244 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 36, NO. 2, FEBRUARY 1989

BIBO Stability of Inverse 2-D Digital Filters in the Presence of Nonessential Singularities

of the Second Kind

Abstract --This paper considers the open problem concerning the BIBO stability of inverse 2-D digital filters in the presence of nonessential singularities of the second kind. It is shown that there exist BIBO stable 2-D filter transfer functions having simple nonessential singularities of the second kind on T 2 , that also admit BIBO stable inverses. A class of such functions is obtained. Another class of BIBO stable 2-D functions that cannot have BIBO stable inverses is also characterized.

I. INTRODUCTION NVERSE multidimensional (n-D) digital filters have I applications in areas such as digital image processing

[l], [2]. For example, inverse 2-D digital filters may be used as tools to restore degraded images [l]. The invertibil- ity of 2-D digital filter (or 2-D system) transfer functions has been investigated by a number of researchers (see, e.g., [l], [3]). In this paper, we study the problem regarding the bounded-input bounded-output (BIBO) stability of inverse 2-D digital filters in the presence of nonessential singulari- ties of the second kind.

Consider the class of n-D (n > 2) linear shift-invariant (LSI) quarter-plane causal digital filters whose region of support is the first quarter-plane and which is described by a real rational transfer function

where P and Q are relatively prime polynomials in n complex variables. If P(0,. e ,0) f 0, then G-'( z,,. . . , zn), the inverse of G(z,,- . -, z,,), also represents an n-D LSI quarter-plane causal digital filter. It is well known [5] that G-'( z,,. . . , z") is BIBO stable if P( z,,. . . , zn ) has no zeros in the closed unit polydisk

U" = { ( z , ; . 0 ) z" ) : lZl l 61 , . * e , lznl 61}.

However, as established in [4], [8], G-'(z,,. . 0 , z n ) may still be BIBO stable even when it has some nonessential singularities of the second kind on the distinguished boundary of the unit polydisk

T" = { ( z , , * . -, z,,): (zll =1,* - e , I Z , ~ =1}.

Manuscript received November 23, 1987; revised July 20, 1988. This

The authors are with the Department of Electrical Engineering, Uni-

IEEE Log Number 8825004.

paper was recommended by Associate Editor D. M. Goodman.

versity of Calgary, Calgary, Alta., Canada T2N 1N4.

The problem concerning the BIBO stability of n-D (n > 2) digital filters in the presence of nonessential singu- larities of the second kind on T" has been the subject of intensive research for a decade (see, e.g., [4]-[15]). In particular, exploiting certain resultants of polynomials in two variables, Roytman et al. [12] have recently developed a useful method for testing the BIBO stability of 2-D digital filter transfer functions having simple nonessential singularities of the second kind on T2. This method can also be used for checking the BIBO stability of inverse 2-D digital filter transfer functions with such singularities on TZ.

An interesting question arises [9]: Does there exist BIBO stable n-D (n > 2) digital filter transfer functions having nonessential singularities of the second kind on T" that also admit BIBO stable inverses? Bose [9, p. 2451 conjec- tured that such a BIBO stable n-D transfer function can- not have a BIBO stable inverse. In this contribution, we show that this conjecture does not hold in general. In fact, as will be seen in Section 111, there are an infinite number of BIBO stable 2-D rational functions having a simple nonessential singularity of the second kind on T 2 that also admit BIBO stable inverses. A class of such 2-D stable functions is parametrized. In Section IV, we give a charac- terization of another class of BIBO stable 2-D digital filter transfer functions which cannot have BIBO stable inverses.

11. PRELIMINARIES Consider the class of 2-D LSI quarter-plane causal digi-

tal filters whose region of support is the first quarter-plane and which is described by a real rational transfer function

where P and Q are relatively prime. A 2-tuple z , such that P(z,) # 0 and Q(z,) = 0 is called a pole or a nonessential singularity of the first kind of G(z,, z2) ; z , is called a nonessential singularity of the second kind of G(z,, z 2 ) if P(z,) = Q(z,) = 0. Because any singularities of a rational function in n variables are necessarily nonessential, we shall, in this paper, abbreviate the term "nonessential singularity of the second kind" as "second kind singular- ity," for the sake of simplicity.

0098-4094/89/0200-0244$01.00 01989 IEEE

LIN AND BRUTON: BIBO STABILITY OF INVERSE 2-D DIGITAL FILXXRS 245

Since Q(zl, z 2 ) in (1) is nonzero in some neighborhood around the origin (O,O), G(z,, z2) is analytic and has a power series expansion in this neighborhood

G(z1, '2) = E gmnzTz2" m , n = O

where g,, is the impulse response of G(z,, z2). The filter is BIBO stable if and only if

c I tL"<~. m , n - 0

In practice, however, it is often more convenient to determine the stability of a filter from its transfer function G(z,, z2). A simple criterion giving a necessary and suffi- cient condition for the BIBO stability of 2-D digital filter transfer functions having simple second kind singularities on T 2 has recently been presented by Roytman er al. [12]. Since this important criterion will be frequently used in this paper, we reproduce it here for convenience of exposi- tion. As in [12], we denote by A(z,, z2) the discrete paraconjugate of a 2-D polynomial A(z,, z2) and by Rz2[A(z l , z2), B(z,, z2)] the z2-resultant of the polynomi- als A(z,, z2) and B(z,, z2).

