PREFACE TO THE SECOND EDITION ................................. xix
PREFACE TO THE FIRST EDITION .................................. xxi
SYMBOLS LIST AND GLOSSARY ..................................... xxv
PART 1 PHASE LOCK WITHOUT NOISE
1 INTRODUCTION ................................................. 3
1.1 What is a Phase-Locked Loop (PLL)? ...................... 3
1.2 Why Use a Phase-Locked Loop? ............................ 3
1.3 Scope of this Book ...................................... 4
1.4 Basic Loop .............................................. 5
1.5 Phase Definitions ....................................... 6
1.6 Phase Detector .......................................... 8
1.7 Combined Gain .......................................... 10
1.8 Operating Range ........................................ 11
1.9 Units and the Laplace Variables ........................ 13
2 THE BASIC LOOP ............................................. 15
2.1 Steady-State Conditions ................................ 15
2.2 Classical Analysis ..................................... 16
2.2.1 Transient Response .............................. 17
2.2.2 Modulation Response ............................. 19
2.3 Mathematical Block Diagram ............................. 21
2.4 Bode Plot .............................................. 24
2.5 Note on Phase Reversals ................................ 26
2.6 Summary of Transient Responses of the First-Order
Loop ................................................... 26
3 LOOP COMPONENTS ............................................. 29
3.1 Phase Detector ......................................... 29
3.1.1 Flip-Flop Phase Detector ........................ 29
3.1.2 Exclusive-OR Gate Phase Detector ................ 30
3.1.3 Charge-Pump Phase Detector ...................... 31
3.1.4 Sinusoidal Phase Detector ....................... 33
3.1.4.1 Phase Detection in a Simple Mixer ...... 34
3.1.4.2 Balanced Mixers ........................ 35
3.1.4.3 Analog Multipliers ..................... 39
3.1.4.4 Integrated Circuit Doubly Balanced
Mixer .................................. 40
3.2 Voltage-Controlled Oscillator (VCO) .................... 41
3.3 Loop Filter ............................................ 44
3.3.1 Passive Loop Filter ............................. 45
3.3.2 Filters Driven by Current Sources ............... 45
3.3.2.1 Integrator ............................. 47
3.3.2.2 Integrator-and-Lead Filter ............. 48
3.3.2.3 Lag (Low-Pass) Filter .................. 48
3.3.2.4 Lag-Lead Filter ........................ 49
3.3.2.5 DC Transimpedance ...................... 49
3.3.3 Active Filters .................................. 51
3.3.3.1 Op-Amp Considerations .................. 52
3.3.3.2 Unintended Poles ....................... 53
3.3.3.3 Reducing the Size of С ................. 54
3.3.3.4 Impractical Loop Filters ............... 55
3.4 Filter Reference Voltage ............................... 55
3.5 Note on the Form of the Filter Equation ................ 56
3.6 Capacitors in Loop Filters ............................. 56
3.7 Higher-Order Filters ................................... 57
i3.A Appendix: Integrated Circuit Doubly Balanced Mixer-
Details ................................................ 57
i3.B Appendix: Op-Amps in Loop Filters ...................... 57
4 LOOP RESPONSE .............................................. 59
4.1 Loop Order and Type .................................... 59
4.2 Closed-Loop Equations .................................. 60
4.3 Open-Loop Equations—Lag-Lead Filter .................... 62
4.4 Loop with a Lag Filter ................................. 65
4.5 Loop with an Integrator-and-Lead Filter ................ 66
4.6 Summary of Equations ................................... 67
4.7 Unity-Gain Bandwidth in Highly Damped Loops ............ 67
5 LOOP STABILITY .............................................. 71
5.1 Observing the Open-Loop Response ....................... 71
5.2 Methods of Stability Analysis and Measures of
Stability .............................................. 72
5.2.1 Bode Plot ....................................... 73
5.2.2 Nyquist Plot .................................... 74
5.2.3 Evans Plot (Root Locus) ......................... 76
5.3 Stability of Various PLL Configurations ................ 76
5.3.1 First-Order Loop ................................ 77
5.3.2 Second-Order Loop ............................... 77
5.3.3 Third-Order Loop ................................ 77
5.3.4 Controlling Stability over Wide Gain Ranges ..... 79
5.3.5 Representing Delay .............................. 80
5.4 Computing Open-Loop Gain and Phase ..................... 81
5.5 Phase Margin Versus Damping Factor ..................... 86
i5.M Appendix: Stability Plots Using MATLAB® ................ 86
6 TRANSIENT RESPONSE .......................................... 87
6.1 Step Response .......................................... 87
