AN INTRODUCTION TO PARAMETRIC DIGITAL FILTERS AND OSCILLATORS
CONTENTS 1 INTRODUCTION BASIS OF DISCRETE SIGNALS AND DIGITAL FILTERS 1 1 1 DISCRETE SIGNALS AND SYSTEMS 1 1 2 DISCRETE SIGNALS 3 1 2 1 TIME DOMAIN REPRESENTATION FOR DISCRETE SIGNALS 3 1 2 2 PRESENTATION OF DISCRETE SIGNALS BY FOURIER TRANSFORM 4 1 2 3 DISCRETE FOURIER TRANSFORM 9 1 2 4 LAPLACE AND Z TRANSFORMS 11 1 3 TIME INVARIANT DISCRETE LINEAR SYSTEMS 16 1 3 1 DIFFERENCE EQUATION AND IMPULSE RESPONSE 17 1 3 2 DLS REPRESENTATION VIA TRANSFER FUNCTION 20 1 4 STABILITY AND CAUSALITY OF DISCRETE SYSTEMS 22 1 5 FREQUENCY RESPONSE OF A DISCRETE LINEAR SYSTEM 23 1 5 1 PROPERTIES OF THE FREQUENCY RESPONSE OF A DISCRETE LINEAR SYSTEM 25 1 5 2 TRANSFER FUNCTION VERSUS FREQUENCY RESPONSE 25 1 6 CASE STUDY LOW ORDER FILTERS 27 1 6 1 PURELY RECURSIVE FILTERS 27 1 6 2 EFFECTS OF WORD LENGTH LIMITATION 37 1 6 3 TRANSVERSAL AND COMBINED FILTERS 37 1 7 SUMMARY 41 1 8 ABBREVIATIONS 42 1 9 VARIABLES 42 1 10 REFERENCES 43 PART ONE LINEAR DISCRETE TIME VARIANT SYSTEMS 45 2 MAIN CHARACTERISTICS OF TIME VARIANT SYSTEMS 47 2 1 DESCRIPTION OF A LINEAR TIME VARIANT DISCRETE SYSTEM THROUGH DIFFERENCE EQUATIONS 48 2 2 IMPULSE RESPONSE 49 2 3 GENERALIZED TRANSFER FUNCTION 52 2 4 SIGNALS ANALYSIS IN FREQUENCY DOMAIN 55 2 5 SAMPLING FREQUENCY CHOICE FOR LINEAR TIME VARIANT DISCRETE SYSTEMS 59 2 6 RANDOM SIGNALS PROCESSING IN LINEAR TIME VARIANT DISCRETE SYSTEMS 61 2 7 COMBINATIONS OF TIME VARIANT SYSTEMS 63 2 7 1 PARALLEL CONNECTIONS 63 2 7 2 CASCADE CONNECTIONS 64 2 7 3 SYSTEMS WITH FEEDBACK 66 2 7 4 CONTINUOUS AND DISCRETE LTV SYSTEMS 68 2 8 TIME VARYING SAMPLING 70 2 8 1 SYSTEMS WITH NON UNIFORM SAMPLING 70 2 8 2 SYSTEMS WITH STOCHASTIC SAMPLING INTERVAL 75 2 9 SUMMARY 77 2 10 ABBREVIATIONS 78 2 11 VARIABLES 78 2 12 REFERENCES 79 3 PERIODICALLY TIME VARIANT DISCRETE SYSTEMS 83 3 1 DIFFERENCE EQUATION 83 3 2 IMPULSE RESPONSE 84 3 3 GENERALIZED TRANSFER FUNCTION AND FREQUENCY RESPONSE 85 3 4 SIGNALS IN PERIODICALLY LINEAR TIME VARIANT SYSTEMS 86 3 4 1 BIFREQUENCY FUNCTION 86 3 4 2 DETERMINISTIC SIGNAL PROCESSING 86 3 4 3 RANDOM SIGNALS PROCESSING 89 3 5 GENERALIZATION OF THE SAMPLING THEOREM 91 3 6 SYSTEM STABILITY 95 3 6 1 GENERAL