DIGITAL FILTER STRUCTURES FROM CLASSICAL ANALOGUE NETWORKS By STUART SIMON LAWSON A THESIS SUBMITTED FOR THE Ph.D. DEGREE IN THE UNIVERSITY OF LONDON. DEPARTMENT OF ELECTRICAL ENGINEERING, IMPERIAL COLLEGE OF SCIENCE AND TECHNOLOGY, LONDON,S.W.7 OCTOBER 1975- -2- A3STRACT decent investigations have shown that a class of digital filter structures exists that poseeeces mudh lower attenuation distortion than the conventional direct or cascade forms. These structures can be derived from classical analogue doubly- terminated lossless networks by using a one-port wave variable description for circuit elements. The basic teChnique, due to Fettweis, consists of expressing the voltage-current relationship of an element in terms of incident and reflected waves and then applying the bilinear transformation to give the digital equivalent. These digital circuits are then interconnected with the aid of 'Adaptors'. An adaptor is simply the digital realization of Kirchhoff's two laws for a parallel or series junction of n ports. The use of waves in the derivation of the digital filter structures has led to the term ''rave Digital ;Filters' being applied to them. In this thesis it is shown that, by considerinf, each clement in the analogue network as a two-port, a true simulation can be achieved for the corresponding digital filter structure. In the new method, the adaptor, which is needed in the one-port description, is not required explicitly but is included as part of the equivalent wave-flow diagram. The relationship between the sensitivity of the attenuation to first-order multiplier variations and the analogue network element sensitivities is derived and it is shown that the multiplier sensitivities are not generally zero at points of maximum pseudopower transfer. A generalization of the Wave Digital filter concept of Fettweis is also examined by the relationship between the wave variables and the voltages and currents as a linear transformation on the _,BCD matrix of the LC two-Port. A particular transformatiOn is studied in detail and the associated signal-flow diagrams are derived. The sensitivity behaviour of structures derived using the general linear transformation is also studied and it is shown that their behaviour is similar to that of Wave Digital filters. We also consider in this thesis the computer-aided analysis of digital filter structures and present a new algorithm for analysis which has many advantages over conventional methods. -j- To Chris -4- Acknowledements The author would like to express his gratitude for the guidance and many helpful comments given by his supervisor, Dr. A.G. Constantinides, over the past three years. Furthermore, the author would like to acknowledge the many stimulating discussions with other staff members and postgraduates of the communication section, in particular Dr. W. Saraga, :ass S. Dasgupta, Mr. D. Haigh, Dr. D. Sud, Mr. I. Colyer and Mr. L. Gusmao. In addition, the author would like to thank Drs. M. Apperley and R.'Snence for the opportunity to use their interactive graphics software for the purpose of analysing digital filter structures. Financial support for the author was provided jointly by Plessey Telecommunications Research Ltd. and the.Science Research Council. The author would especially like to thankl:r. A.J.L. Muir of Plessey. Last, but not least, many thanks to the patient typist Shirley Lawson (the author's sister) who, despite the awesome task, displayed great courage and delivered a finished typed manuscript within four weeks. -5- List of Contents Page No . Title Page Abstract 2 Acknowledgements 4 List of Contents 5 List of Figures 8 Conventions and Symbols 10 Chapter 1 Introduction 1.'1 Theory of Linear Time-Invariant Discrete- - 12 Time Networks 1.2 Design.of Digital Filters 17 1.3 mite-Nord Length Effects in Digital 19. Filters 1.4 Low-Sensitivity Property of Certain 24 Classical Analogue Filters 1.5 Simulation of Analogue Filters with Desirable Properties using Digital 26 Components 1.6 Survey of Literature on Wave Digital Filters 29 1.7 Survey of Literature on Other Low Sensitivity Digital Filters ko 1.8 Approach Taken in the Present Study. 