Multiwavelet bases with extra approximation properties

This paper highlights the differences between traditional wavelet and multiwavelet bases with equal approximation order. Because multiwavelet bases normally lack important properties that traditional wavelet bases (of equal approximation order) possess, the associated discrete multiwavelet transform is less useful for signal processing unless it is preceded by a preprocessing step (prefiltering). This paper examines the properties and design of orthogonal multiwavelet bases, with approximation order >1 that possess those properties that are normally absent. For these “balanced” bases (so named by Lebrun and Vetterli (see Proc. IEEE Int. Conf. Acoust., Speech, Signal Process. (ICASSP), Munich, Germany, vol.3, p.2473-76, 1997 and ibid., vol.46, p.1119-24, 1998)), prefiltering can be avoided. By reorganizing the multiwavelet filter bank, the development in this paper draws from results regarding the approximation order of M-band wavelet bases. The main result thereby obtained is a characterization of balanced multiwavelet bases in terms of the divisibility of certain transfer functions by powers of (z^-2r-1)/(z^-1-1). For traditional wavelets (r=1), this specializes to the usual factor (z+1) ^K     

Multidimensional multichannel FIR deconvolution using Gr?bner bases.

We present a new method for general multidimensional multichannel deconvolution with finite impulse response (FIR) convolution and deconvolution filters using Gr?bner bases. Previous work formulates the problem of multichannel FIR deconvolution as the construction of a left inverse of the convolution matrix, which is solved by numerical linear algebra. However, this approach requires the prior information of the support of deconvolution filters. Using algebraic geometry and Gr?bner bases, we find necessary and sufficient conditions for the existence of exact deconvolution FIR filters and propose simple algorithms to find these deconvolution filters. The main contribution of our work is to extend the previous Gr?bner basis results on multidimensional multichannel deconvolution for polynomial or causal filters to general FIR filters. The proposed algorithms obtain a set of FIR deconvolution filters with a small number of nonzero coefficients (a desirable feature in the impulsive noise environment) and do not require the prior information of the support. Moreover, we provide a complete characterization of all exact deconvolution FIR filters, from which good FIR deconvolution filters under the additive white noise environment are found. Simulation results show that our approaches achieve good results under different noise settings.     

Multidimensional Multichannel FIR Deconvolution Using GrÖbner Bases

We present a new method for general multidimensional multichannel deconvolution with finite impulse response (FIR) convolution and deconvolution filters using GrÖbner bases. Previous work formulates the problem of multichannel FIR deconvolution as the construction of a left inverse of the convolution matrix, which is solved by numerical linear algebra. However, this approach requires the prior information of the support of deconvolution filters. Using algebraic geometry and GrÖbner bases, we find necessary and sufficient conditions for the existence of exact deconvolution FIR filters and propose simple algorithms to find these deconvolution filters. The main contribution of our work is to extend the previous GrÖbner basis results on multidimensional multichannel deconvolution for polynomial or causal filters to general FIR filters. The proposed algorithms obtain a set of FIR deconvolution filters with a small number of nonzero coefficients (a desirable feature in the impulsive noise environment) and do not require the prior information of the support. Moreover, we provide a complete characterization of all exact deconvolution FIR filters, from which good FIR deconvolution filters under the additive white noise environment are found. Simulation results show that our approaches achieve good results under different noise settings.     

Symmetric prefilters for multiwavelets

When applying discrete multiwavelets, prefiltering is necessary because the initial multiscaling coefficients cannot be trivially derived from the samples of scalar signals. There have been many studies on the design of prefilters, and one main approach is to use a superfunction. The idea is to construct a low-pass function from the multiscaling functions that inherits their approximation power for scalar signals. However, none of the existing prefilters give linear phase combined filters, which is important for many practical applications. The authors analyse the conditions on which the prefilters and the combined filters are symmetric. A method is proposed for the design of good multiwavelet prefilters that allow the superfunction to be symmetric, satisfying the Strang-Fix conditions and the resulting combined filters are linear phase. Design examples using DGHM and Chui-Lian multiwavelets are given.     