STABILITY PROBLEM 95 3 6 2 SELECTION OF STABILITY CRITERIA 96 3 6 3 STABILITY EVALUATION 97 3 6 4 STABILITY OF PARAMETRIC RECURSIVE SYSTEMS 99 3 7 STABILITY OF SECOND ORDER SYSTEMS 100 3 8 STABILITY OF STOCHASTIC SYSTEMS 107 3 9 SUMMARY 114 3 10 ABBREVIATIONS 114 3 11 VARIABLES 115 3 12 REFERENCES 116 PART TWO PARAMETRIC SYSTEMS 119 4 PARAMETRIC FILTERS ANALYSIS 121 4 1 NON RECURSIVE PARAMETRIC FILTERS 121 4 2 THE FIRST ORDER RECURSIVE PARAMETRIC FILTER 123 4 2 1 IMPULSE RESPONSE 123 4 2 2 GENERALIZED TRANSFER FUNCTION 126 4 3 A RECURSIVE PARAMETRIC FILTER OF THE SECOND ORDER 129 4 3 1 IMPULSE RESPONSE 129 4 3 2 GENERALIZED TRANSFER FUNCTION 134 4 4 PARAMETRIC FILTERS OF AN ARBITRARY ORDER 136 4 4 1 DIRECT EQUATION SOLUTION 136 4 4 2 EQUATION SOLUTION IN A STATE SPACE 138 4 5 APPROXIMATE METHOD FOR ANALYSIS OF PERIODICAL LINEAR TIME VARIANT DISCRETE SYSTEMS 142 4 6 SUMMARY 146 4 7 ABBREVIATIONS 146 4 8 VARIABLES 146 4 9 REFERENCES 147 5 DESIGN STUDIES FOR PARAMETRIC FILTERS 149 5 1 RECURSIVE PARAMETRIC FILTERS 150 5 1 1 FREQUENCY RESPONSE CORRECTION 150 5 1 2 MULTIPLIER FREE FILTERS 155 5 1 3 HIGH EF CIENCY PARAMETRIC FILTERS 159 5 2 COMBINATIONAL COMPONENTS IN PARAMETRIC FILTERS 161 5 2 1 EVALUATION OF THE LEVEL OF COMBINATIONAL COMPONENTS 162 5 2 2 METHODS OF REDUCING COMBINATIONAL COMPONENTS 164 5 2 3 COMPARISON OF THE COMBINATIONAL COMPONENTS AND NOISE LEVELS IN DIGITAL FILTERS 167 5 3 PARAMETRIC FILTER DESIGN A CASE STUDY 168 5 4 SUMMARY 170 5 5 ABBREVIATIONS 171 5 6 VARIABLES 171 5 7 REFERENCES 172 PART THREE DIGITAL PARAMETRIC OSCILLATORS 175 6 DIGITAL PARAMETRIC OSCILLATORS 177 6 1 REGIONS OF PARAMETRIC OSCILLATIONS 178 6 2 PARAMETRIC RESONANCE IN DIGITAL RESONATORS 183 6 3 APPROXIMATE METHOD OF EVALUATING A REGION OF PARAMETRICAL GENERATION 189 6 4 ANALYSIS OF NON PERIODIC COMPONENTS 193 6 5 ANALYSIS OF THE PERIODIC COMPONENTS 196 6 6 WIDEBAND CONTROL SIGNAL 200 6 7 PERIODIC COMPONENTS SPECTRUM 204 6 8 THE TRANSIENT IN DIGITAL PARAMETRIC OSCILLATORS 205 6 9 SUMMARY 207 6 10 ABBREVIATIONS 208 6 11 VARIABLES 208 6 12 REFERENCES 209 7 PARAMETRIC OSCILLATOR IN STEADY STATE MODE 211 7 1 LIMITING MODE OF PARAMETRIC OSCILLATORS 212 7 2 DPO ANALYSIS IN THE PRESENCE OF NOISE 