Theorem I (121:' Let

where P and Q are relatively prime and t 6 a positive integer. Assume that P/Q has no poles in U 2 , nor any second kind singularities in u2 except for the simple ones' at (a, 8) and (l/a,l/p) on T2. We further assume that P/Q has no simple or multiple second kind singularities of the form (a, y) outside u2. Then, G(z,, z2) is BIBO stable if and only if

tma(Rz2[Q,Ql) <ma(Rz2[P, Ql) (2)

where m,(f(z,)) denotes the multiplicity of the factor (z1- a) in f(z1).

111. EXISTENCE OF STABLE INVERSES FOR A CLASS OF BIBO STABLE 2-D FUNCTIONS

This section provides an answer to the question of whether or not a BIBO stable rational function G(z,, z2) with second kind singularities on T 2 can admit a BIBO stable inverse [12]. Let us begin our discussion with an example.

Example I: Consider the 2-D rational function

P,( z,, z2) 4- 32, - 3z,z2 + 2z;z2 - Gl(Z1, z2) = Q i ( ~ i , ZZ) - 3-2z,-z2

It can be easily checked that Pl(z,, z2) and Ql(zl, z2) are

'TII~S theorem is sufficient for the purpose of this aper although it can be extended to a wider class of 2-D functions as sgown'in [12], [14]. 2As in [12], a second kind singularity (a, 8 ) of the function P / Q is said

to be simple if (aQ/az , ) , , , , , + 0 and (aQ/dz,) , , , , , # 0.

relatively prime, and that G1(z,, z2) has no poles in u2, nor second kind singularities anywhere except for a simple one at (1,l) on T 2 .

Let us compute the discrete paraconjugate of e,(.,, z2)

Q1( zl, z2) = z,z,Q,( zc ', Z; ') = z1z2(3 - 2 ~ ; ~ - z;')

= 3z1z2 - 22, - z,.

The z2-resultant of Q1 and Q,, and that of P, and Q, are

and

from which it follows:

mi(Rz2[Qi,QiI) = 2 and ml(Rz,[P1,QiI) =3 .

Therefore, by Theorem 1, G,(z,, z2) is BIBO stable [12]. Now consider the BIBO stability of the inverse of

Gl(Z1, z2) given by

Since Q,( zl, z2) has no zeros in the region { 3' - (1, l)}, it is necessary for BIBO stability of G-'(z1, z2) that Pl(zl, z2) has no zeros in { u2 -(l,l)}. This is shown as follows.

Rewriting Pl( z,, z2) as

and because

it follows that

from which it is easily established that

P,(z,,z,) zo v l zJa1 , z,+1; lz2la1. (3)

Pl( l , z2) # 0 v z2 Z l . (4)

On the other hand, Pl(l, z2) = 1 - z2 and, therefore,

It follows from (3) and (4) that P,(z,, z2) has no zeros in {c2- ( l , l )} and, therefore, G-'(z,, z2) has no poles in 0 2 .

It is easy to show that G-l(z,, z 2 ) has no second kind singularities anywhere except for a simple one at (1,l) on T 2 , and therefore Theorem 1 can be applied to check the

246 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 36, NO. 2, FEBRUARY 1989

BIBO stability of G-'( z,, z2). Direct calculation gives

PI( Z 1 , Z2) = 4Z;Z2 - 3Z1Z2 - 32, + 2

4z; - 32, - 32, + 2

22; - 32, 4 - 32, = - 6z,( z1 - 1)2 R,[Pl~ 4 1 =

3 R,,[Ql, P1l = - R,,[P1,Qll =4(z1-l) .

Hence,

ml( R,,[P1, P1l) = 2 and ml( R,,[Ql, P1l) = 3.

By Theorem 1, GT1(zl, z 2 ) is BIBO stable. Therefore, it is concluded that both G,( z,, z2) and GT1( z,, z 2 ) are BIBO stable.

Remark I : It is interesting to notice that the inverses of all the BIBO stable 2-D transfer functions having second kind singularities on T 2 discussed in the examples of [4], [7], [12]-[14] are either quarter-plane noncausal, or BIBO unstable. In fact, most of the above-mentioned 2-D func- tions have separable numerators, and consequently have an infinite number of zeros on T '.

