Construction of Arbitrary Dimensional Biorthogonal Multiwavelet Using Lifting Scheme

This paper focuses on the construction of multidimensional biorthogonal multiwavelets and the perfect reconstruction multifilter banks. Based on the Hermite-Neville filter, two lifting structures have been proposed and systematically investigated, and a general design framework has been developed for building biorthogonal multiwavelets and Hermite interpolation filter banks with any multiplicity for any lattice in any dimension with any number of primal and dual vanishing moments. The construction is an important generalization of the Neville-based lifting scheme and inherits all of the advantages of lifting schemes such as fast transform, in-place computation and integer-to-integer transforms. Our multi wavelet systems preserve most of the desirable properties for applications, such as interpolating, short support, symmetry, and high vanishing moments.     

Modulated filter banks with arbitrary system delay: efficientimplementations and the time-varying case

We present a new method for the design and implementation of modulated filter banks with perfect reconstruction. It is based on the decomposition of the analysis and synthesis polyphase matrices into a product of two different types of simple matrices, replacing the polyphase filtering part in a modulated filter bank. Special consideration is given to cosine-modulated as well as time-varying filter banks. The new structure provides several advantages. First of all, it allows an easy control of the input-output system delay, which can be chosen in single steps of input sampling rate, independent of the filter length. This property can be used in audio coding applications to reduce pre-echoes. Second, it results in a structure that is nearly twice as efficient as performing the polyphase filtering directly. Perfect reconstruction is a structurally inherent feature of the new formulation, even for nonlinear operations or time-varying coefficients. Hence, the structure is especially suited for the design of time-varying filter banks where both the number of bands as well as the prototype filters can be changed while maintaining perfect reconstruction and critical sampling. Further, a proof of effective completeness is given, and the design of equal magnitude-response analysis and synthesis filter banks is described. Filter design can be performed by nonconstrained optimization of the matrix coefficients according to a given cost function. Design and audio-coding application examples are given to show the performance of the new filter bank     

Time-varying analysis-synthesis systems based on filter banks andpost filtering

Perfect reconstruction (PR) time-varying analysis-synthesis filter banks are those in which the filters are allowed to change from one set of PR filter banks to another as the input signal is being processed. Such systems have the property that, in the absence of coding, they faithfully reconstruct every sample of the input. Various methods have been reported for the time-varying filter bank design; all of them, however, utilize structures for conventional PR filter banks. These conventional structures that have been applied in the past result in different limitations in each method. This paper introduces a new structure for exactly reconstructing time-varying analysis-synthesis filter banks. This structure consists of the conventional filter bank followed by a time-varying post filter. The new method requires neither the redesign of the analysis sections nor the use of any intermediate analysis filters during transition periods. It provides a simple and elegant procedure for designing time-varying filter banks without the disadvantages of the previous methods     

Symmetric delay factorization: generalized framework forparaunitary filter banks

The symmetric delay factorization (SDF) was introduced to synthesize linear-phase paraunitary filter banks (LPPUFBs) with uniform order (i.e., filter length equal to NM for arbitrary N) and real-valued coefficients. The SDF presents the advantage of decomposing the polyphase transfer matrix (PTM) into only orthogonal matrices, even at the boundary of finite-duration signals, simplifying significantly the design of time-bounded filter banks (TBFBs) or of time-varying filter banks (TVFBs). However, the symmetric delay factorization applies only to LPPUFBs. On the other hand, lattice structures, as well as finite-size lattice structures, are proposed for classes of nonlinear-phase paraunitary filter banks, as the modulated lapped transform (MLT) and the extended tapped transform (ELT). This paper describes a new minimal and complete symmetric delay factorization valid for a larger class of paraunitary filter banks, encompassing paraunitary cosine modulated filter banks, with nonlinear phase basis functions, as well as for a set of LPPUFBs including the linear-phase lapped orthogonal transforms (LOTs) and the generalized tapped orthogonal transforms (GenLOTs). The derivations for filter banks with even and odd numbers of channels are formulated in a unified form. This approach opens new perspectives in the design of time-varying filter banks used for image and video compression, especially in the framework of region or object-based coding     

Entwurf modulierter digitaler Filterbänke

In this paper, we present an improved design method for multirate modulated filter banks with uniform or nouniform frequency spacing. Compared to other design techniques, our method converges very rapidly and allows for the design of excellent prototype filters, especially for audio signal processing applications. The optimization procedure avoids nonlinear function minimization and requires linear equation solving only. We demonstrate our method by a design example of filter banks for critical-band analysis/synthesis.     

