Nonlinear prediction ladder-filters for higher-order stochastic sequences

Lecture Notes in Control and Information Sciences Edited by M.Thoma 73 III IIII IIIIIIIIII IIIII J. Zarzycki Nonli...

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Lecture Notes in Control and Information Sciences Edited by M.Thoma

73 III

IIII

IIIIIIIIII

IIIII

J. Zarzycki

Nonlinear Prediction

Ladder-Filters for Higher-Order Stochastic Sequences I

IIIIIII

II

IIIIIIIIIIIIIIIIIIII

IIIIIIIIIIIIIII

Springer-Verlag Berlin Heidelberg New York Tokyo

Series Editor M.Thoma Advisory Board A.V. Balakrishnan • L. D. Davisson • A. G. J. MacFarlane H. Kwakernaak • J. L. Massey • Ya Z. Tsypkin • A. J. Viterbi Author Jan Zarzycki Institute of Telecommunication and Acoustics The Technical University of Wroclaw ul. B. Prusa 5 3 / 5 5 50-317 Wroclaw - Poland

ISBN 3-540-15635-6

Springer-Verlag Berlin Heidelberg New York Tokyo

ISBN 0-38?-15635-6

Springer-Verlag New York Heidelberg Berlin Tokyo

Library of Congress Cataloging in Publication Data Zarzycki, J. (Jan) Nonlinear prediction ladder-filters for higher-order stochastic sequences. (Lecture notes in control and information sciences; 73) Bibliography: p. 1. Stochastic sequences. 2. Prediction theory. 3. Filters (Mathematics) I. Title. I1. Series. QA274.225.Z37 1985 519.2 85-12668 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under c:354 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © Springer-Verlag Berlin, Heidelberg 1985 Printed in Germany Offsetprinting: Mercedes-Druck, Berlin Binding: LiJderitz und Bauer, Berlin 2161/3020-543910

PREFACE

In t h i s w o r k w e s h a l l b e c o n c e r n e d with t h e p r o b l e m of n o n l i n e a r

least-squares linear

prediction of h i g h e r - o r d e r stochastic s e q u e n c e s

orihogonal

digital

filters. The

nonlinear

problem

as a generalization of the lineaLr least-squares The cond-order

linear l e a s t - s q u a r e s

will be

prediction problem.

for w h i c h

white noise

when

ven sequence)

or s h a p i n g

filters ( w h o s e

modeling

statistically equivalent to the given s e q u e n c e ,

when

sire p r o c e d u r e s ,

in w h i c h

one

will not h a v e

e a c h time the permitted complexity derlies both the ladder-structures ry, a n d

the theory of orthogonal

the gi-

output is

driven b y white noise)

remarkable

implemented

Hence,

expa~sions

means

plemented

using

(namely CORDICS

of a n

any

orthogonal filter w h o s e

via recur-

the w h o l e

the s a m e

in m o d e r n

inherent numerica/

result of this theory is t h a t

alized b y

The

assures

computed

to r e c o m p u t e

is increased.

(Fourier)

be

of the m o s t importa~nt proper~y of the orthogonal

vat[on of 'energy' w h i c h A

driven b y

established° In practice, the linear or£hogonal filters c a n

One

with s e -

the orthogonal prediction or

innovations linear filters ( p r o d u c i n g

can be

considered

estimation theory is a s s o c i a t e d

stochastic s e q u e n c e s ,

as well a s

using non-

filter

idea u n -

digital filters theo-

in Hilber~

spaces.

digital filter is p r e s e t stability of the filter.

transfer function c a n modular

structure c a n

sophisticated 'building-blocks' with V L S I

be be

reim-

integrated circuits

processors).

linear theory results in the o p t i m u m

approximation of s e c o n d - o r d e r

sequences.

(least-squares)

Therefore,

stochastic

the linear estimation

IV filter b e c o m e s properties

the best possible

are completely c h a r a c t e r i z e d

If the undertyin~ s e q u e n c e may

be

In this w o r k least-squares

we

wish

sequence

the s e c o n d - o r d e r

(whose

statistics).

the linear estimation a c c u r a c y

a norz[inear a p p r o a c h

in order to i m p r o v e

to the p r o b l e m

the a c c u r a c y .

to p r e s e n t efficient algorithms of nonlinear

prediction filters for higher-order stochastic s e q u e n c e s ,

sulting in the o p t i m u m

approximate

re-

nonlinear digital filters of the Volterra-

class, ri~hese nonlinear ladder-filters will generalize the linear fil-

ters, p r e s e r v i n g zations, a m o n g order

by

is non-Gaussio.n,

not satisfactory. In that c ~ e ,

shot~Id b e introduced

Wiener

filter for ~ Gatlssi~n

m o s t of their properties

will mention h e r e

reali-

for higher-

stocha.stic s e q u e n c e s .

only t h o s e p a p e r s

ted to the subject of this work, the p a p e r s

modular

others), a n d yielding better estimation a c c u r a c y

(o.nd non-CTaussi~.n)

We

(orthogonality a n d

which

reffering for m o r e

citied ( a n d the r e f e r e n c e s

are closely c o n n e c -

complete

bibliography to

therein),

ACKNOWLEDGMENTS

I am p a r t i c u l a r l y indebted to Professor Patrick DewZlde of the D e l f t University of Technology for ~is helpful suggestions and h i n ~ introduced in many stim~at~ng and f r u i t f ~

disc~sio~,

~ p e c i a ~ l y d~ring my o n e - y e n s t a y

in Delft, which h ~ undoubtedly inspired t h i s work. I am ~ o

g r a t e f u l to P r o f ~ s o r M~ian S. P i e k ~ s k i of the Technical

Unive~Zty of Wro~aw for his valuable commen~ and d ~ c ~ s i o n s

concerning

t h ~ work. I wish to thank ~ . manuscript.

Zdzislawa Zabska for her c ~ e f u l typing of t h i s

CONTENTS

CHAPTER

i, I N T R O D U C T I O N

CHAPTER

2. N O N L I N E A R A UNIFIED

.......,........................................... ............... PREDICTION APPROACH

I

FILTER PROBLEM: .................................................... 1 3

2.1 H i g h e r - o r d e r stochastic sequences ........................................... 1 3 2.2 N o n l i n e a r lea~st-squares prediction: Algebraic a p p r o a c h ......... 20 2.3 N o n l i n e a r least-squares prediction: Geometric a p p r o a c h ........ 24: 2.3.1 S p a c e of t h e r e g u l a r V o l t e r r a f u n c t i o n a l p o l y n o m i a l s .......................... ............................................ 2 4 2.3.2 S p a c e of g e n e r a l i z e d c o e f f i c i e n t - m a t r i c e s .................. 26 2.3.3 S p a c e of g e n e r a l i z e d z - p o l y n o m i a l s ........................... 2 9 2.3.4 I s o m e t r i e s ........................................................................ 3 2 2.3.5 S t o c h a s t i c n o n l i n e a r e s t i m a t i o n .................................... 3 5 2.3.6 O p t i m u m generalized matrix approximaUon ................. 36 2.3.7 O p t i m u m generalized polynomial a p p r o x i m a t i o n .......... 3 8 CHAPTER

3.

GENERALIZED

NONLINEAR

LA/DDER-F'ILTERS

.............

4O

3.1 I n d e x - s e t s a n d their o r d e r i n g ...................................................... 4 1 3.2 N o n l i n e a r filter algorithm: t i m e - d o m a i n a p p r o a c h ....................... 4 8 3.2.1 'Local' e s t / m a t e s a n d errors..• ......................... ............. 5 1 of s u b s p a c e s ......................................... 5 5 3.2.2 D e c o m p o s i t i o n b a s e s .......................................................... 5 7 3.2.3 O r t h o n o r m a l Cholesky factorizat.[ons ............................ 6 0 3.2.4 G e n e r a l i z e d F o u r i e r s e r i e s e x p a / n s i o n ..................................... 6 1 3.2.5 M - D r e c u r s i o n s ................................................ 6 2 3.2.6 O r d e r - u p d a t e approxim~ion of t h e M - D impulse 3.2.7 O p t i m u m responses•..•••....•.••••.•••••....••.•.•.•...•......•..•.....••••...•.••.•.•.....7 0 3.2.8 E s t i m a t i o n a c c u r a c y ........................................................ 7 1 3.3 N o n l i n e a r filter algorithm: t r a n s f o r m - d o m a i n a p p r o a c h .............. 7 4 3.3.1 'Local' e s t i m a t e s a n d e r r o r s ......................................... 7 6 3.3.2 D e c o m p o s i t i o n of s u b s p a c e s , ON bases and M-D ~'~ourier e x p a n s i o n ................................................. 7 9 3.3.3 0 r d e r - u p d a t e r e c u r s i o n s ................................................ 8 3 3.3.4 Optimum ON approximation of t h e s e t of M - D t r a n s f e r f u n c t i o n s ............................................................ 8 5 3.4 N o n l i n e a r time-variax:t ladder~-filter.............................................. 8 6 CHAPTER

4.1

4. T I M E - I N V A R I A N T LADDER-FILTERS Shift-invariance

AND 'QUASI-LINEAR' ................................................................ 9 1

of inner--produc~......... ........................... ...........

91

6.2 T i m e - i n v a r i a n t n o n l i n e a r ladder-filter ~Igorithm......................... 4.3 ' Q u a s i - l i n e a r ' l a d d e r - f i l t e r s............................................................ 4.4 E x p e r i m e n t a l e x a m p l e.....................................................................

93 98 106

CONCLUDING

REMARKS

REFERENCES

..........................................................................

109

............................................................................................. II0

APPENDIX

i .................................................................................... 116 • , * • t • • •,,• •

APPENDIX

2

.................................................................. ........... ....•........ ..... .

127

i. I N T R O D U C T I O N

(a,B,u)

Let

tract set w h o s e sets of

elements

a, a n d

will u n d e r s t a n d which

~s

denote

U

are

12du

<

oo. W e

inner-product

the metric

will b e

d(.~v,v) =

a Hilbert s p a c e .

t + Yt = wt(u) ce if for all {y}. T h e ence

will b e

and

I.q-th o r d e r

w:~

sense

~ + ]lq

on

we

for

IL2(n,B,~ )

llwll 2 = ]E { Iwl 2}

some

index-set.

sequence a

K-th

will b e

order

A

map sequen-

denoted

stochastic

the joint probability

assumed will b e

so

called

k=l,...)I~

known,

by

sequ-

distribu-

that the fiFst

to b e

,

lu2(c,B,u)

a Hilbert stochastic

by

if for

of s u b -

IL2(~ ,B,•)

completeness,

bein~

called

{x}

abs-

I.~ a-

Two

sto-

statistically e q u i v a we

will h a v e

the fol-

]E{ ytl...Ytk } = ~ {xtl...Xtk}

problem

variable

maps

inner-product

called

are

a-algebra

B. B y

the n o r m

described

in a n

bar s t ~ d i n g for complex con-

, T

will b e

f~

on

{Ytl,...,ytk } , k=l,...,K

{y}

equalities

The random

tET

IE {ytl...ytk } , k=l,...,K

lent in a w e a k lowing

introduce

llw-v]I . A s s u m i n g

will b e

tions of the subfamilies

sequences

is a

measure

will i n d u c e

Let

{y }

if its properties

chastic

u, B

]E{ ly t 12 } < co . T h a t

collection

verages

where

by

- ~{~%},

~ IL2(%B,~) t aT,

space

of C - m e a s u r a b l e

(w,v) ~ fa w 0 ) : ~ ( u ) d ~

a~d

denoted

is a probability

a collection

5n~ w ( u )

jugation. T h i s

me

a probabilit~

of prediction

being

'past' subfamily

A Yt = Ytl t-l,t-2,...

is to c o m p u t e

a fixed function

from

{y}

conditional

{yt_l,Yt_2,... } . Statin~ the p r o b l e m

within the Hilbert s p a c e

framework,

the estimate

will b e

upon

, a so-

~eome£rically, optimal

if

A y%

will b e the orthogona]

projectior~

b y %hat 'pas%'. In that case, A ^ et = Yt - Yt

error spanned

by

upon

the closed

the length

will b e

sp~ned

H et]}2 = ]E {] etl 2}

minimized.

{yt_l,Yt_2,...}

subspace

Denoting

, a~nd b y

P

by

~

of the

the s u b s p a c e

the or~hogona/

projection o-

perator taking projection o n the s u b s p a c e ~f , the o p t i m u m estimate ^ _t of Yt will b e Yt = p Yt since the coprojection et = P Yt will be orthogonal

to

~f . Consequent/y,

the p r o b l e m

of prediction

is to corn-

pure t h a t p r o j e c t i o n .

Projection

of

Yt

on

~f

determines

ting o n the 'past' of the s e q u e n c e the least-squares

sense)

a prediction

{y}, a n d yielding the 'best' (in

approximation

of that r a n d o m

notion of 'best' approximation

relies n~turally o n

between

filter

the 'ideal' prediction

a/nd the o p t i m u m The

distance

squares

approximate

between

discrepa/qcy

input collection; ver~ the p r o b l e m able

yt)

'ideal' filter

F)

into the s p a c e e.g., Victor a n d

notion

where

~

indicates

(producing fo b e

a~sser~ed

yt) , ~t ) .

the lea~t-

o v e r the entire

permits

approximation

form

mapping

a vector

one

to c o n -

(of the r a n d o m

approximation

the s p a c e

space

vari-

~f the

choice

the

multiplication

the a v e r a g e

over

a/'e taken)

see

is equivalent

corresponds

of input s e q u e n c e

filters

of excitations

in a natural way,

Addition in %hat s p a c e

of filters, a scalar

between

Fa

observation

from ~Nhich the excitations

o~ ' d i s t ~ c e '

The

of filters.

(1979).

of the filter gain. T h e

babili~ s p a c e

This

as d e v i c e s

of r e s p o n s e s Knight

be

output is just

of 'best' deterministic

in a s p a c e

parallel connection ch~ge

12 }.

of 'best' stochastic

Pilters, c o n s i d e r e d

filter

variable.

a notion of 'distance'

their ou%tputs, a v e r a g e d

]~ ~yt-y t

into the p r o b l e m

(whose

filters c a n

between

i.e.,

F

prediction

the two

filter, o p e r a -

to

to a

(i.e., of the pro-

yields naturally a

il FI-F2112 = E{IFI(') - F2(')I 2} the input probability space.

The

3

inner-product (FI,F2) We

the s p a c e

notice that w e

however, Such

on

A - E{FI(.)~52(-

might h a v e

tion s c h e m e

by

tive to the e n s e m b l e be

completed

ximation p r o b l e m ted a s

~'

to form

= 15"/F °

non-zero

filters,

of excitations.

subspace

will contain

Consequently,

excitations). T h e Hence,

problem

in the s p a c e

of the s p a c e

eus t i m e - i n d e x e d

will c o n s i -

filters (rela-

space

K~'

the stochastic

variables

c~

all

identifica-

we

of the distinguishable

of the r a n d o m

Considering the e l e m e n t s

II FII 2 = ] E { I F ] 2 } .

in the e n s e m b l e

a Hilber~ s p a c e .

in the s p a c e

ats follows

from the zero-filter in a n

of the u n d e r l y i n g

tors or, equivalently,

for s o m e

. That

o

introduced

the n o r m

0

the input e n s e m b l e .

the filter a p p r o x i m a t i o n

(1910)

K~

not distinguishable

implied

=

a~e. z e r o

a subspaee

der the quotient s p a c e

now

will i n d u c e (F,F)

will b e

filters will f o r m are

of filters will b e

) } , and

their outputs

filters w h i c h

~'

now

may

approbe

trea-

of fitters.

of filters

K~'

functionals, it follows

as

opera-

from F r e c h e t

that the r e g u l a r Volterra functional p o l y n o m i a l s

M

GM; t =

Z m=l

Vm; t

(1.1)

where

Vm; t

with

am;jl...j m

0

=

for

2

Jl

...

~

Jm

subspace

ble, a n d

the Volterra functionals

theorem,

in the s p a c e

nomials

will f o r m

is a natural genera/ization

(sufficiently regular) form

functions

a complete

(see

Volterra

of filters. H e n c e ,

stating that p o l y n o m i a l s

span

so

countable

(,.2)

m;Jl'"Jm YJ l'"YJm

Jr >t , r = l ..... m

a dense

sis). T h l s

a

(to functionals)

that the s p a c e set.

that s p a c e

a countable

a dense

(1959))

span

is sepe~ra-

complete

set

(ba-

of the W e i e r s t r ~ s

subspace

in the s p a c e

is s e p a r a b l e ,

and

of

poly-

4 Consequently, exists) c a n

e~ch

filter

be approximated

5'

(for -vvhich s u c h

representation

arbitrarily %yell b y the Vol£erra series

oo 5` =

wT h i s

means

subspaces

~

t h a t the given filter

v {V

}

whose

v

F

(i.3)

m

is a p p r o x i m a t e d in the s u b s e q u e n t

elements

are nonlinear Volterra-type filters

m

of the s u b s e q u e n t be the s u b s p a c e

degrees

of nonlinearity. W e

of linear filters; v {V2}

notice that

v {V 1 }

will

will consists of the s e c o n d -

degree

nonlinear filters, etc. C o m p u t a t i o n

series

(1.3) will therefore consist in determination of the set of multi-di-

mensional

(M-D)

impulse r e s p o n s e s ,

•

,

of the subsequer~t terms in the

see

e.g., S e h e t z e n

(1980),

rn=l,2,...}

(1.4a)

{ at;jl..4m

of the approximate

nonline~.r filter

ly, of the set of M - D

(in time-dom(~/n)

(in f r e q u e n c y - d o m ~ d n ) ,

i@ m)

, m--l,2 .... }

t

is station~y, the approximate

is r e m o v e d

remark

means

f i l t e r will r e s t r i c t

v { v 1}

o n l y . In t h a t c a s e ,

b y the I - D

caSe. If the

nonlinear filter will b e time(1.4)

provided the varia-

(following the shift-invariance of inner-product in sta-

tionary c6use). W e of a l i n e a r

(l.4b)

in the time-variant (nonsfationary)

invaria-nt, a n d will b e r e p r e s e n t e d b y the sets ble

or, equiva/ent-

transfer functions

{ At( eiSl

input s e q u e n c e

F a

members

that approximation of the 'idea/' filter

the

the

considerations

approximate

of the sets

(1.4).

linear

9"

by

to t h e s u b s p a c e

f i l t e r will b e

described

5 The sense

series-expansion

is valid only for the analytic

or Gateatax) elements

of F r e c h e t

5-'or non-analytic

F

filters, the W i e n e r - t y p e

sort of e x p a n s i o n

This

(1.3)

can

be

of the s p a c e

expansion

introduced

via

lization of the Volterra functional polynomials inner-product

considered.

(Gram-Schmidt)

orthogona-

(i.I)

relative to the

Orthogonalization

complete

implying the orthogonal

of

set

orthonormal

{WM , M=I,2,... } , being actually a n ON basis of the s p a c e and

]5"

be

G M

will yield a countable

basis

of filters

should

implied b y the input probability- space.

the Volterra-type

(in the

of filters,

decomposition

(Do

~'

=

Z

•

v

{w M}

(i.s)

M=I

where

(WM'WN)

W M

= 6M,N

of that

ON

basis

being

the

M-th

ments Wiener

cla~s. H e n c e ,

represented

F

will b e

orthogonal

=

co • M=I

M-D

filters, m u c h

class

F

orthogonal

Fourier

=

Z (F,Wm) m= 1

< co , will

series, with

of orthogonal

series

series

That

be

of nonlinear

series ks c o n v e r -

is c o n v e r g e n t

(1.6)

(F,WM)

the 'ideal' fil-

than the cl~ss

expressible

expansion

Volterra-

(1.6)

input s e q u e n c e ) .

of analylic

for a wider

in terms of c o n v e r g e n t can

b e partitioned into

(relative tO the rvl-th term)

components

M

F

2

ele-

series-expansion

of filters (functionals)

the two, mutually orthogonal,

I] F{J

kernel. In other words,

in terms

orthogonal

the

filters of the

functional Fourier

than the functions

series. T h e

with

W M

like the 'usual' Fourier

cleuss of functions

orthogonal

member

delta. E a c h

subspaee

, for w h i c h

(F,WM)

filters (for a given

Kronecker

or£hogona/

nonlinear

auny filter

represented

gent for a wider

Taylor

degree

being the

is actually the stochastic

being the genera/ized ter

will span

b y the W i e n e r - t y p e

F

which

~M,N

'

co

W m

+

Z re=M+ 1

(F,Wm)

W m

(1.7)

6 :l~he first c o m p o n e n t lhogonal

esf/mate

F~

a~ the orthogonal subset

will e x p r e s s

the o p t i m u m

of the 'idea/' filter

projection of

{WI,,.,V~M}°

The

F

second

component

the first R H S

mad M - t h

approximate

nonlinear

F

degree

, and

can

o n the s u b s p a c e

fion error. In other words, desree

M-th

nonlinear

b e interpreted

spanned

will e x p r e s s

term in (i.7)

or-

b y the

the a p p r o x i m a -

will b e the opti-

filter; i.e.,

M

F

Observing optimum

~

FM =

~ m=l

(F',Wm)

Wm

that the output of the 'idea/' filter

M-th

ximate filter

degree F~

nonlinear

estimate

ar filter will yield the o p t i m u m

M-th

de~ree

cha.stic approximation

since

the n o r m

spondin~

to the s e c o n d

computation 2M-th

of (I.8)

order

(i.e.,

It s h o u l d will c o r r e s p o n d F

by

means

RHS

sequence,

Yt

of that approximate

of the error will b e

eM; t

minimized.

will b e

that the linear p r o b l e m

the first

approximation

-

F ,~

Hence,

2M

covariances ~iven.

prediction p r o b l e m

prediction filter

to

M=I

F a In that c a s e i "

v{ W l }

with the c o v a r i a n c e

(1.9)

will b e data

only, so

in (i.7), a n d

= ( m , w I) w I

the linear estimation p r o b l e m

of

of the 'idea/" prediction filter

restricted to the s u b s p a c e

will c o r r e s p o n d

(corre-

of the o p t i m u m prediction of the

provided

of the linear a p p r o x i m a t e

nonline-

solution to the sto-

~{yjl...yjmYkl...yku } , m,u=l,...,IV[) are

to the o p t i m u m

while the

is the output of the appro-

be noted that the linear least-squares

the considerations

Consequently,

equa/s

nonlinear

term in (1.7))

solves the p r o b l e m

stochastic

that sequence

B"

A YM;t

, %re notice that computation

problem

(1,8)

associated

cond-order

sequences

cond-order

statisf/cS are sufficient in order to characterize

with s e -

IE{yJlYkl}.~ . If the sethe underly-

7 ing s e q u e n c e fi/ter F ~

completely

will b e c o m e

and non-C~aussian saHsfactory.

(the G a u s s i a n

case),

the best possible

sequences,

In that c ~ e ,

the linear approximate

filter. In c a s e

the linear estimation

estimation a c c u r a c y

of higher-order

accuracy

may

may

be improved

cing the linear estimation s c h e m e

b y the nonlinear procedure,

higher-order

to

terms

corresponding

fact that in the linear case, Fourier

kernel

M=2,3, .... T h i s

the n o r m

degree

lleM;til2 .

Orthogonal who

i(~. wl)

nonlinea/~ c a s e

llytii2 _

Gram-Schmidt

expansion

Hermi%e polynomials,

and s h o w e d

by Cameron

of the IO2-funcdono/

Ito (1951)

and

~A~a~ c o n s i d e r e d who

introduced

process.

(194:7)

using C O N

following K a c z m a r z

problem

I(F,WM ) 12

Martin. Introducing the Volterra functiona/ polynomials ries, W i e n e r

(1958)

presented

orthogono/ ON

nonlinear

development

of C O N

sys£ems

Bomret

mathemcLtical representation,

The

orthogonal

and

into the latter s e approximation

the theory

(1963)

for a functional of a stationary s e q u e n c e ,

polynomial-functiona/s.

the

and Wie-

of C a m e r o n

form, underlying

with m e m o r y .

of

proposed

his theory of the orthogonal

of a IL2-functional in the W i e n e r - s e r i e s

work

(1935),

a deeper

series

for

set of multi-variate

direct orthogona/

its relation to the Pourier-Hermite

process

The

who

Steinhaus

from

(1.10b)

to W i e n e r

or%hogono/ization

o/nd Martin

preceded

viewpoint b y

the first

(1.1oa)

of a IL2-functional is d u e

Wiener

The

by

wile h a v e

I(F,Wl) I 2 _ ... _

representation introduced

we

of the W i e n e r

ner (1938).

introducing

i 2.

the Volterra functional polynomials

orthogonal

repla-

only; i.e.,

while in the M - t h

was

by

follows from the

of the error is r e d u c e d

[[ el;tl[ 2 ,~ ii ytll2

(194:2,1958)

b e not

proposed

of the the

using £he set

'gate-functions'

approach

8 was

introduced

pansion

by Bose

(1958).

for a functional of P o i s s o n

"l~he orthogonal

development

processes

introduced

was

functional Fourier was

discussed

were

Ogura

series

by Yasui

also c o n s i d e r e d

Knight

(1979).

The

developed

systems

with m e m o r y , class

b y 5egall

the O N

ex-

using Charlier polynomials.

a/qd I<[0/lath (1976).

Stochast/c

for a functional of 'orthogonalizable' (1979).

Or£hogono/

Hida and

in m a n y

(see

process

considered

for a functional of independen£-itlcrement

Volterra a n d

ted a n d

Wiener

by

(1972)

Wiener

works,

later t e r m e d

Schetzen

Ikeda

expansions

(1965),

and

processes

of that sort

b y Victor a n d

theories h a v e

been

investiga-

yieldin~ the theory of the nonlineau ~ the nonlinear

(1980),

and

systems

of the Volterra-

the references

therein).

In t h i s work we w i s h to i n t r o d u c e efficient adgorithms for c o m p u tation of t h e n o n l i n e a r o r t h o g o n a l approximate prediction filters (1.8)

the Volterra-Wiener ved

on

class, for higher-order

a finite time-interval. W e

stochastic

will derive

recursive

sequences

of

obser-

algorithms

of the

n o n l i n e a r digital ladder-filters, in which estimation a c c u r a c y c a n be improved succesively

nonlinear

via updating

the order of the filter. W e

prediction filters with or£hogonal

sections

are introduced

updated,

without aitering the existin~ structure

sequen£1y,

we

wish

to present

filters, yielding better for higher-order -

obeying

to the filter e a c h

structures,

and

a closs

and

only 'new'

its parameters.

of nonlinear

sequences

the attractive properties

where

consider

time the order of the filter is

(tharL in the linear treatment) non-Gaussian

will

of m o d e r n

on

o~hogonal

Con-

prediction

estimation a c c u r a c y

one

hand,

o n the other

linear orthogona/

digital

filters.

