Design Of Recursive Digital Filters Research Articles

Recursive digital filters with optimum equiripple passband magnitude characteristic

The paper investigates, for the first time, the design of recursive digital filters that exhibit optimum equiripple passband magnitude characteristics. After a brief description of the optimum equiripple filters in the analog domain, digital optimum C (Chebyshev) tan and sine filters are derived directly in the digital domain through the minimization of the area under the passband characteristic function. The influence of the truncated coefficients is also considered and its effects are demonstrated through an example proving the robustness of the magnitude function. Furthermore, a design procedure is proposed for the optimum C tan half-band filter. To highlight the effectiveness of the proposed approximation, a comparative analysis is conducted with the staircase digital filter, as both filters optimize passband responses by minimizing the area under the characteristic function. The results of comparison demonstrate that the optimum equiripple approximation surpasses the staircase approximation, not only in the analog but also in the digital domain.

An Optimal Design of Non-Causal Recursive Digital Filters with Zero Phase Shift using Chebyshev approximation and Linear Programming

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Design of Two-Dimensional Recursive Digital Filter Using Multi Particle Swarm Optimization Algorithm

Particle Swarm Optimization (PSO) is an evolutionary algorithm widely used in optimization problems. It is characterized by a fast convergence, which can lead the algorithm to stagnate in local optima. In the present paper, a new Multi-PSO algorithm for the design of two-dimensional infinite impulse response (IIR) filters is built. It is based on the standard PSO and uses a new initialization strategy. This strategy is relayed to two types of swarms: a principal and auxiliaries. To improve the performance of the algorithm, the search space is divided into several areas, which allows a best covering and leading to a better exploration in each zone separately. This solved the problem of fast convergence in standard PSO. The results obtained demonstrate the effectiveness of the Multi-PSO algorithm in the filter coefficients optimization.

An Improved Global-Best-Guided Cuckoo Search Algorithm for Multiplierless Design of Two-Dimensional IIR Filters

Cuckoo search algorithm (CSA) is relatively a new optimization technique with less control parameters and strong exploration ability. Due to the random search associated with CSA, it requires large number of functional evaluations for obtaining optimal solution. An improved algorithm, named as improved global-best-guided CSA, is presented here based on the best solution of previous iteration for the optimal design of multiplierless two-dimensional recursive digital filters. The most important feature of the proposed algorithm is that it is completely self-adaptive with no tuning parameters, whereas in CSA the replacement factor needs to be adjusted. The proposed algorithm exhibits 52% improvement in fitness function evaluation (for p = 2) and the execution time is reduced by 56% in comparison with the existing algorithms. Further, the proposed algorithm has been tested for several benchmark problems and found to exhibit significant performance improvement.

Design of Recursive Digital Filters in Parallel Form by Linearly Constrained Pole Optimization

We present a technique to iteratively optimize poles of a recursive digital filter in parallel form. Only exposing the poles as the variables to optimize, we employ a linearly constrained gradient descent routine in which the numerical estimation of the error gradient involves first obtaining the zeros by projecting the target response over a basis of responses defined by the pole positions at a given step. Example fits are presented for exponentially decaying white noise and measured violin radiativity filters.

Generalized Stability-Triangle for Guaranteeing the Stability-Margin of the Second-Order Digital Filter

In the design of recursive digital filters, the stability of the recursive digital filters must be guaranteed. Furthermore, it is desirable to add a certain amount of margin to the stability so as to avoid the violation of stability due to some uncertain perturbations of the filter coefficients. This paper extends the well-known stability-triangle of the second-order digital filter into more general cases, which results in dented stability-triangles and generalized stability-triangle. The generalized stability-triangle can be viewed as a special case of the dented stability-triangles if the two upper bounds on the radii of the two poles are the same, which is a generalized version of the existing conventional stability-triangle and can guarantee the radii of the two poles of the second-order recursive digital filter below some prescribed upper bound. That is, it is able to provide a prescribed stability-margin in terms of the upper bound of the pole radii. As a result, the generalized stability-triangle increases the flexibility for guaranteeing a prescribed stability-margin. Since the generalized stability-triangle is parameterized by using the upper bound of pole radii, i.e., the stability-margin is parameterized as a function of the upper bound, the proposed generalized stability-triangle facilitates the stability-margin guarantee in the design of the second-order as well as high-order recursive digital filters.

