Realization of NSHP-filters

Realization of NSHP-filters

Citation for published version (APA):

Eising, R. (1978). Realization of NSHP-filters. (Memorandum COSOR; Vol. 7804). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1978

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics

PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP

Memorandum COSOR 78-04 Realization of NSHP-Filters by F. Eising Eindhoven, January 1978 The Netherlands

Realization of NSHP-Filters

by

F. Eising

Abstract

In this note a class of 2-D transfer matrices ~s considered,much larger then the class of so-called causal transfer matrices. We will generalize the state space model of [IJ such that also nonsymmetric-half-plane (NSHP) fil-ters can be state space realized. Surprisingly enough this can be done using realization theory over a field.

- 2

-1. Introduction

First of all we introduce some definitions and notations. lR[s] denotes the set of polynomials in s with real coefficients. lR[s,z] denotes the set of po-lynomials in two variables sand z with real coefficients. It is clear that )R[s,z] =lR[s][z], see also [IJ.

R(s) denotes the set of real rational functions in s. lR(s,z) denotes the set of real rational functions in two variables sand z. Elements of lR.(s, z) will be denoted by p/q with P E lR[s,z] and q E lR[s,zJ. For q E lR[s,z] the degree of q in z will be denoted by deg (q).

z

rxm

lR [s,z] denotes the set of r x m matrices with entries fromlR[s,zJ. rXm

R (s,z) denotes the set of r x m matrices with entries fromR(s,z). rXm

Let T E R (s,z) then T can be written as P/q with q the LCM of the denomi-rxm

nators of the entries of T and P E lR. [s,zJ and q E R[s][z]. Now suppose

that deg (q) = n. He then devide q and the entries of P by the coefficient of

z

zn thus making q a monic polynomial with coefficients fromR(s). Now it is clear that by this procedure we obtained an element of R(s)rxm(z). We define the degree in z of P as the maximum of the degrees in z of the entries of P. W e W1 "II now 1ntroduce the set of proper " 2-D r x ~transfer matr1ces lR • rxm() S,z.

RTxmCs, z) =

p {T E Rrxm(s~z)

I

T

=

P/q, deg (q) z ~ deg z (P)} • rxm

T E R (s,z)

P

This set will

will be called strictly proper if we have deg (q) > deg (P).

z . z

be denoted by Rrxm(s,z). In this paper we will consider mainly sp

strictly proper 2-D transfer matrices. Analogous results with only minor mo-difications can be obtained fpr the proper case.

Qbserve that we now have obtained a straightforward generalization of the I-D notion of a proper transfer matrix. The coefficients of our 2-D transfer matrix are not real numbers as in the case of I-D transfer matrices but they are rational functions (elements of the fieldR(s».

- 3

-2. The realization procedure

. rxm() . .

For a transfer matr~x T E R s,z there exists, as is well known, a m~n~­ sp

mal realization (A(s),B(s),C(s» such that

for some integer n such that

T(s,z) = C(s)[zI-A(s)]-IB(s) •

Minimality is as usual equivalent to (see also [1])

I) (A(s),B(s» is a reachable pair. 2) (C(s),A(s» is an observable pair. Now we have the following theorem.

(2.1) THEOREM. (A(s),B(s» ~s reachable iff (A(sO),B(sO» is reachable for

some complex number sO'

(e (s) ,A(s» is obseI'Vable iff (C (sO) ,A(sO» is observabZe for some complex

nlfllber sO'

Proof. By duality it ~s enough to prove only the reachability part. Suppose (A(s),B(s» is a reachable pair thus by definition we have:

[B(s) 1A(s)B(s) \ •.• IA(s)n-IB(s)] has full rank and is therefore right inver-tible. So for every

So E e

for which B(sO) and A(sO) are defined we have [B(sO)IA(sO)B(So)1 .•. jA(So)n-1 B(sO)] has full rank and therefore

reachabili-I

\

n-l

ty of (A(sO) ,B(sO»' Now suppose [B(s,) A(s)B(s) ••• A (s)B(s)J has not full rank then all n x n minors are zero. Thus for every sl E C for which A(s1)

and B(s1) are defined (A(sl),B(sl» is not a reachable pair.

