Table Of ContentFACTORING PSEUDOIDENTITY MATRIX PAIRS
FLORIAN M. SEBERT AND YI MING ZOU
1
1
0
2
n Abstract. Theproblemoffactorizationandparametrizationofcompactly
a supportedbiorthogonalwaveletswasreducedtothatofpseudoidentityma-
J
trix pairs by Resnikoff, Tian, and Wells in their 2001 paper. Based on
9
a conjecture on the pseudoidentity matrix pairs of rank 2 stated in the
1
same paper, they proved a theorem which gives a complete factorization
] result for rank 2 compactly supported biorthogonal wavelets. In this pa-
A per, wefirstprovideexamplesto showthatthe conjectureis nottrue,then
F we prove a factorization theorem for pseudoidentity matrix pairs of rank
. m≥2. As a consequence, our resultshows that a slightly modified version
h
t of the factorization theorem in the rank 2 case given by Resnikoff, Tian,
a
and Wells holds. We also provide a concrete constructive method for the
m
rank 2 case which is determined by applying the Euclidean algorithm to
[
two polynomials.
1 Key words: BiorthogonalWavelets, Pseudoidentity Matrix Pairs,Fac-
v
torization, Vanishing Moment
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9 AMS subject classifications: 15A23, 42C40
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1. 1. Introduction
0
1 It is well-known that compactly supported orthogonal wavelets cannot be
1 symmetric except for Haar wavelets [5, 11, 13]. In signal processing, the sym-
:
v metry property corresponds to the linear phase condition, and since biorthogo-
i
X nal wavelets permit symmetry, they are preferred over the orthogonal wavelets
r in applications such as digital filter banks. Because of the flexibility of the
a
biorthogonal condition [2, 3, 4, 5, 6, 7, 9, 13], one can add other conditions to
the definition of a biorthogonal wavelet according to the needs. To guarantee
the existence of scaling functions and wavelet functions, a zeroth-order van-
ishing moment condition was imposed on the biorthogonal wavelet matrix in
addition to the usual quadratic condition in [11]. This extra requirement must
be taken into account when analyzing the structure of these wavelets. In par-
ticular,thefactorizationresultsfromtheearlierinvestigations[2,3,6,9,10,12]
do not apply to these wavelets. It was proved in [11] that any biorthogonal
F. Sebert is with d-fine consulting in Frankfurt, email: [email protected]. Y. M. Zou
is with the Department of Mathematical Sciences, University of Wisconsin, Milwaukee, WI
53201,USA, email: [email protected].
1
2 FLORIANM. SEBERT ANDYI MING ZOU
wavelet matrix pair can be decomposed into four components: an orthogonal
component, a pseudoidentity matrix pair (defined in [11]), an invertible ma-
trix, and a constant matrix. Since the factorization and parametrization of the
orthogonal wavelet matrices is well understood [11, 12, 13], the factorization
of a biorthogonal wavelet matrix pair reduces to that of a pseudoidentity ma-
trix pair. To obtain a complete factorization and therefore a parametrization
of the biorthogonal wavelets, a conjecture was proposed for the pseudoiden-
tity matrix pairs of rank 2 in [11], and based on this conjecture, a complete
factorization theorem was proved for the rank 2 case therein.
In this paper, we prove a factorization theorem for a pseudoidentity matrix
pair of rank m ≥ 2, which in particular implies that the factorization theorem
for the rank 2 case stated in [11] is true (with a slightly modified statement)
in spite of the invalidness of the conjecture proposed there. We first provide
examples to show that the conjecture is not true and then proceed to prove a
factorization theorem. Our proof is elementary and constructive: it applies to
any rank and provides an algorithm for the decomposition.
We organize our presentation as follows. In section 2, we first recall the
definition and the basic properties of biorthogonal wavelets as given in [11],
then we discuss some counterexamples to the conjecture stated in [11]. In
Section 3, we prove our factorization result on pseudoidentity matrix pairs and
thus obtain a complete factorization for the biorthogonal wavelet matrix pairs.
In section 4, we consider the rank 2 case in some detail and show that in this
case, the factorization of a pseudoidentity matrix pair is basically determined
by the Euclidean algorithm for finding the greatest common divisor (GCD) of
two polynomials. We conclude the paper in section 5 with some discussion.
