An Algebraic Approach to the Estimation of the Order of FIR Filters From Complete and Partial Magnitude and Phase Specifications

The problems addressed by this paper is the following: Given a set of measurements over the range of normalized frequencies (thetas₁ ,thetas₂) on the magnitude and/or phase of a real FIR but otherwise unknown filter, to estimate the order of the FIR filter. The range (thetas₁,thetas₂) may be partial or it may cover the entire range of frequencies. The purpose of the paper is to propose a new algebraic approach to solve the above collection of problems. Specific new results include FIR order estimation from partial or complete noiseless measurements for a real system from: a) phase alone, from b) magnitude alone (not necessarily piece-wise constant), and from c) joint magnitude and phase. The proposed approach is not only capable of dealing with specifications that go beyond the conventional formulas for the standard piece-wise-constant magnitude FIR filter order estimation, but it also furnishes a nexus for order estimation from phase (or group delay) specifications, areas which have remained hitherto unaddressed. The approach is based on the use of Root Moments. In this context, the novel concept of Fractional Root Moments is used in a key fashion to provide an estimate on the number of zeros inside the unit circle. Open problems and new directions of exploration and research are mentioned in the body of the paper     

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。     

Optimal Design of Nonlinear-Phase FIR Filters With Prescribed Phase Error

Constrained least-squares design and constrained Chebyshev design of one- and two-dimensional nonlinear-phase FIR filters with prescribed phase error are considered in this paper by a unified semi-infinite positive-definite quadratic programming approach. In order to obtain unique optimal solutions, we propose to impose constraints on the complex approximation error and the phase error. By introducing a sigmoid phase-error constraint bound function, the group-delay error can be greatly reduced. A Goldfarb–Idnani based algorithm is presented to solve the semi-infinite positive-definite quadratic program resulting from the constrained least-squares design problem, and then applied after some modifications to the constrained Chebyshev design problem, which is proved in this paper to be equivalent also to a semi-infinite positive-definite quadratic program. Through design examples, the proposed method is compared with several existing methods. Simulation results demonstrate the effectiveness and efficiency of the proposed method.      

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.     

Low-Complexity Design of Variable Bandedge Linear Phase FIR Filters With Sharp Transition Band

This paper presents a very low-complexity design of variable bandedge linear phase finite-impulse-response (FIR) filters with fixed sharp transition width. The idea is to first decompose the input signal into several channels in the frequency domain. The channel(s) involved with the transition band of the variable filter due to the variation of the bandedge is (are) shaped to produce the required transition band, and then summed up with the channels involved with the passband of the variable filter to produce the required frequency response. The proposed variable filter has extremely low complexity when the transition band is sharp, if compared with other techniques such as the Farrow structure. It is possible that the computational complexity of the variable filter is even lower than that of a corresponding fixed filter with the same transition width and ripple specifications implemented in its direct form.      

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.     

Programmable Photonic Microwave Filters With Arbitrary Ultra-Wideband Phase Response

We present a coherent optical signal-processing approach for synthesis of programmable microwave phase filters over an ultra-wideband. Our scheme relies on a programmable optical phase filter implemented in a pulse-shaping geometry incorporating a spatial light modulator and hyperfine (~600-MHz spectral resolution) optical spectral disperser. The user-defined optical phase filter is directly transferred to the electrical domain through heterodyne conversion, and the overall system response is characterized via vector network analyzer measurements. We illustrate our approach by synthesizing linear, quadratic, and cubic spectral phase functions over a 20-GHz band. To our best knowledge, this is the first realization of programmable arbitrary microwave phase filters over such a bandwidth     

Design of Linear Phase FIR Filters in Subexpression Space Using Mixed Integer Linear Programming

In this paper, a novel optimization technique is proposed to optimize filter coefficients of linear phase finite-impulse response (FIR) filter to share common subexpressions within and among coefficients. Existing approaches of common subexpression elimination optimize digital filters in two stages: first, an FIR filter is designed in a discrete space such as finite wordlength space or signed power-of-two (SPT) space to meet a given specification; in the second stage, an optimization algorithm is applied on the discrete coefficients to find and eliminate the common subexpressions. Such a two-stage optimization technique suffers from the problem that the search space in the second stage is limited by the finite wordlength or SPT coefficients obtained in the first stage optimization. The new proposed algorithm overcomes this problem by optimizing the filter coefficients directly in subexpression space for a given specification. Numerical examples of benchmark filters show that the required number of adders obtained using the proposed algorithm is much less than those obtained using two-stage optimization approaches.     

Synthesis of Resonator Filters With Arbitrary Topology Using Hybrid Method

This paper presents a new synthesis method for resonator filters of arbitrary topology using an evolutionary hybrid method. This method consists of a Levenberg-Marquardt algorithm for a local optimizer and genetic algorithm for a global optimizer, respectively. Unlike the conventional hybrid method in which the local optimization is performed after finding appropriate initial values from global optimization, the local optimizer in the proposed method is used as a genetic-algorithm operator to prevent trapping in local minima of the cost function. This method can provide fast convergence and good accuracy to find the final solution from initial population generated by a random number and the known value for the filters with stringent requirements. In addition, multiple coupling matrices to meet the given requirement can be obtained from the initial population based on a random number. Resonator filters with asymmetric eight-pole configurations for single and dual passbands are synthesized using the current method for validation. Excellent agreement between the response computed from characteristic polynomials and the response computed from couplings is obtained from the proposed method.     