The me The

orthogonal

linear ladder-filter

and~or in f r e q u e n c y - d o m a l n s , Levinson

(1947)

using

algorithms

algebraic

algorithm cox~ s e r v e

o.s a n

can

be

derived

o~nd~or geometric example

in ti-

methods.

of time-domain

9 algebraic solution in the autoregressive ( A R )

stationary- case, s e e also

Kailath (1974), If the underlying s e q u e n c e

is

(1982)), the S c h u r pa~ametrization m e t h o d

(see Lev-Ari (1982); Kailath

(1982);

L e v - A r i and Kedlath

efficient realizations generalized salve,

mials,

~chur~Levinson

Genin

Levinson

of the

and

method

and

Kamp

(1983)

to the

cetn be applied in o r d e r to obtoin

innovations/modeling

linear

methods

is closely

related

(1982))

of Deprettere

can

be

connected

Schur

Kailath (1978); Delsax~e, G e n i n

s-statlonary (see Kailath

used

theory

algorithm

and K a m p

and

Lie

of ON

(see

(1979a)).

The

(1980);

in nonstationar~

to the

(1917)

ladder-filters.

case.

Szeg~l

Dewilde, The

DelThe polyno-

Vieira

and

linear filter

problem can be equivalently considered in the s p a c e of functions squareintegrable w.r. to the spectral m e a s u r e

of the underlying s e c o n d - o r d e r sta-

tionary stochastic sequence, u n d e r the I
filters

(see

Deprettere

and

(1981); Deprettere (1981)1 Dewilde

Dewilde

(1979),

(1982)), T h e

realizations of

Dewilde

and

Dym

problems of Darlington

and c a s c a d e synthesis, a n d interpolation with positive real matrices, considered

in

Piekarski (1971,197~=); Piekarski and S a e e d

(1980; Piekarski

and Uruski (1984), are closely c o n n e c t e d to those topics, T h e S z e g ~ theor%& resulting in the approximate rea/izations of the A N ving a v e r a g e

(MR)

Levinson~d

mo-

filters a n d equivalent to the S c h u r method, can be ge-

neralized to the rational ( A R M A ) (1978); De%vilde a n d D y m

case

(see Dewilde, Vieira a n d I
(1981)), This approach, introducing the S c h u r

algorithm in geometric context, underlies the theory/ of the linear A R M A

innovations~modeling

orthogoned filters, which c a n be applied in order to

solve the lossless inverse scattering problem; the Neva/nlinna-Pick interpolation problem; the optimum stochastic modeling problem; the m a x i m u m entropy spectral approximation problem, in discrete- and/or in continuoustime c a s e s

(see Dewilde and O y m

(1981,1984), Dewilde

(1982,1984);

Deprettere (1981); Mort, Vieira, L e e a n d Kailath (1978); I-
I0 Delsarte, G e n i n

and Kemp

theel a n d D e w i l d e (1982),

(1979b); Deprettere a n d D e w i l d e

(1979); P r a b h a k a r a

Rao

a n d the references therein). T h e

and Helmond

(1979), Bul-

(1983); W i d y a

linear ladder-filter algorithms can

b e equivalently derived using projection m e t h o d in time/or in frequencydomains The

(see Delosme

and Morf

(1982);

adaptive properties of the filters c a n

M o r f a n d F'riedlander (1981); Deprettere

(1982)). T h e

Kailath (1982); D e w i l d e also be c o n s i d e r e d

Kallath (1982); Deprettere

(1982)).

( s e e Lee,

a n d Lie (1980);

orthogonal linear ladder-filters c a n b e efficiently

i m p l e m e n t e d using V L S I technology, if the standard filter sections are interpreted in terms of hyperbolic or circular rotations, a n d then realized using C O R D I C S Ahmed

processors

and Morf

Depre£tere

(see Wolder

(1982); Lev-Ari

(1959); T u s z y n s k i

(1983); D e w i l d e

(1983); Deprettere, D e w i l d e

and Udo

(1980);

(1982,1983,1984b);

(1984); Deprettere a n d

~alnand~nsing (1984)). Generalizing the linear ladder-filter algorithms, the nonlinear prediction filter p r o b l e m c a n b e stated a n d solved algebraically ( s e e a~d Dewilde method

(1983); Z a r z y c k i

(Zarzycki

(1983))

(1984a-e,1985a))

Zarzycki

or, equivalent/y, using p r o j e c • o n

yielding or£hogona/ representations

of the nonlinear prediction filters of the Volterra-VJiener class

(see

Za-

rzyck] ( 1 9 8 5 5 ) ) .

T h i s work c a n be outlined a~ follows: In c h a p t e r 2 we introduce the nonlinear least-squa/~es prediction problem algebra/caJ/y, and geometrically in the t h r e e following isomorphic v e c t o r s p a c e s : of the r e g u l a r Volterra functional polynomials; of g e n e r a l i z e d (block, multi-indexed) matrices; ~nd of g e n e r a l i z e d (block, multi-vari-

ate)

z-polynomials. W e

lem in the first s p a c e p r o b l e m of o p t i m u m

can be

that the nonlinear stochastic estimation probequivo/ently c o n s i d e r e d

as a deterministic

approximation of the set of I~-D impulse r e s p o n s e s

the filter in the s e c o n d terministic p r o b l e m

show

space. In the latter space, this will b e c o m e

of polynomial approximation of the set of M - D

of

a de-

transfer

11 functions of that filter, In chapter

3 we

present

recursive

stationary nonlinear prediction spaces

of the generalized

V~e derive nonlinear

solutions to the n o n -

usin s projection m e t h o d

mettrices~ a n d

a nonstationary (Levinson)

problem,

geometric

of the genera/ized

nonlinear ladder-filter

time-variar, t prediction

in

the

z-polynomla/s.

algorithm, yielding the

filter with the orthogonal

stru-

cture°

Chapter

tO the p r o b l e m

4 is d e v o t e d

nonlinear ladder-filter. W e

introduce

following the shif£-inva/~iance ry case. Further

complexity

quence

stocha~stic

reduction

is l o w

(in a s e n s e

In A p p e n d i x

nonlinear

b y introducin s a cl~ss

with the o p t i m u m

whose

of

'dlstance' from the G a u s s i a n

matrix

(mu/ti-indexed)

2 the 'loca/' order-update

filter sections

prediction

se-

defined).

i the generalized

lined while in A p p e n d i x ponding

to b e

in the

in the higher--order stationa-

is a c h i e v e d

associated

sequences

reduction

a time-variant nonlinear filter algorithm,

of inner-products

of 'quasi-linear' ladder-filters, higher-order

of complexity

theory

recursions

i s out-

a/nd corres-

are presented,

USAGE: Capitals

~f,Z,... will usually denote

multi-indexed used

(multi-dimensional)

for their entries, m A

spaces;

matrices

-~viU denote

2A

a 'domain'

will denote

be matrix;

of the

a rect~/ngular

etc, T h e n ,

(ro'vv) matrix %vhose

. Thus,

a m-indexed

block-entries block

]lq . T h e

IA

... M A

ore

(square)

"~ will indicate the unit-circle

will indicate

matrix with entries

mA : DmA

(or 'usual')

{ M }A ,ffi [ I A

stand for a generalized entries.

mA

A,H,...

while lo%ver cause letters will b e

a. , a n d will b e interpreted as at m a p JlO0O]m m-variate index-set DmA into the reals termed

caplta/s

DmA

wil/ b e matrix;

-~ ]R , m a p p i n g

will therefore

a row-matrix 3A

] ~vill b e

(or vector);

a generalized { M×M

matrix with multi-indexed { z:

Iz I =

be

will stand for a cu-

m - i n d e x e d matrices.

'~-

the

i }

while

block }A

will

blockm,,~

12 will b e

the

unit m-tor,ls

integrals a r e gration

are

collection

taken

m.~. =

over

sequences

gular

of the H e r g l o t z

assumed

to b e

are

absent.

will sta~nd for 'lhe s p a n will d e n o t e

will indicate index-sets, dex-set

mL

Hermi%io/% 'outer' and =

ring. ~f'he s y m b o l

yj

and,

j

eJ

(relative

-

tro2%spose;

purely

used

will be

s

product.

will d e n o t e (m-copies),

sequence;

for c o m p l e x (or

Capital

L

a 'symmetric obtained

will sto.nd for expectat/on.

i.e., a All

measure) operators;

is v

conjugation;

orthogonal) will b e

part' of the

due

of inte-

i.e., the sin-

Lebesque

projection

all

index-set).

stochastic;

direct

noted

the limits

some

to the

will indicate

will b e

hence,

(J b e i n g

mea~sure P

othe~ise

a stochastic

to b e

or K r o n e c k e r

IE

~K~

assumed

of';

... x L

. Unless

will d e n o t e

C~pi%al

symmL L×

{ y}

variables

stocha~tlc p~

® ... ® ~

the unit-circle

not indicated.

of r a n d o m

~

~

sum; ®

reserved m-varia%e

to l e x i c o g r a p h i c

for in-

orde-

2. N O N L I N E A R

PREDICTION

In this chapter filter p r o b l e m

me- and

We

responses

the M - t h

can

be

sequences

APPROACH

nonlinear observed

and

predict/on on

a finite

~eometrica~/ly,

in ti-

that the nonlinear lee~st-squa/es

equivalently

approximation

considered

ms:

of the set of M - D

impulse

of gene~'a//zed

(block,

matrices;

of deterministic

approximation

functions of th~£ filter, in %he s p a c e

of the set of M - D

of genera/ized

transfer

(block, multi-varia-

z-polyno mio/s.

2.1 Hi~her-order

Let

[y}

stochastic

denote

sequence

will b e

sequences

a scalar,

cond--oFder Silbert s e q u e n c e This

UNI~'IED

degree

of the nonlinear filter, in the s p a c e

the p r o b l e m

re)

A

adgebr~dcally

showing

of deterministic

multi-indexed) -

state the p r o b l e m

estlmatLon p r o b l e m

the p r o b l e m

PROBLEM:

order stochastic

in frequency-domal/ns,

stoch~£ic -

%re introduce

for 2 M - t h

Ume-inteFvaL

FILTER

zero-mean

observed

represented

on

and

purely

stochastic,

the time-interva/

b y the ra~ndom variables

L = [0,oo) y_j

col

[y_~ ]j E b

.

contai-

n e d in the followin~ column-matrix

Y =

se-

(2.1)

14 The

following t w o - i n d e x e d

H -- I E { Y

whose

(second-order)

® "}}

covariance

meLtrLx is

(2.2)

,,, [ h j , k ] j , k ¢ L

entries are given b y

h j , k = ]E{ Y-j'Y-k }

Given

the c o v a r i a n c e

d~tet (2.3), w e

(2.3)

c a n introduce the following 2-D tq'ou-

tier transform pair

W ( e i O, e i ~ ) =

1

hi, k =

where

W ( e i8 ,eI ~J)

co E j=-oo

5 d 8 [d~

is the 2-D

n=0,...,N

axed let

the finite time-interva/ (offset) a n d then

X

n

hj,k

e - i J O eik~

we

(Z.4b)

{y}, satisfying

. - '/V(e*" ,e'. e)

(2.4c)

x=0,...,N-n . If the s e q u e n c e

L x _A_ { x,...,x+n } , w h e r e n

x

is o b s e r v e d

is ~ reference

will b e repleced b y the submattrix

will associate

covariaz~c e

with

{y}

the

following

(2.5a)

Hermitian,

positive-definite

matrix

Hx E{yX n = n

~ ~x} n

(2.5b) =

[hj,k]

(],k) £ b x x L x n n

on

point

indicates the rax~ge (or 'order' of past observ~ttions),

y X = col [ y_j]

and

(2.4a)

e ije W ( e i e , e i ~) e-ik a

spectral function of

W ( e i e,e i ~)

Let

oo E k=--oo

15 Assuming we

that the s e q u e n c e

will obtain for

is stationary

{r

}

second-order

sense),

j,k~...,-l,0,1,...

hi,k = hj+l,k+ 1 - h k _ j = r

where

(in a w e a k

is t h e

cova~ciance

,

P

sequence.

p=k-j

From

(2.6a)

(2.6a)

it follows tha£

P

Hx = H x+l~ n n

regardless

rely stochastic)

Now

can

noHce

consider

j EL

and

introduce

of %he rat*don v ~ i ~ b l e s

(j,k) c

LO

block-entries

display the Toeplitz proper-

m~trix of the s e c o n d - o r d e r (2.4)

~sociate

stationary

will b e replaced (since

se-

b y %he I-D

the s e q u e n c e

is p u -

sequence.

a 2M-th

order stochastic

sequence

{y}, sa-

m=l,0..,M

ly_jl 2m }

<

(2.7)

oo

the following M - b l o c k a n d £heir products

[my]

(column),

(see

m = 1 .....M

e/~e the followin~ m - i n d e x e d

with t h a t s e q u e n c e

m-indexed

also A p p e n d i x

matrix

i)

(2.8a)

matrices

my = [y_ji...y_j m ] Jl ..... Jm E We con

(2.6b)

×L O n n

%/qat (2.6)

the rela£ions

{ M }y ~. col

whose

]

rela~in~ the spectra/ d e n s i ~

~{

we

hk-j

£o the covariaunce

let u s

#/sfying foF

- [

cova~i~ce

{y}. In tha£ c ~ e ,

Fourier transforms,

Then

n

of the x-shift. W e

ty of £he t w o - i n d e x e d quence

T

the generalized,

(2.8b)

(MxM)-block

(squ-

16 are), ( m + u ) - i n d e x e d c o v a r i a n c e

matrix

N

iB{{M}y ® {M}y}

{MXM} H .

(where

®

Appendix

m~u H

=

=

[meuH]

(2.9a)

m,u=l,...,M

indicates 'outer' product of generm/ized matrices, s e e also I), w h o s e

~{r~®

(m+u)-indexed

block-entries are e x p r e s s e d

u.2} =

. [hj1""Jmkl"'ku]

as

(2.9b)

.

]l,...,]m,kl,...,k6 b

with

h

Each

-

submatrix

(2.9b)

c a n therefore b e interpreted as the (m+u)-th or-

der c o v a r i a n c e

matrix of the 2 M - t h order s e q u e n c e

that

will r e d u c e

{MXM}H

second-order

(2.9c)

-

JI'""jm k i' '"ku ~ ~D { y_j 1 "".y-jm y-k 1"""Y-ku }

sequence

quent p a r a g r a p h s

to the t w o - i n d e x e d

we

if

assume

M-I.

[ y }. Let u s

matrix

We

will

H

prediction problem, m u c h

like the t w o - i n d e x e d

the linear problem. W e then f o r

--

~ll

=

stands for the s u m m a t i o n

pariitioning the (re+u) pairs, s e e

e.g., W i e n e r

the G a u s s i a n

i 0'iH of the

least-squares

matrix (2.2) is associated

sequence

random

}

-k ?

0

symbol

matrix

is

m,u=l,...M

{Y-J1"" " y - j Z - k l " ' ' y - k )

where

the s u b s e -

notice that if the hitcher-order s e q u e n c e

I~llE{y_j --

in

of this chapter that the generedized c o v a r i a n c e

with the M-th d e g r e e nonlinear

Gaussian,

(2.2) of the

show

(2.9) will b e associated

with

observe

, (re+u)

even

, (re+u)

odd

S

{ 2.1o)

o v e r a H distinct w a y s

variables into products

(1958). tl~his m e a n s

of a v e r a g e s

of of

that the 'information' about

is contoined only the t w o - i n d e x e d

submatrix

{MxM}I-I , as the entries of all higher-order (i.e., even-

17 order)

covariance

matrices

meuH

are expressible in terms of the

entries of the left-upper submatrix

101H

Consequently,

to the p r o b l e m of o p t i m u m pr.ediction

a nonlinear a p p r o a c h

of a 2 M - t h order stochastic s e q u e n c e cy if the s e q u e n c e

is n o n - G a u s s i ~ n .

(in the s e n s e

of (2.10)).

will yield better estimation a c c u r a Otherv¢ise, ~he higher-order covari-

a n c e matrices will not cont~dn o.ny 'new' information about the s e q u e n c e , and the linear prediction filter (associated with

l e IH)

will b e c o m e

best possible filter (in the least-squares s e n s e )

for a O a u s s i a n

the

sequence.

Introducing a s h o r t h a n d notation

e

we

can

tom

m

(e

tel ....,eiota )

consider for

[email protected](eiOm

m,ual,...,M

i u ,e

;

co )

=

~

the M - D

co E

j m,= - o o

kl=.-oo

-ij 1 01

1

}l'"Jmkl"'ku

(2z) m + u

IdOl...

i~ I

(e

E

e

h.

.~-

co ..-

j1=-CO

• u

e *~

(2.11)

co ...

E

h.

ku~--oo

-iJn~ m ...e

Id8 m /d%...

.

Jl"'Jmkl'"ku

ikl~ 1 e

• u

•

iWu)

P'ourier transform pair

m e U % v ( e ] em,e ~(~ ) e

The

.....e

iku~ u ,..e

Id~u

-ikl~

e

(2.12a)

iJlSl

i o..e

...e

-ku~ u

~Jmem

(2.12b)

U

m @UVV( e i Om , e 1 0 J )

c a n be interpreted as the ( m + u ) - d i m e n s i o n a /

s p e c t r a / function of the 2M-th o r d e r s e q u e n c e , satisfying

m ~ ) U W ( e i e m , e i u ) ,,~ m® u~V(e i ~u , e ibm)

(2,13)

18 "Dhen w e

c~n

introduce

the f o | / o ~ n g

MxMIN =

[

containing precisely as the generalized second-order

being

Let u s

.

m

information

covaria.nce

matrix

=

for

. We

to M--i)

[ielw(eiei

.....M

(2.3.4)

about the 2 M - t h

we

order s e q u e n c e

notice that for a would

have

' ,e~i)]

actue]/y the two-dimensiona/

introduce

m,u=l

{MxM}H

(corresponding

i×l~v

1 elM/

u'VV(elO "ei~U)]

m ~

the s a m e

sequence

spectral matrix

(2.15)

spectral function

n=0,...,N , x=0...,N-n

and

(2.4a).

m=l,...,M

the

following multi-varia~e index-sets

mLx

n

where

Lx =

=h

{x,...,x+n} . B y

m}

n xn

{(Jl .....Jm )1 ire

the 'symmetric

part'

symmL x

n

set

(2.i6a)

r = 1 .....

of the index-

n

mLx

we

will u n d e r s t a n d

n

•

~ x

,

.

symmLXn "~ { ( J l " ' " J m ) : J1 E ~ n ; J r " J r - 1 ..... x + n , r = 2 ..... m }

We

notice that aunti-lexicographic

ordering

can

be

(2.i~b)

equivadently introduced

in (2.16b). If the 2 M - t h terval

Lx n

me

order s e q u e n c e

{ M }y

(2.8)

will b e

{y}

is o b s e r v e d

replaced

on

a finite time-in-

by

~

(M},~

n

.

c o l [ my:~] n

m=l,...,M

(2.ira)

19 whose

block-entries

n~n

where

x

ance

matrix

=" [ Y_jI.*oY_jm ]

is s o m e

(M~M

are e x p r e s s e d

{MxM]

H

(2.9)

n

m~uHx = E(myX ® n n

will h a v e

for

{y}

= [ m®%:. ]

® (M}~}

n

(2.za~) m,u,,l,...,M

U[/X} .. n

]

(Jl ..... Jm'kl ..... % ) ( symmLX × s y m m L X

is stationary

(in a w e a k

2M-th

order

(2.zab)

sense),

we

jl,...,Jm,kl,...,k u ,= ...,-1,0,1,...

hjl+l...jm+l,kl+i...ku+ I = h.jI "'']'mkl'''ku

so

covari-

given b y

] I.-4 mkl-.-ku

that

the generalized

will b e c o m e

n

with the block-entries

Assuming

(2.17b) (Jl .....Jm ) ¢ s y m m L X

reference point. Consequent/y,

}H* = E{ { M } x x

-[h

as

(2.19a)

that {IVIxM}HX = { M X M } H X + I n n

regardless

of the x-shift. T h u s ,

(2.19)

__A { M × M } T

(2.19b) n

display the Toeplitz structure of

the generetlized cov~rit).nce matrix of the stationary 2M-th order s e q u e n c e . In that case, the block-Hankel, matrix

{MxMJ'H

w i n b e replaced

plitz, positive-definite matrix (2.*9) will r e d u c e

multi-indexed Hermitian, positive-definite

to (2.6).

{M×M

b y the block-Har~kel, }T. W e

multi-indexed "Doe-

a/so notice that a s s u m i n g

M=I,

20

2.2 N o n l i n e a r l e a s t - s q u a r e s

Let u s observed random way,

o~sume

prediction:

approach

that the 2 M - t h o r d e r stochastic s e q u e n c e

o n the time-interval

variables

Algebro#c

L~

and, h e n c e ,

y_i,...,y N M^

YN;o

nonlinear, N-th o r d e r predictor

, we

of

YO

is

is r e p r e s e n t e d b y the

yo,y_l,...,y_N . In o r d e r to predict

given its N-th o r d e r post

{y}

YO

in a nonlineo.r

define the M-t/q d e g r e e a s the M - t h

d e g r e e Vol-

terra functional polynomial

M^

We

notice

assume

M

N

N

~

~

~

that

(2.20)

will

A =

YN;o

N

m-1 J l - 1 j2=,jl

M=I

""

~

e

jm=Jm_l

reduce

to

(2.20)

"" " N d f " J m Y-Jl"'Y'Jm

the

linear

s o that the linear a p p r o a c h

nonlinear treatment of the problem. T h e

N-th

order

estimate

if we

will b e the first step in the

M-th

degree

nonlinear, N-th o r d e r

prediction e r r o r w i l l then b e

M~

A

M^

N;o = Y o

a n d the m e a n - s q u a r e

M

error

(2.21)

IMsN;o

con be

M

eN; o = aN;oY o +

(2.21)

YN;o

error, a s s o c i a t e d

e;N

The

-

~ m=1

}

r e w r i t t e n in a

with

(2.21), will b e

>

expressed

0

(2.22)

r e n o r m a l i z e d form a s

N N N Z ~ ... ~ aN;il...j m y_jl...y_j m J l = l j2---jl jm=~Jm_l

(2.23a)

where

aN; o =

Md~ql

;

aN;Jl.,,jm

In o r d e r to solve the M - t h

= -

degree

as

N;JI...j m

nonlinear, N - t h o r d e r leo.st-squ~res

21 prediction problem, w e

wish to c h o o s e

the coefficients

such w a y that the m e a n - s q u a r e

error (2.22)

achieved if for

u i (kl,...,ku) ¢ s y m L N _ 1

u=i,...,M

and

aN;Jl...j m

in a

is minimized. This will

be

OIVIR c~N

=

(2.24~)

o

8 aN;kf..ku

Conditions

will imply

(2.24a) M

aN;oho,kl...k u +

Moreover,

E m=l jl=l

N ""

j m=Jm_i

aN;jl...j m h.Jl...Jmkl..*k . u

-

0

(2.2,~b)

get

we

M

5q;ohoo +

N

E

N

N

X E ... E h. Md N m = l jl=l jmmJm_l aN;Jl'"Jm Jl""Jm '° =

G e n e r a l i z i n g t h e notion of 'matrix', we c a n i n t r o d u c e t h e following M-block ( r o w ) ,

m - i n d e x e d coefficient-matrix

{M}AN = [ m A N

whoso

each m-indexed

block-entry

from the m-variate index-set

m%

DmAN

con be considered

D mA N

=

numbers

(2.2~b)

can talk e~bout the 'doma/n' of the m-indexed

that m - v a r i a t e index-set, and

~s a m a p

into the real (or complex)

m%. D m%, -~

Hence, w e

(2.25a)

] m = l .....M

matrix es being

we will have

symmLNI-I

if

m=l

if

m=2,...,M

(2.25c)

22 Consequently,

we write

for

m-2,,,,,M

mAN .. [ °N;Jl""Jm ] and for

m

(Jl .....jm)

e

(2.25d)

i

sym LN_1

m=l

1AN -', [ aN;ill JlC LN Thus,

the M-block,

considered

m-indexed

as a m a p

(2.25e)

coefficient-matrix

{M ~AN

(2.25a)

can be

from the vector of simple d o m a i n s

D{M} %

[DmAN] m~,l,00,,M

=

(2.26a)

into the reat (or complex) numbers

{ M}AN. D{M }AN + ~

see

a/so

Appendix

Using

where

1_yo N

1 mYN_ 1 can

i.

(2.25),(2.26)

{M}YN

=

a n d introducing

M

col [ 1 Y No

is e x p r e s s e d

( m - 2 .....M )

(2.26b)

by

(2.17b)

are given b y

rewrite the error (2.23)

with

(2.17b)

re=l, x = 0 with

in a generalized

MeN; ° = {M }AN. {M}YN

where

•

l

YN-1 ]

"'"

x=l

(2.~T~)

and and

n=N

while

n=N-:i , w e

matrix form a.s follows

(2.27b)

indicates product of generalized matrices ( s e e Appendix 1),

23

Introducing the following matrix [ M~ N

where

Md N

i

(2.28a)

ON_ 1 ] { M}~I

is e x p r e s s e d b y ( 2 . 2 2 ) , and

M-block ( r o w ) ,

1

ON-1

u

block-entries

u

=

1 ON_ I

[Uol_1 ]

are u-indexed

(see also A p p e n d i x

mad equat/ons'

in a generalized

rzycki and Dewilde

denotes the

(2.28b)

u=i,...,M

i i D ON_ l - symuLN_l (2.24)

UN-I

zero-matrix

u-indexed

{M}

whose

{M}

zero-mafrices i), w e

with domains

can rewrite the 'nor-

mafrix form a~s follows (see Za-

(1983a))

t =N where {M}Y N

{ M×M}H N

is e x p r e s s e d b y (2.18)

( 2 . 2 7 a ) , Rewriting

{ M x M }HN

(2.29)

with

{M}YXn r e p l a c e d b y

in a generalized

block-column

form

{MXM}HN.,

[{M}x%

] U-1 .....M

(2,30a)

where for u-2,...,M

{M}xUHN .

[ {M} 1 1 HN;kl"'ku] (kl...,ku)¢ myrau LN_

(2.30b)

and {M}×XHN = [ {M}HN;k 1 ]

0 kI ¢ LN

(2,30C)

with { M}H N;kl...ku - [ h.]l...Jmkl...ku . ] (Jl ..... Jm ) ~ D{M}YN

(2.30d)

24 we

c a n express the 'normal equations' (2.29), for

u=l,...,M

and

for

i

( k 1 ..... ku) ~ s y m u L N _ l , a s follows

(M}^

~'4"

{M) H

{M }AN.( M }HN;o

These

(2.31a)

N;kl..,k u = 0

=

MdN

(2.31b)

are the 'normal equations' associated with the M-th degree nonli-

near, N-th order prediction problem. W e

can observe

that they will reduce

to the 'normal equations' corresponding to the linear problem, if Equations

(2.31)

M~I.

can be recurslvely solved via the nonlinear Le-

vinson algorithm, presented in Zarzycki and Dewilde (1983). q)his algorithm computes

(1983a); Z~vzycki

the coefficients of the optimum nonlinear

approximate prediction ladder-filter, a n d implies the generalized C h o l e s k y factoriza~on of the block, multi-lndexed covariance matrix

{ M×M

}HN ,

generalizing the linear case, considered in meprettere a n d Lie (1980).

2.3 Nonlinear ! e ~ t - s q u a r e s

prediction: Geometric

approach

In the s u b s e q u e n t sections of this paragraph w e

will introduce the

nonlinear prediction problem in the following spaces: of the regular Volterra function~/ poIy/qomials; of the generalized coefficient-matrices;

~nd of

generalized z-polynomials.

