Chapter Introduction. Jianzhong Wang
|
|
- Bridget Debra McCormick
- 5 years ago
- Views:
Transcription
1 Chapter Elementary Matrix Decomposition Algorithm for Symmetric Extension of Laurent Polynomial Matrices and its Application in Construction of Symmetric M-band Filter Banks Jianzhong Wang Abstract In this paper, we develop a novel and effective algorithm for the construction of perfect reconstruction filter banks (PRFBs) with linear phase. In the algorithm, the key step is the symmetric Laurent polynomial matrix extension (SLPME). There are two typical problems in the construction: () For a given symmetric finite low-pass filter a with the polyphase, to construct a PRFBs with linear phase such that its low-pass band of the analysis filter bank is a. (2) For a given dual pair of symmetric finite low-pass filters, to construct a PRFBs with linear phase such that its low-pass band of the analysis filter bank is a, while its low-pass band of the synthesis filter bank is b. In the paper, we first formulate the problems by the SLPME of the Laurent polynomial vector(s) associated to the given filter(s). Then we develop a symmetric elementary matrix decomposition algorithm based on Euclidean division in the ring of Laurent polynomials, which finally induces our SLPME algorithm.. Introduction The main purpose of this paper is to develop a novel and effective algorithm for the construction of perfect reconstruction filter banks (PRFBs) with linear phase. In the algorithm, the key step is the symmetric Laurent polynomial matrix extension (SLPME). PRFBs have been widely used in many areas such as signal and image processing, data mining, feature extraction, and compressive sensing [,, 2,, 4]. A PRFB consists of two sub-filter banks: an analysis filter bank, which decomposes a signal into different bands, and a synthesis filter bank, which composes a signal from its different band components. Either an analysis filter bank or a synthesis one consists of several band-pass filters. Assume that an analysis filter bank con- Jianzhong Wang ( ) Department of Mathematics and Statistics, Sam Houston State University, Huntsville, USA, jzwang@shsu.edu 47
2 48 Jianzhong Wang sists of the filter set {H 0,H,,H M } and a synthesis filter bank consists of the set {B 0,B,,B M }, where H 0 and B 0 are low-pass filters. Then they form an M-band PRFB if and only if the following condition holds: M B j ( M)( M)H j = I, (.) j=0 where M is the M-downsampling operator, M is the M-upsampling operator, I is the identity operator, and H j denotes the conjugate filter of H j. Note that the conjugate of a real filter a = (,a,a 0,a, ) is defined as ā = (,a,a 0,a, ). In signal processing, a filter H having only finite non-zero entries is called a finite impulse response (FIR). Otherwise it is called an infinite impulse response (IIR). Since FIR is much more often used than IIR, in this paper we only study FIR with real entries. Recall that the z-transform of a FIR H is a Laurent polynomial (LP) and the z- transform of the conjugate filter of H is H(z) = [ H(/z). We define the M-polyphase ] form of a signal (or a filter) x by the LP vector a [M,0] (z),,a [M,M ] (z)}, where a [M,k] (z) = a(m j + k)z j, 0 k M. j For convenience, we will simplify a [M,k] to a [k] if it does not cause confusion. The polyphase form of a M-band filter bank {H 0,,H M } is the following LP matrix. H [0] 0 (z) H[] 0 (z) H[M ] 0 (z) H [0] H(z) = (z) H[] (z) H[M ] (z)... H [0] M (z) H[] M (z) H[M ] M (z) Using polyphase form, we represent (.) as a LP matrix identity in the following theorem. Theorem.. The filter bank pair of {H 0,,H M } and {B 0,,B M } realizes a PRFB if and only if the following identity holds: H(z)B (z) = I, (.2) M where both H(z) and B(z) are LP matrices, and B (z) denotes the conjugate transpose matrix of B(z). We denote by L the ring of all Laurent polynomials, and call a LP matrix is L - invertible, if its inverse is a LP matrix too. Since MB (z) = H (z) in a PRFB, the polyphases of its analysis filter bank and its synthesis one are L -invertible. By (.2), we also have M MH [ j] 0 (z) B [ j] 0 (z) =. j=0
3 Symmetric LP Matrix Extension 49 In general, we will call a LP vector a(z) = [a (z),,a M (z)] a prime one if there is a LP vector b(z) = [b (z),,b M (z)] such that a(z)b T (z) =. More details of the theory of PRFBs are referred to [0, 5]. The filters with symmetry (also called with linear phases) are more desirable in application [0]. They are formally defined as follows: Definition.2. Let c be an integer. A filter (or signal) x is called symmetric or antisymmetric about c/2 if x(k) = x(c k) or x(k) = x(c k), k Z, respectively. Later, for simplification, we will use the term symmetric to mention both symmetric and antisymmetric. Thus, x is symmetric if and only if x(k) = εx(c k), where ε (= or ) is the symbol of the symmetry-type. Note that we can always shift a signal/filter x such that the shifted one has the symmetric center at c = 0 or c = ±. Hence, without loss of generality, in this paper we always assume that a symmetric filter has the center at c = 0, c =, or c =, and simply call it 0- symmetric, -symmetric, or ( )-symmetric, respectively. Correspondingly, the set of all 0-symmetric filters (-symmetric, or ( )-symmetric ones) is denoted by V 0 (V or V ). Besides, when we need to stress on the symmetry-type, we denote by V +,V 0 +,V + for ε = + and V,V 0,V for ε =. It is clear that if x V 0, then so is x, and if x V, then x V. We also have the following: x V 0 if and only if x(z) = εx(/z), x V if and only if x(z) = εzx(/z), and x V if and only if x(z) = ε/zx(/z). In addition, if a(z) = εb(/z) (a(z) = εzb(/z),a(z) = ε/zb(/z)), we call [a(z),b(z)] a V 0 (V, V ) pair. For a symmetric filter H, we modify its M-polyphase to the following: H [k] (z) = H(M j + k)z j, m k M m, m = j [ M Later, a LP vector is called a S-LP one if it is a polyphase form of a symmetric filter. Similarly, we will call a LP matrix Sr-LP matrix (Sc-LP matrix) if its rows (columns) are S-LP vectors. They will be simply called S-LP matrices if row and column are not stressed. Similarly,, a PRFB is called symmetric if all of its bandfilters are symmetric. Two fundamental problems in the construction of symmetric PRFBs are the following: Problem. Assume that a given symmetric low-band filter H 0 has a prime polyphase. How to construct a symmetric PRFB, in which the first band of its analysis filter bank is H 0? Problem 2. Assume that a dual pair of symmetric low-band filters H 0 and B 0 are given. How to find other symmetric components H,,H M and B,,B M so that they form a symmetric PRFB? By Theorem., we have the following: Corollary.. The symmetric filter banks {H 0,,H M } and {B 0,,B M } form a symmetric PRFB if and only if the following identity holds: 2 ]. H(z)B (z) = I, (.) M