6.1.1 Form of the Equations ........................... 87
6.1.2 Step Response Equations ......................... 89
6.2 Envelope of the Long-Term Step Response ................ 92
6.3 Response to Ramp Input ................................. 96
6.4 Response to Parabolic Input ............................ 98
6.5 Other Responses ....................................... 100
6.6 Note on Units for Graphs .............................. 101
6.7 Equivalent Circuit ................................... 102
6.8 General Long-Term (Steady-State) Response
Characteristics ....................................... 102
6.9 Open-Loop Equations in Terms of Closed-Loop
Parameters ............................................ 103
6.10 More Complex Loops and State-Space Analysis ........... 103
6.10.1 Basic Equations ............................... 104
6.10.2 Initial Conditions ............................ 107
6.11 An Approximate Solution Using State-Space
Variables ............................................. 108
6.12 Effect of an Added Pole ............................... 108
i6.M Appendix: Transient Responses Using MATLAB ............ 109
7 MODULATION RESPONSE ........................................ 111
7.1 Phase and Frequency Modulation ........................ 111
7.2 Modulation Responses ................................. 113
7.3 Responses in a First-Order Loop ...................... 113
7.4 Transfer Functions in a Second-Order Loop ............. 116
7.4.1 Output Response ............................... 117
7.4.2 Error Response ................................. 118
7.4.3 Responses Near ωL ............................. 118
7.4.4 Phase or Frequency at Inputs and Outputs ....... 119
7.5 Transient Responses Between Various Points ............ 120
7.6 Magnitude and Phase of the Transfer Functions ......... 120
7.6.1 Output Responses ............................... 121
7.6.2 Error Responses ................................ 121
7.6.3 Effect of α .................................... 122
7.6.4 Responses for ς ≤ 1 ............................ 123
7.6.5 Responses for ς ≥ 1 ............................ 125
7.7 Related Responses ..................................... 127
7.8 Modulation and Demodulation in the Second-Order
Loop .................................................. 128
7.8.1 Frequency Demodulation ......................... 128
7.8.2 Phase Demodulation ............................. 130
7.8.3 Frequency Modulation ........................... 130
7.8.4 Phase Modulation ............................... 131
7.8.5 Extending the Modulation Frequency Range ....... 131
7.8.5.1 Frequency Modulation .................. 131
7.8.5.2 Phase Modulation ...................... 133
7.9 Measurement of Loop Parameters for α = 0 or 1 from
Modulation Responses .................................. 134
7.10 Effect of an Added Pole ............................... 134
7.M Appendix: Modulation Response Using MATLAB® ........... 135
8 ACQUISITION ................................................ 137
8.1 Overview .............................................. 137
8.2 Acquisition and Lock in a First-Order Loop ............ 141
8.2.1 Transient Time ................................. 143
8.2.2 Acquisition .................................... 144
8.3 Acquisition Formulas for Second-Order Loops with
Sine Phase Detectors .................................. 146
8.4 Approximate Pull-In Analysis .......................... 149
8.4.1 Basic Equations ................................ 150
8.4.2 General Analysis ............................... 153
8.4.3 Pull-In Range .................................. 157
8.4.4 Approximate Pull-In Time ....................... 158
8.5 Phase-Plane Analysis .................................. 161
8.6 Pull-Out .............................................. 164
8.7 Effect of Offsets ..................................... 164
8.8 Effect of Component Saturation ........................ 165
8.9 Hangup ................................................ 166
8.10 Simulation of the Nonlinear Loop ...................... 166
8.A Appendix: Summary of Acquisition Formulas for
Second-Order Loops .................................... 167
i8.M Appendix: Nonlinear Simulation ........................ 167
8.S Appendix: Acquisition Spreadsheet ..................... 167
8.V Appendix: Some Values in Terms of α, ς, and ωn ........ 167
9 ACQUISITION AIDS ........................................... 171
9.1 Coherent Detection—Lock Indicator ..................... 171
9.1.1 During Acquisition ............................. 172
9.1.2 During Sweep, Locked Loop ...................... 173
9.2 Changing Loop Parameters Temporarily .................. 174
9.2.1 Coherent Automatic Gain Control ................ 174
9.2.2 Filter Modification ............................ 175
9.2.3 Comparison of Two Types of Parameter
Modifications .................................. 176
