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.     

Implementation of 2-D denominator-separable digital filters using first-order all-pass sections

It is established that denominator-separable transfer functions which characterize an important subclass of 2-D filters can be expressed as a linear combination of first-order (1-D or 2-D separable) all-pass transfer functions with real or complex coefficients. This type of expansion is referred to as all-pass expansion of the corresponding transfer function. Based on this all-pass expansion, we derive some efficient structures for the realization of 2-D denominator-separable filters using all-pass sections.On leave from S.V. University College of Engineering. Tirupati-517502, India.     

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.     

Design of zero-phase recursive 2-D variable filters with quadrantal symmetries

The digital filters with adjustable frequency-domain characteristics are called variable filters. Variable filters are useful in the applications where the filter characteristics are needed to be changeable during the course of signal processing. In such cases, if the existing traditional constant filter design techniques are applied to the design of new filters to satisfy the new desired characteristics when necessary, it will take a huge amount of design time. So it is desirable to have an efficient method which can fast obtain the new desired frequency-domain characteristics. Generally speaking, the frequency-domain characteristics of variable filters are determined by a set of spectral parameters such as cutoff frequency, transition bandwidth and passband width. Therefore, the characteristics of variable filters are the multi-dimensional (M-D) functions of such spectral parameters. This paper proposes an efficient technique which simplifies the difficult problem of designing a 2-D variable filter with quadrantally symmetric magnitude characteristics as the simple one that only needs the normal one-dimensional (1-D) constant digital filter designs and 1-D polynomial approximations. In applying such 2-D variable filters, only varying the part of 1-D polynomials can easily obtain new desired frequency-domain characteristics.     

Design of recursive variable digital filters with theoretically guaranteed stability

Stability is one of the most concerned issues in designing a recursive variable digital filter (VDF). This is because the coefficients of a recursive VDF constantly vary in the tuning process, and updating the coefficients may incur instability. Thus, an appropriate measure needs to be taken for ensuring its stability. This paper presents a new coefficient transformation (CT) method for transforming the coefficients of a recursive transfer-function denominator into a set of new coefficients. From the viewpoint of conventional constant-coefficient filter (constant filter) design, the new coefficients can take arbitrary values without incurring instability. For designing a stable VDF, we apply this CT to the variable case and approximate each transformed coefficient as a distinct polynomial in the tuning parameter. Thus, we can change the filter coefficients by changing the value of the tuning parameter, and thus tune the magnitude response. Thanks to the proposed CT, updating the filter coefficients will never incur instability. This is the core part of the CT-based design approach. In this paper, we utilise a weighting function to ignore the transition-band errors and thus enhance the design accuracy of important frequency bands (passband and stopband). Moreover, the polynomials use different degrees so as to reduce the VDF complexity. Two design examples (lowpass VDF and bandpass VDF) are provided for verifying the design accuracy and checking the stability.     

A technique for the design of 2-D recursive digital filters with guaranteed stability

In the present technique, the denominator and numerator in the filter are designed separately and recursively. First, the denominator coefficients, derived from the stable matrix of the Roesser model (the Fornasini-Marchesini second model), are found by minimizing a performance index through the alternating variable method. The numerator coefficients are then determined analytically by solving linear simultaneous equations. The above process will be repeated until there is negligible change in the objective function. Finally, a numerical example is given to illustrate the utility of the proposed technique.     

具有任意角度的二维扇形FIR数字滤波器的设计

扇形数宇滤波器由于其良好的方向性而得到广泛的应用.本文导出了一组利用McClellan变换法设计二维零相位扇形数字滤波器的公式.利用该组公式设计的扇形数字滤波器的形状在均方意义下是最佳的.由于设计公式化,避免了设计中的优化过程,因而设计过程简单,设计所需的时间少。     

Implementation of 2-D pseudo-rotated digital filters using a forward lattice structure

The 1-D FDLS shows the localized feedback property and is suitable for modular and concurrent implementation. It is known that the 1-D FDLS shows interesting properties with respect to finite word-length effects. In this paper, a new result is given for the estimation of the lower and upper bound of the variance of the roundoff noise. It is presented how the FDLS can be incorporated to implement 2-D pseudo-rotated digital filters. The 1-D roundoff noise analysis is extended to the 2-D case. It is indicated how 2-D filter banks can be derived from the FDLS.     

On 2-D partial fraction decomposition

In this paper we introduce a mapping technique of two-dimensional (2-D) auto correlation sequences to one dimensional (1-D) auto correlation sequences and obtain certain results useful in the test for existence of the decomposition of the form

$\text{A(Z}{}_1\text{,Z}{}_2\text{)A(Z}{}_1{}^{\mathrm{?1}}\text{)}\text{B(Z}{}_1\text{,Z}{}_2\text{)B(Z}{}_1{}^{\mathrm{?1}}\text{)}\text{=}\text{P(Z}{}_1\text{,Z}{}_2\text{)}\text{B(Z}{}_1\text{,Z}{}_2\text{)}\text{+}\text{P(Z}{}_1{}^{\mathrm{?1}}\text{)}\text{B(Z}{}_1{}^{\mathrm{?1}}\text{)}$

for a stable 2-D transfer function of a recursive digital filter defined by

$\text{H(Z}{}_1\text{,Z}{}_2\text{)=}\text{A(Z}{}_1\text{,Z}{}_2\text{)}\text{B(Z}{}_1\text{,Z}{}_2\text{)}$

These results are simple alternative to that reported in the literature [7] earlier.     

