Filter modeling using gravitational search algorithm

Abstract

This paper is devoted to the presentation of a new linear and nonlinear filter modeling based on a gravitational search algorithm (GSA). To do this, unknown filter parameters are considered as a vector to be optimized. Examples of infinite impulse response (IIR) filter design, as well as rational nonlinear filter, are given. To verify the effectiveness of the proposed GSA based filter modeling, different sets of initial population with the presence of different measurable noises are given and tested in simulations. Genetic algorithm (GA) and particle swarm optimization (PSO) are also used to model the same examples and some simulation results are compared. Obtained results confirm the efficiency of the proposed method.

Introduction

The word "filter" is derived from electrical engineering, where filters are used to transform electrical signals from one form to another, especially to eliminate various frequencies in a signal. This means that the main role of the filter is to keep frequency contents in the desired band and to eliminate other parts when the external input signal passes the filter.

There are two main kinds of filter: analog and digital. An analog filter employs analog electronic circuits made up of components to produce the desired filtering effect. Such filter circuits are widely used in applications such as noise reduction, signal enhancement, etc. A digital filter uses a digital processor to perform numerical calculations on sampled values of the signal. Presently, due to the increase in speed and flexibility of digital systems, most signal processing tasks are being performed in the discrete time domain.

Digital filters are broadly classified into two main categories: linear and nonlinear filters. Two important types of linear filters are finite impulse response (FIR) filters and infinite impulse response (IIR) filters (Oppenheim et al., 1999, Mitra and Kaiser, 1993, Antoniou, 1993). Output of a FIR filter is calculated based on current and previous input values. This means that a FIR filter has a finite impulse response. However, the present output of an IIR filter is dominated not only by present and past inputs but also by past outputs. Current output of an IIR filter depends on previous output values. This means that an IIR filter has an infinite impulse response. It is also termed as a recursive filter. A recursive digital filter is simply a linear difference equation with constant coefficients and nothing more. Nonlinear filters are another type of digital filters used in systems with nonlinear behaviour.

Nowadays, adaptive digital filters are applied in a wide range of areas such as signal processing, control, communication systems, image processing, system identification and modeling, and noise cancellation (Su and Cai, 2009, Saha and Roy, 2009, Farouk and Smith, 2000). The objective of the adaptation is to alter filter coefficients of a digital filter to estimate actual parameters of an unknown system from its inputs and outputs. In this case, minimization of an objective function (typically the mean square error between desired response and estimated filter output) is often followed by gradient based iterative search algorithms.

However, when the error surface (objective function) is multimodal and/or non-smooth, gradient-based methods often cannot succeed in converging to the global minimum. In this condition, heuristic optimization methods that require no gradient and can achieve a global optimal solution offer considerable advantages in solving these difficult optimization problems. Genetic algorithm (GA; Goldberg, 1989), simulated annealing (SA; Kirkpatrick et al., 1983), ant colony optimization (ACO; Dorigo et al., 1996), and particle swarm optimization (PSO; Kennedy and Eberhart, 1995) are four well-known classes of such global optimization methods. The heuristic algorithms are widely used in solving system identification and filter modeling problems (Valarmathi et al., 2009, Chang, 2007, Eftekhari and Katebi, 2008, Chen and Luk, 1999, Howell and Gordon, 2001, Karaboga et al., 2004, Kalinli and Karaboga, 2005, Das and Konar, 2007, Lin et al., 2008).

Filter modeling using these heuristic algorithms like GA, ACO, and PSO are reported in some researches (Chen and Luk, 1999, Howell and Gordon, 2001, Karaboga et al., 2004, Kalinli and Karaboga, 2005, Das and Konar, 2007, Lin et al., 2008). In Chen and Luk (1999), adaptive simulated annealing has been used in signal processing applications, including maximum likelihood (ML) joint channel data estimation and IIR filter design. In Howell and Gordon (2001), continuous action reinforcement learning has been applied in digital IIR filter design. Ant colony optimization has been used for IIR filter design in Karaboga et al. (2004). In Kalinli and Karaboga (2005), immune algorithm (IA) has been proposed for designing IIR filters. In Karaboga et al. (2004) and Kalinli and Karaboga (2005), estimations of a second-order filter with a first-order model and a second-order filter with a second-order model have been reported. Designing a 2-D filter with PSO has been reported for image denoising in Das and Konar (2007). Nonlinear rational filter estimation using PSO and GA has been introduced in Lin et al. (2008).

Recently, a novel heuristic search algorithm, called gravitational search algorithm (GSA), has been proposed motivated by the gravitational law and laws of motion (Rashedi et al., 2009). It is characterized as a simple concept that is both easy to implement and computationally efficient. GSA has a flexible and well-balanced mechanism to enhance exploration and exploitation abilities. In the present work, GSA is proposed to model IIR filters and nonlinear rational filters. To verify the effectiveness of the proposed GSA based filter modeling, different sets of initial population with the presence of different measurable noises are given and tested in simulations. GA and PSO are also used to model the same examples and some simulation results are compared.

The rest of the paper is organized as follows. A brief review of GSA is given in the next section to provide a proper background. This section is followed by problem definition and explanation in Section 3 and experimental results and comparison with other methods in Section 4. Finally, the paper is concluded in Section 5.