1. An Analysis of Phase Noise and Fokker-Planck Equations Hao-Min Zhou School of Mathematics Georgia Institute of Technology Partially Supported by NSF Joint work with Shui-Nee Chow International conference of random dynamical systems, Tianjin, China, June 8-12, 2009

2. An Analysis of Phase Noise and Fokker-Planck Equations Hao-Min Zhou School of Mathematics Georgia Institute of Technology Partially Supported by NSF Joint work with Shui-Nee Chow The Chinese Academy of Sciences July 4, 2005

3. An Analysis of Phase Noise and Fokker-Planck Equations Hao-Min Zhou School of Mathematics Georgia Institute of Technology Partially Supported by NSF Joint work with Shui-Nee Chow SCMA-2005, Auburn University, Dec 4, 2005

4. An Analysis of Phase Noise and Fokker-Planck Equations Hao-Min Zhou School of Mathematics Georgia Institute of Technology Partially Supported by NSF Joint work with Shui-Nee Chow International Conference on Multiscale Modeling and Scientific Computing In Honor of Professor B. Engquist’s 60 th Birthday Peking University, Beijing, June 10-12, 2005

5. An Analysis of Phase Noise and Fokker-Planck Equations Shui-Nee Chow Hao-Min Zhou School of Mathematics Georgia Institute of Technology Partially Supported by NSF

6. Outline Introduction and motivation Moving coordinate transforms Phase noise equations and Fokker-Planck equations Example: van der Pol oscillators and ACD Conclusion

7. Introduction and Motivation A orbital stable periodic solution (limit cycle) (with period ) of a differential system Phase noise is caused by perturbations, which are unavoidable in practice: the solution doesn’t return to the starting point after a period. Phase noise usually persists, may become large. Phase noise is important in many areas including circuit design, and optics.

8. Oscillators Phase noise in nonlinear electric oscillators: Small noise can lead to dramatic spectral changes Many undesired problems associated with phase noise, such as interchannel interference and jitter.

9. Analog to Digital Converter (ADC) 57 ADC is essential for wireless communications. Input: wave (amplitude, frequency). Output: digit computed in real-time, during one single period (number of spikes). Effect of the noise in the transmission system. correct output wrong output 57 58 Bit Error Rate (BER) : ratio of received bits that are in error, relative to the amount of bits received. BER expressed in log scale (dB).

10. ADC Example A piecewise linear ADC model is The input is an analog signal, i.e. The output is the number of spikes in a period, which realizes the conversion of analog signals to digital ones.

11. Our goals Establish a framework to rigorously analyze phase noise from both dynamic system and probability perspectives. Develop numerical schemes to compute phase noise, which are useful tools for system design. Estimate Shannon entropy curves to evaluate the performance of practical systems

12. Approaches Traditional nonlinear analysis based on linearization is invalid: decompose the perturbed solution where is the unperturbed solution and is the deviation, then the error satisfies The deviation can grow to infinitely large (even amplitude error remains small for stable systems, but phase error can be large) The system is self-sustained, and must have one as its eigenvalue.

13. Approaches A conjecture: decompose perturbations into two (orthogonal) components, one along the tangent, one along normal direction, perturbations along tangent generates purely phase noise and normal component causes only amplitude deviation, Hajimiri-Lee (’97). This conjecture is not valid, Demir-Roychowdhury (’98). Perturbation orthogonal to the orbit can also cause phase deviation.

14. Approaches Large literature is available for individual systems, such as pumped lasers by Lax (’67), but lack of general theory for phase noise. Two appealing approaches: 1.Model the perturbed systems by SDE’s and derive the associated Fokker-Planck equations, then use asymptotic analysis to estimate the leading contributions of transition probability distribution function, i.e. in Limketkai (’05), the leading term is approximated by a gaussian: where satisfy a diffusion PDE and are coefficients obtained in asymptotic expansions

15. Approaches 2. Decompose oscillator response into phase and magnitude components and obtain equations for the phase error, for examples: Kartner (’90), Hajimiri-Lee (’98),Demir-Mehrotra- Roychowdhury (’00), i.e. where is defined by a SDE depending on the largest eigenvalue and eigenfunction of state transition matrix in Floquet theory: may grow to infinitely large even for small perturbations

16. Moving Orthogonal Systems A moving orthogonal coordinate systems along Consider solutions of the perturbed systems are small perturbations

17. Equations for the new variables Solutions of the perturbed system can be represented by denoted by For small perturbations, this transform is invertible and both forward and inverse transforms are smooth. Two components and are not orthogonal, which is different from the usual orthogonal decompositions.

