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.     

Variable Digital Filter With Group Delay Flatness Specification or Phase Constraints

In this paper, we consider the design of finite-impulse response variable digital filters (VDFs) with variable cutoff frequency or variable fractional delay. We propose the design of VDFs with minimum integral squared error and constraints on the maximum error deviation in conjunction with flatness group delay specification or phase constraints. These specifications allow the VDFs to have approximately linear phase, especially in the passband. As these specifications are required to be satisfied for all the filters generated by the VDF with controllable spectral characteristics, the linear constraints resulting from the flatness specification are relaxed to inequality constraints. To make the optimization problem tractable for the phase constrained problem, suitable approximations are employed in the paper. The design problem is formulated as an optimization problem with a quadratic cost function and infinite number of constraints. A numerical scheme with adaptive grid step size is proposed for solving the optimization problem.     

Projected least-squares algorithms for constrained FIR filter design

Constrained finite-impulse response (FIR) filter design with time- and frequency-domain linear constraints can be generally transformed into a, or a series of, constrained least-squares problems, which can be generally reformulated as positive definite quadratic programming (QP) problems. This paper presents a novel algorithm referred to as a projected least-squares (PLS) algorithm for the positive definite QP problems. The PLS algorithm essentially projects the unconstrained (least-squares) minimization solution successively onto the boundaries of active constraints that are identified by an active-set strategy. The PLS algorithm has been applied to the constrained least-squares design of FIR filters directly, and to the constrained Chebyshev design of FIR filters in an iterative fashion. The PLS algorithm is compared with the most widely used interior-point methods and an active-set method through design examples of low-pass filters with specified passband and stopband ripples, Nyquist filter constraints and step response constraints. All these examples demonstrate the high efficiency of the PLS algorithm.     

FIR, Allpass, and IIR Variable Fractional Delay Digital Filter Design

This paper presents two-step design methodologies and performance analyses of finite-impulse response (FIR), allpass, and infinite-impulse response (IIR) variable fractional delay (VFD) digital filters. In the first step, a set of fractional delay (FD) filters are designed. In the second step, these FD filter coefficients are approximated by polynomial functions of FD. The FIR FD filter design problem is formulated in the peak-constrained weighted least-squares (PCWLS) sense and solved by the projected least-squares (PLS) algorithm. For the allpass and IIR FD filters, the design problem is nonconvex and a global solution is difficult to obtain. The allpass FD filters are directly designed as a linearly constrained quadratic programming problem and solved using the PLS algorithm. For IIR FD filters, the fixed denominator is obtained by model reduction of a time-domain average FIR filter. The remaining numerators of the IIR FD filters are designed by solving linear equations derived from the orthogonality principle. Analyses on the relative performances indicate that the IIR VFD filter with a low-order fixed denominator offers a combination of the following desirable properties including small number of denominator coefficients, lowest group delay, easily achievable stable design, avoidance of transients due to nonvariable denominator coefficients, and good overall magnitude and group delay performances especially for high passband cutoff frequency ( ges 0.9pi) . Filter examples covering three adjacent ranges of wideband cutoff frequencies [0.95, 0.925, 0.9], [0.875, 0.85, 0.825], and [0.8, 0.775, 0.75] are given to illustrate the design methodologies and the relative performances of the proposed methods.     

Digital Filter Design Using Computer Algebra Systems

Digital filter design can be performed very efficiently using modern computer tools. The drawback of the numeric-based tools is that they usually generate a tremendous amount of numeric data, and the user might easily lose insight into the phenomenon being investigated. The computer algebra systems successfully overcome some problems encountered in the traditional numeric-only approach. In this paper, we introduce an original approach to algorithm development and digital filter design using a computer algebra system. The main result of the paper is the development of an algorithm for an infinite impulse response (IIR) filter design that, theoretically, is impossible to be implemented using the traditional approach. We present a step-by-step procedure which includes derivations of closed-form expressions for (1) the transfer functions of the implemented digital filter which contains the algebraic loop; (2) the closed-form expression for computing the number of requested iteration steps; and (3) the error function representing the difference of the output sample values of the new filter and that of the conventional filter. We demonstrate how one can use some already-known multiplierless digital filter as a start-up filter to design a new digital filter whose passband edge frequency can be simply moved by using a single parameter. As a result, we obtain a multiplierless IIR filter, which belongs to the family of low-power digital filters where multipliers are replaced with a small number of adders and shifters.     

