Theoretical models of voltage controlled oscillators and the effects of non-linearity

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Content THEORETICAL MODELS OF VOLTAGE CONTROLLED OSCILLATORS AND THE EFFECTS OF NON-LINEARITY by Yenming Chen A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful¯llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) December 2007 Copyright 2007 Yenming Chen Dedication To my parents, my family, and my wife, Pei-ling. ii Acknowledgements Iwouldliketoexpressmymostsinceregratitudetomyadvisor,Dr. RobertA.Scholtz,for his guidance, discussions, support, and valuable contributions to the development of this dissertation. I would like to thank Dr. William C. Lindsey and Dr. Kenneth Alexander for their useful comments and contributions in preparing this dissertation. I would like to thank Dr. Scholtz for the ¯nancial support by the Army Research O±ce under MURI Grant No.DAAD19-01-1-0477. I would like to especially express my appreciation to Dr. Peter Baxendale for providing great contributions to my research. I would also want to thank my parents for their continuous support and trusts. iii Table of Contents Dedication ii Acknowledgements iii List of Figures vi Abstract xii Chapter 1: Introduction 1 1.1 Motivation and Problem Statement . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Prior Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Signi¯cance and Contributions of the Research . . . . . . . . . . . . . . . . 3 1.3.1 On a Novel Mathematical Model of Voltage Controlled Oscillators . 4 1.3.2 On Robinson's Oscillator (Noise-Free Condition) . . . . . . . . . . . 5 1.3.3 On Robinson's Oscillator (Noise-Present Condition) . . . . . . . . . 6 Chapter 2: Background on Oscillators and Phase Noise Measures 9 2.1 General System Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 A Modi¯ed Van der Pol Oscillator . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Phase Noise Measures and Models . . . . . . . . . . . . . . . . . . . . . . . 15 Chapter 3: Theoretical Model Introduction and Analysis 22 3.1 Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Model Without Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3 Model With Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3.1 Noise Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3.2 Noise Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3.3 Phase Noise Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.4 Model With External Noise n(t) Present . . . . . . . . . . . . . . . . . . . . 32 3.4.1 Noise Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4.2 Noise Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4.3 Phase Noise Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.5 Simulation Results and Timing Jitter Estimate . . . . . . . . . . . . . . . . 37 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 iv Chapter 4: Analysis of Robinson's oscillator 41 4.1 Existence and Stability of a Solution . . . . . . . . . . . . . . . . . . . . . . 41 4.1.1 Existence of a Solution. . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.1.2 Stability of a Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 More Analysis of Robinson's Oscillator . . . . . . . . . . . . . . . . . . . . . 50 4.2.1 Transformation of the System Equation and Analysis . . . . . . . . 51 4.2.2 Simulation of the Model . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2.3 More Analysis of the Results . . . . . . . . . . . . . . . . . . . . . . 60 4.3 Models with Di®erent Gains . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3.1 E®ects on Non-linearities . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3.2 Model Selections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Chapter 5: Oscillator Models in the Presence of Noise 73 5.1 Oscillator Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2 Existence and Stability of the Solution . . . . . . . . . . . . . . . . . . . . . 76 5.2.1 Existence of the Solution for Oscillator Model I . . . . . . . . . . . . 76 5.2.2 Existence of the Solution for Oscillator Model II . . . . . . . . . . . 79 5.2.3 Existence of the Solution for Oscillator Model III . . . . . . . . . . . 81 5.2.4 Stability of the Solution for Oscillator Model I . . . . . . . . . . . . 83 5.2.5 Stability of the Solution for Oscillator Model II . . . . . . . . . . . . 88 5.2.6 Stability of the Solution for Oscillator Model III . . . . . . . . . . . 91 5.3 Analysis of Oscillator Model I . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.3.1 Transformation of the System Equation . . . . . . . . . . . . . . . . 94 5.3.2 New Phase Noise Model . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.3.3 Analytic Solution of the Conditional Expectation and Application . 100 5.3.4 Simulation of the Oscillator Model . . . . . . . . . . . . . . . . . . . 106 5.3.5 The E®ect of Non-linearity on the Phase Noise . . . . . . . . . . . . 116 5.3.6 Models with Di®erent Gains . . . . . . . . . . . . . . . . . . . . . . . 122 5.4 Analysis of Oscillator Model II . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.4.1 Transformation of the System Equation . . . . . . . . . . . . . . . . 127 5.4.2 Simulation of the Oscillator Model . . . . . . . . . . . . . . . . . . . 130 5.4.3 The E®ect of Non-linearity on the Phase Noise . . . . . . . . . . . . 135 5.4.4 Models with Di®erent Gains . . . . . . . . . . . . . . . . . . . . . . . 138 5.5 Analysis of Oscillator Model III . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.5.1 Transformation of the System Equation . . . . . . . . . . . . . . . . 144 5.5.2 Simulation of the Oscillator Model . . . . . . . . . . . . . . . . . . . 146 5.5.3 The E®ect of Non-linearity on the Phase Noise . . . . . . . . . . . . 151 5.5.4 Models with Di®erent Gains . . . . . . . . . . . . . . . . . . . . . . . 155 Chapter 6: Conclusion and Future Work 162 6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 References 167 v List of Figures 2.1 The phase plane representation of the damped linear oscillator in equation (2.3) for strong damped system with k =100;c=1000 . . . . . . . . . . . . 11 2.2 The phase plane representation of the damped linear oscillator in equation (2.3) for weakly damped system with k =10;c=1000 . . . . . . . . . . . . 12 2.3 Phase plane representation of a modi¯ed Van der Pol oscillator with ² = 2000 and ! 0 =4000¼ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 Phase plane representation of a modi¯ed Van der Pol oscillator with ² = 2:5! 0 and ! 0 =4000¼ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5 A periodic solution of a modi¯ed Van der Pol oscillator in time domain . . 16 3.1 Theoretical model of voltage controlled oscillator with thermal noise and noise from the controlled voltage and tracking loop. . . . . . . . . . . . . . 23 3.2 Equivalent model of VCO (Fig. 3.1) with noise contributions, e F(t), scaled internal noise, and n(t), controller plus tracking loop noise. . . . . . . . . . 26 3.3 General scheme for the tracking loop plus the controller noise, n(t) . . . . . 32 3.4 Averaged upward zero-crossing jitter for a VCO with 2 kHz clock frequency with noise power of ¾ 2 e F =1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5 Cycle-to-cycle jitter statistics for 100 seconds . . . . . . . . . . . . . . . . . 38 3.6 Timing jitters when e F(t) and n(t) are present . . . . . . . . . . . . . . . . . 39 3.7 Power spectral density of the proposed model's output y(t) with di®erent noise levels of F(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.1 Motions of physical dynamics of Robinson's oscillator at di®erent points of the phase plane diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 vi 4.2 Motions of phase paths at di®erent initial points on the phase plane diagram 45 4.3 Phase plane representation of a Robinson oscillator with G 1 = 1000, G 2 = ¼! 0 G 1 4 , and ! 0 =2000¼ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.4 Schematic of Robinson's oscillator from equation (4.2) . . . . . . . . . . . . 56 4.5 Simulink block diagram showing the amplitude and phase di®erential equa- tions (4.12), (4.13) with the clock rest frequency ! 0 =2000¼ . . . . . . . . . 57 4.6 The amplitude function ½(t) of the simulated model shown in Fig. 4.5 . . . 58 4.7 Phase paths of the equation (4.12) at steady state . . . . . . . . . . . . . . 59 4.8 E®ects of non-linearity g(½sinµ) on the amplitude and the phase functions 60 4.9 The phase diagram of the equation (4.13) when ! 0 =2000¼ . . . . . . . . . 61 4.10 Phaseplanerepresentationoftheoscillatoroutputwaveformsshowing½cosµ vs. ½sinµ for G 1 =1000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.11 The steady state amplitude function ½ s (t) and the deviation of the phase drift function µ d (t) for di®erent non-linearities . . . . . . . . . . . . . . . . . 63 4.12 Signal diagrams showing the e®ects of di®erent non-linearities on _ ½ and on _ µ d 64 4.13 Equationsthatareusedtodeterminethephasedriftratedi®erencebetween two oscillators based on equation (4.22) . . . . . . . . . . . . . . . . . . . . 66 4.14 E®ectsofnon-linearitywithdi®erentconstantgainG 1 ontheinstantaneous change of amplitude function _ ½ and the angular frequency drift function _ µ d 69 4.15 Thesteadystateamplitudefunction½ s (t)andthephasedriftfunctionµ d (t) when the same model is simulated with a hard-limiter non-linearity but di®erent constant gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.16 Phase plane representation of the oscillator model with hard-limiter non- linearity when the constant gain G 1 = 100 and the clock rest frequency ! 0 =2000¼ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.17 Phase plane representation of a Robinson oscillator with G 1 = 10000, and ! 0 =2000¼ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.18 A periodic solution of a Robinson oscillator in time domain . . . . . . . . . 73 5.1 Schematic of the oscillator model I from equation (5.1) . . . . . . . . . . . . 75 vii 5.2 Schematic of oscillator model II from equation (5.2) when both internal oscillator noises F 1 (t) and F 2 (t) are present . . . . . . . . . . . . . . . . . . 76 5.3 Oscillator modelIII whenboth internaloscillatornoises F 1 (t) andF 2 (t)are present,andthesoft-limiterg sl (¢)isemployedintheouterloopofthediagram 77 5.4 Schematic of the oscillator model I from equation (5.8) . . . . . . . . . . . . 108 5.5 Simulink block diagram showing the amplitude and phase di®erential equa- tions (5.54), (5.55) with the clock rest frequency ! 0 =2000¼ when internal noise F 1 (t) is present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.6 Cumulativedistributionfunction(cdf)oftheamplitudefunction½(0:02)for 1000 realizations vs. Gaussian cdf. . . . . . . . . . . . . . . . . . . . . . . . 110 5.7 E®ect of noise on the non-linearity function g(½sinµ) for both amplitude and phase functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.8 Sum of the noise term and the non-linearity to the amplitude and total phase stochastic di®erential equations (5.54) and (5.55) . . . . . . . . . . . 112 5.9 Expected phase path of the amplitude function ½(t) at steady state . . . . . 113 5.10 var(µ(t)¡E[µ(t)jµ(0);½(0)]) or var(Á(t)¡Á(0)) for 100 realizations where µ(0)=0, ½(0)=0:125 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.11 Empirical cdf of the new phase noise process Á(0:04)¡Á(0) = µ(0:04)¡ E[µ(0:04)jµ(0)=0;½(0)=0:125] over 1000 realizations . . . . . . . . . . . . 115 5.12 Variance of the amplitude noise process ½(t)¡E[½(t)jµ(0);½(0)] and the modi¯ed innovation process E[½(t)jµ(0);½(0)]¡½(t)j no noise . . . . . . . . . 116 5.13 E[½cosµ] vs. E[½sinµ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.14 (a) normalized standard deviation of µ(t)¡E[µ(t)jµ(0)=0;½(0) = 0:125] for the oscillator model employing a soft-limiter with slope p = 0:001 and (b) normalized RMS frequency deviation [var(Á(t)¡Á(0)) 0:5 ]=(2¼f 0 t) . . . . 120 5.15 Total phase and amplitude noise e®ects on two non-linearities used, one withahard-limiter, theotheronewithasoft-limiterwhoseslope p=0:001, the second term of equation (5.55) and the ¯rst term of equation (5.54) are plotted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.16 (a) Averaged power spectrum of the phase noise process Á(t)¡ Á(0) for the oscillator model employing a hard limiter, (b) averaged phase noise spectrum of Á(t)¡Á(0) for the model employing a soft-limiter with slope p=0:001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 viii 5.17 (a) Normalized standard deviation of the phase noise process Á(t)¡Á(0) when the oscillator model with gain G 1 =100 is used, (b) normalized RMS fractional frequency deviation of the phase noise process . . . . . . . . . . . 125 5.18 (a)Varianceoftheamplitudenoiseprocess½(t)¡E[½(t)jµ(0)=0;½(0)=0:125] at G 1 = 1000 in the oscillator model, (b) variance of the amplitude noise process at G 1 =100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.19 (a)AveragedpowerspectrumofthephasenoiseprocessÁ(t)¡Á(0)whenthe soft-limiterwithslopep=0:001isemployedinthesystematG 1 =1000,(b) averaged power spectrum of the phase noise process when the hard-limiter is used at G 1 =100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.20 Schematic of oscillator model II from equation (5.84) when both internal oscillator noises F 1 (t) and F 2 (t) are present . . . . . . . . . . . . . . . . . . 129 5.21 Simulink block diagram showing the amplitude and phase di®erential equa- tions (5.87), (5.88) with the clock rest frequency ! 0 =2000¼ when internal oscillator noises F 1 (t) and F 2 (t) are present . . . . . . . . . . . . . . . . . . 132 5.22 E®ect of internal noises on the non-linearity function g(½sinµ) for both amplitude and phase functions . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.23 (a)E®ectofµ noiseandnon-linearityon _ µ+! 0 = g(½sinµ) ½ cosµ+ h 1 (t) ½ cosµ¡ h 2 (t) ½ sinµ, (b) E®ect of ½ noise and non-linearity on _ ½ = g(½sinµ)sinµ + h 2 (t)cosµ+h 1 (t)sinµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.24 (a) The e®ect of non-linearity on the normalized standard deviation of the new phase noise process Á(t)¡Á(0), (b) the normalized standard deviation of the phase noise process over time, (c) the normalized RMS fractional frequency deviation in equation (5.83) . . . . . . . . . . . . . . . . . . . . . 135 5.25 (a)Standarddeviationoftheamplitudenoiseprocess½(t)¡E[½(t)jµ(0);½(0)] and (b) modi¯ed innovation process E[½(t)jµ(0);½(0)]¡½(t)j no noise . . . . . 136 5.26 (a)NormalizedstandarddeviationofthenewphasenoiseprocessÁ(t)¡Á(0) when the soft-limiter with slope p = 0:001 is used in the oscillator model, and (b) normalized RMS fractional frequency deviation de¯ned in equation (5.83) for oscillator model II with a soft-limiter . . . . . . . . . . . . . . . . 139 5.27 (a) Averaged power spectrum of the phase noise process Á(t)¡ Á(0) for the oscillator model employed with a hard-limiter, and (b) averaged power spectrum of the phase noise process when the soft-limiter with slope p = 0:001 is used in the oscillator model . . . . . . . . . . . . . . . . . . . . . . 140 ix 5.28 (a) Normalized standard deviation of the phase noise process Á(t)¡Á(0) when the constant gain G 1 = 100 in the oscillator model II, and (b) nor- malized RMS fractional frequency deviation of the phase noise process for the same oscillator model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.29 (a) Standard deviation of the amplitude noise process de¯ned in equation (5.76) when G 1 = 1000 is used in the oscillator model, and (b) standard deviation of the amplitude noise process when G 1 =100 is used . . . . . . . 143 5.30 (a)Averagedspectrumofthephasenoiseprocess Á(t)¡Á(0)whenthesoft- limiterwithslopep=0:001isusedintheoscillatormodel,and(b)averaged spectrum of the phase noise process at G 1 =100 . . . . . . . . . . . . . . . 144 5.31 OscillatormodelIII whenboth internaloscillatornoises F 1 (t) andF 2 (t)are present, and the soft-limiter with slope p=1 is employed in the outer loop of the diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.32 Simulink block diagram showing the amplitude and phase di®erential equa- tions (5.96), (5.97) with the clock rest frequency ! 0 =2000¼ when internal oscillator noises F 1 (t) and F 2 (t) are present and a soft-limiter non-linearity is employed in the outer loop of the oscillator model in Fig. 5.31 . . . . . . 148 5.33 (a) Di®erential phase mismatch term, ¡! 0 [ g sl (½cosµ)cosµ ½ +sin 2 µ], from the phase stochastic di®erential equation (5.97), and (b) _ µ showing the e®ect of non-linearity function g(½sinµ) at G 1 =1000 . . . . . . . . . . . . . . . . . 150 5.34 (a) The e®ect of non-linearity on the normalized standard deviation of the new phase noise process Á(t)¡Á(0) =E[Á(t)jµ(0) = 0;½(0) = 0:125] when G 1 =1000isused, (b)thenormalizedstandarddeviationofthephasenoise process over time, (c) the normalized RMS fractional frequency deviation in equation (5.83) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.35 (a)Standarddeviationoftheamplitudenoiseprocess½(t)¡E[½(t)jµ(0);½(0)] and (b) modi¯ed innovation process E[½(t)jµ(0);½(0)]¡½(t)j no noise . . . . . 152 5.36 a)NormalizedstandarddeviationofthenewphasenoiseprocessÁ(t)¡Á(0) when a soft-limiter with slope p = 0:001 is used in the g(½sinµ) function of the oscillator model III, and (b) normalized RMS fractional frequency deviation de¯ned in equation (5.83) for the oscillator model III with a soft- limiter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.37 (a) Averaged power spectrum of the phase noise process Á(t)¡ Á(0) for the oscillator model III employed with a hard-limiter, and (b) averaged power spectrum of the phase noise process when the soft-limiter with slope p=0:001 is used in the oscillator model III . . . . . . . . . . . . . . . . . . 156 x 5.38 (a) Normalized standard deviation of the phase noise process Á(t)¡Á(0) whentheconstantgainG 1 =500isusedintheoscillatormodelIII,and(b) normalized RMS fractional frequency deviation of the phase noise process for the same oscillator model at G 1 = 500, and (c) normalized standard deviation of the phase noise process at G 1 =100, and (d) normalized RMS fractional frequency deviation of the phase noise process for the same oscil- lator model at G 1 =100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.39 (a) The di®erential phase mismatch term,¡! 0 [ g sl (½cosµ)cosµ ½ +sin 2 µ], from the phase stochastic di®erential equation (5.97) at G 1 = 100, and (b) _ µ at such constant gain when both internal noises are present . . . . . . . . . . . 159 5.40 (a) Standard deviation of the amplitude noise process for the oscillator model II at G 1 = 100, and (b) standard deviation of the amplitude noise process for the oscillator model III at the same G 1 . . . . . . . . . . . . . . 160 5.41 (a) The di®erential amplitude mismatch, ! 0 sinµ[½cosµ¡g sl (½cosµ)], from the amplitude stochastic di®erential equation (5.96) at G 1 = 100, and (b) _ ½ at such constant gain when both internal noises are present . . . . . . . . 161 5.42 (a) Averaged power spectrum of the phase noise process Á(t)¡Á(0) for the oscillator model III at G 1 = 500 , and (b) averaged spectrum of the phase noise process for the oscillator model III at G 1 = 100, and (c) averaged spectrum of the phase noise process for the oscillator model II at G 1 =100 162 6.1 System diagram of two VCOs with switch for synchronization analysis . . . 166 6.2 A possible re¯nement on the simpli¯ed model of a VCO . . . . . . . . . . . 167 xi Abstract Thisthesispresentsmathematicalmodelsandperformanceevaluationsofvoltage-controlled oscillators when noises are included in the model. There are two main parts of the re- search. The ¯rst part introduces a novel mathematical model of a voltage-controlled oscillator (VCO) based on physical dynamics with noise. The e®ects of noise in this sys- tem model are shown and the analytical forms of the resulting phase noise are obtained using the stochastic integrals. It is shown that the VCO has phase noise contributed from the internal noise of the clock and the clock drift caused by tuning plus tracking loop noises. Moreover, a two-pole ¯lter is designed to constrain tracking loop noise. A timing jitterestimateisproposed. Analysisoftheresultingphasenoisealongwiththesimulation suggestthatthetimingjitterprocesshasarandomwalkbehaviorwithrestoringforceand the upward zero-crossing jitter is normally distributed. The cycle-to-cycle jitter statistics are shown to be Gaussian distributed and independent. Thesecondpartoftheresearchisfocusedonoscillatorswithnon-linearity. Theoscilla- tor model proposed is based on Robinson's oscillator [3]. Two cases are studied, noise-free and noise-present conditions. Three types of oscillator models are introduced when noise xii is present. Noise e®ects on the oscillators and the non-linearity e®ect on the noise are an- alyzed. A new oscillator phase noise model which indicates the e®ect of a non-linearity is proposed. Theresultsshowthatthenewlyde¯nedphasenoiseisamodi¯edWienerprocess whose variance increases with time. The phase noise process is shown to be Gaussian dis- tributed. Furthermore, the spread of the variance of the phase noise is reduced if the hard-limiter non-linearity used in the oscillator model is replaced by a soft-limiter with small slope. Similar e®ects are seen by reducing the system gain parameters. Further system performance improvement is achieved by the introduction of a soft-limiter in the outer-loop of the model, which further reduces the degradation of the noise on the steady state response. xiii Chapter 1 Introduction 1.1 Motivation and Problem Statement In communication system analysis, where the voltage controlled oscillator (VCO) of the PLL is modelled as a simple integrator, the model does not capture the essence of the dynamics of oscillation. In addition, instabilities in oscillator models present a major problem in analyzing synchronization performance. When an application in communica- tion requires a oscillator with high oscillating frequency, the mismatch problem between the received signal and the local VCO due to oscillator instabilities would be signi¯cant to the coherent receiver. In ultra-wideband (UWB) communications, where the UWB device is de¯ned having a fractional bandwidth greater than 0.2 or occupying 500 MHz or more of spectrum [13], the pulse width, usually in the sub-nanoseconds, requires a high speed clock. Therefore, oscillator / clock instabilities become an issue in synchronization, especially in UWB communications. Thus, we need a mathematical model of a voltage controlled oscillator that fully incorporates a non-linearity for stabilizing oscillators. 1 1.2 Prior Work Extensive research has been done in the past for oscillators and standards. This includes the theoretical phase noise of oscillators based on structure functions [28]. This paper serves as a standard for understanding how to characterize oscillator phase noise using the structure functions. We will propose a phase noise measure based on structure functions as well. Phase noise analysis is further investigated in electrical oscillators [17] [18] where a general theory of phase noise in di®erent electrical oscillators is developed, including LC, Colpitts, and ring oscillators. Moreover, they propose a phase noise power measure based on circuit parameters. A positive feedback system approach [7] is used on a class of oscillators whose derivation of the power spectral density (PSD) of phase noise is based on a linear periodic time varying (LPTV) system. In addition, a standard for frequency and time that describes the terminology can be found in [1], where the Allan variance for oscillator phase noise is de¯ned. Moreover, the oscillator noise spectrum can be observed by experiments as well as the noise e®ects such as the timing-jitter process [1]. A group from National Institute of Standards and Technology (NIST) [22], [5] uses a statistical regression technique and a compensation technique to analyze the timing jitter processed from the raw data of experiments. The approach to analyzing oscillator noise based on the stochastic integrals can be found in [8] where a perturbation technique is used for noise analysis. However, the derivation of the phase noise term is based on a small noise assumption. The e®ect of largenoiselevelisnotconsideredintheanalysis. Itisnotclearwhatwillhappenwhenwe have large oscillator noise. In addition, in [34] an analysis and simulation of phase noise 2 in a VCO is investigated and it is based on perturbations and noise injections. One more reference worth mentioning is the \Integrated GHz voltage controlled oscillators" in [24], where an intuitive introduction and characterization of phase noise and VCO design are given. Most of the research works are circuit-based and few address the e®ect of noise on a VCO. However, a number of prior works are the cornerstones of my research and they provide strong backgrounds for my research work. In general, we need a mathematical model of voltage controlled oscillator that fully incorporates a means of non-linearity for stabilizing oscillators under any noise level condition. 1.3 Signi¯cance and Contributions of the Research Since most oscillator models are circuit-based, the analysis of the oscillator performance is type-speci¯c and di±cult to generalize. There is no unique theory for characterizing the general oscillator model without going into the speci¯c details of circuit analysis. We are interested in the general mathematical characterization of oscillator models, especially VCO's. Without going into circuit analysis, we start with the physical dynamics of theo- retical oscillators and introduce a model with noise whose performance will resemble the performance of real oscillators. The objective of the research is to study and propose mathematical models of voltage controlledoscillatorsandtounderstandthenoisee®ectsonoscillatorsaswellascharacter- izing the e®ects. The general model of oscillators can be seen as a second-order a linear or non-linear di®erential equation. Furthermore, it can be either a forced or a self oscillating 3 system. To see the dynamics of the physical model, we perform detailed analysis both analytically and numerically. This will give us a basic understanding of the system. 1.3.1 OnaNovelMathematicalModelofVoltageControlledOscillators The¯rstpartoftheresearchisfocusedonanovelmathematicalmodelofvoltagecontrolled oscillatorsthatincludesnoisee®ectsfrominternalnoiseoftheclockandfromthetracking loop plus the controller noise, without going into the VCO circuits. This can be described by a second-order stochastic di®erential equation whose coe±cients consist of the tracking loop plus the controller noise. We are able to describe this type of equation as marginally stable and the noise e®ects on the proposed model are analyzed. ² Noise e®ect on a VCO model The noise sources that contribute to the model include both internal and external noises. The e®ects of these noise sources is manifested in thephase andamplitude noises of oscillators. Analytical expressions for phase and amplitude noise statistics are obtained and detailed analysis are explored. ² Analytical and simulation results The phase noise / timing jitter process is explicitly shown analytically and by sim- ulations. Because of the two-integrator oscillator model, we are able to derive the oscillator output solution in closed form as a sinusoidal waveform with noise contri- butions, using the fundamental matrix solution and the Floquet theorem [6]. The statisticsoftheresultingphasenoiseareobtained, analyzed, andcharacterized. Due tothedi®erentiatorwithintheoscillatormodel,weproposeatwo-pole¯lterthatwill 4 suppresstheexternalnoiseontheVCO'scontrolsignal. Theresultingphasenoiseis non-stationary and has a di®usion-like variance. Simulations demonstrate sta- tisticalpropertiesofthetiming-jitterprocess. Cycle-to-cycle jitterisinvestigated and shown to have a Gaussian distribution. We go one step further by proposing a timing-jitter estimate based on the Markovian property of the timing-jitter process. 1.3.2 On Robinson's Oscillator (Noise-Free Condition) The second part of the research is focused on a stable oscillator with non-linearity whose dynamics can be described by a second-order non-linear di®erential equation. ² Analysis and phase noise model We introduce the general system model of oscillators ¯rst and give a detailed analy- sis of a modi¯ed Van der Pol oscillator (a math model). In addition, phase noise measures and models are studied as a basis for system performance analysis, i.e., an oscillator phase noise model is shown to have a random walk behavior (transfor- mationoftheWienerprocess). WethenintroduceRobinson'soscillator[3](acircuit model) as our system model and give an analysis of its performance. It is proven that the system models a stable oscillator. Moreover, the system model is trans- formed into a pair of di®erential equations governing the dynamics of the amplitude and phase functions and analyzed to see the e®ect of non-linearities on oscillator performance. 5 ² Simulation results and analysis Simulations are done on both the original oscillator model and the transformed model, and the resulting amplitude and phase functions of oscillators are analyzed. System parameters of interest include the e®ect of non-linearity on the steady state amplitude function and on the phase drift function. In addition, more detailed analysis is done on the oscillator model with di®erent constant gains. The e®ect of a non-linearity are observed as system parameters are varied. Furthermore, a modelselectionisintroducedtoimprovesystemperformance,andatrade-o®between choices of system parameters is discussed as well. 1.3.3 On Robinson's Oscillator (Noise-Present Condition) The analysis of non-linear Robinson's model with noise is the main focus of our research. Thee®ectofnoiseonthenon-linearoscillatormodelremainsaninterestingandunexplored area for the purpose of achieving the system stability when a means of non-linearity is incorporated and noise is present. We are interested in characterizing the e®ects of noise on oscillators as well as analyzing the non-linearity e®ect on oscillators and its e®ects on noise. Whether or not the system still remains stable when noise is present is not clear, the existence of the solution would also be another interesting topic to be explored. It is interestingtoknowthatwhenvariousnoisesarepresentinthemodel,whatisthee®ecton the oscillator model performance? Moreover, when the e®ect of noise becomes signi¯cant, whatarethepossiblere¯nementsonthemodelthatcanoptimizethesystemperformance? These types of questions are some of the research areas that one can undertake when the noise is present in the analysis. 6 ² Three proposed oscillator models We propose three models with noise based on Robinson's oscillator. These models can be described by second-order non-linear stochastic di®erential equations. As the noise is present in the oscillator model, analysis is done on the existence and the stability of the solution. For non-linear oscillators with noise present, we prove the stability of the models. Moreover, noise e®ects on the oscillators and the non- linearity e®ect on the noise are analyzed. ² New phase noise model We propose a new phase noise model of oscillators. Results analyzed and simulated canberepresentedbythisnewnoisemeasure. Asa result, thenewphase noisemea- sure clearly indicates the e®ect of non-linearity as compared to theAllan variance [1] (standard phase noise measure). We also de¯ne the amplitude noise model sim- ilarly to the phase noise model. It is evident that the new phase noise model can describe the non-linear e®ect of oscillators. The variance and the mean of the new phase noise are observed. The results show that the phase noise de¯ned is a modi¯ed Wiener process whose variance increases with time. Moreover, the statistics of the phase noise process is shown to be Gaussian distributed. ² E®ects of non-linearity We observe the spread on the variance of the newly de¯ned phase noise process. This phenomenon is investigated and its cause is mainly due to the non-linearity. When di®erent types of non-linearity is employed in the oscillator model, we obtain di®erent degrees of spread on the variance of the new phase noise. This explains 7 the e®ect of non-linearity on oscillator phase noise. As far as the amplitude noise process is concerned, we deduce that the process behaves like a white noise process with a Gaussian distribution. ² Conditional expectation We derive a partial di®erential equation similar to the Kolmogorov backward equation, whose solution is the conditional expectation that we use to de¯ne the new phase noise process. Thus, either by solving the partial di®erential equation or by performing simulations on the oscillator model, we can ¯nd the correct quantities of the conditional expectation. ² Improving oscillator performance Otherinterestingimprovementsontheperformanceoftheoscillatormodelwithnoise areintroducedandanalyzedinthisresearch. Forexample,thespreadofthevariance of the phase noise can be reduced if we change the hard-limiter non-linearity to be a soft-limiter with small slope. A similar e®ect can be seen from the reduction on the system constant gains. Further system performance improvement can be seen by the introduction of a soft-limiter in the outer-loop of the model, which further reduces the degradation of the noise on the steady state response. 