4 50 Jianzhong Wang where both H(z) and B(z) are Sr-LP matrices. Ignoring the factor M on the right-hand side of (.) in Corollary., we can see that the two fundamental problems are equivalent the following symmetric Laurent polynomial Matrix extension (SLPME) problems: SLPME Problem. Assume that a given S-LP row vector a(z) L M is prime. To find an L -invertible Sr-LP matrix A(z) such that A(,:) = a. SLPME Problem 2. Assume that a given pair of S-LP row vectors [a(z),b(z)] satisfies a(z)b T (z) =. To find an L -invertible Sr-LP matrix A(z) such that A(,: ) = a and A (:,) = b T. Laurent polynomial matrix extension (LPME) has been discussed in [, 4, 9]. Having the aid of LPME technique, several algorithms have been developed for the construction of PRFBs [2, 5, 6, 7, 6, 5]. Unfortunately, the methods for constructing LPME usually do not produce SLPME. The main difficulty in SLPME is how to preserve the symmetry. Recently, Chui, Han, and Zhuang in [2] proposed a bottomup algorithm for solving SPLME Problem 2 based on the properties of dual filters. In this paper, we solve the problem in the framework of the algebra of Laurent polynomials. Our approach to SLPME is based on the decomposition of L - invertible S-LP matrix in the LP ring [5]. To make the paper more readable, we restrict our discussion for M = 2,,4. The readers can find that our algorithms can be extended for any integer M without essential difficulty. The paper is organized as follows. In Section 2, we discuss the properties of S-LP vectors and the symmetric Euclidean division in the LP ring. In Section, we introduce the elementary S-LP matrix decomposition technique and apply it in the development of the SLPME algorithms. Finally, two illustrative examples are presented in Section 4..2 S-LP Vectors and Symmetric Euclidean Division For simplification, in the paper, we only discuss LP with real coefficients. Readers will find that our results can be trivially generalized to LP with coefficients in the complex field or other number fields. Let the ring of all polynomials be denoted by P and write P h = P \ {0}. Similarly, let the ring of all Laurent polynomials be denoted by L and write L h = L \ {0}. If a L h, we can write a(z) = n k=m a kz k, where n m and a m a n 0. We define the highest degree and the lowest degree of a L h by deg + (a) = n and deg (a) = m respectively. When a = 0, we agree that deg + (0) = and deg (0) =. We define the support length of a by supp(a) = deg + (a) deg (a). Particularly, when a(z) L h is 0-symmetric, -symmetric, or ( )-symmetric, we have deg (a) = deg + (a), deg (a) = deg + (a) +, or deg (a) = deg + (a), respectively. Let the semi-group G P h be defined by G = {p P h : p(0) 0}. Then, the power mapping π : L h G, π(a(z)) = z deg (a) a(z), defines an equivalent relation in L h, i.e., a b if and only if π(a) = π(b). For convenience, we agree that
5 Symmetric LP Matrix Extension 5 π(0) = 0. Let L m denote the group of all non-vanished Laurent monomials: L m = {m L h ; m = cz l,c 0,l Z}. Then, we have π(m) = c. For a LP vector a = [a,,a s ], we define π(a) = [π(a ),,π(a s )]. Then the greatest common divisor (gcd) of a nonzero row (or column) LP vector a L s is defined by gcd L (a) = gcd(π(a)) G. A LP a(z) L h is said to be in the subset L d if a(z) = εa(/z) and gcd L (a(z),a(/z)) =. A LP matrix A(z) L s s is said to be L -invertible if A(z) is invertible and A (z) L s s too. It is obvious that A(z) is L -invertible if and only if det(a(z)) L m. We now discuss the properties of S-LP vectors. Recall that an M dimensional S-LP vector is defined as the M-polyphase form of a symmetric filter. Let x(z) be an M-dimensional S-LP vector. We list its symmetric properties for M = 2,,4, in Table.. M = 2 c = 0 x [0] (z) = εx [0] (/z),x [] (z) = ε/zx [] (/z) c = x [0] (z) = εx [] (/z) M = c = 0 x [0] (z) = εx [0] (/z),x [] (z) = εx [ ] (/z) c = x [0] (z) = εx [] (/z),x [ ] (z) = εzx [ ] (/z) M = 4 c = 0 x [0] (z) = εx [0] (/z),x [] (z) = εx [ ] (/z),x [2] (z) = ε/zx [2] (/z) c = x [0] (z) = εx [] (/z),x [ ] (z) = εx [2] (/z) Table.: The symmetry of the components in a S-LP vector. Let m = [ M 2 ]. We can verify that, when M is even and c = 0, x[0] (z) V 0,x [m] (z) V, and [x [i] (z),x [ i] (z)],i =,,M, are V 0 pairs; when M is even and c =, (x [i] (z),x [ i+] (z)),i =,,M, are V 0 pairs; when M is odd and c = 0, x [0] (z) V 0, and [x [i] (z),x [ i] (z)],i =,,M, are V 0 pairs; when M is odd and c =, [x [i] (z),x [ i+] (z)],i =,,M, are V 0 pair and x [ M] (z) V. We need the following L -Euclid s division theorem [5] in our discussion. Theorem.4. Let (a,b) L h L h and supp(a) supp(b). Then there exists a unique pair (q,r) L L such that a(z) = q(z)b(z) + r(z) with supp(r) + deg (a) deg + (r) < supp(b) + deg (a), (.4) which implies that supp(q) supp(a) supp(b) and supp(r) < supp(b). By Theorem.4, it is also clear that if deg (a) = deg (b), then q P h and deg(q) deg + (a) deg + (b). From Theorem.4, we derive the symmetric L - Euclid s division theorem to deal with S-LP vectors. Theorem.5. Let a(z) V 0 with supp(a) = 2m, b(z) V with supp(b) = 2k, c(z) V with supp(c) = 2s, and d(z) L h with supp(d) = l be given. Then we have the following:. If m k, then there is p(z) V + with supp(p) 2(m k) and a (z) V 0 with supp(a ) < supp(b) such that a(z) = b(z)p(z) + a (z). If m < k, then there is q(z) V + with supp(q) 2(k m) and b (z) V with supp(b ) < supp(a) such that b(z) = q(z)a(z) + b (z).
6 52 Jianzhong Wang 2. If m s, then there is q(z) V + with supp(q) 2(m k) and a (z) V 0 with supp(a ) < supp(c) such that a(z) = c(z)q(z) + a (z). If m < s, then there is p(z) V + with supp(p) 2(k m) and c (z) V with supp(c ) < supp(a) such that c(z) = p(z)a(z) + c (z).. If supp(a) > supp(d), there is a p(z) P h with deg(p) m [ l+ 2 ] and a (z) V 0 with supp(a ) l such that a(z) = p(z)d(z) + ε p(/z)d(/z) + a (z). 4. If supp(b) > supp(d), there is a q(z) P h with deg(q) k [ l+ 2 ] and b (z) V with supp(b ) l such that b(z) = q(z)d(z) + ε/zq(/z)d(/z) + b (z). Similarly, if supp(c) > supp(d), there is a p(z) P h with deg(p) c [ l+ 2 ] and c (z) V with supp(c ) l such that c(z) = p(z)d(z) + εzp(/z)d(/z) + c (z). Proof. To prove (), we write a(z) = m j= m a jz j and set a t (z) = m j=k a jz j + 2 k j= k+ a jz j so that a t (z) + εa t (/z) = a(z). By Theorem.4, we can find a ˆp(z) P h with deg( ˆp) m k such that a t (z) = ˆp(z)b(z) + r(z), where r L with deg + (r) < k,deg (r) > k. It leads to a(t) = ˆp(z)b(z) + ε ˆp(/z)b(/z) + r(z) + εr(/z). Since b(z) V, we have b(z) = ε/zb(/z), which yields a(z) = ( ˆp(z) + z ˆp(/z))b(z) + (r(z) + εr(/z)). Write p(z) = ˆp(z) + z ˆp(/z),a (z) = r(z) + εr(/z). It is obvious that p(z) V + with supp(p) 2(m k) and a (z) V 0 with supp(r) < supp(b). The proof of the first statement of () is completed. The proofs of the remains are similar.. SLPME Algorithms Based on Elementary S-LP Matrix Decomposition We now discuss SLPME algorithms for M = 2,,4, respectively... The Case of M = 2 We say a(z) = [a (z),a 2 (z)] V 0,2 if a (z) V 0,a 2 (z) V ; and say a(z) V,2 if it is a V 0 pair. We also say b(z) V 0,2 if b (z) V 0,b 2 (z) V. Define S 0,2 = { S(z) = [s i j (z)] 2 i, j=; s ii (z) V + 0,i =,2,s 2(z) V +,s 2(z) V + }. Thus, if S(z) S 0,2, then S(,:)(z) V 0,2 and S(:,)(z) V 0,2.
