A fast matrix iterative technique for the WLS design of 2-D quadrantally symmetic FIR filters

High computational complexity is a major problem encountered in the optimal design of two-dimensional (2-D) finite impulse response (FIR) filters. In this paper, we present an iterative matrix solution with very low complexity to the weighted least square (WLS) design of 2-D quadrantally symmetric FIR filters with two-valued weighting functions. Firstly, a necessary and sufficient condition for the WLS design of 2-D quadrantally symmetric filters with general nonnegative weighting functions is obtained. Then, based on this optimality condition, a novel iterative algorithm is derived for the WLS design problem with a two-valued weighting function. Because the filter parameters are arranged in their natural 2-D form and the transition band is not sampled, the computation amount of the proposed algorithm is reduced significantly, especially for high-order filters. The exponential convergence of the algorithm is established, and its computational complexity is estimated. Design examples demonstrating the convergence rate and solution accuracy of the algorithm, as well as the relation between the iteration number of the algorithm and the size and transition-band width of the filter are given.     

Weighted least mean square design of 2-D FIR digital filters: thegeneral case

This paper solves the weighted least mean square (WLMS) design of two-dimensional (2-D) finite impulse response (FIR) filters with general half plane symmetric frequency responses and nonnegative weighting functions. The optimal solution is characterized by a pair of coupled integral equations, and the existence and uniqueness of the WLMS solution for 2-D FIR filter design are established. Two efficient numerical algorithms using a 2-D fast Fourier transform (FFT) are proposed to solve the WLMS solution. One is based on the contraction mapping and fix point theorem characterizing the coupled integral equation; the other uses conjugate gradient techniques, which guarantees finite convergence. The associated computational complexity is analyzed and compared with existing algorithms. Examples are used to illustrate the effectiveness of the proposed design algorithms. The selection of weighting functions to improve the minimax performance of the filter is also discussed     

A fast design algorithm for elliptic-error and phase-error constrained LS 2-D FIR filters

Two-dimensional (2-D) nonlinear-phase finite impulse response (FIR) filters have found many applications in signal processing and communication systems. This paper considers the elliptic-error and phase-error constrained least-squares design of 2-D nonlinear-phase FIR filters, and develops a matrix-based algorithm to solve the design problem directly for the filter’s coefficient matrix rather than vectorizing it first as in the conventional methods. The matrix-based algorithm makes the design to consume much less design time than existing algorithms. Design examples and comparisons with existing methods demonstrate the effectiveness and high efficiency of the proposed design method.     

Design of 2-D Linear Phase Digital Filters Using Schur Decomposition and Symmetries

In this paper we present a new and numerically efficient technique for designing 2-D linear phase octagonally symmetric digital filters using Schur decomposition method (SDM) and the diagonal symmetry of the 2-D impulse response specifications. This technique is based on two steps. First, the 2-D impulse response matrix is decomposed into a parallel realization of k sections, each comprising two cascaded linear phase SISO 1-D FIR digital filters. It is shown that using the symmetry property of the 2-D impulse response matrix and the fact that the left and right eigenspaces obtained by SDM are transpose of each other, the design problem of two 1-D digital filters is reduced to the design problem of only one 1-D digital filter in each section.     

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。     

Realization of 2-D linear-phase FIR filters by using thesingular-value decomposition

The singular-value decomposition (SVD) technique is investigated for the realization of a general two-dimensional (2-D) linear-phase FIR filter with an arbitrary magnitude response. A parallel realization structure consisting of a number of one-dimensional (1-D) FIR subfilters is obtained by applying the SVD to the impulse response of a 2-D filter. It is shown that by using the symmetry property of the 2-D impulse response and by developing an appropriate unitary transformation, an SVD yielding linear-phase constituent 1-D filters can always be obtained so that the efficient structures of the 1-D linear-phase filters can be exploited for 2-D realization. It is shown that when the 2-D filter to be realized has some specified symmetry in its magnitude response, the proposed SVD realization would yield a magnitude characteristic with the same symmetry. An analysis is carried out to obtain tight upper bounds for the errors in the impulse response as well as in the frequency response of the realized filter. It is shown that the number of parallel sections can be reduced significantly without introducing large errors, even in the case of 2-D filters with nonsymmetric magnitude response     