DESIGN OF POLYPHASE WAVE DIGITAL FILTERS WITH APPROXIMATELY LINEAR PHASE

It is well known that IIR digital filters require quite fewer computations,comparedwith FIR filters,in order to meet stringent magnitude specifications when the phase distortioncan be tolerated.An approximately linear phase,however,can be also obtained with the IIRfilter by making use of a technique without increasing the complexity.Based on a certain numberof attenuation zeros in the pass band,a new approach is developed for the design of polyphasewave digital filters with exact magnitude responses and Chebyshev approximation of the desiredphase responses.The minimum number of attenuation zeros is estimated,and some examples areincluded.     

Lyapunov Equation for the Stability of 2-D Systems

An analogue of the characterization of asymptotic stability of the 1-D systems by the solvability of associated Lyapunov equation is proposed here for 2-D systems. It is shown that internal stability of Roesser model is equivalent to the feasibility of some linear matrix inequality (LMI), related to quadratic Lyapunov functions.     

二维数字系统的系数灵敏度和结构优化

本文从二维数字系统的Roesser模型出发,在Kawamata等人定义的灵敏度的基础上,导出了系统传输函数对全体系数的标量灵敏度的新的表达式,并给出了一种降低灵敏度的优化方法,该优化方法不仅可以降低系数灵敏度,还可以提高结构的效率.举例结果表明,新的表达式与Kawamata等人的表达式相比,能更好地与计算机模拟结果相符.     

线性相位二维跳耦型数字滤波器组设计及其在图像子带编码中的应用

本文提出了一种新的二维递归数字滤波器组－二维跳耦型数字滤波器组的设计与实现方法。这种数字滤波器组的特点为：因果型－可用于实时图像分解与重构；线性相位－可避免传统的二维ＩＩＲ滤波引起的图像的相位畸变，非线性条件下稳定－有限字长实现二维数字滤波器组时，不存在溢出或极限环振荡，实际的图像处理实验证明，本文提出的设计理论及实现方法是正确的，有效的，可行的。     

一种实用高效的二维FIR滤波器

本文提出了一个适用的、高效和的二维FIR数字滤波器的设计新方法,此方法的基本原理是冲击响应舍入法,它与传统的设计方法相比,大大地节省了计算时间,提出了效率。文中详细讨论了新滤波器的实用性,并通过在服装逢缝处理皱褶信号获取的信号处理上的应用,证实了它的可行性。     

基于巴特沃斯逼近的二维IIR数字滤波器的设计

本文给出了一种基于巴特沃斯逼近的二维ＩＩＲ数字滤波器的设计方法，得到了由基本的全通节级联，并联实现的各种二维滤波器函数，包括，镜象对称互补滤波器，扇形滤波器和具有任意矩形通、阻带的滤波器，结果表明，这种实现结构具有通有灵敏度低、滤波器系数少的优点，并且由于巴特沃斯逼近的最大平坦性，得到的滤波器具有良好的相位特性。     

Realization of 2-D Variable IIR Digital Filter Structures with a Small Amount of Calculations for Coefficient Update

This paper proposes 2-D variable IIR digital filter structures with a small amount of calculations for coefficient update. The proposed realization method uses the 2-D parallel allpass structure derived from the separable denominator 2-D filter as the prototype structure for 2-D variable digital filters. In order to reduce the amount of calculations, all the redundant first-order complex allpass sections are combined by modularization of the variable structure. Furthermore, we can realize a very compact variable structure with a minimal number of first-order complex allpass sections by combining complex allpass sections with their complex conjugate allpass sections. Comparison of the calculation loads of the variable structures is presented to demonstrate that the amount of calculations for coefficient update of the proposed variable structure is far less than that of the original and the modular variable structure.     

On the Positive Definite Solutions to the 2-D Continuous-time Lyapunov Equation

The very strict positive real lemma is further developed for nonminimal 1-D continuous-time systems and is used to study the 2-D continuous-time Lyapunov equation. Based on it, an extended condition for the bivariate characteristic polynomial of a matrix to be very strict Hurwitz is proposed for general 2-D analog systems with characteristic polynomials involving 1-D factor polynomials. It is also shown that in such a case the bivariate polynomial can be decomposed into a 2-D bivariate polynomial with the corresponding matrix satisfying certain controllability and observability conditions and into up to two 1-D polynomials. Further, two algorithms for computing the positive definite solutions to the 2-D Lyapunov equation are presented.     

四方向上的任意角度扇形数字滤波器的设计

具有任意角度（扇形所围成的角度）和任意方向（扇形所取方向）的扇形数字滤波器在图像数据压缩和地质、地震等方面的数据处理中有着广泛的应用。目前，大多数的工作多集中于水平及垂直方向的９０°扇形数字滤波器的设计，因而有必要研究任意方向上的任意角度的扇形数字滤波器的设计。本文介绍利用ＭｃＣｌｅｌｌａｎ交换法设计具有垂直，水平及±４５°方向上的任意角度的扇形数字滤波器的设计方法，垂直水平方向上的扇形滤波器的设     

Algebraic necessary and sufficient conditions for the very strict Hurwitz property of a 2-D polynomial

Necessary and sufficient conditions for the bi-variate characteristic polynomial of a matrix to be very strict Hurwitz (VSHP) are presented. These conditions are based on solving the Lyapunov equation for 2-D continuous systems using the Kronecker product and lead to a simple test for the VSHP property. It requires testing the eigenvalues of three constant matrices only and is simpler than existing polynomial tests.This work was supported by NSERC for the first author.