A natural question arises at this point: Does there exist other BIBO stable 2-D transfer functions having second kind singularities on T 2 whose inverses are also BIBO stable? This question is important since it is sometimes desirable to obtain BIBO stable 2-D transfer functions with stable inverses (the so-called minimum phase stable 2-D transfer function) [9]. In the remainder of this section, we provide an affirmative answer to this question by deriving a necessary and sufficient condition for the class of 2-D real rational functions of the form

1 + alzl + elz: + b,z2 + clzlz2 + d,z;z2 1 - a2z1 - b2z2 G,(z,, 1 2 ) = k

to be BIBO stable as well as inverse stable. It is assumed that P2 and Q2 are relatively prime, and that a2 > 0, b2 > 0, a 2 + b2 =1, k # 0.

Following Roytman et al. [12], a necessary and sufficient condition for G2(z1, z 2 ) to be BIBO stable is obtained in the following lemma.

Lemma I: G2(z1, z 2 ) in ( 5 ) is BIBO stable if and only if d , # 0 and

1 + a , + e, + b, + c, + d , = 0 ( 6 4 d, - a2cl + b2e,

= 3 a 2 4

a2b, - b2a, - c1 = 3 (6c)

a 2 4

Proof: Since a2 + b2 = 1 by assumption, Q2(1, 1) = 0. A necessary condition for G2(z,, z 2 ) to be BIBO stable is

that P2(1,1) = 0, i.e., (6a) must hold. Suppose in the fol- lowing that (6a) is satisfied.

It is easily seen that (1,l) is the only simple second kind singularity of G2(z,, z2) in u2, and that G2(z,, z 2 ) has no simple or multiple second kind singularities of the form (1, y) outside 02. Thus by Theorem 1, G2(z1, z 2 ) is BIBO stable if and only if [12]

mi(Rz2[Q2,Q2I) <mi(Rz2[P,~Q21)- (7)

Direct calculation gives

b, + clzl + d,z: 1 + alzl + el.: 1 - a2z, Rzz[P2yQ21 =

= - a2d,zl + ( d , - a2cl + b2e,)z:

- (a2b , - b2a,- c 1 ) z l + ( b 2 + b l ) . (Sa)

Since m,(R,,[Q2, Q2]) = 2, a necessary and sufficient con- dition for (7) to hold is that d , # 0 and

R Z Z [ P 2 , Q21 = x l ( z l - 1)' (Sb)

for some nonzero constant x,. Equations (6b)-(d) now follow as a consequence of comparing the coefficients of (Sa) and (Sb), thereby proving Lemma 1.

Assuming that G2(z1, z 2 ) is BIBO stable, i.e. (6a-d) are satisfied, let us now consider the BIBO stability of its inverse, namely

(9)

Since Q2(zl , z 2 ) # 0 in the region f i 2 e { o2 -(l,l)}, it is necessary for BIBO stability of GT1( z,, z 2 ) that P2( z,, z 2 ) # O i n f i 2 .

Lemma 2: P2(z1, z 2 ) # 0 in f i 2 if and only if

le,l<l (loa) lull- e, (1 (lob)

( 1 - a , + e , ) 2 - ( b , + d , - ~ , ) 2 > 0 (1Oc)

0. (10d)

Proof: That p2(z1, z 2 ) # o in f i 2 implies P,(z,,o) z o for lzll 6 1, which is equivalent to the conditions (loa) and (lob) [16]. We assume hereafter that (loa) and (lob) are already satisfied.

By Lemma 1, the assumption that G2(z,, z 2 ) is BIBO stable implies

1 + a: + e; - b: - c: - d: - 6e, + 6b,d,

b, + c1 + d , = - (1 + a, + e,) # 0 (11) since P2(1,0) = 1 + a, + e, # 0, as implied by (loa) and (lob) [16]. Thus

~ ~ ( 1 , ~ ~ ) = ( 1 + a , + e , ) ( l - z ~ ) # O V z 2 # 1 . (12)

Hence, the condition P2(z1, z 2 ) # 0 in fi' is equivalent to

P2(z,,z2) # O V ( z , , z 2 ) €U2; z l # l . (13)

241 LIN AND BRUTON: BIB0 STABILITY OF INVERSE 2-D DIGITAL FILTERS

if and only if

I f 2 ( Z 1 ) 1 4 V Iz,1=1; z1#1. (16) By means of the bilinear transformation z1 = (s, - l)/(s, + l), (16) is equivalent to

{

2

Rewriting P2( zl, z 2 ) as

where f2(z1) 2 ( b , + clzl t d,z?)/(l + alzl + e&>, (13) is equivalent to

where

P2( z,,z2) = (1 + a,z,+ e,zl)[l+ f2(Z, )Z21 (14)

l.f2(z,)1<1 V Iz,)<l, Z l f l . (15)