二维零相位FIR数字滤波器设计的闭式最小二乘解

本文二维零相位ＦＩＲ数字滤波器的解析最小二乘设计技术。通过建立频域误差差函数的矩阵形式，并运用与设计问题有关的矩阵的一些性质，得到了滤波器系数的闭式解，使得由给定的频响指标可直接计算滤波器系数，而不必对矩阵进行数值示逆，也不需要基于迭代运算的优化过程。文中给出了滤波器实例，其结果证实了该设计方法的简便性与有效性。     

Efficient weighted least-squares algorithm for the design of FIRfilters

The weighted least-squares (WLS) technique has been widely used for the design of digital FIR filters. In the conventional WLS, the filter coefficients are obtained by performing a matrix inverse operation, which needs computation of O(N³). The authors present a new WLS algorithm that introduces an extra frequency response including implicitly the weight function. In the new algorithm, the filter coefficients can be solved just by a matrix vector multiplication. It reduces the computational complexity from O(N³ ) to O(N²)     

具有任意频响的二维FIR数字滤波器设计的闭式最小二乘解

本文研究具有任意频响特性的二维ＦＩＲ数字滤波器的最小二乘设计问题。     

基于神经网络的任意幅频响应二维FIR线性相位数字滤波器的优化设计

本文提出了一种用神经网络算法来设计任意幅频响应二维FIR线性相位数字滤波器的新方法,其主要思想是使频率响应平方误差函数最小化.根据给定的任意幅频响应指标,按该算法可直接获得滤波器系数.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任意幅频响应二维数字滤波器波动小,算法收敛速度快,稳定性强.     

A new two-dimensional block adaptive FIR filtering algorithm andits application to image restoration

This paper presents a new two-dimensional (2-D) optimum block stochastic gradient (TDOBSG) algorithm for 2-D adaptive finite impulse response (FIR) filtering. The TDOBSG algorithm employs a space-varying convergence factor for all the filter coefficients, where the convergence factor at each block iteration is optimized in a least squares sense that the squared norm of the a posteriori estimation error vector is minimized. It has the same order of computational complexity as another 2-D optimum block adaptive (TDOBA) algorithm. Computer simulations for image restoration show that the TDOBSG algorithm outperforms the TDOBA algorithm and other related algorithms in terms of objective and/or subjective measures.     

A computational method for the LU decomposition of rectangularmatrices and its implications to the realization of 2-D digital filters

The realization of 2-D digital filters based on the lower-upper triangular decomposition of the coefficient matrix is investigated. A numerical method based on the QA decomposition, which has some important characteristics, is proposed for reaching the LU structure. The coefficients in the final LU structure have values favorable to fixed-point arithmetic implementation. Furthermore, the QR structure can be used for the realization and possesses good numerical characteristics in terms of the approximate decomposition scheme. The symmetry in the impulse response coefficient matrix of an octagonally symmetric 2-D FIR filter is utilized to reduce the computational effort spent in the decomposition and the total number of multipliers in the final realization structure     

Weighted least squares near-equiripple approximation of variable fractional delay FIR filters

The weighted least squares (WLS) method is a well-known method for designing a finite impulse response (FIR) filter. And some authors have reported that if a suitable frequency response weighting function is used to design the filter, the WLS method can produce an equiripple result. However, the weighting function for minimax optimality of WLS design is hard to derive analytically. By an iterative method with an adjustable elaborately constructed weighting function, this idea is extended to design a near-equiripple variable fractional delay FIR filter. The proposed method is superior to the fixed-weighting WLS design in the peak absolute error by about 6.6874 dB. The algorithm converges very rapidly. From the simulation, it typically produces a design which is only about 1 dB away from the truly equiripple solution in two iterations and converges to within 0.0056 dB in eight iterations.     