2.3.1 S p a c e

o f tlae r e q u l a r V o l t e r r a f u n c t i o n a l . p o l y n o m i a l s

G i v e n the M - b l o c k , m - i n d e x e d matrix of the r a n d o m veu~iables [M} YNI-I

(expressed

by

(2.17) with

x=l

and

n=N-l) , ea~d a s s u m i n g

25 that the entries are linearly independent,

{ M}__I

A

Each

wil/

element from that s p a c e

M

where

<0N

...

{

M

be

the s p a c e

(2.32)

expressed

as

i {M}FN'{M} YN-I

(2.33a)

coefficient-matrix

. [mFN]

}PN

consider

--N-1 }

M-block (row), m - i n d e x e d

the

can

M}yI

{

X N _ 1 =" V {

we

is given b y

(2.33b)

m = l .....M

with

m F N = [ fN;Jl""Jm ] (Jl ..... ]m )

From

Appendix

i it follows that (2.33)

M~

.=

observe

that

M

m

Z; m=,1 M

can

be

(2.33c)

m 1

¢ sym

LN_ 1

rewritten as

:I. • FN " Y N - I

Thus, w e

can

functional

p o l y n o m i a l o f t h e form ( 1 . 2 ) . W e a l s o n o t i c e t h a t a s s u m i n g

we

obtain the s p a c e

diction problem.

M~ N f

(where

replaced

duct o n

{i}~_

Given

G

[ M }yiN_ I

is the regular M - t h

1 , associated

degree

and

Will

be

VoIterra

wihh the lineeuc N-th

two Volterra functional polynomials

the l a t t e r element by

0N

(2.33d)

expressed

g , respectively), w e

by

can

M

(2.33)

introduce

order pre-

~N

and

with

F" a n d

inner-pro-

~s

(M~oN,

M~N)~f

=A IE{M~ N

M-

~N }

M:I

(2.34a)

26 inducing the n o r m

ii M

2.3.2 S p a c e

1,,,jl 2

lowing M-block,

jl...j m

where

m-indexed

=A [ 10N_ll "'"

U O Ni_ I

each

m-indexed

(2.34b)

of .qenera/ized coefficient-matrices

Let u s introduce for

{M}I.

IM ~PNj 2}

E {

=

m=l,...,M

(jl,...,jm) ¢ s y m

m

i

LN_ I

m-l_l UN_ I

.

ml.

M ON_ll ]

m+l_lON_l

jl...jm

is the u-indexed

(2.35a)

"'"

zero-matrix of the form

(2.28b); the

is given b y

J1-0OJm

mi.

(2.35b)

Jf"Jm

with

6

t h e fol-

matrices

ml.

matrix

and

- [ 6 ]'l " 4 m' ; k l ' " k m

] (kl,...,km)

1 1 E s y m m LN_

denoting the m-dimensional

Kronecker

delta

Jl...Jm;kf..km .I 1

if

kl=Jl ,..., km=J m

(2,,35c) Jl"4m;kl'"km

This

means

thai

0

ml.

Jl""Jm

unit-entry with coordinates le ~II remaining

otherwise

~ad, hence,

{ M}I.

contains only o n e

J1"" "jm

(jl,...,jm) , placed in the m - l n d e x e d

entries ore equat[ to zero. Now let for

=

[ { M )l.

] Jl""Jm

block, whi-

m=l,...,M

m 1

(2.36a)

(,i 1 ..... jm ) ~ sym L~_ 1

and, fina/ly, let {M} i = [ m 1 ] IN-I I N - I m=l,...,M

(2.36b)

27 We

can observe

that the entries

{ M }I.jl...jm

ly independent, a n d they C a n b e c o n s i d e r e d

of the

{IV[}_ i_ I IN

are linear-

~s the 'natural' b ~ i s

of

the

space

{M}_I | N - I a v {.{M}IX N-X} whose

{M

elements will b e e x p r e s s e d

}5,N

where space

M

N

=

[mFN] m = l . . . , M

E

Given

by

m-indexed

(2.33c). T h i s

and

means

the generalized c o v a r i a n c e matrix

with

x=l

and

n=N-l)

N

is e x p r e s s e d

that

{M} i

IIN - I

by

{ MxM

}HN_II

{M}I_IN _ I

(expressed

as

A {M}5,N.{MXM}HI N-I .{I~@GN (2.381

with

5'

and

f

Hermitian a n d posltive-definite,

we

can check

({M}FN'{M }GN)Ir = ({M}GN'{M}FN)~~

(2.39a)

replaced by

g , respectively. Recalling that the genera/ized c o v a r i a n c e

multi-indexed

is the

of the 2 M - t h order stochastic s e q u e n c e

define the w e i g h t e d inner-product o n

{M}G

jl...jm

coefficient-matrices.

( {M}FN' {M) GN) ][ where

{M} I

(2.38)

is e x p r e s s e d

of M-block,

{y}, w e

N

E ... ~ fNijl...j m m = l J1=1 j 2 = J l jm,,,,,Jm_l

mF N

by (2.18)

N

a.s

=

E

(2.37)

G

matrix is

that

(2.39b)

(a.{M}AN + b.{M}BN,{M}I~N)I = a({M}AN,{M}FN)II+ b({M}BN,{M}FN) (2.39c)

28

({M}FN {M}FN)Ir = {M}FN"{NIXM}H.NI_I {M}FN z 0 with equality in (2.39d)

If{M}

We

con

II

observe

]if

{M}PN

=

{M} ONi -I " T h e

norm

(2.agd) is then

2 = ({M {M} FNII I }FN, FN)]I

(2.39e)

that

{M}I

Ii 2 J J.'"Jm R

{M}1.

{MxM}. 1 .{M}~ ji...j m" I-IN-i jl...jm

,.,

=

h..

0

>

(2.~0a)

JiJi...JmJm We

also notice that the 'natural' basis

amd

m

(jl,...,Jm,kl,..,,ku) c s y m

{M}I

({M}I

]l...Jm' if the underlying accordance

as we h a v e

basis of t h a t space,

general, an orthogonal

2M-th

{M halN _ 1

of the s p a c e

for

is not, in

m,u=l,...,M

I m i LN_ 1 x sym LN_ 1

.

kl,,,k u) l

h.

.

(2.40b)

]l...]mkl--.ku

order s e q u e n c e

was

Gaussian,

we

would

have

(in

with (2.10) )

m eUHl~-i

Consequently, f o r

(re+u)

({M}I

= m eu~i~N_i

'

(m+u)

odd

(2.41)

odd,

{M}I

]l-..Jm'

kl...k u

)~

0

(2.~2~)

=

and we would have the 'block' orthogonality of the 'natura/' basis in the Gaussian ca~e. Moreover, if the sequence was white, we would write ({M}I']I...jm,{M}Ikl...km)]l = EFI~ . Jr,ks

(2.42b)

29 in

accordance with (2.10). In genera/,

if

({M) FN,{M}GN)I[ = 0

we

shall s a y

that two M - b l o c k

(2.43)

(row), m - i n d e x e d

or£hogonal with respect to the genero/ized matrix of the underlying quence

was

stationary

coefficient-matrices

block, multi-indexed covariance

2M-th order stochastic s e q u e n c e (in a w e a k

2M-th

are

sense),

order

indexed Toeplitz covarlance matrix {MxM}TN_I

{ y }. If the sethe block, multi-

would replace

{ MxM }H

in ( 2 . 3 9 ) , in ~ccordance with (2.Z9).

2.3.3 S p a c e

of

.qenerallzed z-polynomials

Let u s introduce for

m..l .....M

and

(jl,...,jm) e s y m

rn 1~

LN_ 1

the

following row-matrices

Jl [6m,p'Zl""Zm

M zjf..jm

"

=

and

...

[0

0

Jm ] p=l, .... M

Jl Jm zl ...zm

0

...

0]

(2.43a)

let

mXM_l

=

ZN-I

[~.~" o.A~.~ ~'-''/

M

[

zJf"Jm]

(Jl .....Jm ) ¢ s y m

m

1

LN_ 1

so Lhat we c a n introduce

{M)

Then

we

can

1 m×M 1 ZN-1 = [ ZN-1]

consider

the

following

{M}_I A LN_ I =

m=l,...,M

space

{M}_I v {

(2.43c)

~N-I

)

(2.4~)

3O

whose

elements

M#

will be

N

M

.

N

E

j

m-.1

"

expressed

~

M E [0 m-1

as

N N z. . ~ "'" E, fN;jl..j m M 1 12 Jl Jm='Jm-1 ll"'Jm

... 0

¢N(Zi,..0,Zm)

0

=

... 0 ]

(2.4:5a)

= [~N(Zl ..... Zm)] m=:l.,...,M where

~N(Zl, ....Zm)

Given

is the s p a c e

{M}_I ZN_ 1 two elements

is given b y

the foUowing

M ~N

(2.45)

with

of the M-block,

arld F

MS N

and

f

vely), a n d using the spectra/ matrix sequence

m-variate

z-polynomial

N N N Jl Jm E E "' Z z i...z j l = : l j2=1 jm,~Jm.1 fN;jl...j m m

~N(Z"l ..... Zm) "

Thus,

denotes

{y}, w e

(M NMVN)~.. =

introduce

(2.14)

inner-product

.[d0{M} :d~{M} M~

A

M

M

m=l

u=l

E

=

E

m-variale

from that s p a c e

replaced

N

(2.45b)

by

G

z-polynomials. (where

and

g , respecti-

of the underlying on

{M}zI_I

the latter

2M-th

order

as follows

M x M w . M" "~N

.l'dOm [ d ~ u ~ N ( z l ..... z m)

i

(2~) m + u

m ®'~'W(e 'e ,e" ~ ) ~'N (wl ..... w )

1

(2o46a)

iSr , r~--l,...jm zr=e i ~s Ws=e

where

we

~dOm

used

=

the following shorthand

fd81

... f d O m

;

, s=l,...,u

notation

~d~ u =

~d~j....

;doJu

(2.46b)

31

Using

(2.13),

(M¢ N,

M

~N)Z

we

can

.

M ~

show

,hal

M ~ u=l

m-1

1 (2 7) m + u U

:dO m f d u

~ N ( W l ..... W u)

171

m®'~V(e"~ 'eie )~ ~N(=I ..... ~m)

ie Z

r

~e

$ r~it...tm

r W

s

=~e

, Sm1~...tu

(2.46C)

- (M~N ,M ¢N)~.

We

a/so

have

(a-M FN + b,MAN,M~N) Z - a(Mr N , M ~ N ) X + b(MAN,M¢ N) Z

(2.46d)

and, finally,

(M~ N , M ¢ N ) Z, " II M ~ N 112

If

(M

riate) MXMw

REMARK:

¢N,M ,N)Z

= 0 , we

z-polynomials (2.14)

aye

of the

shall s a y

or,hog.hal

2M-th

Let

"~D: {e io }

mnJD

we

m~

.~ ~ e

order

denote

sequence,

(M-block,

to the spectral

on

matrix

the unit ( m + u ) - t o r u s .

will h a v e ... e~jD : {e i8 )® ... ®

{ e ie}

= { e

meu~

we

ie 1

iOm

...e

m

for the unit ( m + u ) - t o r u s

= mT

®u~/D:

m-va-

the unit-circle. "/}hen for the unit m - t o r u s

i OI m @u T

(2.,~6e)

generalized

with r e s p e c t

In

while

that t w o

>-- 0

{e

i 8m ...e } ® ie I

•. { e

i 8i {e

i0 m ...e

i~ 1 e

get

i Ou ...e } =

i~tt ...e

}

}

32

2.3.4 Isometries

We isomorphic

observe

can

spaces;

{M }[i {M "}.y1 N-I N-1 '

thai:

i.e., w e

and

{M

}zl N-1

i~.re

have

y

.

i~.y

-Jl

.

-3m

d {M} 1.

%

(2.47~)

Mz.

~

Jl,,,Jm

Jl.,4m

so that M

q0

N

(2.47b) { M}F N

We

wish to s h o w

4-~

M@ N

that the following isometries hold

(Y_jI'°°Y.jm'Y-kl""Y-ku ) Y z/

( {M}I.

%

~ M}

jl...jm

(2.47c)

(M ikl...k u) II

]3 ...jm

Mzkl...k u) Z

s o that ( M //

(~M} FN( M} GN)~

~N' M ~ N) Y

(2.eva)

%

(M ~N'

M~,~)Z

33 |n o r d e r to do that, let u s o b s e r v e

(following ( 2 . 9 c ) , ( 2 . 3 4 a )

and (2.40b))

that

(Y-jI"'Y-jm ' Y - k l " ' Y - k u

)y

: ~{ Y_jl-..Y_jmY_kl...Y_ku

}

:

= hjl...Jmkl...ku "

= ({M}I.

. ,{M}i

)

(2.48a)

k l , . . k u 1/

]1...$m F r o m (2.lib) and (2.46a) we obtain

Mz (

M

ji...jm'

1

z k l . . . k u) Z :

I dO m

(2 7) m+u

Jl

id u

Jm

zl ""Zm

-k × m®%v(~i ~'ei~) ~I-k I ""~u uI

IZl~:eiOr |ws=e

,

r=1,...,m

• sl: ll...|u

Jl...Jmkl...ku

= ({M} :

{M}

(y_jl...y_jm, y _ k i . . . Y _ k u ) y

s o that ( 2 . 4 7 c )

(Mq°N'M~N)~"

ij1"" j m'

is valid. F r o m ( 2 . 3 ~ a ) , ( a . 3 9 a )

M ]E{ M (PN

~N } =

M

E Z m=l u = l

zk z...k u) ~

and (2.48a)

m_

re@u_ ~'N"

2.

(2.~ 8b)

it follows that

m ~_

MN-I" ~'*N "

" { M }FN'{M×M} t41--N-t'{M }GN~ "~

( {M} F N , {M} GN);I

(2.49a)

34

Finedly, using

(2.12b),(2.3~a),(2.~Sb),(2.46eL)

M (MeN

Z ~ViVN) -

.

×

~

mml

M

$

U-i

(2~) m + u

fdO m

"®~w(Je~,j~)

(2.48b), we o b t a i n

and

/d~u C N ( Z l .....z m) ×

,F-N(~I,...,,%) z]c=e

~ M

M

N

T

T ... ~

m - I u-I

1

x

N

Jl-i

I d8 m

=

M

~

~

m - i u=l

N

kl=l

-iku~ u ...e

N

~

j1=l

...

~

m e U w ( e i e m ,e

u)

;kl...ku

N

jm=Jm_l

~ s=i~...tu

~ .~f~r;jl...jm x ku=ku_ 1

eX]XSl...e iJm e m

-ikl~ I

s

i r ~ ip.••pm

N

~ ...

Id u

e M

---e

N

j m-Jm_i

i8r

N

~

kl--i

...

~

ku=ku_ 1

~.f~,;jl...jm

h.Jl...Jmkl---ku . &;ki...k u M M 2 r m = l u=1

mF. N"

(M}F,N {M×M} •

-,

This proves

m • u.. _i

~-i"

_i

H N _ I,

(M~N,M~N)~f

=

U~

GN

{M}~N

-

=

({M}FN,{M}GN)I

(2.49b)

( 2°~7d).

Prom

(2.47) it follows t h ~ the M-th degree nonlinear, N-th o r d e r {M }_.I stochastic estlma£ion problem considered in the space X N _ l , c ~ be equivalently interpreted ~s the deterministic

problem in the space

{ M } KI~-_

35 of ~eneralized matrices, or as the de£ermlrdst/c p r o b l e m in the s p a c e {M}_IzN_I

of generaJ/zed z-polynomieds. "l~he i s o m o r p h i s m

nera/ization of the i s o m o r p h i s m tima~on p r o b l e m ruing

M=I), Now

c o n s i d e r e d in the linear least-squares es-

the K o l m o s o r o v

isomorphism

are in a position to s h o w

~/m~t[on p r o b l e m minist/c)

is a ge-

( w h i c h c a n b e immediately obtained from (2,47), a~ssu-

and becomes we

(2.47)

optimum

in stationoA-y c~se.

that the nonlinear s t o c h ~ U c

c o n b e eqtdvo/ently c o n s i d e r e d

es-

a s the p r o b l e m of (deter--

approximation of the set of M - D

impulse r e s p o n s e s

and~

or tra/qsfer functions of the nonlinear prediction filter.

2.3.5 StochaStic nonlinear . e s t i m a U o n

Given

{M}_X N1 _ 1

the space

(2.32),

let us

introduce

{ M } ~ N - v{ Yo . {M}yN_,I }

nonlinear, N-t/~ order lea.st-squares predictor

The

M-th degree

Yo

will b e e x p r e s s e d

{ M }p

l Y;N-I

o n the s u b s p a c e

Yo

(W0 Y N*- I

E

{ M ) _X Ni- I

" We

notice t h ~

N-th order prediction error, c o r r e s p o n d i n g

A {M}_l 1 N;o = ~Y;N-I

and can

of

(2.5,a)

is the ortho~onml projection operator, ta/~n~ projection M AY N ; o

linear estimate (2.20), introduced alsebralcallyo T h e

M E

M^ YN;o

as

MgN~o "A {M} P Y ; NI- I

where

(2.50)

be expressed

Yo

-- Y o -

to

M^ YN;o

is precisely the n o n M-hh desree

nonlinear,

M^ Y N ; o ' is given b y

{M}yl ±

in ~ renorma/ized form ~s

N-I

(2.51b)

36

M eN;o '= M C N ; o where

{M}AN

vely. W e

and

It M ¢N;o I1~

{M}YN

{ M}AN'{M }YN

=

(2.51o)

a r e given by (2.25) and (2.27a), r e s p e c t i -

notice that this is precisely fhe nonlinear normalized

prediction

error, introduced algebralca/ly in ( 2.27b). The

orthogonality conditions

(2.51b)

imply for

u=l,...,M

and

(kl,...,ku) ~ s y m U L l _ l

(MeN;o , y_kl...y_ku) Y

Moreover,

the n o r m

Volterra functional polynomials. W e precisely the 'normal equations'

(2.52b)

%vith the M - t h

introduced in the s p a c e can

observe

degree

nonline-

of the regular

that conditions

(2.52)

are

(2.31), derived algebralca//y. M o r e o v e r ,

rema/~k that the 'normal equations' for the linear N-th order predictor

are given b y

(2.52)

2.3.60pfimum

generalized

Considering

with

i%4=i .

matrix approximation

the s p a c e

{ M}_I~N_I

M}~q = v {{M} l o ,

The

as

= ]] Me N;o[I y

are the 'norm~l eque~tions', associated

ar lea~st-squares prediction p r o b l e m

we

(2.52a)

of the error will be e x p r e s s e d

(MEN; O , yo) Y

These

-- 0

N-th order estimate

{ M } ^iN; °

(2.37) , w e

define

iN_l }

of the element

{M} 1 °

will be introdu-

37 ced a.s follows

{M}^

A

{M}_

IN;o =

where

;%'M'P 1 lI;N-3.

space

-"IM)|l -N-I

{M}AN

{M} 1

~

o

(2,54a)

{M}.I

a N-I

indicates the orthogona,1 projection operator on the sub-

°

=~ { M } ~

1

z-'U;N-I

The

N-th order approximation

1 {M}I I;N-1 0 =

{M}I

0

[0

-

error is then given by

{M}IAN| o ]

± {M} 1

KN- I

(e.54b) and can be rewritten in a r.enorma/ized form as

{M}AN = {M}A N II { M } A ~ [ ~ I

Let us observe tatives of the phism

(2.s~c)

that

{M}^IN; O , {M}A N {M} A N and are the represenM ^ M M YN;o ' eN ~nd eN; ° , respectively, in the isomor-

(2.47). Prom

and

w [ m A N ] m = l ..... M

t h e orthogonality conditions l

(k I .....k u ) ~ y m % ~ _

(2.54b), w e obtain f o r

I

( { M } A N ' { M } I k l . . , k u)]I

w h i t e t h e n o r m of t h e e r r o r

(2.54b)

({M}AN'{M}

under

the isomorphism

-, 0

(2.55a)

iS g i v e n by

io) ~ =

We n o t i c e f.hat ~he 'normal equations' (2.52),

U.-I,..,,M

(2.47).

{{{M}A~ I I

(2..55)

are

(2,55b)

actually equivalent to

38 2.3.7 O p t i m u m ~qeneralized p o l y n o m i a l

Given

the space

(2.44), l e t u s c o n s i d e r

{M} ZN_II

{M}ZN

The

a~pproximation

v{ M z

=

°

M^

N-th order estimate

{M}Z Ni_ I }

,

of t h e

ZN; °

element

(2.26)

M

z O -- [I 0 ... 0 ]

will

b e obtained as

M^

{M}p

A

ZN;o =

where

{M}p

1

I

M

Z;N-I

{M}zI

error will then b e e x p r e s s e d

as

A {M}~

N;o =

(2.57a)

-1

d e n o t e s the orthogonal projection operator, taking

z ;N-1

projection o n the subspa~ze

MA

{qVI} z l

e

Zo

1

Z;N-I

M

Zo

N-i

M

=

"

The

Zo "

N-th order approximation

M^

ZN;o

.t

{ M}Z1N_ 1

(2.57b)

or, in o. renormalized form,

M A N A MA :

Let u s o b s e r v e of the

M

-1

N;o II AN;oll Z

that

M^ZN;o ' MA N;o

M^ Y N ; o ' M eN;o ' MeN;o

respectively, in the i s o m o r p h i s m The

(a.~Tc) ** [ ~ ( Z

l .....Z m )

and

~d~or

MA N

of the

] m - - X .....M

are

the representatives

{M}iN;o ^ ' {'M}AN , {M]AN

(2.47).

orthogonality conditions

(2057b)

will imply for

u=l,...,M

and

1

( k i ..... k u ) ¢ s y m U L N _ i

(MAN

' M Zk .... k ) Z

= 0

(2.~8a)

39 Moreover,

we

obtadn

(MA'N' Let u s

observe

=

liMa N;o lI:z

the~ the 'normal equations'

clitlons (2.55),(2,52)

Finally, w e filter p r o b l e m

MZo) Z

and

can

(2.31), u n d e r

conclude

(2.58b)

(2.58)

e q u i v a l e n t to the

are

the i s o m o r p h i s m

that the n o n H n e ~ "

con-

(2.47).

lea.st-squares

prediction

ca/n b e equiva/ent/y stated in the s p a c e s :

- of the r e g u l o / "

Volterra functioned polynomiads;

- o~ the genere~tized coefficient-matrices; - of the gener0dized 'l~his f o l l o w s seen

from

z-polynomieds.

from the fact that the three s p a c e s (2,47). Therefore,

(Le., the p r o b l e m

of o p t i m u m

the nonlinear

}Ylq

can

be

- the o p t i m u m

equivalent/y

deterministic

stochastic

stocha.sfic approximation

'ideal' prediction filter) in the s p a c e

{M

considered

approximation

~

mat/ices

- the o p t i m u m

deterministic

the hi~her-order are

gher-order

sequence

are

of:

{M}K N

covariance

impulse

respon-

of the generali-

dat~ of the underly-

given);

approximation

tions of that filter, in the s p a c e mia/s - in the Alq co.se

of the output of the

of the set of M - D

zed

sequence

estimation p r o b l e m

the p r o b l e m

of the 'ideal' prediction filter in the s p ~ c e

ing stochastic

as it is

of the Volterra functional polynomials

ses

(provided

are isomorphic,

(provided known).

of the set of M - D

{M} Z N the M - D

trarzsfer func-

of the generalized

z-polyno-

spectral flxnctions of the hi-

3. G E N E R A L I Z E D

NONLINEAR

In flue previous prediction

problem

tly introduced

LADDER-FILTERS

chapter

we

for higher-order

algebraically a n d

mains. Algebraic

showed

solution of the problem, algorithh~, w a s

(i983a, b); Z a r z y c k i

(1983);

in Z a r z y c k i

in the s p a c e s :

can

in time- a n d

be

in Z a r z y c k i

derivation

equivalen-

in frequency-dogenera/iza-

o/qd D e w i l d e

of the nonlinear predicti-

of the Volterra functional polynomials

(1984~a-e).

lutions of the nonlinear

sequences

yielding the nonlineom

proposed

geometric

o n filter algorithm in the s p a c e

method

stochastic

geometrically,

f/on of the L e v i n s o n

introduced

that the nonlinear lea.st-squares

Here

least-squares

we

wish

to present

prediction problem,

of the generalized

matrices,

was

recursive

so-

using projection

o2ud of the generalized

z-polynomials. The

nonlinea/" p r o b l e m

will b e p r e s e n t e d This

approach

in c a s e

will s h o w

as

a straightforward

the m o s t i m p o ~ a n t complexity

We riant)

M

while its solution

of the filter nonlinearity.

fealures

of the nonlinear treat-

of the derivations

of the s e c o n d - d e g r e e

estimates

will b e of the s e c o n d - d e g r e e perscript

cO~e

on one

ho.nd, o n

nonlinear prediction filter algorithm will b e

extension

tities (i.e., s u b s p e c e s ,

stated in ~eneral

of the s e c o n d - d e g r e e

menf~ avoiding u n n e c e s s a / ' y the other - the general

was

and

(R4=2)

errors) , we

case.

considered

shall drop

Since

obtained all q u a n -

in this chapter

from n o w

o n hhe su-

in order to simplify notations.

will s h o w

nonlinear

that the solution will result in a generalized

ladder-filter

algorithm, being

o r ~ h o g o n o / linear ladder-filters.

