ABSTRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filter-ing steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists (and expressed by the formula); it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e, non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically re-duces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a wavelet-like transform that maps integers to integers. 1.

This correspondence deals with multiwavelets, which are a recent generalization of wavelets in the context of time-varying filter banks and with their applications to signal processing and especially compression. By their inherent structure, multiwavelets are fit for processing multichannel signals. This is the main issue in which we will be interested here. The outline of the correspondence is as follows. First, we will review material on multiwavelets and their links with multifilter banks and, especially, time-varying filter banks. Then, we will have a close look at the problems encountered when using multiwavelets in applications, and we will propose new solutions for the design of multiwavelets filter banks by introducing the so-called balanced multiwavelets.

We present the characterization and design of multidimensional oversampled FIR filter banks. In the polyphase domain, the perfect reconstruction condition for an oversampled filter bank amounts to the invertibility of the analysis polyphase matrix, which is a rectangular FIR matrix. For a nonsubsampled FIR filter bank, its analysis polyphase matrix is the FIR vector of analysis filters. A major challenge is how to extend algebraic geometry techniques, which only deal with polynomials (that is, causal filters), to handle general FIR filters. We propose a novel method to map the FIR representation of the nonsubsampled filter bank into a polynomial one by simply introducing a new variable. Using algebraic geometry and Gröbner bases, we propose the existence, computation, and characterization of FIR synthesis filters given FIR analysis filters. We explore the design problem of MD nonsubsampled FIR filter banks by a mapping approach. Finally, we extend these results to general oversampled FIR filter banks.

The polyphase representation with respect to sampling lattices in multidimensional (M-D) multirate signal processing allows us to identify perfect reconstruction (PR) filter banks with unimodular Laurent polynomial matrices, and various problems in the design and analysis of invertible MD multirate systems can be algebraically formulated with the aid of this representation. While the resulting algebraic problems can be solved in one dimension (1-D) by the Euclidean Division Algorithm, we show that Grobner bases offers an effective solution to them in the M-D case. 1 Introduction - - Analysis Bank Synthesis Bank H G G p H p = I Figure 1: An analysis/synthesis system with PR property. Department of Mathematical Sciences, Oakland University, Rochester, MI 48309 (park@oakland.edu). y Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA, Eindhoven, The Netherlands (kalker@natlab.research.philips.com). z Department of Electrical Engineering and Computer Sciences,University of C...

The relation between ladder and lattice implementations of two-channel filter banks is discussed and it is shown that these two concepts differ in general. An elementary proof is given for the fact that over any integral domain which is not a field there exist causal realizable perfect reconstructing filter banks that can not be implemented with causal lifting filters. A complete parametrization of filter banks with coefficients in local rings and semiperfect rings is given. 1 Filter Banks Let A be a commutative ring. We assume that all signals and filters are elements of the Laurent polynomial ring B = A[z; z \Gamma1 ]. Note that B is isomorphic to the group algebra A[Z]. Therefore, we refer to the multiplication in B as a convolution or filter operation. We define a downsampling operation [ # 2] on B by [ # 2] a(z) = a e (z), where a(z) = a e (z 2 ) + za o (z 2 ). An upsampling operation [ " 2] on B is defined by [ " 2] a(z) = a(z 2 ). A two-channel filter bank con...

Many problems in digital signal processing can be converted to algebraic problems over polynomial and Laurent polynomial rings, and can be solved using the existing methods of algebraic and symbolic computation. This paper aims to establish this connection in a systematic manner, and demonstrate how it can be used to solve various problems arising from multidimensional signal processing. The method of Gröbner bases is used as a main computational tool. © 2003 Elsevier Ltd. All rights reserved.

Abstract. Let k be a field, and let M be a commutative, seminormal, finitely generated monoid, which is torsionfree, cancellative, and has no nontrivial units. J. Gubeladze proved that finitely generated projective modules over kM are free. This paper contains an algorithm for finding a free basis for a finitely generated projective module over kM. As applications one obtains alternative algorithms for the Quillen-Suslin Theorem for polynomial rings and Laurent polynomial rings, based on Quillen’s proof. I.

Abstract. Recently Umirbaev has proved the long-standing Anick conjecture, that is, there exist wild automorphisms of the free associative algebra K〈x, y, z 〉 over a field K of characteristic 0. In particular, the well-known Anick automorphism is wild. In this article we obtain a stronger result (the Strong Anick Conjecture that implies the Anick Conjecture). Namely, we prove that there exist wild coordinates of K〈x, y, z〉. In particular, the two nontrivial coordinates in the Anick automorphism are both wild. We establish a similar result for several large classes of automorphisms of K〈x, y, z〉. We also find a large new class of wild automorphisms of K〈x, y, z 〉 which is not covered by the results of Umirbaev. Finally, we study the lifting problem for automorphisms and coordinates of polynomial algebras, free metabelian algebras and free associative algebras and obtain some interesting new results. 1.

Many problems in FIR filter banks, via the method of polyphase representation, can be characterized by their transfer matrices. In the case of multidimensional (MD) FIR filter banks, the resulting transfer matrices have Laurent polynomial entries in several variables. For a given analysis filter bank, this method produces a multi-input multi-output (MIMO) system (e.g. an oversampled filter bank). Since such a system does not have a unique perfect reconstruction (PR) pair in general, there exists certain degree of freedom in designing a synthesis system to make the overall system a PR system. In this paper, for a given analysis system, we give a complete parametrization of all the synthesis systems that ensure perfect reconstruction of inputs, which gives us a space to search for an optimal synthesis system. This involves an adaptation of Schreyer's algorithm to carry out syzygy computations with Laurent polynomials. 1. INTRODUCTION Various signal processing problems can be understood ...

We study the invertibility of M-variate Laurent polynomial N × P matrices. Such matrices represent multidimensional systems in various settings such as filter banks, multiple-input multiple-output systems, and multirate systems. Given an N × P Laurent polynomial matrix H(z1,..., zM) of degree at most k, we want to find a P × N Laurent polynomial left inverse matrix G(z) of H(z) such that G(z)H(z) = I. We provide computable conditions to test the invertibility and propose algorithms to find a particular inverse. The main result of this paper is to prove that H(z) is generically invertible when N −P ≥ M; whereas when N −P < M, then H(z) is generically noninvertible. As a result, we propose an algorithm to find a particular inverse of a Laurent polynomial matrix that is faster than current algorithms known to us.