Global Optimal Design of IIR Filters via Constraint Transcription and Filled Function Methods
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Global Optimal Design of IIR Filters via Constraint Transcription and Filled Function Methods B.W.K. Ling · C.Z. Wu · K.L. Teo · V. Rehbock
Received: 20 December 2011 / Revised: 10 October 2012 / Published online: 27 October 2012 © Springer Science+Business Media New York 2012
Abstract In this paper, we consider a globally optimal design of IIR filters. We formulate the design problem as a nonconvex optimization problem with a continuous inequality constraint and a nonconvex constraint. To solve this problem, the constraint transcription method is applied to tackle the continuous inequality constraint. In order to avoid the obtained solution being on the boundary of the feasible set, more than one initial points are used. Moreover, since the objective and the constraints are nonconvex functions, there may be many local minima. To address this problem, the filled function method is applied to escape from the local minima. Some numerical computer simulation results are presented to illustrate the effectiveness and efficiency of the proposed method. Keywords IIR filter design · Constraint transcription · Filled function · Global optimization B.W.K. Ling () Faculty of Information Engineering, Guangdong University of Technology, Guangzhou 510006, China e-mail: [email protected] C.Z. Wu School of Mathematics and Computer Science, Chongqing Normal University, Chongqing 401331, China e-mail: [email protected] K.L. Teo · V. Rehbock Department of Mathematics and Statistics, Curtin University, Perth, Western Australia WA6012, Australia K.L. Teo e-mail: [email protected] V. Rehbock e-mail: [email protected]
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Circuits Syst Signal Process (2013) 32:1313–1334
1 Introduction A globally optimal design of digital filters is important for digital signal processing because the best performance of the filters can be achieved based on a given resource budget. There are two principal approaches for the design of digital infinite impulse response (IIR) filters, namely, the transformation approach and the optimization approach. For the transformation approach, analog IIR filters are pre-designed. Then, they are transformed into digital IIR filters [2]. For the optimization approach, digital IIR filters are obtained by solving an optimization problem with a specified criterion [1, 7, 8, 16], which is either minimax or least square. For the minimax criterion, the emphasis is to minimize the maximum ripple magnitude of the filters. Thus, the minimax filters may have high sidelobe energy. On the other hand, the least square criterion is to suppress the noise gain of the filters. Thus, the obtained filters will have low sidelobe energy, but may have large ripple magnitude near the cutoff frequency. Clearly, it is important to find a proper tradeoff between these two approaches. Thus, a constraint has been imposed on the maximum ripple magnitude [11]. The optimization problem is actually a continuous inequality constrained optimization problem. To tackle the stability issue, there are four major approaches. The first
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