Linear Phase Filter Bank Design by Convex Programming By Hoang Kha Ha B.E (Hons) and M.E in Electrical Engineering and Telecommunications, Ho Chi Minh City University of Technology, Vietnam A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy The University of New South Wales School of Electrical Engineering and Telecommunications Sydney, Australia August 2008 ORIGINALITY STATEMENT I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of material which have been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project’s design and conception or in style, presentation and linguistic expression is acknowledged Signed Date Hoang Kha Ha i Abstract D IGTAL FILTER BANKS have found in a wide variety of applications in data compression, digital communications, and adaptive signal processing The common objectives of the filter bank design consist of frequency selectivity of the individual filters and perfect reconstruction of the filter banks The design problems of filter banks are intrinsically challenging because their natural formulations are nonconvex constrained optimization problems Therefore, there is a strong motivation to cast the design problems into convex optimization problems whose globally optimal solutions can be efficiently obtained The main contributions of this dissertation are to exploit the convex optimization algorithms to design several classes of the filter banks First, the twochannel orthogonal symmetric complex-valued filter banks are investigated A key contribution is to derive the necessary and sufficient condition for the existence of complex-valued symmetric spectral factors Moreover, this condition can be expressed as linear matrix inequalities (LMIs), and hence semi-definite programming (SDP) is applicable Secondly, for two-channel symmetric real-valued filter banks, a more general and efficient method for designing the optimal triplet halfband filter banks with regularity is developed By exploiting the LMI characterization of nonnegative cosine polynomials, the semi-infinite constraints can be efficiently handled Consequently, the filter bank design is cast as an SDP problem Furthermore, it is demonstrated that the resulting filter banks are applied to image coding with improved performance It is not straightforward to extend the proposed design methods for twochannel filter banks to M -channel filter banks However, it is investigated that ii Abstract iii the design problem of M -channel cosine-modulated filter banks is a nonconvex optimization problem with the low degree of nonconvexity Therefore, the efficient semidefinite relaxation technique is proposed to design optimal prototype filters Additionally, a cheap iterative algorithm is developed to further improve the performance of the filter banks Finally, the application of filter banks to multicarrier systems is considered The condition on the transmit filter bank and channel for the existence of zeroforcing filter bank equalizers is obtained A closed-form expression of the optimal equalizer is then derived The proposed filter bank transceivers are shown to outperform the orthogonal frequency-division multiplexing (OFDM) systems Acknowledgements First of all, I would like to express my profound gratitude to my advisor, Associate Professor Tuan D Hoang, whose his constant guidance, support and encouragement have made the completion of this dissertation possible He has introduced me to the attractable field of filter banks and has shared his expertise in the field of global optimization I am also grateful to Professor Truong Q Nguyen of the University of California, San Diego, U.S.A for numerous intellectual discussion on filter banks and wavelets Next, I would like to thank Associate Professor Ba-Ngu Vo of the University of Melbourne, Australia, with whom I had several rewarding collaborations I wish to thank Professor Victor Solo, Professor Tim Hesketh, Professor Andrey V Savkin, and Dr David Clements for serving on my annual progress review panel and for providing valuable comments and suggestions on my work I would like also to thank my lab-mates at the School of Electrical Engineering and Telecommunications for their support, discussion and friendship I gratefully acknowledge the financial supports from the Ministry of Education and Training, Vietnam, through MOET scholarship, the University of New South Wales through Top-Up scholarship, and the Australian Research Council through the discovery research grants for Associate Professor Tuan D Hoang Finally, I would like to thank my parents for their constant love and support, and for encouraging me to pursue higher studies The deepest appreciation is reserved for my wife, Hoang T M Vo whose love, understanding and caring throughout my Ph.D program is a great source of inspiration for completing this dissertation iv To my parents and wife v Contents Abstract ii Acknowledgements iv Contents vi List of Figures x List of Tables xiii Acronyms and Symbols xiv Introduction 1.1 Motivation and Scope 1.2 Related Works 1.2.1 Lattice Structure 1.2.2 Quadratically Constrained Quadratic Program 1.2.3 Spectral Based Method 1.2.4 State Space Approach 1.3 Dissertation Outline 1.4 Contributions and Publications 1.5 Notations 11 Filter Bank and Optimization Theory: An Overview 2.1 13 Filter Bank Fundamentals 13 2.1.1 14 Decimation and Interpolation vi CONTENTS vii 2.1.2 Perfect Reconstruction 16 2.1.3 Linear Phase Filter Banks 18 2.1.4 Two-channel Filter Banks 19 2.1.5 Tree-Structure Filter Banks and Wavelets 22 2.1.6 M -channel Filter Banks 24 2.2 Filter Bank Design Problems 26 2.3 Optimization Theory 28 2.3.1 Convex Optimization 29 2.3.2 Duality Theory 31 2.3.3 Semidefinite Programming Problem 32 2.3.4 LMI Characterization of Nonnegative Polynomials 34 Concluding Remarks 36 2.4 Symmetric Orthogonal Complex-Valued Filter Banks 37 3.1 Introduction 38 3.2 Mathematical Model of Orthogonal and Symmetric Filter Banks 39 3.3 Conversion to Semi-definite Programming 43 3.3.1 Semi-definite Programming Formulation 43 3.3.2 Dimension Reduction 47 3.4 Design Examples 50 3.5 Concluding Remarks 52 Triplet Halfband Filter Bank with K-Regularity and Image Coding Application 54 4.1 Introduction 55 4.2 FIR Triplet Halfband Filter Bank 59 4.2.1 Design of Subfilter T0 (z) 60 4.2.2 Design of Subfilter T1 (z ) 61 4.2.3 Design of Subfilter T2 (z ) 62 4.3 IIR Triplet Halfband Filter Banks 64 4.4 Semi-definite Programming Formulation 68 CONTENTS 4.5 viii 72 4.5.1 FIR and IIR Filter Bank Design Examples 72 4.5.2 4.6 Design Examples and Application to Image Compression Application to Image Compression 75 Concluding Remarks 77 Efficient Design of Cosine-Modulated Filter Banks via Convex Optimization 80 5.1 Introduction 81 5.2 Cosine-Modulated Filter Banks 83 5.3 Nonconvexity Analysis and Semidefinite Programming Relaxation 88 5.4 Cheap Iterative Algorithm 91 5.5 Design Examples 93 5.5.1 Designs Using the SDP Relaxation Method 94 5.5.2 Designs Using the Cheap Iterative Algorithm 96 Concluding Remarks 97 5.6 Optimal FIR Filter Bank Transceivers for Frequency-Selective Channels 104 6.1 Introduction 105 6.2 System Analysis 107 6.3 Optimal Filter Bank Equalizers 110 6.4 Simulation Examples 115 6.5 Concluding Remarks 117 Conclusion and Future Work 121 7.1 Dissertation Summary and Main Contributions 121 7.2 Future Research Directions 123 Appendices 125 A Useful Theorems and Proofs 126 A.1 The proof of 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Processing, vol 14, no 6, pp 760–769, Jun 2005 ... 2.1.3 Linear Phase Filter Banks 18 2.1.4 Two-channel Filter Banks 19 2.1.5 Tree-Structure Filter Banks and Wavelets 22 2.1.6 M -channel Filter Banks... synthesis filters have linear phase Such filter banks are called linear phase filter banks It is well-known that the implementation complexity of linear phase filters can be halved by exploiting the... and phase distortion Systems with small amount of aliasing and distortion are called near-perfect-reconstruction (NPR) filter banks 2.1 Filter Bank Fundamentals 2.1.3 18 Linear Phase Filter Banks