Chebyshev Design of Linear-Phase FIR Filters With Linear Equality Constraints

By using basis transformation, the Chebyshev approximation of linear-phase finite-impulse response (FIR) filters with linear equality constraints can be converted into an unconstrained one defined on a new function space. However, since the Haar condition is not necessarily satisfied in the new function space, the alternating property does not hold for the solution to the resulted unconstrained Chebyshev approximation problem. A sufficient condition for the best approximation is obtained in this brief, and based on this condition, an efficient single exchange algorithm is derived for the Chebyshev design of linear-phase FIR filters with linear equality constraints. Simulations show that the proposed algorithm can converge to the optimal solution in most cases and to a near-optimal solution otherwise. Design examples are presented to illustrate the performance of the proposed algorithm.     

Chebyshev Design of FIR Filters with Frequency Inequality Constraints

In this paper, the constrained Chebyshev design of linear-phase finite impulse response filters with frequency inequality constraints is considered. By converting the constrained design problem into a series of unconstrained Chebyshev design problems, an iterative Remez algorithm is proposed, and the convergence of the algorithm is obtained. The design example demonstrates the effectiveness and the fast convergence of the algorithm.     

带不等式约束的线性相位FIR数字滤波器的Chebyshev设计

本文主要讨论带不等式约束的线性相位FIR数字滤波器的Chebyshev设计问题.我们知道,交错点组定理是Remez算法的理论基础,而此算法很好的解决了无约束线性相位FIR数字滤波器Chebyshev设计问题.本文中给出并证明了一个与交错点组定理类似的定理,并以此定理为依据设计算法求解具有不等式约束的线性相位FIR滤波器Chebyshev设计问题,此算法可通过对Matlab中remez()函数的修改来实现.     

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。     

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。     

Algebraic Design of Some Equiripple Linear-Phase Half-Band FIR Filters

A closed form solution for the approximation of a linear-phase FIR (finite impulse response) filter with equiripple magnitude responsein the passband and stopband was not known. In this letter we present a closed form solution of some equiripple linear-phase half-band FIR filter approximation.     

The Design of Linear-Phase FIR Multiple-Notch Filters with Variable Notch Frequencies

In this paper, a paradigm is developed to design the linear-phase FIR multiple-notch filters with variable notch frequencies. The design procedure can be proceeded through two steps: First, a linear-phase narrow-band low-pass filter met the given bandwidth and stopband ripple specifications is designed. Second, a tuning procedure is applied to the computed low-pass filter to yield the desired multiple-notch filter. When the notch frequencies are varied, the same tuning procedure can be employed to render the multiple-notch filter with the new set of the notch frequencies. The tuning procedure employed reduces the computational complexity of designing the multiple-notch filter with the new set of the notch frequencies. The simulation results demonstrate the effectiveness of our scheme.     

Greedy Algorithm for the Design of Linear-Phase FIR Filters with Sparse Coefficients

In this work, a greedy algorithm for the design of sparse linear-phase finite impulse response filters wherein the coefficients are successively fixed to zero individually is proposed. To meet the filter specifications, the coefficient for which the middle value of its feasible range is closest to zero is selected to be set to zero, whereas all the other unfixed coefficients are free to change. Design examples show that the proposed technique can design FIR filters with higher sparsity than that obtained by existing nonexhaustive algorithms for given specifications. To show the optimality of the algorithm, we design 100 filters, with results showing that the global optimal solution, i.e., the sparsest solution found by exhaustive search, can be achieved in most cases, but with much less computation time.     

二维半平面线性相位FIR数字滤波器的最小二乘设计方法

本文研究了二维线性相位实系数FIR数字滤波器的最小二乘设计问题，导出了滤波器系数的闭式解。运用给出的计算公式可方便地计算滤波器的系数，而不必对矩阵进行数值求逆运算也不需要复杂的优化过程．设计实例表明本文给出的方法程序简单、计算时间极短．     

Restoring Coefficient Symmetry in Polyphase Implementation of Linear-Phase FIR Filters

Polyphase implementation of FIR filters effectively reduces the multiplication rate and data storage in a multirate system. However, the coefficients of the polyphase components are no longer symmetric even though the overall filter has a symmetric (or anti-symmetric) impulse response. In this paper, we introduce a new technique that recasts pairs of the original polyphase components as sums or differences of auxiliary pairs of symmetric and anti-symmetric impulse response filters. The coefficient symmetry of these auxiliary polyphase components can be fully exploited to reduce arithmetic complexity without undue complications. Our new technique makes use of the fact that the impulse responses of the non-symmetric polyphase components exist in time-reversed pairs which can be synthesized from pairs of symmetric and anti-symmetric impulse response filters. This results in a factor-of-two reduction in the number of multipliers required to implement the polyphase components.     