8 Chapter 2 Background on Oscillators and Phase Noise Measures 2.1 General System Models The general model for oscillators can be described by continuous-time systems of di®eren- tial equations which typically take the form dx dt = _ x=f(x) (2.1) where f is a set of functions with domain U, an open subset ofR m , and rangeR m . The vectorx=(x 1 ;x 2 ;:::;x m ) T denotesthephysicalorstatevariablestobestudiedandt2R indicatestime. Whenthefunctionfoftheaboveequation(2.1)doesnotdependexplicitly onthetimevariablet,thisiscalledanautonomoussystemofequation. However,whenthe 9 timevariabletappearsexplicitlyinthefunctionf, thenon-autonomousorforcedequation is formed. We are focusing here on a second-order di®erential equation of the general type Ä x=f(x; _ x) (2.2) which is an autonomous equation without any forcing term. Letusstartwithsomeexamplesofgeneralself-oscillatingsystemofequation. Consider a simple damped linear oscillator [21] having the equation Ä x+k_ x+cx=0 (2.3) where c>0; k >0. The solutions depend on whether the roots of the auxiliary equation m 2 +km+c=0 are real and di®erent, complex, or real and identical. The roots are given by m 1 ; 2 = 1 2 [¡k§ p k 2 ¡4c ] and the discriminant, ¢=k 2 ¡4c, is the parameter which determines the general type of motion of the solution. Three cases can occur. ² Strong damping (¢>0) The solutions are given by x(t)=Ae m 1 t +Be m 2 t ; (2.4) 10 −8 −6 −4 −2 0 2 4 6 8 −100 −80 −60 −40 −20 0 20 40 60 80 100 x dx/dt Figure 2.1: The phase plane representation of the damped linear oscillator in equation (2.3) for strong damped system with k =100;c=1000 where m 1 and m 2 are real and negative, A and B are any constants. The phase plane representation for the system that satis¯es the strong damping condition is shown in Fig. 2.1. There exists a stable equilibrium point at x = 0; _ x = 0, called a stable node. ² Weak damping (¢<0) The exponents are complex with negative real part, and the solutions are given by x(t)=Aexp(¡ 1 2 kt)cos( 1 2 p ¡¢t+®) (2.5) where A; ® are arbitrary constants. This solution represents an oscillation with exponentially decreasing amplitude, decaying more rapidly for larger k. The phase plane representation shown in Fig. 2.2 has an equilibrium point at the origin called a stable spiral. 11 −150 −100 −50 0 50 100 150 200 −6000 −4000 −2000 0 2000 4000 6000 x dx/dt Figure 2.2: The phase plane representation of the damped linear oscillator in equation (2.3) for weakly damped system with k =10;c=1000 ² Critical damping (¢=0) In this case m 1 =m 2 =¡ 1 2 k and the solutions are x(t)=(A+Bt)exp(¡ 1 2 kt): (2.6) The solutions resemble those for strong damping and the phase diagram shows a stable node. Another example that shows a self-oscillating system with non-linearity is a second order autonomous system model Ä x+sgn(x)=0: (2.7) 12 It can be shown that the system _ x = y; _ y =¡sgn(x) has a center at the origin. Suppose the oscillations have the form acos!t approximately. We can therefore approximate the sgn(acos!t) as sgn(acos!t)= 4 ¼ cos!t +H:O:T: Then the approximate equation is Ä x + 4 ¼a x = 0 if we neglect the higher order terms (H.O.T.). The solution can then be found to be x(t)=acos µ 4 ¼a ¶ 0:5 t: (2.8) The above solution has frequency controlled by the amplitude a which is not desirable. Therefore, we need a system model with non-linearity that can stabilize the oscillation with a limit cycle, even in the presence of noise. We next introduce a non-linear oscillator based on a Van der Pol oscillator. 2.2 A Modi¯ed Van der Pol Oscillator Consider the family of equations of the form Ä x+²h(x; _ x)+x=0 ; (2.9) then on the phase plane we have _ x=y; _ y =¡²h(x;y)¡x : (2.10) 13 For a Van der Pol's equation (a mathematical model) of Ä x+²(x 2 ¡1)_ x+x = 0; ² > 0, negative damping occurs in the strip jxj < 1 and positive damping in the half-planes jxj > 1. This type of damping leads to a limit cycle. Now suppose we have a modi¯ed Van der Pol's equation as Ä x+²h(x; _ x)+! 2 0 x=0 : (2.11) Let h(x; _ x) = (x 2 ¡0:25)_ x, we can say there exists a limit cycle and it is unique by the existence and uniqueness theorem [21]. Assuming that the periodic solution is represented approximately by x(t) = acos! 0 t; a > 0, the necessary condition that determines the amplitude a of a limit cycle is Z 2¼ 0 h(acos! 0 t;¡asin! 0 t)sin! 0 tdt=0 : (2.12) The approximate amplitude for the limit cycle of a modi¯ed Van der Pol's equation Ä x+ ²(x 2 ¡0:25)_ x+! 2 0 x = 0 is then found to be 1. The limit cycle of the equation (2.11) is stable if ² > 0. By the averaging method [21], for small positive ², the approximate solutions are found to be x(t)=a(t)cosµ(t)= cos(! 0 t¡µ 0 ) [1¡(1¡1=a 2 0 )exp ¡ ²t 4 ] 1 2 ; (2.13) where µ 0 is the initial polar angle and a(0)=a 0 . The phase plane representation of the modi¯ed Van der Pol's equation further shows the existence of the limit cycle. As seen in Fig. 2.3, a limit cycle with x 1 's amplitude of 1 exists and the approximate solution is cos(! 0 t¡µ 0 ). As the damping coe±cient ²=! 0 =± 14 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −8 −6 −4 −2 0 2 4 6 8 x 10 4 x1 x2 Figure 2.3: Phase plane representation of a modi¯ed Van der Pol oscillator with ²=2000 and ! 0 =4000¼ decreases, it takes longer for the solution to go to the steady state. On the other hand, as the damping coe±cient ² becomes large, i.e., ²>! 0 , the solution is still periodic but it is not close to a sinusoid. This can be seen in Fig. 2.4 and Fig. 2.5. For practical oscillators, features like a sinusoidal waveform and a fast response to the steady state are desired. Therefore, for a nearly sinusoidal solution of the modi¯ed Van der Pol's oscillator with a fast response to the steady state, a proper choice on the damping factor ² should be taken into consideration. 2.3 Phase Noise Measures and Models Oscillator phase noise has always been a major research area in oscillator analysis. This includes the phase noise measure in the frequency domain [30], [16], [36], [20], and RMS fractionalfrequencydeviationandAllanvarianceintimedomain[1],[15]. Characterization 15 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −8 −6 −4 −2 0 2 4 6 8 x 10 4 x1 x2 Figure 2.4: Phase plane representation of a modi¯ed Van der Pol oscillator with ²=2:5! 0 and ! 0 =4000¼ 0 0.5 1 1.5 2 2.5 3 x 10 −3 −1.5 −1 −0.5 0 0.5 1 1.5 time (sec.) amplitude Figure 2.5: A periodic solution of a modi¯ed Van der Pol oscillator in time domain of frequency stability and measurements of frequency stability can be found in [11], [41], [43]. Without going to details of the measurements of oscillator instabilities, we introduce the RMS fractional frequency deviation and Allan variance [31]. Suppose the oscillator output is given by y(t)=a(t)cos(2¼f 0 t+µ d (t)+Á(t)); (2.14) 16 where a(t) is the amplitude function, f 0 is the clock rest frequency, µ d (t) is the phase drift term, and Á(t) is the oscillator phase noise. To characterize oscillator °uctuations caused by the phase noise process Á(t), we use the following measure. The root mean square (RMS) fractional frequency deviation is an IEEE accepted standard which accounts for the °uctuations in the time interval and time interval error due to the oscillator phase noise process Á(t). The normalized RMS fractional frequency deviation is de¯ned as ±f(¿) f 0 = ½ E[(Á(t +¿)¡Á(t)) 2 ] (2¼f 0 ) 2 ¾ 0:5 1 ¿ = · D (1) Á (¿) (2¼f 0 ¿) 2 ¸ 0:5 ; (2.15) where D (1) Á (¿) =E[(Á(t +¿)¡Á(t)) 2 ] is the ¯rst structure function [28] of Á(t), ±f(¿) is RMS fractional frequency deviation, and time interval of length ¿. The Allan variance is used to account for the °uctuations in the time interval stability due to the oscillator phase noise Á(t). It speci¯es the variance of the length of the di®erence in two adjacent time intervals relative to the square of the length 2¿ which would be generated by an ideal oscillator. The Allan variance is de¯ned as ¾ 2 (¿)= Ef[Á(t +2¿)¡Á(t+¿)]¡[Á(t +¿)¡Á(t)]g 2 (2¿) 2 (2¼f 0 ) 2 = D (2) Á (¿) 4(2¼f 0 ¿) 2 ; (2.16) whereD (2) Á (¿) is called the second order structure function of the phase noise process Á(t). We present a phase noise model that will show the random walk behavior for oscillators. 17 Supposewearegivenapairofstochasticdi®erentialequationsgoverningthedynamics of the amplitude ½(t) and total phase µ(t), _ ½ = a 1 (½;µ)+b 1 (½;µ)F(t) (2.17) _ µ = a 2 (½;µ)+b 2 (½;µ)F(t); (2.18) where a 1 ;a 2 ;b 1 ;b 2 are coe±cients that are functions of µ and ½, and F(t) is the internal oscillator white noise process. Let z(t) = (µ(t);½(t)) T be a two dimensional process rep- resenting the phase and amplitude noise processes that are sample-continuous. We de¯ne the increment of the noise process or the mean square unpredictable part of the increment dz(t) as du(t)=dz(t)¡E[dz(t)jz T (t 0 );0·t 0 ·t] (2.19) where u(t) = (u 1 (t);u 2 (t)) T . Let us de¯ne the increment of the phase noise process and the increment of the amplitude noise process as the following dÁ(t)=dµ(t)¡E[dµ(t)jµ(t 0 );½(t 0 );0·t 0 ·t] =µ(t +dt)¡E[µ(t +dt)jµ(t 0 );½(t 0 );0·t 0 ·t] ¯ ¯ dt!0 (2.20) d®(t)=d½(t)¡E[d½(t)jµ(t 0 );½(t 0 );0·t 0 ·t] =½(t +dt)¡E[½(t +dt)jµ(t 0 );½(t 0 );0·t 0 ·t] ¯ ¯ dt!0 ; (2.21) 18 where Á(t) is the phase noise process and ®(t) is the amplitude noise process. By letting dv(t)=E[dz(t)jz T (t 0 );0·t 0 ·t]wherev(t)=(v 1 (t);v 2 (t)) T ,wecanobtainthefollowing dv 1 (t) = a 1 (µ;½)dt dv 2 (t) = a 2 (µ;½)dt (2.22) due to the fact that E[F(t)dtjz T (t 0 );0· t 0 · t] = 0 and dW(t) = F(t)dt is independent of the coe±cient b 1 ;b 2 where W(t) is the Wiener process. We therefore get the increment of the noise process as du(t) = dz(t)¡dv(t) and the increment of the original processes dz(t) = du(t)+dv(t). Now we partition the time interval [0;t] by n+2 points and set t 0 =0;t n+1 =t, and sum the following dz(t k )=du(t k )+dv(t k ) to get z(t)¡z(0)= n X k=0 dv(t k )+ n X k=0 du(t k )=v(t)+u(t): (2.23) Since z(t) and v(t) are sample-continuous processes, u(t) is continuous as well and v(t) is predictable with respect to the process z(t). The predictable part of the random process z(t) is v(t)= n X k=0 dv(t k )= Z t 0 ad¿; (2.24) 19 where the vector a = (a 1 (µ;½);a 2 (µ;½)) T . We have v(0) = 0 and from equation (2.23), we have u(0) = 0. We can verify the results that E[du(t)jz(t);0 · t 0 · t] = 0. For t k ¸t 0 ¸0, we take the conditional expectation on du 2 (t k ) and obtain E[du 2 (t k )jz(t 0 );0·t 0 ·t 0 ]=EfE[du 2 (t k )jz(t 0 );0·t 0 ·t k ]jz(t 0 );0·t 0 ·t 0 g =E[0jz(t 0 );0·t 0 ·t 0 ]=0: (2.25) We now consider a partition t 0 < t 1 < ¢¢¢ < t n+1 = t of interval [t 0 ;t], then u 2 (t) = P n k=0 du 2 (t k )+u 2 (t 0 ). We take the conditional expectation on u 2 (t) and obtain E[u 2 (t)jz(t 0 );0·t 0 ·t 0 ]=E[u 2 (t 0 )jz(t 0 );0·t 0 ·t 0 ]=u 2 (t 0 ); (2.26) since u 2 (t 0 ) is completely determined by the process z(t 0 );0 · t 0 · t 0 . When we take expectation on equation (2.26) we get for t 0 = 0, E[u 2 (t)] = E[u 2 (0)] = 0 and u 2 (t) is a zero mean process. Since u 2 (t) is completely determined by z(t 0 );0 · t 0 · t, by the smoothing property of the conditional expectation applied to equation (2.26), we obtain EfE[u 2 (t)jz(t 0 );0·t 0 ·t 0 ]ju 2 (t 0 );0·t 0 ·t 0 g =E[u 2 (t)ju 2 (t 0 );0·t 0 ·t 0 ]=E[u 2 (t 0 )ju 2 (t 0 );0·t 0 ·t 0 ]=u 2 (t 0 ) (2.27) for 0·t 0 ·t. Thus, u 2 (t);t¸ 0 is a zero-mean martingale. We can also show that u 1 (t) us a zero-mean martingale. We therefore have the Doob-Meyer-Fisk decomposition [27] of z(t) as z(t)¡z(0)=v(t)+u(t) from equation (2.23) such that v(t) is a sample-continuous process predictable with respect to z(t), and u(t) is a sample-continuous martingale with 20 u(0) = 0. With this in mind, we get both phase noise and amplitude noise processes u(t) as the following u(t)= Z t 0 b(µ(¿);½(¿))dW(¿); (2.28) where b = (b 1 ;b 2 ) T and the phase noise process Á(t) = u 2 (t) = R t 0 b 2 (µ(¿);½(¿))dW(¿). Therefore, with this proposed model for phase noise Á(t), we can see that the oscillator phase noise is a transformation of the Wiener process, thus possessing a random walk behavior. 21 Chapter 3 Theoretical Model Introduction and Analysis 3.1 Theoretical Model We propose a theoretical model of a voltage controlled oscillator based on physical dy- namics with noise ( Fig. 3.1). This voltage controlled oscillator can be described by the following stochastic di®erential equation, which includes the internal noise, F(t), of the clock and the tracking loop noise plus the controller noise, n(t). Ä y¡ K vco (_ v+ _ n(t)) ! 0 +K vco (v+n(t)) _ y+[! 0 +K vco (v+n(t))] 2 y =c(t)F(t) (3.1) where we assume that F(t) is a white Gaussian noise, which is independent of n(t). In addition, n(t)2C 1 ([0;1)), y(t) is the oscillator generating waveform, v(t) represents the controlled voltage, c(t) is a scale factor, K vco is the VCO gain, and ! 0 is the clock rest frequency. The output of the proposed model after the hard-limiter is Z(t)=sgn(y(t)): (3.2) 22 x + ∫ ∫ -1 ( ) 2 x + 0 ω dt d K vco + ) (t y 2 pole filter ) (t Z ) ( ) ( t F t c b a a b ) ( ) ( t n t v + + ) ( ~ t v ) (t W Figure 3.1: Theoretical model of voltage controlled oscillator with thermal noise and noise from the controlled voltage and tracking loop. The dash box in Fig. 3.1 encloses the research work that will be proposed and discussed in Section 3.4, where ~ v(t) is the input signal from the tracking loop of a phase-locked loop (PLL), and W(t) is the un¯ltered noise from the tracking loop and the controller. 3.2 Model Without Noise First, we assume that v(t) is known and the noise terms n(t), F(t) are not present. The initial conditions for y(t), v(t) are such that y(0)=0, v(0)=0. The stochastic di®erential equation then becomes an ordinary di®erential equation. Ä y¡ K vco _ v ! 0 +K vco v _ y+[! 0 +K vco v] 2 y =0 (3.3) 23 Two cases are investigated. The ¯rst case is that v(t) is a constant a when the steady state of the system is reached, i.e., when the error input voltage to the VCO is a constant. The equation (3.3) can then be solved to be y(t)=Asin(! new t) (3.4) where ! new = ! 0 +K vco a, and A is any constant. For the case when v(t) is any function other than a constant, equation (3.3) becomes a di®erential equation with time-varying coe±cients. Suppose that v(t) can be expressed as a ramp function v(t)= 8 > > < > > : at if t·t s C if t>t s (3.5) where a, C are constants, and t s is the time of steady state. Here no noise contribution is considered, equation (3.3) therefore becomes Ä y¡ K vco a ! 0 +K vco at _ y+[! 0 +K vco at] 2 y =0 (3.6) for t·t s . Let y 1 =y; y 2 = _ y 1 , a state space equation can therefore be written as _ Y =A(t)Y (3.7) where Y = 2 6 6 4 y 1 y 2 3 7 7 5 ; A(t)= 2 6 6 4 0 1 ¡(! 0 +K vco at) 2 K vco a ! 0 +Kvcoat 3 7 7 5 : 24 The above di®erential equation can be solved in various ways. By the existence and uniqueness theorems [4], the elements of matrix A(t) in (3.7) are continuous on an open interval 0 · t < t s , containing the initial point t = t 0 (e.g., t 0 = 0), then there exists a unique solution, y 1 =Á 1 (t); y 2 =Á 2 (t), of the system of di®erential equations (3.7). This set of solutions also satis¯es the initial conditions, y(0) = 0; v(0) = 0. Furthermore, if the vector functions y (1) ; y (2) are solutions of the system (3.7), then by the superposition principle, any linear combination c 1 y (1) + c 2 y (2) is also a solution for any constants c 1 and c 2 . From the state space equation (3.7), the solution is found having the following form Y =c 1 y (1) (t)+c 2 y (2) (t): (3.8) The solution can be found such thaty (1) (t) andy (2) (t) are linearly independent. They are y (1) (t)= 0 B B @ sin(! 0 t+ R t 0 K vco a¿d¿) (! 0 +K vco at)cos(! 0 t+ R t 0 K vco a¿d¿) 1 C C A ; y (2) (t)= 0 B B @ ¡cos(! 0 t+ R t 0 K vco a¿d¿) (! 0 +K vco at)sin(! 0 t+ R t 0 K vco a¿d¿) 1 C C A : (3.9) It can be shown that they satisfy equation (3.7). The Wronskian of y (1) and y (2) is greater than zero for t ¸ 0. Therefore, the solution exists without discontinuity. For simplicity, when c 1 =c 2 =1, the solution that is veri¯ed to be periodic in the limit is y 1 (t)=sin(! 0 t+ Z t 0 K vco v(¿)d¿): (3.10) 25 ³ A(t, v(t),n(t)) + D Y ) ( ~ t F Figure 3.2: Equivalent model of VCO (Fig. 3.1) with noise contributions, e F(t), scaled internal noise, and n(t), controller plus tracking loop noise. A numerical method is used to demonstrate the behavior of the solution for equation (3.7). It has been veri¯ed that the numerical solution agrees with the solution derived. As a result, for any other controlled voltage waveform with a steady state value in the limit, the solution becomes periodic in the limit. 3.3 Model With Noise In general, the proposed model shown in Fig. 3.1 can be represented by the equivalent model shown in Fig. 3.2. Two cases are considered in the following two sections. One occurs when the internal noise, F(t), of the clock is present while n(t) is excluded from the calculation. A stochastic di®erential equation is thus obtained, _ Y =A(t)Y+D e F(t) ; (3.11) where A(t)= 2 6 6 4 0 1 ¡(! 0 +K vco v) 2 K vco _ v ! 0 +Kvcov 3 7 7 5 ; D= 2 6 6 4 0 1 3 7 7 5 ; e F(t)=c(t)F(t) : 26 The equation (3.11) can be solved using It^ o integral. We considered the case when the controlled voltage v(t)=a, where a is a constant. Similarly, it can be extended to the case of time-varying v(t) as well. The solution to equation (3.11) with initial conditions that y 1 (0)=0; y 2 (0)= ^ a! new , where ^ a2R is y 1 (t)= ^ asin(! new t)+ Z t 0 c(s)sin(! new (t¡s)) ! new dB s (3.12) wherec(s)isascalefactorforthenoise,F(t),andB s isa1-dimensionalBrownianmotion. We can approximate the It^ o integral, I [f](!)= Z T S f(t;!)dB t (!) ; where B t is 1-dimensional Brownian motion, by I[Á] for a class of functions Á. Then for each f 2V, where V is de¯ned in [26], as the class of functions f(t;!) : [0;1)£!R such that the following properties are satis¯ed: 1. (t;!)!f(t;!) isB£F-measurable, whereB denotes the Borel ¾-algebra on [0;1). 2. f(t;!) is Ft -adapted. 3. E[ R T S f(t;!) 2 dt]<1. The function f can be approximated by such Á's and we use this to de¯ne Z fdB = lim Á!f Z ÁdB : 27 Therefore, we can approximate the above integral using elementary functions, Á(t;!)= X j e j (!)¢Â [t j ;t j+1 ) (t) ; where Â denotes the characteristic (indicator) function, and de¯ne the integral as Z T S Á(t;!)dB t (!)= X j¸0 e j (!)[B t j+1 ¡B t j ](!) : The It^ o integral in equation (3.12) can be approximated by t X s i =0 c sin(! new (t¡s i )) ! new ¢B s i ; (3.13) where s i =ih, i2fZ + [0g, h is the time step, and ¢B s i =B s i+1 ¡B s i . 3.3.1 Noise Analysis Without loss of generality, let ^ a=1, then equation (3.12) can then be expressed as y 1 (t)=sin! new t+n 1 (t)sin! new t¡n 2 (t)cos! new t (3.14) where n 1 (t)=K Z t 0 cos! new sdB s ; n 2 (t)=K Z t 0 sin! new sdB s ; 28 and K = c ! new for c(s)=c. Using complex notation we can write for equation (3.14) y 1 (t) = Ref¡j[(1+n 1 (t))¡jn 2 (t)] e j(!newt) g = Ref¡j q (1+n 1 (t)) 2 +n 2 2 (t) e j(!newt+'(t)) g = q (1+n 1 (t)) 2 +n 2 2 (t)sin(! new t+'(t)) (3.15) with '(t)=tan ¡1 µ n 2 (t) 1+n 1 (t) ¶ ; (3.16) where '(t) is the oscillator phase noise contributed by the scaled internal noise, e F(t), of the oscillator. 3.3.2 Noise Processes The noise processes, n 1 (t), n 2 (t), have the following statistical properties. Both noise processes are zero mean, Gaussian distributed, and non-stationary. The correlation func- tions for n 1 (t) and n 2 (t) are found to be R n 1 (t;s) = E[n 1 (t)n 1 (s)] = c 2 2! 2 new · min(s;t)+ sin(2! new min(s;t)) 2! new ¸ ; R n 2 (t;s) = E[n 2 (t)n 2 (s)] = c 2 2! 2 new · min(s;t)¡ sin(2! new min(s;t)) 2! new ¸ ; 29 and the cross-correlation function is found to be R n 1 ;n 2 (t;s) = E[n 1 (t)n 2 (s)] = c 2 4! 3 new [1¡cos(2! new min(s;t))] : In addition, _ n 1 (t) is found to be wide-sense cyclo-stationary, similarly, so is _ n 2 (t). When the 2 £ frequency terms in the correlation functions are ¯ltered, _ n 1 (t), _ n 2 (t) are strict sense stationary. Furthermore, the ¯rst increment process of n 1 (t), n 2 (t), are found to be non-stationary. If we denote the noise e®ect e F(t) on the VCO by N e F (t)= Z t 0 c(s) sin(! new (t¡s)) ! new dB s ; (3.17) the variance of N e F (t) is found to be ¾ 2 N e F = ¾ 2 c 2 4! 2 new · 2t¡ sin(2! new t) ! new ¸ (3.18) where ¾ 2 is the noise power of F(t). This noise generates the amplitude and phase noise of a VCO before the hard-limiter. 30 3.3.3 Phase Noise Process The phase noise process (3.16) induced by the scaled internal noise, e F(t), can be approxi- mated by the following equation '(t)¼ n 2 (t) 1+n 1 (t) ¼n 2 (t)(1¡n 1 (t)) : (3.19) It has been shown that this is a valid approximation by simulation for low noise level. Furthermore, the phase noise '(t) has the following statistical properties. The mean is given by E['(t)] ¼ E[n 2 (t)¡n 2 (t)n 1 (t)] = c 2 4! 3 new [cos(2! new t)¡1] (3.20) with the correlation function R ' (t;s)=E['(t)'(s)]¼R n 2 (t;s) (3.21) when the time step ¢t !0 from the derivation by a stochastic integral [26]. The phase noise process '(t) can also be approximated by n 2 (t) for low noise level and it is shown as a valid assumption by simulation. In this case, the phase noise process '(t) has the same statistical property as n 2 (t), meaning it is Gaussian distributed and _ '(t) is a wide-sense stationary process. 31 3.4 Model With External Noise n(t) Present The original proposed VCO model (3.1) has an additional noise contribution, n(t), from the tracking loop of the phase-locked loop (PLL) plus the controller. In addition, n(t)2 C 2 ([0;1)), and a two-pole ¯lter is proposed to constrain the tracking loop noise due to the di®erentiator as shown in Fig. 3.1. We considered a two pole ¯lter when n(t) is present (shown in Fig. 3.3), where W(t) = W contr (t) + W loop (t) is white assuming noise processes W contr (t) and W loop (t) are independent. 2 pole filter ) (t h ) (t W dt d ) (t n ! ) (t n Figure 3.3: General scheme for the tracking loop plus the controller noise, n(t) The transfer function of a general 2-pole ¯lter is H(s)j s=j! = ! 2 n s 2 +2³! n s+! 2 n ; where ³ is a damping factor, and ! n ¯lter natural frequency. Assuming that the averaged control output voltage after the ¯lter v(t) is equal to a constant, a, the voltage controller is therefore corrupted by the ¯ltered noise, n(t). 32 The ¯ltered noise n(t) can be expressed as the following state equations. Let x 1 (t) = n(t) ! 2 n and x 2 (t) = _ x 1 (t) under the condition that n(t) 2 C 2 ([0;1)), the state equation is shown as _ X= e AX+ e DW(t) (3.22) where e A= 2 6 6 4 0 1 ¡! 2 n ¡2³! n 3 7 7 5 ; e D= 2 6 6 4 0 1 3 7 7 5 : Analytical solution to the above equation (3.22) can be solved using a stochastic inte- gral. It has the following form X(t)=exp( e At)X(0)+ Z t 0 exp( e A(t-s)) e DdB s ; (3.23) where dB s is the increment of a one-dimensional Brownian motion B t . Assuming zero initial conditions for X(t), the correlation function of the output noise n(t) can be found to be R n (t;¿)=E[n(t)n(t+¿)]= Z t 0 ¾ 2 h(l)h(l+¿)dl ; (3.24) whereh(t) is the impulse response of the ¯lter, ¾ 2 is the input noise power of W(t). When t!1, the process n(t) becomes stationary. 33 The analytical solution to the VCO model (3.11) when e F(t), n(t) are present and v(t)=at8 t with initial conditions of y 1 (0)=0, y 2 (0)= ^ a! 0 , where ^ a2R is found using the fundamental matrix solution to be y 1 (t)= ^ asin© new (t)+ Z t 0 c(s)sin ^ b(s;t) ! 0 +K vco (as+n(s)) dB s (3.25) where © new (t) = ! 0 t+ Z t 0 K vco a¿d¿ + Z t 0 K vco n(¿)d¿ ; ^ b(s;t) = ! 0 (t¡s)+ Z t s K vco a¿d¿ + Z t s K vco n(¿)d¿: 3.4.1 Noise Analysis Without loss of generality, let ^ a=1, we can rewrite equation (3.25) as y 1 (t)=sin© new (t)+~ n 1 (t)sin© new (t)¡~ n 2 (t)cos© new (t) (3.26) where ~ n 1 (t) = Z t 0 ~ K(s;n(s))cos© new (s)dB s ; ~ n 2 (t) = Z t 0 ~ K(s;n(s))sin© new (s)dB s ; (3.27) 34 and ~ K(s;n(s)) = c ! 0 +K vco (as+n(s)) for c(s) = c. Again, using complex notation we can write for equation (3.26) as y 1 (t)= q [1+~ n 1 (t)] 2 +~ n 2 2 (t)sin(© new (t)+ ~ '(t)) (3.28) where ~ '(t)=tan ¡1 µ ~ n 2 (t) 1+~ n 1 (t) ¶ : (3.29) We have after the hard limiter as shown in Fig. 3.1, z(t)=sgn(y 1 (t))=sgn[sin(© new (t)+ ~ '(t))] where © new (t)+ ~ '(t) is the total oscillator phase with phase noises caused by n(t) and e F(t). 3.4.2 Noise Processes The noise processes, ~ n 1 (t), ~ n 2 (t), have the following statistical properties. Both noise processes are zero mean and non-stationary. The correlation functions for ~ n 1 (t) and ~ n 2 (t) are found to be R ~ n 1 (t;s) = E[~ n 1 (t)~ n 1 (s)] = 1 2 ½Z min(s;t) 0 E[ ~ K 2 (z;n(z))+cos(2© new (z))]dz ¾ ; R ~ n 2 (t;s) = E[~ n 2 (t)~ n 2 (s)] = 1 2 ½Z min(s;t) 0 E[ ~ K 2 (z;n(z))¡cos(2© new (z))]dz ¾ ; 35 and the cross-correlation function is found to be R ~ n 1 ;~ n 2 (t;s) = E[~ n 1 (t)~ n 2 (s)] = 1 2 Z min(s;t) 0 E[ ~ K 2 (z;n(z))sin(2© new (z))]dz : To evaluate these functions are di±cult, we will show the simulation results in Section 3.5 using the statistical properties of the noise processes ~ n 1 (t), ~ n 2 (t). 3.4.3 Phase Noise Processes For the tracking loop plus controller noise n(t), the phase noise generated is denoted by ' n (t)= Z t 0 K vco n(z)dz: For phase noise generated by internal noise, e F(t), of the VCO, ~ '(t) from equation (3.29) can be approximated by the following equation for small noise level ~ '(t)¼ ~ n 2 (t) 1+~ n 1 (t) ¼ ~ n 2 (t)(1¡~ n 1 (t)) : (3.30) Furthermore, the phase noise ~ '(t) has the following statistical properties. The mean is given by E[~ '(t)]¼E[~ n 2 (t)¡~ n 2 (t)n 1 (t)] =¡ 1 2 Z min(s;t) 0 E[ ~ K 2 (z;n(z))sin(2© new (z))]dz (3.31) 36 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 5 −4 −2 0 2 4 6 x 10 −7 cycle numbers t normalized upward zero−crossing − t ideal upward zero−crossing −4 −3 −2 −1 0 1 2 3 4 5 x 10 −7 0 2000 4000 6000 8000 10000 12000 time (sec.) counts Figure 3.4: Averaged upward zero-crossing jitter for a VCO with 2 kHz clock frequency with noise power of ¾ 2 e F =1 with the correlation function R ~ ' (t;s)=E[~ '(t)~ '(s)]¼R ~ n 2 (t;s) (3.32) when the time step ¢t ! 0 from the derivation by a stochastic integral [26]. The evaluation of these functions are simulated in the following section. 3.5 Simulation Results and Timing Jitter Estimate Weperformed a simulationontheproposedmathematical VCOmodel and weveri¯edthe analytical solutions and noise statistics as well by simulation. The simulation is done for two cases, 1) when the internal noise of the clock, e F(t), is present, 2) when the tracking 37 0 0.5 1 1.5 2 x 10 5 4.9998 4.9998 4.9999 5 5 5 5.0001 5.0001 5.0002 x 10 −4 cycle−to−cycle jitter number t upward zero−crossing (n+1) − t upward zero−crossing (n) 4.9998 4.9999 5 5.0001 5.0002 x 10 −4 0 200 400 600 800 1000 1200 1400 1600 time (sec.) counts Figure 3.5: Cycle-to-cycle jitter statistics for 100 seconds plus the controller noise, n(t), is present as well. It is simulated at a VCO frequency of 2 kHz with noise power ¾ 2 e F = 1, the scale factor c(t) = 1, v(t) = 1, and K vco = 200¼ rad/V for 100 seconds. We obtained the averaged upward zero-crossing jitter statistics shown in Fig. 3.4, which indicates that the timing jitter has a random walk behavior with restoring force. Statistics from simulations also show that the timing jitter has a normal distribution at di®erent times with a di®usion-like variance. In addition to the timing jitter, we are interested in the cycle-to-cycle jitter statistics as shown in Fig. 3.5. We performed the chi-squared goodness of ¯t test as a normality test for the test statistics. We obtained a p value of 0.73537 at 95% con¯dence interval such that we accept the hypothesis that it is normally distributed. We then performed the parametric Pearson correlation test on the test statistics. The results indicate that for p value < 0.05 at 95% con¯dence interval, we found that samples of test statistics are uncorrelated. We then conclude that they are independent since they are normal and uncorrelated. 38 0 0.5 1 1.5 2 x 10 5 −6 −4 −2 0 2 4 6 8 10 12 14 x 10 −5 cycle numbers t normalized upward zero−crossing − t ideal upward zero−crossing −5 0 5 10 15 x 10 −5 0 0.5 1 1.5 2 2.5 x 10 4 time (sec.) counts Figure 3.6: Timing jitters when e F(t) and n(t) are present When both noise sources e F(t) and n(t) are considered in the simulation, we see that at some time in the future, the timing jitter will drift away as shown in Fig. 3.6. It is caused by the tracking loop plus controller noise n(t). The simulation was done for 100 seconds at ¾ 2 W = 0:1 with the Butterworth ¯lter having ³ =1:25 and ! n =2. A power spectral density of the oscillator output y(t) before the hard-limiter shown in Fig. 3.7indicatesthee®ectoftheoscillatorinternalnoise F(t)atdi®erentlevels. Wethen performed a timing jitter estimate based on the Markov property of the phase noise. We denote the total phase as © tot (t)=© new (t)+ ~ '(t)=f(' n (t); ~ '(t)), where f(¢) is any con- tinuousfunction. ItisaMarkovprocesssince' n (t)and ~ '(t)arehiddenMarkovprocesses. We have y 1 (t) = g(© tot (t)) = sin(© tot (t)) as a Markov process as well, where g(¢) is any function. We then take samples of © tot (t) as © tot (t rise (k))=2¼k =h k , where k2Z + . We obtain the normalized upward zero-crossing time estimate as ¢ ^ t(k) = E[¢t(k)j© tot (t)j T 0 ] where ¢t(k)=t rise (k)¡ k fnew and T is the stopping time. 39 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 −140 −120 −100 −80 −60 −40 −20 0 frequency (Hz) dB σ F 2 =1000 σ F 2 =1 Figure3.7: Powerspectraldensityoftheproposedmodel'soutput y(t)withdi®erentnoise levels of F(t) 3.6 Summary This chapter describes a novel mathematical model of a voltage controlled oscillator based on physical dynamic with noise. The e®ects of noise on the proposed model are analyzed and the resulting phase noise is investigated. Analytical forms of the VCO with noise are obtained using a stochastic integral. Moreover, a two-pole ¯lter is introduced for the constraint of tracking loop noise. Simulation of the model is done to further verify the analytic solutions and noise statistics. Analysis of the resulting phase noise along with the simulation suggest that the timing jitter process has a random walk behavior with 40 restoring force and the upward zero crossing jitter is normal distributed. The increment of the timing jitter process, cycle-to-cycle jitter, is shown to behave as normal distributed and independent. We can conclude that from the results obtained, that the tracking loop plus the controller noise n(t) cause the oscillator phase to drift while the internal noise e F(t) tends to cause di®usion on the oscillator phase. 41 Chapter 4 Analysis of Robinson's oscillator 4.1 Existence and Stability of a Solution Let's consider Robinson's oscillator [3] (an electronic oscillator) which is the main interest ofresearchonnon-linear oscillators. Theequationgoverningtheoscillatorsisgivenbythe following, Ä x+G 1 _ x¡G 2 sgn_ x+! 2 0 x=0 ; (4.1) where G 1 = G C , G 2 = ¼mK 0 4C are positive constant gains, described by circuit parameters, C, capacitance of the tank circuit, G = 1 R , shunt conductance of the tank, m = C 1 C 2 , capacitance ratio of the oscillator circuit, and K 0 , a gain constant, and ! 