A general approach for analysis and application of discretemultiwavelet transforms

This paper proposes a general paradigm for the analysis and application of discrete multiwavelet transforms, particularly to image compression. First, we establish the concept of an equivalent scalar (wavelet) filter bank system in which we present an equivalent and sufficient representation of a multiwavelet system of multiplicity r in terms of a set of r equivalent scalar filter banks. This relationship motivates a new measure called the good multifilter properties (GMPs), which define the desirable filter characteristics of the equivalent scalar filters. We then relate the notion of GMPs directly to the matrix filters as necessary eigenvector properties for the refinement masks of a given multiwavelet system. Second, we propose a generalized, efficient, and nonredundant framework for multiwavelet initialization by designing appropriate preanalysis and post-synthesis multirate filtering techniques. Finally, our simulations verified that both orthogonal and biorthogonal multiwavelets that possess GMPs and employ the proposed initialization technique can perform better than the popular scalar wavelets such as Daubechies'D8 wavelet and the D(9/7) wavelet, and some of these multiwavelets achieved this with lower computational complexity     

一类具有线性相位的二重多小波的构造

给出由实单一紧支撑正交的小波构造二重正交多小波的方法。具体地,首先由实单一的紧支撑尺度函数构造出单一紧支撑正交对称的复尺度函数,再由构造出的复尺度函数去构造二重正交紧支撑多尺度函数,然后给出由二重尺度函数构造二重小波的显式公式。紧支撑正交的单一小波除Haar小波外不具有任何对称性,它用作滤波器不可能有线性相位,而由实单一紧支撑正交的尺度函数构造出的二重尺度函数却是对称的,对应的二重小波可以是对称或反对称的,从而使得这种小波在信号处理的过程中具有线性相位。最后给出相应的构造算例。     

Time-invariant and time-varying multirate filter banks : application to image coding

This paper reviews various concepts and solutions of time-invariant and time-varying multirate filter banks. It discusses their performance for image and video coding at low bit rates, and their applicability in the mpeg-4 framework. Time-invariant multirate filter banks, and methods of design with different criteria appropriate for signal compression are first presented. Several procedures of quantization, namely scalar and lattice vector quantization, with bit allocation optimized in the rate-distortion sense, are used for the encoding of the subband signals. A technique of rate-constrained lattice vector quantization (rc-lvq), combined with a three components entropy coding, allow, together with distortion psychovi-sual weighting mechanisms to obtain significant visual improvements versus scalar quantization or the zerotree technique. However, time-invariant multirate filter banks, although efficient in terms of compression, are not well suited for content-based functionalities. Content-based features may require the ability to manipulate and thus encode a given region in the scene independently of the neighbouring regions, hence the use of transformations that can be adapted to arbitrary size bounded supports. Also, to increase the compression efficiency, one may want to adapt the transformation to the region characteristics, and thus use transform switching mechanisms, with soft or hard transitions. Three main classes of transformations can address these problems: shape-adaptive block transforms, transforms relying on signal extensions and transforms relying on time-varying multirate filter banks. These various solutions, with their methods of design, are reviewed. Emphasis is put on an extension of the SDF (symmetric delay factorization) technique which opens new perspectives in the design of time-bounded and time-varying filter banks. A region-adapted rate-distortion quantization algorithm has been used in the evaluation of the transformations compression efficiency. The coding results illustrate the interest of these techniques for compression but also for features such as quality scalability applied to selected regions of the image.     

Filter banks in digital communications

Digital signal processing has played a key role in the development of telecommunication systems over the last two decades. In recent years digital filter banks have been occupying an increasingly important role in both wireless and wireline communication systems. In this paper we review some of these applications of filter banks with special emphasis on discrete multitone modulation which has had an impact on high speed data communication over the twisted pair telephone line. We also review filter bank precoders which have been shown to be important for channel equalization applications.     