7 Symmetric LP Matrix Extension 5... The Case of a V 0,2. To develop our SLPME algorithm, we give the following: Definition.6. Let s(z) V +, t(z) V +, k Z, and r R \ {0}. Then the following matrices [ ] [ ] [ s(z) 0 rz E u (s) = E 0 l (t) =, D(r,k) = k ] 0 (.5) t(z) 0 are called the elementary S 0,2 matrices, and their product is called a S 0,2 - fundamental matrix. It can verify that all of the matrices in (.5) are L -invertible and their inverses are also in S 0,2. Indeed, we have (E u (s)) = E u ( s) (E l (t)) = E l ( t) (D(r,k)) = D(/r, k). (.6) Later, we simply denote by E u,e l,d for the matrices in (.6). We now return the SLPME for a V 0,2. WLOG, we assume supp(a ) > supp(a 2 ). Since gcd L (a) =, By Theorem.5, we can use elementary S 0,2 matrices to make the following: a 0 E l( p ) a E u( q ) a 2 El( p n ) a 2n E u( q n ) 2n D(r,k) a [,0], where a 0 = a, a 2i E l ( p i+ ) = a 2i+,a 2i+ E u ( q i+ ) = a 2i+2,i =,,n. Let E a = E l ( p )E u ( q ) E l ( p n )E u ( q n )D(r,k). (.7) Then E a S 0,2,aE a = [,0], and its inverse A a (z) = E a (z) D(/r, k)e u (q n )E l (p n ) E u (q )E l (p ) (.8) provides a solution for SLPME Problem. We now consider the SPLME Problem 2. WLOG, assuming that the symmetric dual pair [a(z),b(z)] V 0,2 V 0,2 is given. Let E a (z),a a (z) be the matrices given in (.7) and (.8). By a(z)b T (z) = and A a (,:) = a, we have A a (z)b T (z) = [,w(z)] T with w(z) V, which yields E a (z)[,w(z)] T = b T (z). Then the matrices Ã(z) = E l ( w)(z)a a (z) and B(z) = E a (z)e l (w)(z) give the solution....2 The Case of a(z) V,2. If its dual b(z) is not given, then by the extended Euclidean algorithm in [5], we can find LP vector s(z) = [s (z),s 2 (z)], such that as T =. We define b (z) = 2 (s (z) + εs 2 (/z)),b 2 (z) = εb (/z). The vector b(z) is a V 0 pair and ab T =. We now define
8 54 Jianzhong Wang [ ] a (z) a A(z) = 2 (z), b 2 (z) b (z) which is L -invertible and its inverse is B(z) = A (z) = [ ] b (z) a 2 (z). b 2 (z) a (z) Then A(z) provides the solution of SPLME Problem, and the pair [A(z),B(z)] gives the solution of SPLME Problem The Case of M = We say a(z) = [a (z),a 2 (z),a (z)] V 0, if a 2 (z) V 0 and [a (z),a (z)] is a V 0 pair, and say a(z) V, if a 2 (z) V and [a (z),a (z)] is a V 0 pair. We also say b(z) V, if b 2 (z) V and [b (z),b (z)] is a V 0 pair. Note that, in the case of M =,c =, the polyphase form x(z) is not in V,, but [x [0] (z),x [ ] (z),x [] (z)] V,...2. The Case of a V 0, Definition.7. Let q(z) L. The matrices of the following two types are called elementary S 0, matrices: E v (q) = 0 0 q(z) q(/z), E h (q) = q(z) q(/z) In general, we simply denote by E an elementary S 0, matrix, and call their product a Fundamental S 0, one. It is clear that Ev (q) = E v ( q) and Eh (q) = E h( q). By the same argument for M = 2, using the elementary S 0, matrices, we can obtain the following chain: a 0 E a E 2 a 2 En a n where a n has the same symmetry as a and gcd L (a n ) =. Therefore, a n = [p(z),0,ε p(/z)], p(z) L d. Besides, when ε =, it may have another form a n = [0,r,0]. Writing E = E E 2 E n, we have a(z) = a n (z)e (z). Let q(z) L d satisfy p(z)q(z)+ p(/z)q(/z) = and set q(z) = [q(z), 0, εq(/z)]. We define
9 Symmetric LP Matrix Extension 55 0 /2 /2 q(z) 0 ε p(/z) Q (z) = r 0 0 Q 2 (z) = 0 0, (.9) 0 /2 /2 εq(/z) 0 p(z) whose inverses are Q (z) = 0 r p(z) 0 ε p(/z) 2 (z) = 0 0. εq(/z) 0 q(z) Q Finally, we define A(z) = { Q (z)e (z), if a n = [0,r,0], Q 2 (z)e (z), if a n = [p(z),0,ε p(/z)]. It is clear that A(z) is a SLPME of a(z). We now return to SPLME Problem 2. Assume a symmetric dual pair [a(z),b(z)] V 0, V 0, is given so that a(z)b T (z) =. Let E(z) be the LP matrix above. Define w(z) = b(z)(e ) T (z) V 0,. Then a n w T = a n E b T = ab T =. Hence, { [u(z),/r,εu(/z)], if a n = [0,r,0], w(z) = [v(z),v c (z),εv(/z)], if a n = [p(z),0,ε p(/z)], where p(z)v(z) + p(/z)v(/z) =. Write q + (z) = u(z) + εu(/z),q (z) = u(z) εu(/z). Define Q (z) = u(z) v(z) 0 ε p(/z) /r 0 0, Q 2 (z) = v c (z) 0, (.0) εu(/z) εv(/z) 0 p(z) whose inverses are Q (z) = 2 We now define 0 2r 0 rq + (z), Q 2 (z) = rq (z) A(z) = p(z) 0 ε p(/z) v c (z)p(z) εv c (z)p(/z). εv(/z) 0 v(z) { Q (z)e (z), if a n = [0,r,0], Q 2 (z)e (z), if a n = [p(z),0,ε p(/z)]. Then, the pair [A(z),A (z)] is a SLPME of the pair [a(z),b(z)].
10 56 Jianzhong Wang..2.2 The Case of a V, The discussion is very similar to the case of a V 0,. Definition.8. Let q(z) L. The matrices of the following two types are called elementary S, matrices: E v (q) = 0 0 q(z) zq(/z), E h (q) = q(z) zq(/z) The product of elementary S, matrices is called a Fundamental S, one. Using the elementary S, matrices, we can obtain the following chain: a 0 E a E 2 a 2 En a n where a n has the same symmetry as a and gcd L (a n ) =. Therefore, a n = [p(z),0,ε p(/z)], p(z) L d. Note that, because a 2 (z) S,, a n does not have other forms. Writing E = E E 2 E n, we have a(z) = a n (z)e (z). Let Q 2 (z) be the LP matrix in (.9). Then A(z) is a SLPME of a(z). We now consider SPLME Problem 2. In the given dual pair, b(z) V,. Let E(z) be the LP matrix above. Then the LP vector w(z) = b(z)(e ) T (z) V, too. Hence, it has the only form of w(z) = [v(z),v c (z),εv(/z)], v c (z) V, where p(z)v(z) + p(/z)v(/z) =. Let Q 2 (z) be the LP matrix in (.0), and A(z) = Q 2 (z)e (z). Then [(E(z)Q 2 (z)),e(z)q 2 (z)] is a SLPME of the dual pair [a(z),b(z)]... The Case of M = 4 In this case, we say a(z) V 0,4 if a (z) V 0, [a 2 (z),a (z)] is a V 0 pair, and a 4 (z) V ; say a(z) V,4 if both [a (z),a 4 (z)] and [a 2 (z),a (z)] are V 0 pairs. We also say b(z) V 0,4 if b(/z) V 0,4. Note that, if x(z) is the polyphase form of an asymmetry filter in the case of M = 4,c = 0, then [x [0] (z),x [ ] (z),x [] (z),x [2] (z)] V 0,4.
