State models and stability for 2-D filters

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 31, NO. 12, DECEMBER 1990 1509

State Models and Stability for 2-D Filters

Abstract --In this study a generalized 2-D discrete-time filter is considered. Using the concept of a wave advance process the filter equation is converted to a 1-D recursive form. A 1-D state equation is then developed and a canonical form for the state equation is presented.

A norm hound on state transitions is developed, which is then related to the coefficients of the underlying filter equation. This norm bound is used to specify asymptotic stability conditions, the resultant criteria are illustrated through examples.

I. INTRODUCTION N RECENT studies [1]-[3] the authors have intro- I duced a state variable model, called the wave advance

process model (wave model for short) for discrete rn-D processes. This model has proved useful in solving prob- lems of system inversion and decoupling [2]. In conjunc- tion with other developments [4]-[6], the model can be used to transport 1-D results pertaining to optimal con- trols, optimal recursive filters and state estimation, to the rn-D setting.

One distracting feature of the wave model is that the output, state and input vectors can increase in dimension as the recursion moves across the rn-D domain. This implies a nonstationary characteristic, even for systems which are rn-D stationary. On the other hand, the wave model is equally available for nonstationary rn-D systems.

In the earlier studies such well known semi-state models as the G-R model [71 and F-M model [SI were used as intermediate steps in providing examples of the wave model. While it is known that all conic causal maps have wave model representations 151 an efficient procedure for specifying such representations has not been published. The present study supplies such a procedure in the con- text of 2-D stationary filters. With straightforward modification the procedure works for rn-D and/or for nonstationary filters as well.

Our attention then turns to wave model stability. In particular, given the wave model for a 2-D discrete stationary filter, can the filter stability be tested via the wave

Manuscript received February 21, 1989; revised July 14, 1990. This work was supported in part by SDIO/IST under Contract 24962-MA- SDI and by the State of Louisiana under Research Contract LEQSF- RD-A-17. This paper was recommended by Associate Editor D. M. Goodman.

J. L. Aravena and M. Shafiee are with the Department of Electrical and Computer Engineering, Louisiana State University, Bato Rouge, LA 70803-5901.

W. A. Porter was with Louisiana State University, Baton Rouge, LA. He is now with the Department of Electrical and Computer Engineer- ing, The University of Alabama at Huntsville, Huntsville, AL 35899.

IEEE Log Number 9039222.

model? This is a priori a very challenging matter. The voluminous literature on 2-D stability (see [91-[131 as entries) documents the difficulties of 2-D stability issues. Moreover, the time-varying characteristic of the wave model precludes such conventional techniques as eigenvalue analysis of the transition matrix. Nonetheless significant progress is made.

To investigate stability we develop a necessary and sufficient norm bound criterion on the state transitions. This criterion is then related to the coefficients of the underlying filter model. The norm bound criterion is used to define asymptotic stability. We illustrate the resultant filter coefficient criteria through examples.

To illustrate the key concepts with a minimum of notational clutter we dedicate Section 3 to the analysis of a special case. The bivariate equation studied arises from both F-M and G-R semi-stage evolution equations. Later, in Sections IV and V we complete the analysis of the general evolution equation.

11. THE WAVE MODEL FOR 2-D FILTERS Consider now the 2-D recursive equation

y ( k , l ) = [ a i j y ( k - i , l - j ) + b i j u ( k - i , ~ - j ) ] i , j G A

(1)

where A is the index set ( M , N arbitrary)

A = ( ( i , j ) : O < i < N;O< j < M ; ( i , j ) # ( O , O ) } .

The initial conditions on (1) are taken to be

y ( a , b ) = 0, - N < a < - 1, - M < b < - 1

and our interest is with solutions to (1) over, Q,, the first quadrant i, j 2 0. We note that y(0,O) = 0 may be inferred from the above conditions. Moreover if u( i , j ) = 0 on Q , then y ( i , j ) = 0 on Q,.

With the development of [1]-[3] as motivation we define two natural wave variables. For this let n = k + l and let

W,( n ) = CON Y ( 3 0) 9 Y ( n - 1 , 1 1 9 . . .1 Y ( 1, n - I), Y (0, n ) )

W,(n)= col( u(n ,O) , u ( n - 1,l);. . , u ( l , n - l), u ( 0 , n ) ) .

Using these vectors, (1) can be repackaged in a 1-D format. To give insight into this conversion it is useful to consider an example. Further details and discussion are available in the above references.

