Minimax Design Algorithm For IIR Digital Filter Based On GN Strategy

With the rapid development of science and technology, digital signal processing has foundextensive applications in many areas. Digital filters are basic units of digital signal processing sys-tems. Compared with finite impulse response (FIR) digital filters, infinite impulse response (IIR)digital filters have higher frequency selectivities, smaller passband group delays, and can be imple-mented with less arithmetic elements. Therefore, IIR digital filters have attracted more researchers’attention than FIR filters. However, since IIR filter has no precise linear phase and intrinsic stability,and the mathematical models of their optimal designs are usually non-convex programming modelswith nonlinear constraints, there are more challenges and difficulties in the optimal design of IIRdigital filters than FIR digital filters.In this thesis, the optimal design of IIR digital filter is considered with a focus on the minimaxdesign of stable IIR digital filters. After introducing the importance of IIR digital filter designs inpractical applications and academic researches, some basic concepts of IIR digital filter andoptimization methods are given and some past research works on IIR digital filter designs arereviewed, with an emphasis on the advantages and shortcomings of existing methods dealing withthe nonconvexities and stability constraints of the optimal design problems. The main work of thisthesis includes an extensive study on the Gauss-Newton (GN) strategy for converting nonconvexdesign problems to convex ones and combinations of the GN strategy with a sequential constrainedleast-squares (SCLS) method and a second order factor (SOF) based sequential minimization(SMSOF) procedure, which can be described in more details as follows:1. The GN strategy is firstly reviewed and then incorperated with the positive realneas andgeneralized positive realness stability conditions to design stable IIR digital filters. The GN strategyis an effective strategy dealing with the nonconvexities of IIR digital filter design problems. Thestrategy converts the nonconvex problem into a series of convex problems by using the first orderTaylor expansion of the filter’s frequency response. In order to obtain a stable filter, existing GNstrategy based design methods use a stability constraint obtained from the Rouche’s theorem. In thisthesis, the GN strategy is incorperated with the positive realness and generalized positive realnessstability conditions, and then used to solve the minimax design of stable IIR digital filters. Designresults with good performances have been obtained.2. The GN strategy is combined with an SCLS method resulting in an SCLS-GN algorithm.For the minimax design of IIR digital filters, many methods such as linear programming methods,iteratively reweighted least-squares methods, iterative quadratic programming methods, iterative second-order conic programming methods, and iterative semi-definite programming methods hadbeen proposed in past decades. They obtained good design results in various degrees and variousaspects. Recently, an SCLS method combined with the Levy-Sanathanan-Koerner (L-SK) strategyhas been proposed to obtain better filters than several recent methods just mentioned. The methodconverts the design problem into a sequence of constrained least squares (CLS) problems first, andthen into a circularly constrained quadratic programming subproblems using the L-SK strategy. Inthis thesis, the SCLS method is combined with the GN strategy to convert the minimax designproblems into a sequence of CLS problems, and further into a series of circularly constrainedquadratic programming subproblems using the GN strategy. Design results show that the proposedSCLS-GN method has obtained better filters than the L-SK based SCLS method.3. The GN strategy is combined with an SMSOF method resulting in an SOF-and-GN basedsequential minimization method, SMSOF-GN, for the minimax design of stable IIR digital filters.The recently proposed SMSOF procedure method converts the minimax design into a series ofminimax problems of IIR sub-filters with second-order denominators, such that the stability-trianglebased necessary and sufficient stability constraint can be easily applied to the design of thesesub-filters without losing their denominators’linearities with respect to their coefficients. Thenonconvex sub-filter design problems are further converted into convex problems using the L-SKstrategy. In this thesis, the SMSOF method is combined with the GN strategy to convert eachminimax problem of the sub-filter with a second-order denominator into a series of second-orderconic programming problems using the GN strategy rather than the L-SK strategy. The resultingSMSOF-GN method has obtained better filters than SCLS-LSK and SCLS-GN methods.