + F m 0

h = L d + L I. 0

Straightforward calculation yields

y = P R (u - FQ m - LQ £) + (P^ - P R P^) d

r = R (u - FQ m - LQ ii - P^ d)

e = u - y

= d u - P R) u + P R (F^ m + L l) - (? - P R P .) d, H 0 0 0 d

where R = C S, S = (I ^ + F P C)"^ and P^ = F P^ + L.

So

% = (IH - P R) % (IH - P ^ ) *

+ (PQ - P R Pd) Qd ( 0 - P ^d)*

V r ) - ^ M Q , -F^Q^F*.L^Q, L^*.P,Q, P,*) s

•(Pg R)*

ii. Note that the only parameter depending on C is R and ? & R

always appear in product in Q^. Therefore, in order to

67

^^^^ Q(p y,\ compatible, the proper choice for P would be

k P with k be a real constant. For P = k P,

Qe + Q(P^r) = P M Q , + F^ Q^ F^* + L^ Q^ L^* + P, Q, P^*)

•(P R)* (1 + k^) - P R (Q^ + Pj QH P *)

- (Qu ' Po d d*) (P R'* '\*^o % V = P R OQ(P R ) * - P R Q^* - Q^ (P R)* + Q

^^O^dV' Where Q^ = (1 + k^JCQ^ + F^ Q_ F^* + L^ Q^ L^* + P^ Q^ p^*)

and Qi = Q^ + P Q^ P^*.

i i i . QQ is posi t ive de f in i te and onto by assumption, so i t is

also 1-1. Let T: (H, ^E) •> (H, ^E) be the l e f t -

fac tor iza t ion of Q such that (a) . Q = T T*, ^f t

(b ) . ^E X = 0 i f f ME T X = 0, for a l l real number t .

T~ exists and is causal by the same argument as in the

f i l t e r i n g . i v . Qg + Q(p r) " P ^ ^ ^* (P R)* - P R Q^* - Q (P R)* + Q^

+ P Q . P * 0 ^d 0

= [P R T - Q (T* ) "^ ] [T* (P R)* - T"'' Q^*]

+Q + P Q P * - Q (T*) '^ T"^ Q * ^u 0 ^d 0 ^-j ^ / ^1

= [P R T - Q (T*)"" ' ] [P R T - Q (T*)"""]*

^ Qu " ^0 ^d V - ^1 ^o' ^ 1 * -

Q * P Qj P * , Q, Q" Q-,* are posit ive and independent of ^u* 0 ^d 0 ^1 ^0 ^1

68

of C. Therefore, to minimize Q^ + Q,p . is the same as to

def ^

minimize Q(P R) = [P R T - Q^ (T*)"'][P R T - Q^ (T*)"^]*.

V. P R = P C S and Q^ are Hilbert-Schmidt, So Q(P R) is

nuclear. By Lemma 1 to 5 in the filtering, minimal Q(P R)

occurs when P R = B T"\ where B is the causal part of Ql (T*)'\

v i . The next thing is to f ind C. By formula,

P R = P C S = P C ( I ^ + FP C)""*.

So PC = PR + P R F P C ,

( I ^ - P R F) P C = P R,

P C = ( I ^ - P R F)"^ P R.

So P CQ = ( I ^ - B T"^ F)"^ B T " \ i f the inverse of

In - B T " F ex is ts . C denotes an optimal cont ro l le r . H 0

P is not necessarily a 1-1 mapping, but it is onto. This

means there is probably more than one solution for C .

But all we are interested in is "a" solution. The

following shows how we choose to define C :

For all X £ H, define

G X = (I^ - B T"'' F)'"^ B T""* X.

G X £ H, so there exists z £ K such that P z = G x.

Let w = z - P^ z, where P^ is the projection on the

subspace N = {z £ K | P z = 0} — the null space of P.

Define: C X = W = Z - P | M Z .

69

a. CQ is well-defined. For if there exists z^, Z2 e K such

that P z^ = P Z2 = G X, then P (z^ - z^) = 0. So

2-, - 22 £ N and P^ (z^ - Z2) = Z-, - Z2, i.e.

^1 " ^N ^1 = ^2 - ^N H = ^0 •

b. Let CQ (x^ + X2) = z^2 " P^ ZT2' ^ '" 12 ' " ^ ^ ^

P Z]2 " ^ ( 1 + ^2). So

P 2^2 " ^ ^1 + S X2 = P z + P Z2 = P (z^ + Z2), where

z-j, Z2 are such that P z. = 6 x., i = 1, 2.

So CQ ( X ^ + X 2 ) = (z^ + Z2) - P^ (z^ +Z2)

= z - P^ z + Z2 - P^ Z2

= ^0 1 ^ ^0 2-

Similarly, C^ (b x) = b C (x), for all real b.

So C is linear,

c. P CQ X = P (z - P^ z) = P z = G X.

From a, b and c, C is a solution of

p CQ = (IH - B T"'' F)""' B r \

As pointed out in the context of Wiener-Hopf filtering, if we

sacrifice the optimization criterion by minimizing the memoryless part

of Q + Q/p \ instead of the operator itself, the requirement of

being Hilbert-Schmidt could be changed to the requirement of being in

S . The sam.e solution follows.

CHAPTER VII

CONCLUSION

In this dissertation, we have developed Banach resolution space

and factorization theorems for operators that map from reflexive Banach

spaces to their duals. These fundamental definitions and theorems are

then directly applied to the study of scattering operators and Banach

space valued random variables. As yet, however, we do not have

complete theories in these two areas. We then come to the subject of

Wiener-Hopf filtering. According to the development of this disserta-

tation, Wiener-Hopf filtering should have been formulated in reflexive

Banach resolution space. However, there are still problems to be

solved in Banach resolution spaces, e.g., operator decomposition.

Therefore, the Wiener-Hopf filtering is instead formulated in Hilbert

resolution space. In this formulation, the method of taking the trace

of a nuclear operator has been used to get rid of the cross terms

involved in the variance operator. We have mentioned that we could

take the memoryless part of the variance operator to cancel the cross

terms at the expense of accepting a less restrictive optimization

criterion — optimizing the memoryless part of an operator instead of

the operator itself. This is important in two ways. First, it allows

us to work on a more general class of operators which guarantee the

decomposition. The advantage of working with S^ is that S^ is an ideal

in the space S^ of compact operators. This property allows us to make

assumptions directly concerning the noise, the signal and the filter

70

71

instead of only the variance operator of the error. Second, it makes

possible the generalization of the formulation in reflexive Banach

resolution space, because the term "trace" does not make sense for

operators which map from reflexive Banach spaces to their duals whereas

the memoryless part of an operator is meaningful. The only problem

left is the unique existence of the memoryless part of an operator.

This will, presumably, be a future development in Wiener-Hopf filtering.

With the results from filtering, we then look into the problem of

Wiener-Hopf control. What we have done, is to improve the performance

measure by using an optimal controller in a feedback and feedforwark

control system. However, not much effort has been devoted to studying,

the stablization of a nonstable system by the Wiener-Hopf technique.

This will also be a future direction of research for Wiener-Hopf

control theory.

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