0098-4094/90/1200- 1509$01.00 01990 IEEE

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 12, DECEMBER 1990 1510

Example I : Let M = N = 1, then with k + I = n we have, using (1) and its initial conditions,

y(n,O)=a,,y(n -LO)+ bl,u(n - 1,O) y(n - l , l )=a , ,y(n -2,l)+ao1y(n -LO)+ a, ,y(n -2,0)+

b,,u(n -2,1)+ b,,u(n - l,O)+b,,u(n -2,O)

A 2 ( n - 2 ) =

Y O , n - 1) = a,,y(O, n - l)+a,,y(l, n - 2)+ a,,y(O, n - 2)+ b,,u(O,n - l )+b, ,u( l ,n-2)+b1,u(0,n-2)

+ b,,u(O, n - 1) ~ ( 0 , n ) = a,,y(O, n - 1)

- - 0 * * 0

a11

0 - a11

- 0 * * 0 -

A 1 ( n - l ) =

where = 0 whenever ( k - I, I) # A. The entries of a, appear as the first k + 1 entries of the left-hand column of A , which is then completed with zeros. Each nonzero entry of this left-hand column identifies a band of identical values in A, . All other entries of A , are zero.

As an embellishment on our notation we use A, (m) to denote the matrix A,(n - k ) where n = m + k . To illustrate let N = 3, M = 2, and k = 3, then

a3 = { a30, a 2 ] 3 a122 a03) ; a03 = O. Therefore, A,(4) = A3(7 - 3) is the matrix

A, (7 -3 ) =

a 30

a21

a12 0 0 0 0 0

0

30

a21

a12

0 0 0 0

0 0

a 30

a21

a12

0 0 0

0 0 0

30

a21

a12

0 0

0 0 0 0

a 30

a21

a12 0

With these available definitions we state the general form of the 1-D equation. Let d = M + N , then (1) has the equivalent form

w,(n)= [ ~ , ( n - j ) ~ , ( n - j ) + ~ i ( n - j ) ~ ~ ( n - j ) ] .

n 2 l (2)

d

j = l

where the matrices { A j ( n - j ) , B j ( n - j ) ) are defined as in the above discussion. In particular, their dimensions are ( n + l ) x ( n - j + 1).

Equation (2) is the generalized 1-D evolution form of (1). Because the wave variables W,(n> and W,(n) are used in its statement we refer to (2) as the wave evolution equation.

+(1+1, j +1) = J+(i, j + 1) + K+( i + 1, j )

Example 2: We consider the bivariate equation

+ E u ( i , j + 1) + Fu(i + 1, j ) (3) where + may be vector valued. In the spirit of (1) we have alo -, K, a,, + J , b,, -+ F , bo, -+ E , and

1 A , ( n ) = 0 J

K L J 1

E ; 1

ARAVENA et al.: 2-D FILTERS 1511

The evolution equation takes the form

W+(n +1) = A , ( n ) W + ( n ) + B , ( n ) K ( n )

W,(n) = c o l ( 4 ( n 7 0 ) , 4 ( n - 1 , 1 ) , - , 4 ( 0 , n ) )

Y , ( n ) = col( u(n ,O) , u ( n - 1,l);. . ,u (O,n) ) .

where

Example 3: We complete these illustrative examples by revisiting the (G-R) semi-state equation

x " ( i + 1, j ) = A,xh( i, j ) + A,x"( i, j ) + B,u( i , j ) ,

x h € E" x' (1, j + I ) = ~ , x ~ ( i , j ) + A ~ X ' ( i , j ) + B,u(i , j ) ,

x' E E'. Consistent with our assumption about initial conditions we have x"( i , 1) = x ' ( - 1, j ) = 0, and consequently xh(O, j ) = 0, x " ( i , 0) = 0. We select then

+(i +1, j+ 1) = col{x' ( i , j + I ) , x h ( i + IJ)}, i , j = 0,1, . . * .

The G-R equation can now be rewritten as

We have the correspondences

and we can immediately write the wave evolution equation. In particular, we note that the A ( n ) matrices of this example and Example 2 agree in form if

2.1. The State Variable Form

variable format. For this we define The wave evolution equation converts easily to a state

x d n ) = W Y ( 4

then evaluating the wave evolution equation ( 2 ) at n + 1 we get

x , ( n + I > = A , ( n ) x , ( n ) + B , ( n ) W ( n ) + x 2 ( 4

with

x 2 ( n ) = A j ( n + l - j ) W , ( n + l - j ) d

j = 2

+ B j ( . + 1 - j ) K , ( n + 1,j) .

