Further results on the McMillan degree and the Kronecker indices of aArma models

Further results on the McMillan degree and the Kronecker

indices of aArma models

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Janssen, P. H. M. (1987). Further results on the McMillan degree and the Kronecker indices of aArma models.

(EUT report. E, Fac. of Electrical Engineering; Vol. 87-E-175). Eindhoven University of Technology.

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Published: 01/01/1987

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Further Results on the

McMillan Degree and the

Kronecker I ndices of

ARMA Models

by

P.H.M.

Janssen

EUT Report 87-E-175 ISBN 90-6144-175-7 June 1987

ISSN 0167- 9708

Eindhoven University of Technology Research Reports EINDHOVEN UNIVERSITY OF TECHNOLOGY

Faculty of Electrical Engineering

Eindhoven The Netherlands

FURTHER RESULTS ON THE McMILLAN DEGREE AND THE KRONECKER INDICES OF ARMA MODELS

by

P.H.M. Janssen

EUT Report 87-E-175 ISBN 90-6144-175-7

Eindhoven

June 1987

CIP-GEGEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG

Janssen, P.H.M.

Further results on the McMillan degree and the Kronecker indices of ARMA models / by P.H.M. Janssen. - Eindhoven: University of Technology, Faculty of Electrical Engineering.

-(EUT report, ISSN 0167-9708, 87-E-175)

Met lit. opg., reg.

ISBN 90-6144-175-7

SISO 656 UDC 519.71 NUGI832

Abstract

This paper shows that the McMillan degree and the Kronecker

indices of ARMA models can easily be related to the determinantal degree and the row degrees of a suitable submatrix of the polynomial

factors in the ARMA model. For systems having no poles at the

origin, these quantities can even be inferred from the determinantal degree and the row degrees of the polynomial AR factor.

Janssen, P.H.M.

FURTHER RESULTS ON THE McMILLAN DEGREE AND THE KRONECKER INDICES OF ARMA MODELS.

Faculty of Electrical Engineering, Eindhoven University of Technology, 1987.

EUT Report 87-E-175

Address of the author: ire P.H.M. Janssen,

Measurement and Control Group, Faculty of Electrical Engineering, Eindhoven University of Technology, P.O. Box 513,

5600 ME EINDHOVEN, The Netherlands

l . 2. 3. 4. CONTENTS Introduction Mathematical preliminaries

The McMillan degree and the Kronecker indices of MFD and ARMA models

Conclusions Appendix References Page 1 2 4 9 9 15

1. Introduction

The input-output relation of causal time-invariant linear discrete-time systems of finite order can be represented in various ways: e.g. by their transfer function, impulse responses, state space representations, matrix

fraction description (MFD) representations, ARMA representations. We

shall concentrate on the last two representations which render a direct description of the input-output relation in terms of a set of difference equations between input and output components.

Writing the relationship between the m-dimensional input vector u(t) and the p-dimensional output vector yet) in terms of the pxm rational trans-fer matrix function K(z):

y(t) = K(Z) u(t) ( 1. 1)

II.

(Z denotes the forward shift operator: Z u(t):= u(t+t», the left MFO

model represents this input-output relation as a set of differen~

equa-tions in the forward shift operator Z, denoted formally as

P(Z) y(t) = Q(Z) u(t) ( , .2 )

P(z) respectively Q(z) are pxp-, respectively pxm-polynomial matrices in

the indeterminate variable z, with det P(z)

# o.

ARMA (or ARMAX/ARX) models describe the input-output relationship in

terms of difference equations in the backward shift operator 0

II.

(0 u(t):= u(t-t»:

A(D) y(t) = B(O) u(t) (' .3)

Here A(d) respectively B(d) are pxp- respectively pxm-polynomial matrices

in the indeterminate variable d, with det A(d)

+

O.