Design of recursive variable digital filters with theoretically guaranteed stability

ABSTRACTStability is one of the most concerned issues in designing a recursive variable digital filter (VDF). This is because the coefficients of a recursive VDF constantly vary in the tuning process, and updating the coefficients may incur instability. Thus, an appropriate measure needs to be taken for ensuring its stability. This paper presents a new coefficient transformation (CT) method for transforming the coefficients of a recursive transfer-function denominator into a set of new coefficients. From the viewpoint of conventional constant-coefficient filter (constant filter) design, the new coefficients can take arbitrary values without incurring instability. For designing a stable VDF, we apply this CT to the variable case and approximate each transformed coefficient as a distinct polynomial in the tuning parameter. Thus, we can change the filter coefficients by changing the value of the tuning parameter, and thus tune the magnitude response. Thanks to the proposed CT, updating the filter coefficients will never incur instability. This is the core part of the CT-based design approach. In this paper, we utilise a weighting function to ignore the transition-band errors and thus enhance the design accuracy of important frequency bands (passband and stopband). Moreover, the polynomials use different degrees so as to reduce the VDF complexity. Two design examples (lowpass VDF and bandpass VDF) are provided for verifying the design accuracy and checking the stability.

Phase Response Design of Recursive All-Pass Digital Filters Using a Modified PSO Algorithm.

This paper develops a new design scheme for the phase response of an all-pass recursive digital filter. A variant of particle swarm optimization (PSO) algorithm will be utilized for solving this kind of filter design problem. It is here called the modified PSO (MPSO) algorithm in which another adjusting factor is more introduced in the velocity updating formula of the algorithm in order to improve the searching ability. In the proposed method, all of the designed filter coefficients are firstly collected to be a parameter vector and this vector is regarded as a particle of the algorithm. The MPSO with a modified velocity formula will force all particles into moving toward the optimal or near optimal solution by minimizing some defined objective function of the optimization problem. To show the effectiveness of the proposed method, two different kinds of linear phase response design examples are illustrated and the general PSO algorithm is compared as well. The obtained results show that the MPSO is superior to the general PSO for the phase response design of digital recursive all-pass filter.

On the design transitional Legendre–Butterworth filters

This paper presents a new method of the polynomial transfer function approximation both in the analog and the digital domain for the recursive digital filter design. The transfer function of these filters, referred as the transitional Legendre–Butterworth (TLB) filters, has properties that lie between Legendre and Butterworth filters. The so-designed TLB filters exhibit non-equiripple passband and monotonic transition and stopband responses. There are three types of TLB IIR filters which can be approximated in the digital domain. The closed form expressions for the magnitude squared functions coefficient of all types of transfer functions are provided, and a straightforward design technique for each approximation in the digital domain is described. Resulting digital filters are always stable.

Two-Dimensional IIR Filter Design Using Simulated Annealing Based Particle Swarm Optimization

We present a novel hybrid algorithm based on particle swarm optimization (PSO) and simulated annealing (SA) for the design of two-dimensional recursive digital filters. The proposed method, known as SA-PSO, integrates the global search ability of PSO with the local search ability of SA and offsets the weakness of each other. The acceptance criterion of Metropolis is included in the basic algorithm of PSO to increase the swarm’s diversity by accepting sometimes weaker solutions also. The experimental results reveal that the performance of the optimal filter designed by the proposed SA-PSO method is improved. Further, the convergence behavior as well as optimization accuracy of proposed method has been improved significantly and computational time is also reduced. In addition, the proposed SA-PSO method also produces the best optimal solution with lower mean and variance which indicates that the algorithm can be used more efficiently in realizing two-dimensional digital filters.

Design of 2-D Recursive Filters Using Self-adaptive Mutation Differential Evolution Algorithm

This paper investigates a novel approach to the design of two-dimensional recursive digital filters using differential evolution (DE) algorithm. The design task is reformulated as a constrained minimization problem and is solved by an Self-adaptive Mutation DE algorithm (SAMDE), which adopts an adaptive mutation operator that combines with the advantages of the DE/rand/1/bin strategy and the DE/best/2/bin strategy. As a result, its convergence performance is improved greatly. Numerical experiment results confirm the conclusion. The proposed SAMDE approach is effectively applied to test a numerical example and is compared with previous design methods. The computational experiments show that the SAMDE approach can obtain better results than previous design methods.

Design of narrow stopband recursive digital filter

The procedure for design of narrow stopband recursive digital filter realized through parallel connections of two allpass sub-filters is described in this paper. This solution also allows realization of complementary filter, using only one additional adder, and exhibit low sensitivity on coefficients quantization. The method is based on phase approximation of allpass sub-filter. The procedure is very efficient and solution can be obtained within only a few iterations even for large filter order n. Every stopband provides two more equations, one at notch frequency and the other at passband boundary. It is not possible to control attenuation at both passband boundaries, but described procedure provides that achieved attenuations are less or equal to prescribed values. Using this algorithm full control of passband edges is obtained comparing with existing methods where it is not possible.