42 Chapter 2 The Two-Port Approach 2.1 Introduction: Principal Idea of Approach 55 2.2 Scattering Parameters for a Passive Two-Port 55 2.3 Derivation of Wave-Flow Diagrams for Series Elements 58 2.4 Derivation of Wave-Flow Diagrams for Shunt Elements 68 2.5 Derivation of Wave-Flow Diagrams for Lossless Transmission-Line Element 76 2.6 Sources, Terminations and Interconnections 79 2.7 Discussion 83 -6- Page No. Chanter 3 Design and Sensitivity Analysis of WDF Imitating DTLIN 3.1 Design Procedure 97 3.2 A Filter 2xample 100 3.3 The Derivation of MAP Sensitivity Characteristics 102 3.4 Sensitivity Example 115 3.5 Example I: 3rd Order Elliptic LP Filter 118 3.6 Example II: 5th Order Elliptic LP Filter 119 3.7 Example III: 6th Order Elliptic BP Filter 120 3.8 Conclusions 122 Chanter 4 Design and Sensitivity Analysis. of Wave Digital Filters Imitating Doubly-Terminated Lossless Transmission-Line Networks 4.1 Design Procedure 146 4.2 The Derivation of MAP Sensitivity Characteristics 148 4.3 Example I: 3rd Order LP Filter ' 153 4.4 Example II: 7th Order LP Filter 155 4.5 Discussion 156 158 4.6 Conclusions Chapter 5 The General Two-Port Transformation 5.1 Introduction: Principal Idea 166 5.2 Derivation of a--Parameters 168 5.3 Basic Eauations for Series Elements 169 5.4 Basic Equations for Shunt Elements 174 5.5 Basic Equations for Sources, Terminations and Interconnections 177 5.6 Study of Realisability Conditions 181 5.7 Derivation ofiaP Sensitivity Characteristics 198 5.8 Study of Some Special Cases 204 5.9 Discussion of a Condition for Canonic SIB 209 5.10 General Discussion (including table of wave formulations known to yield realisable digital filter structures) 210 -7- Pane Chanter 6 The 'Invariant Volta ;e Ratio' Transformation 6.1 Introduction 216 6.2 Derivation of STD for Basic Elements, Sources, Terminations and Interconnections 216 6.3 Design and Sensitivity Analysis of Digital Filters Imitating Doubly-Terminated Lossless Ladder lietworks 223 6.4 Example I: 3rd Order Chebyshev LP Filter 225 6.5 Example II: 5th Order Elliptic LP Filter 226 6.6 Example III: 6th Order Elliptic BP Filter 227 6.7. The. NZ Lattice Filter, its Derivation and Properties 228 6.8 Comparison of Seven :.:ethods of Synthesising a Digital Filter from a Third-Order Elliptic 233 alter 6.9 Example: 5th Order Chebyshev LP Filter 239 6.10 Conclusions 241 Chapter 7 A Study of Pour Transformations 7.1 Introduction 272 7.2 The 'Invariant Transfer Admittance' Transformation 272 7.3 Study of Similar Canonic Transformations 278 7.4 Study of One Hon-Canonic Transformation 285 7.5 Disc=iicn and Conclusion 288 Chapter 8 Computer-Aided Analysis of Digital Filter. Structures 8.1 Introduction. 294 8.2 Finite Change Sensitivity 295 8.3 New Analysis Algorithm 299 8.4 Experimental Results 300 • 8.5 Discussion 302 Chapter 9 Farther Research 306 Conclusions 309 Appendices 311 References 324 -8- List of 111;;ures Fig. No. Page Fig. No. Page Fig. No. Page 1.1 45 2.20 90 4.1 159 1.2 45 2.21 go 4.2 159 1.3 45 2.22 91 4.3 159 1.4 46 2.23 91 4.4 159 1.5 46 2.24 91 4.5 160 1.6 47 2.25 92 4.6 161 1.7 47 2.26 92 4.7 162 1.8 47 2.27 92 4.8 160 1.9 48 2.28 93 4.9 163 1.10 48 2.29 93 4.10 164 1.11 48 2.30 93 1.12 48 2.31 93 5.1 212 1.13 49 2.32 94 5.2 212 1.14 49 2.33 94 '5.3 212 1.15 49 2.34 94 5.4 212 1.16 49 2.35 94 5.5 212 1.17 49 2.36 94 5,6 212 1.18 49 2.37 94 5.7 212 1.19 49 2,38 95 213 50 50 1.20 2,39' 5.9 213 1.21 50 5.10 213 1.22 50 3.1 123 5.11 213 1.23 5o 3.2 123 5.12 213 1.24 50 3.3 123 5.13 213 50 -), 1.25 Jo'r 124 5.14 213 1.26 51 3.5 125 5.15 214 1.27 51 3.6 125 1.28. 51 3.7 125 6.1 242 1.29 51 3.3 125 6.2 242 1.30 51 3.9 125 6.3 242 1.31 51 3.10 126 6.4 243 1.32 52 3.11 127 6.5 243 1.33 52 3.12 128 6.6 243 1.34 52 3.13 126 6.7 244 1.35 52 3.14 129 6.8 244 1.36 52 3.15 126 6.9 244 1.37 53 3.16 130 6.10 245 1.38 53 3;17 126 6.11 245 3.18 131. 6.12 245. 2.1 85 3.19 132 6.13 245 2.2 85 3.20 133 6.14 246 2.3 85 3.21 134 6.15 247 2.4 85 3.22 132 6.16 248 2.5 85 3.23 132 6.17 249 2.6 86 3.24 135 6.18 249 2.7 86 3.25 136 6.19 250 2.8 86 3.26 137 6.20 251 2.9 86 3.27 138 6.21 249 2.10 87 3,28 139 6.22 249 2.11 87 3.29 140 6.23 252 2.12 87 3.30 140 6.24 253 2.13 88 3.31 .141 6.25 254 2.14 88 3.32 142 6.26 254 2.15 88 3.33 140 6.27 254 2.16 89 3.31 140 6.28 44 2.17 89 3.35 143 6.29 2.18 90.