(time-va-

a natural generalization

of the

41 3.1 Index-sets

Let u s

(see

sets

euad their ordering

consider

a J s o (2.16)

for

n=0,...,N

with

rn=l,2)

ILx= n

2Lxn

and

{Jl} xX+n = {x .....x+n}

x+n

A)

can

for

introduce

the two types

u Lx÷ i n-I

U {x+n+l}

where rence

x

of the index-sets:

point 0),

ber of elements v--0~...,n

Lx,v A=

u s y m 2.~ nx + 1i -

U { ( x + n + l , x + n + l ) .....( x + n + 3 - m , x + n + l ) }

will denote

usly c o m p l e t e d

for

J2ffiJl

m.=O,...,n+2

= Lx A {x} n~m

B)

(3,b)

x+n

J l •X

we

(3.1a)

- Lx~ × Lxn " { q'J2 }jl,ja=~. +n

s y m 2 L x = { jl,j2}

Then

the following index-

x=0,...,N-n

n

a 'globa/' offset ( b a c k w a r d

while

in the l~t, o/ud

set, a n d

of previo-

%%~iliindicate the 'local' order

being completed,

{x} U {(x,x) ....,(x,x+v)}

(num-

column);

u

LX+ I

u

n-i

U {x+n+l}

x

and

m

(number

m = v + l,...,v+n+3

n,m

where

shift from the refe-

will stoJnd for the 'global' order

columns)

(3.2a)

will ago/n

U { ( x + n + l , x + n + l ) , ....( x + n + 4 - m , x + n + l ) }

denote

in the top-row

the 'globa/' offset

(3.2b)

(in the 'uni-v~riate'

of the 'bi-variate' part of the

x~v Ln,m),

v

subwill

42 indicate the last element der while tow

m

~nd

will ind/cate the s u m

C~a~ o b s e r v e

3"LX n

Considering

forward'

will b e

of the n u m b e r

a~edn the '.q/obal' or-

of elements

in the top-

th~

u

s y m 2 L nx

a n d f/nat the ordering is m o d u l o

introduce

n

the last column.

in

We

in the top-row,

x,n

(3.2c)

Ln,n+ 1

(n+2)

as w e

have

LXn,m+n+2

= LX+l,m

(3.3a)

L x'v n,m+n+2

- L x'v n+l,m

(303b)

the element

for fur'£her u s e (B-forward)

~

{ x}

a n d the elements

of the top-row,

the 'linear- for~va/'d' (L-for-ward),

index-sets

and

we

'bilinear -

of the n-th 'global' order, as follows

L-forward

L* - {x} n,o

B-forward

LX+l,n-1 n-i,n

(3.4a)

(for v---0....,n)

{x,x } u

Lx

{x,x+v }

u

njo

Lxl v _ <[ n,v+ i

These

u

index-sets

cal' forward

v=0

if

v=l,...,n

(3.4b)

will b e

approximation

Considering

if

Lx,v_l n~v

applied later, in order to introduce

sets

of 'lo-

errors.

the element

{ x÷n+1 }

and

the elements

of the last

43 column

of the index-sets

ckward

(l,-backward),

L x'v n,m

' we

define

for further u s e

a.~d 'bilinear - b a c k w a r d '

of the n-th 'global' order,

as

the 'lineew - b a -

(B-backward)

index-sets

follows

L-backward

Lx, n - i n,o

B-backward

"

Lx, n

. LX, n-I n,n

The

Lx, n-1 n,m-1

backward

Using

will b e

will b e

(3.4:), w e

can

used

,

m.~l,°..,n

(3.5b)

,

m..n+ 1

(3.5c)

later, w h e n

sets

of 'local' b a c k w a r d

considered.

consider

the following

collection

of the 'local'

[ Lx n,o

Lx, O n,l

collection

Lx,n-1 n,n

""

Lx, n n,n+1 ]

of the 'local' b a % c k w m r d

(3"6'~)

index-sets

results

(3.5)

L x m b;n

ceun o b s e r v e

[ b x'n-I n,o

L x,n-I n,l

cursive

way,

and

°'"

L x,n'1 n,n

that the 'globe/' f o r w a r d

of the (n+1) order (i.e., Lf~+ Ix

L~;n

(3.5a)

index-sets

while the folIowin~

We

{x,x+h }

U

index-sets

ILx ,, f;n

from

{x+n}

u { x+n+l-m,x+n}

a p p r o x i m a t i o n errors

forward

U

(for m--1 .....n + l )

Lx, n - i n,m

n,n+1

.~ LX, n-i n-l,n

~iven

~d

the n-th o r d e r

ILx . The b;n

hlgher-order

and

L x'n n,n+l ]

backwetrd

L~;n+ I) c ~

(3,6b)

index-sets (3.6)

be obtained in a re-

'~lobal' for-ccard a/ad b a c k w a r d '~[obal' i n d e x - s e t s

will b e

index-sets derived

via

44 a 'global' order-update step (3.4) a n d

n

(3.5) of the sets

-+ n + l

, executed

o n the 'local' entries

(3.6), a n d d e s c r i b e d sehemaf/c~l/y in FiS,3.1,

x ]Lb;n+ J_

l,-ba c k w a r d L nl,oX'n +

B-b ackward Lx, n n+ i,m

Lx, n+ 1 n+ l,n+ 2

m = l|...In+1

Lx,n+ Jn+ l,n+ 2

XiV

Ln,v+ 1

LXl v n+ l,v+ 1

v=0~...In ]Lx f;n+l

~[jx f;n

L-forward

L-forward Lx n~o

L:+1,o

B -b ackwo/'d

L-backward LX+1,n -I n~o

LX+l,n-i n,m

Lx+l,n n,n+l

m=lj...tn ILX+ 1 b;n

F i g . 3 . 1 0 r d e r - u p d a £ e step n-+ n + l o~nd L-backwo.rd index-sets.

for the 'global' L-for"ward

45 The

four types

Fig. 3.1 , c a n

be

of the 'local' recursions,

L-forweurd ~ n d -

recursions:

the 'local'

for the 'uni-variate' parts of the

L-backward

index-sets;

order-updates for the 'bi-variate' part of the

L - f o r ~ v ~ d index-set, ~ n d

the 'local' o r d e r - u P d a t e

'uni-vari~te t p~rt of the B - b a c k v c a r d BIu - recursions: the 'loco/' o r d e r - u p d a t e B-forward

recursions:

index-sets, a n d

the 'local' o r d e r - u p d a t e s B-forward

For

example,

from (3.~a)

lows that the L L

'local' o r d e r - u p d a t e

for the

index-set;

for the 'bi-variate' parts of the index-sets.

(with

(3.5a)

index-sets;

the 'loca/' o r d e r - u p d a t e s

eund B - b a c k w a r d

and

for the

for the ~uni--v'ariate' part of the

'bi-voriate' part of the L - b a c k w a r d RB-

of

interpreted m.s follows:

L L - r e c u r s i o n : the 'local' o r d e r - u p d a t e

LIB

indicated in the d i a g r a m

x

replaced

index-set recursion

by can

x+l), be

it

fol-

expressed

LX+l,n-i nwo = Lx Lx nil n,o

u

L x + l'n-I n,o

=

{x} %

u L X + l , n-I u n-l,n ; ..g.......

{x+n*l}

(3.7)

Lx n~o

and

can

be

schematically d e s c r i b e d

a s the following L L

index-set section

Lnxl Lx

n,o

3

~~

(3.8)

r- Lx

n~ 1

!

LX+ 1 , n - 1 n,o

q?he r e m a i n i n g 'local' index-set r e c u r s i o n s

can

be

obtained in a similar w a y .

46

T h e s e recursLons, together with the corresponding index-set section are presented in Appendix I. W e '~ob~9/' order Lndex-sets

notice that the number of entries of the (n+l)

~U~n+l

and

ILb;n+l x

comparison with the n-th order sets

]LXf;n and

B-fozwea'd and B-backweurd index-set

x,n+l 2 Ln+l,n+

'ri.qht-upper' element

is augmented by one i n ~UXb;n ' since the 'new' (a.ssociated with the

{x,x+n+l} ) is arriving to the s c h e m e of Fig. 3.1 al:

the 'global' order-updaLe step

n + n+l. T h e 'local' structure of this order-

update step is presented in 5"[g. 3.Z. Lx,n+ 1 n+ i,n+ 2 >

I2nX~flto Lx-,n ~ - ~ n,n+l T

x~n

xtnl,n Ln+

LnX,fl,o

".__> Lx~n _ ~ .... n+l,n-i ~

Lx,n÷l n+l,n+2

Ln+ I,n+ l

Lx,n + n+l,n

Lnx'+~,n+1

Lx,n-I n,n+ I

Lx,n-I n+ l,n-I

Lx, n-i n+ l,n

LXP 0 n,2

LX, O n+l,o

LX, O n,n+l

f X~O

f

-

. .

-'~ Ln,n+l--~

n,2

Lx n,n+ i Lx

--~

n,o " t Lx+ l,n-i n~o

Lx

nsl

>

--~ L x sin

~

LX~O n+l,l

X~O

1'n+l,o +

Lx n,n+l

Lx+ l,n-i ntn

Lx n+ I,o ~

LX+iio

LX+ l,n n~n+l

Fig. 3.2 'Local' structure of the 'global' order-update index-set step.

n + n+l

47

o

0

t-" ILb;0"% k

•

..

ILb; 1

T[ •

f A

lb ° b ; 2 .............. % A

Lo

b| 3

f

•

A

,%

'

•

~

l

x=l

Ibm3

1

lLf; 2

-J

! x=2

x=3

3 ) Lf; 0

.J Initi~zations:

IL _

'I~Iew" elements:

T'

~

LX oeo

>

(x=O .....3)

5"i~.

3.3

n,n+l

(n=i,2,3 ~ x=O,.,.,3-n)

'Local' structure The symbols 0

cursions

a

of the third-order (N-3) indicate corresponding

of l~'ig. 3.2.

index-set recursions. 'local' index-set re-

48 From

Fig.

the L-forward

3.2 it f o l l o w s t h a t t h e will work

index-set

a)

initialization: L x n)o

b)

'uni-variate ) step: L x + n)o

C) )bi--vaLrlate' steps: L x

n)l

d) For

o)

)bi-variate'

d)

termination: L x)v n+l)v+l*

The

L- ~und B - b ~ c k w a r d

d)

for

follows:

-~ L x n)2 -~ "'"-~ nLX)n+l ~" L xn+l,o

we

steps:

i~U~za~ons:

x)v Ln)v+2-~

Lx)v n,v+2 x,v Ln,v+3

index-sets

L x+1'n-* ntm L x+l'n n,n+l

'uni-varla[e'

get:

(v=0) ....n)

x)v + step: Ln)v÷ 1

'uni-vari~te'

c)

recursion

Lx n) l

the B-for~vard index-sets

b)

b)

order-update

termination: L x n+l)o °

initializat[on: L x'v n,v+l

a)

~s

'global'

+ "'"

Lx,V x)v -~ Lx,v n)n+l ÷ n+1,o ÷ ...-~ L n + l , v + 1

are u p d a t e d

o.s follows:

( m = 0 .....n) (m=n+l)

stepS: L x+l'n-I ~. L x nim n)m+l L x+l'n -~ L x n)n+ l n+ l,o

'bi-variate' steps: L x + n,m+l

L x)O n)m+2

( m = 0 .....n) ( re=n+ i)

-~ L x'l n,m+3

+

"'"

-~ L x'n-I L x)n n + l ) m - i "~ n + l , m

termination: L x)n n+l)m"

We notice that e a c h is associated (N=3)

'local' order-update

with 'label-update'

index-set recursions The

index-set

n e a r ladder-filter

step. T h e

is p r e s e n t e d

recursions,

6dgorithms

step for the b a c k w a r d

of the third-order

in Fig. 3.3.

derived

presented

'Iota/' structure

index-sets

in this section, will underly

in the s u b s e q u e n t

paragraphs

nonliof

this chapter.

3.2 Nonlinear

filter a/~orithm: time-domain

In this p a r a g r a p h degree

we

will derive

nonlinear prediction problem,

of generalized

coefficient-matrices,

approach

a recursive

using projection

introduced

solution to the s e c o n d method

in p a r a g r a p h

in the s p a c e 2.3.2.

49 Let

{y}

denote a fourth-order ( M = 2 )

v e d o n the time-interval riables

[ 0,-l,...,-N ]

stochastic s e q u e n c e ,

and represented

yo,Y_l,...,y_N . C o n s i d e r i n g the index-sets

(3.2b), w e

define for

of the r a n d o m

n-0,...N

and

b y the r a n d o m

Lx n,m

(3.23)

va-

a n d L x'v ntm

the following submatrices

vorlatbles o.nd their products

Y-Jl

nmm yX

=~

(3.9)

=

n,m

2yx

n,m a n d the s u b m ~ t r i c e s L x'v . T h e n n,m

x=0,...,N-n

obser-

we

ly_j~

yX,V ntm

, expressed

(jl,J2) ,xnt m

by

(3.9) with

c o n consider the following ( 2

variance submatrices

Hx n,m

and

H x'v n,m

the former is given b y

le2HX n,m

H xn~m = E { ~ n , m

replaced b y

2)-block, multi-indexed co-

, where

lelHX

Lx nsm

] n,m

@ ~n,m } "

2 • 1HX

2 • 2HX n,mJ

ntm

Ik kk21 J12kl hJlJ2klk2

(Jl,J2,kl,k2)

a n d the latter is e x p r e s s e d

by

(3.10)

with

Lx n,m

replaced b y

(3.1o) ~ ~,m×L xn,m L x'v . n~m

Now let Ix

=

n~m

where

ljl

for

"~

[

(ki,k2)

~kl;j. I

[llX n,m

21x

] =

[i.

n,m

Jl

1..

(3.11~)

] ( j l , J 2 ) • L xn0m

JlJ2

• L xn~rtl

2o ]

~jlj2

[ lo

~

kzk2;jlj2 ]

(3.11b)

50 with

I0

and

20

ctively, w h o s e

being the o n e -

domains

Lx respectively. n,m |

Let us

and

~wo-indexed

are hhe uni- a n d

In a similar w a y

introduce

the

respe-

bi-variate parts of the index-set

we

following,

zero-malrices,

con

introduce

x-labeled

and

the mah-i~

Ix'v n~m

(x,v)-labeled,

sub-

spaces

I nx , m =

We

notice

{IX, mn

11'N-I -N-I,N

thai

by (2.37) with element

v

M.-2)

Fx nmm

}

will b e

se

n,m

domain

Ix'v ntm

is

Ix n~m

will b e

a two-block

F x'v nsm

(row),

with d o m a i n

x

expressed

multi-indexed

Gx

= Fx

n,m) ~x n,m

element

o n the s u b s p a c e s

.H x

n,m

n,m

x=0,...,N-n

the

to

(3.4~)

subsequent

and

(3.5),

subspaces

we

EMh

as

(3.12b)

coefficient-matrix w h o of the s u b s p a c e (2.39a),

(3.12a)

.%x

we

as

(3.13~)

n,m

X,V G X , V ~ --b~X'V°HX'V-G x'v Fn, m ' n,m j ]~x,v n,m n,m n,m ntm

According

(expressed

_o,N KN,N+ 1 °

will b e

D b ~ % v .. b x'v . Following n~m n0m

a family of inner-products

(Fn,m '

{ 2} K ~ _ I

fx ] n'm;jl-x'J2-x (Jl'J2) ¢ LXn,m

DF x - .Lx . Similarly, e a c h n~m nsm

will b e

introduce

v

precisely the s p a c e

of the s u b s p a c e

will b e

F x

.

n~m

while the 'biggest' s p a c e

b"Xn,m m [ f~n,m;Jl_ x

i.e.,

,xv

;

can

of the

(3.13b)

introduce

for

n=0,...,N

and

~o,N

-N,N+I

L-for-Nard

I~n~o = ~ (I~o) i

(3.14~)

51 B-forward

(for

v = O .... , n )

IX, v n,v+l

=

IX'v v { n,v+l }

(3.14b)

itx,n-i n,o

x,n-i = v { In, O }

(3.14c)

L-backward

(for

B-backward

m - i ....,n+l)

N x'n-I n,m

,, v { Ix'n-i } n,m

1[x'n n,n+l

3.2.1 'Local'

v

estimates

Denoting ta/dng projection subsequent

=

by on

,

xln {In,n+1}

and

,

m-.l .....n

(3.14d)

m=n+l

(3.14e)

errors

P l ; nx , m

("--I;n,m" p x,vj

the s u b s p a c e s

'local' estimates

and

the o r £ h o g o n a / ]ix n,m

errors

as

(Ix'vJ " n,m-

'

projection

we

operators,

will introduce

the

follows:

L-forward Following

(3.~a),(3.11),(3.12)

Itx n,o =

v

and

(3.14a),

we

{ix

x+l,n-i ' In-i,n }

can

rewrite

a~

Itx

n~o

(3.i5a)

A

We

define

the L - f o r w a r d

n-th o r d e r

p

estimate

x÷i,n-i i 11;n-l,n

x

'ix

n,o

of the

ix+l,n-1 n-l,n

i

x

as

(3.15b)

52 Let

0jl

denote the zero-entry with 'coordinate'

the L-forward ^ n-lh order approximation error, c o r r e s p o n d i n g to the estimate ix , will n,o

be expressed

as

x =A p l x+i,n-I i - I - [0 An,o --~ ;n-l,n x x o

(since the estimate x ~n,o). This

pace

Jl " T h e n

'%1 x nto

~x I ± n,o ~

is c o n s i d e r e d h e r e ~

~x+l,n-I n-l,n

(3.16a)

etn element of the s u b s -

error caxl b e rewritten in a renormalized form

AX n,o "

x An,o

x

C:

1{ An,o [

=

n,O =

[x

ax ] n,OlJl-X,J2-x (jl,J2) • L x n~o

n,o;Jl-X

in accordance

the errors

with

(3.16)

respec~vely, if

B-forward Using

(3.12b).

can

observe

are precisely the

M~2

( for

We

, x=0

and

that the

estimate

{M) IN;o , { M }A N

~d

c a n rewrite the s u b s p a c e s

I

v { Ix, x

'

Ix } n,o

n,v+l

Let

'

X x'v n,v+ 1

if

Ix'v-I } n,v

car* introduce the B - f o r w a r d

0jl,j2

A

i

(2.54)

(3.14b)

as

v=0 (3.17a)

{v { iX,x+v [

ix, v

{M)A N

and

v=0,...,n)

(3.4b), w e

we

(3,15b)

n=N.

Xx'v s < n,v4-I I

Then

(3.16b)

p I;n,o x Ix,x

c

if

v--l,...,n

estimates

Itxn,o

if

v=0

IIX'V-i n,v

if

v--l,...,n

(3.17b)

p

x,v-i 11;n,v

i

X,X+V

E

stand for the zero-entry- with 'coordinates'

(jl,j2) . T h e n

53 the B-forward

approximation

will be e x p r e s s e d

j

A =

n,'%r%'- ~

to the estimates

(3.17b),

a~

. p±

ax,v

errors, c o r r e s p o n d i n g

x

a

Xx

I[ ;n,o Ix,x

if

v=O

if

v=l,...,n

n,o

(a.18a)

/

[ p l x,v-i

i

1

ITX'V-I n)v

x)x+v

H ;n)v

i.e., Ax ' v

nsv+ 1

as the estimates B-forward

-

Ix,x+v

~x,v n,v+l

(3.18)

errors

AX'V = n,v+ 1

=

[0

~X)V ] n,v+ 1

O)V

are treated here

can

be e x p r e s s e d

(3.18b)

as elements

of

x,v I[n,v+i). The

in the renorma/ized

form as

ax'v llAX'Vl[l-lnv+ n,v+ 1 , llx,v n,v+ 1 x,v

(3.1So)

x,v

= [ an,v+l;Jl-x

an,v+l;Jl-x,J2-xl

(jl,J2)

x,v £ Ln,v+ 1

(3.14c), using

(3.5a), as

L-backward Let us

K~;n-i nuo

rewrite

n" x , n - 1

~' V {In~.q:nl -- )

lq)O

so

that

we

will define

-1 ~x,n n,o

The

n,o

A

x,n-1

= P~;n-l,n

L-backward

k p x,n-i = R;n-l,n i x + n

L-backward

B x,n-1

the

approximation

lx+n

=

-

ix+n}

estimate

(3,19a)

as

l[x,n-i n-l,n

e

error

:1. x+n

!

will then

Vx'n-1

[ln,o

(3.19b)

be

On]

i

I x'n-1

n-l,n

(3,20a)

54 or, in a renormalized

B x'n-I n,o

form,

B x'n-1 IIB x'n-1 -i n,o n,o ]l~x,n-i n,o

=

-- [b x'n-1 n'°;x+n-Jl

(for

B-backward

F'ollowing (3.5)

Hx'n-I

m = l ....,n+l)

and

V

(3.14), w e

{ Ix'n-I

c a n write

'

i x + n + l-m,x+n}

'

mml"°°tn

(3.21a)

x~n . ..xln--I Kntn+l , v [in~ n

,

ix,x+n }

,

m=n+l

(3 21b)

"

the B - b a c k w a r d

~x,n-1 n,m

~= _ x,n-I P~;n,m-i

ix, n A= P x,n-i n, n+ 1 ]I;n,n

Consequently,

estimates will be e x p r e s s e d

C

B x,n nmn+l

IIx'n-I n,m-i

,

ix, x+ n

Nx'n-1 n, n

'

the B - b a c k w a r d

c

approxima£ion

;n,m-I i x + n + l-m,x+n

A= Pli x,n-1 ;n,n

as

i x + n + l-m,x+n

i

x,x+n

±

!

~[x, n-i n,m-1

ix, n-i n,n

i

m=l,...n

(3.21c)

m=n+l

(3.21d)

errors will be defined as

,n-1 n,m

so

(3.20b)

n,m-i

n,m

Hence,

bY.n-1 ] n'°;x+n-Jl'x+n-J2 (jl,J2) E L x'n-I nlo

'

m=ii...,n

(3.22a)

m=n+l

(3.22b)

that Bx,n-1 n,m

~,n

~n,n+l

-

[~x,n-1

= i x + n + l-m,x+n

=" lx,x+n

-

r~ ~ ' n

~ n,n+l

n,m

0

o,n

On+ i-m,n ]

(3.22c)

]

(3.22d)

55 "Dhe B - b a c k w a r d

errors c a n b e e x p r e s s e d

in a renormalized

illx, n_l-i

Bx'n-ln, m I1 Bx'n-ln,m

Sx'n-ln,m "

form

,

m n l .....n

(3.22e)

,

m-n+ l

(3.22f)

n~m Bx,n

x,n n n,n+1 .. Bn, n+ I U 5~n,n+111 -* ~x~n n~n+l

similarly as the L - b a c k w a r d

3.2.2 D e c o m p o s i t i o n

PoUowing

error

of s u b s p a c e s

B x'n-I nmo

(3.20b).

'

the considerations

consider the 'local' decomposlf/ons

o f the previous

paragraph,

we

can

of s u b s p a c e s :

L-for~rard Since

Ax n,o

is in

from (3.16), w e

Ix n,o

'

but is or%hogonal to

n,o

=

fix+ 1,n-I

n-l,n

•

x

v { An, o}

w h i c h implies the 'local' decomposition

x . p x+1,n-1 PII;n,o K;n-l,n

x

P~n,o

denotes of

B-forward

v=O,...,n)

Since

+

(3.23a)

of projectlon operators

x PA;n,o

(3.23b)

the om£hogone/ projection operator, taking projec-

£ion o n the s p a n

(for

, as it follows

c a n write

Kx

where

~x+l,n-i n-l,n

Ax n~o

AX'Vn,v+l belongs

to

~n,v+X'V1

but is orthogonal to:

]iXn, o

(if

v=O),

56 and to

I x'v-I n~v

Ix'v

(if

v--1 ..... n ) ,

I IIx n,o

=

in accordax,c e witkl ( 3 . 1 8 ) , we o b t a i n

o

Ax, o v{ n,l }

v~O

~

x,v v { An,v+ I}

v=l,...In

(3.24a) n,v+ I

This

11x'v- 1 n,v

implies

p

xtv Pl;n,v+ 1

p

where

< l

xtv A;n,v+l

X l[;n,o

+

,

V~0

,

v=l,---,n

(3.24b) X,V--I ][;n,v

p

+

is the projection

operator

on

the s u b s p a c e

spanned

by

Ax, v n,v+ i " L-b ac k w a r d Observing

h a l to

that

I xn-l,n 'n-1

B x'n-I , see

(3.20),

I x'n-I n,o

resulting

PB;n,o

IB-backwvi~rd From

we

to the s u b s p e t c e

]Ix'n-I

n~O

but is o r t h o ~ o -

car* write

= ITx'n-1 n-l,n

(9

v

{B x'n-I n,o }

(3.25a)

in lhe d e c o m p o s i t i o n

p

v~here

belongs

n~o

x,n-i ~;n,o

_ x,n-i = ~]l;n-l,n

is the projection

( for

(3.25b)

+

opereltor o n

the s p a n

of

Bx'n-ln,o "

m=l,...,n+l)

(3.22)

it follows

that

]ix,n-1 n,m

= ~x,n-1 n,m-i

•

v {Bx~n-1 } n~rfl

,

m=l,...,n

(3.26a)

57 ix, n = I x'n-I n,n+ I

This

•

V {B x'n . }

m=n+l

(3.26b)

imp)/es

p

x,n-1 ]I|n,m

p

x,n

= o x,n-1 --~;n,m-i

+

x~n-i I[;n,n

+

~

p

W;n,n+l

p B ; nx,n-i ,m

where spanned

by

We

can

x,n PB;n,n+l

n +i ) ( P B ; n~tn

B n x'n-I .m

3.2.30rthonormal

(Bx:nn+ I )

that the L - a n d

In o r d e r

to s h o w

A x'u n,u+l

'

that, let u s

a/qd let u s

m=n+l

~sume

belongs

£o

we

can

that •x,v n,v+l

show

A x'v nlv÷1

B-forward

set in the s p a c e

consider

A x'v n,v+ 1

Consequen~y,

,

operator

(3.Z6c)

(3.26d)

on

the s u b s p a c e

ba~es

observe

In a similar w a y

m = l,...,n

"

(v--0,...,n) f o r m the O N

A xn ', vv + l

,

is the projection

A x'v n,v+ I

and

,

two

~

errors

A x'u ± n,u+l

a2ud the matrices.

A Xn,,Vv + l Ix'u-I n,u

and

D N x'v n,v+l'

v=0,..,n

A x'u n,u+ I

that for

±

obtain for

A x n~o

of g e n e r a l i z e d

B-foFwca'd

v < u. S i n c e ' we

errors

( 3.27 a)

v:O,...,n

A x n~o

(3.27b)

the entries of

A x = n will f o r m the O N

[A x

n~o set.

AX, o n~o

"'"

An,n x • n - 1 A nx:n ,n+

11

(3.28)

58 If w e

introduce, a c c o r d i n g to (3.15a)

[Ix

iX, x

...

and

(3.1?a), the following set

Ix,x+n_ 1

ix, x+n]

(3.29)

J/~en (3.28) wiU be the orf/,1onormalized version of that set. Using (3.28) we cain write the following 'global' orthogona/ decomposition of subspaces

x,n

~n,n+ i

:

f<+i,n-i n-l,n

{A~}

•

(3.30~)

v

where v{A x}

= v {A x n,o }

~l~he d e c o m p o s i t i o n

(3.30)

•

n ~ v=O

will r e s u l t

x,v V {An,v+l}

•

in t h e

'global'

(3.30b)

decomposition

of p r o j e c -

t / o n operators

p

x,n _ - p x+l,n-I II;n,n+l X;n-l,n

+

x PA;n

(3.30c)

with n p

x p x A|n : A;n,o

+

x,v ~ PA;n,v+ 1 v:0

(3.30d)

T h i s result implies the f o U o w i n g 'global' or£hogonal d e c o m p o s i t i o n of the 'biggest' s u b s p a c e

~,N

N

,N+i =

Z

®

{ x

v ~_~}

(3.31)

x,=,0

Consequent/y, the s p a c e of (3.28) We

(3.28)

ca~% b e interpreted eu~ the 'global' O N

of the two-block, multi-indexed coefficient-m~trices. T h e c a n therefore b e treated a s the 'local' O N

shall call

matrices.

A x n

A x n

the 'forwe~d' O N

b~is

b~sis of entries

basis of that space.

of the s p a c e

of generalized

59 In a s i m U ~ r B x'n-I n, m

(m=0,...,n)

the g e n e r a l i z e d B x'n-I n~w B x'n-1 nlw

way

we

matrices.