MFD models have been most widely used in the control engineering field. In system identification the use of ARMA models prevails (especially

monic ARMA models with A(O)

=

I are popular since they render a

2

Using the fact that (due to (1.1)-(1.3»

p_l (z)Q(z) = K(z) = A-I (z-l) B(Z-l) (1.4 )

the properties and relationships of MFD's p-l(z)Q(z) and ARMA

decomposi-tions A-I (d) B(d) have been studied in detail in Wolovich and Elliott

(1983) and Gevers (1986). Gevers (1986) focuses on the use of ARMA

rnod-els for identification and highlights some difficulties encountered in this context. As he points out an important advantage of MFD models over ARMA models is that the McMillan degree and the left Kronecker indices of

K(z) are easily related to the deterrninantal degree (deg det P(z» and

the row degrees of the denominator matrix p(z) of any row-reduced left

coprime MFD p-l(z)Q(z) of K(z) (see also the subsequent theorem 3.1).

For ARMA models, however, things are less ideal. Gevers (1986) states

(amongst other things) that (i) if A-I (d) B(d) is a left comprime ARMA

decomposition of K(z) with A(d) row-reduced, this does not imply that the

row degrees of A(d) are connected to the left Kronecker indices of K(z);

(ii) the McMillan degree of K(z) cannot be obtained from such A(df. An

easy example illustrating these observations are moving-average models (i.e. A(d) = I).

In this paper these observations are further specified. After presenting some notations and basic definitions in section 2, it is shown in section 3 that the McMillan degree of K(z) is equal to the determinantal degree

of a suitably chosen pxp submatrix SId) of [A(d) : B(d)J where A-l(d) B(d)

is any left coprime ARMA decomposition of K(z). Additionally, the left Kronecker indices of K(z) are equal to the row degrees of Sed), if Sed) is row-reduced. Moreover, it is shown that the submatrix Sed) can be chosen as the denominator matrix A(d) if and only if the system

charac-terized by K(z) has no poles at the origin.

2. Mathematical preliminaries

We shall introduce some notations and recall the basic concepts of McMillan degree and Kronecker indices of rational matrices (see also

3

The row degree of the i-th row of a polynomial matrix F(z) is denoted by ari[F(Z»). Its highest row coefficient matrix is denoted by F

hr• Con-sider a pxm rational matrix K(z) that is proper, i.e.

lim

,..-

K(z)

=

K o (2.1)

exists. The Me-Laurin expansion of K(z) around z~ then gives

K(z)

_L

(2.2)

£=0

where K~ (t = 0,1,2 •. ) are pxm matrices.

Associate with K(z) the Hankel-matrix:

(2.3)

~+1

The rank of H1 ~[K] is called the McMillan degree of K(z), denoted by 6[K(Z)].

H1/~(K]

consists of block rows of size p. Denote the i-th row of the j-th block row of

H1,~[K]

by r(i,j) and

i=1,2, ..• ,p) be an integer such that r(i,n.+1) 1 that is linearly dependent on all rows above it

let n. (for each 1

is the first row r(i,j) in

H1,m[K].

The thus obtained integers n , . . . ,n are called the left Kronecker indices of

1 P

K(z). The following important result holds for those quantities (see e.g. Kailath (1980»:

Lemma 2.1:

Let K(z) be a pxm proper rational matrix. Then the McMillan degree of K(z) is the sum of the left Kronecker indices of K(z):

O[K(Z)] =

f

i=1

(2.4)

4

3. The McMillan degree and the Kronecker indices of MFD- and ARMA models

The McMillan degree and the Kronecker indices of K(z) are easily related to the determinantal degree and the row degrees of P(z) in any row-reduc-ed left coprime MFD p-l(z)Q(z) of K(z). This is formulated in the fol-lowing theorem, which is a slight extension of lemma 2.3 in Gevers (1986).

Theorem 3.1

Let K(z) be a proper pxm rational matrix and let p-l(z)Q(z) be a left coprime fWD of K(z). Then 6[K(Z)] = deg det P(z). Moreover, P(z) is row-reduced if and only if its row degrees are the left Kronecker indices of K(z). In addition these row degrees can be arranged in arbitrary

order.

Proof: See appendix.

***

As we have already noted in the introduction, an analogous result does not need to hold for ARMA models: the determinantal degree and row-degrees of A(d) do not need to be related to the McMillan degree and the Kronecker indices of K(z) (e.g. moving-average models, i.e. A(d)

=

I). However, the following "counterpart" of theorem 3.1 illustrates that things are not as bad as they seem.