0 is the clock rest frequency. We start by rewriting the equation (4.1) using the state space method as the following _ Z= 2 6 6 4 _ z 1 _ z 2 3 7 7 5 = 2 6 6 4 f 1 (Z) f 2 (Z) 3 7 7 5 ; (4.2) 42 x z = 1 x z & = 2 2 G 2 G 2 G 2 G 2 1 2 0 G z − ω 2 1 2 0 G z + ω 2 1 2 0 G z + −ω 2 1 2 0 G z − −ω Figure 4.1: Motions of physical dynamics of Robinson's oscillator at di®erent points of the phase plane diagram where z 1 = x, z 2 = _ x, f 1 (Z) =z 2 , f 2 (Z) =¡G 1 z 2 + G 2 sgnz 2 ¡ ! 2 0 z 1 . We investigate the dynamics of the oscillator by looking at the phase plane diagram. By analyzing the equation(4.2),westartbylookingatthee®ectsoftwoextremes. Whenz 1 andz 2 areboth small, the equation (4.2) governing the dynamics of motions can be seen to exist either a upward or downward motion depending on the sign of the sgn function. Moreover, at the other extreme, when z 1 is large and z 2 is small, the e®ect on the dynamics of motions is a downwardforcethathasdi®erentmagnitudesdependingofthesgnfunctionoftheequation (4.2). ThisisdescribedonthephaseplanediagramasshowninFig. 4.1, wherethearrows represent the directions of the motion °ows, along with the magnitudes of forces at these speci¯ed points. We ¯nd the equilibrium points of the system at the following conditions, _ x=z 2 =0 and _ z 2 =0, resulting an equilibrium point at z 2 =0, and z 1 =x= G 2 ! 2 0 . 43 We next discuss the motions of initial points shown in Fig. 4.1. The motions of these phase paths can be described by the the energy change of the system. For oscillators that are based on the second-order di®erential equation of the following form Ä x+h(x; _ x)+g(x)=0 ; (4.3) it is convenient to use the phase plane representation. The energy E for equation (4.3) is the total energy de¯ned as the sum of potential and kinetic energy [21], E = 1 2 _ x 2 + Z g(x)dx: (4.4) To observe how energy E changes as a particular motion progresses along a phase path, we de¯ne the following [21] E (t)¡ E (t 0 )=¡ Z t t 0 _ xh(x; _ x)d¿ ; (4.5) where t 0 is the initial starting time. It may be possible to say, for a phase path lying in some region of the phase plane, the following two cases can occur. 1. _ xh(x; _ x)>0, in which case E (t)< E (t 0 ), the energy decreases and h has a damping e®ect, contributing to a general decrease in amplitude. 2. _ xh(x; _ x) < 0, in which case E (t) > E (t 0 ), the e®ect is of an internal source injecting energy into the system. 44 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 z 1 z 2 /ω 0 Figure 4.2: Motions of phase paths at di®erent initial points on the phase plane diagram Therefore, from Fig. 4.1, the trajectory of the vector (z 1 ;z 2 ) at these speci¯ed points suggests oscillations on the phase plane with negative oscillating phase assuming the ex- istence of the system's solution. At small initial points, the energy will inject into the system, causing the overall increase in oscillation amplitude. At some large initial points on the phase plane diagram, the amplitude of oscillation may either increase or decrease dependingon theamplitudeofthesystem'ssolution. Weshowthe°owofmotionsastime goes on in Fig. 4.2 at di®erent initial points. 4.1.1 Existence of a Solution The next question arises if there exists any stable solution to the above system. We again use the energy E change property from equation (4.5) to determine the motion of 45 the phase paths and ¯nd a possible stable solution. Denote yh(x;y) = z 2 h(z 1 ;z 2 ) = G 1 z 2 2 ¡G 2 z 2 sgnz 2 , two cases are investigated. 1. when sgn z 2 = 1, z 2 h(z 1 ;z 2 ) = G 1 z 2 2 ¡G 2 z 2 . If z 2 > G 2 G 1 , z 2 h(z 1 ;z 2 ) > 0, then the amplitude of the solution decreases. If z 2 < G 2 G 1 , z 2 h(z 1 ;z 2 )< 0, then the amplitude of the solution increases. 2. when sgn z 2 =¡1, z 2 h(z 1 ;z 2 )=G 1 z 2 2 +G 2 z 2 . If z 2 <¡ G 2 G 1 , z 2 h(z 1 ;z 2 )>0, then the amplitudeofthesolutiondecreases. Ifz 2 >¡ G 2 G 1 ,z 2 h(z 1 ;z 2 )<0,thentheamplitude of the solution increases. Therefore, there exists a limit cycle from the above conditions. We are interested in the existence of such solution and we derive the following approximate solution to the system of equation 4.2. Assuming that x = z 1 = acos! 0 t, _ x = z 2 =¡a! 0 sin! 0 t, for the solution on a limit cycle, the energy balance equation has to be satis¯ed E (2¼)¡ E (0)=¡ Z 2¼ 0 z 2 h(z 1 ;z 2 )dµ =0 (4.6) where µ = ! 0 t. We ¯nd the amplitude a of the approximate solution to be 4G 2 ¼G 1 ! 0 from equation 4.6, making the approximate solution to be x(t) = acos(! 0 t+µ 0 ). Therefore, there exists a sinusoidal solution to the system of equation (4.2). 46 −1 −0.5 0 0.5 1 −6000 −4000 −2000 0 2000 4000 6000 z 1 z 2 Figure 4.3: Phase plane representation of a Robinson oscillator with G 1 = 1000, G 2 = ¼! 0 G 1 4 , and ! 0 =2000¼ A su±cient condition for stability of the limit cycle is when the energy equation E (2¼)¡ E (0)=¡ Z 2¼ 0 z 2 h(z 1 ;z 2 )dµ =g(a) is such that g 0 (a 0 ) < 0, where g(a 0 ) = 0. We therefore show that in our problem that g 0 (a 0 )=¡4G 2 <0 for G 2 >0, the limit cycle is shown to be stable. A more general proof of stability follows later. The phase plane representation of Robinson's oscillator further shows the existence of the limit cycle. As seen in Fig. 4.3, a limit cycle with z 1 's amplitude of 1 exists and the approximate solution is cos(! 0 t+µ 0 ). We will discuss the e®ects (damping e®ects) of the gains G 1 , G 2 , on the oscillator model in Section 4.3. 47 Again we get another approximation to the system equation (4.2) by taking the ¯rst term of the non-linear signum function and ignore the higher order terms (H.O.T.s), we obtain the followings. We assume that z 1 (t) = x(t) = acos! 0 t, z 2 (t) =¡a! 0 sin! 0 t, and we obtain the expression for the non-linear signum function to be sgnz 2 = sgn(¡a! 0 sin! 0 t)=¡ 4 ¼ 1 X k=0 sin[(2k+1)! 0 t] 2k+1 = ¡ 4 ¼ sin! 0 t+H.O.T.s : We choose the ¯rst term and if we could neglect the higher order terms, and substitute back to the equation (4.1), we obtain the following equation Ä x+(G 1 ¡ 4G 2 ¼a! 0 )_ x+! 2 0 x=0 : When G 1 6= 4G 2 ¼a! 0 , we denote k =G 1 ¡ 4G 2 ¼a! 0 , and we solve the equation Ä x+k_ x+! 2 0 x =0 to have the following solution x(t)=Aexp( ¡kt 2 )cos[ p ¡¢t 2 +®] ; (4.7) where ¢ = k 2 ¡4! 2 0 . When k = 0, we return to the original approximate solution. For k6=0, the overall frequency of oscillation is o®set by k, the amplitude of oscillation decays or grows exponentially depending on the sign of k. 48 4.1.2 Stability of a Solution For a formal proof of the stability of a solution to the system equation (4.2), we use Lya- punovfunctions. Thedemonstrationofstabilityorinstabilityrequires¯ndingaLyapunov function for that system [19], [29]. We ¯rst de¯ne what a Lyapunov function is and show how we determine the stability of the solution. De¯nition 4.1 Let V(x) be a scalar function of the n components, x 1 ;x 2 ;:::;x n of x. V(x)ispositive(negative)de¯niteinaneighborhoodLoftheoriginifV(x)>0(V(x)<0) forallx6=0inL,andV(0)=0. V(x)ispositive(negative)semi-de¯niteinaneighborhood L of the origin if V(x)¸ 0 (V(x)· 0) for all x6= 0 in L, and V(0)=0. De¯nition 4.2 The total derivative of a function V(x) along a pathP of the autonomous system x=F(x) corresponding to a solution x(t) is ( dV dt ) P = d dt V[x(t)]= _ V(x)= n X i=1 @V @x i F i (x) : De¯nition 4.3 A function V(x) that is of class C 1 and satis¯es V(0) = 0 is called a Lyapunov function if every open ball B ± (0) contains at least one point where V > 0. If there happens to exist ± ¤ such that the function _ V de¯ned by De¯nition 4:2 is positive de¯nite in B ± ¤(0), then the origin is an unstable critical point of the system. Theorem 4.1 (Lyapunov second theorem on stability [29]) Consider a function V(x):R n !R such that 1. V(x)¸0 with equality if and only if x=0 (positive de¯nite) 2. _ V(x)<0 (negative de¯nite) 49 then the system is asymptotically stable in the sense of Lyapunov. WhentheconditionV(x)!1asx!1issatis¯edinadditiontothesu±cientconditions inTheorem4.1,theglobalstability(stabilityeverywhere)fornon-linearsystemsisensured. Theorem 4.2 The solution of Robinson's oscillator given by the equation (4:1) is stable. Proof Forthesystemequationgivenbyequation(4.2),weneedto¯rstlookforaLya- punovfunctionV(z 1 ;z 2 )suchthatV(z 1 ;z 2 )!1asz 2 1 +z 2 2 !1, and d dt V(z 1 (t);z 2 (t))< 0. We choose the Lynapunov function, V(z 1 ;z 2 ) = 1 2 (! 2 0 z 2 1 +z 2 2 ), therefore, V(z 1 ;z 2 ) is positive de¯nite. We obtain the rate of the Lyapunov function as the following _ V(z 1 ;z 2 )=¡G 1 z 2 2 +G 2 z 2 sgnz 2 : (4.8) Two cases to the above equation are investigated. 1. when sgn z 2 =1, z 2 >0, _ V(z 1 ;z 2 )=z 2 (¡G 1 z 2 +G 2 ), for _ V(z 1 ;z 2 )<0, we need the condition z 2 > G 2 G 1 to be true. 2. when sgn z 2 =¡1, z 2 < 0, _ V(z 1 ;z 2 ) =¡z 2 (G 1 z 2 +G 2 ), for _ V(z 1 ;z 2 ) < 0 we need the condition z 2 <¡ G 2 G 1 to be true. Therefore, we need some conditions on z 2 to make _ V(z 1 ;z 2 ) < 0, and the condition is jz 2 j> G 2 G 1 to achieve stability. Secondly,weneedtolookforanotherLyapunovfunctionW(z 1 ;z 2 )suchthatW(z 1 ;z 2 )! 1 when (z 1 ;z 2 ) ! (0;0), and d dt W(z 1 (t);z 2 (t)) < 0 for z 2 1 (t) + z 2 (t) 2 < ± 2 where 50 ± is some small number approaching zero. Let's de¯ne the Lyapunov function to be W(z 1 ;z 2 )= 1 2(! 2 0 z 2 1 +z 2 2 ) , we ¯nd that _ W(z 1 ;z 2 )= G 1 z 2 2 ¡G 2 z 2 sgnz 2 (! 2 0 z 2 1 +z 2 2 ) 2 : (4.9) For _ W(z 1 ;z 2 ) < 0 with small z 1 ;z 2 or z 2 1 + z 2 2 < ± 2 , we need the following G 1 z 2 2 ¡ G 2 z 2 sgnz 2 <0 to be true. Two cases to the above equation (4.9) are investigated. 1. when sgn z 2 =1, for _ W(z 1 ;z 2 )<0, we need the condition z 2 < G 2 G 1 to be true. 2. when sgn z 2 =¡1, for _ W(z 1 ;z 2 )<0 we need the condition z 2 >¡ G 2 G 1 to be true. Therefore, we need some conditions on z 2 to make _ W(z 1 ;z 2 ) < 0, and the condition is jz 2 j< G 2 G 1 to achieve stability. When we combined V(z 1 ;z 2 );W(z 1 ;z 2 ) associated with the required region of convergence, the limit cycle to the system of equation (4.2) occurs at z 2 = G 2 G 1 . For Robinson's oscillator with noise conditions, the stability of the solution will be analyzedinthelaterchapterwhenweintroducethee®ectsofnoiseintheoscillatormodel. 4.2 More Analysis of Robinson's Oscillator In this section, we will analyze the behavior of Robinson's oscillator and show the e®ect of non-linearity associated with di®erent damping coe±cients, on the performance of the system. We have previously discussed the existence and the stability of the solution to the system equation (4.1). However, to understand the e®ect of non-linearity on the overall amplitude and phase of the solution will help to fully describe the behavior of the system. 51 Therefore,weneedtosomehowtransformthesystemequation(4.1)intoapolarcoordinate system, where we fully describe the dynamics of the system. 4.2.1 Transformation of the System Equation and Analysis We are interested in transforming the system equation (4.1) into a system of equations representing the amplitude and phase dynamics of the system. From the state space equation (4.2), we do the transformation by ¯rst setting y = z 2 ! 0 , we obtain the following new system of equation to be _ z 1 = ! 0 y _ y = ¡! 0 z 1 +g(y) ; (4.10) where g(y)=¡G 1 y+ G 2 ! 0 sgn(! 0 y). Now we transform the new above state space equation into a model in polar coordinates represented by the amplitude ½ and the phase µ. We let z 1 = ½cosµ, and y = ½sinµ (not (¡½sinµ)), we get the following representation in polar coordinates as _ z 1 = _ ½cosµ¡½ _ µsinµ =! 0 ½sinµ _ y = _ ½sinµ+½ _ µcosµ =¡! 0 ½cosµ+g(½sinµ) : (4.11) When we multiply the _ z 1 equation of the equation (4.11) by cosµ and _ y equation by sinµ, and add them, we get the di®erential equation for the amplitude ½ as _ ½=g(½sinµ)sinµ : (4.12) 52 Similarly, we get the di®erential equation for the total phase µ to be _ µ =¡! 0 + g(½sinµ) ½ cosµ : (4.13) From both equations (4.12), (4.13), we see the e®ect of both quantities ½ and µ and the e®ect of non-linearity associated with the g(½sinµ) function. Furthermore, we determine the angular frequency drift from equation (4.13) and we de¯ne the angular frequency drift to be _ µ d = g(½sinµ) ½ cosµ. We now investigate the steady state amplitude and angular frequency from the dif- ferential equations (4.12) and (4.13). When the amplitude di®erential equation (4.12) is set equal to zero, for the amplitude ½ = constant, a steady state amplitude, we need the following equality to be true G 1 ½sinµ = G 2 ! 0 sgn(! 0 ½sinµ)= 4G 2 ! 0 ¼ 1 X k=0 sin[(2k+1)! 0 t] 2k+1 : If we can neglect the higher order terms of the sgn(! 0 ½sinµ) and choose the ¯rst term of the expansion, then we get the approximate steady state solution for the amplitude ½. The approximate steady state solution is ½ s = 4G 2 ¼G 1 ! 0 for G 1 ;G 2 positive, and ½2 (0;1). Therefore,thegainconstantsG 1 andG 2 controltheamplitudeofthesteadystatesolution. Thissolutionobtainedfromthe½di®erentialequationfurtherveri¯estheapproximateam- plitude of the solution shown in the section 4.1.1. To make the steady state amplitude ½ s =a, a a constant, we need to set the gain G 2 = a¼G 1 ! 0 4 . However, this condition only 53 applies to some rest angular frequency ! 0 that makes the total phase µ time-varying. For the case when the ! 0 is small enough that makes the total phase µ become a con- stant, the resulting amplitude ½ of the model and the phenomenon of no-oscillation will be discussed next. The phase di®erential equation (4.13) is examined next, we again expand the non- linearity term and the following di®erential equation is obtained _ µ =¡! 0 ¡ G 1 2 sin2µ+ 2G 2 ¼! 0 ½ 1 X k=0 [sin2kµ+sin(2k+2)µ] 2k+1 : (4.14) Suppose we can neglect the higher order terms other than the ¯rst term of the expansion on sgn(½sinµ), we get the following simpli¯ed di®erential equation _ µ¼¡! 0 +( ¡G 1 2 + 2G 2 ¼! 0 ½ )sin2µ : (4.15) We ¯nd that at the steady state with ½ s computed earlier, the total phase can be approx- imated as µ ¼ ¡! 0 t. To see if the phase di®erential equation (4.13) exists a condition where the phenomenon of no-oscillation can occur, we need to set equation (4.13) · 0. We need to rewrite equation (4.13) with the g(½sinµ) function explicitly written as the following _ µ =¡! 0 + 1 ½ [¡G 1 ½sinµ+ G 2 ! 0 sgn(! 0 ½sinµ)]cosµ : (4.16) Now we ¯nd the lower bound on ! 0 where no-oscillation of the model occurs and the total phase µ is equal to a constant. We set the equation (4.16)·0, there are two cases where 54 the sgn(! 0 ½sinµ) can be evaluated. When sgn(! 0 ½sinµ) is either 1 or -1, we obtain the condition of no-oscillation (lower bound) for ! 0 to be ! 0 ¸ G 1 ja¼cosµ¡2½sin2µj 4½ ; (4.17) where µ is some constant and G 2 = a¼G 1 ! 0 4 . The resulting amplitude ½ can also be determined from equation (4.17) with equality sign given a small value of ! 0 . Generally, when the total phase µ is a constant and ! 0 is at small values obtained from equation (4.17) with equality sign, no oscillation of the model occurs, i.e., a single point is seen on the phase plane diagram. The situation is not desired because oscillators will cease oscillations. When the angular frequency ! 0 is increased where the amplitude function ½ approaches the steady state amplitude ½ s , we will start to see the total phase µ become a time-varyingfunctionagain,meaningthattheoscillatorresumesoscillating. Therefore,we want to pick a large oscillator rest radian frequency ! 0 at ¯xed loop gain G 1 such that it won't fall in the region of no-oscillation. The design of a extreme low frequency oscillator requiresothermodi¯cationsandwedonotpursuefurther. Wewilldemonstratethee®ects of non-linearity on µ by simulations as well as an example of no-oscillation situation, as to be described in Section 4.2.2. The e®ects of non-linearity on the amplitude and total phase of the oscillator model is analyzed in the following sections. 55 ∫ ∫ X 0 ω ( ) 2 − + + - 2 G 1 G x & x Figure 4.4: Schematic of Robinson's oscillator from equation (4.2) 4.2.2 Simulation of the Model Robinson's oscillator described in equation (4.1) with the state space representation in equation (4.2) is described as a block diagram shown in Fig. 4.4. We then perform simulations on this equivalent model and compare the results with the previous described results. To further analyze the e®ects of non-linearity on the amplitude and total phase of the oscillator, we need to evaluate the di®erential equations (4.12) and (4.13). Due to the di±culty of obtaining the analytic solutions of the amplitude and phase functions to the system of equations (4.12) and (4.13), we performed simulations on these equations. For the system in polar coordinate form, we ¯rst simulated the system at the clock rest frequency ! 0 = 2000¼ and assign the coe±cients G 1 and G 2 such that the steady state amplitude of the oscillator is approximately 1. We used Matlab's Simulink to simulate the system of equations. The Simulink's block diagram is shown in Fig. 4.5, 56 ∫ ∫ X X u 1 ) sin(u ) cos(u X 1 G + - 0 ω X X x + - 0 ω ρ ρ & θ θ & θ ρ θ ρ cos ) sin ( g 0 2 ω G Figure4.5: Simulinkblockdiagramshowingtheamplitudeandphasedi®erentialequations (4.12), (4.13) with the clock rest frequency ! 0 =2000¼ where G 1 = 1000;G 2 = G 1 ¼! 0 4 . A complete simulated solution shows the e®ects of non- linearity and constant gains on both amplitude and total phase of the oscillator, and we alsoobservetheinteractionsbetweentheequations(4.12)and(4.13)fromthesimulations. We start the simulation at the initial conditions ½(0)= 0:125;µ(0)= 0 to see how the amplitude function behaves, its transients, and its steady state solution. The total phase function is observed to have negative slope as already shown in Section 4.1. There are several simulated functions from the schematic shown in Fig. 4.5. They include the ½(t), µ(t), _ µ, the non-linearity e®ects on both ½ and µ, and the oscillator output ½cosµ. 57 0 0.005 0.01 0.015 0.02 0.025 0.03 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time (sec.) ρ(t) 0.02 0.0205 0.021 0.0215 0.022 0.0225 0.023 0.0235 0.024 0.0245 0.025 0.98 0.99 1 1.01 1.02 time (sec.) ρ(t) Figure 4.6: The amplitude function ½(t) of the simulated model shown in Fig. 4.5 The amplitude ½(t) function of the oscillator is shown in Fig. 4.6. The transients of the ½(t) as well as the steady state solution are observed. There is a periodic function seen from the steady state solution having a period of approximately 2! 0 and its shape is determined by the g(½sinµ) function that has the e®ect of non-linearity. Furthermore, there is no equilibrium point in Fig. 4.7 showing _ ½ vs ½ meaning that the steady state solution of ½, denoted by ½ s , will continue to oscillate in¯nitely given at such radian frequency ! 0 =2000¼. Thee®ectsofnon-linearityontheamplitudeandthetotalphaseoftheoscillatormodel are shown. In Fig. 4.8, the e®ects of the g(½sinµ) on the amplitude and total phase of the system are observed from equations (4.12) and (4.13). We rephrase the g(½sinµ) function here g(½sinµ)=¡G 1 ½sinµ+ G 2 ! 0 sgn(! 0 ½sinµ) (4.18) 58 0.99 0.995 1 1.005 1.01 1.015 −250 −200 −150 −100 −50 0 50 100 150 200 ρ(t) dρ / dt Figure 4.7: Phase paths of the equation (4.12) at steady state for further explanations. The g(½sinµ) function consists of the gain G 1 that controls the rate of reaching the state state solution and the other gain G 2 associated with the non- linearity sgn(! 0 ½sinµ), which controls the steady state amplitude ½ s . The non-linearity of the oscillator model is what makes the oscillation possible and together with the ¯rst term of the equation (4.18), a limit cycle of the solution exists as already shown in Section 4.1.1. The instantaneous change of the amplitude function _ ½ and the angular frequency drift _ µ d = g(½sinµ)cosµ ½ suggest that ½ and µ have some e®ects on their overall shape re- spectively as well as a function of time. Parameters like the gains G 1 and G 2 , and the non-linearity sgn(½sinµ) play important parts in the overall system performance and they will be explored more in the following section. The previous discussed question in Section 4.2.1 that when the angular rest frequency ! 0 is large such that there exists no equilibrium points on the phase path of the equation 59 0.02 0.0202 0.0204 0.0206 0.0208 0.021 0.0212 0.0214 0.0216 0.0218 0.022 −300 −200 −100 0 100 200 time (sec.) dρ / dt 0.02 0.0202 0.0204 0.0206 0.0208 0.021 0.0212 0.0214 0.0216 0.0218 0.022 −1000 −500 0 500 1000 time (sec.) g(ρ sinθ)cosθ / ρ Figure 4.8: E®ects of non-linearity g(½sinµ) on the amplitude and the phase functions (4.13) is best seen in Fig. 4.9. At ! 0 = 2000¼, the amplitude of the _ µ function is large enough such that no equilibrium points can occur. The _ µ function will simply oscillate toward the negative values of µ inde¯nitely. However, as we reduce the value of ! 0 to a point where no oscillation can occur, an equilibrium point on the _ µ = 0 axis starts to appear. For example, suppose we are given a small angular rest frequency ! 0 = 20¼, the resulting amplitude function is approximately ½ = a¼G 1 4! 0 for some ¯xed equilibrium point µ. Lastly we plot the oscillator output functions ½cosµ and ½sinµ. This can be seen as a phase plane representation shown in Fig. 4.10. There exists a limit cycle with oscillating amplitude of 1 and the solution from the transformation equations (4.12) and (4.13) completely match the ones obtained from the original system of equation (4.2). 60 −100 −98 −96 −94 −92 −90 −88 −86 −84 −82 −80 −7200 −7000 −6800 −6600 −6400 −6200 −6000 −5800 −5600 −5400 θ(t) (radian) dθ(t)/dt Figure 4.9: The phase diagram of the equation (4.13) when ! 0 =2000¼ 4.2.3 More Analysis of the Results In this section, the e®ects of non-linearity on the oscillator output ½cosµ, and on the total phase µ are investigated. There are two types of non-linearity used, they are hard-limiter andsoft-limiterwithdi®erentslopes. Fromtheamplitudeandphasedi®erentialequations (4.12) and (4.13), we replace the hard-limiter within the g(½sinµ) function with a soft- limiter. Furthermore, we evaluate the e®ects of the model employing a soft-limiter with di®erent slopes and compare the results with the one with a hard-limiter non-linearity. First, we evaluate the model employing with a soft-limiter non-linearity by simulation ontheamplitudeandthephaseschematicshowninFig. 4.5withthenon-linearityreplaced 61 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ρ cosθ ρ sinθ Figure 4.10: Phase plane representationof the oscillator output waveformsshowing ½cosµ vs. ½sinµ for G 1 =1000 by a soft-limiter. The new amplitude and phase di®erential equations are then described as the following _ ½ = g 1 (½sinµ)sinµ ; (4.19) _ µ = ¡! 0 + g 1 (½sinµ) ½ cosµ ; (4.20) where g 1 (½sinµ)=¡G 1 ½sinµ+ G 2 ! 0 s l (! 0 ½sinµ), the soft-limiter s l (y)= 8 > > > > > > < > > > > > > : 1 if y¸1 py if¡1<py <1 ¡1 if y·¡1 (4.21) 62 0.1 0.1002 0.1004 0.1006 0.1008 0.101 0.1012 0.1014 0.1016 0.1018 0.98 0.99 1 1.01 1.02 1.03 t (sec.) ρ(t) model with hard−limiter model with soft−limiter| slope=0.01 model with soft−limiter| slope=0.001 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −10 0 10 20 30 40 t (sec.) θ non−linearity − θ 0 (rad.) θ hard limiter −θ 0 θ soft limiter with slope=0.01 −θ 0 θ soft limiter with slope=0.001 −θ 0 Figure4.11: Thesteadystateamplitudefunction½ s (t)andthedeviationofthephasedrift function µ d (t) for di®erent non-linearities and p is the slope of the soft-limiter with the range 0 < p· 1. For the soft-limiter with di®erent slopes p, we can reduce the feedback e®ect from the non-linearity to the system of equations (4.19) and (4.20). Generally, for ¯xed constant gains G 1 and G 2 , we reduce the feedback e®ect of non-linearity while still maintaining the required oscillation steady state amplitude by reducing the slope of the soft-limiter. The signi¯cant e®ects on the total phase µ which includes the phase drift term µ d and the e®ects on the steady state amplitude ½ s are observed in Fig. 4.11, where the ¯rst ¯gure indicates the di®erence of the steady state amplitude ½ s (t) when di®erent non-linearities are employed in the model. Astheslopeofthesoft-limiterisdecreased, weseetheoveralldecreaseonthesteadystate amplitude function. There exists a minimum slope of the soft-limiter that still maintains the required steady state amplitude. The second ¯gure of Fig. 4.11 shows how the phase 63 0.1 0.1002 0.1004 0.1006 0.1008 0.101 0.1012 0.1014 0.1016 0.1018 −300 −200 −100 0 100 200 time (sec.) dρ/dt = g(ρsinθ)sinθ model with hard−limiter model with soft−limiter| slope=0.01 model with soft−limiter| slope=0.001 0.1 0.1002 0.1004 0.1006 0.1008 0.101 0.1012 0.1014 0.1016 0.1018 −1000 −500 0 500 1000 time (sec.) dθ d /dt = g(ρsinθ)cosθ/ρ model with hard−limiter model with soft−limiter| slope=0.01 model with soft−limiter| slope=0.001 Figure 4.12: Signal diagrams showing the e®ects of di®erent non-linearities on _ ½ and on _ µ d drift term µ d di®ers from the ideal total phase µ 0 having a angular rest frequency ! 0 , for di®erent non-linearities. The phase drift µ d of the model is an increasing function of time and the model with the soft-limiter has smaller phase drift as compared to the one with a hard-limiter. As the slope of the soft-limiter decreases, the smaller is the phase drift function µ d . Toseewhythephasedriftfunctionandthesteadystateamplitudechangechangewhen di®erent non-linearities are used in the model, we need to look at the e®ects of di®erent non-linearities on the contributing parameters. These include the angular frequency drift function _ µ d and the instantaneous amplitude change function _ ½ from the system of equa- tions(4.19)and(4.20). AsshowninFig. 4.12,weseethatthedi®erentnon-linearitye®ects for both _ µ d and _ ½. It follows that the smaller the slope p of the soft-limiter while still large enough to maintain the required steady state amplitude, the smoother the slope of the 64 resultingsoft-limiteroutputandthesmallertheenergycontributionsofthenon-linearities on the feedback of the system. Therefore, we expect to see the g 1 (½sinµ) function reduces its signal strength as the slope of the soft-limiter is decreased. Furthermore, since the contributions to the new phase di®erential equation (4.20) is smaller when the soft-limiter with smaller slope is employed, the phase drift function µ d (t) will be smaller, making the total oscillation phase close to the ideal oscillation phase µ 0 . Thus, we combine both ar- guments about the signal strength and the phase drift on the parameters _ µ d and _ ½ with soft-limiters, the phenomenon seen in Fig. 4.12 is explained. Therefore, the e®ect of dif- ferent non-linearities on the steady state amplitude function ½ s (t) and on the phase drift function µ d (t) shown in Fig. 4.11 can be explained by the shapes of functions _ µ d and _ ½ that we observed in Fig. 4.12. To investigate further that how fast is one oscillator output waveform di®ers from the other when di®erent non-linearities are employed in the oscillator model, we do the following analysis. The output oscillator waveform is represented by ½cosµ. Suppose two oscillator outputs from the models with two di®erent non-linearities are represented 65 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −1.5 −1 −0.5 0 0.5 t (sec.) cosφ ρ (t)cosθ δ (t) − sinφ ρ (t) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −1 −0.5 0 0.5 1 t (sec.) cosφ ρ (t)sinθ δ (t) Figure 4.13: Equations that are used to determine the phase drift rate di®erence between two oscillators based on equation (4.22) by sinusoids with di®erent amplitude ½(t) and total phase functions µ(t). We take the di®erence of the two waveforms as the following ½ 1 (t)cosµ 1 (t)¡½ 2 (t)cosµ 2 (t) =( q ½ 2 1 (t)+½ 2 2 (t))[cosÁ ½ (t)cosµ 1 (t)¡sinÁ ½ (t)cosµ 2 (t)] =( q ½ 2 1 (t)+½ 2 2 (t))fcosÁ ½ (t)[cosµ 2 (t)cosµ ± (t)¡sinµ 2 (t)sinµ ± (t)]¡sinÁ ½ (t)cosµ 2 (t)g =( q ½ 2 1 (t)+½ 2 2 (t))fcosµ 2 (t)[cosÁ ½ (t)cosµ ± (t)¡sinÁ ½ (t)]¡cosµ ½ (t)sinµ ± (t)sinµ 2 (t)g (4.22) where Á ± (t) = arctan ½ 2 (t) ½ 1 (t) and µ 1 = µ 2 +µ ± (t). We see the rate of phase drift between 66 oscillatorsemployedwithdi®erentnon-linearitiesbylookingatthelastexpressionofequa- tion (4.22). When the term cosÁ ½ (t)cosµ ± (t)¡sinÁ ½ (t) inside the square bracket is zero and the other term of the last expression cosÁ ½ (t)sinµ ± (t) inside the parenthesis is also equal to zero, the rate of phase drift between two oscillators can therefore be found. Since two oscillator models have di®erent non-linearities employed, we get di®erent amplitude functions and total phase functions respectively. It is di±cult to get an analytic solution for the di®erence of the phase drift functions. We therefore simulated these two models to ¯nd the solution. Without loss of generality, we compare the model with the hard-limiter with the model having a soft-limiter with slope p = 0:001. We simulated the schematic shown in Fig. 4.5 with these two non-linearities and computed the required quantities in equation (4.22). The results are shown in Fig. 4.13 where when both ¯gures have zero values, that's where the phase shift of a complete multiple of 2¼ occurs between these two oscillators. We estimate that the model with a soft-limiter non-linearity with a slope of 0.001 has a phase drift function whose rate is about 1 Hz when compared with the one with a hard-limiter non-linearity. Furthermore, we also ¯nd out the same value for the phase drift rate between these two oscillators from Fig. 4.11, where the phase dif- ference between the model with a soft-limiter whose slope is 0.001 and the one with a hard-limiter at around 1 second is about 2¼. This means that the model employed with a soft-limiter whose slope is 0.001 has a phase drifting of 2¼ when compared to the one with a hard-limiter at 1 second. 67 4.3 Models with Di®erent Gains So far we have looked at the e®ects of di®erent non-linearities on the amplitude and phase drift functions of oscillator model. Other system parameters that may contribute to the system performance are the constant gains G 1 and G 2 as described in the system of equation (4.2). We have previously de¯ned what values G 1 and G 2 take in Section 4.2.1, and they are related by the following equation G 2 = a¼G 1 ! 0 4 ; where a is a constant that determines the steady state amplitude of the oscillator. Recall that G 1 controls the rate to reach the steady state while the other gain G 2 controls the steady state amplitude of oscillations, together with the non-linearity which makes the oscillationspossible. Whentheratetoreachthesteadystateconditionisnotacriticalissue in the system design, we choose lower values of G 1 and G 2 such that the negative e®ects from the non-linearity on the overall amplitude and phase drift will not be signi¯cant. Here we give an example that will describe the situation. 4.3.1 E®ects on Non-linearities Suppose we are given gain constants G 1 = 100 and G 2 = ¼G 1 ! 0 4 , we perform simulations on the model as described earlier in Fig. 4.5. Since the gains are lower than what we previously assigned by a factor of 10, we expect to see the e®ects of the non-linearity on the oscillator model will be reduced. This is seen in Fig. 4.14, where the e®ects of the non-linearity on the system with di®erent constant gains are signi¯cant with a reduction 68 0.1 0.1002 0.1004 0.1006 0.1008 0.101 0.1012 0.1014 0.1016 0.1018 −300 −200 −100 0 100 200 t (sec.) dρ/dt model with hard−limiter G 1 =1000 model with hard−limiter G 1 =100 0.1 0.1002 0.1004 0.1006 0.1008 0.101 0.1012 0.1014 0.1016 0.1018 −1000 −500 0 500 1000 t (sec.) dθ d /dt = g(ρsinθ)cosθ/ρ model with hard−limiter G 1 =1000 model with hard−limiter G 1 =100 Figure 4.14: E®ects of non-linearity with di®erent constant gain G 1 on the instantaneous change of amplitude function _ ½ and the angular frequency drift function _ µ d byafactorof10. Wealsoobservethatthemodelwiththereducedgainshavefasterphase drift than the one with higher gains. This can be explained by the system equation (4.13), lower gains contribute to lower angular frequency drift _ µ d seen in Fig. 4.14, meaning that the overall drift rate is lower. This is a desired feature and the rate of total phase _ µ is closer to the natural angular frequency ! 