An Introduction to Parametric Digital Filters and Oscillators

An Introduction to
 Parametric Digital Filters 
and Oscillators......Page 5
Copyright......Page 6
Contents......Page 9
Preface......Page 13
1.1 DISCRETE SIGNALS AND SYSTEMS......Page 19
1.2.1 Time- Domain Representation for Discrete Signals......Page 21
1.2.2 Presentation of Discrete Signals by Fourier Transform......Page 22
1.2.3 Discrete Fourier Transform......Page 27
1.2.4 Laplace and z - transforms......Page 29
1.3 TIME ¨C INVARIANT DISCRETE LINEAR SYSTEMS......Page 34
1.3.1 Difference Equation and Impulse Response......Page 35
1.3.2 DLS Representation via Transfer Function......Page 38
1.4 STABILITY AND CAUSALITY OF DISCRETE SYSTEMS......Page 40
1.5 FREQUENCY RESPONSE OF A DISCRETE LINEAR SYSTEM......Page 41
1.5.2 Transfer Function versus Frequency Response......Page 43
1.6.1 Purely Recursive Filters......Page 45
1.6.3 Transversal and Combined Filters......Page 55
1.7 SUMMARY......Page 59
1.9 VARIABLES......Page 60
1.10 REFERENCES......Page 61
Part One Linear Discrete Time-Variant Systems......Page 63
2 Main Characteristics of Time-Variant Systems......Page 65
2.1 Description of a Linear Time-Variant Discrete System Through 
Difference Equations......Page 66
2.2 IMPULSE RESPONSE......Page 67
2.3 GENERALIZED TRANSFER FUNCTION......Page 70
2.4 SIGNALS ANALYSIS IN FREQUENCY DOMAIN......Page 73
2.5 SAMPLING FREQUENCY CHOICE FOR LINEAR TIME- VARIANT DISCRETE SYSTEMS......Page 77
2.6 RANDOM SIGNALS PROCESSING IN LINEAR TIME- VARIANT DISCRETE SYSTEMS......Page 79
2.7.1 Parallel Connections......Page 81
2.7.2 Cascade Connections......Page 82
2.7.3 Systems with Feedback......Page 84
2.7.4 Continuous and Discrete LTV Systems......Page 86
2.8.1 Systems with Non- Uniform Sampling......Page 88
2.8.2 Systems with Stochastic Sampling Interval......Page 93
2.9 SUMMARY......Page 95
2.11 VARIABLES......Page 96
2.12 REFERENCES......Page 97
3.1 DIFFERENCE EQUATION......Page 101
3.2 IMPULSE RESPONSE......Page 102
3.3 GENERALIZED TRANSFER FUNCTION AND FREQUENCY RESPONSE......Page 103
3.4.2 Deterministic Signal Processing......Page 104
3.4.3 Random Signals Processing......Page 107
3.5 GENERALIZATION OF THE SAMPLING THEOREM......Page 109
3.6.1 General Stability Problem......Page 113
3.6.2 Selection of Stability Criteria......Page 114
3.6.3 Stability Evaluation......Page 115
3.6.4 Stability of Parametric Recursive Systems......Page 117
3.7 STABILITY OF SECOND- ORDER SYSTEMS......Page 118
3.8 STABILITY OF STOCHASTIC SYSTEMS......Page 125
3.10 ABBREVIATIONS......Page 132
3.11 VARIABLES......Page 133
3.12 REFERENCES......Page 134
Part Two Parametric Systems......Page 137
4.1 NON- RECURSIVE PARAMETRIC FILTERS......Page 139
4.2.1 Impulse Response......Page 141
4.2.2 Generalized Transfer Function......Page 144
4.3.1 Impulse Response......Page 147
4.3.2 Generalized Transfer Function......Page 152
4.4.1 Direct Equation Solution......Page 154
4.4.2 Equation Solution in a State Space......Page 156
4.5 Approximate Method for Analysis of Periodical Linear Time-Variant 
Discrete Systems......Page 160
4.8 VARIABLES......Page 164
4.9 REFERENCES......Page 165
5 Design Studies for Parametric Filters......Page 167
5.1.1 Frequency Response Correction......Page 168
5.1.2 Multiplier- Free Filters......Page 173
5.1.3 High- Efficiency Parametric Filters......Page 177
5.2 COMBINATIONAL COMPONENTS IN PARAMETRIC FILTERS......Page 179
5.2.1 Evaluation of the Level of Combinational Components......Page 180
5.2.2 Methods of Reducing Combinational Components......Page 182
5.2.3 Comparison of the Combinational Components and Noise Levels in Digital Filters......Page 185
5.3 PARAMETRIC FILTER DESIGN ¨C A CASE STUDY......Page 186
5.4 SUMMARY......Page 188
5.6 VARIABLES......Page 189
5.7 REFERENCES......Page 190
Part Three Digital Parametric Oscillators......Page 193
6 Digital Parametric Oscillators......Page 195
6.1 REGIONS OF PARAMETRIC OSCILLATIONS......Page 196
6.2 PARAMETRIC RESONANCE IN DIGITAL RESONATORS......Page 201
6.3 APPROXIMATE METHOD OF EVALUATING A REGION OF PARAMETRICAL GENERATION......Page 207
6.4 ANALYSIS OF NON- PERIODIC COMPONENTS......Page 211
6.5 ANALYSIS OF THE PERIODIC COMPONENTS......Page 214
6.6 WIDEBAND CONTROL SIGNAL......Page 218
6.7 PERIODIC COMPONENTS SPECTRUM......Page 222
6.8 THE TRANSIENT IN DIGITAL PARAMETRIC OSCILLATORS......Page 223
6.9 SUMMARY......Page 225
6.11 VARIABLES......Page 226
6.12 REFERENCES......Page 227
7 Parametric Oscillator in Steady-State Mode......Page 229
7.1 LIMITING MODE OF PARAMETRIC OSCILLATORS......Page 230
7.2 DPO ANALYSIS IN THE PRESENCE OF NOISE......Page 240
7.3.1 Non- Limiting Oscillation Mode......Page 246
7.3.2 Steady- State Oscillation Mode......Page 250
7.3.3 A Digital Parametric Oscillator with Non- Sinusoidal Control Signal......Page 252
7.3.4 Frequency Synthesizer......Page 254
7.6 VARIABLES......Page 257
7.7 REFERENCES......Page 258
Index......Page 261