The discrete wavelet transform for a symmetric-anti symmetric multiwavelet family on the interval

The Chui-Lian multiwavelet family of approximation order three is extended to serve as a basis for an orthogonal discrete multiwavelet transform on the interval. This requires, in addition to the original symmetric and antisymmetric pairs of scaling functions and wavelets, four new functions of each type at each edge of the interval, i.e., one more than required just to preserve the approximation order. The criteria are given that such edge functions must satisfy in order to allow multiresolution analysis of data sequences without special constraints on endpoint behavior or use of periodic boundary conditions, and a specific choice is examined in applications to both noiseless and noisy data. These are accomplished through the development of interval-basis extensions for both an earlier multiwavelet projection-based prefilter/postfilter combination and a multiwavelet vector thresholding algorithm.     

Design of FIR linear phase filters with discrete coefficients usingHopfield neural networks

A novel method is presented for designing FIR linear phase filters with discrete coefficients using Hopfield neural networks. The proposed procedure is based on the minimisation of the energy function of the Hopfield neural network, and can produce a good solution to the design of FIR linear phase filters with discrete coefficients     

3型FIR高阶多通带滤波器的自适应优化设计研究

本文详细讨论了3型线性相位滤波器幅频特性与正弦基神经网络算法间的关系，提出了该算法的收敛定理，给山了有限脉冲响应（FIR）高阶多通带滤波器自适应优化设计算法及实例。计算机仿真结果表明了该算法在FIR高阶多通带滤波器的有效性和优异性能。     

基于多小波分解的多光谱图像矢量融合

在实数域中,对称、正交的紧支集非平凡单小波基不存在,而多小波把紧支性、对称性、正交性完美地结合在一起,使小波理论从标量扩展到矢量范畴。考虑到图像多小波变换系数具有矢量特性,该文将基于像素点和基于区域的标量融合策略推广到矢量情形,提出一种新的、在多小波域中基于矢量融合的图像融合算法,充分利用多小波变换域系数矢量内部各个分量的相关性来提高融合质量。两波段真实多光谱图像融合实验结果表明,与单小波标量融合方法相比,多小波矢量融合算法获得的图像具有较优的视觉效果和客观评价指标,从而证明了用于图像融合时,多小波较之单小波更适合于人类视觉系统,具有广泛的应用前景。     

Discrete wavelet transform based on cyclic convolutions

A filterbank in which cyclic convolutions are used in place of linear convolutions will be referred to as a cyclic convolution filterbank (CCFB). A dyadic tree-structured CCFB can be used to perform a discrete wavelet transform suitable for coding based on symmetric extension methods. This paper derives two types of efficient implementation techniques for the tree-structured CCFB: one using complex arithmetic and one using only real arithmetic. In addition, the present paper analyzes in detail the perfect reconstruction (PR) condition for the two-channel CCFB and shows that this condition is much less restrictive than that of usual two-channel filterbanks. When each of the two-channel CCFBs constituting a tree-structured CCFB is designed to be PR, the whole system is PR. A quadrature mirror filter (QMF) CCFB having the PR property is demonstrated to be easily designed using a standard filter design subroutine. In contrast, designing a linear-phase PR FIR QMF bank that has good frequency response is not possible when filters are realized by linear convolutions.     