0 than the model with higher gains. The e®ects are seen more closely when we compare the steady state amplitude function and the phase drift function of the reduced gain model with the original model with higher gains. Fig. 4.15displaysthesteadystateamplitudefunction½ s (t)andthephasedriftfunction µ d (t) when the model with the same non-linearity (hard-limiter) is employed but with di®erent gains G 1 and G 2 . Here we have the steady state amplitude approximately equal to 1 and the clock rest angular frequency ! 0 =2000¼. It is clear from the ¯gure that the model with a lower gain G 1 = 100 has a better system performance in both steady state 69 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 −1 0 1 2 3 4 t (sec.) θ non−linearity − θ 0 (rad.) model with hard−limiter G 1 =1000 model with hard−limiter G 1 =100 1 1.0002 1.0004 1.0006 1.0008 1.001 1.0012 1.0014 1.0016 1.0018 1.002 0.98 0.99 1 1.01 1.02 t (sec.) ρ(t) model with hard−limiter G 1 =1000 model with hard−limiter G 1 =100 Figure 4.15: The steady state amplitude function ½ s (t) and the phase drift function µ d (t) when the same model is simulated with a hard-limiter non-linearity but di®erent constant gains amplitudeand inthe phasedrift when compared to the one with a higher gain. Therefore, duetothereducedconstantgainsintheoscillatormodel,abetterperformanceisobtained. 4.3.2 Model Selections Due to di®erent applications, the requirements for the selection of models may vary. For the oscillator model with lower constant gains, if we would like to get a better transient performance,ahighergainmaybedesired. Therefore,atrade-o®needstobemadeamong di®erent system performance parameters of interest. In general, we would like to have a good stable steady-frequency oscillator, thus a low-gain type oscillator model is preferred. On the other hand, if in acommunicationsystem that requires a veryfast-stable oscillator that needs to have good transient performance, then a high-gain type system model with 70 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ρcosθ ρsinθ Figure 4.16: Phase plane representation of the oscillator model with hard-limiter non- linearity when the constant gain G 1 =100 and the clock rest frequency ! 0 =2000¼ soft-limiter non-linearity of small slope will make a better choice. Here we plot the phase planerepresentationofthenewmodelthatemployedahard-limiterwiththegainG 1 =100 in Fig. 4.16. Moreover, we are plotting the oscillator outputs as ½cosµ vs. ½sinµ. As seen in the ¯gure, it takes a lot longer than the previous model in Fig. 4.10 to reach the steady state amplitude. There is another condition that when the gains of the oscillator model becomes too large, the oscillator output waveform is no longer a sinusoid. When the constant gain G 1 becomes large, i:e:, G 1 > ! 0 , the oscillator output waveform is still periodic but it is no longer close to a sinusoid. This can be seen in both Fig. 4.17 and Fig. 4.18. The phenomenon of the output oscillator waveform is explained by the following. When large constant gains, i:e:, G 1 > ! 0 , are used in the oscillator model, large energy will be either inserted to or lost from the system, depending on the initial conditions. When the initial 71 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 ρcosθ ρsinθ Figure 4.17: Phase plane representation of a Robinson oscillator with G 1 = 10000, and ! 0 =2000¼ condition is small as compared to the steady state amplitude of oscillation, large gains will contribute to large energy increase of the system, making the transient response to the steady state faster. Moreover, the large energy increase to the system and the e®ect of non-linearity due to large gains are such that the oscillator output waveform will no longer be a sinusoid though it is still periodic. Thus, we will see the e®ects of the results in both Fig. 4.17 and Fig. 4.18. On the other hand, when the initial condition is greater than the steady state amplitude of the system, we will see a large energy loss from the system, resulting in a fast transient response to the steady state solution. The analysis of the energy contribution to the system has been previously discussed in Section 4.1. The above result of having a very large gain is what happens when the system equation is over-damped. 72 0 100 200 300 400 500 600 −1.5 −1 −0.5 0 0.5 1 1.5 t (sec.) ρcosθ Figure 4.18: A periodic solution of a Robinson oscillator in time domain Therefore, we conclude the discussion by specifying a region of system gains and the types of non-linearity for model selections. A soft-limiter non-linearity with small slope is usually desired for the system design for lower phase drift µ d . For the constant gains G 1 and G 2 , we specify the relation to be G 2 = a¼! 0 G 1 4 , where a is the speci¯ed steady state amplitude. In addition, for a sinusoidal oscillator waveform with fast transient response, we need the condition G 1 <! 0 . Furthermore, we need the constant gains to be in a region where reasonable transient response is possible as well as low oscillator phase drifts. In our example, for a 1 kHz oscillator, we need the constant gain G 1 to be around 500, for a acceptable oscillator system performance. 73 Chapter 5 Oscillator Models in the Presence of Noise 5.1 Oscillator Models TheoscillatormodelinthepresenceofnoiseisbasedonRobinson'soscillatorjustdiscussed in Chapter 4. We propose three models with noise and they are described by second-order stochasticdi®erentialequationswith non-linearity. As thenoiseis presentin the oscillator model, analysis needs to be done on the existence and the stability of the solution. We needtoknowthattowhatdegreeofthenoiselevel,theoscillatormodelwithnoisepresent can still remain stable. FortheproposedmathematicalmodelofvoltagecontrolledoscillatorsbasedonRobin- son's oscillator, we consider three cases. The ¯rst case is when only one internal oscillator noise is present, it is represented by Ä x+G 1 _ x¡G 2 sgn_ x+! 2 0 x=F 1 (t) (5.1) 74 ∫ ∫ + X 0 ω ) ( 1 t F ( ) 2 − + + - x 2 G 1 G x & Figure 5.1: Schematic of the oscillator model I from equation (5.1) where G 1 , G 2 are constant gains, ! 0 is the clock rest frequency, and F 1 (t) is an internal oscillatornoiseassumingwhiteandGaussiandistributed. TheoscillatormodelIdescribed in equation (5.1) is described as a block diagram shown in Fig. 5.1. The second case we consider for the proposed model is when both internal oscillator noises are present. This is described by the following stochastic di®erential equation, Ä y+G 1 _ y¡G 2 sgn_ y+! 2 0 x=F 1 (t) (5.2) whereG 1 ,G 2 areconstantgains,! 0 istheclockrestfrequency,F 1 (t)isaninternaloscillator noise considered independent of the second internal noise F 2 (t), and x = y+ R t 0 F 2 (¿)d¿. The oscillator model II described in equation (5.2) is described as a block diagram shown in Fig. 5.2. 75 ∫ ∫ + X 0 ω ) ( 1 t F ( ) 2 − + + ) ( 2 t F + - ) (t x y & 2 G 1 G Figure 5.2: Schematic of oscillator model II from equation (5.2) when both internal oscil- lator noises F 1 (t) and F 2 (t) are present The third case we consider for the proposed model is when both internal noises are present and when a non-linear feedback device is employed. This is described by the following non-linear di®erential equation Ä y+G 1 _ y¡G 2 sgn_ y+! 2 0 g sl (x)=F 1 (t) (5.3) where g sl (x)= 8 > > > > > > < > > > > > > : a if x¸a x if¡a<x<a ¡a if x·¡a (5.4) is a soft-limiter, x, G 1 , G 2 , F 1 (t) are de¯ned earlier, and the approximate steady state amplitude a in the g sl function. The oscillator model III described in equation (5.3) 76 ∫ ∫ + X 0 ω ) ( 1 t F ( ) 2 − + + ) ( 2 t F + - ) (t x y & 2 G 1 G ) ( sl ⋅ g Figure 5.3: Oscillator model III when both internal oscillator noises F 1 (t) and F 2 (t) are present, and the soft-limiter g sl (¢) is employed in the outer loop of the diagram is described as a block diagram shown in Fig. 5.3. These three oscillator models are investigated and analyzed in the following sections. 5.2 Existence and Stability of the Solution Three oscillator models described in the previous section 5.1 are investigated for the ex- istence and stability of the solution when noise is present. For each oscillator model presented, weanalyze the existence of the solution and the stability region for the solution accordingly. 5.2.1 Existence of the Solution for Oscillator Model I For the three proposed oscillator models, we investigate the existence of the solution for eachmodelrespectively. Weintroducetheexistenceanduniquenesstheoremforstochastic di®erential equations from Section 5.2 [26]. Here we state the theorem. 77 Theorem 5.1 (Existenceanduniquenesstheoremforstochasticdi®erentialequa- tions). LetT >0andb(¢;¢):[0;T]£R n !R n , ¾(¢;¢):[0;T]£R n !R n£m bemeasurablefunctions satisfying jb(t;x)j+j¾(t;x)j·C(1+jxj); x2R n ; t 2[0;T] (5.5) for some constant C, (where j¾j 2 = P i;j j¾ ij j 2 ) and such that jb(t;x)¡b(t;y)j+j¾(t;x)¡¾(t;y)j·Djx¡yj; x;y2R n ; t 2[0;T] (5.6) for some constant D. Let Z be a random variable which is independent of the ¾-algebra F (m) 1 generatedbyB s (¢), whereB s isaBrownianmotion, s¸0andsuchthatE[jZj 2 ]<1. Then the stochastic di®erential equation dX t =b(t;X t )dt+¾(t;X t )dB t ; 0·t·T; X 0 =Z (5.7) has a unique t-continuous solution X t (!) with the property that X t (!) is adapted to the ¯ltration F Z t generated by Z and B s (¢); s·t and E[ R T 0 jX t j 2 dt]·1. For the oscillator model I shown in equation (5.1), we use the state space approach to represent the model as _ Z= 2 6 6 4 _ z 1 _ z 2 3 7 7 5 = 2 6 6 4 f 1 (Z) f 2 (Z) 3 7 7 5 + 2 6 6 4 0 1 3 7 7 5 F 1 (t); (5.8) 78 where z 1 =x, z 2 = _ x, f 1 (Z)=z 2 , and f 2 (Z)=¡G 1 z 2 +G 2 sgnz 2 ¡! 2 0 z 1 . The vectorZ is a 2-dimensional Markov process and the function f 2 (Z) is non-linear. We use the existence and uniqueness theorem for SDEs (Theorem 5.1) to show the existence of the solution. We obtain the following inequality from the existence condition (5.5) of the Theorem 5.1. [z 2 2 +(¡G 1 z 2 +G 2 sgnz 2 ¡! 2 0 z 1 ) 2 ] 1=2 +1·C[1+(z 2 1 +z 2 2 )] 1=2 (5.9) We evaluate the above condition (5.9) at steady state where the region of inequality is the smallest at the transition of the signum function. This happens at the region when jz 1 j ! a, where a is the steady state amplitude, z 2 ! 0 + , and jz 1 j ! a, z 2 ! 0 ¡ . For z 2 !0 + , we get the following inequality, jG 2 ¡! 2 0 z 1 j+1·! 2 0 (1+jz 1 j) ; where the constant C =! 2 0 is chosen. As long as the condition G 2 ·! 2 0 +! 2 0 (jz 1 j+z 1 )¡1 is valid, the solution to the oscillator model I (5.1) exists. For the case when z 2 !0 ¡ , we get the following inequality, j¡G 2 ¡! 2 0 z 1 j+1·! 2 0 (1+jz 1 j) ; wheretheconstantC =! 2 0 ischosen. AslongastheconditionG 2 ·! 2 0 +! 2 0 (jz 1 j¡z 1 )¡1is valid, the solution to the oscillator model I (5.1) exists. Therefore, the required condition for the existence of the solution when the existence condition (5.9) is valid is G 2 ·! 2 0 ¡1. 79 The uniqueness condition (5.6) or the Lipschitz condition is used to determine the uniqueness of the solution to the SDE in equation (5.7). Here we evaluate the oscillator model I in state space representation (5.8) and compute the following Lipschitz condition f(z 2 ¡^ z 2 ) 2 +[¡G 1 (z 2 ¡^ z 2 )+G 2 (sgnz 2 ¡sgn^ z 2 )¡! 2 0 (z 1 ¡^ z 1 )] 2 g 1 2 ·D[(z 1 ¡^ z 1 ) 2 +(z 2 ¡^ z 2 ) 2 ] 1 2 : (5.10) At the transition region of the non-linearity, the signum function, where z 2 transits from 0 + to0 ¡ ,wedeterminethatthecondition(5.10)doesnotsatisfytheLipschitzconditionof (5.6). ThereisnouniqueconstantDinthecondition(5.10)thatwillsatisfytheuniqueness condition. 5.2.2 Existence of the Solution for Oscillator Model II For the oscillator model II described in equation (5.2), we again use the state space rep- resentation as the following 2 6 6 4 _ x _ z 2 3 7 7 5 = 2 6 6 4 f 1 (x;z 2 ) f 2 (x;z 2 ) 3 7 7 5 + 2 6 6 4 0 1 1 0 3 7 7 5 2 6 6 4 F 1 (t) F 2 (t) 3 7 7 5 (5.11) where z 1 = y, z 2 = _ y, f 1 (x;z 2 ) = z 2 , f 2 (x;z 2 ) = ¡G 1 z 2 + G 2 sgnz 2 ¡ ! 2 0 x, and x = y+ R t 0 F 2 (¿)d¿. We then test the existence condition (5.5) of Theorem 5.1. We represent the equation (5.11) as the standard form of equation (5.7), where dB t = 2 6 6 4 F 1 (t)dt F 2 (t)dt 3 7 7 5 : (5.12) 80 The existence condition for the equation (5.11) becomes the following [z 2 2 +(¡G 1 z 2 +G 2 sgnz 2 ¡! 2 0 x) 2 ] 1=2 + p 2·C[1+(x 2 +z 2 2 ) 1=2 ] : (5.13) We then evaluate the above condition (5.13) where the region of inequality is the smallest at the transition of the signum function. This happens at the region of transition from z 2 !0 + to z 2 !0 ¡ . For the case when z 2 !0 + , we get the following inequality, jG 2 ¡! 2 0 xj+ p 2·! 2 0 (1+jxj) ; where the constant C =! 2 0 is chosen. When the condition G 2 ·! 2 0 +! 2 0 (jxj+x)¡ p 2 is valid, the solution to the oscillator model II (5.2) exists. For the case when z 2 ! 0 ¡ , we get the following inequality, jG 2 +! 2 0 xj+ p 2·! 2 0 (1+jxj) ; where the constant C =! 2 0 is chosen. When the condition G 2 ·! 2 0 +! 2 0 (jxj¡x)¡ p 2 is valid, the solution to the oscillator model II (5.2) exists. Therefore, the required condition for the existence of the solution to the oscillator model II is G 2 ·! 2 0 ¡ p 2. 81 Weagainusetheuniquenesscondition(5.6)ortheLipschitzconditiontodeterminethe uniqueness of the solution to the oscillator model II (5.2). Here we evaluate the oscillator modelIIinstatespacerepresentation(5.11)andcomputethefollowingLipschitzcondition f(z 2 ¡^ z 2 ) 2 +[¡G 1 (z 2 ¡^ z 2 )+G 2 (sgnz 2 ¡sgn^ z 2 )¡! 2 0 (x¡^ x)] 2 g 1 2 ·D[(x¡^ x) 2 +(z 2 ¡^ z 2 ) 2 ] 1 2 : (5.14) Again at the transition region of the non-linearity, the signum function, where z 2 transits from 0 + to 0 ¡ , we determine that the condition (5.14) does not satisfy the Lipschitz condition of (5.6). There is no unique constant D in the condition (5.14) that will satisfy the uniqueness condition. 5.2.3 Existence of the Solution for Oscillator Model III For the oscillator model III described in equation (5.3), we again use the state space representation as the following 2 6 6 4 _ x _ z 2 3 7 7 5 = 2 6 6 4 f 1 (x;z 2 ) f 2 (x;z 2 ) 3 7 7 5 + 2 6 6 4 0 1 1 0 3 7 7 5 2 6 6 4 F 1 (t) F 2 (t) 3 7 7 5 (5.15) where z 1 = y, z 2 = _ y, f 1 (x;z 2 ) = z 2 , f 2 (x;z 2 ) = ¡G 1 z 2 + G 2 sgnz 2 ¡ ! 2 0 g sl (x), x = y+ R t 0 F 2 (¿)d¿, and the soft-limiter function g sl (x) is de¯ned earlier in equation (5.4). We then test the existence condition (5.5) of Theorem 5.1. 82 Wethen representequationtheequation(5.15)as the standardformof equation(5.7), where dB t is the same as shown in equation (5.12). The existence condition for equation (5.15) becomes the following [z 2 2 +(¡G 1 z 2 +G 2 sgnz 2 ¡! 2 0 g sl (x)) 2 ] 1=2 + p 2·C 2 [1+(x 2 +z 2 2 ) 1=2 ] : (5.16) We then evaluate the above condition (5.16) where the region of inequality is the smallest at the transition of the signum function. This happens at the region of transition from z 2 !0 + to z 2 !0 ¡ . For the case when z 2 !0 + , we get the following inequality, jG 2 ¡! 2 0 g sl (x)j+ p 2·! 2 0 (1+jxj) ; where the constant C =! 2 0 is chosen. When the condition G 2 ·! 2 0 +! 2 0 (jxj+g sl (x))¡ p 2 is valid, the solution to the oscillator model II (5.3) exists. For the case when z 2 ! 0 ¡ , we get the following inequality, jG 2 +! 2 0 g sl (x)j+ p 2·! 2 0 (1+jxj) ; wheretheconstantC =! 2 0 ischosen. WhentheconditionG 2 ·! 2 0 +! 2 0 (jxj¡g sl (x))¡ p 2is valid,thesolutiontotheoscillatormodelIII(5.3)exists. Therefore,therequiredcondition fortheexistenceofthesolutiontotheoscillatormodelIIIisG 2 ·! 2 0 +! 2 0 (jxj¡g sl (x))¡ p 2, which is at the minimum when G 2 ·! 2 0 ¡ p 2. We again use the uniqueness condition (5.6) or the Lipschitz condition to determine the uniqueness of the solution to the oscillator model III (5.3). Here we evaluate the 83 oscillatormodelIIIinstatespacerepresentation(5.15)andcomputethefollowingLipschitz condition f(z 2 ¡^ z 2 ) 2 +[¡G 1 (z 2 ¡^ z 2 )+G 2 (sgnz 2 ¡sgn^ z 2 )¡! 2 0 (g sl (x)¡g sl (^ x))] 2 g 1 2 ·D[(x¡^ x) 2 +(z 2 ¡^ z 2 ) 2 ] 1 2 : (5.17) Again at the transition region of the non-linearity, the signum function, where z 2 transits from 0 + to 0 ¡ , we determine that the condition (5.17) does not satisfy the Lipschitz condition of (5.6). There is no unique constant D in the condition (5.17) that will satisfy the uniqueness condition. 5.2.4 Stability of the Solution for Oscillator Model I We have so far discussed the existence of the solution when noise is present for three non- linear oscillator models, we would like to know whether the solution is stable when noise is present. To proceed further with the stability of the solution for the oscillator models, we need to show the some preliminary results or theorems [26] ¯rst. 84 Theorem 5.2 (The general It^ o formula). Let dX(t)=udt+vdB(t) (5.18) be an n-dimensional It^ o process where X(t)= 2 6 6 6 6 6 6 4 X 1 (t) . . . X n (t) 3 7 7 7 7 7 7 5 ; u= 2 6 6 6 6 6 6 4 u 1 . . . u n 3 7 7 7 7 7 7 5 ; v = 2 6 6 6 6 6 6 4 v 11 ::: v 1m . . . . . . v n1 ::: v nm 3 7 7 7 7 7 7 5 ; dB(t)= 2 6 6 6 6 6 6 4 dB 1 (t) . . . dB m (t) 3 7 7 7 7 7 7 5 : (5.19) Letg(t;x)=(g 1 (t;x);:::;g p (t;x))beaC 2 mapfrom[0;1)£R n intoR p . Thentheprocess Y(t;!)=g(t;X(t)) is again an It^ o process, whose component number k, Y k , is given by dY k = @g k @t (t;X)dt+ X i @g k @x i (t;X)dX i + 1 2 X i;j @ 2 g k @x i @x j (t;X)dX i dX j (5.20) where dB i dB j =± ij dt;dB i dt=0;dtdB i =0. De¯nition 5.1 Let (;N;P) be a probability space and letfN t g t¸0 be an increasing fam- ily of ¾-algebra,N t ½N for all t. A stochastic process N t :!R is called a supermartin- gale (w.r.t. N t ) if N t is N t -adapted, E[jN t j]<1 for all t and N t ¸E[N s jN t ] for all s >t: (5.21) Similarly, if (5.21) holds with the inequality reversed for all s > t, then N t is called a submartingale. And if (5.21) holds with equality then N t is called a martingale. 85 The oscillator model I with added noise given by the state space equation (5.8) is revisited. We reiterate the equation here for discussion. _ Z= 2 6 6 4 _ z 1 _ z 2 3 7 7 5 = 2 6 6 4 f 1 (Z) f 2 (Z) 3 7 7 5 + 2 6 6 4 0 1 3 7 7 5 F 1 (t); where z 1 =x, z 2 = _ x, f 1 (Z)=z 2 , and f 2 (Z)=¡G 1 z 2 +G 2 sgnz 2 ¡! 2 0 z 1 . We are showing that for a SDE, a Lyapunov function V(z 1 ;z 2 ) is a super-martingale thatimpliesaboutstabilityofsolutionstotheSDE.ForaSDE,adi®usionprocessZ,given a second-order partial di®erential operator L, for z 1 ;z 2 \near 1", we want a Lyapunov function V such that LV(z 1 ;z 2 )<0 for being Lyapunov stable. We de¯ne the operator L acting on the function V as the following LV(z 1 ;z 2 )= @V @z 1 f 1 (z 1 ;z 2 )+ @V @z 2 f 2 (z 1 ;z 2 )+ 1 2 @ 2 V @z 2 2 (5.22) When we rewrite the state equation (5.8) as dz 1 = f 1 (z 1 ;z 2 )dt, and dz 2 = f 2 (z 1 ;z 2 )dt+ dW 1 (t), where dW 1 (t) = F 1 (t)dt and W 1 (t) is the Wiener process with variance ¾ 2 1 t, we use the Theorem 5.2, the general It^ o formula, to represent the dV(z 1 ;z 2 ) as the following dV(z 1 (t);z 2 (t))=LV(z 1 ;z 2 )dt+ @V @z 2 dW 1 (t): (5.23) We then integrate on both sides of equation (5.23) to get the following expression V(z 1 (t);z 2 (t))=V(z 1 (0);z 2 (0))+ Z t 0 LV(z 1 (s);z 2 (s))ds+ Z t 0 @V(z 1 (s);z 2 (s)) @z 2 dW 1 (s): (5.24) 86 By taking the expectation on both sides of equation (5.24), we obtain the following E[V(z 1 (t);z 2 (t))]=V(z 1 (0);z 2 (0))+E[ Z t 0 LV(z 1 (s);z 2 (s))ds]: (5.25) When LV < 0, then E[V(z 1 (t);z 2 (t))]· V(z 1 (0);z 2 (0)) and we conclude from De¯nition 5.1 that V(z 1 (t);z 2 (t)) is a supermartingale. For the operator equation (5.22), we choose the Lyapunov function V(z 1 ;z 2 ) to be the following V(z 1 ;z 2 )= 1 2 (! 2 0 z 2 1 +z 2 2 ): (5.26) Then the operator equation (5.22) becomes the following LV(z 1 ;z 2 )=¡G 1 z 2 2 +G 2 z 2 sgnz 2 + 1 2 (5.27) For large z 1 ;z 2 , where z 2 1 +z 2 2 ¸K 2 , for some large K, two cases are investigated. When z 2 > 0, for LV < 0, the condition z 2 > G 2 + p G 2 2 +2G 1 2G 1 is found. On the other hand, when z 2 < 0, the condition z 2 < ¡G 2 ¡ p G 2 2 +2G 1 2G 1 needs to be satis¯ed for LV < 0. When we combine both conditions, we therefore obtain the condition on z 2 to make LV < 0 for large z 1 ;z 2 . The condition is the following, jz 2 j> G 2 + p G 2 2 +2G 1 2G 1 ; (5.28) to achieve stability for V(z 1 (t);z 2 (t)) to be a super-martingale. 87 Inorderto¯ndaregionofthe(z 1 , z 2 )planewherethesteadystatesolutionislocated, let us de¯ne another Lyapunov function e V, given a generator L, for z 1 ;z 2 \near 0", such that e V(z 1 ;z 2 )!1. We are looking for a Lyapunov function e V such that L e V(z 1 ;z 2 )< 0 forbeingLyapunovstable. Wede¯nesimilarlytheoperatorLactingonthenewLyapunov function e V as L e V(z 1 ;z 2 )= @ e V @z 1 f 1 (z 1 ;z 2 )+ @ e V @z 2 f 2 (z 1 ;z 2 )+ 1 2 @ 2 e V @z 2 2 : (5.29) We again proceed with the same method previously to get the following E[ e V(z 1 (t);z 2 (t))]= e V(z 1 (0);z 2 (0))+E[ Z t 0 L e V(z 1 (s);z 2 (s))ds]: (5.30) Again, when L e V < 0, then E[ e V(z 1 (t);z 2 (t))] · e V(z 1 (0);z 2 (0)) and we conclude from De¯nition 5.1 that e V(z 1 (t);z 2 (t)) is a supermartingale. For the operator equation (5.30), we choose the new Lyapunov function e V(z 1 ;z 2 ) to be the following e V(z 1 ;z 2 )= 1 2(! 2 0 z 2 1 +z 2 2 ) : (5.31) Herewelookatthelargestmagnitudeof z 2 thatitcantakewhenz 1 ;z 2 are\near0". This happens when z 1 ! 0 and we perform the following analysis. We evaluate the operator equation (5.29) with the new Lyapunov function in equation (5.31) and we consider two caseswhenz 1 !0andz 2 iseitherpositiveornegative. Asaresult,weobtainthefollowing bound, jz 2 j< G 2 + p G 2 2 ¡6G 1 2G 1 (5.32) 88 to achieve stability. Furthermore, when we combine the required region of convergence associated with the Lyapunov functions V and e V, we obtain the limit cycle region where the maximum magnitude of z 2 exists, i.e., G 2 + p G 2 2 ¡6G 1 2G 1 <jz 2 j< G 2 + p G 2 2 +2G 1 2G 1 : (5.33) 5.2.5 Stability of the Solution for Oscillator Model II For the oscillator model II, we reiterate the state space representation (5.11) as the fol- lowing 2 6 6 4 _ x _ z 2 3 7 7 5 = 2 6 6 4 f 1 (x;z 2 ) f 2 (x;z 2 ) 3 7 7 5 + 2 6 6 4 0 1 1 0 3 7 7 5 2 6 6 4 F 1 (t) F 2 (t) 3 7 7 5 where z 1 = y, z 2 = _ y, f 1 (x;z 2 ) = z 2 , f 2 (x;z 2 ) = ¡G 1 z 2 + G 2 sgnz 2 ¡ ! 2 0 x, and x = y + R t 0 F 2 (¿)d¿. Again, for a SDE, a di®usion process Z, given a generator L, for x;z 2 \near 1", we want a Lyapunov function V such that LV(x;z 2 ) < 0 for being Lyapunov stable. We de¯ne the operator L acting on the function V as the following LV(x;z 2 )= @V @x f 1 (x;z 2 )+ @V @z 2 f 2 (x;z 2 )+ 1 2 @ 2 V @x 2 + 1 2 @ 2 V @z 2 2 (5.34) 89 When we rewrite the state equation (5.11) as dz 1 = f 1 (z 1 ;z 2 )dt + dW 2 (t), and dz 2 = f 2 (z 1 ;z 2 )dt+dW 1 (t), where dW 1 (t)=F 1 (t)dt, dW 2 (t)=F 2 (t)dt, and W 1 (t), W 2 (t) are in- dependentWienerprocesses,weusetheTheorem5.2,thegeneralIt^ oformula,torepresent the dV(x;z 2 ) as the following dV(x(t);z 2 (t))=LV(x;z 2 )dt+ @V @x dW 2 (t)+ @V @z 2 dW 1 (t): (5.35) By following the same procedure in Section 5.2.4, we obtain the following E[V(x(t);z 2 (t))]=V(x(0);z 2 (0))+E[ Z t 0 LV(x(s);z 2 (s))ds]: (5.36) When LV < 0, E[V(x(t);z 2 (t))] · V(x(0);z 2 (0)), we conclude from De¯nition 5.1 that V(x(t);z 2 (t)) is a supermartingale. For the operator equation (5.34), we choose the Lyapunov function V(x;z 2 ) to be the following V(x;z 2 )= 1 2 (! 2 0 x 2 +z 2 2 ): (5.37) Then the operator equation (5.34) becomes the following LV(x;z 2 )=¡G 1 z 2 2 +G 2 z 2 sgnz 2 + 1 2 (! 2 0 +1): (5.38) For large x;z 2 , where x 2 +z 2 2 ¸ K 2 , for some large K, two cases are investigated. When z 2 > 0, for LV < 0, the condition z 2 > G 2 + p G 2 2 +2G 1 (! 2 0 +1) 2G 1 is found. On the other hand, when z 2 < 0, the condition z 2 < ¡G 2 ¡ p G 2 2 +2G 1 (! 2 0 +1) 2G 1 needs to be satis¯ed for LV < 0. 90 Whenwecombinebothconditions,wethereforeobtaintheconditiononz 2 tomakeLV <0 for large x;z 2 . The condition is the following, jz 2 j> G 2 + q G 2 2 +2G 1 (! 2 0 +1) 2G 1 ; (5.39) to achieve stability for V(x(t);z 2 (t)) to be a super-martingale. In order to ¯nd a region where the steady state solution locates, let us de¯ne another Lyapunov function e V, given a generator L, for x;z 2 \near 0", such that e V(x;z 2 ) ! 1. We are looking for a Lyapunov function e V such that L e V(x;z 2 ) < 0 for being Lyapunov stable. We de¯ne similarly the operator L acting on the new Lyapunov function e V as L e V(x;z 2 )= @ e V @x f 1 (x;z 2 )+ @ e V @z 2 f 2 (x;z 2 )+ 1 2 @ 2 e V @x 2 + 1 2 @ 2 e V @z 2 2 : (5.40) We again proceed with the same method previously to get the following E[ e V(x(t);z 2 (t))]= e V(x(0);z 2 (0))+E[ Z t 0 L e V(x(s);z 2 (s))ds]: (5.41) Again,whenL e V <0,thenE[ e V(x(t);z 2 (t))]· e V(x(0);z 2 (0)),weconcludefromDe¯nition 5.1 that e V(x(t);z 2 (t)) is a super-martingale. For the operator equation (5.41), we choose the new Lyapunov function e V(x;z 2 ) to be the following e V(x;z 2 )= 1 2(! 2 0 x 2 +z 2 2 ) : (5.42) 91 Here we look at the largest magnitude of z 2 that it can take when x;z 2 are \near 0". This happens when x ! 0 and we perform the following analysis. We evaluate the operator equation (5.40) with the new Lyapunov function in equation (5.42) and we consider two caseswhenx!0andz 2 iseitherpositiveornegative. Asaresult, weobtainthefollowing bound on z 2 jz 2 j< G 2 + p G 2 2 +2G 1 (! 2 0 ¡3) 2G 1 (5.43) to achieve stability. Furthermore, when we combine the required region of convergence associated with the Lyapunov functions V and e V, we obtain the limit cycle region where the maximum magnitude of z 2 exists, i.e., G 2 + p G 2 2 +2G 1 (! 2 0 ¡3) 2G 1 <jz 2 j< G 2 + p G 2 2 +2G 1 (! 2 0 +1) 2G 1 : (5.44) 5.2.6 Stability of the Solution for Oscillator Model III For the oscillator model III, we reiterate the state space representation (5.15) as the fol- lowing 2 6 6 4 _ x _ z 2 3 7 7 5 = 2 6 6 4 f 1 (x;z 2 ) f 2 (x;z 2 ) 3 7 7 5 + 2 6 6 4 0 1 1 0 3 7 7 5 2 6 6 4 F 1 (t) F 2 (t) 3 7 7 5 where z 1 = y, z 2 = _ y, f 1 (x;z 2 ) = z 2 , f 2 (x;z 2 ) = ¡G 1 z 2 + G 2 sgnz 2 ¡ ! 2 0 g sl (x), x = y+ R t 0 F 2 (¿)d¿, and the soft-limiter function g sl (x) is de¯ned earlier in equation (5.4). We performedthesimilaranalysisforthestabilityofthesolutionasinSection5.2.5. Thus, by using the same super-martingale condition as in equation (5.41), we come to the following 92 Lyapunov stability condition. When LV < 0, then E[V(x(t);z 2 (t))]· V(x(0);z 2 (0)), we conclude from De¯nition 5.1 that V(x(t);z 2 (t)) is a super-martingale. For the operator equation (5.34), we choose the Lyapunov function V(x;z 2 ) to be the following V(x;z 2 )= 1 2 [2! 2 0 Z x 0 g sl (¿)d¿ +z 2 2 ]: (5.45) Then the operator equation (5.34) becomes the following LV(x;z 2 )=¡G 1 z 2 2 +G 2 z 2 sgnz 2 + 1 2 [! 2 0 g 0 sl (x)+1]; (5.46) where g 0 sl (x)= d dx g sl (x)= 8 > > > > > > < > > > > > > : 0 if x¸1 1 if¡1<x<1 0 if x·¡1 for steady state amplitude of 1. For large x;z 2 , where x 2 +z 2 2 ¸ K 2 , for some large K, two cases are investigated. When z 2 > 0, for LV < 0, the condition z 2 > G 2 + p G 2 2 +2G 1 2G 1 is found. On the other hand, when z 2 < 0, the condition z 2 < ¡G 2 ¡ p G 2 2 +2G 1 2G 1 needs to be satis¯edforLV <0. Whenwecombinebothconditions, wethereforeobtainthecondition on z 2 to make LV <0 for large x;z 2 . The condition is the following, jz 2 j> G 2 + p G 2 2 +2G 1 2G 1 ; (5.47) to achieve stability for V(x(t);z 2 (t)) to be a super-martingale. 93 For the operator equation (5.41), we choose the new Lyapunov function e V(x;z 2 ) to be the following e V(x;z 2 )= 1 2(2! 2 0 R x 0 g sl (¿)d¿ +z 2 2 ) : (5.48) Here we look at the largest magnitude of z 2 that it can take when x;z 2 are \near 0". This happens when x ! 0 and we perform the following analysis. We evaluate the operator equation (5.40) with the new Lyapunov function in equation (5.42) and we consider two caseswhenx!0andz 2 iseitherpositiveornegative. Asaresult, weobtainthefollowing bound on z 2 jz 2 j< G 2 + p G 2 2 +2G 1 (! 2 0 ¡3) 2G 1 (5.49) to achieve stability. However, if we re-evaluate the operator function shown in equation 5.46 for small x, we obtain the following bound on z 2 when x 2 +z 2 2 ·K, where K is some large number and the bound is jz 2 j> G 2 + p G 2 2 +2G 1 (! 2 0 +1) 2G 1 (5.50) to achieve stability. Furthermore, when we combine the required region of convergence associated with the Lyapunov functions V and e V, we obtain the limit cycle region where the maximum magnitude of z 2 exists, i.e., G 2 + p G 2 2 +2G 1 (! 2 0 ¡3) 2G 1 <jz 2 j< G 2 + p G 2 2 +2G 1 (! 2 0 +1) 2G 1 : (5.51) 94 5.3 Analysis of Oscillator Model I Inthissection,wewillanalyzethebehavioroftheoscillatormodelIwithnoiseinequation (5.1) and show the e®ect of non-linearity associated with di®erent damping coe±cients, on the performance of the system when noise is present. We have previously discussed the existence and the stability of the solution to the system equation (5.8) when noise is present. Moreover, we obtain the stability region of the solution when noise is present. To further understand the e®ect of non-linearity on the overall amplitude and phase of the oscillator model when noise is present will help to fully describe the behavior of the system. Thus, noise e®ects on the oscillators and the non-linearity e®ect on the noise are analyzed. As the research progresses, we will propose a new phase noise model of oscillators. Results analyzed and simulated can be represented by this new noise measure. As a start, we need to transform the system equation (5.1) into a polar coordinate system, where we fully describe the dynamics of the system. 5.3.1 Transformation of the System Equation We are interested in transforming the system equation (5.1) into a system of equations representing the amplitude and phase dynamics of the system when the noise is present. From the state space equation (5.8), we do the transformation by ¯rst setting y = z 2 ! 0 , we obtain the following new system of equation to be _ z 1 = ! 0 y _ y = ¡! 0 z 1 +g(y)+h 1 (t) ; (5.52) 95 where g(y) = ¡G 1 y + G 2 ! 0 sgn(! 0 y), h 1 (t) = ¾ F 1 ! 0 e F 1 (t), F 1 (t) = ¾ F 1 e F 1 (t) and e F 1 (t) has unit variance. Now we transform the above state space equation into a model in polar coordinates represented by the amplitude ½ and the total phaseµ. Weletz 1 =½cosµ, and y =½sinµ, we get the following representation in polar coordinates as _ z 1 = _ ½cosµ¡½ _ µsinµ =! 0 ½sinµ _ y = _ ½sinµ+½ _ µcosµ =¡! 0 ½cosµ+g(½sinµ)+h 1 (t) : (5.53) When we multiply the _ z 1 equation of the equation (5.53) by cosµ and _ y equation by sinµ, and add them, we get the di®erential equation for the amplitude ½ as _ ½=g(½sinµ)sinµ+h 1 (t)sinµ : (5.54) Similarly, we get the di®erential equation for the total phase µ to be _ µ =¡! 0 + g(½sinµ) ½ cosµ+ h 1 (t) ½ cosµ : (5.55) From both equations (5.54), (5.55), we see the dynamics of both quantities ½ and µ when noise is present, the e®ect of non-linearity associated with the g(½sinµ) function, and the additionaltermofnoisecontribution. However,itisnotclearhowtheamplitude½andthe totalphaseµ behavewhenthenoiseispresentfromthesetwodynamicequations. Wehave previously investigated the non-linearity e®ect on oscillators in Section 4.2 under no noise condition. When noise is present in the oscillator model, the e®ect of noise on oscillators and the e®ect of non-linearity on the noise are areas of interests for investigation. 96 Since noise is present in the system of equations (5.54) and (5.55), the steady state amplitude function driven by the noise F 1 (t) is obtained by taking an ensemble average of the asymptotic amplitude function. Furthermore, there is no equilibrium point on the phaseplaneofthetotalphasefunctionµ(t)forlarge! 0 . AspreviouslydiscussedinSection 4.2.1, there exists a lower bound on ! 0 where oscillation is no longer possible. When noise is present in the oscillator model, the lower bound on ! 0 will be larger. To further demonstrate the e®ect of noise on oscillators and to see the e®ect of non- linearity, we need to evaluate and analyze the system equations (5.54) and (5.55). We will perform simulation on these two system equations. As a start, we will ¯rst introduce the new phase noise model followed by the analytic solution. It is then compared to other noise measures. 