No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Biorthogonal Cosine-Modulated Filter Banks Without Dc Leakage
Abstract
In this paper, we present a structure for implementing the polyphase filters of biorthogonal modulated filter banks that automatically guarantees perfect reconstruction of the filter bank and furthermore allows to specifiy the values of the filters' frequency responses at certain frequencies. Thus, modulated filter banks without DC leakage can be designed. The new stucture is based on lifting schemes for the polyphase filters and DC leakage can be avoided very easily when reducing the number of lifting coefficients that can be freely chosen and used for filter optimization. The great advantage of the new method is that we do not have to take constraints into consideration when optimizing the prototype filter, but PR and specified zeros are structure inherent. 1. INTRODUCTION Cosine-modulated filter banks have been studied extensively in literature within the last 10 years. They have shown to provide a very efficient implementation based on a prototype filter and a fast cosine transfor...
... To overcome this problem, factorizations of the filter bank, which structurally enforces the PR condition, is necessary. In [9], the polyphase components of the CMFB are parameterized by the lifting scheme so that the filter bank is still PR for different choices of the lifting coefficients. The design problem can then be formulated as an unconstrained optimization problem with the lifting coefficients as variables. ...
This paper proposes a factorization for the M-channel perfect reconstruction (PR) IIR cosine-modulated filter banks (CMFB) proposed previously by the authors. This factorization, which is based on the lifting scheme, is also complete for the PR FIR CMFB as well as the general two-channel PR MR filter banks if the determinant of the polyphase matrix is equal to constant multiplies of signal delays. It can be used to convert a numerically optimized nearly PR CMFB to a structurally PR system. Furthermore, the arithmetic complexity of the FB using this structure can be reduced asymptotically by a factor of two. When the forward and inverse transforms are implemented with the same set of SOPOT coefficients, a multiplier-less CMFB can be obtained. Its arithmetic complexity is further reduced and it becomes very attractive for VLSI implementation.
... In a final example, we consider an eight-channel, biorthogonal, low-delay, cosine-modulated filterbank with a filter length of and a system delay of 15 taps. The prototype is designed to have no dc leakage [23]. The signal length is chosen to be an integer multiple of the number of channels. ...
This paper presents boundary optimization techniques for the
nonexpansive decomposition of arbitrary-length signals with multirate
filterbanks. Both biorthogonal and paraunitary filterbanks are
considered. The paper shows how matching moments and orthonormality can
be imposed as additional conditions during the boundary filter
optimization process. It provides direct solutions to the problem of
finding good boundary filters for the following cases: (a) biorthogonal
boundary filters with exactly matching moments and (b) orthonormal
boundary filters with almost matching moments. With the proposed
methods, numerical optimization is only needed if orthonormality and
exactly matching moments are demanded. The proposed direct solutions are
applicable to systems with a large number of subbands and/or very long
filter impulse responses. Design examples show that the methods allow
the design of boundary filters with good frequency selectivity
The construction of spectral filters for graph wavelet transforms is addressed in this work. Both the undecimated and decimated cases will be considered. The filter functions are polynomials and can be implemented efficiently without the need for any eigendecomposition, which is computationally expensive for large graphs. Polynomial filters also have the advantage of the vertex localization property. The construction is achieved by designing suitable transformations that are used on traditional multirate filter banks. It will be shown how the classical Quadrature-Mirror-Filters and linear phase, critically-/over- sampled filter banks can be used to construct spectral graph wavelets that are almost tight. A variety of design examples will be given to show the versatility of the design technique.
In this paper, we discuss the design of prototype filters of cosine modulated filter banks for image compression application. The design problem is formulated as a two stage nonlinear constrained optimization problem i.e. combination of residual stopband energy and coding gain. The quadratic constraints are linearized and the objective function is minimized using constrained optimization by minimizing prototype filter tap weights satisfying dc-leakage free condition. Simulation results show that the proposed design method converges within a few iterations and that high performance cosine modulated filter banks with large stopband attenuation and coding gain are obtained and results analysed on different types of images.
A new formulation for the analysis and design of modulated filter
banks is introduced. The formulation provides a broad range of design
flexibility within a compact framework and allows for the design of a
variety of computationally efficient modulated filter banks with
different numbers of bands and virtually arbitrary lengths. A unique
feature of the formulation is that it provides explicit control of the
input-to-output system delay in conjunction with perfect reconstruction.