2. Biorthogonal wavelets
Let C be the set of complex numbers, let F ⊆ C be a subfield invariant
under the complex conjugation, and let R = F[z,z−1] be the ring of Laurent
polynomials over F. The reason for us to work with F instead of just C is
that it will allow more flexibility in the applications, since the discussions here
apply to a whole range of fields including the rational numbers Q, the real
numbers R, the complex numbers C, as well as other algebraic extensions of
Q. For instance, in some applications of digital filters, it is desirable to work
with rational numbers, since it is easier to implement the filters. We denote
by gl(m,R) the set of all m×m matrices over R and denote by GL(m,R) the
set of all invertible m×m matrices over R. We also denote by gl(m,F) and
GL(m,F) the set of m×m matrices and the set of invertible m×m matrices
FACTORING PSEUDOIDENTITY MATRIX PAIRS 3
over F, respectively. For a matrix A(z) ∈ gl(m,R), we can write
k1
(2.1) A(z) = A z−k = A z−k,
k k
Xk∈Z kX=k0
where A ∈ gl(m,F),k ≤ k ≤ k , and k and k are the smallest and the
k 0 1 0 1
largest indices such that A 6= 0, respectively. We call the expression of A(z)
k
in (2.1) the Laurent series of A(z). The number g = k −k +1 is called the
1 0
genus of A(z). If A(z) has genus g, then we can identify A(z) with the block
matrix
(2.2) A := (A ,...,A ,...,A ),
k0 0 k1
where
(2.3) A = (a ), 0 ≤ i ≤ m−1,0 ≤ j ≤ m−1,k ≤ k ≤ k ,
k i,km+j 0 1
and say that A(z) has size m×mg. We define the adjoint A˜(z) of A(z) by
k1 −k0
(2.4) A˜(z) = A∗(z−1) = A∗zk = A∗ z−k,
k −k
kX=k0 kX=−k1
where A∗ denotes the conjugate transpose of the complex matrix A .
k k
We are now ready to recall the definition of a biorthogonal wavelet matrix
pair and the definition of a pseudoidentity matrix pair from [11].
Definition 2.1. A pair of m × mg matrices (L = (l ),R = (r )) (as in
i,j i,j
(2.2)) is said to be a biorthogonal wavelet matrix pair of rank m and genus g
if the following conditions are satisfied:
˜
(2.5) L(z)R(z) = mI ,
m
and
m, if i = 0,
(2.6) l = r =
i,j i,j (cid:26) 0, if 1 ≤ i ≤ m−1,
Xj Xj
where I is the m×m identity matrix.
m
Intheliterature, thematrixL(z)iscalledtheanalysismatrix andthematrix
R(z) is called the synthesis matrix of the biorthogonal wavelet pair. Condition
(2.5) is called the quadratic or perfect reconstruction condition and condition
(2.6) is called the linear condition. Note that (2.6), which is also referred to as
the zeroth-order vanishing moment condition, is a necessary condition for the
existence of scaling functions and wavelet functions. This is one of the main
differences between wavelets and perfect reconstruction filter banks.
4 FLORIANM. SEBERT ANDYI MING ZOU
Definition 2.2. A matrix pair (C(z),D(z)) is called a pseudoidentity matrix
pair if
kc 0
C(z) = C z−k, D(z) = D z−k, (k ≤ 0 ≤ k )
k k d c
(2.7) Xk=0 kX=kd
C(z)D˜(z) = I , C(1) = D(1) = I .
m m
We remark that it follows from the definition that
(2.8) det(C(z)) = det(D(z)) = 1.
For some other properties of the pseudo identity matrix pairs, please see [11].
Recall that a primitive paraunitary matrix is an m×2m matrix of the form
(2.9) V(z) = I −vv∗ +vv∗z−1,
m
where v ∈ Fm is a unit column vector. Note that a primitive paraunitary
˜
matrix satisfies V(z)V(z) = I
m
The following theorem, due to Resnikoff, Tian, and Wells, reduces the fac-
torization of a biorthogonal wavelet matrix pair to that of a pseudoidentity
matrix pair.
Theorem 2.1. (see [11, Thm. 3.4]) A pair of m× mg matrices (L,R) is a
biorthogonal wavelet pair of rank m if and only if there exist primitive parau-
nitary matrices V ,...,V , d ≥ 0, such that
1 d
1 0
L(z) = z−k0V (z)···V (z)C(z) H,
1 d (cid:18) 0 G (cid:19)
(2.10)
1 0
R(z) = z−k0V (z)···V (z)D(z) H,
1 d (cid:18) 0 (G−1)∗ (cid:19)
where k ∈ Z, d = b−mk , b is the exponent of detL(z), G ∈ GL(m−1,F), H
0 0
is the canonical Haar matrix of rank m, and (C(z),D(z)) is a pseudoidentity
matrix pair such that the genus of C(z) is ≤ g.
