Wavelets and filter banks: theory and design

The wavelet transform is compared with the more classical short-time Fourier transform approach to signal analysis. Then the relations between wavelets, filter banks, and multiresolution signal processing are explored. A brief review is given of perfect reconstruction filter banks, which can be used both for computing the discrete wavelet transform, and for deriving continuous wavelet bases, provided that the filters meet a constraint known as regularity. Given a low-pass filter, necessary and sufficient conditions for the existence of a complementary high-pass filter that will permit perfect reconstruction are derived. The perfect reconstruction condition is posed as a Bezout identity, and it is shown how it is possible to find all higher-degree complementary filters based on an analogy with the theory of Diophantine equations. An alternative approach based on the theory of continued fractions is also given. These results are used to design highly regular filter banks, which generate biorthogonal continuous wavelet bases with symmetries     

The design of 2-D nonseparable directional perfect reconstruction filter banks

In this paper, we develop a directional 2-D nonseparable filter bank that can perfectly reconstruct the downsampled subband signals. The filter bank represents two powerful image and video processing tools: directional subband decomposition and perfect reconstruction. The directional filter banks consist of (1) the input signal and the subband signals modulation, (2) diamond shape prefilter, and (3) four different parallelogram shape prefilters. This paper addresses the design and implementation of a two-band filter bank that is proved to be able to provide perfect reconstruction of the downsampled subband signals. Finally, we use a conventional 1-D half-band filter as a prototype and then apply the McClellan transform for the specific 2-D diamond shape and parallelogram shape subfilters. This method is extremely simple in designing the analysis/synthesis subfilters for the filter bank.     

Nonuniform filter banks: a reconstruction and design theory

A general procedure for the design of analysis-synthesis systems based on nonuniform filter banks is described. The procedure is based on a time-domain analysis of nonuniform systems, which results in a set of conditions for the exact reconstruction of the input signal at the output. These conditions are used as part of a powerful iterative algorithm for designing finite impulse response (FIR) filter banks with an arbitrary nonuniform frequency resolution. This new framework permits the design of systems with arbitrary rational decimation rates in different bands. Systems based on maximally or nonmaximally decimated filter banks, on low and minimum delay systems, and on block decimators are also among the systems that can be designed using this method     

Optimal McClellan transformation and its application to 2-D FIR digial filter design

A new method to choose the coefficients of the McClellan transformation is developed. The coefficients are obtained by using both analytic and nonlinear optimization approaches. The transformation using the proposed method has much better approximation performance for circular symmetry than the existing methods, especially when the cutoff frequency of the 2-D filter is very large. The proposed method also applies to choose the transformation coefficients to approximate elliptic contours. Two design examples of 2-D FIR digital filters demonstrate the good performance of the proposed method.     

A novel nonnegative decomposition method and its application to 2-D digital filter design

In designing two-dimensional (2-D) digital filters in the frequency domain, an efficient technique is to first decompose the given 2-D frequency domain design specifications into one-dimensional (1-D) ones, and then approximate the resulting 1-D magnitude specifications using the well-developed 1-D filter design techniques. Finally, by interconnecting the designed 1-D filters one can obtain a 2-D digital filter. However, since the magnitude responses of digital filters must be nonnegative, it is required that the decomposition of 2-D magnitude specifications result in nonnegative 1-D magnitude specifications. We call such a decomposition the nonnegative decomposition. This paper proposes a nonnegative decomposition method for decomposing the given 2-D magnitude specifications into 1-D ones, and then transforms the problem of designing a 2-D digital filter into that of designing 1-D filters. Consequently, the original problem of designing a 2-D filter is significantly simplified.     

正交镜象滤波器组的理论及应用

正交镜象滤波器组(QMF)是一种新型的滤波系统.与普通滤波器组比较,用它实现对信号的频域分析及综合时可大大减少所需的计算量.本文除阐述了双信道及多信道QMF的概念及频响要求之外,还对有关的设计理论及执行方式作了简述.最后对QMF的主要应用领域作了介绍.     

Some difficulties with the double bilinear transformation in 2-D recursive filter design

It is shown that the double bilinear transformation approach for designing a 2-D IIR digital transfer function from a predetermined 2-D analog transfer function, may in certain cases lead to unstable solutions.     