Design examples are given to illustrate the methodology
Multirate digital signal processing techniques have been practiced by engineers for more than a decade and a half. This discipline finds applications in speech and image compression, the digital audio industry, statistical and adaptive signal processing, numerical solution of differential equations, and in many other fields. It also fits naturally with certain special classes of time-frequency representations such as the short-time Fourier transform and the wavelet transform, which are useful in analyzing the time-varying nature of signal spectra. Over the last decade, there has been a tremendous growth of activity in the area of multirate signal processing, perhaps triggered by the first book in this field [Crochiere and Rabiner, 1983]. Particularly impressive is the amount of new literature in digital filter banks, multidimensional multirate systems, and wavelet representations. The theoretical work in multirate filter banks appears to have reached a level of maturity which justifies a thorough, unified, and in-depth treatment of these topics. This book is intended to serve that purpose, and it presents the above mentioned topics under one cover. Research in the areas of multidimensional systems and wavelet transforms is still proceeding at a rapid rate. We have dedicated one chapter to each of these, in order to bring the reader up to a point where research can be begun. I have always believed that it is important to appreciate the generality of principles and to obtain a solid theoretical foundation, and my presentation here reflects this philosophy. Several applications are discussed throughout the book, but the general principles are presented without bias towards specific application-oriented detail. The writing style here is very much in the form of a text. Whenever possible I have included examples to demonstrate new principles. Many design examples and complete design rules for filter banks have been included. Each chapter includes a fairly extensive set of homework problems (totaling over 300). The solutions to these are available to instructors, from the publisher. Tables and summaries are inserted at many places to enable the reader to locate important results conveniently. I have also tried to simplify the reader’s task by assigning separate chapters for more advanced material. For example, Chap. 11 is dedicated to wavelet transforms, and Chap. 14 contains detailed developments of many results on paraunitary systems. Whenever a result from an advanced chapter (for example, Chap. 14) is used in an earlier chapter, this result is first stated clearly within the context of use, and the reader is referred to the appropriate chapter for proof. The text is self-contained for readers who have some prior exposure to digital signal processing. A one-term course which deals with sampling, discrete-time Fourier transforms, z-transforms, and digital filtering, is sufficient. In Chap. 2 and 3 a brief review of this material is provided. A thorough exposition can be found in a number of references, for example, [Oppenheim and Schafer, 1989]. Chapter 3 also contains some new material, for example, eigenfilters, and detailed discussions on allpass filters, which are very useful in multirate system design. A detailed description of the text can be found in Chap. 1. Chapters 2 and 3 provide a brief review of signals, systems, and digital filtering. Chapter 4, which is the first one on multirate systems, covers the fundamentals of multirate building blocks and filter banks, and describes many applications. Chapter 5 introduces multirate filter banks, laying the theoretical foundation for alias cancelation, and elimination of other errors. The first two sections in Chap. 4 and 5 contain material overlapping with [Crochiere and Rabiner, 1983]. Most of the remaining material in these chapters, and in the majority of the chapters that follow, have not appeared in this form in text books. Chapters 6 to 8 provide a deeper study of multirate filter banks, and present several design techniques, including those based on the so-called paraunitary matrices. (These matrices play a role in the design of many multirate systems, and are treated in full depth in Chap. 14.) Chapters 9 to 12 cover special topics in multirate signal processing. These include roundoff noise effects (Chap. 9), block filtering, periodically time varying systems and sampling theorems (Chap. 10), wavelet trans-forms (Chap. 11) and multidimensional multirate systems (Chap. 12). Chapters 13 and 14 give an in-depth coverage of multivariable linear systems and lossless (or paraunitary) systems, which are required for a deeper understanding of multirate filter banks and wavelet transforms. There are five appendices which serve as references as well as supplementary reading. Three of these are review-material (matrix theory, random processes, and Mason’s gain formula). Two of the appendices contain results directly related to filter bank systems. One of these is a technique for spectral factorization; the other one analyzes the effects of quantization of subband signals. Many of these chapters have been taught at Caltech over the last three years. This text can be used for teaching a one, two, or three term (quarter or semester) course on one of many possible topics, for example, multirate fundamentals, multirate filter banks, wavelet representation, and so on. There are many homework problems. The instructor has a great deal of flexibility in ch∞sing the topics, but I prefer not to bias him or her by providing specific course outlines here. In summary, I have endeavored to produce a text which is useful for the class-room, as well as for self-study. It is also hoped that it will bring the reader to a point where he/she can start pursuing research in a vast range of multirate areas. Finally, I believe that the text can be comfortably used by the practicing engineer because of the inclusion of several design procedures, examples, tables, and summaries.
