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.     

Design of separable-denominator 2-D digital filters using the reduced-dimensional decomposition model

A separable-denominator 2-D digital filter (SD-2DDF) can be decomposed into the cascade form of a pair of 1-D digital filters (1DDFs) with different delay elements. Based on this reduced-dimensional decomposition, in this paper, we propose a new technique for designing SD-2DDFs in the spatial domain. The technique determines the coefficient matrices of 1DDFs by nonlinear optimization techniques first, and then a SD-2DDF can be easily synthesized. In addition, since the existent 1-D linear system realization techniques can be used to choose a good starting point for the optimization, extremely accurate design results can be easily achieved.     

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.     

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。     

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.     

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 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 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.     

一种变时间常数自动增益控制环路与算法设计

针对时间常数恒定的自动增益控制环路的不足,设计了一种时间常数可变的自动增益控制结构,实现了环路响应时间能够自适应地随着输入信号的增大而减小,在一定程度上克服了恒定时间常数自动增益控制环路的不足。给出了其核心算法并进行了详细推导,在理论上证明了该结构的稳定性以及算法的有效性。最后,对提出的结构以及算法进行了仿真。仿真结果表明,本文所设计的结构与提出的算法既能对大电平信号迅速响应,又能保证足够小的反调制失真。     

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。     

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。     

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     

Local controllability and minimum energy control of continuous 2-D linear systems with variable coefficients

Necessary and sufficient conditions for the local controllability of continuous 2-D linear systems with variable coefficients are established. The minimum energy control problem is formulated and solved for the 2-D systems.     

A Spatial-Domain Technique for the Design of 3-D Separable-Denominator State-Space Digital Filters

In this paper,a technique for designing 3-D separable-denominator state-spacedigital filters is developed. The design process is divided intotwo phases. First, the coefficient matrices related to stabilityare constructed for the filter to be stable by using alternatingvariable method. Next, the other matrices are obtained by solvinglinear equations. These phases are repeated until there is nosignificant change in the squared error function.     

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。     

具有任意幅度频响的二维线性相位FIR数字滤波器的设计

本文提出了设计任意幅度频响的二维线性相位ＦＩＲ数字滤波器的解析最小二乘方法，通过最小化频域平方误差函数得到了滤波器系数的闭式解，运用导出的闭式式，可根据给定的任意幅度频响指标直接计算滤波器的系数，从而简化了滤波器设计程序，并大大降低了运算量。     

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.     

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.     

Design for 2-D error feedback networks with identical coefficient sets

In realization of recursive digital filters with fixed point arithmetic, an error caused by roundoff arises. It is known that the level of the roundoff noise of an IIR filter tends to be high when the poles are close to the unit circle. Error feedback (EF) is an effective method to reduce the roundoff noise. It is desirable to design an EF network using as few parameters as possible in order to keep computational costs low. In this paper, we propose a method for designing a 2D EF network with identical coefficient sets. That is, the EF coefficients are divided into several subsets such that all the elements within each set have the same absolute value. In order to optimize the coefficient sets, we propose an algorithm by using the genetic algorithm. In the numerical example, we demonstrate the effectiveness of the proposed method.     

二维线性相位FIR数字滤波器的优化设计

该文提出了一种用神经网络算法来设计二维线性相位数字滤波器的新方法。通过分析二维FIR线性相位滤波器的幅频响应特性,建立了神经网络算法。根据给定的幅频响应指标,按该算法可获得滤波器系数。为保证该算法的稳定性,提出并证明了该算法的收敛定理。文中给出了圆对称和矩形对称二维低通线性相位FIR数字滤波器优化设计实例。计算机仿真结果表明由该方法设计的二维数字滤波器,通带和阻带范围波动小,所需计算量非常少,稳定性强,因而是一种优异的设计方法。