Design Methodology for Nearly Linear-Phase Recursive Digital Filters by Constrained Optimization

A methodology for the design of recursive digital filters having nearly linear phase response is proposed. The underlying design method is of the direct type whereby the filter is designed as a single unit. The design problem is formulated as a cascade of filter sections where each section is represented by a biquadratic transfer function either in the conventional polynomial form or in the polar form. The design problem is then solved using a constrained Newton's method whereby constraints are used to assure the stability of the filter, to control the step size in order to achieve fast convergence, and to eliminate a real-axis pole-migration problem that often interferes with the design process. Several design examples demonstrate that when compared with filters designed using existing state-of-the-art methods, the proposed methodology yields filters having reduced order and/or improved performance.

Design of two-dimensional recursive digital filters with specified magnitude and group-delay characteristics using Taguchi-based Immune Algorithm

This paper presents one modern heuristic optimisation algorithm, named Taguchi-Based Immune Algorithm (TBIA), to solve the problem of designing 2D recursive digital filters with specified magnitude and group-delay characteristics. The algorithm is detailed for the design of three recursive filters' categories, namely filters with predefined magnitude, delay and magnitude and delay. On the basis of minimising the magnitude and group-delay errors, multi-criterion design combination is employed to obtain optimal recursive filters that satisfy the required specifications. Computational experiments show the ability of the proposed algorithm to obtain more robust stable complex filters compared with previously reported design methods.

RECURSIVE DIGITAL FILTER OPTIMIZATION BASED ON SIMULTANEOUS APPROXIMATION IN FREQUENCY AND TIME DOMAINS

This work considers two appropriate optimization approaches for one-dimensional recursive digital filter design when the different requirements for magnitude, group delay and step responses are specified. The simultaneous approximation design in time and frequency domains is stated as a vector optimization problem. The objective functions are defined as a least square error between the approximated characteristic and a desired target function. The other approach used to solve the simultaneous approximation is the semi-infinite optimization problem, where the functional constraints are defined by group delay and step response.

Direct design of sharp cutoff low-pass recursive digital filters

Effects of introducing single or multiple zero pairs on the unit circle on the all-pole transitional Butterworth-Chebyshev filter transfer function are investigated. It is shown that, for the same order n, a filter with zeros on the unit circle provides much sharper cutoff than all-pole filters. Coefficients of the magnitude squared function are given in a closed form. Graphs, helpful in designing such filters, are plotted.

Design of recursive digital filters with prescribed stability margin: a parameterization approach

A major problem in the design of recursive digital filters is stability. Although an unstable recursive filter obtained from a design algorithm can be stabilized by reciprocal substitution of the unstable factors of the filter without changing its amplitude response, this stabilization technique cannot be applied if phase response is a part of the design specifications. In this paper, we propose a new method for the design of recursive digital filters with a prescribed stability margin by parameterizing all stable transfer functions and carrying out unconstrained optimization over this class of transfer functions. Three parameterization techniques are described, and closed-form formulas for the gradient functions and Hessian matrices of several typical objective functions are derived. The design technique is expected to be useful in the cases where both amplitude and phase specifications are required in the design.

Minimax design of recursive digital filters with a lattice denominator

The authors deal with the problem of minimax recursive digital filter design with a lattice structure for the denominator. The design problem is formulated so that the coefficients for the numerator and denominator of a recursive filter can be found by solving the best linear complex Chebȳshev approximation (LCCA). A design technique based on the weighted least-squares algorithm previously proposed by one of the authors is then developed for solving the resulting LCCA problem. During the design process, this technique finds the tap coefficients for the numerator and the reflection coefficients for the denominator simultaneously. The stability of the designed recursive filter is ensured by incorporating an efficient stabilisation procedure to make all of the reflection coefficient values fall between –1 and +1. Computer simulations show that the proposed technique provides better design results than existing techniques.

An Improved Method for Design of Recursive Digital Filters Using Optimization

With the increased capability of today′s personal computers (PC) and software packages, it often is advantageous to formulate engineering problems in such a way as to take full advantage of these resources. In this paper a new approach to recursive filter design is presented which takes advantage of a popular spreadsheet program and a simple formulation of an optimization problem. No code must be written other than the formula definitions in a spreadsheet. Explicit derivatives are not needed and full interaction with the optimization process is possible, including graphical information. Results are given for an eighth-order filter and these are compared to previously published designs.

A systematic methodology for the design of high performance recursive digital filters

A systematic design methodology is described for the rapid derivation of VLSI architectures for implementing high performance recursive digital filters, particularly ones based on most significant digit (msd) first arithmetic. The method has been derived by undertaking theoretical investigations of msd first multiply-accumulate algorithms and by deriving important relationships governing the dependencies between circuit latency, levels of pipelining and the range and number representations of filter operands. The techniques described are general and can be applied to both bit parallel and bit serial circuits, including those based on on-line arithmetic. The method is illustrated by applying it to the design of a number of highly pipelined bit parallel IIR and wave digital filter circuits. It is shown that established architectures, which were previously designed using heuristic techniques, can be derived directly from the equations described.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>