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LMI Characterization for The Convex Hull



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LMI Characterization for The Convex Hull of Trigonometric Curves and Applications H D Tuan T T Son B Vo and T Q Nguyen Abstract In this paper we develop a new linear matrix inequality LMI technique which is practical for solutions of the general trigonometric semi infinite linear constraint TSIC of competitive orders Based on the new full LMI characterization for the convex hull of a trigonometric curve it is shown that the semi infinite optimization problem involving TSIC can be solved by LMI optimization problem with additional variables of dimension just n the order of the the trigonometric curve Our solution method is very robust which allows us to address almost all practical filter design problems Unlike most previous works involving several complex mathematical tools our derivation arguments are based on simple results of the convex analysis and some formal elementary transforms Furthermore many filter filterbank design problems can be reformulated as the optimization of linear convex quadratic objectives over the trigonometric semiinfinite constraints TSIC Based on this reformulation these problems can be equivalently reduced to LMI optimization problems with the minimal size Our examples of designing up to 1200 tap filters verifies the viability of our formulation I I NTRODUCTION A trigonometric curve is the set Ca b 1 cos t cos 2t cos nt T cos t cos a cos b 1 1 Rn 1 1 and its polar is defined as Ca b u Rn 1 u v 0 v Ca b 3 includes several interesting interpretations in signal processing as a particular case For instance the particular case x C 0 4 means that x x0 x1 xn is a positive real sequence H ej n xh cos h 0 0 5 h 0 School of Electrical Engineering and Telecommunication University of New South Wales UNSW Sydney NSW 2052 AUSTRALIA h d tuan unsw edu au Department of Electrical and Computer Engineering Toyota Institute of Technology Hisakata 2 12 1 Nagoya 468 8511 JAPAN Email ttson toyota ti ac jp Department of Electrical and Electronic Engineering University of Melbourne Parkville Vic 3052 AUSTRALIA Email bv ee mu oz au Department of Electrical and Computer Engineering University of California in San Diego 9500 Gilman Dr La Jolla CA 92093 0407 USA Email nguyent ece ucsd edu 0 7803 8874 7 05 20 00 2005 IEEE H z z n x0 n 1 xi z i z i 2 i 1 6 which are expressed as H ej e jn p 0 p H ej s s 7 are indeed x p 1 e1 C p 0 x p 1 e1 C p 0 x s e1 C x s e1 C s s 8 respectively where e1 1 0 0 Rn 1 The simplest and traditional treatment for the TSIC constraints 3 is just to replace it by a finite number of linear constraints Ax d 1 cos ti cos 2ti cos nti T 0 cos ti cos a cos b i 0 1 2 N



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