2-D adaptive CPWL filter for 2-D nonlinear channel equalisation and image restoration

A 2-D adaptive piecewise-linear equaliser is proposed. As an alternative of the 2-D adaptive Volterra equaliser, the proposed equaliser has advantages in its suitability for cases of strong nonlinearity and saving implementation and computation cost. An experiment examples is presented which demonstrates the superior behaviour of this method over either a linear method or the Volterra method in inverse modelling an unknown 2-D channel with blurring and a common point-wise nonlinearity and restoring images degraded by this channel.<>     

General formulation of shift and delta operator based 2-D VLSI filter structures without global broadcast and incorporation of the symmetry

Having local data communication (without global broadcast of signals) among the elements is important in very large scale integration (VLSI) designs. Recently, 2-D systolic digital filter architectures were presented which eliminated the global broadcast of the input and output signals. In this paper a generalized formulation is presented that allows the derivation of various new 2-D VLSI filter structures, without global broadcast, using different 1-D filter sub-blocks and different interconnecting frameworks. The 1-D sub-blocks in z-domain are represented by general digital two-pair networks which consist of direct-form or lattice-type FIR filters in one of the frequency variables. Then, by applying the sub-blocks in various frameworks, 2-D structures realizing different transfer functions are easily obtained. As delta discrete-time operator based 1-D and 2-D digital filters (in \(\gamma \) -domain) were shown to offer better numerical accuracy and lower coefficient sensitivity in narrow-band filter designs when compared to the traditional shift-operator formulation we have covered both the conventional z-domain filters as well as delta discrete-time operator based filters. Structures realizing general 2-D IIR (both z- and \(\gamma \) -domains) and FIR transfer functions (z-domain only) are presented. As symmetry in the frequency response reduces the complexity of the design, IIR transfer functions with separable denominators, and transfer functions with quadrantal magnitude symmetry are also presented. The separable denominator frameworks are needed for quadrantal symmetry structures to guarantee BIBO stability and thus presented for both the operators. Some limitations of having exact symmetry with separable 1-D denominator factors are also discussed.     

LMS自适应滤波器的设计理论研究

主要介绍了自适应滤波器的基本理论思想,具体阐述了自适应滤波器的基本原理、算法及设计方法。首先介绍自适应滤波器的原理,然后对FIR结构滤波器做了详细的阐述,自适应算法是整个系统的核心,重点对LMS算法的设计方法,设计步骤做了分析,最后对LMS算法进行MATLAB仿真,根据性能评价标准和实验结果表明,该自适应滤波器滤波效果优越。     

On the use of continued fraction filter structures for the design of 2-D digital filters

In this paper a method for the realization of 2-D recursive digital filters of second order by means of continued fraction filter structures is presented. The proposed method is based on the interconnection of basic filter structures of first order and gives also the set of all 2-D transfer functions that are realizable by these structures. The general rule for interconnecting these basic structures is described. It is proved that, on the set of second order fraction filter structures, some classes of structures and ordering relations can be defined, so that the set of all the classes form a partially ordered set (poset). Those structures which least constrain the coefficients of the transfer function are defined maximum elements of the poset. It is shown that among all possible continued fraction structures of second order only six are suitable for realization of two dimensional filters. Some results following from the use of spectral transformations are discussed.     

The nonsubsampled contourlet transform: theory, design, and applications.

In this paper, we develop the nonsubsampled contourlet transform (NSCT) and study its applications. The construction proposed in this paper is based on a nonsubsampled pyramid structure and nonsubsampled directional filter banks. The result is a flexible multiscale, multidirection, and shift-invariant image decomposition that can be efficiently implemented via the à trous algorithm. At the core of the proposed scheme is the nonseparable two-channel nonsubsampled filter bank (NSFB). We exploit the less stringent design condition of the NSFB to design filters that lead to a NSCT with better frequency selectivity and regularity when compared to the contourlet transform. We propose a design framework based on the mapping approach, that allows for a fast implementation based on a lifting or ladder structure, and only uses one-dimensional filtering in some cases. In addition, our design ensures that the corresponding frame elements are regular, symmetric, and the frame is close to a tight one. We assess the performance of the NSCT in image denoising and enhancement applications. In both applications the NSCT compares favorably to other existing methods in the literature.     

