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.     

The design of digital all-pass filters using second-order cone programming (SOCP)

This brief proposes a new method for designing digital all-pass filters with a minimax design criterion using second-order cone programming (SOCP). Unlike other all-pass filter design methods, additional linear constraints can be readily incorporated. The overall design problem can be solved through a series of linear programming subproblems and the bisection search algorithm. The convergence of the algorithm is guaranteed. Nonlinear constraints such as the pole radius constraint of the filters can be formulated as additional SOCP constraints using Rouche's theorem. It was found that the pole radius constraint allows an additional tradeoff between the approximation error and the stability margin. The effectiveness of the proposed method is demonstrated by several design examples and comparison with conventional methods.     

METEOR: a constraint-based FIR filter design program

It is proposed to specify a filter only in terms of upper and lower limits on the response, find the shortest filter length which allows these constraints to be met, and then find a filter of that order which is farthest from the upper and lower constraint boundaries in a minimax sense. The simplex algorithm for linear programming is used to find a best linear-phase FIR filter of minimum length, as well as to find the minimum feasible length itself. The simplex algorithm, while much slower than exchange algorithms, also allows the incorporation of more general kinds of constraints, such as concavity constraints (which can be used to achieve very flat magnitude characteristics). Examples are given to illustrate how the proposed and common approaches differ, and how the proposed approach can be used to design filters with flat passbands, filters which meet point constraints, minimum phase filters, and bandpass filters with controlled transition band behavior     

Minimax design of two-dimensional parallelogram filter banks

The authors present a technique for the minimax design of two-dimensional (2-D) parallelogram filter bank (PFB) systems with linear-phase analysis/synthesis filters. To achieve perfect reconstruction, the required analysis filters must have parallelogram-shaped frequency responses. In general, the original design problem is found to be an optimisation problem with nonlinear constraints. The authors present a linearisation approach to reformulate the design problem. As a result, updating the filter coefficient vector at each iteration for the original design problem can be accomplished by searching the gradient of the linearised optimisation problem. They further present an efficient method based on a modified Karmarkar's algorithm for computing the required gradient vector and finding the required step size analytically. Therefore the filter coefficients can easily be computed by solving only linear equations at each iteration during the design process. The effectiveness of the proposed technique is shown by computer simulations     

New approaches to robust minimum variance filter design

This paper is concerned with the design of robust filters that ensure minimum filtering error variance bounds for discrete-time systems with parametric uncertainty residing in a polytope. Two efficient methods for robust Kalman filter design are introduced. The first utilizes a recently introduced relaxation of the quadratic stability requirement of the stationary filter design. The second applies the new method of recursively solving a semidefinite program (SDP) subject to linear matrix inequalities (LMIs) constraints to obtain a robust finite horizon time-varying filter. The proposed design techniques are compared with other existing methods. It is shown, via two examples, that the results obtained by the new methods outperform all of the other designs     

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.     

Quasi-equiripple approximation of minimum phase FIR filters by updating desired response

The authors present a numerical method for the Chebyshev approximation of minimum phase FIR digital filters. This method is based on solving a least squares (LS) problem iteratively. At each iteration, the desired response is transformed so as to have an equiripple magnitude error. This method makes it possible to design minimum phase FIR filters whose magnitude error is quasi-equiripple. Using this method, a quasi-equiripple solution is obtained very quickly. Since the proposed methods do not require any time-consuming optimisation procedure, they require less computational complexity than conventional methods. Finally, some examples to illustrate the advantage of the proposed methods are shown.     

Array pattern synthesis using semidefinite programming and a bisection method

In this paper, we propose an array pattern synthesis scheme using semidefinite programming (SDP) under array excitation power constraints. When an array pattern synthesis problem is formulated as an SDP problem, it is known that an additional rank‐one constraint is generated inevitably and relaxed via semidefinite relaxation. If the solution to the relaxed SDP problem is not of rank one, then conventional SDP‐based array pattern synthesis approaches fail to obtain optimal solutions because the additional rank‐one constraint is not handled appropriately. To overcome this drawback, we adopted a bisection technique combined with a penalty function method. Numerical applications are presented to demonstrate the validity of the proposed scheme.     

Design of optimal minimum phase FIR filters by direct factorization

An improved method of designing optimal minimum phase FIR filters by directly finding zeros is proposed. The zeros off the unit circle are found by an efficient special purpose root-finding algorithm without deflation. The proposed algorithm utilizes the passband minimum ripple frequencies to establish the initial points, and employs a modified Newton's iteration to find the accurate initial points for a standard Newton's iteration. We show, with examples, that the proposed algorithm can be used to design very long filters (L = 325) with very high stopband attenuations.     