Theorem 3.2

Let K(z) be a proper pxm rational matrix and let A-I(d) B(d) be a left coprime ARMA-decomposition of K(z). Then we have:

(i) Let S( d) be a pxp submatrix of [A( d) : B( d)] which has the highest determinantal degree (i.e. deg det Sed»~ amongst all pxp submat-rices of [A(d);B(d)]. Then 6[K(Z)]

=

deg det SId). Moreover, Sed) is row-reduced if and only if its row degrees are the left Kronecker indices of K(z). In addition, these row degrees can be arranged in arbitrary order.

(ii) Let Sed) be a pxp submatrix of [A(d):B(d)] and suppose that Sed) is row-reduced. Then O[K(Z) ]

=

deg det Sed) if and only if

Or.

[A(d); B(d)] ar.[S(d)]

5

(iii) The row degrees of [A(d):B(d)J are the left Kronecker indices of K{z) if and only if [A:B]hr has full row rank.

Proof: See appendix.

***

Remark 3.1:

By using arguments similar to those in the proof of theorem 3.2 we can

show that the properties stated in theorem 3.2 also hold for

MFD

models

p-l(z)Q(z).

For

MFD

models we know even more: the denominator matrix

p(z)

can be chosen as the submatrix of

[P(z):Q(z)]

with the highest

de-terminantal degree. With this extra information the results of theorem

3.2 will specify to those of theorem 3.1.

***

Remark 3.2:

Gevers presents an alternative result on the McMillan degree of monic

(i.e. A(O)

=

I) left coprime ARMA decompositions.

In theorem 5.1 of Gevers (1986) he states that the McMillan degree equals the rank of a suitable matrix M filled with coefficient matrices of A(d)

u

and B(d). This result is, however, incorrect, as illustrated by the following example:

Let A(d) and B(d) be defined as

A(d) B(d) (3.2)

Inspection shows that A-I(d) B(d) is a monic left coprime

ARMA-decomposition of K(z) given by

K(z) [

::~J

(3.3)

One easily establishes that 6[K(Z)] = 2. The rank of the matrix M

u in expression (5.2) of Gevers (1986) is, however, equal to 3. This illustrates the inaccuracy of theorem 5.1 of Gevers (1986).

6

Further insight into the inaccuracy of Gevers' result is obtained by

invoking theorem 2 in Pugh (1976). This theorem tells us that the rank

of matrix Mu' (defined by expression (5.2) of Gevers (1986», is equal to the highest degree of the minors of all orders of [A(d):B(d)j.

On the other hand, our theorem 3.2(i) shows that the McMillan degree O[K(Z)] equals the highest degree of the minors of order p (i.e. the maximum order) of [A(d):B(d)j.

Combining these observations, we obtain that O[K{Z)] ( 6[K(z)j can be smaller than rank M (see A(d) and B(d)

u

an illustration).

rank M and that

u

given in (3.2) for

Finally, we notice that the corollary 5.1 in Gevers (1986) remains valid

notwithstanding the fact that theorem 5.1 is incorrect. (Gevers also

presents an alternative proof of this corollary which does not rely on the incorrect result of theorem 5.1). This corollary can also be

obtained as a special result from our theorem 3.2(i). Observe that the highest determinantal degree amongst all pxp submatrices of [A(d):

I]

is equal to the highest degree of the minors of all orders of A(d). ~ Application of theorem 2 in Pugh (1976) then leads to the result stated in corollary 5.1 of Gevers (1986).

***

proposition (i) in theorem 3.2 shows that the McMillan degree of a left coprime ARMA decompostion A-I(d) B(d) is equal to the highest determinan-tal degree of all pxp submatrices Sed) of [A(d):B(d)]. Moreover, the row degrees of a submatrix Sed) having highest determinantal degree are equal to the left Kronecker indices if and only if Sed) is row-reduced. Prop-osition (ii) gives a simple characterization, for row-reduced sub-mat-rices S(d), of those ARMA models which fulfill 6[K(Z)j

=

deg det S(d). Result (iii) finally establishes a relationship between the row degrees of [A(d):B(d)] and the left Kronecker indices. Using theorem 3.2 we can now easily characterize the systems for which the McMillan degree equals the determinantal degree of the polynomial AR factor A(d) in any left coprime ARMA decomposition of K(z):

I

Corollary 3.1

Let K(z) be a proper pxm rational matrix and let A-l(d) S(d) be a left coprime ARMA decomposition of K(z). Then we have:

( i) ( i i )

degdetA(d)~6[K(Z)]~

f

i=1

6[K(Z)] i=l (3.4)

if and only if [A:B]hr has full row

rank. In this situation the row degrees of [A(d);S(d)] are the left Kronecker indices of K(z).