A new family of nonredundant transforms using hybrid wavelets and directional filter banks.

We propose a new family of nonredundant geometrical image transforms that are based on wavelets and directional filter banks. We convert the wavelet basis functions in the finest scales to a flexible and rich set of directional basis elements by employing directional filter banks, where we form a nonredundant transform family, which exhibits both directional and nondirectional basis functions. We demonstrate the potential of the proposed transforms using nonlinear approximation. In addition, we employ the proposed family in two key image processing applications, image coding and denoising, and show its efficiency for these applications.     

Emerging applications of multirate signal processing and waveletsin digital communications

Multirate systems and filter banks have traditionally played an important role in source coding and compression for contemporary communication applications, and many of the key design issues in such applications have been extensively explored. We review developments on the comparatively less explored role of multirate filter banks and wavelets in channel coding and modulation for some important classes of channels. Some representative examples of emerging potential applications are described. One involves the use of highly dispersive, broadband multirate systems for wireless multiuser communication (spread spectrum CDMA) in the presence of fading due to time-varying multipath. Another is the wavelet-based diversity strategy referred to as “fractal modulation” for use with unpredictable communication links and in broadcast applications with user-selectable quality of service. A final example involves multitone (multicarrier) modulation systems based on multirate filter banks and fast lapped transforms for use on channels subject to severe intersymbol and narrowband interference. Collectively, these constitute intriguing, interrelated paradigms within an increasingly broad and active area of research     

Design of prefilters for discrete multiwavelet transforms

The pyramid algorithm for computing single wavelet transform coefficients is well known. The pyramid algorithm can be implemented by using tree-structured multirate filter banks. The authors propose a general algorithm to compute multiwavelet transform coefficients by adding proper premultirate filter banks before the vector filter banks that generate multiwavelets. The proposed algorithm can be thought of as a discrete vector-valued wavelet transform for certain discrete-time vector-valued signals. The proposed algorithm can be also thought of as a discrete multiwavelet transform for discrete-time signals. The authors then present some numerical experiments to illustrate the performance of the algorithm, which indicates that the energy compaction for discrete multiwavelet transforms may be better than the one for conventional discrete wavelet transforms     

Order statistic filter banks

Filter banks play a major role in multirate signal processing where these have been successfully used in a variety of applications. In the past, filter banks have been developed within the framework of linear filters. It is well known, however, that linear filters may have less than satisfactory performance whenever the underlying processes are non-Gaussian. We introduce the nonlinear class of order statistic (OS) filter banks that exploit the spectral characteristics of the input signal as well as its rank-ordering structure. The attained subband signals provide frequency and rank information in a localized time interval. OS filter banks can lead to significant gains over linear filter banks, particularly when the input signals contain abrupt changes and details, as is common with image and video signals. OS filter banks are formed using traditional linear filter banks as fundamental building blocks. It is shown that OS filter banks subsume linear filter banks and that the latter are obtained by simple linear transformations of the former. To illustrate the properties of OS filter banks, we develop simulations showing that the learning characteristics of the LMS algorithm, which are used to optimize the weight taps of OS filters, can be significantly improved by performing the adaptation in the OS subband domain.     

Efficient iterative design method for cosine-modulated QMF banks

An iterative algorithm for the design of multichannel cosine-modulated quadrature mirror-image filter (QMF) banks with near-perfect reconstruction is proposed. The objective function is formulated as a quadratic function in each step whose minimum point can be obtained using a closed-form solution. This approach has high design efficiency and leads to filter banks with high stopband attenuation and low aliasing and amplitude distortions. The proposed approach is then extended to the design of multichannel cosine-modulated QMF banks with low reconstruction delays, which are often required, especially in real-time applications. Several design examples are included to demonstrate the proposed algorithms, and some comparisons are made with existing designs     

Orthogonal multiwavelets with optimum time-frequency resolution

A procedure to design orthogonal multiwavelets with good time-frequency resolution is introduced. Formulas to compute the time-durations and the frequency-bandwidths of scaling functions and multiwavelets are obtained. Parameter expressions for the matrix coefficients of the multifilter banks that generate symmetric/antisymmetric scaling functions and multiwavelets supported in [O,N] are presented for N=2,...,6. Orthogonal multiwavelets with optimum time-frequency resolution are constructed, and some optimal multifilter banks are provided     