一种高阶线性FIR滤波器设计

为了解决高阶线性FIR滤波器占用查找表资源过多的问题,提出了一种采用对称查找表的分布式结构。利用线性FIR滤波器系数对称的特点,设计了深度更小的对称查找表。采用时分复用技术和流水线技术,有效节约了查找表资源,提高了FIR滤波器的运行频率。在Xilinx XC5VLX110T FPGA芯片上,实现了1 023阶的基于对称查找表的FIR滤波器。结果表明,相比于分段查找表结构,对称查找表结构的FIR滤波器节约了48%的Block Rom资源,提升了15%的最高时钟频率。     

Reduction of Symmetric Complex Filters

Due to their linear-phase property, symmetric filters are an interesting class of finite-impulse-response (FIR) filters. Moreover, symmetric FIR filters allow an efficient implementation. In this paper we extend the classical definition of Hermitian symmetry to a more general symmetry that is also applicable to complex filters. This symmetry is called generalized-Hermitian symmetry. We show the usefulness of this definition as it allows for a unified treatment of even and odd-length filters. Central in this paper is a theorem on the reduction of generalized-Hermitian-symmetric filters to Hermitian-symmetric filters, both with finite precision coefficients. A constructive proof of this theorem is presented and an associated procedure for reducing generalized-Hermitian-symmetric filters is derived. Two of the examples show the application of the reduction procedure and the achieved savings on arithmetic costs. Finally, all three examples show that a special instance of the generalized-Hermitian-symmetric filters with finite precision coefficients, may have lower arithmetic costs than the Hermitian-symmetric filter from which it is derived.     

Optimal design of finite precision FIR filters using linearprogramming with reduced constraints

An algorithm for the design of optimal one-dimensional (1-D) and two-dimensional (2-D) FIR filters over a discrete coefficient space is proposed. The algorithm is based on the observation that the equiripple frequencies of a subproblem (SP) in the branch and bound (BaB) algorithm are closely related to those of neighboring SPs. By using the relationship among the SPs, the proposed algorithm reduces the number of constraints required for solving each SP. Thus, the overall computational load for the design of FIR filters with discrete coefficients is significantly alleviated, compared with the conventional BaB algorithm     

A study of VLSI symmetric FIR filter structures

Symmetric FIR filters, which provide linear phase, are frequently used in digital signal processing. This paper presents a study of structures which take into account the particularities of the symmetry. A novel systolic structure is proposed. An efficient design is presented for the basic module which computes(x±y)h+G and which is only specific to symmetric filters. Special applications of carry-save or signed-digit notations, and tree-type multi-operand adders are shown to significantly enhance the computational speed of symmetric FIR filters.     

Novel Area-Efficient FPGA Architectures for FIR Filtering With Symmetric Signal Extension

This paper presents four novel area-efficient field-programmable gate-array (FPGA) bit-parallel architectures of finite impulse response (FIR) filters that smartly support the technique of symmetric signal extension while processing finite length signals at their boundaries. The key to this is a clever use of variable-depth shift registers which are efficiently implemented in Xilinx FPGAs in the form of shift register logic (SRL) components. Comparisons with the conventional architecture of FIR filter with symmetric boundary processing show considerable area saving especially with long-tap filters. For instance, our architecture implementation of the 8-tap low Daubechies-8 FIR filter achieves ~ 30% reduction in the area requirement (in terms of slices) compared to the conventional architecture while maintaining the same throughput. Two of the above-cited novel architectures are dedicated to the special case of symmetric FIR filters. The first architecture is highly area-efficient but requires a clock frequency doubler. While this reduces the overall processing speed (to a maximum of 2), it does maintain a high throughput. Moreover, this speed penalty is cancelled in bi-phase filters which are widely used in multirate architectures (e.g., wavelets). Our second symmetric FIR filter architecture saves less logic than the first architecture (e.g., 10% with the 9-tap low Biorthogonal 9&7 symmetric filter instead of 37% with the first architecture) but overcomes its speed penalty as it matches the throughput of the conventional architecture.     

基于模块局部可重构FIR滤波器设计

介绍了一种使用动态局部重构技术设计可重构FIR数字滤波器的方法,是一种注重面积效率高、灵活性强,允许动态插入或删除局部模块的方法,并在Xilinx Virtex-4 FPGA上实现.这种设计方法比传统的FIR滤波器的设计方法具有占用资源更少、重构时间更短、高速灵活性等优点.     