(iii) 6[K(Z)]

=

deg det A(d) if and only if there exists a left coprime MFD p-l(z)Q(z) of K(z) with P(O) being non-singular.

(iv) If 6[K(Z)]

=

deg det A(d) then A(d) is row reduced if and only if its row degrees are equal to the left Kronecker indices of K(z). Moreover, these row degrees can be arranged in arbitrary order. (v) If A(d) is row reduced then 6[K(Z)]

=

deg det A(d) if and only if

(3.5) ;

Proof: see appendix.

***

Expression (3.4) gives a lower- and an upper bound for the McMillan de-gree of K(z) on the basis of an arbitrary left coprime ARMA decomposi-tion. Necessary and sufficient conditions are given (see proposition

(ii) and (iii) of corollary 3.1) under which the lower- and upper bounds are attained. Proposition (iii) shows that the McMillan degree can be

expressed in terms of the determinantal degree of the denominator matrix

A(d) of any left coprime ARMA decomposition A-1(d)B(d) of K(z) if and only if the system represented by K(z) has no poles at the origin. More-over, proposition (iv) states that,

in

this case, the row reduced mat-rices A(d) are those polynomial matrices which have row degrees equal to the left Kronecker indices of K(z).

Remark 3.3:

Letting K(z) be a proper rational transfer function having no poles at the origin, one might wonder whether there exists a monic left coprime ARMA decomposition A-1(d)B(d), with A(d) being row-reduced, such that 6 [ K ( z)] = deg det A( d) •

The answer will be, in general, negative, unless O[K(Z)]

=

pxk for some integer k. (Combine corollary 4.1 of Gevers (1986) with corollary 3.1(v)

above}. Recent results of Bokor and Keviczky (1987), however, show that it is always possible (for systems having no poles at the origin) to obtain, on the basis of the constructibility invariants, a monic left

coprime ARMA model A-i(d)B(d), with A(d) being column-reduced, such that

6[K(zl]

=

deg det A(d).

***

Finally, we present a straightforward consequence of theorem 3.2.

Corollary 3.2

Let K(z) be a proper pxm rational matrix and let A-i(d)B(d) be a left

coprime ARMA decomposition of K(z). Let

e

1, c , . . . ,c 2 p+m be the c91umn degrees of [A(d}:B(d)]. order them, in a non-increasing way as

6[K(Z)] ( ) T p+rn

I

i=l T. ~ Then

with equality if and only if there exists a pxp submatrix Sed) of

[A(d):B(d)] which is column reduced with column degrees equal to

Proof: The result is an easy consequence of theorem 3.2(i).

(3.6)

***

Due to this result, the McMillan degree of monic ARMA models which are

specified by prescribing the column degrees of [A(d):B(d)] (see e.g. Deistler (1983), Hannan and Kavalieris (1984» is in general equal to

i=l

T. given in (3.6).

~

(q ) 0, r ) 0), given by

9 A(d) _{A "0} q B "0 r (3.7) (3.8 )

we obtain that the McMillan degree will, in general, be" pemax(q,r) if

m)

P,

respectively, m.max{q,r)+{p-m).q, if m

<

p. (For the situation

with m ) p, see also Gevers (1986».

4. Conclusions

The McMillan degree of a system is equal to the determinantal degree of the denominator matrix P(z) in any left coprime MFD model P-1Cz)Q(z).

For ARMA models a similar property holds: the McMillan degree is equal to

the highest determinantal degree of all pxp submatrices of [A(d):B(d)]

(p

=

number of outputs). Moreover, i t will be equal to deg det A(d) i f

and only if the system has no poles at the origin. In this situation the left Kronecker indices appear to be equal to the row degrees of the poly-nomial AR factor A(d) if A(d) is additionally row-reduced.