An L1-Method for the Design of Linear-Phase FIR Digital Filters

This paper considers the design of linear-phase finite impulse response digital filters using an L₁ optimality criterion. The motivation for using such filters as well as a mathematical framework for their design is introduced. It is shown that L₁ filters possess flat passbands and stopbands while keeping the transition band comparable to that of least-squares filters. The uniqueness of L₁-based filters is explored, and an alternation type theorem for the optimal frequency response is derived. An efficient algorithm for calculating the optimal filter coefficients is proposed, which may be viewed as the analogue of the celebrated Remez exchange method. A comparison with other design techniques is made, demonstrating that the L₁ approach may be a good alternative in several applications.     

带约束的复FIR数字滤波器Chebyshev设计算法

本文主要讨论了复FIR数字滤波器的频域不等式约束Chebyshev设计问题.作者首先把文献[1]中的复交错点组定理扩展到有不等式约束的情况,之后根据扩展定理中对最优解特性的描述,并结合复Remez算法[1]及赖晓平的迭代Remez算法[2][3],提出了一种有效的算法来解决频域带不等式约束的复FIR数字滤波器的Chebyshev设计问题.如果问题的解存在,则算法能保证收敛到最优解.作者用MATLAB语言对上述算法进行了实现并做了仿真分析.     

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.     

复FIR数字滤波器幅值约束Chebyshev设计

L.J.Karam和McClellan最早得到了有关复数域Chebyshev逼近的复交错点组定理,并提出了以此定理为基础的复Remez算法用于复FIR数字滤波器的Chebyshev设计.本文首先给出并证明了复交错点组定理在带不等式约束条件下的扩展定理,之后,根据此扩展定理中对最优解极值频率点特性的描述,提出了一种有效的算法来解决带幅值不等式约束的复FIR数字滤波器的Chebyshev设计问题.这一新算法中还结合了复Remez算法及赖晓平提出的迭代Remez算法,并且如果问题的解存在则保证收敛到此解.作者把上述算法做成了MATLAB语言程序,并进行了大量的实例设计实验,仿真结果表明此算法有效而可靠.     

Two Classes of Frequency-Response Masking Linear-Phase FIR Filters for Interpolation and Decimation

This paper introduces two classes of frequency-response masking (FRM) linear-phase finite (length) impulse response (FIR) filters for interpolation and decimation by arbitrary integer factors M. As they are based on the FRM approach, the proposed filters are low-complexity (efficient) sharp-transition linear-phase FIR interpolation and decimation filters. Compared to previously existing FRM linear-phase FIR filter classes for interpolation and decimation, the new ones offer lower complexity and more freedom in selecting the locations of the passband and stopband edges. Furthermore, the proposed classes of FRM filters can, as special cases, realize efficient Mth-band FRM linear-phase FIR interpolation and decimation filters for all values of M. Previously, only half-band (M = 2) FRM linear-phase FIR filters have appeared in the literature. The paper includes design techniques suitable for the new filters and design examples illustrating their efficiency.     

Synthesis of Narrowband Linear-Phase FIR Filters With a Piecewise-Polynomial Impulse Response

Classes of linear-phase finite-impulse response (FIR) filters with a piecewise-polynomial impulse response are proposed for the four types of linear-phase FIR filters. In addition, very efficient recursive structures to implement these filters in a straightforward and consistent manner are proposed. The desired impulse response is created by using a parallel connection of several filter branches. Only one branch has an impulse response of the full filter length, whereas the impulse responses are shorter for the remaining branches but the center is at the same location. The arithmetic complexity of these filters is proportional to the number of branches and the common polynomial order for each branch, rather than the actual filter order. In order to generate the overall piecewise-polynomial impulse response the polynomial coefficients are found, with the aid of linear programming, by optimizing the responses in the minimax sense, for both narrowband conventional filters and narrowband differentiators. The generation of these structures is based on the use of accumulators so that after using an accumulator, the resulting impulse response is divided into two parts. The first part follows the desired polynomial form, and the second part is what is left after the division, i.e., the nonpolynomial part. This same procedure can be used for all the following accumulators. Several examples are included, illustrating the benefits of the proposed filters, in terms of a reduced number of unknowns used in the optimization and the reduced number of multipliers required in the actual implementation.     

A Sequential Constrained Least-Square Approach to Minimax Design of 2-D FIR Filters

The solution to minimax design of 2-D finite-impulse response filter is not necessarily unique. This paper presents a sequential constrained least-square (SCLS) method to obtain a minimax filter with least total squared error. The method converts the minimax design into a series of constrained least-square problems with the same cost function but different magnitude constraints. By producing the sequence of magnitude error bounds with a binary search, the SCLS method has an exponential convergence rate. Design examples of circular, diamond, and fan filters, and comparison with existing methods show that the SCLS method is efficient and absolutely convergent. The resulted filter is not only a minimax filter but also has least total squared error among minimax filters.     

多带FIR数字滤波器的频域设计

介绍FIR滤波器的FFT快速算法实现方法,推导了FIR数字滤波器的频域直接计算H(k)的计算公式,并进行分析与讨论.根据实际应用需要的滤波器的技术指标,以一个多带FIR线性相位数字滤波器为例进行设计.详细讨论了过渡点的优化设计方法,给出了边界频率点幅度搜索算法,通过用Matlab进行设计和性能分析,结果表明优化后的滤波器能够满足指标要求.