11 Symmetric LP Matrix Extension The Case of a V 0,4 Definition.9. Let q(z) L,s(z) V,t(z) V. The followings are called elementary S 0,4 matrices: q(z) q(/z) 0 Eh 0 (q) = , E0 v (q) = ( 0 0 s(z) Eh 0 (q)) T, Et (s) = , Eh (q) = , E v (q) = ( Eh (q)) T, Eb (t) = q(z) zq(/z) t(z) 0 0 Using the elementary S 0,4 matrices, we can obtain the following chain: a 0 E a E 2 a 2 En a n, where a n has the same symmetry as a and gcd L (a n ) =. Therefore, a n = [0, p(z),ε p(/z),0], p(z) L d. If ε =, it possibly can also have the form (a) n (z) = [r,0,0,0],r 0. Let q(z) L d satisfy p(z)q(z) + p(/z)q(/z) =. Write E = E E 2 E n and define /r Q (z) = 0 /2 /2 0 0 /2 /2 0 Q 2(z) = q(z) 0 0 ε p(/z) εq(/z) 0 0 p(z), whose inverses are r Q (z) = p(z) ε p(z) 0 2 (z) = εq(/z) q(z) 0 Q Set A(z) = { Q (z)e (z), if a n = [r,0,0,0], Q 2 (z)e (z), if a n = [0, p(z),ε p(/z),0]. (.) Then A(z) is a SLPME of a. We now consider SPLME Problem 2. In the given dual pair, b(z) is now in V 0,4. Let E(z) be the LP matrix above. Then the LP vector w = b(z)(e ) T (z) is in V 0,4 too. Hence, if a n = [r,0,0,0], w(z) = [/r,v(z),εv(/z),v (z)], v (z) V,
12 58 Jianzhong Wang else if a n = [0, p(z),ε p(/z),0], w(z) = [v 0 (z),v(z),εv(/z),v (z)], v 0 (z) V 0,v (z) V, where p(z)v(z) + p(/z)v(/z) =. Let /r v 0 (z) 0 0 Q (z) = v(z) /2 /2 0 εv(/z) /2 /2 0 Q 2(z) = v(z) 0 0 ε p(/z) εv(/z) 0 0 p(z), v (z) 0 0 v (z) 0 0 whose inverses are r Q (z) = w + (z) 0 w (z) 0, v (z) p(z) ε p(z) 0 2 (z) = p(z)v 0 (z) ε p(/z)v 0 (z) 0 0 p(z)v (z) ε p(/z)v (z), 0 εv(/z) v(z) 0 Q where w + (z) = s(z) + εs(/z) and w (z) = s(z) εs(/z). Let A(z) be given by (.). Then [A(z),A (z)] is a SLPME of the dual pair [a(z),b(z)]....2 The Case of a V, Let P = be the permutation matrix Definition.0. Let q(z) L. The matrix with the form of q(z) q(/z) 0 E (q) = and E 2 (q) = PE (q)p are called elementary S,4 matrices. Using the elementary S,4 matrices, we can obtain the following chain: a 0 E a E 2 a 2 En a n, where a n (z) = [p(z),0,0,ε p(/z)] with p(z) L d or a n = [0, p(z),ε p(/z),0], p(z) L d. Since [p(z),0,0,ε p(/z)] = [0, p(z),ε p(/z),0]p, we only discuss the first case. Let q(z) L d satisfy p(z)q(z) + p(/z)q(/z) =. Write E = E E 2 E n and define
13 Symmetric LP Matrix Extension 59 q(z) 0 0 ε p(z) Q(z) = 0 /2 /2 0 0 /2 /2 0, εq(z) 0 0 p(z) whose inverse is p(z) 0 0 ε p(/z) Q (z) = εq(/z) 0 0 q(z) The matrix A(z) = Q (z)e (z) is a SLPME of a(z). We now consider SPLME Problem 2. In the given dual pair, b(z) V,4. Let E(z) be the LP matrix above. We still assume that a n = [p(z),0,0,ε p(/z)], p(z) L d. Then the LP vector w(z) = b(z)(e ) T (z) has the form of w(z) = [v(z),s(z),εs(/z),εv(/z)], v(z) L d, where p(z)v(z) + p(/z)v(/z) =. Let w + (z) = s(z) + εs(/z),w (z) = s(z) εs(/z), and v(z) 0 0 ε p(/z) Q(z) = v(z)w + (z) /2 /2 ε p(/z)w + (z) v(z)w (z) /2 /2 ε p(/z)w (z), εv(/z) 0 0 p(z) whose inverse is p(z) 0 0 ε p(/z) Q (z) = s(z) 0 εs(/z) 0. εv(/z) 0 0 p(z) Define A(z) = Q (z)e (z). Then [A(z),A (z)] is a SLPME of the dual pair [a(z),b(z)]..4 Illustrative Examples In this section, we present two examples to the readers for demonstrating the SLPME algorithm we developed in the previous section. Example. (Construction of -band symmetric PRFB). Let H 0 and B 0 be two given low-pass symmetric filters with the z-transforms ( z + + z H 0 (z) = ) 2
14 60 Jianzhong Wang and B 0 (z) = 27 (z + + z) 2 ( + ) We want to construct the -band symmetric PRFB {H 0,H,H 2 },{B 0,B,B 2 }, which satisfies 2 B j ( )( )H j = f raci, j=0 Their polyphase forms are the following: [ H [0] 0 (z), H [] 0 (z), H [2] 0 (z)] = [2 + z,,2 + /z] 9 [ [B [0] 2 0 (z),b[] 0 (z),b[2] 0 (z)] = ,, 2 + z ] 9z 27 9 To normalize them, we set and a = [H [0] 0,H[] 0,H[2] 0 ] = [2 + z,,2 + /z] [ b = [B [0] 2 0,B[] 0,B[2] 0 ] = ; ; 2 + z ] z 9 so that ab T =. We now use elementary S 0, matrix decomposition technique. Let E(z) = /z z We have a (z) = a(z)e(z) = [0,/,0]. To make the SLPME for a(z), we set 0 /2 /2 Q(z) = /2 /2 Then the LP matrix A(z) = Q (z)e (z) = z 9 / 2+/z 9 is a SLPME for a. To obtain the SLPME for the dual pair [a,b], we compute w(z) = b(z)(e (z)) T = [ 2+/z ], which yields,, 2+z 2+/z Q(z) = 0 0, 2+z
15 Symmetric LP Matrix Extension 6 where Q(:,) = w T. Finally, we have and A(z) = (E(z)Q(z)) = B(z) = E(z)Q(z) = 2+z z+2z2 +z 5 2 8z+6z2 +z 5 2+/z 2+/z 9 +z 2 8z +2z+26z2 2z z2 8z +6z 8z2 +2z ) 5 2 4/z+7 /z+z 9 /z+4+z 2+z which are SLPME of (a(z),b(z)). Recovering the filters from their polyphases and applying the normalization factor to B(z), we have the following z-transforms for the -band PRFB: and ( z ) z H 0 (z) =,, H (z) = ( z) ( + 5z + 5z 2 + z + z 4 + 5z 5 + 5z 6 + z 7 ) 5 5, H 2 (z) = (z )2 ( + + 0z z + 6z z 5 + 0z z 8 ) 5 5, B 0 (z) = 27 (z + + z) 2 ( + ), B (z) = 9z + z z z 9, B 2 (z) = 9z + z z z 9. Example.2 (Symmetric LP matrix extension of 4 4 matrix). Let a = 6 [ 2 z + 2 2z, z z, z + 8 z,4 + 4 ] V 0,4 z be a given LP independent vector. We first consider the SLPME Problem. The symmetric Euclidean divisions yields the matrices S = S 2 = ( + z) (7 + z) 4 ( + 7z)
16 62 Jianzhong Wang 2z 2 z z S = S 4 = and a = [,0,0,0]. Let E = S S 2 S S 4 = and Then the SLPME of a is z 2z 2 +z z 2 A(z) = 6z z 2 6z+z z z 2 8z Q = 0 /2 /2 0 0 /2 / z z 2 +8z+z 2 8z 6z +8z z2 6z +z z 2 7+z We now solve the SLPME Problem 2. Let b = 6 +7z 4. [ z + 0 z, 2z + 6,2z + 6,4 + ]. [ ] Then (a,b) is a dual pair. We have w = b(e ) T =, z+ 8z, +z 8, (+z). Set Then, the SLPME for the dual pair is A(z) = (E(z)Q(z)) = z Q(z) = 8 /2 /2 0 +z 8 /2 /2 0. (+z) z z 2 8z +8z+z 2 6z +8z z 2 6z +z 6z+6z z 4 8z+26z 2 +8z +z 4 +8z+26z2 8z +z 4 (z )(z+) z 2 28z 2 2z 2 2z 2 +z 4 +z 4 +80z 2 2z +z z 2 28z 2 +7z+7z2 +z 2z 2 z 0z 2 0z z 4 +z 5 5z+90z 2 0z z 4 z 5 z 0z 2 +90z 5z 4 +z 5 2z (z )2 (+6z+z 2 ) 6z 2 and
17 Symmetric LP Matrix Extension 6 +0z z 2 z 2 +z 2 6z +z z+ B(z) = E(z)Q(z) = 8z z 8 /2 /2 0. +z 5z+5z 2 z 7z 7z 2 +z +6z z 2 4 8z 8z 8z References. C. K. Chui and J.-A. Lian, Construction of compactly supported symmetric and antisymmetric orthonormal wavelets with scale =, Appl. Comput. Harmon. Anal. 2 (995), C. Chui, B. Han, and X. Zhuang, A dual-chain approach for bottom-up construction of wavelet filters with any integer dilation, Appl. Comput. Harmon. Anal. (2) (202), R. Crochiere and L. R. Rabiner, Multirate Digital Signal Processing, Prentice-Hall, Englewood Cliffs, S. Goh and V. Yap, Matrix extension and biorthogonal multiwavelet construction, Linear Algebra Appl. 269 (998), B. Han, Matrix extension with symmetry and applications to symmetric orthonormal complex m-wavelets, Journal of Fourier Analysis and its Applications 5 (2009), B. Han and X. Zhuang, Matrix extension with symmetry and its applications to symmetric orthonormal multiwavelets, SIAM J. Math. Anal. 42 (200), B. Han and Z. Zhuang, Algorithms for matrix extension and orthogonal wavelet filter banks over algebraic number fields, Mathematics of Computation 2 (20), S. Mallat, A wavelet tour of signal processing, Academic Press, San Diego, X. Shi and Q. Sun, A class of m-dilation scaling function with regularity growing proportionally to filter support width, Proc. Amer. Math. Soc. 26 (998), G. Strang and T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambrige, P. Vaidyanathan, Theory and design of M-channel maximally decimated quadrature mirror filters with arbitrary M, having the perfect-reconstruction property, IEEE Trans. Acoust. Speech, Signal Processing 5 (4) (987), P. Vaidyanathan, How to capture all FIR perfect reconstruction QMF banks with unimodular matrices, Proc. IEEE Int l Symp. Circuits Syst., vol., 990, pp P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice-Hall, Englewood Cliffs, M. Vetterli, A theory of multirate filter banks, IEEE Trans. Acoust., Speech, Signal Processing 5 () (987), J. Z. Wang, Euclidean algorithm for Laurent polynomial matrix extension, Appl. Comput. Harmon. Anal. (2004). 6. X. Zhuang, Matrix extension with symmetry and construction of biorthogonal multiwavelets with any integer dilation, Applied and Computational Harmonic Analysis (202), 59 8.