Design of Digital FIR Filters Using Differential Evolution Algorithm

The differential evolution (DE) algorithm is a new heuristic approach with three main advantages: it finds the true global minimum of a multimodal search space regardless of the initial parameter values, it has fast convergence, and it uses only a few control parameters. The DE algorithm, which has been proposed particularly for numeric optimization problems, is a population-based algorithm like the genetic algorithms and uses similar operators: crossover, mutation, and selection. In this work, the DE algorithm has been applied to the design of digital finite impulse response filters, and its performance has been compared to that of the genetic algorithm and least squares method.     

Minimum-Phase FIR Filter Design Using Real Cepstrum

The real cepstrum is used to design an arbitrary length minimum-phase finite-impulse response filter from a mixed-phase prototype. There is no need to start with the odd-length equiripple linear-phase filter first. Neither the phase-unwrapping nor root-finding procedure is needed. Only two fast Fourier transforms and a recursive procedure are required to find the filter's impulse response from its real cepstrum. The resulting filter's magnitude response is exactly the same as the original one even when the filter is of very high order     

Design and Optimization of Multiplierless FIR Filters Using Sub-Threshold Circuits

This paper develops and demonstrates the design and optimization method for fixed coefficient Finite Impulse Response (FIR) filters using sub-threshold circuits to achieve the minimum energy per operation. Sub-threshold circuit current, delay, power consumption, energy per operation and temperature dependence are modeled theoretically and analyzed using Matlab. Then the filter design and optimization are presented. With a frequency characteristic of 80 dB magnitude and 9.6 kHz bandwidth, the 16-bit fixed-point coefficients of the linear phase equiripple low-pass filter are generated from Matlab. Canonical Signed Digit (CSD) arithmetic is used for multiplierless design to improve both cost and performance. The transposed structure and symmetry structure are applied to optimize the delay and cost further. Horner’s rule is used to improve the precision. Tree-height reduction and subexpression sharing at Register Transfer Level (RTL) are used for further delay and cost reduction. Six versions of the filter with the same group of coefficients are designed and synthesized using Design Compiler with a 65 nm process. Synthesis results show that the area of the final version is reduced by 44% compared with the original design at a fixed frequency of 250 MHz, and at the highest frequency of each design, the area is reduce by about 23% while the performance is improved by 60%. These results show the design and optimization method developed in this paper can improve both the area and performance significantly. One adder from the synthesis netlist is simulated at the transistor-level using HSPICE to obtain characteristics of sub-threshold operations. The supply voltage varies from 1.2 to 0.08 V and temperatures from 0 to 110°C. The experiment results verify most characteristics of the sub-threshold models, but also reveal some limitations and defects of the theoretical models and previous results. The observations are discussed carefully with quantitative and qualitative analysis. For 25°C, the minimum energy point for the adder is 0.22 V. Finally, the results of the adder are used to estimate the energy per operation for the filters. For a fixed frequency of 36.4 kHz at 0.22 V, the estimated energy values vary from 4.8 to about 2.7 pJ for the six designed filters.     

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.     

Global Optimum Design of Uniform FIR Filter Bank With Magnitude Constraints

The optimum design of a uniform finite impulse response filter bank can be formulated as a nonlinear semi-infinite optimization problem. However, this optimization problem is nonconvex with infinitely many inequality constraints. In this paper, we propose a new hybrid approach for solving this highly challenging nonlinear, nonconvex semi-infinite optimization problem. In this approach, a gradient-based method is used in conjunction with a filled function method to determine a global minimum of the problem. This new hybrid approach finds an optimal result independent of the initial guess of the solution. The method is applied to some existing examples. The results obtained are superior to those obtained by other existing methods.      

Design and Implementation of Linear Phase FIR Filters for Biological Signal Processing

This paper presents a new method for designing digital linear phase, finite impulse response filters with loose frequency response characteristics, but with good time resolution as is required in biological signal conditioning. The design is very simple and has been used with success in the microcomputer implementation of filters for the automated processing of electroencephalographic (EEG) data. Examples and a discussion of possible filter implementations are included.     

WLS Design of Variable Fractional-Delay FIR Filters Using Coefficient Relationship

In this brief, a new coefficient relationship is proposed for the design of variable fractional-delay (VFD) finite-impulse response (FIR) filters by the weighted least-squares (WLS) method so that the number of filter coefficients to be designed can approximately be halved. To reduce the computational cost, closed-form expressions for the elements of related vectors and matrices are derived. Several design examples are presented, and the comparisons show that the overall performance of the proposed method is comparable with that of the conventional method; however, the number of filter coefficients to be designed for the proposed method is about half of that in the conventional method.      

Weighted-Least-Squares Design of Variable Fractional-Delay FIR Filters Using Coefficient Symmetry

Our previous work has shown that the coefficient symmetry can be efficiently exploited in designing variable finite-impulse-response (FIR) filters with simultaneously tunable magnitude and fractional-delay responses. This paper presents the optimal solutions for the weighted-least-squares (WLS) design of variable fractional-delay (VFD) FIR filters with same-order and different-order subfilters through utilizing the coefficient symmetry along with an imposed coefficient constraint. In deriving the closed-form error functions, since the Taylor series expansions of$sin(omega p)$and$cos(omega p)$are used, the numerical integrals using conventional quadrature rules can be completely removed, which speeds up the WLS design and guarantees the optimality of the final solution. Two design examples are given to illustrate that the proposed WLS methods can achieve better design with significantly reduced VFD filter complexity and computational cost than the existing ones including the WLS-SVD approach. Consequently, the proposed WLS design is the best among all the existing WLS methods so far.