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İki-boyutlu sayısal filtrelerde kararlılık

Stability analysis of two-dimensional digital filters

  1. Tez No: 39119
  2. Yazar: İLTERİŞ DEMİKIRAN
  3. Danışmanlar: DOÇ.DR. A. HAMDİ KAYRAN
  4. Tez Türü: Yüksek Lisans
  5. Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
  6. Anahtar Kelimeler: Kararlılık analizi, Sayısal filtreler, Stability analysis, Digital filters
  7. Yıl: 1993
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Belirtilmemiş.
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: Belirtilmemiş.

Özet

ÖZET 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. v

Özet (Çeviri)

SUMMARY STABILITY ANALYSIS OF TWO-DIMENSIONAL DIGITAL FILTERS 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 vipolynamial 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 oooo 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 oooo H(zi,z2)el2,i.e,lY/^h2(m,n}<'=o(5) jn=0 n=0 we define viiC72={(z1/z2) : |zx <, |z2| <!}(6) to be öpen unit bidisc, and U2={(zltz2) \\Zı\*l, z2\ si}(7) to be closed unit bidisc, and T*={(zi,z2) : zi | =1, zj =1}(8) to be distinguished boundary of the unit bidics. Goodman [3] proved that a two-dimensional transfer function could be stable even with nonessentional singularities of the second kind on the unit bidics. Shanks stability was shown to be not a necessary condition for stability by. Goodman. in Chapter 2, necessary and sufficient conditions for boundedness, and 12 and lx stabilities of a function H(z1,z2)=A(z1,z2y[B(z1,z2)]n, where (A/B) has sinaple nonessential singularities of the second kind on T2, will be obtained. These conditions was found by Roytman [4]. These conditions will be expressed in a very simple way in terms of the multiplicity of the zeros of certain resultants of two-variable polynomials. Many illustrative examples will be given. it will be shown that the fact that the impulse response is square summable, does not guarantee bounded-input bounded output stablity. 0000 H(z1,zz)el1, i.e.,^^ \h(m,n) < ~.(9) m«o n=o 00OO H(z1,z2)elll i.e., ^ £ h2(m,n) < ~.(10) m°Q n=o viiiin Chapter 2 the flowing important results will be proved. If H(z1,z2)=A(zI,z2y[B(z1,z2)]n, wh.ere A/B does not have any polar singularities in Ü2 nor any nonessential singularities of the second kind anywhere except f ör simple ones at (a, P) and, hence at (l/a, l/P) on T2 then a)H(zp Zg) is 12 stable if and only if (2n-i}ma (RZ2 [B',B} ) *2ma (RZ2[A,B} )(11) ör, alternatively (2n-l)m?(Rz:L[B',B] )*2mç(Rgl[A,B}).(12) b)H(z1,z2) is bounded in U2 if and only if nm^(Rz2[B',B]}^m?(Rz2[A,B])(13) ör, alternatively rm?p (Rsl [B1, B}) £fflp (Rzl [A, B] ).(14) c)HCzpZa) is lt stable (BIBO stable) if and only if nma(Rz2[B',B])ıma(Rz2[A,B])(15) ör, alternatively nmp(Rzl[B',B])<mç(Rzl[A,B])(16) where B' is paracongate of B ixEf (Zit Z2) =Zim Z2nB(z^, z;1)(17 ) where m and n are the maximum degrees of zl and z2 in B(z1,z2). m^fCzJ) and mp(f(z2)) denete the multiplicity of the factor (z^a) in f(z^ and (z2-p) in f(z2) respectively. R^I^B] and Rzl[A,B] denote the z2-resultant of the polynomials A and B and the Zj-resultant of the polynomials A and B respectively (Look at appendrs for resultant). Inverse two-dimensional digital filters have applications in area şuch as digital processing. For example inverse two-dimenşional digital filters nıay be used as tools to restore degraded images. in the Chapter 3 we will study the problem regarding the bounded-input bounded-output stability of inverse two-dimensionâl in the presence of nonessential singularities of the second kind. The method given in the Chapter 2 can be used for checking the bounded-input bounded-output stability of inverse two-dimensional digital filter transfer function having nonessential singularity of the second kind on T2. An interesting question arise: Does there exit bounded-input bounded-output two-dimensional functions having nonessential singularities of the second kind on T2 that also admit bounded-input bounded-output stable inverse? Ali of these will be shown in Chapter 3. H(z1>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. When H(z1,z2) is BIBO stable, H^z^) is BIBO stable if and only if m<t(Rz2[A',A]}<maL(RzZ[B,A])(18) where A' (zlt Z2) =z? z? A(z?, z?)(19 ) Xwhere m and n are the maximum degrees of Zj and z2 in B(z1,z2). To find if H"1(z1,z2) is BIBO stable, (18) condition will be used in the Chapter 3 in Chapter 4, it will be assumed two-dimensional digital filter transfer function is devoid of nonessential singularities of the second. The conditions for stability according to Shanks [1], Huang [1] and De Carlo-Strintzis [1] will be given. From these teoreras, the prosedüre that can be used for checking stability of H(z1,z2y=l/B(z1,z2) will be given. it can be seen from this procedure that two-dimensional stability test can be performed by performing many one-dimensional stability test. This will be shown by giving öne example. xi

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