Department of Mathematics and Statistics, Sam Houston State University, 1901 Ave. I, Huntsville, TX , USA;
axioms Article Euclidean Algorithm for Extension of Symmetric Laurent Polynomial Matrix and Its Application in Construction of Multiband Symmetric Perfect Reconstruction Filter Bank Jianzhong Wang Department
More informationWavelets and Filter Banks
Wavelets and Filter Banks Inheung Chon Department of Mathematics Seoul Woman s University Seoul 139-774, Korea Abstract We show that if an even length filter has the same length complementary filter in
More informationWavelet Filter Transforms in Detail
Wavelet Filter Transforms in Detail Wei ZHU and M. Victor WICKERHAUSER Washington University in St. Louis, Missouri victor@math.wustl.edu http://www.math.wustl.edu/~victor FFT 2008 The Norbert Wiener Center
More informationNew Design of Orthogonal Filter Banks Using the Cayley Transform
New Design of Orthogonal Filter Banks Using the Cayley Transform Jianping Zhou, Minh N. Do and Jelena Kovačević Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign,
More informationTwisted Filter Banks
Twisted Filter Banks Andreas Klappenecker Texas A&M University, Department of Computer Science College Station, TX 77843-3112, USA klappi@cs.tamu.edu Telephone: ++1 979 458 0608 September 7, 2004 Abstract
More informationAvailable at ISSN: Vol. 2, Issue 2 (December 2007) pp (Previously Vol. 2, No.
Available at http://pvamu.edu.edu/pages/398.asp ISSN: 193-9466 Vol., Issue (December 007) pp. 136 143 (Previously Vol., No. ) Applications and Applied Mathematics (AAM): An International Journal A New
More informationSymmetric Wavelet Tight Frames with Two Generators
Symmetric Wavelet Tight Frames with Two Generators Ivan W. Selesnick Electrical and Computer Engineering Polytechnic University 6 Metrotech Center, Brooklyn, NY 11201, USA tel: 718 260-3416, fax: 718 260-3906
More informationDesign of Orthonormal Wavelet Filter Banks Using the Remez Exchange Algorithm
Electronics and Communications in Japan, Part 3, Vol. 81, No. 6, 1998 Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J80-A, No. 9, September 1997, pp. 1396 1402 Design of Orthonormal Wavelet
More informationSTABILITY AND LINEAR INDEPENDENCE ASSOCIATED WITH SCALING VECTORS. JIANZHONG WANG y
STABILITY AND LINEAR INDEPENDENCE ASSOCIATED WITH SCALING VECTORS JIANZHONG WANG y Abstract. In this paper, we discuss stability and linear independence of the integer translates of a scaling vector =
More informationAn Introduction to Filterbank Frames
An Introduction to Filterbank Frames Brody Dylan Johnson St. Louis University October 19, 2010 Brody Dylan Johnson (St. Louis University) An Introduction to Filterbank Frames October 19, 2010 1 / 34 Overview
More informationTwo-Dimensional Orthogonal Filter Banks with Directional Vanishing Moments
Two-imensional Orthogonal Filter Banks with irectional Vanishing Moments Jianping Zhou and Minh N. o epartment of Electrical and Computer Engineering University of Illinois at Urbana-Champaign, Urbana,
More informationLapped Unimodular Transform and Its Factorization
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 50, NO 11, NOVEMBER 2002 2695 Lapped Unimodular Transform and Its Factorization See-May Phoong, Member, IEEE, and Yuan-Pei Lin, Member, IEEE Abstract Two types
More informationBasic Multi-rate Operations: Decimation and Interpolation
1 Basic Multirate Operations 2 Interconnection of Building Blocks 1.1 Decimation and Interpolation 1.2 Digital Filter Banks Basic Multi-rate Operations: Decimation and Interpolation Building blocks for
More informationTHEORY OF MIMO BIORTHOGONAL PARTNERS AND THEIR APPLICATION IN CHANNEL EQUALIZATION. Bojan Vrcelj and P. P. Vaidyanathan
THEORY OF MIMO BIORTHOGONAL PARTNERS AND THEIR APPLICATION IN CHANNEL EQUALIZATION Bojan Vrcelj and P P Vaidyanathan Dept of Electrical Engr 136-93, Caltech, Pasadena, CA 91125, USA E-mail: bojan@systemscaltechedu,
More informationINFINITUDE OF MINIMALLY SUPPORTED TOTALLY INTERPOLATING BIORTHOGONAL MULTIWAVELET SYSTEMS WITH LOW APPROXIMATION ORDERS. Youngwoo Choi and Jaewon Jung
Korean J. Math. (0) No. pp. 7 6 http://dx.doi.org/0.68/kjm.0...7 INFINITUDE OF MINIMALLY SUPPORTED TOTALLY INTERPOLATING BIORTHOGONAL MULTIWAVELET SYSTEMS WITH LOW APPROXIMATION ORDERS Youngwoo Choi and
More information446 SCIENCE IN CHINA (Series F) Vol. 46 introduced in refs. [6, ]. Based on this inequality, we add normalization condition, symmetric conditions and
Vol. 46 No. 6 SCIENCE IN CHINA (Series F) December 003 Construction for a class of smooth wavelet tight frames PENG Lizhong (Λ Π) & WANG Haihui (Ξ ) LMAM, School of Mathematical Sciences, Peking University,
More informationClosed-Form Design of Maximally Flat IIR Half-Band Filters
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 49, NO. 6, JUNE 2002 409 Closed-Form Design of Maximally Flat IIR Half-B Filters Xi Zhang, Senior Member, IEEE,
More informationDesign of Biorthogonal FIR Linear Phase Filter Banks with Structurally Perfect Reconstruction
Electronics and Communications in Japan, Part 3, Vol. 82, No. 1, 1999 Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J81-A, No. 1, January 1998, pp. 17 23 Design of Biorthogonal FIR Linear
More informationConstruction of orthogonal multiscaling functions and multiwavelets with higher approximation order based on the matrix extension algorithm
J. Math. Kyoto Univ. (JMKYAZ 6- (6, 75 9 Construction of orthogonal multiscaling functions and multiwavelets with higher approximation order based on the matrix extension algorithm By Shouzhi Yang and
More informationMulti-rate Signal Processing 7. M-channel Maximally Decmiated Filter Banks
Multi-rate Signal Processing 7. M-channel Maximally Decmiated Filter Banks Electrical & Computer Engineering University of Maryland, College Park Acknowledgment: ENEE630 slides were based on class notes
More informationLow-delay perfect reconstruction two-channel FIR/IIR filter banks and wavelet bases with SOPOT coefficients
Title Low-delay perfect reconstruction two-channel FIR/IIR filter banks and wavelet bases with SOPOT coefficients Author(s) Liu, W; Chan, SC; Ho, KL Citation Icassp, Ieee International Conference On Acoustics,
More informationA Higher-Density Discrete Wavelet Transform
A Higher-Density Discrete Wavelet Transform Ivan W. Selesnick Abstract In this paper, we describe a new set of dyadic wavelet frames with three generators, ψ i (t), i =,, 3. The construction is simple,
More informationFast Wavelet/Framelet Transform for Signal/Image Processing.