An improved design method based on polyphase components for digital FIR filters

This paper presents an efficient design of digital finite impulse response (FIR) filter, based on polyphase components and swarm optimisation techniques (SOTs). For this purpose, the design problem is formulated as mean square error between the actual response and ideal response in frequency domain using polyphase components of a prototype filter. To achieve more precise frequency response at some specified frequency, fractional derivative constraints (FDCs) have been applied, and optimal FDCs are computed using SOTs such as cuckoo search and modified cuckoo search algorithms. A comparative study of well-proved swarm optimisation, called particle swarm optimisation and artificial bee colony algorithm is made. The excellence of proposed method is evaluated using several important attributes of a filter. Comparative study evidences the excellence of proposed method for effective design of FIR filter.     

PCLS IIR digital filters with simultaneous frequency responsemagnitude and group delay specifications

This paper presents the peak-constrained least-squares (PCLS) approach to designing IIR digital filters. PCLS IIR digital filters that meet simultaneous specifications on the frequency response magnitude and the group delay are introduced. As a point of reference, we consider the IIR digital filter design problem that appears in Deczky's (1972) classic paper and in the popular textbook by Oppenheim and Schafer (1989). In addition, the same design problem appears in the IIR filter design chapter by Higgins and Munson (1993) in the Handbook for Digital Signal Processing. By using our new algorithm with simultaneous optimization of the frequency response magnitude and the group delay, we obtain a dramatic improvement in the solution of this classic IIR digital filter design problem. Starting from the same filter structure and the same specifications for the frequency response magnitude as in the works of Deczky, Oppenheim and Schafer, and Higgins and Munson, we are able to reduce the group delay ripple by a factor of 35. In another design problem that originated in Deczky's work, we use PCLS optimization to reduce the group delay ripple by a factor of 40 at the same time we reduce the stopband energy by 6 dB, without sacrificing any other performance measure. The group delay ripple in this IIR digital filter example is reduced to only ±0.002 samples     

A design algorithm for optimal low-pass nonlinear phase FIR digital filters

The transfer function of the low-pass nonlinear phase finite impulse response (NLPFIR) digital filter is decomposed into a nonlinear phase part and a linear phase part. An algorithm is proposed to iteratively design the magnitude of the linear phase part and the squared magnitude of the nonlinear phase part by directly calling the Remez algorithm of McClellan, et al. [1]. In the design of the nonlinear phase part, we assume that the linearity constraint on the phase is dropped but the phase response is not specified. A scheme is incorporated into our algorithm so that it can design the filter with the desired ripple ratio. This approach also leads to a method for finding the minimum ripple ratio for the given orders of the two parts and band edges of the filters. The filters with ripple ratio larger than this minimum value can be designed by our algorithm and neither passband nor stopband ripples are required to be prescribed. Analysis of roundoff noise reveals that the cascade filter implementation usually needs higher wordlengths than its direct for counterpart for the same roundoff noise performance.     

A modified algorithm for constrained least square design ofmultiband FIR filters without specified transition bands

In a previous paper, we described a constrained least square approach to FIR filter design that does not use “don't care” regions. In that paper, we described a simple algorithm for the design of lowpass filters according to that approach. In this paper, we describe a modification of that algorithm that makes it converge for many multiband filter designs. Although no proof of convergence is given, the modified algorithm remains simple and converges rapidly in many cases. In this approach, the user supplies a lower and upper bound constraint that is exactly satisfied by the local minima and maxima of the frequency response amplitude. Yet, the constraints can be made as tight as desired-the transition band automatically adjusts (widens) to accommodate the constraints     

Constrained Chebyshev design of FIR filters

In many filter-design problems, additional constraints are often imposed on the optimal filter in the sense of, say, minimal Chebyshev error norm. Based on the characteristic properties of the optimal filter for the Chebyshev design with frequency equation constraints, a modified Remez (MRemez) algorithm is proposed in this paper. The central problem of this paper is the constrained Chebyshev design of finite-impulse response filters with equation and inequality constraints in the frequency domain. By converting the problem into a series of Chebyshev design problems with equation constraints, an iterative MRemez algorithm which uses the MRemez algorithm as the computational core of the iteration is proposed, and the convergence of the algorithm is obtained. Design examples demonstrate the effectiveness and the fast convergence of the MRemez algorithm and the iterative MRemez algorithm.     