18. Equations for the new variables The new phase and amplitude deviation satisfy (Hale (’67)) where notations are Evaluate on the unperturbed orbit Evaluate on the perturbed orbit

19. Stochastic Perturbations Perturbations in oscillators are random, which are often modeled by Where are independent Brownian motions. The transform becomes Theorem 1: if stay close to, then remain as Ito processes and satisfy

20. Stochastic Perturbations Theorem 2: the transition probability of satisfies the Fokker-Planck equation The coefficients are with initial condition

21. Stochastic Perturbations Theorem 4: the transition probability of satisfies the Fokker-Planck equation For a general problem in where The solution can also be transformed into where Theorem 3: if stay close to, then remain as Ito processes and satisfy where can be determined similarly.

22. van der Pol Oscillators Unperturbed van der Pol Oscillators are often described by introduce new variable the equation becomes In practice, noise enters the system, which is model by by introducing the new variable, the system becomes Both and are positive small constant numbers, it is interesting to study the case eventually.

23. van der Pol Oscillators With asymptotic expansion and method of averaging, the equations for new variables are which suggests that the limit cycle is near Substitute into the general theory to obtain The limit cycle has asymptotic expansion

24. van der Pol Oscillators Assume are small (in oscillators, the periodic orbits are stable, and perturbations of amplitude will remain small, i.e. is small). The leading term system is The corresponding Fokker-Planck equation is By the method of averaging for stochastic equations, it is equivalent to

25. van der Pol Oscillators 1.Impuse noise in current at the peak of current (zero voltage), 2.Impose noise in current at the peak of voltage (zero current), Two interesting observations (made by engineers, Hajimiri-Lee(’98), Limketkai(’05 ) ): Perturbation has no impact on amplitude, and maximum impact on phase noise. Noise has no impact on phase, and maximum impact on amplitude error.

26. van der Pol Oscillators If the amplitude deviation remains very small, which is the case in practical oscillators because they are strong orbital stable, then the equation for the phase noise can be further simplified to be to a closed form The corresponding Fokker-Planck equation is We notice that both analysis suggest that phase noise is time variant (which agrees to the practical obversations and play an important rule in the distribution. where is the radius of the limit cycle, and with as the diffusion coefficients. For comparison, we give the phase diffusion equation (under many assumptions) obtained in Limketkai (’05):

27. van der Pol Oscillators The dynamic of amplitude error can be approximated by which leads to the following properties if the initial is small: The mean:. The variance: It is a Gaussian variable. as This implies that if, then for any given The amplitude error also satisfies: where

28. van der Pol Oscillators The corresponding Fokker-Planck equation for the amplitude error is

29. ADC Example Noise enters the system through the input signal (e.g. analog signal is damaged in the transmission process). It is important to know the stability of the number of spikes under stochastic perturbations. As the noise getting larger, the system becomes less reliable. Small perturbations : 5 spikesLarger perturbations: 6 spikes Noise free solution: 5 spikes

30. ADC Example For different levels of perturbations, the probability of getting the correct number of spikes is different. We draw a curve (Shannon curve) of this probability with respect to the noise level in the input, which can be used to evaluate the performance of the system. Two different ways to obtain the probability: 1.Monte Carlo method: many realizations of the SDE’s. 2.Fokker-Planck equations to compute the probability directly. Number of spikes/period = number of circles in phase space/period.

31. ADC Example “unfold” transformation: Equivalent SDE :

32. ADC Example The probability can be computed by Monte Carlo methods (red curve) and Fokker–Planck equations (blue curve) (numerical computations are carried out by Guardiola). Comparing to the monte carlo methods F-P gives a better estimate for the system. Probability of n spikes probability of reaching before

33. Conclusion A general framework, based on a moving orthogonal coordinate system, has been established to rigorously study the phase and amplitude noise. Both dynamic equations and Fokker-Planck equations for the phase noise are derived. The general theory has been applied to the van der Pol oscillators. Derived equations can explain some interesting observations in practice.