CorrespondenceSIMPLE SUFFICIENT TEST FOR STABILITYOF 2-DIMENSIONAL RECURSIVE DIGITALFILTERS

Indexing terms: Filters and filtering, Stability, Polynomials, Digital filters

Abstract: A recently introduced sufficient condi-tion for the stability of polynomials with real coef-ficients is generalised to cover the case when thecoefficients of the polynomials are complexnumbers. Then this new sufficient condition isapplied to checking the stability of 2-dimensionalrecursive digital filters.

Introduction

A 2-dimensional linear shift-invarient filter is describedby its transfer function

G{zx, z2) =i, z2)

u z2)(1)

where A(zu z2) and B{zx, z2) are coprime polynomials inthe independent complex variables zx and z2, and thereare no nonessential singularities of the second kind [1],i.e. there are no 'points' (zx, z2) such that A(zu z2) =B(zlt z2) = 0. The 2-dimensional filter is stable if and onlyif

B(zit z2) ¥^0 in (2)

This stability theorem implies that the filter is BIBOstable [2]. Another important stability criterion for 2-dimensional filters was proposed by Huang [3]:

B(z

(a)

(b)

1, zl) i

B(zx,

B(zx,

*0

0)

Zl)

in \z

0,

11 ^

M1,

< 1

l<

\z2\^

[

U\z2\

1, if and

1

only if

(3)

(4)

Condition 3 reduces to a 1-dimensional stability test andis computationally trivial to implement. However, condi-tion 4 is rather difficult to test.

Main results

A simple sufficient test for checking the stability of 2-dimensional filters is proposed here. First the followinglemma is given [4].

Lemma: The real polynomial B(z) = z" + gnz"~1

+ gl is stable if+

" 1

, = i n

The above result is generalised in the following theoremto cover the case when the coefficients of the polynomialare complex numbers.

Theorem 1: The polynomial B(z) = z" + gnz"~l + • • •+ gx with complex coefficients is stable if

Z I I 2

Proof: Without loss of generality we assume n = 3,B(z) = z3 + g3z

2 + g2z + gv The state-space description

of this system is

X(k + 1) = Ax(k)

A =0 1 0

0 0 1

L-0i -92 -03.

(5)

(6)

Consider now the 1-dimensional Lyapunov equationwhich has complex coefficients

Q = P-A*PA (7)

where A* — AT and the bar denotes the conjugatematrix.

A recently proposed lemma [5] gives a condition for acomplex matrix A to be stable. Thus, the matrix A isstable if, for any given positive definite Hermitian matrixP, the matrix Q is positive definite.

Let

(8)

Obviously the matrix P is positive definite Hermitian.Then

100

020

o"03

-3< j 2 01 l - 3 £ 2 02 - 3 0 2- 3 0 3 01 - 3 0 3 02 1 - 3 0 3

(9)

This matrix is Hermitian and has real principal minors.Therefore the matrix Q is positive definite if all its prin-cipal minors are positive, i.e.

dj = det [Ij - 3GjGj] > 0, ; = 1, 2, 3

where

0i

02 j = 1.91 02 0j]

The values of the determinants for; = 1, 2, 3 are

and this completes the proof.Now, we will apply this condition to test the condition

4. Consider the reciprocal polynomial Bx(zx, z2) of theB(zx, z2) which is defined as

X\Zx, Z2) — ZJ2 D\ZX, Z2 ) l * " J

where z22 is the maximum power of z2 in B{zx, z2). Then,

B(zx, z) # 0 in | z j = l, | z 2 | ^ l if and only if#i(zi> zi) ^ 0 m lzi I = 1» \Z2\ ^ 1- The polynomialBx{zx, z2) can be written as a polynomial in z2 with coef-ficients which are polynomials in zx:

or equivalently

*i(*i. zj =N2

bN2(Zl)

(11)

(12)

246 IEE PROCEEDINGS, Vol. 134, Pt. G, No. 5, OCTOBER 1987

where Application of theorem 2 gives

(13)

and bN2(zx) ^ 0 in \zx | ^ 1 because of eqn. 3. Thus, thefollowing theorem follows directly.

Theorem 2: B(zx, z2) # 0 in | zx | = 1, | z21 < 1 if

Zl

2zx + 5

2

+

1*1

2zx^

= 1

1

h5

2

<f

^223 + 16x > 0

- 1

X = t +zf

The above inequality is satisfied. Thus, by theorem 2 thefilter is stable.

Example 1: Consider the polynomial

B(zx, z2) = 2zxz2 + zx + z2 + 4

Clearly condition 3 is satisfied. Therefore condition 4 willdetermine the stability. After forming B1(z1, z2) as ineqns. 11 and 12, we obtain

tSX\Zx, Z2f — Z2

Theorem 2 gives

+2zt < 1 with | zx | # 1

This is equivalent to

" l + 2 z 1 | 2 < | 4 + z1 12 + 4x > 0

- 1 ^ x ^ 1

The above inequality is satisfied. Thus we conclude thatthe filter is stable.

Example 2: Let J3(zl5 z2) = zxz\ + z\ + zvz2 + 2zt + 5.Again condition 3 is satisfied. Condition 4 will determinethe stability. Form £i(zl5 z2) as in eqns. 11 and 12, i.e.

x{zu z2)

Conclusion

A simple stability testing method for 2-dimensionaldigital filters has been proposed. The implementation ofthe method has been discussed and the results illustratedby two examples.

A.G. KANELLAKIS 16th April 1987N.J. THEODOROUNational Technical University of AthensDepartment of Electrical EngineeringElectric Power Division42 28th October Street106 82 Athens, Greece

References

1 GOODMAN, D.: 'Some stability properties of two-dimensionallinear shift invariant digital filters', IEEE Trans., 1976, CAS-24,pp. 201-208

2 SHANKS, J.L., TREITEL, S., and JUSTICE, J.H.: 'Stability and syn-thesis of two dimensional recursive filters., ibid., 1972, AU-20,pp. 115-128

3 HUANG, T.S.: 'Stability of two dimensional recursive filters', ibid.,1972, AU-20, pp. 158-163

4 BERGER, C.S.: 'Proof of a certain conjecture on the stability oflinear discrete systems', Int. J. Control, 1982,36, pp. 545-546

5 LU, W.S., and LEE, E.B.: 'Stability analysis for two dimensionalsystems via a Lyapunov approach', IEEE Trans., 1985, CAS-32,pp. 61-68

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