5.3.2 New Phase Noise Model We propose a new phase noise model for oscillator phase noise analysis. Given the total phase di®erence over the time interval (t m ;t), we de¯ne the di®erence of the phase noise process as the following Á(t)¡Á(t m )=µ(t)¡µ(t m )¡E[µ(t)¡µ(t m )jµ(t 0 );½(t 0 );t 0 ·t m ] ; (5.56) where total phase function µ(t) and the amplitude function ½(t) are determined from system equations (5.54) and (5.55). This process in equation (5.56) is interpreted as the predictionerrorofthetotalphaseµ(t)atcurrenttimetgiventhepasthistoryoftotalphase 97 and amplitude up to some time t m < t. Moreover, we are interested in the mean and the variance of the di®erence of the phase noise process as the measure for oscillator phase noise. We evaluate the mean of the phase noise process in equation (5.56) by taking the conditional mean as the following E[Á(t)¡Á(t m )jµ(t 0 );½(t 0 );t 0 ·t m ]=0 (5.57) and we obtain zero mean for the phase noise process by taking the expectation on the equation (5.57). Is the new phase noise process a martingale? If it is a martingale, we can simplify the analysis. Let us ¯rst de¯ne a martingale [27]. De¯nition 5.2 A stochastic process X(t), t2[0;1), such that 1. X(t) is X(t)-measurable for all t 2. E[X(t)]<1 for all t 3. E[X(t 2 )jX(t 0 );t 0 ·t 1 ]=X(t 1 ) for all t 1 ·t 2 from the time domain [0;1) is called a martingale. Using the martingale de¯nition 5.2, we prove the following corollary for the new phase noise process. 98 Corollary 5.3 The phase noise process Á(t)¡Á(t m )=µ(t)¡µ(t m )¡E[µ(t)¡µ(t m )jµ(t 0 );½(t 0 );t 0 ·t m ] is not a martingale. Proof LetF s =¾fµ(t s 0);½(t s 0);s 0 ·sgbethe¾-algebrageneratedbythepasthistory of (µ(t s 0);½(t s 0)) for s 0 · s. Without loss of generality, we assume that the starting time is at t m =0. We therefore have the following Á(t)¡Á(0)=µ(t)¡µ(0)¡E[µ(t)¡µ(0)jµ(0);½(0)]: (5.58) Now let us take the conditional expectation on equation (5.58) conditioned on the ¾- algebra F s and we divide the time series into in¯nitestimal spacings of n segments, we obtain the following for 0<t s <t E[Á(t)¡Á(0)jF s ] = n X k=0 E[dÁ(t k )jF s ] = n X k=0 E[dµ(t k )jF s ]¡ n X k=0 E[dµ(t k )jµ(0);½(0)] = µ(t s )¡µ(0)+ n X k=s E[dµ(t k )jF s ]¡ n X k=0 E[dµ(t k )jF 0 ] (5.59) To see if equation (5.59) is a martingale, we need E[Á(t)¡Á(0)jF s ] = Á(t s )¡Á(0) to be true. As compared to equation (5.58) with t=t s , we need the following to be true E[µ(t)¡µ(t s )jF s ]¡E[µ(t)¡µ(0)jµ(0);½(0)]=¡E[µ(t s )¡µ(0)jµ(0);½(0)]: (5.60) 99 If E[µ(t)¡µ(t s )jF s ] =E[µ(t)¡µ(t s )jF 0 ], then the equality holds on equation (5.60) and Á(t) is a martingale. Since the new phase noise process is not a martingale, its statistical property needs to be explored further. Let us return to our problem on a set of system equations (5.54) and (5.55), we verify the statistical property of the new phase noise. We obtain the new phase noise process as the following Á(t)¡Á(t m )= Z t tm g(½(¿)sinµ(¿))cosµ(¿) ½(¿) ¡E[ g(½(¿)sinµ(¿))cosµ(¿) ½(¿) j µ(t 0 );½(t 0 );t 0 ·t m ]d¿ + ¾ F 1 ! 0 Z t tm F 1 (¿) ½(¿) cosµ(¿)d¿: (5.61) We then take the conditional expectation on equation (5.61) and the result follows E[Á(t)¡Á(t m )jµ(t 0 );½(t 0 );t 0 ·t m ] (5.62) = ¾ F 1 ! 0 Z t t m E[ cosµ(¿) ½(¿) dW 1 (¿)jµ(t 0 );½(t 0 );t 0 ·t m ] = ¾ F 1 ! 0 Z t tm E[ cosµ(¿) ½(¿) jµ(t 0 );½(t 0 );t 0 ·t m ]E[dW 1 (¿)jµ(t 0 );½(t 0 );t 0 ·t m ]=0; where W 1 (t) is the standard Wiener process for the white noise process F 1 (t). Therefore, when taking the expectation on the equation (5.62), the mean of the new phase noise process Á(t)¡Á(t m ) is veri¯ed to be zero. The variance of the new phase noise process is di±cult to obtain analytically when evaluating the conditional variance directly from 100 equation (5.61). Without going into evaluating the variance of the phase noise from equa- tion (5.61), we instead ¯nd the variance of the new phase noise process from equation (5.56) as the following var(Á(t)¡Á(t m )) = E[fµ(t)¡µ(t m )¡E[µ(t)¡µ(t m )jµ(t 0 );½(t 0 );t 0 ·t m ]g 2 ] = E[fµ(t)¡E[µ(t)jµ(t 0 );½(t 0 );t 0 ·t m ]g 2 ] = EfE[µ 2 (t)jµ(t 0 );½(t 0 );t 0 ·t m ]¡E 2 [µ(t)jµ(t 0 );½(t 0 );t 0 ·t m ]g (5.63) 5.3.3 Analytic Solution of the Conditional Expectation and Application Toevaluatetheconditionalexpectationanalytically,wetakeanapproachsimilartosolving theKolmogorov'sbackwardequation. Westartwiththede¯nitionoftheMarkovproperty for It^ o di®usion [26] that will be used throughout the proof. De¯nition 5.3 (The Markov property for It^ o di®usions). Let f be a bounded Borel function fromR n toR. Then, for t;h¸0 E x [f(X t+h )j ~ F (~ m) t ] (!) =E Xt(!) [f(X h )] (5.64) Here E x denotes the expectation w.r.t. the probability measure Q x , where Q x is the proba- bility law of X t starting at x (X 0 =x). The right hand side means the function E y [f(X h )] evaluated at y =X t (!). 101 We have used the following ¾-algebra representation where ~ F (~ m) t = ¾fB r ;r · tg, M t = ¾fX r ;r ·tg, B r is ~ m-dimensional Brownian motion, and X r is n-dimensional. We estab- lish (see Theorem 5.1) that X t is measurable with respect to F (~ m) t , so M t µ F (~ m) t . We obtain the following conditional expectation E x [f(X t )jM t m ]=E x [f(X t )jX [0;tm] ;t m <t] =E x [E x [f(X t )jF ~ m t m ]jM t m ]=E x [E X t m [f(X t¡t m )]jM t m ]=E X t m [f(X t¡t m )]: (5.65) For a time-homogeneous It^ o di®usion X t inR n , dX t =v(X t )dt +¾(X t )dW t where X t is n-dimensional and W t is m-dimensional. We use the general It^ o formula of equation (5.20) from Theorem 5.2 on the above equation and integrate, we obtain the following f(X t ) = f(X tm )+ Z t t m X i v i (X s ) @f @x i (X s )+ 1 2 X i;j (¾¾ T ) i;j (X s ) @ 2 f @x i @x j (X s )ds + Z t tm X i @f @x i (X s )(¾dW) i : (5.66) 102 We then take the conditional expectation on equation (5.66), the result is the following E x [f(X t )jX [0;t m ] ;t m <t]=E x [f(µ t ;½ t ) j µ(t 0 );½(t 0 );t 0 ·t m <t] =f(X tm )+ Z t t m E x [ X i @f @x i (X s )+ 1 2 X i;j (¾¾ T ) i;j (X s ) @ 2 f @x i @x j (X s ) ¯ ¯ X [0;t m ] ;t m <t]ds: (5.67) Wethende¯nethegeneratorAofanIt^ odi®usion,whereasecondorderpartialdi®erential operator A is the generator of the process X t . De¯nition 5.4 Let X t be a (time-homogeneous) It^ o di®usion inR n . The (in¯nitesimal) generator A of X t is de¯ned by Af(x 0 ) ¯ ¯ X tm =x 0 = lim t#tm E x [f(X t )jX [0;t m ] ;t m <t]¡f(x 0 ) (t¡t m ) ; x 0 2R n : (5.68) The set of functions f :R n !R such that the limit exists at x is denoted byD A (x 0 ), while D A denotes the set of functions for which the limit exists for all x 0 2R n . The condition expectation in equation (5.67) is then evaluated in the generator A of the It^ o di®usion as the following Af(x 0 ) ¯ ¯ X tm =x 0 =lim t#t m 1 (t¡t m ) Z t t m E[ X i ¾ i (X s ) @f @x i (X s )+ 1 2 X i;j (¾¾ T ) i;j (X s ) @ 2 f @x i @x j (X s ) ¯ ¯ X [0;t m ] ;t m <t]ds = X i ¾ i (x 0 ) @f @x 0 i (x 0 )+ 1 2 X i;j (¾¾ T ) i;j (x 0 ) @ 2 @x 0 i @x 0 j (x 0 ): (5.69) We de¯ne the following theorem using the results just obtained. 103 Theorem 5.4 Let X t be the It^ o di®usion dX t =v(X t )dt +¾(X t )dW t If f 2C 2 0 (R n ) then f 2D A and Af(x)= X i ¾ i (x) @f @x i (x)+ 1 2 X i;j (¾¾ T ) i;j (x) @ 2 @x i @x j (x): (5.70) We then come to a modi¯ed Dynkin's formula for time t non-random as the following theorem. Theorem 5.5 (Modi¯ed Dynkin's formula). Let f 2C 2 0 (R n ), then E x [f(X t ) j X [0;tm] ;t m <t]=E x 0 [f(X t¡t m )] ¯ ¯ X tm =x 0 =f(X t m )+ Z t tm E x [Af(X s ) j X [0;tm] ;t m <t]ds: (5.71) Proof We use the results of the operator of the It^ o di®usion in equation (5.70) with x replaced by X s of the Theorem 5.4 and substitute back in equation (5.67). By using the Markov property, we obtain the stated result for the modi¯ed Dynkin's formula. We now let u(t¡t m ;x 0 )=E x 0 [f(X t¡tm )] in the modi¯ed Dynkin's formula in equation (5.71), and take a ¯rst derivative with respect to t, we get the following equation @ @t u(t¡t m ;x 0 )=E x [Af(X t ) j X [0;t m ] ;t m <t]: (5.72) 104 It turns out that the right hand side of equation (5.72) is expressed in terms of u also: Theorem 5.6 (Kolmogorov's backward equation). Let f 2 C 2 0 (R n ). De¯ne u(t¡t m ;x 0 ) =E x 0 [f(X t¡t m )]. Then u(t¡t m ;¢)2D A for each t and @ @t u(t¡t m ;x 0 )=Au(t¡t m ;x 0 ); X t m =x 0 ; t>0; x2R n (5.73) u(0;x 0 )=f(X t m ); x2R n (5.74) proof Let h(x 0 ) = u(t¡t m ;x 0 ) and since t! u(t¡t m ;x 0 ) is di®erentiable, we have the following 1 (t r ¡t m ) E x [h(X t r ) j X [0;t m ] ;t m <t r ]¡h(x 0 ) = 1 (t r ¡t m ) E x [E Xt r [f(X t¡t m )] j X [0;tm] ;t m <t r <t]¡E x 0 [f(X t¡t m )] = 1 (t r ¡t m ) fE x [E x [f(X t+t r ¡t m ) j X [0;t r ] ;t r <t] j X [0;t m ] ;t m <t r ] ¡E x [f(X t ) j X [0;tm] ;t m <t]g = 1 (t r ¡t m ) fE x [f(X t+t r ¡t m ) j X [0;tm] ;t m <t r ]¡E x [f(X t ) j X [0;tm] ;t m <t]g = u(t+t r ¡2t m ;x 0 )¡u(t¡t m ;x 0 ) (t r ¡t m ) ! @u @t as t r #t m By de¯nition Au=lim t r #t m E x [h(X tr ) j X [0;t m ] ;t m <t r ]¡h(x 0 ) (tr¡tm) exists and @u @t =Au, as asserted. Therefore, the conditional expectation E x 0 [f(X t¡t m )] orE x [f(X t )j X [0;tm] ;t m <t] can be solved by the Kolmogorov's backward equation of @u @t =Au with the initial condition 105 u(0;x 0 ) = f(X tm ) and the operator Au satis¯es equation (5.70). We then apply this important result to solving the conditional expectation on the evaluation of the variance of the new phase noise process Á(t)¡Á(t m ). Here we reiterate the new phase noise process as follows Á(t)¡Á(t m )=µ(t)¡µ(t m )¡E[µ(t)¡µ(t m )jµ(t 0 );½(t 0 );t 0 ·t m ] ; whose variance is evaluated in equation (5.63). Thus, we are interested in looking for the analytic solutions of two conditional expectations, E x [µ(t) j X [0;tm] ;t m < t] and E x [µ 2 (t) j X [0;tm] ;t m < t]. By letting u(t¡ t m ;x 0 ) = E x [f(X t ) j X [0;tm] ;t m < t], we can solve for the partial di®erential equation (5.73) given the initial condition in equation (5.74), where the operator Au(t¡t m ;x 0 ) is given by Au(t¡t m ;x 0 )= X i v i (x 0 ) @ @x 0 i u(t¡t m ;x 0 )+ 1 2 X i;j (¾¾ T ) i;j (x 0 ) @ @x 0 i @x 0 j u(t¡t m ;x 0 ): To¯ndtheconditionalmeanandthesecondmomentofthenewphasenoiseprocessforthe oscillatormodelIshowninequations(5.54)and(5.55),wecandothefollowingformulation and solve the partial di®erential equation of the backward Kolmogorov's equation @ @t u(t¡t m ;µ 0 ;½ 0 )=[¡! 0 + g(½ 0 sinµ 0 )cosµ 0 ½ 0 ] @ @µ 0 u(t¡t m ;µ 0 ;½ 0 )+[g(½ 0 sinµ 0 )sinµ 0 ]¢ @ @½ 0 u(t¡t m ;µ 0 ;½ 0 )+ 1 2 ¾ 2 F 1 cos 2 µ 0 ! 2 0 (½ 0 ) 2 @ 2 @(µ 0 ) 2 u(t¡t m ;µ 0 ;½ 0 )+ ¾ 2 F 1 sinµ 0 cosµ 0 ! 2 0 ½ 0 @ 2 @µ 0 @½ 0 u(t¡t m ;µ 0 ;½ 0 )+ ¾ 2 F 1 sin 2 µ 0 2! 2 0 @ 2 @(½ 0 ) 2 u(t¡t m ;µ 0 ;½ 0 ); (5.75) 106 with initial condition u(0;x 0 ) = u(0;µ 0 ;½ 0 ) = f(X tm ) = µ 0 for the evaluation of the con- ditional mean of µ(t), where g(½ 0 sinµ 0 ) =¡G 1 ½ 0 sinµ 0 + G 2 ! 0 sgn(! 0 ½ 0 sinµ 0 ). Similarly, the evaluation of the conditional second moment of µ(t) can be proceeded with the initial condition u(0;x 0 ) = u(0;µ 0 ;½ 0 ) = (µ 0 ) 2 . From the backward equation (5.75), more prop- erties are explored such as the boundary condition. We have the boundary condition u(t¡t m ;½ 0 ;µ 0 +n¼)=u(t¡t m ;½ 0 ;µ 0 ) since the coe±cients of equation (5.75) are periodic in µ 0 . Therefore, we can ¯nd the solution to the partial di®erential equation (5.75) with the following boundary conditions, u(t¡t m ;½ 0 ;¼)=u(t¡t m ;½ 0 ;¡¼) at boundary (¡¼;¼) of µ 0 , and u(t¡t m ;1;µ 0 )=u(t¡t m ;0;µ 0 )=0. To solve the PDE (5.75) with the speci¯ed initial and boundary conditions, there is no easy method. We use the numerical methods [39], [10], [12], [37], [32] to solve for such 3-dimensional, second-order partial di®erential equation. However, due to the issue of numerical instability and the sensitivity of numerical method, we can not obtain a reasonable result. Hence, other methods are explored such as performing simulations on the system equations (5.54) and (5.55). 5.3.4 Simulation of the Oscillator Model The oscillator model I described in equation (5.1) with the state space representation in equation (5.8) is described as a block diagram shown in Fig. 5.4. We perform simulations on this oscillator model and verify the existence and stability of the solution. To further analyzethee®ectsof non-linearityonthe amplitudeandtotalphaseofthe oscillatorwhen noise is present, we need to evaluate the di®erential equations (5.54) and (5.55). 107 ∫ ∫ + X 0 ω ) ( 1 t F ( ) 2 − + + - x 2 G 1 G x & Figure 5.4: Schematic of the oscillator model I from equation (5.8) Due to the di±culty of obtaining the analytic solutions of the amplitude and phase functions to the system of equations (5.54) and (5.55) when noise is present, we perform simulations on these equations. For the system in polar coordinate form, we ¯rst simulate the system at the clock rest frequency ! 0 = 2000¼ and assign the coe±cients G 1 and G 2 such that the steady state amplitude of the oscillator is approximately1. We use Matlab's Simulink to simulate the system of equations. The Simulink's block diagram is shown in Fig. 5.5, where G 1 =1000;G 2 = G 1 ¼! 0 4 , and the internal oscillator noise F 1 (t) being white Gaussian with variance ¾ 2 F 1 . Other simulation technique for SDE can be found in [25]. A completed simulated solution show the e®ect of noise, non-linearity, and constant gains on both amplitude and phase of the oscillator, and we observe the the e®ect of non-linearity 108 ∫ ∫ X X u 1 ) sin(u ) cos(u X 1 G + - 0 ω X X + - 0 ω ρ ρ & θ θ & θ ρ θ ρ cos ) sin ( g 0 2 ω G X + + X 0 1 ω X X + + ) ( 1 t F Figure5.5: Simulinkblockdiagramshowingtheamplitudeandphasedi®erentialequations (5.54),(5.55)withtheclockrestfrequency! 0 =2000¼ wheninternalnoiseF 1 (t)ispresent on the new phase noise process as well. The interaction of the amplitude and phase functions from equations (5.54) and (5.55) when noise is present is also seen and the noise e®ect is investigated. We start the simulation at the initial conditions ½(0) = 0:125;µ(0) = 0, at the noise variance ¾ 2 F 1 = 100! 2 0 , we see the noise e®ect on the amplitude function, total phase, steady state oscillation, and the non-linearity. There are several simulated functions that we are interested in from the schematic shown in Fig. 5.5. They include functions ½(t), µ(t), amplitude and phase noise e®ect on non-linearity, the total noise e®ect on _ ½ and on _ µ, and the oscillator output ½cosµ. 109 0.992 0.9925 0.993 0.9935 0.994 0.9945 0.995 0.9955 0.996 0.9965 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ρ(0.02) F(ρ(0.02)) empirical cdf cdf of normal distribution Figure 5.6: Cumulative distribution function (cdf) of the amplitude function ½(0:02) for 1000 realizations vs. Gaussian cdf We begin by showing the empirical cumulative distribution function for the amplitude function ½ where we start the simulation at time t m = 0 for 1000 runs. By the Berry- Ess¶ een theorem [14], we see from Fig. 5.6 that at 1000 sample size of the internal noise realizations, the rate of convergence to normality is reasonably quite fast, making the amplitude function ½(t) Gaussian distributed. We next show the e®ect of noise on non- linearity including the amplitude and the phase noise. AsalreadydiscussedinSection4.2.2aboutthee®ectofnon-linearityontheamplitude and the total phase of the oscillator model withoutnoise present, we will thusdescribe the e®ect of noise on non-linearity. When the internal noise is present, the simulated results for 100 runs shown in Fig. 5.7 indicate that noise has somehow changed the shape of non- linearity g(½sinµ) in the system of equations (5.54) and (5.55). Thus the instantaneous changeoftheamplitudefunction _ ½whennoiseispresentandthenoisee®ectontheangular frequency drift _ µ d = g(½sinµ)cosµ ½ suggest that the noise on ½ and µ have some e®ects on 110 0.1 0.1001 0.1002 0.1003 0.1004 0.1005 0.1006 0.1007 0.1008 0.1009 0.101 −1000 −500 0 500 1000 time (sec.) g(ρsinθ)cosθ/ρ 0.1 0.1001 0.1002 0.1003 0.1004 0.1005 0.1006 0.1007 0.1008 0.1009 0.101 −300 −200 −100 0 100 200 time (sec.) g(ρsinθ)sinθ Figure 5.7: E®ect of noise on the non-linearity function g(½sinµ) for both amplitude and phase functions their overall shape respectively as well as a function of time. When we combine the noise contribution with the non-linearity term, we see the total feedback e®ect to the system of stochastic di®erential equations (5.54) and (5.55). This is seen in Fig. 5.8, where _ µ+! 0 and _ ½ are plotted and the feedback is distorted and intensi¯ed when the noise is present as compared to Fig. 5.7. To further analyze the results, we do the following. We start by plotting the steady state phase path of the expected amplitude function, i:e:;E[_ ½] vs. E[½] in Fig. 5.9 where the approximate steady state solution of the expected amplitude function is obtained by taking an average on the realizations. As compared to Fig. 4.7, the uncertainty region of the steady state oscillation increases due to noise. 111 0.1 0.1001 0.1002 0.1003 0.1004 0.1005 0.1006 0.1007 0.1008 0.1009 0.101 −1000 −500 0 500 1000 time (sec.) dθ/dt + ω 0 0.1 0.1001 0.1002 0.1003 0.1004 0.1005 0.1006 0.1007 0.1008 0.1009 0.101 −300 −200 −100 0 100 200 time (sec.) dρ/dt Figure 5.8: Sum of the noise term and the non-linearity to the amplitude and total phase stochastic di®erential equations (5.54) and (5.55) We then study the new phase noise process Á(t)¡Á(t m ) de¯ned earlier in equation (5.56) reiterated here as the following Á(t)¡Á(t m )=µ(t)¡µ(t m )¡E[µ(t)¡µ(t m )jµ(t 0 );½(t 0 );t 0 ·t m ] : The variance of the new phase noise process is simulated as shown in Fig. 5.10 where the starting time t m = 0. The mean of the new phase noise process is veri¯ed to be zero and the variance of the process has some interesting behaviors. The variance of the new phase noiseprocessincreaseswithtimeanditsshapeismodi¯edbythenon-linearity. Thespread of the variance increases over time and the e®ect of non-linearity on the phase noise are clearly seen as shownonthe ¯rst ¯gureof Fig. 5.10. As compared to the traditional phase noise of a Wiener process, the new phase noise process behaves like a modi¯ed Wiener 112 0.985 0.99 0.995 1 1.005 1.01 1.015 1.02 −250 −200 −150 −100 −50 0 50 100 150 200 E[ρ] E[dρ/dt] Figure 5.9: Expected phase path of the amplitude function ½(t) at steady state process with non-linearity e®ect. Thus, the new measure of phase noise captures the e®ect of non-linearity for non-linear oscillators. Next we look into the distribution of the new phase noise process. The simulation result with 1000 realizations suggests that the new phase noise process Á(t)¡Á(t m ) is Gaussian distributed as shown in Fig. 5.11. Furthermore, we verify that the process is time-homogeneous from simulations starting at di®erent initial times. Moreover, the process is found to have independent increments from the simulation results of the phase noisevariance. Sinceweobservethetime-increasing(unbounded)propertyofthevariance of the process in Fig. 5.10 with the shape modi¯ed by the non-linearity and the fact that the process is zero mean, weconclude that the new phase noise process behaves like a time-homogenousmodi¯edWienerprocesswhosevarianceismodi¯edbythenon-linearity. This is what makes the process not a martingale. 113 0 0.005 0.01 0.015 0.02 0.025 0.03 0 0.5 1 1.5 2 2.5 3 x 10 −5 time (sec.) rad 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.5 1 1.5 2 2.5 3 x 10 −4 time (sec.) rad 2 Figure 5.10: var(µ(t)¡E[µ(t)jµ(0);½(0)]) or var(Á(t)¡Á(0)) for 100 realizations where µ(0)=0, ½(0)=0:125 The next thing of interest is that if we can apply the same method as we de¯ned the new phase noise process in equation (5.56) to the amplitude noise process as the following ¢½(t;t m )=½(t)¡½(t m )¡E[½(t)¡½(t m )jµ(t 0 );½(t 0 );t 0 ·t m ]: (5.76) We also de¯ne a notation similar to the innovation process as the following E[½(t)jµ(t 0 );½(t 0 );t 0 ·t m ]¡½(t) ¯ ¯ no noise ; (5.77) that we name as a modi¯ed innovation process. The amplitude noise process and the modi¯ed innovation process are simulated as shown in Fig. 5.12. The variance of the amplitude noise process is quite jittery but with an approximately constant value, while 114 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 φ(0.04) −φ(0)= θ(0.04) − E[ θ(0.04) | θ(0), ρ(0)] F(φ(0.04)−φ(0)) empirical cdf cdf of normal distribution Figure 5.11: Empirical cdf of the new phase noise process Á(0:04)¡ Á(0) = µ(0:04)¡ E[µ(0:04)jµ(0)=0;½(0)=0:125] over 1000 realizations the modi¯ed innovation process and the amplitude noise process is veri¯ed to be Gaussian distributed. We conclude that the amplitude function is bounded with a small prediction error shown in the second ¯gure of Fig. 5.12 for equation (5.77). Furthermore, for a small approximately constant variance of the amplitude noise process in equation (5.76), the expected amplitude function is therefore bounded and this further veri¯es the stability condition shown in Section 5.2.4. The next thing we are interested in pursuing is the oscillator output phase plane rep- resentation ½cosµ vs. ½sinµ when noise is present. We will verify the stability condition for the oscillator model discussed in Section 5.2.4. The constant gains G 1 and G 2 satisfy the existence condition of the solution in equation (5.9), we take the average of the sim- ulated outputs ½cosµ and ½sinµ to show the stability of the solution. When we take the expectation on the Lyapunov stability condition such that E[LV] < 0 or E[L e V] < 0 in 115 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.2 0.4 0.6 0.8 1 1.2 x 10 −6 Var(ρ(t) − E[ρ(t) | ρ(0), θ(0)]) time (sec.) 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 −2 0 2 x 10 −4 E[ρ(t) | ρ(0), θ(0)] − ρ(t)| no noise time (sec.) Figure 5.12: Variance of the amplitude noise process ½(t)¡E[½(t)jµ(0);½(0)] and the modi¯ed innovation process E[½(t)jµ(0);½(0)]¡½(t)j no noise Section 5.2.4 by relaxing the requirement that LV < 0 or L e V < 0, we get the region of limit cycle where the maximum normalized E[ z 2 ! 0 ] can exist, i.e., G 2 + p G 2 2 ¡6G 1 2G 1 ! 0 <jE[ z 2 ! 0 ]j< G 2 + p G 2 2 +2G 1 2G 1 ! 0 : (5.78) Again the simulation results obtained are used to compute the E[½cosµ] vs. E[½sinµ] shown in Fig. 5.13 which indicates the stability of the expected oscillator output phase path diagram. Thus, we have veri¯ed the stability of the oscillator model I when noise is present. 116 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 E[ρcosθ] E[ρsinθ] Figure 5.13: E[½cosµ] vs. E[½sinµ] 5.3.5 The E®ect of Non-linearity on the Phase Noise In this section, we are interested in knowing the e®ects of non-linearity on the new phase noise process de¯ned in equation (5.56). As seen in Fig. 5.10 that the variance of the phase noise process is modi¯ed by the non-linearity employed in the oscillator model, we would like to alleviate the e®ect to some degree. Thus, we will investigate the e®ect of di®erent types of non-linearity on the phase noise process. Two types of non-linearity are investigated similarly as performed in Section 4.2.3 and they are hard-limiter and soft-limiter with di®erent slopes. From the amplitude and phase stochastic di®erential equations (5.54) and (5.55), we replace the hard-limiter within the g(½sinµ) function with a soft-limiter. Furthermore, we evaluate the e®ects of the model employing a soft-limiter with di®erent slopes and compare the results with the one with the hard-limiter non-linearity. 117 Similarly, we ¯rst evaluate the model employing with a soft-limiter non-linearity by simulation on the amplitude and the phase schematic with noise in Fig. 5.5 with the non- linearity replaced by a soft-limiter. The new amplitude and phase stochastic di®erential equation are described as the following _ ½ = g 2 (½sinµ)sinµ+h 1 (t)sinµ (5.79) _ µ = ¡! 0 + g 2 (½sinµ) ½ cosµ+ h 1 (t) ½ cosµ ; (5.80) where h 1 (t) = ¾ F 1 ! 0 e F 1 (t), g 2 (½sinµ) =¡G 1 ½sinµ+ G 2 ! 0 s sl (! 0 ½sinµ), F 1 (t) = ¾ F 1 e F 1 (t), the soft-limiter s sl (y)= 8 > > > > > > < > > > > > > : 1 if y¸1 py if¡1<py <1 ¡1 if y·¡1 (5.81) and p is the slope of the soft-limiter with the range 0 < p · 1. As shown previously in Section 4.2.3, for the soft-limiter with di®erent slopes p, we can reduce the feedback e®ect from the non-linearity to the system of SDEs (5.79) and (5.80). Generally, for ¯xed constant gains G 1 and G 2 , we reduce the feedback e®ect of non-linearity while still maintaining the required oscillation steady state amplitude by reducing the slope of the soft-limiter even in the presence of noise. We have previously shown the e®ect of di®erent non-linearities on the total phase µ which includes the phase drift term µ d and the e®ects on the steady state amplitude ½ s in Fig. 4.11 when noise is not present. However, when noise is present in the oscillator model, we still see the noise e®ect on the phase drift for models employed with di®erent non-linearities as well as on the approximate steady state 118 amplitude. Here, we focus on the e®ect of di®erent non-linearities on the oscillator model by investigating the e®ect on the new phase noise process Á(t)¡Á(t m ). Beforegoingintothesimulationresultsfordi®erentnon-linearitiesusedintheoscillator model, we characterize the following de¯nitions [31]. De¯nition 5.5 Given the new phase noise process Á(t)¡Á(t m ), the frequency deviation is de¯ned as ±f(t¡t m )= · var(Á(t)¡Á(t m )) [2¼(t¡t m )] 2 ¸ 0:5 : (5.82) De¯nition 5.6 Given the new phase noise process Á(t)¡ Á(t m ), the normalized RMS frequency deviation is de¯ned as ±f(t¡t m ) f 0 = · var(Á(t)¡Á(t m )) (2¼f 0 ) 2 ¸ 0:5 1 (t¡t m ) = ½ D (1) Á (t¡t m ) [2¼f 0 (t¡t m )] 2 ¾ 0:5 ; (5.83) where D (1) Á (t¡t m ) is the ¯rst structure function of Á(t). We next show the simulation results for the oscillator model shown in Fig. 5.5 with the non-linearity replaced by a soft-limiter. The simulation is performed with di®erent slopes p of the soft-limiter from equation (5.81). At the clock rest frequency f 0 =1000, the slope ofthesoft-limiterp=0:001, weobtainthenormalizedstandarddeviationofthenewnoise process Á(t)¡Á(t m ) for t m = 0, and the normalized RMS fractional frequency deviation de¯ned in equation (5.83) as shown in Fig. 5.14. We see from Fig. 5.14 that the spread of the normalized standard deviation of the phase noise process decrease as compared to the oscillator model employed with a hard-limiter non-linearity shown in Fig. 5.10 and the normalized RMS fractional frequency deviation decreases over time to an approximate 119 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 x 10 −5 time (sec.) radian/Hz 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 1 2 3 4 5 6 7 x 10 −5 time (sec.) b. a. Figure 5.14: (a) normalized standard deviation of µ(t)¡E[µ(t)jµ(0)=0;½(0)=0:125] for theoscillatormodelemployingasoft-limiterwithslopep=0:001and(b)normalizedRMS frequency deviation [var(Á(t)¡Á(0)) 0:5 ]=(2¼f 0 t) constant value and the spread of the shape decreases as well. The above results due to change of non-linearity are explained below. To see why the spread of the simulation results decrease in Fig. 5.14, we need to look atthee®ectsofdi®erentnon-linearitiesontheamplitudeandtotalphasesystemequations (5.79) and (5.80). As shown in Fig. 5.15, we see both the total phase and the amplitude noise e®ects on two di®erent non-linearities. The noise e®ect on non-linearity has been previously demonstrated for the hard-limiter non-linearity, moreover, we show the noise e®ectonthesoft-limiteraswell. Theresultsfollowthatthesmallertheslope pofthesoft- limiter while still large enough to maintain the required expected steady state amplitude of oscillation when noise is present, the smoother the slope of the resulting soft-limiter output and the smaller the energy contributions of the non-linearities on the feedback of 120 0.1 0.1001 0.1002 0.1003 0.1004 0.1005 0.1006 0.1007 0.1008 0.1009 0.101 −1000 −500 0 500 1000 time (sec.) g(ρsinθ)cosθ/ρ 0.1 0.1001 0.1002 0.1003 0.1004 0.1005 0.1006 0.1007 0.1008 0.1009 0.101 −300 −200 −100 0 100 200 time (sec.) g(ρsinθ)sinθ due to soft−limiter due to hard−limiter due to soft−limiter due to hard−limiter Figure 5.15: Total phase and amplitude noise e®ects on two non-linearities used, one with a hard-limiter, the other one with a soft-limiter whose slope p=0:001, the second term of equation (5.55) and the ¯rst term of equation (5.54) are plotted the system. Therefore, we expect to see the g 1 (½sinµ) function reduces its signal strength as the slope of the soft-limiter is decreased when the noise strength is smaller than the maximumofg 1 (½sinµ)function. Thus, atthesamenoiselevel, constantgains G 1 , G 2 , the oscillator model employed with a soft-limiter with a small slope contributes less feedback tothesystemascomparedtotheonewithahard-limiter. Thisexplainswhythespreadof the new phase noise process and the normalized fractional frequency deviation is smaller shown in Fig. 5.14 as we use the soft-limiter in the oscillator model. OnemorethingofinterestsisthespectrumofthenewphasenoiseprocessÁ(t)¡Á(t m ). In Fig. 5.16, averaged power spectrums of Á(t)¡Á(0) are plotted for oscillator model em- ploying di®erent non-linearities. We observe the phase noise spectrum has periodic spikes at frequencies around multiples of 2 kHz due to non-linearities. The sub-harmonics of the 121 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4 −140 −120 −100 −80 −60 −40 frequency (Hz) dB 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4 −140 −120 −100 −80 −60 −40 frequency (Hz) dB a) b) Figure 5.16: (a) Averaged power spectrum of the phase noise process Á(t)¡Á(0) for the oscillatormodelemployingahardlimiter,(b)averagedphasenoisespectrumofÁ(t)¡Á(0) for the model employing a soft-limiter with slope p=0:001 averaged power spectrum for the model with a hard-limiter non-linearity are diminished when a soft-limiter with small slope is used in the model. The noise level is lower at these periodic spikes for the soft-limiter with smaller slope. This phenomenon is explained by the lower feedback on the system due to smaller signal strength by the non-linearity like a soft-limiter with small slope. Thus, we expect to see the smaller periodic spikes at the phase noise spectrum. Therefore, with a soft-limiter non-linearity employed in the oscilla- tor model, the smaller the slope p, the smaller the spread of the variance of the new phase noise process and the smaller the periodic spikes at the phase noise spectrum caused by non-linearity. In general, a soft-limiter is preferred over the hard-limiter non-linearity for theoscillatormodelwithnoise, thisalsohasthee®ectofreducingtheclockdrift µ d caused by the non-linearity and the internal noise. 122 5.3.6 Models with Di®erent Gains So far we have looked at the e®ects of di®erent non-linearities on the amplitude and total phase functions of oscillator model when noise is present. In addition, there are other system parameters that may contribute to the system performance. These are the constant gains G 1 and G 2 as described in the state equation (5.8). We have previously de¯ned what values G 1 and G 2 take and they are related by the following equation G 2 = a¼G 1 ! 0 4 ; where a is a constant that determines the approximate steady state amplitude of the oscillator. We have previously discussed the the e®ects of both gains in Section 4.3 when noise is not present, we restate that G 1 controls the rate to reach the steady state while theothergainG 2 controlstheapproximatesteadystateamplitudeofoscillations,together with the non-linearity which makes the oscillations possible. When noise is present in the oscillatormodel, weneedtoevaluatethee®ecttothesystemperformancewhenwechange the gains G 1 , G 2 . When the rate to reach the steady state condition is not a critical issue in the system design, we choose lower values of G 1 and G 2 such that negative e®ects from the non- linearity on the overall amplitude and phase will not be signi¯cant. However, the noise strength becomes a issue when the constant gains are decreased. Suppose we are given gain constants G 1 = 100 and G 2 = ¼G 1 ! 0 4 at the noise variance ¾ 2 F 1 = 100! 2 0 , we perform simulations again on the model as described earlier in Fig. 5.5. Since the gains are lower 123 thanwhat wepreviously assignedbyafactorof 10, weexpectto seethee®ectsof the non- linearity on the oscillator model will be reduced. However, since the e®ect of non-linearity is reduced signi¯cantly by a factor of 10 as seen in Fig. 4.14 when noise is not present, the internal noise with the same variance ¾ 2 F 1 = 100! 2 0 as before will make signi¯cant distortions to the system of equations (5.54) and (5.55). The simulation results for the normalized standard deviation of the new phase noise process and the normalized RMS fractionalfrequencydeviationinthisoscillatormodelwith G 1 =100isshowninFig. 