9.3 Automatic Tuning of ωc, Frequency Discriminator ....... 177
9.4 Acquisition Aiding Logic .............................. 179
9.5 Sweeping ωc, Type 2 Loop .............................. 180
9.5.1 Maximum Sweep Speed, Closed-Loop Sweeping ...... 180
9.5.2 Open-Loop Sweeping ............................. 181
9.5.3 Combined Techniques ............................ 185
9.6 Sweep Circuits ........................................ 185
9.6.1 Switched Current Source ........................ 185
9.6.2 Automatic Sweep Circuit—Sinusoidal ............. 186
9.6.3 Automatic Sweep Circuit—Nonsinusoidal .......... 186
9.A Appendix: Maximum Sweep Rate, Open-Loop
vs. Closed-Loop ....................................... 187
10 APPLICATIONS AND EXTENSIONS ................................ 189
10.1 Higher-Order Loops .................................... 189
10.2 Generalized Voltage-Controlled Oscillator ............. 189
10.2.1 Frequency Synthesis, Frequency Division ........ 190
10.2.1.1 Stability ............................. 191
10.2.1.2 Transient Response .................... 191
10.2.1.3 Response to Noise ..................... 192
10.2.2 Heterodyning (Frequency Mixing) ................ 192
10.3 Long Loop ............................................. 193
10.4 Carrier Recovery ...................................... 195
10.4.1 Biphase Costas Loop ............................ 195
10.4.2 N-Phase Costas Loop ............................ 196
10.4.3 Multiply and Divide ............................ 196
10.5 Data Synchronization .................................. 197
10.5.1 Early-Late Gate Bit Synchronizer ............... 197
10.5.2 Synchronizing to a Pseudorandom Bit Sequence ... 198
10.5.3 Delay-and-Multiply Synchronizer ................ 200
10.6 Clock and Data Timing Control ......................... 200
10.6.1 Phase-Locked Loops ............................. 200
10.6.2 Delay-Locked Loops ............................. 202
10.6.2.1 For Clock Synchronization .................... 202
10.6.2.2 For Data Synchronization ..................... 203
10.6.3 Combined Loops ................................. 204
10.7 All-Digital Phase-Locked Loop (ADPLL) ................. 204
10.7.1 Basic Digital Implementation ................... 205
10.7.1.1 The Loop .............................. 205
10.7.1.2 Sampling and Stability ................ 206
10.7.1.3 Choice of Values ...................... 209
10.7.1.4 Higher-Order Loops .................... 210
10.7.2 OA,NCO, DDS .................................... 211
10.7.3 Implementing an ADPLL by Pulse Addition and
Removal ........................................ 212
10.7.3.1 Transfer Function ..................... 213
10.7.3.2 Tuning Range .......................... 214
10.8 Summary ............................................... 214
10.A Appendix: Exact Analysis of a Special-Case Third-
Order Loop ............................................ 215
10.A.l Loop Response ................................. 217
10.A.2 Final Values .................................. 219
10.A.3 Triple Roots .................................. 220
10.A.4 Step Response ................................. 220
10.A.5 Modulation Response ........................... 224
10.B Appendix: Costas Loop for N Phases .................... 227
10.C Appendix: Symbol Clock Recovery ....................... 227
10.D Appendix: ADPLL by Pulse Addition and Removal,
Additional Material ................................... 228
10.F Appendix: Fractional-TV and Sigma-Delta ............... 228
i10.M Appendix: MATLAB® Simulations ........................ 228
i10.T Appendix: Combined PLL and DLL ....................... 229
PART 2 PHASE LOCK IN NOISE
11 PHASE MODULATION BY NOISE ................................. 233
11.1 Representation of Noise Modulation .................... 233
11.2 Processing of Noise Modulation by the Phase-Locked
Loop .................................................. 236
11.3 Phase and Frequency Variance .......................... 237
11.4 Typical Oscillator Spectrums .......................... 238
11.5 Limits on the Noise Spectrum—Infinite Variances ....... 240
11.6 Power Spectrum ........................................ 242
11.6.1 Spectrum for Small m ........................... 242
11.6.2 Single-Sideband Density ........................ 243
11.6.3 When is the Modulation Small? .................. 245
11.6.4 Modification of Spectral Shape at Higher
Modulation Index ............................... 246
11.6.5 Scripts  ..................................... 247
11.7 Frequency Multiplication and Division ................. 247
11.8 Other Representations ................................. 248
ill.H Shape of Output Spectrum ............................. 249
11.S Appendix: Spreadsheets for Integrating Densities ...... 249
12 RESPONSE TO PHASE NOISE .................................... 251
12.1 Processing of Reference Phase Noise ................... 251