We consider an example of pseudoidentity matrix pairs.
Example 2.1. For any sequence of nonzero numbers
(2.11) a ,a ,...,a ∈ F,
1 2 k
let 0 < m < ··· < m be k positive integers, let
1 k
k k
(2.12) a = a , u(z) = a z−mi,
0 i i
Xi=1 Xi=1
FACTORING PSEUDOIDENTITY MATRIX PAIRS 5
let
k
1−a a a −a
(2.13) C(z) = 0 0 + i i z−mi,
(cid:18) −a0 1+a0 (cid:19) Xi=1 (cid:18) ai −ai (cid:19)
and let
k
1+a¯ a¯ −a¯ −a¯
(2.14) D(z) = 0 0 + i i zmi,
(cid:18) −a¯0 1−a¯0 (cid:19) Xi=1 (cid:18) a¯i a¯i (cid:19)
where a¯ is the complex conjugate of a . Then
i i
C(1) = D(1) = I , C(z)D˜(z) = I ,
2 2
(2.15)
det(C(z)) = det(D(z)) = 1.
The first equality follows from the definition, and the second and the third
equalities can be easily seen since
1−a +u(z) a −u(z)
(2.16) C(z) = 0 0
(cid:18) −a +u(z) 1+a −u(z) (cid:19)
0 0
has inverse
1+a −u(z) −a +u(z)
(2.17) (C(z))−1 = 0 0 ,
(cid:18) a −u(z) 1−a +u(z) (cid:19)
0 0
and since D˜(z) = (C(z))−1.
For instance, if we take k = 2, a = a = 1, and m = 1,m = 2, then
1 2 1 2
−1 2 1 −1 1 −1
(2.18) C(z) = + z−1 + z−2,
(cid:18) −2 3 (cid:19) (cid:18) 1 −1 (cid:19) (cid:18) 1 −1 (cid:19)
and
3 2 −1 −1 −1 −1
(2.19) D(z) = + z1 + z2.
(cid:18) −2 −1 (cid:19) (cid:18) 1 1 (cid:19) (cid:18) 1 1 (cid:19)
Remark 2.1. Since
2
a −a
(2.20) i i = 0,
(cid:18) ai −ai (cid:19)
these pseudoidentity matrix pairs provide counterexamples to the following con-
jecture (Conjecture 1) in [11]:
˜
Conjecture: For m = 2 and C(z) and D(z) as in (2.7), if C(z)D(z) = I
2
and p is the smallest positive integer such that C 6= 0, then both C and
kc−p kc−p
D must be invertible matrices.
kd+p
6 FLORIANM. SEBERT ANDYI MING ZOU
Remark 2.2. A weaker condition than the condition stated in the conjecture
that guarantees the factorization of a pseudoidentity matrix pair was also given
in [11] (see the remark after the proof of Thm. 4.4 in [11]). We note that our
examples do not satisfy this weaker condition either, since for the matrices in
(2.18) the condition
−C N +C = 0
kc−p kc
becomes
1 −1 1 −1 1 −1
− N + = (I −N) = 0.
(cid:18) 1 −1 (cid:19) (cid:18) 1 −1 (cid:19) (cid:18) 1 −1 (cid:19) 2
But this last equality cannot be true, since if N is such that N2 = 0, then
I −N is invertible with (I −N)−1 = I +N.
2 2 2
Remark 2.3. We note that the matrix in (2.18) can be factored as
1 0 1 −1 1 −1
C(z) = − + z−1
(cid:18)(cid:18) 0 1 (cid:19) (cid:18) 1 −1 (cid:19) (cid:18) 1 −1 (cid:19) (cid:19)
(2.21)
1 0 1 −1 1 −1
· − + z−2 .
(cid:18)(cid:18) 0 1 (cid:19) (cid:18) 1 −1 (cid:19) (cid:18) 1 −1 (cid:19) (cid:19)
Note also that the degrees do not add up.
3. Complete factorization
We now prove the following theorem.