5.17. We see that the normalized standard deviation of the phase noise process Á(t)¡Á(0) at G 1 = 100 is higher than the one employed with the soft-limiter at G 1 = 1000 as shown in Fig. 5.14. Similar results are observed for the normalized RMS fractional frequency deviation as well. This is due to the reduction of the constant gain G 1 in the oscillator model while the noise strength remains the same as before, thus contributing more to the phase noise process. In addition, the spread of the results shown in Fig. 5.17 is smaller than the ones in Fig. 5.14 due to a reduction of 10 by the constant gain G 1 used in the non-linearity function g(½sinµ) of the system. In addition to the phase noise process, we look at the amplitude noise process ½(t)¡ ½(t m )¡E[½(t)¡½(t m )jµ(t 0 );½(t 0 );t 0 · t m < t] as shown in equation (5.76). The variance of the amplitude noise process is one of the main interests shown in Fig. 5.18. Clearly from Fig. 5.18, the variance of the amplitude noise process at G 1 =100 is approximately 10 times more than the one at G 1 = 1000. This is due to the reduction of a factor 10 by G 1 which reduces the e®ect of feedback non-linearity function g(½sinµ) of the system while the internal noise remains the same. Although the variance of the amplitude noise process increases at G 1 = 100, the non-linearity e®ect on the variance of the amplitude 124 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.5 1 1.5 2 x 10 −5 time (sec.) radian/Hz 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.2 0.4 0.6 0.8 1 1.2 x 10 −4 time(sec.) a.) b.) Figure5.17: (a)NormalizedstandarddeviationofthephasenoiseprocessÁ(t)¡Á(0)when the oscillator model with gain G 1 =100 is used, (b) normalized RMS fractional frequency deviation of the phase noise process noise process decrease. Furthermore, we expect to see the decrease in periodic spikes (harmonics) of the new phase noise process due to the reduction of the spread of the normalized standard deviation of the phase noise process seen in Fig. 5.17. In Fig. 5.19, a comparison of the averaged power spectrums of the phase noise process Á(t)¡Á(0) is plotted for di®erent G 1 . We see that at the same internal noise, the phase noise spectrum at G 1 = 100 has smaller periodic spikes than the one at G 1 = 1000. Regardless of the non-linearityused,thereductionofthegainG 1 intheoscillatormodelreducesthee®ectof non-linearity at the same internal noise, however, the variance of the phase and amplitude noise are increased. Moreover, due to the reduction of the mean function of the amplitude ½(t) at steady state when G 1 = 100 is used as compared to the one at G 1 = 1000, we expect to see a better system performance in steady state amplitude (less variations in 125 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.2 0.4 0.6 0.8 1 1.2 x 10 −6 time (sec.) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 1 2 3 4 5 6 7 x 10 −6 time (sec.) a.) b.) Figure5.18: (a)Varianceoftheamplitudenoiseprocess½(t)¡E[½(t)jµ(0)=0;½(0)=0:125] at G 1 = 1000 in the oscillator model, (b) variance of the amplitude noise process at G 1 =100 the average sense) and better performance in the phase drift similar to the discussion in Section 4.3.1 (no noise case). Thus, although the variance of the phase and amplitude noise are increased at the same internal noise for reduced constant gain G 1 , the e®ect of non-linearity is signi¯cantly reduced. As already discussed in Section 4.3.2 about the selection of models when no noise is present, due to the above ¯ndings, we ¯nd similar conclusions about the selection of oscillator models when noise is present. To avoid the increase of the variance of the phase and amplitude noise processes, a suitable selection for the constant gain G 1 needs to be chosen. Thesmallerthegain,thelessthee®ectofnon-linearity,butthelargerthevariance of the phase / amplitude noise at the same internal noise when compared with the one with a larger gain G 1 . A soft-limiter non-linearity with small slope is desired for the 126 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4 −140 −120 −100 −80 −60 −40 Hz dB 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4 −140 −120 −100 −80 −60 −40 Hz dB a) b) Figure 5.19: (a) Averaged power spectrum of the phase noise process Á(t)¡Á(0) when the soft-limiter with slope p = 0:001 is employed in the system at G 1 = 1000, (b) averaged power spectrum of the phase noise process when the hard-limiter is used at G 1 =100 system design of lower phase drift µ d . For the constant gains G 1 and G 2 , we specify the relation to be G 2 = a¼! 0 G 1 4 , where a is the speci¯ed steady state amplitude. In addition, for a sinusoidal oscillator waveform with fast transient response, we need the condition G 1 < ! 0 . Furthermore, we need the constant gains to be in a region where reasonable transient response is possible as well as low oscillator phase drifts and low variance of phase / amplitude noise. In our example, for a 1 kHz oscillator, we need the constant gain G 1 to be around 500, for a acceptable oscillator system performance. 5.4 Analysis of Oscillator Model II Inthissection,wewillanalyzethebehavioroftheoscillatormodelIIwithnoiseinequation (5.2) and show the e®ect of non-linearity associated with di®erent damping coe±cients, 127 on the performance of the system when noise is present. We have previously discussed the existence and the stability of the solution to the system equation (5.11) when two internal noise sources are present. Moreover, we obtain the stability region of the solution as well. Here, we consider for the proposed model is when both internal oscillator noises arepresent. Thisisdescribedbythefollowingstochasticdi®erentialequationasreiterated from equation (5.2). Ä y+G 1 _ y¡G 2 sgn_ y+! 2 0 x=F 1 (t) (5.84) whereG 1 ,G 2 areconstantgains,! 0 istheclockrestfrequency,F 1 (t)isaninternaloscillator noise considered independent of the second internal noise F 2 (t), and x = y+ R t 0 F 2 (¿)d¿. Both internal oscillator noises F 1 (t), F 2 (t) are assumed white, Gaussian and independent. The diagram for the system equation (5.84) is represented in Fig. 5.20, where two internal oscillator noises F 1 (t) and F 2 (t) are present. Similar to the Section 5.3, noise e®ects on the oscillators and the non-linearity e®ect on the noise are analyzed. Again, we need to transform the system equation (5.84) into a polar coordinate system, where we fully describe the dynamics of the system. 5.4.1 Transformation of the System Equation We are interested in transforming the system equation (5.84) into a system of equations representingtheamplitudeandphasedynamicsofthesystemwhenbothinternaloscillator 128 ∫ ∫ + X 0 ω ) ( 1 t F ( ) 2 − + + ) ( 2 t F + - ) (t x y & 2 G 1 G Figure 5.20: Schematic of oscillator model II from equation (5.84) when both internal oscillator noises F 1 (t) and F 2 (t) are present noisesare present. Fromthe state space equation (5.11), wedo the transformation by¯rst setting v = z 2 ! 0 , we obtain the following new system of equation to be _ x = ! 0 v+h 2 (t) _ v = ¡! 0 x+g(v)+h 1 (t) ; (5.85) where g(v) =¡G 1 v+ G 2 ! 0 sgn(! 0 v), h 1 (t) = ¾ F 1 ! 0 e F 1 (t), h 2 (t) = ¾ F 2 e F 2 (t), F 1 (t) = ¾ F 1 e F 1 (t), F 2 (t) =¾ F 2 e F 2 (t), e F 1 (t) and e F 2 (t) are independent with unit variance. Now we transform 129 the above state space equation into a model in polar coordinates represented by the am- plitude ½ and the total phase µ. We let x = ½cosµ, and v = ½sinµ, we get the following representation in polar coordinates as _ x = _ ½cosµ¡½ _ µsinµ =! 0 ½sinµ+h 2 (t) _ v = _ ½sinµ+½ _ µcosµ =¡! 0 ½cosµ+g(½sinµ)+h 1 (t) : (5.86) When we multiply the _ x equation of the equation (5.86) by cosµ and _ v equation by sinµ, and add them, we get the di®erential equation for the amplitude ½ as _ ½=g(½sinµ)sinµ+h 1 (t)sinµ+h 2 (t)cosµ : (5.87) Similarly, we get the di®erential equation for the total phase µ to be _ µ =¡! 0 + g(½sinµ) ½ cosµ+ h 1 (t) ½ cosµ¡ h 2 (t) ½ sinµ : (5.88) From both equations (5.87), (5.88), we see the dynamics of both quantities ½ and µ when both internal oscillator noises are present, the e®ect of non-linearity associated with the g(½sinµ) function, and the additional terms of noise contribution. However, it is not clear how the amplitude ½ and the total phase µ behave when the noise is present from these two dynamic equations. Again, when noise sources are present in the oscillator model, the e®ect of noises on oscillators and the e®ect of non-linearity on the noises are areas of interests for investigation. 130 To further demonstrate the e®ect of noise on oscillators and to see the e®ect of non- linearity, we need to evaluate and analyze the system equations (5.87) and (5.88). We will perform simulations on these two stochastic system equations. 5.4.2 Simulation of the Oscillator Model The oscillator model II described in equation (5.84) with the state space representation in equation(5.11)isdescribedasablockdiagramshowninFig. 5.20. Weperformsimulations on this oscillator model and verify the existence and stability of the solution. To further analyzethee®ectsof non-linearityonthe amplitudeandtotalphaseofthe oscillatorwhen both internal oscillator noises are present, we need to evaluate the stochastic di®erential equations (5.87) and (5.88). Due to the di±culty of obtaining the analytic solutions of the amplitude and phase functions to the system of equations (5.87) and (5.88) when both internal oscillator noises arepresent,weperformsimulationsontheseequations. Forthesysteminpolarcoordinate form, we ¯rst simulate the system at the clock rest frequency ! 0 = 2000¼ and assign the coe±cients G 1 and G 2 such that the steady state amplitude of the oscillator is approxi- mately 1. We use Matlab's Simulink to simulate the system of equations. The Simulink's block diagram is shown in Fig. 5.21, where G 1 =1000;G 2 = G 1 ¼! 0 4 , the internal oscillator noises F 1 (t), F 2 (t) being independent white Gaussian with variance ¾ 2 F 1 , ¾ 2 F 2 respectively. We start the simulation at the initial conditions ½(0) = 0:125;µ(0) = 0, at the noise vari- ance ¾ 2 F 1 =100! 2 0 and ¾ 2 F 2 =100, we see the noise e®ects on the amplitude function, total phase,steadystateoscillation,andthenon-linearity. Thereareseveralsimulatedfunctions 131 ∫ ∫ X X u 1 ) sin(u ) cos(u X 1 G + - 0 ω X X + - 0 ω ρ ρ & θ θ & θ ρ θ ρ cos ) sin ( g 0 2 ω G X + + X 0 1 ω X X + + ) ( 2 t F X X + - + + X ) ( 1 t F Figure 5.21: Simulink block diagram showing the amplitude and phase di®erential equa- tions(5.87),(5.88)withtheclockrestfrequency! 0 =2000¼ wheninternaloscillatornoises F 1 (t) and F 2 (t) are present that we are interested in from the schematic shown in Fig. 5.21. They include func- tions ½(t), µ(t), amplitude and phase noise e®ect on non-linearity, the total noise e®ect on _ ½ and on _ µ, and the oscillator output ½cosµ. SimilartowhatwehavediscussedinSection5.3.4, whenbothinternaloscillatornoises are present, the simulated results for 100 runs shown in Fig. 5.22 indicate that noises have somehow changed the shape of the non-linearity g(½sinµ) in the system of equations (5.87) and (5.88). When we combine the noise contributions with the non-linearity term, we see the total feedback e®ect to the system of stochastic di®erential equations (5.87) 132 0.2 0.2001 0.2002 0.2003 0.2004 0.2005 0.2006 0.2007 0.2008 0.2009 0.201 −800 −600 −400 −200 0 200 400 600 800 time (sec.) g(ρsinθ)cosθ/ρ 0.2 0.2001 0.2002 0.2003 0.2004 0.2005 0.2006 0.2007 0.2008 0.2009 0.201 −300 −200 −100 0 100 200 time (sec.) g(ρsinθ)sinθ Figure 5.22: E®ect of internal noises on the non-linearity function g(½sinµ) for both amplitude and phase functions and (5.88). This is seen in Fig. 5.23, where _ µ+! 0 and _ ½ are plotted and the feedback is distorted and intensi¯ed when the internal noises are present as compared to Fig. 5.22. As expected, the functions _ µ+! 0 and _ ½ are distorted more than the case when only one internal noise is present as shown in Fig. 5.8. Similarly, we obtain the normalized standard deviation of the new noise process Á(t)¡ Á(t m ) for t m = 0, and the normalized RMS fractional frequency deviation de¯ned in equation (5.83) as shown in Fig. 5.24. The normalized standard deviation of the new phase noise process increases with time and its shape is modi¯ed by the non-linearity as shown in Fig. 5.24. The spread of the normalized standard deviation of the phase noise process increases over time and the e®ect of non-linearity on the phase noise are clearly seenasshownonFig. 5.24(a). Again, thenormalizedRMSfractionalfrequencydeviation decreases over time to an approximate constant value. The results shown in Fig. 5.24 133 0.2 0.2001 0.2002 0.2003 0.2004 0.2005 0.2006 0.2007 0.2008 0.2009 0.201 −1000 −500 0 500 1000 time (sec.) dθ/dt +ω 0 0.2 0.2001 0.2002 0.2003 0.2004 0.2005 0.2006 0.2007 0.2008 0.2009 0.201 −300 −200 −100 0 100 200 time (sec.) dρ/dt a.) b.) Figure 5.23: (a) E®ect of µ noise and non-linearity on _ µ+! 0 = g(½sinµ) ½ cosµ+ h 1 (t) ½ cosµ¡ h 2 (t) ½ sinµ, (b) E®ect of ½ noise and non-linearity on _ ½ = g(½sinµ)sinµ + h 2 (t)cosµ + h 1 (t)sinµ indicate that the performance is worse than the one with one internal oscillator noise as seen in Fig. 5.14. With the internal noise F 2 (t) present in the oscillator model as well, the e®ect to the system performance is more severe than the one with the internal noise F 1 (t). Thus, the internal noise F 2 (t) causes more system degradation than the noise F 1 (t). As compared to the traditional phase noise of a Wiener process, the new phase noise process behaves like a modi¯ed Wiener process with non-linearity e®ect. Similarly, we evaluate the amplitude noise process as de¯ned in equation (5.76) for the oscillator model II as well as the modi¯ed innovation process in equation (5.77). The am- plitude noise process and the modi¯ed innovation process are simulated as shown in Fig. 5.25, where the standard deviation of the amplitude noise process ½(t)¡E[½(t)jµ(0);½(0)] shown in (a) is slightly more than the one in the oscillator model I, and so is the modi¯ed 134 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.5 1 1.5 2 2.5 x 10 −5 time (sec.) radian/Hz 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.5 1 x 10 −4 time (sec.) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0 2 4 6 8 x 10 −6 time (sec.) radian/Hz a.) b.) c.) Figure 5.24: (a) The e®ect of non-linearity on the normalized standard deviation of the new phase noise process Á(t)¡Á(0), (b) the normalized standard deviation of the phase noiseprocessovertime,(c)thenormalizedRMSfractionalfrequencydeviationinequation (5.83) innovation process E[½(t)jµ(0);½(0)]¡½(t)j no noise . The standard deviation of the ampli- tude noise process is quite jittery but with an approximately constant value, while the modi¯ed innovation process and the amplitude noise process is veri¯ed to be Gaussian distributed. We conclude that the amplitude function is bounded with a small prediction error shown in Fig. 5.25 (b) for equation (5.77). Furthermore, for a small approximately constantstandarddeviationoftheamplitudenoiseprocessinequation(5.76),theexpected amplitude function is therefore bounded and this further veri¯es the stability condition shown in Section 5.2.5. We verify the stability condition for the oscillator model discussed in Section 5.2.5. The constant gains G 1 and G 2 satisfy the existence condition of the solution in equation 135 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.5 1 1.5 x 10 −3 time (sec.) amplitude 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 −2 0 2 x 10 −4 time (sec.) a). b). Figure5.25: (a)Standarddeviationoftheamplitudenoiseprocess ½(t)¡E[½(t)jµ(0);½(0)] and (b) modi¯ed innovation process E[½(t)jµ(0);½(0)]¡½(t)j no noise (5.13),wetaketheaverageofthesimulatedoutputs½cosµ and½sinµ toshowthestability of the solution. When we take the expectation on the Lyapunov stability condition such thatE[LV]<0 orE[L e V]<0 in Section 5.2.5 by relaxing the requirement that LV <0 or L e V <0, we get the region of limit cycle where the maximum normalized E[ z 2 ! 0 ] can exist, i.e., G 2 + p G 2 2 +2G 1 (! 2 0 ¡3) 2G 1 ! 0 <jE[ z 2 ! 0 ]j< G 2 + p G 2 2 +2G 1 (! 0 2 +1) 2G 1 ! 0 : (5.89) 5.4.3 The E®ect of Non-linearity on the Phase Noise Two types of non-linearity are investigated similarly as performed in Section 5.3.5 and they are hard-limiter and soft-limiter with di®erent slopes. From the amplitude and phase stochastic di®erential equations (5.87) and (5.88), we replace the hard-limiter within the 136 g(½sinµ) function with a soft-limiter. Furthermore, we evaluate the e®ects of the model employing a soft-limiter with di®erent slopes and compare the results with the one with the hard-limiter non-linearity. Similarly, we ¯rst evaluate the model employing with a soft-limiter non-linearity by simulation on the amplitude and the phase schematic with internal oscillator noises in Fig. 5.21 with the non-linearity replaced by a soft-limiter. The new amplitude and phase stochastic di®erential equations are described as the following _ ½ = g 2 (½sinµ)sinµ+h 1 (t)sinµ+f 2 (t)sinµ (5.90) _ µ = ¡! 0 + g 2 (½sinµ) ½ cosµ+ h 1 (t) ½ cosµ¡ h 2 (t) ½ sinµ ; (5.91) where h 1 (t) = ¾ F 1 ! 0 e F 1 (t), h 2 (t) = ¾ F 2 e F 2 (t), g 2 (½sinµ) = ¡G 1 ½sinµ + G 2 ! 0 s sl (! 0 ½sinµ), F 1 (t) = ¾ F 1 e F 1 (t), F 2 (t) = ¾ F 2 e F 2 (t), e F 1 (t) and e F 2 (t) are independent with unit variance, and the soft-limiter s sl (y) is de¯ned in equation (5.81). Generally, for ¯xed constant gains G 1 and G 2 , we reduce the feedback e®ect of non-linearity while still maintaining the required oscillation steady state amplitude by reducing the slope of the soft-limiter even in the presence of noise. Again, we focus on the e®ect of di®erent non-linearities on the oscillator model by investigating the e®ect on the new phase noise process Á(t)¡Á(t m ). We next show the simulation results for the oscillator model shown in Fig. 5.21 with the non-linearity replaced by a soft-limiter. The simulation is performed with di®erent slopes p of the soft-limiter from equation (5.81). At the clock rest frequency f 0 = 1000, the slope of the soft-limiter p=0:001, we obtain the normalized standard deviation of the new noise process Á(t)¡Á(t m ) for t m = 0, and the normalized RMS fractional frequency 137 deviation de¯ned in equation (5.83) as shown in Fig. 5.26. We see from Fig. 5.26 that the spread of the normalized standard deviation of the phase noise process decrease as comparedtotheoscillatormodelemployedwithahard-limiternon-linearityshowninFig. 5.24 and the normalized RMS fractional frequency deviation decreases over time to an approximate constant value and the spread of the shape decreases slightly as well. The results follow that the smaller the slope p of the soft-limiter while still large enough to maintain the required expected steady state amplitude of oscillation when both internal noises are present, the smoother the slope of the resulting soft-limiter output and the smaller the energy contributions of the non-linearities on the feedback of the system. Thus, at the same noise level, constant gains G 1 and G 2 , the oscillator model employed with a soft-limiter with a small slope contributes less feedback to the system as compared to the one with a hard-limiter. We then explore the spectrum of the phase noise process Á(t)¡Á(t m ). In Fig. 5.27, averaged spectrums of Á(t)¡Á(0) are plotted for the oscillator model employing either a hard-limiter or a soft-limiter at the same internal noise levels. We observe the phase noise spectrum has periodic spikes at frequencies around multiples of 2 kHz due to non- linearities. The side-lobes of the periodic spikes of the averaged spectrum of the phase noise for the model with a hard-limiter non-linearity are further reduced when we use the soft-limiter with a small slope as seen in Fig 5.27 (b). Moreover, the strength of the periodic spikes of the averaged spectrum reduce more as frequency increases for the model employed with a soft-limiter non-linearity. This phenomenon is again explained by the lower feedback on the system due to smaller signal strength by the non-linearity like a soft 138 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.5 1 1.5 2 2.5 x 10 −5 time (sec.) radian/Hz 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.2 0.4 0.6 0.8 1 x 10 −4 time (sec.) a). b). Figure 5.26: (a) Normalized standard deviation of the new phase noise process Á(t)¡ Á(0) when the soft-limiter with slope p = 0:001 is used in the oscillator model, and (b) normalized RMS fractional frequency deviation de¯ned in equation (5.83) for oscillator model II with a soft-limiter limiter with small slope. Therefore, with a soft-limiter non-linearity employed in the oscillator model, the smaller the slope p, the smaller the spread of the normalized stan- dard deviation of the new phase noise process and the smaller the periodic spikes at the phase noise spectrum caused by non-linearity. 5.4.4 Models with Di®erent Gains So far we have looked at the e®ects of di®erent non-linearities on the amplitude and total phase functions of the oscillator model when both internal noises are present. In addition, there are other system parameters that may contribute to the system performance. These are the constant gains G 1 and G 2 as described in the state equation (5.11). We have 139 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4 −140 −120 −100 −80 −60 −40 Hz dB 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4 −140 −120 −100 −80 −60 −40 Hz dB a). b). Figure 5.27: (a) Averaged power spectrum of the phase noise process Á(t)¡Á(0) for the oscillator model employed with a hard-limiter, and (b) averaged power spectrum of the phase noise process when the soft-limiter with slope p = 0:001 is used in the oscillator model previously de¯ned what values G 1 and G 2 take and they are related by the following equation G 2 = a¼G 1 ! 0 4 ; where a is a constant that determines the approximate steady state amplitude of the oscillator. When the rate to reach the steady state condition is not a critical issue in the system design, we choose lower values of G 1 and G 2 such that negative e®ects from the non-linearity on the overall amplitude and phase will not be signi¯cant. However, the noise strength becomes a issue when the constant gains are decreased. This is due to the factthatinternalnoiseshasmoreimpactwhenthefeedbacksignalstrengthisreduceddue to the decrease of the constant gains. 140 Suppose we are given gain constants G 1 = 100 and G 2 = ¼G 1 ! 0 4 at the noise variance ¾ 2 F 1 =100! 2 0 ,¾ 2 F 2 =100,weperformsimulationsagainonthemodelasdescribedearlierin Fig. 5.21. The simulation results for the normalized standard deviation of the new phase noise process and the normalized RMS fractional frequency deviation in this oscillator modelwithG 1 =100isshowninFig. 5.28. Weseethatthenormalizedstandarddeviation of the phase noise process Á(t)¡Á(0) at G 1 = 100 is higher than the one employed with the soft-limiter at G 1 = 1000 as shown in Fig. 5.26. Similar results are observed for the normalized RMS fractional frequency deviation as well. This is due to the reduction of the constant gain G 1 in the oscillator model while the noise strength remains the same as before, thus contributing more to the phase noise process. In addition, the spread of the results shown in Fig. 5.28 is smaller than the ones in Fig. 5.26 due to a reduction of 10 by the constant gain G 1 used in the non-linearity function g(½sinµ) of the system. In addition to the phase noise process, we look at the amplitude noise process ½(t)¡ ½(t m )¡E[½(t)¡½(t m )jµ(t 0 );½(t 0 );t 0 · t m < t] as shown in equation (5.76). The variance or standard deviation of the amplitude noise process is one of the main interests shown in Fig. 5.29. Clearly seen from Fig. 5.29, the standard deviation of the amplitude noise process at G 1 = 100 is approximately 3 times more than the one at G 1 = 1000. This is duetothereductionofafactor10byG 1 whichreducesthee®ectoffeedbacknon-linearity function g(½sinµ) of the system while the internal noises remain the same. Although the variance of the amplitude noise process increases at G 1 = 100, the non-linearity e®ect on the variance of the amplitude noise process decrease and the standard deviation of the amplitude noise process is bounded. Thus, the oscillator model is asymptotically stable. 141 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.5 1 1.5 2 2.5 x 10 −5 time (sec.) radian/Hz 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.5 1 1.5 2 x 10 −4 time (sec.) a). b). Figure5.28: (a)NormalizedstandarddeviationofthephasenoiseprocessÁ(t)¡Á(0)when the constant gain G 1 =100 in the oscillator model II, and (b) normalized RMS fractional frequency deviation of the phase noise process for the same oscillator model Furthermore, we expect to see the decrease in periodic spikes of the new phase noise process due to the reduction of the spread of the normalized standard deviation of the phase noise process seen in Fig. 5.28. In Fig. 5.30, a comparison of the averaged spectrum of the phase noise process Á(t)¡ Á(0) is plotted for di®erent G 1 . We see that at the same internal noise, the phase noise spectrum at G 1 = 100 has smaller periodic spikes than the one at G 1 = 1000, but the noise°oor isincreasedslightlyat G 1 =100duetotheincrease in thenormalizedstandard deviation in Fig. 5.28 (a). Regardless of the non-linearity used, the reduction of the gain G 1 in the oscillator model reduces the e®ect of non-linearity at the same internal noises, however, the variance or standard deviation of the phase and amplitude noise processes areincreased. Althoughthevarianceofthephaseandamplitudenoiseareincreasedatthe 142 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.5 1 1.5 x 10 −3 time (sec.) amplitude 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 1 2 3 4 x 10 −3 time (sec.) amplitude a). b). Figure 5.29: (a) Standard deviation of the amplitude noise process de¯ned in equation (5.76) when G 1 = 1000 is used in the oscillator model, and (b) standard deviation of the amplitude noise process when G 1 =100 is used same internal noise for reduced constant gain G 1 , the e®ect of non-linearity is signi¯cantly reduced and better oscillator output performance is obtained. Therefore, similar to what we have discussed in Section 5.3.6, to avoid the signi¯cant increase of the variance of the phase and amplitude noise processes, and to reduce the periodic spikes of the phase noise spectrum, a suitable selection for the constant gain G 1 needs to be chosen at the presence of both internal oscillator noises. 5.5 Analysis of Oscillator Model III In this section, we will analyze the behavior of the oscillator model III with noise in equation (5.3) and show the e®ect of non-linearity associated with di®erent damping co- e±cients, on the performance of the system when noise is present. We have previously 143 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4 −140 −120 −100 −80 −60 −40 Hz dB 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4 −140 −120 −100 −80 −60 −40 −20 Hz dB a). b). Figure 5.30: (a) Averaged spectrum of the phase noise process Á(t)¡Á(0) when the soft- limiter with slope p=0:001 is used in the oscillator model, and (b) averaged spectrum of the phase noise process at G 1 =100 discussedtheexistenceandthestabilityofthesolutiontothesystemequation(5.15)when twointernalnoisesourcesarepresentandanon-linearfeedbackdeviceisemployed. More- over, we obtain the stability region of the solution as well. Here, we assume both internal noises are present and a non-linear feedback device is employed. This is described by the following non-linear stochastic di®erential equation as reiterated from equation (5.3) Ä y+G 1 _ y¡G 2 sgn_ y+! 2 0 g sl (x)=F 1 (t) (5.92) where g sl (x)= 8 > > > > > > < > > > > > > : a if x¸a x if¡a<x<a ¡a if x·¡a (5.93) 144 ∫ ∫ + X 0 ω ) ( 1 t F ( ) 2 − + + ) ( 2 t F + - ) (t x y & 2 G 1 G ) ( sl ⋅ g Figure 5.31: Oscillator model III when both internal oscillator noises F 1 (t) and F 2 (t) are present, and the soft-limiter with slope p=1 is employed in the outer loop of the diagram is a soft-limiter, G 1 , G 2 de¯ned earlier, F 1 (t) is an internal oscillator noise considered independentofthesecondinternalnoiseF 2 (t),x=y+ R t 0 F 2 (¿)d¿,andaistheapproximate steady state oscillator amplitude. Both internal oscillator noises F 1 (t), F 2 (t) are assumed white, Gaussian and independent. This is described by the following diagram as shown in Fig. 5.31. Similar to the Section 5.4, noise e®ects on the oscillators and the non-linearity e®ect on the noise are analyzed. Again, we need to transform the system equation (5.92) into a polar coordinate system, where we fully describe the dynamics of the system. 5.5.1 Transformation of the System Equation We are interested in transforming the system equation (5.92) into a system of equations representingtheamplitudeandphasedynamicsofthesystemwhenbothinternaloscillator 145 noises are present. Fromthe state space equation (5.15), wedo the transformation by¯rst setting v = z 2 ! 0 , we obtain the following new system of equation to be _ x = ! 0 v+F 2 (t) _ v = ¡! 0 g sl (x)+g(v)+ F 1 (t) ! 0 ; (5.94) where g(v) =¡G 1 v+ G 2 ! 0 sgn(! 0 v), and both internal oscillator noises F 1 (t) and F 2 (t) are independent. Now we transform the above state space equation into a model in polar coordinates represented by the amplitude ½ and the total phase µ. We let x=½cosµ, and v =½sinµ, we get the following representation in polar coordinates as _ x = _ ½cosµ¡½ _ µsinµ =! 0 ½sinµ+F 2 (t) _ v = _ ½sinµ+½ _ µcosµ =¡! 0 g sl (½cosµ)+g(½sinµ)+ F 1 (t) ! 0 : (5.95) When we multiply the _ x equation of the equation (5.95) by cosµ and _ v equation by sinµ, and add them, we get the di®erential equation for the amplitude ½ as _ ½=! 0 sinµ[½cosµ¡g sl (½cosµ)]+g(½sinµ)sinµ+ F 1 (t) ! 0 sinµ+F 2 (t)cosµ : (5.96) Similarly, we get the di®erential equation for the total phase µ to be _ µ =¡! 0 [ g sl (½cosµ)cosµ ½ +sin 2 µ]+ g(½sinµ) ½ cosµ+ F 1 (t) ! 0 ½ cosµ¡ F 2 (t) ½ sinµ : (5.97) 146 From both equations (5.96), (5.97), we see the dynamics of both quantities ½ and µ when both internal oscillator noises are present with the soft-limiter g sl controlling the oscil- lating amplitude, the e®ect of non-linearity associated with the g(½sinµ) function, and the additional terms of noise contributions. However, it is not clear how the amplitude ½ and the total phase µ behave when both internal oscillator noises are present from these two dynamic equations while the soft-limiter g sl controls the oscillating amplitude. Again, when noise sources are present in the oscillator model, the e®ect of noises on oscillators and the e®ect of non-linearity on the noises are areas of interests for investigation. To further demonstrate the e®ect of noise on oscillators and to see the e®ect of non- linearity with the soft-limiter in the outer loop of the oscillator model III, we need to evaluate and analyze the system equations (5.96) and (5.97). We will perform simulations on these two stochastic system equations. 5.5.2 Simulation of the Oscillator Model TheoscillatormodelIIIdescribedinequation(5.92)withthestatespacerepresentationin equation(5.15)isdescribedasablockdiagramshowninFig. 5.31. Weperformsimulations on this oscillator model and verify the existence and stability of the solution. To further analyzethee®ectsof non-linearityonthe amplitudeandtotalphaseofthe oscillatorwhen both internal oscillator noises are present, we need to evaluate the stochastic di®erential equations (5.96) and (5.97). Due to the di±culty of obtaining the analytic solutions of the amplitude and phase functions to the system of equations (5.96) and (5.97) when both internal oscillator noises arepresent,weperformsimulationsontheseequations. Forthesysteminpolarcoordinate 147 ∫ ∫ X X u 1 ) sin(u ) cos(u X 1 G + - 0 ω X X + - 0 ω ρ ρ & θ θ & θ ρ θ ρ cos ) sin ( g 0 2 ω G X + + X 0 1 ω X X + + ) ( 2 t F X X + - + + X ) ( 1 t F X - + X X + + X X 2 ) ( ⋅ - - X + + Figure 5.32: Simulink block diagram showing the amplitude and phase di®erential equa- tions(5.96),(5.97)withtheclockrestfrequency! 0 =2000¼ wheninternaloscillatornoises F 1 (t) and F 2 (t) are present and a soft-limiter non-linearity is employed in the outer loop of the oscillator model in Fig. 5.31 form, we ¯rst simulate the system at the clock rest frequency ! 