Fast Wavelet/Framelet Transform for Signal/Image Processing. The following is based on book manuscript: B. Han, Framelets Wavelets: Algorithms, Analysis Applications. To introduce a discrete framelet transform,
More informationQuadrature-Mirror Filter Bank
Quadrature-Mirror Filter Bank In many applications, a discrete-time signal x[n] is split into a number of subband signals { v k [ n]} by means of an analysis filter bank The subband signals are then processed
More informationLecture 16: Multiresolution Image Analysis
Lecture 16: Multiresolution Image Analysis Harvey Rhody Chester F. Carlson Center for Imaging Science Rochester Institute of Technology rhody@cis.rit.edu November 9, 2004 Abstract Multiresolution analysis
More informationQuadrature Prefilters for the Discrete Wavelet Transform. Bruce R. Johnson. James L. Kinsey. Abstract
Quadrature Prefilters for the Discrete Wavelet Transform Bruce R. Johnson James L. Kinsey Abstract Discrepancies between the Discrete Wavelet Transform and the coefficients of the Wavelet Series are known
More informationONE-DIMENSIONAL (1-D) two-channel FIR perfect-reconstruction
3542 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 55, NO. 11, DECEMBER 2008 Eigenfilter Approach to the Design of One-Dimensional and Multidimensional Two-Channel Linear-Phase FIR
More informationSDP APPROXIMATION OF THE HALF DELAY AND THE DESIGN OF HILBERT PAIRS. Bogdan Dumitrescu
SDP APPROXIMATION OF THE HALF DELAY AND THE DESIGN OF HILBERT PAIRS Bogdan Dumitrescu Tampere International Center for Signal Processing Tampere University of Technology P.O.Box 553, 3311 Tampere, FINLAND
More informationarxiv: v2 [math.fa] 27 Sep 2016
Decomposition of Integral Self-Affine Multi-Tiles Xiaoye Fu and Jean-Pierre Gabardo arxiv:43.335v2 [math.fa] 27 Sep 26 Abstract. In this paper we propose a method to decompose an integral self-affine Z
More informationWavelet Bi-frames with Uniform Symmetry for Curve Multiresolution Processing
Wavelet Bi-frames with Uniform Symmetry for Curve Multiresolution Processing Qingtang Jiang Abstract This paper is about the construction of univariate wavelet bi-frames with each framelet being symmetric.
More informationDESIGN OF ALIAS-FREE LINEAR PHASE QUADRATURE MIRROR FILTER BANKS USING EIGENVALUE-EIGENVECTOR APPROACH
DESIGN OF ALIAS-FREE LINEAR PHASE QUADRATURE MIRROR FILTER BANKS USING EIGENVALUE-EIGENVECTOR APPROACH ABSTRACT S. K. Agrawal 1# and O. P. Sahu 1& Electronics and Communication Engineering Department,
More informationCOMPACTLY SUPPORTED ORTHONORMAL COMPLEX WAVELETS WITH DILATION 4 AND SYMMETRY
COMPACTLY SUPPORTED ORTHONORMAL COMPLEX WAVELETS WITH DILATION 4 AND SYMMETRY BIN HAN AND HUI JI Abstract. In this paper, we provide a family of compactly supported orthonormal complex wavelets with dilation
More informationOn the reduction of matrix polynomials to Hessenberg form
Electronic Journal of Linear Algebra Volume 3 Volume 3: (26) Article 24 26 On the reduction of matrix polynomials to Hessenberg form Thomas R. Cameron Washington State University, tcameron@math.wsu.edu
More informationTWO VARIATIONS OF A THEOREM OF KRONECKER
TWO VARIATIONS OF A THEOREM OF KRONECKER ARTŪRAS DUBICKAS AND CHRIS SMYTH ABSTRACT. We present two variations of Kronecker s classical result that every nonzero algebraic integer that lies with its conjugates
More informationVECTORIZED signals are often considered to be rich if
1104 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 54, NO 3, MARCH 2006 Theoretical Issues on LTI Systems That Preserve Signal Richness Borching Su, Student Member, IEEE, and P P Vaidyanathan, Fellow, IEEE
More informationENEE630 ADSP RECITATION 5 w/ solution Ver
ENEE630 ADSP RECITATION 5 w/ solution Ver20209 Consider the structures shown in Fig RI, with input transforms and filter responses as indicated Sketch the quantities Y 0 (e jω ) and Y (e jω ) Figure RI:
More informationAN ALGORITHM FOR MATRIX EXTENSION AND WAVELET CONSTRUCTION
MATHEMATICS OF COMPUTATION Volume 65, Number 14 April 1996, Pages 73 737 AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET CONSTRUCTION W LAWTON, S L LEE AND ZUOWEI SHEN Abstract This paper gives a practical
More informationConstruction of Multivariate Compactly Supported Orthonormal Wavelets
Construction of Multivariate Compactly Supported Orthonormal Wavelets Ming-Jun Lai Department of Mathematics The University of Georgia Athens, GA 30602 April 30, 2004 Dedicated to Professor Charles A.
More informationModule 4 MULTI- RESOLUTION ANALYSIS. Version 2 ECE IIT, Kharagpur
Module MULTI- RESOLUTION ANALYSIS Version ECE IIT, Kharagpur Lesson Multi-resolution Analysis: Theory of Subband Coding Version ECE IIT, Kharagpur Instructional Objectives At the end of this lesson, the
More informationREGULARITY AND CONSTRUCTION OF BOUNDARY MULTIWAVELETS
REGULARITY AND CONSTRUCTION OF BOUNDARY MULTIWAVELETS FRITZ KEINERT Abstract. The conventional way of constructing boundary functions for wavelets on a finite interval is to form linear combinations of
More informationMulti-rate Signal Processing 3. The Polyphase Representation
Multi-rate Signal Processing 3. The Polyphase Representation Electrical & Computer Engineering University of Maryland, College Park Acknowledgment: ENEE630 slides were based on class notes developed by
More information1 The Continuous Wavelet Transform The continuous wavelet transform (CWT) Discretisation of the CWT... 2
Contents 1 The Continuous Wavelet Transform 1 1.1 The continuous wavelet transform (CWT)............. 1 1. Discretisation of the CWT...................... Stationary wavelet transform or redundant wavelet
More information! Downsampling/Upsampling. ! Practical Interpolation. ! Non-integer Resampling. ! Multi-Rate Processing. " Interchanging Operations
Lecture Outline ESE 531: Digital Signal Processing Lec 10: February 14th, 2017 Practical and Non-integer Sampling, Multirate Sampling! Downsampling/! Practical Interpolation! Non-integer Resampling! Multi-Rate
More informationFilter Banks II. Prof. Dr.-Ing. G. Schuller. Fraunhofer IDMT & Ilmenau University of Technology Ilmenau, Germany
Filter Banks II Prof. Dr.-Ing. G. Schuller Fraunhofer IDMT & Ilmenau University of Technology Ilmenau, Germany Page Modulated Filter Banks Extending the DCT The DCT IV transform can be seen as modulated
More informationMultidimensional Orthogonal Filter Bank Characterization and Design Using the Cayley Transform
IEEE TRANSACTIONS ON IMAGE PROCESSING Multidimensional Orthogonal Filter Bank Characterization and Design Using the Cayley Transform Jianping Zhou, Minh N Do, Member, IEEE, and Jelena Kovačević, Fellow,
More informationEhrhart polynomial for lattice squares, cubes, and hypercubes
Ehrhart polynomial for lattice squares, cubes, and hypercubes Eugen J. Ionascu UWG, REU, July 10th, 2015 math@ejionascu.ro, www.ejionascu.ro 1 Abstract We are investigating the problem of constructing
More informationc 1999 Society for Industrial and Applied Mathematics
SIAM J. MATH. ANAL. Vol. 30, No. 3, pp. 678 697 c 999 Society for Industrial and Applied Mathematics ARBITRARILY SMOOTH ORTHOGONAL NONSEPARABLE WAVELETS IN R EUGENE BELOGAY AND YANG WANG Abstract. For
More informationBin Han Department of Mathematical Sciences University of Alberta Edmonton, Canada T6G 2G1
SYMMETRIC ORTHONORMAL SCALING FUNCTIONS AND WAVELETS WITH DILATION FACTOR d = Bin Han Department of Mathematical Sciences University of Alberta Edmonton, Canada T6G 2G1 email: bhan@math.ualberta.ca Abstract.