These results indicate, in contrast to the last statement of the

tonclu-sions in Gevers {1986}, that inspection of, and control over the McMillan

degree of systems having no poles at the origin can simply be performed

when using left coprime ARMA models.

Appendix

In proving the results of the paper we shall need the following lemma:

Lemma A.1

Let F(z) be a pxs (s > p) polynomial matrix with rank Fhr

F(z) be a pxs polynomial matrix with

p and let

F(z)

=

U(z) F(z) (A.l)

for certain unimodular matrix U(z). Order the row degrees of F{z) (res-pectively F(z» in a non-increasing way and denote them by R

(respective-ly R) (Le. R = (i'I'A

2.:, ••• ,Ap) where Al ) A2 ) ••• ) Ap are row degrees

of F(z). Similar for R). Then we have

10

Proof:

The implication 1t+1t in expression (A.2) follows from applying theorem 43

and note 49 in appendix A2 of Blomberg and Ylinen (1983). To prove the implication "+11, we take a pxp submatrix F(z) of F(z) with (such a sub-matrix exists)

(A. 3)

P

hr is non-singular (A.4 )

Denoting the corresponding submatrix of F(z) by P(z) we have (due to (A.1» that P(z)

=

U(z)P(z). As a consequence

deg det P(z)

=

deg det P(z) (A. 5)

Using (A.3) and (A.4) we conclude from (A.5) that

deg det P(z) =

f

ar.[F(Z)]

i=l ~

(A.6)

IfR R we therefore have that

i=l

ar.

[F(

z)]

~

deg det P(z)

From this we easily deduce that P

hr is a non-singular submatrix of Fhr -This proves that F

hr has full rank. This completes the proof.

***

Now we present the proof of the theorems 3.1 and 3.2 and of corollary

Proof of theorem 3.1:

It is a standard result (see e.g. Kailath (1980» that the McMillan degree of a proper rational matrix K(z) equals deg det P(z) in any left coprime MFD P-1(z)Q(z) of K(z). Moreover, there exists a left coprime MFD, say [P*(Z)]-lQ*(Z), of K(z) with P*(z) being row-reduced, having row

11

degrees equal to the left Kronecker indices (see e.g. Beghelli and

Guidorzi (1976), Guidorzi (1975, 1981».

Since any left coprime MFD P-1(z)Q(z) of K(Z) is related to [p*(z)j-1Q*(z) as

u(z)[ p* (z) :Q*(z) j (A.8 )

for certain unimodular matrix U(z), application of lemma A.1 to P*(z) and p(z) gives that P(z) is row-reduced if and only if its row degrees are the left Kronecker indices. The proof is completed by observing that the row degrees can be arbitrarily arranged by permutation of the rows of

[P(z);Q(z)j.

***

Proof of theorem 3.2:

First we state some useful preliminary results which will be used in proving the statements of the theorem. According to Beghelli and

Guidorzi (1976) there exists a canonical MFD [p*(z)j-1Q*(z) of K(z) which

*

has the following properties (see also Guidorzi (1975, 1981» (P .. (z),

*

~J

respectively Q .. (z) denote the i,j-th entry of P*(z), resp. Q*(z»: ~J

*

(i) _{Pij(z) is a monic polynomial with}

*

deg PH (z) = n.

~ (A.9a)

* *

(H) deg P ij (z)

<

deg PH (z) j

<

i (A.9b)

* *

deg P ij (z)

<

deg PH (z) j

>

i (A. 9c)

* *

deg Pji(z)

<

deg PH (z) j

"

i (A.9d)

*

(iii) deg Qij(z) ( deg Pii(z) (A.ge)

12

Here n

i denotes the i-th left Kronecker index of K(z). Observe that P*(z) is row-reduced and column-reduced with p* being a lower triangular

hr

matrix having unit diagonal elements. Define now

n

n

where M(d):= diag (d.1 , ••• ,d

P).

(A.10)

[A*(d)]-lS*(d) forms a canonical ARMA decomposition of K(z) which is sometimes called the reversed echelon form (see Gevers , 1986).

