Two-dimensional Recursive Digital Filters Research Articles

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.

Hybrid Approach to Design of Two Dimensional Stable IIR Digital Filter

In this paper, the magnitude optimization for the design of two dimensional digital filter is attempted and new design method for a two-dimensional (2-D) recursive digital filter is proposed via appropriate genetical swarm optimization with the introduction of the desired stability constraints. The method is tested on a numerical example and the obtained results are presented with previously published methods.

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.

Lattice Structure Realization for The Design of 2-D Digital Allpass Filters With General Causality

This paper presents a lattice structure for the realization of two-dimensional (2-D) recursive digital allpass filters (DAFs) with general causality. We employ four basic lattice sections to realize 2-D recursive DAFs with wedge-shaped coefficient support region like a nonsymmetric half-plane (NSHP) support region. The theory and transfer functions of the realized 2-D lattice DAFs are derived. Some variations of the 2-D lattice structure are also presented. We use the Roesser state space model to verify the minimal realization of the proposed 2-D recursive lattice DAF. We present a least-squares design technique and a minimax design technique to solve the nonlinear optimization problems of the proposed 2-D lattice DAF structure. The novelty of the presented lattice structure is that it not only inherits the desirable attributes of 1-D Gray-Markel lattice allpass structure but also possesses the advantage of better performance over the existing 2-D lattice allpass structures. Finally, several design examples are provided for conducting illustration and comparison.

The Phase Characteristics for the Stability of 2-D Nonsymmetric Half-Plane Digital Allpass Filters

The phase-response property for the bounded-input bounded-output (BIBO) stability of a two-dimensional (2-D) causal recursive digital allpass filter (DAF) with nonsymmetric half-plane (NSHP) support region is presented. Both cases of filters with real coefficients and complex coefficients are explored. Moreover, the effect of the numerator polynomial of a 2-D NSHP DAF on stability is also considered. The presented phase-response property has several applications. A remarkable application is that it can be utilized to enforce stability for a 2-D NSHP DAF design by choosing an appropriate phase specification. The eigenfilter design of 2-D NSHP DAFs for this application is also presented. Design examples are given to show these applications and the stability of a 2-D NSHP DAF is also inspected by the 2-D spectral factorization-based algorithm and a zeros-examining method presented by Ekstrom and Woods for confirmation.

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 digital allpass-based two-channel quincunx QMF banks

This paper presents a novel structure for constructing two-channel quincunx quadrature mirror filter (QQMF) banks. The analysis and synthesis filters of the QQMF bank are composed of two-dimensional (2-D) recursive digital allpass filters with symmetric half-plane support regions. Using the 2-D doubly complementary half-band property possessed by the analysis and synthesis filters, we facilitate the design of the digital allpass-based two-channel QQMF banks. For finding the coefficients of the 2-D recursive symmetric half-plane digital allpass filters, the design problem is appropriately formulated to result in an optimization problem that can be solved by using a weighted least-squares algorithm in the minimax (L∞) optimal sense. The novelty of the proposed design method is that the designed 2-D QQMF bank achieves perfect magnitude response and possesses satisfactory phase response without requiring extra phase equalizer. Simulation results are also provided for illustration and comparison. Copyright © 2012 John Wiley & Sons, Ltd.

Noise Reduction in Two-Dimensional Recursive Digital Filters with Discrete Optimized Error Feedback Coefficients

This paper treats the minimization problem for roundoff noise in two-dimensional (2-D) recursive digital filters with an error feedback (EF). The EF is known as an available scheme for the reduction of roundoff noise, and this can be achieved by extracting a roundoff error from the quantizer and feeding it back to the adder. Since the EF circuit is an additional device of the digital filter, the bit length of the coefficients should be as short as possible for the fast arithmetic and low-cost realization. In this paper, we propose an algorithm to optimize the EF coefficients in a discrete space. The proposed algorithm is based on the branch and bound method by estimating the lower bound of the optimal solution, and the optimality of the obtained solution can be guaranteed. Finally, we give a numerical example to demonstrate the effectiveness of the proposed method, and show the optimal solution can be found without using the round-robin algorithm.

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.

Two-Channel Parallelogram QMF Banks Using 2-D NSHP Digital All-Pass Filters

This paper presents a novel structure for the analysis and synthesis filters of two-channel parallelogram quadrature mirror filter (PQMF) banks. Two-dimensional recursive digital all-pass filters (DAFs) with nonsymmetric half-plane (NSHP) support regions are used as the basic sections for building this structure. With an appropriate combination of the 2-D NSHP DAFs, the resultant analysis/synthesis filters possess a 2-D doubly complementary symmetry that facilitates the design and implementation of two-channel PQMF banks. Moreover, the proposed PQMF bank provides approximately linear-phase response without magnitude distortion. The design problem of the proposed PQMF bank is appropriately formulated to result in a minimization of the linear-phase error associated with the 2-D NSHP DAFs in the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$p$</tex></formula> th-norm <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$(L_{p})$</tex></formula> sense. Efficient design techniques are developed to solve the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$L_{p}$</tex> </formula> optimization problems. Simulation results for illustration and comparison are also provided.