Clearly, we can write

x 2 ( n + 1) = A , ( n ) W , ( n ) + B , ( n ) W ( n ) + x3(n)

with

x 3 ( n ) = A j ( n + 2 - j ) W y ( n + 2 - j ) d

j = 3

+ B j ( n + 2 - j ) W u ( n + 2 - j ) .

The process is repeated d times in the same manner. Letting

while @(n) is the block matrix x( n) = col(x,( n ) ; . . , X d ( n))

@( n) =

A , ( n ) I 0 0 A 2 ( n ) 0 I 0

0 I

A d ( n ) 0

and B(n) = col[B,(n), B2(n); . * , B&)] our equation set takes the vectorial form

x( n + 1 ) = (D (n )x ( n) + E ( n)W,( n), n 2 0 W y W = xdn). ( 5 )

Remark I: For future reference we note the dimensionality of the state vector x(n). A brief examination shows

d i m x j ( n ) = n + j d d ( d + l )

dim x ( n ) = E( n + j ) = dn + 7. 1 L

In this calculation, of course, d = N + M. Remark 2: In our development we have treated y ( k , I),

u ( k , l ) as scalar valued. The examples show that the analysis remains valid in the vector valued case. Indeed we need only interpret the symbols a j j , b j j as subblock matrices to extend (2) and (5 ) to the vector valued case. Of course dimensionality scales up with block size.

At this juncture in the development we proceed in parallel along two directions. In Section IV we continue the discussion of the general model with the objective of determining a canonical form and affiliated properties. This development, although both abstract and involved, culminates in results which are argueably intuitive extensions of the results we pursue in the next section of the paper.

In the following sections we focus attention on a special case which illustrates the general techniques without the attendant abstraction and notional clutter. This special case is described by Examples 2 and 3 above.

111. AN EXTENDED EXAMPLE Referring to Example 3 we note that @(n) = Al(n) and

$(i, j ) E Em+'. To simplify notation we shall retain the A notation without the subscript. Without ambiguity we drop the subscript from W&). We focus also on the free response, in short the solution of the equation:

where A h ) is of the form of A l ( n ) of Example 2. W ( n + 1) = A( n)W( n), n 2 0 ( 6 )

1512 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 12, DECEMBER 1990

The result in fRmma 2 provides the basic tool for our stability analysis. Consider the matrix V with the scalars, Pi, Satisfying

Defining .\Il(n) = A(n)A(n - 1) . . . A(1) the free re- Theorem I: If ?(n) is the transition matrix sponse of (6) satisfies

We refer to "(n) as the transition matrix from the origin. This matrix is of dimension (n + 1) ( m + I ) X(m + I ) . For brevity we shall say that its block dimension is (n + l)X 1.

Let q,(n), i = 0,. * , n denote the column blocks of q(n) . That is, 1Ir,Tn) E E(m+r)x (m+r) and

*( n) = A( n ) A ( n - 1) * * A( 1)

W ( n + l ) =*(n)W(l) . then 1 n + l

n + l i = l 9 * ( n ) W n ) = - c e n * ( P , ) e " ( P , )

with

*( n) = col[ q0( n),. * * ,qn( n)] . P,=exp ( j25r- y). Our first result takes note of a convenient connection between the blocks *,(n) and the matrix K + P J .

We now turn our attention to the general n-step transition matrix. Let

V( n, k ) = A( n ) A ( n - 1) * . A ( k ) . Lemma I : Let = ( K + P J ) for scalar P, then

*d4)

*d4)

q2(4) = A(4)A(3)A(2)A( l ) ='€'(4).

For example, the matrix $(5,2) will have the form i = O

This relationship is made apparent by an examination of WO, W2), W3). A proof by induction is then trans- qt(5,2) = parent.

We introduce now a Vandermonde matrix (see [ l l ] )

(7)

Computing the indicated products for each block column of the matrix V(5y2) we get

. . .

where the P, are pairwise distinct. Consider the direct (Kronecker) product, Z 0 V, with Z the identity. It is well known that Z 0 V can be written in the block partitioned form as

[ IoV]=[Pf - 'Z] (8)

furthermore, Z o V is nonsiqgular if and only if V is nonsingular. The next result follows immediately.