More informationMULTIRATE DIGITAL SIGNAL PROCESSING
MULTIRATE DIGITAL SIGNAL PROCESSING Signal processing can be enhanced by changing sampling rate: Up-sampling before D/A conversion in order to relax requirements of analog antialiasing filter. Cf. audio
More informationLifting Parameterisation of the 9/7 Wavelet Filter Bank and its Application in Lossless Image Compression
Lifting Parameterisation of the 9/7 Wavelet Filter Bank and its Application in Lossless Image Compression TILO STRUTZ Deutsche Telekom AG, Hochschule für Telekommunikation Institute of Communications Engineering
More informationParametrizing orthonormal wavelets by moments
Parametrizing orthonormal wavelets by moments 2 1.5 1 0.5 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0 1 2 3 4 5 Georg Regensburger Johann Radon Institute for Computational and
More informationaxioms Construction of Multiwavelets on an Interval Axioms 2013, 2, ; doi: /axioms ISSN
Axioms 2013, 2, 122-141; doi:10.3390/axioms2020122 Article OPEN ACCESS axioms ISSN 2075-1680 www.mdpi.com/journal/axioms Construction of Multiwavelets on an Interval Ahmet Altürk 1 and Fritz Keinert 2,
More information4214 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 11, NOVEMBER 2006
4214 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 11, NOVEMBER 2006 Closed-Form Design of Generalized Maxflat R-Regular FIR M th-band Filters Using Waveform Moments Xi Zhang, Senior Member, IEEE,
More informationCourse and Wavelets and Filter Banks. Filter Banks (contd.): perfect reconstruction; halfband filters and possible factorizations.
Course 18.327 and 1.130 Wavelets and Filter Banks Filter Banks (contd.): perfect reconstruction; halfband filters and possible factorizations. Product Filter Example: Product filter of degree 6 P 0 (z)
More informationFilter Banks with Variable System Delay. Georgia Institute of Technology. Atlanta, GA Abstract
A General Formulation for Modulated Perfect Reconstruction Filter Banks with Variable System Delay Gerald Schuller and Mark J T Smith Digital Signal Processing Laboratory School of Electrical Engineering
More informationApplied and Computational Harmonic Analysis 11, (2001) doi: /acha , available online at
Applied and Computational Harmonic Analysis 11 305 31 (001 doi:10.1006/acha.001.0355 available online at http://www.idealibrary.com on LETTER TO THE EDITOR Construction of Multivariate Tight Frames via
More informationInfinite Sequences, Series Convergence and the Discrete Time Fourier Transform over Finite Fields
Infinite Sequences, Series Convergence and the Discrete Time Fourier Transform over Finite Fields R M Campello de Souza M M Campello de Souza H M de Oliveira M M Vasconcelos Depto de Eletrônica e Sistemas,
More informationA new interpretation of the integer and real WZ factorization using block scaled ABS algorithms
STATISTICS,OPTIMIZATION AND INFORMATION COMPUTING Stat., Optim. Inf. Comput., Vol. 2, September 2014, pp 243 256. Published online in International Academic Press (www.iapress.org) A new interpretation
More informationLecture notes: Algorithms for integers, polynomials (Thorsten Theobald)
Lecture notes: Algorithms for integers, polynomials (Thorsten Theobald) 1 Euclid s Algorithm Euclid s Algorithm for computing the greatest common divisor belongs to the oldest known computing procedures
More informationGroups and Symmetries
Groups and Symmetries Definition: Symmetry A symmetry of a shape is a rigid motion that takes vertices to vertices, edges to edges. Note: A rigid motion preserves angles and distances. Definition: Group
More informationON FRAMES WITH STABLE OVERSAMPLED FILTER BANKS
ON FRAMES WITH STABLE OVERSAMPLED FILTER BANKS Li Chai,, Jingxin Zhang, Cishen Zhang and Edoardo Mosca Department of Electrical and Computer Systems Engineering Monash University, Clayton, VIC3800, Australia
More informationName: Solutions Final Exam
Instructions. Answer each of the questions on your own paper. Be sure to show your work so that partial credit can be adequately assessed. Put your name on each page of your paper. 1. [10 Points] All of
More informationHomework #2 solutions Due: June 15, 2012
All of the following exercises are based on the material in the handout on integers found on the class website. 1. Find d = gcd(475, 385) and express it as a linear combination of 475 and 385. That is
More informationPOINT VALUES AND NORMALIZATION OF TWO-DIRECTION MULTIWAVELETS AND THEIR DERIVATIVES
November 1, 1 POINT VALUES AND NORMALIZATION OF TWO-DIRECTION MULTIWAVELETS AND THEIR DERIVATIVES FRITZ KEINERT AND SOON-GEOL KWON,1 Abstract Two-direction multiscaling functions φ and two-direction multiwavelets
More information3-D Directional Filter Banks and Surfacelets INVITED
-D Directional Filter Bans and Surfacelets INVITED Yue Lu and Minh N. Do Department of Electrical and Computer Engineering Coordinated Science Laboratory University of Illinois at Urbana-Champaign, Urbana
More informationNotes on n-d Polynomial Matrix Factorizations
Multidimensional Systems and Signal Processing, 10, 379 393 (1999) c 1999 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. Notes on n-d Polynomial Matrix Factorizations ZHIPING LIN
More informationImage Denoising using Uniform Curvelet Transform and Complex Gaussian Scale Mixture
EE 5359 Multimedia Processing Project Report Image Denoising using Uniform Curvelet Transform and Complex Gaussian Scale Mixture By An Vo ISTRUCTOR: Dr. K. R. Rao Summer 008 Image Denoising using Uniform
More informationTitle Perfect reconstruction modulated filter banks with sum of powers-of-two coefficients Author(s) Chan, SC; Liu, W; Ho, KL Citation IEEE International Symposium on Circuits and Systems Proceedings,
More informationMultirate signal processing
Multirate signal processing Discrete-time systems with different sampling rates at various parts of the system are called multirate systems. The need for such systems arises in many applications, including
More informationLEGENDRE S THEOREM, LEGRANGE S DESCENT
LEGENDRE S THEOREM, LEGRANGE S DESCENT SUPPLEMENT FOR MATH 370: NUMBER THEORY Abstract. Legendre gave simple necessary and sufficient conditions for the solvablility of the diophantine equation ax 2 +
More informationSOLUTIONS OF NONHOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS WITH EXCEPTIONALLY FEW ZEROS
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 23, 1998, 429 452 SOLUTIONS OF NONHOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS WITH EXCEPTIONALLY FEW ZEROS Gary G. Gundersen, Enid M. Steinbart, and
More informationCitation Ieee Signal Processing Letters, 2001, v. 8 n. 6, p
Title Multiplierless perfect reconstruction modulated filter banks with sum-of-powers-of-two coefficients Author(s) Chan, SC; Liu, W; Ho, KL Citation Ieee Signal Processing Letters, 2001, v. 8 n. 6, p.
More informationEQUIVALENCE OF DFT FILTER BANKS AND GABOR EXPANSIONS. Helmut Bolcskei and Franz Hlawatsch
SPIE Proc. Vol. 2569 \Wavelet Applications in Signal and Image Processing III" San Diego, CA, July 995. EQUIVALENCE OF DFT FILTER BANKS AND GABOR EPANSIONS Helmut Bolcskei and Franz Hlawatsch INTHFT, Technische
More informationVARIOUS types of wavelet transform are available for
IEEE TRANSACTIONS ON SIGNAL PROCESSING A Higher-Density Discrete Wavelet Transform Ivan W. Selesnick, Member, IEEE Abstract This paper describes a new set of dyadic wavelet frames with two generators.
More informationGRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory.
GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. Linear Algebra Standard matrix manipulation to compute the kernel, intersection of subspaces, column spaces,
More informationThe Application of Legendre Multiwavelet Functions in Image Compression
Journal of Modern Applied Statistical Methods Volume 5 Issue 2 Article 3 --206 The Application of Legendre Multiwavelet Functions in Image Compression Elham Hashemizadeh Department of Mathematics, Karaj
More informationFast Transforms: Banded Matrices with Banded Inverses
Fast Transforms: Banded Matrices with Banded Inverses 1. Introduction Gilbert Strang, MIT An invertible transform y = Ax expresses the vector x in a new basis. The inverse transform x = A 1 y reconstructs
More informationQuivers of Period 2. Mariya Sardarli Max Wimberley Heyi Zhu. November 26, 2014
Quivers of Period 2 Mariya Sardarli Max Wimberley Heyi Zhu ovember 26, 2014 Abstract A quiver with vertices labeled from 1,..., n is said to have period 2 if the quiver obtained by mutating at 1 and then
More informationOctober 7, :8 WSPC/WS-IJWMIP paper. Polynomial functions are renable
International Journal of Wavelets, Multiresolution and Information Processing c World Scientic Publishing Company Polynomial functions are renable Henning Thielemann Institut für Informatik Martin-Luther-Universität
More informationPRODUCT OF OPERATORS AND NUMERICAL RANGE
PRODUCT OF OPERATORS AND NUMERICAL RANGE MAO-TING CHIEN 1, HWA-LONG GAU 2, CHI-KWONG LI 3, MING-CHENG TSAI 4, KUO-ZHONG WANG 5 Abstract. We show that a bounded linear operator A B(H) is a multiple of a
More informationa b (mod m) : m b a with a,b,c,d real and ad bc 0 forms a group, again under the composition as operation.
Homework for UTK M351 Algebra I Fall 2013, Jochen Denzler, MWF 10:10 11:00 Each part separately graded on a [0/1/2] scale. Problem 1: Recalling the field axioms from class, prove for any field F (i.e.,
More informationResearch Article Minor Prime Factorization for n-d Polynomial Matrices over Arbitrary Coefficient Field
Complexity, Article ID 6235649, 9 pages https://doi.org/10.1155/2018/6235649 Research Article Minor Prime Factorization for n-d Polynomial Matrices over Arbitrary Coefficient Field Jinwang Liu, Dongmei
More informationECEN 5022 Cryptography
Elementary Algebra and Number Theory University of Colorado Spring 2008 Divisibility, Primes Definition. N denotes the set {1, 2, 3,...} of natural numbers and Z denotes the set of integers {..., 2, 1,
More informationGröbner bases and wavelet design
Journal of Symbolic Computation 37 (24) 227 259 www.elsevier.com/locate/jsc Gröbner bases and wavelet design Jérôme Lebrun a,, Ivan Selesnick b a Laboratoire I3S, CNRS/UNSA, Sophia Antipolis, France b
More informationMath 121 Homework 5: Notes on Selected Problems
Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements
More informationTWO DIGITAL filters and are said to be
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 5, MAY 2001 1013 Biorthogonal Partners and Applications P. P. Vaidyanathan, Fellow, IEEE, and Bojan Vrcelj, Student Member, IEEE Abstract Two digital
More informationAlgebra Exam Syllabus
Algebra Exam Syllabus The Algebra comprehensive exam covers four broad areas of algebra: (1) Groups; (2) Rings; (3) Modules; and (4) Linear Algebra. These topics are all covered in the first semester graduate
More informationCOMPLEX WAVELET TRANSFORM IN SIGNAL AND IMAGE ANALYSIS
COMPLEX WAVELET TRANSFORM IN SIGNAL AND IMAGE ANALYSIS MUSOKO VICTOR, PROCHÁZKA ALEŠ Institute of Chemical Technology, Department of Computing and Control Engineering Technická 905, 66 8 Prague 6, Cech
More information2D Wavelets for Different Sampling Grids and the Lifting Scheme
D Wavelets for Different Sampling Grids and the Lifting Scheme Miroslav Vrankić University of Zagreb, Croatia Presented by: Atanas Gotchev Lecture Outline 1D wavelets and FWT D separable wavelets D nonseparable
More informationPARAMETRIC OPTIMIZATION OF BIORTHOGONAL WAVELETS AND FILTERBANKS VIA PSEUDOFRAMES FOR SUBSPACES
PARAMETRIC OPTIMIZATION OF BIORTHOGONAL WAVELETS AND FILTERBANKS VIA PSEUDOFRAMES FOR SUBSPACES SHIDONG LI AND MICHAEL HOFFMAN Abstract. We present constructions of biorthogonal wavelets and associated
More informationDecomposition of Riesz frames and wavelets into a finite union of linearly independent sets
Decomposition of Riesz frames and wavelets into a finite union of linearly independent sets Ole Christensen, Alexander M. Lindner Abstract We characterize Riesz frames and prove that every Riesz frame
More informationMEROMORPHIC FUNCTIONS AND ALSO THEIR FIRST TWO DERIVATIVES HAVE THE SAME ZEROS
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 30, 2005, 205 28 MEROMORPHIC FUNCTIONS AND ALSO THEIR FIRST TWO DERIVATIVES HAVE THE SAME ZEROS Lian-Zhong Yang Shandong University, School of Mathematics
More informationCHARACTERIZATIONS OF LINEAR DIFFERENTIAL SYSTEMS WITH A REGULAR SINGULAR POINT
CHARACTERIZATIONS OF LINEAR DIFFERENTIAL SYSTEMS WITH A REGULAR SINGULAR POINT The linear differential system by W. A. HARRIS, Jr. (Received 25th July 1971) -T = Az)w (1) where w is a vector with n components
More informationM. VAN BAREL Department of Computing Science, K.U.Leuven, Celestijnenlaan 200A, B-3001 Heverlee, Belgium
MATRIX RATIONAL INTERPOLATION WITH POLES AS INTERPOLATION POINTS M. VAN BAREL Department of Computing Science, K.U.Leuven, Celestijnenlaan 200A, B-3001 Heverlee, Belgium B. BECKERMANN Institut für Angewandte
More informationAnalysis Of Ill-Conditioning Of Multi-Channel Deconvolution Problems
Analysis Of Ill-Conditioning Of Multi-Channel Deconvolution Problems Ole Kirkeby, Per Rubak, and Angelo Farina * Department of Communication Techology, Fredrik Bajers Vej 7, Aalborg University, DK-9220
More informationFRACTIONAL BIORTHOGONAL PARTNERS IN CHANNEL EQUALIZATION AND SIGNAL INTERPOLATION
FRACTIONA BIORTHOGONA PARTNERS IN CHANNE EQUAIZATION AND SIGNA INTERPOATION Bojan Vrcelj and P. P. Vaidyanathan Contact Author: P. P. Vaidyanathan, Department of Electrical Engineering 136-93, California
More informationCompression on the digital unit sphere
16th Conference on Applied Mathematics, Univ. of Central Oklahoma, Electronic Journal of Differential Equations, Conf. 07, 001, pp. 1 4. ISSN: 107-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
More informationINTEGRATION WORKSHOP 2003 COMPLEX ANALYSIS EXERCISES
INTEGRATION WORKSHOP 23 COMPLEX ANALYSIS EXERCISES DOUGLAS ULMER 1. Meromorphic functions on the Riemann sphere It s often useful to allow functions to take the value. This exercise outlines one way to
More informationStationarity condition for. Fractional sampling filters
INDIAN INSTITUTE OF TECHNOLOGY DELHI Stationarity condition for Fractional sampling filters by Pushpendre Rastogi Report submitted in fulfillment of the requirement of the degree of Masters of Technology
More informationPerfect Reconstruction Two- Channel FIR Filter Banks
Perfect Reconstruction Two- Channel FIR Filter Banks A perfect reconstruction two-channel FIR filter bank with linear-phase FIR filters can be designed if the power-complementary requirement e jω + e jω
More information