Multi-dimensional Filter Banks and Wavelets—A System Theoretic Perspective
01 Nov 1998-Journal of The Franklin Institute-engineering and Applied Mathematics (Pergamon)-Vol. 335, Iss: 8, pp 1367-1409
TL;DR: In this paper, the authors view the current status of multi-dimensional filters bank and wavelet design from the perspective of signal and system theory, and provide a flavor of techniques germane to this development by considering a few specific problems.
Abstract: Wereview the current status of multi-dimensional filters bank and wavelet design from the perspective of signal and system theory. The study of wavelets and perfect reconstruction filter banks are known to have roots in traditional filter design techniques. On the other hand, the field of multi-dimensional systems and signal processing has developed a set of tools intrinsic to itself, and has attained a certain level of maturity over the last two decades. We have recently noted a degree of synergy between the two fields of wavelets and multi-dimensional systems. This arises from the fact that many ideas crucial to wavelet design are inherently system theoretic in nature. While there are many examples of this synergy manifested in recent publications, we provide a flavor of techniques germane to this development by considering a few specific problems in detail. The construction of orthogonal wavelets can be essentially viewed as a circuit and system theoretic problem of design of energy dissipative (passive) filters, the multi-dimensional version of which has very close ties with a classic problem of lumped distributed passive network synthesis. Groebner basis techniques, matrix completion problems over rings of polynomials or rings of stable rational functions, i.e., Quillen Suslin (31) type problems are still other examples, which feature in our discussion in an important manner. A number of open problems are also cited. ©1998 The Franklin Institute. Published by Elsevier Science Ltd.
Citations
More filters
TL;DR: An algorithm based on Gröbner bases for computing complete systems of solutions (“syzygies”) for linear diophantine equations with multivariate polynomial coefficients is described, showing how many fundamental problems of systems theory can be reduced to the problem of syzyGies computation.
Abstract: We present the basic concepts and results of Grobner bases theory for readers working or interested in systems theory. The concepts and methods of Grobner bases theory are presented by examples. No prerequisites, except some notions of elementary mathematics, are necessary for reading this paper. The two main properties of Grobner bases, the elimination property and the linear independence property, are explained. Most of the many applications of Grobner bases theory, in particular applications in systems theory, hinge on these two properties. Also, an algorithm based on Grobner bases for computing complete systems of solutions (“syzygies”) for linear diophantine equations with multivariate polynomial coefficients is described. Many fundamental problems of systems theory can be reduced to the problem of syzygies computation.
76 citations
TL;DR: In this paper, a multivariate polynomial matrix factorization algorithm for computing a globally minimal generating matrix of the syzygy of solutions associated with a Polynomial Matrix Factorization (PMF) is presented.
Abstract: A multivariate polynomial matrix factorization algorithm is introduced and discussed. This algorithm and another algorithm for computing a globally minimal generating matrix of the syzygy of solutions associated with a polynomial matrix are both associated with a zero-coprimeness constraint that characterizes perfect-reconstruction filter banks. Generalizations, as well as limitations of recent results which incorporate the perfect reconstruction as well as the linear-phase constraints, are discussed with several examples and counterexamples. Specifically, a Grobner basis-based proof for perfect reconstruction with linear phase is given for the case of two-band multidimensional filter banks, and the algorithm is illustrated by a nontrivial design example. Progress and bottlenecks in the multidimensional multiband case are also reported.
72 citations
TL;DR: A reasonably detailed review is given of several fundamental theoretical issues that occur in the use of Grobner bases in multidimensional signals and systems applications, including the primeness of multivariate polynomial matrices, multivariate unimodularPolynomial matrix completion, and prime factorization of multidity matrices.
Abstract: This paper is a tutorial on Grobner bases and a survey on the applications of Grobner bases in the broad field of signals and systems. A reasonably detailed review is given of several fundamental theoretical issues that occur in the use of Grobner bases in multidimensional signals and systems applications. These topics include the primeness of multivariate polynomial matrices, multivariate unimodular polynomial matrix completion, and prime factorization of multivariate polynomial matrices. A brief review is also presented on the wide-ranging applications of Grobner bases in multidimensional as well as one-dimensional circuits, networks, control, coding, signals, and systems and other related areas like robotics and applied mechanics. The impact and scope of Grobner bases in signals and systems are highlighted with respect to what has already been accomplished as a stepping stone to expanding future research.
60 citations
TL;DR: The method casts the design problem as a linear minimization of filter coefficients such that their value at ω = π /2 M is 0.707, which results in a simpler, more direct design procedure.
Abstract: This paper presents a simple and efficient design method for cosine-modulated filter banks with prescribed stopband attenuation, passband ripple, and channel overlap The method casts the design problem as a linear minimization of filter coefficients such that their value at ω = π /2 M is 0707, which results in a simpler, more direct design procedure The weighted constrained least squares technique is exploited for designing the prototype filter for cosine modulation (CM) filter banks Several design examples are included to show the increased efficiency and flexibility of the proposed method over the exiting methods An application of the proposed method is considered in the area of sub-band coding of the ECG and speech signals
44 citations
TL;DR: An eigen filter-based approach for the design of two-channel linear-phase FIR perfect-reconstruction (PR) filter banks and shows that, by an appropriate choice of the length of the filters, it can ensure the existence of a solution to the constrained eigenfilter design problem for the complementary-synthesis filter.