Lemma 2: If V is the Vandermonde matrix of (7) then

lowing lkmrna.

1 = 0 then Lemma 3: If V is given in (7) with P , satisfying P:+' -

i) V * V = ( n + l ) Z ; ii) (1 0 V*XZo V ) = ( n + 01, + '.

Note in i) the identity is of dimension m + 1, while in ii) the matrix I,,+' is of block dimension n + 1 (i.e., of dimension (n + 1Xm + I ) ) .

Using Lemmas 2 and 3 we derive:


n a U 0 3

If (pi, i = l , - . . , n + k } is a collection of scalars we define the matrix

V ( n + k , l ) =

The Vandermonde matrix V, of size n + k, corresponds to V(n + k, n + k ) when the scalars are all distinct.

We can now establish Lemma 4. Lemma 4: The transition matrix *(n + k - 1, k ) satis-

fies the identity [ I o V( n + k , n + k ) ] *( n + k - 1, k ) = diag(On( p i ) )

- [ I o V ( n + k , k ) ] . (10)

Proof: The left-hand side of the identity has the form

Carrying out the product we get

Consider now a vector

~ ( k ) = c o l [ w , ( k ) , . - * , w k ( k ) ] , w j ( k ) E Ern+'

and define

n(p) = [ I o V ( n + k , k ) ] w ( k ) .

We can write

p ) p1) 9 * * 9 +k( p n +k ) with

k

ai( p i ) = W j ( k ) p p . (11) j = l

Also, from Theorem 2 we conclude that

( W P ) , W P ) ) = (. + k ) ( w ( k ) , w ( k ) )

(ai(Pi),ni<Pi>) ~ k ( w ( k ) , w ( k ) ) .

while

Using the definition for n(p ) we can establish.

step transition in the wavefront model then Theorem 3: If w(k + n) = *(k + n - 1, k ) w ( k ) is an n-

= diag (On( pi)) [ I 0 V( n + k , k ) ] .

Choosing again the scalars, p,, i = 1; . -, n + k, as roots

Theorem 2: The transition matrix T ( n + k - 1, k ) satis- of pn+k - 1 = 0 we obtain the following theorem.

fies the identity

( n + k)**(n + k - l , k ) * ( n + k - 1 , k )

= [ I o V * ( n + k , k ) ] diag[On*(p,)8"(p,)]

* [ I o V ( n + k , k ) ]

where p, = e x p ( j 2 d i - l)/(n + k) ) , i = 1,2; .- , n + k. Furthermore, the matrix V(n + k , k ) satisfies

V * ( n + k , k ) V ( n + k , k ) = ( n + k ) Z k .

3.1. Stability Analysis We study here the asymptotic behavior of the homoge-

neous wave model arising from Examples 2 and 3. We shall say that the wavefront model is stable if and only if given k > 0 and Ilw(k)ll finite, there exist C > 0 such that Ilw(n + k)ll Q C for all n > 0. For similarity with the conventional 1-D case we say that the wavefront model is asymptotically stable if and only if it is stable and for any finite Ilw(k)ll initial condition, one has I l w h + k)ll+ 0 as n + W. Since w(n + k + 1) = W n + k, k ) w ( k ) a direct ex- tension of 1-D results yields the following.

Lemma 5: The homogeneous wave model in asymptotically stable if and only if IlWn + k, k)ll + 0 as n + c ~ .

1514 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEM5, VOL. 37, NO. 12, DECEMBER 1990

Our next stability result is a necessary condition. From

1 n + l

Theorem 2 we have

q * ( n ) q ( n ) =- en*(p i )en(p, ) n + l ; = I

with pi = exp (j27di - l ) / n + 1). Consider a rational number, r = p / q , in [0,1). If I tk = kq - 1 and ik = kp + 1 we see that

Furthermore if p, = e x p ( j 2 ~ r ) then

If O(p,) has an eigenvalue, A,, outside the unit circle (i.e., l A , l = 1 + 6, 6 > 0) then there exist an initial condition w(0) such that

Since nk = kq - 1 can be made arbitrarily large we conclude that the wave state

w ( n + l ) = V ( n ) w ( O ) grows unbounded. Since the number n is arbitrary we show the following.

Theorem 4: The wave model is unstable if for at least one rational number r, in [0,1] the matrix O(p,) has eigenvalues outside the unit circle.

