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   -> Volume 6, Issue 4


Preprint: New preprints available from wlawton@iss.nus.sg
 
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wlawton@iss.nus.sg (Wayne Lawton)
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PostPosted: Fri Mar 07, 1997 4:58 am    
Subject: Preprint: New preprints available from wlawton@iss.nus.sg
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#3 Preprint: New preprints available from wlawton@iss.nus.sg

1) Title: Construction of Conjugate Quadrature Filters With Specified Zeros
(to appear in Numerical Mathematics)

Authors:

Wayne Lawton
Institute of System Science
National University of Singapore
Heng Mui Keng Terrace, Kent Ridge
Singapore 119597

Charles A. Micchelli
Department of Mathematical Sciences
IBM T. J. Watson Research Center
Yorktown Heights, N.Y. 10598

Abstract: Let C denote the complex numbers and L denote the ring of
complex-valued Laurent polynomial functions on C {0}. Furthermore,
we denote by _R, L_N the subsets of Laurent polynomials whose
restriction to the unit circle is real, nonnegative, respectively. We
prove that for any two Laurent polynomials P_1, P_2 in L_N, which have
no common zeros in C {0}, there exists a pair of Laurent polynomials
Q_1, Q_2 in L_N satisfying the equation Q_1 P_1 + Q_2 P_2 = 1. We
provide some information about the minimal length Laurent polynomials
Q_1 and Q_2 with these properties and describe an algorithm to compute
them. We apply this result to design a conjugate quadrature filter
whose zeros contain an arbitrary finite subset Lambda of C {0} with
the propertythat for any x, y in Lambda, x not equal to y implies x is
not equal to either -y or -1/(conjugate y).

2) Title: Design of Conjugate Quadrature Filters Having Specified Zeros
(to appear in Proceedings ICASSP97)

Authors:

Wayne Lawton
Institute of System Science
National University of Singapore
Heng Mui Keng Terrace, Kent Ridge
Singapore 119597

Charles A. Micchelli
Department of Mathematical Sciences
IBM T. J. Watson Research Center
Yorktown Heights, N.Y. 10598

Abstract: Conjugate quadrature filters with multiple zeros at 1 have
classical applications to unitary subband coding of signals using
exact reconstruction filter banks. Recent work shows how to construct,
given a set of n negative numbers, a CQF whose degree does not exceed
2n-1 and whose zeros contain the specified negative numbers, and
applies such filters to interpolatory subdivision and to wavelet
construction in Sobelov spaces. This paper describes a recent result
of the authors which extends this construction for an arbitrary set of
n nonzero complex numbers that contains no negative or negative
reciprocal conjugate pairs. Detailed derivations are to be given
elsewhere. We design several filters using an exchange algorithm to
illustrate a conjecture concerning the minimal degree and we discuss
an application to coding transient acoustic signals.

3) Title: Convergence of Multidimensional Cascade Algorithm
To appear in Numerische Mathematik

Authors:

Wayne Lawton
Institute of System Science
National University of Singapore
Heng Mui Keng Terrace, Kent Ridge
Singapore 119597

Seng Luan Lee and Zuowei Shen
Department of Mathematics
National University of Singapore
10 Kent Ridge Crescent, Singapore 119260

Abstract: A necessary and sufficient condition on the spectrum of the
restricted transition operator is given for the convergence in
$L^2(R^d)$ of the multidimensional cascade algorithm for any starting
function $phi_0$ whose shifts form a partition of unity.

4) Title: Stability and Orthonormality of Multivariate Refinable Functions
To appear in SIAM Journal of Mathematical Analysis

Authors:

Wayne Lawton
Institute of System Science
National University of Singapore
Heng Mui Keng Terrace, Kent Ridge
Singapore 119597

Seng Luan Lee and Zuowei Shen
Department of Mathematics
National University of Singapore
10 Kent Ridge Crescent, Singapore 119260

Abstract: This paper characterizes the stability and orthonormality of
the shifts of a multidimensional $(M,c)$ refinable function $phi$ in
terms of the eigenvalues and eigenvectors of the transition operator
$W_{c_{au}}$ defined by the autocorrelation $c_{au}$ of its refinement
mask $c, $ where $M$ is an arbitrary dilation matrix. Another
consequence is that if the shifts of $phi$ form a Riesz basis, then
$W_{c_{au}}$ has a unique eigenvector of eigenvalue $1 ,$ and all its
other eigenvalues lie inside the unit circle. The general theory is
applied to two-dimensional nonseparable $(M,c)$ refinable functions
whose masks are constructed from Daubechies' conjugate quadrature
filters.
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