Design of a two-dimensional recursive filter using the bees algorithm

This paper presents the first application of the bees algorithm to the optimisation of parameters of a two-dimensional (2D) recursive digital filter. The algorithm employs a search technique inspired by the foraging behaviour of honey bees. The results obtained show clear improvement compared to those produced by the widely adopted genetic algorithm (GA).

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.

A Comparative Study of Stability Testing Approaches of Two-Dimensional Recursive Digital Filters

There are many problems in science and engineering whose solution is applied in the design of Multi-Dimensional (MD) digital filters. Digital filtering finds an important position in the field of digital signal and image processing. Recently there had been a great deal of interest in the design and stability analysis of Two-Dimensional (2-D) recursive digital filters. The design techniques for stable One Dimensional (1-D) digital filters are relatively well developed; but their extension to 2-D had been tried with difficulties. The stability problem of 2D filters had been the subject of intensive research during the recent decades. The trend is towards finding sufficient conditions, because necessary and sufficient conditions are difficult to find or to test. In this study, we make a detailed comparison of the various approaches available for testing the stability of 2-D recursive digital filters. Though researchers had done a lot on the stability testing methods, a comparative study by listing the computational complexities of various methods had not been done so far. This study would help the researchers to choose proper stability testing methods based on the computational complexities.

Design of two-dimensional recursive filters by using Taguchi-based immune algorithm

The Taguchi-based immune algorithm (TBIA), based on both the features of a biological immune system and the systematic reasoning ability of the Taguchi method, is applied in this work to solve the design problem of two-dimensional (2D) recursive digital filters. In the TBIA, the clonal proliferation within hypermutation for several antibody diversifications and the recombination by using the Taguchi method for the local search are integrated to improve the capabilities of exploration and exploitation. The systematic reasoning ability of the Taguchi method is implemented in the recombination operation to select the better antibody genes to achieve the potential recombination, and consequently enhance the TBIA. The design of the 2D filter is reduced to a constrained minimisation problem, the solution of which is achieved by the convergence of the presented TBIA. The computational experiments show that the presented TBIA approach can obtain better results than some previous design methods.

An Overview of Modified Shanks’ Conjecture and Comments on its Validity

This study reports the validity of the modified Shanks’ conjecture on the planar least squares inverse (PLSI) method of stabilizing two-dimensional (2-D) recursive digital filters. A theoretical procedure proposed based on the Lagrange multiplier method of mathematical optimization. The results indicate that the modified Shanks’ conjecture reported by Jury was valid for special classes of 2-D polynomials.

Evolutionary design of 2-dimensional recursive filters via the computer language GENETICA

In this paper, we present a new design method for a class of two-dimensional (2-D) recursive digital filters using an evolutionary computational system. The design of the 2-D filter is reduced to a constrained minimization problem the solution of which is achieved by the convergence of an appropriate evolutionary algorithm. In our approach, the genotypes of potential solutions have a uniform probability within the region of the search space specified by the constraints and zero probability outside this region. This approach is particularly effective as the evolutionary search considers only those potential solutions that respect the constraints. We use the computer language GENETICA, which provides the expressive power necessary to get an accurate problem formulation and supports an adjustable evolutionary computational system. Results of this procedure are illustrated by a numerical example, and compared with those of some previous designs.

The PLSI Method of Stabilizing Two-Dimensional Nonsymmetric Half-Plane Recursive Digital Filters

Two-dimensional (2D) recursive digital filters find applications in image processing as in medical X-ray processing. Nonsymmetric half-plane (NSHP) filters have definitely positive magnitude characteristics as opposed to quarter-plane (QP) filters. In this paper, we provide methods for stabilizing the given 2D NSHP polynomial by the planar least squares inverse (PLSI) method. We have proved in this paper that if the given 2D unstable NSHP polynomial and its PLSI are of the same degree, the PLSI polynomial is always stable, irrespective of whether the coefficients of the given polynomial have relationship among its coefficients or not. Examples are given for 2D first-order and second-order cases to prove our results. The generalization is done for theth order polynomial.

Design of two-dimensional recursive filters using genetic algorithms

In this paper, we examine a new design method for two-dimensional (2-D) recursive digital filters using genetic algorithms (GAs). The design of the 2-D filter is reduced to a constrained minimization problem the solution of which is achieved by the convergence of an appropriate GA. Theoretical results are illustrated by a numerical example. Also, comparison with the results of some previous design methods is attempted.

Identification of 2-D Discrete Systems Using Neural Network

Abstract A Neural Network (NN) architecture is proposed for identification of two-dimensional (2-D) recursive digital filters in spatial domain using arbitrazy inputs and their corresponding output data. The method utilizes a recurrent neural network to obtain various parameters of a Local State-Space (LSS) 2-D model by solving the nonlinear optimization problem in l2 norm. The proposed identification technique could be applied to a general 2–D Roesser’s LSS model as well as a particular class of 2–D digital filters such as separable-in-denominator structure. Simulation results indeed demonstrate the effectiveness of the proposed methods.

Design of two-dimensional recursive filters by using neural networks

A new design method for two-dimensional (2-D) recursive digital filters is investigated. The design of the 2-D filter is reduced to a constrained minimization problem the solution of which is achieved by the convergence of an appropriate neural network. The method is tested on a numerical example and compared with previously published methods when applied to the same example. Advantages of the proposed method over the existing ones are discussed as well.