By stating the converse proposition we get Corollary: If the wave model is stable then the matrix,

e@,) = K + p ,J , cannot have eigenvalues outside the unit circle.

We now turn to the analysis of a general n-step transition. From Theorem 3 we have

I I W ( ~ + n ) 11, (mai118n*(Pi)en(pi) II)IIw(k) I12. (12) The quantity Il~"*(p,)~"(p,>ll is the spectral norm of the matrix On(pi>. Assume now that for every p on the unit circle, the eigenvalues of e(p) are strictly inside the unit circle, i.e.,

I m < P > 1 I P < 1, all P where A[e(p)] denotes an eigenvalue for e@). In this case, the collection of matrices

m

M ( P ) = C e i * (p )e i (p ) , lpi = 1 1 = 0

is uniformly bounded in norm. Hence given E > 0, w ( k ) , and k, there exist i(E, dk)) such that for n > i(E, dk))

I l~"* (P)e"(P) I I<~/ I IW(k) l l , IPI=1.

II w ( k + n ) II < E .

Therefore, from (12) we have

Thus we have shown

Theorem 5: The wavefront model is asymptotically stable if all the eigenvalues of

are strictly inside the unit circle for all lpl= 1. The necessity of the condition is not apparent. In fact it

seems that for a finite number of points, the spectral norm of e(@) can reach the value one and we could still have asymptotic stability. The next example illustrates this point

Example 4: We consider here the G-R semistate model given by (homogeneous case)

e ( p ) = ~ + p ~

In an earlier study [ l ] (see also Example 3) it is shown that the equivalent wave model has a transition matrix of the form A,(n) of Example 2. In addition J , K have the special structure

This example considers the case

The matrix e(@) takes the special form A , = A , = A , = A , = a

By direct computation we determine the eigenvalues of e(@) as

while A , = 0, A , ( P ) = 4 1 + P )

e y p ) =a.( 1 + p ) " - ' e ( p ) and

e n * ( p)efl( p ) = 2aznl1 + I 2'.-')[ ;*

Hence the maximum eigenvalue of e"*(p>e"(p) when IpI = 1 is

a( n , p ) = 4a2"ll + p12("-l)

= 4a21A2( p ) 1 2 ( n - 1 ) . For future use we note that a(n ,p> = tr{e"*(p)O"(p)).

this let p = exp( j27rf ), and Consider now the variation of A2(p) when IpI = 1. For

I A 2 ( p ) l 2 = 41aI2 COS' n-t.

Clearly,

lA2(p)12 d 4bI2 and reaches the maximum at p = 1 ( t = 0). Using Theo- rem 5 we see that the wave model is asymptotically stable when la1 < 1/2. When la1 = 1/2 the maximum eigenvalue of e(p) is on the unit circle. In this case (12) guarantees at least stability.

In the following we shall show that even when la1 = 1/2 the model satisfies the asymptotic stability condition in


Lemma 5. Since all matrix norms are equivalent (for finite dimensional case) we use the definition lqI2 = tr{'P*q} to show IIWn + k - 1, k)1I2 + 0.

We consider IpI = 1 and use the identity (1 + P * ) =

p*(l+ p ) to establish

[( 1 + p*)( 1 + p ) ] " - ' = ( p * ) " - ' ( 1 + p)2 'n -1 )

This extended example is based on the special form for A(n) derived in Examples 2 and 3. The results show the power of the wave model approach. In the next section we continue the abstract development and extend the stability analysis to the general wave evolution equations.

IV. A GENERAL CANONICAL FORM

In this section we continue the development of Section I1 for the general case. In this regard the transition matrix @ ( n ) and the input constraint matrix B(n) as identified in (4)-(5) are noted. Recall that @(n), B ( n ) are rectangular.

Since d n , p i ) = tr{en*(pj)en(p,)}, using Theorem 1 we Our first objective is the specification of a canonical form. have Let a = {aO,al , - * denote arbitrary scalars and

Am(0 the ( m + 1 + 1) X (1 + 1) matrix

- 2n - 2 ( 2nI-2)p1-n+1 -

I = O

= a(n ,p ) /4a2" .

n t k

an = u ( n , P i ) = ( n + k ) t r ( q * ( n + k ) q ( n + k ) } i = l

If the pj's are solution of p n + k = 0 each of the summation terms is zero except when 1 - n + 1 = 0. Thus

Using the identity

I = ( ) J

we can see that when la1 = 1/2

Hence 1

lim - U, = 0. n + m n + k

On the other hand, by direct computation we can show k

t r (Y*(n + k - 1, k ) q ( n + k - l ) , k } = - n + k'"'

Therefore, we conclude that the wave model is asymptotically stable even when la1 = 1/2.