CAS-CAS transform for 2-D signals: A few applications

A few applications of a separable Hartley-like (CAS-CAS) transform in two-dimensional (2-D) signal processing is presented. The applications discussed include (i) the interpolation of signals, (ii) the computation of Hilbert transform, and (iii) the complex cepstrum computation. The computational advantage of the proposed methods over the algorithms using 2-D FFT are discussed.     

Transient image theory for 2-D and 3-D conducting wedge problems

In the present paper, two- and three-dimensional transient scattering from a perfectly conducting wedge is studied. The exciting sources analyzed are an infinite impulsive line current parallel to edge of the conductor and an arbitrarily oriented electric dipole. The analysis presented is based on the time-harmonic image solution found for the scattered field. The frequency-domain image current corresponding to the field contribution from the wedge is an exact function expressed in terms of simple trigonometric functions. It is dependent only on the coordinates and the amplitude of the source, but it is independent of the observation point. It is shown that the transient scattering can be interpreted as arising from an image source by generalizing the time domain Green's function to complex source points     

2-D periodic leaky-wave Antennas-part II: slot design

The far-field radiation patterns of a two-dimensional (2-D) periodic slot leaky-wave antenna (LWA) are studied. The antenna consists of a two-dimensional periodic array of slots in a conducting plane that is printed on top of a grounded dielectric slab. The antenna is excited by a simple source such as a dipole inside the slab. Reciprocity along with the spectral-domain method is used to calculate the far-field pattern, and the radiation characteristics of the structure are investigated. A comparison between the present periodic slot LWA and a 2-D periodic patch LWA discussed in Part I is given to show the advantages of the slot antenna for certain applications. The slot LWA can achieve high directivity patterns, and a circularly-polarized version of the antenna can achieve good circular-polarization at broadside.     

General IIR optical filter design for WDM applications usingall-pass filters

A general design algorithm is presented for infinite impulse response (IIR) bandpass and arbitrary magnitude response filters that use optical all-pass filters as building blocks. Examples are given for an IIR multichannel frequency selector, an amplifier gain equalizer, a linear square-magnitude response, and a multi-level response. Major advantages are the efficiency of the IIR filter compared to finite impulse response (FIR) filters, the simplicity of the optical architecture, and its tolerance for loss. A reduced set of unique operating states is discussed for implementing a reconfigurable multichannel selection filter     

RoF中AWG周期滤波特性应用的Optisystem设计

采用光通信仿真软件Optisystem,对基于阵列波导光栅(AWG)周期滤波特性的RoF系统用复用器和解复用器进行了仿真设计,表明基于单一AWG可实现无线(或混合)宽带RoF系统的复用和(或)解复用功能.     

二维DCT图像处理器的低功耗实现

本文提出了一种基于矩阵向量乘法器的低功耗二维DCT结构,该结构通过最大限度地共享矩阵向量乘法中的乘积因子降低二维DCT中的乘法计算量,实现低功耗计算.此外,该二维DCT设计支持对矩阵向量乘法器的计算精度控制,从而实现对二维DCT处理器的低功耗调整.FPGA硬件平台的实际验证结果表明,与传统的基于移位累加乘法器的二维DCT设计相比,本设计可以节省35%以上的功耗.     

Analytical design of 3-D wavelet filter banks using themultivariate Bernstein polynomial

The design of 3-D multirate filter banks where the downsampling/upsampling is on the FCO (face centred orthorhombic) lattice is addressed. With such a sampling lattice, the ideal 3-D sub-band of the low-pass filter is of the TRO (truncated octahedron) shape. The transformation of variables has been shown previously to be an effective technique for designing M-D (multidimensional) filter banks. A design technique is presented for the transformation function using the multivariate Bernstein polynomial which provides a good approximation to the TRO sub-band shape. The method is analytically based and does not require any optimisation procedure. Closed form expressions are obtained for the filters of any order. Another advantage of this technique is that it yields filters with a flat frequency response at the aliasing frequency (ω₁, ω₂ , ω₃)=(π, π, π). This flatness is important for giving regular discrete wavelet transform systems