An Analysis of Phase Noise and Fokker-Planck Equations

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

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

3. 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.

4. 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.

5. Analog to Digital Converter (ADC) • 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. 5 7 5 8 5 7 wrong output correct output 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).

6. 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.

7. 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

8. 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 system is self-sustained, and must have one as its eigenvalue. • The deviation can grow to infinitely large (even amplitude error remains small for stable systems, but phase error can be large)

9. 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.

10. 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: • 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

11. 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

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

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

14. 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

15. 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

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

17. Stochastic Perturbations • For a general problem in 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. • Theorem 4: the transition probability of satisfies the Fokker-Planck equation where

18. 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.

19. 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 By the method of averaging for stochastic equations, it is equivalent to The corresponding Fokker-Planck equation is

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

21. 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 The amplitude error also satisfies: where This implies that if , then for any given

22. 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.