However, geometric convergence cannot be established for this case. This is the most striking difference between I-D and 2-D systems. It makes evident why conventional Lyapunov type methods cannot provide necessary and sufficient conditions.

V( s , m + I + 1)A,( 1 ) =

a m - ] k t p = {pl,. . . , p,) denote arbitrary scalars and V b , m)

denote the s X m matrix

Remark 3: If the scalars {pl; .,P,} are chosen as the roots of P s - l = O then we can prove the identity V*(s, m)V(s, m) = s Zm for s z m.

Using the scalars a we define the polynomial m - 1

P ( Y ) = aiy' . i = O

Using also the scalars p we define the s X s diagonal matrix

O m ( s) = diag [ P( P I ) 7 . . ., P( P s ) ] . Lemma 6: V(s, m + I + l)Am(I) = Dm(s)V(s, 1).

Pro08 The following identities summarize the steps of a proof by inspection:

. . .

1516 I E E F TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 12, DECEMBER 1990

Consider now the matrices A j ( n ) of (4). Inspection confirms that each A ,(n> is a banded matrix of dimension

For example, if N + M = 2 and s = 3 the matrices A, f have the form

( n + j + 1 ) x ( n + 1 ) . .&I immediate result is the corollary. Corollary 1:

We are now prepared to describe a canonical form for the wave state equation. We consider first the form of @(n) as depicted in equation (4). We note that all block entries are column banded. With this in mind we introduce the diagonal matrix. n

I , =

U ( s , n ) = diag[ V(s,n + 1);. * , V ( s , n + d ) ] .

By direct calculation we have

This expression can be written as a left operation on For the next result we need to introduce the d d U(s,n). For this we define

r D l ( s ) I 0

companion matrix

V P ) =

the column vector Corollary 2: U(s , n + l ) @ ( n ) = A,U(s, n). In this case we arrive at the corollary:

b( P ) = col (41( P I >. . . 9 q d ( P ) 1 We turn now to state variable equation (5). For conve- nience we introduce the notation

and the elementary matrix E, with x , ( n ) = W s , n > x ( n )

U(.) = V(s ,n)W,(n) .

By inspection of the structure for A and Then operating with U ( s , n ) on ( 5 ) we have

U ( s , n + l ) x ( n + l ) = U ( s , n + l ) @ ( n ) x ( n ) prove the following.

+ U( s , n + 1 ) B ( n)W,( n )

(13)

which, by inspection becomes

x S ( n + 1) = ~ , x , ( n ) + r S u ( n ) .

rS = col ( D;( SI, D;( SI,. . . , D;( s)

Where rS is the block matrix

with

0; = diag [ qj( P I ) , . . * G,( P , ) ] , j = 1 , . . . , d .

The matrix DA(s) has the same definition as D J s ) except that the coefficients of B ( m , 1 ) rather than A(m, 1 ) are used. We note that AS, r have only diagonal blocks present and that A, is in block companion form.

f we can

Lemma 7: The matrices A s , f s can be written as S 5

A, = c E, o r ( P , ) , f, = c E, b ( P , ) , = 1 ,=1

where 0 denotes the Kronecker or direct matrix product. Hence

S

A', = c E, O W P , )

J = 1

and

A1*Al S s = i E , O W P , ) * ~ ' ( P , ) . / = 1


V. ASYMPTOTIC STABILITY OF THE WAVE STATE Using the Kronecker product we have EQUATION d

The free response of the wave state satisfies the equa- U( s, k ) x ( k ) = V,( k + i ) x i ( k ) 0 e j i = l tion

and also,

k q U ( s, k ) x ( k ) =

x( n + 1) = a( n ) x ( n ) P d

where x ( n ) is the wavefront state. Using the canonical

n + 1 we have

E,%( k + i ) x , ( k ) 0 Ts(p , )e i . embedding developed in the previous sections, with s > q = l i = 1

Since E , = e,e,*, by defining U ( s , n + l ) x ( n + 1) = A,U(s,n)x( n )