, i f 2 a 2 - 1 < e , < - a 2 < d , < -

a2 - e , - -

-- a2 + e , < d , < - a2 + e ,

3a2 - 1 1- a2

a2 + e , 1+ a2 1- a 2

if e, 2 - a2 and <dl<-----, e , > 2a2-1

if - a 2 < e , < 2 a 2 - 1

a - -P~(Z, ,Z~) dZ2

Recalling (11) and simplifying yields (Sufficiency): Suppose that a,, b,, c,, d , and e, sat- isfy (18a)-(18e). Then, (6a)-(6d) and (lOa)-(lOd) hold (see the Appendix). By Lemma 1, G,(z,, z2) is BIBO stable. It remains to show that G;'(Z,, z,)-is also BIBO stable. By Lemma 2, P2(z1, z 2 ) f 0 in U', which implies that G;'(zlLz2) has no poles in U', nor second kind singulari- ties in U 2 except for one at (1,l). We also notice from (12) that P2(l, z2) # 0 V z2 # 1, i.e., G;'(z,, z2) has no second kind singularities of the form (1, y ) outside U '. Therefore,

2

2[ 1 + a: + e: - b: - c: - d: - 6e , + 6b,d,]

[ (1 + a, + e , ) w: + (a , - 1 - e,)] + 4(1- e,)2wi =I-

2

(1 - a, + e,)'- ( b , + d, - c1)2 -

= b , + c , + d , = - ( l+a ,+e , ) Z O (1 3 1)

2 [ (1 + a, + e,)w: + (a , - 1 - e,)] +4(1- e,)2w: *

It is now easy to see that (17) holds if and only if (1Oc) and (1Od) are satisfied, thereby proving Lemma 2.

Remark 2: Several methods for testing a 2-D polyno- mial P(z,, z2) # 0 in u2 are available in the literature (see, e.g., [5], [16]). However, these tests may not be applicable to the case where P(z1,z2) has some zeros on T 2 , as discussed in the above lemma.

We are now in a position to state the main result of this section.

Theorem 2: Let

where P2(z1, t2 ) and Q2(z1, z2) are defined as in (5). Then, G2(z,, z 2 ) and G;'(z,, z2) are both BIBO stable if and only if a,, b,, c,, d,, and e, satisfy

b, = a2d, + ( a 2 -1) (18b) 1

a2 c1= -[(1-3a2)d1+(1- u,)e,] (18c)

to show that G; '( z,, z2) is BIBO stable, it suffices to show that (I, 1) is a simple second kind singularity of G-'(z,, z2) , and that condition (2) in Theorem 1 is satisfied. This will be done in two steps.

Step 1:

= a, + c1 + 2e, + 2d, 1

a2 = - - [ a : + ( u 2 - 112d, + e,]

1

a2 + - [(1-3a2)d, +(I - a,)e,] +2e1 +2d1

= - [ ( a 2 - w, + ( a 2 - 4 1 2 0

since d , < ( a 2 - e,)/(l - a 2 ) (see (A13)):

(from (11)). Thus according to [12], (1,l) is a simple second kind

248 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 36, NO. 2, FEBRUARY 1989

-1 -0.5 I

Fig. 1. The domain in the a2-e , plane for (18e).

singularity of G; '( z,, z2), and, therefore, Theorem 1 can be applied to check the BIBO stability of G;'(z,, z2 ) .

Lemma 1 Step 2: Since R Z 2 [ Q 2 , P21 = - Rz2[P2 , Q21, by

mi(Rzz[Q2,f'zI) =mi(RZ2[f'2,Q2I) =3. Let us compute the z2-resultant of P2 and P2

z: + alzl + e, b, + clzl + d,z:

biz: + clzl + d, 1 + alzl + e,z: RZ2EP29P21 =

= ( e , - b,d,)z,4 + ( a, + alel - b,c, - cld,)zl

+ (1 + a: + e: - b: - c: - d:)z:

+ ( a, + alel - b,c, - cld,)z , + ( e , - b,d,) .

(19) To show that G;'(z,, z2) is BIBO stable, it suffices to show that m,( RLZ[ P2, P2] ) < 3, which is ensured if

= 2[6( e , - b,d,) + 3( a, + alel - b,c, - cld,)

+ ( 1 + a : + e : - b : - c : - d , 2 ) ] (20)

is not equal to zero. Substituting (A15) in the Appendix into (20) leads to

= 2[6( e, - b,d,) + 3( a, + alel - b,c, - cld,)

+ 2( b,c, + b,d, + cld, - a, - e , - ale,)]

= - 2[( b, + d, )c , - (1 + e , ) a , +4( b,d, - e , ) ]

= - - [ ( a 2 - l ) d l + ( a 2 - e , ) ] 2 2 (from (A29)) a2

# O

82

Fig. 2. The admissible regions in the az -d l plane given by (18e) for e, = 0.5.