± Zx, n-I nlw-1

=

%hat

can

and

w

, we

1

c~n

errors

set in %he s p a c e errors

B x'n-1 npm

observe

ge% for

and

£ha£

m,w..l,...,n

B x'n-1 ntw

that (3.32a)

of

(3.32a)

holds

for

m,w,=0,1,...,n

, and

moreover,

m=0,...,n

for

Therefore,

]Bx n

the entries

=

I

B x'n n,n+l

Bx, n-i n,l

set. C o n s i d e r i n g

[ Ix+ n

notice that (3.33)

(3.32b)

of

[ Bx, n-I n,o

will f o r m the O N

f/y, w e

m

B-baokwaFd

ON

B-backward

Bx, n-I q~hus, w e n~m "

a

B x'n-I n,m

we

WiU form a n o t h e r

assumin 8

~[x,n-1 npm

show

%h~t the L - a n d

C o n s i d e r i n s two

B x'n-I n~m

Similarly, w e

show

BX, n n,n+ I

~nd

(m,w=l,...,n),

can

Ix+n,x+ n

""

Bx, n-I n,n

(due

to (3.19)

...

nmn+l

version

'global' or+~hogonal

n-l,n

~

and

Ix+l,x+ n

is the o r t h o n o r m a l l z e d

obt~dn the following

x,n . Bn~n+l ]

(3.33)

(3.21))

the set

ix, x + n ]

(3.34)

of (3.34).

decomposition

Consequen-

of s u b s p a c e s

v

where

%' { B x } n

=

n ~ m--O

x,n-i ~) V{ Bn, m }

~)

x,n V{Bn,n+l}

(3.35b)

6O The

(3.35)

decompositions

i m p l y the 'g/ob~l' d e c o m p o s i t i o n

of the projection

opera~ors p

x,n = p x,n-i l[{n,n+l ll;n-l,n

p +

x B;n

(3.35c)

where n X

=

PBIn

Hence,

we

~ m=0

obtain the s e c o n d

'biggest' s u b s p a c e ,

shadl call

neralized

B x n

ON

3.2.4 G e n e r a / i z e d

Let u s

kind

=

N Z n=O

entries

basis

x,n B;n,n+l

(3.35d)

of the 'global' d e c o m p o s i t i o n

of the

~

V {B:}

Cholesks/

introduce

(3.33)

of

(3.36)

ON B x n

basis

of the s p a c e

will therefore

be

of the g e termed

the

of that s p a c e .

factorizations

%he following multi-indexed

= c o 1 [ ~ _ x ] x = o .....~ =

and

p +

the global ' b a c k w a r d '

matrices. T h e

'local' b a c k w a r d

x,n-i B;n,m

in the f o r m

]i°,N N,N+I

We

p

coI [ ~

'upper-triangular'

matrLx

...

Axn "'" ANo ]

(3.374

"'"

B ° n

(3.37b)

let

B N -- COl [ B ° ] n n=O,...,N

=

COl

[B:

"'"

o ] BN

be the corresponding multi-indexed 'lower-tri~ungula~' matrix. Recalling the inner-product,

introduced

in

the 'biggest' s u b s p a c e

of g e n e r a l i z e d

matrices

61 and expressed

by

(3.13b)

with

x--0 , v - N

, n=N

, reraN+l), w e

can

obser-

v e that

(AN'

AN)_o,N ~N,N+l

o,N " AN*HN,N+I

.T ~N

-

"

~

-

•

(3.38a)

"N,N+I

where

~

is the unit (2x2)-block, multi-indexed matrix (i.e., the unit-ent-

ries of the

I[

will h a v e

L N°'N ,N+I

madn equals

writing the matrices ving that the L H S

x

coordinates

L NO'N ,N+I

A N

" The

I[ . E q u a t i o n

ConsequenUy,

(3.38b)

the O N

can be shown

matrix while the R H S

s o that the R H S

caJn b e s h o w n

conditions

'backward' generalized C h o l e s k y

(jl,Jl,J2,J2)) w h o s e

(3.38)

of s e c o n d - o r d e r b l o c k - e n t ~ of

is u p p e r -

in a similar w a y . express

the 'forward', resp.

factorization of the block, multi-indexed co-

c a n b e generalized to the 2 M - t h order s e q u e n c e s

forward w a y . W e

obser-

is the unit multl-inde-

variance matrix of the fourth-order stochm.stic s e q u e n c e . W e thai (3.38)

do-

by

'on the paper' a n d then,

is a s y m m e t r i c

h'iangulam (with unit diagonal-entries) x e d matrix

and

relations (3.38a)

HO, N ,NN + I

~ud

of (3.38a)

(jl,jl)

also r e m a r k that, a s s u m i n g

c~n

observe

in a straight-

thai the underlying s e q u e n c e

is

(and, hence, is r e p r e s e n t e d b y the t w o - i n d e x e d left-upper

o~N

H N , N + I ) , conditions

(3.38)

wiU reduce

to the C h o l e s k y

fac-

£orizat[on of the t w o - i n d e x e d covaria/qce m~trix associated with the linear least-squares prediction p r o b l e m

3.2.5 M - D

We

Fourier series e x p a n s i o n

can observe

that the 'biggest' s u b s p a c e

to the biggest 'estimation space' The

( s e e Deprettere a n d Lie (1980)).

latter s p a c e

has been

_ I,N-I YN-I,N

considered

_I,N-I RN_I, N

will c o r r e s p o n d

, u n d e r the i s o m o r p h i s m

(2.47).

in order to obtain the generalized O N

62 expansion

(in the form of the stochastic

for the r a n d o m

variable

Yo

functional M - D

, in the s p a c e

lynomials, yielding the o p t i m u m predicUon

that the

is actually the represento.£ive

(2.47) above

i°

problem

(see

s o that the s u b s p a c e

of

development

of the element

Yo

nonlinear

(1984b)).

We

not/ce

in the i s o m o r p h i s m

will play the role of the m e n t i o n e d

'estimation space'; i.e., it will b e u s e d

red O N

(second-de.~ee)

e.g., Z a r z y c k i

_I,N-I ~N-I,N

series)

of the Volterra functional p o -

solu1~on to the

least-squares

Fourier

i

in order to o b ~ n

. B'ollowln~

o

(3.35), w e

the desiget

N-I

~I,N-I

-i,N

"

~ ~ v { B nI } nmO

(3.39a)

s o that p

This

decomposi~don

ties e x p a n s i o n

i,N-i I[;N-I,N =

c~n

N-I ~ n=0

be used

for the

1

PB;n

1

(3.39b)

in order to obtain the M - D

Fourier

, in the form

o

N-i

i

We

shall s e e

~

O

later that (3.39)

for the set of M - D

impulse

filter, yielding the o p t i m u m tion problem. solved

once

is obtedned,

and

w i U imply a generalized

responses

we ON

that the basis

will s h o w basis can

sulting in the nonlinear ladder-filter

3.2.6 O r d e r - u p d a t e

We

wish

of the

be

Fourier

(second-de~ree)

solution to the non)/near

In other words, the 'backward'

(3.39c)

B i~) ~ in

(io'

~'

n=0

se-

expansion nonlinear

least-squares

predic-

that the nonlinear p r o b l e m

of the s p a c e computed

of generalized

in a recursive

is

matrices

way,

re-

algorithm.

recursions

to s h o w

Lhat the 'glob~[' forward

as well as b a c k w a r d

ON

63 bases

be

can

recursively, via a set of appropriately defined

introduced

'Ioc~Ll' e~nd '.~oba1' o r d e r - u p d a t e show

how

the

(n+l)

and

of

A x n+l

xtn (In,n+1) A x n+1

'glob~l' o r d e r t f o r w a r d

A x n+l

B x+l n

. Let u s

( B nx+ l )

observe

should

be

that the

B x n÷l

can

'local' o r d e r - u p d a t e backward

w i s h to

eund b a c k w a r d

(mutually orthonorma/)

all oi~hogoned to the s u b s p a c e

. Using the decompositions (3,35)

and

we

]~.x n+l

b e obtained recursively f r o m the n-th 'global' o r d e r solutions

solutions c a n A x n

recuLrsions. In other w o r d s ,

the index-set

and ( 3 . 3 0 ) , we will s h o w

executed

A x n

and

on

the L - a n d

]~x+l • These n

recursions, c o n s i d e r e d in p a r a g r a p h

errors A nx+ I

entries of the

that

B-, f o r w a r d a n d

recursions 3.3)

after e a c h

'local' o r d e r - u p d a t e

]~x n+l

and

(based

will p r e s e r v e

ed orthonormality of the 'local' entries of the 'globe/' f o r w a r d approximation

Kx + i ' n n,n+l

b e obtained via a set of appropriately introduced

recursions,

entries of the

entries

and

on mutu-

backward

step. C o n s e q u e n t l y ,

the

' obtained that w a y , wi]/ form the O N

seLs age/n. Reinterpreting the index-set s c h e m e order-update

step c a n

be

described

as

of Fig. 3.1, the d e s i r e d 'globe/'

follows

~x n+1

'new'---).

A x n

x

(3.40)

An+l

Bx+l n

Considering (3.28)

and (3.33)

that the number of entries of one,

during

the

order-update

with Ax n

step

n

replaced by

and of

Bx n

n -~ n + l

, yielding

n+l

, we notice

will be augmented b y Ax n+1

and

Bx n+l

'

64 ( m u c h like in the index-set s c h e m e

of Fig. 3.1). Thus,

e a c h '~lobal' order-

update step will introduce the following 'new' B-for%yard a n d B - b a c k w a r d approximation error

A x'n+l n+ l,n+ 2

whose

=

B x'n+l n+ l,n+ 2

unnorma/dzed version will be e x p r e s s e d

Ax,n+l n+1,n+2 =

as

Bx,n+1 = pl x,n n+l,n+2 If;n+l,n+l ix,x+n+l

(~s i£ follows from (3.18a) from (3022b)

The

(3.41a)

with

n

with

n

replaced b y

I

x,n In+l,n+ 1

replaced b y

n+l , a n d

(3.41b)

v=n+l; a n d

n+l).

four types of the 'local' order-update recursions, indicated in

the diasram

(3.zi0), car* be interpreted as follows:

LL-recursion: 'local' order-update for the submatrices

LB-recursions:

iAx

and

~x+l,n-I

'locai' order--updates for the submatrices 2AX and nlm 2ix n,n+l a n d

1Bx+l'n-i ntm iBx+l'n n,n+ ~i

(m=l,,00,n) (m=n+l)

BL-recursions: 'local' order-updates for the submatrices iAx'° n,1

arid

IAX'V n~v+1 and

2 B Xn,l

(v,-0)

2BX'V'l n,v÷l

(v=$, ....n)

BB-recursions: 'local' order-updates for the submatrices 2AX'° n,m 2AX'V n,m

As

and and

an example, w e

2BX n,m 2BX'V-i n,m

(v=0

(v=l,...,n a/nd

shall discuss in s o m e

update recursion while the remainin~ LB, B L ted in A p p e n d i x

2.

and

m = 2 .....n+2) m = v + 2,...,v+n+2)

detail the LL 'local' order-

and BB

recursions are presen-

65 PROPOSITION

Given we

3.1

(q)he L L

the L - [ o z w a r d

have

and

=

Sx = n,l

x

{i-

[ Pn,1

]2

}

-~

x ]2 - ½ {I- [Pn, l }

{[

recursion)

L-backward

the following r e c u r r e n c e

Ax

n,1

(3.16)

normalized

(3.20)

approximation

errors,

BX+l,n-i ] } n,o

(3.42a)

relat/ons

A x n,o

x

0n+l]

x A x {-P n,1 [ n,o

- Pn,1 [ 0 o

0n+l

] - [0

BX+l'n-l]} n,o

o

(3.42b)

where

x l = Pn,

([

A xn,o

0n+l ] ' [ 0o

(3.42c)

Bx+l'n-l]) n~o - ix n,l

PROOF. LL-fo~vard Let u s

recursion

consider

]TX

n,l

This

subspace

Ix n,l =

~

x

{ In,l }

v

caJn b e rewritten, in a c c o r d a n c e

v {i x

,

[x+l'n-3"} n,o

=

v {i x

'

with (3.7), a.s

Ix + l ' n ' l n-i,n

SO that it contains the 'new' uni-variate e l e m e n t with

Kx n~o

(3.15). T h e

variate' submatrix;

i.e.,

L-forward

estimate of the

AX In, 1 , wi~l b e e x p r e s s e d

^ = p x+l,n-1 IX, l H;n,o

IX

E

v{I x+l'n-I } n,o

' ix+n+1}

ix+n+ 1 I as

x

, in c o m p a r i s o n

with u p d a t e d

'uni-

The

error

~,A~I , a s s o c i a t e d

Ax s pl x+i,n-i n,l K ; n,o

and

D Ax

n,1

(3,25b)

=

m,

Ix

1

66 A i~, I~

with

x

-[ 0

, will b e

^x ] in, l

o

Lx ,= L x U {x+n+l} n~i m,O

, in

expressed

as

Ex+l,!d-I n,o

l

accordance

with

(3.7).

Prom

~ve o b t ~ n

Ax

=

(I-

P

n,1

=

- P

(I

x+l,n-1 ~|n-l,n

"

x+l,n-l% i x _ p x+i,n-i ix K;n-l,n " B;n,o ,% ix n,o

_

= ix

_ x+l,n-l~ - • B ; n,o ] ix

[0o

=

x+l,n-i - PB;n,o

0n+l ]

1

x

Therefore,

ax = n,l

~o

DI

= Lx : n,l

x

Since

p 0n+i]

by

-

x + l,n-i B;n,o

(3.16a). S i n c e

{x} u L x + l ' n - i n,o

, we

x

=

x+l,n--i

Rx + l , n - I n-l,n

but

L x+l'n-1 nlo

but

have

X

AX £ in, o

1

DB x+l'n-i n~o

[G,o % + i ] - (Ix'[°o B ,o

a ,l "

we

expressed

Ax

with

[A x n,o

BX+l,n-$]

rlsll

IB x+l'n-i n,o

(in & c c o r d a n c e

%vith (3.20)),

obtedn

ax

n)l

= [ax

n,o

According

([a~, °

on+ i]

-

( Ax

to (3.16), w e

o+ 1],[o o

[ n)o

' [0o

x+l,n-i Bn,o ] )lx [0o n,l

get

x+lm-i ] ) I x

Sn, o

0n+l]

=

x

IIA n,o

llo xn,1

Bx+l,n-i] n,o

67 where

we

defined

x Pn,1 "A ( [ A ~ , o

B Xn,o + 1,n-l] ) 11x

On+l]'[Oo

nll and we

can observe

Ax,1 =

x [p n,1 ~ -< i. C o n s e q u e n t l y ,

that

flAX n,o II {[A xn,o

Renormalizing

A xn,l ' w e

observing

B x+l'n-I ] } n)o

0n,x [0 o

get

Ax x -I x A x n,l = ]IAn,II] [IAn,o H {[ n,o

Finally,

x

On+ 1] -

0n+l]

B x+l'n-I ] } n)o

x - p n,l [ 0 o

that

x 1 , A:,l)llX (An,

x lll-2 IIAXn,o llAn,

= I =

II2 (i-[ Pn, lX ]2)

n,l we

obtc~dn t h e

LL-backward

normalized L L forwox'd recursion

recursion

Let u s r e w r i t e

In)l x

=

v

ITx ) in accordance n)3.

{I x

,

n,o

ix+n+l

} =

v {

with (3.7)i as

+l'n-I i x ' 1xn-l,n

so that it contains the 'new' element (3.1Q). T h e

L-backward

estimate

1

x

~:,i

Vx ~ p x ln, l X;n,o i x + n + l

The

(3.42a).

, in c o m p a r i s o n of the

~

' Ix+n+l )

ix+n+ I

with

is given b y

v{ lXn,o }

approximation error, c o r r e s p o n d i n g to that estimate, is then

X I x 8n, 1 = P W ;n,o l x + n + l

=

Ix+n+l

Vx - [In,l

0n+l ]

Kx+l'n-3" nlo

±

ix n,o

68 and

]9 Bx = Lx = n,1 n,l

From

(3.23b)

5~n,l

(I - P

(I-

=

=

U L x+l'n-I n,o

in a c c o r d a n c e

'

with

(3.7)°

get

we

=

{X}

x+l,n-i x ~;n-l,n - PA;n,o ) ix÷n÷l

P

x+l,n-l~

K;n-l,n

ix+n+l

"

-

p

" Ix+n+l

--=

x

A|n,o I x + n + l

[x+l,n-i

[0o

=

0n+l ]

n,o

_ p

x

A;n,o

ix+n+l

ConsequenUy,

~n,l =

where

B x+l'n-I

=

=

BX+l,n-I n,o

Lx

:

n,l

"

nlo

p

x A;n,o

ix+n+l

(3.20). S i n c e

Lx n,o u { x + n + l }

BX+l'n-l]

[0 °

n, i

] -

is g i v e n b y

rlsO

Dlx+n÷l

Bx

[ 0o

(lx+n+

, we

1

'[ A

= Lx n~o

DA x

n~o

but

h~ve

xn,o

0n+l])~x n,1

Since

~x+l,n-I n~o

(3,20)),

Bx

n,l

=

we

[0°

but

i A x nto

(~cco[ding

to (3.16)

.a/qd

obtain

[ Ax

B x+1'n-1 ] - ([0 ° B x+1,n-l] nso

nio

!

n~o

°~+1])

x [AX,o n,l

:

II~ +I'n-I il { [ 0 n,o

Renormalizing

o

sx+1,n-1 ] n,o

-

B xn,l ' 6und o b s e r v i n ~

x x (Bn, 1 , Sn, l)]Tx

1

=

IIB x

x

Pn,l [ A X , o

0n+l] }

that

x+l,n-i 2

n,1 II-2 llBn, o

II

n~l we

obtain the n o r m a l i z e d

LL

backward

recursion

x

(l_[pn,1

(3.4~2b).

]2)

0 n + l ]=

69 rl~he remaining 'local' order-updette recursions c~.n be obtained in a similar m a n n e r , We

can observe

(1983a);

a n d o/~e s u m m a r i z e d

in A p p e n d i x

2.

that these recurslons will constitute the generalized

nonlinear L e v l n s o n Dewilde

LIB, IBL a n d BIB ,

ed~orithm, introduced a/~ebraicaJ/y in Z a r z y c k ~ Zarzycki

vinson algorithm h a ~

been

and

(1983). Here, the generalized nonlinea/" L e introduced geometrica/ly, using projection m e -

thod, a n d c a n therefore b e interpreted e~ a m e t h o d for ( ~ r a m - S c h m i d t ) orthogonalization of the ba.sis of the s p a c e ent-matrices. W e

of the generalized coeffici-

notice that, neglecting all 'bi-variate' terms

entries of the t w o - i n d e x e d

submatrices

quently, reducing the s c h e m e obtain the generalized

and

to the L L recursions

(nonstationary)

ciated with s e c o n d - o r d e r

2A...

sequences),

(i.e., the

2B...) and, only, w e

linear L e v i n s o n

conse-

immediately

algorithm ( a s s o -

reported in Deprettere

a n d Lie

(1980).

'I'he p a r a m e t e r s

p, c o m p u t e d

as the reflection coefficients. F o r

b y the algorithm, c o n b e interpreted

example, the

x P n,1

calty equal to the L L reflection coefficient c o m p u t e d sented in Ze~rzycki a n d D e w i l d e

(1983a)

This follows from the i s o m o r p h i s m

b y the algorithm pre-

(1984:b), a n d similarly

b y the L B , B L

and BB

'global' order-update

x-labeled 'level' ( w h e r e

x=0,...,N-n

step and

of the reflection coefficients is c o m p u t e d gorithm:

recursions.

(2.47).

C o n s i d e r i n g the 'local' order-update recursions, w e that during e a c h

is numeri-

and/or b y the time-variant n o n -

linear ladder-filter algorithm p r e s e n t e d in Z a r z y c k i for the reflection coefficients c o m p u t e d

(3.420)

n

+ n+l

can observe

, executed

at the

nf0,...,N), the following set b y the nonlinear L e v i n s o n

al-

70 x~n

/~n n+l,n

pX, n

...

P

xln-1 P n,n+ 1

x,n-1 On+ i,o

""

x,n-I P n+ 1,n-I

°

"

P n+l,o

BL

n+1,1

BB

P

xsn n+1,n+3.

x,n-i Pn+ 1,n

:

(3.43)

i

X~O

xlo

P n,2

Pn,3

x Pn,1

P n,2

xlo

P

"'"

P n,n+l

x

xto P n+1~1

n+1,o

"'"

x

LL

x P n÷l,o

LB

3.2.7 O p t i m u m

approx/maH0n

Considering

'O f ~ h e M - D

impulse

responses

'we obtain the following oz'£hogonal e x p a n -

(3.39b),

sion for the N-lh 'global' o r d e r L-forward approximation e r r o r A o N,o Ao ( a s s o c i a t e d with the d e s i r e d N-th order estimate IN, ° of the I o )

o

_±

I,N-I

A N,o " lJI[;N-I,N

1o =

o B 1 -- l o - Po,1 o,o N-1 -

Interpreting f/mum

Z n=l

A °N,o

-

N-I ~ o _ l,n-1 n,=l Pn, l ~:~n,o

n o Bl,n-I ~ Pn,m+l n,m m=1

nonlinear

io

(equiv~lenf/y - b y

o 1,n Pn+l,o B n , n + l

impulse

approximate

~hat (3.4:4) is ach/ally the genera/ized

(3.4&a)

(3.&&b)

n=O

~s the set of the M - D

(second-degree)

element

N -i ~ -

+

prediction

Fourier

the r a n d o m

responses

of the op-

filter, .Are notice

series, g e n e r a t e d

variable

Yo

, under

by the

the

71 isomorphism words,

(2.47)),

in t h e s p a c e

this is the desired

reflactlon coefficientS, rithm, a r e (3.39c)

actually

ana/or We

gorithm,

of t h e

expansion

computed

@eneralized

(3.39c).

We

also

by the nonlinear

the generalized

Fourier

matrices. remark

prediction

coefficients

In other that the

filter algo-

in t h e

expansion

(3.44).

can

therefore

introduced

nonlinear L e v i n s o n

conclude

in this chapter algorithm, c a n

that the nonlinear ~nd be

prediction tilter al-

equivalent to the generalized

interpreted

~s the m e t h o d

cursive (actually Grarn-Schrnidt)

o~ho~ona/iz~/on

ce of the generalized

resulting in the o r t h o n o r m a l

matrices,

for re-

of the ba~sis in the s p a exparlsion

for the desired

estimate. In other %,vords, the n o n l i n e ~

le~t-squ~ires

diction p r o b l e m

for higher.-order stochastic

ca/n b e

fly c o n s i d e r e d M-D

impulse

as the p r o b l e m responses

of o p t i m u m

of the desired

nonlinear filter a/gorithm, p r e s e n t e d method for computation the Volterra-Wiener

Let u s

class, since the ON

for the I-D

observe

impulse

'bi-variate' terms

response

(3.44~b) in the O N

of P r o p o s i t i o n

prediction

of the o p t i m u m

by

an

a4oproximate

expansion if

of the set of

treated as

o~hogonal

(1.8)

(3.~)

Uqe

effective filter of

is actually a n

M=2.

to the Fourier line~

expansion

pcedictlon filter, if

in the nonlinear

the existence

equiva/en-

filter. M o r e o v e r ,

be

will r e d u c e

will be improved

son with the linear a p p r o a c h )

Recalling

approximation

are ne~_/ected. In the next p a r a S r a p h

estimation a c c u r a c y

3.2.8 Estimation

expansion

that (3.4~)

ON

here, c a n

of the nonlinear

explicite form of the Fourier

sequences,

pre-

we

wiJ/ s e e

case

that

(in c o m p a r i -

of the s e c o n d

component

expaJnsion.

accuracy

(3.42a))

and

3.1) w e n o t i c e

comparing

with equation

that the norms

(i)

in t h e p r o o f

of t h e L - f o r w a r d

approxima-

72 tion errors in the L L 'local' order-update

recursion

can

be related as

follov~s

(3.-[p x

IIA x I1 --11A x II n,1 n,o

Considerin~

the L B

similar w a y

that

flAx, roll =

'local' recursions

Combining

II

(3.45&)-(3.45c),

'global' order-update

step

llA~+1,oll =

where

= ]1Anx , n + l

x lqLL:n+l

we n

(3.45a)

of A p p e n d i x

(J_- [Pn,m ] 2 ) ½

IIAnX, m_111

x I]An,n+ 2 II ,~ IIA xn + l , o

]2)½

n,l

,

2, w e

m=2 .....

n+1

]2) ½

,

11( 1 - [ p x + l , o

obtain the error-norm + n+l

can show

(3.,~5b)

re=n+2

(3.45c)

relation in the

as

IIA~n,o IIR LxL ; n + I R xLI3~n+l

is the factor associated

(3.4~a)

with the L L 'local' recursion

~L~n+x ~ ~ (i- [~,i]2) ½

while

RLB;n+IX

RLB;n+I

order-update

{3,46)

express

case,

]fAn+iX II =

'local' recursions

the

(l-[pX+l,o]2) ½

error-norm

step, in the ( s e c o n d - d e g r e e )

In the linear L e v i n s o n

to the L B

as

Lm=a (1- [pnXm]2) ½

"£'hus, e q u a t i o n s

(3.¢6b)

is the faLctor c o r r e s p o n d i n g

(3.45b,c), a n d is e x p r e s s e d

in a

(3.4=6) w o u l d

reduction

(3.46e) in t h e

nonlinear L e v i n s o n be replaced

II A xn H (i- [Pn+iX ]2)½

'global'

scheme.

by

(3.4=7)

73 see a/so Deprettere a n d Lie (1980). Therefore, better estimation a c c u racy is a c h i e v e d

d u e to the factor (3.4~6c), associated with the L B 'lo-

cal' recursions. Considering

(3.~6)

for

x II AN_x, o II =

where

R LxL

x~0,...,N-x , w e

obtain

x x x II A o,o III~LL lqLs

(3.383)

is the factor associated with all L L recursions

at the x-labeled 'level' in the N-th order s c h e m e ,

x

RLL

while the

N-x-1 n Rx = n=0 LL;n+ 1

R LxB

=

N-X-1 II n=0

a n d is given b y

x

2

( 1 - [Pn,1 ] )

f~ctor is associated with all L B

executed

(3.48b)

"

'local' steps, e x e e u -

ted at that 'level', a n d c a n b e w r R t e n as

x RLB

"

N-x-i H n=0 N-x-i

11

[ n+1

n=0

Equations

(3.48)

x

(1-[Pn,m])

2 ½ ]

]2)½

(3.48c)

in the

N-th order, s e c o n d - d e g -

at the x-labeled 'level'. Interpreting the p a r a m e -

as a backward

b=0, the relalions (3.~8)

shift (i.e., time-delay) from the reference point will c o r r e s p o n d

Since the desired estimate

I

o

to the time-instant

corresponds

racy for that estimate will b e e x p r e s s e d In the linear N-th order s c h e m e ,

c o n s i d e r e d for

x

( 1 - [Pn+l,o

m=2

relate the error-norms

ree nonlinear s c h e m e , ter

x

II

by we

to

t=x .

t=0 , estimation a c c u -

(3.48)

with

x=0

.

get (accordin~ to (3.47)

n=.0,...,N-x)

N-x-1

o [I IIAN- xll = IIAX

n

( 1 - [%+,]2) x

(3.~9)

n~o so that better es~dmation a c c u r a c y

in the nonlinear s c h e m e

will b e achie-

74 d u e to the factor (3.4=8c)

ved

recursions

(since the n o r m s

associated with the L B

(i.e,, nonlinear)

of ~l/ reflection coefficients c o m p u % e d

by

the nonlinear filter algorithm are less than l).

algorithm, introduced here

*l~he generoli~ed nonlinear L e v i n s o n

geome£ric~lly using projection m e t h o d in the s p a c e matrices, ~ d

of the generalized

interpreted in terms of the o p t i m u m o ~ h o n o r m ~ l

malion of the set of M - D

impulse r e s p o n s e s ,

will imply the time-variant

orthogonad nonlinea/~ digital prediction filter. Its s t r u c ~ r e ties will b e d i s c u s s e d

in p a r a g r a p h

n=0,...,N

zx n,m

[izX n,m

o~nd

a n d proper-

3.4.