0

Consider the NSHP

S

(see also [5J)

S

=

{(h,k) E

z2

1 h >

a

or (k >

a

and h

=

O)} •

Now consider the I/O (input/output) description of a 2-D system

(2.2) _Yhk

₌

I

_{Th · k . u ..}

( . . 1 , ] E ) S -1, -J 1J

(h,k) E S

where u .. E Rm, Yhk ERn, T has its support in

S.

This I/O system will be

1J pq

called a NSHP filter. Applying the 2-D Z-transform (or (z,s) transform) (see

4

-9(s,z)

=

I

_YhkZ -h -k s (h,k)ES 9(s,z) =

_I

_~kZ -h -k s (h,k)ES T(s,z)

=

L

_Thkz -h -k s (h,k)ES and (2.3) 9(s,z) = T(s,z)G(s,z)

From now on we will assume that this T(s,z) is a strictly proper 2-D trans-fer matrix (for conditions see [IJ, [2J). We will now derive a state space description of this I/O map. Using (2.3) we obtain:

(2.4) ~+I(S) = A(s)~(s) + B(S)~(s) h=O,I, ... where ~(s) is defined by ~(s)

=

and ~k E JRn (h,k) E S.

~(s) and Yh(s) are defined analogously.

Remark. Observe that the state space is infinite dimensional. Writing the equations for every ~k' ~k' Yhk we obtain an infinite dimensional system where the system matrices are doubly infinite block-Toeplitz matrices [3J. This way of representation can be used in filtering [4J.

Remark. (2.4) is a generalization of the first level realizati~n defined in

[IJ.

The generalized first level realization (2.4) can be realized itself to ob-tain a generalized second level realization. Compare also [2J. First of all we observe that there exists an integer d such that

- d - d ~ d

A(s)

=

A(s)/s , C(s)

=

C(s)/s , B(s)

=

B(s)/s are proper l-D transfer matrices.

Now consider the equations:

(2.5)

5

-Now take minimal realizations of A(s), B(s) and C(s) • Let

A(s) AD + AC[sI- MJ-1AB B(s) = BD + BC[sI - BAJ-1BB C(s) = CD + GC[ sI - GAJ-J GB

Then a generalized second level realization is: b h,k+l BAbhk + BB~,k+d a = _{AAah,k + AB~,k+d} h,k+l ~ "" _"" (2.6) _~+l ,k = AD~,k+d + ACahk + BD~,k+d + BCbhk '" ch,k+l CAchk + CB~,k+d Yh,k

=

CD~,k+d _{+ CCchk •} Initial conditions are zero.b

hk, ~k' chk are intermediate state space va-riables of appropriate dimensions.

Remark. Observe that we have taken the number d to be the same for A(s), B(s), C(s). Of course this is not necessary but because of simplicity we have chosen to do so.

We will now assume that there exists a positive integer I such that

u

(2.7) ~k = 0 for k < -luh, (h,k) E

S .

Introducing this condition it can easily be seen that equations (2.6) can be evaluated in a recursive way.

Remark. The condition (2.7) is not necessary but because of simplicity we have chosen this condition. Note also that (2.7) is a kind of uniformity condition.

Remark. For more details on equations like (2.6) see [2J, where also 2-D systems with support in a certain subset of a half plane,. not necessarily 1n the first and fourth quadrant, are taken into consideration. This is done by using spectral transformations.

6

-3. Conclusion

In this note a generalized first level realization and a generalized second level realization of a (strictly) proper 2-D transfer matrix has been intro-duced. TIlis approach leads to equations comparable to those of [2J. The ad-vantage of this approach is that one can work over a field while in [2J one has to work over a ring. However [2J gives an a priori bound on the number d

(see (2.5)). It is still an open question how one can reduce the number d by a state space isomorfism.

7

-References

[IJ F. Eis • Realization and Stabilization of 2-D systems. CaSaR-Memoran-dum 77-16.

[2J F. Eising. Recurrence and Realization of 2-D systems. CaSaR-Memorandum 77-26.

[3J F. Eising. Toeplitz matrices and 2-D systems, to appear.

[4J M.F. ter Horst, F.C. Schoute, J.C. Willems. Digital Image Enhancement. IEEE trans. CAS. 1977.

[5] J.W. Woods. ~farkov Image Modeling. IEEE ]976 Decision & Control Confe-rence, pp. 596-601.