222 7 3 MODELLING OF A DIGITAL PARAMETRIC OSCILLATOR USING MATLAB A CASE STUDY 228 7 3 1 NON LIMITING OSCILLATION MODE 228 7 3 2 STEADY STATE OSCILLATION MODE 232 7 3 3 A DIGITAL PARAMETRIC OSCILLATOR WITH NON SINUSOIDAL CONTROL SIGNAL 234 7 3 4 FREQUENCY SYNTHESIZER 236 7 4 SUMMARY 239 7 5 ABBREVIATIONS 239 7 6 VARIABLES 239 7 7 REFERENCES 240 1 INTRODUCTION BASIS OF DISCRETE SIGNALS AND DIGITAL FILTERS THE THEORY AND PRACTICE OF DIGITAL SIGNAL PROCESSING DSP ARE CURRENTLY IN A MATURE STAGE IT IS DIF CULT TO IMAGINE ANY MODERN ELECTRONIC SYSTEM WITHOUT WIDE APPLICATION OF DSP AND IN PARTICULAR LINEAR TIME INVARIANT ALGORITHMS FOR LTERING EQUALIZATION CHARACTERISTIC CORRECTION AND SO ON THE MAJOR GOAL OF THIS CHAPTER IS TO INTRODUCE THE THEORETICAL BASIS OF DISCRETE SIGNALS AND TIME INVARIANT DIGITAL SYSTEMS TO HELP READERS MORE EASILY UNDERSTAND THE MAIN TEXT DEDICATED TO TIME VARIANT SYSTEMS AND TO MINIMIZE THE NECESSITY TO CONSULT OTHER TEXTS WHILE READING THIS BOOK THIS INTRODUCTION PROVIDES A SUPER CIAL OVERVIEW OF DSP CONCEPTS SAMPLING AND QUANTIZATION IMPULSE AND FREQUENCY RESPONSES FOURIER LAPLACE AND Z TRANSFORMS SYSTEM STABILITY AND CAUSALITY AND NITE AND IN NITE IMPULSE RESPONSE IIR DIGITAL LTERS DFS FOR THOSE FAMILIAR WITH DSP AND RELATED SUBJECTS THIS INTRODUCTION WILL HELP REFRESH THEIR KNOWLEDGE FOR THOSE WHO ARE UNFAMILIAR THIS CHAPTER CAN BE USED AS THE RST STAGE OF STUDY OF DISCRETE SIGNALS AND SYSTEMS OF COURSE THIS INTRODUCTION DOES NOT AND CANNOT REPLACE SPECIAL LITERATURE AND TEXT BOOKS DEDICATED TO DSP PROBLEMS AMONG THE LATEST TEXTBOOKS IN THIS AREA THE AUTHOR RECOMMENDS REFERENCE 1 1 1 DISCRETE SIGNALS AND SYSTEMS MOST SIGNALS USED IN INFORMATION SYSTEMS ARE SIMILAR TO ANALOG PROCESSES IN THE GEN ERAL CASE THEY ARE FUNCTIONS OF CONTINUOUS TIME DIGITAL LTERS BELONG TO THE GROUP OF DISCRETE SYSTEMS OF SIGNAL PROCESSING WHICH OPERATE WITH DISCRETE INPUT PROCESSES THUS AN ANALOG INPUT SIGNAL IS REPRESENTED BY DISCRETE SAMPLES OBTAINED IN TIME MOMENTS PROPORTIONAL TO THE SAMPLING INTERVAL T AN ANALOG WAVEFORM CAN BE TRANS FORMED INTO AN APPROPRIATE DISCRETE SIGNAL WITHOUT INFORMATION LOSSES IF SAMPLING FREQUENCY FS IS DETERMINED AS FS S 2 1 T 2FO MAX 1 1