82

-1

Fig. 3. The admissible regions in the az -d l plane given by (18e) for e, = 0.

since d , < (a2 - e l ) / ( l - a,) (see (A13)). Therefore,

mi(R2z[P2, 41) < 3 = m 1 ( R Z l [ Q 2 , 41)- By Theorem 1, G; '( z,, z2) is BIBO stable, and the proof

is completed. Remark 3: It can be seen from Lemma 1 that for

G2(z,, z 2 ) to be BIBO stable, the absolute values of a,, b,, c,, d,, and el are not necessarily bounded. However, it can be shown from Theorem 2 that for both G2(z1, z2) and G;'(z,, z 2 ) to be BIBO stable, the absolute values of b,, d,, and e, are bounded by 1, while those of a, and c1 by 2. The derivation is straightforward but rather tedious, and is omitted here.

The domain in the a2 - e , plane for (Me) is sketched in Fig. 1, in which (18d) is given by the union of the regions E,, E, and E3. Equation (Me) is depicted in Figs. 2-4 for e , = 0.5,0, and -0.5, respectively; and in Figs. 5 and 6 for

LIN AND BRUTON: BIBO STABILITY OF INVERSE 2 - D DIGITAL FILTERS 249

Fig. 4. The admissible regions in the a2-dl plane given by (18e) for e, = - 0.5.

Fig. 5. The admissible regions in the el-dl plane given by (Be) for a2 = 0.2.

u2 = 0.2 and 0.8, respectively. It is interesting to see from Figs. 1 and 4 that for a fixed e, such that - 1 < e , < - 1/3, the regions in the u2 - d , plane such that both G2(z1, z 2 ) and G;'(z,, z 2 ) are BIBO stable are disjointed.

Before closing this section, let us illustrate Theorem 2 by reconsidering Example 1. Rewrite G,(z,, z2) as

3 3 1 4 4 2

4 1 - -2 , - -z1z2 + - 4 z 2

2 1 3 3

G,(z,, 2 2 ) = - 1- - 2 , - - z2

3

which is of the same form as in (5). Since the point (a2 , e,) = (2/3,0) is in E2 of Fig. 1, (18d) holds. Observing Fig. 3, we see that the point (2/3,1/2) lies inside D,, i.e., (18e) is satisfied. Straightforward calculation shows that a, = - 3/4, b, = 0 and c1 = - 3/4 satisfy (18a)-(18c) for

\ /

Fig. 6. The admissible regions in the el-d, plane given by (Be) for a2 = 0.8.

u2 = 2/3, d , = 1/2 and e , = 0. Therefore, by Theorem 2, G,(z,, z 2 ) and G;l(z,, z 2 ) are both BIBO stable, as has been established previously.

IV. CHARACTERIZATION OF A CLASS OF BIBO STABLE 2-D FUNCTIONS HAVING NO

STABLE INVERSES It has been shown in the previous section that some

BIBO stable 2-D transfer functions having simple second kind singularities on T 2 can admit BIBO stable inverses. In this section, we concentrate on a special class of 2-D real transfer functions, and show that a BIBO stable function in this class cannot have a BIBO stable inverse. The following theorem contains the main result of this section.

Theorem 3: Let

(1 + UIZl + b,z, + c1z1z2) = k (21)

(1 + u2z1 + b2z2 + c,z,z~)' where P3 and Q3 are relatively prime, k # 0 and r , t are positive integers. Assume that G3( z,, z 2 ) has some second kind singularities on T 2 and is BIBO stable. Then, G; '( z,, z 2 ) cannot be BIBO stable.

The proof of the above theorem is essentially based on Lemmas 3-5 given as follows.

Lemma 3: G3(z,, z2) as defined in Theorem 3 has a unique second kind singularity in v2 at (* 1, & 1).

Proof: Rewrite Q3(z1, z 2 ) in (21) as

where f,(z,) (b, + c2z1)/(1 + u2z1). The assumption that G3(z,, z 2 ) is BIBO stable implies that la,[ < 1 and Q3(zl,z2)#0 in the region {e2--'}, which in turn

250 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 36, NO. 2, FEBRUARY 1989

a az2 -Q3(zl, z2)

implies

If3(z1) Id1 v IziI (23) By means of the bilinear transformation z1 = (sl - 1)/ (sl + l), (23) is equivalent to

= b, + c2 # 0 (from (35)). (1 > 1)

2 lf3(Js)l - O O < W ~ < + O O . (24)

Direct calculation gives

There are four possible cases where (24) can be satisfied. Case 1: (b, - c21 < 11 - a2( and Ib, + c21 < 11 + u21. From (25), we have

Hence,

Ih(z1) I<1 tf IZ i I=1 . (26)

I f 3 ( Z 1 ) I<1 v IZiI (27)

By the maximum modulus principle [17], (26) implies that

It follows from (22) that Q3(zl, z 2 ) # 0 in U’, contradict- ing the assumption that G3(zl, z2) has some second kind singularities on T 2 .