0 = 2000¼ and assign the coe±cients G 1 and G 2 such that the steady state amplitude of the oscillator is approxi- mately 1. We use Matlab's Simulink to simulate the system of equations. The Simulink's block diagram is shown in Fig. 5.32, where G 1 =1000;G 2 = G 1 ¼! 0 4 , the internal oscillator noises F 1 (t), F 2 (t) being independent white Gaussian with variance ¾ 2 F 1 , ¾ 2 F 2 respectively, andthesoft-limiterwithaslopep=1withapproximatesteadystateamplitudea=1. We 148 start the simulation at the initial conditions ½(0) = 0:125;µ(0) = 0, at the noise variance ¾ 2 F 1 =100! 2 0 and¾ 2 F 2 =100,weseethenoisee®ectsontheamplitudefunction,totalphase, steady state oscillation, and the non-linearity. There are several simulated functions that we are interested in from the schematic shown in Fig. 5.32. They include functions ½(t), µ(t), amplitude and phase noise e®ect on non-linearity, the total noise e®ect on _ ½ and on _ µ, and the oscillator output ½cosµ. The soft-limiter with slope p = 1 employed in the outer loop of the oscillator model in Fig. 5.31 intends to further restrict the amplitude growth when both internal oscillator noises are present. Therefore, the simulation results further improve the performance of the oscillator model II. Similar to what we have discussed in Section 5.4.2, when both internal oscillator noises are present, the simulated results for 100 runs shown in Fig. 5.33 indicate that noises have changed the shape of the _ µ that includes the non-linearity function g(½sinµ) in the system of equations (5.96) and (5.97), and the mismatch in the ¯rst term,¡! 0 [ g sl (½cosµ)cosµ ½ +sin 2 µ], of equation (5.97) does not have much e®ect on the _ µ function at this constant gain G 1 =1000 in the oscillator model. The mismatch term in equation (5.97) caused by the soft-limiter has non-negative variance seen from Fig. 5.33 (a) at times multiples of 1 2f 0 and can e®ect the phase noise process. We will discuss the simulation results in the following sections about the issue of frequency mismatch due to the soft-limiter. Similarly, we obtain the normalized standard deviation of the new noise process Á(t)¡ Á(t m ) for t m = 0, and the normalized RMS fractional frequency deviation de¯ned in equation (5.83) as shown in Fig. 5.34 at G 1 = 1000. The normalized standard deviation 149 0.1 0.101 0.102 0.103 0.104 0.105 −6285 −6280 −6275 −6270 −6265 −6260 −6255 −6250 time (sec.) radian 0.1 0.1001 0.1002 0.1003 0.1004 0.1005 0.1006 0.1007 0.1008 0.1009 0.101 −7500 −7000 −6500 −6000 −5500 −5000 time (sec.) radian a). b). Figure 5.33: (a) Di®erential phase mismatch term, ¡! 0 [ g sl (½cosµ)cosµ ½ +sin 2 µ], from the phase stochastic di®erential equation (5.97), and (b) _ µ showing the e®ect of non-linearity function g(½sinµ) at G 1 =1000 of the new phase noise process increases with time and its shape is modi¯ed by the non- linearity as shown in Fig. 5.34. The spread of the normalized standard deviation of the phase noise process increases over time and the e®ect of non-linearity on the phase noise areclearlyseenasshownonFig. 5.34(a). Again,thenormalizedRMSfractionalfrequency deviationdecreasesovertimetoanapproximateconstantvalue. TheresultsshowninFig. 5.34 indicate that the performance is similar to the one without the soft-limiter non-linear device (oscillator model II) as seen in Fig. 5.24. Similarly, we evaluate the amplitude noise process as de¯ned in equation (5.76) for the oscillator model III as well as the modi¯ed innovation process in equation (5.77). The amplitudenoiseprocessandthemodi¯edinnovationprocessaresimulatedasshowninFig. 5.35, where the standard deviation of the amplitude noise process ½(t)¡E[½(t)jµ(0);½(0)] shownin(a)issimilartotheoneintheoscillatormodelII,andsoisthemodi¯edinnovation 150 0 0.005 0.01 0.015 0.02 0.025 0.03 0 2 4 6 8 x 10 −6 time (sec.) radian/Hz 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.5 1 1.5 2 2.5 x 10 −5 time (sec.) radian/Hz 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.2 0.4 0.6 0.8 1 x 10 −4 time (sec.) a). b). c). Figure 5.34: (a) The e®ect of non-linearity on the normalized standard deviation of the new phase noise process Á(t)¡Á(0) = E[Á(t)jµ(0) = 0;½(0) = 0:125] when G 1 = 1000 is used, (b) the normalized standard deviation of the phase noise process over time, (c) the normalized RMS fractional frequency deviation in equation (5.83) processE[½(t)jµ(0);½(0)]¡½(t)j no noise . Thus, at the constant gain G 1 =1000 used in the oscillator model III when the soft-limiter device is employed does not make a signi¯cant di®erencethantheoscillatormodelII.Weconcludethattheamplitudefunctionisbounded with a small prediction error shown in Fig. 5.35 (b) for equation (5.77). Furthermore, for a small approximately constant standard deviation of the amplitude noise process in equation (5.76), the expected amplitude function is therefore bounded and this further veri¯es the stability condition shown in Section 5.2.6. We again verify the stability condition for the oscillator model discussed in Section 5.2.6. The constant gains G 1 and G 2 satisfy the existence condition of the solution in equation (5.16), we take the average of the simulated outputs ½cosµ and ½sinµ to show 151 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.5 1 1.5 x 10 −3 time (sec.) 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 −2 0 2 x 10 −4 time (sec.) a). b). Figure5.35: (a)Standarddeviationoftheamplitudenoiseprocess ½(t)¡E[½(t)jµ(0);½(0)] and (b) modi¯ed innovation process E[½(t)jµ(0);½(0)]¡½(t)j no noise the stability of the solution. When we take the expectation on the Lyapunov stability condition such thatE[LV]<0 orE[L e V]<0 in Section 5.2.6 by relaxing the requirement that LV < 0 or L e V < 0, we get the region of limit cycle where the maximum normalized E[ z 2 ! 0 ] can exist, i.e., G 2 + p G 2 2 +2G 1 (! 2 0 ¡3) 2G 1 ! 0 <jE[ z 2 ! 0 ]j< G 2 + p G 2 2 +2G 1 (! 0 2 +1) 2G 1 ! 0 : (5.98) 5.5.3 The E®ect of Non-linearity on the Phase Noise Similar to what we have discussed in Section 5.4.3, we analyze the e®ect of non-linearity on the phase noise for the oscillator model III. From the amplitude and phase stochas- ticdi®erentialequations(5.96)and(5.97),wereplacethehard-limiterwithinthe g(½sinµ) 152 functionwithasoft-limiter. Furthermore,weevaluatethee®ectsofthemodelemployinga soft-limiterwithdi®erentslopesandcomparetheresultswiththeonewiththehard-limiter non-linearity. Similarly, we¯rstevaluatethemodel employingwith a soft-limiter non-linearityin the g(½sinµ) function by simulation on the amplitude and the phase schematic with internal oscillator noises in Fig. 5.32 with the hard-limiter replaced by a soft-limiter. The new amplitude and phase stochastic di®erential equations are described as the following _ ½ = ! 0 sinµ[½cosµ¡g sl (½cosµ)]+g 2 (½sinµ)sinµ+h 1 (t)sinµ+f 2 (t)sinµ (5.99) _ µ = ¡! 0 [ g sl (½cosµ)cosµ ½ +sin 2 µ]+ g 2 (½sinµ) ½ cosµ+ h 1 (t) ½ cosµ¡ h 2 (t) ½ sinµ; (5.100) where h 1 (t) = ¾ F 1 ! 0 e F 1 (t), h 2 (t) = ¾ F 2 e F 2 (t), g 2 (½sinµ) = ¡G 1 ½sinµ + G 2 ! 0 s sl (! 0 ½sinµ), F 1 (t) = ¾ F 1 e F 1 (t), F 2 (t) = ¾ F 2 e F 2 (t), e F 1 (t) and e F 2 (t) are independent with unit variance, and the soft-limiter s sl (y) is de¯ned in equation (5.81). Generally, for ¯xed constant gains G 1 and G 2 , we reduce the feedback e®ect of non-linearity while still maintaining the required oscillation steady state amplitude by reducing the slope of the soft-limiter even in the presence of noise. Again, we focus on the e®ect of di®erent non-linearities on the oscillator model by investigating the e®ect on the new phase noise process Á(t)¡Á(t m ). WenextshowthesimulationresultsfortheoscillatormodelshowninFig. 5.32withthe non-linearitying(½sinµ)replacedbyasoft-limiter. Attheclockrestfrequency f 0 =1000, the slope of the soft-limiter p=0:001, we obtain the normalized standard deviation of the new noise process Á(t)¡Á(t m ) for t m = 0, and the normalized RMS fractional frequency 153 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.5 1 1.5 2 2.5 x 10 −5 time (sec.) radian/Hz 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.2 0.4 0.6 0.8 1 x 10 −4 time (sec.) a). b). Figure 5.36: a) Normalized standard deviation of the new phase noise process Á(t)¡Á(0) when a soft-limiter with slope p = 0:001 is used in the g(½sinµ) function of the oscillator model III, and (b) normalized RMS fractional frequency deviation de¯ned in equation (5.83) for the oscillator model III with a soft-limiter deviation de¯ned in equation (5.83) as shown in Fig. 5.36. We see from Fig. 5.36 that the spread of the normalized standard deviation of the phase noise process decrease as compared to the oscillator model employed with a hard-limiter non-linearity shown in Fig. 5.34 and the normalized RMS fractional frequency deviation decreases over time to an approximate constant value and the spread of the shape decreases slightly as well. Thus, at the same noise level, constant gains G 1 and G 2 , the oscillator model employed with a soft-limiter with a small slope contributes less feedback to the system as compared to the one with a hard-limiter. Moreover, the performance of the oscillator model III with the soft-limiter non-linearity is similar to the oscillator model II with the soft-limiter non-linearity shown in Fig. 5.26 at the high constant gain G 1 =1000. 154 We then explore the spectrum of the phase noise process Á(t)¡Á(t m ). In Fig. 5.37, averaged spectrums of Á(t)¡Á(0) are plotted for the oscillator model employing either a hard-limiter or a soft-limiter at the same internal noise levels. We observe the phase noise spectrum has periodic spikes at frequencies around multiples of 2 kHz due to non- linearities. The side-lobes of the periodic spikes of the averaged spectrum of the phase noise for the model with a hard-limiter non-linearity are further reduced when we use the soft-limiter with a small slope in the g(½sinµ) function as seen in Fig 5.37 (b). Moreover, the strength of the periodic spikes of the averaged spectrum reduce more as frequency increases for the model employed with a soft-limiter non-linearity. This phenomenon is again explained by the lower feedback on the system due to smaller signal strength by the non-linearity like a soft-limiter with small slope. Therefore, with a soft-limiter non- linearity employed in the oscillator model, the smaller the slope p, the smaller the spread of the normalized standard deviation of the new phase noise process and the smaller the periodic spikes at the phase noise spectrum caused by non-linearity. 5.5.4 Models with Di®erent Gains Similarly to Section 5.4.4, there are other system parameters that may contribute to the system performance. These are the constant gains G 1 and G 2 as described in the state equation (5.15). We have previously de¯ned what values G 1 and G 2 take and they are related by the following equation G 2 = a¼G 1 ! 0 4 ; 155 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4 −140 −120 −100 −80 −60 −40 Hz dB 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4 −140 −120 −100 −80 −60 −40 Hz dB a). b). Figure 5.37: (a) Averaged power spectrum of the phase noise process Á(t)¡Á(0) for the oscillator model III employed with a hard-limiter, and (b) averaged power spectrum of the phase noise process when the soft-limiter with slope p=0:001 is used in the oscillator model III where a is a constant that determines the approximate steady state amplitude of the oscillator. When the rate to reach the steady state condition is not a critical issue in the system design, we choose lower values of G 1 and G 2 such that negative e®ects from the non-linearity on the overall amplitude and phase will not be signi¯cant. However, the noise strength becomes a issue when the constant gains are decreased. This is due to the factthatinternalnoiseshasmoreimpactwhenthefeedbacksignalstrengthisreduceddue to the decrease of the constant gains. Here we are investigating the e®ects of di®erent constant gains G 1 on the system performance. Suppose we are given two cases of gain constants G 1 = 500 and G 1 = 100, for G 2 = ¼G 1 ! 0 4 at the noise variance ¾ 2 F 1 = 100! 2 0 , ¾ 2 F 2 = 100, we perform simulations again on the model as described earlier in Fig. 5.32. The simulation results for the 156 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 1 2 x 10 −5 time (sec.) radian/Hz 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.5 1 1.5 x 10 −4 time (sec.) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 2 4 6 x 10 −5 time (sec.) radian/Hz 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 2 x 10 −4 time (sec.) a). b). c). d). Figure5.38: (a)NormalizedstandarddeviationofthephasenoiseprocessÁ(t)¡Á(0)when the constant gain G 1 = 500 is used in the oscillator model III, and (b) normalized RMS fractional frequency deviation of the phase noise process for the same oscillator model at G 1 =500, and (c) normalized standard deviation of the phase noise process at G 1 =100, and (d) normalized RMS fractional frequency deviation of the phase noise process for the same oscillator model at G 1 =100 normalized standard deviation of the new phase noise process and the normalized RMS fractional frequency deviation in this oscillator model with G 1 = 500 and with G 1 = 100 are shown in Fig. 5.38. We see that the normalized standard deviation of the phase noise process Á(t)¡Á(0) at G 1 =100 in Fig. 5.38 (c) is approximately 3 times higher than the one at G 1 =500 as shown in Fig. 5.38 (a). Similar results are observed for the normalized RMS fractional frequency deviation as well. Moreover, the normalized standard deviation of the phase noise process at G 1 = 500 is identical to the one at G 1 = 1000 shown in 157 Fig. 5.34, but with reduced spread, and it is slightly higher than the results when a soft- limiter non-linearity with small slope is employed in the model as shown in Fig. 5.26. As we decrease the gain even more to G 1 = 100 in the oscillator model, a much higher normalized standard deviation of the phase noise process will result as shown in Fig. 5.38 (c), thus the performance degrades. Therefore, there is a range of the constant gain G 1 that can produce good system performance. This is due to the reduction of the constant gain G 1 in the oscillator model while the noise strength remains the same as before when thefeedback signal strengthis reduced, thus contributing moreto the phasenoise process. Moreover,whencomparingtotheoscillatormodelIIatG 1 =100,thenormalizedstandard deviation of the phase noise process for the oscillator model III at the same gain G 1 is higher. In addition, the spread of the results shown in Fig. 5.38 (c) is smaller than the onesinFig. 5.36duetoareductionof10bytheconstantgainG 1 usedinthenon-linearity function g(½sinµ) of the system. The increase of the normalized standard deviation of the phase noise process at lower constant gain G 1 =100 is explained more as the following. When we decrease the gain G 1 from 1000 to 500, there is enough gain to produce the feedback signal with high enough gain to overcome the mismatch problem associated with the phase stochastic di®erential equation (5.97) and the internal oscillator noises. The di®erential phase mismatch term, ¡! 0 [ g sl (½cosµ)cosµ ½ +sin 2 µ], discussed earlier is the one produced by the amplitude limiting soft-limiter device in the oscillator model III. When at the gain G 1 = 500, the di®eren- tial phase mismatch term does not produce signi¯cant e®ect, thus we see better system performance than the one with G 1 = 1000. As the gain is decreased more to G 1 = 100 while the internal noises still maintain at the same level , the di®erential phase mismatch 158 0.1 0.101 0.102 0.103 0.104 0.105 −6285 −6280 −6275 −6270 −6265 −6260 −6255 −6250 time (sec.) radian 0.1 0.1001 0.1002 0.1003 0.1004 0.1005 0.1006 0.1007 0.1008 0.1009 0.101 −6400 −6350 −6300 −6250 −6200 −6150 time (sec.) dθ/dt a). b). Figure 5.39: (a) The di®erential phase mismatch term, ¡! 0 [ g sl (½cosµ)cosµ ½ +sin 2 µ], from the phase stochastic di®erential equation (5.97) at G 1 = 100, and (b) _ µ at such constant gain when both internal noises are present term starts to make a di®erence as explained from the results shown in Fig. 5.39. The non-negative values of the di®erential phase mismatch term at multiples of 1 2f 0 are large enough such that _ µ shows signi¯cant e®ect due to the large reduction of the gain G 1 . Therefore, although the reduction of the gain G 1 produces less spread in the normalized standard deviation of the phase noise process, when the gain is too low, the phase mis- match problem starts to degrade the system performance at the presence of the internal oscillator noises. Thus, a proper choice of the constant gain G 1 needs be be selected. In addition to the phase noise process, we look at the amplitude noise process ½(t)¡ ½(t m )¡E[½(t)¡½(t m )jµ(t 0 );½(t 0 );t 0 · t m < t] as shown in equation (5.76). The variance or standard deviation of the amplitude noise process is one of the main interests shown in Fig. 5.40 for t m = 0. This is a comparison of the standard deviation of the amplitude noise process between model II and model III at G 1 = 100. Although the normalized 159 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.5 1 1.5 2 2.5 3 3.5 x 10 −3 time (sec.) amplitude 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.5 1 1.5 2 2.5 3 3.5 x 10 −3 time (sec.) amplitude a). b). Figure5.40: (a)Standarddeviationoftheamplitudenoiseprocessfortheoscillatormodel IIatG 1 =100,and(b)standarddeviationoftheamplitudenoiseprocessfortheoscillator model III at the same G 1 standard deviation of the phase noise process increases, the standard deviation of the amplitude noise process is bounded in the oscillator model III as shown in Fig. 5.40 (b). Therefore, when the soft-limiter is employed in the outer loop of the oscillator model in Fig. 5.31, the amplitude limiting mechanism starts to take e®ect, making a tighter bound on the amplitude noise process as compared to the one without such device as in the oscillator model II. This situation is explained by the results shown in Fig. 5.41. The di®erential amplitude mismatch term, ! 0 sinµ[½cosµ¡ g sl (½cosµ)], from the amplitude stochastic di®erential equation (5.96) has both positive and negative values at locations close to multiples of 1 2f 0 as shown in Fig. 5.41. The e®ect is the amplitude limiting mechanism in the presence of internal oscillator noises. Thus, there is no degradation on the amplitude noise process, but rather a improvement on the bound of the process. 160 0.1 0.1005 0.101 0.1015 0.102 0.1025 0.103 0.1035 0.104 0.1045 −1.5 −1 −0.5 0 0.5 1 1.5 time (sec.) amplitude 0.1 0.1001 0.1002 0.1003 0.1004 0.1005 0.1006 0.1007 0.1008 0.1009 0.101 −60 −40 −20 0 20 40 60 time (sec.) dρ/dt a). b). Figure5.41: (a)Thedi®erentialamplitudemismatch,! 0 sinµ[½cosµ¡g sl (½cosµ)],fromthe amplitude stochastic di®erential equation (5.96) at G 1 = 100, and (b) _ ½ at such constant gain when both internal noises are present Although the value of the standard deviation of the amplitude noise at G 1 =100 is higher than the one at G 1 =1000 shown in Fig. 5.35(a), the non-linearity e®ect on the variance of the amplitude noise process decrease and the standard deviation of the amplitude noise process is bounded. Thus, the oscillator model is asymptotically stable. Furthermore, we expect to see the decrease in periodic spikes of the new phase noise process due to the reduction of the spread of the normalized standard deviation of the phasenoiseprocessseeninFig. 5.38. InFig. 5.42, acomparisonoftheaveragedspectrum of the phase noise process Á(t)¡Á(0) is plotted for di®erent G 1 for the oscillator model III and II. We see that at the same internal noises, the phase noise spectrum at G 1 =100 has smaller periodic spikes than the one at G 1 = 500, but the noise °oor is increased at G 1 = 100 due to the increase in the normalized standard deviation in Fig. 5.38 (c). Moreover, when comparing to the averaged power spectrum of the phase noise process 161 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4 −150 −100 −50 0 Hz dB 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4 −140 −120 −100 −80 −60 −40 −20 Hz dB 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4 −140 −120 −100 −80 −60 −40 −20 Hz dB a). b). c). Figure 5.42: (a) Averaged power spectrum of the phase noise process Á(t)¡Á(0) for the oscillator model III at G 1 = 500 , and (b) averaged spectrum of the phase noise process for the oscillator model III at G 1 = 100, and (c) averaged spectrum of the phase noise process for the oscillator model II at G 1 =100 for the oscillator model II at the same gain G 1 , the noise °oor is increased as shown in Fig. 5.42 while the periodic spikes are at the same level. This is due to the increase of the normalized standard deviation of the phase noise process for the oscillator model III shown in Fig. 5.38 (c) as comparing to the much lower value for the oscillator model II shown in Fig. 5.28 (a) at the same gain G 1 = 100. Therefore, the soft-limiter amplitude limiting device in the oscillator model III increases the averaged spectrum of the phase 162 noise process at such low gain G 1 = 100. Thus, similar to what we have discussed in Section 5.4.4, to avoid the signi¯cant increase of the variance of the phase and amplitude noise processes, and to reduce the periodic spikes of the phase noise spectrum, a suitable selection for the constant gain G 1 needs to be chosen at the presence of both internal oscillator noises and the soft-limiter device used in the oscillator model III. 163 Chapter 6 Conclusion and Future Work 6.1 Conclusion In this thesis, theoretical models of voltage controlled oscillators (noise vs. no noise case) are proposed and the e®ects of non-linearity are investigated as well as introducing a new phasenoiseprocess. Anovelmathematicalmodelofvoltagecontrolledoscillatorsprovides the following conclusions. ² the timing jitter process has a random walk behavior with restoring force and the upward zero crossing jitter is normal distributed seen in Section 3.5. ² The cycle-to-cycle jitter statistics are shown to be Gaussian distributed and inde- pendent seen in Section 3.5. ² the tracking loop plus the controller noise n(t) causes the oscillator phase to drift while the internal noise e F(t) tends to cause di®usion on the oscillator phase seen in Section 3.5. 164 Our investigations of these oscillator models modi¯ed from Robinson's original work, suggest the following conclusions. ² Soft limiters are better than hard limiters for (a) minimizing harmonic generation as seen in Sections 5.3.5, 5.4.3, and 5.5.3, and (b) reducing oscillator drift as seen in Section 4.2.3. ² Oscillator models with hard limiters are easier to analyze than the ones with soft limiters. ² High gain loops are better for fast transient response to the steady state solution as seen in Section 4.3. ² A good compromise to achieve reasonable transient response to the steady state solution, lower oscillator drift, lower phase noise harmonics, and lower noise °oor is to choose G 2 = a¼! 0 G 1 4 , where a is the approximate steady state amplitude, and a moderate gain G 1 , as seen in Sections 4.3.2, 5.3.6, 5.4.4, and 5.5.4. ² A soft limiter in the outer loop of the oscillator model bounds the amplitude noise, but increases the phase noise °oor as seen in Section 5.5.4. ² The newly de¯ned phase noise process has the following properties. (1) It is a modi¯ed Wiener process whose variance increases with time as seen in Section 5.3.4. (2)Itcapturesthee®ectofnon-linearityonoscillatorsasseeninSections5.3.4,5.4.2, and 5.5.2. 165 RF zonal LIMIT x loop filter VCO + ) (t W RF zonal LIMIT x loop filter VCO + ) (t W Figure 6.1: System diagram of two VCOs with switch for synchronization analysis The results in the second part of the research can be extended to oscillators with voltage-controlled frequency. The assumptions of the white internal oscillator noises can be extended to colored noises as well. 6.2 Future Work We have completed the system models for voltage-controlled oscillators or non-linear os- cillators with a means of non-linearity for stabilizing oscillators in the presence of internal oscillator noises. For the next step, we would like to apply the results to other areas. These may include but not limited to the following applications. ² Synchronization of two VCOs with switch For the UWB communications, systems of transceivers cannot transmit and receive simultaneously, therefore, a technique like time-division multiple-access (TDMA) 166 ∫ ∫ + + X 0 ω ) ( 1 t F ) ( 2 t F ( ) 2 − ) (t y Figure 6.2: A possible re¯nement on the simpli¯ed model of a VCO technique must be employed. We are interested in understanding the e®ect of net- worksynchronizationunderthiscondition. Wewouldliketoinvestigatethesynchro- nization of two VCOs with switch shown in Fig. 6.1. The channel is assumed to be AWGN initially. Various aspects of synchronization can be investigated. When the switch is set to transmit, the transmitter transmits its clock signal, while the switch of the receiver is set to listen. Similarly, the same procedure is performed for the otherVCO.Thishalf-duplexcommunicationtechniqueistobeanalyzedforthecon- dition that the two oscillators remain locked. Parameters such as mean-time-to-lose lock or mean-time-between-lock will be an research interest. The two-oscillator network synchronization [40], [23], [9] can be further investigated under UWB multipath channel conditions. Further research in the area of UWB pulse can be applied for the two transceivers with switch. 167 ² Other Re¯nement of Models on VCO's We are interested in re¯ning the 2-pole oscillator model that has a limit cycle and an amplitude limiting mechanism within the oscillator. A possible re¯nement on the simpli¯ed VCO model is to introduce a hard-limiter between integrators and a soft-limiter in the feedback loop. In Fig. 6.2, a re¯nement scheme is presented whereF 1 (t),F 2 (t)areinternalnoisesoftheoscillator, however, theoutputwaveform may not be a sinusoid and the output amplitude and phase are e®ected by the non-linearities employed in the system. Other non-linear system models of voltage controlled oscillators can be explored as well. 168 References [1] D.Allan,P.Kartascho®,J.Vanier,J.Vig,G.Winkler,andN.Yannoni. Standardter- minology for fundamental frequency and time metrology. In 42nd Annual Frequency Control Symposium, 1988. [2] A. Blaquiµ ere. Nonlinear System Analysis. Academic Press Inc., 1966. [3] A.Blaquiµ ereandP.Grivet. Nonlineare®ectsofnoiseinElectronicclocks. Proceedings of the IEEE, 51:1606{1614, 1963. [4] W. E. Boyce and R. C. DiPrima. Elementary di®erential equations and boundary value problems. John Wiley and Son, Inc., 3rd edition, 1977. [5] K.J.C.C.M.Wang, P.D.HaleandT.S.Clement. Uncertaintyofoscilloscopetime- base distortion estimate. IEEE Transactions on Instrumentation and Measurement, 51(1):53{58, 2002. [6] E.CoddingtonandN.Levinson. TheoryofOrdinaryDi®erentialEquations. McGraw- Hill Book Company, Inc., 1st edition, 1955. [7] A.Dec, L.Toth, andK.Suyama. Noiseanalysisofaclassofoscillators. IEEE Trans- actions on Circuits and Systems-II: Analog and Digital Signal Processing, 45(6):757{ 760, 1998. [8] A. Demir, A. Mehrotra, and J. Roychowdhury. Phase noise in oscillators: unifying theory and numerical methods for characterization. IEEE Trans. on Circuits and Systems, 47(5):655{674, May 2000. [9] K. Dessouky and W. C. Lindsey. Phase and Frequency Transfer Between Mutually Synchronized Oscillators. IEEE Transactions on Communications, 32(2), 1984. [10] D. Dubin. Numerical and Analytical Methods for Scientists and Engineers using Mathematica. John Wiley and Sons, Inc., 1st edition, 2003. [11] J.A.B.etal. CharacterizationofFrequencyStability. IEEE Transactions on Instru- mentation and Measurement, 20:105{120, May 1972. [12] S. Farlow. Partial Di®erential Equations for Scientists and Engineers. Dover Publi- cations, Inc., 1st edition, 1993. [13] Federal Communications Commission. Revision of Part 15 of the Commisions rules Regarding Ultra-Wideband Transmission Systems: First report and order. Technical Report FCC 02-48 (adopted February 14, 2002; released April 22, 2002). 169 [14] W. Feller. An Introduction to Probability Theory and Its Applications, Volume II. John Wiley and Sons, Inc., 2nd edition, 1972. [15] E. Gerber and A. Ballato. Precision Frequency Control, Volume 2, Oscillators and Standards. Academic Press, Inc., 1st edition, 1985. [16] S. Goldman. Phase Noise Analysis in Radar Systems Using Personal Computers. John Wiley and Sons, Inc., 1st edition, 1989. [17] A.HajimiriandT.Lee. Ageneraltheoryofphasenoiseinelectricaloscillators. IEEE Journal of Solid-State Circuits, 33(2):179{194, Jan. 1998. [18] A. Hajimiri and T. H. Lee. The Design of Low Noise Oscillators. Kluwer Academic Publishers, 1st edition, 1999. [19] C. Hayashi. Nonlinear Oscillations in Physical Systems. McGraw-Hill, Inc., 1st edi- tion, 1964. [20] C. V. Jan Westra and A. van Roermund. Oscillators and Oscillator Systems, Classi- ¯cation, Analysis and Synthesis. Kluwer Academic Publishers, 1st edition, 1999. [21] D. W. Jordan and P. Smith. Nonlinear ordinary di®erential equations. Oxford: Ox- ford University Press, 2nd edition, 1987. [22] P. D. H. K. J. Coakley, C.-M. Wang and T. S. Clement. Adaptive characterization of jitter noise in sampled high-speed signals. IEEE Transactions on Instrumentation and Measurement, 52(5):1537{1547, 2003. [23] K. Kiasaleh and W. C. Lindsey. Time and Frequency Transfer Between Master and Slave Clocks. IEEE Transactions on Communications, 38(10), 1990. [24] P. Kinget. Analog circuit design: (X)DSL and other communication systems, RF MOST models, integrated ¯lters and oscillators. Kluwer Academic Publishers, 1999. [25] P. Kloeden. Numerical Solution of SDE Through Computer Experiments. Springer- Verlag, 1st edition, 1994. [26] B. O. ksendal. Stochastic Di®erential Equations: an introduction with applications. Germany: Springer, 6th edition, 2003. [27] H.J.LarsonandB.O.Shubert. Probabilistic Models in Engineering Sciences Volume II: Random Noise, Signals, and Dynamic Systems. John Wiley and Sons, Inc., 1st edition, 1979. [28] W. C. Lindsey and C. Chie. Theory of oscillator instability based upon structure functions. Proceedings of the IEEE, 64(12):1652{1666, Dec. 1976. [29] A. M. Lyapunov. Stability of Motions. Academic Press, New-York and London, 1st edition, 1966. [30] V.Manassewitsch. Frequency Synthesizers Theory and Design. JohnWileyandSons, Inc., 3rd edition, 1987. 170 [31] H. Meyr and G. Ascheid. Synchronization in Digital Communications volume 1: Phase-, Frequency-Locked Loops, and Amplitude Control. John Wiley and Sons, Inc., 1990. [32] P. P. P. Kythe and M. SchÄ aferkotter. Partial Di®erential Equations and Boundary Value Problems with Mathematica. Chapman and Hall / CRC, 2nd edition, 2003. [33] R. Pawula. Generalization and Extensions of the Fokker-Planck-Kolmogorov Equa- tions. IEEE Trans. Information Theory, 13(1):33{41, 1967. [34] B. Razavi. Analysis, modeling, and simulation of phase noise in monolithic voltage- controlledoscillators. InIEEECustomIntegratedCircuitsConference,pages323{326, 1995. [35] B. Razavi. A study of phase noise in CMOS oscillators. IEEE Journal of Solid-State Circuits, 31:331{343, 1996. [36] U. Rohde. Microwave and Wireless Synthesizers. John Wiley and Sons, Inc., 1st edition, 1997. [37] G. D. Smith. Numerical Solution of Partial Di®erential Equations, Finite Di®erence Methods. Oxford University Press, 3rd edition, 1985. [38] S. S. Soliman and R. A. Scholtz. A Technique to Approximate the Eigenvalues of Autocorrelation Functions. GLOBECOM, 3:1463{1467, 1988. [39] S. Stojanovic. Computational Financial Mathematics using Mathematica. Birkhuser Boston, 1st edition, 2002. [40] A. V. K. W. C. Lindsey and A. Dobrogowski. Network Synchronization by Means of a Returnable Timing System. IEEE Transactions on Communications, 26(6), 1978. [41] F. L. Walls and D. W. Allan. Measurements of Frequency Stability. Proceedings of the IEEE, 74(1):162{168, 1986. [42] J. Wilbur B. Davenport and W. L. Root. An introduction to the Theory of Random Signals and Noise. McGraw-Hill Book Company, Inc., 1958. [43] R. Witte. Electronic Test Instruments, Theory and Applications. Prentice Hall P T R, 1st edition, 1993. 171