12.2 Processing of VCO Phase Noise ......................... 254
12.3 Harmful Effect of Phase Noise in Radio Receivers ...... 255
12.4 Superposition ......................................... 256
12.5 Optimum Loop With Both Input and VCO Noise ............ 257
12.6 Multiple Loops ........................................ 260
12.7 Effects of Noise Injected Elsewhere ................... 260
12.8 Measuring Phase Noise ................................. 261
12.8.1 Using a Phase Detector ......................... 263
12.8.1.1 Calibration ........................... 263
12.8.1.2 Obtaining a Measurement Reference ..... 265
12.8.2 Using a Frequency Discriminator ................ 269
12.8.3 Using a Spectrum Analyzer or Receiver .......... 270
13 REPRESENTATION OF ADDITIVE NOISE ........................... 271
13.1 General ............................................... 271
13.2 Phase Modulation on the Signal ........................ 273
13.3 Multiplicative Modulation on Quadrature Carriers ...... 275
13.4 Noise at the Phase Detector Output .................... 276
13.5 Restrictions on the Noise Models ...................... 277
13.6 Does the Loop Lock to the Additive Noise? ............. 280
13.7 Other Types of Phase Detectors in the Presence of
Noise ................................................. 281
13.7.1 Triangular Characteristic ...................... 282
13.7.2 Sawtooth Characteristic ........................ 283
13.8 Modified Phase Detector Characteristic with Phase
Noise ................................................. 283
i13.A Appendix: Decomposition of a Single Sideband ......... 286
14 LOOP RESPONSE TO ADDITIVE NOISE ............................ 287
14.1 Noise Bandwidth ....................................... 287
14.2 Signal-to-Noise Ratio in the Loop Bandwidth ........... 290
14.3 Loop Optimization in the Presence of Noise ............ 291
14.3.1 The Problem ................................... 292
14.3.2 Measures to be Used ............................ 292
14.3.3 Optimum Loop for a Phase Step Input ............ 293
14.3.4 Optimum Loop for a Frequency Step Input ........ 294
14.3.5 Optimum Loop for a Frequency Ramp Input ........ 294
i14.A Appendix: Integration of Eq. (6.4a) .................. 296
i14.B Appendix: Loop Optimization in the Presence of
Noise ................................................. 296
15 PHASE-LOCKED LOOP AS A DEMODULATOR ......................... 297
15.1 Phase Demodulation .................................... 297
15.1.1 Noise .......................................... 297
15.1.2 Distortion of the Demodulated Signal ........... 299
15.1.3 Demodulation with a Linear Phase Detector
Characteristic ................................. 300
15.2 Frequency Demodulation, Bandwidth Set by a Filter ..... 301
15.3 Frequency Discriminator, First-Order Loop ............. 305
15.4 Frequency Discriminator, Second-Order Loop ............ 306
15.5 Expected Phase Error .................................. 307
15.6 Summary of Frequency Discriminator S/N ................ 308
15.7 Standard Discriminator and Click Noise ................ 310
15.8 Clicks with a PLL ..................................... 313
15.9 Noise in a Carrier Recovery Loop ..................... 314
15.C Appendix: Spectrum of Clicks .......................... 316
15.E Appendix: erfс ........................................ 317
16 PARAMETER VARIATION DUE TO NOISE ........................... 319
16.1 Preview ............................................... 319
16.1.1 Automatic Gain Control (AGC) ................... 319
16.1.2 Limiter ........................................ 320
16.1.3 Driving the Phase Detector Hard from the
Signal ......................................... 321
16.1.4 Effects of Variations .......................... 322
16.2 AGC ................................................... 322
16.2.1 Types of Detectors ............................. 322
16.2.2 Square-Law Detection ........................... 323
16.3 Limiter ............................................... 325
16.3.1 Limiting in the Presence of Small Noise ........ 325
16.3.2 Limiting in the Presence of Large Noise ........ 326
16.4 Effects of Gain Variation on Loop Parameters .......... 329
16.5 Effect of AGC or Limiter on an Optimized Loop ......... 331
16.A Appendix: Modified Bessel Functions ................... 333
17 CYCLE SKIPPING DUE TO NOISE ................................ 335
17.1 Phase ................................................ 336
17.1.1 Fokker-Plank Method ............................ 336
17.1.1.1 Assumption Regarding the Nature of
the Noise ............................. 337
17.1.1.2 First-Order Loop ...................... 337
17.1.1.3 Second-Order Loop ..................... 339
17.1.2 Experimental Results ........................... 341
17.1.3 Simulation ..................................... 342
17.1.3.1 First-Order and α = 0 ........................ 342