Case 2: Ib, - c2( = 11 - a21 and (b, + c2J = 11 + a,(. It follows from (25) that

Thus

l f 3 ( Z i ) I=1 v lZll =l. (28) However, this implies from (22) that Q3(z1, z2) has an

infinite number of zeros on T 2, contradicting the assump- tion that G3(z1, z2) is BIBO stable.

Case 3: Ib, - c21 < 11 - a21 and Ib, + c21 = 11 + a,!. Then,

and

That is,

and

It follows from (22) that

Q 3 ( ~ 1 , ~ 2 ) # 0 v ( z 1 , ~ 2 ) E U’, 21 (32) On the other hand, taking into account that G3(~1, z,)

has real coefficients, we have from (30a) that f3(l) = f 1. Hence,

Q3(l ,z2) = (1+a2)[ l+f3(l )zz] # o v z z # - f 3 ( 1 ) . (33)

Combining (32) and (33) yields

Q3(zl ,z2) = 0 , i Q3( zl, z2) # 0,

if z l = l , z 2 = -j3(l) = f l ;

elsewhere in U’. Case 4: Ib, - c21 = 11 - a21 and Ib, + c21 < 11 + azl. Similarly as in Case 3, it can be shown that

Q3(z1,z,) =0 , Q3( zl, z 2 ) # 0,

It follows from the above discussion that Q3(~1, z2) has a unique zero in U 2 at ( f 1, f l), which is the only second kind singularity of G3(z1, z 2 ) in U 2 .

Because of Lemma 3, it may be assumed in the rest of this section that G3(z1, z,) in (21) has (1,l) as its only second kind singularity on T ’.

if z l= -1, z 2 = - f3(-I) =&I; elsewhere in 0’. i

Lemma 4: Let

where P3 and Q3 are defined as in Theorem 3. Assume that (1,l) is the only second kind singularity of G3(zl, z2) on T 2 , and G, is BIBO stable. Then, G3(~1, z 2 ) has no simple or multiple second kind singularities of the form (1, y ) outside U’, and (1,l) is a simple second kind singu- larity of the function [ P3(zl , z2)‘/[Q3(z1, z2)].

Proof: The assumption that (1,l) is the second kind singularity of G3(z1, z2) implies Q3(l , 1) = 1 + a2 + b2 + c2 = 0, from which it follows

a2 + c2 = - (1 + b2) # 0 (34)

b 2 + c 2 = - ( 1 + a Z ) # O (35)

and

since la21 <1, lb21 <1 for stability of G3(z1, z2 ) . Thus

Q 3 ( l , z 2 ) = (1 + a 2 ) ( l - z2) # 0 V ~2 # 1.

Therefore, G3(z1, z2) has no second kind singularities of the form (1, y ) outside U’.

Direct computation leads to

251 LIN AND BRUTON: BIBO STABILITY OF INVERSE 2-D DIGITAL FILTERS

b, + clzl b, + c2zl

1 + alzl 1 + u2z1

presence of nonessential singularities of the second kind on T2. Using a recently proposed method [12] for testing

= j z , + a , b ,z ,+c, b, + c2z1 1 + a2z1

= (a , - b , c , ) z : + ( l + a ; - b z - c ; )z l t (a2 - b,c,).

The assumption that P3 and Q3 are relatively prime im- plies that RZ2[ P3, Q 3 ] f 0. Hence,

since the degree of Rz2[P3, Q3] in z, is less or equal to two. Next, the fact that (1 , l ) is a common zero of Q3 and Q 3

implies

(37)

m1(Rz2[P39 Q 3 I ) Q 2 (38)

RZ2 [ Q3 7 Q 3 I = ( z i - 1)(xiz i - ~ 2 ) (39) for some real numbers x , and x, . Comparing the coeffi- cients of (37) and (39) gives x , = x 2 = ( a , - b2c2), or RZ2[Q3, Q, ] = ( a , - b,c,)(z, -1),. The assumption that G3(z, , z , ) is BIBO stable requires that Q3 and Q3 are relatively prime [12], which implies ( a , - b,c,) # 0. There- fore,

mi(Rr2[Q3,Q31) = 2 - (40) Since we have shown in Lemma 4 that G3(z1, z,)-has no

second kind singularities of the form (1, y) outside U,, and that (1 , l ) is a simple second kind singularity of the func- tion [P3(z1, z2) ]r / [Q3(z1 , z ,)] , according to Theorem 1, G3(z1, z , ) is BIBO stable if and only if

tmi(Rz2[Q3,Q31) <mi(Rz2[P3",Q31)

tmi(Rz2[Q3,Q31) < m i ( R z 2 [ P 3 , e ,]) . From (38) and (40), this is possible only if r > t , and the proof is completed.