A singular value decomposition derivation in the discrete frequencydomain of optimal noncentro-symmetric 2-D FIR filters

A new derivation is presented for the least squares solution of the design problem of two-dimensional (2-D) finite impulse response (FIR) filters by minimizing the Frobenius norm of the difference between the matrices of the ideal and actual frequency responses sampled at the points of a frequency grid. The mathematical approach is based on the singular value decomposition (SVD) of two complex transformation matrices. Interestingly, the designed filter is proved to be zero-phase if the ideal filter is so without assuming any kind of symmetry     

Optimal Design of 2D FIR Filters with Quadrantally Symmetric Properties Using Fractional Derivative Constraints

In this article, an optimal design of two-dimensional finite impulse response (2D FIR) filter with quadrantally even symmetric impulse response using fractional derivative constraints (FDCs) is presented. Firstly, design problem of 2D FIR filter is formulated as an optimization problem. Then, FDCs are imposed over the integral absolute error for designing of the quadrantally even symmetric impulse response filter. The optimized FDCs are applied over the prescribed frequency points. Next, the optimized filter impulse response coefficients are computed using a hybrid optimization technique, called hybrid particle swarm optimization and gravitational search algorithm (HPSO-GSA). Further, FDC values are also optimized such that flat passband and stopband frequency response is achieved and the absolute \(L_1\)-error is minimized. Finally, four design examples of 2D low-pass, high-pass, band-pass and band-stop filters are demonstrated to justify the design accuracy in terms of passband error, stopband error, maximum passband ripple, minimum stopband attenuation and execution time. Simulation results have been compared with the other optimization algorithms, such as real-coded genetic algorithm, particle swarm optimization and gravitational search algorithm. It is observed that HPSO-GSA gives improved results for 2D FIR-FDC filter design problem. In comparison with other existing techniques of 2D FIR filter design, the proposed method shows improved design accuracy and flexibility with varying values of FDCs.     

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.      

Frequency sampling method for designing 2-D FIR real-coefficientdigital filters using Hermite interpolation

A novel frequency-sampling method for designing 2-D real-coefficient FIR filters, given the values and slope estimates of the desired frequency response at each of the node points of a rectangular grid, is presented. Based on a new class of bivariate Hermite-type polynomials suitable for interpolating at complex conjugate points, and using Kronecker products, the original 2-D filter design problem is reduced to the solution of two 1-D systems of linear equations. Additional advantages of the method are the securing of the existence and uniqueness of the solution of the design problem, computational efficiency, the use of simple and recursive 1-D algorithms; the guarantee of real accurate results; and the inherent parallelism. The method is also applied to design 2-D symmetric FIR filters and can be extended to m-D design problems     

A Design Method for 3-D FIR Cone-Shaped Filters Based on the McClellan Transformation

This paper presents an efficient technique for designing three-dimensional (3-D) finite-impusle-response (FIR) digital filters having cone-shaped magnitude response. The McClellan transformation method is used as basis of the design. The new approach proposes a closed form solution for the transform coefficients of the mapping function. A simultaneous determination of the 1-D cutoff frequency of the prototype filter is also considered. The newly obtained relations for the transform parameters are very simple and depend on the angle of inclination of the cone. Additionally, the analytical expressions for the impulse response coefficients of the transformation subfilter are also derived. The behavior of the transformation function is examined, followed by several detailed examples with different input specifications of designed 3-D cone filters.      

Design of 2-D FIR narrow transition band filters by double transformations using GA approach

In this investigation, subfilters are cascaded in the design of a 2-D narrow transition band FIR digital filter with double transformations, a transformation from wide transition band subfilter into 1-D narrow transition band filter and a McClellan transformation from 1-D filter into 2-D filter. The traditional method for designing a 2-D FIR digital filter with a narrow transition band yields very high orders. The difficulty of the design and implementation will increase with orders exponentially. Numerous identical low-order subfilters are cascaded together to simplify the design of a high-order 2-D filter compared to traditional design method. A powerful genetic algorithm (GA) is presented to determine the best coefficients of the McClellan transformation. It can be used to design any contours of arbitrary shape for mapping 1-D to 2-D FIR filters very effectively. A generalized McClellan transformation is presented, and can be used to design 2-D complex FIR filters. Various numerical design examples are presented to demonstrate the usefulness and effectiveness of the presented approach.

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。