Asset Metadata

Creator Chen, Yenming (author)

Core Title Theoretical models of voltage controlled oscillators and the effects of non-linearity

School Andrew and Erna Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 10/11/2009

Defense Date 09/13/2007

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag OAI-PMH Harvest,oscillator model,phase noise,timing jitter,voltage controlled oscillator

Language English

Advisor Scholtz, Robert A. (committee chair), Alexander, Kenneth S. (committee member), Lindsey, William C. (committee member)

Creator Email yenmingc@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m862

Unique identifier UC1163596

Identifier etd-Chen-20071011 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-580145 (legacy record id),usctheses-m862 (legacy record id)

Legacy Identifier etd-Chen-20071011.pdf

Dmrecord 580145

Document Type Dissertation

Rights Chen, Yenming

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email uscdl@usc.edu

Abstract (if available)

Abstract This thesis presents mathematical models and performance evaluations of voltage-controlled oscillators when noises are included in the model. There are two main parts of the research. The first part introduces a novel mathematical model of a voltage-controlled oscillator (VCO) based on physical dynamics with noise. The effects of noise in this oscillator model are shown and the analytical forms of the resulting phase noise are obtained using the stochastic integrals. It is shown that the VCO has phase noise contributed from the internal noise of the clock and the clock drift caused by tuning plus tracking loop noises. Moreover, a two-pole filter is designed to constrain tracking loop noise. A timing jitter estimate is proposed. Analysis of the resulting phase noise along with the simulation suggest that the timing jitter process has a random walk behavior with restoring force and the upward zero-crossing jitter is normally distributed. The cycle-to-cycle jitter statistics are shown to be Gaussian distributed and independent.