İki boyutlu sayısal filtrelerde kararlılık

Abstract

Sayısal filtre tasarımında kararlılık çok önemli bir kavramdır. Kararlı olmayan sayısal filtrenin pratikte kullanılma olanağı yoktur. Bu nedenle, tüm tasarlanan sayısal filtreler kararlı olmak zorundadır. Yapılan bu tezde, daha çok ikinci tipten tekilliği bulunan 2-boyutlu sayısal filtre transfer fonksiyonun pay ve payda polinomlarını kullanarak, 2-boyutlu sayısal filtrelerin sınırlı giriş-sınırlı çıkış (SGSÇ) anlamında kararlılığı incelenmiştir. Bu tip filtrelerde sınırlı giriş-sınırlı çıkış anlamında kararlılık var iken, ters 2-boyutlu sayısal filtrenin sınırlı giriş- sınırlı çıkış anlamında kararlı olup olmadığı ele alınmıştır. Kararlılık araştırması gerekli teoremler verildikten sonra, örnekler üzerinde açıklanmaya çalışılmıştır. Tezin son kısmında, ikinci tip tekilliğin bulunmadığı durumda, 2-boyutlu sayısal filtreler için önemli bazı teoremler verildi ve son aşamada da bu teoremlerden yararlanılarak bir kararlılık prosedürü geliştirilmiştir.

The problem bounded-input bounded-output is important for the design of recursive digital filters. The purpose of this thesis is to discuss certain stability properties of two-dimensional linear shift invariant digital filters. Most of these properties have no anologs in the one- dimensional case. in the one-dimensional case, the problem of testing the stability of a causal system is quite straightforward. Since a one- dimensional polynomial can always be factored straightforwardly as a product of first-order polynomials, we can easily determine the poles of H(z). The transfer function H(z) is devoid of poles in the entire unit disk for bounded-input bounded output stability and this condition which is necessary and sufficient can easily be tested. The extension of one- dimensional to the two-dimensional is known Shanks's teorem [1], [2]. Shanks theorem: stability - denominator of H(z1,z2')^0 for any |zj si, \z2\ sl.(l) We will be concerned with two-dimensional linear shift invariant filters which are causal (i.e., have a first quadrant impulse response) and have rational transfer function. Hence our transfer functions will be form H(z Z}=^L^ (2) 1 2 B(zitZ2) where A(z1,z2) and BCz^Zg) are two-variable polynomials in zl and z2. it is well known that a two-variable polynamial is not in general factorable in to first-order polynomials; rather, a two-variable polynomial can be factored in to irreducible factors which are themselves two-variable polynomials but which cannot be further factored [3]. (of course a given vi polynamial may itself be irreducible). These irreducible polynomials are unique up to multiplicative constants. Two polynomials which have no irreducible factor in common are said to be co-prime, such as A(z1,z2) and B(z1,z2)- it will be used the symbol 3 to stand for "such that". A point (zx,z2) ^ B(z1,z2)=0 but A(z1,z2)'tO will be called a pole ör a nonessential singularity of the first kind (such a point is analogous to a pole in the öne variable case). A point (zj,z2) 3 A(z1,z2)=B(z1,z2)=0 will be called a nonessential singularity of the second kind (such points have no öne variable anolog). Clearly, If (z^Zg) is nonessential singularity of the first kind, H(z1,z2)=°o. If (z1,z2) is a nonessential singularity of the second kind, the value of H(z1,z2) is undefined. Assumming that B(z1,z2)'tO at the origin, by continuity argument we can derive that B(z1,z2)'tO in. some neighborhood around the origin; thus, H(z1,z2) can be expanded into a power series in this neighborhood as co » H(z^lz2)=Y, E h(m,n) z?z? (3) m-O n=0 where h(m,n) is the impulse response of H(zpz2) The filter is bounded-input bounded-output stable if and only if oo oo H(z^,z2)eli, i.e.,]£ ^ h(m,n) <~. (4) fll=0 /3=0 We say that the impulse response is square summable if'and only if oo oo H(zi,z2)el2,i.e,lY/^h2(m,n}<'=o (5) jn=0 n=0 we define vii C72={(z1/z2) : |zx <, |z2| z2)=A(z1,z2yB(z1,z2) where A,B are co-prime. Assume that A/B has no poles in Ü2, nor any second kind singularities in Ü2 except for simple ones at (a,p) and T2, it will be assumed that A/B has no simple ör multiple second kind singularities of the form (a;y) outside.