Elsevier

Applied Soft Computing

Volume 13, Issue 1, January 2013, Pages 329-338
Applied Soft Computing

Fixed-point digital IIR filter design using two-stage ensemble evolutionary algorithm

https://doi.org/10.1016/j.asoc.2012.09.004Get rights and content

Abstract

The research on optimal design of infinite-impulse response (IIR) filter design based on various optimization techniques, including evolutionary algorithms (EAs), has gained much attention in recent years. Previously, the parameters of digital IIR filters are encoded with floating-point representations. It is known that a fixed-point representation can effectively save computational resources and is more convenient for direct realization on hardware. Inherently, compared with the floating-point representation, the fixed-point representation would make the search space miss much useful gradient information and therefore, surely rises new challenges for continuous EAs. In this paper, we first analyze the fitness landscape properties of optimal digital IIR filter design. Based on the fitness landscape investigation, a two-stage ensemble evolutionary algorithm (TEEA) is applied to digital IIR filter design with fixed-point representation. In order to fully evaluate the performance of TEEA, we experimentally compare it with five state-of-the-art EAs on four types of digital IIR filters with different settings. Based on the experimental results, we can conclude that TEEA has higher convergence speed, better exploration, and higher success rate. In order to benchmark TEEA further, we apply it to some more difficult problems with shorter word length or higher order. We can find that TEEA can provide satisfying performance on these hard tasks as well.

Highlights

► Fitness landscape properties of LP, HP, BP and BS. ► Procedure of TEEA. ► Filter performance comparison on four types of fixed-point digital IIR filters. ► Scalability test of TEEA.

Introduction

Considered as an important but very hard task in digital signal processing, digital IIR filter design has attracted much research attention in recent decades [4], [5], [6], [7], [8], [9], [10], [11], [12]. Digital IIR filter is one of the most frequently used computation tools in digital signal processing systems. In many applications, such as high-speed and low-power communication transceivers, it is routinely employed as a custom designed digital block [1].

In previous works, some classical methods have been proposed to tackle such a hard task. The bilinear transformation approach is one of the early techniques [5] and has been widely adopted. Via this approach, a digital filter is transformed to a corresponding analog low-pass (LP) filter. Then, the well-known LP filter design methods, such as Butterworth, Chebyshev Type I, and Chebyshev Type II, can be used to accomplish the design of the analog LP filter. Finally, the analog LP filter is transformed back to the digital filter by again using a bilinear transformation [14]. However, this procedure needs too much pre-knowledge and shows poor performance in most cases [6]. Besides, the filters designed by these methods are always encoded with floating-point representations. Using floating-point numbers is impractical in hardware design and requires higher computational power. Therefore, this stimulated the research on more effective optimization approaches with less prior knowledge and higher accuracy to obtain digital IIR filter with fixed-point representation [7], [12].

Since the seminal work published in [4], diverse evolutionary algorithms (EAs) have been developed for digital IIR filter design [2], [3]. The major advantages of these EAs over other methods can be summarized in [8], [19], [20]: (1) pre-knowledge of the problems is not necessary for the application of EAs, while the highly nonlinear characteristic must be approximated firstly for transformation methods and other mathematical optimization approaches; (2) EAs usually work with a population of candidate solutions and can handle the constraints adaptively under the strategy set beforehand in a single run. The current research of applying EAs to design digital IIR filters can be mainly categorized into the following three classes:

  • Parameter estimation for single-objective digital IIR filters with floating-point representation: This design process can be described as follows: given the settings of the digital IIR filter, the method is expected to provide a final digital IIR filter to meet all requirements. Important contributions of this class include the hybrid genetic algorithm [11], the hierarchical genetic algorithm (HGA) [12] and the Taguchi-strategy enhanced GA (HTGA) [6]. Due to the fact that the fitness landscapes of digital IIR filter design problems contain many local optima, the motivation of these algorithms is to develop specific operators to strengthen the exploration ability [20].

  • System identification for single-objective digital IIR filters with floating-point representation: The major difference from this class to the first class is that the fitness value of one individual changes from time to time. The reason can be ascribed to the different input test sequences, which are always randomly or probabilistically generated. Therefore, it is apparent that this problem belongs to the noise-induced optimization domain. The typical methods for solving it are ant colony optimization (ACO) [22], seeker optimization [21], artificial immune systems [23], tabu search [24] and structured stochastic optimization [25]. Since these optimization algorithms have to work in uncertain environments, the accuracy cannot be effectively guaranteed. Hence, all the above algorithms are just applied to very low order digital IIR filter design (not higher than 7 orders) and furthermore, all algorithms in comparison in [21] cannot obtain satisfactory solutions for some hard problems.

  • Multi-objective digital IIR filter design with floating-point representation: In the above two classes, digital IIR filter design is treated as a single-objective optimization problem of minimizing the magnitude response error with supplementary conditions. However, the lack of considering the linear phase response error and the order may result in the loss of control on the structural flexibility, the distortion of output, and the dependency on pre-knowledge. Therefore, two recently published works [7], [20] implemented multi-objective evolutionary algorithms to simultaneously optimize three objectives: the magnitude response error, the linear phase response error and the order. Especially in [20], without pre-knowledge, the three objectives are considered equally important during the optimization procedure. Based on the experimental results on four types of low order digital IIR filter design, the multi-objective digital IIR filter design is promising.