An iterative algorithm for optimal design of non-frequency-selective FIR digital filters

This paper proposes a novel iterative algorithm for optimal design of non-frequency-selective Finite Impulse Response （FIR） digital filters based on the windowing method. Different from the traditional optimization concept of adjusting the window or the filter order in the windowing design of an FIR digital filter, the key idea of the algorithm is minimizing the approximation error by successively modifying the design result through an iterative procedure under the condition of a fixed window length. In the iterative procedure, the known deviation of the designed frequency response in each iteration from the ideal frequency response is used as a reference for the next iteration. Because the approximation error can be specified variably, the algorithm is applicable for the design of FIR digital filters with different technical requirements in the frequency domain. A design example is employed to illustrate the efficiency of the algorithm.     

多通带低群延迟误差FIR滤波器设计的迭代算法

常数低群延迟有限冲击响应（FIR）滤波器在通信等领域得到了广泛应用,尤其是要求无波形失真、信号延迟小的场合。而低群延迟的FIR滤波器,其相位响应只能做到近似线性相位,其群延迟只能做到近似常数。为了减小与期望常数群延迟之间的误差,最近提出的通过迭代更新相位误差上界函数来逐步减小群延迟误差的方法,只考虑了单通带滤波器的minimax设计。本文将把该方法推广至多通带FIR滤波器的minimax设计和约束最小二乘设计,先对各通带单独处理使每个通带的最大群延迟误差有效降低后,再考虑各通带之间平衡,对各子带的最大群延迟误差进行折中,进而使整个通带上的最大群延迟误差继续减小。对约束最小二乘设计还特别考虑了通过修改收敛参数来解决相位误差约束过紧时设计问题无解的问题。仿真实例表明,该方法能有效减小多通带滤波器的最大群延迟误差。      

L(p) norm design of stack filters.

Addresses the problem of designing optimal stack filters by employing an L(p) norm of the error between the desired signal and the estimated one. It is shown that the L(p) norm can be expressed as a linear function of the decision errors at the binary levels of the filter. Thus, an L(p)-optimal stack filter can be determined as the solution of a linear program. The conventional design of using the mean absolute error (MAE), therefore, becomes a special ease of the general L(p) norm-based design developed here. Other special cases of the proposed approach, of particular interest in signal processing, are the problems of optimal mean square error (p=2) and minimax (p-->infinity) stack filtering. Since an Linfinity optimization is a combinatorial problem, with its complexity increasing faster than exponentially with the filter size, the proposed L(p ) norm approach to stack filter design offers an additional benefit of a sound mathematical framework to obtain a practical engineering approximation to the solution of the minimax optimization problem. The conventional MAE design of an important subclass of stack filters, the weighted order statistic filters, is also extended to the L(p) norm-based design. By considering a typical application of restoring images corrupted with impulsive noise, several design examples are presented, to illustrate the performance of the L(p)-optimal stack filters with different values of p. Simulation results show that the L(p)-optimal stack filters with p=/>2 provide a better performance in terms of their capability in removing impulsive noise, compared to that achieved by using the conventional minimum MAE stack filters.     

PLS Algorithms for Equality-Constrained Linear-Phase FIR Filter Design

The least-squares design of linear-phase finite impulse response (FIR) filters with equality constraints in the time and frequency domains can be formulated as an equality-constrained quadratic programming (QP) problem. This paper presents an effective and robust novel algorithm, the projected least-squares (PLS) algorithm, for equality-constrained QP problems. The algorithm eventually projects an unconstrained minimization solution successively onto the feasible hyperplane of the problem. An additional term is added into the Hessian matrix of the cost function, thus ensuring the positive definiteness of the Hessian matrices during the iterative procedure for the design of constrained minimax filters. To demonstrate its effectiveness and robustness, the PLS algorithm is applied to design optimal linear-phase Nyquist filters that may satisfy some frequency domain constraints besides the zero-crossing impulse response.     