Discrete Zak transforms, polyphase transforms, and applications

We consider three different versions of the Zak (1967) transform (ZT) for discrete-time signals, namely, the discrete-time ZT, the polyphase transform, and a cyclic discrete ZT. In particular, we show that the extension of the discrete-time ZT to the complex z-plane results in the polyphase transform, an important and well-known concept in multirate signal processing and filter bank theory. We discuss fundamental properties, relations, and transform pairs of the three discrete ZT versions, and we summarize applications of these transforms. In particular, the discrete-time ZT and the cyclic discrete ZT are important for discrete-time Gabor (1946) expansion (Weyl-Heisenberg frame) theory since they diagonalize the Weyl-Heisenberg frame operator for critical sampling and integer oversampling. The polyphase representation plays a fundamental role in the theory of filter banks, especially DFT filter banks. Simulation results are presented to demonstrate the application of the discrete ZT to the efficient calculation of dual Gabor windows, tight Gabor windows, and frame bounds     

Multirate digital filters, filter banks, polyphase networks, andapplications: a tutorial

The basic concepts and building blocks in multirate digital signal processing (DSP), including the digital polyphase representation, are reviewed. Recent progress, as reported by several authors in this area, is discussed. Several applications are described, including subband coding of waveforms, voice privacy systems, integral and fractional sampling rate conversion (such as in digital audio), digital crossover networks, and multirate coding of narrowband filter coefficients. The M-band quadrature mirror filter (QMF) bank is discussed in considerable detail, including an analysis of various errors and imperfections. Recent techniques for perfect signal reconstruction in such systems are reviewed. The connection between QMF banks and other related topics, such as block digital filtering and periodically time-varying systems, is examined in a pseudo-circulant-matrix framework. Unconventional applications of the polyphase concept are discussed     

Subband decomposition of signals with generalized sampling

The authors generalize down and up-sampling operations by proposing block sampling. Perfect reconstruction conditions for two-band subband coding with block sampling are derived. By generalizing the sampling operation, new degrees of freedom are introduced and as a result, filter banks which were not previously possible become possible. Generalized down-sampler introduces different aliasing components than that of the traditional down-sampler. This can be used to ease some the requirements of the filter bank design problem. A constructive sampling method is proposed so that coprimeness of the transfer functions of the analysis filter banks is not only a necessary but also sufficient condition for perfect reconstruction. The results are extended to the case where the filter banks are linear periodically time-varying. The multichannel case is analyzed and the relation between unimodular matrices and perfect reconstruction filter banks is discussed     

Recursive time-varying filter banks for subband image coding

Filter banks, subband/wavelets, and multiresolution decompositions that employ recursive filters have been considered previously and are recognized for their efficiency in partitioning the frequency spectrum. This paper presents an analysis of a new infinite impulse response (IIR) filter bank in which these computationally efficient filters may be changed adaptively in response to the input. The new filter bank framework is presented and discussed in the context of subband image coding. In the absence of quantization errors, exact reconstruction can be achieved. By the proper choice of an adaptation scheme, it is shown that recursive linear time-varying (LTV) filter banks can yield improvement over conventional ones.     

Oversampled filter banks from extended perfect reconstruction filter banks

Oversampled filter banks are currently being proposed for robust transmission applications. In this paper, we completely characterize multidimensional doubly finite-impulse-response (FIR) filter banks, that is, oversampled filter banks whose dual is FIR. Then, we consider the problem of extending perfect reconstruction critically sampled multidimensional filter banks in order to obtain doubly FIR (DFIR) filter banks. As a result, very simple criteria for constructing DFIR filter banks as extensions of orthogonal filter banks are obtained. This paper also analyzes the problem of constructing totally FIR filter banks, i.e., DFIR filter banks that remain DFIR even when some channels are removed. It is shown that any totally FIR filter bank can be implemented as the cascade of a critically sampled DFIR filter bank whose number of channels is equal to the subsampling factor, a redundant finite-dimensional transform, and a suitable set of delays.