A weighted least squares algorithm for quasi-equiripple FIR and IIRdigital filter design

It has been demonstrated by several authors that if a suitable frequency response weighting function is used in the design of a finite impulse response (FIR) filter, the weighted least squares solution is equiripple. The crux of the problem lies in the determination of the necessary least squares frequency response weighting function. A novel iterative algorithm for deriving the least squares frequency response weighting function which will produce a quasi-equiripple design is presented. The algorithm converges very rapidly. It typically produces a design which is only about 1 dB away from the minimax optimum solution in two iterations and converges to within 0.1 dB in six iterations. Convergence speed is independent of the order of the filter. It can be used to design filters with arbitrarily prescribed phase and amplitude response     

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.     

Equiripple FIR filter design by the FFT algorithm

The fast Fourier transform (FFT) algorithm has been used in a variety of applications in signal and image processing. In this article, a simple procedure for designing finite-extent impulse response (FIR) discrete-time filters using the FFT algorithm is described. The zero-phase (or linear phase) FIR filter design problem is formulated to alternately satisfy the frequency domain constraints on the magnitude response bounds and time domain constraints on the impulse response support. The design scheme is iterative in which each iteration requires two FFT computations. The resultant filter is an equiripple approximation to the desired frequency response. The main advantage of the FFT-based design method is its implementational simplicity and versatility. Furthermore, the way the algorithm works is intuitive and any additional constraint can be incorporated in the iterations, as long as the convexity property of the overall operations is preserved. In one-dimensional cases, the most widely used equiripple FIR filter design algorithm is the Parks-McClellan algorithm (1972). This algorithm is based on linear programming, and it is computationally efficient. However, it cannot be generalized to higher dimensions. Extension of our design method to higher dimensions is straightforward. In this case two multidimensional FFT computations are needed in each iteration     

Analytical design method for optimal equiripple comb FIR filters

A novel analytical design method for highly selective digital optimal equiripple comb finite-impulse response (FIR) filters is presented. The equiripple comb FIR filters are optimal in the Chebyshev sense. The number of notch bands, the width of the notch bands and the attenuation in the passbands can be independently specified. The degree formula and the differential equation for the generating polynomial of the filter is presented. Based on the differential equation, a fast simple algebraic recursive procedure for the evaluation of the impulse response of the filter is described. Its arithmetic robustness outperforms, by far, the known analytical design method. Highly selective equiripple comb FIR filters with thousands of coefficients can be designed. One example demonstrates the efficiency of the filter design.     

FIR数字滤波器幅频响应约束最大加权相位误差最小化设计

多数信号滤波应用,对滤波器幅频响应的要求高于相频响应.本文研究了满足幅频响应约束的有限脉冲响应（Infinite Impulse Response,FIR）数字滤波器设计,提出了最大加权相位误差最小化方法.用凸的椭圆误差约束代替非凸的幅值误差约束,将设计问题转化为凸问题;通过与二分技术结合,提出了给定权函数的幅值误差约束最大加权相位误差最小化设计的求解算法.以此算法为核心,构建了迭代重加权最大加权相位误差最小化算法,其中的权函数不再固定,而是基于修改的群延迟误差包络线在迭代中不断更新.权函数收敛后,所得滤波器具有近似等纹波的群延迟误差,最大群延迟误差得到了有效减小.仿真实验表明,与现有相位误差约束最大幅值误差最小化方法相比,得到的FIR滤波器具有更小的最大相位误差和最大群延迟误差.     

Efficient 2-D based algorithms for WLS designs of 2-D FIR filters with arbitrary weighting functions

The impulse response coefficients of a two-dimensional (2-D) finite impulse response (FIR) filter naturally constitute a matrix. It has been shown by several researchers that, two-dimension (2-D) based algorithms that retain the natural matrix form of the 2-D filter’s coefficients are computationally much more efficient than the conventional one-dimension (1-D) based algorithms that rearrange the coefficient matrix into a vector. In this paper, two 2-D based algorithms are presented for the weighted least squares (WLS) design of quadrantally symmetric 2-D FIR filters with arbitrary weighting functions. Both algorithms are based on matrix iterative techniques with guaranteed convergence, and they solve the WLS design problems accurately and efficiently. The convergence rate, solution accuracy and design time of these proposed algorithms are demonstrated and compared with existing algorithms through two design examples.     

A weighted least squares approach to the design of FIR filters synthesized using the modified frequency response masking structure

This paper presents an application of the weighted least squares (WLS) method to the design of sharp linear phase finite-impulse response (FIR) digital filters synthesized using a modified frequency-response masking (FRM) structure. In our approach, the original minimax design problem is converted into a WLS problem. The WLS problem is highly nonlinear with respect to the coefficients of the filter. However, it can be decomposed into four linear least squares (LS) problems, each of which can be solved analytically. The design problem is then solved iteratively by using an alternating variable approach. The effectiveness of the method is demonstrated through solving a low-pass linear phase sharp FIR digital filter example.     

Minimax design of two-channel nonuniform-division FIR filter banks

The paper deals with the minimax design of two-channel nonuniform-division filter (NDF) banks. Based on a linearisation scheme, the design problem is formulated as an optimisation problem with linear constraints. The authors present a method to design a two-channel NDF bank using a modified dual-affine scaling variant of Karmarkar's (1984) algorithm. This method provides the optimal results that the linear-phase FIR analysis and synthesis filters have equiripple stopband response and the resulting NDF bank also shows equiripple reconstruction error behaviour. The effectiveness of the proposed design technique is demonstrated by several simulation examples     

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)     

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.     