17.1.3.2 Second-Order, High-Gain ...................... 343
17.2 Cycle Skipping, Mean Time ............................ 347
17.2.1 First-Order Loop ............................... 347
17.2.2 Second-Order Loop, α = 0 ....................... 348
17.2.3 Second-Order Loop, α = 1 ....................... 348
17.2.4 Second-Order Loop, 0 < a < 1 ................... 350
17.2.5 Probability of Cycle Skipping in a Given
Time ........................................... 350
17.3 Cycle Skipping, Mean Frequency ........................ 350
17.4 Cycle Skipping with Mistuning ......................... 353
17.4.1 Effect of α .................................... 353
17.4.2 Comparison to Other Results .................... 356
17.5 Summary ............................................... 356
17.5.1 Phase Variance ................................. 356
17.5.2 Cycle Skipping ................................. 357
17.5.3 Skip Frequency ................................. 357
17.5.4 Mistuning in High-Gain Loops (ζ = 0.7) ......... 357
il7.D Appendix: Additional Data ............................ 357
il7.M Appendix: MATLAB® Scripts ............................ 357
18 NONLINEAR OPERATION IN A LOCKED LOOP ....................... 359
18.1 Notation .............................................. 359
18.2 Phase-Detector Output u1 .............................. 360
18.3 Changes in the Output Spectrum ........................ 361
18.4 Gain Suppression, Quasi-Linear Approximation .......... 361
18.4.1 Basics ........................................ 363
18.4.2 Phase Variance with Additive Noise ............ 365
18.4.3 Tracking the Carrier ........................... 368
18.4.3.1 With Phase Modulation ................. 368
18.4.3.2 With Phase Modulation and VCO Noise ... 369
18.4.4 Effect on Phase Offset ......................... 372
18.4.4.1 With High-Frequency Phase
Modulation ............................ 372
18.4.4.2 With Additive Noise ................... 373
18.4.4.3 With Limiting ......................... 375
18.4.5 Summary of Suppression .................. 375
i18.S Appendix: Additional Offset Simulation Data .......... 376
19 ACQUISITION AIDS IN THE PRESENCE OF NOISE .................. 377
19.1 Sweeping with Plain Closed-Loop ....................... 377
19.1.1 Maximum Sweep Rate in Noise .................... 377
19.1.2 Effect of Initial Mistuning .................... 380
19.1.3 Determining Successful Acquisition ............. 382
19.1.4 Optimum Sweep Parameters ....................... 383
19.2 Reduction of Coherent Detector Output (Closed-Loop
Sweeping) ............................................. 385
19.3 Closed-Loop Sweeping in Noise with Coherent
Detector .............................................. 386
19.3.1 Successful Acquisition ......................... 386
19.3.2 False Stops .................................... 388
19.3.3 False Restart .................................. 389
19.3.4 False Stop versus False Restart ................ 390
i19.M Appendix: MATLAB® Script, swpi ....................... 393
il9.S Appendix: Optimum Sweep for a Fixed Noise Density .... 393
20 BANDLIMITED NOISE .......................................... 395
20.1 Signals Centered in a Noise Band ...................... 395
20.1.1 Clicks with a First-Order PLL .................. 395
20.1.2 Simulation Results Compared to Measured Data ... 396
20.1.3 Skip Rate ...................................... 398
20.1.4 Output Phase Variance .......................... 403
20.1.5 Effect of a Limiter ............................ 404
20.1.6 Comparison to AGC .............................. 406
20.1.7 Higher-Order Loops ............................. 407
20.1.8 Summary, Symmetrical Narrowband Noise .......... 407
20.2 Eccentric Signals ..................................... 407
20.2.1 Expected Performance .......................... 408
20.2.2 Simulating Eccentric Noise ..................... 409
20.2.3 Some Simulation Results ........................ 409
20.2.3.1 Cycle Skipping ........................ 410
20.2.3.2 Phase Offset .......................... 411
20.2.3.3 Output Variance ....................... 411
20.3 Extension to Other Types of Interference .............. 411
i20.M Appendix: MATLAB® Scripts ............................ 413
i20.O Appendix: Offset Interference Data Correlation ....... 413
i20.S Appendix: Band Limited Simulation Data ............... 413
21 FURTHER INFORMATION ........................................ 415
21.1 Sources for Additional Studies in Phase Lock .......... 415
21.2 Sources Covering Phase-Locked Frequency Synthesis ..... 416
21.A Appendix: Modulations and Spectrums ................... 416
21.В Appendix: Getting Files from the Wiley Internet
Site .................................................. 421
REFERENCES .................................................... 423
INDEX ......................................................... 429
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