3.3 Nonlinear filter o/~orithm: t r a n s f o r m - d o m a i n

Let for

approxi-

approach

x=0,...,N-n

2zX ] n,m

[z

z

Jl

]

.

(3.50a)

(jl,j2)¢ L x

JlJ2

him where

r ]I -x = Lz I

z. J1

;

z..

z x

=

n,m

= [0

Z x'v . "ghen w e n0m

v {z x } n,m

sho/l denote b y

_I,N-I ~N-I,N

Pz;n,m

;

zx, v . v {zX, v} n,m n,m

"~-~Z;n,m )

_o,N raN,N+ I .

the

(3.51)

orthogonal projec%ion o p e -

Z xn,m

will b e the 'estimation s u b s p a c e '

u n d e r %he i s o m o r p h i s m be

(3.50b)

c a n consider the following s u b s p a c e s

rator, t~xking projection o n the s u b s p a c e that

zJ#-XzJ22-x]

Jl]2

exnd simil~rly for

We

0 ]

'

( Z nx,v , m). W e (see

notice

(2.44) wi%h

(2.47), while the 'biggest' s u b s p a c e

M=2),

will actually

75

Each element

cx n,m

~:,m(Z )

from the s u b s p a c e

[ }:,m(Zl)

=

Z x n, rn

wiU be expressed

%:,m(Zl.Z2) ]

as

(3.52a)

where

(>X'm(Zl)

-- Jl ~,

(3.52b)

f~n,m;Jl_x zJll-x

and

~x

(~I,=2)-

n,m

E

Jl

w h e r e the summations

E

O

J2

.

zi

zJ~-x

(a.52c)

n'm;]l-X'J2-x

a r e over the uni-variate, respectively,

in (3.52b,c)

over the bi-variate parts of the index-set introduce inner~-product on

jl_x •

.

~x n,m

Lx . Pollowing ntm

(2.46a), w e

as

( ~,.(z), ~x.,. (z))

. ~

(2") 2

;d@l

fd~l Cx

n,m(Zl)

i

e~V(e

iel

-x

"

,eI°~I) ~n,m(Wl)

+

1 x 2 e l W ( ie 1 i8 iw~ + --(2.)3 ~d81 Sd@ 2 [ d ~ 1 ~n,m(Zl,Z2) e ,e 2,e u.) -:,re(w1 ) + •

fde i fd~ I Sd~ 2 ~nX,m ( Z l ) l e 2 w ( e * 8 1 , e

i~

1

i~

,e

2

-x

) ~n,m(Wl,W2)

+

(2-) ~

(2.) 4

m e I f de 2 Sd~ 1 m ~ 2 ~ , m ( = l , z2) 2 ~ 2 W ( e i ~ l , e

i@2

,e

i~ 1 i~2) ×

× "~Xn,m(Wl,W2)

where

..m . i~ m e U w ( e ' 8 ,elm ) , re,u=1,2

(3.53)

a r e the entries of the ( 2 × 2 )

ral matrLx (2.14) of the fourth-order stochastic s e q u e n c e relations can be written for the s u b s p a c e s

~x,v nlm

,e

spect-

{y}. Similar

76 Pollowing

the o r d e r i n g

scheme

of p a r a g r a p h

3.1, let u s

introduce

the 'loc~l' s u b s p a c e s :

L-forward

z xn,o B-forward

(for

v {z

X

'

Z x + l,n-1 } n-~i~n

(3.54a)

v-0,...,n)

Iv Z x'V n,v+ i

=

{ z

Zx X,X '

l

v { zx,x+ v

}

v=0

n,o x,v-1 , Zn, v }

'

(3.5~b)

,

v=l,...,n

L-backward

X x,n-I v { Z x'n-I n,o --" n-l,n ' Z x + n + l

B-backward

(for

m = l,...,n+l)

Z x'n-I V { Z x'n-1 n,m = n,m-1

' Zx+n+l-m,x+n)

Z x,n n,n+l

' Zx,x+n }

= v { Z x'n-I n,n

3,3.1 'Local' estimates

Following forward

and

}

(3.54)

and

,

m=l

.... ,n

,

m=n+l

(3,55b)

(3.55c)

errors

and

(3.55), w e

back~v~J:'d estimates

and

can

errors

introduce

the L - a n d

IB-,

of the n-th '~lobad' order:

J_,-fo r v v a x ' d A

"]~he L-for-~ard estimate

zx n,o

of %he e l e m e n t

^x A P x + l,n-I Zn,o = Z;n-l~n Zx

£

z

x

~ x + 1,n-i n-l,n

will b e

defined

as

(3.56a)

77

while the corresponding

L-forward

Ax h _ i x+l,n-1 n,o = ~'~Z;n-l,n

Its normalized

version

AX, o ( Z ) = ix

I]Ax

njo

We

can observe

n=N

B-for~vomd The

]]-1

F~m°

n,v+l

(for

Ax n,o

Zn, °

(z_,z.) ] .L

a n d the errors

and

(3.56c)

-"

n,o '

(2.57), if

MAN; °

,o

1.-2 , x=.0

v=0,..0,n)

=A

^x,v Zn,v+ I

I p_ X z x~;n,o x,x p

x,v-1 Z;n~v

approximation

h

of the elements

C

~x n,o

z x~x+v

errors

Z

Pz;n,o

£

are

X~X

will be

V==0 (3.57a) v=l,...,n

by

Zx

v=O

nso

(3.57b)

l

Z x'v-1 nsv

'

v=l,...,n

form,

o

,

z x,x+v

'

Z x'v-I n~v

given

l

p ± x,v-I Z;n~v Zx~x+v

or, in a r e n o r m a l i z e d

n,v+ 1

as

[AXo(Zl)

M AZN; ° , M AN; °

± x

A X,V

(3.56b)

.

n,v+l

while the

expressed

=

error will be

zx+l,n-i n-l,n

±

Z

B-for-ward estimates

zAX,V

Zx

that the estimate

are precisely the and

will be

approximation

n,v+ 1 I[Z

cA

:L1(z1

x v

An,v+l(zl,z2)]

(3.57c)

78 L-backward The

L-backward

zVX,n-i n,o

estimate of

~= p

z

x,n-i Z;n-lpn Z x + n

a n d flee c o r r e s p o n d i n g

(for

"l~he B - b a c k w a r d

( 3.5 sa)

Z x , n-i n-l,n

±

(3.58b)

version

[~.Xln--it

= r x,n-I IF x'n-iI -i njo npo Z

B-backward

a~

error is given b y

£x,n-I =A p i x,n-1 z n,o 2;n-i,n x+n

B x'n-I (Z) nto

expressed

z x , n-i n-l,n

e

L-backward

EogeEher with its normalized

will b e

x+n

= ~n,o

s:,o (zi,~2)] (3.58~)

~Zl)

m = l .....n+l) estimates of the elements

zx+n,x+ n ~...i Zx~x÷ n

a/~e

given b y

zVX,n-I ~ _ x,n-i n,m l"Z;n,m-i Z x + n + l - m , x + n

VzX,n & p x,n-I n,n+l Z;n,n Zx,x+n

"l~he c o r r e s p o n d i n g

F x,n-i n~

B-backward

~ _ I x,n-i P Z;n,m-i

F x,n A= p ± x,n-i n,n+l Z;n~n

T h e i r normalized v e r s i o n s a n d are e x p r e s s e d

are

Z x'n-I n,m-i

m=l,...,n

(3,59a)

m=n+l

(3.59b)

'

Z x'm-I n,n

~

errors are

Zx+n+ l-m,x+n

Zx~x+n

(

l

1

Z x'n-I nlm- I

,

Z x'n-I n,n

'

Bx'n-~z) n|m

similarly ~s in ( 3 . 5 8 ) .

a/ad

m=$,...,n

(3.59c)

mmn+l

(3.59d)

B x'n _ ( Z ) n,n+l

respectively,

79

Let us o b s e r v e tes

that the L-- a n d B-, forward

z~n,m ' zAx'Vn, m ' vX,Vn,m a n d

Ax A x'v B x'v njm ~ nlm * njm

) ' A nx,v .m(Z)

AXm(Z

of the estimates

the representatives errors

errors

-I X m

of p a r a g r a p h

s

~ n d b~ck%vard

~_x,v n#m

'

~ xn,m ,v

3.2, u n d e r

estima-

, B xn,m 'v(z)

and

are

of the

the isomorphism

¢2.~v).

3,3,2 Decompositio n of s u b s p a c e s !

x x,v ' l--I_ x,v I-IA; ,m ' l-'I~ ~;n,m ~:~;n,m

Let

opera~r~ o~ ~

a n d NI-D Fourier

denote

~ub~p~es span~.a by

respectively. T h e (3.59c,d)

uON b a s e s

the ort/nogon~l projection

A~,m(Z)

orthosonalit'y conditions

expansion

'

A ~n , m (Z)

B~'V(z) n,m '

'

(3.56b) ,(3.57b),(3.58b)

will imply the following ort/nogonal decompositions

and

of s u b s p a c e s

and of projection operators:

L-forward

z x

n,o

= z ~+I,n-I n-l,n

X Pz;n,o

S-forward

~x,v

(for

v

X+ l,n-i Z;n-l,n

x z)} {An,o (

(3.~0a)

l-I x A;n,o

(3.60b)

+

v=0,...,n)

---

n,o

n,v+1

| z x'v-I

~.

I =

(3.61a) e

v ( A x'v

n~v

p

x,v PZ;n,v+I

= p

®

p

(Z) }

n~v+ 1

v = l .....n

'

x Z;n,o

+

I-I x,o A;n,1

'

v=0

x,v-I g;n,v

+

r-I x,v A;n,v+l

'

(3.61b) v=l,...,n

8O L-backward

Z x , n-I ~x,n-I n,o = n-l,n

p

x,n-1 ~ p x,n-I Z;n,o Z;n-l,n

(for

B-b~ekward

• V {

(3.62a)

I-I x,n-i B;n,o

(3.62b)

+

re=l, .... n+l)

2x,n-i Zx,n-I n,m = n,m-i

~

x,n-i V{Bn,m (Z)}

Z x'n = Z x'n-I n,n÷1 n,n

$

V {B:::

p

Bx, n-l( Z ) } n,o

,

+l(Z)}

m=1,..n

(3.63a)

m--n+l

(3,63b)

x,n-i p x,n-i Z;n,m = Z;n,m-i

+

x,n-i lqB;n,m

,

m=l,...,n

(3.63c)

p.. x~n _ = p x~n-I Z;n,n+l Z;n,n

+

x,n I-IB•,n+ 1

,

m=n+l

(3.63d)

We

can observe

A~(Z) - [ A~,o(Z)

that the entries of

Ax'o(Z) n,~

"'"

A x'n

(3.64a)

n,n+i (z) ]

as well as of

~(z)

- [B

x,n-I n, o

will form the O N much

Bx, n-1 (z)

(z)

n,n

sets in the s p a c e

s~.n

n~n+1

lized v e r s i o n s

(3.64b)

of the generalized z-polynomials,

like their time-domain counterparts form the O N

of the generalized matrices. T h e

(z)]

sets (3.64)

sets in the s p a c e

are actually the o r ~ h o n o r m a -

of

[zx

Zx, x

...

Zx, x + n ]

(3.64c)

81 and of

[z

z,

x+n

respectively. Therefore, backward

...

x+ntx+n

o~hogonal

we

z

x~x÷n

]

(3.64d)

c a n write the following '~Iobal' forward

decomposi£ions

and

of s u b s p ~ c e s

zxn',nn+l m zx+l'n-ln-l,n •

v {AnX(Z)}

(3.65a)

z~, n

v {B~(z)}

(3.65b)

. zx, n-i

n,n+ 1

®

n-l,n

where n

~{A~(z)}-

~ {A:,o(Z)}

•

Z

• ~ { A X'~ .(Z)}

(3.~Sc)

~ {B X'n _(z)}

(3.65a)

n

~{B<(z)}.-

~

•

~ {B~'~-i(z)}

m=0

These

n,n+l

orthogonal decomposition

decomposi£ions

•

n,m

of s u b s p a c e s

imply the following 'global'

of projec£ion operators

= p

x+i,n-i

p

x,n Z;n,n+l

p

x,n = ID x,n-i Z;n,n+l --Z;n-l,n

Z;n-l,n

+

+

x

l-IA;n

x 1-1B;n

(3.65e)

(3.65f)

where n

x

x

I-IA;n ~

r-IA;n,o

x I-'IB;n =

n ~ m=0

Consequently, ~,N ,N+I =

we

+

~ v-,O

l'q x,n-i B;n,m

+

m

x,v

A;n,v+ 1

l-I x,n B;n,n+l

ge£ the following orthogonal

N x ~ • v {~_x(Z) x~O

}

=

(3.65g)

(3.65h)

decomposifion

N o ~. @ v { B N _ n ( Z ) } n,,,O

(3°66)

82

,~

Hence,

se~

A~(Z)

and

bal' forward, resp. b a c k w a r d

ON

mbdti-variate z-polynomia/s, Ms

B~(Z)

o~

b~sis

be in~erp~te~ as

of the s p a c e

square-integrable

We

stocha~/c

sequence

'~lo-

of the generalized,

o n the generalized

with respect to the spectral matrix of the underlying

(Le., four~-order)

the

unit-to-

hi,her-order

{y}.

also obtain the following or£hogonal

decomposiL%on

of the 'es-

t/matio n space'

N-I

zI,N-I

Z ® v {B~(Z)}

N-l,N

which

=

(3.6w)

n~0

implies

p

Let u s

observe

representative

that h e r e of the

"l~hus, the Fourier

the element

Yo

Following

z

and~or of the

series for the

terpart of Lhe o p t i m u m problem.

N-1 Z i = I-IB; n n=,0

I,X-I Z;N-I,N

z

(3.67b)

o

~

[i

10

0]

will actually b e the

in t h e i s o m o r p h i s m

will b e the tra/nsform-domadn

o

solution to the nonlinear least-squares

(3.67b),

that O N

expansion

will b e

(2.47). coun-

prediction

expressed

as

N-1

%

~

z

(%.~,~(z))

z ~(z)

(3.67c)

rx=O

We

will s e e

of o p t i m u m

later that the a b o v e approximation

ON

series

can

be

interpreted in terms

of the set of transfer functions

of the nonlinear

predication f/Iter. We (3.67a)

notice that the O N

shoudd

be

derived

In the next p a r a g r a p h the tra/asform-domaln

we

backward

polynomial

basis

in order to get the Fourier will s h o w

version

that the desired

of the nonlinear

of the s p a c e

expaJnsion

basis

(3.67c).

is c o m p u t e d

prediction filter a/gorithm.

by

83 3.3.3 O r d e r - u p d a t e

recursions

In this poragrexph w e ce of the generalized

w i s h to s h o w

polynomials

c~

that the O N

basis of the spa-

b e recursively

computed

b y the

transform--domain counterpart of the nonlineo.r filter algorithm, derived in paragraph

3.2.6. In order to do that, it is sufficient to s h o w

higher-order sets of the generalized

polynomials

~:+I(Z)

can be derived recurslvely from the lower-order B:+I(z), be d o n e

preserving

that the

ON

and

sets

~x+l(Z

A:(Z)

and

the mutual orthonormolity of their entries. This

can

b y introducing a set of appropriately derived 'local' ordeF-upda-

te recursions,

executed

sets

and

~k:(Z)

the L L , L B ,

BL

o n the multi-variate z-polynomial

IBx+l(Z)

and BB

much

'local'

entries of the

(3.40)

l i k e in t h e s c h e m e

recursions

of that scheme,

will

Here,

be inter-

p r e t e d as follows:

LL-recursion: order-update x An, ° ( z I) LB-recursions:

step for the polynomials _x+l,n-l, ~ n,o [ zl )

and

orders-update steps for the polynomials x An, m(Zl,Z2)

x+1,n-i 8.r'~d Br],m (Zl)

BL-recursions: order-update x~o

An, l ( % ) A

X$V

BB-recursions: order-update

m = l .....n

,

m=n+l

steps for the polynomials 2£

and

.i(~ ~)

~x+l,nz ~ ~n,n+lkZl)

and

A : , n + l (Zl,Z2)

,

Bn, a(zl,~ 2)

~d

,

~-0

B~;+~(~IZa)n

v l .....n

steps for the polynomials

An, m(Zl,z 2)

and

B~,m(z1,~ 2)

X,V An, m(Zl,Z2)

and

B X'v-I (Zl,Z2) n,m

, v=0 ,

and

V= l,...,n

m= 2,...,n+ 2 and

re=v+ l,...,v+n+ 2.

"I'hese 'local'

recursions

can

be introduced

by considering

the

)

84

orthogona/ decompositions

of the sub.paces

and of projection operators,

m u c h like in the tlme-domaRn solution considered

in the previous para-

graph. For example, the LL 'local' recur.ion is introduced via

PROPOSITION Gi~

m~

3.2

>fo~=d

error

A~

(Z)

(~.~6c), ~ d

the L - b a c ~ d

~O (3.58c), w e have the following recurrence

Bx+I'n-I(z) nmo

~r~o~

relations

(3.68a)

where

@n,1

2 )-~ A (i-[0 x n,1 ]

1IX :I

(3.68b)

Pn,1

,

x ~n,i : (AX,o(Z) ' Z'BX+I'n-i(Z))zn,o

(in accordance

and

(3.68C)

with (3.53)), with

Z'BX'v(Z)n,m

XtV

=A [ zl.Bn,m(Zl )

•

X)V t

\

ZlZ2 Bn,miZl,Z2 ) ]

PROOF. Pm-a/lels the proof of Proposition 3.1, and is omitted.

(3.68d)

85 We

can observe

ly equal t O

that the reflection coefficient (3.68c)

(3.4~2C), u n d e r the i s o m o r p h i s m

(2.47), a n d t h a t

transform-domain counterpart of the L L L e v i n s o n mo/ning transform-domadn versions be obtained in a similar way,

recursion

of the LB, B L

in numerical(3.68)

iS the

(3.42). "i~he re-

a n d BIB recursions

a n d are p r e s e n t e d in A p p e n d i x

can

2. C o n s i d e -

red accordingly together, they w i l . t constitute the transform-domain version of the generalized nonlinear L e v i n s o n

algorithm, w h i c h

c a n b e interpreted

in the tro.nsform-domadn context, o~ the m e t h o d for recursive or£ho~onalizaUon of the basis in the s p a c e

3.3.4 O p t i m u m

ON

Recalling

of the genera/ized z-polynomials.

approximation of the set of M - D

tro.nsfer functions

(3.67b), o~nd considering the 'local' fro.helots-domain re-

cursions of A p p e n d i x

2, w e

ca/n write the following O N

expo/nsion for the

AN,o(Z )

error

~,o(Z)

_ ! I,N-I = I'JZ;N_I, N

zo N-:l.

ffi Z o - pOo,1 Bo,o( 1 Z)

+

N-I •

n ~

n--i m = l

I'n-I (Z) • P no, 1 B n,o n= i

-

o B i'n-I (Z) Pn,m+1 n,m

+

+

(3.69a)

N-l ~ po i,n n+l,o B n , n + 1 (Z) n-0

(3.69b) which is the genero/ized z

o

=

[1

0 ]

(M-D)

Fourier series, g e n e r a t e d b y the element

with respect to the O N

backward

polynomial basis of the

' e s U m a U o n space' in the transform-doma/n. In other words, this is the expi[cite version of the e x p a n s i o n va/ent to the Fourier e x p a n s i o n

(3.67c). W e

notice that (3°69)

(3.44), and~or to the stochastic functional

Fourier serles for the nonlinear estimate in the s p a c e ctional polynomials

(see

is equi-

e.g., Z a r z y c k i

of the Volterra fun-

(1984:b)), u n d e r the i s o m o r p h i s m

86 (2.&7). W e

can

therefore

nonlinear lea~t-squares sequences,

lea~st-squares

lc~th ( 1 9 7 8 ) ,

can

be

approximaf/on

the time- a n d

of

stochoistic

filter.

transform-domain considered

version

of the set of M - D

all bi-vari~te terms, this result will r e d u c e

predict/on problem,

to the c o n -

solutions to the linear

in Dewilde,

D e w i l d e a n d D y m (1983-), D e w i l d e

3.4 Nonlinear

We

ON

for higher-order

of the nonlinear prediction

Ne~lectin~ between

that the transform-domain

prediction p r o b l e m

is actually the o p t i m u m

transfer functions

nection

conclude

Vieira a n d

I~ai-

(1982)o

time-variant ladder-filter

can

observe

interpreted

that the L L

'local' order-update

a.s the following L L

recursion

(3.68)

section

B ~n,l (z)

Ax

(z)

n,~J

~- A~,I(Z)

(3.70)

B x + l,n-1 (Z) nlo

of the nonlinear

digital prediction filter. T h e

date recursions

can

a/qd B B

sections

b e interpreted

of that filter ( s e e

remaining

in a similar way, also A p p e n d i x

'local' order-upyielding the LIB, B L

2). C o n n e c t e d

dingly together, they will constitute the 'global' section nonlinear

filter (at the x-labeled

'~lobal' filter section

is p r e s e n t e d

n ÷

accor-

n+l

'level'). q~his 'local' structure

of the

of the

in Fig. 3.4 while the 'local' structure

87 of the t/~ird-order (N=3)

nonlinear prediction ladder-filter is s h o w n

in

Fig. 3.5. IBx,n+ 1 n÷l,n+2 Bx,n n+ 1,o n,n+l -f

x,n B n + 1,n n+l,o -)""'" -'~

n+l,n-l- ~

Bx,n n+ l,n+ 3_

An+l,n'~'~" An+3-,n+l

Bx, n-1 n+l,n-i

Bx,n-i n,n+ 1

~_ -- Ax, n+l n+ l,n+ 2

Hx, n-i n+l,n

:

f Ax,o n~2

Ax,o n,l

I

BXIO n+l,o

Bx, O n,2

x° + A x ° n+l,o ~% Bx n n+l

A~.~

nto

-~...-

.~ a~. n

..........×... ~

ax

Bx, v n,m+l ;

Ax n+l,o

,n+l

Bx+l,n n,n+l

Bx,v A x,v n,m+l

Bx,v_ 1 n~m

An+l,l

Bx n+l,o

Bx+l,n-I n,n

B X + l,n-Â nto

Ax, v n,m

x°

--'~"" ' > An,n+l

Bx Ax

xiO Bn+l,l

. AX, v n,m

AX,V n,m+l B x,v-I n,m

Fig. 3.~ 'Local' structure of the 'global' section of the timevariant nonlinear ladder-filter.

88

"l

,--Bo°(Z)-~

¢---

B

(

'(z)-m

[I

~~(z)

--J

u ---.~ ~a(z)

I

x--1

J ~(z) x=2

J ~3(z) o

x=3

> Initia/izations:

lu

~=

T

.)

'New' elements:

B2;2(z)

'

ao~,o(Z)

u

Bx'n _ (z)

'

n ~ n-l- 3

>

x=O,...,3)

Fig. 3.5 'Local' structure of the third-order (N=3) time-variant prediction filter. T h e symbols ponding 'local' 8-recursions of Fig. 3,4.

(n---i,2,3 ~ x=0,.,.,3-n)

nonlinear (Levinson) O indicate corres-

89 Let u s nested

observe

'local' 8-transformations

the s u b s e q u e n t m6dn

that the filter structure consists

sets of the forward

mutually orthonorma/

hal' order-update solutions

(actually - ~ i v e n ' s

step

in Figs,

a.s we]/ ~

after e a c h

(see

3,4: a n d

3.5). T h i s

means

of the 'local" ol~hogonal

Vieira a n d

De~vilde

(1982)).

cients

is associated

Kailath

These

(with n o r m s

(1978),

matrices

being

will re-

after e a c h

'~lo-

resp. horizontal

orthogon~9/it'y requirements, a/nd norma/ized

aJnd D y m

are specified b y

less than one),

errors

with the J-lossless

Dewilde

s o that

that the structure of this

as the filter consists

Dewilde,

backward

the subsequent-vertical,

will satisfy the desired

section

rotations)

'local' a/qd, hence,

nonlinear ladder-filter

sections. E a c h

of a cluster of

'local'

8-matrix

(see

(1981,1984),

the reflection coeffi-

actually the Fourier

coeffici-

enhS. Since the p a r a m e t e r observation, ar prediction

the reflection coefficients x , being the b a c k w a r d which

We

can

input s e q u e n c e

notice t h a t ON

ar prediction p r o b l e m

that

interpreted

(not Toeplltz)

well a~ back~vard

obtained

be

to the generalized

in meprettere

and

Lie

be solved

of the generalized the s p a c e

is reflected in the H e r m i cova/'iar*ce matrix. computes

the forward

~s

the solution to the N-th order nonlinein the innovations

context)

is

@/so r e m a r k

terms, the filter of B~i~. 3.5 Will immediately

(time--vari~/nt) linear L e v i n s o n

filter c o n s i d e r e d

(1980).

~Ve car, c o n c l u d e lem c a n

, and

'level' in the filter structure. W e

ne.qlectin~ all nonlinear

reduce

{y}

(actually c o n s i d e r e d

at the 0-1abeled

point of

is implied b y the nonstationarity

the nonlinear ladder-filter and

on

~s the 'current' time, this nonline-

of the genera/ized

b~es,

)galns )) d e p e n d

shift from the reference

filter is time-variant. T h i s

of the higher-order tian propet~y

(i.e.)the f i l t e r

Uqat the nonlinear

geometrically, (block,

using

multi-indexed)

of the genera/ized

least-squares

projection

method,

prediction

in the s p a c e

coefficient-matrlces,

(block, multi-variate)

prob-

and#or

z-polynomials

in

(provi-

9O d e d the higher-order c o v a r i a n c e

dmt~ or, equdva/ent|y, the higher-order

spectral functions of the underlyln~ s £ o c h ~ t i c former a p p r o a c h fine M - D

results in the o p t i m u m O N

impulse r e s p o n s e s

sequence

approximation of the set of

of the nonlinear prediction filter of the Volte-

rra-Wiener class. In the latter case, the o p t i m u m O N mation of the set of M - D

polynomial approxi-

transfer functions is obta/ned. W e

the results p r e s e n t e d he~'e ~ e s e n t e d in Z a r z y c k i

are given). "imhe

and Dewilde

equlv~/ent to [he algebraic solution pre(1983a),

a n d to the ~eometric solution

of the stochasf/c esf/mation problem, d i s c u s s e d in Z a r z y c k i

The

non]/near a p p r o a c h

(1984a, b)o

to the lea.st-squares prediction p r o b l e m

(for higher-order stoch~.~tic s e q u e n c e s ) the linear treatment)

r e m a r k that

may

estimation a c c u r a c y

result in better (than in

(if the s e q u e n c e

is n o n - G a u s -

sign), ho'vvever, complexity of the genera]/zed nonlinear filter p r e s e n t e d here [ n c r e ~ e s

rapidly ( s y n c h r o n o u s l y

step), a n d b e c o m e s

rather big e v e n

with e a c h

'global' order-update

in re~tively low-order nonlinear

filters ( c o m p a r i n g to the complexity of the linear filter), as it c a n b e seen

in Fig. 3.5. "l~herefore, the complexity reduction p r o b l e m will b e

the subject of the next chapter, w h e r e

time-invariant as well as 'quasi-

linear' ladder-filter algorithms will b e presented.

