IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 53, NO. 3, JUNE 2006 867 Optimal Design of Digital IIR Filters by Using Hybrid Taguchi Genetic Algorithm Jinn-Tsong Tsai, Jyh-Horng Chou, Senior Member, IEEE, and Tung-Kuan Liu Abstract—A hybrid Taguchi genetic algorithm (HTGA) is ap- plied in this paper to solve the problem of designing optimal digital infinite-impulse response (IIR) filters. The HTGA approach is a method of combining the traditional GA (TGA), which has a powerful global exploration capability, with the Taguchi method, which can exploit the optimum offspring. The Taguchi method is inserted between crossover and mutation operations of a TGA. Based on minimizing the L p -norm approximation error and min- imizing the ripple magnitudes of both passband and stopband, a multicriterion combination is employed as the design criterion to obtain the optimal IIR filter that can fit different performance requirements. The proposed HTGA approach is effectively applied to solve the multiparameter and multicriterion optimization prob- lems of designing the digital low-pass (LP), high-pass (HP), band- pass (BP), and bandstop (BS) filters. In these studied problems, there are many parameters and numerous local optima so that these studied problems are challenging enough for evaluating the performances of any proposed GA-based approaches. The compu- tational experiments show that the proposed HTGA approach can obtain better digital IIR filters than the existing GA-based method reported recently in the literature. Index Terms—Digital infinite-impulse response (IIR) filters, genetic algorithms (GAs), multiple criteria, Taguchi method. I. I NTRODUCTION D IGITAL filter design is always an important issue in digital signal processing. In digital infinite-impulse re- sponse (IIR) filter design, there are principally two approaches, namely: 1) transformation approach and 2) optimization ap- proach. The transformation approach to the design of digital IIR filters involves the transformation of an analog filter into a digital filter at a given set of prescribed specifications. In general, a bilinear transformation is adopted in the transfor- mation approach [1]. But the performance of digital IIR filters designed by using the transformation approach is not good in most cases. In the optimization approach, with some criteria, various optimization methods have been proposed to obtain optimal filter performances to some extent, where the p-error, mean-square-error, and ripple magnitudes (tolerances) of both passband and stopband are usually used as criteria to measure Manuscript received December 3, 2003; revised November 18, 2005. Abstract published on the Internet March 18, 2006. This work was supported in part by the National Science Council, Taiwan, R.O.C., under Grants NSC94- 2218-E327-001 and NSC94-2213-E037-002. J.-T. Tsai is with the Department of Medical Information Management, Kaohsiung Medical University, Kaohsiung 807, Taiwan, R.O.C. J.-H. Chou and T.-K. Liu are with the Department of Mechanical and Automation Engineering, National Kaohsiung First University of Science and Technology, Kaohsiung 824, Taiwan, R.O.C. (e-mail: choujh@ccms.nkfust. edu.tw). Digital Object Identifier 10.1109/TIE.2006.874280 the performance of the designed digital IIR filters (for example, see [2]–[7] and references therein). In fact, digital IIR filter design is essentially a multiparameter and multicriterion opti- mization problem with multiple local optima. When we come to the design of a digital IIR filter, the following constraints should be strictly imposed in order that the overall design criteria are met in a satisfactory fashion: 1) determination of the lowest filter order; 2) filter stability; and 3) fulfillment of the tolerance settings that are determined by minimizing the ripple magnitudes of both passband and stopband in the optimization procedure. These constraints always pose great difficulty in the process of optimization. So, it is very neces- sary to develop efficient optimization algorithms to deal with digital IIR filter design problems. Therefore, recently, some researchers [8]–[14] have proposed various genetic-algorithm (GA)-based methods to solve digital IIR filter design problems. The GA is not only capable of searching multidimensional and multimodal spaces but is also able to optimize complex and discontinuous functions that are difficult to analyze mathemat- ically. The use of the GA for digital IIR filter design is prac- tical and attractive because of the following advantages [11]: 1) filter can be constructed in any form, such as cascade, parallel, or lattice; 2) low-pass (LP), high-pass (HP), band- pass (BP), and bandstop (BS) filters can be independently de- signed; 3) classical analog-to-digital transformation is avoided; 4) multiobjective functions can be simultaneously solved; and 5) obtained model can be of the lowest order. From the pre- sented results of using GA-based approaches to design optimal digital IIR filters in the literature, it can be seen that 1) design cases studied by Tang et al. [11] are more complete than those studied in [8]–[10] and [12]–[14], and 2) design results proposed by Tang et al. [11] are better than those given in [8]–[10] and [12]–[14]. In the multiparameter and multicriterion optimization prob- lem of designing digital IIR filters, the particular challenge is that the GA-based methods may be trapped in the local optima of the multiobjective functions when the number of parameters is large and there are numerous local optima. Therefore, the purpose of this paper is to apply a robust approach, which is motivated by the work in [15] and is named the hybrid Taguchi GA (HTGA) [16], to solve the problem of designing optimal digital IIR filters. The HTGA approach is a method of combin- ing the traditional GA (TGA) [17] with the Taguchi method [18]–[20]. In the HTGA approach, the Taguchi method is inserted between crossover and mutation operations. Then, the systematic reasoning ability of the Taguchi method is incorpo- rated in the crossover operations to select better genes to tailor the crossover operations in order to generate the representative 0278-0046/$20.00 © 2006 IEEE