U( s, n + 1 ) x( n + 1 ) = A',U( s, n ) x( n ) . y ( q , i ) = e , * V , ( k + i ) x , ( k + l )

y , = col ( y ( q , i ) , i = 1; , d ) Choosing the scalars P to satisfy p s - 1 = 0 and hence we get

U * ( s , n + j ) U ( s , n + j ) = s l , s d

~ ' s ~ ( s , k ) x ( k ) = C C y ( q , i ) e q O T ' ( P q ) e i (with Zj the identity of appropriate size), we have the q = l i = l inequality U

From this expression we conclude that the spectrum of A':A's is the union of the spectra of {T'(pq)*T'(pq), q = 1,2,. . . , s). Thus we have

= 1,2,. . . , p , denote spectral norms of the indicated matrices. If

q = l

Lemma 8: Let l l~ , l l , ~ ~ ~ ( ~ , ) ~ ~ , Using the definition for y ( q , i ) we can see that

ly(q,i) 1' G ( k + i + l ) I I X i ( k ) 1 1 2

I IA~I I = m ~ ( ~ ~ ~ ' ( D q ) ~ ~ : 1 < 4 G s].

Assume now that for every p, the matrix T(p,) has eigenvalues strictly inside the unit circle. In this case the sequence of functions {ll'Y'i(p)ll) will be uniformly bounded and converge uniformly to zero. Thus the matrix AIx approaches zero. Furthermore the rate of convergence will be geometric (exponential). Hence we show the following.

Theorem 6: If the matrices {T(p), llpll= 1) have eigenvalues strictly inside the unit circle then the wave state model is (exponentially) asymptotically stable.

For the conventional 1-D time-invariant system exponential stability is the only class of asymptotic stability. The situation is different for the wave state model. In order to see this difference, consider again the equation

U ( s , k + l ) x ( k + I ) = [A,]'U(s, k ) x ( k ) .

This last inequality can be rewritten as 1 s

I M k + 1 ) 11' G ( k + d)llx(k) II', c pv,)~~'. q = l

We note that the inequality is valid for all values of s satisfying the constraint s > k + 1 + d . On the other hand the summation is recognized as a partial Cauchy sum. In order to make this connection more apparent we write

and the identities p, = exp ( j 2 7 r t ) , t E [ 0,1]. P Considering the limit as s approaches infinity we have

fies the bound & = c E , O W P , ) Lemma 9: The homogeneous wave state solution satis-

q = 1

x( k ) = col(x,( k ) ; . . , X d ( k ) )

U ( s , k ) x ( k ) = col ( . * , l / ( s , k + i ) x , ( k ) , . . . ).

1518 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 12, DECEMBER 1990

Using the Cauchy dominated convergence theorem we can establish.

Theorem 7: The wave state model is asymptotically stable if the sequence of functions

is uniformly bounded and

for all but a finite set of values of t .

Proofi The sequence of functions satisfies the conditions of the Cauchy dominated convergence theorem. Since the limit function is piecewise continuous we have

lim j 1 T S ( t ) dt = / l lim T,( t ) dt = 0.

Hence, given E > 0 there exists S ( E ) such that for s > S ( E )

we can write

S - m 0 0 s-m

and consequently,

Remark: This theorem allows for a finite number of matrices, ‘Up) , to have eigenvalues on the unit circle. In these cases, however, it is not possible to retain exponential asymptotic stability. In general then the impulse response will not be absolutely summable. Thus we may lose BIBO stability even though we have asymptotic stability in the sense of Lyapunov.

VI. COMMENTARY The development has considered scalar valued station-

ary filters of the form specified in (1). The methodology however extends beyond these limitations. Indeed if y ( - , - ) and U(-, 0 ) are vector valued and aij, bjj are matrices of compatible dimensions then the entire development with only elementary and obvious modification remains valid. We have demonstrated such extensions in the examples. The methodology may also be extended to cover the nonstationary case. Of course results such as Lemma 1 and its sequels have tacitly used stationarity and hence a loss of detail is incurred in such extensions.

The development provides several open questions for further work. For example controllability, observability and minimality constitute one direction of inquiring. The time dependent dimension of the state and input vectors presents a nontrivial complication which is finessed by our embedding technique. We note, however, that alternative conventional 2-D minimality literature is not in entirely satisfactory condition either. Indeed it has been recently noted [5] that stationary filters can actually have minimal realizations which are nonstationary.