As a consequence of Lemma 5, a proof of Theorem 3 is now sketched. The inverse of G3(~1, z , ) in (21) is given by

or

1 + alzl + e,.: + b,z, + clzlz2 + d,z:z, 1 - a2z1 - b2z2 G ( z , , . , ) = k

to be BIBO stable as well as inverse stable, where a , > 0, b, > 0 and a , + b, =l. It is believed that this result can be useful in the investigation of minimum phase stable 2-D transfer functions [9], as well as in the synthesis of stable inverse 2-D digital filters.

We have also shown that the class of BIBO stable 2-D real transfer functions of the form

(1 + alzl + b,z, + c1z1z2)

(1 + a2z1 + b2z2 + C , Z ~ Z , ) ~ G ( z , , z 2 ) = k

having nonessential singularities of the second kind on T cannot admit BIBO stable inverses.

APPENDIX DERIVATION OF (18a)-(18e)

In this appendix, we derive all the solutions of a,, b,, c,,

(Al)

= 3 (A2)

d,, and e , that satisfy the following conditions:

1 + a, + b, + c, + d , + e , = 0

d, - a2cl + b,e,

a 2 4

a,b, - b , ~ , - C, = 3 (A3)

a 2 4

-- -1 (A4) b2 + b, a 2 4

le,l<l ('45)

14- e1 (A61

(1 - U , + e , ) , - ( b , + d , - c1), > 0 (A7)

1 (1 + a2z1 + b,z, + C , Z ~ Z , ) ~

k (1 + ulzl + b,z, + c1z1z2) ' 1 + U : + e: - b: - c: - df -6e, +6b1d1 2 0 (A8) = -

where a , > 0, b, > 0, a , + b, = 1, and d , Z 0.

252 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 36, NO. 2, FEBRUARY 1989

Substituting b, = 1 - a, into (Al-4) and simplifying Hence, gives 4

F 1 =-(( ~ , + 1 ) ( 1 - 3 ~ , ) d : a2

a1 1 1 1 1 1 0 0 - a 2 1-3a2

1;a2][ f]=[ +2(a ,+ e,)(l- a,)d,+(a,+e,)2}. (AM)

We now investigate in detail the conditions for (A13) and (A14) to hold. The condition (A5), or (ell <1, is implicitly assumed in the following discussion. There are three cases to consider.

case 1: 0 < a 2 < 113 We have from (A18)

$ 1. - 1

( ~ 9 )

Using standard algebraical techniques, the solutions to (A9) can be obtained as follows:

U 2 - 1 c l 2 -1 -3U2 0 1 0 - a 2 0

e,

b ,= a2d, + ( a 2 -1)

C, = - [ (1 - 3 a 2 1 4 + (1 - a,)e,l.

(All)

(A12)

The ranges of d, and e, will be determined such that

There are three subcases to discuss. 1

a2 Case la: e, + a, < 0 It can be easily seen from (A19) that f‘, > 0 if and only

if (A5)-(A8) are satisfied. < d, < + bo. (A20)

a2 + e, or ~

It can be seen from (A10) that condition (A6) is equiva- - a2 + e, < d, < lent to 1+ a2 3a2-1

la: +(I- ~ , ) ~ d , + e, I < (I+ el)u2 Straightforward computation gives

a2 -e , a , + e , d-

a 2 - e , a ,+e , (A21b) if e, d 2a2 - 1

if e, > 2a2 - 1

1 - U , 3U2-1’

>- l - a , 3a2 -1 ’

Letting F, lent to

(1 - a, + e,), - (b, + d, - cl),, (A7) is equiva-

F, > 0. (A141 From (Al), we have

(b, + c,+ d,)’= (1+ a,+ e,)’

or

1 + U: + e: - b: - c: - d: = 2( b,c, + b,d, + c,d, - a, - e, - alel). (A15)

Thus

I;; = (I + U: + e: +2e, -2a, -2a,e,)

- (b : + d: + C: + 2b1d1 - 2b1c1 - 2cld1)

=1+ a: + e: - b: - c: - d: + 2( b,c, - b,d, + cld, - a, + e, - alel)

+ 2( b,c, - b,d, + cld, - a, + e, - ale,) = 2( b,c, + b,d, + cld, - a, - e, - ale1)

= 4[ ( b , + d, )c , - (1 + e,)a,] . (A161 Substituting (AlO)-(A12) into (A16) and simplifying yields

( b, + dl 1 c1- (1 + el) a1 1

= [ ( U , +1)(1-3a2)d; a2

+2(a,+ e,)(l- a,)d,+(a,+e,)2}. (A17)

in this subcase. It follows from (A21a) and (A21b) that for both (A13) and (A14) (or (A20)) to hold, e, and d, must satisfy

a2 - e, <d,<-. a2 + e, e1>2a,-1 and -

3a2 -1 1- a,

Case lb: e, + a 2 = 0 We have

4

a2 F = - ( , a, +1)(1- 3a,)d;.