Lost knowledge refound: sharpened FIR filters

A scenario is presented in which an engineer in the field finds that there is a problem with the system specifications and a symmetric finite impulse response (FIR) filter in the software does not do the job; it needs reduced passband ripple or, maybe, more stopband attenuation. We present a simple method for transforming an FIR filter into one with better passband and stopband characteristics, while maintaining phase linearity. While filter sharpening may not be used often, it does have its place in an engineer's toolbox. An optimal filter has a shorter impulse response than a sharpened filter with the same passband and stopband ripple, and thus is more computationally efficient. However, filter sharpening can be used whenever a given filter response cannot be modified, such as a software code that makes use of an unchangeable filter subroutine. The scenario described is hypothetical, but all practicing engineers have been in situations where a problem needs to be solved without the full arsenal of normal design tools. Filter sharpening could be used when improved filtering is needed, but insufficient ROM space is available to store more filter coefficients, or as a way to reduce ROM requirements. In addition, in some hardware applications using filter ASICs, it may be easier to add additional chips to a design than it is to design a new ASIC.     

Envelope-constrained H2 FIR filter design

In this paper, we consider an envelope-constrained (EC)H ₂ optimal finite impulse response (FIR) filtering problem. Our aim is to design a filter such that theH ₂ norm of the filtering error transfer function is minimized subject to the constraint that the filter output with a given input to the signal system is contained or bounded by a prescribed envelope. The filter design problem is formulated as a standard optimization problem with linear matrix inequality (LMI) constraints. Furthermore, by relaxing theH ₂ norm constraint, we propose a robust ECFIR filter design algorithm based on the LMI approach.     

Efficient techniques for widening the passband of a wavelengthrouter

The wavelength response of the waveguide grating router plays an important role in optical networks, for it determines the maximum number of channels, the minimum channel separation, and the level of isolation between channels. One would like in general the router to approximate a rectangular response without exceeding a specified loss. Here, we show how the best approximation obtained for a given loss depends on the filter complexity, the total number of arms, and the specified tolerance to fabrication errors. We derive explicity the minimum loss required to produce a given approximation, and show that the filter design is primarily determined by two parameters, the ratio of channel spacing to passband width and the passband flatness. We determine explicitly the optimum design conditions. We show that the loss can be entirely eliminated by concatenating two routers, synchronized so as to produce a unitary transformation over a finite passband     

A new method for designing multiplierless two-channel filterbank using shifted-Chebyshev polynomials

This paper presents an efficient design method for a digital multiplierless two-channel filterbank using the shifted-Chebyshev polynomials and common sub-expression elimination (CSE) algorithm for reducing hardware requirements such as adders and multipliers. For designing a two-channel filterbank, the design problem is constructed as minimization of integral mean square error between the desired and designed response of a prototype filter in the passband and stopband. For controlling the performance in passband and stopband, two parameters (K_P, and K_S) are used, whose optimum values are determined by swam optimization techniques such as differential evolution algorithm, artificial bee colony optimization, particle swarm optimizations, cuckoo search algorithm and hybrid method using a fitness function, constructed by perfect reconstruction condition of a filterbank. The number of polynomials used for approximation depends upon the order of a prototype filter. A new hybrid CSE is proposed for further reduction of hardware requirement. A comparative study of various CSE techniques such as horizontal, vertical and proposed hybrid CSE is also made. Numerical examples illustrate the effectiveness of the proposed algorithm in the reduction of adders with comparisons accomplished using existing methods. It has been found that almost 43% adder gain can be achieved when a filter is designed with N = 32 and wordlength (WL) as 12 using proposed methodology.     

Design of optimal digital lattice filter structures based on genetic algorithm

In this paper, a new class of lattice-based digital filter structures is derived. The optimum structure problem is formulated in terms of minimizing the signal power ratio with respect to the two sets of free parameters in the proposed structures. An efficient genetic algorithm is proposed to solve the optimum structure problem with the constraint on the structure parameters to be represented in signed power-of-two format. Two design examples are given, in which the optimized structures show excellent finite wordlength properties such as very low parameter sensitivity and very uniform signal powers across signal nodes and outperform the classical lattice structures.     