Noticing that [A*;B*]hr = [P*(O):Q*(oJ] and [A*(O):B*(O)] [p*;Q*]hr'

one can easily deduce from (A.9)-(A.10) that

(i) dr.[A*(d); B*(dJ] = _1. 1 ( i , P

n 1

(ii) [A*:B*]hr is full rank

(iii) A*(d) and B*(d) are left coprime

From (A.11a)-(A.llb) we deduce

(1) The highest determinental degree of any pxp submatrix

( U)

S*(d) of [A*(d): B*(d)] is equal to

I

i=l

n. = 6[K(Z)]

~

Any pxp submatrix S*(d) of [A*(d): B*(d) ].'with

deg det S*(d) = 6[K(ZJ] is row-reduced with 3r.[S*(d)]

~ 1 ( _i ( P (A.lla) tA.llb) (A.llc) (A.12a) n. 1. (A.12b)

Using these properties we now proceed with proving the ~atementB of the

theorem. Let A-l(d) B(d) be a left coprime ARMA decomposition of K(z).

Then there exists a unimodular matrix U(d) such that (all left coprime ARMA models are related to each other in this way, see e.g- Kailath

(1980»

13

Let S*(d) be any pxp submatrix of [A*(d):B*(d)]. Denoting the

corres-ponding submatrix of [A(d):B(d») by S(d), we obtain from (A.13)

S(d)

=

U(d) S*(d) (A.14)

Therefore

deg det S(d)

=

deg det S*(d) (A.1S)

Using lemma A.l we additionally obtain:

If S*(d) is row-reduced, then Sed) is row-reduced if and only

if the row degrees of Sed) are equal to the row degrees of S*{d) (A.16)

Using (A.12), (A.1S) and (A.16) proposition (i) of theorem 3.2 follows

eaSily.

In order to prove (ii) we assume that Sed) is a row-reduced pxp submatrix

of [A(d):B(d»). I f ar.[A(d):B(d)] _{, .} = ar,[S(d)] (1 ₁

~

i

~

p), then i t is

obvious that the highest determinantal degree of any pxp submatrix of

[A(d);B(d)] is equal to

f

i=l

Or . [

,

S ( d) J = deg det S ( d) •

Using theorem 3.2(i), we conclude that

8[K(zl] =

deg det SCd). Thus we have proved the "if-part" of ( i i ) . For the proof of the "only if-part"

we assume that 6[K(Z») = deg det S(d). Now let S*(d) be the

correspond-ing submatrix of (A*(d):B*(d)]. Using (A.15) we conclude from (A.12b) that S*(d) is row-reduced with

(A.17)

Since S{d) and S*{d) are both row-reduced, the unimodular matrix U(d} in (A.14) will have a special structure, as shown in appendix A2 of Blomberq' and Ylinen (1983) (see theorem 43 and the subsequent notes). Inspection teaches now, due to the special structure of U(d), that (A.17) gives rise

to

ar.[U(d)S*(d)j

~

14

(A.18)

Therefore (3~1) holds. This completes the proof of proposition (ii). Proposition (iii) is an easy consequence of (1). According to (i) there

exists a pxp submatrix S(d) with deg det S(d)

=

6[K(z)j. If the row

degrees of (A(d):B(d)] are the left Kronecker indices of K(z) we can

therefore conclude that S(d) is row-reduced with ar.[S(d)j

=

~

3ri[A(C):B{d)]. Thus Shr is a non-singular pxp submatrix of [A:B]hr. So [A;B]hr has full row rank. Conversely, suppose that [A:B]hr has full row rank. Then there exists a pxp submatrix S of [A~BJhr which is

non-hr

singular. Denoting the corresponding submatrix of [A(d):B{d)J by Sed) we observe that

sed)

is row-reduced with the same row degrees as

[A(d):B(d)j. Therefore S(d) has the highest determinantal degree of all

pxp submatrices of [A(d);B(d)j. Applying proposition (i) we learn from

the foregoing that the row degrees of [A(d):B(d)j are the left Kronecker

indices of K(z). This proves proposition (iii) of theorem 3.2.

.**

Proof of Corollary 3.1:

Observing that deg det S(d) (

f

ari[A(d):B(d)j. expression (3.4) is an

i=l

easy consequence of theorem 3.2(i). Moreover proposition (ii) in

corol-lary 3.1 then follows easily (consider also theorem 3.2(iii». Corollary

3.1(iv) and (v) are straightforward consequences of theorem 3.2(i) and (ii). Now proposition (iii) of the corollary remains te be proved.

