Regular Paper
Artificial bee colony algorithm to design two-channel quadrature mirror filter banks

https://doi.org/10.1016/j.swevo.2014.12.001Get rights and content

Abstract

Artificial bee colony (ABC) algorithm has been introduced recently for solving optimization problems. The ABC algorithm is based on intelligent foraging behavior of honeybee swarms and has many advantages over earlier swarm intelligence algorithms. In this work, a new method based on ABC algorithm for designing two-channel quadrature mirror filter (QMF) banks with linear phase is presented. To satisfy the perfect reconstruction condition, low-pass prototype filter coefficients are optimized to minimize an objective function. The objective function is formulated as a weighted sum of four terms, pass-band error, and stop-band residual energy of low-pass analysis filter, square error of the overall transfer function at the quadrature frequency and amplitude distortion of the QMF bank. The design results of the proposed method are compared with earlier reported results of particle swarm optimization (PSO), differential-evolution (DE) and conventional optimization algorithms.

Introduction

The design of two-channel quadrature mirror filter (QMF) banks has been comprehensively studied in recent years [1], [2], [3], [4], [5], [6], [7], [8]. Interest in efficient design of QMF banks is due to their significant applications in many engineering fields, such as analog to digital conversion [9], design of wavelet bases [10], [11], image processing [12], [13], digital trans-multiplexers [14], discrete multi-tone modulation systems [15], 2-D short-time spectral analysis [16], antenna systems [17], [18], biomedical signal processing [19], wideband beamforming for sonar [20] and in wireless communication for noise cancellation [21].

Relevant previous state-of-the-art work on the design of linear phase two-channel QMF banks [1], [2], [3], [4], [5], [6], [7], [8], [22], [23], [24], [25], [26], [27], [28], [29] can be classified into a number of different approaches. The least-squares [22], [23], [24] and weighted least-squares (WLS) [25], [26], [27] design methods had been applied previously. In [23], [24], an eigenvector-eigenvalue approach was presented to find the optimum prototype filter tap weights in time domain. Chen and Lee [25] proposed a WLS method for QMF bank in frequency domain. This method uses a linearization technique to reformulate the highly nonlinear design problem into quadratic form and obtain optimized filter coefficients. Lu et al. [27] developed a method based on self convolution technique to reformulate a fourth order objective function as a quadratic function. But due to complex optimization techniques these methods are not suitable for higher order filter banks. Various iterative methods [6], [7], [8], [28], [29] have been applied for the design problem of two-channel QMF bank based on single objective or multi-objective, and constraint or unconstraint nonlinear optimization. Authors in [6] have developed an efficient technique by considering filter responses in transition band as well as in pass-band and stop-band regions. Reference [8] presented a modified field function method for the design of QMF bank by finding the global minimum of the nonconvex optimization problem.

The conventional design methods may fail to achieve the optimal design for highly nonlinear and complex objective functions. Gradient based methods [6], [26], [27], [28], [29] may easily be trapped at local minima on search space and some methods [25], [29] requiring intensive matrix inversion calculations therefore, not suitable for designing QMF bank in real-time. Consequently, nowadays researchers have been attempting the design methods for QMF bank based on modern global optimization algorithms. The authors in [30] applied a genetic algorithm for the design of multiplier-less lattice QMF. Neural networks [4], differential-evolution [3], [31] and swarm intelligence [1], [2] based approaches have been presented for the design of optimum QMF bank.

In recent years, swarm intelligence has become very popular among researchers for solving optimization problems from various engineering fields. The swarm intelligence models the population of interacting agents that are able to self-organize [32]. Typical examples are: an ant colony, an immune system, a flock of birds, fish schooling and bees swarming around their hive. Particle swarm optimization (PSO) algorithm has emerged as a powerful tool for solving non-linear equations in multi-dimensional space and it has been applied successfully for the design of two-channel QMF bank [2]. Rafi et al. [1] proposed an improved particle swarm optimization method for designing linear phase QMF banks. Ghosh et al. [3] presented an approach based on adaptive-differential-evolution algorithm for the design of two-channel QMF banks.

ABC algorithm is a newly introduced optimization algorithm for both constrained and unconstrained problems based on intelligent foraging behavior of honeybee swarms and has many advantages over earlier swarm intelligence algorithms [33]. The scout bee phase is a peculiar stage of ABC algorithm in comparison to PSO and DE algorithms that provides diversity in the population. In [34], Karaboga successfully applied ABC algorithm for design of digital IIR filters. The authors in [35] used ABC algorithm to design multiplier-less nonuniform filter bank transmultiplexer.

In this paper, a novel method based on ABC algorithm is described for designing two-channel QMF bank. The results of proposed method are also compared with existing algorithms based on PSO, modified PSO and DE. The organization of rest of paper is as follows. Section 2 reviews the design problem of QMF bank based on Marquardt optimization method presented in [6]. Section 3 describes the ABC algorithm and modified ABC algorithm. Section 4 presents ABC algorithm based design of prototype filter for QMF bank. Section 5 discusses the design results of the filter bank and comparison with already existing methods. Finally, conclusions are drawn in Section 6.

Section snippets

Design of nearly perfect reconstruction QMF bank

Fig. 1 shows a typical two-channel QMF bank. The reconstructed signal x^(n) suffers from three types of errors: aliasing distortion (ALD), phase distortion (PHD), and amplitude distortion (AMD). ALD can be canceled totally by selecting the synthesis filters adroitly in terms of the analysis filters and PHD eliminated by using the linear phase finite impulse response (FIR) filters [36], [37]. The overall transfer function of such an alias and phase distortion free system turns out to be a

Artificial bee colony (ABC) algorithm

Artificial bee colony (ABC) algorithm [33] was proposed by Karaboga in 2005, based on foraging behavior of real honey bee swarms. The collective intelligence model of honeybee swarms consists of four essential elements: food sources, employed bees, onlooker bees and scouts [34], [38].

  • Food sources: A forager bee evaluates various properties to select a food source, i. e., its distance from the hive, nectar amount, taste of nectar, and difficulties in extracting the nectar.

  • Employed bees: Bees

Minimization of objective function using ABC algorithm

The ABC algorithm, discussed in Section 3, is used to solve the unconstrained optimization problem formulation for two-channel QMF bank. The prototype filter coefficients h0 (n), n=0, 1, 2….. (N/2−1) are optimized using the computational procedure illustrated in Fig. 2. The main structure of the algorithm contains the evolution of best food source by placing the employed bees, onlooker bees and scout bees in every iteration, until the maximum number of cycles or end condition reached.

In the

Simulation results and discussion

This section presents results obtained by applying the proposed algorithm to three design examples of two-channel QMF bank. Step by step design procedure for prototype filter described in the previous section has been implemented using MATLAB and tested on an Intel Core 2 Duo CPU @ 2.10 GHz with 1 GB RAM. We examine the effectiveness of the proposed ABC based algorithm in terms of five significant parameters: (i) measure of ripple (εR) in dB=|ωmax10log|T0(ejω)|||ωmin10log|T0(ejω)||, (ii) mean

Conclusion

In this paper, a novel design method based on artificial bee colony algorithm has been proposed for two-channel QMF banks with linear phase. The developed algorithm has been tested on three examples of QMF bank design problem, which is a nonlinear multidimensional optimization problem. The proposed objective function for the design problem is minimized by optimizing the low-pass prototype filter tap weights. The performance of the ABC based algorithm was compared with that of PSO,

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