Theorem 3.1. Let (C(z),D(z)) be a pair of matrices of rank m ≥ 2 defined
by
kc 0
(3.1) C(z) = C z−k, D(z) = D z−k, k ,k > 0.
k k c d
Xk=0 kX=kd
Then (C(z),D(z)) is a pseudoidentity matrix pair if and only if there exist a
positive integer r and nilpotent matrices
N ∈ gl(m,F), N2 = 0, 1 ≤ i ≤ r,
i i
such that
C(z) = L (z)···L (z)L (z), D(z) = R (z)···R (z)R (z),
Nr N2 N1 Nr N2 N1
with
L (z) = I −N +N z−ki, R (z) = I +N∗ −N∗zki, 1 ≤ i ≤ r.
Ni m i i Ni m i i
FACTORING PSEUDOIDENTITY MATRIX PAIRS 7
Proof. Itisclearthattheconditionsaresufficient. Toprovethenecessary part,
note that since D(z) is just the adjoint of the inverse of C(z), we need only
to prove the statement for C(z). To avoid writing the negative exponents,
we make a change of variable (not essential) by letting t = z−1. We abuse
notation and write
kc
(3.2) C(t) = C tk.
k
Xk=0
Since the entries of C(t) are elements of the Euclidean domain F[t], by the
Division algorithm, we can use elementary row and column operations of the
form (type I elementary operations):
add a multiple of a row or column to another row or column,
to reduce C(t) to a diagonal matrix
d
1
d
(3.3) C′(t) = 2 ... ,
d
m
where
(3.4) 0 6= d ∈ F[t], 1 ≤ i ≤ m.
i
This is possible since: (a) we can use elementary row and column operations
to reduce C(t) to the given form (see for example [1, pp. 459-460]–here,
the statement is actually weaker, since we do not require that d /d ); (b)
i i−1
the operations that interchange two rows or columns can be obtained from
type I elementary operations together with scalar multiplications by nonzero
numbers to rows and columns; and, (c) in our diagonal form, we allow the
elements d ,1 ≤ i ≤ m, to be defined up to nonzero scalar multiples (e.g. we
i
do not care about negative signs), and thus multiplication by nonzero numbers
to rows and columns are not needed here.
Writing our reduction process as matrix multiplications, we have
(3.5) C′ := C′(t) = P (t)···P (t)C(t)P′(t)···P′(t),
u 1 s 1
where the P (t),1 ≤ k ≤ u, and P′(t),1 ≤ l ≤ s, are elementary matrices of
k l
the form:
(3.6) I +f(t)E , i 6= j, f(t) ∈ F[t],
m ij
where E is the m×m matrix with 1 at the ijth entry and 0 elsewhere.
ij
Since det(C(t)) = 1 (see (2.8)), by taking determinants on both sides of
(3.5), we conclude that all d ∈ F,1 ≤ i ≤ m, that is, C′ is a constant diagonal
i
matrix.
8 FLORIANM. SEBERT ANDYI MING ZOU
Let
(P (t))−1, if 1 ≤ k ≤ u,
(3.7) E (t) = k
k (cid:26) C′(P′ (t))−1(C′)−1, if u+1 ≤ k ≤ u+s.
k−u
Then all E (t) are elementary matrices of the form (3.6), and
k
(3.8) C(t) = E (t)···E (t)C′.
1 u+s
Consider elementary matrices of the form (3.6). Let
(3.9) E(t) = I +f(t)E , i 6= j,
m ij
with
(3.10) f(t) = a tn +a tn−1 +···+a ∈ F[t].
n n−1 0
We can assume that f(t) 6= 0, since f(t) = 0 gives the identity matrix, and
factor E(t) as
(3.11)E(t) = (I +(a tn −a )E )···(I +(a t−a )E )(I +b E ),
m n n ij m 1 1 ij m 0 ij
where a term exists only if the corresponding coefficient a is not 0, and
k
n
(3.12) b = a .
0 i
Xi=0
Note that the factors in (3.11) involving t can be written as
(3.13) I −a E +a E tk = I −N +Ntk,
m k ij k ij m
where N := a E is nilpotent and N2 = 0. If we conjugate the right hand
k ij
side in (3.13) by G ∈ GL(m,F), we have
(3.14) G(I −N +Ntk)G−1 = I −N′ +N′tk,
m m
with (N′)2 = (GNG−1)2 = 0. Therefore, upon plugging (3.11) and (3.13) into
(3.8) and moving all the constant matrices (e.g. the last term in (3.11)) to the
end, we obtain
(3.15) C(t) = L (t)···L (t)L (t)QC′ for some r > 0,
Nr N2 N1
where the factors L (t) are as defined in (3.13) or (3.14), and Q ∈ GL(m,F).