4. T I R I E - I R V A R I A N T

We

AND

noticed in the

'QUASI-LINEAR'

~DER-F'ILTERS

previous chapter that complexity of the ~enerali-

z e d nonline~gx' ladder-filter i n c r e a s e s

rapidly ( s y n c h r o n o u s l y

'global' order-update step), a n d b e c o m e s nonlinear filters. Consequently,

with e a c h

rel;~tively 'big' e v e n

in this chapter w e

in low-order

w i s h to c o n s i d e r the

problem of complexity reduction in nonlinear ladder-filters. In order t o obtadn efficient nonlinear filter algorithms, w e

w i n first d i s c u s s the nonlinear

least-squares prediction problem for stationary (in the higher-order s e n s e ) stochastic s e q u e n c e s . W e

~Nill s h o w

near Ume-invariant filter w h o s e

that the solution results in the nonli-

complexity is m u c h

reduced

(comparing

to

the genera/ized algorithm). Purther complexity reduction will b e a c h i e v e d by introducing simplified nonlinear estimation s c h e m e s ,

ca/led 'quasi-linear'

filters a n d associated with the o p t i m u m prediction of higher-order stochastic s e q u e n c e s

whose

'distar,ce' from the G a u s s i a n

s e n s e to b e defined). T h a t Zarzyckl and DewJ/de

problem

(1983b),

has been

is l o w

(in a

introduced algebraically in

and considered

ce of the Volterra functional polynomials)

sequence

geometrically

in Z a r z y c k i

(in the s p a -

(1984c,e).

4.1 Shift-invaria~nce of inner-products

Let u s

a~sume

stationary (in a w e a k

Ulat the underlying stochastic s e q u e n c e four~h-order s e n s e ) ,

Then,

{ y }

is

following (2.19), w e

92 will

obtain

I-Ix n,m

H x'v n,m

regardless

of t h e x - s h i R

with d o m a i n s

respectively.

- H e = "9 n,m n,m

-- H x + l ' v , , n,m.

are the g e n e r e d i z e d ces

= H x+l n,m

nom

Applying

T v n,m

(4.1b)

(i.e., the time-shift), where

(block, DT

H O'v ~ n,m

(4.la)

multi-indexed)

= L° x LO n~m n~m

(4.1a)

x x (b-~n,m ' G n , m ) xx

Toeplltz and

in (3.13),

we

"1"

n,m

covariance

DT v n,m can

and

= L °'v nlm

T v n,m

subma%rix L O'v nlm

,

write

= (_x+l _x+i~ - b ' n , m ' ('Zn,m) E x + l ~"

n,m

n,m

=

(FOn, m ,

A

F

G°m)

.T n,m

=

o

.~_T nmm

n,m

= ( F ~ , m ' Sn,~)~o

(~.2~)

n~m where

F

[fn,m;jl

=

f . . ] L° n'm;Jl'J2 (Jl'J2) ~ n,m

n,m

Applying

(4.1b),

we obtain similar relations

for t h e

(x,v)-labeled

(4.2b)

quanti-

ties

.,v

x,v

~ Fv

(Fn,m'Gn,m)ix,v

.~ v

n,m

.Sv

n,m

n,m

v

n.m "

(Fn, m ' G n ,

v

(4.z~) m)i[o,v n,m

wiLh

F v = l-l,m

[ fv . • n'm;Jl

fV . . ] L O'v n'm;JlJ2 (Jl']2) ¢ n , m

(~.2d)

93 Equations product next

(4~.2) e x p r e s s

the x-shift (i.e., time-shift)

in the h i g h e r - o r d e r

paragraph

simplifications

we

(i.e., fourth-order)

will show

of the nonlinear

4.2 "l~ime-lnvarlan% n o n l i n e a r

Following

Ao

n,m

A A

~__-

a

significant

inner-product

(4.2), w e

will satisfy the following

=

.

]

.

n'm;J1]2

notice

relations

[a v n'm;Jl

(¢.3a) (jl,J2) c L °

n,m

(v=O,...,n), w e

: A °,v n~m

Similarly, for the L- a n d

obtain

=

a

] n'm;]lJ2

B-backward

(jl,j2) E

errors,

we

L O'v n,m

can

write

= B x + I , n-I = B°, n-I = n~m n~m

A Bn-1 n,m

m=0~...,n , a n d

~':,~ n,n+l

in

=

errors

A x,v = A X + l , v n,m n,m

if

Of the

errors

n'mlJl

the B - f o ~ a r d

Bx, n-I njm

will result

In the

ladder-filter e d ~ o r i t h m

[a

==

A A v : n,m

property

case.

n,m

n,m

For

stationary

of inner-

ladder-filter algorilhm.

approximation

~ Ax+I=

n,m

this

the shift-invariance

that the L-forwc~rd

Ax

that

invariance

[bn-I n,m;n-Jl

for

~ Bn n,n+l

bn-1 n,m;n-Jl,n-J2

(4.40.)

] (jl,J2) ( Lo,n-i n,m

m-n+l

-_ [ b n , ~ + l ; n _ q ~

bn . . 1 n,n+ l;n-] l,n-] 2

(jl,ja)

e Lo, n n,n+l

94 Consequently,

the forward a n d b a c k w a r d

'global' O N

bases

will

satisfy

A x = Ax+£= n n

A ° £ A n n

(4.5a)

B *n - ~ +n * =

B ° ~: B

(4.5b)

n

n

wihh

A

B

We

n

-

n "

[A

A ° n,l

n,o

[ 13 n - I

n,o

"'"

"""

A n ntn+l ]

B n-I n,n

B n n n +.I]

notice that in stationary case, the entries of

initializations in the 'g/obal' order-update step higher-order forward a n d b ~ c k w a r d

solutions

(4.5c)

(4.5d)

B n

n ~ Z~n+ l

will be u s e d

n+l

as

, yielding the

and

l~n+ 1 . C o n -

sequentiy, in the stationoxy case: a)

there

is n o 'nesting' b e t w e e n

the x-labeled 'levels' in the structure of

the nonlinear ladder-filter; b) the nonlinear filter atgorithm c a n b e e x e c u t e d

at e a c h

x-labeled 'level'

s ep arately; c)

it is sufficient to run the a/gorithm at the (x::0)-labeled 'level' only,

following (4.5). Hence,

the stationary version of the generalized nonlinear ladder-filter

algorithm -#vil/b e obta/ned if w e

c o n s i d e r the 'loca/' LL, LB, B L

recurslons ~t the (x=0)-labeled 'level'. F o r sion of the L L "local' ordeP--updatte recursion

An,1 = ( 1 - [ P n , 1 1 2 ) - ½

Bn, I

( l - [ p n,1] 2) -P~

([An, o 0n+l]

(-Pn,l[An,o

and BB

example, the st~ttionary ver(3.42) will take the form

- Pn,1 [ 0o Bn-1])n,o

0n+l ] +[0o

B nn,o -l])

(4.6a)

(4.6b)

95 with

P~,I" ([A,,,o o . i] . [0o Bn-:t])Io~,o

(4.6~)

n,1 The ssed

transform-domain as

counterpart of the L L recurSion

(4.6) will b e expre-

(following (3.68))

InlzI 0. Fnozl

(4,7~)

with

69 n,l =" (I- [Pn, l]2) -½

(4.7b) -P n,l

and ~,i

The

remaining LB, B L

tionary c a s e

"

(A~,o(Z)'z'~.oi(z))z

and BB

'local' order-update

will be the 'local' recursions

(4.7=) recursions

of A p p e n d i x

in the sta-

2, provided the

x-label is r e m o v e d . The

L L 'local' recursion

[ion of the corresponding

(4.7)

cart b e interpreted a~ the L L - s e c -

nonlinear ladder-filter

B~,I(Z) A o(Z)

~

-- A i(Z)

B~-i(z) n

(~.Td)

g6 "l~he remaining

'local' LL, L B

and

filter will again b e

expressed

v i d e d the x-labels

~/'e r e m o v e d .

BB

sections

of the stationary nonlinear

a s the 'local' sections These

of A p p e n d i x

sections, c o n n e c t e d

together, w i U constitute the 'global' section

accord[n~y

n÷l

of the filter. W e

notice that the "set o f reflection coefficient~ c o m p u t e d

b y the alsorithm

the 'global' step x-labels Pn,m

n -~ n + l

are again

removed.

P vn,m

~qd

paraJneter

x

, will b e

(being

Observing

me

(N=3)

rithm c ~ n

can

observe

~s

lying stochastic On

sequence

(block,

mu/ti-indexed)

moving

all nonlinea~r terms, w e

sequences,

~

Toeplitz

considered

Comparing (Fig. 4,1)

in the time-variant

of the generalized

Kailath

(in a w e a k

higher-order

Levinson

matrix. W e

(1982),

Vieira a n d

Deprettere

ladder-filters, has

been

be con-

also notice that re-

a~nd Lie

we

achieved

ase

with e a c h achieved

sense).

the classical linear stationary

i~[ai%ath (1978),

of the time-variant

of the quaint[ties p r o c e s s e d

WiLl b e

the u n d e r -

algorithm c a n

will immediately o b t a i n

in Dewllde,

algo-

factorization of the generalized

Uqough, the n u m b e r

ty reduction

ca~e.

a/qd/or of the

prediction filter for s e c o n d - o r d e r

nonlinear

synchronously

matrices

provided

covari~nce

the structures

tion of the filter complexity

is

or%hogonaliza-

z-polynomials,

for C h o l e s k y

(Levinson)

(1981),

'local'

nonlinear ladder-filter

for ( G r a m - S c h m i d t )

the stationa/q~" nonlinear

a~ the fast m e t h o d

variant

considered

is stationa/'y

sidered

time-invariant A R

o n the

conclude

is time-invariant. T h e

t[me-invariant

of the generalized

the other hamd,

a.nd D y m

o n 'current' time), w e

b e treated as the fast m e t h o d

in the s p ~ c e

do not d e p e n d

that the time-invariant nonlinear ladder-filter

tlon of the ba.sis in the s p a c e basis

the

notice that the filter satisfies precisely the sa-

or£hogonality requirements, We

(3.43), p r o v i d e d

at

that the reflection coefficients

prediction ladder-filter

in Fig, zi.l. W e

by

actual/y the filter gains)

structure of the third-order presented

expressed

(i.e., they d o not d e p e n d

that the nonlinear

n +

2, pro-

Dewilde

(1980).

(Fig. 3.5)

and

time-in-

notice that significant reducin the stationary case,

al-

in the filter ~ill still incre-

'global' order-update

step. Further

in 'quasi-linear' prediction filters.

complexi-

have

structure

All s y m b o l s

F i g . 4.1 ' L o c a l '

III

IL]I the s a m e

meaning

of t h e t h i r d - o r d e r

time-invarlant nonlinear a s in Fig. 3.5.

(N=3)

ladder-filter.

(D

98 4.3 'quasi-linear' ladder-fiite[s

In this p ~ r ~ g r ~ p h ladder-filter

algorithms

we

w J ~ consider

which

we

ters will yields better estimation le their complexity considered

nonllneam

Let u s ce

{y }

riables

will b e

a~sume

accuracy

(than in the linemr cruse)

in c o m p a r i s o n

filwhi-

with the previously

algorithms.

that

is r e p r e s e n t e d

of simplified nonlinear

w J ~ c~i[ 'qu~si-lhqear' filters. T h e s e

reduced

filter

a class

the underlying

fourth-order

b y the following s u b m a t r i c e s

stochastic

sequen-

of the r a n d o m

va-

( a n d their productS)

yl,n n,n+l

.

__

_

n=0,...,N-i

Y-JlY-J2J

(4.8a)

(Jl,J2) ¢ L n,n+ l,n 1

where L l'n = LI 2Ll,n n,n+ 3. n u n,n+l

with

LI = n

{ 3,...,n+l }

2Ll'n n,n+l

Now win~

let u s

(4.Sb)

and

= sym2L I n

introduce

for

× sym2Ln1

n=0,...,N-1

(4.8c)

and

~ =0,...,n+l

the foUo-

index-sets

~(~) n

& L I u 2L(~) n

n

(4.9~)

99

where

the bi-varia~e part of the i n d e x - s e t

2L(n~)

"l~hen w e

if ~ = n + l

- i f

If w e

0

<

n

eL(~)

-

by

(4.9b)

~}

then

(~));

(since

2L(n+l)

n

n,n+l

L(°) n

c

L (~) n

c

n

L (n+l) n

="

2_ 1,n

,

D n , n + l );

.

the i n d e x - s e t

=

=

-

j 2 - j l , ....

2L(°)-

=. L l ' n

n

A 2L~,n

n

-

L (n+l)

B < n+l

L n1 i

(since

n

then

Jle

is g i v e n

that:

L ( ° ) - L1

introduce

then w e

{(jl,j2):

observe

then

~='0

-if -

can

~

(4.9a)

2L(~)

n,n+l

\

n

=

((kl,k2) : k l = B + i .....n , k 2 = k 1 ....,n }

(4.10)

notice that:

if

B =0

if

15 = n + l

if

0 < B < n+l

Following

then

eL(°) n

then

=

2Ll'n n,n+l

eL(n+1) n

= ~ ;

then

(4.9), w e

eL(B) n

can

c

consider

;

2Ll'n n,n+l for

"

n.=0,...,N-i

and

~ =0,...,n+l

the

s u b m a t r i ce s

y(n~)

A

(4.il)

~

[~-~-~

J

(~.J~) ~ 4 ~

a~nd we mention that:

- if -

if

~-0 ~mn+l

Following

then then

y(O)

-[

n

y(n+l) n

(4.10), w e

]

=

L1;

Jl E

n

_- y l , n n,n+1

will i n t r o d u c e

ey(8) n

Y-J1

"

the s u b m a t r i c e s

[Y-JlY-J2 ] (Jl'J2) ~ eL(B)n

(4.12)

I00 If the fourth-order (2.10)

with

M=2

for the submatrix

stochastic

would

hold. N o w

sy(5)

sequence let u s

only; i.e., w e

{y }

suppose

was

Gaussian,

that (2.10)

then

applies

nave

n

IF.{ y o Y _ k l Y _ k 2 }

This

means

ssian', a n d

.

~o~

0

that t h e s e q u e n c e its G a u s s i a n

(kJ.'k2) ~ %(~)n

{y}

(4.13)

is 'partially' G a u s s i a n ,

port is determined

or, ' 5-C~au-

b y the submatrix

ey(~)

•

n

From (4.9) -

and

(4.10)

it f o l l o w s that:

the s e q u e n c e

is just G a u s s i a n

if

B=0

(since

the s e q u e n c e

is n o n - C T a u s s i a n

if

B--n+l

eyn(e)

(since

=

% 2 I~ n Yn, n + l ) ;

D ' Y (n+x)

= (~);

n

-the

sequence

is ' 5-Gaussi~-u'

indicating the n o n - G a u s s i a . n

if

0

<

B < n+l

, with

part of that s e q u e n c e ,

and

with

O Y (B) n

being its G a u s s i a n If the value

part.

of the p a r a m e t e r

3

is low, w e

will s a y

that the s e q u e n c e

is 'qua~i-G a u s sian'.

PoUowing (2.~7))

(4.8)-(&.1~),

the s u b s p a c e s

(for

we

con

consider

n = 0 .....N - 1

and

(under

the i s o m o r p h i s m

~ = 0 .....n+l)

n,n+ 1

,,(B) =~ v {i(B)} n

(4.15b)

o~(B) =A v (el(B)} n n

(¢.isc)

where

~(~) n

~ h. J1

1.. ] J1J2 (jv.i2) ~ L(~)

(4.15o)

101

%(~) £ Then

we

can

[iklk2

observe

= ~(~)

n,n+l

n

(4.15),(4.16),

(4.13)

be

can

and

rewritten

define

the

An,o (~) and let

A (~) nee

(4.16)

n

according

no'~ orthogono/)

P

(~) H;n

]I(~)). Then ~q

we

±

%(B) n

following

approximation

J

its normalized

is the o r [ h o g o n a l show

i n(#)

version;

projection

-

% (n~ )

let

i n,o (~)

error

(4.18~)

i.e.,

(4.~8~) operator o n the s u b s p a c e

that

A(~) n,o Indeed,

conditions

(4.17)

~ p~ ;n(~) i o

denote

can

o£ s u b s p a c e s .

to (2.47), the o ~ h o g o n ~ U ~

~(~) ~ A n(~), A (n,~o),-1 n~o io ]I (where

sum

as

i0 let us

(4.i5e)

n

• %(~)

q) sto~nds for direct (o/though

Usin~

Now

(kl'k2)

that

?.n

where

~ %(~)

]

n

=~ P ]l';n(~)

1o

(4.19)

102 b-'Pom (4.3.7) it follows tha~

(•)

i

~(~) n~o

" PX|

O

An,o (B)

- I o _ ¢(~) n,O

1,n P~[;n,n÷l 1 o

"

T h i s implies

•

~i,~ n,n+l

~nd, consequently,

A(~)

•

e~(~)

n,o

as

e=(S) ~

,i,n

n

n~n+l "

n

Hence,

A( ~ ) n,o

This

means

tha% the u s e

estima~don accuracy,

iklk 2

of the s u b s p a c e

the o p U m u m

in the G a u s s i a n slan part

e~ O )

estimation s c h e m e

S. = pe~'t ~ N - I This

means

~)

case

into the

(corresponding

2_l,N-i RN_I, N )

for that s e q u e n c e

will be

( e x p r e s s i n g the n o n - G a u s s i a n

u n d e r the i s o m o r p h i s m

=

will not imply better

fop the underlying ' ~-Graussian' s e q u e n c e .

~ssocia~ed with the s u b s p a c e of the s e q u e n c e ,

e~B,(% n

a n d it is u s e l e s s to include that s u b s p a c e

nonlinear estimation s c h e m e Consequently,

for' (k1.,k2) ¢ eL(~ ) n

(2,47)). W e

to

~ m0

can observe

part that

and, hence, to the G a u s -

it is sufficient to c o n s i d e r the 'uni-variate'

v {i , jS=I,.,.,N} Jl

in

the o p t i m u m

that the best filter fop a G a u s s i ~ n

estimation s c h e m e .

sequence

is the linear

ladder-filter (being actually the 'most simple nonlinear filter'). ThePefore) it c ~

b e e x p e c t e d that% the o p t i m u m nonlinear ladder-filter, associated

with a ' ~-Cvaussia.n' s e q u e n c e complex

(where

0

< S < n+l)

s h o u l d b e less

than the general nonlinear filter, while estimation a c c u r a c y

s h o u l d b e still betteP than in the linear tPeatment.

103 In o r d e r to s h o w

thai, it is c o n v e n i e n t to introduce a notion of the 'nonli-

n e a r ro.nk' of the filter, a ~ the n u m b e r ling in the filter structure after observe

(see

Zarzycki

' 6-Gaussi~.n' s e q u e n c e T h i s filter will b e S

is low, w e

N

(1984~c))

of the v-labeled

'levels'

'.global' o r d e r - u p d a t e that the o p t i m u m

iS the filter w h o s e

, exis-

steps. W e

cam

nonlinear filter for a

nonlinear r a n k

equals

~ .

called the ' B-linear' filter, If the v a l u e of the p a r a m e t e r

shall s a y

that the filter is 'quasi-linear'. T h e n

we

c~

ob-

s e r v e that: - if

S =0

then w e

obto/n the linear ladder-filter a/gorithm

(of the smallest

complexity) ; -

if

~ =lq

then w e

get the g e n e r a l t~me-invariant nonlinear ladder-filter al-

gorithm, introduced in the p r e v i o u s - if

0

< ~

< 1"4

paragraph

(of biggest complexity);

then the filter complexity will b e

algorithm but smaller than in the g e n e r a l

bigger than in the linear

nonlinear case, s i n c e the o r d e -

ring in the ' ~-]/near' filter algorithm is

modulo

(n+2)

if

n-0,..., ~-i

modulo

(8 +i)

if

n-s ....N

(~.2o)

This

means

is g r o w i n g date step

that the n u m b e r from (for

1

up

to

~

n--0,...,B-l)

and B-backward

of the v-labeled 'levels' in the f i l t e r structure synchronously

s i n c e at e a c h

v-labeled 'levels' is kept constant 'length' of the filter. T h u s ,

for

starting with (and

equals

n = ~ ,.,,,N

much

n-B

'global' o r d e r - u p 'new' B - f o r w a r d like in the g e n e -

, the n u m b e r

scheme.

This

ters (S-0,1,Z,S) Now

let u s

reduced,

of the

S ), regeLrdless of the

the filter algorithm c o m p u t e s

reflection coefficients p e r 'global' order-update. Therefore,

filter c o m p l e x i t y is c o n s i d e r a b l y (8=N)

step the o n e

error is introduced to the s c h e m e ,

ral nonlinear filter algorithm. T h e n ,

(~+I) 2

with e a c h

in c o m p a r i s o n

is illustrated in Fig. 4.2 w h e r e

the

the

with the g e n e r a /

the 'qu~i-linear' fil-

are presented, evaluate estimation a c c u r a c y

l:~ollowing (3.48), w e

can

observe

in the 'S-llnear' filters.

th;~t the e r r o r - n o r m

relations in the g e -

104

D)

C)

B)

A)

F i g . 4.2 ' Q u ~ i - t i n e ~ c '

A) ~ - o B) ~ - 1 t3=

3

l~dder-filters.

105

neral time-invariant nonlinear ladder-filter are given by

tl ,4 N,o II

=

II A o,olI RI.,I.,RLB

(4.21,~)

where

IqLL

= N-1 n [n~l

Then

we

(1- [Pn,:l.]2) ½

can observe

(4:.21b)

]2) ½ 1 ( i - [ P

( i - [P

n,m0 L m = 2

RLB

N-I I] n=O

=

n,m

n+l,o

tha.t in t h e

' ~-linear' s c h e m e

]2~ "

(~.21c)

II A(~)N,oIt = llAo,oll RLL "'LB'C'(B)

(4-.22a.)

where

with

RI(B) = BI~I F n ~ 1 (l-[p (B)]2)½] (l-[p (~) ]2-~% n=0 Lm=2 n,m n÷l~o"

(4.22c)

"

(13) R2

We

N~I F ~ = n=~ Lm --2

(1_ [_(~)]2 ) %1 (1-[p (B) Pn,m

~]

n+l,o

]2)%

(4:.22d)

notice that the norm of the error in the '~-linear' case is reduced

(with respect to the linear case) can observe

by the factor

R L(~) B

(4.22b), and w e

that:

- if

~=0

then

R/uLR

-if

~=N

then

RLLR(~)

Consequently,

we

) = lqL = = RLLRLB

N-I If n=0

(1-[p n+ 1 (with

RLB

can associexte with each

j2) % given by

(4.21c)).

' B-Cxaussia/a' s e q u e n c e

the optimum ) B-linear' filter and) working with not too complex nonlinear orthogon~d structures

(whose

complexity m a y

be successively

incre~used

106 until

the

desired

accuracy

estimation

than in

accuracy

the linear

is

achieved),

we

will

obtain

better

treatment.

4.4 Experimented example

The sian a n d

' P-linear' ladder-filters h a v e non-Gaussian

20 m s

1.0 0.9 0.8

(MSE)

t)

k:

of the input G a u s s i a n

|

0.9 0.8

t

0.7

0.6

0.6

0.5 0.4 0.3

0.5 0.4 0.3

0.2

0.2

0.1 0.0 0

0.i 0.0 i0

A)

(N=8),

non-C, a u s s i a n

' S-linear'

a s s o c i a t e d with time-series.

1.0

beta=@

--4--

0.7

a~d

present compu-

in the adaptive

innovations filters of the eight-order

samples

H5£(@

tested u s i n g p s e u d o - G a u s -

excitations. In Figs. 4.3 - 4.6 w e

ter plots of the m e a n - s q u a r e - e r r o r s (~=0,i,2,3)

been

20 m s

0

I0

20

ms

B)

Fig. 4.3 M e a n - s q u a r e e r r o r in the 'O-linear', 8-th o r d e r ladder-filter, inputted with: A ) O aussia~n B) non-G aussian excitations.

107 bCta:l

t)

1.0

o.9

~

0.8

....

1.0 ]

mmmmmm-mmmm m I mmmmmmmmmmm • mmmmmmmmmum wmmmmmmmmm 11! , m m m m m m m m m 1 ,~ , m - , m m m m m m m

0.9 0.8

o7

07

0.6

0.6

o.~ [ 0.4 i J 0.3-'L,i

~

0.5 o.~ 0.3

L f

0.2 0.I

mmmlmm,---

0.2 0.i

0.0

Q.0

0

i0

20

~ 0

ms

A)

i0

20

ms

B)

5'ig. 4.4 M e a n - s q u a r e - e r r o r in the 'l-linear', 8-th o r d e r ladder-filter, inputted with: A) G aussian B ) non-Gaussia/n excitation.

1.0

t)

beta=~

1.0 0.9

0.9 ~ 0.8

0.8

0.7

-

0.6

~ ~

-

A

.

.... , . . . . .

;

,,

0.7 0.6

,.

]

o.5)

_

o.4Ji ) 0.3 0.2 0.1 0°0 0

10

A)

20 ms

0.5 0.4 0.3 0.2 0.1 0.0 0

I0

20

ms

B)

Fig. 4.5 M e a n - s q u a r e - e r r o r in the '2-I/near', 8-th o r d e r ladde~-fJ/ter, inp u ~ e d with: A) Gaussian B) non-G aussian excitations.

108 1.0

,MS~ (8

t)

beta =3

1.0

0.9

0.9 I

O.8 0.7

i1 i

0.5

O.8 0.7 ....

00

I

0.4

I

0.5

0.~ k

0.3 i_ 0.2 0.I

0.3 O.2 0.i

0.0

0.0 0

i0

20

ms

i0

0

A)

20

ms

B)

Fig. 4.6 M e a n - s q u a r e - e r r o r putted with: A) Gaussian B) non-Gaussian excitations.

in the '3-1inear', 8-th o r d e r

ladder-filter, in-

Comparing Fi~s. 4.3a and 4.3b, we c a n o b s e r v e that the linee~C esti-

mat[on

accuracy

the G a u s s i a / ~

in the non--Gaussi~Ln

excitation. T h i s

filter o p e r a t e s

on

Comparing schemes This

do

Figs.

is m u c h

to c h a r a c t e r i z e

not i m p l y better estimation

non-Gaussian

the m o s t vement

simple)

Figs.