This paper presents methods for the efficient realization of
prototype filters for modulated filter banks. The implementation is
based on the lattice structure of the polyphase filters. The lattice
coefficients, representing rotations, are approximated by a small number
of simple μ-rotations each of which can be realized by some shift and
add operations instead of a multiplication. Since the lattice structure
is robust against coefficient quantization we do not loose the perfect
reconstruction (PR) property of the filter bank when doing this
approximation. The frequency responses of the original and approximated
prototype filters are compared in terms of complexity and stopband
attenuation
We present the lifting scheme, a new idea for constructing compactly supported wavelets with compactly supported duals. The lifting scheme uses a simple relationship between all multiresolution analyses with the same scaling function. It isolates the degrees of freedom remaining after fixing the biorthogonality relations. Then one has full control over these degrees of freedom to custom design the wavelet for a particular application. The lifting scheme can also speed up the fast wavelet transform. We illustrate the use of the lifting scheme in the construction of wavelets with interpolating scaling functions.
Cet article presente des implantations optimisées de filtres prototypes de bancs de filtres modulés à reconstruction parfaite, à retard quelconque. Les implantations proposées sont basées sur des "schémas de liftage". Ici, l'approximation des coefficients "de liftage" est faite de manière à ce que le filtrage n'emploie que peu d'operations de décalage et addition. Comme le "schéma de liftage" est robuste aux erreurs de quantification ainsi introduites, la propriété de recontruction parfaite du banc de filtres est conservée. Le prototype d'origine et celui obtenu par cette méthode sont comparés en terme de complexité d'implantation et d'atténuation hors-bande.
We connect the design of biorthogonal modulated filter banks with
the idea of lifting schemes, which were proposed for the construction of
biorthogonal wavelets. Based on this formulation, we derive a
lattice-like structure for the polyphase components. As the
rotation-based structures for the design and implementation of
paraunitary modulated filter banks, our structure automatically
guarantees the perfect reconstruction (PR) property of the filter bank.
The coefficients can be freely chosen and used for filter optimization
A new design method for biorthogonal modulated filter banks is presented. It is based on a cascade of simple matrices, and it has some properties that have not been reported before. It represents filter banks with arbitrary overall system delay and filter length, it is shown that almost all cosine modulated filter banks can be described by this structure, and that it leads to a more efficient implementation than previous structures. Imposing certain symmetries on the matrices can be used to design low delay filter banks with identical (except for the sign) baseband impulse responses for the analysis and synthesis filter bank.
Cosine-modulated filter banks have been studied extensively because of their design ease and efficient implementation. These filter banks either have restricted lengths or assume the paraunitary property for the polyphase matrices. In this paper, the biorthogonal cosine-modulated filter bank with arbitrary length is considered and the perfect reconstruction (PR) conditions are derived. These conditions are the general form of the PR conditions reported in the literature. Examples of PR systems with variable overall delay are designed using the quadratic constrained least squares formulation
This paper presents a general framework for maximally decimated
modulated filter banks. The theory covers the known classes of cosine
modulation and relates them to complex-modulated filter banks. The
prototype filters have arbitrary lengths, and the overall delay of the
filter bank is arbitrary, within fundamental limits. Necessary and
sufficient conditions for perfect reconstruction (PR) are derived using
the polyphase representation. It is shown that these PR conditions are
identical for all types of modulation-modulation based on the discrete
cosine transform (DCT), both DCT-III/DCT-IV and DCT-I/DCT-II, and
modulation based on the modified discrete Fourier transform (MDFT). A
quadratic-constrained design method for prototype filters yielding PR
with arbitrary length and system delay is derived, and design examples
are presented to illustrate the tradeoff between overall system delay
and stopband attenuation (subchannelization)
Formulate the filter bank design problem as an
quadratic-constrained least-squares minimization problem. The solution
of the minimization problem converges very quickly since the cost
function as well as the constraints are quadratic functions with respect
to the unknown parameters. The formulations of the
perfect-reconstruction cosine-modulated filter bank, of the
near-perfect-reconstruction pseudo-QMF bank, and of the two-channel
biorthogonal linear-phase filter bank are derived using the proposed
approach. Compared with other design methods, the proposed technique
yields PR filter banks with much higher stopband attenuation. The
proposed technique can also be extended to design multidimensional
filter banks