Using (A.13) we have that 6[K(Z)j

=

deg det A(d) if and only if 6[K(z)j

=

deg det [A*(d)j. Application of theorem 3.2(i) and lemma 2.1 shows that

this will be the case if and only i f A*(d) is row-reduced, i.e. i f and

only if P*(O) is non-singular. Since all left coprime MFD's P-1(z)Q(z)

of K(z) are related to [p*(z)j-lQ*(z) as [P(z):Q(z)j

=

U(z)[p*(z);Q*(z)j.

with U(z) unimodular. the result of corollary 3.1(iii) follows easily.

I~

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RANDOM ACCESS MEMORY TESTING: Theory and pr"cticc. The gains of fault modelling. EUT Rt'Jlort 86-E-161. 1986. ISBN 90-6144-161-7

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TMS32lilQ EVALUATION MODULE CONTROLLER. EUT Report 86-E-162. 1986. ISBN 90-6144-162-5 (163) Stok, L. and R. van den Born, G.L.J.M. JanSSen

HIGHER LEVELS OF A SILICON COMPILER. ---EDT Report 86-£-163. 1986. ISBN 90-6144-163-3 (i,,.1) Enqelshoven, R.J. van and J.F.M. Th ... cuWerl

GENERATING LAYOUTS FOR RANDOM LOGIC: Cell generalion schemes. EUT Report 86-E-164. 1986. ISBN 90-6144-164-1

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~IMAL CMOS STRUCTURE FOR THE DESIGN OF A CELL LIBRARY. EUT Report 87-E-l06. 1987. lSFlN 90-t,144-t(,6-8

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ESK.ISS;--;;; ~r{)yram for uptlmdl stilt\~SSlCjnment. EUT Report 87-E-167. 19~7. IS8N 90-6144-167-6

(16H) Llnnartz, J.P.M.G.

SPATIAL DISTRIBUTION Of TRAFF'IC IN A CELLULAR MOBlLE DATA NETWORK. EU'I'Report 87-£-168. 1987. ISBN 90-6144-168-4

(169) Vlnck, A.J. and Pineda de Gyvez, K.A. ~

1."IPLEMENTATION AND EVALUATION OF A COMBINED TEST-ERROR CORRECTION PROCEDURE FOR MEMORIES WITH DEFECTS. EUT Report 87-E-169. 1987. ISBN 90-6144-169-2

(l,u) lluu Ylbln

DASM: A tool for decomposition and analySIS of sequential machines. EUT Report 87-Eo-170. 1987. ISBN 90-6144-110-6

(171) Monnee, P. and M.H,A.J. Herben

MULTIPLE-BEAM GROUNDSTAT~FLECTOR ANTENNA S¥STEM: A preliminary study. EUT Report 87-E-171. 1987. ISBN 90-6144-171-4

(J72) Bastiaans, M.J. and A.H.M. Akkermans

ERROR REDUCTION IN TWO-DIMENSIONAL PULSE-AREA MODULATION, WITH APPLICATION TO COMPUTER-GENERATED TRANSPARENCIES.

EUT Report 87-E-172. 1987. ISBN 90-6144-172-2 (173) ~ YU-Cdl

ON A BOUND OF THE MODELLING ERRORS OF BLACK-BOX TRANSFER FUNCTION ESTIMATES. EUT Report 87-£-173. 1987. ISBN 90-6144-173-0

(}74) BeO:elaar, M.R.C.M. and J.F.M. Theeuwen

TECHNOLOGY MAPPING FROM BOOLEAN EXPRESSIONS TO STANDARD CELLS. EUT Report 87-E-174. 1987. ISBN 90-6144-174-9

(17~) JJnSSen, P.H.M.

FURTHER RESULTS ON THE McMILLAN DEGREE AND THE KRONECKER INDICES OF ARMA MODELS. EUT Report 87-E-175. 1987. ISBN 90-6144-175-7

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MODEL STRUCTURE SELECTION FOR MULTI VARIABLE SYSTEMS BY CROSS-VALIDATION METHODS. EUT Report 87-£-176. 1987. ISBN 90-6144-176-5