基于雷米兹交换算法设计线性相位对数FIR滤波器

根据线性相位对数FIR滤波器幅度响应与线性相位FIR滤波器幅度响应的关系 ,将线性相位对数滤波器设计转换为线性相位FIR滤波器设计。该方法直接采用雷米兹交换算法即可获得线性相位对数滤波器通带和阻带的等纹波特性。另外 ,该方法既可基于频域均匀采样也可基于频域非均匀采样 ,具有一定的通用性和灵活性     

A fast algorithm for FIR digital filtering with a sum-of-triangles weighting function

A doubly recursive algorithm for time domain convolution with a piecewise linear weighting function is presented that combines the speed of a recursive (IIR) digital filter with the flexibility and ease of design of a nonrecursive (FIR) digital filter. The approach approximates the desired FIR weighting function by a sum-of-triangles weighting function. ForL triangles (or triangle pairs for a linear phase filter) the algorithm is of orderLN. The approximation improves with the number of triangles. A significant advantage of the algorithm compared to FFT filtering or direct convolution is that there is no necessity of a tradeoff between frequency response accuracy and computation time per output point as the data spacing decreases in the filtered signal. The computational complexity is dependent on the number of triangles chosen, not the width of the weighting function, so the algorithm is especially effective for filters with an inherently wide FIR weighting function.     

一种设计对数FIR数字滤波器的方法

本文提出了一种设计具有等波纹对数幅度响应的线性相位ＦＩＲ数字滤波器的方法,该设计方法以多次交换算法为基础。在给定通带与阻带误差比、通带误差和阻带误差三种情况下讨论该设计方法。介绍几个低通对数ＦＩＲ滤波器的设计例子,来说明该设计方法的效率。     

Fast analytical design algorithms for FIR notch filters

Fast analytical design procedures for finite impulse response (FIR) maximally flat (MF) and optimal equiripple (ER) notch filters are introduced. The closed form solution provides recursive computation of the impulse response coefficients of the filter. The ER FIR filters are optimal in the Chebyshev sense. The relation between the MF and ER notch filter is presented in order to emphasize the superior performance of the ER narrow-band filters over their MF counterparts. The discrete nature of the notch frequency in both filter types is emphasized. Four design examples are included to demonstrate the efficiency of the presented approach.     

Time-frequency distribution kernels using FIR filter designtechniques

In this correspondence, time-frequency distribution (TFD) kernels are obtained using finite impulse response (FIR) filter design methods, namely, the windowing method and the equiripple approximation method based on Chebyshev criterion. It is shown that the class of the window-designed kernels are simple to obtain and can handle most time-varying environments     

Optimal design of complex FIR filters with arbitrary magnitude and group delay responses

This paper presents a method for the frequency-domain design of digital finite impulse response filters with arbitrary magnitude and group delay responses. The method can deal with both the equiripple design problem and the peak constrained least squares (PCLS) design problem. Consequently, the method can also be applied to the equiripple passbands and PCLS stopbands design problem as a special case of the PCLS design. Both the equiripple and the PCLS design problems are converted into weighted least squares optimization problems. They are then solved iteratively with appropriately updated error weighting functions. A novel scheme for updating the error weighting function is developed to incorporate the design requirements. Design examples are included in order to compare the performance of the filters designed using the proposed scheme and several other existing methods.     

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.      

Optimal design of two-channel nonuniform-division FIR filter bankswith -1, 0, and +1 coefficients

This paper deals with the optimal design of two-channel nonuniform-division filter (NDF) banks whose linear-phase FIR analysis and synthesis filters have coefficients constrained to -1, 0, and +1 only. Utilizing an approximation scheme and a weighted least squares algorithm, we present a method to design a two-channel NDF bank with continuous coefficients under each of two design criteria, namely, least-squares reconstruction error and stopband response for analysis filters and equiripple reconstruction error and least-squares stopband response for analysis filters. It is shown that the optimal filter coefficients can be obtained by solving only linear equations. In conjunction with the proposed filter structure, a method is then presented to obtain the desired design result with filter coefficients constrained to -1, 0, and +1 only. The effectiveness of the proposed design technique is demonstrated by several simulation examples     

Generalized WLS Method for Designing All-Pass Variable Fractional-Delay Digital Filters

This paper presents an optimal weighted least squares (WLS) method for designing low-complexity all-pass variable fractional-delay (VFD) digital filters. Instead of using a fixed range for the VFD parameter p and same-order constant-coefficient filters (subfilters), both the VFD parameter range p isin [p _Min,p _Max] and subfilter orders are optimized such that a low-complexity all-pass VFD filter can be achieved for the LS design. To suppress the peak errors of variable frequency response, weighting functions are adopted and optimized such that the boundary peak errors can be further reduced but without noticeably increasing the total error energy (integral of squared error) of variable frequency response. After optimizing the variable range of the VFD parameter, weighting functions, and subfilter orders, an all-pass VFD filter can be designed by using a generalized noniterative WLS method, which yields a closed-form solution. Design examples are given to illustrate that utilizing different-order subfilters, along with the optimal range and optimal weighting functions, can yield an all-pass VFD filter with significantly reduced complexity and design errors as compared with existing ones.     

Note on the Design of an Equiripple DC-Notch FIR Filter

An extremely robust analytical procedure for the effective evaluation of the impulse response of a highly selective optimal equiripple DC-notch finite-impulse response (FIR) filter is presented. The DC-notch filter is optimal in the Chebyshev sense. The computational superiority of the presented procedure over the standard approach is emphasized