The development has featured a state response stability. This is a natural first objective for the state-variable format and we have established significant differences in

the behavior of 2-D and 1-D systems. In particular, the homogeneous response may be asymptotically stable but the rate of convergence is slower than geometrical. As a result, the concepts of asymptotic stability and absolute integrability of the impulse response are not equivalent. A similar phenomena has been recognized in the BIBO stability literature. In this regard we refer to the well known [ 141 almost paradoxical properties of 2-D systems with singularities of the second kind.

REFERENCES W. A. Porter and J. L. Aravena, “1-D models for m-D processes,” IEEE Trans. Circuits Syst., vol. CAS-31, Aug. 1984. -, “On the inversion of 2-D systems,” IEEE Trans. Circuits Syst., vol. CAS-32, Dec. 1985. -, “State estimation in discrete m-D systems,” IEEE Trans. Automat. Contr., vol. AC-31, Mar. 1986. J. L. Aravena and W. A. Porter, “State Representations for m-D systems with generalized causality structures,” Math. Syst. Theory,

-, “Results in state and minimality for m-D system,’’ in Linear Circuits, Systems, and Signal Processing, New York: Elsevier Science, 1988. -, “Recursive forms of optimal multidimensional filters,” Proc. Inst. Elect. Eng., vol. 134, pt. G, no. 3, June 1987. D. D. Givone and R. D. Roesser, “Multidimensional linear itera- tive circuits-general properties,” IEEE Trans. Computers, vol. C-21, Oct. 1972. E. Fornasini and G. Marchesini, “State-space realization theory of two-dimensional filters,” IEEE Trans. Automat. Contr., vol. AC- 21, Aug. 1976. R. M. Merserau and D. E. Dudgeon, “Two dimensional digital filtering,” Proc. IEEE, vol. 63, pp. 610-623, Apr. 1975. E. I. Jury, Inners and Stability of Dynamic Systems. New York: Wiley-Interscience, 1974; 2nd ed., Melbourne, FL: Krieger, 1982. T. S. Huang, Two Dimensional Digital Signal Processing I: Linear Filters, vol. 42, M. Marcus and H. Minc, Introduction to Linear Algebra. New York: Macmillan 1965. W. U. Lu and E. B. ,p, “Stability analysis for 2-D systems via A Lyapunov approach, IEEE Trans. Circuits Syst., pp. 61-68, Jan. 1985. B. R. Anderson, P. Agathoklis, E. I . Jury, and M. Mansour, “Stability and the matrix Lyapunov equation for discrete 2-D system,” IEEE Trans. Circuits Syst., vol. CAS-33, pp. 261-267, Mar. 1986. D. Goodman, “Some study by properties of two-dimensional linear shift-invariant digital filters,” IEEE Trans. Circuits Syst., vol. CAS-24, Apr. 1977.

vol. 20, pp. 155-168, 1987.

Berlin, Germany: Springer-Verlag. 1981.

Jorge L. Aravena (M’89) received the degree of Electrical Engineer from the University of Chile, Santiago, and the Ph.D. degree in computer, information and control engineering from the University of Michigan at Ann Arbor.

He has served as member of the faculty at the University of Chile and the University of Santi- ago. He also headed the Process Engineering Group at the University of Concepcion. He is now an Associate Professor of Electrical and Compute Engineering at Louisiana State Uni-

versity, Baton Rouge, where he also serves as Coordinator for the Graduate Program. His current research interests include, mathematical


system theory, m-D systems and signal processing, parallel computing, and algorithm-dependent computing structures.

Masoud Shafiee (S’84-M’88) received M.S. degrees in mathematics and systems engineering form Wright State University, Dayton, OH, and M.S. and Ph.D. degrees in electrical engineering from Louisiana State University, Baton Rouge.

He is currently a faculty member with the Department of Electrical Engineering at Amir Kabir University, Tehran, Iran. His research interests include m-D systems, automatic control, and signal processing.

ests include computer fast algorithmic forms.

William A. Porter (S’56-M’61-SM’70) received the B.S. degree in electrical engineering from Michigan Technological University, and the M.S. and Ph.D. degrees in electrical engineering from the University of Michigan.

He has served on the faculties of the Univer- sity of Michigan, The University of Southern California, Louisiana State University, and is currently Eminent Scholar of Electrical and Computer Engineering at The University of Al- abama in Huntsville. His current research inter-

architecture, signal processing, neural computing, , and distributive memory recognition devices.

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State models and stability for 2-D filters