Ni
Setting t = 1 in (3.15) and noticing that L (1) = I ,1 ≤ i ≤ r, we have
Ni m
QC′ = I . Substituting back z−1 = t, we have proved the theorem. (cid:3)
m
Remark 3.1. Since the factors are formed by nilpotent matrices, we should
not expect that the degrees of the factors add up to the degree of C(z) as (2.21)
shows.
Remark 3.2. In general, the factorization depends on the reduction process
and it is not unique.
FACTORING PSEUDOIDENTITY MATRIX PAIRS 9
Example 3.1. Here we consider a 3×3 example. Let
−z−2 −z−1 +3 −z−2 −z−1 +2 z−2 +z−1 −2
C(z) = 2z−2 +2z−1 −4 2z−2 +2z−1 −3 −2z−2 −2z−1 +4 ,
z−2 +z−1 −2 z−2 +z−1 −2 −z−2 −z−1 +3
1 0 1 1 0 1 1 0 0
E = 0 1 −2 = 0 1 0 0 1 −2 ,
1
0 0 1 0 0 1 0 0 1
1 0 0
E = 0 1 0 ,
2
−z−2 −z−1 +2 0 1
1 0 0
E = 0 1 0 ,
3
0 −z−2 −z−1 +2 1
1 0 −1 1 0 −1 1 0 0
E = 0 1 2 = 0 1 0 0 1 2 .
4
0 0 1 0 0 1 0 0 1
Then
E E E C(z)E = I .
3 2 1 4 4
Thus after moving the E ’s to the right hand side and simplifying, we obtain
i
C(z) = (I −N +N z−1)(I −N +N z−2)(I −N +N z−1)(I −N +N z−2),
4 1 1 4 1 1 4 2 2 4 2 2
where
−1 0 −1 0 −1 2
N = 2 0 2 , N = 0 2 −4 .
1 2
1 0 1 0 1 −2
As a consequence of Thm. 3.1, we have a refinement of Thm. 2.1:
Theorem 3.2. (biorthogonal wavelet factorization theorem) For m ≥ 2, a pair
of m×mg matrices (L,R) is a biorthogonal wavelet pair of rank m if and only
if there exist primitive paraunitary matrices V ,...,V , d ≥ 0, and nilpotent
1 d
matrices
N ∈ gl(m,F), N2 = 0, 1 ≤ i ≤ r,
i i
10 FLORIANM. SEBERT ANDYI MING ZOU
such that
1 0
L(z) = z−k0V (z)···V (z)L (z)···L (z) H,
1 d Nr N1 (cid:18) 0 G (cid:19)
(3.16)
1 0
R(z) = z−k0V (z)···V (z)R (z)···R (z) H,
1 d Nr N1 (cid:18) 0 (G−1)∗ (cid:19)
where k ∈ Z, d = b−mk , b is the exponent of detL(z), G ∈ GL(m−1,F),
0 0
H is the canonical Haar matrix of rank m, and L ,R ,1 ≤ i ≤ r, are as
Ni Ni
defined in Thm. 3.1 such that the genus of
C(z) = L (z)···L (z)
Nr N1
is ≤ g.
4. The rank 2 case
For a rank 2 pseudoidentity matrix pair (C(z),D(z)), the factorization can
be described in a concrete way. We will see that the computational com-
plexity in the factorization is basically determined by the operations involved
in carrying out the Euclidean algorithm to determine the GCD of two poly-
nomials with coefficients in F. Again, to avoid writing the negative expo-
nents, we use the variable t instead of z in our discussion as before. Let
a = a(t),b = b(t),c = c(t),d = d(t) ∈ F[t] be such that
a b
(4.1) C(t) = .
(cid:18) c d (cid:19)
Since C(1) = I , ad 6= 0, and b(1) = c(1) = 0. If one of b and c is 0, then
2
det(C(t)) = 1 implies that a and d are constants and hence a = d = 1. If, say
c = 0, then we can write
b(t) = (b tn −b )+...+(b t−b ),
n n 1 1
and factor C(t) as
C(t) = L (t)···L (t)L (t),
Nn N2 N1
where
0 b 0 b
L (t) = I − i + i ti, 1 ≤ i ≤ n.
Ni 2 (cid:18) 0 0 (cid:19) (cid:18) 0 0 (cid:19)