Abstract: We present an eigenfilter-based approach for the design of two-channel linear-phase FIR perfect-reconstruction (PR) filter banks. This approach can be used to design 1-D two-channel filter banks, as well as multidimensional nonseparable two-channel filter banks. Our method consists of first designing the low-pass analysis filter. Given the low-pass analysis filter, the PR conditions can be expressed as a set of linear constraints on the complementary-synthesis low-pass filter. We design the complementary-synthesis filter by using the eigenfilter design method with linear constraints. We show that, by an appropriate choice of the length of the filters, we can ensure the existence of a solution to the constrained eigenfilter design problem for the complementary-synthesis filter. Thus, our approach gives an eigenfilter-based method of designing the complementary filter, given a ldquopredesignedrdquo analysis filter, with the filter lengths satisfying certain conditions. We present several design examples to demonstrate the effectiveness of the method.
31 citations
References
More filters
TL;DR: The author describes the mathematical properties of such decompositions and introduces the wavelet transform, which relates to the decomposition of an image into a wavelet orthonormal basis.
Abstract: The author reviews recent multichannel models developed in psychophysiology, computer vision, and image processing. In psychophysiology, multichannel models have been particularly successful in explaining some low-level processing in the visual cortex. The expansion of a function into several frequency channels provides a representation which is intermediate between a spatial and a Fourier representation. The author describes the mathematical properties of such decompositions and introduces the wavelet transform. He reviews the classical multiresolution pyramidal transforms developed in computer vision and shows how they relate to the decomposition of an image into a wavelet orthonormal basis. He discusses the properties of the zero crossings of multifrequency channels. Zero-crossing representations are particularly well adapted for pattern recognition in computer vision. >
2,109 citations
Book•
01 Jan 1988
TL;DR: In this article, the stable factorization approach is introduced to the synthesis of feedback controllers for linear control systems, where the controller is designed as a matrix over a fraction field associated with a commutative ring with identity, denoted by R, which also has no divisors of zero.
Abstract: This book introduces the so-called "stable factorization approach" to the synthesis of feedback controllers for linear control systems The key to this approach is to view the multi-input, multi-output (MIMO) plant for which one wishes to design a controller as a matrix over the fraction field F associated with a commutative ring with identity, denoted by R, which also has no divisors of zero In this setting, the set of single-input, single-output (SISO) stable control systems is precisely the ring R, while the set of stable MIMO control systems is the set of matrices whose elements all belong to R The set of unstable, meaning not necessarily stable, control systems is then taken to be the field of fractions F associated with R in the SISO case, and the set of matrices with elements in F in the MIMO case The central notion introduced in the book is that, in most situations of practical interest, every matrix P whose elements belong to F can be "factored" as a "ratio" of two matrices N,D whose elements belong to R, in such a way that N,D are coprime In the familiar case where the ring R corresponds to the set of bounded-input, bounded-output (BIBO)-stable rational transfer functions, coprimeness is equivalent to two functions not having any common zeros in the closed right half-plane including infinity However, the notion of coprimeness extends readily to discrete-time systems, distributed-parameter systems in both the continuous- as well as discrete-time domains, and to multi-dimensional systems Thus the stable factorization approach enables one to capture all these situations within a common framework The key result in the stable factorization approach is the parametrization of all controllers that stabilize a given plant It is shown that the set of all stabilizing controllers can be parametrized by a single parameter R, whose elements all belong to R Moreover, every transfer matrix in the closed-loop system is an affine function of the design parameter R Thus problems of reliable stabilization, disturbance rejection, robust stabilization etc can all be formulated in terms of choosing an appropriate R This is a reprint of the book Control System Synthesis: A Factorization Approach originally published by MIT Press in 1985 Table of Contents: Introduction / Proper Stable Rational Functions / Scalar Systems: An Introduction / Matrix Rings / Stabilization
1,840 citations
Book•
[...]
01 Jun 1992
TL;DR: In this article, a mathematical notation and review of state equation representation state equation solution transition matrix properties two important cases internal stability Lyapunov stability criteria additional stability criteria controllability and observability realizability minimal realization input-out-put stability controller and observer forms linear feedback state observation polynomial fraction description polynoial fraction applications geometric theory applications of geometric theory.
Abstract: Mathematical notation and review state equation representation state equation solution transition matrix properties two important cases internal stability Lyapunov stability criteria additional stability criteria controllability and observability realizability minimal realization input-out-put stability controller and observer forms linear feedback state observation polynomial fraction description polynoial fraction applications geometric theory applications of geometric theory.
1,575 citations