Optical half-band filters

This paper proposes two kinds of novel 2×2 circuit configuration for finite-impulse response (FIR) half-band filters. These configurations can be transformed into each other by a symmetric transformation and their power transmittance is identical. The configurations have only about half the elements of conventional FIR lattice-form filters. We derive a design algorithm for achieving desired power transmittance spectra. We also describe 2×2 circuit configurations for infinite-impulse response (IIR) half-band filters. These configurations are designed to realize arbitrary-order IIR half-band filter characteristics by extending the conventional half-band circuit configuration used in millimeter-wave devices. We discuss their filter characteristics and confirm that they have a power half-band property. We demonstrate design examples including FIR maximally flat half-band filters, an FIR Chebyshev half-band filter, and an IIR elliptic half-band filter     

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     

Two-Dimensional Farrow Structure and the Design of Variable Fractional-Delay 2-D FIR Digital Filters

In this paper, a 2-D Farrow structure is proposed and used to implement variable fractional-delay (VFD) 2-D FIR digital filters. Compared with the existing literature, the desired response of a VFD 2-D digital filter is analyzed in detail, and it is found that there are four types of 2-D symmetric/antisymmetric sequences that need to be used for the design of VFD 2-D FIR digital filters. Moreover, due to the orthogonality among the approach functions, the four types of 2-D sequences can be determined independently, such that the dimension for each computation can be reduced drastically. For simplicity, only the designs of even–even- and odd–odd-order VFD 2-D filters are presented in this paper, and the other cases can be achieved in the same manner. To reveal the coefficient characteristics, the symmetric/antisymmetric properties of filter coefficients and the relationships between coefficients are all tabulated. Also, design examples such as nonseparable circularly symmetric low-pass VFD filters are presented to demonstrate the effectiveness of the proposed method.      

FIR digital filter design techniques using weighted Chebyshev approximation

This paper discusses the various approaches to designing FIR digital filters using the theory of weighted Chebyshev approximation. The different design techniques are explained and compared on the basis of their capabilities and limitations. The relationships between filter parameters are briefly discussed for the case of low-pass filters. Extensions of the theory to the problems of magnitude and complex approximation are also included, as are some recent results on the design of two-dimensional FIR filters by transformation.     

A Novel Diversity-Controlled Genetic Algorithm for Rapid Optimization of Bandpass FRM FIR Digital Filters Over CSD Multiplier Coefficient Space

The conventional frequency response masking (FRM) approach is one of the most well-known techniques for the design of sharp transition band finite impulse response (FIR) digital filters. The resulting FRM digital filters permit efficient hardware implementations due to an inherently large number of zero-valued multiplier coefficients. The hardware complexity of these digital filters can further be reduced by representing the remaining (non-zero) multiplier coefficient values by using their canonical signed-digit (CSD) representations. This paper presents a novel diversity-controlled (DC) genetic algorithm (GA) for the discrete optimization of bandpass FRM FIR digital filters over the CSD multiplier coefficient space. The resulting bandpass FIR digital filters are permitted to have equal or unequal lower and upper transition bandwidths. The proposed DCGA is based on an indexed look-up table of permissible CSD multiplier coefficients such that their indices form a closed set under the genetic operations of crossover and mutation. The salient advantage of DCGA over the conventional GA lies in the external control over population diversity and parent selection, giving rise to a rapid convergence to an optimal solution. The external control is achieved through the judicious choice of a pair of DCGA optimization parameters. An empirical investigation is undertaken for choosing appropriate values for these control parameters. The convergence speed advantages of the DCGA are demonstrated through its application to the design and optimization of a pair of bandpass FRM FIR digital filters with equal or arbitrary lower and upper transition bandwidths. In both cases, an increase of about an order of magnitude in the speed of convergence is achieved as compared to the conventional GAs.