University of Wollongong Research Online University of Wollongong Thesis Collection University of Wollongong Thesis Collections 1990 The implementation of multidimensional discrete transforms for digital signal processing Hong Ren Wu University of Wollongong Recommended Citation Wu, Hong Ren, The implementation of multidimensional discrete transforms for digital signal processing, Doctor of Philosophy thesis, Department of Electrical and Computer Engineering, University of Wollongong, 1990. http://ro.uow.edu.au/theses/1353 Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: [email protected] THE IMPLEMENTATION OF MULTIDIMENSIONAL DISCRETE TRANSFORMS FOR DIGITAL SIGNAL PROCESSING A thesis submitted in fulfilment of the requirements for the award of the degree DOCTOR OF PHILOSOPHY from THE UNIVERSITY OF WOLLONGONG by WU, HONG REN, B.E., M.E. THE DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING. FEBRUARY 1990. "Entertaining someone with fish, uou could only serve him once,', but if you teach him the, art of fishing, it vt>iCC serve fvim for a Cije time,." —cAn ancient Chinese wise man and, philosopher. 1 CONTENTS ACKNOWLEDGEMENTS vi ABSTRACT viii LIST OF ACRONYMS AND SYMBOLS x CHAPTER ONE: INTRODUCTION 1 1 -1 Introduction to Multidimensional Digital Signal Processing 1 1-2 Applications 2 1-3 History and New Achievements in Fast Signal Processing Algorithms 3 1-4 Objectives 6 1-5 Thesis Review and Contributions 8 1-6 Publications, Submitted Papers and Internal Technical Reports 11 PART I. MULTIDIMENSIONAL DISCRETE FOURIER TRANSFORMS 14 CHAPTER TWO: 1-D DISCRETE FOURIER TRANSFORM AND FAST FOURIER TRANSFORM ALGORITHMS 15 2-1 Definitions 15 2-2 Matrix Representations for 1-D Cooley-Tukey FFT Algorithms 16 2-3 Computational Considerations 23 2-4 Summary 31 ii CHAPTER THREE: 2-D DFT AND 2-D FFT ALGORITHMS 32 3-1 Introduction to 2-D Discrete Fourier Transforms 32 3-2 Definitions 37 3-3 Row-Column FFT Algorithms 38 3-4 Vector Radix FFT Algorithms 39 3-5 Matrix Representations for 2-D Vector Radix FFT Algorithms 43 3-6 Structure Theorems 46 3-7 Structural Approach via Logic Diagrams 55 3-8 2-D Vector Split-Radix FFT Algorithms 63 3-9 Comparisons of Various 2-D Vector Radix FFT Algorithms 67 3-10 Vector Radix FFT Using FDP™ A41102 69 3-11 Summary 72 CHAPTER FOUR: A PERSPECTIVE ON VECTOR RADLX FFT ALGORITHMS OF HIGHER DIMENSIONS 72 4-1 Definitions 74 4-2 Matrix Representations and Structure Theorems 75 4-3 Diagrammatical Presentations 78 4-4 Computing Power Limitations 84 Ill PART II. MULTIDIMENSIONAL DISCRETE COSINE TRANSFORMS 87 CHAPTER FIVE: INTRODUCTION TO MULTI­ DIMENSIONAL DISCRETE COSINE TRANSFORMS 88 5-1 Definitions of 1 -D DCT and Its Inverse DCT 91 5-2 Definitions of 2-D DCT and Its Inverse DCT 93 5-3 Applications of 2-D DCTs in Image Compression 95 5-4 2-D Indirect Fast DCT Algorithms 100 CHAPTER SIX: 2-D DIRECT FAST DCT ALGORITHMS 103 6-1 2-D Direct Fast DCT Algorithm Based on Lee's Method 103 6-1-1 1-D Lee's algorithm in matrix form 103 6-1-2 Derivation of 2-D fast DCT algorithm from Lee's algorithm 108 6-2 2-D Direct Fast DCT Algorithm Based on Hou's Method 114 6-2-1 1-D Hou's algorithm in matrix form 114 6-2-2 Derivation of 2-D fast DCT algorithm from Hou's algorithm 118 6-3 Comparison of Arithmetic Complexity of Various DCT Algorithms 124 6-4 Comparison of Computation Structures of 2-D Direct VR DCTs and VR FFTs 125 6-5 Summary 126 IV CHAPTER SEVEN: HARDWARE IMPLEMENTATION OF 2-D DCTS FOR REAL-TIME IMAGE CODING SYSTEMS 128 7_ l Description of Hardware Implementation of Modified 2-D Makhoul DCT Algorithm Using FDP™ A41102 129 7-2 Discussion of 2-D DCT Image Coding Systems Using VLSI Digital Signal Processors 132 CHAPTER EIGHT: THE EFFECTS OF FINITE-WORD- LENGTH COMPUTATION FOR FAST DCT ALGORITHMS 136 8-1 Introduction 136 8-2 Simulation Design 138 8-2-1 Structure of the simulation program 138 8-2-2 Error model for the basic computation structure 140 8-2-3 DCT in infinite-word-length 141 8-2-4 Data collection 141 8-3 Simulation Results 143 8-3-1 Floating-point computation of 1-D DCTs 143 8-3-2 Floating-point computation of 2-D DCTs 147 8-4 Summary 151 CHAPTER NINE: CONCLUSIONS 152 9-1 Conclusions 152 9-2 Suggestions for Future Research 154 BIBLIOGRAPHY 156 APPENDIX A: PRELIMINARY BACKGROUND ON THE TENSOR (KRONECKER) PRODUCT AND THE LOGIC DIAGRAM 172 APPENDIX B PROOF OF STRUCTURE THEOREMS 175 APPENDIX C THE COMBINED FACTOR METHOD 176 APPENDIX D DERIVATION OF VECTOR RADIX 2-D FAST DCT BASED ON LEE'S ALGORITHM 181 APPENDIX E: ARITHMETIC COMPLEXITY OF THE VECTOR SPLIT-RADIX DIF FFT ALGORITHM 187 vi ACKNOWLEDGEMENTS The author wishes to express his deepest appreciation to his Supervisor, Dr. FJ. Paoloni, Associate Professor of the Department of Electrical and Computer Engineering, The University of Wollongong, for his guidance, support and encouragement and also for his understanding and confidence in the author throughout this research. His professional