It s h o u l d ally d e p e n d s

on

be

filter is u s e d .

4.3b - 4.6b, w e

accuracy

which

signa/s. estimation

in the G a u s s i ~

noted

con

observe

estimation p r o c e d u r e s

accuracy

case.

filter is the b e s t p o s s i -

that the i m p r o v e m e n t

ca/q b e

that the u s e

introduce

in cause of n o n - G a u s s i a / q

the h i g h e r - o r d e r

ever, that i m p r o v e m e n t

of

signal.

nonlinear

of estimation

in c a s e

notice that the n o n l i n e a r

resttlts f r o m the fact that the linear estimation

Comparing

th~n

statistics of the input time-series:

4 . 3 a - 4.6a, w e

ble filter for a G a u s s i a n

worse

follows from the fact that the linear l a d d e r -

the s e c o n d - o r d e r

a r e not sufficient in o r d e r

case

significant impro-

excitation.

of estimation

accuracy

statistics of the u n d e r l y i n g

achieved

if a

of ( e v e n

(suitably c h o s e n )

sequence,

actuhow-

nonlinear

5. C O N G L U D I N G

The can

be

REMARKS

nonlinear

prediction filter a/gorithms,

directly i m p l e m e n t e d

modular

memory.

A

in this work,

in a soft- a/nd/or in a h a r d - w a r e

structure of the nonlinear

w@/'e reetlizations,

presented

way.

orthogonetl laddeP-filters implies soft-

requiring relatively small capacity of the operational

hard-ware

realization follows from the fact that the basic

thogonetl 'building-block'

of the nonlinear

a s in the linear ladder-filters.

That

block' c a n

using V L S I

(namely

DICS

implemented

processors),

introduced

hal filters ( s e e

e.g., A h m e d

ret£ere, D e w i l d e

and

nonlinear cessors

Udo

(1984);

a s well, taking a d v ~ t a g e

versions

of those

troduced

(see

having

Zarzycki

like in the linear c a s e

(1981).

Consequent/y,

in this work,

(1983b);

Consequently,

of the para/lel computations. orthogonat

by

the

adaptive

directly o n

capability, c a n

q'he nonlinear

considered

Dep-

It s h o u l d

filters. M o r e o v e r ,

filters, operating

least-squares'

COR-

'building-blocks' a s s u -

adaptive

a

also b e

in-

filter algorithms

solution at e a c h

time-ins£artt,

Lee, M o r f a/nd FriedlaJnder

the nonlinear prediction filter algorithms,

are a/so suitable for on-line nonlinear

order time-series.

'building-

b e realized with those pro-

parameter-tracking

result in the 'exact nonlinear much

here, c a n

ro-

of the linear orthogo-

(1983)).

Dewilde

prediction

(1984d)).

circuits

Deprettere

stability of the nonlinear

nonlinear

of data, a n d

(1982);

of the norma/ized

res inherent numerical

stream

Morf

digital filters, c o n s i d e r e d

b e noted that the u s e

integrated

in the rea/izettions

and

or-

filter (actually the G i v e n ' s

tor) is precisely the s a m e be

The

processing

introduced of higher-

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R.H. a n d

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MAR"I~IN W . ' I '

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1979 a

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1983

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b

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E. a n d D E ; W I L D E

P.

Genere/ized ort4hogonal fJ/ters for stochastic prediction a n d modeling, in Digital signo/ processing, Ed.V.CapeUlnL~ A c a d . Press, N.Y.

1979

DEPRETTERE

E., D E W I L D E

P., a n d U D O

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Pipelined C O R D I C architectuz'es for fast VLSI filtering m~nd array pz'ocesslnS, P r o c . I C A S S P ' 8 4 .

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DEPRETTERE

E. and J A I N A N D U N S I N G

K.

1984

D e s i g n and VLSI implementation of a concurrent solver for N coupled systems of linear equations, Techn.Rept., Delft Univ, T e c h n .

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E. a n d LIE S.C. Generalized Schur-Darllngton algorithms for lattice-structured matrix inversion end stochastic modeling, ~echn.Rept., Delft Univ. T e c h n .

DEWILDE

P.

1982

Stochastic modeling with orthogone/ filters, in Outils et toodeles mathematiques pour l'automatiquem l'anedyse de s y s tames et le tra/tement du si~ne/j E d . C N R S , Paris, vol.2, pp.331-398.

1983

O ~ h o g o n e d filters: Pipelining a n d VLSI implementation, Proc. E C C T D ' 83, Stuttgart.

1984a

Spectral approximation a n d estimation with scattering functions, in Mathematice/ T h e o r y of Ne£v~orks a n d Systems, Lecture Notes in Control a~nd Information Sciences, vol.ziS, Ed.P.A.Fuhrma~n, Springer-Verlag, pp.234-252.

1984b

O£thogone/ filters: A Ibid., pp.253-267.

DEWILDE 1979

DEWILDE 1984

DEWILDE 1981a

P. a n d B U L T H E E L

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Orthogonal functions related to the Nevanlinna-Pick problem, h Mathematical "l~heor~ of N e t w o r k s a n d Systems, Ed.P.Dewilde, vol.3, Delft~ pp.207-212.

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ParaJ/el ~ d pipelined VLSl implementations of signa/ processing algorithms, in V L S I e n d signal p r o c e s s i n ~ Ed.S.Y. Kung.

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S c h u r recursions, error formulcms e n d c o n v e r g e n c e of railone/ e s t i m a t o r for stationaz'y stochastic processes, I E E E Trans. on IT-27, pp.44=6-461.

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P., V I E I R A

1978

A.C. a n d K A I L A T H

T.

O n a generalized S z e g b - L e v i n s o n realization algorithm for optimal linear predictors b a s e d o n a n e t w o r k synthesis approach, I E E E T r a n s . o n C A S - 2 5 . pp.663-675.

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S u r l e s fonctione~es c o n ~ n u e s , Sup. 3-me, S e n V . 2 7 .

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Integral, J . M ~ h . S o c . , Japan, vol.13, nr i,

T.

1974

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D.T.L., M O R F

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LEV-ARI

M. and FRIEDLANDER

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1982

Parametrization a n d modeling of nonstationary p r o c e s s e s , Ph.D.Thesis, S t ~ f o r d Univ.

1983

M o d u l a r architectures for adaptive mulfichannel lattice a/gorithms, P r o c . I C A S S P ' 8 3 .

LEV-ARI

H. a n d K A I L A T H

1982

LEVIN SON 1947

T.

Lattice filter parametrization a n d modeling of nonstationary p r o c e s s e s , T e c h n . R e p L , Stanford Univ.

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M., VIIi;IRA A.C., L E E

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0GURA

Recursive tion, I E E E

D.T.L. a n d

KAILATI4

T.

mu/tichannel m a x i m u m entropy spectra/ Trans. o n O E - 1 6 , pp.85-94.

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Orthogon~II functionals for the P o i s s o n Tra/ns. o n IT-18, pp.4:73-481.

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M.S., S A E E D

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A

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M.S.

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M.

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M.

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114 VICTOR

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KNIGHT B .

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1959

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Trans.

S.

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ZAIqZYCKI

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1983

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1984a

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115 ZARZYCKI

J. a n d D E W I L D E

P.

1983a

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ran-

predic-

APPENDIX

i

MUI/~I-INDEXED

Let

I

MATRICES

denote

a

real n u m b e r s .

AND

contiguous

We

define

GENERALIZED

subset

of integers,

a m-indexed

m A : ml ~

where ml

mI = I × . . . × I

can

will b e

be

c~/led

denoted

domain

DmA.

Lo£ n

mL°n =A LOn x . . . .

matrix

let

mA

THEORY

~ as

be a

the set of

map

(A.1)

~

( m - c o p i e s ) . A c c o r d i n g to ( A . 1 ) , the i n d e x - s e t

the

~s

MATRIX

of the

Let u s

(j } n . o

m-indexed

introduce

matrix

the

mA,

This

index-sets

{0,I .....n }

L On = { ( J l ..... Jm ) :

domain

(A.2)

Jk E L :

, k = l ..... m }

(A.3)

m A m-indexed

matrix

mR ~n

DmA

This

matrix

entries

as

ca/n b e

W e will c o n s i d e r operations and

type

(A.4),

n-th order

m a t r i x if

(A,4a)

expressed

in t e r m s

of its m - i n d e x e d

follows

[ a.

=

-n

order

the

rnL° n

~

--n

equivalently

mA

duce

will be called

here

some

on those

domain unless

]

~l""Jm

of t h e

(A.4b) ( J l . . . . . 4 . ) ~ m~o

n

properties

matrices. matrices,

otherwise

of m u l t i - i n d e x e d

W e will u s u a l l y assuming

stated.

matrices,

drop,

and

intro-

for simplicity, the

that all matrices

are

of t h e

117

matrix

Symmetric

A

m-indexed

matrix will b e

co/led s y m m e t r i c

if for a n y

permutation

ul,..., m

of integers

l,...,m w e

shadl h a v e

a.

_51...jm

sm

--J nl...j m

Consequently, symmetric ~jl...jm

matrix instead denote

corresponding

n m

o

Ln

a,

-ll...jm

of equai

elements

(jl,...,jm) . W e

matrix

= [ a, . )l...]m

- m -

'different' entries of the

%n

by

mA

of the s y m m e t r i c shall denote

let

matrix,

the

'symmet-

, where

]

(A.Sb) (j:l,...,jm)

to lexicographic

~ symmbno

port' of the m-varlate

or anti-lexicographic

a

of m A c a n then b e e x p r e s s e d Jl...jm n of the s y m m e t r i c matrix mA as --n

ajl...jm

matrix. N o w

entries of n o n - s y m m e t r i c

denoting the 'symmetric

obtained a c c o r d i n g entries

n

t o the s e q u e n c e

mA

sym

of

the n u m b e r

ric part' of a m - i n d e x e d

with

(re+n)

it is sufficient to consider m

(A.Sa)

= 7jl...jm

a. , -Jl...Jm

index-set,

ordering. T h e

in terms

of the entries

(A.5c)

"!~ranspose matrix

Let

r~ b e

represented

a permutaf/on

of the index-set

{ (Jl'""Jm) ¢ m L °n }. u

may

by a map

1 , a ,..., ~ ) ( ~ l ' ~ 2 .... ' ~ m

(A.6~)

be

118 where xed

( nl' ~2 .....nm)

matrix

n

tion

mA~

is a permutation

A

{1,2 .....m } . T h e n ,

will b e co/led the transpose

a m-inde-

matrix due to the permuta-

if

( mAn )jl...jm

Z

of

:

( mA )rt (jl,...,jm)

(A.6b)

ero-m~trix

m-indexed

indices

matrix

will be called a zero-matrix if for e a c h s e q u e n c e

(jl,...,jm) c D m A

we

have

ajf..jm

0 . This

of

matrix will be de-

noted b y

mo

=

[0jl...jm ]

where

O.

.

J1..,Jm

(A.7)

(Jl .....jm) c D m O n

n

will b e the z e r o - e n t r y

wlth 'coordinates'

(jl,...,jm) .

Unit-matrix

A

2m-indexed

and

matrix will be called the unit-matrix if for e a c h

(k1'""km)

¢ D2mAn

~.

(Jl .....Jm )

we have

.

=

]1.,,]mkl,,,km

~

(A.Sa)

Jl,0.Jm;kl---k m

where X

5

.

]f " ] m ; k l " ' k m

=

I

0

if

Jl=kl

,..., J m = k m (A.Sb)

otherwise

119 This matrix will b e d e n o t e d

as

2m I

n

=

[

.

.

6 ]l...3m;kl...km

]

(jl,...,Jm,kl,...,km)

D2mi[n

(A.8c)

Block-matrices

A block-matrix w h o s e

block-entries are m-indexed, n-th orde• mmtrices

{ M } A n = [ m a n ] m = l ..... M

will b e called a M - b l o c k

row d o m ~ d n

D {1%4}A

n

(A.9a)

(row), m-indexed, n-th order matrix. Its block-

will b e a vector of simple d o m a i n s

D{M}An

"

[DmAn]

(A.gb)

m=l,...,M

Similarly, a block-matrix

{M}Bn

= col [ m B

will b e called a M - b l o c k block-column

(column),

n

]

(A.10a)

m = 1,...raM

m-indexed,

n-th order matrix with the

domain

D {M} B n

= col [ D m B n

] m=l,...,M

(A.10b)

Finally, a block-mc~trlx

{M×M }H

= [ meu H n

whose

block-entries are

] n

(m+u)-indexed

m,u= I,°..,M

matrices

(A,11a)

120

meuI-I

will b e c a l l e d

[h.

=

n

" k

Jl'"Jm

k ]

i"' U

a (MxM)-block

Its b l o c k - s q u a r e

(jl,...,Jm,kl ....,ku) ¢

(square),

D{M×M}H

domain

=

[DmeUH

observe

'block-column'

that the matrix form. T o

DUll n , w h e r e

(A.11)

can

order

matrix.

(A.:lc)

be described

d o that, l e t u s s u p p o s e

( J l ..... Jm ) ¢ D m H n

n-th

(m+u)-indexed,

n ] m,u=l,...,M

n

Let us

(A,1lb)

is g i v e n b y

n

D { M x M }H

DmeUHn

and

that

in a generalized

DmSuH ~ DmI-I x

( k l ..... k u ) E D u l l

n

n

. ~hen

n

we can

write {MXM) H

=

[{M}xu

H

n

] n

u=I,.o0~M

(A.laa)

where

{M}x UHn -

col [meuH n]

m = l ..... M

(A.12b)

or, equivalently,

{M} xu H

-

[{M} H

n

] n;ki...k u

(A.12~) (kl,...,ku) ~ DUll n

with

(M} Hn;kl.o.k u = [ 51***Jmkl,..ku ]

(A.12d) (jl,..,,jm) ¢ DmH n

Equal. matrices

Two

m-indexed

mA = [a.

matrices

] Jl""Jm

; (Jl ..... Jm ) ~ DmA

ms = [b.

] (A.13) "~l""Jm (Jl ..... Jm) ~ Drab

121 will b e called e q u a l matrices if of indices

..(Jl,*-*,Jm] ¢

DmA

=

Given the matrices

if for e a c h

sequence

(A.14)

b.

ll...Jm

ll...Jm

of multi-indexed

, and

DmA

a.

Sum

= DraB

matrices

we shall s a y that the m - i n d e x e d

(A.13),

.

mG

[gjl...j m

matrix

(A.15a)

]

(Jl

.....jm ) c D m G

is the s u m mG

Drag

,= D m A

- Dn~

=

mA

, and

mB

+

(A.15b)

if for e a c h

(jl,...,jm) ¢ D m A

A

gjl...j m = ajl...j m

Sum

of bl.ock, ' multi-indexed

Given two M-block (row),

matri.ce.s

m-indexed m a t r i c e s

{M}A = [ mA] m=l ..... M

we

sh~U

say

;

that the M - b l o c k

{M} G

{M }m = [ m B ] m=l ..... M

(row),

=

m-indexed

[raG] m=$,..,M

where

mG

is given by

(A.15c)

b.Jl...Jm

+

( A . 1 5 a ) , is the sum

(A.16a)

matrix

(A°16b)

122 {M} G .

if for ce

m~l,...,M

of indices

we

have

{M} A

DmA

+ {M} B

= Drab

(jl,...,jm) E D m A

(A.16d)

= DmG-,

the entries

and

gjl...jm

if for e a c h are

sequen-

expresed

by

(A.1~o).

Product

Given say

of a s c a l a r

a scalar

and

c ~ ]~q

and

that the m - i n d e x e d

lar ~ n d

the m - i n d e x e d

a m-indexed

a m-indexed

matrix

mG

matrix

(A,15a)

mA

(A.13),

is the p r o d u c t

we

shall

of the s c a -

m~£rix

mG

if for e a c h

matrix

= c-mA

(Jl .....Jm ) ¢ D r a g

(A.17a)

(where

Drag

= DmA)

we

have

A

gjl...j m = C.ajl...j m

Product

Let

of mul#/-indexed

mA

replaced

be

given

by

by

s , where

(A.iYb)

matrices

(A.13), m

and

let

< s . Let

sB

be

u, ~ , 9

given be

(A.13)

by

some

given

with

m

integers,

satisfying u

and

moreover

and

the

s

+

~

let indices

= m

r ;

;

(u + ~))

of the

sB

~

*

+

Partitioning

9

= s

the

in accordance

(A.18a)

m with

indices (A.18a),

of the

mA

and

assu-

t

123 rning that

mA

DmA

=

= DUA

×D~A

and

DSB

= D~tB x D 9 B

We

c a n write

[~ki...k~l...j ]

(A.18b) (kl)...,ku) ~ D U A

SB =

, we

[b ] Jl'"J~ii'"i9 (Ji ..... J~) ~ D ~ B

shall s a y that the

r =

; (jl)...,j~) E D ~ A

; ( i i ..... i~) ~ D9B

(u + 9) -indexed

matrix

] r'(u+ whose

(~ lSd)

9) G = [ g k l . . . k u i l . . . i 9 domain

is

DrG

= DUA

(A.18c)

(k 1 .....k u ) ~ D U A x DqB

; (i i ..... i 9) ~ D g B

, is the

~t-product of the matrices

ue~ A" ~ e g B

(A.18e)

(,aL,18b,c)

rG

if

Dt~A = D ~ B

= UD

= mA.s B

=

, a n d if for e a c h

(kl,...,ku) ¢ D U A

, a n d for e a c h

(iI,....i 9 ) ~ D g B

gki'"k uil "''i9

where

the s u m

(jl,...,jt~)

Product

Given

=

~~D

in (A.i8f)

over the

~i...ku Jl...J ~

denotes

square-matrix u-indexed

(row)

{ M x M }H

matrix

~-fold s u m m a t i o n

~-variate index-set

of block) multi-indexed

the M - b l o c k

the

matrix

b. . il...i~ J1 '''J p.

(A.i8f)

with respect

~D .

matrices

{M }A

(A.il), w e

(A.9),

shall s a y

a n d the

(MxM)-block

that the M - b l o c k

(row),

to

124 {M}

G

=

(where

[uG]

for

u..1 , . , M

u=l,...,M

UG

we

have

: [ g k l .....k u]

DUG

= Dull

is the block m - p r o d u c t of the matrices

{M}G

for

:

{ M }A

(k 1 .....k u)(

(A.19a)

DUG

, with

D m~u H

and

{MXM} H

= DmH,DUH)

{M}A'{MXM}H

(A.19b)

u=l,...,M

uG

M )-] m A ' m S UH

A

(A.19C)

m=l with

"

denoting

9 =u , we

the

product

.(A.18e).

t e n rewrite (A.19)

Using

(A.18)

with

u :0

,

k =m and

e~

M

gkl...k u =

where

=

mD

(A.12d), w e

DmA

~

:

mD

mffil

- DmH

'Outer' or K r o n e c k e r

m

mA

(kl,...,ku) ~ D U G

{M} A {M} •

, Equiva/ently, using

(A.19e)

Hk1,..k u

SB

be expressed

by

shall s a y that the

(m+s)-indexed

matrix

(A. 13), a n d let

s . We

=

domain

is

DmesG

(A.13)

with

(A 20~)

]

[gJl'"Jmkl"'ks (whose

~nd

(A.19d)

h j .... l j m k .... i ku

product of mudti-indexed matrices

b e given b y

replaced b y

m *s G

jl...jm

c a n write

gkl...k u =

Let

a.

(Jl ..... jm ) c DmA ; (k 1 ..... ks) ~ DSB :. D m A ×

DSB)

is the 'outer' (or lqronecker)

125

product of the matrices

mA

m • s G

if for e a c h

and

mA ®

.

( J l ..... jm ) c D m A

SB

and

(A.2Ob)

sB

( k 1 ..... k s ) ¢ D S B

we have

A g j l . . . J m k l o . . k s = %l...jm b k l * . ) k s

From

(A.20)

(A.20¢)

it f o l l o w s t h a t if

iy

= [yj]

(A.21a)

J CDIY

then m

my=

,~ i y

=[yjl...yjm ]

(A.21b) ( J l .....jm ) E D m y

where

Dmy

= DIy

(A.21C)

×...× D I y m

'Outer'

product of b l o c k ,

multi-indexed

matrices

Let

{M}y = [my]

(A.22~)

m=l,..o)M

where

my

is given

by

(A.21),

and

{M×M} G = [ m ~ u G ]

moreover

m)u=l,.oo)M

let

(A,22b)

126 with

m ~ u G = [gJl°"]mkl"'ku]

We

(Jl ..... jm ) c D m y ; (k 1 ..... ku) ¢ DUy

shall say that the matrix

{MxM} G

if for

{MxM}G

(A.22c)

is the block, outer-product

=

{M}y

(~ { M } y

m@u G

} my

~ Uy

(A.22d)

re,u= 1,...,M

or, equivalently, if for each

gji...Jmkl°..ku

(]i.....Jm ) ¢ D m y

=

(A.22e)

and

yjl..°yjmYkl.--Yku

(k i .....ku) E D u y

(A.22f)

APPENDIX

2

MULTI-VARIATE

INDEX-SET

LB-recursions:

RECURSIONS

'loc~' o r d e r - u p d a t e s

l,-for-vvard index-set, a n d index-sets; i.e., for

for the 'bi-variate' part of the

for the 'uni-va~iate' part of the B - b a c k w a r d

m=2,...,n+l

Lx+l,n-I n,m-I Lx . Lx n,m n,m-I

u

L x + I'n-I n,m-i

=

u LX+l, n-I nsm-2 2 T

(x} k

u {x+n+ 3-re,x+ n + I}

(A.2a~)

Lx n,m-I and

for

m=n+2

LX+l,n n,n+l

Lx . Lx n+l,o n,n+l

L x+l'n U n~ n'~l-- m

f U L x+l'n-I n,n

{x}

• U {x+l,x+n+l}

(A.23b)

Lx

n,n+l

These

recursions

can

b e schematically d e s c r i b e d

a s the L B

index-set

sections

Lx

x

n,m

nmm_ 1

~

LX+l,n-i n,m-i

Ln+ljo

>

Lx

n=m

Lx

~

nln+l

f

LX+l,n n,n+l

Lx

n+ljo

(A.23~)

128 BL-recursions:

'loco/' order-upda£es

for the 'uni-variate' part of the

B-fo~va/~d index-se£s, a n d for the 'bi-varia£e' part of %he L - b a c k w o 2 d index-se~

i.e., for

v=0

n~l L x'O nt2

=

L x'O n~l

u

Lx n,l

.

{x,x} %,....

u Lx U {x+n+l} n,o ~, 2

( A . 2Zia)

LX~ O n,l

a n d for

v=l,...,n

Lx,v-I n,v+ 1 LXt v

n,V+2

= L x'v L x'v-I ~ { x , x + v } n,v+l U n~V+1

u Lx, v-I ntv

u {x+n+1}

(A,2a~b)

T LX, v *%~V+ i

These

recurs[ons will b e in£erpre£ed as the B L

Lx, o n, 2

Lx~° n,l

~I .~

LX, v n,v+ 2

~ r

LX'° n,2

Lx'v n,v+l

'"

Lx n, 1

BB-recursions:

index-set sec%ions

~ "~

"2

(A.2~c)

LX, v 2 n,V+

Lx, v-I rl,v+1

'local'

order-updates

B-foF%va~d a n d B - b a c k w a r d

for

the

Jbi-v~iate

index-sets; i.e., for

v=0

~ parts

and

of

the

m=3,...,n+3

129 LX

n,m-1 Lx. ° _- Lx, ° n,m n,m-I

u

A

f

Lx n,m-I"

{x,x} k

...............

u L=,m_ 2 U {x+n+4-m,x+n+l} J

%

(A.25a.)

Lxs O

n,m-i

arld for

v~.l,...,n e ~ n d m m v + 3,...,v+n+ 3

ijxlv'l n,m--i [ L x'v n,m

m

L x'v n,m-I

u L xn ,' vm--12

- { x,x+v}

U

•

u {x + n + 4 + v - m , x + n + 1 }

Y LXl v

n,m-1

These

recursions will result in the B B

index-set sections

L~,o

LXi v r'isi-n

ntm

Lx, O n,m-1

~ ~

(A.e5b)

>

L x'° n,m

Lx ' v n Im - 1

-~ "~ |

Lx n,m-i

Lx,v-I n,m-I

~

Lx'v n~m

(A.25c)

130

'LOCAL' 0 R D E R - U P D A T E

LB-recursions:

for

RECUI~SIONS

m=2,...,n+l

A~,m_z(z) ]

AX, re(Z)

lB~,m(Z)J =

X

n,m

and for

,.

@

x n,,m

(A~,m(Z)

(A.26a) z.Bx+I,n-L(Z) [ n, m-i ~ j

x+i,n-l.

, Z'Bn,m_ l

(Z))Z

(A.26b)

re=n+2

(A. 26C:)

Sn,n+i(z) (A.26d)

x P

BL-recursions-

for

n+l,o

v=O

~"~(~)] . 0~.o XsO = Pn,2

( A X~, , I (O z)

-A::~(=~I

, B~,I(Zl) z

(A.2Va)

(A.27b)

131 for

v=l,...,n

[]:::+~(~1 .o.,v [ n,v+2

px,o ~ n,m

BB-recursions:

for

v=0

(AX, O (Z) n,v+l

and

(A.27c) B x'v-I ( Z ] I n,v+ 1 ~

"J

B x'v-I (Z~) n,v+l ~ " Z

'

(A.27d)

m=3,...,n+3

Ax'O ~ (Z)l n,m-I | =

<::(~)J p~,o . n,m

for

~-- l~...,n

and

n~,m

CAX,O ~

(A.20,,)

0x, °

~:.,,,_l(=)J

~z~

n,m-i ~

B:,~_1(z))

(Ao28b)

Z

" '

re=v+ 3,...,v+n+3

A:::,(~) 0x~v ntm

i

IB ~'v (z' m

n, rn

~,v n,m

'

CA~,V (Z)

"- -n,m-i

(A.28~)

L<: v=.t(~)-'J '

B ~ ' v - I ~z Z,~,

(A.28d)

n,m-i

where

e~n , m

= (i_[o~

n,m

12)-½ -

(A.29a)

x n,m

o~''

nsIT).

= (1-[p

gl~lffl

12) X~g Pn, m

1

(A.29b)

132

'LOCAL' FILTER

SECTIONS

LB-sec[ions: BX n~rn AX ~ n,m-i

+

AX n~m

AM n+l,o

Ax

n,n+l

BX+l,n n,n+l

BX+l,n-1 n~m

for re=n+2

for m-=2,...,n+ 1 BL-s ections:

Bx~v n,v+2

n,2 AX, O

n,l

_

~

A x,o ~ n,2

Ax'v

~

AXsV

n,n+ 1

n,v+

BX,V-I n,V+1

Bx n,1

for v=l,...,n

for v=0 BB-sections: BX~O rl~lll A x'° ~ n,m-i

B x~v nmln

A x'° n,m

~ Bx n,m-I

for v=0

and

m=3,...,n+3

AX'V ~ nsrn-i

AX'V n,m BX,V-i n,m-1

for v=l,..,n and m=v+3,...,v+n+3

2