Minimax design of two-channel low-delay perfect-reconstruction FIRfilter banks

The authors deal with the design problem of low-delay perfect-reconstruction filter banks for which the FIR analysis and synthesis filters have equiripple magnitude response. Based on the minimax error criterion, the design problem is formulated in such a manner that the coefficients for the FIR analysis filters can be found by minimising the weighted peak error of the designed analysis filters, subject to the perfect-reconstruction constraints. A design technique based on a modified dual-affine scaling variant of Karmarkar's (1989) algorithm, in conjunction with approximation schemes, is then developed for solving the resulting nonlinear optimisation problem. The effectiveness of the proposed design technique is demonstrated by several simulation examples     

Efficient Design of Cosine-Modulated Filter Banks via Convex Optimization

This paper presents efficient approaches for designing cosine-modulated filter banks with linear phase prototype filter. First, we show that the design problem of the prototype filter being a spectral factor of $2M$th-band filter is a nonconvex optimization problem with low degree of nonconvexity. As a result, the nonconvex optimization problem can be cast into a semi-definite programming (SDP) problem by a convex relaxation technique. Then the reconstruction error is further minimized by an efficient iterative algorithm in which the closed-form expression is given in each iteration. Several examples are given to illustrate the effectiveness of the proposed method over the existing ones.      

An iterative approach for minimax design of multidimensional quadrature mirror filters

In this paper, a nest of iterative techniques is proposed for the minimax design of quadrature mirror filter (QMF) banks. The method can be generalized such that multidimensional QMF banks can be designed by the proposed algorithm. For a given weighting function, an iterative method is used to minimize the objective error function in the inner iterations. To further reduce the peak error of overall magnitude response, an iterative weighting-updated technique is adopted in the outer iterations. Comparing with the existing works concern the design of perfect-reconstruction QMF banks, only one of the filters is needed to be designed under the cost of magnitude distortion, but the system complexity can be reduced drastically. Several examples, including design of 2-D and 3-D QMF banks, will be presented to demonstrate the effectiveness of the proposed method.     

Linear periodic systems and multirate filter design

We present a systematic procedure for the design of filters intended for multirate systems. This procedure Is motivated by viewing the equiripple design of filters in linear time-invariant systems as a process of obtaining optimum minimax filters for a class of bounded energy input signals. The philosophy of designing optimum minimax filters for classes of input signals is extended to multirate systems, which are not time-invariant. We develop a generalized Fourier analysis appropriate for linear periodic systems and use it to derive new error criteria for multirate filter design. Using such criteria yields optimum minimax multirate filters for the input signal class. The utility of our method is demonstrated by using it to analyze several multirate systems. We give numerical results on the design of a multirate implementation of a narrowband filter and compare our work to previous work on multirate filter design. Our numerical analysis is based upon a new formulation of the design as a semi-infinite linear programming problem     

Improved clustered look-ahead pipelining algorithm with minimumorder augmentation

The author compares the overall performance of clustered look-ahead (CLA) filters with minimum order augmentation and scattered look-ahead (SLA) filters. To optimize the domain searching procedure and to improve the numerical performance, an improved CLA algorithm with minimum order augmentation, which is especially beneficial for high-Q high-speed digital filters, is proposed. This algorithm is optimized in the aspect of minimizing both the augmented pipelining order and undesirable quantization effects simultaneously. The proposed CLA structure with minimum order augmentation turns out to be very suitable for the applications that require high-speed and numerical stability     

The design and multiplier-less realization of software radio receivers with reduced system delay

This work studies the design and multiplier-less realization of a new software radio receiver (SRR) with reduced system delay. It employs low-delay finite-impulse response (FIR) and digital allpass filters to effectively reduce the system delay of the multistage decimators in SRRs. The optimal least-square and minimax designs of these low-delay FIR and allpass-based filters are formulated as a semi-definite programming (SDP) problem, which allows zero magnitude constraint at /spl omega/=/spl pi/ to be incorporated readily as additional linear matrix inequalities (LMIs). By implementing the sampling rate converter (SRC) using a variable digital filter (VDF) immediately after the integer decimators, the needs for an expensive programmable FIR filter in the traditional SRR is avoided. A new method for the optimal minimax design of this VDF-based SRC using SDP is also proposed and compared with traditional weight least squares method. Other implementation issues including the multiplier-less and digital signal processor (DSP) realizations of the SRR and the generation of the clock signal in the SRC are also studied. Design results show that the system delay and implementation complexities (especially in terms of high-speed variable multipliers) of the proposed architecture are considerably reduced as compared with conventional approaches.     

Time-domain design of FIR multirate filter banks

A new time-domain methodology for designing FIR multirate filter banks is proposed. The conditions for perfect reconstruction systems can only be met by a limited number of systems, and consequently one of the major problems is to design analysis and synthesis filters which reduce the reconstruction error to a minimum. A recursive technique is proposed which uses the synthesis filters from one iteration to update the analysis filters for the next. The Letter shows that this is computationally simpler than previously proposed time-domain methods and produces filter banks in which the reconstruction error is reduced to practically acceptable levels.<>