and optimistic attitude towards the research have made this research challenging, interesting, productive and enjoyable. The author wishes to thank Professor Huang, Ruji, of the Department of Industrial Automation, University of Science and Technology, Beijing (formerly Beijing University of Iron and Steel Technology), who, as his Masters' Supervisor, had a great influence on shaping the author's research skills and abilities as an independent as well as cooperative researcher. Sincere thanks are also extended to Professor B.H. Smith who introduced the author to this Institution and made this study possible in the first place. The author wishes to thank the following people for their generous help, patience and useful discussions at various stages of this program: Dr. G.W. Trott and Dr. T.S. Ng, Department of Electrical and Computer Engineering; Mr. I.C. Piper, Computer Services; Mr. J.K. Giblin, formerly with Computer Services and now with Network Technical Services, B.H.P. Steel International Group; Mr. G. Andersson, Computer Services; Dr. N. Smyth and Dr, K.G. Russell, Department of Mathematics, The University of Wollongong; Professor J.H. McClellan, School of Electrical Engineering, Georgia Institute of Technology, formerly with Schlumberger Well Services; Dr. J.D. O'Sullivan, Dr. DJ. McLean, Dr. C.E. Jacka and Mr. K.T. Hwa, Division of Radio Physics, CSIRO in Epping, New South Wales; Dr. MJ. Biggar and Dr. W.B.S. Tan, Telecom Research Laboratories (Australia); Professor K.R. Rao, Department of Electrical Engineering, The University of Texas at Arlington; Professor M. Vetterli, Department of Electrical Engineering, Columbia University; Mr. P. Single, Austek Microsystems (Australia); Dr. M.A. Magdy, Mr. J.F. Chicharo and Mrs. C. Quinn, Department of vii Electrical and Computer Engineering, The University of Wollongong; Mr. P.J. Costigan and all technical staff in the Department. The author is deeply grateful to his friend and English teacher Mrs. B. S. Perry for her generous help and professional assistance in the author's understanding of English and Australian culture, and her and her husband's, Mr. E.J.W. Perry, understanding, friendship and encouragement which made the author's stay in Wollongong worthwhile, most pleasant and enjoyable. The assistance from Miss. M.J. Fryer, of the Department of Electrical and Computer Engineering, The University of Wollongong, throughout this research and particularly in reading the final manuscript of this thesis is warmly appreciated. Financial support received from the Department of Electrical and Computer Engineering and the Committee of Post-Graduate Study, The University of Wollongong, by means of the Departmental Teaching Fellowship and the Post-Graduate Research Scholarship respectively, which made this research possible, is sincerely acknowledged. Financial supports from Australian Telecommunication and Electronics Research Board and from Telecom Research Laboratories (Australia) through R&D contract No.7066 are also acknowledged. Finally, the author wishes to express his deepest gratitude to Mei Mei, his best friend, colleague and wife, without whose patience, understanding, appreciation and continuous support, encouragement and inspiration, this work would not have been accomplished. The continuous support and understanding from his parents, from whom he has been separated for the cause is also greatly appreciated. vm ABSTRACT A structural approach to the construction of multidimensional vector radix fast Discrete Fourier Transforms (DFTs) and fast direct Discrete Cosine Transforms (DCTs) is presented in this thesis. The approach features the use of matrix representation of one- dimensional (1-D) and two-dimensional (2-D) FFT and fast DCT algorithms along with the tensor product, and the use of logic diagram and rules for modifications. In the first pan of the thesis, the structural approach is applied to construct 2-D Decimation-In-Time (DIT), Decimation-In-Frequency (DIF) and mixed (DIT & DIF) vector radix FFT algorithms from corresponding 1-D FFT algorithms by the Cooley- Tukey method. The results are summarized in theorems as well as examples using logic diagrams. It has been shown that the logic diagram (or signal flow graph) as well as being a form of representation and interpretation of fast algorithm equations, is a stand­ alone engineering tool for the construction of fast algorithms. The concept of "vector signal processing" is adapted into the logic diagram representation which reveals the structural features of multidimensional vector radix FFTs and explains the relationships and differences between the row-column FFT, the vector radix FFT reported previously and the approach presented in this thesis. The introduction of the structural approach makes the formulation of a multidimensional vector radix FFT algorithm of high radix and dimension easy to evaluate and implement by both software and hardware.