Theory and design of signal-adapted FIR paraunitary filter banks

We study the design of signal-adapted FIR paraunitary filter banks, using energy compaction as the adaptation criterion. We present some important properties that globally optimal solutions to this optimization problem satisfy. In particular, we show that the optimal filters in the first channel of the filter bank are spectral factors of the solution to a linear semi-infinite programming (SIP) problem. The remaining filters are related to the first through a matrix eigenvector decomposition. We discuss uniqueness and sensitivity issues. The SIP problem is solved using a discretization method and a standard simplex algorithm. We also show how regularity constraints may be incorporated into the design problem to obtain globally optimal (in the energy compaction sense) filter banks with specified regularity. We also consider a problem in which the polyphase matrix implementation of the filter bank is constrained to be DCT based. Such constraints may also be incorporated into our optimization algorithm; therefore, we are able to obtain globally optimal filter banks subject to regularity and/or computational complexity constraints. Numerous experiments are presented to illustrate the main features that distinguish adapted and nonadapted filters, as well as the effects of the various constraints. The conjecture that energy compaction and coding gain optimization are equivalent design criteria is shown not to hold for FIR filter banks     

Gradient flow approach to discrete-time envelope-constrained filterdesign via orthonormal filters

Using digital orthonormal filters and Lagrangian duality theory, the envelope-constrained (EC) filtering problem has been formulated as a dual quadratic programming (QP) problem with simple constraints. Applying the barrier-gradient and barrier-Newton methods based on the space transformation and gradient flow technique, two efficient design algorithms are constructed for solving this QP problem. An adaptive algorithm, based on the barrier-gradient algorithm, is developed to solve the EC filtering problem in a stochastic environment. The convergence properties are established in the mean and mean square error senses. To demonstrate the effectiveness of the proposed algorithms, a practical example using the Laguerre networks is solved for both the deterministic and stochastic cases     

Optimal FIR filter for suppression of power line frequency component

In this paper a design technique for multiband FIR filter for DC and power line frequency component suppression is described. The digital filter has a linear phase characteristic and equiripple amplitude characteristic in the passband. The amplitude characteristic of this filter satisfies the alternation theorem and consequently has the lowest complexity in obtaining a prescribed maximum deviation in the passband and the fastest possible rate of attenuation near the cutoff frequency. In this sense this filter is optimal. Closed-form expressions for direct calculation of filter coefficients, cutoff frequency and maximal passband attenuation are derived.     

Constrained eigenfilter design without specified transition bands

The design of a finite-impulse response (FIR) filter with constraints in the frequency domain and/or time domain is considered. We further consider the design specification without explicitly specified transition band bandedges. An iterative algorithm without transition band specification is proposed to design FIR filters with various design constraints. We suggest the possible design tradeoff between transition band bandwidth and the ripple size of the filter. The proposed algorithm can be used to design filters with an optimal tradeoff from the design specification. The eigenfilter formulation further allows the propose algorithm to incorporate time-domain constraints simultaneously. Various design examples are presented to illustrate the versatility of the digital filter obtained by the propose algorithm. Although we have not proven the convergence of the proposed algorithm, it is found to converge efficiently in all the simulations.     

针对级联型ⅡR滤波器群延时的均衡优化技术

为了减小由非恒定群延时所引起的滤波器的输出信号失真,本文提出一种适用于级联型无限长脉冲响应数字滤波器的群延时均衡优化方法.通过在级联型ⅡR数字滤波器每一级的输出插入全通均衡器,减小群延时在通带范围内的变化,进而减小滤波器的输出信号失真.对于本文提出的群延时优化方法,当采用1阶和2阶均衡器进行电路优化时,在0~100Hz的通带范围内,分别将群延时的变化量减小了28.19%和49.93%.基于0.18μm CMOS标准单元库进行逻辑综合与版图设计,最终得到整个滤波电路IP核版图的面积为0.1747mm².相比于已有文献方法,本文方法在群延时优化上效果显著,电路实现上功耗和面积较小,非常适合片上系统应用.