Although there have been a number of successful EA-based digital IIR filter design works as summarized above, most of them only consider parameters that take on value from a continuous domain. Very Only very little research [9], [26], [27] takes fixed-point representations into consideration. However, in the development and application of embedded systems, fixed-point implementations of digital IIR filters are more and more common in real applications. Therefore, optimal design based on fixed-point representations quickly gains significance. The previous limited works on fixed-point digital IIR filter design just tackled some relatively simple problems and are lack of necessary comparison analysis. In this paper, we try to fill this gap by applying a specific evolutionary algorithm to higher order and more difficult digital IIR filter design problems.

It has been acknowledged that digital IIR filter design is a multi-modal optimization problem with a complex fitness landscape [21], which would inevitably cause premature convergence of most conventional EAs. However, as far as the authors know, no research or investigation of the optimization problems has been reported before. In this paper, the fitness landscape properties are depicted in a 3-D space and discussed from the view of heuristic methods. Based on the investigation, the difficulties of optimal digital IIR filter design are clearly revealed.

To handle such kind of challenging problems, the conflict between exploitation and exploration abilities of EAs must be balanced in a more effective and efficient way, which has been a challenging problem for decades. To cope with this challenge, we resort to a two stage ensemble evolutionary algorithm (TEEA), which was originally proposed in [28] and has been extended to large-scale global optimization in [29]. Different to the existing methods that use multiple population generation strategies in each iteration, e.g. [40], [37], [39], the main idea of TEEA is to divide the optimization procedure into two stages: (1) the global shrinking stage and (2) the local exploration stage. The objective of the first stage is to shrink the searching scope to the promising area as quickly as possible, and the objective of the second stage is to explore the limited area extensively to find as good as possible solutions. Generally speaking, the exploration (stage 2 in TEEA) may consume too much computational cost and cannot efficiently detect the promising area for complex multi-modal problems. Differently, the technique with good exploitation capability can quickly converge to an optimal solution. The execution of stage 1 surely has some risk of converging to a local optimal solution. However, by only using some seed solutions provided by stage 1 and randomly generating the other individuals, the stage 2 can make up this demerit by its strong exploration capability. If stage 1 fails, the problem is proven to be very complex that more exploration capability is needed. The advantages of TEEA, compared with several classical continuous EAs, are experimentally verified on four types of digital IIR filters with different settings. In order to evaluate TEEA further, we apply it to some more difficult problems with shorter word length or higher order. It is interesting to note that TEEA can provide a success rate of 100% on these hard tasks.

The remainder of this paper is structured as follows: in Section 2, the digital IIR filter design problem is introduced first. Then, the fitness landscape properties of the digital IIR filter design problems are investigated. Section 3 presents the TEEA. Section 4 provides an experimental evaluation of the TEEA on designing four types of digital IIR filters with different settings, including low-pass (LP), high-pass (HP), band-pass (BP) and band-stop (BS) filters. Five state-of-the-art continuous EAs, are utilized to provide comparisons. Moreover, scalability analysis is carried out to illustrate the scalable characteristics of the TEEA. In Section 5, a brief conclusion is given and the future work is outlined.

Section snippets

Problem statement and analysis

The cascade form of an infinite-impulse response (IIR) filter can be described as follows [4], [5], [6], [7], [8], [9], [10], [11], [12]:H(z)=Kk=1n1+bkz11+akz1i=1m1+di1z1+di2z21+ci1z1+ci2z2where K is the gain, ak and bk for k = 1, 2, …, n are the first-order coefficients, and ci1, ci2, di1 and di2 for i = 1, 2, …, m are the second-order coefficients. In this paper, all of the coefficients are fixed-point numbers, whose word length is set beforehand.

Two-stage ensemble evolutionary algorithm

In order to solve the conflict between exploration and exploitation in digital IIR filter design, a serial two-stage optimization framework is used in our new two-stage ensemble evolutionary algorithm (TEEA). In the previous research, many hybrid algorithms have been developed to achieve better balance between exploration and exploitation, such as in [30], [31], [32]. Unlike the above three works that simultaneously adopt different new offspring creating strategies to balance exploration and

Experiment

In order to provide comprehensive impression of the utility of our TEEA, we adopt twelve test problems belonging to four types of digital IIR filters with different settings, including low-pass (LP), high-pass (HP), band-pass (BP) and band-stop (BS). The detailed settings of the test problems are summarized in Table 4.

For the digital IIR filters represented by Eq. (1), we restrict all of the parameters in [− 2, 2], but most of the parameters locate in [− 1, 1]. Therefore, for a word length of nL,

Conclusion

In previous research, the parameters of digital IIR filters are always encoded with floating-point representations in most continuous EA based works. Since the fixed-point representation can effectively save computational resources and is more convenient for the direct realization on hardware application, we design a two-stage ensemble evolutionary algorithm (TEEA) for fixed-point digital IIR filter design. In order to fully evaluate the performance of TEEA, we designed two experiments: (1) In

Acknowledgements

The authors would like to thank Prof. P.N. Suganthan for providing the codes of his research group. The work is partially supported by the National Natural Science Foundation of China under Grand Nos. 61071024 and U0835002, and the USTC Innovation Fund for Young Researchers.

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