Hence, (A13) and (A14) are satisfied if and only if a2 - e,

O<d,<-. 1- a,

Case IC: e, + a, > 0 It can be seen from (A19) that F, > 0 if and only if

< d, < + bo. (A22) a2 + e, or --

a, + e, - ~ < d , < - 3a2 - 1 1+ a,

Since a 2 + e, ( a2 + .,)(I + a,) a 2 + e, a2 - e,

< - <--<- 3a2-1 (1 - 1 + a , 1 -a ,

LIN AND BRUTON: BIB0 STABILITY OF INVERSE 2-D DIGITAL FILTERS

Again, there are three subcases to discuss.

It is easy to see from (A19) that (A14) is equivalent to a2 + e, < d , < - - a2 + e,

3a2-1 1 + a , .

Case 3a: e, + a2 < 0 ‘ (A29

253

a2 - e, 3a2 - 1 1- a2 a2 + e, a2 - el if e l> - a 2 and 1 + a 2 1- a2

a2 + e, <d,<- a2 + e,

<d,<- , i f2a2 -1<e ,< -a2

e,>2a2-1 <d,<-,

if - a < e, g 2 a - 1 -~ , 1+a , 3a2-1’

in this subcase, (A13) and (A14) together require only if d, satisfies

a2 - e, a2 - e, -- <d,<-.

1+ a2 1- a2

Case 2: a2 =1/3 In this case, (A13) and (A14) become

1 + 3e, -(1+3e,) <d,<- 2

and 4 3 Fl = - (1 + 3e1)(4d, + 1 + 3e,) > 0. (A24)

It is straightforward to show that (A23) and (A24) hold if and only if e, and d, satisfy

1 1 + 3e, <d,<-

2 . - - < e , < l and -- 3 4

if e, Q 2a2 - 1 a2 + e, <d,<- a2 + e, 1+ a2 3a2-1’

<d,<- a2 + e, 1+ a , 1- a2

--

a2 - e, , if e,>2a2-1. --

Summarizing the results obtained in the above case studies, we now derive close-form solutions of a,, b,, c,, d,, and e, that satisfy (Al)-(A7), i.e., if and only if a,, b, and c1 satisfy (AlO)-(A12) where e, and d, satisfy (A27) and (A28) given as follows:

1 if 0 < a2 < - 3

if - g a 2 < 1

2a2-1 < e, <I,

- a2 < e, <1, (A271 1

3

and

Case 3b: e, + a2 = 0 We have

F2 = 2( b,c, + b,d, + q d , - a, - e, - alel) - 6e1+ 6b,d, = 2[ (b, + d,)c, - (1 + e,) a, +4( b,d, - e,)].

4

a2 Fl= -(l-3a2)(l+ a2)d: Q 0 V d, From (All) and (A17), we have

(b1+ d1)ci - (1 + e1)a1+4( bid1 - el) 1

a2

and thus (A14) cannot be satisfied. Case 3c: e, + a, > 0 = - [ ( a 2 +1)(1-3a2)d: +2(a2+ e,)

In this subcase, (A14) is equivalent to

a2 + e, <d,<--- a2 + e, 1+ a, 3a2 -1 (A26) * (1 - a21 dl + ( a2 +

= - [( a2 -1)’d: +2( a2 - el)(a2 - l )d ,+( a2 - e,)2]

+ 4[( a2d1 + a2 - 1) d, - e,] --

1

Since a2

and

1

a2 = -[(a2 -l)d, + ( a 2 - e,)]2.

Hence, 2

a2 F2 = - [( a2 - l)d, + ( a2 - e , ) ] 2 >, 0.

a2-e, a2+e , 1-a, 3a2-1’

Therefore, we have shown that a,, b,, c,, d,, and e, satisfy (Al)-(A8), with the constraint d, # 0, if and only if they satisfy (AlO)-(A12) and (A27), (A28) or (18a)-(18e) in Theorem 2.

I-<- if e ,>2a2-1

in this subcase, it follows that (A13) and (A26) hold if and

254 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 36, NO. 2, FEBRUARY 1989

151

161

191

1101

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Leonard T. Bruton (M71-SM80-F‘81) is a Pro- fessor of Electrical Engineering and Dean of Engineering at The University of Calgary, Cal- gary, Alta., Canada. His research interests are in the areas of analog and digital signal processing. He is particularly interested in the design and implementation of microelectronic digital filters and the applications of multidimensional circuit and systems theory to digital image processing.